Studies in Systems, Decision and Control 454
Artur Zaporozhets Editor
Systems,
Decision and
Control in
Energy IV
Volume I. Modern Power Systems and
Clean Energy
Studies in Systems, Decision and Control
Volume 454
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Artur Zaporozhets
Editor
Systems, Decision
and Control in Energy IV
Volume I. Modern Power Systems and Clean
Energy
Editor
Artur Zaporozhets
General Energy Institute
National Academy of Sciences of Ukraine
Kyiv, Ukraine
ISSN 2198-4182
ISSN 2198-4190 (electronic)
Studies in Systems, Decision and Control
ISBN 978-3-031-22463-8
ISBN 978-3-031-22464-5 (eBook)
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Preface
Reforming the energy sector remains a key factor in Ukraine’s sustainable growth.
Ukraine is a strategic player in the transportation of energy resources and one of the
largest regional producers of hydrocarbons.
Despite the open energy market, state-owned enterprises maintain a dominant
role in the energy sector. The largest energy producers are nuclear and hydropower
enterprises. “Naftogaz Ukrayiny” and its subsidiaries play a key role in the supply of
oil and gas to Ukraine and neighboring countries. However, new private companies
are gradually entering to the market, mainly in the field of thermal generation, as well
as companies involved in the distribution of electricity and natural gas. A significant
number of new private enterprises in renewable energy should also be noted.
In 2020, the total supply of primary energy to Ukraine was 86.402 million toe.
The largest particles in its structure were natural gas (≈27.6%), coal and peat
(≈26.4%), and nuclear energy (≈23.1%). Biofuels and waste (≈4.9%), wind, solar,
and geothermal energy (≈0.9%) have relatively low particles so far. Thus, Ukraine
has a great potential for the development of renewable energy sources.
Despite the social and economic difficulties Ukraine has faced, it has shown a
commitment to reforming the energy sector, which will put it on a path of sustainable
growth. The occupation of the Crimean peninsula and part of the Donbass by Russia
in 2014 broke the energy supply chain to Ukraine, since most of the mines are
located in the Donetsk and Lugansk regions. However, after signing the Association
Agreement with the European Union in 2014 and taking on international obligations,
Ukraine began to work on promoting energy efficiency. A significant role was played
by scientific developments carried out by Ukrainian scientists, including the authors
of this book. Various approaches were also used to the economic deregulation of
energy enterprises and economic incentives for end users of energy resources.
Despite the war activities caused by the Russian invasion on February 24, 2022,
and the destruction of a large number of energy generation and transmission enterprises, NPC “Ukrenergo” disconnected the Ukrainian energy system from Russia
and Belarus and joined the unified energy system of continental Europe ENTSO-E.
The physical interconnection operations were completed on March 16, 2022. The
v
vi
Preface
process of union of energy systems in conditions of military aggression, the introduction of active war activities, the destruction of critical infrastructure facilities,
including energy enterprises, became possible only due to highly qualified specialists in the energy sector and modern scientific theoretical and applied developments
that formed the basis for the work of many energy companies.
This book consists of two volumes, and this volume consists of five parts: Cybersecurity and Computer Science, Electric Power Engineering, Heat Power Engineering,
Fuels, and Renewable Power Engineering.
Scientists from more than 20 leading scientific, educational, governmental and
private Ukrainian institutions took part in the creation of the book. Among them
are National Academy of Sciences of Ukraine (Kyiv), General Energy Institute of
NAS of Ukraine (Kyiv), Institute of Engineering Thermophysics of NAS of Ukraine
(Kyiv), State Institution “The Institute of Environmental Geochemistry” of NAS
of Ukraine (Kyiv), Institute of Physics of NAS of Ukraine (Kyiv), Institute of
Telecommunications and Global Information Space of NAS of Ukraine (Kyiv), Taras
Shevchenko National University (Kyiv), National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute” (Kyiv), Lviv Polytechnic National University (Lviv), National Aviation University (Kyiv), National Technical University
”Kharkiv Polytechnic Institute” (Kharkiv), Kharkiv National University of Radio
Electronics (Kharkiv), Ivano-Frankivsk National Technical University of Oil and
Gas (Ivano-Frankivsk), National University of Life and Environmental Sciences
of Ukraine (Kyiv), Mykolayiv National Agrarian University (Mykolaiv), Cherkasy
Bohdan Khmelnytsky National University (Cherkasy), Kyiv International University
(Kyiv), Donetsk National Technical University (Pokrovsk), International Scientific
and Educational Center for Information Technologies and Systems (Kyiv), Kharkiv
National Air Force University (Kharkiv), Military Academy (Odesa), Verkhovna
Rada of Ukraine (Kyiv), NPC “Ukrenergo” (Kyiv), and PJSC “Ukrnafta” (Kyiv).
Also, scientists from Rzeszów University of Technology (Rzeszów, Poland) joined
for the creation of this book.
A major role in the preparation and creation of this volume of the book was played
by Academician of the National Academy of Sciences of Ukraine, Doctor of Technical Sciences, Professor, Director of the General Energy Institute of the National
Academy of Sciences of Ukraine (1997–2022) Kulyk Mykhailo Mykolayovych, and
Corresponding Member of the National Academy of Sciences of Ukraine, Doctor
of Technical Sciences, Professor, Acting Director of the General Energy Institute of
the National Academy of Sciences of Ukraine Babak Vitalii Pavlovych.
This book is for scientists, researchers, engineers, as well as lecturers and postgraduates of higher education institutions dealing with energy sector, power systems,
ecological safety, etc.
Kyiv, Ukraine
March 2022
Artur Zaporozhets
Contents
Cybersecurity and Computer Science
Development of Top-down and Bottom-up Methodology Using
Risk Functions for Systems with Multiplicity of Solutions . . . . . . . . . . . . .
Anatoly Zagorodny, Viacheslav Bogdanov, Yurii Ermoliev,
and Mykhailo Kulyk
3
Basic Matrix Forms of the System Input–Output and Their
Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mykhailo Kulyk
25
Development and Application of New Price Models in the System
of Means Input–Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mykhailo Kulyk
43
Mathematical Models and Software for Studying the Elasticity
of Building Structures and Their Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vitalii Babak, Artur Zaporozhets, Vladyslav Khaidurov,
Leonid Scherbak, Ihor Bohachev, and Tamara Tsiupii
Application of Discrete Hilbert Transform to Estimate
the Characteristics of Cyclic Signals: Information Provision . . . . . . . . . . .
Vitalii Babak, Artur Zaporozhets, Mykhailo Kulyk, Yurii Kuts,
and Leonid Scherbak
63
93
Using of Big Data Technologies to Improve the Quality
of the Functioning of Production Processes in the Energy Sector . . . . . . . 117
Viktoria Dzyuba and Artur Zaporozhets
Parametric Identification of Dynamic Systems Based on Chaotic
Synchronization and Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Artem Zinchenko
vii
viii
Contents
Detection Method of Augmented Reality Systems Mosaic
Stochastic Markers for Data-Centric Business and Applications . . . . . . . 145
Hennadii Khudov, Igor Ruban, Oleksandr Makoveichuk,
Vladyslav Khudov, and Irina Khizhnyak
Method for Converting the Output of Measuring System
into the Output of System with Given Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Elena Revunova, Volodymyr Burtniak, Yuriy Zabulonov,
Maksym Stokolos, and Volodymyr Krasnoholovets
Electric Power Engineering
Analysis of UAVs and Their Technical Parameters for Overhead
Power Lines Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Serhii Babak, Artur Zaporozhets, Oleg Gryb, and Ihor Karpaliuk
Determination of Energy Characteristics for Choice of Surge
Arresters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Sergii Shevchenko, Dmytro Danylchenko, Stanyslav Dryvetskyi,
Natalia Savchenko, and Serhii Petrov
Heat Power Engineering
Methodology for Designing Precision Sensors Which Using
in Thermal Conductivity Measurement Systems . . . . . . . . . . . . . . . . . . . . . . 223
Zinaida Burova, Svitlana Kovtun, Leonid Dekusha,
and Valentina Vasilevskaya
Methods of Ecologization of Gas-Consuming Industrial Furnaces
by Using Waste Heat Recovery Technologies . . . . . . . . . . . . . . . . . . . . . . . . . 239
Nataliia Fialko, Vitalii Babak, Raisa Navrodska, Svitlana Shevchuk,
and Nataliia Meranova
Simulation Modeling of Vapor Compression Refrigeration Unit
Temperature Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Andrii Bukaros, Oleg Onishchenko, Alexander Herega,
Herman Trushkov, and Konstantin Konkov
Methods for Diagnosing the Technical Condition of Heating
Networks Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Vitalii Babak, Oleg Dekusha, Artur Zaporozhets, Leonid Vorobiov,
and Svitlana Kovtun
Thermal Power Plants’ Coal Stock Short Term Projection Method
for Ensuring National Energy Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Sergii Shulzhenko, Borys Kostyukovskyi, Olena Maliarenko,
Vitalyi Makarov, and Maryna Bilenko
Contents
ix
Use of Improved Methodology to Determine the Total Power
Efficiency of Energy Products in Their Co-production at Combined
Heat and Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Vitalii Horskyi and Olena Maliarenko
Physical Model of Structural Self-organization of Tribosystems . . . . . . . . 309
Vitalii Babak, Nataliia Fialko, Vitalii Shchepetov, and Sergii Kharchenko
Fuels
Effect of Diethyl Ether Addition on the Properties
of Gasoline-Ethanol Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Viktoriia Ribun, Sergii Boichenko, Anna Yakovlieva,
Lubomyr Chelaydyn, Dubrovska Viktoriia, Shkyar Viktor,
Artur Jaworski, and Pawel Wos
Efficiency of Electric Logging in Thin-Layer Sections
of Hydrocarbon Deposits (Gas Fields of the Precarpathian
Depression) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
Oleksiy Karpenko, Mykyta Myrontsov, Yevheniia Anpilova,
and Oleksii Noskov
Formation Mechanisms and Overcoming Methods to Reducing
Natural Gas Consumption in the Residential Sector . . . . . . . . . . . . . . . . . . 353
Olexandr Yu. Yemelyanov, Tetyana O. Petrushka,
Anastasiya V. Symak, Kateryna I. Petrushka, and Oksana B. Musiiovska
Research of Characteristics of Solid Waste as Energy Resource . . . . . . . . 371
Artur Voronych, Teodoziia Yatsyshyn, Petro Raiter,
Lubomir Zhovtulya, and Serhii Maksymiuk
Renewable Power Engineering
Geothermal Heat Supply Development Pathways in Ukraine . . . . . . . . . . 385
Yulia Shurchkova, Sergii Shulzhenko, Anna Pidruchna,
Volodymyr Deriy, and Vitaly Dubrovsky
Environmental Aspects of Geothermal Energy . . . . . . . . . . . . . . . . . . . . . . . 397
Anna Pidruchna and Yulia Shurchkova
Straw Pellets for Heat Supply in the Countryside: Economic,
Environmental and Circular Economic Indicators . . . . . . . . . . . . . . . . . . . . 411
Valerii Havrysh and Vasyl Hruban
Comparative Analysis of Energy-Economic Indicators
of Renewable Technologies in Market Conditions and Fixed
Pricing on the Example of the Power System of Ukraine . . . . . . . . . . . . . . . 433
Mykhailo Kulyk, Tetiana Nechaieva, Oleksandr Zgurovets,
Sergii Shulzhenko, and Natalia Maistrenko
x
Contents
Prospects and Energy-Economic Indicators of Heat Energy
Production Through Direct Use of Electricity from Renewable
Sources in Modern Heat Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Volodymyr Derii, Oleksandr Teslenko, Eugene Lenchevsky,
Viktor Denisov, and Natalia Maistrenko
Cybersecurity and Computer Science
Development
of Top-down and Bottom-up Methodology
Using Risk Functions for Systems
with Multiplicity of Solutions
Anatoly Zagorodny , Viacheslav Bogdanov , Yurii Ermoliev ,
and Mykhailo Kulyk
Abstract There is a wide and important class of objects and systems with a hierarchical structure, which by their nature should be applied methodology TOP-DOWN–
BOTTOM-UP (TD–BU), but in the current state of the most important problems of
its study can not be solved existing models TD–BU. This class of tasks, first of
all, includes forecasting the production of almost all types of products and services,
demand for them in all sectors of the economy, social sphere and in the country as
a whole, ensuring unambiguous performance indicators of upper and lower levels
of government, banks, trade, transport networks, etc. For these systems, the key
problem is the discrepancy between the corresponding indicators for the upper and
their sum for the lower hierarchical levels. The problem of discrepancy was solved
by developing special analytical dependencies for indicators of both upper and lower
levels. However, the solution of the problem of discrepancy led to the problem of
ambiguity of solutions, these analytical solutions have n − 1 (n—dimension of the
system) modification, each of which provides the necessary equality of indicators
of these levels. There was a problem of choosing the best solution from many. The
problem of building a mathematical model and a corresponding method that would
simultaneously solve the problems of both divergence and ambiguity was overcome
in the work by combining the TD–BU methodology with the analytical apparatus
of general risk theory. At the same time, using the TD–BU methodology, analytical
dependences were obtained that provide a solution to the problem of discrepancy.
Based on these dependencies, in combination with the apparatus of risk theory, solutions were determined for both levels that have zero risks. As a function of the risk
function, the functionality of the classical least squares method was used, which
(method) gives better results (zero risks) for this problem in comparison with others.
A. Zagorodny · V. Bogdanov · Y. Ermoliev
National Academy of Sciences of Ukraine, Kyiv, Ukraine
M. Kulyk (B)
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: info@ienergy.kiev.ua
Y. Ermoliev
International Institute for Applied Systems Analysis, Laxenburg, Austria
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_1
3
4
A. Zagorodny et al.
As a result of combining TD–BU methodology and general risk theory, a comprehensive method of researching a new, important class of problems in the field of
TD–BU problems was synthesized.
Keywords Top-down · Bottom-up · TD–BU · Risk theory · Hierarchy ·
Discrepancies · Risks
1 Introduction
The interplay of Top-Down and Bottom-Up technologies, information processing
strategies used in many and varied management and organizational tasks, enables
researchers to exploit the strengths of both approaches. Most of the proposed methods
of synthesis of Top-Down and Bottom-Up as a holistic methodology are a combination of analysis of the components of the analyzed hierarchical systems with iterative
procedures and are characterized by high labor costs and convergence problems that
arise. Despite these difficulties, researchers are constantly turning to the use of TopDown–Bottom-Up (TD–BU) methodology for analysis, optimization or synthesis
of systems and structures of hierarchical nature, which indicates the high relevance
of improving the TD–BU methodology. Here are just a few examples of the use of
TD–BU in specific areas. Thus, in [1] a comprehensive analysis of energy policy
was conducted, for which the technological details of the ascending models and
the economic wealth of the descending ones were combined. At the same time, a
combination of different mathematical formats—mixed complementarity and mathematical programming—was carried out, which allowed to overcome the limitations
of the dimensionality of the analyzed systems. To develop cost-effective ways to
combat karst rocky desertification (KRD)—a serious environmental problem threatening southwest China—a spatial model was developed to model KRD dynamics
using TD–BU, which allowed to predict its potential expansion or contraction [2].
The need to develop effective measures to respond to rapid changes in the environment has led the authors [3] to use TD–BU methods to combine downstream
large-scale software approaches with rising, initiated and managed at the community level, which together with indigenous and local knowledge compliance with
the monitoring program and community priorities, as well as respect for indigenous
peoples’ intellectual property rights. In [4], the combined TD–BU approach allowed
the authors to create a model for assembling zeolite frames, according to which
the fusion of zeolite minerals may be the result of several possible ways of crystal
growth. The authors [5] used hybrid modeling of TD–BU to analyze problems related
to energy systems, indicating that linking upstream industry (engineering) models
with downward (macroeconomic) models is an important contribution to the design
of energy systems compatible with steady economic growth. Thus, a number of
problems in various fields are currently successfully solved by TD–BU methods.
However, there are a wide range of important systems and facilities that, by their
nature, should be subject to the TD–BU methodology, but this could not be done
Development of Top-down and Bottom-up Methodology Using Risk …
5
using existing TD–BU models. This class of tasks, first of all, includes the simultaneous forecasting of production volumes both at the national level and at the level
of industries of almost all types of products and services, demand for them in all
sectors of the economy, social sphere and the country as a whole. For such systems
and facilities, the key issue is the discrepancy (mismatch) of the indicators for the
upper and their sum for the lower hierarchical levels. The publication [6] shows that
this discrepancy problem for this new class of problems can be solved by developing
a generalized model of the TD–BU class and special analytical dependences for indicators of both upper and lower hierarchical levels. However, solving the problem of
discrepancy leads to the problem of ambiguity of solutions, namely—these analytical solutions have n − 1 modification (n—dimension of the system), each of which
provides the necessary equality of the upper and their sum for the lower levels. That
is, there is a problem of choosing the best solution from many. This paper proposes
the solution of the above complex problem by combining the TD–BU methodology
using the above-mentioned special mathematical dependencies with the analytical
apparatus of general risk theory and the formation of a generalized mathematical
method. Currently, a wide range of risk assessment algorithms has been developed,
and the choice of a specific one directly depends on the nature of the problems that
constantly arise in various fields of technology, economics, finance, and others. In
particular, to assess credit risk, which is the main focus of the financial and banking
sectors in connection with recent financial crises, the function of the least squares
method of reference vectors is used as a risk function [7], as it can turn quadratic
programming into linear programming, thereby reducing computational complexity.
The same method of risk assessment, namely, the method of least squares of reference
vectors used in [8] to analyze the factors influencing the design of the construction
industry in developing countries. To assess the risk of planning electrical networks,
a model of least squares of reference vectors was created [9], based on the clipping
algorithm. This is due to the fact that the application of the least squares method of
reference vectors leads to a loss of sparseness, and the clipping algorithm makes the
corresponding matrix sparse. The study [10] develops and tests a model to increase
the accuracy and facilitate decision-making on project selection for international
construction firms, while the data analysis uses the method of partial least squares,
i.e., finds a model of linear regression by projecting predicted variables and existing
variables into new space. For epidemiological studies, risk differences were estimated using modified least squares regression, which is a useful analytical tool for
rare binary results on the number of distortion factors, which gave reliable results for
such systems [11]. In [12] shows how the use of the least squares method of Monte
Carlo for long-term modeling of economic balance can be implemented in practice.
As can be seen from the analysis of literature sources, most of the problems of
risk theory in the role of functionals use various modifications of the least squares
method, or this method in its classical form. It will be shown below that the object of
study of this work in the intermediate form is a redefined system of linear algebraic
equations. It turned out that the best results in terms of risk indicators for such an
object are provided by the risk function in the form of the least squares functional in
its (method) classical interpretation.
6
A. Zagorodny et al.
Mathematical model and methods of analytical determination of indicators of
upper and lower levels in these problems that solve the problem of ambiguity were
presented in [6]. The mathematical model is formed in such a way that provides an
opportunity to find solutions for the upper and each of the lower (sectoral) levels
in a unique, analytical form. Therefore, the search for solutions is non-iterative. It
is carried out in two stages. On the first of them, using known (standard) methods,
preliminary forecasts are developed for indicators of the upper and lower levels.
At the second stage a special system of algebraic equations is formed, from which
analytical dependences for calculation of refined indicators of both levels are defined.
This ensures a complete match between the upper indicator and the sum of the lower
levels indicators.
These mathematical model and methods can be used quite reasonably to reconcile the reporting indicators of the upper and lower levels of the respective objects
(management structures, banks, trade network, etc.). In this case the consensual
decisions are formed in one stage.
2 Mathematical Model
The source information in this study is the vector of predictive functions f , formed
at a given period of time using certain mostly known forecasting methods
f = [ f 1 , f 2 , f i , f n ]; ,
(1)
where f 1 —Top-level forecast, f i , i = 2 − n—Down-level forecasts.
According to the problem
f 1 /=
n
.
fi .
(2)
i=2
The purpose of the study in mathematical terms is to find a solution to the system
of equations
x = f1,
x2 = f 2 ,
xi = f i ,
xn = f n ,
(3)
in which x a —T-level solution, and x i , i = 2, n—D-level solutions, and these
solutions are related by an equation
Development of Top-down and Bottom-up Methodology Using Risk …
xa −
n
.
7
xi = 0.
(4)
i=2
The system of equations (3) and (4) in the matrix–vector form is
Ax = F,
(5)
or
1
2
i
n
n+1
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 2 Ai n
1
⎤
⎥
⎥
⎥
1
⎥
⎥
1 ⎦
1 −1 −1 −1
1
⎡
⎤
f1
xa
⎢ ⎥
f2 ⎥
⎢x ⎥ ⎢
⎥
⎢ 2⎥ ⎢
⎢
,
× ⎢ ⎥ = ⎢ fi ⎥
⎣ xi ⎦ ⎢ ⎥
⎥
f
⎣ n⎦
xn
0
⎡
⎤
(6)
;
in which the vector [f 1 , f 2 , f i , f n , 0] denote by F.
The system of equations (5), (6) has n + 1 equation and n unknowns. Such a
system is redefined and therefore does not have an exact solution. To find the best of
the approximate solutions, we use the Gaussian transformation in the form
A' Ax = A' F.
(7)
For the right-hand side of Eq. (7), the dependence is valid
A' F = f ,
(8)
which is confirmed by multiplying A' by F.
Therefore, Eq. (7) will be considered in the form
Bx = f , B = A' A,
(9)
in which
⎡ 1 2 Ai n ⎤ ⎡ ⎤ ⎡
1 2 Ai n n + 1
xa
⎡
⎤
1
1 1
1
⎢ 1
⎥ ⎢ x2 ⎥ ⎢
⎢
⎥ ×⎢ ⎥=⎢
2 ⎢
−1 ⎥
⎥ ⎢ ⎥ ⎢
⎢ 1
⎥ ×⎢
1
⎢
⎥ ⎣ xi ⎦ ⎣
⎣
⎦
⎢
⎥
i
1 −1
⎣
1 ⎦
xn
n
1 −1
1 −1 −1 −1
⎤
f1
f2 ⎥
⎥
⎥,
fi ⎦
fn
or, in the final version, the mathematical model of the problem has the form
8
A. Zagorodny et al.
1
2
i
n
12
2 −1
⎢ −1 2
⎢
⎣ −1 1
−1 1
⎡
Bi
−1
1
2
1
⎡ ⎤ ⎡
n
xa
⎤
−1
⎢x ⎥ ⎢
⎢ 2⎥ ⎢
⎥=⎢
1 ⎥
⎥ ×⎢
⎣ xi ⎦ ⎣
⎦
1
xn
2
⎤
f1
f2 ⎥
⎥
⎥.
fi ⎦
(10)
fn
3 Analytical Solutions
In algebraic equations (9) and (10), the matrix B is square with dimension n, the
vectors x and f also have dimension n, the determinant |B| /= 0, that is, the system
of equations (9) and (10) has one solution. This system has a structure that provides
a unique opportunity to find this solution in an analytical form. To do this, we apply
another Gaussian transformation to system (10), reducing the matrix B to a triangular
form and limiting itself to the dimension n = 3. As a result, we obtain an algebraic
system (11)
⎤
⎡ ⎤ ⎡
1 2 3
f1
xa
⎤
1 2 −1 −1
⎥
⎢ ⎥ ⎢
× ⎣ x2 ⎦ = ⎣ f 1 + 2 f 2
⎦.
2 ⎣
3 1⎦
x3
−2 f i + 2 f 2 − 6 f 3
3
−8
⎡
(11)
We find analytical solutions for the two dimensions of the system (11), namely,
for n = 2 and n = 3.
For n = 2 we have:
x2 = ( f 1 + 2 f 2 )/3 = f 2 + ( f 1 − f 2 )/3,
2xa = f 1 + x2 = f 1 + f 2 + f 1 /3 − f 2 /3,
xa = 2/3 f 1 − 1/3 f 2 = f 1 − ( f 1 − f 2 )/3.
For the case n = 3 we receive:
−8x3 = −2 f 1 + 2 f 2 − 6 f 3 ,
4x3 = f 1 − f 2 + 3 f 3 ,
x3 = f 3 + ( f 1 − f 2 − f 3 )/4;
3x2 = f 1 + 2 f 2 − f 3 − ( f 1 − f 2 − f 3 )/4,
x2 = (1/3 − 1/12) f 1 + (2/3 + 1/12) f 2 − (1/3 − 1/12) f 3
= f 2 + ( f 1 − f 2 − f 3 )/4;
2xa = f 1 + x2 + x3 ;
xa = ( f 1 + f 2 + f 3 )/2 + ( f 1 − f 2 − f 3 )/4 = f 1 − ( f 1 − f 2 − f 3 )/4.
Development of Top-down and Bottom-up Methodology Using Risk …
9
Analysis of the received decisions x a and x i , i = 1, 2 for both cases shows that
these solutions have the same structure, namely, the top-level solution has the form
xa = f 1 −
1
r,
n+1
(12)
and sectoral decisions in this case take shape
xi = f i +
1
r, i = 2, n,
n+1
(13)
where
r = f1 −
n
.
fi ,
(14)
i=2
is the difference between the upper level indicator and the sum of sectoral data, n
– dimension of the system (10), i = 2, n.
To confirm the dependences (12)–(14) we will show that they are valid not only for
the dimension n, but also for the dimension n + 1. To do this, consider the structure
(10) with dimension n + 1, shown in Eq. (15).
⎡
⎤ ⎡
xa
⎡ 1 2 Bi n n + 1 ⎤
⎢
⎥ ⎢
2 −1 −1 −1 −1
x2 ⎥ ⎢
⎢ −1 2 1 1 1 ⎥ ⎢
⎢
⎢
⎢
⎥ ×⎢x ⎥
=⎢
⎢
⎥ ⎢ i ⎥
⎢
⎥
⎢ −1 1 2 1 1 ⎥ ⎢
⎢
⎢
⎥ ⎣ xn ⎥
n
⎣
⎦
⎣ −1 1 1 2 1 ⎦
n+1
xn+1
−1 1 1 1 2
1
2
i
⎤
f1
⎥
f2 ⎥
⎥
fi ⎥
⎥.
⎥
fn ⎦
f n+1
(15)
It is easily verified that the matrix B in the system (15) has the structure of the
matrix B from Eq. (10) and differs only in the dimension.
Analytically determine the unknown x a and x i , i = 2, n + 1 in the system (15)
taking into account the dependences (12)–(14), in which the dimension n is increased
by one.
New unknown x n+1 determined from the last equation of the system (15)
2xn+1 − xa +
n
.
xi = f n+1 ,
i=2
or
2xn+1 = f n+1 + f 1 − r/(n + 2) −
n
.
i=2
f i − r (n − 1)/(n + 2).
10
A. Zagorodny et al.
Seeing that
f1 −
n
.
f i = r + f n+1 ,
i=2
we get the result
xn+1 = f n+1 + r/(n + 2).
(16)
Unknown x a is determined from the first equation of the system (15)
2xa −
n+1
.
xi = f 1 ,
i=2
which, with considering (12)–(14) and (16), is transformed into a form
2xa = f 1 +
n+1
.
f i + r n/(n + 2),
i=2
or
2xa = 2 f 1 − r + r n/(n + 2),
and as a result
xa = f 1 − r/(n + 2).
The dependence for x i in system (15) is determined from equation
n
.
2xi − xa +
xk = fi ,
k=2; k/=i
which is converted taking into account (12)–(14) and (16), (17) to the form
n+1
.
2xi = f 1 −
f k − r n/(n + 2) + f i .
k=2; k/=i
Applying the identity
f1 −
n+1
.
k=2; k/=i
fk = r + fi
(17)
Development of Top-down and Bottom-up Methodology Using Risk …
11
we get the final
xi = f i + r/(n + 2).
(18)
Dependencies for x m , m = 2, n, m /= i are obtained analogously to (18) with
insignificant differences in their definition, and as a result, taking into account (16),
formula (18) is true for all i = 2, n + 1.
The validity of the dependences (16)–(18) is also confirmed by their direct
substitution into the system (15). In particular, for the equation I, we have
− f 1 + r/(n + 2) +
n+1
.
f k + r n/(n + 2) + f i + r/(n + 2) = f i .
k=2
Taking into account (14) we obtain the expression
−r + f i + r (n + 2)/(n + 2) = f i ,
which is an identity.
Thus, the dependences (12)–(14) are the solution of the system of Eq. (10).
However, their substitution in Eq. (4) does not satisfy him and gives an error
xa −
n
.
xi = r/(n + 1).
(19)
i=2
This is quite natural, because the system (3), (4) does not have, as noted, an exact
solution.
However, it is noteworthy that the error r in the forecasts f, i = 1, n due to
transformations (7)–(9) in the system (10) decreases according to (19) by n + 1
times. Therefore, it seems appropriate to organize the iterative process to ensure
Eq. (4)
Bx (1) = f , Bx (2) = x (1) , . . . ,
(m−1)
Bx (m) = x (m−1) , ||x||
(m)
− ||x||
≤ ε,
(20)
where E—permissible error, m—number of iterations.
Decision x(1) after the first iteration in the form (12)–(14) already found. After
the second iteration, it looks like
)
(
1
1
r,
+
x (2)
=
f
−
1
a
n + 1 (n + 1)2
)
(
(21)
1
1
x (2)
+
r,
i
=
2,
n.
=
f
+
i
i
n + 1 (n + 1)2
12
A. Zagorodny et al.
The solution of the iterative process (20) for the m-th iteration is determined using
the method of complete induction. Considering (21), there is reason to assert that
after the m-th iteration process (20) will provide a solution
xa(m) = f 1 − s(m)r,
(22)
xi(m) = f i + s(m)r, i = 2, n,
(23)
s(m) =
m
.
1/(n + 1)k .
(24)
k=1
According to the method of complete induction, the solution after the m + 1
iteration should have the same form as in (22)–(24) with the difference that in it
instead of the value of m will appear m + 1.
According to (12)–(14) and using (20), (22)–(24), the solution after the m + 1
iteration is represented in the form
)
(
n
.
1
(m)
(m)
xi
=
−
,
x −
n+1 a
i=2
)
(
n
.
1
(m+1)
(m)
(m)
(m)
xi
xi
= xi +
.
x −
n+1 a
i=2
xa(m+1)
xa(m)
(25)
(26)
Dependence (25) is revealed using (22)–(24):
xa(m+1) = f 1 − s(m)r − (1/(n + 1))
)
(
n
.
× f 1 − s(m)r −
( f i + s(m)r )
i=2
= f 1 − r/(n + 1) − (1 − 1/(n + 1) − (n − 1)/(n + 1))s(m)r.
Seeing that
(1 − 1/(n + 1) − (n − 1)/(n + 1)) = 1/(n + 1),
and
s(m)/(n + 1) =
m+1
.
k=2
dependence (27) is transformed into form
1/(n + 1)k ,
(27)
Development of Top-down and Bottom-up Methodology Using Risk …
13
x (m+1)
= f 1 − s(m + 1)r,
a
which is according to the method of complete induction and confirms correctness of
(22). The correctness of the dependence (23) is proved similarly.
To further use solutions (22), (23) it is necessary to determine the sum (24) with
an unlimited increase in the number of iterations m, i.e., it is necessary to establish
a convergence limit
c(n) = lim s(m) = lim
m→∞
m→∞
m
.
(
)
1/(n + 1)k .
(28)
k=1
)
. (
k
Series m
coincides with all the attribute.
k=1 1/(n + 1)
The limit of convergence is determined by the assumption with its subsequent
verification. Suppose that such a limit is a quantity
c(n) = 1/n.
(29)
This assumption is justified, in particular, by the fact that the amount
4
.
1/(n + 1)k =0.0999931 at n = 10.
k=1
To prove (29) we form according to (24) the difference between c(n) and the
partial sum s(m), which at m → ∞ must turn into zero
p(m) = c(n) − s(m).
(30)
(Value p(m)) at m = 1 is equal to p(1) = 1/(n(n + 1)) and at m = 2 p(2) =
1/ n(n + 1)2 .
By the method of complete induction we assume that the value of p(m) has the
form
)
(
p(m) = 1/ n(n + 1)m ,
(31)
and prove that this expression is valid for a series with m + 1 members, i.e.,
)
(
p(m + 1) = 1/ n(n + 1)m+1 .
Denote n + 1 = ω then according to (30) we establish
(
p(m + 1) = ω
(
m+1
− (ω − 1)
m
.
k=0
))
ω
k
)
(
/ (ω − 1)ωm+1 .
(32)
14
A. Zagorodny et al.
Revealing the amount
.m
k=0
ωk , we receive
(ω − 1)
m
.
ωk = ωm+1 − 1.
k=0
As a result
)
(
p(m + 1) = 1/(ω − 1)ωm+1 = 1/n(n + 1)m+1 .
This dependence proves that expression (31) is valid for all positive integers m.
In this regard lim p(m) = lim 1/(n + 1)m = 0, that is, the dependence (29) is
m→∞
m→∞
true. Thus, the dependence (31) is valid for all positive integer m. Then lim p(m) =
m→∞
1
lim
m = 0 at all positive integers n and m, and the dependence (21) is true.
m→∞ n(n+1)
The consequence of this is that the dependences (22), (23) take the form
xa = f 1 − r/n,
(33)
xi = f i + r/n, i = 2, n.
(34)
Substitution of dependences (33), (34) satisfies Eq. (4).
However, although the problem of discrepancy for this set of problems has been
overcome, the model (10), (33), (34), which solves it (the problem), provides not
one but n − 1 solution. Indeed, if in system (3) with dimension n it is equivalent to
combine any two equations of level
. DOWN, we obtain a system with dimension n
− 1, in which the values f 1 and ni=2 f i will be unchanged, i.e., r will not change
either. But the decision xa will be determined not by dependence (33), but by another,
xa = f 1 − r/(n − 1). Repeating this procedure until exhaustion, we obtain a system
of n − 1 solution
⎫
⎪
⎪
⎪
⎪
= f 1 − r/(n − 1),⎪
⎪
⎬
x (n)
a = f 1 − r/n,
x (n−1)
a
···
,
(35)
x (k)
a = f 1 − r/k, k = 2, n.
(36)
x (3)
a
x (2)
a
= f 1 − r/3,
= f 1 − r/2,
⎪
⎪
⎪
⎪
⎪
⎪
⎭
or in a compact form
Thus, as a result of transformations (10)–(14), (20), (33), (34) of the original
system of Eqs. (3)–(4) n − 1 solution for the indicator is obtained xa top level. That
Development of Top-down and Bottom-up Methodology Using Risk …
15
is, there is a problem of choosing the best of them from a set of solutions (36). A
similar problem applies to sectoral indicators xi (34).
According to the review of literature sources related to these problems of choice,
they (problems) can be solved quite effectively by minimizing a certain mathematical
form that is a function of risk [7–12]. This approach provides an approximation of
the set of solutions, in particular, (36) one value of xa , which may differ from each
solution x (k)
a , k = 2, n. The module of such discrepancy serves as a measure of the
risk of applying the selected risk function and provides an opportunity to determine
the best solution from the set of acceptable ones. The performed literature analysis
shows that the most popular risk function is the least squares functional and its
modifications.
Therefore, in this study the determination of the optimal values of the indicators
as the top xa , and lower x i , i = 2, n levels is performed on the basis of the risk
function by the classical method of least squares.
4 Determining the Best Solutions for the Upper and Lower
Levels Based on the Least Squares Risk Function
According to the method of least squares to determine the best solutions from the
set (36) must first find a solution x, which minimizes functional in the form of the
sum of squares of inconsistencies
ϕ(x a ) =
n
.
(
x a − xa(k)
)2
→ min, k = 2, n.
(37)
k=2
For a fixed n, the functional (37) will have a minimum value when derived
dϕ(x a )/d x a will be zero. This feature allows you to find the value x as
x a = f 1 − S(n)r,
(38)
S(n) = C(n)/(n − 1),
(39)
where
C(n) =
n
.
(1/k).
(40)
k=2
Having a dependency for x in the form (38), we can determine the overall
dependence for sectoral indicators x. We will look for this dependence in the form
x i = f i + a(n)r, i = 2, n.
(41)
16
A. Zagorodny et al.
Then condition (4) takes the form
xa −
n
.
( f i + a(n)r ) = 0,
i=2
or
f 1 − S(n)r −
n
.
f i − (n − 1)a(n)r = 0,
i=2
whence we have finally
(n) = (1 − S(n))/(n − 1).
(42)
The direct substitution of solutions (38) and (42) satisfies Eq. (4). That is, the decisions received on the basis of method of least squares, satisfy the basic requirement
of the set task concerning equality of the T-level indicator of the sum of indicators
of the DOWN level.
Constants S(n) and a(n) are used not only to determine the values x a (38) and x i
(41). In the future it will be shown that they are also needed when calculating the
values of risks. Therefore, these constants are tabulated and listed in Table 1 for n
from 2 to 20 inclusive.
Dependencies x a (38) and xa(k) (36) provide an opportunity for the upper level
indicators to determine the discrepancies from the functional (37) at a fixed n, namely.
.a(k) = x a − xa(k) = (1/k − s(k))r, k = 2, n.
(43)
The set of discrepancies (43) has n − 1 values. From this set we choose the
maximum modulo
| (k) |
|. |
a max(k) = Ra (n) = |(S(n) − 1/n)r |.
(44)
In dependence (44) there is always (Table 1) inequality S(n) > 1/r , but r can be
negative. In the future, the value Ra (n) will be used as a measure of risk (risk) when
choosing the best solution from the set (36).
The set of discrepancies for sectoral indicators is similar (43)
.i(k) = x i − xi(k) = (a(n) − 1/k)r, k = 2, n , i = 2, n,
(45)
and the measure of risk (risk) for each of n − 1 of these sets is similar to the form
(44)
|
|
|
|
R(n) = |.i(k) |
= |(a(n) − 1/n)r |, i = 2, n.
(46)
max(k)
0.202
0.0798
S(n)
a(n)
11
–
–
a(n)
1
0.5
0.5
0.0735
0.1912
12
2
Number of forecasts (n)
S(n)
Constants
Table 1 Constants S(n), a(n)
0.0682
0.1817
13
0.2917
0.4167
3
0.3611
0.213
0.0636
0.1732
14
4
0.0596
0.1656
15
0.1698
0.3208
5
0.29
0.142
0.0561
0.1587
16
6
0.2655
0.1224
0.053
0.1525
17
7
0.2454
0.1078
0.0502
0.1468
18
8
0.2286
0.0964
0.0477
0.1415
19
9
0.0454
0.1367
20
0.0873
0.2143
10
Development of Top-down and Bottom-up Methodology Using Risk …
17
18
A. Zagorodny et al.
Using the dependencies for risks (43), (46) and indicators of Table 1 for S(n)
and a(n). such important generalizations can be made. When using the least squares
approximation for solutions of both the upper and sectoral levels, there are two values
on the numerical axis n in which these risks are zero. These are points n = 2 and n
= ∞. Point n = ∞ provides a formal. degenerate solution. because at this point the
constants S(n) and a(n) reach zero values. As a result. a top-level decision is made
x = f 1 . and sectoral decisions— x i = f i , i = 2, n , that is, at this point system
(3), (4) degenerates into the initial system (3). At the point n = 2 according to Table
1 constant S(n = 2) = a(n = 2) = 1/2, and therefore risks Ra (n = 2) = R2 (n = 2)
= 0. Thus, when approximating solutions of both upper and sectoral levels by the
method of least squares, many solutions are obtained x(n) and x(n) power ( n − 1),
among which the best in terms of risk are solutions x a (n = 2) and x i (n = 2), at the
same time their risks Ra (2) and R2 (2) equal to zero.
In the range of values 2 < n < N < ∞ the risk measures of these indicators are
greater than zero. Therefore, to obtain the most reliable indicators of both upper and
lower levels, you need to use the following comprehensive method.
1. On the first stage the original system of dimension n is aggregated into a
system of dimension m = 2.
2. Top level indicator x a it is fixed as
x a = f 1 − r/2 .
(47)
3. On the second stage the system is disaggregated with dimension m = 2 into the
original system with dimension n. The following operations are performed. Sectoral
adjustment factors are determined
μi = f i / f s , f s =
n
.
f i , i = 2, n.
(48)
i=2
4. The values of sectoral corrections are calculated
pi = μi r/2 , i = 2, n .
(49)
5. Sectoral decisions are determined
x i = f i + pi , i = 2, n .
(50)
Solutions (47), (48) provide the basic Eq. (4). Indeed, making their substitution
in (4), we obtain
n
.
xi =
i=2
given (47) we have
n
.
i=2
fi +
n
.
i=2
μi r/2 = f s + (r/2 f s )
n
.
i=2
f i = f s + r/2;
Development of Top-down and Bottom-up Methodology Using Risk …
19
Table 2 The influence of the dimension of the system n on its solution x a (n), x i (n) and risks
Ra (n), Ri (n) (f 1 = 200, r = 30, f i = 15)
n
S(n)
x(n)
x(n)
Ra (n)
Ra (n), %
a(n)
x(n)
x(n)
Ri (n)
Ri (n), %
2
0.5
185
185
0
0
0.5
30
30
0
0
3
0.4167
187.5
190
2.5
1.3
0.2917
23.75
25
1.25
5.3
4
0.3611
189.2
192.5
3.3
1.7
0.213
21.39
22.5
1.11
5.2
5
0.3208
190.4
194
3.6
1.9
0.1698
20.09
21
0.91
4.5
6
0.29
191.3
195
3.7
1.9
0.142
19.26
20
0.74
3.8
10
0.2143
193.6
197
3.4
1.8
0.0873
17.62
18
0.38
2.2
15
0.1656
195
198
3
1.5
0.0596
16.79
17
0.21
1.3
20
0.1367
195.9
198.5
2.6
1.3
0.0454
16.36
16.5
0.14
0.9
f 1 − r/2 = f s + r/2; f 1 = f s + r ; f 1 = f 1 .
This substitution gives identity, i.e. requirement (4) is satisfied. The obtained
identity also gives grounds to claim that the application of the least squares method
to approximate the solutions of the initial system (3), (4) provides solutions (47).
Equation (50) with zero risks.
At the end of the study, it is also advisable to analyze the risks and behavior of
upper and lower level decisions in cases where the dimension of the system lies
in a wide range 2 < n < ∞. In the Table 2 shows the results of such calculations
for systems with dimensions from 2 to 20. For comparative analysis for all n the
same data were selected: f 1 = 200, r = 30 and one sectoral indicator f i with value
f i = 15. In the whole range n = 3 ÷ 20 decision x a (n) is less important than the
decision x a (n). This is due to the fact that in this range is a constant S(n) exceeds 1/n.
Comparison of solutions x i (n) and x i (n) in the same range n shows the same trend,
x i (n)<x i (n) throughout the range n = 3 ÷ 20. Although in this case a(n) < 1/n,
but the summation depending on (41) is conducted with a plus sign, so this trend
continues.
Not only sound behavior but his alertness and dedication too are most required
x a (n) and x i (n), as well as their risks in the given range of change n.
As noted, when n = 2 risks for both upper and lower levels are zero, which is
recorded in Table 2.
For completeness of the analysis in Table 2 shows the relative values of risks
R(n, %) = R(n)/x(n) and R(n, %) = R(n)/x(n). Pays attention to what is in the
range n = 3 ÷ 20 risks R(n) and R(n, %) have maxima at n = 6. Absolute values of
risks R(n) lie in the range of 2.5–3.7 units, which is 1.3–1.9% of the corresponding
values x(n). Absolute values of risks R(n) sectoral indicator x(n) are in the range of
0.14–1.25 units, which in relative terms reaches 0.9–5.3%, respectively. Calculations
of decisions are carried out x(n) and x(n) when changing the dimension of the system
in the range n = 3 ÷ 20 lead to minor deviations of both absolute (especially) and
relative risks. The fact that when n = 2 risk indicators x(n) and x(n) equal to zero,
and when n is in the range n = 3 ÷ 20 the risks are insignificant, give grounds to argue
20
A. Zagorodny et al.
Table 3 Forecast of electricity consumption in Ukraine in 2040 (MWh)
№
1
Sector
Indicator
Before use
TD–BU
Correction factor
Correction
Ukraine, TOP
225,288
–1
–19,930.5
205,357.5
Ukraine, DOWN
185,427
1
19,930.5
205,357.2
521.4
5,372.4
Agriculture
and others
4,851
0.02616
After application
TD–BU
2
Mining industry
11,772
0.06349
1,265.4
13,037.4
3
Manufacturing
industry
28,207
0.15212
3,031.8
31,238.8
4
Production and
distribution of
electricity, gas,
heat, water, etc.
45,515
0.24546
4,892.1
50,407.1
5
Another industry
17,577
0.09479
1,889.2
19,466.2
6
Transport
11,881
0.06407
1,276.9
13,157.9
7
Other sectors
31,577
0.17029
3,394
34,971
8
People
34,047
0.18361
3,659.4
37,706.4
that the approximation of solutions x and x using the least squares risk function is
justified and effective.
Example. Table 3 shows the sequence and results of the application of the
two-stage method (47)–(50) for the correction of the real forecast of electricity
consumption in the country for the long term.
The initial information for the correction of the forecast of electricity consumption in Ukraine at the level of 2040 was the real data given in the column “Before
the application of TD–BU”. These data were obtained by using forecast macroeconomic indicators (electricity capacity and gross domestic product, TOP level)
and determining the demand for electricity in sectors of the economy based on the
projected electricity intensity of sectoral production and forecasts of its volumes.
These indicators were determined using the methods of identifying dependencies.
The sum of sectoral electricity demand indicators (DOWN level) is less than the
TOP level indicator by r = 39,861 MWh, which is about 18% of the TOP level, and
therefore the entire demand vector of this column needs to be corrected.
The correction was performed according to the complex method (47)–(50).
Column (3) of Table 3 with dimension n = 9 was aggregated into a column with
dimension n = 2. From the Table 1 constants were selected S(n = 2) = 1/2 and
a(2) = 1/2, according to (47) the T-level indicator is determined x(2) = 205,357.5.
Correction coefficients (column 4) were determined according to (48) and correction
volumes according to (49) (column 5). Dependence (50) provides calculations of all
sectoral indicators (column 6). In this column, in the line “Ukraine, TOR” there is an
indicator corresponding to (47), and in the line “Ukraine, DOWN”—an indicator that
Development of Top-down and Bottom-up Methodology Using Risk …
21
is the sum of indicators of sectors 1–8. Comparison of these two indicators shows
their coincidence in six decimal places. Combined with the fact that they have zero
risk, it is sufficient to ensure the reliability of the results.
5 Analysis of Results and Conclusions
An analysis of the literature has shown that the TD–BU methodology in its current
state covers a wide range of objects and systems of various natures and with specific
research objectives. Their common feature is that they have a hierarchical structure
and feature systems with extremely large dimensions. This leads most researchers
to use part analysis methods in combination with iterative procedures, which are
associated with convergence problems and high labor costs. Therefore, the relevance
of research on the development of TD–BU methodology remains high. In addition,
there is a wide range of objects and systems to which, by their nature, the TD–
BU methodology should be applied, but in the current state, the crucial tasks of
its study cannot be solved by existing TD–BU models. This class of tasks, first of
all, includes forecasting the production of almost all types of products and services,
demand for them in all sectors of the economy, social sphere and in the country as
a whole, ensuring unambiguous performance indicators of upper and lower levels
of government, banks, trade and transport networks, forecast and reporting data on
the production and consumption of electricity, heat, coal, oil, gas, products of their
processing, etc. For these systems and facilities, the key issue is the discrepancy
between the relevant indicators for the upper and their sum for the lower hierarchical
levels.
The problem of discrepancy was solved by developing special analytical dependencies for indicators of both upper and lower levels. However, the solution of the
problem of discrepancy has led to the problem of ambiguity of solutions, namely,
these analytical solutions have n − 1 modification (n—dimension of the system),
each of which provides the necessary equality of indicators of these levels. That
is, there was a problem of choosing the best solution from many. In this paper, the
problem of constructing a mathematical model and an appropriate method that would
simultaneously solve problems of both discrepancy and ambiguity was overcome by
combining the TD–BU methodology with the analytical apparatus of general risk
theory. Were obtained using the TD–BU methodology xa (33) and xi (34), which
provide a solution to the problem of disagreement. On the basis of these dependencies in combination with the apparatus of risk theory, indicators were determined
x(n) (38) and x(n) (41) and appropriate risk measures. The least squares functional
was used as a risk function, which (method) gives better results for this problem with
the specified initial conditions in comparison with other.
The analysis of measures (indicators) of risks obtained by this method (Table 2)
shows unique results. Their feature is that with the dimension of the system n =
2 the risks of both upper and lower levels are zero, R(n = 2) = Ri (n = 2) = 0.
Neither in the cited literature [7–12], nor in many others known to the authors, such
22
A. Zagorodny et al.
results are found. The mathematical explanation for this positive phenomenon is
equality S(2) = a(2) = 1/(n = 2) = 1/2, which according to (44) and (46) causes
the transformation of the values of these risks to zero. The informal justification for
this feature is as follows. In this work, the least squares functional is applied to the
quantities described by analytical dependences (33), (34), and therefore residuals
(45) are determined exactly in contrast to, in particular, problems [7–12] and many
others, where they are according to certain estimates with methodological errors. In
addition, in this problem, by minimizing the risk function by the method of least
squares, scalar values are determined in contrast to the vector, which occurs in many
other problems and can lead to additional coarsening of the results.
Based on these conditions and approaches, it was determined that the best solutions
for both upper and lower levels are obtained when the original system (3) with
dimension n is aggregated into a system with dimension n = 2. Only with this
dimension risks for upper and lower levels are zero. Therefore, the best method for
determining the approximate solutions of the initial system (3), (4) is a complex
method (47)–(50), according to which after aggregation to the dimension n = 2 is
determined by the upper level x according to (47), then the system is disaggregated
to the initial dimension n, the coefficients are determined, the amount of correction
according to (48) and (49) and according to (50)—all indicators x for the lower
level. In the end, the indicator of the upper level x equal to the sum of the lower
x i , i = 2, n, as required by condition (4), and all these indicators have zero risks.
Acknowledgements This research and the publication were formed in the process of joint research
between the Committee on Systems Analysis under the Presidium of the NAS of Ukraine and the
International Institute for Applied Systems Analysis (IIASA) under the joint project “Integrated
modeling for safe management of food, water and energy resources for sustainable social, economic
and environmental development”, 2022–2026.
References
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K., Iversen, L., Pulsifer, P.: Connecting top-down and bottom-up approaches in environmental
observing. Bioscience 71(5), 467–483 (2021)
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(2020)
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Bottom-Up Hybrid Modelling”. Trondheim (2016)
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of energetics. Am. J. Electrical Comput. Eng. 5(1), 25–31 (2021)
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7. Lai, K.K., Yu, L., Zhou, L., Wang, S.: Credit risk evaluation with least square support vector
machine. In: Rough Sets and Knowledge Technology, pp. 490–495 (2006)
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(RAITEA) (2019). https://ssrn.com/abstract=3353125
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least squares support vector machines. In: International Conference on Machine Learning
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risk assessment—Oberlain Nteukam Teuguia (HSBC) (2014). http://www.isfa.fr/la_recherche
Basic Matrix Forms of the System
Input–Output and Their Fundamental
Properties
Mykhailo Kulyk
Abstract It was investigated the fundamental properties of the matrices of three
systems of algebraic equations describing the key problems of intersectoral balance:
determination of output by the data of final demand, determination of output by the
data of added value, and establishing the interrelation between equilibrium prices and
output. All these problems have been solved with using the same method proposed
here and called the method of extrapolation to zero determinant. It was shown that the
system of equations recommended by numerous authors for establishing the interrelation between equilibrium prices and output volumes in output units is homogeneous. It was proved that, in this matrix, there exist at least n positive minors with
a dimension r = n − 1, where n is the dimension of matrix. This important feature
envisions the fact that, in the continual set of solutions of the corresponding system,
it is impossible to find even if a single vector that would correspond to the meaning
content of tables “Input–Output”. Therefore, this system cannot be used not only for
determining equilibrium prices and output in output units, but also for the solution of
other problems of intersectoral balance. It was proved that the matrix of the system
of equations for finding output by the data of final demand has a rank r = n, and its
determinant is always positive for any dimension of this matrix and non-negativeness
of the elements of final demand, and the last condition not always is necessary. It
was established that the system of equations for output determination by the data of
added value has a matrix whose rank is r = n, and its determinant is positive for all
values of variables satisfying the condition of input balance.
Keywords Matrix · Determinant · Rank · Vector · Output · Input · Balance · Price
1 Introduction
The theory of intersectoral balance (Input–Output) during its almost 100-year development became widespread and was used for various applications practically in
M. Kulyk (B)
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: info@ienergy.kiev.ua
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_2
25
26
M. Kulyk
all countries of the world as well as in different international economical, financial,
business, scientific, and other structures, organizations, and formations. As the information basis used in the development of the corresponding numerous mathematical
models, which are applied in the solution of a wide complex of the problems of
intersectoral balance, it is customary to use statistical tables (Input–Output), whose
present-day configuration can be written as follows:
X
f
x
Sector
1
i
j
n
Final
consumption
Output
1
x11
x1i
x1j
x1n
f1
х1
і
xi1
xii
xij
xin
j
xj1
xji
xjj
xjn
fj
хj
n
xn1
xni
xnj
xnn
fn
хn
vj
vn
;
zn
.
;
fi
;
хi
;
(1)
v
Added
value
v1
vi
z = x'
Total
input
z1
zi
zj
Here, i, j = 1, n are the numbers of sectors, x ij are the elements of the matrix of
intermediate sales X; f , x, ν, and z are the vectors of final demand, output, added
value, and total input, respectively.
V. Leontiev and his followers on the basis of tables Input–Output developed
three main systems of equations for two key problems of intersectoral balance,
namely: determination of output according to final demand data or data on value
added [1, 2] and establishment of interdependence between equilibrium prices and
output volumes in units of output [3–8]. In the theoretical and applied investigations
in the field of intersectoral balance, these models play the key part.
In the present work, we investigate and establish the main properties and
interrelations of matrices, appearing in the mentioned systems of equations.
Basic Matrix Forms of the System Input–Output and Their Fundamental …
27
2 System of Equations for the Interrelation of Equilibrium
Prices and Output Volumes
This model is based on the input balance
zj = xj =
n
.
xi j + ν j ,
j = 1, n.
(2)
i=1
With the use of (1), (2), transformations similar to those described in [3–8], and
also the relation
xj = x j pj,
j = 1, n,
(3)
where p j , x j are the equilibrium prices and output in output units, respectively, of the
j-th sector, the following vector–matrix equation of equilibrium prices was obtained
in [4–7]:
Q p + γ = p,
(4)
where the matrix
1
1
i
Q=
.
j
n
x11
x1
x1i
x1
x1 j
x1
x1n
x1
i
j
n
xi1
xi
xii
xi
xi j
xi
xin
xi
x j1
xj
x ji
xj
xjj
xj
x jn
xj
xn1
xn
xni
xn
xn j
xn
xnn
xn
(5)
and γ is the vector with elements
γj =
νj
,
xj
j = 1, n .
(6)
Since the unknown quantity x j figures in expression (6), the Eq. (4) with using
(6) is transformed to
K p = 0,
(7)
K = I − Q − G,
(8)
where the matrix
and G is the diagonal matrix with elements
28
M. Kulyk
gjj =
νj
,
xj
j = 1, n.
(9)
System (7) is homogeneous and has nonzero solutions because, as shown in [9], the
determinant of matrix (8) is equal to zero. The character of these solutions depends
cardinally on the rank of matrix (8). As shown in [9], the rank of this matrix is
r = n − 1, where p is the number of sectors in this system. This conclusion was
drawn on the basis of estimation of the (n − 1)-order minors of matrix (8). In [9], this
estimate was defined as a lower estimate. In what follows, we present the complete
proof that the rank of matrix (8) is equal to n − 1.
The determinant .s of matrix (8) with using (1), (6), and (9) can be written as.
.s = . n
1
i=1
xi
.x ,
(10)
where
|
| x −x −d
1
11
1
i ||
.x =
−x1i
xi
|
j |
|
−x1 j
|
|
−x1n
i
−xi1
− xii − di
−xi j
xj
−xin
j
−x j1
−x ji
− xjj − dj
−x jn
xn
−xn1
−xni
−xn j
− xnn − dn
|
|
|
| . (11)
|
|
|
|
|
With regard for (2), (5), and (6), it may be written
xiis = xi − xii − vi =
n
.
x ji , i = 1, n,
(12)
j=1 i/= j
and, as a result, determinant (11) will take the form
1
i
.x =
j
n
1 i
j n
|
| x s −xi1 −x j1 −xn1
|
11
| −x
xiis −x ji −xni
|
1i
|
| −x1 j −xi j x sj j −xn j
|
| −x1n −xin −x jn x s
nn
|
|
|
|.
|
|
|
|
|
(13)
It is reasonable to find the rank of matrix (8) by means of the analysis of its
minors of dimension n − 1, obtained by cancellation of the i-th row and column in
determinant (13).
It first performs this analysis for a system of dimension n = 4 with subsequent
generalization for arbitrary p. Determinant (13) for the case n = 4 has the form
Basic Matrix Forms of the System Input–Output and Their Fundamental …
. x(4)
|
| x s −x −x −x
21
31
41
|
11
|
s
−x32 −x42
| −x12 x22
=|
s
| −x13 −x23 x33
−x43
|
s
| −x14 −x24 −x34 x44
29
|
|
|
|
|
|.
|
|
|
(14)
After cancellation of the first row and column in (14), we arrive at the third-order
minor
|
|
| x s −x −x |
32
42
|
|
22
|
|
s
. x(4)m = | −x23 x33
(15)
−x43 |.
|
|
s
| −x24 −x34 x44
|
With the use of (2), we may represent determinant (15) as follows:
. x(4)m
|
| x s0 + x
−x32
−x42
12
| 22
|
s0
=|
−x23
x33 + x13
−x43
|
s0
|
−x24
−x34
x44
+ x14
|
|
|
|
|,
|
|
(16)
where
xiis0 = xiis − x1i , i = 2, 4,
(17)
and wherein, as shown in [9], it obtains the property
. x(4)m0
|
| x s0 −x −x
32
42
|
22
|
s0
= | −x23 x33
−x43
|
s0
| −x24 −x34 x44
|
|
|
|
| = 0.
|
|
(18)
Determinant (16) can be calculated with the use of relation given in [10]:
|
| a +b a +b
11
12
12
| 11
|
a21
a22
|
|
|
·
·
|
|
an1
an2
· a1n + b1n
·
a2n
·
·
·
ann
| |
| | a a
| | 11 12
| |
| | a21 a22
|=|
| | ·
·
| |
| | an1 an2
· a1n
· a2n
· ·
· ann
| |
| | b b
| | 11 12
| |
| | a21 a22
|+|
| | ·
·
| |
| | an1 an2
· b1n
· a2n
· ·
· ann
(19)
Applying successively equality (19) to all rows of determinant (16), we receive
at the calculating procedure for determing its value in the general case, presented in
Fig. 1.
In the first step of this procedure, we remove element x 12 from the first row of
determinant .x(4)m , in the second, element x 13 is removed from the second row, and,
in the third, element x 13 from the third row.
As a result, we obtain the equality
|
|
|
|
|
|.
|
|
|
30
M. Kulyk
Fig. 1 Calculating procedure for finding the value of determinant .x(4)m
.x x(4)m =
8
.
.i .
(20)
i=1
In this equality, the determinant .1 = 0 because it has the structure of determinant
S0
S0
· x33
> x23 · x32 ; by similar causes
(18); the determinant .2 is positive since x22
the determinants .3 and .5 are also positive; the determinants .4 , .6 , .7 , and .8
are equal to the products of their diagonal elements and are positive by virtue of the
positiveness of elements xi j , i, j = 1, n in structure (1).
So, the determinant .x(4)m is positive for arbitrary values of the elements of
structure (1), and the rank of matrix (8) at its dimension p = 4 is equal to r =
n − 1 = 3.
It is important to note the following specific features of determinant .x(4)m . All
matrices presented in the second and third tiers of Fig. 1 also have positive determinants, which can be seen from its structure. For the positiveness of .x(4)m , it is
necessary and sufficient that even if one of its elements x1i , i = 2, 4 should be
positive.
Further, we will prove that matrices (8) possess this property (r = n − 1) for
arbitrary values of their dimensions. For this purpose, we use the method of complete
induction, namely, assuming that the rank of matrix (8) at n ≤ m is equal to
r = m − 1,
(21)
we prove that, if the dimension of matrix (8) is n = m + 1, its rank
r = m.
(22)
Basic Matrix Forms of the System Input–Output and Their Fundamental …
31
In the case where the matrix K has a dimension n = m, determinant (13) can be
transformed to the form
|
|
| x S −x21 −x31 −xi1 −xm1 |
|
| 11
|
| −x
S
12 x 22 −x 32 −x i2 −x m2 |
|
|
|
S
| −x13 −x23 x33 −xi3 −xm3 |
(23)
.x(m) = |
|
| −x1i −x2i −x3i xiiS −xmi |
|
|
S
|
| −x1m −x2m −x3m −xim xmm
|
|
|
|
and it is equal to zero, and the corresponding (m − 1)—order minor, obtained by
analogy with the previous case, has the form
.x(m)M
| s
| x22
|
| −x23
|
= || −x2i
| −x
| 2m
|
−x32
s
x33
−x3i
−x3m
−xi2
−xi3
xiis
−xim
|
−xm2 |
|
−xm3 ||
−xmi ||.
|
s
xmm
|
|
(24)
After transformation of the determinant .x(m)M , similar to (16), we reduce it
to the form suitable for applying relation (19) in order to estimate its properties
(Fig. 2). In this and subsequent tables, all tabular forms for the simplification and
better visualization of the drawing should be perceived as determinants.
According to the assumptions made by the method of complete induction, all
determinants of this table (except .5 , which is equal to zero) are positive and possess
properties inherent in the determinants of Fig. 1.
In the case where the dimension of matrix (8) is n = m + 1, determinant (13) can
be reduced to the form of (m − 1)-th order
.x(m+1)
|
| xS
11
|
| −x
12
|
| −x
|
13
|
= | −x1i
|
| −x1m
|
| −x1m+1
|
|
−x21
S
x22
−x23
−x2i
−x2m
−x2m+1
−x31
−x32
S
x33
−x3i
−x3m
−x3m+1
−xi1
−xi2
−xi3
xiiS
−xim
−xim+1
−xm1
−xm2
−xm3
−xmi
S
xmm
−xmm+1
and it also is equal to zero and has a minor of the m-th order:
|
−xm+11 ||
−xm+12 ||
−xm+13 ||
|
−xm+1i |.
|
−xm+1m |
|
S
|
xm+1m+1
|
|
(25)
32
Fig. 2 Calculating scheme of the analysis of the determinant .x(m)M properties
M. Kulyk
Basic Matrix Forms of the System Input–Output and Their Fundamental …
.x(m+1)M
|
| xS
|
22
| −x
23
|
|
| −x2i
=|
| −x2m
|
| −x2m+1
|
|
−x32
S
x33
−x3i
−x3m
−x3m+1
−xi2
−xi3
xiiS
−xim
−xim+1
−xm2
−xm3
−xmi
S
xmm
−xmm+1
|
−xm+12 ||
−xm+13 ||
|
−xm+1i |
|.
−xm+1m |
|
S
|
xm+1m+1
|
|
33
(26)
After transformations of determinant (26), similar to those performed in the
previous cases, we carry out the calculating process presented in Fig. 3.
Here, as before, determinant (26) is equal to a sum of determinants in which .6
= 0.
.x(m+1)M =
6
.
.i ,
(27)
i=1
The dimension of all determinants .i , i = 1, 5 in Table 3 is equal to n = m.
They have identical structure because are formed by the multiplication of positive
element x1i , i = 3, m + 1 by the determinant of (m–1)-th order, obtained by the
cancellation of i-th row and column from the determinant . x(m+1)M with subsequent
transformations similar to those described above. Such determinants (of order m −
1) are positive by the condition of the method of complete induction. Therefore,
determinant (27) is positive, the rank of matrix (8) for its dimension n = m + 1
corresponds to (22), and, at an arbitrary value of n, its rank is
r = n − 1.
(28)
Example. To verify the adequacy of transformations (19)–(27), we apply them
for the tables Input Output on the example of Ukraine for 2012. For this purpose, we
use tables given in [9], preliminarily aggregating them to four sectors (Fig. 4).
According to Fig. 4 data, the determinant .x(4) (14) has the form and determinant
(15) is
.x(y)
|
| 123461
|
| −1282
|
= || −348
| −71006
|
|
−7454
188348
−35913
−286965
−11977
−25087
98801
−98986
|
−104030 |
|
−161979 ||
−62540 ||.
456957 ||
|
(29)
34
Fig. 3 Calculating algorithm of finding the properties of determinant .x(m)M
M. Kulyk
Basic Matrix Forms of the System Input–Output and Their Fundamental …
35
Fig. 4 Aggregated indicator Input–Output in Ukraine for 2012, millions of hryvnas, prices of 2012
.x(y),M
|
|
188348
|
−25087
−161979
| (187066 + 1282)
|
|
98801
|
−35913
−62540
|
=|
(98453 + 348)
|
|
456957
|
−286965
−98986
|
(385951
+ 71006)
|
|
|
|
|
|
|
|
|
|
|.
|
|
|
|
|
|
(30)
Application of equality (19) to the determinant .x(y)M leads to the process
illustrated in Fig. 5.
Here, as earlier, we perform with the use of (19) step-by-step cancellation of the
second term in diagonal elements of the initial and newly formed determinants. As
a result, we receive the equality
.x(y)M =
8
.
.i .
(31)
i=1
In Table 1, we present the values of 8 determinants-terms of sum (31) and the
determinant .x(y)M , calculated according to formula (31) as well as (for verification)
by the classical method. We observe here a practically complete coincidence between
the values of this determinant, calculated by different methods, which confirms the
validity of procedure (19)–(27).
36
M. Kulyk
Fig. 5 Calculating process of finding the determinant .x(y)M
3 System of Equations for Determining Output by the Data
of Final Demand
This system of equations is based on the output balance
n
.
xi j + f i = xi , i = 1, n.
(32)
j=1
V. Leontiev transformed system (32) to matrix–vector form
(I − A)x = f ,
(33)
where
ai j =
xi j
, i, j = 1, n.
xj
(34)
It is reasonable to represent the determinant of matrix I–A from (33) in the
following way:
.ν = . n
1
i=1
xi
.xν ,
(35)
Value
0
.3
.4
.5
.6
.7
.8
.x(y)M
Verification
1.24375952 0.0089491701 0.00462241732 0.04077715009 0.00896214627 0.0001721866353 0.0000316783328 1.3072749 1.307275
Determinant .1 .2
Table 1 Values of determinants from Fig. 4, (×1015 )
Basic Matrix Forms of the System Input–Output and Their Fundamental …
37
38
M. Kulyk
where
. xv
|
| x1 − x11
|
| −xi1
|
= || −x j1
| −x
n1
|
|
|
−x1i
−x1 j
−x1n |
|
xi − xi j −xi j
−xin ||
−xi j x j − xi j −x jn ||.
−xni
−xn j xn − xnn ||
|
(36)
Subtracting and adding the quantities f i , i = 1, n , to the diagonal elements of
(36), after transformations similar to those used in constructing (16), we obtain a
determinant analogous by its value to (36):
. xv
| s
| x v + f 1 −x1i
−x1 j
−x1n
| 11
| −x
sv
xii + f i
−xi j
−xin
|
i1
=|
| −x j1
−x ji x sjvj + f j −x jn
|
sv
| −xn1
−xni
−xn j xnn
+ fn
|
|
|
|
|
|,
|
|
|
(37)
where
xiisv = xi − xii − f i , i = 1, n.
(38)
The structures of determinants (37) and (16) are identical, and, therefore, if the
elements of the vector of final demand fi , i = 1, n, are nonnegative, then determinant
(37) is nonzero and positive for arbitrary values of the dimension n of system (33),
elements of the matrix A, and vector of final demand f .
4 System of Equations for Determining Output by the Data
of Added Value
The system is based on the input balance (2), which can be rewritten in matrix-vector
form as follows:
(I − Q)x = ν,
(39)
where the elements of matrix Q are
qi j =
x ji
, i, j = 1, n.
xj
As earlier, we represent the determinant of matrix Q by the product
(40)
Basic Matrix Forms of the System Input–Output and Their Fundamental …
.ν = .n
1
i=1
xi
39
. xν ,
(41)
where
. xν
|
| x −x
−xi1
−x j1
−xn1
11
| 1
|
xi − xii −x ji
−xni
| −x1i
=|
| −x1 j
−xi j x j − x j j −xn j
|
| −x1n
−xin
−x jn xn − xnn
|
|
|
|
|
|.
|
|
|
(42)
Further, as in the derivation of (37), we transform relation (42), using, however,
instead of the vector f , elements of the vector v. As a result, we obtain
. xv
| s
| x v + v1 −xi1
−x j1
−xn1
| 11
| −x
sv
xii + vi −x ji
−xni
|
1i
=|
| −x1 j
−xi j x sjvj + v j −xn j
|
sv
| −x1n
−xin
−x jn xnn
+ vn
|
|
|
|
|
|,
|
|
|
(43)
where
xiisv = xi − xii − νi , i = 1, n.
(44)
Determinant (43), by analogy with determinant (37), has the structure of determinant (16). Therefore, the decomposition of Table 1 can be applied to (43). This fact
gives us all reasons to assert that determinant (43) is nonzero and positive for arbitrary values of the dimension n of matrix I–Q and elements xi j and νi , i, j = 1, n ,
satisfying condition (2).
5 Analysis and Comments
All three problems considered above were solved with using the same method, which
can be called the method of extrapolation to zero determinant. We showed that the
system of equations (4) is homogeneous and has a singular matrix (8). We proved
that this matrix has, as a minimum, n positive minors with a dimension r = n − 1,
where n is the dimension of matrix (8). This important feature envisions the fact that,
from the continual set of the solutions of system (7), it is impossible to isolate even if
a single vector that would correspond to the meaning content of tables Input–Output.
It is known from the theory of homogeneous systems (see, e.g., [10]) that, in the case
where the rank of matrix of the system similar to (7) is less by one than its dimension,
the ratios between k-th and l-th elements of any vector of its solutions is determined
rigidly and unambiguously. As shown in [9], these ratios as applied to system (7)
have the form
40
M. Kulyk
Pk
xk
= ,
Pl
xl
(45)
where Pk , xk , P, and xl are the equilibrium prices and outputs in output units of
the sectors k and l, respectively. Obviously, the ratio of equilibrium prices of two
arbitrary sectors cannot be equal to the ratio of their output volumes in «physical»
units. This means that the system of equations (7) gives erroneous or degenerate
solutions. Therefore, it is impossible to use systems (4) and (7) not only for finding
equilibrium prices and outputs in output units, but also for the solution of other
problems of intersectoral balance.
The matrix I–A of system (33) for finding output by the data of final demand has
a rank r = n, where n is its dimension, and its determinant is always positive for any
dimension of the matrix and the non-negativeness of elements of the final demand.
Note that the last condition, as can be seen even from Table 1 and Figs. 2, 3, 4 and
5, is necessary not always.
The system of equations (39) for finding output by the data of added value has
the matrix I−Q, its rank is r = n, and its determinant is positive for all values of
xi , xi j , ν j , satisfying the condition of input balance (2). Note that the property of
positiveness of the determinant of matrix I−Q was proved by R. Bellman in [11]
with the use of Gram determinants. This proof is more compact than the method
of extrapolation to zero determinant, but the latter can be applied to all problems
considered here and, which is quite important, enables one to obtain not only the
sign, but also the values of determinants under study.
We should especially emphasize the unique property of systems (33) and (39),
where the matrices I−A and I−Q have determinants whose values depend on the
right-hand sides of these systems, namely, on the values of elements of the vectors
of final demand f i and added value ν j , i, j = 1, n, respectively. It is easy to make
certain of this by comparing the determinant . x(m+1)M from Table 3 with determinants (37) and (43). The initial cause of such property lies in the fact that systems
(33) and (39) are based on the input (2) and output (32) balances. Therefore, the
proof of the positiveness of the determinant of the matrix I−Q, which is given in
[11] and does not take into account these properties, needs additional justification.
References
1. Leontiev, W.: Studies in the Structure of the American Economy: Theoretical and Empirical
Explorations in Input-Output Analysis, Oxford (1953)
2. Ghosh, A.: Input-output approach to an allocation system. Economica 25, 58–64 (1958)
3. De Mesnard, L.: Price consistency in the Leontief model. Dans Cahiers d’économie Politique
71, 181–201 (2016)
4. Carter, A.P.: Structural Change in the American Economy. Harvard University Press,
Cambridge (1970)
5. Handbook of Input-Output Table Compilation and Analysis. UN, New York (1999)
6. Eurostat Manual of Supple, Use and Input-Output Tables, Eurostat, European Commission,
Luxembourg (2008)
Basic Matrix Forms of the System Input–Output and Their Fundamental …
41
7. Handbook on Supply and Use Tables and Input Output-Tables with Extensions and Applications. United Nations Publication, New York (2018)
8. Miller, R.E., Blair, P.D.: Input-Output Analysis, Foundations and Extensions, 2nd edn.
Cambridge University Press, Cambridge (2009)
9. Kulyk, M.M.: Revision of the possibilities of the models of equilibrium prices and outputs
in the theory of intersectoral balance. In: Problemy Zahal’noi Enerhetyky—The Problems of
General Energy, vol. 4, issue 47, pp. 5–22 (in Russian, in English). https://doi.org/10.15407/
pge2016.04.005
10. Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers. McGraw-Hill,
New York (1961)
11. Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill, New York (1960)
Development and Application of New
Price Models in the System of Means
Input–Output
Mykhailo Kulyk
Abstract Theoretical analysis of the known monetary price models of intersectoral
balance and the corresponding digital experiments have shown that these models have
methodical errors, which are manifested in the application of real data in calculations.
The article analytically proves that the input model based on the input balance does
not provide a such balance in real cases, namely, when all sectoral prices are not the
same in modulus. Leontiev price model provides such balance only in an extremely
unrealistic case, when all sectoral prices are equal to one. For real cases, when
sectoral prices may differ significantly, the errors in these models reach unacceptably
high values. Therefore, currently known monetary price models can be used only
in theoretical studies as degenerate cases. The paper presents two new price models
(price model based on output balances and the corresponding model of price indices).
It is mathematically proved that both new models satisfy the output balances, i.e. do
not have methodical errors. Their application provides zero imbalances of output
when using both theoretical and realistic data packets. The generalized model of
price indices allows to use as a base state of the Input–Output system its arbitrary
state, in which only the appropriate conditions of balance are provided. It is proved
that the Leontief price model is a special case of the generalized model of price
indices proposed in the paper.
Keywords Price model · Leontiev model · Generalized model of price indices ·
Input balance · Output balance · Error
1 Introduction
Price models in the theory of intersectoral balance in mathematical and applied implications are a unique phenomenon in the environment of systems analysis models.
Mathematically, these models are indeterminate systems of algebraic equations, the
M. Kulyk (B)
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: info@ienergy.kiev.ua
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_3
43
44
M. Kulyk
dimension of which is twice less than the number of unknowns. Under the conditions of the problem, the researcher is given the opportunity (more precisely, the
requirement) to redefine the algebraic system based on the specifics of the problem.
Otherwise, the model provides an infinite number of solutions.
Even more importantly, price models, which have long been widely used in Input–
Output tools, have been put into practice without strict mathematical justification.
Attempts of the author to build a mathematically reasonable grounds for the adequacy
of existing price models led him to the conclusion that they contain methodological
errors. In this work the correctness of this conclusion will be proved, however, despite
the existence of methodical errors, the models of equilibrium prices in combination
with the Leontiev models [1] and Ghosh models [2] have received a very wide and
diverse application (especially recently).
There is a significant attention in recent years publications to the direct impact
of the price factor on the functioning of both national and interregional economics
[3–9]. Here are some typical examples. So, it was investigated and found by Flaaen
et al. [3] that since the beginning of 2018, after an unprecedented tariffs increase in
the US manufacturing sector, there has been (contrary to expectations) a reduction
in employment, rising producer prices and production resources costs.
As well using Leontiev price model it has been shown [4] that the abolition of
fuel subsidies in Malaysia’s manufacturing sectors has led to an average increase in
producer prices of 32%, which has a serious impact on inflation in the country. Sharp
changes in world oil prices could jeopardize Malaysia’s economic stability.
Recently, an important theoretical and practical result has been obtained [5] to
take into account the impact of price changes on the matrix of direct (intersectoral)
costs, wherein it is proposed to adjust the matrix coefficients by using Cobb–Douglas
functions or elasticity coefficients.
Theoretically and practically important studies by Wiebe et al. [6] was performed
using a closed input–output model that analyzes the economy of the planet as a whole.
Compared to the usual scenario, this model shows a decrease in global production
of materials by about 10%, while the impact on employment is small but positive.
Noting that the transition from the resource sector to the service sector provides more
opportunities for highly qualified professionals and women.
Deviations between direct prices based on labor costs, production and market
prices according to the Input–Output tables in China in the period from 1990 to 2012
were studied by Li [7]. It was found that cross-deviations between direct and market
prices are average 17–18%. Variations in direct prices can explain about 70% of
changes in market prices.
A study of the comparison of market and shadow prices in the US economics and
social sphere using the means of intersectoral balance was performed in [8]. This
research allows to identify areas in which it is necessary to change the combination
of results or costs to improve social welfare.
Based on the means of intersectoral balance and tools of integrated network analysis, global energy flows in international trade researched by Chen et al. [9], taking
into account the vulnerability of the environment at the global, regional and national
levels. In this study at a global level the nature of the small world is revealed, in which
Development and Application of New Price Models in the System …
45
economies are closely interconnected through the transfer of energy. It is shown that
at the national level, key countries (USA, China, Germany) are at the forefront of
network centralization, and the security of the implemented energy supply is assessed
for each economy.
Research on the daily economic costs of control strategy of the COVID-19 mitigation effect for informing the governments of Brazil and Colombia was conducted
in a short time using the methodology and means of cross-sectoral balance [10].
In work [11] an important study was conducted to model the economic impact of
COVID-19 on the economies of affected countries. A roadmap has been developed to
assess the vulnerability of the supply chain through disease outbreaks at the company
level, national and global levels.
Increased attention in recent publications is paid to the study using the means
of intersectoral balance of interaction and interconnection in the economic and
environmental spheres of certain groups of countries, their sectors and regions
[12–20].
In particular, population growth and climate change have made food, energy and
water security a global issue. To address this issue, a problem-oriented Input–Output
model has been developed by Tabatabaie et al. [12], which provides linkages between
the food, energy and water (FEW) for the Northwest Pacific. The results showed that
agricultural crops have the highest sensitivity to water and energy consumption. To
minimize costs and environmental impact, more use should be made of surface water,
hydroelectric power and wind energy.
Research on the relationship between water, energetics and carbon emissions in
China has been conducted on the basis of intersectoral balance tools [13]. It was
found that the results of the interaction between water, energy and carbon emissions
in light, heavy industry and services were comparable, agriculture accounted for
about 64% of the country’s water supply. It is also shown that indicators of water
and energy consumptions and carbon emissions can significantly affect the country’s
sustainable development strategies.
Due to a number of objective factors, Iran-Iraq and Turkey have joined forces
to address the strategic security issue of Water-Energy-Food (WEF). According to
the UN, the demand for water, energy and food in these countries has grown significantly over the past 20 years. This can exacerbate the conflict probability, especially across transboundary water resources. The interrelationships of the WEF as a
holistic approach to finding regional solutions to common problems in these countries
have been studied by Zarei [14]. Cooperation and interaction between the scientific
community and decision-makers are vital to the complex challenges of WEF security
management and development.
The transnational interregional input–output approach is used in work [15] to
analyze the relationship between East Asia WEF (Japan, China, and South Korea) to
assess competing this resource needs and environmental performance. This analysis
demonstrates the hidden virtual flows of water, energy and food embodied in intraregional and transnational trade. China has been shown to be a purely virtual exporter
of WEF resources due to its trade in low value-added and high-pollution sectors.
46
M. Kulyk
It turns out that the Input–Output methodology can be effectively applied even for
the analysis of such complex processes as Brexit. The research of Giammetti et al.
[16] provides an opportunity to identify areas that are key in the structure of relations
between the United Kingdom and the European Union. It is possible to assess which
tariffs are the most influential in the negotiations, which export sector needs to be
intensified, and which imports should be protected. The results show that Brexit will
be a problem not only for Britain, but any form of it can affect the global production
system.
The possibilities of the Input–Output toolkit are used quite effectively in the
analysis of trade integration of the economies of the Western Balkans [17]. Their
results allows to identify industries associated with high economic effects and to
form an idea of sectoral interdependence of economics. The multi-regional data set
was used to study the international integration of the region’s economics in order to
participate in global value chains. Shown, that although this indicator has recently
tended to grow, some economies benefit from this is more than others.
The use of methodology and means of intersectoral balance often deals with the
action of counter-orientation factors. In Brazil, in particular, greenhouse gas (GHG)
emissions have been reduced by 12% over the last three decades due to reduced
deforestation. At the same time, GHG emissions without this factor increased by
18%, and gross domestic product increased by 17%. As GHG emission reduction
activities are quite costly, there is a problem of ensuring sustainable development of
the country. A comparison of GHG emission multipliers in the Brazilian economics
with employment and income multipliers (especially in agriculture) provided an
opportunity to develop appropriate solutions to this problem [18].
The analysis shows that the use of structures, capabilities and means of intersectoral balance over time significantly expands geographically and increases in quantitative and qualitative terms. Characteristically, these positive phenomena occur
despite the existence (as mentioned above) of certain methodological errors in the
key models of the input–output apparatus, namely, in equilibrium price models. We
can be sure that the removal of these errors will contribute to even greater dissemination and increase the quality of research results provided by the methodology of
intersectoral balance.
The purpose of this publication is mathematical and computational confirmation
of the fact of methodical errors that occur in existing pricing models (IO), identifying
their causes, development, description and research of new IO models for this purpose
and the possibilities of their application.
2 Analysis of Existing Price Models of Intersectoral
Balance
The methodological basis for building equilibrium price models in the Input–Output
system is a set of IO matrices shown in (1).
Development and Application of New Price Models in the System …
1 i Xj n
⎤
x11 x1i x1 j x1n
⎢ xi1 xii xi j xin ⎥ ;
⎢
⎥
⎣ x j1 x ji x j j x jn ⎦
xn1 xni xn j xnn
[
]
v1 vi v j vn v;
[
]
z1 zi z j zn z = x ' .
1
i
j
n
⎡
⎡ f ⎤
f1
⎢ fi ⎥ ;
⎢ ⎥
⎣ fj⎦
fn
⎡x⎤
x1
⎢ xi ⎥ ;
⎢ ⎥
⎣ xj⎦
xn
47
(1)
Here i, j = 1, n—sector numbering, x ij —elements of the matrix of intermediate
sales X; f , x, v, z—vectors of final demand, output, value added and total input,
respectively. In balanced tables (1) are always provided, as is known, dependencies:
z i = xi ,
n
.
j=1
vj =
n
.
(2)
fi ,
(3)
i=1
as well as the balance of output
n
.
xi j + f i = xi ,
(4)
xi j + v j = x j .
(5)
j=1
and input balance
n
.
i=1
We draw attention to the fundamental need to ensure the dependences (2)–(5) in
the construction of correct Input–Output models.
Currently, in the theory and practice of intersectoral balance, a significant number
of price models have been developed, which are naturally divided into two groups,
namely, models whose matrices are formed on physical data, and monetary models.
As points out De Mesnard [19] physical models are usually used only in theoretical
research, because the formation of their matrices is associated with difficulties in
providing statistical information.
The available price models of the intersectoral balance use value added in one
form or another as initial information. Therefore, these models are based on the
input balance (5) [19–24]. The most transparent of the available schemes for the
formation of price models is given in Handbook of Input–Output Table Compilation
and Analysis, UN [21]. Its developers use the input balance as a basis (5). In this
case, each of its j-th column is divided into output x j , j = 1 − n in units of output
48
M. Kulyk
and get a system of equations
p = (I − A' )−1 γ ,
(6)
where p, γ —price vectors, parts of value added per unit of output
γ j = v j /x j ,
(7)
[
]
[ ]
A = ai j = xi j /x j —matrix from the Leontiev output model for its (model)
monetary form.
As can be understood from this Handbook, the developers of model (6), (7)
consider it a monetary model of equilibrium prices in the system of models of
intersectoral balance. Carter [20] and a number of authors in remote publications
formulate this statement clearly and unambiguously. We will show that this statement is not true. First, we show that the model (6), (7) in the general case does not
satisfy the input balance Eq. (5). To do this, consider model (6), (7) in an expanded
form
pj −
xi j
xjj
xn j
vj
x1 j
p1 −
pi −
pj −
pn = ,
xj
xj
xj
xj
xj
j = 1, n,
(8)
and taking into account the obvious dependencies
xj = pjx j,
j = 1, n.
x ji = pi x ji , i, j = 1, n,
(9)
(10)
we get the expression for the input balance
x j = x1 j
p1
pi
pn
+ xi j
+ x j j + xn j
+ vj,
pj
pj
pj
(11)
or
xj =
n
.
i=1
xi j
pi
+ vj,
pj
j = 1, n.
(12)
Dependence (12) gives grounds to conclude that model (6), (7) is not a model from
the class of Input–Output models, because it does not satisfy the input balance (5) in
the general case. This balance is satisfied according to (12) only in one degenerate
case, namely when prices in all sectors have the same value
pi = p j , i, j = 1, n.
(13)
Development and Application of New Price Models in the System …
49
It is clear that model (6), (7) cannot have any practical application, because in
real calculations case (13) (equality of prices in all sectors) is nonsense. A very
limited consideration of this model is found in purely theoretical analysis, although
not always with positive results, as will be discussed below.
We show further that the scheme of obtaining the price model described in [21–
23] does not lead to the model (6), (7), but to a completely different result. To do
this, according to [21], first, using the input balance (5) and the dependence (10), we
obtain a system
p1 x 11 + pi x i1 + p j x j1 + pn x n1 + v1 = p1 x 1 ,
p1 x 1i + pi x ii + p j x ji + pn x ni + vi = pi x i ,
p1 x 1 j + pi x i j + p j x j j + pn x n j + v j = p j x j ,
p1 x 1n + pi x in + p j x jn + pn x nn + vn = pn x n .
(14)
Dividing each of the equations of system (14) by the corresponding x j , we obtain
the final system in an expanded form
p1 xx111 + pi xxi11 + p j xj11 + pn xxn11 + xv11 = p1 ,
x
p1 xx1ii + pi xxiii + p j xjii + pn xxnii + xvii = pi ,
x1 j
xi j
x
x
v
p1 x j + pi x j + p j x jjj + pn xnjj + x jj = p j ,
x
p1 xx1nn + pi
x in
xn
+ pj
x jn
xn
+ pn xxnnn +
vn
xn
(15)
= pn ,
which is obviously presented in matrix form
'
(I − A ) p = γ , γ j = v j /x j ,
j = 1, n,
(16)
where the matrix
]
[ ] [
A = a i j = x i j /x j , i, j = 1, n
(17)
is a matrix of coefficients of intermediate sales in physical form.
Thus, model (6), (7) cannot satisfy the input balance (5), because (as can be seen
from (14)) it is satisfied by a completely different model (16), (17).
Although outwardly the models (6), (7) and (16), (17) are similar, their essences
differ radically due to the striking discrepancy, as is known, of the values of the
elements of the matrices A and A. Note that the dependences (14)–(16) are obtained
by identical transformations, i.e., model (16), (17) does not contain methodical errors.
Thus, the scheme of transformation of the monetary model IO into the monetary
price model (6), (7) given in [21–23] cannot be carried out, because it leads to the
price model in physical form (16).
As shown by the calculations of equilibrium prices on examples with real monetary
data (Appendices A1–A6), the methodical errors of model (6), (7) can reach such
values that they can’t be ignored.
50
M. Kulyk
To ensure the possibility of working with input data in monetary form, Leontiev
developed (1986) a model of price indices which is given, in particular, in [19, 22–24].
The essence of the model of price indices is that its two states are considered—basic
and current. This model differs from model (6), (7) only by the right part (7), namely,
it (model) has the form
(I − A' )β = vc ,
(18)
where β j —the price index of the j-th sector,
vcj = v j /x j ,
j = 1, n,
(19)
0
vcj
= v 0j /x 0j ,
j = 1, n,
(20)
vcj = v j /x 0j ,
j = 1, n.
(21)
And besides for the base state
and for the current state
By direct substitution
of (22) into (18), one can verify that for the base state the
| |
| 0|
price modules | p j | satisfy the input balance (5).
| 0|
| p | = 1,
j
j = 1, n.
(22)
Therefore, the solution of system (18) directly provides the vector of price indices
β j , j = 1, n the current state of the model in relation to its base state.
This model of price indices has a very important drawback. It does not have
(ignored) the output in its physical units, which sharply narrows the application
scope of this model. In addition, ignoring its value often leads (shown below) to
unacceptably large errors.
De March et al. proposed [22] a generalized pricing model in the form of
p = (I − A' )−1 qv,
(23)
]
[ ] [
which at A = ai j = xi j /x j and [qv] j = vcj according to (21) becomes a model of
]
[ ] [
equilibrium price indices (18), and when A = a i j = x i j /x j and [qv] j = v j /x j
turns into a price model (16). Model (23) is used unchanged in Handbook on Supply
and Use Tables and Input Output-Tables with Extensions and Applications, UN
[23]. The use of model (23) does not provide additional advantages or disadvantages
compared to models (16) and (18).
De Mesnard [19] in fact confirms the validity of the conclusions regarding the
model of price indices (18)–(21).
Development and Application of New Price Models in the System …
51
Thus, the analysis of present monetary price models of the intersectoral balance
gives grounds to claim that they all have certain defects and need to be clarified or
developed.
3 New Monetary Price Models in the Input–Output System
This section presents two new monetary price models, namely, strictly speaking the
price model and the model of price indices that do not contain methodical errors.
Unlike the models discussed in the previous section, these models are not based
on the input balance (5), but on the output balance (4). To build such a model, we
first use the balance (4) in expanded form and, dividing each of its equations by the
corresponding output in physical form x i , i = 1, n, we obtain a system of equations
(24)
x11
x1
xi1
xi
x j1
xj
xn1
xn
+ xx1i1 +
+ xxiii +
x
+ x jij +
+
xni
xn
+
x1 j
x1
xi j
xi
xjj
xj
xn j
xn
+
+
+
+
x1n
x1
xin
xi
x jn
xj
xnn
xn
+ xf11 = xx11 = p1 ,
+ xfii = xxii = pi ,
f
x
+ x jj = x jj = p j ,
+
fn
xn
=
xn
xn
(24)
= pn .
Using the dependence (9), the system of equations (24) is transformed into the
system (25)
x
x11
p + xx1i1 p1 + x11j p1 + xx1n1 p1 +
x1 1
xi j
xi1
xii
p + xi pi + xi pi + xxini pi + xfii
xi i
x
x
x
x j1
p j + xjij p j + xjjj p j + xjnj p j +
xj
x
xn1
p + xxnin pn + xnnj pn + xxnnn pn +
xn n
= p1 ,
= pi ,
fj
= pj,
xj
f1
x1
fn
xn
(25)
= pn .
The system of equations (25) is a price monetary model in an expanded form.
After entering the notation
sii =
n
.
xi j /xi , i = 1, n
(26)
j=1
we obtain a monetary price model in matrix form
(I − S) p = μ,
(27)
in which the matrix S has a diagonal structure with non-zero elements (26), p—price
vector, μ—vector particles of final demand μi per physical unit of output of the i-th
sector
52
M. Kulyk
μi = f i /x i , i = 1, n.
(28)
To verify the conformity of model (27) to the structure of IO, its i-th equation
⎞
⎛
n
.
pi − ⎝
xi j ⎠ pi = f i /x i , i = 1, n
j=1
is sufficient multiply by x i , , as a result, it is converted into the equation of the balance
of output (4). This indicates that the new model has no methodical errors.
The main destination of the new model is to determine equilibrium prices and
solve related problems based on the Input–Output methodology without methodical
errors and limitations. No less important possibilities of its application are connected
with the fact that the matrix (I − S) in the system (27) has a diagonal structure. This
feature allows to find solutions of this system in analytical form. In turn, this provides
an opportunity to determine the equilibrium prices and their changes in the current
state through their value in the base state. That is, the prospect of building a new
model of price indices opens up without the limitations and shortcomings that have,
in particular, models (16)–(18).
Using the output balance (4), we obtain the diagonal element of the matrix S in
the form
sii = (xi − f i )/xi .
(29)
After that, the analytical solution of the system (27) is determined as
pi = μi xi / f i , i = 1, n.
(30)
The presence of dependence (30) makes it possible to determine price indices in
monetary form.
For the current state, the analytical solution of the system of equations (27) has
the form
pi = μi xi / f i , i = 1, n
(31)
pi0 = μi0 xi0 / f i0 , i = 1, n.
(32)
and for the base state
Determination of price indices β i , i = 1, n carried out as the ratio of their value
pi in the current state to the value pi 0 in the base state
βi = pi / pi0 = μi xi f i0 /μi0 xi0 f i , i = 1, n.
(33)
Development and Application of New Price Models in the System …
53
With the involvement of (28) and after the introduction of symbols .xi = xi − xi0
and .x̄ i = x̄ i − x̄ i0 we obtain the final dependence for price indices
βi = (1 + . xi /xi0 )/(1 + . x i /x i0 ), i = 1, n.
(34)
The generalized model of price indices also satisfies the balance of output (4). To
prove this, we use the i-th equation of system (27) in the form
⎛
⎞
n
.
βi pi0 − ⎝
xi j /xi ⎠βi pi0 = f i /x i ,
j=1
or
⎛
⎞
n
.
pi − ⎝
xi j /xi ⎠ pi = f i /x i ,
j=1
or
xi −
n
.
xi j = f i .
j=1
The last expression is the modified balance of output (4) in monetary form. Thus,
the generalized model of price indices also does not contain methodical errors.
Model (34) is called a generalized model of price indices, because when . x i = 0
it becomes Leontiev price model. This can be seen, in particular, by comparing the
prices of Appendix A4 at . x i = 0 and prices of Appendix A2. That is, the Leontiev
price model (18)–(21) is a special case of model (34).
When performing both theoretical and applied research, it is advisable to have not
only dependencies for price indices, but also expressions for the deviation (change)
of equilibrium prices compared to the baseline. Such expressions are obtained, in
particular, by forming dependencies
(
)
pi − pi0 / pi0 = βi − 1,
(35)
. pi = (βi − 1) pi0 , i = 1, n.
(36)
or
Formula (36) is needed, in particular, to determine price multipliers.
It is worth emphasizing that the dependence (30) provides another proof that the
model (27) has no methodical errors. It is enough for this to substitute (28) in (30),
after which we obtain an obvious dependence pi = xi /x i , i = 1, n.
54
M. Kulyk
In contrast to the Leontiev price model, the generalized model of price indices
allows to use as a base state of the Input–Output system its arbitrary state, in which
only the appropriate conditions of balance are provided.
4 Examples
The capabilities and indicators of the above price models were demonstrated by
calculations in which the initial data were formed on the basis of information provided
in Eurostat Manual of Supple, Use and Input–Output Tables [22] (Table 1). This is
a system of real reporting data in the Input–Output format (Germany 1995).
In order to study as widely as possible the properties and capabilities of the
studied models on the basis of information Table 1 and other additional sources, two
universal input data packets were formed. The first of them (theoretical) contains
input data, which as a result of their application give the values of equilibrium prices
in the district of the unit. The second packet (realistic) provides data that cause
the resulting sectoral equilibrium prices to differ several times depending on the
technological nature of the sector, which is close to reality.
In this chapter, the equilibrium prices were calculated according to the four models
discussed above.
4.1 Equilibrium Price Model in the Input–Output System,
Built on the Basis of Input Balance
p = (− A' )−1 γ , γ j = v j /x j , j = 1 − n
This model requires the availability of output indicators in physical units, which are
not in [22]. Therefore, the condition of equality according to the modulus of monetary
and physical indicators of output in the base state was used for calculations (Table
1)
| 0| | 0 |
| x | = | x |,
j
j
j = 1, n,
(37)
which analytically provides equality (22).
These features in no way limit the content of the conclusions in the comparative
analysis. Deviations of initial values vj and x j are chosen small, but such (± 30%) that
the deviation of equilibrium prices was noticeable. All data and results of calculations
of equilibrium prices on this model according to a theoretical packet are provided in
Appendix A1. Equilibrium prices locate in the district of the unit.
1,079,446
43,910
96,115
Input, total
Other services
6
3,637
7334
72,717
14,986
Business services
5
426
3,559
558,230
Trade
4
25,480
304,584
1,552
Construction
3
7,930
1,131
25,675
Manufacturing
2
Value added
Agriculture
1
2
1
245,606
130,599
1,747
31,027
14,190
3875
64,167
540,063
341,699
11,225
65,755
74,399
5,296
41,082
607
4
3
1
Trade
Construction
Intermediate sales Z (millions of euro)
Manufacturing
Agricul-ture
Table 1 Reporting system Input–Output
710
692,487
437,270
15,058
193,176
10,835
23,457
11,981
5
Business
services
762
508,918
391,340
22,070
34,223
21,008
9,155
30,360
6
Other
services
15,219
z=x'
v
442,280
268,554
343,355
196,063
619,342
f
7
Final
demand
43,910
508,918
692,487
540,063
245,606
1,079,446
x
8
Output
Development and Application of New Price Models in the System …
55
56
M. Kulyk
4.2 Leontiev Price Model
(− A' )β = vc, p j = β j p0j ,
j = 1, n
During calculating prices for this model, the base state was considered to be the
state shown in Table 1. The interrelation between monetary and physical output was
determined by the dependence (37), so there was an equality (22). This led, in turn,
that the equilibrium prices for this model modulo coincide with price indices
| |
|pj| = βj,
j = 1, n.
The initial data according to the theoretical packet and indicators of price indices
and equilibrium prices according to the specified model are given in Appendix A2.
It is noteworthy that the equilibrium prices for the considered models (items 1 to 2)
differ significantly.
4.3 Equilibrium Price Model Based on the Output Balance
(−S) p = μ, μ i = f i /x i , i = 1, n
Price calculations for this model were performed on the basis of the Table 1 and taking
into account the dependences (37), (22) for the possibility of further comparison of the
obtained indicators with the indicators of other models. Unlike the models according
to items 1 to 2, this model allows to check the execution of the output balance (4)
in monetary form. In addition, this model does not use value added v as the initial
vector, but final demand f . Because in the calculations according to items 1 to 2 used
a vector v that differs from the vector v0 according to Table 1, to ensure compliance
with the requirements of the balance of the system of tables IO (1) it is necessary to
change the vector f . This was done according to the dependence
f = (−A)(−B ' )−1 v,
because f = (−A)x and x = (−B ' )−1 v, where B—Ghosh matrix. The calculation
of equilibrium prices according to this model and verification of the balance of output
with their use are given in Appendix A3.
Development and Application of New Price Models in the System …
57
4.4 Generalized Model of Price Indices and Equilibrium
Prices
β i = (1 + Δxi /xi0 )/(1 + Δ x i /x 0i ), pi = β i p0i , i = 1, n
In the calculations for this model were also used the data of Table 1 and dependences
(37), (22). Calculations of price indices and equilibrium prices for this model are
provided in Appendix A4. It is noteworthy that the values of prices for this model and
the model based on the output balance (4) (Appendix A3) are completely the same.
It is very important that the equilibrium prices obtained for these models exactly
satisfy the balance of output according to the theoretical data packet.
Significant results were obtained by digital modeling of these models using a
realistic data packet. Combined with similar indicators for equilibrium prices in the
theoretical packet, this provides an opportunity to carry out an extended comparative
analysis of the results. A comparative analysis of the results for the theoretical packet
is provided in Appendix A5 and for the realistic packet in Appendix A6.
5 Analysis of Results
Theoretically, all known price models, built on the basis of input balance, are empirical, developed without a strict mathematical justification and have methodical errors.
In particular, model (6), (7), the right part of which uses the share of value added
per physical unit of output (7), can be considered a model of the class Input–Output
only if prices in all sectors are the same. Otherwise, this model according to (12)
does not provide a input balance (5), which is an integral requirement for IO class
models. In practical application this model will provide a methodical error, because
in real calculations price equality in all sectors is impossible.
The model of price indices (Leontiev price model) (18)–(21) is forced to use as
its base state the initial data, which are even less realistic than in model (6), (7). If
in model (6), (7) methodical errors will be absent at equality of prices in all sectors,
then to achieve such result in model of price indices in its basic state the prices in all
sectors should be not only identical, but also equal units. The use of such a basic state
of the model narrows the scope of its possible applications to unrealistic options.
Another very important disadvantage of this price indices model is that the equations of its current state do not take into account the output in units of output, despite
the fact that this indicator has a decisive influence as on prices as on their |indices.
|
Therefore,
with a significant (but realistic) difference between the modules |x j | and
| |
|x j | price errors p j according to this model can reach unacceptable values.
This chapter proposes and investigates in detail two monetary price models,
namely, the price model based on the output balance and the price indices model
formed on the basis of this price model.
It is theoretically proved that both the new price model and the generalized model
of price indices satisfy the balance of output (4), i.e., they do not have methodical
58
M. Kulyk
errors. Both series of calculations carried out in the course of research (Appendix
A1–A6) confirm and concretize the theoretical conclusions and provisions given
above. Appendices A1–A4 show the equilibrium prices for the respective models,
and Appendices A5, A6—their errors for the theoretical and realistic packets, respectively. For comparative analysis, the equilibrium price indicators obtained using
the price models (26)–(28) and (34)–(36) were chosen as a reference for both data
packets, as they satisfy the balances of output in monetary form. In addition, these
models provide a complete match of the significatives of the obtained equilibrium
prices, which also confirms their accuracy (Appendices A3–A6). Deviations of input
data at value added in the calculations according to the theoretical packet were chosen
in the range of ±30% in all Appendices A1–A6. The same deviations for this packet
were synchronously selected for physical output in Appendices A1–A5. These deviations for value added are quite realistic, but the deviations of physical output in
real cases can be much larger. In particular, for high-tech sectors (industry) monetary output in modulus may be several times higher than the physical, which is not
typical, for example, for agriculture.
To bring the equilibrium prices closer to reality, a analogous series of calculations
was performed according to the realistic data packet (Appendix A6). In this packet,
the ratio of sectoral monetary output to physical output in modules was in the range
of 1.5–5.9 depending on the technological characteristics of the sectors.
A comparative analysis of the obtained results of determining the equilibrium
prices in the intersectoral balance on four mathematical models and two packets of
initial data allows us to make the following generalizations.
All existing mathematical models of equilibrium prices contain in their structure
certain methodical errors, which depend on the nature of the input data. In particular,
for Leontiev price model, these errors are absent when prices in all sectors of the
module are equal to one. The price model, built on the input balance, does not provide
errors only if the modulo prices in all sectors are equal. However, these conditions
in the practical application of these models can not be met. Even in the conditions
of application of the theoretical data packet, when the obtained equilibrium prices
for these models are close to one (Appendices A1, A2), the error rate for the model
according to the input balance model is 18.24% and for Leontiev price model—
30% (Appendix A5). When calculating a realistic data packet, these models provide
catastrophic errors, ||namely,
|| the vector of errors for the model according to the||input
||
balance has a norm ||δp j || = 116.8% and according to Leontiev price model—||δp j ||
= 83.2% (Appendix A6). If the results of the application of the theoretical data packet
these two models should be assessed as grossly inaccurate, then the results of the use
of a realistic data packet should be classified as incorrect. The accuracy of the two
new price models, namely, the model based on the output balance and the generalized
model of price indices was checked by calculating the imbalance of output
δxi = xi −
n
.
j=1
pi x i j − f i .
(38)
Development and Application of New Price Models in the System …
59
According to Appendices A5 and A6, the imbalance (38) is zero both when
calculating prices for the theoretical data packet and when using a realistic packet
for this purpose. It is also important that the eponymous indicators of price obtained
using these models coincide in five to six decimal places.
6 Conclusions
Theoretical research and digital experiments have shown that the currently known
monetary price models of the intersectoral balance theory do not contain methodical
errors only in some cases that are degenerate. In particular, Leontiev price model has
no methodical errors only in the case when all sectoral equilibrium prices are modulo
one. The monetary price model, built on the input balance, will not have methodical
errors, provided that the equilibrium prices obtained according to it will all be the
same. Failure to comply with these requirements leads to a violation of the monetary
input balance, and, as a consequence, to the emergence of methodical errors in
decisions. By digital modeling, arrays of methodical errors are obtained when using
these models depending on the input information. When using the theoretical data
packet (changes in value added and physical outputs are in the range of ±30%, the
resulting prices are placed near the unit) the error rate for Leontiev price model
reaches 30% and for the model formed on the input balance—more than 18%. Both
of these significatives are unacceptable for practical use.
When a realistic data packet was used (the ratio of monetary and physical output
modulo increased several times), the error rates for these models increased catastrophically: for the Leontief price model it was about 83% and for the model formed
on the basis of input balance—almost 117%. According to such error rates, the
relevant models should be classified as incorrect.
Therefore, the currently existing price models and models derived from them
analyzed in the paper are not suitable for practical use according to realistic data, as
such use leads to unacceptably large errors. They can be used only in situations where
the equilibrium sectoral prices of the module are equal to each other, or (moreover)
equal to one. That is, they can be used only in theoretical research as degenerate
cases.
Proposed and comprehensively studied in the work of two new price models (price
model, built on the basis of output balance, and the corresponding model of price
indices) are devoid of these shortcomings. It is mathematically proved that both new
models satisfy the balances of output. The consequence of this is that when they are
used in digital modeling, zero output imbalances are provided for both theoretical
and realistic data packets. In contrast to the Leontief price model, the generalized
model of price indices allows us to use as the base state of the Input–Output system
its arbitrary state, in which only the appropriate conditions of balance are provided.
It is proved that the Leontiev price model is a special case of the generalized
model of price indices proposed in the work.
60
M. Kulyk
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9. Chen, B., Li, J.S., Wu, X.F., Han M.Y., Zeng L., Li Z., Chen G.Q.: Global energy flows embodied
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for Assessing the Impacts of Lockdown Measures, TD NEREUS 1–2020, Núcleo de Economia
Regional e Urbana da Universidade deSãoPaulo(NEREUS) (2020).
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Process Integr. Optimization Sustain. 4, 183–186 (2020)
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and Turkey. Water-Energy Nexus 3, 81–94 (2020)
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Cambridge University Press, Cambridge (2009)
Mathematical Models and Software
for Studying the Elasticity of Building
Structures and Their Systems
Vitalii Babak , Artur Zaporozhets , Vladyslav Khaidurov ,
Leonid Scherbak , Ihor Bohachev , and Tamara Tsiupii
Abstract The main task of mathematical and computer modeling is detailed analysis of the studied object and/or processes in various conditions and environments
using modern supercomputers. Modern problems of science and technology require
significant computing resources, as well as the development of appropriate software, which can be used to simplify and efficiently investigate such processes. This
chapter proposes mathematical computational models for studying the elasticity of
building structures and their systems, which can be used in the energy sector, as well
as the corresponding basic fragments of software implementation in the MATLAB
computer mathematics system. Mathematical models are represented by both ordinary differential equations and partial differential equations. Computational experiments for elastic deformation of one-dimensional and two-dimensional building
structures are presented.
Keywords Software · Partial differential equations · Mathematical and computing
modelling · Simulation · MATLAB
1 Introduction
At present, it is difficult to imagine modern science without the active use of mathematical modeling and the use of software for solving problems and performing tasks
in various fields of science and technology.
Mathematical modeling is a theoretical and practical study of an object, in which
not the object itself is studied, but some artificial or natural system that describes
V. Babak · A. Zaporozhets (B) · V. Khaidurov · L. Scherbak · I. Bohachev
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: a.o.zaporozhets@nas.gov.ua
A. Zaporozhets
State Institution “The Institute of Environmental Geochemistry” of NAS of Ukraine, Kyiv,
Ukraine
T. Tsiupii
National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_4
63
64
V. Babak et al.
its work as much as possible. Mathematical modeling is one of the most effective
methods for testing and predicting the operation of various devices and technological processes. Modeling elements have been used since the very beginning of the
appearance of the exact sciences.
Technical, economic, biological and other systems studied in modern science
cannot be studied by common theoretical and empirical methods. A direct classical
experiment can often be dangerous or can be quite expensive, since most of the real
studied systems exist in a single copy.
The available numerous methods for solving such non-stationary problems lead,
of course, to great computational difficulties associated with the solution of large
algebraic equations, which are not always efficient. These circumstances significantly
complicate the modeling of elastic–plastic cylindrical bodies under the influence of
temperature and force loads.
Therefore, the scientific and technical task of creating mathematical models,
software, as well as the application and development of methods and means of
computer modeling of the elastic deformation of building structures and their systems
is relevant.
2 Problem Setting
The main task sets before the authors of the work is the analysis of physical and technical phenomena and processes that are relevant in modern conditions in building
objects and structures, as well as their description in the form of mathematical models,
in particular, in the form of ordinary differential equations and equations of mathematical physics. The next task is to develop appropriate flexible software that allows
to quickly obtain approximate solutions, as well as to present the results obtained in
forms convenient for further analysis. To do this, it is necessary to consider the basic
mathematical models in one-dimensional and two-dimensional space.
3 One-Dimensional Case in Elasticity Problems
3.1 Construction of Difference Scheme for Solving Problems
For an example of modeling, let’s take an equation describing the deflection of
continuous beam. It looks like this [1–3]:
y I V = f (x).
Let write the fourth-order derivative by the finite difference method. First, we
write the expression for the central derivative of the first order [4–6]:
Mathematical Models and Software for Studying the Elasticity …
yi' =
65
yi+0.5 − yi−0.5
.
h
We write the second derivative as follows
yi'' =
'
'
yi+0.5
− yi+0.5
=
h
yi+1 −yi
h
−
h
yi −yi−1
h
=
yi−1 − 2yi + yi+1
.
h2
The third order derivative will look like this:
yi−0.5 −2yi+0.5 +yi+1.5
''
''
−
− yi−0.5
yi+0.5
h2
=
h
h
yi+1.5 − 3yi+0.5 + 3yi−0.5 − yi−1.5
=
.
h3
yi''' =
yi−1.5 −2yi−0.5 +yi+0.5
h2
The fourth derivative will look like this:
yi+2 −3yi+1 +3yi −yi−1
'''
'''
−
yi+0.5
− yi−0.5
h3
=
h
h
yi+2 − 4yi+1 + 6yi − 4yi−1 + yi−2
=
.
h4
yiI V =
yi+1 −3yi +3yi−1 −yi−2
h3
Therefore, the numerical scheme for this equation has 5 points. To solve it, you
can use an analog of sweep for systems of equations with a 5-diagonal matrix. Our
equation can be written like this [1, 6, 7]:
ai yi−2 + bi yi−1 − ci yi + di yi+1 + ei yi+2 = f i , i = 1, n.
(1)
Now we write the solution of this system of equations in the form:
yi = K i yi+1 + L i yi+2 + Mi .
(2)
Using Eq. (2), it can be removed the first two components in the formula (1). To
do this, we write that
yi−1 = K i−1 yi + L i−1 yi+1 + Mi−1 , yi−2 = K i−2 yi−1 + L i−2 yi + Mi−2 .
Now let’s substitute the last two formulas in (1):
ai (K i−2 yi−1 + L i−2 yi + Mi−2 ) + bi (K i−1 yi + L i−1 yi+1 + Mi−1 ) − ci yi
+ di yi+1 + ei yi+2 = f i , i = 1, n;
ai (K i−2 (K i−1 yi + L i−1 yi+1 + Mi−1 ) + L i−2 yi + Mi−2 )
+ bi (K i−1 yi + L i−1 yi+1 + Mi−1 ) − ci yi + di yi+1 + ei yi+2 = f i , i = 1, n.
Then
66
V. Babak et al.
yi (ai K i−2 K i−1 + ai L i−2 + bi K i−1 − ci ) + (ai K i−2 L i−1 + bi L i−1 + di )yi+1
+ ei yi+2 = f i − ai K i−2 Mi−1 − ai Mi−2 − bi Mi−1 , i = 1, n.
The last equation can be written in the form (2):
ai K i−2 L i−1 + bi L i−1 + di
yi+1
ci − (ai K i−2 K i−1 + ai L i−2 + bi K i−1 )
ei
yi+2
+
ci − (ai K i−2 K i−1 + ai L i−2 + bi K i−1 )
ai K i−2 Mi−1 + ai Mi−2 + bi Mi−1 − f i
.
+
ci − (ai K i−2 K i−1 + ai L i−2 + bi K i−1 )
yi =
Then, we have
ei
L i−1 (ai K i−2 + bi ) + di
; Li =
;
ci − K i−1 (ai K i−2 + bi ) − ai L i−2
ci − K i−1 (ai K i−2 + bi ) − ai L i−2
(ai K i−2 + bi )Mi−1 + ai Mi−2 − f i
, i = 1, n.
Mi =
ci − K i−1 (ai K i−2 + bi ) − ai L i−2
Ki =
We also take into account that
xi+1 = xi+2 = x0 = x−1 = 0, a1 = a2 = b1 = en−1 = en = dn−1 = 0.
The next step after the obtained mathematical models is the software implementation, the description of which is given below.
3.2 Fragments of Difference Numerical Method Realization
As signed above, we need to solve the system:
ai yi−2 + bi yi−1 − ci yi + di yi+1 + ei yi+2 = f i , i = 1, n.
Below is a software realization of the sweep for systems with 5-diagonal matrices,
obtained during mathematical description of one-dimensional objects and structures
in a discrete representation of ordinary differential equations of the 4-th order, which
are underlie in the modeling of elastic deformation of the studied objects.
%%% Call is made either from the command line, or using another function and the input data passed to it
function x = prog5dig(a,b,c,d,e,f,n)
K(1) = d(1)/c(1);
L(1) = e(1)/c(1);
M(1) = −f(1)/c(1);
zn
= c(2)−b(2)*K(1);
Mathematical Models and Software for Studying the Elasticity …
67
K(2) = (d(2)+b(2)*L(1))/zn;
L(2) = e(2)/zn;
M(2) = (b(2)*M(1)−f(2))/zn;
for i = 3:n
zn = c(i)−a( i )*L(i−2)−(a(i)*K(i−2)+b(i))*K(i−1);
K(i) = (d(i)+(a(i)*K(i−2)+b(i))*L(i−1))/zn;
L(i) = e(i)/zn;
M(i) = ((a(i)*K(i−2)+b(i))*M(i−1)+a(i)*M(i−2)−f(i))/zn;
end
x(n) = M(n);
x(n−1) = K(n−1)*x(n)+M(n−1);
for i = n−2:−1:1
x(i) = K(i)*x(i+1)+L(i)*x(i+2)+M(i);
end
Let us now describe a more detailed account of the limiting conditions of the
problem. Let the following conditions be set at the left end in the problem of deflection
of a continuous homogeneous beam [6, 8]:
y(a) = y A , y ' (a) = y 'A .
Since we have limit conditions at the left end, we need to take them into account
in the first and second equations of the system as follows:
a1 y−1 + b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 .
If, y(a) = y A , then y−1 = y A and
a1 y A + b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 ;
b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A .
If y ' (a) = y 'A , then
y0 − y−1
y0 − y A
= y 'A ,
= y 'A , y0 = hy 'A + y A .
h
h
We substitute the last equation into the first equation of the system after taking
into account the conditions of the first kind (Dirichlet conditions) at the left end of
the beam [8–11]:
( '
)
b1 hy A + y A − c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A .
Then
(
)
−c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A − b1 hy 'A + y A .
Lets write the second equation of the system:
68
V. Babak et al.
a2 y0 + b2 y1 − c2 y2 + d2 y3 + e2 y4 = f 2 .
Taking into account that
y0 = hy 'A + y A ,
we have
(
)
a2 hy 'A + y A + b2 y1 − c2 y2 + d2 y3 + e2 y4 = f 2 ;
(
)
b2 y1 − c2 y2 + d2 y3 + e2 y4 = f 2 − a2 hy 'A + y A .
During developing a software realization, taking into account the limit conditions
at the left end will look like this:
y2 = DyA*h + yA;
f(1) = f ( 1) − a(1)*yA;
a(1) = 0;
f(1) = f ( 1) − b(1)*y2;
b(1) = 0;
f(2) = f ( 2) - a(2)*y2;
a(2) = 0;
From a physical point of view, the combination of conditions of this type means
that the left end is hard fixed and it is immovable.
Similarly, we derive formulas for taking into account the same combination of
limiting conditions at the right end. In this case, the last and penultimate equations
of the system [5, 6, 8, 11] are corrected.
If at the right end we have the following conditions:
y(b) = y B , y ' (b) = y B' ,
then we correct the equation
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en yn+2 = f n .
Then yn+2 = y B ,
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en y B = f n ;
an yn−2 + bn yn−1 − cn yn + dn yn+1 = f n − en y B .
Now, in this equation, we take into account that
n+1
n+1
= y B −y
= y B' ; yn+1 = y B − hy B' ;
y ' (b) = y B' ; y ' (b) = yn+2 −y
h
h
(
)
an yn−2 + bn yn−1 − cn yn + dn y B − hy B' =( f n − en y B);
an yn−2 + bn yn−1 − cn yn = f n − en y B − dn y B − hy B' .
Mathematical Models and Software for Studying the Elasticity …
69
Now let’s take into account the limit conditions at the right end in the penultimate
equation of the equations system:
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn + en−1 (yn+1 = f n−1
);
'
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn + en−1 y B − hy
( B = f n−1);
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn = f n−1 − en−1 y B − hy B' .
During developing a software realization, taking into account such a combination
of limiting conditions at the right end will look like this:
yn_1 = yB − DyB*h;
f ( n) = f ( n) − e ( n) * yB;
e ( n) = 0;
f ( n) = f ( n) − d( n) * yn_1;
d ( n) = 0;
f ( n-1) = f ( n − 1) - e( n − 1) * yn_1;
e ( n−1) = 0;
Now lets consider the case when the following limit conditions are given:
y ' (b) = y B' , y '' (b) = y B'' .
Similarly, the last two equations of the equations system will be adjusted:
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en yn+2 = f n ;
yn+2 − yn+1
= y B' ; yn+2 = hy B' + yn+1 ;
h
(
)
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en hy B' + yn+1 = f n ;
an yn−2 + bn yn−1 − cn yn + yn+1 (dn + en ) = f n − hy B' en ;
yn+2 −yn+1
h
− yn+1h−yn
yn+2 − 2yn+1 + yn
=
;
h
h2
yn+2 − 2yn+1 + yn
''
yn+2
= y B'' ;
=
h2
hy B' + yn+1 − 2yn+1 + yn
hy B' − yn+1 + yn
= y B'' ;
= y B'' ;
2
h
h2
hy B' − yn+1 + yn = h 2 y B'' ; yn+1 = hy B' + yn + h 2 y B'' ;
)
(
an yn−2 + bn yn−1 − cn yn + hy B' + yn + h 2 y B'' (dn + en ) = f n − hy B' en ;
)
(
an yn−2 + bn yn−1 − cn yn = f n − hy B' en − hy B' + yn + h 2 y B'' (dn + en ).
''
yn+2
=
Similarly, we take into account the limiting conditions in the penultimate equation
of the equations system:
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn + en−1 (yn+1 = f n−1 ;
)
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn + en−1 hy B' + (yn + h 2 y B'' = f n−1);
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn = f n−1 − en−1 hy B' + yn + h 2 y B'' .
70
V. Babak et al.
Taking into account the above information, the limiting conditions can be written
as follows:
f(n) = f(n) − e(n)*h*DyB;
d(n) = d(n) + e(n);
e(n) = 0;
f(n) = f(n)−d(n)*(h^2*D2yB+h*DyB);
c(n) = c(n)−d(n);
d(n) = 0;
f(n−1) = f(n−1)−e(n−1)*(h^2*D2yB+h*DyB);
d(n−1) = d(n−1)+e(n−1);
e(n−1) = 0;
The combination of these conditions means that when the beam is loaded, the
right end is fixed in horizontal plane, but it can move vertically [2, 3, 12, 13].
Now lets consider the case when the following limit conditions are given:
y(a) = y A , y '' (a) = y ''A ;
y(b) = y B , y '' (b) = y B'' .
Let’s correct the first equation of the equations system according to the given
conditions on the left end:
a1 y−1 + b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 , y−1 = y A ;
a1 y A + b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 ;
b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A ;
y−1 − 2y0 + y1 2 ''
y A + y1 − h 2 y ''A
''
y−1
;
=
; h y A = y A − 2y0 + y1 ; y0 =
2
h
2
(
)
y A + y1 − h 2 y ''A
b1
− c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A ;
2
(
)
)
(
y A − h 2 y ''A
b1
y1 + d1 y2 + e1 y3 = f 1 − a1 y A − b1
.
− c1 +
2
2
Similarly, we will do this in the second equation of the system:
(
)
y A + y1 − h 2 y ''A
+ b2 y1 − c2 y2 + d2 y3 + e3 y4 = f 2 ;
a2
2
(
)
(
y A − h 2 y ''A
a2 )
b2 +
y1 − c2 y2 + d2 y3 + e3 y4 = f 2 − a2
.
2
2
We will carry out the same procedure on the right end. We first correct the last
equation of the system:
Mathematical Models and Software for Studying the Elasticity …
71
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en yn+2 = f n ; yn+2 = y B ;
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en y B = f n ;
an yn−2 + bn yn−1 − cn yn + dn yn+1 = f n − en y B ;
y B − h 2 y B'' + yn
yn+2 − 2yn+1 + yn
2 ''
;
;
y
−
2y
+
y
=
h
y
;
y
=
B
n+1
n
n+1
B
h2
2
(
)
y B − h 2 y B'' + yn
an yn−2 + bn yn−1 − cn yn + dn
= f n − en y B ;
2
(
)
)
(
y B − h 2 y B''
dn
yn = f n − en y B − dn
.
an yn−2 + bn yn−1 − cn −
2
2
''
yn+2
=
Now let’s correct the penultimate equation:
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn + en−1 yn+1 = f n−1 ;
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn
(
)
y B − h 2 y B'' + yn
= f n−1 ;
+ en−1
2
(
en−1 )
yn
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 +
2
(
)
y B − h 2 y B''
.
= f n−1 − en−1
2
Let us consider the last case when the following limiting conditions are given:
'''
'''
y(a) = y A , y ''' (a) = y '''
A , y(b) = y B , y (b) = y B .
Then
y0 −2y1 +y2
0 +y1
''
− y−1 −2y
y0'' − y−1
y2 − 3y1 + 3y0 − y−1
h2
h2
=
=
;
h
h
h3
yn+2 −2yn+1 +yn
n +yn−1
''
− yn+1 −2y
y '' − yn+1
h2
h2
=
= y B''' = n+2
h
h
yn+2 − 3yn+1 + 3yn − yn−1
=
.
h3
'''
y−1
= y '''
A =
'''
yn+2
Let’s correct the first equation of the equations system according to the given
conditions on the left end:
a1 y−1 + b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 ; y−1 = y A ;
a1 y A + b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 ;
b1 y0 − c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A ;
y2 − 3y1 + 3y0 − y−1
'''
y−1
; y2 − 3y1 + 3y0 − y A = h 3 y '''
= y '''
A;
A =
h3
72
V. Babak et al.
h 3 y '''
A − y A − y2 + 3y1
;
y0 =
3
)
( 3 '''
h y A − y A − y2 + 3y1
− c1 y1 + d1 y2 + e1 y3 = f 1 − a1 y A ;
b1
3
( 3 '''
)
)
(
h yA − yA
b1
y2 + e1 y3 = f 1 − a1 y A − b1
.
− (c1 − b1 )y1 + d1 −
3
3
We will do the same in the second equation of the system:
(
)
h 3 y '''
A − y A − y2 + 3y1
+ b2 y1 − c2 y2 + d2 y3 + e3 y4 = f 2 ;
3
(
h 3 y '''
a2 )
A − yA
y2 + d2 y3 + e3 y4 = f 2 − a2
.
(b2 + a2 )y1 − c2 +
3
3
a2
We will carry out the same operation on the right end. We first correct the last
equation of the system:
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en yn+2 = f n ; yn+2 = y B ;
an yn−2 + bn yn−1 − cn yn + dn yn+1 + en y B = f n ;
an yn−2 + bn yn−1 − cn yn + dn yn+1 = f n − en y B ;
yn+2 − 3yn+1 + 3yn − yn−1
'''
yn+2
;
= y B''' =
h3
y B − h 3 y B''' + 3yn − yn−1
;
y B − 3yn+1 + 3yn − yn−1 = h 3 y B''' ; yn+1 =
3
(
)
y B − h 3 y B''' + 3yn − yn−1
= f n − en y B ;
an yn−2 + bn yn−1 − cn yn + dn
3
(
)
)
(
y B − h 3 y B'''
dn
yn−1 − (cn − dn )yn = f n − en y B − dn
.
an yn−2 + bn −
3
3
Now let’s correct the penultimate equation:
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn + en−1 yn+1 = f n−1 ;
an−1 yn−3 + bn−1 yn−2 − cn−1 yn−1 + dn−1 yn
(
)
y B − h 3 y B''' + 3yn − yn−1
+ en−1
= f n−1 ;
3
(
en−1 )
yn−1 + (dn−1 + en−1 )yn
an−1 yn−3 + bn−1 yn−2 − cn−1 +
3
(
)
y B − h 3 y B'''
.
= f n−1 − en−1
3
Similarly, other combinations of limit conditions can be derived for the given
problem of deflection of a continuous multi-span beam.
Mathematical Models and Software for Studying the Elasticity …
73
3.3 Testing and Analysis of the Results
Below is a software realization of the single-span beam model with 1 m long, the
ends of which are hard fixed. The load w(x) = −10−4 sin 10x acts on the beam. It
is necessary to find the deflection line of this beam as a result of such a load.
Mathematical model of the task:
⎧ IV
⎨ y = 104 sin 10x;
(3)
y(0) = 1, y ' (0) = 0;
⎩
y(1) = 1, y ' (1) = 0.
Realization fragment looks like this:
%%% limiting conditions
y2 = DyA*h + yA;
f(1) = f(1) − a(1)*yA;
a(1) = 0;
f(1) = f(1) − b(1)*y2;
b(1) = 0;
f(2) = f(2) − a(2)*y2;
a(2) = 0;
yn_1 = yB − DyB*h;
f(n) = f(n) − e(n)*yB;
e(n) = 0;
f(n) = f(n) − d(n)*yn_1;
d(n) = 0;
f(n−1) = f(n-1) − e(n−1)*yn_1;
e(n−1) = 0;
The analytical solution of problem (3) has the next form:
x 2 (20 cos 10 − 6 sin 10 + 40)
2
x 3 (60 cos 10 − 12 sin 10 + 60)
+ 1.
−
6
y(x) = sin 10x − 10x +
According to Fig. 1, it can be seen quite well that the limit conditions are
performed, since the values at the ends are 0. The derivatives are also equal to zero,
since the graph at the extreme points is parallel to the x-axis.
Now we have an example of a single-span beam, which is in a quiet state. A certain
point force acts on the beam. As a result, the point of the beam to which this force
was applied deviated from its initial position by a certain amount. In this problem, it
also needs to find the deflection line of a given beam under a given load.
Now we demostrate an example of another type problem. A beam with hard fixed
ends is in an equilibrium position. The length of the beam is 1 m. A force acts on
a point contained at a distance of 1/3 m from the left end, deviating this point from
its base position on 1 m. It is necessary to calculate the deflection of the entire beam
74
V. Babak et al.
Fig. 1 Numerical task (3)
solution
as a result of such point action on it. The mathematical model of the problem looks
like this:
⎧ IV
y = 0;
⎪
⎪
⎨
y(0) = 0, y ' (0) = 0;
⎪ y(1) = 0, y ' (1) = 0;
⎪
⎩ (1)
y 3 = −1.
(4)
The software realization of the task has the following form.
%%% Accounting for the limiting and internal conditions of the task
i = round((n+4)/3);
a(i) = 0;
b(i) = 0;
c(i) = −1;
d(i) = 0;
e(i) = 0;
f(i) = −1;
plot(x(i),f(i),’*’);
hold on;
y2 = DyA*h + yA;
f(1) = f(1) − a(1)*yA;
a(1) = 0;
f(1) = f(1) − b(1)*y2;
b(1) = 0;
f(2) = f(2) − a(2)*y2;
a(2) = 0;
yn_1 = yB − DyB*h;
f(n) = f(n) − e(n)*yB;
e(n) = 0;
f(n) = f(n) − d(n)*yn_1;
d(n) = 0;
f(n-1) = f(n−1) - e(n−1)*yn_1;
e(n−1) = 0;
According to the obtained graphical results (Fig. 2), it can be seen that there is
satisfaction of the limit conditions, as well as the value at the x = 1/3 point (the point
is indicated by asterisk in Fig. 2) is equal to (−1)—the deviation of this point from
the initial position (problem condition). The graph also passes through this point.
Mathematical Models and Software for Studying the Elasticity …
75
Fig. 2 Numerical task (4)
solution
The next task will be in which the left end of a single-span beam with 1 m long
is hard fixed on a fixed support, and the right end is also hard fixed, but the support
moves along the vertical (as a result of the load on this beam). The load acting on
the beam is given by the formula:
w(x) = −100(1 − x)2 .
Next, it is necessary to calculate the deflection of the beam as a result of the impact
of this load on it. The mathematical model for this task is as follows:
⎧ IV
2
⎨ y = −100(1 − x) ;
'
(5)
y(0) = 0, y (0) = 0;
⎩ '
y (1) = 0, y '' (1) = 0.
The fragment of the software realization of this task looks as follows:
% Dy4=−100*(1−x)^2, y(0)=0, Dy(0)=0, Dy(1)=0, D2y(1)=0.
Specifying the ends of the integration interval
yA = 0; % beam position value at left end
DyA = 0; % beam slope value at the left end
DyB = 0; % beam position value at right end
D2yB = 0; % beam slope value at the right end
h = (xb−xa)/(n+3); % spacing step
x = xa:h:xb; % creating an integration range
Setting the coefficients of the equations system
y2 = DyA*h + yA; % second point from left
f(1) = f(1) − a(1)*yA; % correction of the right side of the 1st
equation
a(1) = 0;% resetting the used coefficient
f(1) = f(1) − b(1) y2; % correction of the right side of the 1st
equation (for the derivative)
b(1) = 0; % resetting the used coefficient
f(2) = f(2) − a(2)*y2; % correction of the right side of the 2nd
equation
a(2) = 0;% resetting the used coefficient
f(n) = f(n) − e(n)*h*DyB; % second point from right
d(n) = d(n) + e(n); % correction of the right side of the n equation
76
V. Babak et al.
Fig. 3 Numerical task (5)
solution
e(n) = 0;% resetting the used coefficient
f(n) = f(n)−d(n)*(h^2*D2yB+h*DyB); % correction of the right side
of the n equation (using the derivative as a limit condition)
c(n) = c(n)−d(n);
d(n) = 0;
f(n−1) = f(n−1)−e(n−1)*(h^2*D2yB+h*DyB);
d(n−1) = d(n−1)+e(n−1);
e(n−1) = 0;
Y = prog5dig(a,b,c,d,e,f,n);
yn_1 = h^2*D2yB + h*DyB + Y(n);
yn_2 = h*DyB + yn_1;
Y = [yA y2 Y yn_1 yn_2];
for i=1:length(x)
t = x(i);
yy(i) = − (5*t^6)/18 + (5*t^5)/3 − (25*t^4)/6 + 5*t^3
− (5*t^2)/2;
end
plot(x,Y,’−−’,x,yy,’−’);
grid on;
legend(‘Numeric Solution’, ‘Analitic Solution’)
Analytical and numerical solutions of problem (5) are presented in Fig. 3.
The analytical solution of problem (5) has the next view:
y(x) = −
5x 5
25x 4
5x 2
5x 6
+
−
+ 5x 3 −
.
18
3
6
2
Now let’s modify the problem for multi-span beams. Suppose we have a beam
with three spans.
⎧ IV
4
⎨ y = −10
( 1 )x; ( 2 )
y(0) = y 3 = y 3 = y(1) = 0;
⎩
y(0) = y ' (1) = 0, y(1) = y ' (1) = 0.
(6)
The numerical solution of problem (6) together with the internal conditions are
presented in Fig. 4.
We also present the task of a beam with three spans, which is fixed on moving
boundary supports, while the internal supports are fixed. The mathematical model
of the problem looks like follows:
Mathematical Models and Software for Studying the Elasticity …
77
Fig. 4 Numerical task (6)
solution
⎧ IV
4
⎨ y = −10
)
( 1 )x(1 −( x);
'
y (0) = y 3 = y 23 = y ' (1) = 0;
⎩ ''
y (0) = y '' (1) = 0.
Fragments of the software raealization of the task (7):
• consideration of the first internal support:
i = round(xb/3*n);
a(i) = 0;
b(i) = 0;
c(i) = −1;
d(i) = 0;
e(i) = 0;
f(i) = 0;
plot(x(i),f(i),’*g’);
hold on; grid on;
• consideration of the second internal support:
i = round(2*xb/3*n);
a(i) = 0;
b(i) = 0;
c(i) = −1;
d(i) = 0;
e(i) = 0;
f(i) = 0;
plot(x(i),f(i),’*r’);
• consideration of limit conditions on the left side:
f(1)
b(1)
a(1)
f(1)
c(1)
b(1)
f(2)
b(2)
a(2)
=
=
=
=
=
=
=
=
=
f(1)
b(1)
0;
f(1)
c(1)
0;
f(2)
b(2)
0;
+ a(1)*h*DyA;
+ a(1);
+ b(1)*(h*DyA+h^2*D2yA);
− b(1);
+ a(2)*(h*DyA+h^2*D2yA);
+ a(2);
(7)
78
V. Babak et al.
• consideration of limit conditions on the right side:
f(n) = f(n) − e(n)*h*DyB;
d(n) = d(n) + e(n);
e(n) = 0;
f(n) = f(n)−d(n)*(h^2*D2yB+h*DyB);
c(n) = c(n)−d(n);
d(n) = 0;
f(n−1) = f(n−1)−e(n−1)*(h^2*D2yB+h*DyB);
d(n−1) = d(n−1)+e(n−1);
e(n−1) = 0;
Y = prog5dig(a,b,c,d,e,f,n); % Correction of the first two points of
the integration area
Y0 = Y(1) − h*DyA − h^2*D2yA;
Y_1 = Y0 − h*DyA; % Correction of the last two points of the integration area
yn_1 = h^2*D2yB + h*DyB + Y(n);
yn_2 = h*DyB + yn_1; % Formation of a numerical solution with extreme points
Y = [Y_1 Y0 Y yn_2 yn_1];
plot(x,Y, ’−−’);
legend(’Internal Support _1’, Internal Support _2’,’Numerical Solution)
The numerical solution of problem (7) together with the internal conditions are
presented in Fig. 5.
The last is the task of a beam with 1 m long with one hard fixed end and the other
free. The following load is applied to the beam
w(x) = −104 x(1 − x).
Mathematical model has the next form
⎧ IV
⎨ y = −104 x(1 − x);
y(0) = y ' (0) = 0;
⎩ ''
y (1) = y ''' (1) = 0.
The task analytical solution has the following form:
Fig. 5 Numerical task (7)
solution
(8)
Mathematical Models and Software for Studying the Elasticity …
y analit x
0.025 x 6
9
0.025 x5
3
0.25 x 3
9
79
0.125x 2
.
3
The software realization of the task looks the next.
function main3prolyot_end
clc;
n = 1000; xa = 0; xb = 1;
yA = 0;
DyA = 0;
D2yB = 0;
D3yB = 0;
h = (xb−xa)/(n+3);
x = xa:h:xb;
a = ones(1,n)/h^4;
b = −4*ones(1,n)/h^4;
c = −6*ones(1,n)/h^4;
d = −4*ones(1,n)/h^4;
e = ones(1,n)/h^4;
f = −x(3:end−2).*(1−x(3:end−2));
y2 = DyA*h + yA;
f(1) = f(1) − a(1)*yA;
a(1) = 0;
f(1) = f(1) − b(1)*y2;
b(1) = 0;
f(2) = f(2) − a(2)*y2;
a(2) = 0;
f(n) = f(n) − e(n)*h^2*D2yB;
c(n) = c(n) + e(n);
d(n) = d(n) + 2*e(n);
e(n) = 0;
f(n) = f(n) − d(n)*(h^2*D2yB−h^3*D3yB);
b(n) = b(n) − d(n);
c(n) = c(n) − 2*d(n);
d(n) = 0;
f(n−1) = f(n−1) − e(n−1)*(h^2*D2yB−h^3*D3yB);
c(n−1) = c(n−1) + e(n−1);
d(n−1) = d(n−1) + 2*e(n−1);
e(n−1) = 0;
Y = prog5dig(a,b,c,d,e,f,n);
yn_1 = h^2*D2yB + 2*Y(n) − Y(n−1) − h^3*D3yB;
yn_2 = h^2*D2yB + 2*yn_1 − Y(n);
Y = [yA y2 Y yn_1 yn_2];
t = x;
yy = (0.025*t.^6)/9 − (0.025*t.^5)/3 + (0.25*t.^3)/9
(0.125*t.^2)/3;
plot(x,Y,‘−−’,t,yy);
hold on; grid on;
legend(‘Numerical Solution’, ‘Analytic solution’)
−
As we can see, the numerical solution of the formulated task (8) coincides with
the analytical solution of the same task (Fig. 6).
80
V. Babak et al.
Fig. 6 Numerical task (8)
solution
4 Two-Dimensional Case in Elasticity Tasks
4.1 Mathematical Model of the Task
Similarly to the one-dimensional case, an equation is obtained for the deflection of
a rectangular continuous multi-span beam of a stable section. The general mathematical model of the task for such beam with rigidly fixed ends looks like this [5, 6,
8]:
⎧
( 4
)
∂ U
∂ 4U
∂ 4U
⎪
⎪
E
J
= w(x, y);
+
2
+
⎪
⎪
⎪
∂x4
∂ x 2∂ y2
∂ y4
⎪
⎪
⎪
⎨ U (x, 0) = f 1 (x), U (x, l x ) = f 2 (x);
⎪
U (0, y) = f 3 (y), U (l x , y) = f 4 (y);
⎪
⎪
⎪
⎪
⎪
U y (x, 0) = g1 (x), U y (x, l x ) = g2 (x);
⎪
⎪
⎩
Ux (0, y) = g3 (y), Ux (l x , y) = g4 (y).
The equation that describes the beam deflection in the two-dimensional case is
called the biharmonic equation.
4.2 Construction of the Task Difference Equation
For obtaining the classical difference scheme, the classical Taylor series is used.
∂ 4U
∂ 4U
∂ 4U
+2 2 2 +
= f (x, y);
4
∂x
∂x ∂y
∂ y4
|
Ui+2, j − 4Ui+1, j + 6Ui, j − 4Ui−1, j + Ui−2, j
∂ 4 U ||
=
;
∂ x 4 |(xi ,y j )
h 4x
|
Ui, j+2 − 4Ui, j+1 + 6Ui, j − 4Ui, j−1 + Ui, j−2
∂ 4 U ||
=
;
|
4
∂ y (xi ,y j )
h 4y
Mathematical Models and Software for Studying the Elasticity …
81
|
Ui+1, j − 2Ui, j + Ui−1, j
∂ 2 U ||
=
;
∂ x 2 |(xi ,y j )
h 2x
|
|
2 |
∂ 2U |
|
− ∂∂ xU2 |
2 |
3
∂
x
∂ U ||
(xi ,y j+0.5 )
(xi ,y j−0.5 )
=
|
2
∂ x ∂ y (xi ,y j )
hy
=
|
∂ U ||
∂ x 2∂ y2 |
|
4
=
−
Ui+1, j+0.5 −2Ui, j+0.5 +Ui−1, j+0.5
h 2x
(xi ,y j )
=
∂ 3U |
∂ x 2 ∂ y | x ,y
( i j+0.5 )
−
hy
|
3
|
− ∂∂x 2U∂ y |
Ui+1, j−0.5 −2Ui, j−0.5 +Ui−1, j−0.5
h 2x
(xi ,y j−0.5 )
hy
Ui+1, j+1 −2Ui, j+1 +Ui−1, j+1
U
−2U +U
− i+1, j h 2i, j i−1, j
h 2x
x
h 2y
U
−2U
+U
Ui+1, j −2Ui, j +Ui−1, j
− i+1, j−1 hi,2j−1 i−1, j−1
h 2x
x
h 2y
(
;
−
1
1
1
1
=
Ui+1, j+1 + 2 2 Ui−1, j−1 + 2 2 Ui+1, j−1 + 2 2 Ui−1, j−1
2
2
hx hy
hx hy
hx hy
hx hy
)
(
2
2
2
2
+ − 2 2 Ui, j+1 − 2 2 Ui, j−1 − 2 2 Ui+1, j − 2 2 Ui−1, j
hx hy
hx hy
hx hy
hx hy
4
+ 2 2 Ui, j .
hx hy
)
Ui+2, j − 4Ui+1, j + 6Ui, j − 4Ui−1, j + Ui−2, j
∂ 4U
∂ 4U
∂ 4U
+
2
+
=
4
2
2
4
∂x
∂x ∂y
∂y
h 4x
Ui, j+2 − 4Ui, j+1 + 6Ui, j − 4Ui, j−1 + Ui, j−2
+
h 4y
)
(
1
1
1
1
U
+
U
+
U
+
U
+2
i+1, j+1
i−1, j−1
i+1, j−1
i−1, j−1
hx 2 hy 2
hx 2 hy 2
hx 2 hy 2
hx 2 hy 2
)
(
2
2
2
2
+ 2 − 2 2 Ui, j+1 − 2 2 Ui, j−1 − 2 2 Ui+1, j − 2 2 Ui−1, j
hx hy
hx hy
hx hy
hx hy
(
)
4
4
1
4
−
+ 2 2 Ui, j =
U
+
U
−
i+2, j
i+1, j
hx hy
hx 4
hx 4
hx 2 hy 2
(
)
(
)
6
4
8
6
4
+
U
−
+ Ui, j
+
+
−
i−1, j
hx 4
hx 2 hy 2
hy 4
hx 4
hx 2 hy 2
(
)
4
1
4
+ 4 Ui, j+2 + Ui, j+1 − 4 − 2 2
hy
hy
hx hy
(
)
4
2
4
2
+ Ui, j−1 − 4 − 2 2 + 2 2 Ui+1, j+1 + 2 2 Ui−1, j−1
hy
hx hy
hx hy
hx hy
82
V. Babak et al.
Fig. 7 Type of difference scheme for the biharmonic equation
+
2
hx 2 hy 2
Ui+1, j−1 +
2
hx 2 hy 2
Ui−1, j−1 = f i, j .
It can now be seen that this equation, with using the difference method, will
contain 13 points in its scheme, as shown below.
Below is the main fragment of the program for modeling problems described by
biharmonic equations, using the scheme shown in Fig. 7.
function main
clc;
tic;
nx = 46; %%% set the number of grid nodes along OH
ny = 46; %%% set the number of grid nodes along OU
hx = 1/(nx-1); %%% set the step along OH
hy = 1/(ny-1); %%% set the step along OU
x = 0:hx:1;% form grid nodes
y = 0:hy:1; % form grid nodes
% set the equation coefficients
for i=1:nx
for j=1:ny
ax(i,j) = 1/hx^4;
bx(i,j) = -4/hx^4 - 4/(hx*hy)^2;
cx(i,j) = 6/hx^4;
dx(i,j) = -4/hx^4 - 4/(hx*hy)^2;
ex(i,j) = 1/hx^4;
ay(i,j) = 1/hy^4;
by(i,j) = -4/hy^4 - 4/(hx*hy)^2;
cy(i,j) = 6/hy^4;
dy(i,j) = -4/hy^4 - 4/(hx*hy)^2;
ey(i,j) = 1/hy^4;
c (i,j) = cx(i,j) + cy(i,j) + 8/(hx*hy)^2;
a00(i,j) = 2/(hx*hy)^2;
a01(i,j) = 2/(hx*hy)^2;
a10(i,j) = 2/(hx*hy)^2;
a11(i,j) = 2/(hx*hy)^2;
d (i,j) = f(i,j);
end
Mathematical Models and Software for Studying the Elasticity …
83
end
After one of the main tasks of the work (development of software for analyzing
the state of building elements and structures) is completed, the next step is to test
the appropriate software for analyzing the operating modes of objects, taking into
account various limiting conditions.
4.3 Testing and Analysis of Obtained Results
Let’s consider the task, the mathematical formulation of which is given below and
is described by a linear biharmonic equation [14, 15]:
⎧ 4
∂ U
∂ 4U
∂ 4U
⎪
⎪
+
2
+
= −100;
⎪
⎪
⎪
∂x4
∂ x 2∂ y2
∂ y4
⎪
⎪
)
(
⎪
⎨
1
, y = 0;
U (x, 0) = U (x, 1) = U (0, y) = U (1, y) = 0, U
2
⎪
⎪
⎪
⎪
⎪
U y (x, 0) = U y (x, 1) = Ux (0, y) = Ux (1, y) = 0;
⎪
⎪
⎪
⎩
(x, y) ∈ [0; 1]2 .
(9)
The task (9) implies the study of deflection (elastic deformation) that may occur in
a building structure for a two-dimensional model. It should be noted that among the
conditions used in the task, there is an internal condition that simulates a continuous
fixed beam. The beam is in the center of the study area. The results of computational
experiments are presented below.
Figure 8 shows the numerical solution of the problem (9).
Now let’s consider another task:
⎧ 4
4
4
∂ U
⎪
+ 2 ∂ x∂2 ∂Uy 2 + ∂∂ yU4 = −100;
⎪
∂x4
⎪
⎨
U (x, 0) = U (x, 1) = U (0, y) = U (1, y) = 0;
(10)
⎪
=
U
1)
=
U
y)
=
U
y)
=
0;
U
(x,
(0,
(1,
y((x, 0)
y
x
x
⎪
⎪
⎩ 1 1)
U 2 , 2 = 0.
Fig. 8 Image of the
deflection of a continuous
2-span two-dimensional
beam with rigidly fixed
ends—task (9)
84
V. Babak et al.
Fig. 9 Task (10) numerical
solution
Task (10) provides for taking into account such a condition that imitates some
support, which is set at one point of the studied object. This condition is internal and
is located in the middle of the computational domain. The results of computational
experiments are presented below.
Figure 9 shows the numerical solution of the problem (10).
Let’s consider an applied problem, which is described mathematically as follows:
⎧ 4
4
4
∂ U
⎪
+ 2 ∂ x∂2 ∂Uy 2 + ∂∂ yU4 = 0;
⎪
∂x4
⎪
⎪
⎪
⎨ U (x, 0) = U (x, 1) = U (0, y) = U (1, y) = 0;
U y((x, 0)
) = U y (x,(1)1 =2 )Ux (0, y) = Ux (1, y) = 0;
⎪
⎪
1 1
⎪
= −1, U
,
U
⎪
⎪
( 3 ,)3 = −1;
⎩ ( 32 31 )
U 3 , 3 = 2, U 23 , 23 = −1.
(11)
The program results for task (11) are presented below.
Figure 10 shows the numerical solution of the problem (11).
After the analysis of mathematical models in the Cartesian coordinate system, it
is possible to pass to the analysis of models, for example, in the polar coordinate
system.
Fig. 10 Task (11) numerical
solution
Mathematical Models and Software for Studying the Elasticity …
85
5 Passing to Polar Coordinate
5.1 Task Mathematical Model
For example, let’s solve a task that has the following mathematical formulation:
⎧ 4
∂3ϕ
∂ ϕ
∂2ϕ
∂ϕ
⎪
⎨ ∂r 4 + r2 ∂r 3 − r12 ∂r 2 + r13 ∂r = 0;
ϕ(r = ra ) = ϕ A , ϕr' (r = ra ) = ϕ 'A ;
⎪
⎩ ϕ(r = r ) = ϕ , ϕ ' (r = r ) = ϕ ' .
b
B
b
r
B
(12)
Task (12) can have different limit conditions, the choice of which depends on the
specific technical problem [16, 17].
5.2 Construction of the Difference Equation of the Task
Model
Let us write the fourth-order derivative by the finite difference method. First, we
write the equation for the central derivative of the first order:
ϕi' =
ϕi+0.5 − ϕi−0.5
.
h
The second derivative write as follows:
ϕi'' =
'
'
ϕi+0.5
− ϕi−0.5
=
h
ϕi+1 −ϕi
h
−
h
ϕi −ϕi−1
h
=
ϕi−1 − 2ϕi + ϕi+1
.
h2
The third derivative has the following form:
ϕi''' =
ϕi−0.5 −2ϕi+0.5 +ϕi+1.5
h2
+ϕi+0.5
− ϕi−1.5 −2ϕhi−0.5
2
h
ϕi+1.5 − 3ϕi+0.5 + 3ϕi−0.5 − ϕi−1.5
=
.
h3
''
''
ϕi+0.5
− ϕi−0.5
=
h
The fourth derivative looks like the next:
ϕiI V =
ϕi+2 −3ϕi+1 +3ϕi −ϕi−1
h3
i−1 −ϕi−2
− ϕi+1 −3ϕi +3ϕ
h3
h
ϕi+2 − 4ϕi+1 + 6ϕi − 4ϕi−1 + ϕi−2
=
.
h4
'''
'''
ϕi+0.5
− ϕi−0.5
=
h
For the third derivative ϕi''' , we will use the following difference scheme:
86
V. Babak et al.
ϕi''' =
ϕi −2ϕi+1 +ϕi+2
h2
i−1 +ϕi
− ϕi−2 −2ϕ
h2
2h
ϕi+2 − 2ϕi+1 + 2ϕi−1 − ϕi−2
=
.
2h 3
''
''
ϕi+0.5
− ϕi−0.5
=
h
Therefore, the difference scheme for this equation has five points. To solve this,
it can be used an analog of sweep for systems of equations with a 5-diagonal matrix.
Our equation can be written like this:
ai ϕi−2 + bi ϕi−1 − ci ϕi + di ϕi+1 + ei ϕi+2 = f i , i = 1, n.
(13)
Now we represent the solution of this system of equations in the form:
ϕi = K i ϕi+1 + L i ϕi+2 + Mi .
(14)
Using Eq. (14), the first two components in formula (13) can be removed. To do
this, we write that
ϕi−1 = K i−1 ϕi + L i−1 ϕi+1 + Mi−1 ;
ϕi−2 = K i−2 ϕi−1 + L i−2 ϕi + Mi−2 .
So using the last 2 formulas, we have:
ai (K i−2 ϕi−1 + L i−2 ϕi + Mi−2 ) + bi (K i−1 ϕi + L i−1 ϕi+1 + Mi−1 )
− ci ϕi + di ϕi+1 + ei ϕi+2 = f i ;
ai (K i−2 (K i−1 ϕi + L i−1 ϕi+1 + Mi−1 ) + L i−2 ϕi + Mi−2 )
+ bi (K i−1 ϕi + L i−1 ϕi+1 + Mi−1 ) − ci ϕi + di ϕi+1 + ei ϕi+2 = f i , i = 1, n.
Then, we have
ϕi (ai K i−2 K i−1 + ai L i−2 + bi K i−1 − ci ) + (ai K i−2 L i−1 + bi L i−1 + di )ϕi+1
+ ei ϕi+2 = f i − ai K i−2 Mi−1 − ai Mi−2 − bi Mi−1 , i = 1, n.
The next equation we write in the form:
ai K i−2 L i−1 + bi L i−1 + di
ϕi+1
ci − (ai K i−2 K i−1 + ai L i−2 + bi K i−1 )
ei
ϕi+2
+
ci − (ai K i−2 K i−1 + ai L i−2 + bi K i−1 )
ai K i−2 Mi−1 + ai Mi−2 + bi Mi−1 − f i
.
+
ci − (ai K i−2 K i−1 + ai L i−2 + bi K i−1 )
ϕi =
Then, we have
Mathematical Models and Software for Studying the Elasticity …
87
L i−1 (ai K i−2 + bi ) + di
ei
, Li =
;
ci − K i−1 (ai K i−2 + bi ) − ai L i−2
ci − K i−1 (ai K i−2 + bi ) − ai L i−2
(ai K i−2 + bi )Mi−1 + ai Mi−2 − f i
, i = 1, n.
Mi =
ci − K i−1 (ai K i−2 + bi ) − ai L i−2
Ki =
We also take into account that
ϕi+1 = ϕi+2 = x0 = x−1 = 0, a1 = a2 = b1 = en−1 = en = dn−1 = 0.
Similarly, any other type of limit condition can be taken into account both inside
the disk or tube and outside. We write the final difference scheme for the biharmonic
equation as follows:
ϕi+2 − 4ϕi+1 + 6ϕi − 4ϕi−1 + ϕi−2
2 ϕi+2 − 2ϕi+1 + 2ϕi−1 − ϕi−2
+
−
4
h
ri
2h 3
1 ϕi+1 − 2ϕi + ϕi−1
1 ϕi+1 − ϕi−1
− 2
= 0.
+ 3
r
2h 2
r
2h
The coefficients of the difference equation can be written as follows:
1
1
1
1
4
2
1
1
−
, ei = 4 +
, bi = − 4 +
− 2 2− 3 ;
h 4 ri h 3
h
ri h 3
h
ri h 3
2r h
2r h
)
(
2
1
1
6
1
4
, f i = 0.
,
b
di = − 4 −
−
+
=
−
−
−
i
h
ri h 3
2r 2 h 2
2r 3 h
h4 r 2h2
ai =
Therefore, it is possible to simulate various applied problems, which are reduced
to solving a biharmonic equation in polar coordinates.
5.3 Program Development for Numerical Simulation
Below is the software implementation of the sweep for systems with 5-diagonal
matrices:
%%% Call is made either from the command line, or by another function and the input data
function x = prog5dig(a,b,c,d,e,f,n)
K(1) = d(1)/c(1);
L(1) = e(1)/c(1);
M(1) = −f(1)/c(1);
zn = c(2)−b(2)*K(1);
K(2) = (d(2)+b(2)*L(1))/zn;
L(2) = e(2)/zn;
M(2) = (b(2)*M(1)−f(2))/zn;
for i = 3:n
zn = c(i)−a( i )*L(i−2)−(a(i)*K(i−2)+b(i))*K(i−1);
88
V. Babak et al.
K(i) = (d(i)+(a(i)*K(i−2)+b(i))*L(i−1))/zn;
L(i) = e(i)/zn;
M(i) = ((a(i)*K(i−2)+b(i))*M(i−1)+a(i)*M(i−2)−f(i))/zn;
end
x(n) = M(n);
x(n−1) = K(n−1)*x(n)+M(n−1);
for i = n−2:−1:1
x(i) = K(i)*x(i+1)+L(i)*x(i+2)+M(i);
end
Let us now describe a more detailed account of the task limiting conditions. Let
the following conditions be set at the left end:
ϕ(a) = ϕ A , ϕ ' (a) = ϕ 'A .
Since we have conditions at the left end, we need to take them into account in the
first and second equations of the system as follows:
a1 ϕ−1 + b1 ϕ0 − c1 ϕ1 + d1 ϕ2 + e1 ϕ3 = f 1 .
If ϕ(a) = ϕ A , then ϕ−1 = ϕ A and
a1 ϕ A + b1 ϕ0 − c1 ϕ1 + d1 ϕ2 + e1 ϕ3 = f 1 ;
b1 ϕ0 − c1 ϕ1 + d1 ϕ2 + e1 ϕ3 = f 1 − a1 ϕ A .
If ϕ ' (a) = ϕ 'A , then
ϕ0 − ϕ−1
ϕ0 − ϕ A
= ϕ 'A ,
= ϕ 'A , ϕ0 = hϕ 'A + ϕ A .
h
h
We substitute the last equation into the first equation of the system after taking
into account the condition of the first kind at the left end of the beam:
(
)
b1 hϕ 'A + ϕ A − c1 ϕ1 + d1 ϕ2 + e1 ϕ3 = f 1 − a1 ϕ A .
Then
(
)
−c1 ϕ1 + d1 ϕ2 + e1 ϕ3 = f 1 − a1 ϕ A − b1 hϕ 'A + ϕ A .
Let’s write the second equation of the system:
a2 ϕ0 + b2 ϕ1 − c2 ϕ2 + d2 ϕ3 + e2 ϕ4 = f 2 .
Taking into account that
ϕ0 = hϕ 'A + ϕ A ,
Mathematical Models and Software for Studying the Elasticity …
89
we have
(
)
a2 hϕ 'A + ϕ A + b2 ϕ1 − c2 ϕ2 + d2 ϕ3 + e2 ϕ4 = f 2 ;
(
)
b2 ϕ1 − c2 ϕ2 + d2 ϕ3 + e2 ϕ4 = f 2 − a2 hϕ 'A + ϕ A .
During developing a software implementation, accounting the limit conditions at
the left end will look like this:
y2 = DyA*h + yA;
f(1) = f(1) − a(1)*yA;
a(1) = 0;
f(1) = f(1) − b(1)*y2;
b(1) = 0;
f(2) = f(2) − a(2)*y2;
a(2) = 0;
From a physical point of view, a combination of this type conditions means that
the inner part of the disk is fixed rigidly and immobile.
The mathematical model of the task of the first problem looks like this:
⎧ ∂4ϕ 2 ∂3ϕ
2
+ r ∂r 3 − r12 ∂∂rϕ2 + r13 ∂ϕ
= 0;
⎪
∂r 4
∂r
⎪
⎨
r ∈ [1; 2];
⎪
ϕ(r = 1) = 1, ϕr' (r = 1) = 0;
⎪
⎩
ϕ(r = 2) = 0, ϕr' (r = 2) = 0.
(15)
The software implementation fragment looks like this:
%%% taking into account limiting conditions
y2 = DyA*h + yA;
f(1) = f(1) − a(1)*yA;
a(1) = 0;
f(1) = f(1) − b(1)*y2;
b(1) = 0;
f(2) = f(2) − a(2)*y2;
a(2) = 0;
yn_1 = yB − DyB*h;
f(n) = f(n) − e(n)*yB;
e(n) = 0;
f(n) = f(n) − d(n)*yn_1;
d(n) = 0;
f(n−1) = f(n−1) − e(n−1)*yn_1;
e(n−1) = 0;
The following example has the form:
⎧ ∂4ϕ 2 ∂3ϕ
2
+ r ∂r 3 − r12 ∂∂rϕ2 + r13 ∂ϕ
= 0;
⎪
∂r 4
∂r
⎪
⎨
r ∈ [1; 2];
⎪
ϕ ' (r = 1) = ϕr'' (r = 1) = 0;
⎪
⎩ r
ϕ(r = 2) = ϕr'' (r = 2) = 1.
(16)
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V. Babak et al.
Fragments of the software implementation of this task.
%% Setting the ends of the integration interval
yA = 0; % beam position value at left end
DyA = 0; % beam slope value at the left end
DyB = 0; % beam position value at right end
D2yB = 0; % beam slope value at the right end
h = (xb−xa)/(n+3); % spatial coordinate step
x = xa:h:xb;
% creating an integration gap
%% Setting the coefficients of the system of equations
y2 = DyA*h + yA;
% second dot from left
f(1) = f(1) − a(1)*yA; % correction of the right
side of the 1st equation
a(1) = 0; % resetting the used coefficient
f(1) = f(1) − b(1)*y2; % correction of the right side of the 1st equation (for the derivative)
b(1) = 0;
% resetting the used coefficient
f(2) = f(2) − a(2)*y2; % correction of the right
side of the 2nd equation
a(2) = 0;
% resetting the used coefficient
f(n) = f(n) − e(n)*h*DyB;
% second dot from right
d(n) = d(n) + e(n);
% correction of the right side of
the n−th equation
e(n) = 0;
% resetting the used coefficient
f(n) = f(n)−d(n)*(h^2*D2yB+h*DyB); % correcting the right
side of the n−th equation (using the derivative as a limit condition)
c(n) = c(n)−d(n);
d(n) = 0;
f(n−1) = f(n−1)−e(n−1)*(h^2*D2yB+h*DyB);
d(n−1) = d(n−1)+e(n−1);
e(n−1) = 0;
Y = prog5dig(a,b,c,d,e,f,n);
yn_1 = h^2*D2yB + h*DyB + Y(n);
yn_2 = h*DyB + yn_1;
plot(x,Y,‘−−’,x,yy,‘−’);
grid on;
legend(‘Numerical Solution’, ‘Analitical Solution’)
The above software allows to take into account the conditions of various kinds
for arbitrary processes that occur in construction objects and structures.
6 Conclusions
In this chapter, the main applied physical and technical tasks of modeling the
deflection of a continuous beam in the MATLAB R2021b environment were implemented. The chapter contains the main results for the one-dimensional case, the
two-dimensional case, and the case in polar coordinates.
Mathematical Models and Software for Studying the Elasticity …
Table. 1 Main
characteristics of the personal
computer on which the
developed programs
implemented in MATLAB
R2021b were tested
Parameter
Value
Central processor unit
AMD Phenom X4 Black Edition,
3.4 GHz
Video card
NVidia, GeForce GTX 1050 Ti
Random access memory
16 Gb
Hard disk drive
Western Digital Black 500 Gb
Operating system
Windows 10 × 64
Software
MatLab 2021b
91
Various situations are modeled in the problems of elasticity theory, depending on
the technical task. Also, the models of applied problems considered in the chapter
do not have an analytical solution or their solution can only be represented using an
infinite functional series.
Testing of the developed programs was carried out on a personal computer, the
main characteristics of which are presented in Table 1.
Analysis of simulation results gave high performance in the process of obtaining
numerous solutions to the tasks. For multidimensional models, the modification of
the developed software is insignificant.
References
1. Leipholz, H., Hutchinson, J.W.: Theory of elasticity. J. Appl. Mech. 42(4): 911 (1 page) (1975).
https://doi.org/10.1115/1.3423754
2. Richard, B.H., Ignaczak J.: Mathematical theory of elasticity, pp. 505–506 (2006). https://doi.
org/10.1080/01495730500495751
3. Lazopoulos, K.A., Ogden, R.W.: Nonlinear elasticity theory with discontinuous internal
variables (1998). https://doi.org/10.1177/108128659800300103
4. Babak, V. P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Principles
of construction of systems for diagnosing the energy equipment. In Diagnostic Systems for
Energy Equipments (pp. 1–22). Springer, Cham (2020. https://doi.org/10.1007/978-3-030-444
43-3_1
5. Zaporozhets, A.O.: Experimental research of a computer system for the control of the fuel
combustion process. In: Control of Fuel Combustion in Boilers, pp. 89–123. Springer, Cham
(2020). https://doi.org/10.1007/978-3-030-46299-4_4
6. Zaporozhets, A.O., Khaidurov, V.V.: Mathematical models of inverse problems for finding the
main characteristics of air pollution sources. Water Air Soil Pollution 231, 563 (2020). https://
doi.org/10.1007/s11270-020-04933-z
7. Zaporozhets, A.O.: Methods and means for the control of the fuel combustion process. In:
Control of Fuel Combustion in Boilers, pp. 1–33. Springer, Cham (2020). https://doi.org/10.
1007/978-3-030-46299-4_1
8. Zaporozhets, A., Khaidurov, V., Tsiupii, T.: Optimization Models of Industrial Furnaces and
Methods for Obtaining Their Numerical Solution. In: Zaporozhets, A., Artemchuk, V. (eds.)
Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, vol. 346.
Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69189-9_7
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9. Huang, C.H., Dong, S.B.: Analysis of laminated circular cylinders of materials with the most
general form of cylindrical anisotropy. Int. J. Solids Struct. 38(34–35), 6163–6182 (2001).
https://doi.org/10.1016/s0020-7683(00)00374-7
10. Lin, H.-C., Dong, S.B.: On the Almansi-Michell problems for an inhomogeneous, anisotropic
cylinder. J. Mech. 22(01), 51–57 (2006). https://doi.org/10.1017/s1727719100000782
11. Horgan, C.O., Chan, A.M.: The pressurized hollow cylinder or disk problem for functionally
graded isotropic linearly elastic materials. J. Elast. 55(1), 43–59 (1999). https://doi.org/10.
1023/A:1007625401963
12. Sophie Germain’s Early Contribution to the Elasticity Theory. Published online by Cambridge
University Press. MRS Bulletin, vol. 24, no. 11, pp. 70–71 (2013). https://doi.org/10.1557/S08
83769400053549.
13. Akhmedov, N., Akbarova, S., Ismayilova, J.: Analysis of axisymmetric problem from the theory
of elasticity for an isotropic cylinder of small thickness with alternating elasticity modules.
Eastern-Eur. J. Enterpr. Technol. 2(7(98)), 13–19 (2019). https://doi.org/10.15587/1729-4061.
2019.162153
14. Abdulhadi, Z., Muhanna, Y.A., Khuri, S.: On some properties of solutions of the biharmonic
equation. Appl. Math. Comput. 177(1), 346–351 (2006). https://doi.org/10.1016/j.amc.2005.
11.013
15. Li, J., Cheng, Y.: Barycentric rational method for solving biharmonic equation by depression
of order. Numer. Methods Part. Diff. Equ. 37(3), 1993–2007 (2021). https://doi.org/10.1002/
num.22638
16. Kuts, Y.V., Shengur, S.V., Shcerbak, L.N.: Circular measurement data modeling and statistical
processing in LabView. In: 2011 Microwaves, radar and remote sensing symposium, pp. 317–
320 (2011). https://doi.org/10.1109/MRRS.2011.6053664
17. Babak, V.P., Babak, S.V., Eremenko, V.S., Kuts, Y.V., Myslovych, M.V., Scherbak, L.M.,
Zaporozhets, A.O.: Examples of using models and measures on the circle. In: Models and
measures in measurements and monitoring, pp. 127–156 (2021). https://doi.org/10.1007/9783-030-70783-5_5
Application of Discrete Hilbert
Transform to Estimate
the Characteristics of Cyclic Signals:
Information Provision
Vitalii Babak , Artur Zaporozhets , Mykhailo Kulyk , Yurii Kuts ,
and Leonid Scherbak
Abstract Cyclic signals are an important source of information about the processes
and phenomena occurring in the surrounding world and technical systems. This
chapter considers the methodology for measuring and analyzing the characteristics
of cyclic signals based on the discrete Hilbert transform (DHT). Peculiarities of DHT
application in the problems of measurement of signal characteristics are shown. The
methodological error in determining the characteristics of signals, due to the limited
time of signal realizations, is considered. The possibility of reducing this error due
to additional window processing of signal realizations is shown. The advantages of
circular median filtering of the phase characteristics of signals in the presence of
noise of significant intensity are presented.
Keywords Discrete Hilbert transform · Signal processing · Phase · Phase
measurement · Circular statistics
1 Introduction
A significant number of processes and phenomena occurring in the surrounding
world are of a cyclic nature. The word “cycle” comes from the Greek “kyklos”—
circle. A cycle is understood as a set of interrelated states of phenomena or processes
that form a cycle for a limited period of time. Such phenomena and processes give
rise to cyclic or rhythmic (rhythm is created by a cycle) signals that have certain
repeatability signs. Cyclic signals can be of both natural and artificial origin. For
example, in the energy sector, the source of cyclic signals is most of the energy
V. Babak · A. Zaporozhets (B) · M. Kulyk · Y. Kuts · L. Scherbak
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: a.o.zaporozhets@nas.gov.ua
A. Zaporozhets
State Institution “The Institute of Environmental Geochemistry of NAS of Ukraine, Kyiv, Ukraine
Y. Kuts
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv,
Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_5
93
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V. Babak et al.
machines [1, 2]. Information carriers in such signals are energy parameters, such
as power and energy. Information carriers in such signals are not only their energy
parameters, such as amplitude, power or energy, but also the phase.
In measurements, periodic harmonic signals are used to determine the transmission coefficients of linear and non-linear links of radio-electronic systems. The phase
as an informative parameter (characteristic) makes it possible to make conclusions
not only about the state of the studied process at the current moment, but also to evaluate the development of the process in the past and predict its course in the future. This
feature contributes to the maintenance of constant interest in the phase measurement
method and the improvement of the corresponding measuring instruments [3–10].
Changes in the parameters of harmonic signals (their amplitudes, frequencies and
phases) contain information about the characteristics of the studied systems. This
approach is also widely used to study objects of different physical nature.
The term “cyclic signal” is used both for natural (physical) interpretation and
for defining mathematical models of such signals. Models of signals, depending on
the formulation of the problem, acquire a more concrete form. One of the models
of a cyclic signal is a periodic signal with a fixed period T r of the form u(t) =
u(t + Tr ), t ∈ R = (−∞, ∞).
A significant part of actual measurement tasks is related to the analysis of cyclic
signals. The solution of these problems led to the development of appropriate information technologies based on various methods of signal analysis. For example, using
the Fourier analysis method, such signals are decomposed into harmonic components. The dependence of the amplitudes and initial phase shifts of the harmonics
on their frequencies makes it possible to determine the amplitude-frequency and
phase-frequency characteristics of the signals.
Another approach is based on the representation of the model of the cyclic signal
u(t), obtained from the measurement results, in the form of a narrow-band signal,
the model of which has the form
u(t) = U (t) cos[.(t)], U (t) > 0,
d.(t)
> 0, t ∈ R,
dt
(1)
where U (t), .(t)—respectively bypass and phase of the signal, presented as a
function of time.
The functions U (t), .(t) of the model (1) have a specific physical content, but
their determination only from the known values of the function u(t) is impossible,
since it is impossible to determine two unknowns from one equation. This problem
can be eliminated by the integral Hilbert transform (IHT) and the concept of an
analytical signal introduced on its basis [11, 12]. IHT was proposed by the German
mathematician D. Hilbert in 1912. This transformation and the concept of “analytical signal” introduced on its basis is used in the theory of oscillations, communication technology, telecommunication systems, etc. for theoretical studies of cyclic
processes, phenomena and solving practical problems of decoding modulated signals.
The term “analytical signal” was proposed by D. Gabor (1946) and D. Will (1948). It
was used as a theoretical basis for solving problems of determining estimates of the
Application of Discrete Hilbert Transform to Estimate …
95
bypass, phase and frequency of vibrations in mechanics, optics, radio engineering,
telecommunications, etc. However, in measurements, this transformation has not yet
found a proper practical application for obtaining the values of signal characteristics
with the appropriate accuracy.
The aim of the work is to create information support based on the use of DHT and
suitable for assessing the characteristics of information cyclic signals in technical
measurement, control and diagnostic systems for various purposes.
2 Formulation of the Problem
Based on the IHT of continuous functions given on R, it is necessary to consider the
features of the practical application of the DHT to evaluate the characteristics of cyclic
information signals according to their time-limited realizations (samples); consider
the features of the DHT use in information-measuring technologies of signals and
give examples of use.
The mathematical model of the studied cyclic processes is generally represented
by the sum of the informative component u(t) and the noise of the form ξ (ω, t):
[
]
u ξ (ω, t) = u(t) + ξ (ω, t) = Uξ (ω, t) cos .ξ (ω, t) , ω ∈ ., t ∈ R,
(2)
where ξ (ω, t)—stationary random process; ω—elementary event from event area ..
In most practical cases, ξ (ω, t) has a Gaussian distribution with zero mean and
variance σ 2 .
In the case of the formation of test signals by a technical system, their model is
known
u r (t) = Ur cos[2π fr t + ϕr ], Ur > 0, t ∈ Ta < ∞, ϕr ∈ [0, 2π ),
(3)
where U r —amplitude, ϕr —initial phase, fr —signal frequency, Ta —final time
interval.
3 Main Part
3.1 Cyclic Signals and Related Measurement Tasks
Types of information cyclic signals and tasks related to measuring their characteristics
are shown in Fig. 1.
According to the place of signal generation, two groups can be distinguished—
signals generated by cyclic physical processes, and signals generated by technical
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V. Babak et al.
Fig. 1 Tasks of phase measurements of cyclic signals (PFC—phase-frequency characteristic,
PDT—phase delay time, GDT—group delay time)
means. These signals are used as test signals, the phase of which is modulated by the
phenomenon or studied process.
The next sign of classification is different demonstration of the cyclicity property.
On this basis, almost periodic processes can be distinguished in the first group. These
processes are represented by almost periodic functions, for which the concept of
almost period is defined. For a uniform almost periodic function f (t), −∞ < t < ∞
the T = T f (ε) number is called the ε-almost period of the f (t) function if the
| f (t + T ) − f (t)| < ε equality performs for all t.
[
)
By determining T f (ε), it is possible to determine the delay τ ∈ 0, T f [ε] and
the corresponding phase shift of the signal within its limits
/
ϕ f (ε) = 2π τ T f (ε).
The measurement of phase relationships for almost periodic processes has not
found wide application in practice, since the conditions for conducting physical
experiments do not imply the formation of a reference signal.
The second group of signals is formed by signals generated by technical means.
As a rule, they are reference signals in phase measurements, and their phase is influenced by the studied physical phenomenon, process or quantity in the sensor. The
largest number of phasemetry tasks is associated with periodic harmonic signals.
Such tasks include reconstruction of phase shifts of signals, measurement of phase
stability (phase noise) of oscillation sources, measurement of phase shifts of signals
ϕ ∈ [0, 2π ), measurement of cumulative phase shifts (CPS). The CPS is understood as the phase difference .2 (t) − .1 (t), which, when during observing over
a time interval Ta , change by more than 2π , that is, ..(Ta ) = .2 (0) − .1 (0) −
(.2 (t1 = Ta ) − .1 (t1 = Ta )) >> 2π . The process of accumulation of the phase
Application of Discrete Hilbert Transform to Estimate …
97
Fig. 2 Graph of the phase
change of signals and the
CPS formation
shift during the measurement of the CPS is shown in Fig. 2. Phase shift has changed
from .(0) < 2π to .(t1 ) >> 2π by time t1 .
The use of polyharmonic signals in phasometry (or the performance of phase
measurements sequentially in time for harmonic signals of different frequencies)
makes it possible to determine the PFC of various devices and systems, to measure
PDT and GDT. Measurements of PDT and GDT are used to solve the problems of
quality control of communication lines, conduct a physical experiment and determine
various physical properties of substances that are indirectly related to the group
and phase velocities of propagation of ultrasonic waves—concentration, density,
temperature, pressure, flow, level, etc.
In digital data transmission systems, piecewise periodic signals are often used,
which, for example, are formed using phase manipulation.
3.2 Formation of Cyclic Information Signals
The general structure of the formation of cyclic signals in technical systems is shown
in Fig. 3. The program production unit generates a digital test signal (digital signal
prototype (3)), which is converted into a physical signal (most often voltage) using
a digital-to-analog converter (DAC) and acts on the research object (RO). The initial
information is converted by the sensor into an electrical signal of the type u r (t) =
Ur cos[2π fr t + ϕr ], Ur > 0, t ∈ Ta , ϕr ∈ [0, 2π ) (most often into voltage) by
means of signals of various physical nature.
The analog block provides signal amplification and matching with the ADC,
which performs the operations of sampling and quantizing the signal u ξ (t) as an
implementation of the u ξ (ω, t) (2) process (hereinafter, if non-random implementations of the random processes under study are considered, the ω argument in the
signal record will be omitted). These operations are performed by the clock generator G and the reference voltage generator REF. Practically, at the output of an n-bit
98
V. Babak et al.
Fig. 3 Interaction between RO and the measurement system
ADC, at each( fixed point in
) time, the information signal is represented by the code
n
u ξ [ j] = ent u ξ ( j Td ) U2r e f . But for measuring the amplitude characteristics of the
signal, it is important to match the value of Ur e f with the unit of voltage measurement,
and for measuring the frequency characteristics, it is important to match the sampling
period Td with the unit of time. The program-production unit (PPU) performs the
processing of digital signals and the calculation of the discrete characteristics of the
signals displayed by the Visualizer block.
The possibility of measuring the phase difference exists due to the fact that the
reference signal of the form (3) is generated by the PPU.
The structure of the formation of cyclic signals in the analysis of natural
phenomena and processes differs from that shown in Fig. 3 structures for technical
systems in that it lacks DAC and the formation of a reference signal.
Next, we concretize the measurement problem in a discrete formulation.
Let the lattice be given on the observation interval Ta :
}
{
S = t1 , t2 , . . . , t j , . . . t n ,
(4)
the set of elements of which is ordered and the inequality 0 ≤ t1 < t2 < · · · t j <
· · · < tn ≤ Ta is satisfied for it. The elements of the S lattice are evenly spaced and
form an arithmetic progression t j = t1 + (( j − 1)Td , j ∈ [1,) n].
On the lattice S, a sample of values u ξ [ j], j = 1, n is given, which is an
implementation of a signal of the form (2) with a discrete argument. The analysis
interval contains at least
( ent(Ta f ) >> )1 periods of the signal.
For the sequence u ξ [ j], j = 1, n , we use a discrete transformation with the
Hilbert kernel.
(
)
Based on the results of u ξ [ j], j = 1, n observation, it is necessary to propose
algorithms for determining the discrete characteristics of the signal and consider the
errors in their determination.
Application of Discrete Hilbert Transform to Estimate …
99
3.3 Hilbert Transform and Properties for Characterizing
Signals
For a real random stationary process
. ∞ u ξ (ω, t), ω ∈ ., t ∈ (−∞, ∞), belonging
to the class of processes L 2 , i.e. −∞ u 2ξ (ω, t)dt < ∞, there is a Hilbert transform
[
]
that gives a new stationary random process û ξ (ω, t) = H u ξ (ω, t) . The Hilbert
transform operator and the inverse operator are represented by improper integrals
[10, 11]
⎤
⎡
.∞
[
]
u ξ (ω, τ ) ⎦
1⎣
dτ ;
û ξ (ω, t) = H u ξ (ω, t) =
v. p.
π
t −τ
(5)
−∞
⎤
⎡
.∞
[
]
û
τ
1
(ω,
)
ξ
dτ ⎦,
u ξ (ω, t) = H−1 û ξ (ω, t) = ⎣v. p.
π
t −τ
(6)
−∞
chere π (τ1−t) —kernel of the Hilbert transform, v. p.—designation of the principal
value of the Cauchy integral.
Obtaining û ξ (ω, t) allows to create a complex random process ζ̇ (ω, t) =
|
|
u ξ (ω, t) + i û ξ (ω, t) = |ζ̇ (ω, t)|ei arg ζ̇ (ω,t) , which is also stationary and belongs
to the class of Hilbert processes. Its amplitude response and unwrapped phase (or
phase response) are defined as
/
|
|
|ζ̇ (ω, t)| = Uξ (ω, t) = u 2 (ω, t) + û 2 (ω, t);
ξ
ξ
(7)
arg ζ̇ (ω, t) = .ξ (ω, t)
[
]}
û ξ (ω, t) π {
= arctg
2 − signû ξ (ω, t) 1 + signu ξ (ω, t)
+
u ξ (ω, t)
2
]
[
+ K û ξ (ω, t), u ξ (ω, t)
[
]
[
]
= L u ξ (ω, t), û ξ (ω, t) + K u ξ (ω, t), û ξ (ω, t) ,
(8)
[
]
where L u ξ (ω, t), û ξ (ω, t) —operator for uniquely determining the phase in the
interval [0, 2π ); K[·]—phase unfolding operator that eliminates jumps in its values
at points 2π n, n ∈ N . Functions (7) and (8) are taken as estimates of the information
characteristics of the signal u(t) (1).
“Unwrapped phase” or “true phase”—continuous function of the t argument in
case the u(t) function is a continuous function of time. The process of determining
an unwrapped phase from the moment it is represented by a winding phase is called
the process of phase unwrapping (reconstruction, “unwinding”). The wrapped and
unwrapped phases are interconnected by a nonlinear transformation
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V. Babak et al.
Fig. 4 Graphical
representation of the phase
unwrapping of a harmonic
signal
ϕξ (t) ≡ .ξ (t) mod 2π,
(9)
where x mod 2π —operation of determining the remainder of the number x ∈ R
modulo 2π .
Expression (9) explains another widely used name for the function ϕξ (t) ∈
[0, 2π )-modulo-2π phase value.
The process of phase unwrapping on the example of a harmonic signal with
frequency f is shown in Fig. 4. Here modulo-2π phase value is defined as ϕ(t) =
(2π f t + ϕ0 ) mod 2π .
The idea of phase unwrapping of a harmonic signal is simple and clear (Fig. 4).
For signals of a more complex structure, the determination of the unwrapped
phase provides much broader possibilities, in particular, the possibility of studying
the dynamics of the phase development of modulated signals, analyzing signal phase
fluctuations, and analyzing the phase of signals in the noise presence. However, the
noise presence complicates the process of phase unwrapping and requires the use
of additional statistical methods for processing the phase, taking into account the
signal-to-noise ratio.
3.4 Hilbert Transform on Finite Time Intervals
The main theoretical results related to the use of the Hilbert transformation have been
obtained for continuous functions given on infinite time intervals. For the practical
implementation of the Hilbert transformation in non-destructive testing systems, it
is important to use the transformation on a finite (finite) time interval.
In the general case, the Hilbert transform is physically unrealized, since it assumes
the existence of a signal on an infinite time interval. Therefore, real measurements
Application of Discrete Hilbert Transform to Estimate …
101
of signal characteristics based on this transformation can only be performed approximately. Indeed, the amplitude, phase and frequency (time derivative of the phase)
characteristics of the u(t) signal are of an integral nature. The determination of these
characteristics requires signal processing over the entire time interval of its determination. For some cases, it is possible to obtain approximate estimates of characteristics from the results of time-limited observations of the signal. Such cases are possible
when a significant contribution to H[u(t)] is made by the corresponding finite values
of the studied characteristics. Such a possibility exists in the case of processing
narrow-band signals, which makes it possible to apply “sliding” processing to determine the current estimates of the characteristics of the studied signals. Let’s dwell
on this in more detail.
Let’s give the following definition.
Definition 1 Integral transformation of a continuous function u(t) of the form
.t
û W (t) = HW [u(t)] = −
t−TW
u(τ)
dτ,
π(t − τ)
(10)
will be called the sliding Hilbert transform, where TW > 0—aperture of the time
window.
The term “sliding” means that the time window can be chosen arbitrarily on the
time axis t. Using a finite time window, we have
.
W (t) =
1, t ∈ [t − TW , t);
0, t ∈
/ [t − TW , t).
(11)
As a rule, it is advisable to choose a sliding time window TW > T0 , where T0 —
period of the u(t) signal. In this case, the sliding Hilbert transform of the signal u(t)
according to (10) is determined as follows
û W (t) = H[W (t)u(t)] ≈ W (t)H[u(t)] ≈ W (t)û(t).
(12)
Since the Fourier spectra of W (t) and u(t) usually overlap, the values of u W (t) and
û(t) do not coincide and differ by a certain methodological error .m , i.e. û W (t) =
û(t) + .m .
Sliding signal processing makes it possible to obtain signal parameters in real
time, however, the price for this possibility is the appearance of a corresponding
methodological error in measuring .m .
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3.5 DHT
The practical implementation of the sliding Hilbert transform (10) in a continuous
(analogue) form causes certain difficulties, which can be overcome in the case of
using the DHT. The use of DHT allows the terms “phase response” and “phase shift”
to be extended to a class of discrete sequences derived from cyclic signals.
A given sequence of a discrete signal u[ j], j = 1, n is associated with a complex
sequence (discrete analytical sequence)
ż[ j] = u[ j] + i û[ j], j = 1, n.
(13)
For
.n ż[ j], the following conditions are satisfied: (1) Reż[ j] = u[ j], (2)
o=1 u[ j] · û[ j] = 0. The fulfillment of these conditions corresponds to the properties of an analytical signal, and, in fact, justifies the use of the term “analytical”
to a discrete sequence (13) and allows to consider Imż[ j] as a discrete analog of the
integral transform (5).
An efficient method for calculating the analytic sequence ż[ j] from the sequence
u[ j], j = 1, n via the DHT was considered in [13]. This method is based on the
connection of signal spectra: the Fourier transform of an analytical signal has a
one-sided spectrum (only positive frequencies), and the spectral components of the
analytical signal are equal to twice the values of the spectral components of the
original signal. This method involves the following steps:
• calculation of the n-point DFT U̇ (m) of the sequence u[ j]
U̇ (m) =
)
(
2π
jm .
u[ j] exp −i
n
j=1
n
.
(14)
/
U̇ (m) value is calculated for discrete frequencies f m = m nTd ,0 ≤ m ≤ n − 1;
• calculation of the n-point DFT Ż (m) of the sequence ż[ j]
⎧
U̇ [0],
⎪
⎪
⎪
⎪
⎨ 2U̇ [m],
Ż (m) =
⎪
U̇ [m],
⎪
⎪
⎪
⎩
0,
at
m = 0,
/
at 1 ≤ m < n 2 − 1,
/
at m = n 2,
/
at n 2 + 1 ≤ m ≤ n − 1.
(15)
• calculation of ż[ j] from Ż (m) value by inverse DHT
)
(
n−1
1.
2π
jm .
ż[ j] =
Ż [m] exp i
n m=0
n
(16)
Application of Discrete Hilbert Transform to Estimate …
103
The ż[ j] sequence is complex, in which the imaginary part of the û[ j] is the DHT
of the u[ j] sequence, i.e. û[ j] = Reż[ j].
The characteristics of a discrete signal u[ j], j = 1, n are defined as:
1. discrete amplitude characteristic
A[ j] =
.
u 2 [ j] + û 2 [ j];
(17)
2. discrete phase characteristic
[
]
[
]
.[ j] = arg ż[ j] = L u[ j], û[ j] + K u[ j], û[ j] ;
(18)
3. discrete frequency characteristic
F[ j] =
.[ j] − .[ j − 1]
;
2πTd
(19)
4. difference between the discrete phase characteristics of two given discrete
sequences u 1 [ j] and u 2 [ j], each of which is associated with a complex sequence
of the form (13)
ϕ[ j] = .2 [ j] − .1 [ j].
(20)
If discrete sequences u 1 [ j] and u 2 [ j] are embedded in harmonic signals of the
same frequency, then expression (17) corresponds to the phase shift between these
signals.
The possibilities of the considered method for determining the characteristics of
signals are illustrated by the following model experiment.
The simulation is performed for the following families of curves, represented by
discrete values at n = 104 , .t = 10−5 s.
Example 1 Given a 1 kHz sine wave signal with amplitude-frequency modulation
(
)
u[ j] = A[ j] · [1 + 0.5 cos(2π 50nTd )] · sin 2π 103 j Td + 20 cos(2π 40 j Td ) ,
.
Aj =
1 B, j ∈ [500, 9500],
0, j ∈
/ [500, 9500].
Modulation was performed according to harmonic laws: amplitude modulation—
with a frequency of 50 Hz and a depth of 0.5, and frequency modulation—with
a frequency of 40 Hz and a depth of 0.8; sample size n = 104 , sampling period
.t = 10−5 s.
It is necessary to determine the modulation laws and evaluate their errors.
The simulation results are shown in Fig. 5: Fig. 5a shows the u[ j] and
û[ j] sequences; Fig. 5b—reconstructed phase values in the [0, 2π ) interval, i.e.
(.[ j]) mod 2π ; Fig. 5c—value of the unwrapped phase .[ j].
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V. Babak et al.
Fig. 5 Graphs of sequences: a u[ j] and û[ j], b (.[ j]) mod 2π , c .[ j]
Figure 6a shows the value of the bypass U [ j] calculated in accordance with (17); Fig. 6b—calculation error of bypass .U [ j] = U [ j] −
A[ j](1 + 0.5 cos(2π 50 j Td )); Fig. 6c—value of the instantaneous frequency f [ j]
according to (19); Fig. 6d—error in calculating the instantaneous frequency . f [ j] =
f [ j] − 1000[1 + 0.8 cos(2π 40 j Td )].
Figure 6 shows that, firstly, the errors .U [ j] and . f [ j] are sign-based, and
secondly, the general tendency of their change is to increase the maximum values at
the beginning and end of the window (or in the vicinity of the signal characteristics
manipulation).
3.6 Peculiarities of Using DHT in Measurements of Physical
Quantities
Despite the fact that DHT has long been used in digital signal processing, the issues of
using DHT in measuring technology for measuring physical quantities are still insufficiently covered. Let us consider the most important features of the use of information
signal processing algorithms based on DHT [13] for measuring equipment.
• The output signal of the sensor (Fig. 3) has a dimension (usually volts). It can
be represented by the product of a numerical function u(t) per unit [U] of the
Application of Discrete Hilbert Transform to Estimate …
105
Fig. 6 Graphs of sequences: a u[ j], b û[ j], c (.[ j]) mod 2π , d .[ j]
•
•
•
•
corresponding physical quantity U. Therefore, to represent discrete amplitude
(17) and frequency (19) characteristics of a physical quantity in accepted units (in
volts and hertz), it is sufficient to perform an analog-to-digital conversion with
measures of time and voltage units. In the diagram (Fig. 3) units of time and
voltage reproduced by blocks G and REF. The phase characteristic (18) in radians
is determined by the ratio of two quantities of the same kind— û(t) · [U] and
u(t) · [U], therefore, if only this characteristic is determined, the analog-to-digital
conversion can occur with an arbitrary choice of the unit [U], that is, it does not
require the use of a special physical measure of phase measurement signals.
Ability to obtain and analyze discrete amplitude, phase and frequency characteristics of information signals as a function of time, given by samples of a certain
size.
Possibility to obtain a sample of characteristics of information signals of significant volumes, which creates the prerequisites for a more correct use of statistical
methods for processing their characteristics.
Possibility to synchronously calculate the amplitude, phase and frequency characteristics of information signals for joint use to highlight new information features
in monitoring and diagnostics problems.
Ability to determine local changes in the parameters of information signals even
at time intervals comparable to the period of the carrier signal.
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• Ability to minimize the analog part of the signal processing circuit by expanding
digital processing, which increases the flexibility of measurement, control and
diagnostic systems and the possibility of improving them through software
improvements.
3.7 Using of Double Window Function in the Processing
of Information Signals
In Sect. 3.5, it is noted that in the case of using the Hilbert transform on finite
time intervals, methodological errors in measuring signal characteristics arise. For
example, during estimating the phase characteristics of harmonic signals and the
duration of the TW = (5...10)T time window, a methodological error arises, the
value of which can reach an unacceptably large value ~(0.16…0.56) rad.
To reduce the methodological error in determining the characteristics of information signals, it was proposed in [14] firstly, to choose the duration of the window
as a multiple of the studied signals period (if the signal frequency is known), i.e.
TW = kT , k ∈ N , and, secondly, values û W [ j], u W [ j] must be chosen from the
middle part of the time window W 1 , where the errors are the smallest, that is, from
the window W 2 with a smaller aperture (Fig. 7).
The methodological error in determining the signal characteristics has the smallest
value in the middle part of the window W 1 . Therefore, it is desirable to discard at
least 0.25k samples from each edge of the window as burdened with unacceptable
errors. In addition, it was found that the methodological error is significantly reduced
if the window aperture W 1 is chosen as a multiple of the signal period T.
In the case of sliding window processing, in addition to the aperture TW = kT of
the window W 1 , the aperture st of the window W 2 is also introduced. The value actually determines the step of the window system movement. Thus, double windowing
of the sequence u[ j] is performed: the outer window W 1 is used to determine the
signal’s Hilbert image û W [ j] and signal phase characteristic (SPC), and the inner
rectangular window W 2 with a much smaller aperture is used to select SPC values
from the middle part of the window. The W 1 and W 2 windows move synchronously
with respect to the u[ j].
Fig. 7 Double sliding
window of signal sampling
(W 1 ) and selection of the
signal characteristic section
with the smallest
methodological error
Application of Discrete Hilbert Transform to Estimate …
107
In [15], it is proposed to use window W 1 of a shape other than rectangular. In
general, window processing of signals with windows of various shapes (Hamming,
Hann, Triangular, Kaiser, Chebyshev, etc.) is known and used in spectral analysis
to improve the accuracy of estimating signal spectra [16]. Below are the results of
modeling the process of SPC determining and evaluating the effectiveness of the
weight window processing in the SPC analysis.
A harmonic signal with frequency f with a linear SPC (.0 (t) = 2π f t + ϕ) was
chosen as a test signal. The research methodology consisted of the following stages:
• formation of a signal sample of the type u(t) = U cos(2π f t + ϕ) on the time
interval Ta with a sampling period Td << f −1 << Ta , i.e. the process sample
formation u[n] = U cos(2π f nTd + ϕ), j = 1, n, nTd = TW ;
• formation of the weight function W [ j] with sampling period Td for different types
of windows (Hamming, Triangular, Kaiser, Chebyshev);
• determination of samples of the/signal’s Hilbert image with weight processing of
the form û[ j] = H(u[ j]W [ j]) W [ j];
• determination of the .[ j] SPC assessment on the Ta time interval according to
the algorithm (18);
• calculation of the methodological error of the SPC assessment by the expression
.ϕ[ j] = .[ j] − .0 [ j];
(21)
• comparative analysis of the obtained results for different types of windows.
Table 1 shows the analytical expressions for windows, graphics and methodological errors in determining the SPC for a window of a certain type and a rectangular
window.
The efficiency of using different types of windows was evaluated due to comparison with the methodological error in determining the SPC when using a rectangular
. [ j]
was considered, where .ϕ. [ j]—
window. For this purpose, the ratio K [ j] = .ϕ
.ϕ[ j]
methodological error in determining the SPC of the product of the output signal and
. [ j]
graph for different windows
a rectangular window. A fragment of the K [ j] = .ϕ
.ϕ[ j]
is shown in Fig. 8.
In the experiment, a decrease in the error .ϕ[ j] was obtained: for Hamming
window—by 40%; for triangular window—by 25%; for Kaiser window – by 63%;
for Chebyshev window—by 64%. Consequently, the best results were obtained for
Kaiser and Chebyshev windows, which make it possible to reduce the methodological
error in determining the SPC without using the second window W 2 by more than
60%.
Chebyshev window
Type
0 ≤ | j| ≤ n − 1
[
[
( )]]
cos n · arccos β cos π nj
[
]
W [ j] = (−1) j
;
ch nch −1 (β)
Type, analytical view of the window and
methodological error
Kaiser window
Type
Table 1 Characteristics of methodological error .ϕ[ j] for different types of windows
W [ j] =
I0 (β)
)
(
)
(
n−1
n−1
≤ j≤
−
2
2
;
(continued)
( .
)
I0 β 1 − [2 j/(n − 1)]2
Type, analytical view of the window and
methodological error
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V. Babak et al.
⎧ j
N
⎪
; j = 0; 1; . . . ; ;
⎨
n/2
2
W [ j] =
⎪
⎩ W (n − j); j = n ; . . . ; n − 1
2
Type, analytical view of the window and
methodological error
Hamming window
Type
0≤ j ≤n−1
W [ j] = 0.54 − 0.46 cos
[
]
2π j
;
n−1
Type, analytical view of the window and
methodological error
where β—constant that defines the relationship between the maximum levels of the side lobes and the width of the main lobe; I 0 (x)—zero-king Bessel function
of the first order
Triangular window
Type
Table 1 (continued)
Application of Discrete Hilbert Transform to Estimate …
109
110
V. Babak et al.
Fig. 8 Graphs of K [ j] functions for different windows: 1—rectangular, 2—triangular, 3—
Hamming, 4—Kaiser, 5—Chebyshev
3.8 Circular Median Filtering in the Problem of Analyzing
the Phase of Harmonic Signal Based on DHT
in the Presence of Noise of Significant Intensity
The principle of unwrapping of SPC on the example of a harmonic signal is considered in Sect. 3.3 (Fig. 4). Under the action of noise, with a decrease in the signalto-noise ratio, the probability of a jumps series in the modulo-2π phase value in the
vicinity of the ϕ[ j] mod 2π ≡ 0 increases. This fact is illustrated in Fig. 9. The presence of jumps series leads to the appearance of gross errors in the unwrapped phase.
Under these conditions, the problem arises of filtering the ϕ[ j] sequence before the
unwrapping procedure, which eliminates multiple changes in the 2π −0−· · ·−2π −0
phase in the vicinity of ϕ[ j] mod 2π ≡ 0 values. Since this question is not widely
represented in the measurements’ literature, we will consider it in more detail.
In general, median filtering as an effective method of nonlinear signal processing
for the purpose of statistical smoothing and suppression of impulse noise has been
known for more than 30 years [17–19]. It was proposed by J.W. Tukey in 1971 as
a method for time series analysis. Median filtering is implemented when a sliding
window of a certain width (aperture) moves along the time series by replacing the
value of the series element in the center of the aperture with the median of those values
of the series that are selected from the series using the aperture. If we designate the
one-dimensional median filtering operator as MF2k+1 , where 2k + 1—aperture of
the digital filter, then the response of the median filter to the result of independent
observations {x1 , . . . xn } will be defined as
Application of Discrete Hilbert Transform to Estimate …
111
Fig. 9 Graph of sum of harmonic signal and: Gaussian noise (a), modulo-2π phase value (b)
{ }
{ }
y j = MF2k+1 x j ,
(22)
)
(
where y j = Me x j−k , . . . x j , . . . x j+k ,Me—median operator.
Median filtering is used for random variables and has the following characteristics:
(1) it preserves sharp changes in the input sequence, while linear low-pass filtering
smooths them out; (2) it is effective in smoothing impulse noise.
Since the unwrapped phase and phase shifts of signals belong to the class of
angular quantities with distributions on a circle, special distributions and numerical
characteristics—circular statistics are used to describe their probabilistic properties
[20, 21], then it seems more natural for them in the problem of filtering phase data
to use circular median [22], and circular median filtering. The circular median of a
continuous distribution on a circle with density p(θ ), θ ∈ [0, 2π ) is understood as
the value of the angle θm ∈ [0, 2π ), which is one of the solutions of the equation
Me+π
.
p(θ )dθ = 0.5,
(23)
Me
and p(Me) value is maximum.
For unimodal distributions, the median is always uniquely determined. The
considered property was formulated and proved in [20] by means of the following
statement. If a continuous distribution on a circle with a probability density
distribution p(θ ) is unimodal and
p(θ ) /= p(θ + π ), ∈ [0, 2π )
(24)
is for almost all θ , then the circular median in the interval [0, 2π ) is uniquely
determined.
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V. Babak et al.
Another important property of the circular median concerns the circular mean
deviation: in the case of a unimodal distribution, the circular mean deviation reaches
its minimum at the Me point. Actually, this property of the circular median is the
basis for the use of the circular median for statistical smoothing of the results of
phase measurements.
{
}
In the case of estimating the circular median from the sample ϕ j , j = 1, n , the
phase data values use the sample circular median [Mardia] according to the following
Definition 2.
Definition 2 The sample circular median is the phase angle that corresponds to the
point P on a circle with the following properties:
– half of the sampling points lie on one side of the diameter P Q,
– most sampling points are closer to P than to Q.
This definition is illustrated in Fig. 10, on which on a circle of unit radius the
values of a certain sample of phases of the volume n = 21 are indicated by dots. The
PQ diameter divides the circle into two parts so that there are 10 phase data values
on each side of the diameter.
Circular median filtering of phase data involves determining
}
{ the sample circular
median while moving the rectangular window relative to the ϕ[ j], j = 1, n set
and replacing the element in the center of the filter aperture with the circular sample
median of the values selected by the window.
Denoting the operator of one-dimensional median filtering on the circle as
MFC2k+1 , where 2k + 1—aperture of the circular median filter, the result at its
output can be represented as
{ }
{ }
ϕ j = MFC2k+1 ϕ j ,
where
Fig. 10 Graphical
illustration of the definition
of the term “sample circular
median”
(25)
Application of Discrete Hilbert Transform to Estimate …
113
Fig. 11 Graphs illustrating the modulo-2π phase value filtering process for various median filters
)
(
ϕ j = Mec ϕ j−k , . . . ϕ j , . . . ϕ j+k ,
(26)
where Mec—circular sample median operator that implements the processing of
phase data according to the definition.
Example 2 Check the effectiveness of the process of circular median filtering
modulo-2π phase value in the problem of unwrapping the phase characteristic of
a harmonic signal observed against the background of noise.
The following parameters of the signal and averaging modes were used in the
experiment: U = 1, f = 100 Hz, Td = 10−4 s, n = 1000, σ = 1.3, MW = 23. Graphs
illustrating the filtering process for different types of statistical smoothing are shown
in Fig. 11. Figure 11 shows: (b) MF{ϕ{ j}}—median filtering, (c) MFC{ϕ[ j]}—
median filtering on the circuit. In the graphs on Fig. 11, the number 1 marks the
sample {ϕ[ j]} for a harmonic signal without noise.
It can be seen from the graphs that ϕ[ j] filtering by the median leads to a significant
decrease in the magnitude of the filtered ϕ[ j] values and a significant decrease in the
decay rate when their values change from maximum to minimum (i.e., smoothing out
phase jumps). In addition, the simulation results showed that an increase in the MW
window aperture worsens these characteristics, although it leads to a certain decrease
in the dispersion of values ϕ[ j]. Against, the use of circular median filtering allows,
within certain limits, to combine the contradictory requirements of a small dispersion
of values after filtering and keeping the phase jumps close to the 2π value necessary
for the correct SPC unwrapping.
All types of considered additional processing of signal samples and phase data
are performed in the presented in Fig. 11 scheme programmatically in the Program
production unit.
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V. Babak et al.
4 Conclusions
The obtained results expand the possibilities of practical use of the DHT in the
implementation of information-measuring technologies for processing cyclic signals
in power engineering, radiophysics and other industries.
1. The characteristic features of the practical use of DHT in the problems of
measuring the characteristics of cyclic signals, primarily the characteristics
associated with their phase, are given.
2. According to the simulation results, the methodological error in determining the
characteristics of signals based on the DHT, due to the completed interval of their
analysis, is a function of an oscillating nature, the values of which increase in
the vicinity of the boundaries of the analysis time window.
3. Additional window processing of time-limited signals makes it possible to reduce
the methodological error in SPC determining the. It has been established that the
greatest gain (up to 60%) in reducing this error comes from the use of Kaiser and
Chebyshev windows, which can be recommended for use in high-speed precision
phase-measuring devices.
4. In the case of statistical processing of the cyclic signals phase, the selective
circular median can be effective for constructing digital filters for the modulo-2π
phase value under the action of noise of considerable intensity.
5. The given version of the structure of the system implementation for obtaining
and analyzing the characteristics of cyclic signals based on the use of DHT
can be used as a typical one in the development of new information-measuring
technologies.
References
1. Babak, V.P., Babak, S.V., Eremenko, V.S., Kuts, Y.V., Myslovych, M.V., Scherbak, L.M.,
Zaporozhets, A.O.: Models and Measures for the Diagnosis of Electric Power Equipment.
In: Models and Measures in Measurements and Monitoring, vol. 360, pp. 99–126 (2021).
https://doi.org/10.1007/978-3-030-70783-5_4
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Using of Big Data Technologies
to Improve the Quality
of the Functioning of Production
Processes in the Energy Sector
Viktoria Dzyuba
and Artur Zaporozhets
Abstract The chapter is devoted to the use of the latest Big Data technologies in
the study of production processes for energy facilities. The review of publications of
Ukrainian and foreign scientists was carried out, the forecasts of leading IT companies on the relevance of the use and practical significance of the latest Big Data
technologies were considered. The world and domestic experience of the practical
use of Big Data technologies is analyzed, which made it possible to emphasize their
leading role in accelerating the digital transformation of the energy infrastructure.
It is shown that the popularization of the concept of extensive data and the use of
modern digital technologies leads to the need to create a global electronic environment. The current directions of digitalization of the energy industry and methods for
analyzing Big Data in the energy industry are being explored. The advantages and
disadvantages of existing models for the presentation of Big Data are analyzed. A
binary search algorithm is presented in a given range for a wide circle of problems
of the energy complex. It has been established that in most modern organizations,
systems designed for processing and storing Big Data are a common component of
the data management architecture. The accumulation and use of Big Data leads to the
improvement of operational processes, improved service, the development of unique
marketing campaigns, taking into account consumer preferences, which generally
leads to an increase in profitability. It is assumed that the optimization of the organization of production activities will open up the latest approaches to the development
of the energy industry, since the effective Big Data using will make it possible to
make quick and high-quality business decisions and, accordingly, have a competitive
advantage over others.
V. Dzyuba
Cherkasy Bohdan Khmelnytsky National University, Cherkasy, Ukraine
A. Zaporozhets (B)
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: a.o.zaporozhets@nas.gov.ua
State Institution “The Institute of Environmental Geochemistry of NAS of Ukraine”, Kyiv,
Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_6
117
118
V. Dzyuba and A. Zaporozhets
Keywords Big data · Information technology · Data science · Mathematical
model · Artificial intelligence · Energy objects
1 Introduction
Big Data (BD) technologies allow to see, understand and connect large-scale volumes
of information, where the main role is given to the design of data storage and the
successful use of information technologies.
The main sources for generating BD are:
• information from the Internet (social networks, blogs, media, forums, websites);
• device metrics (sensors, video recorders, “smart” gadgets, smartphones);
• corporate data (archives, internal information of companies and organizations,
etc.) [1].
BD has been at every stage of human development, however, they did not have
such value as in our time with the emergence of the latest technology. According to
IDC forecasts, the total amount of data in the world will grow by about 61% annually,
that is, we live in a world where information is generated at an ultra-fast pace, where
the main role is given to its processing and analysis. BD is a collection of structured,
semi-structured and unstructured data that can be used for modeling, forecasting,
and machine learning [2].
The modern possibilities of BD allow to solve diverse tasks in most areas of human
activity, namely: retail, medicine, industry, energy, ecology, banking, tourism, and
the gaming industry. In particular, the main task of today for the energy sector is the
collection, transmission and visualization of data for decision-making, forecasting of
production processes. Successful processing and analysis of large amounts of information leads to an improvement in the quality of goods and services and automation
of the energy industry as a whole [3, 4].
2 Literature Analysis and Problem Formulation
The study of the latest technologies and information systems is related to the BD
concept. The basic concepts, principles and approaches of BD are reflected in the
works of K. Lynch, W. Mayer-Schenberger, K. Kukier, J.-P. Dix. The impact of BD
on production processes and business as a whole was studied by B. Franks, E. Siegel,
D. Foreman and many others. It should be noted the results of a study by scientists
from the Warsaw Institute S. Buchholz, M. Bukowski, A. Shnegolski, which was
aimed at studying the relationship of BD with forecasting the development of the
energy complex and other leading economic sectors [5, 6].
Using of Big Data Technologies to Improve the Quality …
119
International research companies (Forrester, IDC, Gartner) predict the rapid
growth of the BD market over the next 5 years. An increase in spending on
infrastructure, software, and mass-use services is expected [7].
Analyzing the existing approaches to managing production processes in the energy
sector, one can notice that outdated management models are not functional and lose
their relevance. Along with this, there is a need to optimize the network structural
elements of the energy market in order to provide high service, confidentiality of
operations, control over transactions, etc.
It should be noted that foreign projects on technical diagnostics of the structural
components of energy systems are being actively developed. For example, Westinghouse, using BD technologies, has developed a platform for monitoring the technical
condition of equipment for energy systems. The US Department of Energy launched
a pilot project using BD to detect and predict vibrations for selected types of fittings
[8, 9].
Korea has developed BD system that diagnoses, analyzes and predicts the operation of energy facility equipment in real time (about 16,000 pieces of equipment).
This occurs due to wireless sensors.
Similar systems are being developed in Ukraine, which used for diagnosing the
state of thermal power equipment using BD and machine learning [10, 11]. Also in
Ukraine, studies are being conducted on the implementation of BD technologies to
predict the economic efficiency of projects in the heat power engineering [12–14].
A company of scientists led by M. Gilburnt and S. Strinivas proposed a new
approach and software tools to search for the necessary arrays with data that are
grouped by certain characteristic properties and located side by side. At the same
time, the task of mapping between data models from different environments remains
relevant. Alejandro Maté describes the benefits of using multidimensional models
and mapping to a relational model. However, the use of multidimensional models is
impossible when using a single “key-value” database [15].
V. Borkar, Yingyi Bu offer a qualitatively new approach based on object-oriented
programming, but with a number of restrictions on the number of links between
objects. That is, there is no single universal approach to working with BD today.
According to McKinsey Global Institute forecasts, a new frontier of innovation
and competition is approaching, as data storages are constantly growing, and organizations cannot generate information arrays that go beyond traditional databases
(web logs, machine code, geodata, video and audio materials, etc.). That is, companies have access to their BD, which is regularly updated, but do not have software
tools for analyzing it. This is the basis for the development and use of the latest BD
technologies [16].
The latest trends in BD research are related not only to data descriptions, but also
to sematics. It should be noted the progressive achievements (software for collecting
various types of data and their storage) in the field of energy, oil and gas industry,
and administrative management.
Leading information technology companies (IBM, Oracle, Microsoft, HewlettPackard, EMC) use the concept of BD to build business strategy and productive
organization of workflows.
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V. Dzyuba and A. Zaporozhets
However, for most companies there are difficulties associated with poorly structured data, which makes it impossible to try to analyze and process BD. There is
a need to develop new approaches for quantitative and qualitative analysis of BD.
Based on this, a common problem for users of this BD architecture design can be
identified, since such databases need to be adapted to the needs of each company
using a specific set of tools.
3 Research Goals and Objectives
The study is carried out in order to improve the quality of the functioning of production processes for energy facilities through the use of new approaches to the BD
processing.
To achieve these goals, it is proposed to solve the following tasks:
1. to consider the existing areas of digitalization of the energy industry and ways
to BD analyze in the energy industry;
2. to analyze models and tools for various types of BD processing;
3. to develop and implement a binary search algorithm in a given range of energy
complex problems.
4 Research Methods
In the course of the study of BD technologies, an analysis of the world and domestic
experience in BD processing was carried out. In the process of considering mathematical models for representing different types of BD, the method of comparative
analysis was used with the involvement of system analysis methods. The methods
of artificial intelligence, object-oriented analysis and design were used to establish patterns, predict values, and evaluate the parameters of production processes of
energy facilities.
5 Research Results
5.1 Main Approaches of the BD Concept in the Energy
Industry
Modern continuous control of the pipelines operation, electrical networks, real-time
risk management, optimization of delivery routes, emergency response, building
smart networks can be implemented using BD technologies.
Using of Big Data Technologies to Improve the Quality …
121
The key characteristics of BD include the following: (1) a large amount of data
(Volume); (2) regular updating and processing of data (Velocity); (3) simultaneous
combination of different types of information processing (Variety). However, over
time, the IBM proposed to add Veracity, and IDC added Viability and Value [15].
Using BD analytics it is possible, in an optimally short period of time, to visualize information, find patterns and predict the further development of this area. The
simplest example of using BD technologies is the analysis of the consumer energy
sector in order to provide potential users with the necessary services.
BD, by itself, does not carry meaningful information for a person, in connection
with this, existing ones are constantly being improved and new approaches to BD
analysis are arise. The main BD analysis methods successfully used in the energy
industry include [2, 6, 9]:
• classification—to predict consumer behavior in a particular market segment;
• cluster analysis—for classifying energy facilities into groups by identifying their
common features;
• crowdsourcing—to collect information from a large number of sources;
• data mining—to identify previously unknown and useful information that will be
useful for decision-making in various areas of the energy sector;
• machine learning—to create self-learning neural networks, as well as high-quality
and fast information processing;
• mixing and integration—for converting data into a single format (for example,
converting audio and video files to text);
• signal processing—for recognition of signals against the background of noise and
further analysis;
• unsupervised learning—to reveal hidden functional relationships in data arrays;
• visualization—to present the results of the analysis in the form of diagrams and
animations.
The main means for placing BD systems are cloud technologies, in which, if
necessary, the number of servers can be increased, and payment is made for the
spent time on processing and storage. This is explained by the fact that most enterprises give off insignificant funds to maintain their own system of extensive server
and storage infrastructure. In order to improve service, cloud providers offer the
latest approaches to BD transfer, through managed services: Amazon EMR (formerly
Elastic MapReduce); Microsoft Azure HDInsight Google Cloud Dataproc.
The use of cloud technologies allows to store BD in such systems as: Hadoop
(HDFS—distributed file system); Amazon Simple Storage Service (S3—low cost
cloud object storage); NoSQ databases; relational databases.
Cloud technologies make it possible to develop a virtual infrastructure and its
distribution according to profile characteristics to support production processes. That
is, the more criteria we analyze, the more accurate the final result will be (Table 1).
It is worth noting that the digitalization of the energy industry is intensifying
every year, however, there remains a need to build an operating model that will
be sustainable and efficient. To do this, it is necessary to analyze large-scale data
arrays in detail, which will allow a clear understanding of production processes and
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V. Dzyuba and A. Zaporozhets
Table 1 Directions of digitalization in the energy industry
No
Name
Description
1
Blockchain
Distributed energy indicators monitoring
database
2
Digital analogs
Digital mathematical models of real energy
facilities
3
Big data
Multifunctional applications
4
API and SaaS
Synchronized management of engineering
and analytical models in energy
5
Mobile devices
Automated data collection of local
parameters
6
Artificial intelligence
Improvement and self-learning of
production forecasting models
7
Community platform
Synchronization and data exchange
8
Remote information transfer technologies
Automation of emergency systems
processes, data mapping
track the change in parameters or predict their value. In addition, the analysis and
identification of potential drilling places in the energy industry can be carried out
by processing large amounts of data. All this will allow to provide services in the
energy sector at a high level.
5.2 Mathematical Formalization of Energy Objects
The use of innovative approaches (automated sensors, controllers, specialized software) in the energy industry has contributed to the BD accumulation (state of substations, network indicators, value of production process coefficients, changes in the
value of limiting parameters, etc.), which require processing for further successful
use and analysis.
The mathematical representation of BD arrays that arise in the energy sector
is most conveniently presented in the form of multidimensional models with the
following components: data hypercube rel, dimension V, attribute A, cell X, value
rel (V, A) [1, 5, 7].
A hypercube structurally consists of dimensions and is a correctly ordered set of
cells. A cell can be designated by only one set of dimension values—attributes, but it
is allowed that the cell can be empty. A dimension is an array of attributes that make
up one of the faces of the hypercube. An example of a measurement in the energy
sector is a set of energy facilities: stations, substations, networks, boiler house, main
heating network, treatment facilities, etc.
Let’s consider the main stages of BD describing and processing in practice using
a multidimensional model.
Using of Big Data Technologies to Improve the Quality …
123
Let V be a set of dimensions of a hypercube, then the set of dimension attributes
for V i will look like
}
{
A Vi = A1i , A2i , ..., Aki ,
where A = A V1 ∪ A V2 ∪ ... ∪ A Vn —hypercube attribute set at i = 1, 2, ..., n. The
following inclusions are allowed: V ' ⊆ V —set of fixed dimensions, A' ⊆ A—set
of fixed attributes. The hypercube is defined as a correspondence to the sets V, A:
rel (V, A), and the subsets of the hypercube, respectively, are r el ' (V ' , A' ). That is,
for each cell of the data hypercube, a single set of dimensions attributes Ar el ⊆ A
is associated. Based on this, in order to access the data, it must be specified a set
of fixed dimensions and attributes V ' ⊆ V , A' ⊆ A. Obviously, the dimension
key is an attribute that specifies the hypercube dimension string. The formation of a
multidimensional database consists of many hypercubes that support a hierarchy of
dimensions and formulas.
The task of BD processing is reduced to making a decision about grouping certain
objects e ∈ E and establishing characteristic relationships f ∈ F between them,
taking into account existing associations a → n e, f . For the existing f characteristics
of the general set of characteristics |F| and objects e from the set of the total number
of objects |E|, the following relations will take place:
{
}
e( f ) = e ∈ E : n e, f > 0 ,
{
}
f (e) = f ∈ F : n e, f > 0 .
If there are several objects that depend on one characteristic, then binary search
should be used to find the required object. In the case of searching for characteristics
of N alternatives, the number of binary queries q can be given by the formula
q = log2 (N ).
Establishing associated relationships between objects e and characteristics f
minimizes the number of requests:
(
q = log2
)
|E|
.
|e( f )|
(1)
The presence of additional associations n e, f can be expressed in terms of the
number of additional binary queries in order to further establish links with the
required object. The increase in additional requests can be expressed as follows
(
)
k = 1 + log2 n e, f .
(2)
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V. Dzyuba and A. Zaporozhets
Thus, taking into account (1) the characteristic f significance for the object e and
the importance factor (2), a significant number of binary queries can be represented
as (3):
(
(
))
I (e, f ) = 1 + log2 n e, f · log2
(
)
|E|
.
|e( f )|
(3)
Using (3) allows one to predict the number of questions of existing objects e with
different characteristics f . During studying the objects proximity degree E 1 and E 2 ,
it is necessary to use the basic laws of vector algebra, consider the distance between
the vectors (V (e1 , f ), V (e2 , f ), ...), because each object e can be assigned weight
V (e, f ). V (e, f ).
Assessment of the distance between objects for each characteristic can be
represented as follows
.
f
d(e1 , e2 ) = .
f
|V (e1 , f ) − V (e2 , f )|
max(V (e1 , f ), V (e2 , f ))
.
(4)
Distance (4) depends on the number of existing characteristics, if it is normalized
on the interval by dividing by the largest value of the distance, the dependence can
be avoided [2, 4, 8].
Thus, a large-scale set of objects (domains, users, energy resources, reports,
geolocations, etc.) and a data features repository (values of indicators, parameters,
documents for data processing, dictionaries, etc.) are formed.
5.3 New Approach to the Search for Parameter Values
in the Analysis of Energy Sector Data
During BD processing in practice, the problem arises in maximizing the minimum
value of the array after performing the specified operations. Let’s consider the operations of triple and double multiplication often used in the analysis of data in the
energy sector [17, 18]. This is because when parameter values and their backups are
stored, the distance between objects increased in 2 or 3 times.
A successful solution of this problem lies in the use of software tools based on
binary search. Let’s consider a binary search algorithm in the range [1, max(array)]:
(1) let f = 1, while l—maximum element of the array, and res as INT _MIN;
(2) perform a binary search at f ≤ l;
(3) check if mid is the minimum element. For this its necessary to perform the
“is_possible_min()” operation;
(4) in the “is_possible_min()” function, it is necessary to iterate over the elements
from the end (N-1) of the array to index 2 and check if “arr [i] < mid”. Then
return 0 if the condition is performed. Otherwise, it’s necessary to calculate
Using of Big Data Technologies to Improve the Quality …
125
Fig. 1 Software
implementation of the
algorithm in the Python
environment
the value 3 × which is attached to arr[i-1] as x and arr[i-2] as 2x. Next, it’s
necessary to check the truth of the arr [0] ≥ mid and arr [1] ≥ mid conditions
and output 1. Otherwise, it’s nesessary to return 0.
The implementation of the described algorithm is shown in Fig. 1.
If the “is_possible_min()” function evaluates to true when the algorithm is
performed, then the mid value exists. Since the minimum value of max(res, mid)
is stored in the res variable, it can be maximized the minimum value by moving
to the right on step f = mid + 1. Otherwise, it can be tried moving left on step
l = mid − 1.
The computational complexity of mathematical operations for the proposed algorithm is defined as O(N*log(maxval)), where N—dimension of the original array,
and maxval—maximum element of the array. Adequate processing power must be
used to achieve the required speed of operations. In practice, this can be hundreds
or thousands of servers that are capable to distribute data and to work together in a
cluster architecture (Hadoop, Apache Spark). To date, establishing a high speed of
computing in a cost-effective way is a rather problematic task.
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V. Dzyuba and A. Zaporozhets
6 Discussion
Conducting research within the framework of the topic of this section has shown that
in connection with the rapid development of the latest technologies, a large-scale
information accumulation is taking place. Using methods of BD analysis, humanity
can qualitatively and quickly exctract benefit from this data array. The common
advantage of BD is the provision of better services for the population, since each
existing link in the public or private structure has the ability to optimize its processes.
It has been established that BD arrays, in most cases, remain in an unprocessed
form, and if further use is necessary, processing is carried out with the involvement
of software or intelligent data processing. As confirmation, a binary search algorithm
is presented to maximize the minimum value of the array.
The mathematical representation of BD arrays arising in the energy sector in the
form of a multidimensional hypercube model is detailed.
The process of BD consolidating for analyzing and forecasting the development
of the energy complex allows generating the following tasks:
• growth of optimization of data receipt, processing and further use for making
managerial decisions on energy facilities;
• building new strategies for the development of the energy industry through the
data analysis that are not included in the reports and are not taken into account
during decision making;
• regular monitoring of negative development trends for their further elimination.
To ensure an integrated approach to the information analysis on energy facilities,
it is necessary:
(1) to store and to manage BD;
(2) to process information from different databases (relational, multidimensional,
XML, NoSQL, structured, unstructured, etc.);
(3) to use combined approaches to obtain information data.
The need for regular processing and BD high-speed transmission has led to a
number of requirements for the original computing infrastructure. To avoid overloading a server or server cluster, the computing power provided for BD processing
and transmitting must meet the established requirements.
To create a local BD system, it is appropriate to use Apache open source technologies in addition to Hadoop and Spark, which contain the following structural
elements: YARN (built-in resource manager and Hadoop work scheduler); MapReduce programming program, which is also the main component of Hadoop; Kafka
(messaging platform and data transfer from program to program); HBase databases;
SQL-on-Hadoop query systems such as Drill, Hive, Impala and Presto.
For example, the use of the latest Hadoop-based appliances allows you to create
a customized computing structure for implementing BD projects, where hardware
and software are kept to minimum.
Thus, the transfer of local BD sets and their processing is a rather cumbersome
process for energy companies. This is due to ensuring the availability of data for
Using of Big Data Technologies to Improve the Quality …
127
analysts, in particular in distributed environments, which are a combination of diverse
platforms and data storages. The solution is possible through the development of data
catalogs containing metadata management functions and data line functions. That is,
data quality and data management must be a priority to ensure that BD sets are used
in a trustworthy manner.
7 Conclusions
The chapter provides a rationale for the relevance of using BD technologies to
improve the quality of the functioning of production processes in relation to energy
facilities. The main directions of digitalization of the energy industry are considered and the main approaches to the further development of the energy complex are
identified.
Mathematical tools for BD representing and processing are analyzed, in particular,
a multidimensional model in the form of a hypercube is considered in detail. A
software implementation of the binary search algorithm for maximizing the minimum
value of an array is presented. It has been established that the allowable dimensional
values define the cells of the hypercube, which in practical use can be located densely
or sparsely. Sparsity of the hypercube arises in the case of an increase in the number
of dimensions and the establishment of measurement values.
It should be noted the advantages and disadvantages of existing models for BD
representing, namely: a multidimensional model allows to visualize and analyze data,
along with this, the sparsity of a hypercube with heterogeneous data is a significant
disadvantage during calculations; object model requires modification for the use
of extensive data; the graph model is used to analyze and establish links between
objects of a small number, since the computational complexity of search algorithms
increases.
Thanks to the successful analysis and BD processing, it is possible to achieve ultraaccurate forecasts to assess the effectiveness of production activities in a particular
area. In addition, it can be determined the current geolocation for energy facilities
and the necessary equipment; calculate the minimum and maximum loads in the
network; effectively allocate the use of energy resources, etc.
The conducted studies suggest that one of the topical areas of using BD technologies in the energy complex of Ukraine can be monitoring the functional state of power
units to improve their operational properties. In general, the creation of a multi-level
system for diagnosing and predicting the technical condition of energy facilities
(turbines, pumps, substations, etc.) opens up new prospects for the development of
the industrial infrastructure of the country.
Therefore, the use of BD technologies in the energy sector is not only a tool for
data processing and well-predicted management decisions, but also a promising way
to transform the energy complex as a whole.
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Parametric Identification of Dynamic
Systems Based on Chaotic
Synchronization and Adaptive Control
Artem Zinchenko
Abstract In this chapter algorithm of parametric identification based on chaotic
synchronization and adaptive control is offered. It’s shown that noise influence
on the nonautonomous dynamic system which is near to border of a synchronous
mode establishment, leads to appearance in time realization intervals of synchronous
dynamics interrupted with asynchronous intervals. Numerical simulations on Rössler
system are presented to demonstrate the effectiveness of the proposed approach.
Furthermore, the principal possibility of use of a method on small samples during
observing a function from all system coordinates is shown. It also demonstrates
results of comparison from a time delay method on which basis full reconstruction
of Lorenz dynamic system is made.
Keywords Parametric identification · Chaotic synchronization · Adaptive control ·
Rössler system · Lorenz system
1 Introduction
Investigation of nonlinear dynamics in spatially distributed systems of different physical nature (in thermal physics, chaotic advection, hydromechanics, geophysics,
chemistry, etc.) and identification of parameters and structures of mathematical
models of complex processes and systems, based on accurate and incomplete
measurements, is actively studied and attracts the attention of many researchers
in recent times [1–4]. In this area, the interest of research is due to the great fundamental and practical value of this issue due to the fact that most important systems are
dissipative, distributed and demonstrate complex, including chaotic modes of oscillation. Many problems of radiophysics, plasma physics, nanoelectronics, ecology,
etc. are reduced to the analysis of space-continuous models that demonstrate space–
time chaos and the processes of structure formation. The task of the investigation
A. Zinchenko (B)
Kyiv International University, Kyiv, Ukraine
e-mail: artem005@yahoo.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_7
129
130
A. Zinchenko
is quite complicated if the information about the studied object is limited to a onedimensional implementation of one of the coordinates of the system or the presence of
only the observed scalar time sequences (incomplete measurements). In this case, an
algorithm of global reconstruction was proposed in [4] to build mathematical models
which is implemented in several stages. At the first stage, it was visualized the series,
highlight the trend, identify the transition mode, the process of data accumulation,
the steady process of chaotic fluctuations, identify and analyze chaotic dynamics.
In the second stage, we determined the dimension of the embedding space, time
delay and reconstruct closed trajectories using Takens’s and Whitney’s embedding
theorems for the scalar realization xi = x(i.t), i = 1, . . . , N of the system. In the
third stage, we a priori defined a system of ordinary 1st-order differential equations
of the selected structure and determine the evolution operator, for example, by the
least squares method (LSM). For accurate data obtained from the chaotic mode of
a dynamic system, we can skip the first step. It is clear that reconstruction algorithms (for example, the method of sequential differentiation) significantly depend
on the error of approximation by regression analysis (for example, the recurrent
LSM), the accuracy of derivative calculations, the presence of noise in data limiting
the use of algorithms for large embedding spaces. Therefore, the reconstruction of
nonlinear dynamic systems is not always successful, and the most common problems
are the tasks of automatic nonlinear control—parametric identification with a known
structure of the system.
This paper considers the application of chaotic synchronization [5, 6] and adaptive
control [7] to the problem of parametric identification. To accelerate the synchronization of unidirectional oscillators, it is proposed to use a relay algorithm which
is a partial case of the standard high-speed pseudo-gradient algorithm. In addition,
instead of the standard vector function of the rate of change of a smooth objective function, it is proposed to use a class of functions (for example, trigonometric)
that fully satisfies the conditions of pseudo-gradient algorithm, the conditions of
reachability, the conditions of existence of "ideal control" and the convexity of the
feedback function on the corresponding state vector of the controlled system. The
importance of the study of the zero Lyapunov exponent is shown. The example of
the Lorentz system shows the efficiency of the vector function in comparison with
the standard one. In addition, on the example of this system, parametric identification was performed by observing not only one coordinate of the system but function
from all coordinates. For this case, the formulas are derived and the conditions of
convergence of the method are given. Furthermore, on the example of the Lorentz
system, we are showed the possibility of applying the algorithm for small samples, in
particular, the results of the convergence of the method when observing the function
from all coordinates of the system. To test the effectiveness of the algorithm, the
results are presented to demonstrate comparing the proposed method with the standard based on approximations. In addition, based on the latter method, an example of
a complete reconstruction of the dynamic Lorentz system is showed when observing
the function from all coordinates of the system.
The structure of the article is as follows: the first part provides general information
about the chaotic synchronization of two oscillators and about the zero Lyapunov
Parametric Identification of Dynamic Systems Based on Chaotic …
131
exponent, modifications are proposed to accelerate the convergence of the method;
the second part presents the structure of the algorithm with proposals and it considers
the standard method of identification from the family of approximation methods; the
third part presents the results of the study.
2 Chaotic Synchronization of Two Unidirectional Coupled
Oscillators
Recently, there has been great interest in the study of synchronization in complex
nonlinear systems consisting of a large number of elements with different connections. Currently, the basic laws of the phenomenon of chaotic synchronization
are well studied for systems with a small number of degrees of freedom. There
are several different types of synchronous behavior of connected chaotic systems:
complete, generalized, noise-induced, asymptotic, phase, time-scale synchronization, and others. However, the use of chaotic synchronization for parametric identification of spatially continuous high-order self-oscillating systems is a new area of
research, little studied and first proposed in [8–10]. That is why there is a need to give
own definition of synchronization of dynamic
systems for parametric identification.
.
Let’s consider k dynamic systems i (i = 1, . . . , k), each of which is formally
described by 6 parameters:
.
= {T , Ui , X i , Yi , φi , h i }, i = 1, . . . , k,
i
where T – total set of time points; Ui , X i , Yi – set of inputs, states and outputs,
respectively; φi : T × X i ×Ui → X i – display of transitions; h i : T × X i ×Ui → Yi —
display of outputs. Assume that given l functionals g j : ϒ1 ×ϒ2 ×· · ·×ϒk ×Ti → R1 ,
i = 1, . . . , l. Here ϒi are the sets of all functions on T with values in ϒi , so
ϒi = {y : T → Yi }. We will assume that the set of moments T is either a positive
axis T = R1 (continuous time) or a set of natural numbers T = 1, 2, . . . (discrete
time). For any τ ∈ T we define στ as a shift operator on τ, so στ : ϒi → ϒi is defined
y(t +τ) for any y ∈ ϒi and t ∈ T. Let yi (·) denotes the output function
as (στ y)(t) = .
t ∈ T , i = 1, . . . , k. Let x (1) (t), . . . , x (k) (t)
of the system i :yi (t) = h(x
.i (t), t),.
are solutions of the system 1 , . . . , k defined for all t ∈ T with initial states
x (1) (0), . . . , x (k) (0), respectively. Then we can define the synchronization of dynamic
systems.
.
.
Definition 1 In systems 1 , . . . , k , processes x (1) (t), . . . , x (k) (t) will be called
synchronized (and systems will be called fully synchronized) in relation to functionals g1 , . . . , gl , if the identities of g j (στ1 y1 (·), . . . , στk yk (·), t) ≡ 0, j = 1, . . . , l
are true for all t ∈ T and some τ1 , . . . , τk ∈ T .
132
A. Zinchenko
.
.
Definition 2 We will also call the
1,...,
k systems asymptotically
synchronized in relation to functionals g1 , . . . , gl , if for some τ1 , . . . , τk ∈
T lim g j (στ1 y1 (·), . . . , στk yk (·), t) = 0, j = 1, ..., l.
t→∞
.
.
Definition 3 Let’s call systems 1 , . . . ,
k approximately synchronized with
respect to functionals g1 , . . . , gl , if there exists ε > 0 and τ1 , . . . , τk ∈ T , such that
inequalities
||
||
||g j (στ y1 (·), . . . , στ yk (·), t)|| ≤ ε, j = 1, . . . , l
1
k
performed for all t ∈ T .
In our further studies, we will focus on the problem of asymptotic coordinate
synchronization or global exponential synchronization, which is optimal with respect
to convergence and synchronization errors [11, 12] for the parametric identification
of dynamical systems. We will understand it as the fulfillment of the relation
||
||
lim ||x (1) (t) − x (2) (t)|| = 0,
t→∞
(1)
where x (1) , x (2) —state vectors of synchronized dynamic systems. Let us describe a
general method for solving such problem. For simplicity, let’s assume that there are
two subsystems described by affine models:
ẋ1(1) = f 1 (x1(1) ) + g1 (x1(1) )u 1 ,
ẋ1(2) = f 2 (x1(2) ) + g2 (x1(2) )u 2 .
(2)
The original systems are not interconnected. We pose the problem of asymptotic coordinate synchronization of subsystems—to find a control algorithm u i =
Ui (x1(1) , x1(2) ), i = 1, 2 to ensure the control goal (1). The solution of the problem
is trivial if the right parts of (2) can be changed arbitrarily and independently, that
is, if the following conditions are satisfied m = n,g1 (x1(1) ) = g2 (x1(2) ) = In , where
In —identity matrix n × n. Then, taking, for example, u 1 = 0, u 2 = K (x1(1) − x1(2) ),
where K > 0—amplification factor, we obtain the synchronization error equation in
the form:
ė = f (x1(1) (t)) − f (x1(1) (t) − e) − K e,
(3)
where x1(1) (t)—solution of the first Eq. (2) on a given time interval at u 1 = 0.
If the Jacobi matrix A(x (2) ) = ∂ f(2) (x (2) ), j = 1, . . . n is bounded in some zone .
∂x j
containing a solution to system (2), then for sufficiently large K > 0 the eigenvalues
of the symmetric matrix A(x (2) ) + A T (x (2) ) − 2K In lie to the left of the imaginary
axis at x (2) ∈ .. In this case, the system itself will have the property of convergence
in .,, that is, all its trajectories lying in ., coincide at t → ∞ to a single limited
solution. And since e(t) ≡ 0 is such a solution, then all trajectories converge to it.
Parametric Identification of Dynamic Systems Based on Chaotic …
133
Thus, the synchronization of two systems takes place when the feedback between
the subsystems whose parameters should be estimated is strengthened. Note that
the smoothness of the right-hand sides and the existence of the Jacobian matrix are
not required
of
| systems: |it is sufficient to require the Livshitz
|
| for synchronization
|
|
|
(1)
(2) |
condition | f (x1 ) − f (x1 )| ≤ L |x1(1) − x1(2) | to be satisfied for some L > 0 [12].
As a control algorithm (adaptation method) in this work, a relay (sign) algorithm
was chosen, which is a special case of the standard velocity pseudogradient algorithm
[12]. The latter is written like this:
u(t) = u 0 − γψ(x(t), u(t)),
where γ > 0—scalar step multiplier (amplifier factor or feedback constant),
u 0 —some initial (reference) control value (usually u 0 = 0), and vector-function
ψ(x, u) satisfies the pseudogradient condition ψ(x, u)T ∇u w(x, u) ≥ 0, where
w(x, u)—velocity change of a smooth objective function (feedback function), and
∇u = { ∂u∂ 1 , . . . , ∂u∂ n }T —gradient vector. Since to estimate the parameters of the
leading system in the state vector xi , i = 1, . . . , n, the control is u = xi' , where xi' —
corresponding state vector of the controlled system, then the relay (sign) algorithm
will take the form:
ẋ i' (t) = f i (x ' ) − γ sign∇xi' w(x(t), x ' (t)),
(4)
where sign for a vector is understood as by component, for state vector x ' =
col(x1' , . . . , xn' ) we have sign(x ' ) = col(sign(x1' ), . . . , sign(xn' )). For the convergence of this algorithm, a number of conditions are required [12], the main of
which are the convexity of the function w(x, x ' ) at xi' and the existence of an
“ideal control”—vector x∗' such that the condition w(x, x∗' ) ≤ 0 for all x is satisfied
(accessibility condition).
In numerical simulation of synchronization for the parametric identification of
the Ressler, Lorentz, Van der Pol, Rayleigh and other systems, it was found that
the use of some feedback functions that satisfy the above conditions accelerates
synchronization and improves adaptation.
So, at choosing the feedback function γsign∇xi' ar ctg(x ' (t) − x(t)), in contrast
to the standard [5, 6] γ(x ' (t) − x(t)) for synchronization and proposed in [8–10]
for parametric identification using synchronization, with the same choice of initial
conditions, the method coincides approximately at 2 times faster. The simulation
results for the example of the Lorenz system are presented below (Figs. 3 and 4).
Let’s consider now the behavior of two unidirectional coupled oscillators
ẋ = H (x, α),
ẋ ' = G(x ' , α' ) + γw(x, x ' ),
(5)
where x = (x1 , x2 , . . . , xn ), x ' = (x1' , x2' , . . . , xn ' )—state vectors of the leading and
controlled “drive-response” systems, respectively, H and G define the vector field
134
A. Zinchenko
of the considered systems, α and α' are parameter vectors, component w is responsible for unidirectional communication between systems, and parameter γ—feedback
constant that determines the strength of the connection between these systems.
The behavior of unidirectional coupled oscillators (5) for the dimensions of the
phase spaces n and n ' , respectively, can be characterized using the spectrum of
Lyapunov characteristics λ1 ≥ λ2 ≥ · · · ≥ λn+n ' . Since the behavior of the leading
system does not depend on the state of the controlled oscillator, the spectrum of
Lyapunov characteristic exponents can be divided into two parts: Lyapunov indicators of the leading system λ1 ≥ λ2 ≥ · · · ≥ λn and conditional Lyapunov indicators
λ'1 ≥ λ'2 ≥ · · · ≥ λ'n ' , characterizing the behavior of the controlled system. When
the feedback parameter γ is changed, the Lyapunov indicators of the leading system
remain unchanged, since the dynamics of the leading system does not depend on the
intensity of the connection, while the values of the conditional Lyapunov indicators
change. Obviously, this approach can also be applied to describe the non-autonomous
behavior of an oscillator under external influence: in this case, it is appropriate to
consider only the spectrum of conditional Lyapunov indicators λ'1 ≥ λ'2 ≥ · · · ≥ λ'n ' ,
responsible for the behavior of systems (5), and the feedback value γ should be interpreted as a controlled parameter, which determines the amplitude of the external
action. Although, as shown above, the main synchronization condition is the convergence condition: the eigenvalues of the Jacobi matrix A(x ' ) = ∂∂xf' (x ' ), j = 1, . . . , n '
j
are uniformly negative for all values x ' , and this condition is much easier to verify than
the condition for the negativity of all conditional Lyapunov indicators, however, the
research of the Lyapunov indicators spectrum of coupled systems near the synchronization limit (of any kind) is important. Interacting systems can be characterized by
both chaotic and periodic dynamics. If we consider chaotic dynamical systems, then
the leading Lyapunov exponents of each of them (at least λ1 and λ'1 ) are positive.
In any case, in the absence of a connection between the systems (γ = 0), in each
of the spectra of the Lyapunov exponents, there necessarily exists a zero Lyapunov
exponent (λi = 0 and λ'j = 0, respectively), which is responsible for the evolution
of a small perturbation, which describes the displacement of the graphic point along
the phase trajectory in the phase space of the considered system. In the case of periodic systems, these zero Lyapunov exponents are older (i.e. i = 1, j = 1). With an
increase of the connection parameter γ between systems, the zero Lyapunov exponent of the leading system remains zero, and the conditional zero Lyapunov exponent
characterizing the behavior of the controlled system can change. It is known [6], that
for a system with periodic behavior in the absence of noise .0 becomes negative
precisely when the controlled system is synchronized by the periodic signal (control)
acting on it, which is acting on it from the side of the leading system. A more complicated situation arises when the leading system is affected by noise (deterministic or
random). In this case, as shown in [13], the Lyapunov exponent .0 becomes negative even before the start of the synchronous regime, and its value depends on the
feedback parameter as follows:
Parametric Identification of Dynamic Systems Based on Chaotic …
.
.0 (γ) ≈
135
1
− |γ−γ
,
γ < γc ,
| c| √
|
|
|
ln 1 − β γ − γc , γ > γc ,
where γ—feedback parameter between interacting oscillators; γc parameter corresponds to the bifurcation value of the feedback parameter, at which, in the absence
of noise, the synchronous mode is established; β parameter is determined by the
properties of the studied systems.
The change in the sign of the Lyapunov exponent indicates, in general, the qualitative changes that have taken place in the dynamics of the system. The transition
of one Lyapunov exponent to the zone of negative values is associated with the
occurrence of synchronous behavior, for example, in the case of synchronization
of periodic oscillations or the establishment of complete chaotic synchronization.
At the same time, for coupled chaotic oscillators, when the regime of asymptotic
coordinate synchronization is established, the conditional Lyapunov exponent .0
is already essentially negative [14]. Therefore, taking into account the negativity of
the zero Lyapunov exponent, it must be assumed that in this case, below the phase
synchronization limit, some properties of synchronous behavior should appear themselves, although the synchronization mode itself has not yet been established and,
therefore, the use of parametric identification in such cases is inappropriate. That is
why studies of the zero Lyapunov exponent are often used as a criterion for the established synchronous coupling and for finding the feedback parameter corresponding
to the synchronization boundary.
3 Assessment of Unknown Parameters
Despite the ability to successfully synchronize two unidirectional dynamical systems
according to formula (4) with the fulfillment of all synchronization conditions, it is
impossible to assess the unknown parameters of the leading system. For this purpose,
∂g
adaptive control by an unknown parameter is used αl' = h((xi' (t) − xi (t), ∂α
' ), i =
l
'
'
1, . . . n, i /= l, n = n , where αl —unknown parameter, which must be estimated. In
works [9, 10, 13] it is proposed to use adaptive control (adaptive control equation) of
.'
∂g
the following form: αl = −δ(xi' (t) − xi (t)) ∂α
' , where δ – adaptation parameter. In
l
this paper, we will follow to algorithm (4). Then, in the general case, system (5) using
the proposed algorithm (4) for synchronization and the equation of adaptive control
over the unknown parameter αl , l = 1, . . . , n, l /= i will be written as follows:
ẋ = H (x, α),
ẋk' = f k (x ' , {α j | j /= l}, αl' ) − γsign∇xk' w(xk , xk' )
= f k (x ' , {α j | j /= l}, αl' ) − γsign∇xk' .(xk' (t) − xk (t)),
ẋi' = f i (x ' , {α j | j /= l}, αl' ), i = 1, . . . , n, i /= k
136
A. Zinchenko
α̇l' = −δsign∇xk' ar ctg(xk' (t) − xk (t))
∂ fk
,
∂αl'
(6)
where xk (t)—k-th coordinate of the leading system that we observe (control or
controlled variable [10]), ( f 1 , . . . , f k , . . . , f n ) = G from system (5)—controlled
system.
In cases of assessing unknown parameters during controlling other coordinates
ẋ, differentiation difficulties arise. In [10], the following formula was proposed for
this case. Let us consider the growth of the controlled variable in one step of time
discretization of the controlled system under the conditions of algorithm convergence
(4):
∂ fs
'
2
'
.xk' ≈ . f k dt ≈ ∂∂ xfk' .xs' dt ≈ ∂∂ xfk' . f s (dt)2 ≈ ∂∂ xfk' ∂α
' .αl (dt) , where x s —ss
s
s
l
'
th coordinate of xk (t), equation of which consist unknown parameter αl . Then we
∂ fk
∂ fk ∂ fs
can write that ∂α
. In the case when the unknown parameter appears in
' ≈
∂ x ' ∂α'
s
l
l
∂ fk
different coordinates of the vector x by m times, the calculation ∂α
' is made as
l
follows [10]. Let’s consider again the growth of the controlled variable per one step
of time discretization of the controlled system under the conditions of algorithm
convergence (4):
.xk' ≈ . f k dt ≈
≈{
. ∂ fk ∂ fk ∂ fs
}
.αl' (dt)3 .
'
' ∂α'
∂
x
∂
x
s
k
l
m
∂ fk
Then ∂α
' ≈ {
l
. ∂ fk ∂ x ' ∂ fs
∂ fk
∂ fk
m
'
2
.x
dt
≈
.
f
(dt)
≈
{
}
.αl' (dt)2
s
s
' ∂ x ' ∂α'
∂ xs'
∂ xs'
∂
x
m
s
l
m
.
m
∂ fk ∂ fm ∂ fs
} . Further, in order to numerically assess of the unknown
∂ xm' ∂ xs' ∂αl'
parameter αl of the leading system, it is necessary to simultaneously solve two
systems (6) by an iterative method, for example, the fourth or fifth order Runge–Kutta
method with a variable numerical integration step, for example, with the DormandPrince corrective procedure, which makes it possible to ensure local accuracy order
of magnitude at O(10−12 )–O(10−15 ). Note that the synchronization error (3) for
this method is defined as e(αl' , t) = .(xk' − xk )2 , where xk (t)—control variable.
From this equation, we can write that the error in assessing the unknown parameter
∂e(α' ,t)
∂x'
α̇l' ≺ −sign∇xk' .(xk' − xk ) ∂αk' , because α̇l' ≺ − ∂αl' . For small orders dt the
l
l
gain of the controlled variable can be written as .xk' =
∂ ẋ '
∂ ẋk'
.αl' dt.
∂αl'
From this α̇l' ≺
−γ sign∇xk' .(xk' − xk ) ∂αk' .
l
Since the chaotic synchronization of two unidirectional coupled oscillators essentially depends on the initial conditions, the proposed scheme of the parametric identification algorithm coincides for sufficiently small sample sizes of observations of
the controlled variable (about 30 basic oscillations of the Lorentz system). In this
case, the final conditions for the divergence of the method are assumed by the initial
Parametric Identification of Dynamic Systems Based on Chaotic …
137
conditions, and the method is started again. The results of the research are shown in
Fig. 6.
If instead of one coordinate of the leading system xk (t), we observe the function
F(x1 , x2 , . . . , xn ),F : Rn → R1 , then for the equation of adaptive control and
synchronization, the author proposes to use a vector-function of the following form:
ψ(x, u) = γsign∇x ' .(F ' − F), where F ' = F(x1' , x2' , . . . , xn' ). In this case, in
addition to the general conditions of algorithm (4) and synchronization, it’s add a
'
condition ∂∂ Fx ' /= 0, where xk' takes from the system (6).
k
To compare the effectiveness of the method, let’s compare it with the standard
time delay method [15–17], which uses approximation (recursive LSM) for identification. The only clear advantage of this method is the guaranteed possibility of a
global reconstruction of a dynamical system, provided that the system functions are
successfully expanded into the required series.
Let’s set the structure of the dynamical system by ordinary differential equations
of the 1-st order ẋ = F j (x), j = 1, . . . , n. Then, in order to obtain a specific form
of the evolution operator of the function F j is represented as a decomposition by
some basis, while being limited to a finite number of decomposition components.
In a simpler case, the F j specification can be carried out by a polynomial of some
degree v:
v
.
F j (xi ) =
C j,1 ,l2 ,...,ln
l1 ,l2 ,...,ln =0
n
.
n
.
lk
xs,i
, j = 1, . . . , n,
s=1
ls ≤ v,
s=1
where C j,1 ,l2 ,...,ln —unknown coefficients necessary to be found. To calculate these
coefficients, it is necessary to solve a system of N linear algebraic equations:
x j,i+1 =
v
.
l1 ,l2 ,...,ln =0
C j,l1 ,l2 ,...,ln
n
.
lk
xs,i
, i = 1, . . . , N , j = 1, . . . , n,
(7)
s=1
with unknowns C j,1 ,l2 ,...,ln , in which N—the number of points in the pseudo-phase
reconstruction of the scalar time series xi (t) used to approximate the right-hand sides,
and v—polynomial degree. Legendre polynomials may be used for the approximation, or a more complex technique may be used. For given n and v, the number of
polynomials coefficients K (7) in the general case can be determined by the formula:
K = (n + v)!/(n!v!). Usually N ≥ K , therefore, to specify the evolution operator, the system of equations (7) is solved by the LSM. In the end, the mathematical
model is complex, but with a good choice of the general form of nonlinear functions,
its solution is successful. It is clear that for parametric identification this algorithm
always coincides and is much more accurate, but the approximation error remains.
138
A. Zinchenko
4 Research Results: Ressler and Lorenz Models
Let us consider the behavior of two unidirectionally coupled oscillators—the Ressler
system, which is on the verge of the asymptotic coordinate synchronization regime.
The equations of interacting oscillators at controlling the first coordinate will be
written as:
ẋ ' = −ω' y ' − z ' − γsign∇x ' ar ctg(x ' (t) − x(t)),
ẋ = −ωy − z,
ẏ ' = ω' x ' + ay ' ,
i '
ẏ = ωx + ay,
ż = b + z ' (x ' − c),
ż = b + z(x − c),
ȧ ' = −δsign∇x ' ar ctg(x ' (t) − x(t)),
(8)
where (x, y, z) i (x ' , y ' , z ' )—the vectors of the leading and controlled oscillator,
respectively; γ—feedback parameter; δ—adaptation parameter, ω—parameter that
determines the main natural oscillation frequency (it was chosen ω = 0.93, ω' =
0.95); other standard options were chosen as follows a = 0.15, b = 0.2, c = 10.
With this choice of controlled parameters, coupled systems show coordinate-coherent
synchronization. As it was written above, the spectrum of Lyapunov indicators of
the leading system (λ1 > λ2 > λ3 ) does not depend on the value of the connection
parameter γ, , while the conditional Lyapunov exponents (λ'1 > λ'2 > λ'3 ) change
with increase coefficient γ. Figure 1 shows the dependences of 4 Lyapunov exponents
on the value γ.
The other two Lyapunov exponents are essentially negative (≈−10), so they do
not affect on synchronization. To calculate the Lyapunov exponents, the Bennettini
algorithm was used [1, 18, 19]. The exponents λ1 > 0 and λ2 = 0 correspond to
the behavior of the host system, so their values are constant. If connection between
the systems is absent (γ = 0), then we have λ'1 > 0 and λ'2 = 0. Since it is the
first exponent that characterizes the chaotic dynamics of the controlled system, then
λ'2 = .0 —zero exponent with an increase in which already at γ = 0.03, as can be
seen from Fig. 2, the partial chaotic synchronization mode is established. A negative value λ'2 = .0 causes synchronization of the oscillators (8), although not all
conditional Lyapunov exponents are already negative. Coordinate synchronization
Fig. 1 Dependence of the
Lyapunov exponents on the
feedback parameter γ
Parametric Identification of Dynamic Systems Based on Chaotic …
139
occurs at some time intervals near the steady state limit, but the asymptotic coordinate
synchronization mode is not fully established. Therefore, we apply parametric identification to the second coordinate of the vector (x, y, z). The simulation results are
shown in Fig. 2. Modeling of all drawings was carried out in the Java programming
language.
Fig. 2 Asymptotic coordinate synchronization of two unidirectional Ressler oscillators (8) and
estimation of an unknown parameter a at γ = 1 and δ = 1, b = 0.2, c = 10, ω = 0.93, ω' =
0.95.
Fig. 3 Asymptotic coordinate synchronization of two unidirectional Lorentz oscillators (9) and
assessment of the unknown parameter σ at γ = 1 and δ = 1 with a standard choice of the function w(x, x ' ) = γ(x ' (t) − x(t)) for the proposed algorithm (4) (observing the control function
F(x1 , x2 , x3 ) = 1/2x 2 + 1.1y)
140
A. Zinchenko
Fig. 4 Asymptotic coordinate synchronization of two unidirectional Lorentz oscillators (9) and
assessment of an unknown parameter σ at γ = 1 and δ = 1 with the proposed choice of function w(x, x ' ) = γsign∇xi' ar ctg(x ' (t) − x(t)) for algorithm (4) (observing the control function
F(x1 , x2 , x3 ) = 1/2x 2 + 1.1y)
Let us now consider the behavior of two unidirectionally coupled Lorentz systems
at observing the control function F(x1 , x2 , x3 ) = 1/2x 2 + 1.1y:
ẋ = σ (y − x),
ẏ = r x − y − x z,
ż = x y − bz
ẋ ' = σ (y ' − x ' ) − γ sign∇x ' ar ctg(1/2x '2 + 1.1y '
− (1/2x 2 + 1.1y)),
'
ẏ = r x ' − y ' − x ' z ' ,
ż ' = x ' y ' − bz ' ,
σ̇ ' = −δsign∇x ' ar ctg(1/2x '2 + 1.1y '
− (1/2x 2 + 1.1y))(y ' − x ' ).
Jacobi matrix of this system can be writtem as follows:
⎛
⎜
⎜
⎜
J =⎜
⎜
⎜
⎝
'
−σ − γsign(x ' ) 1+(1/2x '2 +1.1yx' −1/2x 2 −1.1y)2
r −z
y
−δsign(x ' )(−ar tng(1/2x '2 + 1.1y ' − 1/2x 2 − 1.1y)
1
+
(y ' − x ' )
1 + (1/2x '2 + 1.1y ' − 1/2x 2 − 1.1y)2
σ − γsign(x ' ) 1+(1/2x '2 +1.1y1,1' −1/2x 2 −1.1y)2
−1
x
−δsign(x ' )(ar tng(1/2x '2 + 1.1y ' − 1/2x 2 − 1.1y)
1, 1
+
(y ' − x ' )
'2
1 + (1/2x + 1, 1y ' − 1/2x 2 − 1, 1y)2
⎞
0 y' − x '
−x
0 ⎟
⎟
⎟
−b 0 ⎟.
⎟
⎟
0
0 ⎠
(9)
Parametric Identification of Dynamic Systems Based on Chaotic …
141
Fig. 5 Convergence of the
method depending on the
parameters for system (9). In
the zone (−) there is
convergence, in the zone
(+)—no convergence
Figures 3 and 4 show the parametric identification of the unknown parameter σ of
the Lorenz system during observing the function F(x1 , x2 , x3 ) = 1/2x 2 + 1.1y
using algorithm (4) with the standard choice of the vector-function w(x, x ' ) =
γ(x ' (t) − x(t)) (Fig. 3) and with the proposed choice of the vector-function
w(x, x ' ) = γsign∇xi' ar ctg(x ' (t) − x(t)) (Fig. 4). As can be seen from the graphs
above, the method coincides with the proposed choice of the vector-function w(x, x ' )
approximately by 10 times faster (with t ≈ 1 in dimensionless time units).
At the same time, the assessment accuracy using the fifth-order Runge–Kutta
method with the Dormand-Prince corrective procedure for a variable step of numerical integration was 10–7 . Figure 5 shows the selection ranges of the controlled
parameters (adaptation and feedback) for which the method coincides, that is, when
all real parts of the eigenvalues of the Jacobi matrix J are less than zero.
Let’s present the structural and parametric identification of dynamical systems
using the example of the Ressler system for the scalar implementation of system (8)
and the chosen parameters a = 0.15, b = 0.2, c = 10 under the initial conditions
x(t0 ) = y(t0 ) = z(t0 ) = 0.001. In this case, the method of successive differentiation,
the recurrent LSM and the mentioned Runge–Kutta method of the fifth order with
the corrective procedure of Dormand-Prince were used. The results of the recurrent
LSM assessment are shown in Table 1. From the above table, it is observed that the
best assess of the unknown parameter was 0.15008583965282281 at 25,000 values
of the scalar realization of the first coordinate of the Ressler system (8).
Assessing the unknown coefficients of the system (8), which is shown on the
left, we obtain the Ressler system reconstructed from observations (only 100,000
observations) (on the right):
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A. Zinchenko
Fig. 6 Ressler attractor for the reconstructed system (8) (a) and the convergence of the method for
system (9) on a small sample of observations (20 basic oscillations) at replacing the final values of
the divergence with the initial ones (b)
Table 1 Assessment of the unknown parameter a of the Ressler system
.
Number of
observations, n
Assessment of the
unkmown parameter a
Student’s t-test
(yi −yiM )
n
2
5000
0.15279989073271564
16.11550727420384
0.00051349525955
10,000
0.15179031214712582
21.7848638879715
0.00037518457237
15,000
0.15023587771954774
24.780482729799697
0.00021788606654
20,000
0.14991215402599043
28.15722213497546
0.00013270174521
25,000
0.15008583965282281
31.755325001013514
0.00007546570353
30,000
0.14958778332228523
33.92477404405509
0.00034885192906
35,000
0.14975804834261507
36.9612833901198
0.00023395394712
⎧
⎪
⎨ ẋ = −ωy − z,
ẏ = ωx + ay,
⎪
⎩
ż = b + z(x − c).
⎧
ẋ = y,
⎪
⎪
⎪
⎨ ẏ = z,
⎪ ż = −0.2 − 10x + 0.5y − 9.85z
⎪
⎪
⎩
+1.0225x y + yz − 0.15x z − 0.15y 2 − 0.15x 2 + 0z 2 .
Figure 6b shows the convergence of the method for system (9) with a small sample,
observing the function F(x1 , x2 , x3 ) = 1/2x 2 + 1.1y using algorithm (4) with the
selected vector-function w(x, x ' , t) = γsign∇xi' ar ctg(x ' (t) − x(t)). The simulation
found that even with 20 basic steady state oscillations in the sample, the method
coincides.
Parametric Identification of Dynamic Systems Based on Chaotic …
143
5 Conclusions
In this chapter, for the problem of parametric identification, a new algorithm based
on chaotic synchronization and adaptive control is proposed. Thus, the acceleration
of synchronization of unidirectional oscillators occurs due to the use of a relay algorithm as an adaptation method, which is a special case of the standard speed pseudogradient algorithm. In addition, it is proposed to use, instead of the standard vector
function of the rate of change of a smooth objective function, a class of functions
(for example, trigonometric) that satisfies the conditions of pseudo-gradient, reach,
existence of “ideal control” and convexity of the feedback function with respect to
the corresponding state vector of the controlled system. It was found that using a
vector-function ar ctg(x), that fully satisfies these conditions, with the same choice
of initial conditions of the dynamic system, the method coincides approximately
faster at 2 times. This feedback function was also used in the adaptive control equation to estimate unknown parameters. Using the Ressler system as an example, the
influence of the zero Lyapunov exponent on the synchronization of two unidirectional coupled oscillators of this system under the control of the first coordinate
is shown, and the spectrum of Lyapunov characteristic indicators is calculated. In
this case, although asymptotic coordinate synchronization occurs due to the negativity of the zero Lyapunov exponent, however, it is interval-phase, so identification
is impossible. The example of the Lorenz system shows the effectiveness of the
vector-function ar ctg(x) in comparison with the standard one. In addition, using
the example of this system, parametric identification was made during observing not
only one coordinate of the system, but also a function of all coordinates. For this case,
formulas are derived and conditions for the convergence of the method are given.
Also, on the example of the Lorentz system, the possibility of applying the algorithm
for small samples is shown, in particular, the results of the convergence of the method
during observing a function of all coordinates of the system are given. To check the
effectiveness of the algorithm, the results of comparing the proposed method with
the standard one based on approximation are presented. In addition, on the basis of
the last method, an example of a complete reconstruction of the dynamical Lorentz
system is given at observing a function of all coordinates of the system.
References
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Part 2. Mechanic 15, 21–30 (1980)
Detection Method of Augmented Reality
Systems Mosaic Stochastic Markers
for Data-Centric Business
and Applications
Hennadii Khudov , Igor Ruban , Oleksandr Makoveichuk ,
Vladyslav Khudov , and Irina Khizhnyak
Abstract The paper proposes the method of detection of mosaic stochastic markers
of augmented reality systems for data-centric business and applications. A description of the well-known markers of augmented reality systems, their advantages and
disadvantages is given. Determined that the finding of angles or special areas of reference of well-known markers is a quick method, but requires unambiguous detection
of all four points of well-known markers. The detection method of mosaic stochastic
markers for augmented reality systems has been improved. The main stages of the
method are: the preprocessing of the input image; the finding the marker area; to
determine the bit container. Results of the experiments of the method of detection of
mosaic stochastic markers for augmented reality systems has been given.
Keywords The mosaic stochastic marker · The detection · The method · Bit · The
system of augmented reality · Data-centric business application
1 Introduction
Nowadays the virtual and augmented reality technologies capture the imagination.
But we perceive them, rather, as an entertainment component of life, gaming applications are by far the largest and most interested investors in this area. But thanks
to the development of augmented and virtual reality technologies at the expense of
gaming giants, technologies are also being developed that allow augmentation of
reality to help other areas of business [1].
The technology of creating augmented reality (AR) in any field includes three
mandatory components [2–5]:
. mobile device (smartphone, tablet) with a built-in video camera;
. a specially designed application containing virtual information;
H. Khudov (B) · I. Khizhnyak
Kharkiv National Air Force University, Kharkiv, Ukraine
e-mail: 2345kh_hg@ukr.netl
I. Ruban · O. Makoveichuk · V. Khudov
Kharkiv National University of Radio Electronics, Kharkiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_8
145
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H. Khudov et al.
. a label, Global Position System (GPS) coordinates, or a marker (a real object, its
part or a printed image), which launch the mobile application.
A marker or tag allows the program to bind augmented reality information to the
outside world and transfer the image to a smartphone display or to special glasses.
But these three components are not always enough; various fields of application of
AR technologies have their own specifics [6].
To see an AR object, you need to point the camera at a mark or marker, and the
image will appear on the screen of a smartphone or tablet. AR can be formed not
only on the basis of an image, but also include, for example, readings of a compass,
gyroscope, 3D model—without this, it is impossible to create a three-dimensional
object visible from different angles [1].
Augmented reality allows to make a presentation anywhere, show how the object
will look from all sides, and even make the perception of the object interactive. To
do this, you only need a program (application), a display with a camera and a marker
printed on a sheet. When you point the camera at the printout, an object will appear
on the screen. True, for the full effect of presence, spectators will need special glasses
or a helmet [1, 4].
In medical diagnostics, there is not always direct access to the information base
on which additional reality is built. In this case, AR technology uses data from highprecision diagnostic devices, such as magnetic resonance imaging, computed tomography, ultrasound diagnostics, X-rays, etc. They are the basis to which augmented
reality is tied, and certain organs or points are markers. An image of the patient’s
internal state appears on the doctor’s monitor [7, 8].
AR-devices that help patients cope with illness exist in psychiatry and
psychotherapy. Barcelona-based Psious Inc. has developed a technology that allows
to simulate situations that provoke phobias, for example, the fear of enclosed spaces,
flying in an airplane or spiders [7, 8]. Under the supervision of specialists, a person
suffering from a phobia undergoes a phased adaptation to a stressful situation.
For the field of distance education, AR technologies are a universal tool [9]. There
are certain best practices for conducting classes in a virtual laboratory or applications
that create the effect of being present at a real operation in a remote operating room
or on a battlefield in the distant past. Japanese publishing house Tokyo Shoseki has
prepared a special AR English textbook. It has built-in cameras that project onto the
pages of animated characters [10, 11].
AR allows to translate sign or digital information into a more easily perceived
visual and make the process of perceiving information interactive. In the use of
AR-devices when monitoring the environment while the car is moving, feedback is
provided. Data management is carried out in voice mode [12, 13].
Architectural visualization is a demonstration of an object under construction
from any point of view and from the inside. This requires special markers in the
building itself. By pointing a video camera at them, you can get a variety of images,
and with the help of simple manipulations on the display, turn the object at any angle,
see the internal structure or disassemble the building by floors [14].
Detection Method of Augmented Reality Systems Mosaic Stochastic …
147
The marker and unmarked AR systems are presented in approximately the same
way, with unmarked systems most often used in gaming and travel applications. In
unmarked systems, the location of the user (his geographical coordinates) obtained
by GPS is used as a marker. Additional information is most often downloaded from a
remote server, where the coordinates and orientation of the smartphone’s camera are
transmitted. This naturally binds the system to one place and makes it inaccessible for
use to moving objects, which narrows its scope [13, 15]. In order to obtain additional
information about arbitrary objects, augmented reality marker systems will be used
in the vast majority of cases, which will be considered in this paper.
The main existing types of AR markers are given in [2, 3, 11–13, 16, 17]:
. the template markers—black and white markers that have a simple image inside
a black frame (Fig. 1);
. the 2D barcode markers—which consist of black and white cells that code data
bit by bit, and sometimes frames or synchronization areas (Fig. 2). Most often,
quick response (QR) codes are used as barcode AR-markers;
. the circular markers—similar to barcode markers (Fig. 3). But the bits are encoded
not in rectangular cells, but in black and white circular slices;
. the images (image markers)—ordinary color images are used as markers (Fig. 4).
Its may contain a frame or other landmarks to identify and find a position. The
image markers are usually identified by searching by pattern or by image feature.
Fig. 1 The template marker
Fig. 2 The 2D barcode
marker
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H. Khudov et al.
Fig. 3 The circular marker
Fig. 4 The image marker
Theoretically, an AR marker can be any figure (object), but in practice we are
limited by the resolution of the camera, the features of color reproduction, lighting
and computing power of the equipment. Therefore, for work in real time, a simple
black and white marker of a simple shape is usually chosen. This is usually a rectangle or square with an inscribed image inside. The paper [18] describes the main
types of template markers and compares the recognition performance of different
implementations of markers (Fig. 5). A typical method of processing a template
marker consists of the following steps [18, 19]:
.
.
.
.
.
.
the transition to grayscale;
the threshold determination and image binarization;
the finding closed areas;
the selection of contours;
the finding the angles of the marker;
the finding the parameters of projective transformation and coordinate transformation.
Analyzing the known types of AR-markers, we can conclude that each of them
has its advantages and disadvantages:
(1) all AR-markers allow to determine the position of the camera, but this uses
different methods:
Detection Method of Augmented Reality Systems Mosaic Stochastic …
149
Fig. 5 The main types of template markers [18]: a ArToolKit (ATK); b Institut Graphische
Datenverarbeitung (IGD); c Siemens Corporate Research (SCR); d Hoffman marker system (HOM)
. finding image angles (template markers);
. finding special areas of reference (bar code and circular markers);
. finding special points of the image and their descriptors (image markers).
(2) some of them (bar codes and circle markers) contain additional information
(messages), such as links to information resources, which is a clear advantage
because it allows to expand the scope.
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H. Khudov et al.
Fig. 6 The mosaic
sustainable marker for
augmented reality systems
[14, 17]
Finding angles or special areas of reference is a quick method, but requires unambiguous detection of all 4 points, finding special points of the image and building
their descriptors requires more computing resources, but is much more stable, part
of the image can be obstructed, however, this method allows to correctly determine
the position of the camera.
In [14, 17] a new type of mosaic sustainable markers of augmented reality systems
is proposed. Its form is shown in Fig. 6.
So, we will develop the mosaic sustainable marker detection method for
augmented reality systems.
2 The Method of Detection of Mosaic Stochastic Markers
for Augmented Reality Systems
To detect a mosaic stochastic marker of augmented reality, a method is proposed,
the block diagram of which is shown in Fig. 7.
2.1 The Preprocessing of the Input Image
The preprocessing of the input image (Fig. 6) involves the transition from color to
grayscale. Formally, this can be written as (1):
g=
1
(0.2989R + 0.587G + 0.114B),
255
(1)
Detection Method of Augmented Reality Systems Mosaic Stochastic …
Fig. 7 The block diagram of
the mosaic stochastic marker
detection method for
augmented reality systems
151
The image
The processing
The filtration
The segmentation
The morphological processing
The mask of
AR-marker
where R, G, B—the corresponding color components of the original image in RGBrepresentation; g—grayscale image; conversion factors are the same as those for the
Y-channel (luminance) in the transition from RGB to NTSC-representation; coefficient 1/255 is introduced for convenience—to normalize the dynamic range of
brightness from 0.0.255 to 0.0.1.
The grayscale image is shown in Fig. 8.
Fig. 8 The grayscale image
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H. Khudov et al.
2.2 The Finding the Marker Area
It is proposed to use the operation of finding the local standard deviation over the
square area for the AR-marker of detection on the image f (Fig. 8) because the AR
marker contains only 3 grayscale values {0, 1/2, 1}. At the border between the ARmarker cells, the local standard deviation will be maximum and small for smooth
image areas. Since all coordinates in the images are set in integers, it is convenient
(for symmetry) to choose the size of the region as an odd number.
Figure 9 shows an image of σ, this is the result of calculating the local standard
deviation over an area of size (3 × 3).
For a square area of size (2a + 1)x(2a + 1) centered at a point with coordinates
(x, y), we have expression (2):
σ (x, y) =
a
a
.
.
1
(f(x + m, y + n) − μ(x, y))2 ,
(2a + 1)2 m=−a n=−a
(2)
where μ(x, y) is the local average, which is calculated according to expression (3):
μ(x, y) =
a
a
.
.
1
(f(x + m, y + n)).
(2a + 1)2 m=−a n=−a
(3)
The next step is to binarize the image (Fig. 9). Since the histogram of the local
standard deviation image is unimodal in the general case (Fig. 10), then the standard
binarization by Otsu’s method.
It works well for bimodal distributions. But this will give an overestimated
threshold, which will lead to the loss of some useful information (Fig. 11).
We will carry out binarization using k-means segmentation. The simplest way
is segmentation into k = 2 classes (background and object) [20, 21]. But in this
Fig. 9 The image of local
standard deviation
Detection Method of Augmented Reality Systems Mosaic Stochastic …
153
Fig. 10 The histogram
image of the local standard
deviation
Fig. 11 The result of
binarization by Otsu’s
method (some information is
lost)
case it will give results that will be close to binarization by Otsu’s method [20, 21].
Therefore, it is proposed to use k = 3 classes—background, intermediate values and
object (Figs. 7 and 12).
Then the binary image is obtained as a union of class masks that are not related to
the background. Background is defined as the class that has the lowest average luminance value. Since the background occupies the largest area, the best is determined.
An increase in the number of classes k > 3 does not significantly change the results
and is impractical. The result of binarization by the proposed method is shown in
Fig. 13.
To compare the results in Fig. 14 shows fragments of binary masks obtained by
the Otsu method and the proposed method. You can see that the proposed method
gives significantly better results—all cell contours are well distinguished.
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H. Khudov et al.
Fig. 12 The index
image—the result of
segmentation into 3 classes
Fig. 13 The result of
binarization by the proposed
method
The resulting binarized image is processed using operations of mathematical
morphology. It is proposed to use the morphological closure operation with a square
window for to fill the inner areas. In this case, the size of the window sets the
maximum size of the voids that will be filled. In this work, it is proposed to use a
window (63 × 63). The results are shown in Fig. 15.
Next, the largest 4-connected area is found (Fig. 16).
In the future, it makes sense to discard uninformative image areas. And as the
area of the AR-marker, take a rectangular fragment in which its mask is inscribed
(Fig. 17).
All further operations will be performed only for the part of the image that is
highlighted by this mask (Fig. 18).
Detection Method of Augmented Reality Systems Mosaic Stochastic …
Fig. 14 The comparison of
binarization results: a Otsu’s
method (some of the
contours are missing); b the
proposed method
Fig. 15 The result of filling
the inner areas
Fig. 16 The most connected
area
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Fig. 17 The mask of
AR-marker
Fig. 18 The image of
AR-marker
2.3 To Determine the Bit Container
To determine the bit-containers (information elements cells, the colour of which
encodes the information bits), it is proposed to use the segmentation of the AR-marker
image into k = 3 classes using the k-means algorithm (similar to the definition of
the AR-marker area). The result of segmentation is shown in Fig. 19.
It should be noted that the k-means algorithm assigns class indices in an arbitrary
manner. And to obtain a picture that will look more similar to the original image, the
resulting indices should be sorted according to the growth of the average brightness
of each class. The result of this ordering is shown in Fig. 20. In this case, the dark
cells (encoding bits 0) will belong to the class with the minimum index value, which
is equal to 1. Gray (border)—of the class with index = 2. White (coding bit 1)—class
with index = 3.
Detection Method of Augmented Reality Systems Mosaic Stochastic …
157
Fig. 19 The index image of
segmentation of AR-marker
Fig. 20 The result of
ordering of class index (to
compare with Fig. 18)
If there are correctly ordered class indices, then we can select the cell masks that
correspond to each bit. Bit 0 is coded in black. It will correspond to the class with
the lowest index, which is 1 (Fig. 21).
Bit 1 is coded white. It will correspond to the class with the highest index, which
is 3 (Fig. 22).
The next step is filtering the bit container masks. This operation is efficiently
performed with a square window morphological opening operation. The problem is
the choice of the window size. It should be such that artifacts are filtered out as much
as possible. This did not remove information items. To select the optimal window
size, the following is proposed. Let’s count the number N(a) of 4-connected areas
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Fig. 21 The mask for cells
with index 1 (bit 0)
Fig. 22 The mask for cells
with index 3 (bit 1)
in the binary image that remain after the morphological opening operation. We will
count it as a function of the filtering window size a (Fig. 23).
The function N(a) first falls. As the size of the window increases, more and more
areas are filtered out. Further, for a certain range of sizes, we reach a plateau—
the number of areas remains unchanged. Since the window size is smaller than the
typical cell size. With a further increase in the window size, the function N(a) will
fall again. Since the cells will start to filter out. Thus, to determine the size of the
window, it is necessary to find the point at which the function N(a) reaches a plateau.
This will mean that the noise has already been filtered, and the cell is not yet. Since
the function N(a) is nonincreasing, it is sufficient to find the first maximum of the
Detection Method of Augmented Reality Systems Mosaic Stochastic …
159
Fig. 23 The function N(a) for each bit of container mask
derivative dN/da to find this point (Fig. 24). The results of morphological filtration
are shown in Figs. 25 and 26.
Taking into account the perspective distortions of the image, the described
behavior of the function N(a) is made less certain. Since some of the cells in the
far area of the image will be smaller than the noise in the near area. However, as
the experiments have shown, the proposed method for determining the optimal filter
window size gives good results. It will be shown further for the proposed algorithms
that the loss of a certain number of information cells is less critical than the presence
of noise areas.
Morphological filtration effectively removes only those areas that are smaller than
the cell size. To filter areas that are significantly larger than this size—you must use a
different method. If small artifacts arise through binarization defects, then the nature
of this noise is different. It is either interference that obscures part of the container’s
area (such as a hand or other object). Or a part of the cells that “stuck” together due
to uneven lighting. To eliminate such noise, it is proposed to use statistical filtering
by the size of connected regions. All areas with an area greater than three standard
deviations from the mean are filtered out. To increase efficiency, this method is used
iteratively. The number of iterations is 3. The results of statistical filtration for each
mask are shown in Figs. 27 and 28.
The result of the merge of the masks is shown in Fig. 29.
Thus, we have obtained the mosaic sustainable marker detection method for
augmented reality systems. It is based on the binarization of the local variance,
detects the marker area in the original image and finds the masks of bitcontainers.
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Fig. 24 The plots of derivative dN/da for each mask of bit-container. The circle indicates the point
of the first maximum: a for 0; b for 1
Detection Method of Augmented Reality Systems Mosaic Stochastic …
161
Fig. 25 The result of
morphological filtration of
the mask of bit-container (bit
0)
Fig. 26 The result of
morphological filtration of
the mask of bit-container
(bit 1)
This is done by segmentation and subsequent morphological filtration of the masked
area of the image.
3 Conclusions
In this paper we have obtained the method of detecting mosaic stochastic markers of
augmented reality systems for data-centric business and applications. It is based on
the binarization of the local variance, detects the marker area in the original image
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H. Khudov et al.
Fig. 27 The result of
statistical filtration of the
mask of bit-container (bit 0)
Fig. 28 The result of
statistical filtration of the
mask of bit-container (bit 1)
and finds the masks of bit-containers. This is done by segmentation and subsequent
morphological filtration of the masked area of the image.
Areas for further research are:
. the development of a method for determining the parameters of projective transformation. This is necessary to align the image and determine the position of the
camera;
. the development of a method for decoding the mosaic sustainable marker of
augmented reality systems.
Detection Method of Augmented Reality Systems Mosaic Stochastic …
163
Fig. 29 The result of the
marge of the mask
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(2017)
Method for Converting the Output
of Measuring System into the Output
of System with Given Basis
Elena Revunova , Volodymyr Burtniak , Yuriy Zabulonov ,
Maksym Stokolos , and Volodymyr Krasnoholovets
Abstract The chapter reviews the methods of data processing aimed at improving
the accuracy of measurements in radiation monitoring systems. The accuracy of
radionuclide activity determination using model selection criteria has been studied.
It is shown how the output of a linear measured system to a system with specified
properties can be converted. The mentioned specified properties are classified and
considered. Preliminary processing has been performed by the method of converting
the output of the measuring system into the output of the system with a given basis.
The method has successfully been applied for the study of spectra of radionuclides
137
Cs, 134 Cs and 60 Co.
Keywords Model selection criteria · Measuring system · Transformation of the
output · Radionuclide activity
1 Introduction
Identification and determination of the activity of weak sources of radioactive radiation is an urgent task of radiation monitoring [1]. This article reviews the methods of
data processing aimed at improving the accuracy of measurements in radiation monitoring systems, as well as the accuracy of determining the activity of radionuclides
using the output transformation method of the measuring system.
E. Revunova (B)
International Research and Training Center for Information Technologies and Systems of the NAS
of Ukraine and the MES of Ukraine, Kyiv, Ukraine
e-mail: egrevunova@gmail.com
V. Burtniak · Y. Zabulonov · M. Stokolos
State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences
of Ukraine”, Kyiv, Ukraine
V. Krasnoholovets
Institute of Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_9
165
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E. Revunova et al.
2 Methods of Data Processing in Radiation Monitoring
Systems and Factors Complicating Processing
The requirement for the mobility of monitoring systems, together with the need to
measure sources with low levels of radioactivity, determines the choice of a scintillation detector as a detecting element, which does not require a cooling system and has
a high detection efficiency (compared to semiconductor detectors). The disadvantage
of scintillation detectors is their low energy resolution compared to semiconductor
detectors. The choice of these types of detectors affects the requirements for methods
of processing the spectrum of ionizing radiation. As is known, the full spectrum of
gamma radiation includes three main characteristic regions (zones of interest): the
total absorption peak, the backscatter peak, and the Compton part. The shape and
severity of the characteristic regions of the gamma spectrum is determined by the
properties of the detector and the measurement geometry. Thus, the low energy
resolution of a scintillation detector, together with the requirement to measure the
activity of objects with a complex (previously unknown) spectral composition, leads
to the need to process spectra that have such features as overlapping peaks, complete
masking of the peak of one of the elements by the peak or ‘Compton’ part of the
other.
On the other hand, in almost all monitoring activities aimed at ensuring the radiation safety of nuclear fuel cycle facilities, the problem arises of processing gamma
radiation spectra measured in complex (non-fixed) geometry.
To process spectra of complex composition measured in non-fixed geometry, we
have developed full spectrum processing methods [2–4], which take into account the
number of gamma quanta recorded in the entire measured energy range. The essence
of the full spectrum processing methods is as follows. The measured spectrum is
presented as a combination of the response functions of the radionuclides that make
up the radiation source, weighted by the activities. The task of processing is to
determine from the measured spectrum which response functions and with which
weighting factors are activities formed the observed spectrum.
The initial information is a set of detector response functions (DRF) to the impact
of gamma quanta with energies in the range of 100 keV–2 meV. Programs have
been developed that make it possible to obtain DRF by simulating the process of
propagation of gamma radiation. The measured spectrum of gamma radiation is
modeled as the sum of DRF weighted by coefficients proportional to the activity.
The basic hypothesis about the composition of the spectrum is the assumption
that the spectrum includes all possible spectral lines in the range of 100 keV–2 meV:
Ax+ε=b where A is the DRF matrix of size m × n, x is the vector of weights
proportional to the activity of radionuclides, ε is the intrinsic noise vector of the
measuring path, b is the output vector of the measuring system of size m. When
digitizing the energy range with a step of 16 keV and n = 125 the observed emission
spectrum is modeled as a weighted sum of 125 DRF (with a step of 7.8 keV and 256
DRF). The weighting coefficients corresponding to the radionuclides present in the
Method for Converting the Output of Measuring System into the Output …
167
measured spectrum are proportional to the activity of these nuclides, and all other
weighting coefficients are zero.
The use of a preliminary hypothesis that the spectrum includes all possible spectral
lines of the measured range is a hallmark of our approach. In the traditional approach,
a preliminary hypothesis about the composition of the spectrum is the hypothesis
that the spectrum is composed of nuclides of a certain group, for example, thoriumradium. Using the assumption that the spectrum includes all possible spectral lines of
the energy range under study makes it possible to avoid the situation when a nuclide
that is not included in the model appears in the measured spectrum.
When the spectrum is processed by traditional methods, the appearance in the
spectrum of a nuclide that is not included in the model leads to an increase in the
error in determining the activity of all nuclides present in the spectrum, and if the
activity of a nuclide that is not included in the model is high, it also leads to errors
in identifying the nuclides present.
However, a large number of terms in the preliminary spectrum model, together
with the presence of additive intrinsic noise in the measured spectrum, leads to the fact
that the determination of activity, for example, by the least squares method (LSM) is
unstable. The instability manifests itself in the fact that zero weights (corresponding
to lines of nuclides that are not present in the spectrum) are assigned certain values
(both positive and negative) by the LSM. As a result, nuclides that are not actually
present in the spectrum are erroneously identified. This behavior of the LSM is due
to the fact that the multicomponent model tries to approximate not only the real
function of the spectrum, but also the additive noise.
Modern methods that work stably in the presence of noise are the methods of
model selection [5] and sparse approximation [6]. Model selection methods, through
the use of model selection criteria [7, 8], provide a balance between the approximation
accuracy (spectrum functions) and the number of basic functions (response functions)
included in the model, thereby preventing the model from “tuning” to noise. The use
of the methods of this group makes it possible to avoid such a situation when the
spectrum model includes the response functions of elements that are not actually
present in the spectrum; which, in fact, is an attempt to approximate the diurnal
fluctuations of the background radiation spectrum by the model.
The model selection criteria are formulated in such a way that they automatically
reduce the model dimension with increasing noise level. For various model selection
criteria, a study was made of the dependence of the model dimension and activity
determination accuracy of the noise level.
The comparison has shown that the best accuracy is provided by the criterion
that retains the true dimension of the model longer than others with increasing noise
level. Suppose the measured spectrum includes four monochrome sources of gamma
radiation, in this case the true dimension of the model is four. However, the true
dimension is unknown, and a preliminary hypothesis about the composition of the
spectrum is the hypothesis that it includes 125 DRF.
The use of model selection criteria makes it possible to exclude from the model
lines that are absent in the measured spectrum. However, as the noise level increases,
the criterion for choosing the model begins to reduce the dimension of the model,
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E. Revunova et al.
making it less true, due to the exclusion from the model of lines of nuclides with
low activities that are not much higher than the noise level. Thus, the value of the
minimum detectable activity (MDA) is overestimated. Relevant is the development
of methods for processing gamma spectra, free from this drawback.
In work [9], a model selection criterion (the l0 -optimality criterion) has been
proposed, which does not explicitly link the model dimension with the noise level,
but allows testing the validity of the hypothesis about the composition of the model.
We have carried out computational experiments, the purpose of which is to compare
the accuracy of determining the weight coefficients of the model using the criteria:
Cr , MDL and l0 -optimality. Experiments have shown that with an increase in the
noise level, the error in determining the weight coefficients of the model according
to the l0 -optimality criterion is much less than the error for the model selection criteria
Cr and MDL [10].
However, the l0 -optimality test has drawbacks because it is not applicable to any
system of basic functions. For example, the system of basic functions formed from
the responses of a scintillation detector does not meet the requirements for the test
for l0 -optimality. The signal model can be tested for l0 -optimality if the value of the
cumulative connectivity function μ [9] for the system of basic functions that form
the model is less than one. Otherwise, the test for l0 -optimality is not applicable.
To overcome such a shortcoming, we can use the output conversion method. The
output of a linear measuring system is a spectrum measured using a scintillation
detector. The system of response functions of the detector has μ > 1 and can be
converted into the output of a measuring system having a cumulative connectivity of
less than one.
3 Method for Converting the Output of Measuring System
into the Output of System with Given Basis
Let some object emit a signal x. The linear measurement system A converts the
signal emitted by the object into the measured output b by linear transformation
using a matrix A (the matrix of basic functions) Ax = b0 and addition with the noise
vector ε: b = b0 + ε. The observed output b may not meet user requirements or be
incompatible with further processing methods.
Let some other measurement system C have a set of basic functions (detector
response functions) that provide the required output. In this case, we can set the
problem of finding the transformation of the observed output b into the output of the
system C.
We will look for the output transformation as a linear transformation. For the
case when the noise vector is known and its covariance matrix is not degenerate, and
also the matrix of basic functions A, weighted by the noise covariance matrix, is not
degenerate, it has been proposed [11] to obtain the desired transformation using the
inversion A. However, if A has a high condition number and the series of its singular
Method for Converting the Output of Measuring System into the Output …
169
Fig. 1 Complication of the measurement system
values smoothly decreases to zero, the solution obtained using the inverse matrix (the
result of transformation into the output of the system C) is unstable. The instability
manifests itself in the fact that small changes in b correspond to large changes in the
solution, and the error in the solution is large. So a kind of decomposition should be
introduced (Fig. 1).
The approach we are developing to a stable solution of the output transformation
problem is based on the use of a truncated singular value decomposition [12–14].
The estimate of the output of system C obtained using the k-component of the
singular value decomposition of A, looks as below.
dk' = CA+
k b = Tk b;
Tk =
CA+
k
)
ϕi
UT dk' ,
= CVdiag
σi
(1)
(
(2)
where C is the matrix that performs the transformation Cx = d0 ,
A+
k = V diag
(
)
ϕi
UT ,
σi
(3)
where ϕi = 1 if i ≤ k, otherwise ϕi = 0.
−1 T
Here, A+
k = V S U is the pseudoinverse matrix (n × m) obtained of k (k < n)
components of the singular value decomposition, U = (u1 , ..., uk ) is the matrix
of left singular vectors, V = (v1 , ..., vk ) is the matrix of right singular vectors,
S = diag(σ1 , ..., σk ) is the singular value matrix. The optimal number k of singular
value decomposition components can be found using the model selection criteria.
Figure 2 shows the approach is function in principle.
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E. Revunova et al.
Fig. 2 Operation of the matching method
4 Improving the Accuracy of Estimating the Vector
of Parameters by Converting a Linear System
to a System with Specified Properties
When solving a class of problems related to the processing of information received
from various sensors (problems of protection against interference, identification,
diagnosis, interpretation, etc.) there is a problem of effective analysis of noisy signal
mixtures. In a number of such problems, the measured data are the result of summation of the effects generated by the physical process and weighted
.by the coefficients,
which leads to the use of linear parameters of the form y = Nj=1 ϕ j (z)β j where
(β1 , β2 , ..., β N ) is the vector of parameters β ∈ R N , (ϕ1 (z), ϕ2 (z), ..., ϕ N (z)) is
the vector of values of basic functions ϕ ∈ R N . The input vectors ϕ i form the matrix
of inputs . ∈ R L ×N , the output values yi form the output vector y ∈ R L .
If a possible set of basic functions is known (for example, a set of detection system
response functions to known influences), but it is not known which of them formed
the observed output, the solution of the approximation problem can be obtained by
sparse approximation methods (see, e.g. Refs. [12–14]).
For the output vector y0 not distorted by noise, the problem of sparse approximation is set as the problem of minimizing the number of nonzero components in
the parameter vector under the condition y0 = . β. If the output vector is distorted
by noise, the problem of sparse approximation is set as the problem of minimizing
the number of nonzero components in the vector of parameters under the condition
||y − .β|| ≤ δ, where δ is a (small) value proportional to the noise vector ε.
In connection with the solution of the problem of sparse approximation of the
noisy output vector, the concept of “l0 -optimal solution” was introduced a solution
that provided both the minimum approximation error and the maximum possible
sparseness. However, the disadvantage of the approach to solving the problem of
Method for Converting the Output of Measuring System into the Output …
171
sparse approximation using the test for l0 -optimality is that the test cannot be applied
to any system of basic functions. It is necessary to develop methods that allow the
use of a wider class of basic functions for sparse approximation.
However, the approach to solving the problem of sparse approximation using the
l0 -optimality test has a drawback: the test cannot be applied to any system of basic
functions. It is necessary to develop methods that allow the use of a wider class of
basic functions for sparse approximation.
4.1 The Matching Method with the Conversion of the Output
of a Linear System to a System with Specified Properties
To solve the problem of sparse approximation with a noisy output vector, a modified
matching method (MMM) is proposed. The method works as follows. Starting with
k = 0 and f0 = 0, at the (k + 1)th pass the selection of the vector ϕ k+1 ∈ R L
(columns of the matrix .) and calculation of the parameter βk+1 are performed, which
minimizes the square of the residual norm: (βk+1 , ϕ k+1 ) = arg min ||rk − β ϕ|| 2
where rk = y − fk . After that the next appoximation fk+1 = fk + βk+1 ϕ k+1 is
calculated.
The vector of parameters β∗k obtained at the kth pass is checked for l0 -optimality.
If the conditions of l0 -optimality are satisfied, the method ends.
The test for l0 -optimality is as follows: the value of β∗k is a solution with the
maximum possible sparsity and the smallest approximation error if
d1 + d2K < 0.5 × (1 − μ(2k − 1)) × max |βi | and μ(2k − 1) < 1,
(4)
.
where d K = ( j=K |<r, ϕ j >|)1/2 , K is the number of largest scalar products of the
remainder r with all ϕ j , μ(s) is the function of cumulative connectivity.
Note that with respect to the l0 -optimality test, expression (4) uses the value of
the maximum (by module) component of the vector of parameters (max j |β j |).
The cumulative connectivity function is calculated for normalized vectors ϕ j by
the rule:
.
μ(s) = max max
|<ϕ j , ϕ i >|,
(5)
/I
card(I ) ≤ s j ∈
i∈I
where s is the number of nonzero parameters, I is the set of indices of functions that
form the subspace under consideration, i indicates an element from the subspace for
all possible decompositions of card(I )-members of the output vectors y (card(I ) =
1, 2, ..., s),i = 1, ... , card(I ), i ∈ I andcard(I ) ≤ s. These conditions mean
that the power of the set of indices (subspace dimension) varies from 1 to s.
Since the ‘basis connectivity condition’ μ(2k − 1) < 1 is not satisfied for any
system of basic functions, we propose to stop the MMM according to the criterion
of model selection. Comparative studies of the accuracy of estimating the vector of
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parameters of MMM with a stop on the criterion of model selection and the test for
l0 -optimality showed that the accuracy of restoration of the vector of parameters on
the criterion of model selection is worse than the l0 -optimality test. This forces us to
look for ways to extend the class of basic functions used for sparse approximation.
In the context of expanding the class of basic functions, we propose to transform
the existing output vector to the output of a linear system formed by a system of
basic functions that satisfy the condition of connectivity of the basis.
The MMM algorithm with the conversion of the output vector consists of the
following sequence of actions.
Step 1.1. Form the matrix of inputs A ∈ R L × N , L << N .
Step 1.2. For the initial set of basic functions A calculate the connectivity function
μ(s).
Step 1.3. Check the condition of connectivity of the basis: μ(2s − 1) < 1 for
s = 1, ... , 0.5N . If the connectivity condition is not met, go to step 1.5.
Step 1.4. Initialize the input matrix . = A and go to step 2.1.
Step 1.5. Form a matrix of inputs B ∈ R L × N , L << N .
Step 1.6. Decompose the matrix A by singular values.
Check the number of conditionality and behavior of the series of singular
values.
If the conditionality number is small, go to step 1.6.1.
If the number of conditionality is large and the series of singular values gradually
decreases to zero, go to step 1.6.2.
Step 1.6.1. Calculate the transformation matrix Tk by the equation Tk = BA+
k
and go to step 1.7.
Step 1.6.2. Calculate the transformation matrix Tk as below.
Tk =
BA+
k
(
ϕi
= BV diag
σi
)
UT dk' .
Step 1.7. Carry out the transformation of the output vector to the system of basic
functions B that satisfy the condition of connectivity of the basis.
Step 1.8. Initialize the matrix of inputs . = B.
Step 2.1. Initialize f0 = 0 and rk = y. Normalize the columns of the matrix of
inputs.
Step 2.2. In the matrix of inputs . find the basis function i for which the scalar
product of the vector ϕ i and the vector of the current residue rk is maximum:
γk = arg max |<.(∗, i ), rk >|,
i = 1,...,N
where γk is the index of the basis function in the matrix of inputs γk ∈
{1, ..., N }.
Step 2.3. Form .k = {.k−1 , ϕγk }. Check the conditionality number .Tk .k and
the behavior of a series of eigenvalues.
If the conditionality number is small, go to step 2.4.
Method for Converting the Output of Measuring System into the Output …
173
If the conditionality number is large and the number of singular values gradually
decreases to zero, go to step 2.5.
Step 2.4. Calculate the values of the vector of parameters
βk = (.Tk .k )−1 .Tk rk .
Step 2.5. Determine the value of the vector of parameters βk using the regularization approach.
Step 2.6. Calculate the new balance vector
rk+1 = rk − βγk .(:, γk ).
Step 2.7. Calculate
d1 + d2k
⎛
⎞1/2
.
|
|
)
(
1/2
|<rk+1 , ϕ i >|2 ⎠ ,
= |<rk+1 , ϕ i >| 2
+⎝
i ∈ I2k
where I2k is the set of indices 2k of the largest scalar products <rk+1 , ϕ i >.
Step 2.8. Calculate
0.5 × (1 − μ(2k − 1)) × max |βi |,
i
where i is the index of the minimum module parameter; k is the number of
selected basic functions.
Step 2.9. Check performance
d1 + d2 < 0.5 × (1 − μ(2k − 1)) × max |βi |.
i
If the inequality is satisfied, the resulting linear model of k terms gives the solution
with the maximum possible sparseness and the smallest approximation error based on
the optimal sparseness test. Otherwise, continue the formation of the model, moving
on to the next iteration, namely, Step 2.2.
Let us investigate experimentally the accuracy of the parameters recovery by the
MMM with the transformation of the output of a linear system to a system with given
properties.
5 Application of the Output Conversion Method in Gamma
Spectrometry
The method of converting the output of a linear system into the output of a system
with the required properties can work in gamma spectrometric measuring systems
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in which the detector of the linear measuring system A has a lower resolution than
the detector of the system C.
Discretely specified detector response functions, which form the output of
measuring systems A and C, are shown in Figs. 3a, 4a. The outputs of the measuring
systems and (gamma radiation spectra) are shown in Figs. 3b, 4b. The spectra are
formed by the following radionuclides: cesium-137 (137 Cs), cesium-134 (134 Cs) and
cobalt-60 (60 Co).
The full set of 256 DRFs for the original measuring system is shown in Fig. 5a.
The set of DRFs for the measuring system, to the output of which the conversion
will be performed, is shown in Fig. 5b.
The transformation matrix T obtained by the formula (Step 1.62) is shown in
Fig. 6.
The vector of weights x (proportional to the activities of radionuclides) in the
experiments have been as follows: xCs-137 = 1.5, xCs-134 = 0.5, xCo-60 = 0.26, xCo-60
Fig. 3 Spectra of the system A: a discretely given functions that form the matrix A; b output of
the system A: signal b
Fig. 4 Spectra of the system C: a discretely defined basis functions forming the matrix C; b the
output of the system C: signal b
Method for Converting the Output of Measuring System into the Output …
175
Fig. 5 Set of DRFs: a discretely defined basic functions forming the matrix A; b discretely defined
basic functions that form the matrix C
Fig. 6 Transformation
matrix T
= 0.25. We have considered the estimation error of the vector of weights based on
the output of a real measuring system with DRF (Fig. 5b) and the output obtained as
a result of converting the output of the measuring system with DRF (Fig. 5a). At two
levels of intrinsic noise (0.01 and 0.02), real spectra have been measured, the output
has been converted, and the accuracy of the estimation of the vector of weights has
been calculated: e = ||x − x∗ ||2 where x is the true vector of the weights and x∗ is
the estimate of the vector of the weights. The results are shown in Table 1.
We have compared the estimation error of the vector of weights using the Mallows
criteria (Cp ), the minimum description length (MDL), and the l0 -optimality test (L 0 ).
The accuracy of the weight vector estimation from the output obtained as a result of
the transformation is marked with an asterisk.
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Table 1 Calculation results
Noise level
Estimation error of the vector of weights
Cp
Cp∗
MDL
MDL*
L0
L ∗0
True
True*
0.01
0.011
0.021
0.011
0.013
0.011
0.013
0.011
0.013
0.02
0.260
0.250
0.025
0.024
0.025
0.024
0.025
0.024
The spectra have been measured at noise levels of 0.01 and 0.02 under laboratory
conditions. When measuring in the field, the noise level tends to increase due to
changes in ambient temperature. Since it is difficult to measure the intrinsic noise of
the measuring path in the field, we modeled the increase in intrinsic noise by adding
noise to the measured spectra, simulating an increase in intrinsic noise in the range
of 0.03–0.09.
The results of the study are presented in Fig. 7. With an increase in the level
of intrinsic noise, the error in estimating the vector of weights increases for all
the methods of parameter estimation. The estimation error of the vector of weights
(EEVW) using the Mallows criterion is the largest, the EEVW for the real measuring
system (Cp ) and for the transformed one (Cp∗ ) are close. The EEVW for the criterion
of the minimum description length is less than for the Mallows criterion, but it is also
large; the EEVW for the real measuring system (MDL) and for the transformed one
(MDL*) are close. The least EEVW is given by the l0 -optimality test. The estimation
error of the vector of weights according to the l0 -optimality test for a real measuring
system (L 0 ) is close to the estimation of the vector of weights by the true model; the
EEVW for the transformed measuring system (L ∗0 ) is close to that obtained in the
framework of the true model with the transformation.
Fig. 7 Dependence of the
estimation error of the vector
of weights (EEVW) versus
the level of self-noise
Method for Converting the Output of Measuring System into the Output …
177
6 Conclusion
The application of the output conversion method in the spectrometry problem makes
it possible to use the l0 -optimality test and thereby improve the accuracy of estimating the vector of weights (activities) of the radionuclides that formed the radiation
spectrum. The good results of applying the output conversion method in a practical
problem prove the relevance of further development (analytical and experimental
research) of the method of converting the output of measuring systems.
References
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2. Hendriks, P.H., Limburg, J., de Meijer, R.J.: Full-spectrum analysis of natural gamma-ray
spectra. J. Environ. Radioact. 53(3), 365–380 (2001)
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Meijer, R.J., Hlatshwayo, I.N.: Determination of soil, sand and ore primordial radionuclide
concentrations by full-spectrum analyses of high-purity germanium detector spectra. Appl.
Rad. Isotopes. 66, 855–859 (2008)
4. Rachkovskij, D.A., Revunova, E.G.: Intelligent gamma-ray data processing for environmental monitoring. In: Intelligent Data Processing in Global Monitoring for Environment and
Security—ITHEA, pp. 136–157, Kyiv-Sofia (2011)
5. Hansen, M., Yu, B.: Model selection and minimum description length principle. J. Amer. Statist.
Assoc. 96, 746–774 (2001)
6. Donoho, D.L., Elad, M., Temlyakov, V.: Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise. Technical report, Department of Statistics, Stanford University
(2004)
7. Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control
19(6), 716–723 (1974)
8. Mallows, C.L.: Some comments on CP. Technometrics 15(4), 661–675 (1973)
9. Gribonval, R., Figueras, I., Ventura, R.M., Vandergheynst, P.: A simple test to check the
optimality of sparse signal approximations. Tech. Rep., IRISA PI-1661 (2004)
10. Zabulonov, Y.L., Lysychenko, G.V., Odukalet, L.A., Revunova, E.G.: Increasing the accuracy
of radionuclide identification by the matching method. Model. Inf. Technol. 53, 108–114 (2009)
11. Pytiev, Yu.P.: Mathematical Methods of Interpretation of the Experiment. Vysshaya shkola,
Moscow (1989)
12. Revunova, E.G., Rachkovskij, D.A.: Stable transformation of a linear system output to the
output of system with a given basis by random projections. In: The 5th International Workshop
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13. Revunova, E.G.: Stable transformation of the output of a linear system into the output of a
system with a given basis. Control Syst. Mach. 6, 28–35 (2013)
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of converting the output of a linear system into the output of a system with a given basis. Control
Syst. Mach. 2, 28–33 (2013)
Electric Power Engineering
Analysis of UAVs and Their Technical
Parameters for Overhead Power Lines
Monitoring
Serhii Babak , Artur Zaporozhets , Oleg Gryb , and Ihor Karpaliuk
Abstract This chapter is devoted to possibilities of using unmanned aerial vehicles
for high-voltage power lines monitoring. Traditionally, unmanned aerial vehicles in
power industry are used to monitor extended objects. And mostly, such monitoring
is associated with optical devices (photo and video), which ensures the fulfillment
of the requirement of a superficial examination. Chapter presents classification of
Ukrainian and foreign unmanned aerial vehicles and their technical characteristics.
The possibility of their use for electric power industry tasks is estimated. For each
class of unmanned aerial vehicles, the most suitable tasks are described.
Keywords UAVs · Overhead power lines · Monitoring · Control · Energy system
1 Introduction
UAV is a basic component of remote monitoring system. It is an aircraft, which flight
is made under monitoring or direct control of an operator located in ground (or air)
control center, using two-way communication channels, or an autopilot according to
flight tasks. Recently, UAVs have been used to solve much more problems in various
fields of human activity, from space exploration and military application to agriculture [1–7]. This is due to a number of significant advantages of this type of aircraft:
no crew, low capital cost and low operating costs [8, 9]. Alongside with significant
progress in the development of computer technology and especially its miniaturization and energy efficiency, as well as development and practical application of
S. Babak
Verkhovna Rada of Ukraine, Kyiv, Ukraine
A. Zaporozhets (B)
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: a.o.zaporozhets@nas.gov.ua
State Institution “The Institute of Environmental Geochemistry of NAS of Ukraine”, Kyiv,
Ukraine
O. Gryb · I. Karpaliuk
National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_10
181
182
S. Babak et al.
new algorithms and methods for using UAVs, it makes it possible to increase the
efficiency of solving complex scientific and practical problems related to logistics,
monitoring, control and safety.
Most successful in development and application of UAVs are companies from
USA, France, Germany, Great Britain, China and Israel. UAVs are also being developed in countries that are not aviation industry leaders: Australia, Austria, Belgium,
Bulgaria, Croatia, Czech Republic, Finland, Greece, Holland, India, Iran, North
Korea, Norway, Pakistan, Poland, Portugal, Slovenia, South Korea, Spain, Sweden,
Switzerland, Thailand, Tunisia, Turkey [10, 11].
At the moment, Ukraine is implementing the «Strategy for the Development of the
Domestic Aviation Industry and Civil Aviation for the Period until 2020», approved
by the Cabinet of Ministers on December 27, 2008. This strategy defines the conceptual provisions for the formation and implementation of state policy in development,
manufacture, sale and after-sales service of aviation technique [12].
For civilian UAVs, basic tasks could be defined as follows [8, 13, 14]:
.
.
.
.
.
.
.
.
.
monitoring and observation of areas and facilities;
remote control of objects (including hazardous);
meteorological survey;
identification of people, places and objects;
patrolling and early detection of unauthorized access, protection of objects and
borders;
cartography;
retransmission of signals;
chemical treatment of agricultural land;
cargo delivery.
In energy sector, UAVs allow to solve the following tasks [15–20]:
.
.
.
.
.
.
.
.
.
.
.
assessment of condition of overhead power lines (OPLs);
aerial photography of OPLs and transmission towers;
thermal imaging control of power elements of OPLs;
measurement of wire slack;
control of the permissible height of trees in high-voltage lines zones;
analysis of corridor overgrowth;
identification of unauthorized construction sites in security zone;
search for new routes of power lines and creation of a digital relief model;
creation of a photographic plans of power facilities construction sites;
analysis of damages, accidents;
forecast and modeling of impact on nature.
We should also note the problems that arise during UAVs operation. Most
significant are [21, 22]:
. ensuring the transfer of information through communication channels between
the UAV and control point, with required width and without distortion;
. object recognition based on registered information;
Analysis of UAVs and Their Technical Parameters for Overhead Power …
183
. energy efficiency and autonomy;
. safety and trouble-free operation.
UAVs are also classified according to construction scheme [23]:
.
.
.
.
aerodynamic (aircraft type: fuselage, “flying wing”);
aerostatic and aerostatically unloaded;
reactive;
helicopter and multicopters (3, 4, 6 and 8 rotary).
Most common UAVs are aircraft and multicopter (helicopter) types. Each of them
is better at solving a certain range of problems. Considering the issues of electric
power monitoring, we will restrict ourselves to these two classes, as well as unmanned
balloons and attached UAVs.
Aircraft-type UAVs are used to create plans, digital terrain models and monitor
extended objects. Their advantages are high speed, significant range and autonomy.
Helicopter-type UAVs and multicopters are used to survey complex (small in
length) structures or lidar surveys. Basic advantages are small size, launch from any
platform, possibility of hanging.
Balloons and attached UAVs are used to monitor stationary objects. Main
advantages are simplicity and reliability, unlimited working time.
Also, combined schemes are used, for example, aircraft-type with propellers to
ensure vertical take-off and landing (hanging).
Also, UAVs are distinguished by:
. mass dimensions: micro, mini, etc., depending on size and payload;
. launch type: ground (from a catapult, from a runway or a platform, from hands),
air, aerospace, space;
. type of landing: with a run (with skis), vertically, with a parachute;
. application: disposable and reusable (in reusable UAVs, the possibility of landing
with a parachute, in a special braking network, on the fuselage or on the runway
is assumed).
2 Main Part
Majority of light UAVs use electric engines (EEs), powered from a battery, thus,
engine characteristics and battery capacity set maximum range and flight time
(autonomy). In most cases, autonomy depends on UAV size and batteries capacity.
Typical autonomy time very from half an hour to several hours, while the flight range
reaches 300–500 km. Fuel engines, as a rule, are installed on heavy UAVs (20 + kg),
while flight duration reaches 10 h, and flight distance is up to 1000 km. For extended
objects monitoring, it is advisable to use models with internal combustion engines
(ICEs).
Also, UAVs can be classified by purpose and engine charging schemes, by
autonomy and by types of channels used for control. Table 1 shows the conditional
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Table 1 UAV classification
Class
Weight (kg)
Autonomy (h)
Flight altitude (km)
Flight radius (km)
Micro
to 3
to 1
to 0.5
to 5
Ultralight
to 10
to 1.5
to 1
to 40
Light
to 50
1–3
1–3
to 100
Normal
to 1000
5–12
3–8
100–500
Heavy
more 1000
12 and more
to 20
more 1200
division of UAVs into classes, depending on their parameters [23]. Typical UAV
speed is: for light models 50–60 km/h, and for large ones—up to 100 km/h or more.
The review will consider UAVs that could be used to solve problems of three
following types:
. automatic monitoring, location and mapping of power lines;
. search and localization of accident sites, as well as logistic support for repair of
power lines;
. ensuring control and safety of power facilities.
Despite the fact that most UAVs are developed for military use, in practice they
could be effectively used for OPL monitoring.
Next, we will consider UAV models suitable for OPLs monitoring.
1. Trimble UX5 (Trimble, USA, 2018) is a lightweight UAV for monitoring and
mapping (Fig. 1).
Takeoff is carried out using a catapult. Autonomy is from 50 min, range—up
to 5 km, speed—80 km/h. The UAV is equipped with one pushing EE (0.7 kW).
Wingspan—1.0 m, length—0.6 m, height—0.1 m. Flight altitude—up to 5 km.
Takeoff weight—3–5 kg, payload—2.5 kg. Equipment: camera 16.1 MP.
2. DJI Mavic 2 (Da-Jiang Innovations Science and Technology Co., China) is a
commercial quadcopter designed for video filming and monitoring (Fig. 2).
Autonomy is 15–25 min, range is up to 500 m. The UAV uses EEs for movement,
which rotate 4 rotors and provide vertical take-off and landing. The flight speed is
Fig. 1 Trimble UX5
Analysis of UAVs and Their Technical Parameters for Overhead Power …
185
Fig. 2 DJI Mavic 2
up to 72 km/h. Dimensions: 0.322 × 0.242 × 0.084 m. Weight—0.907 kg. Flight
altitude—up to 6 km (programmatically limited to 120 m from the start point).
Takeoff weight—up to 1 kg, payload—0.4 kg. Among the features of the device,
low price and an automatic return (and landing) system should be noted. Equipment:
video camera.
3. WATT 200 (Drone Aviation Corp, USA, 2014) is attached quadcopter designed
for observation, monitoring and control of the perimeter (Fig. 3).
Autonomy—8 h, detection radius—up to 6 km (vehicle type target). Could work
in rainy and strong wind conditions. UAV uses an EE for movement, powered by
a 220 V network, which drives 4 rotors into rotation. Operating height up to 80 m.
Takeoff weight—up to 15 kg, payload—2 kg. Equipment: optical and IR sensors.
Fig. 3 WATT 200
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4. Trimble ZX5 (Trimble, USA, 2018)—a quadcopter designed for geodetic and
agricultural tasks, monitoring and control of the perimeter (Fig. 4).
Autonomy is 20 min, range is up to 3 km. It could operate at wind speeds up to
36 m/s. UAV uses 6 EEs for movement, which drive 6 rotors. Flight altitude—up to
3000 m. Takeoff weight—up to 5 kg, payload—2.3 kg. Equipment: camera Olympus
16 MP.
Further, Ukrainian UAVs are considered as the most probable candidates for
solving the problems of monitoring objects of the energy system of Ukraine [24].
5. Strepet-S (State Enterprise “Chuguevsky Aviation Repair Plant”, Ukraine, 2006)
is a multipurpose UAV designed for monitoring various objects, surveillance and
reconnaissance, special operations, as well as patrolling the perimeter (Fig. 5).
Autonomy is up to 6 h (maximum range 300 km), speed is 180 km/h. Equipped
with an ICE with power of 19 hp. Takeoff and landing are carried out from the runway
(up to 120 m long), there is also the possibility of emergency landing by parachute.
Dimensions: length 3.2 m, wingspan 4.2 m. Flight altitude—up to 4 km. Takeoff
weight—up to 90 kg, payload—up to 35 kg. The advantages of the UAV include the
ability to fly at night and in adverse weather conditions, as well as the presence of
an automatic system that allows to fly along a given route. Equipment: video camera
and IR sensor.
6. Observer SM1 (Yumiko Aerospace, Ukraine, 2013) is an aircraft-type UAV with
a pusher propeller designed for monitoring airspace, the earth’s surface and
industrial facilities, patrolling areas, observing crops and controlling fire safety
(Fig. 6).
Fig. 4 Trimble ZX5
Analysis of UAVs and Their Technical Parameters for Overhead Power …
187
Fig. 5 Strepet-S
Fig. 6 Observer SM1
Autonomy is up to 6 h (maximum range—500 km), speed—80–120 km/h. UAV
is equipped with ICE. Takeoff and landing are carried out from the runway (50–
70 m long), there is also possibility of emergency landing by parachute. Dimensions:
length—2.47 m, wingspan—6.8 m. Flight altitude—up to 5 km. Takeoff weight—
up to 240 kg, payload—up to 40 kg. UAV advantages include functions for fixing
coordinates, range and dimensions of objects, as well as capture and tracking of
moving objects. Equipment: video camera, IR sensor and laser rangefinder.
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Fig. 7 Viper SM3
7. Viper SM3 (Yumiko Aerospace, Ukraine, 2014) is a three-propeller multicopter designed for object monitoring, chemical and radiation analysis, geodetic
reconnaissance (Fig. 7).
Autonomy period is from 0.3 to 0.9 h (maximum radius 6 km), speed—80–
120 km/h. It is equipped with EEs that rotate 3 rotors, which provides vertical takeoff and landing. Length—0.65 m. Flight altitude—up to 2 km. Takeoff weight—up
to 10 kg, payload—up to 5 kg. UAV advantages include ability to operate in strong
winds (up to 20 m/s). Equipment: optical and IR sensor, gas analyzer and dosimeter.
8. A-2 Synytsa (Design Bureau “Zlit”, Ukraine, 2004) is a small-sized UAV made
according to the “duck” scheme (Fig. 8). UAV is designed for environmental
monitoring, reconnaissance of areas of major accidents and disasters, control of
the perimeter of large objects. In context of environmental monitoring, device
allows to calculate the places of accumulation of water of insufficient purity.
Autonomy time is up to 1 h (range 20 km), speed—80 km/h. UAV is equipped
with a 0.54 hp EE. Takeoff is performed with a catapult. Dimensions: length 0.95 m,
Fig. 8 A-2 Synytsa
Analysis of UAVs and Their Technical Parameters for Overhead Power …
189
wingspan—1.8 m. Takeoff weight—up to 5 kg, payload—up to 1 kg. Equipment:
two video cameras and IR sensor.
Tornado-2 (Geokom, Ukraine) is a UAV developed according to an airplane
scheme and is used for aerial photography and topographic location of objects, environmental monitoring and perimeter control, as well as reconnaissance of areas of
accidents and disasters (Fig. 9).
Autonomy period is up to 4 h (range—200 km), speed—150 km/h. The UAV is
equipped with a 3.6 kW ICE, which drives the rotor. Takeoff is performed using a
catapult or by hand. Dimensions: length 2 m, wingspan 2 m. Takeoff weight—up to
12 kg, payload up to 5 kg. Equipment: FPV camera.
9. Tornado EL (Geokom, Ukraine) is a UAV developed according to an airplane
scheme, is used for aerial photography and topographic location of objects, environmental monitoring and perimeter control, as well as reconnaissance of areas
of accidents and disasters (Fig. 10).
Autonomy time is up to 90 min (range—17 km), speed—85 km/h. The UAV is
equipped with a 1.2 kW EE, which drives the rotor. Takeoff is performed using a
catapult or by hand. Dimensions: length 1.6 m, wingspan 2.05 m. Takeoff weight—up
to 5 kg, payload—up to 2 kg. Equipment: camera SONY RX 1.
Table 2 shows the comparative characteristics of Ukrainian UAVs. For a UAV,
the most important parameter is payload weight that it can lift into the air and total
take-off weight. On the one hand, weight determines the set of permissible equipment (communication, processing, etc.) and sensors (lidars, IR, optical) that can
Fig. 9 Tornado-2
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Fig. 10 Tornado EL
be installed on the UAV, and on the other hand, its autonomous characteristics and
engine power.
3 Conclusions
Summing up, it should be noted the significant advantages that is given by UAV as
main component of monitoring system:
. efficiency: 30 km2 /h for areal monitoring and up to 35 km for linear objects
(regardless of location);
. objectivity: single “digital map” based on high-precision photo and video filming,
in electronic form;
. cost: UAVs are more economically efficient than manned aircraft, satellite imagery
or ground survey.
From the point of view of the tasks being solved, the most suitable are UAVs:
1. for automatic monitoring, location and mapping of power lines, the main element
of the monitoring system—aircraft type, or combined. It is advisable to use solar
panels to increase autonomy, or systems for recharging from the mains (for
combined);
2. to search and localize accident points, as well as logistical support for the repair
of OPLs—helicopter type or multicopter. It is advisable to use all-weather UAVs
that do not require special take-off and landing sites, while high autonomy is not
a significant factor;
Aircraft type with
pusher propeller,
ICE
Aircraft type with
pusher propeller,
ICE
Strepet-S
Observer SM1
0.95/1.8
2/2
2.05/1.6
Aircraft-type, IEC
Aircraft-type, EE
Tornado 2
Tornado EL
17
400
(20)
(6)
Multicopter (3), EE 0.65/−
Aircraft-type, EE
Viper SM 3
A-2 Synytsa
500
300
Flight range
(radius) (km)
2.47/6.8
3.2/4.2
Scheme and engine Length/Wingspan
(m)
Model
Table 2 Comparative characteristics of Ukrainian UAVs
0.8
5
12
5
3
10
2
240
90
Takeoff
weight (kg)
−
5
4
Flight altitude
(km)
2
5
1
5
40
35
Payload (kg)
85
150
80
120
80
180
Speed
(km/h)
1.5
4
1
0.3–0.9
2–5
6
Autonomy (h)
Analysis of UAVs and Their Technical Parameters for Overhead Power …
191
192
S. Babak et al.
3. to ensure control and safety of power facilities—aircraft or helicopter type, or
tethered systems (for control of power facilities). It is advisable to use a UAV with
high autonomy and all-weather capability to ensure round-the-clock control.
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2021.44.08
Determination of Energy Characteristics
for Choice of Surge Arresters
Sergii Shevchenko , Dmytro Danylchenko , Stanyslav Dryvetskyi ,
Natalia Savchenko , and Serhii Petrov
Abstract This monograph is devoted to the definition of energy characteristics for
the selection of surge arresters. The scheme of AR substitution is considered, which
allows to take into account the possibility of resonant phenomena in the calculations
of AR operation modes and to estimate the values of leakage currents on their surface.
Experimental studies of the electrophysical characteristics of ARs in the assembled
state have been carried out. A mathematical model was obtained that reflects the physical processes occurring in the varistor material and allows calculations of the power
lost in the arrester when exposed to the highest operating voltage of the network.
Calculations using the obtained mathematical model of the arrester, showed that the
change in the maximum allowable voltage of the network changes significantly the
percentage of harmonic amplitude in the presence of which the thermal balance of
the arrester is disturbed. Experimental studies of VAC of varistors from different
manufacturers have been performed.
Keywords Surge arresters · Non-linear overvoltage limiter · Volt-ampere
characteristics · Electric networks · Quality of electricity · Varistor · Active power
losses · Mathematical model
1 Introduction
The main and almost the only means of protection of power supply systems from
overvoltages, in accordance with regulations is a non-linear overvoltage limiter (AR).
When choosing the parameters of the protective device to limit overvoltages in power
supply systems of all classes of rated voltage, it is assumed that the modes of the
power supply system with short-term exceedances of the maximum operating voltage
caused by various types of overvoltages are acceptable. In this case, for non-linear
S. Shevchenko · D. Danylchenko (B) · S. Dryvetskyi · S. Petrov
National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine
e-mail: 555dd555@ukr.net
N. Savchenko
Donetsk National Technical University, Pokrovsk, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_11
195
196
S. Shevchenko et al.
overvoltage limiters (ARs) as the largest operating voltage is taken to be close to the
linear voltage of the power supply system.
The protective device is selected in such a way that the total current flowing
through its varistor column under the action of a voltage equal to the linear voltage
of the power supply system does not exceed a few milliamperes. At the same time in
the body of the varistor in a continuous mode power is released, which leads to the fact
that the temperature of the arrester is several degrees higher than the ambient temperature. However, this approach to the choice of arrester does not allow to determine its
parameters in the occurrence of harmonic oscillations in the current voltage because
the operating conditions of the overvoltage limiter become completely different.
With a sufficiently high content of higher harmonics and the time of their action in
the power supply system, the power released in the arrester varistors can increase
significantly. The amount of power dissipated by the arrester in this case may exceed
the manufacturer’s standard for varistors, which will lead to a violation of the thermal
balance of the arrester and its failure.
2 Literature Review
Currently, the main way to protect electrical equipment from overvoltages of 6–
750 kV is the use of nonlinear surge arresters (ARs). The issue of selection and
operation of arresters in power supply systems at different times has received much
attention from leading scientists in Ukraine and abroad. However, all researchers
focused on the operation of arresters under the action of overvoltages and did not
consider the modes of their operation under the action of low quality voltage. This
is confirmed by the content of current regulations on the selection and operation of
AR.
The main task when choosing an arrester is to limit the overvoltages to a safe level
for the protected electrical equipment and at the same time ensure the resistance of the
limiters to dangerous for them overvoltages. AR for 6–750 kV electrical networks
[1–6]. Presented on the Ukrainian market, are produced by different plants (both
domestic and foreign) on the basis of their own technical solutions, so the arresters of
different manufacturers, designed for one voltage class, may differ in characteristics
that must be considered when choosing them.
Electric networks of Ukraine 6–35 kV operate mainly with isolated or grounded
through the arc-quenching reactor neutral, so the operating conditions of arresters in
these networks differ from the operating conditions in 110–750 kV networks with
large values of switching and quasi-stationary overvoltages [1–6]. In this regard,
the review of methods for selecting the main characteristics of the AR should be
conducted for two types of networks: 6–35 kV and 110–750 kV.
Determination of Energy Characteristics for Choice of Surge Arresters
197
3 The Scheme of Replacement of Overvoltage Limiters
in the Area of Leakage Currents of Volt-Ampere
Characteristics (VAC)
As is known, the known schemes of substitution of arresters (AR) are considered
from the point of view of its work in the modes of overvoltage limitation. In our work
we have made an attempt to determine the scheme of replacement of arresters in the
modes of long-term application of operating voltage, which will allow the analysis
of their work in conditions of violations of the quality of electricity.
Given that the capacitance and value of tg δ of dielectrics may depend on
the frequency in the substitution circuit of the varistor, to account for resonant
phenomena, it is necessary to add inductance, which is due to the inductance of
busbars connected to arresters and internal inductance of the device. This inductive
resistance is presented in the diagram of Fig. 1 inductance L. Appropriate calculations should determine the length of the bus that will lead to resonance at a frequency,
and issued recommendations for the prevention of such phenomena.
The magnitude of power losses from currents flowing on the wet surface of the
arrester during operation in areas with high levels of air pollution can be quite significant, but it can not be decisive in choosing the arrester because it affects its heat
balance only through surface temperature. large values of leakage current can increase
by several degrees. Such losses in the substitution scheme of Fig. 1, takes into account
the parallel resistance Rlos . Shown in Fig. 1, scheme is a refined scheme of substitution of the arrester takes into account all the components necessary for the analysis
of the mode of operation of the surge arrester as a whole in the region of volts of the
ampere characteristic concerning leakage currents. This analysis allows to clarify the
choice of arresters, taking into account its operation in the area of leakage currents
of the ampere characteristic, where it works most of the time during operation in
electrical networks.
Fig. 1 Complete scheme of
replacement of arrester
replacement
198
S. Shevchenko et al.
This scheme allows to take into account the possibility of resonant phenomena
when calculating the operating modes of surge arrester and estimate the magnitude of
leakage currents on their surface, which allows to estimate the magnitude of moisture
discharge voltages and calculate leakage current through varistors during operation
in areas with varying degrees of air pollution [7, 8]
Unfortunately, in the presence of a large number of robots to study. characteristics of zinc oxide ceramics is quite difficult to determine the parameters of the
surge arrester in general, because each manufacturer has its own technology for their
production [9, 10]. Which leads to large differences in parameters. An example is
the capacitance, which will depend on the design of the electrodes and the method
of contact between the electrodes and varistors. In our opinion, these circumstances
illustrate the need for a detailed study of the parameters of the surge arrester in
the assembled state of different manufacturers to determine the parameters of the
substitution scheme.
4 Experimental Studies of Electrophysical Characteristics
of Surge Arresters
To analyze the power losses in the arrester it is necessary to know their electrophysical
characteristics, which will determine the parameters of the substitution scheme. In
addition, for a correct analysis of the influence of harmonic voltage fluctuations on the
thermal regimes of arresters, it is necessary to have information about the dependence
of the frequency of the parameters of the substitution scheme and the tangent of the
dielectric loss angle. Surge limiters in the assembled state of production of Tavrida
Electric Ukraine LLC were used for researches of electrophysical parameters. 6/6,9
UHL1 training copy (with contacts made of aluminum), overvoltage limiter TEL
6/6,9 UHL1 serial number 4810, overvoltage limiter TEL 6/6,9 UHL1 serial number
4812, overvoltage limiter TEL 10/12 UHL1 serial number 4731, overvoltage limiter
TEL 10/12 UHL1 serial number 4732, overvoltage limiter TEL 10/12 UHL2 serial
number 4005 and varistors manufactured by ABB and Ersos.
The research was performed at the Department of Electricity Transmission of NTU
“KhPI”. E7-14 type immetance meter and E7-22 type digital immetance meter were
used in the experiments. Three series of experiments were performed to determine
the capacitance and tangent of the dielectric loss angle for surge arresters in the
assembled state.
The results of experimental studies are shown in Figs. 2, 3 and 4. [11].
Capacitance and tg δ measurements were performed for three values of frequency
100, 1000, 10,000 Hz. This choice of frequencies is due to the fact that the electric
networks of Ukraine take into account harmonic oscillations up to frequencies of 41
harmonics, equal to 2050 Hz.
As can be seen in Fig. 3, the dependence of tg δ on frequency is significant
for frequencies significantly greater than 1000. In the frequency range required to
Determination of Energy Characteristics for Choice of Surge Arresters
199
Fig. 2 Dependence of the tangent of the dielectric loss angle on frequency
Fig. 3 The dependence of the capacitance of the surge arrester on the frequency for 10 kV arresters
perform analysis of harmonics (up to 2050 Hz) tg δ different samples of AR may
differ several times at low frequencies. However, when the frequency increases to
1 kHz, this difference is significantly reduced, for higher frequencies—almost absent.
Therefore, the limit values of tg δ from 1.5 to 4 percent can be used to calculate the
power loss in the AR. To solve the problem of choosing the AR, as a rule, you should
use the upper limit of the oscillation interval tg δ to obtain larger values of power
acting on it. This assumption will allow you to choose the arrester with a margin for
energy performance. Based on the fact that all varistors that complete the studied AR
200
S. Shevchenko et al.
Fig. 4 Dependence of the capacitance of the surge arrester on the frequency for the surge arrester
6 kV
are manufactured by ABB, it can be argued that to analyze the effects of harmonic
oscillations on the AR of this manufacturer can be used an average value of tg δ
equal to 0.04 [11].
The nature of the dependence of the capacity on the frequency shown in Figs. 3.
Figure 4 shows that in the range of frequencies required to perform the analysis of
the influence of harmonics (up to 2050 Hz), the dependence of the capacitance on
the frequency is almost absent. The difference between the capacitance values at
frequencies of 100 and 10,000 Hz is a maximum of 3.5%. It should be noted that
Figs. 3 and 4, shows the dependences of the capacitance on the frequency for two
different classes of ARs, which have a capacitance in the range of 140–120 and 65
pF, respectively. This arrangement of curves is related to the design of the studied
ARs. Structurally, they are made of the same varistors, only AR TEL 6/6,9 has one
varistor, and AR TEL 10/12—two of the same connected in series. This correlates
well with the values of their capacities, which differ almost twice. The capacitance
of the arrester can be calculated by the expression for a flat capacitor [11]:
C=
εε0 πr 2
,
h
(1)
where r —the radius of the cover AR; ε—relative dielectric constant of AR; h—
column height of AR varistors; ε0 —absolute dielectric constant.
The results obtained when measuring the capacities allow us to conclude that for
the calculation of the energy characteristics of the arresters under the influence of
Determination of Energy Characteristics for Choice of Surge Arresters
201
harmonic oscillations, they can be performed at a fixed value of their capacitance
for the entire frequency range under consideration. For arresters with ABB varistors
manufactured by Tavrida Electric Ukraine LLC, they will be 125 pkF for arresters
with a nominal voltage of 6 kV, and 66 pkF for arresters with a nominal voltage of
10 kV, respectively [11].
The obtained results make it possible to perform calculations of the relative dielectric constant of the material from which the varistors are made. Determining the value
of the relative dielectric constant for zinc oxide ceramics will allow to use it for engineering calculations of the array capacity when considering the effects of harmonic
voltage fluctuations on it at the stage of its selection.
From equation
P = U I cos φ = U I sin δ = U
U
Ia
sin δ = U tgδ = U 2 ωC2 tgδ,
cos δ
xa
(2)
we obtain a relation that will make it possible to obtain the values of relative dielectric
constant for the studied ARs
ε=
Ch
.
ε0 πr 2
(3)
The results of the calculation of the relative dielectric constant for different
frequencies of the applied voltage are shown in Fig. 5.
The nature of the dependence of the dielectric constant, which is presented in Fig. 5
allows us to conclude that in the studied frequency range, the value of the relative
dielectric constant of the arrester has a weak frequency dependence. This fact shows
that it is possible to use for engineering calculations of AR parameters a single
average value for all frequencies characteristic of harmonic oscillations existing in
Fig. 5 Dependence of relative dielectric constant on frequency
202
S. Shevchenko et al.
electrical networks, which is equal to 585. The maximum error in determining the
relative dielectric constant is less than 10% [11].
5 Mathematical Model for the Selection of Energy
Characteristics of Arresters with Low Quality
of Electricity in the Network
In order to make the correct choice of arresters at the design stage of surge protection,
it is necessary to determine its criteria. One such criterion may be power loss. In the
previous section, examples of calculations of such power were given.
The calculation of power losses in the arrester under the influence of harmonic
oscillations must be performed if the customer has information about the presence in
the power supply system in which it will be installed, there are harmonic oscillations
that exceed the allowable values. To choose the arrester, in this case, you need to
know the harmonic composition of the operating voltage and the amount of energy
it is able to absorb.
We have developed a method of such a choice based on the above sections of
the work. One of the main issues of the methodology is the question of determining
the limit value of the power of losses that AR can withstand. This value can be
taken as the amount of active power that the arrester can dissipate when exposed
to the maximum operating voltage of the network throughout the operation time
without loss of thermal equilibrium. Leading manufacturers of varistors, such as
ABB and Ersos, provide information on the magnitude of such power in the technical
documentation for varistors [12–15]. In addition, some of them give the value of such
power when marking the varistor. Therefore, it is quite easy to obtain information on
the magnitude of this power for the arrester assembled from varistors of a known type.
However, in the absence of GOST on ARs, the labeling of devices manufactured in
Ukraine does not contain information on the amount of active power that the AR can
dissipate when exposed to the maximum operating voltage of the network throughout
operation without loss of thermal balance. In our opinion, it is expedient to indicate
the value of such power in the catalogs of arresters together with the energy that can
dissipate arresters when exposed to high current pulses.
All of the above shows that the following relationship should be used when
choosing AR [16]:
Pa. p. ≥ U 2 ωC2 tgδ,
Pa. p. ≥
U 2 ωπr 2 εε0 tgδ
,
h
(4)
(5)
Determination of Energy Characteristics for Choice of Surge Arresters
203
where Pa.p. —the amount of active power that the arrester can dissipate when exposed
to the maximum operating voltage of the network throughout the operation time
without loss of thermal equilibrium.
The obtained expressions reflect the physical processes occurring in the material
of the varistors and make it possible to calculate the power lost in the arrester when
exposed to the highest operating voltage of the network. These expressions represent
a mathematical model of the thermal stability of the arrester in the mode of longterm application of the operating voltage, which reflects the processes during the
operation of the arrester in the area of leakage currents.
This mathematical model allows us to easily estimate the effect on the AR of any
spectral composition of harmonic oscillations, taking into account the expression of
Percival’s law [17] and the results obtained when measuring the capacitance and.
Let’s write an expression for determining the total power loss in the arrester in the
presence of harmonic influences:
Pa. p. ≥
∞
.
(Uk2 ωk Ck tgδk ),
(6)
k=0
where k—number of the harmonic component; Pa.p. —permissible active power that
the arrester can dissipate during the service life without loss of heat balance; U k —
voltage of the k-th harmonic; ωk —circular frequency of the k-th harmonic; C k —
capacitance of the k-th harmonic; tgδk —tangent of the dielectric loss angle of the
k-th harmonic.
Taking into account the results obtained above, we will rewrite the expression
Pa. p. ≥ C · tgδ
∞
.
(Uk2 ωk ).
(7)
k=0
We obtain a mathematical model of thermal stability of arresters:
∞
Pa. p. ≥
πr 2 εε0 tgδ . 2
×
(Uk ωk ).
h
k=0
(8)
For a certain arrester model, tgδ, r, h and ε are not variables, so the first component
of expression (8) becomes a constant, multiplying which by the voltage and circular
frequency of the corresponding harmonic allows to determine the active power loss
for each of the existing harmonic voltage components. Which makes it possible to
determine the effect of different harmonics on the thermal balance of the arrester.
Given that, tgδ, r, h and ε are not variables, we write expression (8) as follows
Pa. p. ≥ K × t
∞
.
k=0
(Uk2 ωk ),
(9)
204
S. Shevchenko et al.
where K – πr εεh 0 tgδ .
The value of the value of the loss of active power of the k-th harmonic is attributed
to the loss of the 1st harmonic to can be determined using the following expression:
2
.Pk =
Pk
Uk
=
× k,
P1
U1
(10)
where .Pk —loss of active power from the k-th harmonic in relative units; Pk —loss
of active power from the voltage of the k-th harmonic; P1 —loss of active power
from the voltage of the 1st harmonic; U1 —maximum value of the voltage of the 1st
harmonic; Uk —maximum value of the voltage of the k-th harmonic.
The calculations showed that when meeting the requirements of GOST for the
quality of electricity, the presence of harmonic voltage fluctuations does not affect
the thermal balance of the arrester.
However, in the presence of increased voltage relative to the values specified in
GOST can lead to a significant increase in power losses. For example, if there are 3,
5 and 7 harmonic components in the network with a value of 10% of the voltage of
the 1st harmonic, the loss of active power will increase by 15%, which may exceed
the value of losses defined by the varistor manufacturer violation of heat balance.
Figure 6 shows that in the presence of harmonic components, the value of the
allowable active power that the AR can dissipate throughout its service life may be
exceeded. Also in Fig. 6 you can see a big difference between the normalized values
of the active power of different manufacturers of varistors. It should be noted that
the calculations were performed for the network with a maximum allowable voltage
of 6.9 kV. Structurally, the arresters made of ABB varistors differ from the arresters
made of EPCOS varistors in that the latter contain two varistors in their design,
and the former only one. This determines the difference between their maximum
allowable voltage. Calculations for 7.2 kV voltage are shown in Fig. 7.
A comparison of Figs. 6 and 7 clearly shows that when the maximum allowable
mains voltage changes, the percentage of harmonic amplitude changes significantly,
in the presence of which the thermal balance of the arrester is disturbed. This conclusion indicates the need to take into account the presence of harmonic oscillations in
the network at the design stage of the surge protection scheme and the availability
of information on the quality of electricity in the city of installation of arresters.
In the absence of such information, it is necessary to study the composition of the
equipment of the electrical power supply system and determine the possibility of
harmonic oscillations.
As can be seen from expression (7), the thermal stability of the arrester depends on
its geometric and electrophysical parameters. The effect of the varistor column height
on the thermal stability of the arrester is quite unexpected. It should be noted that
there is a great influence on the value of the loss of active power of the varistor radius,
this is due to the decrease in current density under the same voltage. The obtained
model clearly shows ways to increase the magnitude of active power losses due to
geometric and electrophysical parameters of the arrester. To date, a large number of
ceramics have been synthesized on the basis of zinc oxide with impurities that have a
Determination of Energy Characteristics for Choice of Surge Arresters
205
3
4
5
2
1
Fig. 6 Estimated power dependences of different harmonic oscillations at the maximum allowable
mains voltage of 6.9 kV: 1—power of the 2nd voltage harmonic; 2—power of the 3rd voltage
harmonic; 3—power addition of the 1st and 3rd voltage harmonics; 4—rated power of varistors
produced by EPCOS; 5 – rated power of varistors produced by ABB
3
4
5
2
1
Fig. 7 Estimated power dependences of different harmonic oscillations at the maximum allowable
mains voltage of 7.2 kV: 1—power of the 2nd voltage harmonic; 2—power of the 3rd voltage
harmonic; 3—power addition of the 1st and 3rd voltage harmonics; 4—rated power of varistors
produced by EPCOS; 5—rated power of varistors produced by ABB
fairly wide range of electrophysical parameters. Additives and production conditions
significantly affect the dielectric constant of varistor ceramics, which allows it to be
manufactured taking into account the requirements for electrophysical parameters,
such as dielectric constant. Methods of production and compositions of ceramics are
the know-how of varistor manufacturers. Based on the given mathematical model,
they have the opportunity, by changing the production technology or the composition
of impurities, to increase the energy that the arrester dissipates during operation.
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The results of calculations show that the quality of electricity has a significant
impact on the energy performance of arresters. The presence of harmonic oscillations in the network leads to the probability of violation of the thermal balance of
the arrester and its failure. In cases where the number of harmonic oscillations is
not sufficient to disrupt the thermal balance due to overheating of the arrester, its
operating temperature rises. This in turn can lead to the failure of the arrester due to
the absorption of surge energy at elevated operating temperatures.
6 Experimental Studies of Volt-Ampere Characteristics
of Arresters in the Zone of Leakage Currents
Varistors from two leading manufacturers of ABB and EPCOS varistors were used for
experimental studies. All varistors studied had appropriate labeling unique to each of
them, unlabeled varistors were not studied. Marking on the end of the varistor using
laser printing, which is a guarantee that the varistor is manufactured in the factory and
it is not counterfeit. The label usually contains information about the manufacturer,
the magnitude of the qualifying current (different for different manufacturers) and
voltage, the residual voltage at a current pulse of 5, 10 or 20 kA, the maximum active
power that the varistor can dissipate throughout life without loss of heat balance and
factory number [18].
To test varistors and arresters in the assembled state and individual varistors, a
laboratory test installation “Test Surge Arrester up to 25 kV” (TSA-25) was developed
and created. This installation is intended for carrying out high-voltage tests, voltage
of industrial frequency of 50 Hz, limiters of surges of nonlinear (AR) like ARKR, AR-RT, etc. Any AR with U mlt no more than 20 kV, in accordance with the
requirements of the international standard IEC 99-4 and GOST 16357-83 in the part
relating to:
• measurement of the amplitude and (or) effective value of the conduction current
at the maximum long-term allowable operating voltage of the arrester (U mlt );
• measurement of classification current (amplitude of the largest half-life of the
active component of the current);
• measurement of the amplitude and (or) effective value of the classification
voltage when flowing through the arrester of the active component of the current
(amplitude of the largest half-life) at maximum voltage.
The installation belongs to the category of special purpose equipment and is made
in a single copy. The appearance is shown in Fig. 8.
Experimental studies of VAC varistors in the leakage current zone have been
performed for more than 1000 varistors of different types used by AR manufacturers
for voltage classes 6–10 kV. The design of the arresters for these voltage classes may
include several varistors so the VAC characteristics were measured for the varistor
pairs used in production. A characteristic view of the I–V characteristics is shown in
Determination of Energy Characteristics for Choice of Surge Arresters
207
Fig. 8 Appearance of the
TSA-25 installation
Figs. 9, 10, 11 and 12. All graphs are shown in the coordinates of current—voltage
ratio. The ratio of the voltage acting on the varistor to the highest operating voltage
is used as the voltage ratio.
Comparing VAC of varistors from different manufacturers, we can see a significant
difference in the amount of voltage at which their nonlinear properties begin to
Fig. 9 VAC of 5 kV varistors produced by ABB
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Fig. 10 VAC of 6 kV varistors produced by ABB
Fig. 11 VAC of 5 kV varistor pairs produced by ABB
appear. This may be due to the different content of impurities in zinc oxide or the
technology of varistors. The obtained results of measurements of VAC of varistors of
different manufacturers very well demonstrate the differences in the characteristics
of ARs produced on the basis of different varistors. This state of affairs makes it
very difficult or almost impossible to correctly choose the AR in networks with low
quality of electricity in the absence of information about their VAC in the area of
Determination of Energy Characteristics for Choice of Surge Arresters
209
Fig. 12 VAC of 6 kV varistor pairs produced by ABB
leakage currents. There is a need to generalize the obtained results and develop a
method for calculating the modes of operation of the arresters in the area of leakage
currents.
It should be noted that the shape of the current flowing through the arrester when
exposed to the highest operating voltage of the industrial frequency for varistors based
on zinc oxide is not sinusoidal. The first half-current is greater than the second. This
difference can be no more than 20% according to the IEC standard [19], this is typical
for varistors produced by ABB, EPCOS varistors have almost no such difference. All
varistors have markings on one of the ends. When measuring the VAC, the varistors
were oriented upwards, in this case, as mentioned above, the first half-current is
greater than the second. If the varistor is oriented by marking downwards, its VAC
will change. We measured the VAC of varistors oriented to each other. Comparison
of the results of VAC measurements of varistors with different orientation to each
other is shown in Fig. 14.
In Fig. 14 it is clear that there is a significant difference between the currents
flowing through the varistors for differently oriented varistors. Thus, for the VAC
shown in this figure for differently oriented 5 kV varistors, the difference in currents
at the same voltage reaches 25%. This phenomenon can be called the effect of
varistor orientation. The difference in currents of differently oriented varistors makes
it possible to adjust the parameters of the arrester during production and greatly
improves its energy performance. Which, in turn, makes it possible to reduce the
maximum operating and residual voltage of the arrester. Reducing the residual
voltage of the arrester will reduce the values of voltages that can affect the equipment
of the electrical network and will reduce the overvoltage coefficients. The presence
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of the effect of the orientation of the AR varistors emphasizes the need to obtain
information on how the varistor is oriented at the stage of its selection in the design
of the electrical network.
From the experimentally obtained VAC [16] (Figs. 9, 10, 11, 12 and 13), it is clear
that they have two characteristic segments in the zone of leakage currents. The first,
which is characterized by almost linear slow current rise with increasing voltage,
and the second, which is characterized by rapid current rise with low voltage rise.
This nature of the VAC can be explained by the fact that in the first segment the
processes characteristic of capacitors take place, namely the current is capacitive
in nature with a low content of active current. In the second segment of the VAC,
the active component of the current grows quite rapidly, which becomes decisive
in the total current. This is due to the fact that in the structure of the varistor there
are conductive circuits, the number of which increases avalanche when the voltage
increases. The typical structure of zinc oxide varistor is shown in Fig. 15 [13–15, 18,
20].
The arrows in the figure show the conduction circles in the varistor structure. As
you can see from the figure, the structure of each varistor is unique, so each of them
has unique properties. This circumstance significantly complicates the analysis of
varistor parameters when the active component of the current flows through them and
makes it almost impossible to accurately determine the current at a given voltage.
The analysis of the obtained results and literature sources shows that on each of
the characteristic segments of VAC of the varistors the approximating function has
different coefficients, which makes it very difficult to use in practice when choosing
the required AR model.
Fig. 13 VAC of varistors and their pairs produced by EPCOS
Determination of Energy Characteristics for Choice of Surge Arresters
211
Fig. 14 VAC of differently oriented varistors: 1—varistors are oriented by marking to meet each
other; 2—varistors are oriented upwards
Fig. 15 Typical structure of varistor ceramics based on zinc oxide
Expressions
U = AI α ,
(11)
where α—coefficient of nonlinearity of the material (the value of β = 1−α is called
the coefficient of ventilation); A—constant, which depends on the material and size
of the varistor sample; I—current, A; U—voltage, V, and
I = K U α.
(12)
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For zinc oxide varistors, the nonlinearity coefficient α is usually 20–60 units.
Expressions (11) and (12) are used for engineering calculations, which approximate the volt-ampere characteristic of varistors. However, their use is possible only
in the presence of appropriate values of current and voltage at two characteristic
points, which will determine the coefficients of approximation in these expressions.
In the process of selecting ARs, engineering and technical personnel do not have
the opportunity to obtain the necessary data to determine the coefficients of approximation, because they are not listed in any catalog of the manufacturer of ARs. As a
rule, such catalogs contain information on residual voltages and the corresponding
current values when operating on the AR of different types of pulse voltages, which
can be used to determine the coefficients of the approximating curve only on the
VAC segment corresponding to the working area. The working area of the VAC is
called its segment on which the AR limits the value of overvoltages to the residual
values.
It is almost impossible to determine the coefficients of the approximating expression in the zone of leakage currents of the VAC because there is no data on the
dependence of voltage and current on this segment. Figures 9, 10, 11, 12 and 13
make it possible to determine the necessary values for current and voltage to calculate the coefficients of the approximating expression. The nature of the dependence
of current on the voltage in the leakage zone of the VAC must be known to analyze the
behavior of the arrester during long-term exposure to the highest operating voltage
of the electrical network. Such an analysis will allow to obtain information about the
heat balance of arresters in electrical networks with low quality of electricity and to
clarify the methods of their selection in the presence of harmonic oscillations in the
network.
7 Method for Determining Active Power Losses in AR
To obtain the values of the energy passing through the arrester, it is necessary to
be able to determine the value of the current by the value of the voltage in the area
of leakage currents of the VAC. We have calculated the energy passing through the
arrester for certain periods of time on the basis of experimentally obtained VAC.
Typical results are shown in Fig. 16 [16].
The limit value of active power that the arrester can dissipate is calculated in
this case according to the manufacturer’s catalog data according to the following
expression [16]:
Wmpl = Wsp × Uho ,
(13)
where Wmpl —maximum power under which the AR loses heat balance; Wsp —
specific energy of AR; Uho —the highest operating voltage of the network.
Determination of Energy Characteristics for Choice of Surge Arresters
213
Fig. 16 Dependence of active power on the value of phase voltage relative to the highest phase
operating voltage for arresters with a maximum operating voltage of 12 kV: 1—under the action of
voltage for 1000 s; 2—under the action of voltage for 10,000 s; 3—under the action of voltage for
36,000 s; 4—the limit value of power that the arrester can dissipate
As can be seen from Fig. 16 in 36,000 s (10 h) the thermal balance of the arrester
will be disturbed, which will lead to its destruction and the emergence of an emergency situation in the electricity grid. Experience of operation of arresters in electric
networks of the world and Ukraine shows that similar cases at observance of conditions of quality of electric energy in a network happen seldom enough. The estimated
time by analogy with the voltage–time graph (Fig. 17) can be considered 100,000 s
or more [16, 18].
The values of the calculated capacities are much higher than those obtained using
expression (7). This fact prompted us to look for factors that could affect the results.
Fig. 17 Typical characteristic of “voltage–time” arresters 6–35 kV (I n = 10 kA) with the previous
action of two normalized pulses of current capacity of 2000 μs
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S. Shevchenko et al.
Fig. 18 Dependence of active power on the value of phase voltage relative to the highest phase
operating voltage for arresters with a maximum operating voltage of 12 kV, taking into account the
dielectric properties of varistor ceramics: 1—under the action of voltage for 100,000 s; 2—under
the action of voltage for 200,000 s; 3—under the action of voltage for 600,000 s; 4—the limit value
of power that the arrester can dissipate
In the first place, among others, is the fact that the dielectric properties of varistor
ceramics in the area of leakage currents were not taken into account when performing
energy calculations. Taking into account the data of capacitance measurements and
tg δ, the calculations of the energy dissipated in the arrester are obtained, taking into
account the dielectric properties of varistor ceramics. The results of such calculations
are shown in Fig. 18 [16].
Figure 18 clearly shows that the energy values are much lower in the case of taking
into account the dielectric properties of varistor ceramics, and the time before the
loss of thermal balance is significantly increased. Thus, for the type of AR shown
in the figure, it is 600,000 s (166.67 h), which can be considered almost infinity.
The obtained results do not take into account the fact that when performing the
calculations we did not consider the phenomenon of energy radiation from the surface
of the arrester during long-term exposure to the highest operating voltage of the
electrical network during its operation. Experimental studies of the cooling time of
arresters after exposure to high current pulses indicate a rather slow decrease in their
temperature (10–30 min) [19]. However, the cooling time during such tests of the
arrester is much less than that obtained by us during the calculation of the energy
that affects it when working in the area of leakage currents of the VAC.
It should be noted that the obtained values of the calculated energies are poorly
correlated with the values of the powers obtained during the analysis of the substitution circuits of the AR. For example, the value of active power loss of 0.1193 W
was obtained for AR TEL 10/12, and the value calculated according to expression
without taking into account the dielectric properties of varistor ceramics is 1.3528 J s,
which is much higher than that calculated by
Determination of Energy Characteristics for Choice of Surge Arresters
215
Fig. 19 Examples of volt-farad characteristics of varistors of different voltage classes
P=
U 2 ωπr 2 εε0 tgδ
.
h
(14)
Taking into account the dielectric properties of varistor ceramics led to a decrease
in the value of active power to a value of 0.05411 J s, which is significantly less than
that obtained by expression (14). These results indicate the imperfection of the used
array model in the calculation of energy by VAC. This indicates the need to improve
the calculated mathematical model of the arrester in the area of leakage currents of
the VAC and the method of determining the active power that it dissipates.
Further analysis of the experimentally obtained VAC of varistors and literature
sources [21–26] allowed us to draw an important conclusion that the dielectric properties of varistor ceramics affect the amount of active power loss in the arrester only
when the active component of leakage current is very small. This is confirmed by
the volt-farad characteristics of varistors of different voltage classes. Characteristic
volt-farad characteristics of varistors are shown in Fig. 19. [17, 27–33].
The value of voltage in the marking of varistors is equal to the maximum allowable
operating voltage of this type of varistor. As can be seen from Fig. 19 capacitance
of varistors remains unchanged in almost the entire range of operating voltages.
However, when the voltage values approach the maximum allowable values, the
capacitance of the varistors decreases very quickly to zero values. This kind of voltfarad characteristics of varistors is due to the fact that in the structure of varistor
ceramics with increasing voltage there are circuits that conduct electricity. Thus, the
varistor becomes a conductor and the loss of active power in it is determined by the
internal resistance and operating voltage. This property of varistors and arresters in
general necessitates the need to take into account both the dielectric and conductive
properties of varistor ceramics in the analysis of the operation of arresters in the
leakage currents of the VAC. These results demonstrate the need to refine the mathematical model and method for determining the energy that dissipates the arresters
in the area of leakage currents of the VAC.
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Fig. 20 VAC of 5 kV varistor pairs produced by ABB (in range from 0.6 to 0.73 U/Umit)
For such improvement of the specified mathematical model it is necessary to
define size of pressure at action by which the varistor turns to the conductor. A
detailed study of the VAC of varistors in the area of leakage currents allowed us to
obtain the value of this voltage in relative units. To determine this value of voltage,
we give the characteristic VAC obtained as a result of the experiment.
From Fig. 20 it is clear that the value of the ratio of the operating voltage to
the largest operating network, equal to 0.65. The VAC of all studied varistors are
almost identical, and after this value they begin to differ significantly. This type of
VAC was observed for all types of studied varistors of different types, sizes and
manufacturers. Differences of VAC occur because the structure of varistor ceramics
in each specific varistor has an individual appearance. In some varistors, the number
of circuits conducting the active current may be greater or less, so the amount of
current in the leakage current zone of the VAC of AR can vary greatly and has a
unique appearance for each individual varistor [16].
Based on the above, there is a possibility to improve the mathematical model
and method for determining the energy that dissipates the arrester in the leakage
currents of the VAC, taking into account that at a voltage of 65% of the maximum
operating voltage of the arrester network is converted into a conductor. Therefore,
the mathematical model for determining the energy that dissipates the arresters in
the zone of leakage currents of the VAC can be written as follows [16]:
.t
U < 0.65Umlt , w(t) = tgδ ·
u(t) · i(t)dt;
0
.t
U ≥ 0.65Umlt , w(t) =
u(t) · i (t)dt.
0
(15)
Determination of Energy Characteristics for Choice of Surge Arresters
217
This improvement of the mathematical model for determining the energy dissipated by the arrester in the zone of leakage currents will take into account both states
of the arrester, namely when it is a dielectric and when it becomes a conductor, and
develop a method for estimating the ability of arresters The essence of this method is
that to determine the ability to operate the arrester without loss of heat balance, the
energy loss of active power in it for one second must be calculated (16). The obtained
value of active power losses must be less than the allowable losses specified in the
catalog of the varistor manufacturer. In the case when the AR consists of several
varistors, the catalog power data for them should be added to each other [16].
.1
U < 0.65Umlt , w(t) =
tgδ · u(t) · i (t)dt;
0
(16)
.1
U ≥ 0.65Umlt , w(t) =
u(t) · i (t)dt.
0
The use of this method of determining the ability of ARs to do without loss of
heat balance will allow at the stage of selection to obtain its appropriate type. At the
stage of manufacture, the manufacturer, using the above method, can determine in
which cases this type of AR can be used, as well as, if there is information about the
operating conditions to change the necessary design and properties.
According to our calculations, using the above method and the correspondingly
improved mathematical model, for different types of arresters, they must all work
properly in networks with electricity quality that meets GOST. In the case of low
quality electricity in the electricity grid, the situation changes radically. Expression
(16) in this case can be written as follows [16]:
U < 0.65Umlt , w(t) =
.1 .
k
0
U ≥ 0.65Umlt , w(t) =
tgδk · pk (t) =
0
.1 .
k
0
k .
.
0
0
pk (t) =
k .1
.
0
1
tgδk · u k (t) · i k (t)dt;
0
(17)
u k (t) · i k (t)dt.
0
where k—number of the harmonic component.
Expression (17) is an improved mathematical model for calculating the sum of
active power losses, which are due to the harmonic voltage components operating
in the network. This model allows to implement the method of determining the
ability of arresters not to lose heat balance during the entire period of operation in
networks with low quality electricity at the design stage of its protection against
overvoltages. The ability to assess the ability of ARs to withstand the effects of
harmonic oscillations at the stage of selection and design is very important for the
correct choice of energy characteristics of the device. The correct choice of the
218
S. Shevchenko et al.
characteristics of the arrester allows you to significantly increase the operational
reliability of the electrical network.
8 Conclusions
1. The scheme of arrester replacement is given, which allows to take into account
the possibility of resonant phenomena when calculating the modes of arresters
and to estimate the magnitude of leakage currents on their surface, which allows
to estimate the magnitude of moisture discharge voltages and calculate leakage
current through varistors during operation in areas atmospheric pollution.
2. Experimental studies of the electrophysical characteristics of ARs in the assembled state. The obtained results allowed to assert that the dependence of the
electrophysical parameters of the arrester on the frequency of the applied voltage
is weak. This fact allows you to use for the entire range of frequencies considered
a single value of such parameters, which will allow you to choose the arrester
with a margin for energy performance.
3. The obtained mathematical model that reflects the physical processes occurring
in the material of varistors and make it possible to calculate the power lost in the
arrester when exposed to the highest operating voltage of the network. This model
can be called the model of thermal stability of the arrester in the mode of longterm application of operating voltage, which reflects the processes during the
operation of the arrester in the area of leakage currents. This mathematical model
allows you to easily estimate the effect on the AR of any spectral composition
of harmonic oscillations.
4. Calculations performed using the obtained mathematical model of arresters
showed that when changing the maximum allowable mains voltage, the
percentage of harmonic amplitude changes significantly in the presence of which
the thermal balance of arresters is disturbed. This indicates the need to take into
account the presence of harmonic oscillations in the network at the design stage
of the surge protection scheme and the availability of information on the quality
of electricity in the city of installation of arresters. In the absence of such information, it is necessary to study the composition of the equipment of the electrical
power supply system and determine the possibility of harmonic oscillations.
5. Thermal stability of AR depends on its geometric and electrophysical parameters.
As the height of the varistor increases, the power emitted in it decreases, which
is due to the decrease in the capacity and strength of the electric field in the body
of the varistor column, which reduces the loss of polarization, other things being
equal. It should be noted that there is a great influence on the value of the loss of
active power of the varistor radius, this is due to the decrease in current density
under the same voltage.
6. The obtained mathematical model clearly shows ways to increase the magnitude
of active power losses due to geometric and electrophysical parameters of the
arrester. On the basis of the given mathematical model manufacturers of AR
Determination of Energy Characteristics for Choice of Surge Arresters
219
have an opportunity, due to change of technology of production or structure of
impurity, to increase energy which AR dissipates in the course of operation.
7. The results of calculations show that the quality of electricity has a significant
impact on the energy performance of arresters. The presence of harmonic oscillations in the network leads to the probability of violation of the thermal balance
of the arrester and its failure. In cases where the number of harmonic oscillations
is not sufficient to disrupt the thermal balance due to overheating of the arrester,
its operating temperature rises. This in turn can lead to the failure of the arrester
due to the absorption of surge energy at elevated operating temperatures.
8. Experimental studies of VAC of varistors from different manufacturers have been
performed. The obtained experimental VAC determined the voltage at which the
transition of the AR from the dielectric state to the conductor state.
9. Based on an improved mathematical model, a method has been developed to
assess the ability of ARs to maintain thermal balance throughout its service life.
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1729-4061.2015.47123
Heat Power Engineering
Methodology for Designing Precision
Sensors Which Using in Thermal
Conductivity Measurement Systems
Zinaida Burova , Svitlana Kovtun , Leonid Dekusha ,
and Valentina Vasilevskaya
Abstract In this chapter, the heat flux distortions in the sample are considered
and possible methodical errors in the thermal conductivity and thermal resistance
measuring process are evaluated. It is shown that using the results of analytical
studies in designing a measuring system implementing a stationary hot plate method
we can choose such basic thermophysical and geometric factors combinations of
primary heat flux sensors taking into account the characteristics of samples that enable
methodical errors to minimum. Temperature and thermal fields distortions study due
to the heat flux sensors design features shown that it is necessary to minimize the
difference between the thermal conductivity values of sensor sensitive and guard
zones. A proper thermoelectric pair for such sensors is a constantan-nickel couple
that provides temperature and time stability of characteristics simultaneously. A technique for designing precision sensors has been developed. An analytical study results
implementation ensures the optimization sensors design and allows to use them in
modern thermal conductivity measurement systems for non-standard, dimensional
and inhomogeneous samples testing without accuracy losses.
Keywords Thermal conductivity · Heat flux sensors · Methodical error ·
Information measurement system
S. Kovtun (B) · V. Vasilevskaya
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: KovtunSI@nas.gov.ua
Z. Burova
National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine
L. Dekusha
Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_12
223
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1 Introduction
The problem of improving the accuracy of thermal conductivity information measurement systems is closely related to the heat and energy saving. Using of high-quality
heat insulators in construction, industry and energy allows a rational approach to
solving this problem. And the main technical characteristics of thermal insulation
materials are their thermal conductivity and thermal resistance. The study of these
parameters is now standardized in a number of international and regional documents
[1–3]. They provide typical design schemes of measuring cells and regulate the main
geometrical parameters of their elements. According the standard reequipments, the
test sample thickness shall be at least five times less than the length of its edge face
or diameter.
The main measuring elements are heat flux sensors (HFS), and the measuring
cell of the system, according to regulatory documents, may contain one or two such
sensors. The HFS sensitive zone should be located in the central part of its front face.
Its area should be at least 10% and not more than 40% of the total front face area.
For the precision information measurement systems designing the accuracy capabilities of a thermometric device assembled according to a symmetrical scheme as
it regulated by [1–3] have been studied. Distortions of the heat flow in the sample
are considered and an assessment of possible methodical errors the thermal conductivity and thermal resistance measuring by the stationary plate method using HFS
are implemented. The heat flow meter system measuring cell structure is presented
in Fig. 1.
According to this scheme the heat flux q is unidirectional with a uniform surface
density passes through the sample central zone and the sensitive elements zones of
both identical HFS simultaneously. Under idealized measurement conditions, the
entire heat flux generated by the heater passes through the sensors of the upper plate,
the test sample and the sensors of the lower plate without distortion. However, in the
practical implementation of the measurement process there are physical effects that
lead to heat loss, as well as deviations in the reproducibility of the measurement of
temperature, heat flux and geometric parameters. As a result, there are components
Fig. 1 Symmetrical heat
flow meter cell structure
equipped with two HFS
Methodology for Designing Precision Sensors Which Using in Thermal …
225
Fig. 2 Causal diagram of the formation the thermal conductivity measuring results error by the
protected hot plate method
of measurement error, which can be taken into account using statistical methods of
experimental data processing or compensated by appropriate corrections [4–6].
The structural model of the formation the thermal conductivity measuring results
error in a thermometric device with a symmetrical scheme of execution of the thermal
unit is presented in Fig. 2 in the form of a causal diagram [7]. The elements of the
diagram are the input factors and quantities that are directly measured to determine
the thermal conductivity, as well as the physical effects that influence the end result
- the accuracy of measurement [8].
To assess the errors that arise when measuring the thermal conductivity of materials and thermal resistance by the stationary plate method using HFS, in this work
are considered the distortions of the heat flux when passing through the sample. It is
also necessary to study and minimize the methodological errors associated with the
thermophysical and geometric factors of the heat flux sensors themselves [9–11].
2 Measuring Errors in Distribution of Integrated Heat
Flux in the Thermal Conductivity Study
The problem of heat transfer is considered to determine the stationary distribution
of the heat flux density in a flat cylindrical (or square) sample installed between
two identical HFS coaxially with a heater and a cooler (see Fig. 1). Assuming the
independence the thermophysical properties of HFS from temperature and the axial
symmetry, the solution reduces to solving the stationary heat conduction equation in
a cylindrical coordinates system under the third kind boundary conditions [12, 13].
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Z. Burova et al.
In this study, the following influence factors on the methodological error, as shown
in Fig. 2, have been established and researched:
• sample geometric factor—ratio its diameter to height, DS / h S = 3, 4, 5, 10 and
20;
• system geometric factor—ratio the HFS sensitive element diameter to the sample
diameter DHFS /DS in the range from 0 to 1;
• system thermophysical factor—ratio the thermal resistances of the sample and
the HFS RHFS /RS in the range from 0 to 1;
• heat transfer conditions along the lateral surface of the sample—the sample
thermal resistance ratio to the thermal resistance to heat transfer along its lateral
surface, that makes it possible to estimate heat losses through the sample lateral
surface.
To minimize lateral heat losses, it was decided to use active lateral thermal
insulation (see Fig. 1)—a shield where two modes of temperature regime may be
implemented:
I mode TLI = TH (orTLI = TC );
II mode TLI = 0.5 · (TH + TC ).
The distribution of the specific average integral heat flux density q characterizes
one of the methodical error components in measuring the total heat flux passing
through the sample. The calculations results of this distribution studies at the input
q in and at the output qout of the low-conductivity sample for two shield temperature
modes with the influencing factors variations are shown in the graphs Fig. 3.
The results analysis demonstrates that for the II mode of shield temperature
control:
1. the undistorted heat flow zone in the radial direction is much wider compared to
similar sections of the corresponding graphs for the I mode;
2. the component of the methodical error is 10 times smaller.
Therefore, for minimizing lateral heat loses it is advisable to use just II mode
of shield temperature control in the measuring cell of precision devices for thermal
conductivity study.
Next, the analysis of the criteria for choosing the ratio of the sample and HFS
geometric dimensions was carried out. To do this the specific average integral heat
flux density q was calculated for the II mode of the shield temperature control and
fixed HFS thermal resistance value RHFS = 0.01 m2 K/W with a variation the sample
thermal conductivity λS = 0.03, 0.3 and 3 W/(m K) and its geometric factor DS / h S .
The calculations results are shown in the graphs Fig. 4.
The results obtained confirm the requirements for the geometric factors of lowconductivity samples and the size of the HFS sensitive zone determinate in [1–3]:
DS / h S ≥ 5 for λS = 0.03 W/(m K) (see Fig. 4, graph 3). This means that for a fixed
transverse size of the sample, due to the measuring cell dimensions and equal to DS
= 300 mm, the height of the sample should not exceed h S = 50 mm.
Methodology for Designing Precision Sensors Which Using in Thermal …
227
Fig. 3 Specific average integral heat flux density measurement error distribution
Fig. 4 Specific average integral heat flux density measurement error study for low-conductivity
samples
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Z. Burova et al.
However, currently in building there is a tendency to increase the thermal resistance of enclosing structures by increasing the heat-insulating layer thickness to
100 mm or more. In this regard, the geometric factor of heat-insulating and building
materials samples reaches to DS / h S = 3.
Analysis the graphs 1, 2 (Fig. 4) for insulating materials samples with an overall
dimension DS = 0.3 m proves the possibility of studying their effective thermal
conductivity on samples cut from finished blocks without removing the outer layers.
At the same time, to study homogeneous materials without loss of accuracy, an HFS
with a smaller size of the sensitive zone should be used.
The required size of HFS sensitive zone increases for testing samples containing
significant inhomogeneous inclusions. To maintain high accuracy and universalization of the thermal conductivity measuring devises in a wide range of materials it
may be advisable to use HFS contained of two zones made with the same thermophysical characteristics and located coaxially: a central one with a geometric factor
DHFS /DS ∼
= 0.2 for homogeneous materials, and an additional an annular thermopile
connected additively, with a geometric factor corresponding to the maximum value
regulated by the standards DHFS /DS = 0.4.
According to the standards [1–3] requirements the HFS sensitive element area
must be at least 10% and not more than 40% of the total front face area. But in this
case, a distortion of the temperature and thermal fields may occur both in the test
sample and in the HFS due to the inconsistency the thermophysical characteristics
of the sensitive element and the HFS guard zone. This is also one of the important
sources of methodical error in thermal conductivity measuring. A study of these
possible distortions for the second mode of the side shield temperature control was
carried out. The task of heat transfer is solved, where the HFS is considered as an
object installed in an unlimited plate. Graphs of the distribution the measurement
error of thermal conductivity are presented in Fig. 5 received at variation of the
geometric factor h S /DHFS and the ratio of thermal conductivities the HFS sensitive
and guard zones λHFS /λGZ in the practical range from 1.03 to 1.06.
The results of the computational experiment show when designing HFS intended
for use in precision thermal conductivity measuring systems it is necessary to provide
the closest values of the effective thermal conductivity its sensitive and guard zones,
especially for non-standardly thick samples studying (see graphs 1–4 in Fig. 5).
3 HFS Design Types and Technological Parameters
Research
Heat flow sensors used in thermometric measuring systems and devices are converters
of the heat flow value passed through them into an electrical signal. They are
based on the thermocouples thermometric effect and consist of identical thermoelements connected in series according to the generated electrical signal and in parallel
according to the determined heat flow. The characteristics and design of such sensors
Methodology for Designing Precision Sensors Which Using in Thermal …
229
Fig. 5 Estimation the methodical error dependence in thermal conductivity measuring on the
geometric dimensions and characteristics of the sample and HFS
are normalized according to [14]. As it shown in Fig. 6, a multi-element bimetallic
HFS is a battery of thermoelements made as a flat tape-like spiral from the main
thermo-electrode wire 1 wound on a frame electrical insulating tape 4. A thermoelectric couple material 2 is periodically applied on it as a galvanic coating. The transition boundaries from the main thermoelectrode to the coating areas are thermocouple
junctions 3.
The prepared spiral is placed in a special matrix, formed as a disk or a square plate
and filled with electrical insulating material to make it solid. As a molding material
can be used: epoxy compounds for HFS used at moderate and low temperatures,
special purpose epoxy resins for HFS with heat resistance up to 500 K, cements and
enamels if heat resistance is need to be up to 1000 K. Using of various fillers for
epoxy resins and varying their concentrations makes it possible to provide a HFS
thermal conductivity values in a range 0.3…1.2 W/(m K).
To unify further calculations, we introduce several dimensionless complexes
characterizes the main HFS components physical properties:
• form-factor − the ratio of main HFS elements geometric factors:
.=
2 f1 + f2 + f3
= 2 + f 21 + f 31 ,
f1
(1)
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Z. Burova et al.
Fig. 6 Galvanic
thermoelements battery
scheme
where f − cross-sectional area; f i1 = f i / f 1 − thermoelement specific area; indices
i = 1, 2, 3 refer to the thermoelements area: the main thermoelectrode 1, the
electroplated coating material 2 and the molding electrical insulating compound
3;
• specific thermal conductivity λi1 = λi /λ1 , i = 2, 3;
• specific electrical resistance ρ21 = ρ2 /ρ1 ;
• thermoelectrode couple thermoelectric sensitivity α1−2 = α1 − α2 .
It is necessary to predetermine the effective thermal conductivity coefficient of
HFS for using it correctly in a measurement system. Based on the concept that
HFS is a heterogeneous body having a structure with closed inclusions elongated
in the direction of the heat flow with contrasting thermal conductivity, the method
of isothermal and adiabatic fragmentation the HFS into unit cells may be used. The
average results of such calculations give the maximum approximation to the true
value of thermal conductivity. The calculation equations are:
λHFS =
λGZ
2
(
)
−h SP
SP
SP
2. · (. + a − b) · a − hhHFS
· (a − b) − hhHFS
· h HFS
(a − b)2 · (. − b)
h HFS
(
) (
)
·
,
−h SP
−h SP
. · . + h HFS
− b) · a − h HFS
− b)
h HFS (a
h HFS (a
(2)
were a = 2λ13 + λ23 · f 21 ; b = 2 + f 21 .
Equation (2) analysis follows that the HFS effective thermal conductivity depends
not only on the thermophysical properties of its components, but also on its formfactor F and specific height h HFS / h SP , were h SP is the thermoelements spiral height.
Consider to it calculations were made according to Eq. (2) to determine dependence
the thermal conductivities ratio of sensitive and guard zones λHFS /λGZ the ratio
Methodology for Designing Precision Sensors Which Using in Thermal …
231
Fig. 7 The optimal HFS
form-factor determination
h HFS / h SP in practically reasonable ranges. Computation was proceeded in varying
the form-factor F for the value λGZ = 1 W/(m K) ensured in practice using an epoxy
compound filled with powdered corundum. Resulting graphs are presented in Fig. 7.
The results, presented graphically in Fig. 7, show that for the previously obtained
optimal thermal conductivity ratio λHFS /λGZ (the zone located between the two dotted
lines) the HFS form-factor must range in limits F = 400…500.
It should be noted that the form-factor characterizes the HFS as a system with
inclusions only partially and determines the geometric factors ratio in its crosssection along the thickness. The HFS main technical characteristics are its sensitivity
to the measured heat flux density as well as electrical and thermal resistance and
dimensions. To ensure the required values of these characteristics firstly it is necessary
to determine the initial factors of the thermoelement spiral: the main wire diameter,
the galvanic coating optimal thickness, HFS thermoelements filling density, and
its thermophysical properties: thermoelement materials thermal conductivity and
electrical resistivity and the molding compound thermal conductivity.
For predictive calculations of the HFS characteristics the following equation was
obtained that relate their sensitivity to the geometric and thermophysical factors of
all elements included:
• sensitivity to heat flux density formula
Sq =
α1−2 · F · h SP
· (2 + λ21 · f 21 + λ31 · (. − 2 − f 21 ))−1 ;
λ1 · (1 + ρ21 / f 21 ) · f 1
(3)
• heat flux sensitivity formula (volt-watt sensitivity)
SF =
Sq
;
F
(4)
• specific sensitivity formula (sensitivity to heat flux density per unit volume HFS)
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Z. Burova et al.
SV =
Sq
,
(h · F)
(5)
where F—HFS sensitive area for which form-factor F is also valid as:
.=
1
F
=
,
(Z · f 1 )
(n · f 1 )
(6)
were Z and n—the total number of thermocouples and their filling density.
To optimizing the HFS characteristics for the maximum sensitivity to the measured
heat flux density, a formula was found to determine the optimal electroplated coating
specific cross-sectional area ( f 21 )opt :
/
( f 21 )opt =
ρ21
(2 + (. − 2) · λ31 ).
λ21 − λ31
(7)
In the heat flow measurements practice using thermoelectric HFS it is more convenient to use a value inversely proportional to their sensitivity − the conversion factor
K HFS the measured heat flux density q into an electrical signal E HFS . This coefficient
may be calculated, as follows from (3), by the formula:
K HFS =
λ1 · (1 + ρ21 / f 21 ) · f 1
1
=
· (2 + λ21 · f 21 + λ31 · (. − 2 − f 21 )). (8)
Sq
F · h SP · α1−2
If the HFS sensitivity is independent of temperature, K HFS is a constant value and
the surface heat flux density calculation formula is:
q = K HFS · E HFS .
(9)
In general, when measurements are carrying out under the temperature variation
conditions in a wide range of values, it must be taken into account that the HFS conversion factor optimized for maximum
sensitivity is a function of several
(
) temperaturedependent factors: K HFS = f ρ21 (T ); α1−2 (T ); λi (T ); ( f 21 )opt (T ) , i = 1, 2, 3.
Presently the most capable thermoelectric materials used in the HFS manufacture for the thermal conductivity study are constantan-copper and constantan-nickel
couples. Let us consider in more detail the nature of the temperature dependencies
for these materials’ physical characteristics.
• The specific electrical resistance ρ21 (T ) and thermoelectric sensitivity α1−2 (T )
temperature dependences for constantan-copper and constantan-nickel HFS are
shown in Fig. 8.
The graphs Fig. 8 comparison shows that in the HFM operating temperature range
from 250 to 450 K the constantan-nickel thermoelectric couple has a higher resistivity
but a lower temperature dependence the thermoelectric sensitivity and, accordingly,
greater stability.
Methodology for Designing Precision Sensors Which Using in Thermal …
233
Fig. 8 The specific electrical resistance (a) and thermoelectric sensitivity (b) temperature dependences for constantan-copper and constantan-nickel thermoelectric couples
Fig. 9 The thermal
conductivity temperature
dependences for HFS
thermoelectric materials
• The thermal conductivity temperature dependence graphs built on the reference
data for initial thermoelectric materials HFS λ1,2 (T )—constantan, nickel and
copper are shown in Fig. 9.
• The thermal conductivity of the molding compound λ3 depends not only on the
type of filler, but also on its concentration. The studies were carried out for a
molding compound based on a high-temperature special purpose epoxy resin with
two types of fillers: ground fused quartz and powdered corundum with varying
their volume concentration m in a wide temperature range. The thermal conductivity calculating results received by the V. I. Odelevsky method [15, 16] are
presented in Fig. 10.
According to the graphical data, using the corundum filler provides an almost
double value of thermal conductivity—up to 1.2 W/(m K) (Fig. 10b) while maintaining good electrical insulating qualities, therefore the resulting HFS will have less
thermal resistance.
For the corundum filler at its mass concentration m = 0.5 the thermal conductivity
experimental studies were carried out, the results are shown as a dotted line in Fig. 9b.
As we can see, the experimental data are overestimated compared to the calculated
234
Z. Burova et al.
Fig. 10 Thermal conductivity temperature dependences for epoxy resin filled with quartz (a) and
corundum (b)
ones, and the thermal conductivity of such a compound decreases with increasing
temperature. This can be explained by the contacts between the filler particles, which
has a higher but significantly decreasing thermal conductivity depends of temperature
growing compared to epoxy resin.
According to the research results, we choose high-temperature epoxy resin with
corundum filling as the main forming material for the HFS manufacture.
Let us study the temperature dependences of the optimal cross-sectional area
ratio ( f 21 )opt the main thermoelectrode wire and the galvanically besieged coating
of paired thermoelectrode material. Calculations are made according to formula
(7) by varying the form-factor F for two studied couples—constantan-copper and
constantan-nickel. The results are presented in Fig. 11.
• To select the optimal thermoelectrode couple for the HFS manufacture the conversion factor K HFS temperature dependences were researched according to the
formula (8) depending on the specific cross-sectional area f 21 for constantancopper and constantan-nickel thermoelectric couples in the temperature range
from 173 to 473 K. The calculations were carried out using all the data about
the temperature dependences of various parameters presented in Figs. 7, 8, 9, 10
and 11, with the following initial data: form-factor F = 500; constantan wire
initial diameter 0.06 mm, thermoelements packing density 0.7 pcs/mm and HFS
Fig. 11 The optimal specific cross-sectional area temperature dependences
Methodology for Designing Precision Sensors Which Using in Thermal …
235
sensitive zone diameter 120 mm; thermocouple spiral height h SP = 1 mm. The
results are represented as a family of curves in the Fig. 12.
Curve analysis demonstrates a clear advantage of HFS based on a constantannickel couple (Fig. 12b). This pair of materials is characterized by a slight temperature
dependence of the conversion factors at a larger thickness of the galvanized coating.
That is more convenient in practice and provides both temperature and time stability
of its conversion factor. As a result, there is no need to calibrate the measuring
system with working standards before and after each experiment, as recommended
in regulatory documents [1–3].
Taking into account all the presented results of research and calculations two
HFS design types were developed for using in thermal conductivity measurement
systems. They have the same sensitive zone with a diameter of 120 mm made from
a constantan-nickel couple tape-like spiral and differ by a guard zone structure
(Fig. 13).
Fig. 12 HFS conversion
factor temperature
dependences
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Z. Burova et al.
Fig. 13 HFS design types for using in thermal conductivity measurement systems: specialized
HFS (a-type); precision HFS (b-type)
1. specialized HFS (a-type) with a guard zone made by the molding compound;
2. precision HFS (b-type) with a guard zone formed from the same thermoelements
spiral as the sensitive element, to maximize equaling their thermal conductivity
and measurement accuracy increasing.
These sensors are also equipped with temperature sensors based on platinum
resistance thermometers (PRT). One PRT is installed in the center of the each HFM
sensitive zones to measure the temperature difference between the test sample faces,
as it recommended in the regulatory documents. To control the temperature dispersion within the sensor sensitive zone one (a-type) or two (b-type) additional PRT
are installed in the developed HFM designs, as it shown in Fig. 13. The difference
between their signals and the central one is measured and taken into account as a
correction in the measurement information processing by the measuring system electronic unit. This makes it possible to reduce the total error in the thermal conductivity
measuring of the inhomogeneous material test sample.
Further, each HFS from the manufactured batch undergoes its conversion factor
preliminary assessment. For use in the measuring system, a pair of HFS with the
closest possible characteristics would be selected. The final HFS calibration as a part
of the measuring cell is carried out by a ‘two measurements method’. As a result, an
individual static transformation function will be determined for each of two identical
HFS and their signals will transfer to the statistical data processing module [17].
HFS based on the constantan-nickel couple became the basis of the measuring
units installed in the information measuring system for the precision thermal conductivity of solid low-heat-conducting materials studies [18]. Hi-precision b-type HFS
are installed as s part of metrological support system that can be used to certify standard reference materials and transfer standard used to certify working devices for
measuring thermal conductivity in the range from 0.02 to 1.5 W/(m K) at temperature
variety from 240 to 400 K [17–19].
Methodology for Designing Precision Sensors Which Using in Thermal …
237
4 Conclusions
This paper deals with the problem of increasing the accuracy of thermal conductivity
measurement systems. It is shown that during the practical implementation of the
measurement process, physical effects arise that lead to heat losses, as well as deviations in the reproducibility of measuring temperature, heat flux, and geometric parameters. As a result, measurement error components appear, which can be taken into
account using statistical methods for processing experimental data or compensated
by appropriate corrections.
The ways to reduce the methodological errors of the measuring system due to
the adiabatization of the side surface by active thermal insulation are substantiates.
The temperature regime of the later insulation, equal to the arithmetic mean of the
temperatures of the heater and cooler TLI = 0.5 · (TH + TC ), leads to a significant
reduction in the methodological error component.
The main measuring element is a heat flux sensor (HFS), and the measuring cell of
the system, according to regulatory documents, may contain one or two such sensors.
Therefore, it is necessary to minimize the methodological errors associated with the
thermophysical and geometric factors of these sensors themselves.
The ratios of the geometric factors of samples of low thermal conductivity materials and the sensitive zone of sensors are studied. To maintain high accuracy and
universalization of the thermal conductivity measuring devises in a wide range of
materials it may be advisable to use HFS contained of two zones made with the same
thermophysical characteristics and located coaxially: a central one with a geometric
factor DHFS /DS ∼
= 0.2 for homogeneous materials, and an additional an annular thermopile connected additively, with a geometric factor corresponding to the maximum
value regulated by the standards DHFS /DS = 0.4.
Thermophysical properties of materials for manufacturing heat flow sensors have
been studied. It is proposed to introduce the concept of a form-factor for designing
and calculating sensor parameters. Taking into account all the presented results two
HFS design types were developed for using in thermal conductivity measurement
systems. They have the same sensitive zone with a diameter of 120 mm made from
a constantan-nickel couple tape-like spiral and differ by a guard zone structure.
Precision HFS (b-type) with a guard zone formed from the same thermoelements
spiral as the sensitive element, provides the measurement accuracy increasing.
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Methods of Ecologization
of Gas-Consuming Industrial Furnaces
by Using Waste Heat Recovery
Technologies
Nataliia Fialko , Vitalii Babak , Raisa Navrodska ,
Svitlana Shevchuk , and Nataliia Meranova
Abstract Calculation studies have been carried out to improve the environmental
safety of chimneys operation of regenerative-type glass furnaces by using recovery
technologies of the waste heat of exhaust gases using water and air-heating heat
exchangers and chimneys of various designs. In order to improve the environmental efficiency and operational reliability of chimneys, it is proposed using the
thermal method of bypassing part of the exhaust gases from regenerators past the
heat recovery equipment in heat recovery technologies. The effectiveness of this
method for improving the dispersion in the surface layer of such harmful emissions
of furnaces as sulfur dioxide, nitrogen oxides and dust when using the proposed
heat recovery systems has been analyzed. It is established that the application of this
method provides at the chimney orifice a relative increase in the temperature of gases
by 1.13–1.16 times, and their velocity by 1.05–1.09 times for systems with air-heating
heat recovery equipment and by 1.6–1.8 and 1.2–1.3 times respectively, when using
water-heating heat recovery equipment. The maximum ground-level concentrations
of Cm of these harmful emissions in the environment were determined. It is shown
that the lower values of Cm correspond to higher ambient temperature, heat recovery
systems with air-heating heat recovery equipment, chimneys with better heat insulation properties of the shell, most of the bypass of hot gases from the furnace regenerators past the heat recovery equipment. In this case, the efficiency of the impact of gas
bypass method is higher for heat recovery systems with water-heating heat recovery
equipment. The results obtained can be used in the development of energy-efficient
technologies for gas-consuming heat plants of technological purposes.
Keywords Industrial furnaces · Heat recovery systems · Chimneys · Harmful
emissions from glass furnaces · Ground-level concentration · Environmental
efficiency
N. Fialko (B) · R. Navrodska · S. Shevchuk · N. Meranova
Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine
e-mail: nmfialko@ukr.net
V. Babak
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_13
239
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N. Fialko et al.
1 Introduction
The operation of modern fuel-consuming heat plans for various technological
purposes (for example, glass furnaces, smelting, kilns and other furnaces) is currently
characterized by two main trends: the creation of competitive products and the
increase in efficiency of fuel using [1–3]. This increase is usually realized by
optimizing combustion processes [4], improving the processes of regeneration and
recovery of combustion products [3, 5], creating heat recovery technologies of waste
heat [6–8], etc. The realization of these trends may be associated with the deterioration of the environmental safety of these plants [2, 3, 9]. Namely: technological
improvements in the production of more competitive products can lead to increased
levels of harmful emissions and increase the chemical aggressiveness of exhaust
gases; increasing fuel efficiency along with a decrease in fuel consumption and
temperature potential of emissions can lead to deterioration of the chimney modes
regarding the dispersion of harmful substances contained in the combustion products
of these heat plants.
Toxicity of harmful emissions and the level of their concentration in flue gases
depends significantly on the technological purpose of industrial furnaces. In particular, for glass furnaces, nitrogen and sulfur oxides and technological dust are typical
harmful emissions [10]. Among these emissions, the greatest atmospheric pollution
from glass furnaces (up to 80% and more) is caused by nitrogen oxides (NOx):
nitrogen monoxide (NO) and its dioxide (NO2 ) [10].
This work is devoted to the development and study of the effectiveness of measures
to improve the environmental safety of gas-consuming industrial furnaces in the
realization of heat recovery technologies. These technologies realize the useful use
of part of the waste heat of exhaust gases to reduce fuel consumption by heating the
combustion air and meeting the needs of enterprises operating industrial furnaces in
heat energy for heating, technology and hot water supply.
For useful use of waste heat from flue gases, heat recovery systems of various
schematic designs can be applied: using air-heating equipment (regenerators and
recuperators), as well as using heat recovery systems for waste heat from exhaust
gases after regenerators. Volumes of this heat are quite significant because of the
usually high temperature of the flue gases after the regenerators (400–450 °C and
can reach 700 °C [5]).
Recovery of exhaust gases waste heat after the regenerators can be carried out
using water-heated and air-heated heat recovery exchangers of various schematic
and structural designs [7].
Traditionally, water-heated heat recovery equipment was used to recovered heat
of the exhaust gases after the regenerators. Analysis of experience in operating such
equipment and our own experience in implementing heat recovery systems with
water-heated heat equipment behind glass furnaces indicates that this equipment
usually provides recovery of about 30% of the technically possible waste heat potential of the furnaces. The small volume of waste heat used is mainly due to the
insignificant volumes and seasonal needs of enterprises operating industrial furnaces
Methods of Ecologization of Gas-Consuming Industrial Furnaces …
241
for heat energy in the form of hot water. Therefore, recently, for useful use of waste
heat, so-called end recuperators have been applied. These recuperators are used to
preheat cold air before it enters the furnace regenerators [7].
The type of heated heat-transfer-agent used in the heat recovery system and
its temperature characteristics have a significant influence on the temperature and
velocity of the exhaust gases [7] in the chimney, and, consequently, on the conditions of dispersion in the environment of harmful emissions of industrial furnaces.
These indicators are also significantly affected by the type of chimney (metal, brick,
reinforced concrete, etc.) and its design features.
In addition to the indicated operating characteristics and structural differences
in chimneys, the environmental safety of the environment significantly depends on
the toxicity of the harmful emissions themselves and their mass concentrations in
exhaust gases. These characteristics, as already noted, depend significantly on the
purpose and type of industrial furnace.
2 The Purpose and Methods
The purpose of this work is to improve the conditions for dispersion of harmful
emissions of gas-consuming glass furnaces based on the use of effective means of
their ecologization with the application of modern technologies for flue gas waste
heat recovery.
Analysis of the operating experience of these furnaces indicates their high energy
intensity, and a significant potential of waste heat [6, 9] and the content of harmful
substances in the combustion products.
For many years, the authors have been studying the composition of flue gases of
regenerative-type glass furnaces. Gas analysis was performed during commissioning
tests of prototypes of the developed heat recovery equipment [7] at different glass
production plans in Ukraine and the Russian. To perform the measurements, the
certified methods of the commissioning services of these plants were used. Research
was carried out on furnaces designed for the production of different glass containers
and for the melting of medical glass. The studies conducted included:
• determination of the content of oxygen, carbon monoxide and dioxide, sulfur
oxides, the total concentration of nitrogen oxides NOx ;
• determination of the quantitative and qualitative composition of dust.
The results obtained indicated a wide range of changes in both qualitative and
quantitative composition of the flue gases of glass furnaces, including to harmful
emissions. Thus, the content of sulfur dioxide in flue gases ranged from 90 to
1150 mg/m3 , nitrogen oxides from 1100 to 4000 mg/m3 , technological dust was
120–220 mg/m3 for furnaces producing glass containers and 570–620 mg/m3 for
melting medical glass. The flue gases temperature after the regenerators was 330–
480 ºC. Carbon oxides were not detected in most measurements, but there were single
cases with fixing the concentration of carbon oxides in the exhaust gases 750 mg/m3 .
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N. Fialko et al.
Fig. 1 Schematic circuit of the heat recovery system of an industrial regenerative furnace using
the heat method of partial flue gas bypass past the heat recovery equipment
To improve the conditions for the dispersion of harmful emissions contained in
the flue gases of gas-consuming boiler plants, heat methods of heat and humidity
treatment of flue gases after heat recovery are used [11]. Of these methods, the bypass
method is the most suitable for furnaces. It consists in passing part of the exhaust
gases from the furnace regenerators past the heat recovery equipment to increase the
temperature and velocity of the flue gases in the chimneys.
The realization of this method is always assumed when designing exhaust gases
heat recovery systems. The schematic circuit solution for applying the bypass method
for industrial regenerative furnaces is shown in Fig. 1.
It should be noted that the application of the bypass method worsens the heat
performance of the heat recovery systems used due to a decrease in the use volume
of waste heat. The level of this deterioration is directly related to the bypass part of
exhaust gases. Therefore, the use of this method can be justified only to the extent
that it provides the operating modes of chimneys necessary to improve the dispersion
of harmful flue gas emissions.
The paper considered water-heating and air-heating heat recovery equipment
for glass furnaces of regenerative type, developed at the Institute of Engineering
Thermophysics of NAS of Ukraine [10].
To assess the efficiency of application of the heat method of bypassing exhaust
gases, their temperatures tori and velocities Vori in the orifice of different chimneys at
various modes of operation of water- and air-heating heat recovery equipment were
calculated. Traditional chimneys (reinforced concrete, brick, metal) and a reinforced
concrete chimney with three inserted gas waste trunks were considered (Table 1).
At the same time, the part of bypassed gases in the gas mixture before the chimney
varied from 0 to 40%.
Methods of Ecologization of Gas-Consuming Industrial Furnaces …
243
Table 1 Characteristics of chimneys
Chimney type
Values of geometric parameters
Height, m
Diameter at the
orifice of chimney,
m
Diameter inserted
gas waste trunk, m
Wall thickness,
orifice-base,
10−2 m
Reinforced
concrete
55
4.2
–
50
Reinforced
concrete with
inserted gas waste
trunks
55
4.8
1.8
60
Brick
55
3.1
–
60
Metal
55
3.1
–
10
The maximum ground-level concentrations Cm in the environment of such harmful
emissions from glass furnaces as oxides of sulfur, nitrogen and technological dust
were determined. The calculation studies were based on the use of the classical
formula (1).
Cm =
AM Fmnη
,
√
H 2 3 V1 .t
(1)
where A is a coefficient depending on the temperature gradient of the atmosphere;
M is the mass of a harmful substance that is emitted into the atmosphere per unit
time, g/s; coefficient F, which takes into account the sedimentation rate of harmful
substances in the atmosphere for gases; η is the coefficient of terrain influence, m
and n are dimensionless coefficients, taking into account the conditions for the exit
of flue gases from the chimney orifice; H is chimney height, m; V1 is the volume
flow of waste gases, m/s; .t is the difference between the temperature of the flue
gases from the chimney orifice tori and the ambient temperature tamb , °C.
When calculating the maximum ground-level concentrations Cm of the considered
harmful substances, the coefficient A was taken as the maximum (200). The coefficients of the settling rate of harmful substances in the atmosphere and the terrain were
taken equal to one, the calculated dimensionless coefficients, taking into account the
conditions for the exit of flue gases from the chimney orifice, were m = 0.98–1.23
and n = 1.14–1.8.
The initial data for performing computational studies, taken on the basis of an
analysis of the characteristics of glass furnaces, the heat recovery technologies used,
as well as the most typical types of harmful emissions and their concentrations, are
presented in Tables 1 and 2.
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Table 2 Initial data
Indicator values
Name of the indicator, dimension
Process entrainment concentration in flue gases, mg/m3
Sulfur dioxide
1150
Nitrogen oxides
1300
300
Dust
Gas consumption per furnace,
m3 /h
2350
Excess air ratio in flue gases
1.4
Flue gas consumption, kg/s
12.24
Flue gas temperature after furnace regenerators, °C,
400–450
Initial temperature of heated air, °C
−20 to + 20
Heating surface area of air-heating heat exchanger, m2
480
Initial temperature of heated water during the heating period with ambient
temperature from −20 to +10 °C, °C
70–35
Heating surface area of water -heating heat exchanger, m2
440
Consumption of pollutants per unit of combusted gas, mg/s per 1
m3
Sulfur dioxide
4.7
Nitrogen oxides
5.3
Dust
1.2
3 Results
The main obtained results of calculations of heat and aerodynamic modes of operation
of different types chimneys of with a heat recovery system (HR) and without it when
using the developed air-heating heat exchangers (end recuperators) are shown in
Fig. 2.
Analysis of the research results shows that the use of heat recovery systems with
air-heating heat exchangers, the chimney type and heat insulation properties of its
casing have a significant influence on the cooling processes of exhaust gases in
chimney, hence on their temperature tg ori and velocity Vg ori . At the same time, the
values of these quantities increase with rise in the ambient temperature tamb due to
a decrease of heat losses from the surface of the chimney casing. The use of heat
recovery systems causes a significant decrease in all of the studied values.
Taking into account the significant cooling of exhaust flue gases when using their
heat recovery systems, the temperatures of the inner surface ts ori of the chimney
orifice were calculated. The data obtained indicate that the dew point of the exhaust
gases in the chimney did not change and amounted to 54 °C with and without the
heat recovery system. Moreover, the use of heat recovery systems for all considered
chimneys, except for metal ones, does not lead to a decrease in temperature ts ori
below the flue gas dew point. For the metal chimney, such a decrease occurred and
corresponded to the ambient temperature below 0 °C. This fact indicates the danger
Methods of Ecologization of Gas-Consuming Industrial Furnaces …
245
Fig. 2 Dependences of
ambient temperature of the
velocity Vg ori and
temperature tg ori of exhaust
gases in the chimney orifice
during operation of
air-heating heat exchangers
(1–4) and without them (5–8)
for the chimney. 1, 5—brick;
2, 6—reinforced concrete; 3,
7—metal; 4, 8—three-trunks
of increased corrosion of the metal chimney, even of alloy structural steel, given the
content of corrosive substances in the exhaust gases.
As for the velocity of exhaust gases in the chimney, its change (namely, decrease)
was insignificant depending on the ambient temperature for each of the studied
chimney options and with the use of a heat recovery system and without it. Without
a system of HR, the velocity value of flue gas was determined mainly by the size
of the cross sections for their passage and the type of chimney. At the same crosssections of chimneys, the decrease in velocity with increasing ambient temperature
was somewhat perceptible only for metal chimneys (see Fig. 2).
Application of the heat recovery system with air-heating exchangers has a more
significant effect on reducing the velocity of exhaust gases in the chimney. If it is
possible to change the cross-section (when using a three-trunks chimney) in the case
of using the HR system for flue gas evacuation can be used two trunks. This fact
allows to keep the calculated values of the velocity at the outlet from the chimney
orifice at the level (or higher) values of this velocity before the installation of heat
recovery equipment.
The change in flue gas temperatures depending on the ambient temperature tamb
was more significant than for the velocity in the considered variants of the furnace
operation. In three-trunks chimneys, characterized by the smallest heat loss to the
environment (due to the heat insulation properties of the chimney casing and the
air gap between the three trunks), the flue gas cooling was insignificant for each of
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N. Fialko et al.
the options considered. The temperature difference .t between the inlet and outlet
of flue gases from the chimney did not exceed 2 °C throughout the entire range of
ambient temperature (from −20 °C to +20 °C).
Brick and reinforced concrete single-trunk chimneys with relatively high heat
insulation properties of the casing have somewhat worse indicators regarding the
cooling of flue gases. For these chimneys, the maximum value of the indicated
difference .t in the same temperature range of the ambient temperature was 10 and
18 °C, respectively, for the operation of the furnace with a heat recovery system and
18 and 34 °C without it. Higher flue gas temperatures tori during the warm period of
the year are explained by:
(a) an increase in flue gas temperature at the chimney inlet due to a decrease in
the efficiency of regenerators and recuperators with an increase in the initial
temperature of the heated air;
(b) a decrease in heat losses from the chimney shell body at this time of the year.
As for the metal chimney, here the cooling of gases during their passage through
it was the largest for both considered options of furnace operation. The maximum
value of the temperature difference .t between the inlet and outlet of gases from the
chimney when using the heat recovery system was 27 °C and without it 53 °C. At
the same time, for all chimneys, the lowest value of the gas temperature tg ori at the
chimney orifice corresponded to the lowest of the considered ambient temperatures
(−20 °C). For a metal chimney, the value of tg ori was 196 °C in the case of using the
HR system and 397 °C without it.
Calculated research of chimneys operation modes of glass furnaces when they are
equipped with water-heating heat exchangers has also been carried out. The main
results are shown in Fig. 3.
Analysis of the obtained results concerning the temperature tg ori and velocity Vg ori
of flue gases at the outlet from the chimney orifice for water-heating heat exchangers
showed the same trends in changing these values depending on the design features
of the chimney, as for air-heating heat exchangers. However, the absolute values of
the studied quantities compared to air-heating heat exchangers were lower due to
deeper cooling of exhaust gases in water-heating heat recovery equipment.
Different, namely opposite, was also the nature of the change in the temperature
tg ori and flue gas velocity Vg ori with an increase in the ambient temperature tamb .
Thus, in the case of application of air-heating heat exchangers, the values of tg ori and
Vg ori increase with the growth of tamb , and for water-heating heat exchangers, on the
contrary, these values decrease. This is due to a decrease in the temperature of the
heated heat-transfer-agent (water) with an increase in the ambient temperature tamb .
Regarding the character of changes in the surface temperature of ts ori at the
chimney orifice during the growth of ambient temperature when using water-heating
heat exchangers, it corresponds to the character of changes in tg ori and Vg ori for the
three considered types of chimneys, except for the metal one. For a metal chimney,
there is an increase in ts ori with a growth in tamb . This is due to the predominant
influence of the tamb temperature on the change in ts ori as compared to the influence
of the gas temperature tg ori .
Methods of Ecologization of Gas-Consuming Industrial Furnaces …
247
Fig. 3 Dependences of
ambient temperature of the
velocity Vg ori and
temperature tg ori of exhaust
gases in the chimney orifice
during operation of
water-heating heat
exchangers (1–4) and
without them (5–8) for the
chimney. 1, 5—brick; 2,
6—reinforced concrete; 3,
7—metal; 4, 8—three-trunks
The calculated studies showed that the dew point of water vapor in the flue gases in
the chimney for both variants of application of heat recovery systems did not change
and was 54 °C, which corresponded to the dew point of flue gases without the use of
heat recovery system.
The obtained results also indicate that for both considered heat recovery systems
using water or air-heating heat exchangers, the most efficient chimneys in terms of
indicators of heat and aerodynamic of their operation modes are chimneys with
inserted gas waste trunks. Chimneys made of brick are characterized somewhat
worse indicators by an even worse from reinforced concrete one, and the use of
metal chimneys for evacuation of exhaust gases is problematic without the use of
systems of heat protection against condensation formation.
Using the results obtained, calculations of maximum ground-level concentrations
Cm of characteristic harmful emissions of flue gases from glass furnaces for different
chimneys when using the considered heat recovery systems were performed. For
systems with air-heating heat recovery equipment, the values of these concentrations
at different ambient temperatures tamb throughout the year and depending on the part
χ of gases bypassing past the heat recovery equipment are shown in Fig. 4.
As can be seen from the presented data, the values of maximum ground-level
concentrations Cm depend significantly on the type of chimney used. The lowest Cm
values correspond to chimneys with better heat insulation properties, a larger part of
χ hot gas bypasses from furnace regenerators and higher ambient temperature tamb .
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Fig. 4 Dependence of maximum ground-level concentrations of sulfur dioxide, nitrogen oxides
and dust on bypass part for different chimneys at minimum winter temperature of −20 °C (a)
and maximum summer temperature of +20 °C (b) and when using air-heating heat exchangers:
1—brick; 2—reinforced concrete; 3—metal; 4—three-trunks chimney
Methods of Ecologization of Gas-Consuming Industrial Furnaces …
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The obtained data also indicate that when the part of bypassed gases increases
from 0 to 40%, the value of maximum ground-level concentrations Cm decreases
within 5–7% for sulfur oxide and nitrogen oxide emissions, and within 13–15% for
dust. In this case, lower values correspond to chimneys with better heat insulation
properties of their casing.
As for influence of ambient temperature tamb on Cm value, according to the results
of performed studies, it has lesser influence than χ value. The decrease of Cm values
by 1–3% is observed with increase of tamb from −20 to +20 °C.
The results of the performed studies of the maximum ground-level concentrations
for water-heating heat exchangers are shown in Fig. 5.
Analysis of the obtained results shows that, as in the situation with the use of
air-heating heat exchangers, the values of maximum ground-level concentrations Cm
of all the considered harmful emissions (sulfur dioxide, nitrogen oxides and dust)
decrease with increasing the gas bypass part past the water-heating heat exchangers.
However, for heat recovery systems with water-heating equipment, the concentration levels of all harmful emissions Cm are higher due to deeper cooling of exhaust
gases in these systems, which reduces the efficiency of the applied heat method at
the same bypass part χ of exhaust gases. Under these conditions, the application
of the bypass method is more effective than when using air-heating heat recovery
exchangers. The obtained data indicate that with the part of bypassed gases increases
from 0 to 40%, the value of maximum ground-level concentrations Cm decreases by
about 19–25% for sulfur oxide and nitrogen oxides emissions and 21–36% for dust.
In this case, lower values also correspond to the chimneys with good heat insulation
properties of the chimney shell.
The influence of the ambient temperature tamb on the levels of ground-level concentrations Cm decrease for all types of atmospheric pollution under consideration is
also different. The obtained data indicate that the values of maximum ground-level
concentrations Cm of the considered harmful emissions increase by 9–19% with the
increase in tamb . This is also explained by deeper cooling of the flue gases with an
increase in temperature tamb , which corresponds to a decrease in the temperature of
the heated water.
4 Conclusions
1. A set of computational studies to improve the environmental safety of chimneys operation of regenerative type glass furnaces when using technologies
for recovery the waste heat of exhaust gases using the developed water and
air-heating equipment was performed.
2. The mode parameters (temperature tg ori and velocity Vg ori of flue gases) at the
outlet from the chimneys orifice of different types when using heat recovery
systems with water-heating and air-heating equipment were investigated. It is
shown the use of these systems worsens the technological modes of chimneys
due to a decrease in indicated parameters. Systems with the considered air-heating
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Fig. 5 Dependence of maximum ground-level concentrations of sulfur dioxide, nitrogen oxides
and dust on bypass part for different chimneys at minimum winter temperature of −20 °C (a)
and maximum summer temperature of +10 °C (b) and when using water-heating heat exchangers:
1—brick; 2—reinforced concrete; 3—metal; 4—three-trunks chimney
Methods of Ecologization of Gas-Consuming Industrial Furnaces …
251
heat exchangers, compared with systems with water-heating equipment, are characterized by higher values of tg ori and Vg ori at the outlet from the chimneys orifice
under the same initial conditions has been established.
3. An analysis of the effectiveness of using the method of bypassing exhaust
gases past the heat recovery equipment in heat recovery systems to improve
the operating conditions of chimneys was carried out. It has been established
that this method, with an increase in the bypass part from 0 to 40%, provides
in the chimney orifice a relative increase in the temperature of gases by
1.13–1.16 times, and their velocity by 1.05–1.09 times for systems with airheating heat exchangers and 1.6–1.8 and 1.2–1.3 times, respectively, when using
water-heating heat recovery equipment.
4. The maximum ground-level concentrations Cm in the environment of characteristic harmful emissions of glass furnaces, such as sulfur oxides, nitrogen and
technological dust when using the bypass method for different heat recovery
systems and chimneys are determined. It is shown that the lower values of Cm
correspond to heat recovery systems with air-heating heat exchangers, chimneys
with better heat insulation properties of the shell, a larger part of bypass of hot
gases from the furnace regenerators past the heat recovery equipment and higher
ambient temperature. In this case, the efficiency of the bypass method is higher
for heat recovery systems with water-heating heat recovery exchangers.
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11. Fialko, N.M., Navrodska, R.O., Shevchuk, S.I., Stepanova, A.I.: Improvement of environmental
conditions by applying heat recovery technologies of boiler plants. Sci. Bull. Natl. Min. Univ.
6, 148–152 (2021). https://doi.org/10.33271/nvngu/2021-6/148
Simulation Modeling of Vapor
Compression Refrigeration Unit
Temperature Modes
Andrii Bukaros , Oleg Onishchenko , Alexander Herega ,
Herman Trushkov , and Konstantin Konkov
Abstract Based on the analysis of the processes occurring in the refrigeration
chamber and evaporator of the vapor compression refrigeration unit with smooth
control of the compressor’s cooling capacity, a mathematical description of the
boiling temperature and cooling object temperature dynamics has been obtained.
The structural scheme of the simulation model of the vapor compression refrigeration unit as a control object has been offered. The main difference of the developed
model is taking into account changes in time constants and transfer coefficients of
the evaporator and the cooled object depending on the ambient temperature and the
controller settings which will allow developing an energy efficient control strategy for
such units. As an example, at different ambient temperatures, a study of the dynamic
properties of a vapor compression refrigeration unit with a cooling capacity of 14 kW
of refrigerated vessel intended for the transportation of citrus has been carried out.
The analysis of the influence of evaporator icing on the dynamic parameters of the
investigated refrigeration unit has been carried out. A subsystem for determining the
evaporator thermal conductance coefficient in the conditions of ice plaque formation
on the heat exchange surface according to the relevant temperature and electrical
sensors data has been developed. Based on this subsystem, an algorithm for diagnosing evaporator icing with a possible automatic start of the defrosting process
has been proposed. Ways to improve the simulation model of vapor compression
refrigeration unit to increase its energy efficiency, remote monitoring and predictive
diagnostics by means of digital twins have been outlined.
Keywords Vapor compression refrigeration unit · Control · Simulation model ·
Thermal conductance coefficient · Evaporator icing · Digital twin
A. Bukaros (B) · O. Onishchenko · A. Herega · H. Trushkov · K. Konkov
Department of Electrical Engineering and Missile and Artillery Weapons Systems, Odesa Military
Academy, Odesa, Ukraine
e-mail: andrey.bucaros@gmail.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_14
253
254
A. Bukaros et al.
1 Introduction
Vapor compression refrigeration units (VCRU) are used in almost all spheres of
human activity: in everyday life, industry, transport, medicine, military equipment—
for refrigeration and food storage, preparation of cold water, ice, comfortable and
technical air conditioning, gas liquefaction, cooling technological units, weapons
and other applications and are notable consumers of electricity [1, 2]. It is impossible to ensure high quality of VCRU maintenance (stabilization of the cooling set
parameters, energy efficiency, diagnostics) without the use of modern automation
tools [3–5].
One of the tasks of VCRU automation involves the development and implementation of energy efficient cooling capacity control systems. The creation of such
systems is quite complicated without taking into account the dynamic properties of
VCRU as a control object. Such consideration is possible by means of simulation
and use of thermodynamics laws.
An analysis of recent work on VCRU modeling shows that:
• the most effective and convenient representation of VCRU mathematical models is
carried out by means of the structural schemes containing transfer functions with
the concentrated and/or distributed parameters, in object-oriented environments
of simulation modeling [6, 7];
• in the design of control systems, cooled objects and elements of VCRU are
traditionally presented as objects with concentrated parameters without taking
into account changes in thermal conductance coefficients and environmental
parameters, which calls into question the adequacy of such models [8–10].
The urgency of developing simple and effective for practical application (in the
creation of control systems) of simulation models is due to the widespread using
VCRU, and increasingly stringent energy efficiency requirements for such units
[11–15].
In general, the automation of VCRU involves the solution of the main task and
a number of auxiliary tasks. The main task [16, 17] is to stabilize the temperature
of the cooling object under the influence of external perturbations. Auxiliary tasks
[16–19] include tasks to increase energy efficiency, stabilize condensing pressure, fill
evaporators, protect against dangerous conditions etc. On the basis of the automation
main task analysis we will carry out an estimation of the basic thermodynamic
processes, a cycle of work and features of boiling temperature stabilization in VCRU.
This analysis is a prerequisite for creating a simulation model of VCRU.
Simulation Modeling of Vapor Compression Refrigeration Unit …
255
2 Thermodynamic Analysis of the Work Cycle of Vapor
Compression Refrigeration Unit
In Fig. 1 a block diagram of a single-stage VCRU is shown. From the block diagram
it follows that for removing heat QL from the cooling chamber (CC) by using the
evaporator, the cooling machine (CM) consumes electrical energy W e . In this case,
heat Q0 is supplied from the environment to the CC through the barriers, and heat
QH is removed from the system to the environment by using the condenser.
The main energy and thermodynamic relations that characterize the operation of
the VCRU for stabilizing the temperature t cc in the CC (cooling object) per unit time
are:
⎧
Q̇ H = Q̇ L + Ẇc = kcon · Fcon · (t H − t0 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪ Q̇ L = Q̇ cc−ev + Q̇ ev = a · t L + c
⎪
⎪
⎨ Q ev = Cev · .t L
,
(1)
⎪
Q̇ cc−ev = Q̇ 0 + Q̇ cc = kev · Fev · (tcc − t L )
⎪
⎪
⎪
⎪
⎪
Q cc = Ccc · .tcc
⎪
⎪
⎪
⎩
Q̇ 0 = kcc · Fcc · (t0 − tcc )
where Q̇H —rate of heat removal from the condenser to the environment, W; Q̇L —
rate of heat removal from the cooling object (cooling capacity), W; Ẇ c —compressor
compressive power, W; Q̇cc-ev —rate of heat removal from the cooling object (CC)
to the evaporator, W; Qev , Q̇ev —respectively heat (J) and its removal rate (W) from
the evaporator and refrigerant, W; Qcc , Q̇cc —respectively, heat (J) and its removal
rate (W) from CC; Q̇0 —heat flow rate from the environment to CC, W; t H , t L —
respectively condensation and boiling point, °C; t 0 —ambient temperature, °C; t cc —
temperature in CC, °C; kev · Fev , kcon · Fcon —respectively the thermal conductance
Fig. 1 Block diagram of VCRU
256
A. Bukaros et al.
coefficients of the evaporator and condenser, W/°C [20]; kcc · Fcc —thermal conductance coefficient of CC walls, W/°C; C ev —total heat capacity of the evaporator
and boiling refrigerant, J/°C; C cc —CC heat capacity [21], J/°C; a, c—compressor
constants [21].
From the analysis of Eq. (1) it follows that the task of stabilizing the cooling
object temperature can be solved by ensuring the equality of heat flow rates Q̇cc-ev =
Q̇0 . In this equation, both heat flow rates depend significantly on several factors, as
shown in (1).
The heat flow rate Q̇0 is proportional to the ambient temperature t 0 , which may
have diurnal and seasonal fluctuations. VCRU used in transport and military equipment can also undergo significant changes in ambient temperature during the crossing
of climatic zones. In view of this, it can be argued that VCRU often operate in conditions of constant changes in heat load from the environment, which in turn can lead
to significant changes in their energy efficiency.
The rate of heat removal from the cooling object to the evaporator Q̇cc-ev is proportional to the thermal conductance coefficient of the evaporator kev · Fev , which can also
change during the operation of the VCRU, for example, when a snow coat appears
on the surface of the evaporator. In such a situation, the delay in cooling process, the
increase in energy losses in VCRU and the decrease in its energy efficiency will be.
These examples show the need for energy efficient control of VCRU cooling
capacity, which can be achieved by creating an appropriate simulation model that will
take into account changes in internal (kev · Fev ) and external (t 0 ) dynamic parameters.
3 Simulation Model of the Temperature Modes Control
System of the Vapor Compression Refrigeration Unit
To create such a model, we differentiate Eq. (1) by time τ and make elementary
transformations. As a result, we obtain a system of differential equations:
⎧
⎪
⎪
⎨
Cev
dt L (τ )
kev · Fev · tcc (τ ) − c
·
+ t L (τ ) =
kev · Fev + a
dτ
kev · Fev + a
.
C
dt
kev · Fev · t L (τ ) + kcc · Fcc · t0 (τ )
(τ
)
⎪
cc
cc
⎪
⎩
+ tcc (τ ) =
·
kev · Fev + kcc · Fcc
dτ
kev · Fev + kcc · Fcc
(2)
The obtained Eq. (2) describe the temperature processes in the evaporator and CC
provided the compressor work. With on–off control of the VCRU cooling capacity
for some time the compressor is switched off, which leads, firstly, to exclude the
coefficients a and c from Eq. (2), and secondly, to significantly reduce the thermal
conductance coefficient of the evaporator kev · Fev due to the lack of refrigerant
boiling process. As a result, during the period of stopping the compressor, Eq. (2)
take the form:
Simulation Modeling of Vapor Compression Refrigeration Unit …
⎧
Cev
dt L (τ )
⎪
⎪
·
+ t L (τ ) = tcc (τ )
⎨k
dτ
ev_s · Fev
,
dtcc (τ )
Ccc
⎪
⎪
⎩
·
+ tcc (τ ) = t0 (τ )
kev_s · Fev + kcc · Fcc
dτ
257
(3)
Where kev_s · Fev —thermal conductance coefficients of the evaporator when the
compressor is switched off, W/°C.
Equations (2) and (3) allow building on their basis a model of evaporator and
CC temperature modes in on–off control of VCRU cooling capacity [22]. However,
many VCRU manufacturers (Carrier Transicold, Daikin, Danfoss, Copeland, Haier,
etc.) provide smooth control of cooling capacity, especially in the range of positive
refrigerant boiling points, by changing the speed of the compressor built-in electric
motor. In this case, Eq. (2) does not take into account the dependence of the constant
a and c of the compressor and the thermal conductance coefficient of the evaporator
kev · Fev on the compressor motor shaft rotation frequency.
Changing the rotation frequency . of the VCRU compressor motor leads to a
proportional change in the refrigerant mass flow rate mR through the compressor,
which in turn causes, with constant other parameters of the refrigeration cycle
(enthalpy, temperature, pressure, etc.), proportional changes in cooling capacity
QL and compressor compressive power Ẇ c as the specific cooling capacity qL and
compression work wc , will not change:
.
mR
Q L = qL · m R
Ẇc = ẇc · m R
=
=
=
= Ẇc∗ .
.n
m Rn
Q Ln = q L · m Rn
Ẇcn = ẇc · m Rn
(4)
where Ẇ * c —relative compressive power of the compressor, the index n indicates the
nominal value of the corresponding quantity.
Analysis of expression (4) taking into account (1) allows us to state that the
coefficients a and c of the compressor performance characteristics in a rather small
range of boiling temperatures proportionally depend on the relative compressive
power of the compressor Ẇ * c :
a ' = a · Ẇc∗ ; c' = c · Ẇc∗
(5)
The dependence of the thermal conductance coefficient of the evaporator kev · Fev
on the change in the refrigerant mass flow rate, as shown by studies [23], can be
described by a power function with exponent k, which depends on the refrigerant
thermophysical properties:
) ( )k
(
'
kev · Fev = kev_s · Fev + kev · Fev − kev_s · Fev · Ẇc∗ .
(6)
As mentioned above, the exponent k depends on the using refrigerant. For
example, for R22 and R404A, k is 1.45 and 1.3, respectively [23].
258
A. Bukaros et al.
Fig. 2 Simulation model of CC temperature automatic control system
Thus, substituting the coefficients obtained in (5) and (6) in Eq. (2), we finally
get:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
'
Cev
dt L (τ )
kev · Fev · tcc (τ ) − c · Ẇc∗
+
t
·
=
(τ
)
L
'
'
dτ
kev
· Fev + a · Ẇc∗
kev
· Fev + a · Ẇc∗
'
Ccc
dtcc (τ )
k · Fev · t L (τ ) + kcc · Fcc · t0 (τ )
·
+ tcc (τ ) = ev
'
'
kev
· Fev + kcc · Fcc
dτ
kev
· Fev + kcc · Fcc
.
(7)
The obtained system of equations allows building on its basis a simulation model
of the CC temperature automatic control system, taking into account changes in
internal and external parameters of the VCRU (Fig. 2).
In the model in Fig. 2 the Controller unit is introduced and simulates the frequency
regulator of the temperature. While the inertia of the compressor electric drive is
neglected, because they are several orders of magnitude less than the inertia of
the evaporator and CC. A signal equal to the relative compressive power of the
compressor Ẇ * c is sent from the output of the Controller unit. In addition, an
adder and a CC temperature reference signal t s are introduced into the model for
the formation of feedback and setting action.
To adjust the controller, the value of Ẇ * c is proposed to be defined as follows:
Ẇc∗ =
Ẇc
M c · . · ηc
=
,
Mcn · . · ηcn
Ẇcn
(8)
where Ẇ cn —nominal compressive power of the VCRU compressor; M cn —nominal
compressor shaft load torque; .n —nominal rotor angular rotation frequency of the
drive motor; ηc , ηcn —current and nominal value of compressor efficiency.
Simulation Modeling of Vapor Compression Refrigeration Unit …
259
The method of determining the compressor compressive power is described in
[24]. It should be noted that the nominal compressor compressive power significantly
depends on the VCRU operating temperature, namely the values of temperatures t L
and t H , and must be determined before starting the unit according to the building
refrigeration cycle.
Research and verification of the simulation model shown in Fig. 2 has been carried
out in Matlab for VCRU with a cooling capacity of 14 kW. This is the VCRU of a
refrigerated vessel intended for the transport of citrus (storage conditions correspond
to the transport of mandarins) with passport data [22]:
• heat capacity of the evaporator C ev = 220 kJ/°C;
• heat capacity of the loaded cooling chamber C cc = 250 kJ/°C;
• thermal conductance coefficients of the evaporator: kev · Fev = 580 W/°C, kev_s · Fev
= 290 W/°C;
• thermal conductance coefficients of the CC walls kcc · Fcc = 250 W/°C;
• constant coefficients of the compressor characteristic: a = 2800 W/°C, c = 30 kW;
• mandarin storage temperature t cc = 4 … 7 °C;
• the initial temperature of the evaporator and the cooling object is taken equal to
the ambient temperature.
R22 data have been used as refrigerant data [23]. The set temperature in CC has
been set at 5.5 °C. The settings of the proportional-integral-derivative controller have
been set by Matlab using the pidtuner function. As a result of simulation, graphs of
changes in boiling point and cooling object temperature with smooth regulation of
compressor cooling capacity at ambient temperatures of 20 and 30 °C for the three
hours model time have been obtained (Fig. 3).
Analysis of Fig. 3 shows that at the regulator set settings the processes of boiling
point and cooling object temperature change occur smoothly without overshoot. The
steady-state temperature value t cc corresponds to the set one. The steady-state value
t L corresponds to the average value at on–off control [22].
Fig. 3 Transient processes in the VCRU with smooth regulation of productivity at t 0 = 20 °C (a)
and t 0 = 30 °C (b)
260
A. Bukaros et al.
Fig. 4 Change the relative compressive power of the compressor at t 0 = 20 °C (a) and t 0 = 30 °C
(b)
The dynamics of changes in the relative compressive power of the compressor is
shown in Fig. 4.
Analysis of Fig. 4 shows that at an ambient temperature of 20 °C the relative
compressive power of the compressor is 0.24, and at an ambient temperature of
30 °C is 0.73, which agrees well with the data obtained in [21, 22], and shows the
adequacy of the obtained model to real temperature processes.
4 Diagnostic Algorithm of the Frosting Vapor Compression
Refrigeration Unit Evaporator
The obtained model can serve as a basis for a digital twin [25] of cooling processes,
that is to model the “reference” temperature modes of the evaporator and CC.
Comparison of “reference” and actual cooling processes according to the relevant
temperature sensors data allows identifying the causes and predict possible deviations
of the VCRU maintenance parameters from normal operation.
One of the important reasons for the disruption of the VCRU normal operation is
the frosting evaporator, which leads to a significant decreasing thermal conductance
coefficient kev · Fev , worsening heat transfer conditions from the cooling object to
the evaporator and reducing energy efficiency. To combat this phenomenon, VCRU
manufacturers provide periodic defrosting evaporator with electric heaters or hot
liquid refrigerants [26]. The defrosting frequency is either set manually or automatically. At the same time, the automation starts the defrosting process in accordance
with the recommendations of a particular VCRU manufacturer depending on, for
example, the ambient temperature, but does not take into account the actual frosting
processes.
Given the above, it is important to find new methods for determining the evaporator
frosting degree and the timely start of the defrosting process. Via the developed model
it is impossible to determine in real time and predict changes in the value of kev · Fev ,
Simulation Modeling of Vapor Compression Refrigeration Unit …
261
because it does not provide for the dependence of this parameter on the evaporator
frosting degree. Therefore, we will consider the possibilities of improving this model,
for which we will form an algorithm for diagnosing evaporator frosting.
To determine the algorithm for the diagnosis of evaporator frosting, it is necessary
to determine the effect of the value kev ·Fev on the operating temperature of the VCRU.
For this purpose we will use means of simulation modeling and the developed model.
Assume that the frosting evaporator, which must start the defrosting process, leads to
a decrease in the value of the thermal conductance coefficient kev · Fev by 10%. The
actual value can be set by the manufacturer depending on the type and purpose of
the VCRU, operating conditions, etc. Figure 5 shows graphs of temperature modes
of the previously studied VCRU.
In Fig. 5 the “reference” processes of stabilizing the cooling object temperature t cc
and boiling point t L at the nominal thermal conductance coefficient of the evaporator
kev · Fev are shown by solid lines. The same processes when the value kev · Fev is
reduced by 10% due to evaporator frosting are shown by dashed lines.
Analysis of Fig. 5 shows that the reducing thermal conductance coefficient due
to the evaporator frosting leads, firstly, to an increase in the temperature difference
(t cc −t L ) between the evaporator and the cooling object; secondly, to increase in
Fig. 5 Transient processes in VCRU with on–off (a, b) and smooth (c, d) control at t 0 = 20 °C (a,
c) and t 0 = 30 °C (b, d)
262
A. Bukaros et al.
the operating time of the compressor with on–off control or to increase the relative
compressive power of the compressor Ẇ * c with smooth control in accordance with
(4).
From Fig. 5a, b, it is clear that when kev · Fev decreases, the process of warming the
refrigeration object during the compressor stopping is faster than cooling, despite
increasing the corresponding time constants [22]. This is explained by the deterioration of the heat removal conditions from the cooling object by the evaporator
during non-operating hours of the compressor. In turn, the cooling process during
compressor operation is slower due to increasing time constants [22] and reducing
cooling capacity of the compressor QL (1).
It should be noted that the magnitude of these changes differs significantly
depending on external conditions. For example, when the ambient temperature
increases to 30 °C, evaporator frosting leads to significant increasing the operating
time (relative compression power) of the compressor and increasing the cycle time
of the VCRU as a whole (Fig. 5b, d). At the same time, there is a slight increasing
the temperature difference between the evaporator and the cooling object. However,
at t 0 = 20 °C the evaporator frosting, despite not some increasing the compressor
operating time, leads to decreasing the cycle time of the VCRU and a more noticeable
increasing the difference (t cc −t L ).
All this necessitates the search for an integrated criterion for estimating the thermal
conductance coefficient kev · Fev in variable modes of VCRU operation. To determine
such a criterion, we will integrate the first equation of system (7) and after the
elementary transformations we obtain:
'
kev · Fev =
a · Ẇc∗ ·
.
t L (τ )dτ + c · Ẇc∗ · τ + Cev · t L (τ )
.
.
[tcc (τ ) − t L (τ )]dτ
(9)
The integration time interval τ is chosen from the point of view of ensuring the
required rate of evaporator frosting identification, but not less than the time constants
of the cooling object and the evaporator.
The above provisions allow forming the following algorithm for diagnosing the
VCRU evaporator frosting.
1. Data are entered: boiling point t L and cooling object t cc sensor; values of the
compressor characteristics constants a, c; evaporator nominal (reference) thermal
conductance coefficients kev · Fev and kev_s · Fev ; exponent k depending from the
using refrigerant; the value of the evaporator heat capacity C ev .
2. According to the method described in [24], the current value of the relative
compressive power of the compressor Ẇ * c is determined.
3. Calculate the current value of the evaporator thermal conductance coefficient by
expression (9).
4. The obtained value is compared with the “reference”, calculated by expression
(6).
5. If the “reference” value of the evaporator thermal conductance coefficient does
not exceed the current, for example, on 10%, then go to step 1. If it exceeds, then
Simulation Modeling of Vapor Compression Refrigeration Unit …
263
Fig. 6 Subsystem for calculating the current value of the thermal conductance coefficient of the
VCRU evaporator
the signal of automatically start the evaporator defrosting process is generated
or diagnostic message for manual start is issued.
As already mentioned the reduction of the evaporator thermal conductance coefficient value by 10% due to frosting is approximate and can be set by the manufacturer,
for example, depending on the VCRU temperature limits.
The implementation of the developed algorithm requires the installation of two
temperature sensors in the VCRU of the cooled object and the boiling point of the
refrigerant, the voltage and current sensors of the compressor motor to determine
the relative compression power [24]. The subsystem of the simulation model for
calculating the current value of the thermal conductance coefficient of the evaporator
by expression (9) is shown in Fig. 6.
In the presented subsystem in Fig. 6 information inputs are signals from temperature sensors t L , t cc and value of relative compressive power of the compressor Ẇ * c .
The output signal of the subsystem must be applied to the corresponding blocks of the
model in Fig. 2, thus forming a refined value of the thermal conductance coefficient
of the evaporator k'ev · Fev using in Eq. (7).
Practical implementation of the evaporator frosting algorithm on the microcontroller does not cause any difficulties. To eliminate the accumulation of additive
measurement error when integrating sensor signals, integrators in Fig. 6 are recommended cover by single negative feedback while maintaining their properties in the
operating frequency range. Verification of the algorithm by means of simulation
modeling proved the efficiency of the proposed method.
264
A. Bukaros et al.
5 Conclusions
The investigations carried out in this chapter allow forming the following conclusions.
A simulation model of VCRU evaporator and cooling object temperature modes
control has been developed. It allows analyzing the main properties of the VCRU as
a control object in variable operating modes and can be the basis for designing closed
refrigeration control systems. Further improvement of the control system model is
possible through the use of adaptive self-tuning controllers that will provide energy
efficient control of temperature modes in conditions of permanent change of VCRU
internal and external dynamic parameters.
A subsystem for determining the thermal conductance coefficient of the evaporator
has been developed as part of a simulation model. It can be used for predicted diagnostics of evaporator frosting and automatic start of the defrosting process. Further
research may be aimed at developing a digital twin of VCRU temperature processes,
for which it is necessary to determine and describe the impact on the created model
behavior of changes in internal parameters of VCRU such as thermal conductance
coefficient of walls and heat capacity of cooling chamber.
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Methods for Diagnosing the Technical
Condition of Heating Networks Pipelines
Vitalii Babak , Oleg Dekusha , Artur Zaporozhets , Leonid Vorobiov ,
and Svitlana Kovtun
Abstract In district heating systems, a significant share of heat loss (according to
some data up to 20 … 30%) is accounted for by heating networks. Since the repair
and relocation of heating networks is very expensive, determining the need and
priority of such work is of paramount importance. A reliable quotative criterion for
the condition of the heating network is the actual heat loss. But heating network diagnosing assume not only quantitative methods but qualitative methods for diagnosing
main defects of pipelines. The chapter covers the diagnosing of the heating network
by quantitative methods for determination the actual specific heat loss and qualitative methods for diagnosing main defects of pipelines. The quantitative methods
and means allow determining integral and specific heat losses on the sections of
heating networks, both equipped with thermometer shell casings and without shell
casings, at the mode of operation of the heating network, close to the operating one,
without switching off heat consumers. For the case of pipelines without shell casings
proposed method of two measurements to determine the thermal resistance. The use
of heat flux sensors in systems with an overhead combined sensor demonstrates the
possibility of improving the accuracy of measurement by considering thermal resistance of the pipeline. The qualitative method for diagnosing technical condition of
pipelines of heating networks, which makes it possible to monitor the state by using
thermal aerial imaging, based on micro air vehicles. Method is especially effective
in accidents in areas of spatially branched heating networks for diagnosing main
defects of pipelines such as crack, rupture of the metal, thinning of the wall due to
mechanical stress, the effects of corrosion or delamination.
V. Babak · O. Dekusha · A. Zaporozhets (B) · S. Kovtun
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: a.o.zaporozhets@nas.gov.ua
A. Zaporozhets
State Institution “The Institute of Environmental Geochemistry of NAS of Ukraine”, Kyiv,
Ukraine
L. Vorobiov
Institute of Engineering Thermophysics NAS of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_15
267
268
V. Babak et al.
Keywords Heating networks pipelines · Specific heat loss · Integral heat loss ·
Heat flux sensors · Thermal resistance · Thermal aerial imaging · Micro air vehicles
1 Introduction
In district heating systems, a significant share of heat loss (according to some data
up to 20 … 30%) is accounted for by heating networks. This is due to the fact
that a third of all heating networks are worn out and in a state of emergency. Since
the repair and relocation of heating networks is very expensive, determining the
need and priority of such work is of paramount importance [1–4]. Manufacturers
of pipes for heating networks and construction and installation organizations are
interested in increasing the volume of work, not always taking into account the real
state of replacement heating networks and the economic efficiency of the work [2–
6]. Correct determination of the order of relocation of heating networks will save
energy resources and minimize costs [5, 6]. The duration of operation of the heating
network is not the main criterion for the need for relocation, as the condition of the
pipeline and insulation is influenced by many factors, from route design, soil type
and groundwater level to quality of installation, insulation and operation. A reliable
criterion for the condition of the heating network is the actual specific (per 1 m of
length) heat loss in the network [2, 7]. Permissible specific heat losses for pipelines of
different diameters are standardized, however, since these standards were developed
several decades ago [7], do not take into account the capabilities of modern insulation
materials and energy prices, they can hardly be the only and objective criterion for
relocation.
In the main sections of heating networks with high values of transmitted energy,
the relative heat loss is much less than the measurement error of modern industrial
heat meters. Therefore, expensive periodic tests of heating networks are carried out,
in which individual sections are connected in a ring and at low consumption of heat
coolant (water) temperature differences are measured at each of the sections. In this
regard, it is very important to develop measurement methods and equipment that
allow you to control heat loss on sections of heating networks in modes close to
operating.
The heating network diagnosing assume not only quantitative methods but qualitative methods [8] for diagnosing main defects of pipelines. Defects such as crack,
rupture of the metal, thinning of the wall due to mechanical stress, the effects of corrosion or delamination [9, 10] should be found as fast as possible to reduce potential
losses.
Methods for Diagnosing the Technical Condition of Heating Networks …
269
2 The Method for Determining the Integral Heat Losses
on Heating Pipeline Segment
The power of integrated heat losses in the section of the heating network can be
calculated as the ratio of the difference of total energies of the coolant, the beginning
and end of the controlled area, to the duration of measurements [10]:
QT B
⎡ τ
⎤
τ.
2 .τ2
.2
= ⎣ TM1 · g M1 · C V · dτ −
TM2 · g M2 · cv · dτ ⎦/(τ2 − τ1 ),
τ1
(1)
τ1 +.τ1
where τ1 and τ2 —initial and final moments of measurements at the initial point of
segment; .τ1 and .τ2 — duration of the passage of the segment by the elementary
volume of water, respectively, at the initial and final moments of measurement; TM1
and TM2 —instantaneous values of water temperature at the beginning and end of
the site; g M1 and g M2 —instantaneous values of volumetric water flow rate at the
beginning and end of the segment; cv —volumetric heat capacity of water.
In the general case, the variables are not only the temperature TM1 and TM2 , but
also flow rate g M1 and g M2 consequently not the same. Heat capacity is a function of
temperature and pressure. Thus, to accurately determine the power of heat loss, it is
necessary to measure not only the temperature but also the flow rate at the beginning
and end of the segment and determine the time of passage.
With an apparently small change in temperature, which is typical for heating
networks, the heat capacity can be taken constant. With a constant flow rate g M1 =
g M2 = g, the time of segments passage is equal to .τ1 = .τ1 = .τ . For this simple
way, the calculation of heat losses can be done using the formulas:
. the integral heat losses in the segment:
Q T B = (T1 − T2 ) · g · cv ,
(2)
. consumption of heat (per unit length of the pipeline):
Q P R = Q T B /L ,
(3)
where L—length of the segment; T 1 —average temperature of the coolant at the
beginning of the segment for the time from τ1 to τ2 ; T 2 —average temperature of the
coolant at the end of the segment for the time from (τ1 + .τ ) to (τ2 + .τ ).
Equation (2) allows to reliably determine heat loss only in the case of precise
measurements of the differences in the average temperature of the coolant at the
beginning and end of the segment. To carry out such measurements, it is necessary to
create appropriate equipment that allows to perform measurements with the required
accuracy during the operation of the heating networks.
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3 The Method Integral Heat Losses on Heating Pipeline
Without Standard Shell Casings
The described above method can be used to determine heat loss in areas of heating
networks equipped with standard shell casings in pipelines. However, most pipelines
are not equipped with such shell casings. Attempts to use thermometers of various
designs to determine the temperature of the coolant by measuring the temperature
of the outer surface of the pipe were unsuccessful due to the fact that the errors of
such measurements are commensurate or exceed the decrease in coolant temperature
due to heat loss. Therefore, it is proposed to create specialized devices that can
accurately measure the temperature of the coolant on the basis of joint measurements
of temperature and heat flux on the surface of the pipeline [11].
It is proposed to use an overhead combined sensor of heat flux and temperature,
which is made in the form of an elastic (flexible) plate with a thickness of approximately 2 … 3 mm. This design will allow you to easily apply the sensor on the metal
surface of the pipeline (Fig. 1).
The value of the thermal resistance of the pipe wall may change during the operation of the pipeline—increase due to lime deposition, or decrease due to abrasive
particles in the coolant on the wall. The use of inaccurate values of thermal resistance
leads to an increase of the error of determining heat losses.
Overhead combined sensor must contain two sensitive elements: a small highresistance platinum resistance thermometer and heat flux sensor of the auxiliary wall
type of bimetallic galvanic coil of thermocouples. The results of joint measurements
of temperature and heat flux on the surface of the pipeline allow to determine the
temperature of the coolant by the formula:
Tm H = Tm + q × (Rk + RT P + Ra ),
(4)
where T m —measured pipe surface temperature; q—measured heat flux from the
surface of the pipeline; Rk —contact thermal resistance between the sensor and
Fig. 1 Application of the
overhead combined sensor
on the surface of the pipeline
Methods for Diagnosing the Technical Condition of Heating Networks …
271
the pipeline surface; RTP —thermal resistance of the pipeline wall; Rα —thermal
resistance of the convective heat transfer between the coolant and the pipeline;
The contact thermal resistance between the sensor and the pipeline surface can be
reduced and stabilized through the use of thermally conductive greases. The thermal
resistance of the metal wall of the pipeline depends on the thickness and material of
the wall and can be easily calculated for the entire range of pipes used. The thermal
resistance of convective heat transfer between the coolant and the pipe is a value
inverse to the heat transfer coefficient, which can be calculated by known formulas
[12] depending on the speed of the coolant and its properties. Thus, the values of
all thermal resistances used to calculate the temperature of the coolant in the pipe
can be easily determined by information about the type of pipe, its temperature and
velocity of the coolant. Calculations show that the values of the corrections due to
the difference between the measured temperature of the pipe surface and the coolant
temperature are in the range from 0.05 to 0.5 K.
The order of the method application is as follows:
. Install sensors using heat-conducting greases on the cleaned metal surface of the
pipelines at the beginning and end of the segment. The installation is carried out
on the side surface of the pipeline to eliminate the impact on the measurement
results of bottom sediments and air bubbles. Thermal insulation is installed on
top of the sensor;
. connect the output circuits of the sensors to the electronic units and for several
days to record the signals of temperature and heat flux;
. carry out the verification of the temperature sensors of in a calibration thermostat
with a temperature whose value is in the range of measurements, and determine
the corrections to the readings of the temperature;
. enter the values of corrections into the data, as well as information about the type
of pipeline, calculated values the speed of movement (flow) of the coolant, the
value of thermal resistance, determine the temperature of the coolant for (4) taking
into account the corrections determined when comparing the sensors;
. determine the temperature differences at the beginning and end of the test area
for the coolant going to the consumer and the coolant coming from the consumer,
calculate the integral and specific heat losses in the test area.
For determine the total thermal resistance, a two-measurement method is
proposed, which consists in installing two sets of surface temperature and heat flux
sensors on the pipeline before the tests at the beginning and end of the study area
(Fig. 2).
Preliminary measurements are performed in two modes with significantly
different values of heat flux through surface for example in the presence of thermal
insulation on the sensor and in its absence.
According to the results of measurements in two modes make a system of linear
equations of heat transfer through the wall of the pipeline:
TT H 1 = T1−1 + q1−1 · R1T P ;
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V. Babak et al.
Fig. 2 The method of two measurements to determine the thermal resistance
TT H 1 = T2−1 + q2−1 · R2T P ;
TT H 2 = T1−2 + q1−2 · R1T P ;
(5)
TT H 2 = T2−2 + q2−2 · R2T P ,
where TT H 1, TT H 2—value of the coolant temperature during measurements in the
first and second modes; T1−1 , T1−2 —temperature values determined by the first
sensor in the first and second modes; T2−1 , T2−2 —temperature values determined
by the second sensor in the first and second modes; q1−1 , q1−2 —heat flux values
determined by the first sensor in the first and second modes; q2−1 , q2−2 —heat flux
values determined by the second sensor in the first and second modes; R1T P , R2T P —
thermal resistance of the pipe wall in the places of installation of the first and second
sensors.
The solution of the system of Eq. (6) allows to determine the value of the total
thermal resistance in the places of installation of the sensors.
R1T P =
q2,2 (T1,1 − T2,1 ) + q2,1 (T2,2 − T1,2 )
;
q1,2 · q2,1 − q1,1 · q 2,2
R2T P =
q1,1 (T2,2 − T1,2 ) + q1,2 (T2,1 − T1,1 )
.
q1,2 · q2,1 − q1,1 · q 2,2
(6)
Methods for Diagnosing the Technical Condition of Heating Networks …
273
Subsequently, during the tests, the values found are used to determine the
temperature of the coolant in accordance with (5).
The considered method of determining heat loss based on the results of measuring
the temperature change of the heat carrier and its consumption, in fact, reproduces
the principle of operation of the heat exchange calorimeter or heat meter. The use
of heat flux sensors in systems with an overhead combined sensor demonstrates the
possibility of improving the accuracy of measurement.
A patent of Ukraine for an invention was obtained for the method of determining
heat losses in the segment of the heating network with the determination of the
thermal resistance of the pipeline wall by the method of two measurements [13].
4 The Use of Thermal Imaging for Diagnosing
the Technical Condition of Pipelines of Heating Networks
One of the most promising qualitative method for diagnosing technical condition of
pipelines of heating networks is thermal aerial imaging based on micro air vehicles.
The essence of the method of thermal aerial imaging is that first the heat network
is divided into areas, some of which are reference, without deviations from normal
operating conditions, and then in the autumn (or early spring) when heating systems
conduct simultaneous thermal ground and aerial imaging of reference areas [14].
For each of them build two graphs one of which reflects the functional relationship
between temperature contrast and the background of the earth’s surface above the
heat pipe and the laying depth. The other—the relationship between the ratio of
temperature contrasts with the background of the earth’s surface over reference heat
pipes and the depth of the gasket in different states of thermal insulation.
By comparing, the thermal fields of the reference and controlled heat pipelines
according to the data of simultaneous thermal aerial imaging on calibration graphs
determine the actual state of the controlled heat pipelines and the presence of
violations of the state of their insulating structures.
To implement the method, it is necessary to perform thermal aerial imaging and
ground thermal imaging, which are performed simultaneously, and the interpretation
of the obtained data.
It should be noted that the normal organization of work in this area is possible only
in the presence of special diagnostic services department of the heating networks,
which should perform strict certification of all heating pipelines in operation. Until
recently, this method was quite expensive and its implementation required significant
material costs. However at the moment thermal aerial imaging is the way to detect
emergency and potentially defective sections of heating network pipelines in a short
period of time. With thermal aerial imaging, it is possible to survey quickly large
areas of the urban landscape and with high probability to record anomalous areas of
the temperature field on the soil surface.
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V. Babak et al.
Usually thermal aerial imaging is performed at an altitude of 300–400 m on a
system of parallel routes with a distance of 300–500 m, which provides at least 40%
overlap of the image to obtain a picture of the distribution of thermal energy on the
plane. Thermal aerial imaging is carried out in early spring or late autumn in the
absence of snow cover, when the heating networks are operating. To eliminate the
distortion of thermal effects from solar insolation, aerial imaging should be carried
out at night, at least during the day with continuous clouds. Aerial imaging is not
performed in fog, precipitation and wind speeds greater than 10 m/s [8, 14].
Hidden places of leakage of the heat coolant, zones of destruction of thermal
insulation, sites of flooding of heat pipelines are reliably fixed on the thermograms
received during thermal aerial imaging.
The main task of thermal aerial imaging is not only to identify emergency areas.
As a rule, in case of rupture of the pipeline, such places are quickly localized and
the necessary measures are taken. One of its tasks is to forecast the development of
emergencies to prevent their occurrence.
At this stage of development of thermal aerial imaging it became possible to
improve the thermal monitoring system of heating networks using a set of hardware
and software for thermal imaging by using modern unmanned aerial vehicles (quadcopters), which significantly reduced the cost and speed up this method is available
for a number of municipal and industrial heat companies [15–17].
In Fig. 3 presented example of thermal imaging images of segments of the heating
network on which experimental studies were conducted. Shooting was performed
in November at 18 o’clock in the cloudless sky at a temperature of minus 4 °C.
According to the results of experimental studies, the technical condition of the
pipelines of thermal networks is assessed.
For heat power facilities, first of all underground heating networks, it is an assessment of the level of heat loss to the environment, more effective monitoring of the
state of heating networks, an objective assessment of the ways of development of
Fig. 3 Thermal image of the
studied areas of heating
networks
Methods for Diagnosing the Technical Condition of Heating Networks …
275
city heat supply. Thus, according to the latest data, the range of estimates of heat
loss is 15–40%, and the latter figure seems more realistic. Sometimes even a value
of 60% is indicated, which may well occur in many areas of abnormal heat loss.
To achieve a significant reduction in heat loss, you need to know their real values
at any facility. Thermal aerial imaging of the city or its individual districts and
facilities can provide this information. First of all, it is necessary to identify areas of
abnormal losses in heating networks, where the anomalies exceed the predetermined
temperature contrast, thus increasing the likelihood of diagnosis. Quantitative data are
also needed for other urban infrastructure, industrial, residential and office buildings,
etc.
The following solutions can be used to calibrate thermal imaging devices [14–16].
. installation of reference calibrators in the optical part of the thermal imager. The
use of this method significantly complicates the optical unit, there are problems
with the certification of calibrators and does not take into account the influence
of the atmosphere;
. binding to the emitter of the internal elements of the optical unit. Provides for the
installation of thermal sensors that detect changes in temperature during flight.
This method is not highly accurate, it also does not take into account the effects
of the atmosphere;
. installation of a radiometer on board the aircraft carrier, which registers the overhead temperature graph. Its data are extrapolated by the area of thermal aerial
imaging. This method gives good results for homogeneous temperature fields,
such as water surface.
. the use of reference objects on the earth’s surface. The method is well known
in aero geophysics, such as aeromagnetic surveying. For thermal aerial imaging,
it is necessary to study the transfer function of the optical and electronic path
of the aviation thermal imager, select the most representative reference objects
within the area of thermal aerial imaging and develop programs for processing
the results. When calibrating the reference objects, it is necessary to conduct
laboratory studies of the transfer function of the thermal imaging device in order to
determine influence on the calibrated characteristic of the thermal imaging device
of the temperature regime; influence on the measurement of analog-to-digital
conversion parameters (data bit rate, range of input signals).
5 Conclusions
A reliable diagnosing of the heating network required quantitative methods for determination the actual specific heat loss and qualitative methods for diagnosing main
defects of pipelines.
The proposed quantitative methods and means allow determining integral and
specific heat losses on the sections of heating networks, both equipped with thermometer shell casings and without shell casings, at the mode of operation of the
heating network, close to the operating one, without switching off heat consumers.
276
V. Babak et al.
For the case of pipelines without shell casings proposed method of two measurements
to determine the thermal resistance. The use of heat flux sensors in systems with an
overhead combined sensor demonstrates the possibility of improving the accuracy
of measurement by considering thermal resistance of the pipeline.
The qualitative method for diagnosing technical condition of pipelines of heating
networks, which makes it possible to monitor the state by using thermal aerial
imaging, based on micro air vehicles. Method is especially effective in accidents in
areas of spatially branched heating networks for diagnosing main defects of pipelines
such as crack, rupture of the metal, thinning of the wall due to mechanical stress, the
effects of corrosion or delamination.
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Thermal Power Plants’ Coal Stock Short
Term Projection Method for Ensuring
National Energy Security
Sergii Shulzhenko , Borys Kostyukovskyi , Olena Maliarenko ,
Vitalyi Makarov , and Maryna Bilenko
Abstract Ukrainian National Power System, like others in the world, is in the transition period due to several fundamental factors: rapid penetration of renewables to the
generation mix; implementation of market principles for electricity supply, transmission, distribution, and end-use; the increasing role of ecological concerns, first of all
in the electricity production; a necessity steadily decrease greenhouse gases (GHG)
emissions at the national level which are essentially depend on coal use for electricity
production and hence require accurate operation planning of coal-burning thermal
power plants (TPP) and even shutting down them permanently. But from another point
of view, the fundamental requirement of Sustainable Goal 7 is to “ensure access to
affordable, reliable, sustainable and modern energy for all”, where one of the key
points is “reliable” which currently could be provided only by conventional fuelburning power plants, e.g. coal-burning ones. Moreover in Ukrainian Power System,
the coal-burning TPPs are important players providing all types of required reserves
to insure overall grid stability, and the question of which electricity producers could
provide a similar level of reserves currently does not have a well-grounded answer.
Besides, the price for electricity produced by existing coal-burning plants compared
to gas-burning ones which could replace them is lower, and this factor is very important both for the production sector to ensure its global competitiveness, and social
sphere. That is why the stable operation of coal-burning TPPs in Ukraine will play
an important role at least in the short term perspective—about the nearest 10 years,
and the use of well-suited methods allowing accurate projection of TPPs’ coal-stock
and hence the stable TPPs operation are important not only for grid resilience but
for national energy security too.
Keywords National power system · Projected electric energy balance ·
Mathematical modeling technique · Thermal power plant · Projected available
stock of fuel
S. Shulzhenko (B) · B. Kostyukovskyi · O. Maliarenko · V. Makarov
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: mail2ua@gmail.com
M. Bilenko
NPC “UKRENERGO”, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_16
279
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S. Shulzhenko et al.
1 Introduction
The United Nations Sustainable Development Goal 7: Affordable and Clean Energy
[1, 2] requires to “ensure access to affordable, reliable, sustainable and modern energy
for all” by 2030. The “most” modern energy in the contemporary World is electricity,
as it could be widely end-used by various industries, tertiary sector and households.
Currently, the state-of-the-art electricity generation and storage technologies are still
expensive, and hence the technological measures to cover local demand in electricity
allow generate electricity with production cost higher than in the case of obtaining
electricity from a centralized Power System. On the other hand, conventional technologies that are widely used in Power Systems usually have a high negative impact
on the environment, and hence on human health. That is why an important goal for
electricity generation is to make it essentially more environmentally friendly, which
is only possible by gradually changing the generation mix implementing ecologically neutral generators, usually renewable energy sources (RES). This is leading to
politically supported penetration of a large amount of renewable generation, mostly
wind and photovoltaic, into conventional existing Power Systems, and this is the
case also for Ukraine as renewables according to the Ukrainian Law on Electricity
Market (Articles 3, 68) [3], are preferably dispatched (operated without restrictions
if it potentially will not cause an accident (Sect. VII, Chap. 6 of the Grid Code) [4])
in the Ukrainian Power System.
Since renewable energy sources during a day generate uncontrolled or intermittent
electric power the Power System to be balanced requires the operation of generators which capable change the output according to the current demand for electric
power and to compensate an unstable RES generation. Such controlled or balancing
generators in the case of Ukraine are hydroelectric and coal-burning thermal power
plants. The operation modes of hydroelectric generation depend on the amount of
water in the river which is changing through the year, month, and week and this is an
important limiting factor, thus the balancing capabilities of this type of generation
vary during the year and are limited. The TPPs instead of hydroelectric generation
are seasonally independent and depend only on the availability of coal that could
be burned in the boilers, and because of that, they are important “participants” that
could ensure the Power System balancing at any time of the year, in the day- or night
time. That is why the projection of the availability of coal at TPPs’ stock through a
whole year taking into account the level of required balancing service, a maintenance
schedule of all other generators in the Power System, campaigns of Nuclear Power
Plants’ refueling, availability of water in the river, forecasted profiles of wind and PV
generation, and finally necessity to cover actual demand for electricity is a vital task
for Power System resilience. Since this task is important for national energy security the appropriate projections are made by the Ministry of Energy of Ukraine on
a regular basis according to the legal document “Procedure for Projected Electricity
Balance Formation” [5].
The official Procedure [5] prescripts fulfillment of such main steps/requirements
regarding fuel on stock:
Thermal Power Plants’ Coal Stock Short Term Projection Method …
281
According to the Article 2 of Section 2 the design period is a calendar month;
According to the Article 4 of Section 2 to determine fuel provision to be
considered:
• for coal power plants: formation of guarantied coal reserves, corresponding to 10or 20-day amount of consumption, depending on the remoteness of the point of
its mining; preventing the reduction of reserve fuel (fuel oil) below the 10-day
volume required to provide power unit start-ups and the necessity to use it as
additional high-calorific fuel;
• for combined heat and power plants: generation of guarantied coal reserves, corresponding to 20-day average daily consumption, necessary to ensure a heat load
schedule in accordance with the concluded contracts for heat supply in the forecast month, at power stations for which the main fuel is coal; prevention of the
reduction of the reserve fuel (fuel oil) reduction below the 10-day amount required
for the start-up of the units and the necessity to use it as additional high-calorific
fuel at the power stations for which the main fuel is.
The procedure of balance formation, according to the above-mentioned points,
is enough detailed, but actually, it does not provide any particular mathematical
formulas to fulfill it. Therefore, the mathematical linear programming model for
projecting the amount of coal on TPPs’ stock through the year is proposed.
2 Literature Review
There are a lot of advanced mathematical approaches, methods, and tools which
could be used to solve particular fuel supply chain optimization, optimal inventory management, and resource planning tasks. Those models are based on a
wide range of approaches, including classical linear programming, mixed-integer
linear programming, stochastic programming methods, fuzzy logic, simultaneous
modeling, etc.
Since the inventory planning problem regards power sector operation this kind of
task is solved with classical approaches traditionally used for power sector studies
[6–9]. But as objects of research consist also of coal suppliers which have their
behavior and technological structure more comprehensive approaches are used, e.g.
distributed models [10], which could be multicriteria ones [11]. Also, the important
and usual issue is uncertainty, which is hard to overcome since the system (object)
is an open one but which could be “reduced” with the use of appropriate stochastic
models [12, 13], and some of them even are using multi-stage stochastic optimization
[14]. There are types of models that are based on fuzzy logic algorithms [15] and
machine learning [16], which allow obtaining results based on prehistory, which does
not always consists much enough input (learning) data.
Taking into account the complicity of some approaches and models the traditional models are also “on the table desk” [17]. These traditional approaches and
models have an important advantage from the point of view of an administrator as
282
S. Shulzhenko et al.
the result obtained could be clearly explained and quickly double-checked. That is
why the proposed mathematical model is based on the traditional approach already
used for power sector studies including some methods used by classical inventory
optimization models.
3 Model Formulation
It is assumed that before starting the calculation with the proposed model the amount
of electricity to be produced by TPPs for each month of the year has been already
defined; hence the obvious constraint is a strict balance between TPPs electricity
production and demand for it for each month:
I \K
.
i=1
Wit +
K
.
Wkt = Bt ; ∀t = 1 . . . 12,
(1)
i=1
Bitmin ≤ Wit ≤ Bitmax ; ∀t = 1 . . . 12, ∀i ∈ I,
(2)
where index i correspond to the particular group of the TPP units with similar technical and economic parameters, e.g. coal-burning units (both CHP, and TPP) with an
installed capacity equal to the 300 MW; index k corresponds to the units listed in the
National Emission Reduction Plan for Large Combustion Plants [18], the set K is
subset of set I; index t corresponds to the number of the month of the year; parameter
Bt is an exogenously defined amount of electricity which should be produced by all
TPP units which are available for operation during the appropriate month, GW hour;
variables Wit and Wkt correspond to amount of electricity produced by each group of
the TPP units, GW hour; parameters Bminit and Bmaxit is an exogenously defined
minimum and maximum allowed amount of electricity which could be produced
by each TPP unit which is available for operation during the appropriate month,
GW hour.
According to the National Emission Reduction Plan for Large Combustion
Plants [18] CHP and TPP are restricted to emitting some defined amounts of nitrogen
oxides (NOx ), sulfur dioxide (SO2 ), and dust then the next constraint is ensuring
fulfillment of this:
12 .
K
.
ekpt · Wkt − H p ≤ B p ; ∀ p ∈ {NOx ; SO2 ; dust},
(3)
t=1 k=1
where index p corresponds to the particular emission matter, i.e. NOx , SO2 or dust; ekpt
is a specific emission of particular pollutant for the combustion unit, ton/GW hour;
parameter Bp is an allowed upper limit for a monthly amount of particular pollutant
emission, ton; artificial variable H p allows fulfill constraint in the case amount of
emission is higher than an allowed monthly limit, ton.
Thermal Power Plants’ Coal Stock Short Term Projection Method …
283
Because the Ukrainian TPPs’ units burn two types of coal, i.e. anthracite (A) and
bituminous coal (BC) they should be accounted separately:
I
.
ai jt · Wit − F I jt − F D jt − S D jt + S I jt − H jt
i=1
≤ F j ; ∀t, ∀ j ∈ {A; BC},
(4)
F I jt ≤ I jt ; ∀t = 1 . . . 12, ∀ j ∈ {A; BC},
(5)
F D jt ≤ D jt ; ∀t = 1 . . . 12, ∀ j ∈ {A; BC},
(6)
where index j corresponds to the particular type of coal—A or BC; aijt is a specific
consumption of a particular type of coal by the combustion unit, ton/GW hour;
parameter F j is an allowed upper limit for a monthly amount of a particular type of
coal consumption, ton; variable FI jt is an monthly imported amount of particular coal
type, ton; variable FDjt is a monthly domestically produced amount of particular coal
type, ton; variable SDjt is a monthly amount of particular coal type use (a decrease on
stock), ton; variable SI jt is a monthly amount of particular coal type stock increase,
ton; artificial variable H jt allows fulfill constraint in the case amount of particular coal
type use is higher than an allowed monthly limit, ton; parameter I jt is an allowed upper
limit for a monthly imported amount of a particular type of coal, ton; parameter Djt
is an allowed upper limit for a monthly domestically produced amount of a particular
type of coal, ton.
The important requirement of to the legal document “Procedure for Projected Electricity Balance Formation” [5] is to ensure enough coal amounts at stock including
some guaranteed reserve amount. Next equations are describing the state of the stock
and its dynamics. The required guaranteed amount of coal at stock at the beginning
of the month, FGjt , ton:
F G jt =
I
.
i=1
ai jt Wit k jt
D jt
; ∀t = 1 . . . 12, ∀ j ∈ {A; BC},
M D jt
(7)
where parameter k jt is an exogenously defined parameter (according to [5]) of guaranteed coal at stock change during a month; parameter Djt is a number of the days
for which the reserved guaranteed amount of coal at stock should be enough for unit
operation, days; parameter MDjt is a number of the days in the month, days.
The allowed change of the reserved guaranteed amount of coal at stock during a
month, FGC jt , ton:
F GC jt = F G j (t+1) − F G jt + F G I jt ; ∀t = 1 . . . 11, ∀ j ∈ {A; BC},
(8)
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S. Shulzhenko et al.
where variable FGI jt is an increase of the reserved guaranteed amount of coal at
stock during a month, ton.
Another important indicator describing the actual state of the coal stock is the
operational coal stock that in the ideal case only one is used for electricity generation
by the unit during a month. The next equations of the model ensure enough amount of
operational coal stock for each month of the year. The available amount of operational
coal stock for the beginning of each month (FOjt , ton) should be grate than the use
of the coal during this month (FOC jt , ton) plus increase of operational coal stock
(FOI jt , ton):
F O jt − F OC jt ≥ 0; ∀t = 1 . . . 12, ∀ j ∈ {A; BC},
F O j (t+1) = F O j (t) − F OC jt + F O I jt ; ∀t = 1 . . . 11, ∀ j ∈ {A; BC}.
(9)
(10)
Finally, the increase and decrease of the amount of coal at the stock including the
guaranteed and operational amount of coal is described by the next two equations:
S D jt = F OC jt + F GC jt ; ∀t = 1 . . . 12, ∀ j ∈ {A; BC},
(11)
S I jt = F O I jt + F G I jt ; ∀t = 1 . . . 12, ∀ j ∈ {A; BC}.
(12)
The objective of the mathematical linear programming model is the minimum of
expenditures for electricity production by all TPPs units for whole year:
12 .
.
.
.
(
ci · Wit +
(cij F I jt + cdj F D jt + M j H jt ) +
M p H pt ) → min (13)
t=1
i
j
p
where parameter ci is a fixed expenditures per GW hour of electricity produced by
the TPP unit, USD/GW hour; parameter cj i and cj d are prices for imported coal and
domestically produced coal, USD/ton; parameter M j is a large value corresponding
to the price of penalty amount of the coal consumed, USD/ton; parameter M p is
a large value corresponding to the price of penalty amount of particular pollutant
emission, USD/ton.
4 Calculation Results
The input data for calculations (Table 1) have been obtained from “The Projected
Electricity Balance of the Integrated Power System of Ukraine for 2021” developed
by the Ministry of Energy of Ukraine [19]. The specific coal consumption by TPP
units in Ukraine according to the actual statistics is 0.405 kg tce per MWh in 2020,
and this value is used as the input parameter to calculate coal consumption by TPPs
Thermal Power Plants’ Coal Stock Short Term Projection Method …
Table 1 Input data for the
calculations (based on [19]),
MWh
285
Month of the year Electricity generated Electricity generated
by TPPs
by CHPs
1
4067
1549
2
3479
1443
3
3154
1215
4
2268
720
5
2664
614
6
2848
631
7
3288
729
8
3333
684
9
3089
720
10
3740
1039
11
4132
1486
12
3855
1603
39,917
12,433
Total
for the trough the year, and month by month calculation. The assumed amount of
coal at the TPPs’ stocks at the beginning of 2021 was set to 1 (one) and 2 (two)
million tons, which is corresponding to statistics to assess the required amount of
imported coal for the year.
The amount of electricity generation according to the Ministry of Energy of
Ukraine projection varies month by month for both TPPs, and CHPs. The difference between the highest and lowest amounts of electricity generated by TPPs is
about 25%, while for CHPs the generation during wintertime (highest generation) is
more than 2.5 higher than during summertime. This potentially allows to fill power
plants coal stocks during summertime to be prepared for the peaking coal consumption during wintertime, but the productivity of domestic coal mines is limited, and
also the stock storing volumes is limited, therefore in real life not so evident and
simple to determine how much coal on stock should be for each period.
The results of calculations (Tables 2 and 3) for two options of initial coal stock
states demonstrate that the approach used by the Ministry of Energy of Ukraine is
rather conservative—it is required to keep higher than the necessary amount of coal
on TPPs’ and CHPs stocks, hence in real-life coal-fired power plants not always
keeping the projected requirements, what was absolutely evident during the end of
2021.
5 Discussion and Conclusions
The results of the calculation show that a very important effect on a stable situation
with coal stock has the initial state of the stock. In the case of initial stock volume
–
Required Import to fulfill the Minenergo
projection
1.11
–
On stock deficit (−) / surplus(+) to cover
required guaranteed + operational amount of
coal at the end of the month (based on
Minenergo projection)
1.18
−1.11
−1.18
–
1.03
1.59
1.09
1.82
Required Import according to the calculations –
Coal on stock at the end of the month (the
Minenergo projection[20])
−1.03
0.9
−1.09
On stock deficit (−) / surplus(+) to cover
required guaranteed + operational amount of
coal at the end of the month
0.48
0.64
–
2.1
Required coal on stock at the end of the
month (guaranteed + operational)
1.0
1.94
1.78
1.51
1.1
Required operational amount of coal at the
end of the month for the next two weeks
0.77
2
1.74
1.1
1.0
Coal on stock at the end of the month
Required guaranteed amount of coal at the
end of the month
0.64
2.26
1.80
–
–
Coal consumption
1
Average domestic coal production
−1
1.75
0.96
0.6
0.33
0.49
1.76
0.88
−0.88
1.37
0.46
−0.46
3
1.25
1.11
0.7
0.38
1.00
1.76
0.23
−0.23
1.23
0.11
−0.11
4
0.00
0.06
1.24
0.00
0.11
1.19
0.8
0.41
1.30
1.76
1.46
5
1.56
1.49
0.9
0.59
1.48
1.74
0.00
0.19
1.30
0.01
−0.01
6
1.80
1.51
0.9
0.60
1.42
1.73
0.00
0.08
1.34
0.09
−0.09
7
1.82
1.75
0.00
0.06
1.29
0.8
0.45
1.35
1.75
0.40
−0.4
8
1.69
1.71
1.0
0.68
1.41
1.75
1.31
−1.31
2.71
0.30
−0.30
9
1.94
−1.94
3.05
0.87
−0.87
1.98
1.1
0.83
1.11
1.76
2.06
10
Table 2 Results of calculated monthly balances for coal on Ukrainian TPPs and CHPs stocks for 2021 (initial stock is 1.1 mil ton), mil ton
2.01
−2.01
2.63
1.31
−1.31
1.92
1.1
0.85
0.62
1.80
2.29
11
2.04
−2.04
2.31
1.56
−1.56
1.83
1.0
0.84
0.27
1.80
2.15
12
286
S. Shulzhenko et al.
1.51
−0.03
1.74
−0.09
1.0
1.1
2.1
–
–
–
Required operational amount of coal at the end of the
month for the next two weeks
Required coal on stock at the end of the month
(guaranteed + operational)
On stock deficit (−)/surplus(+) to cover required
guaranteed + operational amount of coal at the end of
the month
Required Import according to the calculations
Coal on stock at the end of the month (the Minenergo
projection[20])
On stock deficit (−)/surplus(+) to cover required
guaranteed + operational amount of coal at the end of
the month (based on Minenergo projection)
Required Import to fulfill the Minenergo projection
−0.11
−0.18
0.18
–
–
0.11
1.59
0.03
0.9
0.64
1.82
0.09
1.0
0.77
1.78
1.48
Required guaranteed amount of coal at the end of the
month
1.64
–
1.94
2.1
2
Average domestic coal production
1.80
1
2.26
−1
–
Coal on stock at the end of the month
Coal consumption
0.00
0.12
1.37
0.00
0.54
0.96
0.6
0.33
1.49
1.76
1.75
3
0.00
0.77
1.23
0.00
0.89
1.11
0.7
0.38
2.00
1.76
1.25
4
0.00
1.06
1.24
0.00
1.11
1.19
0.8
0.41
2.30
1.76
1.46
5
0.00
1.19
1.30
0.00
0.99
1.49
0.9
0.59
2.48
1.74
1.56
6
0.00
1.08
1.34
0.00
0.91
1.51
0.9
0.60
2.42
1.73
1.80
7
0.00
0.60
1.75
0.00
1.06
1.29
0.8
0.45
2.35
1.75
1.82
8
2.71
0.00
0.70
1.71
1.0
0.68
2.41
1.75
1.69
0.31
−0.31
9
0.94
−0.94
3.05
0.00
0.13
1.98
1.1
0.83
2.11
1.76
2.06
10
Table 3 Results of calculated monthly balances for coal on Ukrainian TPPs and CHPs stocks for 2021 (initial stock is 2.1 mil ton), mil ton
1.01
−1.01
2.63
0.31
−0.31
1.92
1.1
0.85
1.62
1.80
2.29
11
1.04
−1.04
2.31
0.56
−0.56
1.83
1.0
0.84
1.27
1.80
2.15
12
Thermal Power Plants’ Coal Stock Short Term Projection Method …
287
288
S. Shulzhenko et al.
at 1 mil ton of coal, a higher amount of imported coal is required compared to the
initial amount at 2 mil ton, even if domestic coal production is the same for both
options. This result directly influences the state energy security—less initial stock
means larger import, which is not so easy always to deal with, especially during
winter time when the consumption is peaking. This issue became evident enough at
the end of 2021.
Another important result the existing coal mines’ production capacities are not
enough to cover the demand for electricity generation by coal burning TPPs and
CHPs. Since coal-burning TPPs are important for grid flexibility, at least in the next
several years increasing domestic coal production to supply TPP units with enough
amount of fuel is a critical task for state energy security, more exactly to ensure grid
stability, not only during wintertime, when renewable generation is low but also in
the summertime, when renewable generation is high but intermittent.
The existing approach of the Ministry of Energy of Ukraine regarding actual
requirements to TPPs to keep some oversized excess of coal on stock is rather conservative but allows to ensure overall state energy security. Since TPPs are operated
according to the electricity market principles they are not interested to keep higher
than actually necessary coal on stock, and that is why they do not always meet
the Ministry of Energy of Ukraine projected requirements regarding stock volume.
The market drives all players to advance the existing approach requiring appropriate
modeling, which could be performed with a proposed mathematical model.
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purchase using genetic algorithms. Eur. J. Oper. Res. 126(1), 218–230 (2000). ISSN 0377-2217,
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M., Duffuaa, S.O., Raouf, A. (eds.) Maintenance, Modeling and Optimization. Springer,
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ugol/control/uk/publish/officialcategory?cat_id=245255478 (in Ukrainian)
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for 2021. Ministry of Energy of Ukraine. http://mpe.kmu.gov.ua/minugol/control/uk/publish/
article?art_id=245566184&cat_id=245183250 (in Ukrainian)
Use of Improved Methodology
to Determine the Total Power Efficiency
of Energy Products in Their
Co-production at Combined Heat
and Power Plant
Vitalii Horskyi
and Olena Maliarenko
Abstract On the example of energy production at a combined heat and power plant,
an improved method of determining the total energy consumption of products and
its components has been considered: direct energy consumption—at the level of the
technological unit or shop; technological—at the level of the technological chain
of production in the production facility or group of facilities; full factory—at the
enterprise level, also includes energy intensity of fixed assets, labour costs, in-plant
transportations, except for the technological energy intensity; total power efficiency
of products—at the level of the country as a whole, in which the total power efficiency
of extraction and transportation of raw materials to the enterprise is added to the total
factory power efficiency. The main methods of energy distribution in co-production
of energy have been analysed. For the four most common methods, calculations of
the direct energy efficiency of energy products have been performed and the use
of the thermodynamic aloccation method has been chosen. The chain of the main
components of total energy consumption in co-production of energy products has
been compiled and the energy intensity of electric and thermal energy for coal-fired
power plants with unit costs per 1 ton of steam produced by energy boilers has
been calculated. The total energy consumption has been calculated and distributed
between the generated heat and electricity. The predicted energy intensity of heat and
electricity production for coal-fired CHP has been determined with the introduction
of the latest technologies of preparation, combustion of coal and waste disposal and
neutralization of discharges and emissions.
Keywords Energy · District heating · Energy supply · Energy efficiency · Direct
energy intensity · Technological energy intensity · Total energy intensity
V. Horskyi (B) · O. Maliarenko
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: witalij.3d@gmail.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_17
291
292
V. Horskyi and O. Maliarenko
1 Introduction
Combined generation of electricity and heat is a major trend in modern development
of energy supply systems in the world. An overview of trends in domestic and world
production of electricity and heat at CHP is provided in the following works: [1, 2].
Increase the energy efficiency of the main and auxiliary equipment of a CHP plant is
an important task, since the National Emission Reduction Plan for large combustion
plants [3] includes number of public CHPs plants and powerful industrial CPHs.
Reducing fuel use is the main direction for reducing greenhouse gas emissions [4],
therefore, a step-by-step analysis of fuel and energy costs in order to identify ways
to reduce it is the crucial task.
An important indicator of energy efficiency, that characterizes the complete technological cycle of production, is the total energy intensity of products [5], which
allows to calculate the energy efficiency of replacement, modernization, reconstruction of technological equipment with detail, which is absent in the calculation of
other indicators of energy efficiency [6].
2 Literature Review and Problem Statement
Distribution coefficients depend on the specific technology. Let us consider the
existing approaches below.
At present, one of the main tasks in the power industry is to increase its level of
reliability and competitiveness. To solve this problem it is necessary to determine
reasonable tariffs for the production of electric and thermal power. An important
indicator of energy efficiency assessment is the total energy intensity of products
and their components. An overview of existing approaches and own developments
in the methodology for estimating the total energy intensity is given in [1].
The current DSTU 3682-98 “Energy efficiency methods for the determination
of power consumption of production, works and services” provides a methodology
for determining the components of the total power consumption of products without
distributing the total energy costs in many food industries. An improved methodology
that takes into account the need to distribute total power costs in multicommodity
production is provided in [7]. On the one hand, tariffs for served energy should
reflect all types of production costs and ensure a certain level of profitability of
energy supply organizations. On the other hand, tariffs should encourage consumers
to reduce energy consumption and optimize the power supply regime [8]. Controlling
the prices of the monopolist’s company is a difficult task facing the state. At energy
companies, the calculation of costs associated with production and energy resources
transmission, is based on the following components:
• fuel costs;
• purchased electricity costs,
• costs of paying for services of third parties;
Use of Improved Methodology to Determine the Total Power Efficiency …
•
•
•
•
•
•
293
costs of raw materials and supplies;
fixed asset renovation costs;
labour costs and social security contributions;
depreciation costs of fixed assets and intangible assets;
sustained activity costs;
other (shop) costs.
One of the most important methodological issues in energy is the optimal
distribution of costs between generation and transmission of electric and thermal
energy.
At present, there are a number of methods for allocating costs by product. The
most common among them are the following:
• Calculation in accordance with energy validation of heat.
• Depreciable value methods:
(a) charging off the residual costs over electric power;
(b) charging off the residual costs over thermal power.
•
•
•
•
Energy value method.
Physical method.
Method of reducing the production of electrical energy.
Methods of distribution of savings:
(a) the method of equal savings;
(b) the method of proportional savings;
(c) the method of the total profit distribution.
For a long time, the cost-cutting method was first used with the cost reduction
factor of 1 ton of extracted steam compared to the cost of live steam [8]:
(1)
are cost of 1 ton of
where y is cost reduction factor of 1 ton of steam;
extracted and live steam (UAH/t) respectively.
It was proposed to determine the cost reduction factor in the following ways:
(1) by the ratio of the enthalpy of extracted steam hextr to the enthalpy of upstream
of the turbine—h0 [8]:
y=
h extr
.
h0
(2)
Since the enthalpy of extracted and live pairs is close to each other, the cost of
extracted steam differs little from the cost of live steam and the cost reduction factor
of extracted steam is close to one, i.e. almost both steams have the same value.
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V. Horskyi and O. Maliarenko
Therefore, all the benefits of cogeneration (heat-and-power supply) with this method
of calculation are charged off to electricity;
(2) by the value of heat utilization of steam flows in the turbine (capacity factor)
[8]:
y=
h extr − h cond
,
h 0 − h cond
(3)
where hcond —enthalpy of downstream steam fed to a condenser.
In this case, the enthalpy drop, underused in the low-pressure turbine cylinder
(hextr − hcond ), and the available enthalpy drop (h0 – hcond ) differ significantly in
value, resulting in excessively reducing the cost of extracted steam and most of the
savings from combined generation of electricity and heat is accounted for served
heat;
(3) by the average value of the above factors (engineer Rumiantsev’s formula) [8]:
(
y = 0.5 ·
)
h extr − h cond
h extr
.
+
h0
h 0 − h cond
(4)
This calculation formula was valid until 1937, when the “thermodynamic” method
of allocating costs at CHP was replaced by a physical or balance method, as advised
by A. S. Gorshkov (Mosenergo), where the total costs allocation is proportional to
the amount of fuel consumed for the production of each product [9, 10]. This method
was once approved by the scientific and technical community and recommended by
the energy management as an official one and is used practically to this day.
It should be noted that the decision of the Scientific and Technical Council, held by
G. Krzhyzhanovskyi Power Engineering Institute together with the Moscow Scientific and Engineering Society of the Energy Industry (1952) [11], who adopted the
physical (balance) method of cost allocation at CHP read as follows: “Methods for the
fuel savings allocation in cogeneration of thermal and electrical energy between these
types of energy received cannot be a consequence of the laws of thermodynamics,
and all attempts to directly thermodynamically substantiate one way or another to
partition fuel savings between the types of energy received are devoid of scientific
foundation”, which is currently being criticized.
The advantage of the balance (physical) method is the unambiguity in the allocation of savings and the simplicity of practical calculation by CHP employees.
This method is not economically justified. In the balance (physical) method, all the
savings from the cogeneration of electric and thermal energy at the CHP relate only
to electricity, so its cost is underestimated, and the cost of heat is inflated.
The use of this method leads to the following disadvantages [12]:
(1) the transition to higher initial steam parameters at CHP leads to a reduction in
the cost of electricity and an increase in the cost of heat, because the total capital
costs increase, and operating cost savings are mostly charged off to electricity.
Use of Improved Methodology to Determine the Total Power Efficiency …
295
Therefore, with increasing initial parameters of steam at the CHP, the efficiency
of heat generation decreases;
(2) the fuel component of the cost of heat at the CHP does not depend on the pressure
in the steam draw-off and therefore reducing the steam pressure in the draw-offs
does not lead to a reduction in the cost of heat;
(3) an increase in blow off the steam from the turbines of the CHP does not lead to
a decrease in the cost of heat.
Some of these shortcomings were eliminated by a special heat tariff [6]. Thus,
this method is not right for the essence of the technological process at the CHP
and its economic results and does not meet the requirements of cost allocation in
cogeneration. Therefore, there has always been a task to improve the method of cost
allocation at the CHP.
Except for to the balance method, in a real-case scenario, the method of “shutdown” was used, where the total cost of cogeneration excluded the cost of byproducts, estimated at the cost of their generation at other enterprises or the set
price (tariff).
When applying this method in power engineering, the so-called L. L. Ginter’s
triangle was used [7]. When it is constructed, on one side of the right-angled triangle,
the cost of 1 kWh is marked off, and on the other, the cost of 1 GJ (1 Gcal) (Fig. 1).
The sides of the triangle CA and CB are determined by the maximum value of the
cost of electricity and heat at a given annual operating costs.
Providing that [8]:
(5)
The highest cost of 1 kWh will be at Qser = 0, and 1 GJ—E an = 0.
Given the cost of one type of energy, you can determine the cost of the second
type.
The disadvantage of the Ginter triangle method is the impossibility of simultaneously determining the cost of heat and electricity. The Ginter triangle can be used
Fig. 1 Ginter’s triangle [8]
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V. Horskyi and O. Maliarenko
in design conditions, for example when comparing combined and separate power
supply schemes.
In 1963, a compromise method of cost allocation was proposed, based on the
allocation of profits in the cogeneration of electricity and heat at the CHP. This
method implies that the ratio of the cost of electricity to the cost of heat in their
cogeneration should be the same as the ratio of the cost of CPP electricity and the
cost of heat produced in a specialized boiler house [8]:
(6)
In due time, methods of fuel cost allocation between products in cogeneration were
widely developed, including both electricity and heat at CHP, through the use of the
concept of exergy, which allows one to represent both quantitative and qualitative
characteristics of energy in one quantity [10, 13], etc.
The paper [10] concludes not in favour of using exergy for such calculations:
“The entropy method, the method of efficiency (exergy method), has the logical
justification that all losses of real cycles do not mean the disappearance of energy,
but the loss of its energy value, measured by thermal potential and entropy. From
a thermodynamic point of view, this justification is correct. However, in energy
production there are not only thermal processes and not all thermal processes have
the ultimate goal of getting work value”.
The exergy method of cost allocation was proposed in 1956 by Z. Rant. The use
of the energy method is based on the exergy balance of CHP [13]. It is assumed
that the cost of the fuel, charged off to the generation of electricity and heat, must
be determined by dividing the fuel consumption in accordance with the ratio of the
electric power to the decrease in the exergy of the heat-transfer fluid. Taking into
account all EFs, we get a decrease in the cost of electricity.
All rational methods must meet the following test criteria: with a decrease in
the pressure while the intermediate steam bleeding, the cost of this steam generation must constantly decrease; under the limiting conditions, when the pressure of
the intermediate steam bleeding reaches the value of the pressure that exists in the
condensers of the condensing turbines, the calculated cost of generating this steam
should be zero or close to zero.
Neither the physical nor the compromise methods meet these verification criteria,
at the same time the exergetic method meets them, because it assesses the quality of
the steam not by its enthalpy, but by its efficiency.
Since CHP operates as part of a power system, when choosing a cost allocation
method for such a CHP, except for the thermodynamic criteria, it would be worth
considering its impact on capital costs and the cost of electric energy transmission
in the power system. Such accounting was proposed in 1965 as the development of
the exergetic method [13].
This method of cost allocation for CHP takes into account that the exergetic cost of
electricity generation is higher than for its production at CPP because of the supply of
Use of Improved Methodology to Determine the Total Power Efficiency …
297
heat at CHP, and therefore extra costs should be borne by heat consumers. Therefore,
according to Wagner’s method, the generation of electricity at CHP should consume
as much fuel as it does at CPP. Fixed costs in the prime cost (depreciation charges,
salaries, etc.) of electricity at CHP should be the same as in the power system. Then
the unit cost of electricity generation found by this method will be less than that
found by the exergy method.
Widely used until recently (1998) normative document GKD 34.09.103-96
provides for the allocation of CHP costs between thermal and electrical energy by
the physical (balance) method, which has certain disadvantages: all savings due to
combined generation of electrical and thermal energy are charged off to electricity,
and fuel consumption per served unit of heat 1 GJ (1 Gcal) at CHP was higher than
in boilers designed to supply heat only [14].
Therefore, in connection with a significant increase in the cost of fuel and a corresponding increase in the tariff for served thermal energy on behalf of the National
Electricity Regulatory Commission (NERC) OJSC, in 1997, “LvivORGRES” developed a methodology “Distribution of fuel consumption in thermal power plants for
served electric and thermal energy at their cogeneration” (GKD 34.09.108-98) [14].
It was a supplement to GKD 34.09.103-96. In this case, the calculation of all indicators of the thermal efficiency of power plants is carried out in accordance with
the specified methodology, with the exception of fuel consumption and specific fuel
consumption for served electric and thermal energy, are determined by the new
method—GKD 34.09.108-98.
This technique is based on the principle of equal benefit, in which fuel savings
due to the combined generation of electricity and heat at CHP are distributed equally
between them—a factor of 0.5.
In this method, the allocation of fuel between types of energy, its consumption
for served thermal energy are determined taking into account the factors of the value
of heat supplied to external consumers from the extraction of CHP steam turbines.
As a result, fuel consumption for electricity generation increases compared to the
calculation by the physical method, and decreases for the supply of thermal.
This allows to increase the estimated heat supply from CHP and increase economic
interest in the combined generation of electricity and heat.
In 2009, S. V. Dubovskyi and O. O. Hortova [15] presented the theoretical features
and the main results of the calculation of energy efficiency of steam turbines by
thermodynamic method.
According to the thermodynamic method, the heat on the supply of thermal and
electrical energy from steam turbines is calculated by the formulas [15]:
Q el = Q ·
E ser y
Q T er m = Q ·
E ser y
,
+ ω · Q ser y
ω · Q ser y
,
E ser y + ω · Q ser y
(7)
(8)
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V. Horskyi and O. Maliarenko
where Qel —heat consumption to electricity supply; Q—actual heat consumption per
the steam turbine unit; E serv —actual electricity supply; Qserv —actual heat supply;
QT —heat consumption for thermal energy supply.
The specific heat consumption for served electric and thermal energy is determined
by the following ratio [15]:
Q el
;
E ser y
qel =
qT er m =
Q T er m
.
Q ser y
(9)
(10)
The average factor of thermodynamic value of heat is the ratio of the specific
consumption of primary heat [15]:
ω=
qT er m
.
qel
(11)
This method, unlike the empirical ones, uses real, rather than conditional, values
of the turbine operation parameters as initial values. As you know, the main difficulty
of STP energy estimates, like other cogeneration plants, is due to the physical inseparability of the working fluid flow at the turbine inlet into components associated with
getting work and heat. The thermodynamic approach makes it possible to make such
a separation using an objective law that follows directly from the first and second
principles of thermodynamics and establishes a relationship between energy inputs
and outputs of combined processes.
It should also be noted that recently the number of mini-CHP and large boilers,
retrofitted with build-up gas turbine engines. And as shown by a comparative analysis of the thermal efficiency of cogeneration units in operation [16], the results
of calculations, obtained by different methods, differ significantly from each other
(Table 1).
According to the normative method (No.1), electricity generation EF is extremely
high, and under conditions of constant thermal power, it shrinks to one with a decrease
in electric load. Meanwhile, the heat generation EF is lower than the EF of the
boiler, which produces this heat capacity. According to “Energoproekt” Scientific
Research Institute’s method (No.2), heat generation EF exceeds one, which goes
against common sense. According to other methods, EF values for heat generation
make greater values than the boiler EF, which is also inconsistent with the physical
nature. Given the above, it is hardly possible to recognize the existing methods of fuel
allocation between types of energy products as satisfactory for cogeneration plants
of this type, due to their significant difference from the existing CHP.
Use of Improved Methodology to Determine the Total Power Efficiency …
299
Table 1 Comparison of calculations using different methods, based on mini-CHP [16]
Methodology number
Indicators
1
2
3
4
1
Reference fuel
consumption, kg/s:
for electricity
generation;
for heat generation
0.10867
1.14393
0.2416
1.0354
0.2416
1.0354
0.17395
1.0786
2
Specific consumption of 0.13727
reference fuel:
38.132
for electricity
generation, kg/(kWh);
for heat generation,
kg/GJ
0.3052
33.763
0.2734
34.512
0.2197
35.954
3
Gross efficiency:
for electricity
generation;
for heat generation
0.403
1.011
0.4483
0.993
0.5591
0.95
0.896
0.895
3 Purpose and Objectives of the Study
The purpose of the study is to adapt an improved method for determining the total
energy intensity of energy—electric and thermal energy, simultaneously produced
at a combined heat and power plant from a steam turbine unit during the combustion
of coal and natural gas, and include both the direct use of fuel to obtain steam, and
support costs of energy resources for the preparation and supply of fuel to the burners
of steam generators (own needs of the plant), reduction of which directly affects efficiency of energy production, use of recycled energy resources (RER), consideration
of energy efficiency costs for environmental protection measures, improved algorithms for determining the labour costs energy consumption, fixed assets, production
and transportation of fuel to the power plant.
The objectives of the study are the analysis and selection of the method of distribution of total energy consumption in combined energy production at CHP, determination of the total energy intensity of joint production of heat and electricity throughout
the technological chain of their production.
4 Improved Methodological Approach to Determination
of Total Energy Intensity of Products
The definition [7] of several types of energy efficiency: direct, technological, full
factory and full energy efficiency with algorithms for their calculation has been put
forward in publication.
300
V. Horskyi and O. Maliarenko
The chain of consumption of energy resources from fuel supply to CHP to heat
generation with the allocation of two options of fuel management to determine the full
energy intensity of energy carriers from CHPP and its components on the example
of a coal-fired plant with a heating turbine type “T” with heating steam extraction for
the needs of municipal consumers is considered. Assessment of the energy-saving
potential at coal-fired TPPs with implementation of innovative technologies was
analyzed in [4tezi], and the results were taken into account in the initial data for
the calculation and selection of auxiliary equipment of each of the options.
According to [7]:
• the direct energy intensity is defined as follows:
edn = kn'
.
es bs ,
(12)
where kn'' —partition coefficient choice for a n-th specific multi-product manufacturing technology; s is the index of the type of energy resources; es —full energy
intensity of s-th type of energy resources; bs —specific consumption of s-th type of
energy resources in the main production;
• the technological (component of the total) energy intensity of the energy carriers
produced at the CHP, is defined as follows:
etecn =
kn'
.
s
(
es bs +
.
)
'
ai' bis
+ eenv ,
(13)
i
where in addition to the above, i—index of the type of auxiliary production; ai' —
'
—specific consumption
specific consumption of i-th type of auxiliary production; bis
of s-th type of energy resources for the production of i-th type of auxiliary production, eenv —full energy intensity of costs used for environmental protection in the
production of products (services);
• the total factory energy intensity is defined as follows:
et f n = kn' (etecn + ez ),
(14)
where in addition to the above, ez —energy intensity of auxiliary factory energy
consumption (energy intensity of fixed assets, energy intensity of labor costs, energy
intensity of in-plant transportation);
Use of Improved Methodology to Determine the Total Power Efficiency …
301
• the total energy intensity is defined as follows:
(
)
etotn = kn' eext + etr + ei pt ,
(15)
where in addition to the above, eext —energy intensity of raw material extraction [5],
etr —energy intensity of raw material transportation to the enterprise [5], ei pt —energy
intensity of in-plant transportation [17].
Calculations of technological energy intensity are made according to the next
alternatives. An information database was created and a mathematical model was
developed to calculate the total energy intensity throughout the whole technological chain of costs at CHP, the calculations were performed using the Microsoft
Excel application [18]. The results are shown in Fig. 2. Energy intensity of environmental protection measures in the technological energy intensity was calculated by
the method [19] and by the initial data given in [20].
The first option is a coal-fired power plant technology with the most common basic
and auxiliary equipment currently operating at Ukrainian thermal power plants, such
as:
•
•
•
•
gas turbines for defrosting fuel in winter;
ball mills for milling;
dust feeding with a normal concentration of dust in the air stream;
flaring in chamber furnaces with hydro ash removal and wet electric filters for
cleaning from solids;
• semi-dry lime technology desulfurization and selective catalytic reduction to
decrease nitrogen oxide emissions.
Fig. 2 Energy intensity of energy production at CHP together depending on the type of equipment
and type of fuel, excluding burned fuel [21]
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V. Horskyi and O. Maliarenko
Improved coal-fired power units, with technologies partially used in selected
thermal power plants and promising at present, including the following main and
auxiliary equipment, such as:
• gas-fired fuel defrosting radiant panels;
• roller or hammer mills;
• dust feeding with high concentration of dust in the air flow and boilers with circulating fluidized bed technology, which allow to organize measures for reduction of
nitrogen oxides right in the furnace during combustion without additional energy
costs;
• dry bottom ash removal;
• purification from solid particles was chosen in the form of dry electrostatic precipitators, and desulfurization technology used semi-dry ammonia, were considered
as a second option.
For determination of the energy intensity of electricity and heat generated at the
CHP (Table 1), it is required to find the operating parameters of the turbine unit with
the help of the mode diagram. From the diagram of the modes of operation of turbine
unit T-110/120-130 we can get:
•
•
•
•
•
the steam consumption by the turbine—482.5 t/h;
nominal electric capacity of 110 MW;
the nominal heat output of 208.7 MWh;
the steam consumption for fuel in the amount of 10 t/h;
the steam consumption for own needs—7.5 t/h.
During entering the obtained data into the developed model, we can obtain the
energy intensity of energy production at the CHP with a partitioning by the two types
of energy produced by the four modes of distribution:
•
•
•
•
50%/50%, developed by “LvivORGRESS”;
by the cost of energy carriers, taking into account the prices as of 10.2019 [21];
by specific fuel consumption for the production of a given type of energy carrier;
adapted to the methodology of total energy intensity of production by thermodynamic method [15] (Table 2).
5 Analysis of Results Obtained
The results of the calculation of the total energy intensity of energy sources produced
at the coal-fired power plant for different technologies are given in Table 3.
Use of Improved Methodology to Determine the Total Power Efficiency …
303
Table 2 Allocation of energy consumption for heat and electricity supply
Distribution method Power intensity of CHP
energy supply
Option 1
MJ/MW
%
MJ/MW
%
Without distribution Energy together
2634.19
100
2621.88
100
50/50% [14]
by the cost of
energy [21]
by specific fuel
consumption [13]
Thermodyna-mic
method [15]
Option 2
(a) thermal power
1724.99
65.48
1716.93
65.48
(b) electric power
909.20
34.52
904.95
34.52
(a) thermal power
1207.80
45.85
1202.15
45.85
(b) electric power
1426.39
54.15
1419.73
54.15
(a) thermal power
1056.60
40.11
1051.67
40.11
(b) electric power
1577.59
59.89
1570.21
59.89
(a) thermal power
476.00
18.07
473.78
18.07
(b) electric power
2,158.19
81.93
2148.10
81.93
6 Conclusions
Cogeneration of electricity and heat is the main trend of modern development of
energy supply systems in the world. The share of electricity production by CHP in
Ukraine correlates with the share of cogeneration in G8+5 countries and equals to
11–19%. The coefficient of fuel heat consumption by CHP in EU countries reaches
75% [22]. An important indicator of energy efficiency, which characterizes the full
technological cycle of production, is the total energy intensity of production. To determine the technical and economic performance of CHPs, the cost of energy products,
and justified tariffs, it is necessary to determine the approach to the distribution of
energy consumption for the output of each of the products. Currently there are a
number of methods for the distribution of energy consumption by product type in
cogeneration. All methods give different results in the assessment, and the discrepancy is quite significant. By analyzing and comparing them, it is possible to identify
both the advantages and disadvantages of each method, depending on the task of
assessment. Thermodynamic distribution method was chosen as the most reasonable
one among the studied methods. The whole chain of energy costs at coal-fired TPPs
was considered with the reduction of specific energy intensity per 1 ton of direct
steam, and distribution of these energy costs on the produced electricity and heat for
steam turbine TPPs with different technologies of coal preparation and combustion
was performed using the thermodynamic method. Analysis of the obtained results
showed that the fuel preparation and combustion technology can have a significant
impact on the overall energy intensity of production.
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V. Horskyi and O. Maliarenko
Table 3 Calculation of total energy consumption
Type of energy
resources, other
resources and energy
saving indicators
Resource
costs for
traditional
technologies
of coal
preparation
and
combustion
(var.1),
(n.u./t steam
Resource
costs for the
latest *
technologies
of coal
preparation
and
combustion
(var.2),
(n.u./t
steam)
Total energy
consumption
of the
resource
(MJ/n.u.)
1. Energy
consumption in the
fuel economy—Total,
Including:
Total energy
consumption
of products
with
traditional
technologies
(MJ/t of
steam)
Total energy
consumption
of products
with the
latest *
technologies
(MJ/t of
steam)
2634.19
2621.88
96.91
96.91
27.0
2616.57
2616.57
Gas (for defrosting
coal in winter), m3
0.19
0.09
31.0
5.89
3.16
Electricity (for
grinding and
transportation of coal
to the boiler plant),
kWh
3.24
0.59
11.73
2.15
59.89
51.41
59.89
51.41
24.76
24.76
24.76
24.76
Coal, kg
3.62
2. Energy
consumption in the
boiler room—Total,
Including:
Electricity (for dust
supply, traction and
blasting equipment,
smoke extractors,
feed pumps,
regenerative water
heater, chemical
water treatment, slag
removal), kWh
16.54
14.21
3.62
3. Energy
consumption in the
turbine
department—Total,
Including:
Electricity
(circulating pumps,
mains pumps,
drainage pumps),
kWh
6.84
6.84
3.62
(continued)
Use of Improved Methodology to Determine the Total Power Efficiency …
305
Table 3 (continued)
Type of energy
resources, other
resources and energy
saving indicators
Resource
costs for
traditional
technologies
of coal
preparation
and
combustion
(var.1),
(n.u./t steam
Resource
costs for the
latest *
technologies
of coal
preparation
and
combustion
(var.2),
(n.u./t
steam)
Total energy
consumption
of the
resource
(MJ/n.u.)
Direct energy
consumption of
energy
carrier—Total (1 +
2 + 3)
4. Energy
consumption at the
purification
plant—Total,
Including:
Total energy
consumption
of products
with
traditional
technologies
(MJ/t of
steam)
Total energy
consumption
of products
with the
latest *
technologies
(MJ/t of
steam)
2718.84
2698.05
30.26
11.54
Electricity, kWh
3.96
2.09
3.62
14.33
7.56
Lime (sorbent), kg
3.3
1.1
3.62
11.95
3.98
Catalyst, kg
1.1
0
3.62
3.98
Technological
energy intensity of
energy
carriers—Total (1 +
2 + 3 + 4)
5. Total energy
intensity of fixed
assets
6. Total energy
intensity of labor
costs, man-hours
0.18
149
Total factory energy
consumption—Total
(1 + 2 + 3 + 4 + 5
+ 6), Including:
0
2749.11
2709.59
164.95
135.47
27.49
27.09
2941.55
2872.15
Electricity, MJ
0.599
1761.69
1720.42
Thermal energy, MJ
0.401
1179.86
1151.73
28.09
29.18
7. Energy intensity of
raw material
extraction
Electricity, kWh
7.76
8.06
3.62
(continued)
306
V. Horskyi and O. Maliarenko
Table 3 (continued)
Type of energy
resources, other
resources and energy
saving indicators
Resource
costs for
traditional
technologies
of coal
preparation
and
combustion
(var.1),
(n.u./t steam
Resource
costs for the
latest *
technologies
of coal
preparation
and
combustion
(var.2),
(n.u./t
steam)
Thermal energy, Mcal
3.0
3.12
8. Energy intensity of
raw material
transportation to the
enterprise
0.23
0.239
Total energy
consumption
of the
resource
(MJ/n.u.)
Total energy
consumption
of products
with the
latest *
technologies
(MJ/t of
steam)
12.55
13.05
6.74
7.00
2988.93
2921.38
Electricity, MJ
1793.82
1753.79
Thermal energy, MJ
1195.11
1167.59
Total energy
consumption—Total
(1 + 2 + 3 + 4 + 5
+ 6 + 7 + 8)
4.184
Total energy
consumption
of products
with
traditional
technologies
(MJ/t of
steam)
29.3
Including:
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Physical Model of Structural
Self-organization of Tribosystems
Vitalii Babak , Nataliia Fialko , Vitalii Shchepetov,
and Sergii Kharchenko
Abstract The results of the study of the physical model of the structural evolution of
the processes of self-organization of the surface structures of materials in rhenium are
presented. It is noted that the conditions of self-organization counteract the increase in
entropy. And the fundamental phenomenon of structural adaptability is an important
manifestation of self-organization for the theory of practice. The implementation of
which in the established range causes the manifestation of physical and chemical
processes and mechanisms of structural ordering in such a way that the production
of entropy and the inevitable dissipation of energy tend to a minimum. At the same
time, secondary structures are reborn on the contact surfaces with rhenium, which
are extremely resistant to destruction. Based on experimental and theoretical data on
fine structures and final states of surface layers, a hypothetical scheme of a physical
model of self-organization of surface structures under friction conditions is proposed.
The scheme of self-organization under wear is considered. It is shown that selforganization as a phenomenon of rhenium is a logical expression of the universal
phenomenon of structural adaptability. It is noted that the process of regeneration and
destruction of secondary structures is described by a first-order differential equation
with a retarded argument.
Keywords Self-organization · Secondary structures · Entropy · Friction · Physical
model · Structural adaptability
1 Introduction
The search for the most general approaches to explaining tribological processes
based on the fundamental laws of nature has been going on for a long time and
intensively. The greatest achievement in this area is the application of methods of
V. Babak (B) · V. Shchepetov · S. Kharchenko
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: vdoe@ukr.net
N. Fialko
Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_18
309
310
V. Babak et al.
non-equilibrium thermodynamics and the theory of self-organization. The scientific
basis of which was the phenomenon of structural adaptability of materials during
friction, which is of exceptional importance in technology [1].
In the field of functional interaction, the key point in considering the patterns
of manifestation of statistically averaged interdependent relationships is the analysis of energy parameters. According to the energy interpretation, the fundamental
sequence of tribological processes is realized due to the main physical mechanism—
the phenomenon of structural adaptability of materials under friction conditions. This
fundamental conclusion is based on an experimentally established fact, namely, for
all materials and working media there are ranges of loads and relative displacement
velocities, within which the friction and wear indicators are stable and several orders
of magnitude lower than outside these ranges, which are defined as critical values
activation and passivation energies, as well as the corresponding conditions for the
formation of protective ordered dissipative structures.
The patterns of processes under friction loading make it possible to determine that
structural-thermal activation, which causes the interdependent flow of tribophysical and tribochemical reactions that stimulate the regularity and adequacy of the
formation of secondary structures, shows eternal opposition and competition to the
conditions of their dynamic equilibrium. Nevertheless, the phenomenon of structural
adaptation of materials during friction, established by fundamental physics, related
to the phenomenon of self-organization, allows us to speak about the fundamental
possibility of realizing the maximum permissible (theoretical) wear resistance.
The relationship between the processes of self-regulation of destruction and
restoration of secondary structures consists in the ability of the thermodynamic friction system, starting from the end of running-in, to maintain a constant level of all
tribological parameters characterizing the normal process over time.
The purpose of this work is to present a physical model of the structural evolution
of the processes of self-organization of protective surface films from the standpoint
of the structural-energy concept of friction.
2 Main Part
The development of modern hardening technologies is due not only to the requirements for obtaining new types of products with previously unknown sets of properties, but also to the need to save resources (materials and energy) and environmental
cleanliness. And for the most part, these requirements are met by technologies based
on the principles of nonlinear thermodynamics. In accordance with the properties of
solutions of non-linear equations describing the indicated technologies, the processes
occurring at high levels of excitations are very unstable. Therefore, they can be
controlled with the help of small energy impacts. Hence the logical requirements for
knowledge of the micro mechanisms underlying them, the exact observance of the
sequence of technological processes, the accuracy of control and the qualifications
of personnel. All of the above fully applies both to materials science in general and
Physical Model of Structural Self-organization of Tribosystems
311
to the problem of contact interactions of solid surfaces and wear resistance, in particular. Traditional methods of obtaining and processing structural materials have now
exhausted themselves. The tasks are both the development of new resource-saving
and environmentally friendly technologies, and the creation of new materials. At the
same time, accents are changing in relation to various characteristics of materials, if
earlier the main ones were average characteristics, such as tensile strength, ductility,
tensile strength, etc., now the requirements for a small dispersion of all properties
come first.
The most important characteristic of most parts and assemblies that are in contact
interactions is wear resistance. During friction, the surface layers of the material
are under conditions of high energy impacts, and their rapid destruction under these
conditions can only be prevented if so-called dissipative structures appear in the
material, which have the ability to convert almost all of the supplied energy into
heat. In wear-resistant materials, latent energy, which then turns into fracture energy,
can be significantly reduced.
Many modern trends in increasing wear resistance are based on non-linear
processes. Coating methods are also undergoing significant changes, so there is a
strong point of view that this should be a multi-stage cycle, in which a number of
stages are desirable to be carried out at a non-equilibrium state of the substrate.
Thus, at the present stage, the physics of friction, wear and increase in wear resistance has come to the need to include in the consideration of nonlinear phenomena
occurring in the material at several interconnected scale structural levels, which
requires a deep study of the physical processes occurring in the surface structures of
solids under high excitation conditions.
The state of self-organization of matter in open systems is as fundamental as
the spontaneous transition of closed systems to an equilibrium state with maximum
entropy. When self-organizing, materials and systems counteract the increase in
entropy, which opens up new reserves of scientific and applied orientation. At the
same time, the possibilities of achieving self-organization are not special, since the
state of stability and metastability is realized, which are much more than equilibrium
ones.
The established phenomenon of structural adaptability during friction represents a
wide class of exceptionally striking manifestations of self-organization. The possibilities for implementing a wide range of physicochemical processes and mechanisms
of structural reordering are created in such a way that the production of entropy and
the inevitable dissipation of energy are as small as possible. In fact, new structures
appear on the contact surfaces, while friction acts as the creator of new materials that
are extremely resistant to destruction under friction loading—secondary structures.
And the phenomenon of adaptability during friction can be interpreted (for a given
combination of materials and boundary conditions) in a certain range of loads and
speeds of movement, all types of interactions are localized in a minimum volume
and such a spectrum of dissipative metastable structures and such a distribution of
their volumes and dissipative flows between them is realized that the total entropy
production was minimal. Because of this, the achievement of a steady state at the
macro level occurs as a result of the flow of a complex of physicochemical and
312
V. Babak et al.
mechanical phenomena in the surface layers of the coatings, caused by an activating
factor, due to which the original structure changes, undergoing a transformation.
As a result of these transformations, the regularities and nature of which are determined by friction modes, secondary structures are formed on the contact surfaces,
which characterize the irreversible residual state of the surface layers. Using the
experimental results, ideas about particle sizes and the number of defects, dispersion
and orientation processes, data on the role of intermediate and final states of thinfilm structures, a hypothetical scheme of the mechanism for the formation of surface
layers is proposed (Fig. 1).
According to the structural-energy theory, formed in the works [2–5], the work
of friction forces (A) in accordance with the first law of thermodynamics is mainly
converted into heat (Q) and is partially stored by the rhenium node (.E). The release
of heat causes thermal activation of processes during friction. In the general energy
balance under normal friction, the second term (.E) is insignificant (less than 1%).
However, this energy, stored in the thinnest submicrolayer of the contact zone with a
Friction loading
Elastic-plastic deformation of the
surface layer
Structural
thermal
activation
Anomalous
processes of
adsorption
Dispersion and
structure
orientation
Passivation due
to interaction
with active elements of the en-
Solid state
topochemical
reactions,
mechanochemica
alloying
Formation of
submicrorelief
and optimal topography
vironment
Formation of surface films of
secondary structures
Fig.1 Scheme of the physical model of the transformation of surface layers during friction
Physical Model of Structural Self-organization of Tribosystems
313
thickness of the order of tens and hundreds of nanometers, determines such a limiting
energy density per unit deformed volume that metal in the solid phase can absorb.
The high density of stored energy causes anomalous effects of increased activity
of contact submicrovolumes and the formation of structures that are characterized
by a specific mechanism of plastic deformation and an oriented ultrafine structure.
The maximum possible strength in this case can be achieved under the condition of
maximum energy saturation surface volumes and optimization of energy dissipation
under friction loading.
The study of the manifestation of normal wear and the conditions for the occurrence of damage found that the dynamic equilibrium of the processes of destruction
and restoration of secondary structures is realized under the condition under the
condition Vp = Vb , where Vp is the rate of destruction of secondary structures and
Vb is the rate of their recovery.
The presentability of the quasi-stable state, as a consequence of dynamic equilibrium, is ensured if Spl = const, where Spl is the total area of the contact surface
shielded by secondary structures. Stability of dynamic equilibrium (Vp = Vb ) and
a certain range of friction parameters and environmental conditions is ensured at: p
< εkr Vi , Ci ; V < νkr Vi , where ρkr Vi , Ci are critical values of velocity Vi and friction
parameters Ci , Vkr , Ci are the critical speed values for some fixed value of the load and
parameters Ci . It is possible to single out three main groups of passivation reactions
during friction, which are realized under strictly defined conditions, the first—in the
interaction with the active components of the medium, the second—with the counter
body material, and the third—due to the internal restructuring of the surface layers.
The passivated state of the working surface, due to the ordered dynamics of the
shielding structures, corresponds to the minimum values of friction parameters and
corresponds to normal mechanochemical wear. However, as a result of the impact of
external conditions (plastic deformation, temperature, etc.), the dynamic equilibrium
shifts towards an increase in structural-thermal activation, which causes a change in
conditions, and the wear process changes qualitatively [6–8].
The process occurring in the thinnest surface layers can be conditionally divided
into three stages, the first one is deformation and activation, the second one is the
formation of secondary structures, friction is the destruction of secondary structures.
Thus, structural-thermal activation determines the peculiar course of physical and
chemical reactions and has a decisive influence on the emergence and development
of processes under external friction.
The second side of this most complex interaction in nature is related to the fact
that the state of self-organization of matter in open systems is as fundamental as
the spontaneous transition of closed systems to an equilibrium state with maximum
entropy [9–11]. When self-organizing, materials and systems counteract the increase
in entropy, which opens up reserves for their scientific and applied application. At
the same time, it should be noted that the possibilities of achieving self-organization
are not unique, since the states of stability and metastability of materials are realized,
which are much more than equilibrium ones.
One of the parameters of self-organization in rhenium is running-in, i.e., the quasirelaxation of the tribosystem structure from equilibrium to a stable state, the former
314
V. Babak et al.
being subject to the condition of minimum free energy, while stability is controlled
by the minimum entropy production [12–15].
To consider the mechanism of self-regulation of the processes of destruction and
restoration of secondary structures, the physical model of normal wear is represented
by a block diagram (Fig. 2.), in which the following designations are accepted:
S—total contact area, Spl —area covered with a film, Sp —part of Spl subjected to
destruction, Sb is the part of the juvenile surface on which the film was restored, Z
is the film thickness, i = kZSp is the wear rate, k is the coefficient of proportionality,
f is the mismatch equal to the area of the juvenile surface at each moment of time,
i.e. ε = Sp − Sb ; load p and sliding speed V combined into one vector (g) [16, 17].
As a result of deformation, the activated layer and the active components of the
medium (in particular, oxygen) present at the friction point form secondary structures
during physicochemical interaction. As a result of repeated loading and the presence
of internal stresses in the film of secondary structures, the formation, accumulation
and development of microdefects occurs, and on the interface between the film and
the base metal, the bonds are weakened and settled. Subsequent mechanical impacts
(vector g) cause the destruction and wear of the film fragments. On the juvenile
areas of the friction surface, the process is repeated. Moreover, the area on which the
destruction occurred depends on the strength and thickness Z of the film, therefore:
Sp = a(g, z)Spl (block a(g, z)).
On each elementary section, the moments of destruction and restoration of the film
are separated by a non-zero-time interval, which is dictated by the discreteness of
the contact and the finiteness of the sliding speed. In other words, at each moment of
time, the film is restored only on the area of the juvenile surface. So, considering, for
example, the parameter characterizing the discreteness of the contact under steady
wear, constant, we get: Sin = β(V)Sp β(V) ≤ 1(block β(V)). Thermodynamically
stable is the state when the entire contact surface is covered with a film, therefore ε
Sb
β(V)
ε
i
k
α(q,z)
S
Spl
Sp
z
q
V(q)
Fig. 2 Structural diagram of self-regulation of processes during wear of metals
Physical Model of Structural Self-organization of Tribosystems
315
→ 0. But due to the delay in the restoration of the film, only the condition ε = ε0
> 0 is satisfied, which corresponds to the dynamic equilibrium of the processes of
destruction and restoration of secondary structures. The thickness of the destroyed
film is a function of the vector g (block V(g)).
The block diagram corresponds to the following system of equations: Spl = S
− f ; f = Sp − Sb ; Sp = α1 (q)Spl ; SB = β(V)Sp ; i = khSp ; z = γ(g); α1 (g) = α[q,
z(g)]. Therefore, we can write: i = [kSγ (q1 C)α1 (q1 C)]/1 + α1 (q1 C)[1 − β(v, C)].
The expression explicitly contains the vector C, the components of which are the
parameters of materials and working media.
Self-regulation as a phenomenon of friction is a logical expression of the universal
phenomenon of structural adaptability. In fact, new phases appear on rubbing surfaces
under conditions of adaptability (as a result of mass transfer with the medium and
structural-chemical transformations), while friction acts as the creator of new materials that are extremely resistant to destruction during friction (surface structures)
[18, 19].
To describe the regularity of the phenomenon of structural adaptability, which
consists in the fact that in a certain (for a given combination of materials and boundary
conditions of the scale, external temperature, physical and chemical means of the
medium) range of loads and movement speeds, all types of interactions are localized
in a minimum volume and such a spectrum of dissipative metastable structures and
such a distribution of their volumes and dissipative flows between them, in which the
total production of entropy would be minimal, a physical model of self-regulation is
proposed, which is shown in Fig. 3.
The following designations are used to describe the model under consideration:
Sf is the area of actual contact, Sy is the area of juvenile areas on the surface of
f1
τ-ψ
Wоb
WЕ
∫
Sy
Spl
Wр
Sf
f2
q(t)
Fig. 3 Dynamic model of self-regulation of metals and environment in the friction zone
316
V. Babak et al.
actual contact, formed as a result of destruction and wear of films, Wp is the rate of
destruction of films (an increase in their area per unit time), g(τ) is the generalized
loading parameter, depending on the value of the specific load p(τ), f1 and f2 are
functions expressing the dependences of the rates of formation and destruction of
films of secondary structures on the corresponding parameters, ψ is the time between
the moments of destruction and wear of films (the formation of juvenile areas) and
the appearance of new films in these areas. The parameter ψ is mainly characterized
by the penetrating ability of the medium and the rate of physicochemical interaction
between the medium and the juvenile metal surface. The following dependences
correspond to the considered physical model: dSpl /dτ = Wob (τ) − Wp (τ) = WE (τ);
Wob (τ) = f1 [V(τ), Sy (τ − ψ)]; Wp (τ) = f2 [g(τ)]; Sy (τ) = Sf − Spl .
The film formation rate is proportional to the sliding velocity and the area on which
their formation is possible and can be written as Wob = kW(τ)Sy (τ − ψ) where k
is the film formation intensity factor. Taking into account the independence of the
mechanical effects of the specific load and the speed of movement, it can be written
that Wp [g(τ)] = a1 (V(τ) + a2 p(τ)), where a1 and a2 are coefficients depending on
film strength [20].
Thus, the process of formation and destruction of secondary structures is described
by a first-order differential equation and a retarded argument dSpl /dτ + k(τ)Spl (τ −
ψ) = V(τ)(kSf − a1 ) − p(τ)a2 , which for p, v = const describes the state of selfregulation and stationarity of the processes of destruction and formation of films of
secondary structures.
Based on the analysis of the physical model of the transformation of surface
layers during friction and the results of structural changes, taking into account the
thermodynamic processes of mechanical energy dissipation, it was found that the
physical and chemical wear mechanisms are invariant with respect to materials during
friction. In addition, the indicated processes and mechanisms determine the nature
and patterns of wear of both metal and metal-polymer friction units, regardless of
the conditions of their loading and lubrication.
3 Conclusions
1. The evolution of the structure of surface layers of metallic materials and coatings
under contact interactions is considered. A physical model of self-organization
of surface layers during friction and an analysis of the state of surface structures
subjected to energy impact are presented.
2. It is shown that one of the fundamental theoretical and applied achievements
underlying the fundamental sequence of friction processes and the implementation of the physical wear mechanism is the structural adaptability of materials
under friction loading.
3. The process of self-organization of materials under conditions of structural adaptability, which stimulates the regularity and adequacy of the formation of protective hardened dissipative structures, is considered. On the basis of the structural
Physical Model of Structural Self-organization of Tribosystems
317
scheme of self-organization of wear processes, it is presented that the regeneration
and destruction of secondary structures is described by a first-order differential
equation with a retarded argument.
4. The mechanism of self-organization, which causes the appearance of normal
wear and the occurrence of damage, is considered, the condition of dynamic
equilibrium of the processes of destruction and restoration of secondary structures during friction is established. It is emphasized that the state of the contact
surface, due to the ordered dynamics of the shielding structures, corresponds
to the minimum values of the friction parameters and corresponds to normal
mechanochemical wear.
5. The activated changes in the structure of a thin surface layer in the process of
self-organization under friction loading are described, on the basis of which a
sufficiently complete and consistent physical model is constructed that meets the
fundamental principles of thermodynamics of irreversible processes.
6. Based on the analysis of the physical model of self-organization of surface layers
during friction and the results of their structural changes, taking into account
the thermodynamic processes of dissipation of mechanical energy, it has been
established that the physicochemical wear mechanisms are invariant with respect
to the materials of friction pairs.
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Fuels
Effect of Diethyl Ether Addition
on the Properties of Gasoline-Ethanol
Blends
Viktoriia Ribun , Sergii Boichenko , Anna Yakovlieva ,
Lubomyr Chelaydyn , Dubrovska Viktoriia , Shkyar Viktor ,
Artur Jaworski , and Pawel Wos
Abstract This paper presents experimental studies carried out to investigate the
effect of diethyl ether on the properties of gasoline-ethanol blends. The solubility of
ethyl alcohol of different dehydration degrees in gasoline and stability of gasolineethanol blends was studied. It is shown that higher degree of dehydration provide
better stability of gasoline-ethanol blends. The anti-knock properties of ethanolcontaining gasolines with different content of ethanol and diethyl ether additive was
studied. The synergistic effect of anhydrous ethanol/diethyl ether mixtures on the
properties of composite gasoline is shown. A mathematical model for calculating the
octane number of gasoline-ethanol-diethyl ether blends has been developed. Amounts
of some exhaust gases emission of gasoline and ethanol-containing fuels were studied
experimentally and compared.
Keywords Anhydrous ethanol · Diethyl ether · Stability · Synergistic effect ·
Octane number · Exhaust gases · Emissions
V. Ribun
Chemical-Analytical Laboratory of the PJSC Ukrnafta, Kyiv, Ukraine
S. Boichenko (B) · D. Viktoriia · S. Viktor
Institute of Energy Saving and Energy Management, National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine
e-mail: chemmotology@ukr.net
A. Yakovlieva
Ukrainian Research and Educational Center of Chemmotology and Certification of Fuels,
Lubricants and Technical Liquids, National Aviation University, Kyiv, Ukraine
L. Chelaydyn
Department of Environmental Protection Technology, Ivano-Frankivsk National Technical
University of Oil and Gas, Ivano-Frankivsk, Ukraine
A. Jaworski · P. Wos
Department of Automotive Vehicles and Transport Engineering, Rzeszów University of
Technology, Rzeszów, Poland
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_19
321
322
V. Ribun et al.
1 Introduction
Modern transport sector is strongly dependent on non-renewable energy resources
(oil, coal, gas), which are used to produce motor fuels—gasoline, diesel fuel and
liquefied petroleum gas. However, taking into account scarcity of fossil fuels and
negative impact of transport on environment (transportation is responsible for about
14% of global carbon dioxide emissions), during last decades countries actively
promote introduction of alternative motor fuels. Among the great variety of existing
today biofuels, bioethanol is considered as one of the most promising alternatives to
substitute petroleum-derived gasoline completely or partially in a form of gasolineethanol blends (GEBs).
For Ukraine GEBs are currently considered as a way to reduce the consumption
of light petroleum products and to improve the ecological characteristics of the
environment. Actually, more than half of ethanol, produced worldwide is used as
an additive to fuels for internal combustion engines. Many countries, including the
United States, Brazil and Sweden, use fuels with an ethanol content of up to 85%,
therefore road vehicle manufacturers produce cars equipped with engines, which are
already adapted to gasoline-ethanol fuels. For example, in France fuel containing 5%
ethanol is widely used [1]. In general, the dynamics of biofuel usage in the world is
constantly growing. Consuming about 200 million tons of fuel and energy resources
annually, Ukraine is considered an energy-deficient country because it does not fully
cover the needs for energy resources and imports up to 85% of petroleum products.
Such a state of the energy economy causes dependence of Ukraine on oil and gas
exporting countries and threatens the country’s national energy security. Operating
only 30% of its total capacity, the alcohol industry of Ukraine fully satisfies the
domestic needs for the alcoholic beverage production [2]. That is why the study of
properties of gasoline-ethanol blends (GEBs) containing various additives, finding
out and elimination of the main drawbacks of these fuels is an important scientific
and applied problem.
2 Literature Overview
Internal combustion engines (ICEs) can operate with different types of fuel if the
temperature is high enough to initiate fuel ignition at the end of the compression stroke
[3]. So the use of blended fuels for internal combustion engines is appropriate and
efficient [4, 5]. Oxygen-containing additives such as diethyl ether (DEE) and ethanol
are used in the internal combustion engine (ICE) to eliminate the drawbacks of fuel
combustion in the internal combustion engine in order to increase engine efficiency
and control the combustion. Moreover, mixing DEE and ethanol is a promising way
to prepare additives for blended fuels as DEE is higher reactive than most fuels, has
appropriate evaporation heat, high oxygen content, low ignition temperature, which
boosts ignition and cold start of the engine [7, 8]. Ethanol (E) is high octane biofuel
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
323
made from crops with a higher heat of vaporization, rate of laminar flame spread and
additional oxygen atoms compared to gasoline [9, 10].
Several studies show that DEE and ethanol, due to their higher oxygen content
in its chemical structure, reduce CO, unburned hydrocarbons and soot emissions
[11, 12]. The authors [13] showed that DEE contributes to the completeness of fuel
combustion. In addition, some authors claim that at the engine speed of 1500 rpm
butanol-gasoline mixtures have an earlier ignition time than ethanol-gasoline mixture
containing the same fraction of alcohol. Results of the studies on using ethanol
mixtures with gasoline are considered in [14].
However, there is a lack of researches on the use of DEE and ethanol mixtures
in different proportions as additives to gasoline for improving the octane number of
gasoline as anti-knock property and the stability of blended fuels for spark ignition
ICE. Taking into account the mentioned above the study of cumulative effect of DEE
and ethanol use for improving octane rating of gasoline seems to be relevant.
Over the past decade, transport’s greenhouse gases emissions have increased at
a faster rate than any other energy using sector. Rising of road traffic causes the
need in improvement of fuel efficiency and decreasing exhaust gases emissions
from motor transport [15]. Taking into consideration this urgent problem, the use
of oxygen-containing additives in motor fuels may significantly contribute to reduction of exhaust gases during fuels combustion. Despite the numerous studies devoted
to evaluation of emissions during ethanol-containing fuels combustion, there is a lack
of studies of DEE effect on emissions reduction.
The aim of the work is to study the synergistic effect of DEE and ethanol on the
operation and physical–chemical properties of blended gasoline fuels.
To achieve the aim of the study the following tasks should be fulfilled:
. to study experimentally physical stability of GEBs;
. to study the effect of DEE additives on physical stability of GEBs;
. to study the influence of DEE and ethanol additives on the octane number (ON)
of blended gasoline fuels;
. to propose optimal composition of GEBs with DEE additives;
. to study experimentally amounts of exhaust gases emission in a result of GEBs
combustion.
3 Materials and Methods of the Study
For determining physical stability of GEBs gasoline of grade A-92-Euro 5 and
samples of ethyl alcohol of different purity were used. For blending with gasoline
we have used industrially produced rectified alcohol (96% vol.)—E96, industrially
produced alcohol (90% vol.)—E90 and anhydrous ethyl alcohol (100% vol.)—E100.
Anhydrous ethyl alcohol was synthesized with calcium oxide and distilled using
calcium chloride tube [13]. To determine the composition of ethanol samples we
performed infrared spectral analysis of the original 96%, industrial 90% and absolute
ethanol received in a result of synthesis.
324
V. Ribun et al.
The following GEBs were studied 100/0, 90/10, 80/20, 70/30, 60/40, 50/50, 0/100
(were first number is ration of gasoline, and second is ration of ethyl alcohol).
To study the influence of ethanol and DEE on ON of GEBs the samples were
prepared using gasoline of grade A-80 and anhydrous ethyl alcohol containing
2.343% of diethyl ether. The following samples were prepared: 0.5% diethyl ether
+ 0.5% ethanol + 99% gasoline, 1% diethyl ether + 1% ethanol + 98% gasoline,
1.5% diethyl ether + 1.5% ethanol + 97% gasoline.
Physical stability of GEBs was studied by means of optical methods by parameters
of optical density, light transmission and refractive index. Measurements were done
using photoelectric colorimeter KFK-2.
Density of GEBs was determined using standard method for determination of
density of oil products using set of areometers.
Anti-knock properties of GEBs were studied by the parameter of ON. ON was
measured using octane/cetanometer Shatox 100, research and empirical methods for
ON determining as well. The principle of operation octane/cetanometer is to determine the detonation resistance of gasoline on the basis of measuring its dielectric
constant and resistivity. Determination of the ON of gasoline mixtures was performed
by the research method using reference mixtures of isooctane and n-heptane. Empirical determination of ON of GEBs was performed using the calculation method
according to the formula:
ONGEB = [26.44−0.29(ONo )] ln CE + [1.32(ONo )−29.49],
(1)
where ONGEB —ON gasoline-ethanol blend, ONo —ON of base gasoline, CE —
ethanol content in gasoline-ethanol blend, % (vol.)
To evaluate the amount of exhaust gases emissions formed during the complete
combustion of gasoline with the amount of gases formed during complete combustion
of the developed ethanol-containing fuels, the theoretical volume of air required for
complete combustion of 1 kg of fuel was calculated according to the formulae 2.
The theoretical volume of the content of products of complete fuel combustion was
calculated according to the formula 3–7.
V =
(
)
H
S
O
mol C
+ +
−
,
0.21 12
4
32 32
(2)
where C, H, S, O—mass content of elements in fuel; V mol —volume of 1 mol of air;
0.21—volumetric content of oxygen in air.
VCO2 = 22.4
C
;
12
(3)
VO2 = 0.21(α − 1)V ;
(4)
VN2 = 0.79 · α · V ;
(5)
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
325
H
VH2 O = 22.4 ;
2
(6)
S
,
32
(7)
VSO2 = 22.4
where V —volume of air necessary for complete combustion of 1 kg of fuel.
The theoretical content (%) of chemical elements in ethanol, DEE, and ethanolcontaining fuel was calculated according to the formula (8). Data on the content of
chemical elements in gasoline are taken from the reference literature.
ωelement =
n · Ar element
,
Mr f uel
(8)
where n—number of atoms of the element; Ar —atomic mass of element; M r —molar
mass of element.
The theoretical content (%) of chemical elements in ethanol-containing fuel was
calculated according to the formula (9):
E blend =
.
K i Ei,
(9)
where E blend —content of element in fuel blend, %; K i —content of i-th component
in fuel blend; E i —content of element in i-th component of blend.
4 Results and Discussion
4.1 Analysis of Physical–Chemical Properties
of Gasoline-Ethanol Blends
The physical–chemical properties of the GEBs were analyzed. Samples were
prepared by blending gasoline with ethyl alcohol (E100, E96 and E90) in different
ratios.
In order to understand the solubility of ethyl alcohol of different dehydration
degrees (E100, E96 and E90), photo colorimetric and refractometric analysis of
gasoline-ethanol blends was performed. The curves of the optical density, light
transmittance and refractive index of the blends are shown in the Figs. 1, 2 and
3 respectively. Some trends can be observed from Figs. 1, 2 and 3, namely, adding
small quantities of ethyl alcohol to gasoline (up to 10%) leads to the separation of
gasoline-ethanol blends.
Gasoline, alcohol and water form an emulsion, because in the presence of fine
water droplets the stability of gasoline-ethanol mixtures is disturbed. Due to the alkyl
residue and high polarity, ethanol molecules have affinity to both water and gasoline
326
V. Ribun et al.
Fig. 1 Dependence of optical density of GEBs on the ethanol dehydration degree and its content
in the blends
Fig. 2 Dependence of light transmission coefficient of GEBs on the ethanol dehydration degree
and its content in the blends
and can serve as stabilizers. However, if the amount of ethanol molecules is low, they
cannot provide emulsification of water and gasoline molecules. Visually, this can be
seen as turbidity.
Deviation from the linear dependence of optical density and light transmission
coefficient on the content of hydrous ethanol in GEBs at an ethanol fraction of 10
vol. % (Figs. 1 and 2) proves the inability of small amounts of ethyl alcohol to
emulsify the water-gasoline system. However, the use of anhydrous ethanol solves
this problem and a tendency to linear dependence on the curve of anhydrous alcohol
can be observed.
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
327
Fig. 3 Dependence of refractive index of GEBs on the ethanol dehydration degree and its content
in the blends
Additionally, the refractive index of GEBs with different content of ethanol and
in different ratios was studied (Fig. 3). The high content of ethanol (50% vol.) significantly changes the refractive index of the original gasoline (1435). However, the
introduction of small amounts of alcohol (up to 20% vol.) does not significantly
change the refractive index of GEBs.
As can be seen from Figs. 1, 2 and 3, addition of small amounts of ethyl alcohol
to gasoline (up to 10%) leads to the separation of GEBs. The use of absolute ethanol
does not provide such effect, and the curve of absolute alcohol can be traced to linear
dependence of GEBs stability on ethanol content.
Since density is an important physicochemical parameter for fuels, the effect of
ethanol concentration and content on the density change of gasoline-ethanol fuels
was studied.
From Fig. 4 it can be concluded that the addition of 90% and 96% ethyl alcohol
in a volume up to 30%, and absolute alcohol up to 50% does not cause a significant
change in density.
The curves of the dependences of the GEBs density on the content of ethanol and
its concentration express a linear relation. The higher the ethanol content the higher
the density of ethanol-containing gasoline as the alcohol density is higher than the
gasoline density.
4.2 Analysis of Anti-knock Properties of Gasoline-Ethanol
Blends
After studying the physical–chemical parameters of GEBs, a study of the anti-knock
properties of ethanol-containing fuels was conducted. Synthesized by the authors [13]
328
V. Ribun et al.
Fig. 4 Dependence of density of GEBs on the ethanol dehydration degree and its content in the
blends
dehydrated ethanol containing 2.343% diethyl ether (DEE) was mixed with gasoline
and its effect on the octane number (ON) of blended fuels was investigated. The
physicochemical characteristics of some oxygen-containing additives for gasoline
were previously analyzed (Table 1).
As it can be seen from Table 1, DEE is similar in density and molecular weight to
A-92 gasoline, and the boiling point of DEE (Tb = 34.6 °C) is close to the boiling
point of A-95 gasoline (initial boiling point Tb = 40 °C). These properties are really
useful for operating boosted gasoline engines in winter, when ignition is complicated
because of low temperatures (< −10 °C).
Therefore, to evaluate the combustion of GEBs prepared with anhydrous ethanol
containing DEE, the ON of such mixtures (with an anhydrous ethanol content of 10–
95 vol. %) was determined by experimental and empirical methods (Fig. 5) [14]. The
blends were prepared using gasoline of grade A-80 and anhydrous ethanol containing
DEE.
Table 1 Physical–chemical characteristics and ON of some oxygen-containing additives and
gasoline
No
Oxygen-containing
additives/gasoline
Molecular weight,
g/mol
Density ρ, kg/m3
Boiling temperature
tb , °C
ON
1
Dimethyl ether
46.07
0.002
−24.9
105
2
Diethyl ether
74.12
713
34.6
110
3
Methanol
32.04
792
64.5
156
4
Ethanol
46.07
789
78.29
132
5
Gasoline A-92
72
730–780
40–205
95
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
329
Fig. 5 Dependence of ON of gasoline A-80 on the volume of anhydrous ethyl alcohol containing
DEE (99.95%)
ON of GEBs, determined experimentally, are significantly higher comparing to
those determined by the empirical method (Fig. 5). This difference is explained by
the fact that the empirical method for determining the ON takes into account only
the content of gasoline and ethanol in the GEBs [14]. Based on experimental data,
it can be assumed that the increase in ON of GEBs is due to the presence of DEE in
those blends.
Further experimental studies were performed to determine the ON of GEBs
without DEE, GEBs containing DEE and pure DEE. It was found that ethercontaining anhydrous ethanol increases the ON of GEBs.
Based on experimental data, a mathematical model for determining the ON of
GEBs containing DEE was developed.
.
4
C D E E )))
.
· ln(C E + (1.32 · (ON0 ) − 29.49)
ONG E B = (26.44 − 0.29 · (ON0 − (4.6 ·
(10)
Thus, ether-containing ethyl alcohol enhances the gasoline ON more intensely. At
a content of 20–40% vol. of such ethanol, the gasoline ON increased to 91–95 units
(Fig. 6). At a content of 20–40% of such ethanol, the gasoline ON increased to 91–95
units. At the same time, according to empirical calculations, this content of dehydrated ethanol should increase the octane number of A-80 gasoline to 85–88 units
(Figs. 4 and 5) therefore not so intensely. At the maximum possible content of anhydrous ethanol (80–90%) in gasoline A-80 octane number according to calculations
should reach 91–93 units (Figs. 4 and 5), however, introducing the same amount
of ether-containing anhydrous alcohol into gasoline causes much more effective
increasing in the octane number and it reaches 97–97.5 units (Figs. 4 and 5).
330
V. Ribun et al.
Fig. 6 ON of blends determined by experimental and empirical methods
Empirical model adequacy Sad was verified using Fisher’s test (F), which can be
calculated by the following formula:
.
F=
2
2
Sad
/S y2 , i f Sad
> S y2
2
2 ,
2
2
S y /Sad , i f S y > Sad
(11)
2
2
where Sad
—adequacy variance; Sad
—adequacy of experiment.
The adequacy variance is determined by following formula:
.N
2
Sad
=
1=1 (yic
− yie )2
f
,
(12)
where yic —calculated values of the parameter; yie —experimental values of the
parameter; f —the number of degrees of freedom.
The number of degrees of freedom is determined by the formula:
f = N − K.
(13)
The number of experiments is chosen within 10–99% of the ethanol content in the
blend. For this study N = 13. The number of approximation coefficients depending
on the ON of ethanol was calculated as the number of all numerical coefficients in
the formula, taking into account the exponents and bases of logarithms (1). In this
case, k = 7. Then the number of degrees of freedom f = 13–7 = 6.
The variance of the experiment was calculated from the values obtained in a series
of repeated experiments following:
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
331
.N
S y2
=
2
i=1 (yi− y)
f
,
(14)
where yi —values obtained in each experiment; y—the average values of the
measured parameters; n—the number of repeated experiments.
The number of repeated experiments at each point was 3. The arithmetic mean
was taken as a result of the experiment at each point. The model can be considered
adequate with a corresponding reliable probability, if the calculated value of the
Fisher criterion does not exceed the tabular data. The reliable probability is 95%.
Due to the fact that GEBs prepared on the basis of absolute ethanol can get some
water during storage, transportation and operation, the effect of diethyl ether (DEE)
on the stability of blended fuels and their octane numbers was examined.
Since the lowest stability is found in GEBs containing less than 20% of ethanol,
they were chosen to study the effect of diethyl ether on the stability of gasolineethanol fuels.
As we can summarize from Figs. 7 and 8 diethyl ether has a positive effect on
the stability of gasoline-ethanol fuels, causing increasing the light transmittance
and decreasing the optical density of the examined compositions. Namely, these
properties characterize the stability of GEBs: the higher the transmittance and lower
optical density, the more stable GEBs are.
Diethyl ether as well as ethyl alcohol has better performance than gasoline. In
particular, the octane number of both oxygen-containing additives exceeds the octane
number of gasoline by 20–30 units. However, ethyl alcohol has a significant disadvantage, namely, low saturated vapor pressure. To address these weaknesses of diethyl
ether containing ethyl alcohol was blended into gasoline.
Fig. 7 Dependence of light transmission of GEBs containing 5% diethyl ether: 1—100% gasoline;
2—a blend containing 90% gasoline and 10% additives; 3—a blend containing 80% gasoline and
20% additives; 4—a blend containing 70% gasoline and 30% additives
332
V. Ribun et al.
Fig. 8 Dependence of optical density of GEBs containing 5% diethyl ether: 1—100% gasoline;
2—a blend containing 90% gasoline and 10% additives; 3—a blend containing 80% gasoline and
20% additives; 4—a blend containing 70% gasoline and 30% additives
To study the effect of diethyl ether on the ON of GEBs, several reference mixtures
were prepared. ON of blends were measured using a Shatox 100 octanometer (Fig. 9).
DEE and ethanol have a synergistic effect on engine performance. The addition
of only pure ethanol or only DEE in the amount of 1, 2 and 3% increases the ON of
fuel blends by 2–3 units, and the addition of DEE/E mixtures containing ethanol E
(0.5% DEE + 0.5% E; 1% DEE + 1% E and 1.5% DEE + 1.5% E) increases ON
by 4–6 units.
Fig. 9 The effect of the oxygenated additives on the octane number of gasoline
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
333
4.3 Analysis of Exhaust Gases Emissions
from Gasoline-Ethanol Blends Combustion
Automobile exhaust gases are a complex mixture of toxic components, including
nitrogen oxides, carbon dioxide and carbon monoxide, sulfur oxides, unburned
hydrocarbons, soot, aldehydes, and others. The composition of the exhaust gases
is not constant and may vary depending on the fuel composition, type of internal
combustion engine, its operating mode, load and technical condition of the vehicle
[15–19]. Complete combustion of fuel results in emissions of carbon dioxide, sulfur
dioxide, water, and nitrogen oxides [3, 6, 7].
To compare the amount of exhaust gases produced during the complete combustion of gasoline with the amount of exhaust gases produced during complete combustion of the developed GEBs, the theoretical volume of air required for complete
combustion of 1 kg of fuel and the theoretical volume of products of complete fuel
combustion were calculated.
We have used a GEB sample that was composed of 75% gasoline (G) and 25%
absolute ethanol (E) containing 2.34% of DEE was selected. This amount of DEEcontaining absolute ethanol increases the ON of A-80 gasoline by 12.4 units to 92.4.
Table 2 shows the component composition of the studied fuel samples.
Table 3 provides results of calculation of the theoretical amount of air required for
complete combustion of 1 kg of fuel sample and the volume of products of complete
fuel combustion.
After theoretical calculations, the experimental study of CO2 and SO2 in exhaust
gases was performed using gas analyzers. Portable gas analyzers of the “Dozor” type
Table 2 Elemental composition of studied fuel samples
Content, %
Fuel sample
C
H
S
O
Gasoline
85
14.95
0.05
Ethanol
52.17
13.04
–
13.79
DEE
64.84
13.51
–
21.65
GEB (75% gasoline + 25% DEE containing ethanol)
76.87
17.57
0.02
0.05
5.54
Table 3 Composition of exhaust gases during complete fuel combustion
Fuel sample
Volume of air, Vair , m3
Volume of combustion products V, m3
CO2
H2 O
SO2
O2
N2
Gasoline
11.43
1.59
1.67
0.0004
0.96
3.36
Ethanol
6.96
0.97
1.46
–
1.02
9.33
DEE
8.64
1.21
1.51
–
1.64
3.45
GEB (75% gasoline + 25%
DEE containing ethanol)
7.464
1.22
1.26
0.0004
0.98
4.83
334
V. Ribun et al.
Fig. 10 The amount of air required for the combustion of gasoline (G) and its mixtures with
DEE-containing ethanol (E) and the amount of carbon dioxide emissions in the exhaust gases
were used to analyze the composition of exhaust gases during the operation of the
carburetor engine on A-95 and GEBs (75% gasoline + 25% DEE containing ethanol
and 50% gasoline + 50% DEE containing ethanol). The studies were performed at
the minimum crankshaft speed nmin = 800 min−1 ± 100 min−1 and at the maximum
crankshaft speed nmax = 2200 min−1 ± 100 min−1 . Figure 10 presents the comparative analysis of theoretical and experimental amounts of CO2 in the exhaust gases
of gasoline and GEBs and the amount of air required for their combustion.
Therefore, the presence of ethanol and DEE in motor fuels reduces the amount
of air required for fuel combustion and the amount of carbon dioxide in the exhaust
gases. Moreover, the higher the percentage of additives, the more significant its effect.
This effect is explained by the presence of oxygen atoms in ethanol molecules.
5 Conclusions
In the result of this study the physical–chemical and operation properties of gasolineethanol fuels were studied. The obtained results allow us concluding the following:
1. The use of mixtures of DEE and absolute ethanol has further research prospects
for improving the operational and physical-chemical characteristics of blended
oxygenated fuels.
2. It is shown that ethyl alcohol blending into gasoline reduce the stability of ethanolgasoline blended fuels. However, increasing ethanol concentration and the DEE
introduction stabilize blended fuels.
3. The DEE additives cause a synergistic effect on the ON of GEBs.
4. It has been developed the mathematical model for determining the ON of GEBs
blends containing diethyl ether.
Effect of Diethyl Ether Addition on the Properties of Gasoline-Ethanol …
335
5. It is shown that ethyl alcohol blending into gasoline allows improving environmental properties of fuels. Mainly, it allows reduction of carbon dioxide
emissions and amount of air required for fuel combustion
The fulfilled study allows us solving an important scientific and applied problem
of the rational use of gasolines with ethanol content. The results of the study may
be used during developing technology of production of oxygen-containing additives
from domestic raw materials and new compositions of motor fuels with improved
operational and environmental properties.
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Efficiency of Electric Logging
in Thin-Layer Sections of Hydrocarbon
Deposits (Gas Fields of the Precarpathian
Depression)
Oleksiy Karpenko , Mykyta Myrontsov , Yevheniia Anpilova ,
and Oleksii Noskov
Abstract The non-trivial task of searching for and diagnosing oil and gas deposits
in the sections of wells represented by thin-layer deposits remains relevant. Existing
traditional methods of interpreting logging data are focused on other types of sections.
In thin-layer deposits, the effectiveness of logging methods is much lower. Anisotropy
and the mutual influence of the characteristics of neighboring strata eliminate anomalies in geophysical curves. This leads to numerous gaps in productive formations and
objects in the well sections. Thickness values for the thickness of thin single layers
sufficient to determine their geophysical characteristics, and further—and reservoir
properties, are not suitable for bundles or layers of thin-layer thickness of the section.
Despite significant advances in the theory and practice of interpreting these electrical
methods data, geophysical characteristics or logging curves in front of thin-layer
intervals of well sections in most cases do not allow direct effective quantitative and
qualitative geophysical and geological interpretations for individual strata. Only the
integration of these geophysical methods, the use of new approaches to the interpretation of logging data can increase the effectiveness of exploratory research. The
statistical approach to the creation of new synthetic parameters allows the formation
of contrasting anomalies in front of gas or oil-saturated formations. The research is
devoted to this and the results are presented below.
Keywords Thin-layered section · Well logging · Electrologging · Gas deposit ·
Electrical resistivity · Interpretation
O. Karpenko (B)
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
e-mail: karpenko.geol@gmail.com
M. Myrontsov · Y. Anpilova · O. Noskov
Institute of Telecommunications and Global Information Space of the National Academy of
Sciences of Ukraine, Kyiv, Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_20
337
338
O. Karpenko et al.
1 Introduction
The thin-layered type of section of sedimentary rocks is interpreted differently
by specialists—representatives of different geological disciplines. The concept of
“layer” in general geology is “an elementary unit of layered texture of sedimentary
rocks, which differs from adjacent similar units in material composition, particle
size, mineralogy, rock structure, nature of inclusions, staining, etc.“; in field geology,
“layer” is defined as “an elementary unit in the description of a section.“ In stratigraphy, it is “a term of free use to denote small lithostratigraphic subdivisions, which
often have local development” [1, p. 544]. “Layering” is characterized as “the texture
of sedimentary rocks, which is expressed in the alternation of thin layers with a thickness of 1 mm to 1 cm” [1, p. 544]. Common in these definitions is the property of
the rocks of the layer to differ from the rocks that contain it, some characteristics
that are the subject of study of a particular geological science. Thus, the widely
used term—“thin-layer section” should have distinctive features in different geological and geophysical disciplines. In geophysics, the term “fine-grained” is associated
with the geometric resolutions of research methods. In the field of well logging, “thinlayering” is the property of a section consisting of a sequence of individual layers of
rocks that differ from each other in lithological and reservoir characteristics, which
creates anomalies in the curves of geiphysical methods, the width of which is to
quantify the geophysical and geometric parameters of these layers. Naturally, when
geologically interpreting the results of well logging on such anomalies, it is impossible to accurately estimate the lithological, reservoir or industrial characteristics of
the layers (using standard methods of interpretation).
Thus, Siberian scientists V. G. Mamyashev and I. G. Glazunov set the lower
limit for estimating the electrical resistivity of layers in the layered thickness of
about 2 m [2]. They divide the layering into levels: the first—macrolayer and the
second—microlayer. The first level of stratification is characterized by layers of
rocks from 2 m and less—to the lower limit of the resolution of the methods—0.4–
0.6 m. If the inhomogeneity is significantly less than the vertical resolution of the
method (stratification is not distinguished by the shape of the curves), then use the
“method of anisotropic formation” according to V. P. Zhuravlyov, also using the
data of electrical methods [3]. When applying the above methods, it is possible to
obtain only approximate values of geophysical parameters (electrical resistivity) of
individual lithological components of the layered strata.
The often used expression “thin-layer section” without reference to a specific
geological or geophysical field of study creates difficulties in comparing the results
of the diagnosis of rocks by different methods. Thus, even for different logging
methods in the study of oil and gas wells, the vertical resolution varies significantly—from the first tens of centimeters (micromethods, LL and caliper) to 2–4 m
(electrical gradient probes of large size). For most conventional measuring devices of
radioactive, acoustic, electric focused methods, this value is 0.4–0.8 m. The figures
are approximate, because the resolution property, in each case, depends on the well
conditions, the ratios of the measured parameters of closely spaced layers and layers,
Efficiency of Electric Logging in Thin-Layer Sections of Hydrocarbon …
339
the frequency of alternation of layers with different properties. For example, for a
single layer it is possible to determine the value Rz (resistivity of the invaded zone) by
the data only of small gradient probes with a thickness of at least 1 m [4, p. 103]. This
is the maximum size of the layer sufficient to obtain the optimal value of resistivity.
Determining the resistivity Rt of a layer using large gradient probe readings requires
an even greater limit thickness of a single layer. In the sonic log method (SL), it is
advisable to determine the value of the interval time when the thickness of the single
layer is greater than the base of the probe, usually—0.4 m.
These limit values of the thickness of thin single layers, sufficient to determine
their geophysical characteristics, and then—and capacitive properties, are not suitable for bundles or layers of thin-layer thickness of the section. Due to the mutual
influence on the probe readings (especially large, with a significant radius of the study
area) of a number of adjacent layers and strata with different values of geophysical
parameters, the resulting log curve will have a smoothed appearance; the integral
characteristic of the thin-layer section of the well will be observed and registered.
Thus, in the above we can highlight the following: it is necessary to specify the
definition of the term “thin layer” for geological objects studied by well logging
and use it for such types of sections, the geometric properties of which do not allow
using conventional (standard) methods of interpretation to carry out a quantitative
assessment of geophysical parameters and geological properties separately for each
layer (layer) of small thickness of a certain lithological affiliation.
From the analysis of the content of numerous scientific publications and production reports, it should be noted that, despite significant advances in the theory and
practice of interpretation of electrical logging methods, geophysical characteristics
or logging curves opposite thin-layer intervals of well sections in most cases do
not allow directly perform qualitative geophysical and geological interpretations for
individual layers of rocks of small thickness. This also applies to the data of other,
non-electric methods.
2 Purpose and Objectives of the Study
Issues related to the study of thin-layer rock deposits are considered. Such rocks are
common in the sections of numerous gas fields of the Outer Zone of the Precarpathian
Depression. Rocks of terrigenous composition, macro- and microlayered, predominate in the deposits of the Neogene system, as well as in the Neocomian and Senonian
sections of the Cretaceous system.
Lower Sarmatian sediments (Lower and Upper Dashava subsuites) are composed
of layers of gray and dark gray shale, argillite-like clays and light gray, gray, greenishgray multigrained calcareous sandstones and siltstones and thin layers of tuffs. Tuffs
and tuffites have been found to be largely pyritized, as a result of which they are
easily distinguished on logging diagrams by very low values of electrical resistivity.
In fact, these deposits, as well as the rocks of the Kosiv suite, are the main objects
of research in this paper.
340
O. Karpenko et al.
It has traditionally been believed that gas reservoirs in thin-layer Neogene deposits
of the Outer Zone of the Precarpathian Depression are the thin layers, layers of sandstones and siltstones, which are often lenticular in shape and lie among impermeable
layers and clay strata. Some researchers (O. O. Orlov, A. V. Loktev, etc.) develop a
hypothesis about the gas-bearing reservoir of the clay layer directly, in which with
increasing content of psammitic material appear collector conditioning properties
[5].
General well-logging geophysical characteristics of typical thin-layer deposits.
Most methods of interpreting well-logging data are based on factual information
about the properties of geological objects obtained in a petrophysical laboratory or
(often more reliably) in formation tests. The quality of quantitative or qualitative interpretation depends largely on the presence and magnitude of systematic and random
errors in the recording of geophysical parameters, most often—on the accuracy of
registration of a single parameter. Serious shortcomings that affect the effectiveness
of most methods of interpretation of well logging data are:
. common errors about higher accuracy and reliability of laboratory analyzes of
core material in comparison with geophysical data;
. unrepresentative diversity of lithotypes of rocks in the section of the well core
material;
. insufficient amount of core material to build reliable petrophysical models of
section rocks;
. discrepancy between the physical volumes of research in core analysis and
geophysical research (corresponding to different hierarchical levels of the
structure of the geological object from the standpoint of systems analysis);
. methodological differences in measurement technologies by laboratory and well
installations;
. others.
In addition to the above, traditional methods of interpretation, like any other, have
their natural limitations in terms of accuracy and reliability of the final results.
In the conditions of thin-layered terrigenous sections of gas fields of the Outer
Zone of the Precarpathian Deflection, the efficiency of using the electrical resistivity Rt of formations, its ratio to the resistivity of the invasion zone Rt/Rz, or the
saturation parameter Fs = Rt/Ro (where Ro—electrical resistivity of water saturated
rock/formation) as a diagnostic sign of gas bearing reservoir formation, is low due
to the manifestation of the known effect of anisotropy of thin-layer strata, increased
clayness, the large invasion zones and too small thicknesses of individual layers
[6–9].
The Dashava and Kosiv suites are very complex objects for geophysical research,
and special methods and techniques of research and interpretation must be used
here. Unfortunately, due to various reasons, logging methods in the open wellbore of
exploration and prospecting wells are not very suitable for detecting reservoirs and
productive strata in the low-resistivity thin-layer section.
With the increase in the specific content of clayey sand-siltstone and clay layers,
which is typicaly for most gas deposits of the Outer Zone, direct separation of
Efficiency of Electric Logging in Thin-Layer Sections of Hydrocarbon …
341
water-saturated, gas-saturated and “dry” intervals or strata by geophysical parameters
becomes impossible. Only the experience and intuition of interpreters and geologists
allows, according to the standard set of geophysical studies of wells to establish the
nature of the saturation of individual parts of the sections and offer them for testing.
However, there are practically no quantitative geophysical signs of gas-bearing reservoir of thin-layered strata, ie, significant differences in the values of readings of
individual methods by the nature of saturation.
According to the results of research to identify the limit values of the parameters
for the separation of water-saturated and gas-saturated rocks of the Upper Dashava
subsuite and low-sandy intervals of the Lower Dashava, the following is established. Practically for almost every field, vague boundaries of geophysical parameters
have their own meanings. The coefficient of separation efficiency by the nature of
saturation for each parameter does not exceed 60–70%.
Histograms of distribution of characteristics for rocks (strata) with different nature
of saturation practically oveRtap. A small difference in the average values (or median)
of the electrical resistivity, the intensity of radiation according to GR or NGR, the
interval time for sonic log does not always allow to reliably determine the saturation
of rocks.
The thickness of individual layers and layers, as noted above, is much smaller than
the size of gradient probes of resistivity log. As a result, the effect of parallel inclusion
of thin layers is cleaRty manifested here—the curves of resistivity log probes are
pooRty differentiated, the electrical resistivity of productive and water-saturated
strata is quite low, in the range of 2–4 .·m.
Studies of the spread over the area of gas-saturated strata within individual fields
indicate a predominantly lenticular structure of deposits. Many productive or watersaturated strata opened by wells, when detailed by well-logging methods, are the
“brushes” made of thin layers of reservoirs and clays. Externally hidden cyclic structure of strata and horizons is revealed by mathematical processing of curves of
well-logging methods by mathematical filtering of data of separate methods on a
well section [10–12]. At the same time contrasting anomalies of a high-frequency
component on the logging curves recorded by probes with rather low vertical resolution on electrical resistivity R, for example, 2.25 or 4.24 m gradient probes are
shown.
The use of the method with the number of resistivity gradient probes (BKZ) in
order to identify productive and water-saturated reservoirs in this type of section does
not give noticeable positive results. Partly low efficiency of BKZ is connected with
a thin-layered structure of a thickness, partly—with application of the traditional
processing techniques calculated for homogeneous layers of considerable thickness [13, 14]. Unfortunately, geophysical surveys in wells in the Precarpathians
use standard sets of methods and techniques suitable for qualitative and quantitative
processing of well logging data in simpler types of sediments.
342
O. Karpenko et al.
3 Research Methods
In recent years, work has been done to identify geological and geophysical conditions,
the feasibility of effective application of new methods of rapid interpretation of
well logging data with the involvement of specialists of industrial and geophysical
organizations in order to establish in thin layers of productive objects. It should be
emphasized that these works were supported by the management and specialists of a
number of geophysical and oil and gas production organizations. When developing
new methods and techniques, the following tasks were set: firstly, to use the data of
conventional well logging methods, which are included in the mandatory complex of
well logging; secondly, to develop quantitative criteria for the detection of productive
(oil and gas-saturated) objects—strata and bundles of layers in thin-layered deposits
of different types.
Using the relaxation parameter of electrical resistivity to detect gas-saturated
layers and section intervals. An analogue of the following approach to the detection
of productive thin-layer parts of the section is a known method of determining the
nature of fluid saturation of the reservoir rock in the well section by determining the
value of electrical resistivity or resistivity increase parameter (parameter of saturation)—Fs. As already noted, the efficiency of separation of thin-layered rocks by the
value of electrical resistivity is extremely low, as with other geophysical parameters
and known criteria, so you should look for other ways to solve this problem. The
resistivity increase parameter (or saturation parameter) is calculated as follows:
Fs =
Rt
Rt
=
,
Ro
F · Rw
(1)
where Rt—electrical resistivity of the rock, determined according to electrical
logging methods; Ro—calculated electrical resistivity of a similar rock under the
condition of 100% filling of the pore space with formation water; F—formation
resistivity factor—relative parameter that characterizes the porosity of the rock;
Rw—electrical resistivity of formation water.
In the practice of well logging or geophysical works, the value of the electrical
resistivity of rocks R is determined only by resistivity gradient probes (BKZ) or by
other electrical methods using probes of large radius of study and, accordingly, low
vertical resolution. It follows that the ability to determine the values of R layers of
different thickness is limited by the size of large probes of electrical logging, and in
thin layers it is impossible to determine the electrical resistivity of individual layers
to assess the saturation of specific layers of reservoir rocks by (1). Practically critical
value of the Fs parameter in homogeneous low-clay oil and gas saturated rocks in
the separation of productive and water-saturated intervals is 6–8. From (1) it follows
that the separation efficiency of reservoir rocks by saturation is determined by the
difference between electrical resistivity.
This method of detecting productive gas-bearing and oil-bearing formations and
intervals according to existing electrical research methods is ineffective in thinlayer sections of wells. The values of electrical resistivity measured in the well
Efficiency of Electric Logging in Thin-Layer Sections of Hydrocarbon …
343
in productive and water-saturated thin-layer strata are close in magnitude to the
values characteristic of the general background of clay rocks and are, as a rule, units
of Ohm·m. Significant oveRtap of the ranges of electrical resistivity of productive
and water-saturated rocks in clays or thin-layer sections does not allow the use of
electrometry methods to detect oil or gas saturated intervals in well sections using
Fs or absolute values of electrical resistivity.
The main purpose of the new method is to allocate intervals with different nature
of fluid saturation in the sections of wells of thin-layer types according to the values of
electrical resistivity of rocks, which are registered by conventional probes of electrical
well studies. To do this, based on the data of R logging probes of different sizes of
the BKZ method, the transformation of well logging information is performed in
order to obtain parameters that characterize the presence of productive (oil or gas
saturated) intervals in terms of clay layers in the presence of layers with high sand
content.
The essence of the new method of detecting productive intervals is that the interpretation uses not the absolute values of the electrical resistivity of individual probes,
and the statistical characteristics of high-frequency components of the curves R, we
called the curves of “residual apparent resistivity” Rf . The parameter Rf at the depth
zi of the section of the well is determined by the procedure of digital filtering of
logging curves, for example by the method of a moving strip:
R f (z i ) = R(z i ) − R(z i ),
(2)
where zi —the depth of the well section at the point i taking the electrical resistivity
of the probe; R f (z i )—residual electrical resistivity of the probe at depth z i ; R(z i )—
counting the electrical resistivity of the probe at depth z i ; R(z i )—smoothed (average)
value of the electrical resistivity of the probe in the depth range from (zi —0.5·Δz) do
(zi + 0.5·Δz), calculated for depth zi (.z—the depth interval in which the averaging
of the electrical resistivity of the probe is performed; for a typical thin-layer section,
it is recommended to choose about 2 m).
Figure 1 shows a typical interval of a thin-layered section of the Kosiv suite, where
the repetition of almost all anomalies from thin layers (less than 1 m) on BKZ pribe
curves, which are more contrasting after filtering by moving strip, is cleaRty visible.
This pattern is observed in all, without exception, curves of BKZ gradient probes
with a length of 0.45–4.25 m inclusive, recorded in wells of thin-layer sections of all
types in the hydrocarbon deposits covered by our research.
Example of Fig. 1 is a confirmation of the possibility that local layers of small
size form noticeable anomalies on the curves of not only small but also large of
the gradient probes, which is often underestimated in the detailed interpretation of
well logging data. This makes it possible to determine the bundles of productive
layers and layers of rocks because in front of water-saturated or clayey impermeable thin-layer intervals there is a rapid damping of oscillations of this component
with increasing size or radius of the logging probe. In contrast, in productive thinlayered intervals, significant differentiation of the Rf parameter is preserved on the
curves of large probes with a significant study radius. For comparison: in a thin-layer
344
O. Karpenko et al.
Fig. 1 An example of the
reflection of anomalies of
electrical resistivity from
individual layers of rocks on
the curves of the gradient
probes BKZ in the sediments
of the Kosiv suite in the well
of Bohorodchany gas field:
R1–R4—curves of electrical
resistivity 0.45–4.25 m
gradient probes BKZ;
R1f –R4f —curves of residual
electrical resistivity after
filtering the curves of the
gradient probes
section according to R values taken from the curves of imaginary apparent resistivity
of electric logging probes of different sizes, it is impossible to effectively divide
the section into aquifers and gas-bearing intervals, and Rf curves already have a
marked degree of attenuation with increasing probe size saturation of rocks. This is
explained by the fact that due to the presence of the ivasion zone in reservoir rocks
(characterized by increased sandiness in clay strata) on the curves of small probes,
the differentiation of Rf values is determined only by the ratios of thicknesses and
resistivity values of impermeable clay and permeable layers, filled with filtrate of
drilling mud fluid. If the section is represented by alternating layers of clay rocks and
permeable reservoirs, then on the residual apparent resistivity curves of large probes
significant fluctuations in Rf values are observed only opposite the productive intervals, when reservoir layers outside the penetration zone due to oil or gas saturation
have increased electrical resistivity values. in relation to the electrical resistivity of
clay rocks (Fig. 2). In the presence of layers of water-saturated rocks, the electrical
resistivity of which differs little from the resistivity of clay rocks, the differentiation of Rf curves of large probes will be much smaller than in the presence of oil
or gas-saturated layers [2, 9, 15]. Figure 2a also shows the case of alternation of
clayey rocks of sandstones and compacted rocks-non-reservoirs with high electrical
resistivity. Similar intervals are distinguished on the curves of small gradient probes
by a high degree of differentiation, which also naturally decreases in the direction of
increasing the probe size.
Efficiency of Electric Logging in Thin-Layer Sections of Hydrocarbon …
345
Fig. 2 Examples of reducing the differentiation of curves of gradient probes with increasing probe
size in a thin-layered clay-sand section of the well: a—interval with compacted layers of sandstone;
b—interval with water-saturated layers of sandstones; c—interval with gas or oil-saturated layers
of sandstones; R1f, R3f, R4f —curves of residual electrical resistivity of 0.45 m, 2.25 m and 4.25 m
of gradient probes, respectively
The phenomenon of preserving significant differentiation of residual resistivity
curves with increasing probe size (for example, gradient probes of electric logging—
up to 2.25, 4.25 m with an average thickness of layers of rocks 0.3–0.8 m opposite the
productive intervals) is reliably confirmed in typical thin layers Neogene deposits
in gas fields of the Outer Zone of the Precarpathian Depression, where currently
revealed gas-saturated horizons, which were missed in the past in some areas due
to the impossibility of their allocation by conventional methods of interpretation of
logging data in well sections.
To assess the degree of differentiation of the residual electrical resistivity curve at
each depth point zi of the logging probe curve, statistical evaluation of the variability
of Rf values, such as variance, is performed in a certain depth window on both sides
of this point zi —0.5·Δz (for a thin-layer type of section, it is desirable to set the
value of Δz about 2 m). Thus, the residual electrical resistivity curves are converted
into a parameter curve called the “residual electrical resistivity variance DRf or
the standard deviation of the residual electrical resistivity σ Rf ”. Table 1 shows the
average values of the distributions of the specified parameter σ Rf depending on the
size of the probe and the nature of the saturation of the formations, established by
industrial tests in exploration wells of Rubanivske gas field, Orkhovitske oil and gas
field, Lyubeshivske, Vereshivske, Khidnovitske, Teysarivske, Bohorodchanske gas
fields.
346
O. Karpenko et al.
Table 1 Distribution of σ Rf values depending on the nature of saturation of thin-layer layers and
the size (L) of the gradient probes BKZ
The nature of
saturation
Values of σ Rf , Ohm·m of gradient probes:
L = 0.45 m
L = 1.05 m
L = 2.25 m
L = 4.25 m
Number of test
intervals
Gas
0.225
0.426
0.522
0.429
20
Water
0.487
0.458
0.087
0.074
23
12
“Dry”
0.495
0.833
0.241
0.176
Gas + water
0.033
0.056
0.031
0.031
5
All groups
0.363
0.489
0.258
0.209
60
4 Research Results
From the data in Table 1 it is seen that in all cases of saturation of thin-layer strata
or the layers except gas-saturated, there is a decrease in the differentiation of Rf
curves on large 2.25 and 4.25 m probes relative to 0.45 m. Data on 1.05 m gradient
probe were excluded from consideration and use due to the fact that its readings and
configuration of the curve are significantly influenced not only by the features of the
washed formation zone or invariant part, but also by the variability of invasion zone
diameter. its neither as a “detector of lithology” in the near zone, nor as a “detector
of saturation of the formation”—under the influence of an invariable part of the
formation.
In the calculations in order to use quantitative indicators of the presence of the
productive interval, it is proposed to introduce a value called “parameter of radial
relaxation of residual electrical resistivity” or “parameter of relaxation of residual
electrical resistivity” Pr [12]:
Pr = f
σ (R f l)
,
σ (R f sm)
(3)
where Pr—relaxation parameter of the residual electrical resistivity; σ Rfl and
σ Rfsm—respectively, the value of the standard deviation of the residual electrical
resistivity of the high-frequency component of the readings of the logging probes
curves of large and small size in a certain window depth of 0.5 · Δz on both sides of the
observation point; f —function or constant value to improve the visual rePrentation
of the Pr curve.
The limit value of the relaxation parameter of the residual electrical resistivity is
set using reference samples. When using the decimal logarithm of the ratio of the
corresponding standard deviations of the residual electrical resistivity instead of the
function f :
Pr = lg
σ (R f l)
.
σ (R f sm)
(4)
Efficiency of Electric Logging in Thin-Layer Sections of Hydrocarbon …
347
The Pr parameter is usually in the range of −2 to +2 for typical terrigenous
thin-layered sediments. It is established that gas-bearing productive horizons in thinlayered sections exist where the values of the relaxation parameter are greater than
0. The value 0 is the limit value Pr1, Pr2, which is set for Dashava and Kosiv suites
deposits in the depths of 500–2500 m) (Figs. 3 and 4) using cumulative curves of
parameter distributions for rocks with different nature of saturation.
Comparison with the test results of the intervals of thin-layered sections of wells
found that the efficiency of separation of rocks into water-saturated and productive
using standard methods of interpretation of the usual complex of well-logging and
geophysical research averages no more than 59% in Neogene deposits for different
gas fields of Precarpathian. During the experimental use of the relaxation parameters
of the residual electrical resistivity Pr at the Orkhovitske and Lyubeshivske gas fields
(gradient probe data of 4.25 and 0.45 m were used), the efficiency of separation into
productive and aquifer intervals in these areas increased on average to 87%. When
using the same Pr parameters for two pairs of probes with sizes of 4.25 and 0.45 m
and 2.25 and 0.45 m, the efficiency of separation of rocks into water-saturated and
gas-saturated was 92%.
Fig. 3 Distribution of
average mean square
deviations of residual
electrical resistivity σ Rf for
curves of BKZ gradient
probes and relaxation
parameters of residual
electrical resistivity Pr
depending on the nature of
saturation in the intervals of
thin-layered sections of gas
fields
Fig. 4 Determination of limit values of residual electrical resistivity parameters Pr for Neogene
rocks
348
O. Karpenko et al.
Figure 5 shows diagrams of electrical resistivity, residual electrical resistivity and
relaxation parameters of residual electrical resistivity of gradient probes of different
sizes: in the range of depths of thin-layered water-saturated rocks and in the range
of depths of gas-saturated rocks of the Kosiv suite of the well Bohorodchanske gas
field. It should be noted that the formation with a positive anomaly of electrical
resistivity (1503.0–1503.8 m in Fig. 5) within the interval of the section, where the
gas flow of 36 thousand m3 /day, can be attributed to the tight lay in accordance with
the anomalies Pr1, Pr2, and the main productive reservoir here should be considered
a thin-layer pack at a depth of 1500–1505 m. The inflow of formation water from the
upper perforation interval is confirmed by the Prence of small values of the relaxation
parameter of the residual resistivity.
5 Discussion and Conclusions
A detailed analysis of the field of effective application of the parameters Pr1, Pr2
notes that it can already be noted that studies indicate the possible existence of false
positive extremes of the parameter opposite the layers, where there is a sharp change
in electrical resistivity. For example, in the presence of individual high-resistivity
strata thicker than 0.6–0.8 m, or—pyritized tuffs and tuff with very low values of
R, even against the background of low-resistivity clay-sandy rocks Dashava and
Kosiv suites. That is, the use of the method of radial relaxation of residual electrical
resistivity as with all other existing methods and techniques, competent analysis of
the results in order to prevent or reduce the likelihood of erroneous conclusions about
the existence or absence of oil and gas intervals in the section of the well.
Using a new method of the radial relaxation parameter of the residual electrical
resistivity, thin-layered potentially gas-saturated strata of rocks were recommended
for testing horisons VD-13, VD-14 of well 12-Makunivska, where industrial gas
inflow was obtained from the recommended depth intervals and, thus, a new gas
deposit was discovered in the formations of Upper Dashava subsuite.
Earlier we found that the limit value of Pr1, Pr2, which separates water-saturated
or impermeable clay thin-layered intervals from gas-saturated, is 0. For typical thinlayered sediments with frequent alternation of layers of clays, siltstones and sandstones mostly less 0.4–0.6 m, the specified value of the relaxation parameter is
quite stable for different depth intervals. However, with increasing sand content
of the section and the thickness of individual layers (0.8–1.6 m), a sharp decrease
in the value of Pr1 or Pr2 is often observed, even in gas-saturated intervals. This
phenomenon is due to the fact that the layer ceases to be thin-layered in relation to
the size of the applied gradient probes BKZ (0.45; 2.25 m). Anomalies in the curves
of electrical resistivity in front of gas-saturated permeable layers increase sharply,
including for small gradient probes. Under such conditions, the values (or curves) of
the relaxation parameter of the electrical resistivity are in the negative area, which is
not due to the nature of the saturation of the stratum, but to the fact that the sandy-clay
stratum ceased to satisfy the thin layer. In connection with the above, this thickness
Efficiency of Electric Logging in Thin-Layer Sections of Hydrocarbon …
349
Fig. 5 Example of detection of gas-saturated thin-layered rocks in the well section of Bohorodchanske gas field
is the part of the well section, which clearly stands out individual layers on all curves
BKZ; curves are characterized by sharp differentiation. The preconditions on which
the determination and use of the relaxation parameter were previously based (for
thin-layered strata, poorly differentiated by electrical resistivity) in this case are not
met.
The issue of identifying the area of effective application of the parameters Pr1,
Pr2, has led to additional research related to the existence of different types of
thin-layered well sections. As mentioned eaRtier, under a thin-layered section (in
350
O. Karpenko et al.
relation to the geometric characteristics of probe devices of well-logging methods)
we consider layering of layers with thicknesses less than or equal to 0.7–0.6 m, with
small amplitudes of geophysical anomalies, which in most cases cannot be used for
quantitative geological interpretation according to standard methods. However, just
such noted features of the thin-layered section are a prerequisite for the effective
application of the parameters Pr1, Pr2 in the detection of productive thin-layered
strata in the thin-layered terrigenous section.
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azhinnoj-zony-plasta-kollektora-po-geofizicheskim-dannym
Formation Mechanisms and Overcoming
Methods to Reducing Natural Gas
Consumption in the Residential Sector
Olexandr Yu. Yemelyanov , Tetyana O. Petrushka ,
Anastasiya V. Symak , Kateryna I. Petrushka ,
and Oksana B. Musiiovska
Abstract The authors simulated the barrier formation mechanisms on the way to
improving the natural gas use efficiency in the residential sector and suggested effective ways to overcome these barriers. Regularities of barriers formation on the way
to reduction of consumption of natural gas in apartment houses are considered.
Grouping of these obstacles was carried out. Factors affecting the level of these
barriers have been identified: shortage of necessary resources, insufficient level of
resource quality, insufficient level of investors competence (i.e. persons who decide
on the implementation of investment measures to save natural gas consumption
in residential buildings), political and institutional factors, as well as insufficient
level socio-economic results from the implementation measures to save natural gas
consumption in the residential sector. A number of barrier model mechanisms have
been built to improve the efficiency of natural gas use in the residential sector. The
quantitative measurement method of the financial and economic barriers level that
arise during the measure’s implementation for the purpose of natural gas saving by
households is proposed, in case of financing these measures by borrowed funds.
A grouping of ways to overcome barriers on the course to reducing natural gas
consumption in the residential sector has been made. An optimization model for the
state programs formation of financial support for measures for the purpose of reducing
natural gas consumption in residential buildings has been developed. According to
the analysis of the sample of Ukrainian households, the most significant barriers to
the measures implementation with the object of reducing natural gas consumption
by the surveyed households are obstacles caused by shortage of necessary resources
and obstacles caused by shortage of investor competence. The forecast indicators
of the state financial supporting program of those households seeking to implement
measures with the object of reducing natural gas consumption on the basis of thermal
modernization of residential buildings were calculated.
Keywords Energy consumption · Barrier · Modeling · Overcoming · Residential
building · Thermal modernization · Natural gas
O. Yu. Yemelyanov · T. O. Petrushka · A. V. Symak · K. I. Petrushka · O. B. Musiiovska (B)
Lviv Polytechnic National University, Lviv, Ukraine
e-mail: Oksana.B.Musiiovska@lpnu.ua
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_21
353
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O. Yu. Yemelyanov et al.
1 Introduction
For many countries around the world, the problem of ensuring proper economic
growth is acute. Solving this problem is a necessary condition for improving the
welfare of the population [1], improving employment [2] and reducing the budget
deficit [3]. At the same time, significant rates of economic growth are often accompanied by increased use of various energy resources types, especially non-renewable
energy sources [4]. Such an increase usually has a negative impact on the environmental situation [5], worsens the energy security of countries [6] and reduces their
opportunities for sustainable development [7].
Therefore, exists a contradiction between the urgent need for sustainable economic
growth and the need for economical consumption of fossil energy resources. This
contradiction is resolved by the transition to sustainable energy-saving economic
development [8], which is accompanied by long-term economic growth while
reducing the consumption of non-renewable energy. Whereas the transition to this
economic development type requires the implementation of various technical, technological, organizational and other measures [9, 10] aimed at improving the efficiency
of non-renewable energy [11], particularly by replacing them with renewable energy
[12].
In many countries around the world a significant share of the energy consumption places such non-renewable energy resources as natural gas [13]. Therefore,
reducing the use of natural gas is one of the main ways to ensure energy-saving
economic development [14]. In recent years, there has been a significant reduction
in the consumption of this energy resource in a number of countries. As a consequence, further reduction of its use with the simultaneous growth of gross domestic
product may be associated with significant difficulties [15]. One of the prospective
overcoming methods of these difficulties is to accelerate reduction rate of natural
gas consumption in those economic sectors that do not have a significant impact
on the value of gross domestic product. Particularly, this includes residential sector.
At the same time, as for any other economic sector, the projects implementation to
reduce natural gas consumption in residential buildings faces various barriers [16].
Identifying overcoming methods of these barriers requires preliminary study of their
formation mechanisms [17] and the development of effective assessing tools for the
level of these barriers [18].
2 Literature Review and Setting Research Objectives
A significant amount of scientific work has been devoted to the assessment and overcoming of barriers to the implementation of energy-saving projects in both enterprises
and households. Meanwhile, there are some differences between the authors’ views
on the peculiarities of the formation of these barriers. In particular, scientists identify
different types of the most important of them. Including in particular, in [19] the main
Formation Mechanisms and Overcoming Methods to Reducing Natural …
355
obstacles to the implementation of energy-saving technological change projects are
economic barriers. A similar opinion is expressed in [20], where the delay in the
implementation of energy saving projects is explained by the lack of appropriate
financial incentives.
Some other researchers point out that management barriers play a crucial role.
Particularly, in the study [21] has been showed that due to lack of rationality, lack of
energy saving among the main goals and shortcomings in information support, businesses abandon energy saving projects, which are instead quite effective. The importance of information barriers as a factor that determines the slow implementation of
energy-saving projects is also noted in the study [22].
Also, some authors, particularly in the study [23], among the barriers to energy
efficiency, pay special attention to the significant level of risk in the implementation
of many energy-saving programs and projects. However, the implementation of such
programs and projects often requires large investments [24], while many businesses
and households lack the necessary financial resources. Meanwhile, lending as the
most common external source of financing for energy saving measures is often not
attractive enough for the persons who plan to implement these measures [25].
One of the barriers that can hinder the implementation of energy reduction projects
is the low level of energy prices, which makes projects unprofitable. At the same time,
scientists are ambivalent about the significance of this barrier. Thus, in [26] it was
found that the volume of electricity consumption changes with changes in its prices,
however, for example, in [27] this pattern is not found.
In general, we can agree with the statement made in [28] that an exhaustive list
of barriers to energy efficiency has not yet been provided and the possibility of such
a presentation is questionable.
The fact that different scientists identify different types of barriers that arise in the
process of implementing energy saving measures may be due to the lack of attention
paid by most researchers to the relationships that exist between certain types of these
barriers. Therefore, it should be noted the study [29], which distinguishes three types
of such relationships, namely: causal, hidden and synergistic. In the meantime, the
formation regularities of these barriers, particularly their sequence formation, have
not been fully studied. The same applies to the barrier’s assessment, as there are
currently no generally accepted tools for measuring them.
At the same time, a significant number of scientists are limited to qualitative
barriers analysis to energy efficiency, as was done, for example, in the study [30],
which studies the Finnish construction sector. Researchers also used the results of a
survey of energy managers to assess these obstacles [31]. Among the more objective
methods of estimating barriers to the implementation of energy-saving projects is
the measurement of these barriers in the study [32] using graph-analytical models
and a hierarchical approach.
The presence of a significant number of factors that hinder the implementation of
measures to save non-renewable energy resources, naturally leads to the existence of
various ways to overcome barriers to such implementation. According to the context
of the study [30] the main way is to provide complete and up-to-date information
on energy efficiency measures. The study [33], which researches Swedish municipal
356
O. Yu. Yemelyanov et al.
energy saving programs, notes the need for both comprehensive information support
for those seeking energy saving measures and the mastery of these individuals’ skills
in processing such information.
A number of scientists consider providing energy saving projects with the necessary amounts of financial resources as one of the most important ways to reduce
the consumption of non-renewable energy sources. To this end, scientists propose
measures to improve the investment climate [34], the implementation of programs
to subsidize projects for the transition to clean energy [35], the use of soft loans [36],
in particular the introduction of such loans for small businesses [37]. However, the
relationship between the necessary parameters of soft loans and the level of barriers
that arise in the implementation of energy saving measures, which are expected to
be overcome through the provision of soft loans, remains unexplored.
Some authors consider overcoming barriers to energy efficiency on the basis of
improving management, in particular, due to improved methods of energy audit [38]
and improved motivation mechanisms [39].
Regarding the formation mechanisms of certain barriers’ types on the way to
reducing the consumption of natural gas in residential sector, the modeling of these
mechanisms requires the specifics consideration of each of them. Particularly, the
most common are barriers associated with insufficient financial resources and (or)
insufficient cost-effectiveness of measures to improve the use of natural gas in
residential buildings. Hereinafter, we will call these barriers financial and economic.
It should be noted that the formation mechanism of financial and economic barriers
depends on what sources of funding for energy saving measures can be potentially
used. For many households, bank credit is almost the only source. Then, in order for
a certain household to be interested in taking a loan to implement measures to save
natural gas consumption, two basic conditions must be met. First, the cost saving of
paying for thermal energy must be not less than the amount of interest on the loan.
Second, households must have sufficient income to repay the loan and pay interest
on it. For the case of natural gas consumption and considering the possible change
in its prices, these two conditions can be formalized as follows:
E g · l · T pg ≥ Cig · (1 − α) · l;
(1)
Cbg · (1 − α) ≤ Rh − E g · l · T pg ,
(2)
where E g —annual household expenditures for heat energy at the basic level of
natural gas prices, monetary units; l—the reduction share of thermal energy consumption after the measure implementation to reduce natural gas consumption; T pg —the
growth rate of natural gas prices compared to the base price level; C ig —investment
expenditures required for the measure implementation to reduce the consumption of
natural gas, monetary units; α—the share of investment expenditures for the measure
implementation by the household to reduce the consumption of natural gas, which
may be hypothetically covered by certain third parties, the share of the unit; r—
annual interest rate, unit share; C bg —annual expenses for servicing and repayment
Formation Mechanisms and Overcoming Methods to Reducing Natural …
357
of the loan, provided that there is no partial compensation by third parties for household expenses for the measures implementation to save natural gas, monetary units;
Rh —the maximum possible part of the household income that it can use to pay for
thermal energy and to service and repay the loan, monetary units.
From expressions (1) and (2) we obtain:
E g · l · T pg
;
Cig · l
(3)
Rh − E g · l · T pg
,
Cbg
(4)
αmin 1 = 1 −
αmin 2 = 1 −
where α min 1 —the minimum possible value of α, for which inequality (1), the fraction
of one; α min 2 —the minimum possible value of α, for which inequality (2), the fraction
of one.
Given that the indicator α can’t be negative, we finally get:
αmin = max{0; αmin 1 ;αmin 2 },
(5)
where α min —the minimum share of investment expenditures for a certain natural gas
saving measure, which is reimbursed by third parties, at which it is economically
advantageous for the household to obtain a loan to finance this measure.
Therefore, expression (5) can be used to quantify the level of financial and
economic barriers to the measures implementation to save natural gas by households, provided that these measures are financed through borrowed funds. On the
other hand, the above formulas (1)–(5) are essentially a model of the formation
mechanism of these barriers.
3 Justification of Ways to Overcome Barriers on the Way
to Reducing Natural Gas Consumption in the Residential
Sector
There are many ways to overcome barriers to natural gas efficiency in the residential
sector. At the same time, the implementation subjects of these methods can be both
households and state or local authorities (Table 1).
As follows from the data presented in Table 1, important ways to overcome
barriers on the course to increase the natural gas efficiency use in the residential
sector are: improving the information and financial support for the development and
implementation of measures, namely:
(1) improving the provision of households—consumers of natural gas with
complete, accurate and relevant information on future measures to reduce its use.
For this purpose, the following main tasks need to be solved: to find and generate
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O. Yu. Yemelyanov et al.
Table 1 Grouping methods of overcoming barriers on course to increasing natural gas efficiency
use in the residential sector by types of these barriers and the implementation subjects of appropriate
methods to overcome them
Barriers types on course to
increasing natural gas
efficiency use in the
residential sector
Methods to overcome barriers according to their implementation
subjects
Households
State and local authorities
1. Obstacles associated with
shortage of necessary
resources volume
Upgrading the ability to find
sufficient resources needed to
improve the natural gas
efficiency in the residential
sector
Households assisting in finding
sufficient resources to improve
the natural gas efficiency use in
the residential sector (including
providing financial and
information support to
households)
2. Obstacles associated with
insufficient quality of
necessary resources
Upgrading the ability to find
the resources of proper quality
needed to improve the natural
gas efficiency in the residential
sector
Households assisting in finding
the resources of proper quality
needed to improve the natural
gas efficiency in the residential
sector (including providing
financial and information
support to households)
3. Obstacles associated with
insufficient competence of
investors
Upgrading the investors
competence in the
development and
implementation of measures to
improve the natural gas
efficiency in the residential
sector
Households assisting in
upgrading their competence by
the development and
implementation of measures to
improve the natural gas
efficiency in the residential
sector
4. Political and institutional
obstacles
Improving the investors
competence in cases when the
political and institutional
factors in the development and
implementation of measures to
improve the natural gas
efficiency in the residential
sector should be considered
Perfection the legal framework
for the implementation of
measures to improve the natural
gas efficiency in the residential
sector; improving the state
energy saving policy; increasing
the availability of loan
financing measures to improve
the natural gas efficiency use in
the residential sector
5. Obstacles caused by
insufficient level of
socio-economic results of
implementation measures to
improve the efficiency of
natural gas use in the
residential sector
Improving the investors
competence in choosing the
best options for implementing
measures to improve the
natural gas efficiency in the
residential sector
Financial support for measures
to improve the natural gas
efficiency in the residential
sector, implemented by
households; natural gas price
regulation; improving the
regulation of energy-saving
materials and equipment
manufacturers
Formation Mechanisms and Overcoming Methods to Reducing Natural …
359
data sets on prospective measures to reduce the use of natural gas in the residential sector; to structure the array of source information on socio-economic
results of measures to reduce the use of natural gas in the residential sector;
identify the best channels for transmitting information on prospective measures
to reduce the use of natural gas in the residential sector for different consumer
groups of this information; determine the rational composition and architecture
of the developed web resources, which contain input information on prospective
measures to reduce the use of natural gas in the residential sector; to determine
the rational composition and architecture of the developed web resources, which
provide consumers (households) with information on the socio-economic results
of the measures implementation to reduce the use of natural gas in the residential
sector;
(2) improving the competencies of households in the development and implementation of measures to improve the natural gas efficiency use in the residential
sector. Such improvement requires conducting initial courses, seminars, trainings, etc. to raise public awareness of the methods and techniques of developing and implementing measures to reduce the consumption of natural gas in
residential buildings;
(3) stimulating the state and local authorities in the process of implementing
measures by households to improve the efficiency of natural gas use. With
this end in view, it is necessary to solve the following main tasks: promotion
of relevant measures by state and local authorities; advising state and local
authorities on households—consumers of natural gas on prospective areas and
specific measures to reduce the use of this type of energy; search and attraction of additional financial resources by state and local authorities to finance
measures to reduce household gas consumption; financial support from state
and local authorities of those households that seek to implement investment
measures to reduce natural gas consumption, on the basis of soft loans; financial support from the state and local authorities of those households that seek
to implement investment measures to reduce natural gas consumption, on the
basis of non-repayable funding for these measures.
Particularly, an important method to overcome barriers on the course to reduce
natural gas consumption in the residential sector should be recognized as preferential
lending for measures to reduce such, which shell be carried out at the expense of the
state budget (similar programs can be implemented at the local government level).
Then the task to substantiate the program preferential crediting of measures to reduce
household gas consumption can be formulated as follows: let there be a set of households divided into classes according to the average income of the members of these
households. Also, let there be a set of measures aimed at reducing the consumption
of natural gas in the residential sector (in this case, the same type of measures can be
combined into groups; under such conditions, the measures indicators are averaged
within each of their groups). Then it is necessary to establish such a share of reimbursement by the state of household expenditures for the implementation of each
group of measures (the rest of the expenditures shall be financed by loans provided
360
O. Yu. Yemelyanov et al.
by state banking institutions), to maximize the reduction of natural gas consumption
in the residential sector:
E = I1 · e1 + . . . + Ii · ei + . . . + In · en =
n
.
Ii · ei → max,
(6)
i=1
where E—is the expected total natural volumes of reduction of natural gas consumption in the residential sector due to the implementation of the state program of
preferential lending; I i —total expected volumes of investment expenditures in the
implementation of the measures group to reduce natural gas consumption, which
shall be financed by concessional government lending, monetary units; ei —natural
volumes of reduction of natural gas consumption in the residential sector by a group
of measures for such reduction per one monetary unit of investment expenditures
in the implementation of these measures; n—number of measures groups to reduce
natural gas consumption in the residential sector.
The indicator I i can be presented as follows:
Ii =
m
.
Ii j ,
(7)
j=1
where m—the number of household groups differentiated by the average income of
their members; I ij —the expected amount of investment expenditures in the implementation of measures group to reduce natural gas consumption, which shall be
financed by concessional government lending, a group of households, monetary
units.
The following restrictions must also be met:
(1) on the total amount of compensation from the state of household expenditures
for the implementation of measures to reduce natural gas consumption:
I1 · α1 + . . . + Ii · αi + . . . + In · αn =
n
.
Ii · αi ≤ B,
(8)
i=1
where α i —the generalized level of financial and economic barriers to the implementation of measures group to save natural gas consumption, the share of the unit;
B—the general limit of compensations from the state of households’ expenses on
measures implementation for reduction of natural gas consumption, monetary units.
In this case, α i shall be determined using the following expression:
{
αi = max αi1 , . . . , αi j , . . . , αim } ,
(9)
where α ij —is the generalized level of financial and economic barriers to the measures
implementation to save natural gas consumption by the group of households (this
level is determined using expression (5));
Formation Mechanisms and Overcoming Methods to Reducing Natural …
361
(2) the value of the indicator I ij (condition of their inseparability):
Ii j ≥ 0.
(10)
The use of the proposed optimization model (6)–(10) in the practice of public
authorities shall provide an opportunity to increase the state programs validity of
financial support for reducing measures of natural gas consumption in residential
buildings.
4 Empirical Barriers Analysis on the Course
to the Measures Implementation for the Purpose
of Reducing Natural Gas Consumption in the Residential
Sector
In order to assess barriers to the measures implementation with the object of reducing
natural gas consumption by households, a survey of 400 Ukrainian households was
conducted. It turned out that out of the total number of respondents, 128 tried to implement measures with the object of saving natural gas during 2020–2021. However, not
all of these households overcame certain barriers to the successful implementation
of these measures (Table 2).
Based on the data presented in Table 2, it is possible to assess the relevant barriers
level on the course to the measures implementation with the object of reducing
natural gas consumption by the surveyed households. This level, as mentioned above,
can be defined as the ratio of the households’ number that couldn’t overcome the
barrier to the households’ number that approached it. The results of the corresponding
calculations are given in Table 3.
As follows from the data presented in Table 3, the most significant barriers to the
measures implementation with the object of reducing natural gas consumption by the
surveyed households include obstacles due to shortage of necessary resources and
obstacles due to insufficient investor competence. The removal of barriers of the first
type according to formula (1) would increase the level of measures implementation
for the whole set of households from 0.246 to 0.246/(1 − 0.491) = 0.483 for measures
to install heat-saving windows on balcony doors and from 0.258 to 0.258/(1 − 0.548)
= 0.571 for measures to insulate the exterior walls of buildings.
As mentioned above, the level of financial and economic barriers on the course
to the measures implementation with the object of saving natural gas by households
can be significantly affected by its price. Using expression (5), the generalized level
of financial and economic barriers on the way to the implementation of natural gas
saving measures by the surveyed households was calculated. At the same time, the
baseline was the average level of prices for this type of energy resources for household
consumers, which developed in Ukraine as of December 31, 2021. The results of the
calculations are presented in Table 4. According to the data presented in this table,
362
O. Yu. Yemelyanov et al.
Table 2 Number of households that, according to the survey results, overcame the relevant type
of barriers to the measures implementation for the purpose of reducing natural gas consumption
Barriers types
on the course to
the measures
implementation
for the purpose
of reducing
natural gas
consumption
Measures groups to reduce natural gas consumption
Installation of
heat-saving
windows on
balcony doors
Insulation
of external
walls of
buildings
Replacement
of gas boilers
for solid fuel
boilers
Installation of
air
temperature
controllers
Installation of
heat
recuperators
of ventilation
air
1. Obstacles
associated with
shortage of
necessary
resources
volume
29
14
11
3
5
2. Obstacles
associated with
insufficient
quality of
necessary
resources
27
13
11
3
5
3. Obstacles
associated with
insufficient
competence of
investors
21
10
8
2
4
4. Political and
institutional
obstacles
18
9
7
2
4
14
5. Obstacles
caused by
insufficient level
of
socio-economic
results of
implementation
measures to
improve the
efficiency of
natural gas use
in the residential
sector
8
6
2
3
(continued)
Formation Mechanisms and Overcoming Methods to Reducing Natural …
363
Table 2 (continued)
Barriers types
on the course to
the measures
implementation
for the purpose
of reducing
natural gas
consumption
The total
number of
surveyed
households that
sought to
implement a
relevant
measures group
for the purpose
of reducing
natural gas
consumption
Measures groups to reduce natural gas consumption
Installation of
heat-saving
windows on
balcony doors
Insulation
of external
walls of
buildings
Replacement
of gas boilers
for solid fuel
boilers
Installation of
air
temperature
controllers
Installation of
heat
recuperators
of ventilation
air
57
31
24
7
9
Source By authors
the increase in natural gas prices for most of the measures to save it in this case does
not cause a reduction in the general level of financial and economic barriers.
The existence of a certain level of financial and economic barriers on the course
to the measures implementation with the object of reducing natural gas consumption
by the surveyed households requires state financial support from these households.
This support may take the form of preferential lending relevant measures by stateowned banks with reimbursement of a certain share of the principal loan amount.
Based on the data on the surveyed households and using the optimization model
developed above (6)–(10), some forecast indicators of the financial support program
of households seeking to implement measures with the object of reducing natural gas
consumption were calculated. Particularly, this applies to the expected efficiency of
public expenditures to reimburse the initial loans amount for thermal modernization
of residential buildings and the proposed shares of such reimbursement (Table 5).
As follows from the data presented in Table 5, despite the rather significant
proposed share of state reimbursement of the initial loans amount received by households on the course to implement measures with the object of natural gas saving,
the expected effectiveness of such reimbursement is quite high (ranging from 4.07
to 8.54 m3/USD).
364
O. Yu. Yemelyanov et al.
Table 3 The barriers level on the course to the measures implementation with the object of reducing
natural gas consumption according to a survey of households in Ukraine
Barriers types
on the course to
the measures
implementation
for the purpose
of reducing
natural gas
consumption
Measures groups to reduce natural gas consumption
Installation of
heat-saving
windows on
balcony doors
Insulation
of external
walls of
buildings
Replacement
of gas boilers
for solid fuel
boilers
Installation of
air
temperature
controllers
Installation of
heat
recuperators
of ventilation
air
1. Obstacles
associated with
shortage of
necessary
resources
volume
0.491
0.548
0.542
0.571
0.444
2. Obstacles
associated with
insufficient
quality of
necessary
resources
0.069
0.071
0.000
0.000
0.000
3. Obstacles
associated with
insufficient
competence of
investors
0.222
0.231
0.273
0.333
0.200
4. Political and
institutional
obstacles
0.143
0.100
0.125
0.000
0.000
0.222
5. Obstacles
caused by
insufficient level
of
socio-economic
results of
implementation
measures to
improve the
efficiency of
natural gas use
in the residential
sector
0.111
0.143
0.000
0.250
(continued)
Formation Mechanisms and Overcoming Methods to Reducing Natural …
365
Table 3 (continued)
Measures groups to reduce natural gas consumption
Barriers types
on the course to
the measures
implementation
for the purpose
of reducing
natural gas
consumption
Installation of
heat-saving
windows on
balcony doors
The actual level 0.246
of a measure
implementation
by respondents
for the purpose
of improving the
natural gas
efficiency in the
residential
sector
Insulation
of external
walls of
buildings
Replacement
of gas boilers
for solid fuel
boilers
Installation of
air
temperature
controllers
Installation of
heat
recuperators
of ventilation
air
0.258
0.250
0.286
0.333
Source By authors
Table 4 Results of assessing the changes impact in natural gas prices for household consumers on
the general level of financial and economic barriers to the measures implementation by households
with the object of saving this energy resource
The growth
rate of natural
gas prices
relative to
their base
level
Generalized level of financial and economic barriers by household groups on
the way to the implementation of the surveyed households’ measures to save
natural gas by groups of these measures
Installation of Insulation of
heat-saving
external walls
windows on
of buildings
balcony doors
Replacement
of gas boilers
for solid fuel
boilers
Installation of
air
temperature
controllers
Installation of
heat
recuperators
of ventilation
air
0.6
0.39
0.41
0.30
0.33
0.32
0.8
0.31
0.30
0.17
0.27
0.20
1.0
0.23
0.19
0.22
0.21
0.26
1.2
0.15
0.24
0.30
0.15
0.32
1.4
0.33
0.29
0.38
0.27
0.38
Source By authors
5 Conclusions
In the current scientific literature, the mechanisms of creating obstacles to the
implementation of energy-saving projects, including measures to reduce natural gas
consumption in the residential sector, remain incompletely studied. Accordingly,
most of the presented ways of overcoming these obstacles require a more thorough
366
O. Yu. Yemelyanov et al.
Table 5 Expected efficiency of public expenditures on state reimbursement of the initial loans
amount received by households in order to implement measures with the object of natural gas
saving, and the proposed share of such reimbursement
Indicator name Measures groups to reduce natural gas consumption
Installation of
heat-saving
windows on
balcony doors
Expected
efficiency of
public
expenditures
to reimburse
the initial
amount of
loans received
by households
in order to
implement
measures to
save natural
gas, m3/USD
7.12
Proposed state 28.1
reimbursement
share of the
initial loans
amount
received by
households in
order to
implement
measures to
save natural
gas, %
Insulation of
external
walls of
buildings
6.46
23.5
Replacement of
gas boilers for
solid fuel
boilers
8.54
26.4
Installation of
air
temperature
controllers
5.39
24.8
Installation of
heat
recuperators of
ventilation air
4.07
30.1
Source By authors
justification, which would be based on knowledge of the laws of their formation.
Having this in view, there is a need to model the barriers mechanisms to improve
the efficiency of natural gas in the residential sector and develop scientifically sound
ways to overcome these barriers.
The process of obstacles modeling to improving the efficiency of natural gas
use in residential buildings should be based on pre-grouping the types of such
barriers. Particularly, this grouping can be carried out by stages of the development
process and measures implementation to save natural gas in the residential sector.
Under such conditions, the formation mechanism of these obstacles is described by
a certain sequence of their occurrence. Another grouping way the studied obstacles
follows from those presented in the work five groups of factors that directly affect the
formation of these barriers. These groups include: shortage of necessary resources,
Formation Mechanisms and Overcoming Methods to Reducing Natural …
367
insufficient level of resource quality, insufficient level of investors competence (i.e.
persons who decide on the implementation of investment measures to save natural
gas consumption in residential buildings), political and institutional factors, as well
as insufficient level socio-economic results from the implementation measures to
save natural gas consumption in the residential sector.
An important way to overcome barriers on the course to reducing natural gas
consumption in the housing sector is to recognize concessional lending for such
reductions, which will be carried out at the expense of the state budget. The use
of the optimization model proposed in this paper for the state programs formation
of financial support for measures to reduce natural gas consumption in residential
buildings in the practice of public authorities shall provide an opportunity to increase
the parameters validity of these programs. Calculations using this model on a sample
of households seeking to implement measures with the object of natural gas saving
showed that, despite the rather significant proposed size of the reimbursement shares
of the initial loans amount received by households to implement measures for the
purpose of natural gas saving, the expected efficiency such compensation is quite
high.
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2012.04.012
Research of Characteristics of Solid
Waste as Energy Resource
Artur Voronych , Teodoziia Yatsyshyn , Petro Raiter ,
Lubomir Zhovtulya , and Serhii Maksymiuk
Abstract In Ukraine, about 10 million tons of municipal solid waste (MSW) are
generated annually, of which less than 10% is recycled. On average, the formation
of MSW per capita is 220–250 kg/yr, and in large cities up to 380 kg. Thus, MSW
is a major environmental problem. Accordingly, landfills are currently overcrowded
and cause a difficult environmental situation in the surrounding areas. Solving this
problem is a multi-stage process: starting with producers of various products and
creating the conditions for them to prevent waste generation, public awareness of the
importance of reducing waste flows, as well as consideration of already established
solid waste as a resource. The morphological composition of MSW is analyzed. An
approximate percentage of MSW suitable for energy recovery from the total mass of
MSW entering landfills has been established. The level of reduction of mass of MSW
by burning is determined. Some characteristic parameters of heat treatment of MSW
by experimental method are determined. Thus, the emissions during combustion of
the samples, fuel consumption for their combustion, excess oxygen and combustion
temperature were analyzed. Determination of the calorific value of the samples was
the basis for determining the energy potential of solid waste in Ivano-Frankivsk
region.
Keywords Municipal solid waste · Waste management strategy · Recycling waste
management · Separate waste
A. Voronych · T. Yatsyshyn (B) · P. Raiter · L. Zhovtulya · S. Maksymiuk
Ivano Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
e-mail: teodoziia.yatsyshyn@nung.edu.ua
T. Yatsyshyn
State Institution “The Institute of Environmental Geochemistry” of NAS of Ukraine, Kyiv,
Ukraine
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_22
371
372
A. Voronych et al.
1 Introduction
Ivano-Frankivsk region is located in southwestern Ukraine. The area of the region is
13.9 thousand km2 , which is 2.4% of the area of Ukraine. Ivano-Frankivsk—regional
center of Ivano-Frankivsk region, economic and cultural center of Prykarpattia.
Municipal solid waste is waste that is generated in the course of human life and
activities and accumulates in residential buildings, social and cultural institutions,
public, educational, medical, commercial and other institutions (this is food waste,
household items, garbage, fallen leaves, waste from cleaning and current repair of
apartments, waste paper, glass, metal, polymer materials, etc.) and have no further
use at the place of their formation.
Currently, there are about 18 landfills and dumps where municipal solid waste is
collected in the Ivano-Frankivsk region. But only 8 such facilities have passports.
There are also a large number of unauthorized landfills. In Fig. 1 the main landfills
and dumps in the region are given. The Ivano-Frankivsk landfill near Rybne village
is the largest in the Ivano-Frankivsk region.
According to the Ministry of Development of Communities and Territories of
Ukraine, as of September 1, 2019, separate collection of solid waste has been introduced in 65 settlements. Glass, paper, and plastic are collected separately (all three
fractions, or only some of them, depending on the settlement). Biomass is collected
separately in Halych. In 2019, the share of settlements with separate collection of
solid waste to the total number of settlements in the region is 8%.
At the same time, according to the additional monitoring of the Ministry of
Regional Development, only 7 landfills meet the state construction requirements for
landfills. Only the landfill in Ivano-Frankivsk has a filtrate collection and purification
system, landfill gas collection and utilization of seven landfills, and three landfills
have only filtrate collection systems.
Fig. 1 Operating landfills in Ivano-Frankivsk region
Research of Characteristics of Solid Waste as Energy Resource
373
Fig. 2 Handling MSW in Ivano-Frankivsk region [1]
In accordance with the data of the Ministry for Communities and Territories
Development of Ukraine, from 2015 to 2017, there was a tendency to an increase in
the volume of solid waste generation in Ivano-Frankivsk region. In 2019, the volume
of municipal waste generated was 1,027,000 m3 , which was 29% more than in 2011.
Municipal solid waste generated in Ivano-Frankivsk region is currently a major
environmental problem. Imperfect system of solid waste management causes their
constant accumulation and burial in landfills (Fig. 2).
According to the calculated data, and taking into account the fact that the service
for the removal of household waste in 2019 covered only 78.2% of the region’s
population, the estimated average waste generation rate will be 0.96 m3 /person/yr
or 180 kg/person/yr. The actual volumes of household waste generation are larger,
since usually the volumes of waste removal are equated to the volumes of waste
generation, and waste generated in villages, where the service is not provided, is not
accounted to it. Therefore, according to expert estimates, the real volume of solid
waste generation in Ivano-Frankivsk region may be about 1,256,000 m3 . According
to the annual environmental passports of Ivano-Frankivsk region, the average value
of solid waste generation is 207829.1 tons/yr [2–6]. The data were analyzed for 2017
and 2018, as data on MSW generation from previous years are not available.
The morphological composition of MSW is quite diverse and variable both in time
and geographically. Municipal solid waste that is taken to the landfill is waste from
residential buildings—food waste, room and yard waste, glass, leather, rubber, paper,
metal, waste from apartment renovations, ash and slag, large household items, as well
as household waste of trade enterprises and cultural and welfare institutions, waste
of catering enterprises, waste of markets, medical institutions, street waste, industrial
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A. Voronych et al.
Fig. 3 Determining the percentage composition of MSW
and construction waste of hazard class IV. The average indicators of the composition
of MSW in Ivano-Frankivsk are shown in Fig. 3) [7]. It significantly depends on the
season due to the increase in food waste content from 20–25% in spring to 40–55%
in autumn. Therefore, the percentage ratio between various components of MSW
can be given only conditionally or for a specific batch of waste.
Establishing the mass of MSW suitable for energy recovery makes it necessary to
analyze the morphological composition of solid waste. Not all landfills of the region
have data from the study of the morphological composition of MSW, therefore, there
has been studied the information on the receipt of solid waste of the largest operating
landfill in the region, located in Rybne village near Ivano-Frankivsk. This solid
waste landfill serves the settlements of Ivano-Frankivsk City Council, Tysmenytsya,
Nadvirna, Kosiv and Kolomyia districts. In 2020, the enterprise plans to accept
110 thousand tons of household waste for disposal, of which 1.0 thousand tons of
recyclable materials will be sorted, and the rest will be buried [8].
There was determined the approximate percentage of solid waste suitable for
energy recovery from the total mass of solid waste supplied to landfills, which was
67.61%. These components of MSW include: paper (cardboard), rubber and leather
waste, plastic, wood, biowaste and unsorted residues suitable for incineration. About
32.4% of MSW is unsuitable for energy production—it is unsorted (non-combustible)
residue, glass, metal.
Thus, based on the obtained value of the total amount of solid waste, which is
207,829.1 t/yr, there has been established an approximate value of solid waste suitable
for energy recovery by incineration in Ivano-Frankivsk region, which accounts for
140,513.2 t/yr. It should be taken into account that the composition of MSW varies
Research of Characteristics of Solid Waste as Energy Resource
375
Fig. 4 The composition of wastes used for incineration
throughout the year and depends on the area. Figure 4 shows the percentage of solid
waste that is suitable for energy recovery in the total mass of waste.
2 Research Methods
According to the standard method for determining the moisture content, the MSW
samples were weighed and then placed in a drying cabinet, where the temperature did not exceed 105 °C. The weight of the samples was monitored periodically
and there was established the moment when the decrease in their weight stopped.
According to the found masses of wet and absolutely dry sample, the relative humidity
is determined for each sample.
Determination of the pH of MSW samples was carried out by analyzing the water
extract of the waste using a pH meter.
The electrical conductivity of the test samples was determined by impedance
spectroscopy using an Autolab PGSTAT 12/FRA-2 modular potentiostat at room
temperature [9, 10].
The study of the chemical composition of the studied samples was carried out by
the method of X-ray fluorescence analysis in Laboratory of gamma-resonance spectroscopy with analysis of electron conversion, gamma and X-ray radiation (Vasily
Stefanik Precarpathian National University) [11]. The method is based on the analysis of the fluorescence spectra of radiation elements during the adsorption of highenergy radiation. The method allows obtaining data on the chemical composition of
a substance in a wide range with an accuracy of 1–10 ppm. The experiments were
376
A. Voronych et al.
carried out on an EXPERT 3L Precision Analyzer with a constant supply of helium
to the collimator channels.
Analytical methods
The assessment of the morphological composition of MSW was carried out on
the basis of the averaged data of MSW indicators in 2017–2018, provided by the
Municipal Solid Waste Landfill and obtained according to the recommendations
[12].
MSW sampling was carried out at the landfill near Rybne village. This landfill is
equipped with a sorting line with a capacity of 50 tons/day, which has been operating
since 2018.
The expedition to the landfill was carried out in the winter, therefore, at the time
of sampling, the sorting line was not working due to the lack of a cover over the
conveyor. 5 samples of 10 kg each were taken from the waste collection vehicles of
MSW.
At the next stage, there was performed sorting by morphological composition of
municipal solid waste.
1 group of 5 samples were prepared according to the established composition,
shown in Fig. 4 in a form of 1 kg. It was prepared for analyses to determine the
physical and chemical characteristics, as well as to establish the calorific value of
MSW using a calorimeter. To carry out the analyses envisaged by the project, the
constituent samples were grounded by the component with a knife grinder to a particle
size of no more than 0.1 mm.
The grounded samples were stored in closed glass containers. Depending on the
requirements for the analysis, the samples were subjected to subsequent processing:
• a water extract was prepared to define the pH;
• to determine the chemical composition, the sample preparation needed heating
in a muffle furnace at a temperature of 700 °C until the waste was completely
converted into ash [13];
• for analysis in the calorimeter, MSW samples were compacted using a press to
one-gram Tablets, provided by the method.
Determination of energy recovery potential from MSW was carried out in two
stages:
stage I—experimental determination of the heat released by MSW combustion
suitable for energy recovery using IKA C1 calorimeter;
stage II—calculation method for determining the lower heat of combustion and
calculating the energy potential of the MSW mass.
There was formed a sample of fuel from the selected and appropriately prepared
samples of MSW using a press. All components of the sample were formed in
accordance with the percentage composition of MSW, suitable for energy recovery
according to the data shown in Fig. 4. The mass of the filling corresponded to 1 g ±
0.05. 1 ml of distilled water was introduced into the bomb and placed in a calorimeter
for conducting an experiment to determine the heat of combustion.
Research of Characteristics of Solid Waste as Energy Resource
377
3 Research Results
The study of MSW characteristics is a prerequisite for establishing the energy potential of MSW in the region. Physical and chemical analysis of MSW samples provided
for the determination of the following indicators:
• humidity and pH;
• electrical conductivity and redox potential;
• elemental composition of solid waste.
The total humidity for the five samples ranges from 48.97 to 55.24%. High
humidity values are associated with a significant content of bio-waste, as well as
the ingress of external moisture from precipitation, since containers for collecting
MSW are mostly without shelter.
The average value of obtained pH values is 5196 with deviation 0.09. This
corresponds to the moderately acidic reaction.
As a result, there were obtained the dependences of the electrical conductivity of
the samples on the frequency (Fig. 5).
In general, the properties of all samples are close to those of dielectrics. As can
be seen in Fig. 5, all test samples have a conductivity in the range of 1*10–4 –3*10–4
.−1 m−1 at constant current. As the frequency rises to 100–150 Hz, the conductivity
increases rapidly (which is typical for dielectrics). At frequencies exceeding 150 Hz,
the dynamics of the growth of conductivity decreases, which indicates a complex
case of superposition of various types of conductivity, which is characteristic of both
semiconductors and metals.
-3
2,5x10
-3
2,0x10
1
2
3
4
5
-3
σ'
1,5x10
-3
1,0x10
-4
5,0x10
0,0
-3
10
-2
10
-1
10
0
10
1
10
2
10
Frequency, Hz
Fig. 5 Relation of test sample conductivity to frequency
3
10
4
10
5
10
6
10
378
A. Voronych et al.
Table 1 Chemical composition of MSW, %
Chemical element
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
O
23.201
26.021
23.706
27.069
25.069
Al
0.198
0.905
0.524
0.469
1.021
Si
1.886
3.143
1.524
4.164
4.516
P
0.0
0.0
0.0
0.0
0.289
S
0.762
1.132
0.546
1.089
0.942
Cl
19.307
13.259
16.722
13.116
17.017
K
12.568
13.683
11.353
9.171
17.289
Ca
34.921
30.803
42.07
34.541
28.16
Ti
1.285
1.523
0.391
1.855
0.911
Cr
0.275
0.805
0.094
0.479
0.206
Mn
0.234
0.26
0.183
0.256
0.162
Fe
4.521
7.083
2.448
6.505
3.389
Ni
229 × 10–5
0.0
259 ×
Cu
316 ×
Zn
0.538
Br
0.199
10–5
10–5
198 × 10–5
152 × 10–5
333 ×
322 ×
10–5
10–5
3.389
0.14
0.892
0.319
0.742
0.551
0.316
0.0
0.4
0.149
Sr
495 ×
0.113
0.067
0.057
0.067
Nb
0.0
381 × 10–5
0.0
0.0
0.0
Ag
0.0
0.0
0.0
405 × 10–5
0.0
Cd
0.0
0.0
0.0
0.0
197 × 10–5
Pb
0.0
0.0
0.0
0.0
0.082
10–5
The analysis of chemical composition was performed for 5 samples of MSW. As a
result of processing the spectra, the average chemical composition of the experimental
samples was obtained (Table 1).
As can be seen from the data of Table 1 the samples differ greatly in their chemical
composition. So, basically, the samples contain such chemical elements as Ca, Si,
Fe. The amount of other chemical elements included in the sample products is less
than 0.01%.
The main chemicals and compounds that are encountered in the analysis of
these samples can probably be attributed to construction waste (SiO2 , Fe2 O3 , CaO).
Another group of chemical compounds are the remains of organic compounds, probably food and plant products (K2 O, P2 O5 ). Also, a significant part is made up of
compounds that can be components of various paints and dyes (TiO2 , Fe2 O3 ).
As a result of the experiments determination of the heat of MSW combustion,
there was obtained the calorific value (higher heat of combustion) of MSW five
samples, given in Table 2.
The obtained experimental data do not take into account the humidity values,
since the use of dried samples is envisaged for the experiment with the calorimeter.
Research of Characteristics of Solid Waste as Energy Resource
Table 2 Data of
experimental determination
of MSW calorific value
379
Sample no
q_(V,gr,d), [J/g]
Sample No 1 Ua
24,333
Sample No 2 Ua
24,772
Sample No 3 Ua
26,195
Sample No 4 Ua
25,942
Sample No 5 Ua
24,624
Thus, it becomes necessary to take this factor into account by carrying out additional
calculations.
The energy recovery potential of a territory or region is defined as the product of
the amount of MSW produced in a specified region during the year by the value of
the net calorific value of the specified MSW. The value of the lowest calorific value of
experimental MSW samples is obtained using the experimentally determined values
of their calorific value (higher calorific value) according to the conversion formula
[14, 15]:
qp, net, m = {q V , gr, d − 206W H, d} · (1 − 0.01M T ) − 23.05M T ,
(1)
where qp,net,m—lower heating value, J/g; qV,gr,d—calorific value (higher calorific
value), J/g; WH,d—hydrogen content in MSW, %; MT —humidity of MSW, %.
The values of humidity for each component of MSW and the calculated values
of humidity for MSW for each of the experimental samples are experimentally
determined during physical and chemical analysis of MSW.
The moisture that enters the firebox is non-combustible and does not give heat.
But, in the process of high-temperature combustion of solid waste in the furnace,
hydrogen is released, which must be added to the main component of hydrogen,
which is already present in the components of MSW, as one of the chemical elements,
chemical reactions during the combustion of which lead to the production of thermal
energy. Therefore, it is necessary to carry out calculations of the total amount of
hydrogen (percentage), which is present in MSW, the calorific value of which is
being investigated. For this purpose, on the basis of typical data on the content of
“combustible” chemical elements—carbon, hydrogen, oxygen, sulfur—in different
components of MSW, there were performed calculations of the amount of these
chemical elements in grams.
However, these calculations include the mass of hydrogen contained in the socalled “dry” part of MSW.
To determine the mass of hydrogen that is released from water during the
incineration of MSW with a certain humidity, there was applied the formula [16]:
H ydr ogen mass (H ) = H ydr ogen in Dr y Mass+
2
+ · (W et Mass o f M SW − Dr y Mass o f M SW ),
18
(2)
380
A. Voronych et al.
Table 3 Calculation results
of the hydrogen content
Sample no
Hydrogen mass(H2), g
Hydrogen mass(H2)
part in MSW, %
Sample 1 Ua
461.9
21.45
Sample 2 Ua
456.6
21.13
Sample 3 Ua
452.7
18.41
Sample 4 Ua
462.3
20.97
Sample 5 Ua
459.9
21.43
where Hydrogen mass (H) is the mass of the hydrogen element in MSW, g; Hydrogen
in Dry Mass is the hydrogen contained in the so-called “dry” part of MSW, g; Wet
Mass of MSW is the mass of wet MSW, g; Dry Mass of MSW is the mass of dry
MSW.
Table 3 shows the results of calculating the values of hydrogen content in each of
the MSW samples.
According to the formula (1), there were calculated the values of the lower calorific
value of experimental MSW samples, given in Table 4. These values made it possible
to calculate the average value of the lower heat of MSW combustion obtained from the
results of the study of the calorific value of five samples of MSW in Ivano-Frankivsk
region.
Energy recovery potential of the territory or district of Ivano-Frankivsk region was
defined as the product of the amount of solid waste produced in the specified region
during 2018, by the average value of the lower calorific value of the mentioned MSW
indicated in Table 5. The results of calculations of the Energy recovery potential for
the Ivano-Frankivsk region are shown in Table 5.
Table 4 Calorific value and Net Calorific Value
Sample no
Calorific value,
qV,gr,d, [J/g]
Total moisture,
MT [%]
Hydrogen content
of the sampel,
WH,d, [%]
Net Calorific Value
(NCV), qp,net,m,
[J/g] or [kJ/kg]
Sample 1 Ua
24,333
55.24
21.45
7640.1
Sample 2 Ua
24,772
54.65
21.13
8000.5
10302.2
Sample 3 Ua
26,195
48.98
18.41
Sample 4 Ua
25,942
54.35
20.97
8617.9
Sample 5 Ua
24,624
55.20
21.43
7782.3
Average
8468.6
Research of Characteristics of Solid Waste as Energy Resource
381
Table 5 Energy recovery potential of MSW per year and Energy potential of electricity and thermal
energy per year in Ivano-Frankivsk region
Mass of waste per
year suitable for
energy recovery by
incineration
Mass of waste per year Energy recovery
suitable for energy
potential of MSW per
recovery by
year
incineration—without
ash
Energy potential of
electricity and thermal
energy per year
kg
kg
[MJ/yr]
[kWh/yr]
80%,
[kWh/yr]
140,513,200
134,976,980
1143062.8
317 518
254014.4
4 Discussion and Conclusions
The work is carried out study the possibilities of completing the management of
municipal solid waste (MSW) in the Ivano-Frankivsk region by thermal treatment
methods for recovery of energy.
Physical and chemical analysis of MSW was performed. It included determination of the humidity and pH, electrical conductivity and redox potential, elemental
composition of solid waste.
According to the research of the humidity of solid waste samples, the percentage
of humidity in the samples is high (in the range from 48.97 to 55.24%), which is
caused by the lack of protection of solid waste from atmospheric moisture at the
stage of their collection.
Analysis pH of MSW samples determine that the average pH is 5.2, which
corresponds to the moderately acidic reaction.
Based on the nature of the electrical conductivity dependence on the frequency
and chemical composition of the test MSW samples, we can say that the samples
contain different types of conductivity, due to the multicomponent composition of
the samples.
According to the research of chemical composition of MSW, the share of heavy
metals and chemical elements harmful to human health does not exceed the permissible limits. The chemical composition of MSW is a rather variable characteristic;
therefore, it can be given only in the form of estimated values.
The study of the energy potential of solid waste in Ivano-Frankivsk region showed
that average Net Calorific Value is equal 8468.6 J/g, and energy potential of electricity
and thermal energy in region per year is 254014.4 kWh/yr.
Acknowledgements This research was co-financed by the European Union within the framework
of Hungary-Slovakia-Romania-Ukraine ENI CBC Programme 2014-2021 under the project “Energy
Recovery from Municipal Solid Waste by Thermal Conversion Technologies in Cross-border
Region” HUSKROUA/1702/6.1/0015.
382
A. Voronych et al.
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org/10.15242/dirpub.er0517030
Renewable Power Engineering
Geothermal Heat Supply Development
Pathways in Ukraine
Yulia Shurchkova , Sergii Shulzhenko , Anna Pidruchna ,
Volodymyr Deriy , and Vitaly Dubrovsky
Abstract The chapter considers the trends in the development of geothermal heat in
the world and the situation with the use of renewable energy sources in Ukraine. The
reasons for the country’s lag in this area in the presence of resource and scientific
base, experience in construction and operation of thermal geothermal stations are
analyzed, as well as economic prerequisites for creating geothermal heat supply
systems based on deep wells and near-surface geothermal resources.
Keywords Geothermal energy · Heat supply technology · District heating ·
Economic feasibility · Geological wells
1 Introduction
Since the world’s ecological challenges, the reduction of world reserves of fossil
fuels, and the rising fuel prices, the question of the development of the alternative
energy industry is acute. The world is moving from the traditional combustion of
fossil fuels for heat supply to the use of energy-efficient technologies, including
geothermal. Geothermal energy is developing in two main areas: electricity generation and heat production. Geothermal electricity is developing mainly in countries
located in areas of modern volcanism, where the coolant has high parameters, available on the Earth’s surface, the cost of building geothermal power plants is minimal,
and energy costs are competitive in the energy market. Geothermal heat is geographically more widespread, as it requires thermal resources with lower temperatures.
Analysis of trends in the development of geothermal energy shows that in the coming
decades, the most intensive development of geothermal heat supply. The production
of geothermal heat accounts for 85% of the total capacity of the world’s geothermal
energy by today. The total installed capacity of thermal geothermal plants in the
world accounted for 107,727 MW at the end of 2019. According to WGC2015 [1],
the increase in capacity in the period from 2010 to 2015 was about 45%, and in
Y. Shurchkova · S. Shulzhenko · A. Pidruchna (B) · V. Deriy · V. Dubrovsky
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: pidruchna@gmail.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_23
385
386
Y. Shurchkova et al.
the period from 2015 to 2019—52%. In terms of volumes of all types of renewable
energy sources for heat supply, geothermal energy ranks second in the world after
solar [2].
Geothermal district heating currently is used in 28 countries in Europe, Asia
and America. The leaders are China, Iceland, France and Germany. There are more
than 5000 geothermal district heating systems in Europe, which is about 10% of
the total heating market. The countries of Eastern and Central Europe—Hungary,
Poland, Slovakia, Slovenia, the Czech Republic and Romania—are also active in
developing geothermal heat supply. Ukraine is not still belonging to the countries that
develop geothermal energy, despite the obvious urgency of the problems, both fuel
and environmental, and despite the fact that the country has a fairly high geothermal
potential. In terms of the scale of geothermal energy use, it is critically lagging behind
not only the leading countries in this field, but also the neighboring countries that
have similar or even less potential for geothermal resources. Table 1 shows the data
for the use of geothermal energy in Ukraine’s neighboring countries in 2015 [3].
The use of geothermal energy for the purposes of heat supply, ventilation and
air conditioning can reduce energy consumption by 25–50% compared to traditional systems [4]. According to the International Geothermal Agency, the use of
geothermal heat in 2015 saved 52.5 million tons of oil equivalent per year and
significantly reduces the consumption of traditional fuels and reduces environmental
pollution. The study has been conducted by the US Department of Energy in the field
of geothermal heating has shown that geothermal heating system to reduce carbon
emissions by 46 million tons compared to the use of fuel oil.
The intensive development of geothermal energy in the world is largely determined
by the fact that a number of industrialized countries are investing heavily in this
Table 1 The use of geothermal energy in Ukraine’s neighboring countries
Country
Installed capacity,
MW
Annual
consumption TJ/yr
Coefficient
GWh/yr
4.73
113.53
31.54
Hungary
905.58
10 268.06
2 852.47
Moldova
–
–
–
488.84
2 742.60
761.89
Belorussia
Poland
Load Factor*
0.76
0.36
–
0.18
Russia
308.20
6 143.50
1 706.66
0.63
Romania
245.13
1 905.32
529.30
0.25
Slovakia
149.40
2 469.60
686.05
0.52
10.90
118.80
33.00
0.35
** Ukraine
(according to the
statistical data for
2005)
*
Load Factor—power factor: (annual energy consumption in TJ/year)/(installed capacity in MW)
× 0.03171
** data for 2005 are given for Ukraine, as they are not available for the following years
Geothermal Heat Supply Development Pathways in Ukraine
387
industry. Thus, in the period from 2010 to 2014, 49 countries invested about $20
billion in geothermal energy, which is twice as much as in 2005–2009. Turkey,
Kenya, China, Thailand, the United States, Switzerland, New Zealand, Australia,
Italy and South Korea have invested more than $500 million.
2 Ukraine’s Available Geothermal Energy Resources
and Technologies
The explored reserves of geothermal energy in Ukraine are about 200 GW. They are
represented by thermal waters with a temperature from 50 to 220C and are located in
almost all regions of the country. According to the State Energy Efficiency Agency
of Ukraine, the long-term potential of geothermal energy is about 90 TWh/yr, which
can provide annual fuel savings for heat production of about 10 billion cubic meter
of natural gas.
The Geothermal Atlas of Ukraine [5] presents temperature fields at depths from
0 to 75 km. According to these data, at a depth of 0.5 km temperatures vary widely:
from 130C for the Ukrainian Shield (Fig. 1) to 19–320C for the Donbass and the
Dnieper-Donetsk basin and up to 430C for the Transcarpathian Depression.
At a depth of up to 1 km in the area of the Ukrainian Shield temperatures do
not exceed 19–220 °C; in the Donetsk region—in the range of 23–500 °C; in the
Carpathian region—30 to 500 °C; in the Transcarpathian depression from 70 to 1000
°C; in the Pre-Carpathian zone—from 45 to 700 °C. In the territory of Crimea in the
central and western part, and also on the Kerch peninsula waters with a temperature
of 60–900 °C could be found out.
At depths of 3–3.5 km there is a higher background temperature with large differences. In the Donetsk region (in the extreme west of the country), and Crimea,
temperatures reach 100–1400 °C, and in the Uzhgorod region—1600 °C. In the
area of the Dnieper-Donetsk basin, which covers Chernihiv, Sumy, Kharkiv, Poltava,
Luhansk regions, as well as in the Carpathians and Prykarpattia, the temperature of
the overlying rocks is 70–900 °C.
At a depth of 5 km there are large volumes of coolant with a temperature of 84–950
°C almost throughout the country.
As can be seen, high-temperature waters at accessible depths are available in
limited quantities in several small deposits. Mostly in most parts of the country there
are waters with temperatures from 50 to 1000 °C. According to the calculations of
the ITTF of the NAS of Ukraine [7], the technically available energy potential of
geothermal energy sources in Ukraine is 51.14 million MWh/yr, which can provide
annual fuel savings for heat production of 6.65 million tons per ton.
Technologies for the use of geothermal energy for heat supply. Modern geothermal
systems use the principle of forced circulation of the coolant through underground
permeable layers, when heated geothermal water, which is in a natural or artificial
underground permeable reservoir, is extracted to the surface through a production
388
Y. Shurchkova et al.
Fig. 1 State Geological Map of Ukraine [6]: 1—Ukrainian shield; 2—slopes of the Ukrainian
Shield and the Voronezh Massif; 3—shield framing: Volyn-Podilsk and Scythian plates, DnieperDonetsk depression and Pripyat depression; 4—south-eastern outskirts of the Western European
platform; 5—Black Sea basin; 6—Donetsk folded region; 7—folded systems of the Carpathians,
Dobrudzha and Crimea; 8—Carpathian and Pre-Dobrudzha depressions
well, fed to the consumer and then pumped back into the downhole. This method
of removing heat from the deep layers of the Earth—the creation of geothermal
circulation systems (GCS)—was first proposed in the mid-50 s of the last century
at the Institute of Heat Power Engineering of the Academy of Sciences of Ukraine
by Academicians A. N. Scherban and O. O. Kremnev. In the scientific team under
the leadership of Doctor of Technical Sciences AV Shurchkov, the scientific basis
for the functioning of GCS systems was created, a large amount of research and
development work was carried out. A number of technologies and installations,
including for geothermal heat supply of settlements, industrial, agricultural, social,
communal and other facilities, were brought to the research and industrial stage.
The emergence of new technologies for the extraction and use of coolant, as well
as methods for forecasting geothermal resources, have provided a significant increase
in the last 25 years of consumption of thermal geothermal energy. In the world, a
multivariate technology for the use of geothermal resources has been developed and
millions of existing heat supply systems have been built [8].
A typical heat supply system consists of two main parts: an underground complex
that includes production and injection wells, and a complex of ground structures that
includes pumping stations, heat exchangers, heat transformers (heat pumps), heating
Geothermal Heat Supply Development Pathways in Ukraine
389
mains, water treatment and treatment plants, peak boilers. Schematic diagram of the
station is shown in Fig. 2 [9].
For the conditions of Ukraine, several variants of schemes for creating geothermal
heat supply systems are possible:
1. Based on specially drilled wells in the area of the geothermal field, when the
water temperature exceeds 750 °C. The implementation of such technology is
possible in some areas of the Carpathian region and the Donetsk-Dnieper Basin,
in the Crimea at a depth of wells up to 3500 m.
2. On the basis of specially drilled wells with water temperature below 750 °C. The
depth of wells, as a rule, does not exceed 2500 m. As the water temperature is
Fig. 2 Schematic diagram of a geothermal heat supply station: 1-injection well; 2-pump installation; 3-system of water and gas purification and water treatment; 4-heat exchangers; 5-peak heaters;
6-mains pump; 7-main heating mains; 8-12—potential heat consumers; 13-heat pump; 14-depth
pumps; 15-production (water-lifting) well; 16-filter system
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Y. Shurchkova et al.
insufficient for supply to the heat supply system, heat pumps are used to increase
its temperature.
3. Based on existing wells. These can be spent and preserved wells of gas and oil
fields or various types of exploratory wells containing thermal waters. There are
more than 20,000 unused wells in Ukraine that could potentially be used for
these purposes. The results of their survey are shown in Table 2. / 10 / The largest
number of wells is located in densely populated regions in Donetsk, Chernihiv,
Sumy, Poltava, Kharkiv regions. Depth of thermal waters from 3.5 to 5 km,
temperature range from 35 to 1700 °C.
4. Based on near-surface heat resources. Due to the use of heat pump technologies, low-temperature near-surface geothermal resources at depths from a few
meters to 100–300 m are becoming increasingly important. Despite the fact that
the properties and processes occurring in the near-surface zone are currently
insufficiently studied, there is no substantiated data for the selection of sites for
geothermal systems, the existing recommendations are indicative and preliminary, world practice shows that the use of low-potential Geothermal resources are
economically viable for heat supply of low power facilities. Shallow geothermal
resources have a number of advantages, such as practical inexhaustibility, ubiquity, proximity to the consumer, safety, economic competition for traditional
boilers, environmental friendliness. The essence of the technology of using the
heat of the near-surface zone is to create a downhole or horizontally located
underground heat exchanger connected to the heat pump. Figure 3 shows the
schematic diagrams of heat supply systems using the heat of the surface layers
of the earth [9].
The heat pumping equipment of most world companies is currently present on
the Ukrainian market. We offer mainly systems for private homes and cottages. The
payback period of such systems is from 2 to 5 years.
Table 2 Lawn and oil wells suitable for thermal water production
Regions
Crimea
Zakarpattia
Dnieper-Donetsk Rift
Number of existing wells
36
14
141
Depth, m
1200…1600
1000…2000
3500…5000
Flow rate, m3/day
500…1200
500…1000
500…1200
Temperature, oC
50…70
55…75
90…170
Mineralization, g/l
10…20
15…25
150…200
The nature of productio
self-deprecation
pump
pump
Geothermal Heat Supply Development Pathways in Ukraine
391
Fig. 3 Geothermal system. A—with horizontal channels: 1–heat pump, 2–heat accumulator, 3–
ground heat exchanger;B—with vertical wells: 1–heat accumulator, 2–heat pump, 3–columns of
pipes
3 Geothermal Heat Supply Economic Feasibility
Assessment for Ukraine
The main consumers of thermal energy in Ukraine are households and communal
services. The needs of households and communal services in the total energy balance
of the country accounts about 55% of heat produced and more than 27% of fuel
consumed.
Heat is currently produced at 14 large coal-fired thermal power plants and 31,000
boilers, 24% of which are equipped with boilers operated for more than 20 years,
with an efficiency of less than 82% and a low level of gas cleaning. Coal and gas for
thermal energy production are largely imported from abroad. The difference between
the needs of the industry and own resources exceeds 30%.
The housing sector is considered to be technically backward with a number of
economic and environmental problems. Large-scale use of geothermal energy for
housing and the private sector could be a good alternative to partially replace traditional fuels, significantly reduce greenhouse gas emissions and improve the environment. The main advantage of geothermal energy compared to other renewable
sources is that its use is possible around the clock, all year round, unlike, for example,
solar or wind, which can generate energy only about one third of the time. In addition,
the direct use of geothermal energy is one of the most environmentally friendly. The
main disadvantage of the geothermal system is the need for significant investments at
the initial stage of development, in the design and construction of stations in combination with a high level of risk. But the specifics of projects for the construction of
geothermal stations is that with a fairly high capital investment, operating costs are
sharply reduced.
Total costs for the construction of a geothermal thermal power plant based on
deep wells include costs for preparation for construction, preparation of design and
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Y. Shurchkova et al.
estimate documentation, topographic and engineering surveys, construction of an
underground complex, construction of surface structures. The largest capital expenditures are for the construction of an underground complex—for drilling exploration
and production wells or reconstruction of existing ones. They account for 50–90% of
the total investment, depending on the depth of the productive formation containing
thermal water, its temperature, effective power, permeability and formation pressure
[9]. According to [10] in Ukraine, the average cost of drilling 1 m of oil and gas wells
on land is about 2000 USD/meter. The cost of drilling geothermal wells is within the
same limits. A new gas well could cost up to $5.5 million USA. For comparison, in
the USA the average cost of drilling of 1 m of wells depending on depth lies within
the same limits—from 500 to 2000 dollars USA/m.
When using existing wells, capital costs for the construction of an underground
complex are significantly reduced. For example, according to [11], the amount of
capital investment in the construction of the underground HRT complex in Beregovo
in the Transcarpathian region on the basis of a specially drilled well with a depth
of 1300 m amounted to 3.226 million US dollars, and capital expenditures for the
reconstruction of canned wells of the same depth $57 million USA.
The costs of construction or reconstruction of terrestrial infrastructure consist of
the cost of pumping stations, heat exchangers, thermal transformers, heating mains.
The total cost of building a ground complex depends on many factors and varies
widely.
The specific cost of construction of thermal power plants depends on the depth of
wells, their type, the configuration of surface structures, the location of the station
relative to the consumer and others.
The expediency and efficiency of the use of geothermal heat supply systems are
determined mainly by the amount of production profit and payback periods, which,
in turn, depend on heat tariffs. In [11] the results of researches of dependence of
payback period on heat tariffs on materials of 20 projects of geothermal thermal
power plants developed in Institute of thermal physics of National Academy of
Sciences of Ukraine during 1998–2003 are shown at Fig. 4.
With existing tariff for the heat energy (14.1–15.91 USD/MWh), the payback
period of the projects averaged about 15 years. Therefore, despite the high assessment
of technical, environmental, social solutions, the projects were not implemented, as
economic parameters made them unprofitable. Projects could be considered feasible
at rates of $27/MWh and above, with a payback period of 7 years or less.
During the period from 1998 to 2020 in Ukraine, tariffs for heat and electricity
have increased many times, both in hryvnia and in dollar terms.
It is of interest to compare the economic efficiency of these projects in terms
of tariffs and prices in 2003 and 2020. For this purpose, 4 projects were selected,
which passed the technical expert evaluation of foreign experts and were approved
for implementation. Table 3 presents the technical parameters of these projects.
While maintaining all the technical decisions made in these projects, calculations
of economic indicators for the conditions of 2020 were made. The results of the
calculations are presented in Table 4. The calculations showed that the specific capital
investment in geothermal heating system for the conditions of 2020 compared to 2003
Geothermal Heat Supply Development Pathways in Ukraine
393
Fig. 4 The payback period depending on the value of tariff [11]
Table 3 The list of the projects and their technical characteristics
Regions
Lviv region,
Mostyska
Chernihiv region
Crimea
Zakarpattia region,
Beregovo
Depth of wells, m
3500
3500
2230
1300
Number of well
3 (2/1)*
2 (1/1)*
3 (2/1)*
5 (3/2)*
Type of wells
NW**
RW**
NW**
NW/RW**
Water temperature, °C 95
90
90
60
Heat load, MW
12.57
1.63
20
6
Annual heat
consumption, MWh
42,075
7304.7
51,200
18,148
*
number of production / number of absorbing wells
** NW —new wells, RW —restored wells
increases by 42% for newly drilled wells and 40% for restored wells. The cost of
heat production for systems based on newly drilled wells increased by an average
of 62%, for restored wells—by 49.8%. But, at the same time, due to high tariffs for
heat production profit increased by an average of 580%. Payback periods of projects
have been reduced to 3–7 years.
It is evident that the listed projects are feasible in modern conditions, and could
be implemented with high economic benefits [12]. If we compare geothermal heat
supply systems with fuel boilers, the analysis presented in [11] shows that in terms
of profitability they correspond to the economic indicators of heat supply projects
based on small fuel boilers. But environmental benefits and independence from fuel
market conditions and pricing make geothermal heating systems more cost-effective
than fuel boilers.
*30.6/26.9
_
Over 15
USD/kWyr
USD
Years
Cost
Production profit
Payback period
7/4
*614.8/584.5
*45.9/40.3
*702/365
2020
Over 15
366.0
9.5
460
3
5372212.8
19.5
519.7
2020
2003
*/
261
1998
USD/kWyr
Chernihiv region
Lviv region. Mostyska
Regions
Specific capital expenditures
Units
Table 4 The financial efficiency of the projects
13
508.2
3.9
433
2003
Crimea
3
2737.6
5.9
872
2020
Over 15
39.8
7.7
625
2001
6
360.2
11.5
767
2020
Zakarpattia region
Beregovo
394
Y. Shurchkova et al.
Geothermal Heat Supply Development Pathways in Ukraine
395
4 Ukrainian Legal Base and State Support
In Ukraine, the main legal documents in the field of alternative energy sources are:
Law of Ukraine «On Alternative Energy Sources», Law of Ukraine «On Energy
Conservation», Code of Subsoil of Ukraine, Law of Ukraine «On Heat» Law of
Ukraine «On Energy Lands and Legal Regime» special zones of energy facilities
«and a number of others. The development of geothermal energy is envisaged by the
National Renewable Energy Action Plan until 2020. According to Ukraine’s energy
strategy in the field of geothermal energy, it is planned to reach 0.19 GW of installed
capacity by 2020 and 0.7 GW by 2030. The Law of Ukraine on Heat Supply provides
for the use of «non-traditional and renewable energy sources, including geothermal
waters». A number of documents declare state support in accordance with the amount
of funds provided by the law on the State Budget of Ukraine, as well as funds
for research work to improve heat supply and energy saving systems. In 2015, a
Memorandum of Cooperation and Understanding in the Development of Geothermal
Energy was signed between Ukraine and Iceland. The Tax Code of Ukraine provides
benefits for the taxation of energy-saving and energy-efficient projects, including
those related to geothermal energy. It would seem that favorable conditions have
been created for the development of geothermal heat supply. However, according to
the State Agency for Energy Efficiency, as of the end of 2020, geothermal energy is
not among the recently commissioned renewable energy facilities.
World experience shows that the successful development of geothermal energy
requires government support, detailed information on geothermal deposits, sufficient
funding, and involvement of modern technologies.
5 Conclusions
The analysis of trends in the development of geothermal energy in the world shows
that in the coming decades, the most intensive development of geothermal heat
supply. There is a transition from traditional combustion of organic fuels to the use
of energy efficient technologies, including geothermal. In terms of volumes of all
types of renewable energy sources for heat supply, geothermal energy ranks second
in the world after solar. The use of geothermal energy can significantly reduce the
cost of traditional fuels and reduce environmental pollution, ensures independence
from the situation and pricing in the fuel market. The main advantage of geothermal
energy compared to other renewable sources is that its use is possible around the
clock all year round, unlike, for example, solar or wind, which can generate energy
only about one third of the time.
Ukraine is critically lagging behind not only the leading countries in this field in
terms of the scale of geothermal energy use, but also the neighboring countries that
have similar or even lower potential for geothermal resources.
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Y. Shurchkova et al.
In the presence of fossil fuel shortages, low efficiency of most old boilers, high
levels of greenhouse gas pollution, the development of geothermal energy in the field
of housing and communal services may be an alternative to partial replacement of
traditional fuels. Ukraine has all the prerequisites for the development of geothermal
heat energy: there are significant reserves of geothermal energy of medium potential,
spread almost throughout the territory; has scientific potential; scientific and technical
experience in the design and implementation of geothermal thermal power plants.
The country has extensive legislation regulating the use of alternative energy sources,
declaring state support and allocating funds for research.
Calculations show that geothermal projects of thermal power plants in the conditions of existing tariffs and prices can be economically feasible, they are highly
profitable with short payback periods. In terms of profitability, they correspond to
the economic indicators of heating projects based on small fuel boilers. But environmental benefits and independence from fuel market conditions and pricing make
geothermal heating systems more cost-effective than fuel boilers.
World experience shows that the successful development of geothermal energy
requires government support, detailed information on geothermal deposits, sufficient
funding, and involvement of modern technologies.
References
1. World Geothermal Congress, 2015, Melbourne, Australia, International Geothermal Association (2015). https://www.geothermal-energy.org/…/world-geothermal-co
2. International Energy Agency. Statistics. https://www.iea.org/
3. Lund, J.W., Boyd, T.L.: Direct utilization of geothermal energy 2015 worldwide review. In:
Boyd Geo-Heat Center, Oregon Institute of Technology, Klamath Falls, OR 97601, USA, retired
(hidden) (Last Accessed 07 Aug 2020)
4. Geothermal Heating and Cooling Technologies. https://www.epa.gov/rhc/geothermal-heatingand-cooling-technologies. (Last Accessed 23 Apr 2020)
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html. (Application date 25 Feb 2020)
6. State geological map of Ukraine. geoinf.kiev.ua/wp/kartograma_rep.php?listn=m35-4. (Application date 21 Feb 2020)
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Ukraine. K.: ITTF NAS of Ukraine, p. 211 (2002)
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Accessed 19 Aug 2020
9. Boguslavsky, E.I.: Development of thermal energy of the subsoil. M.: Sputnik + Publishing
House, p. 448 (2018)
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11. Zabarny, G.M., Shurchkov, A.V., Barilo, A.A.: Feasibility study of the feasibility of using heat
pumps in geothermal heat supply systems using thermal waters of the Miocene thermal aquifer
complex of the Transcarpathian region. Kyiv ITTF NAS of Ukraine, p. 230 (1999)
12. Kostyukovsky, B.A., Shulzhenko, S.V., Maksimets, E.A. A system of mathematical models for a
comprehensive assessment of the prospects for the development of the fuel and energy complex.
In: International Scientific and Practical Conference «Energy Efficiency-2008», p. 37–38. Gas
Institute of the National Academy of Sciences of Kyiv, Ukraine, October 6–8 2008
Environmental Aspects of Geothermal
Energy
Anna Pidruchna
and Yulia Shurchkova
Abstract The chapter deals with the problems associated with the global environmental energy conservation, the causes of its occurrence and possible ways out of it.
Issues of international cooperation in the field of development of renewable energy
sources are discussed. An analysis of the life cycles of various types of renewable
energy, features of the life cycle of geothermal stations, possible geological consequences of the impact on water and land resources, environmental problems and risks
associated with the implementation of geothermal projects are given. The prospects
of using geothermal energy in Ukraine are shown.
Keywords Ecology · Renewable energy sources · Life cycle · Emissions ·
Greenhouse gases · Environment
The average temperature on the planet in the period from 2008 to 2018 increased by
1.00 C. If this rate of temperature growth continues, then by 2050 global warming
will reach the level of 1.50 C. The Nuclear Energy Agency of the Organization for
Economic Co-operation and Development (OECD/NEA) has published a study «The
Cost of Decarbonization: The Cost of Systems with a High Share of Nuclear and
Renewable Generation», which showed that in order to prevent a rise in temperature
by 2050, it is necessary to reduce CO2 emissions in electricity sector of the OECD
countries by 90%. Currently, this figure averages 430 g/1 kWh, and by 2050 should
be reduced to 50 g/1 kWh.
The energy industry, which provides 40% of total atmospheric emissions and
housing and communal services are the largest air polluters, since the main share of
energy, both electrical and thermal, is produced by burning fossil fuels.
Figure 1 shows the growth rate of emissions into the atmosphere by regions of
the world over the past 50 years. As shown China, the USA, and India produce the
largest volumes of CO2 emissions.
A. Pidruchna (B) · Y. Shurchkova
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: pidruchna@gmail.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_24
397
398
A. Pidruchna and Y. Shurchkova
Fig. 1 Emissions of fossil CO2 (CDIAC/GCP/BP/USGS data)
Developed countries currently spend approximately 1–2% of GDP on environmental protection, while the cost of environmental damage annually is 4–6%
[1].
Against the backdrop of the environmental, with the reduction of world reserves
of fossil fuels and rising prices for it, the question of developing the industry of
alternative energy carriers has become acute. In the middle of the last century, broad
international cooperation on environmental issues began, important environmental
agreements were reached, and most countries adopted important environmental laws.
In 2019, the European Commission launched the European Green Deal (EGD)—a
roadmap to ensure the resilience of the EU economy by overcoming the climate
crisis by reducing CO2 emissions, efficient use of resources, moving towards a clean
economy and slowing climate change [2]. The main goal of the European Green
Deal is to reduce emissions by 50–55% by 2030 and reduce greenhouse gas emissions to zero by 2050. It concerns all sectors of the economy, in particular, energy,
metalworking, transport, construction, agriculture, chemical industry, etc. In 2020,
the share of renewable energy sources (RES) in the generation of EU countries for
the first time in history overtook all other energy sources: RES accounted for 38% of
total generation against 37% of the share of traditional electricity. In March 2021, a
historical maximum of electricity production from wind-based generation facilities
was recorded in Europe. Renewable wind power provided 28.9% of daily electricity
demand.
The Government of Ukraine has announced its intention to join the Green Deal as
it is a practical implementation of the European integration vector of the country’s
development.
The main advantage of using renewable energy sources in comparison with other
types of energy carriers is their environmental friendliness and minimal impact on
Environmental Aspects of Geothermal Energy
399
Table 1 Environmental parameters of power plants
Type of
power plant
Emissions
in
atmosphere
m3 /MWh
Fresh water
consumption
m3 /MWh
Waste
water
discharge
m3 /MWh
Volume
of solid
waste
kg/MWh
Withdrawal
lands
ha/MWh
Conservation
spending
%total cost
solar
–
–
0.02
–
2–3
–
Wind
–
–
0.01
–
1–10
1
geothermal
1
–
–
–
0.2
1
biomass
2–10
20
0.2
0.2
0.2–0.3
–
TPP–coal
20–35
40–60
0.5
200–500
1.5
30
TPP–gas
2–15
2–5
0.2
0.2
0.5–0.8
10
HPP
–
–
–
–
100
2
NPP
–
70–90
0.2
0.2
2.0
50
the environment. The one option to compare the environmental impact of different
power generation technologies is to analyze their environmental parameters, but it is
necessary taking into account e.g. whole power system operational modes.
Table 1 shows comparative data on the environmental performance of power plants
operating on both renewable energy sources and conventional fuels [3].
In almost all most often, power plants operating on renewable energy sources have
significant advantages over conventional fuel power plants, since energy generation
in this case occurs without burning hydrocarbons and without emitting greenhouse
gases into the atmosphere.
However, if we consider from the standpoint of ecology not only the period of
energy generation, but also the preparatory stages of project development, then it
will be necessary to take into account the side effects accompanying these stages.
To ensure the operation of stations on renewable energy sources, it is necessary to
carry out a number of activities related to the operation of machine-building, metallurgical and other enterprises that use energy obtained from traditional sources that
generate greenhouse gases and other pollution. If we consider the full period of the
existence of a renewable energy facility project—from the idea to the disposal of
used equipment (“from the cradle to the grave”), including preparation, exploration,
infrastructure creation, equipment manufacturing, provision of raw materials and
materials, construction work, waste and equipment disposal at the end of the life of
the facility, it is no longer possible to speak of “zero CO2 emissions”. Therefore,
the transition to renewable energy does not always give the effect that is determined
only by the period of energy production. To assess the impact on the environment,
it is necessary to take into account the impact of all stages of the object’s existence.
Life cycle analysis takes into account the complete life cycle of the system, from the
receipt of materials during construction to operation and end-of-life waste management, and helps to identify the key stages that affect the effectiveness of the chosen
technology. Research on side effects from the creation and operation of renewable
energy facilities is currently insufficient and they are often contradictory. In the work
400
Table 2 Indicators of CO2
emission in life cycles for
various types of power plants
A. Pidruchna and Y. Shurchkova
Technology
Life cycle emissions,
gCO2 eq/kWh
Wind
12
Tidal
15
Hydraulic
20
Ocean Wave
22
Geothermal
35
Solar (photovoltaic) batteries
40
Solar Concentrators
10
Bioenergy
230
Coal
820
Gas
490
Atomic
12
of a researcher at the Western Norway Research Institute, WNRI Otto Andersen
“Unintended consequences of renewable energy. Problems to be Solved” [4] provides
the results of generalization of information on studies of the negative environmental
impacts of renewable energy on various types of energy and regions of the world,
considers the unintended impact of renewable energy sources on human health and
the environment, and also provides an analysis of the full “life cycle” renewable
energy facilities and assessment of the so-called «reverse effects» (rebound effects).
According to Andersen, different types of renewable energy differ significantly in
the intensity of green color, if they are evaluated from the standpoint of the entire life
cycle. The indicator of the intensity of greenhouse gas emissions in the production of
energy is the amount of gram-equivalent of CO2 per unit of energy produced, taking
into account the time interval and the installed capacity utilization factor.
The Intergovernmental Panel on Climate Change—IPCC—released a report in
2014 on climate change mitigation [5]. The energy systems chapter provides lifecycle emissions data for various types of power plants, both renewable and fossil
fuels (Table 2).
As can be seen, the total CO2 emissions of the life cycles of power plants operating
on renewable energy sources are an order of magnitude lower compared to those
operating on fossil raw materials. At the same time, it should be emphasized that the
lowest emission rate for traditional plants is in the nuclear power industry−12—i.e.
at the level of the lowest indicator of energy from renewable sources.
It also provides data on the distribution of emissions over the life cycle, broken
down by source (Fig. 2).
It is obvious that the distribution of greenhouse gas emissions by stages of the life
cycle of production for different types of energy is fundamentally different. In the
case of wind, solar, geothermal and hydropower, the main environmental burden falls
on the stage of production of materials, equipment and construction of stations. The
nuclear power industry has a similar structure. Fossil fuel-based power generation
Environmental Aspects of Geothermal Energy
401
Fig. 2 Own estimations according to cycle greenhouse gas emissions from electricity supplied
using fossil fuels, renewable sources and nuclear power [5]
accounts for the bulk of emissions during the operation of the plant, which requires
fuel combustion. The same is true for bioenergy. The reasons why greenhouse gas
emissions can reach high values for the life cycles of hydroelectric, solar, bioenergy
and geothermal plants are different, as much depends on the technologies used and
the specific production conditions.
Thus, the development of energy from renewable energy sources requires a simultaneous increase in fossil fuels for the operation of enterprises that produce materials
and equipment for the creation and operation of these stations, i.e. Increasing the
production of energy through renewable energy, respectively, leads to an increase in
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A. Pidruchna and Y. Shurchkova
the consumption of traditional resources. And only when full production cycles are
created that ensure the production of renewable energy without the participation of
traditional fuels, it will be possible to talk about zero emissions and the bright green
color of energy from renewable energy sources.
Ecological problems of geothermal energy. The impact of geothermal energy on
the environment depends on the form of its use or transformation: either directly in
the form of heat production, or for the production of electricity in geothermal power
plants.
Power generation. Geothermal power plants differ in the type of technologies that
are used to convert heat into electricity, such as direct steam, flash evaporation, binary
technologies. Various cooling technologies are also used—water or air. The environmental impact will vary depending on the conversion and refrigeration technology
used.
The most significant possible adverse effects of geothermal energy on the environment include the following: discharge of waste water and condensate contaminated with chemical impurities; change in the level of groundwater, soil failures,
waterlogging; gas emissions (methane, hydrogen, nitrogen, ammonia, hydrogen
sulfide); pollution of groundwater and aquifers, soil salinization; brine emissions
from pipeline ruptures; heat emissions into the atmosphere or surface water, which
create a local increase in air humidity; change in temperature fields of underground
horizons; land alienation.
Impact on water systems. One of the serious problems in the use of underground
thermal waters is their high mineralization, increased gas content, tendency to salt
deposition when temperature and pressure conditions change, and high corrosive
aggressiveness to structural materials. The discharge of such brines into natural water
systems can lead to irreversible environmental consequences. In this regard, the waste
thermal waters are in most cases pumped back into the underground aquifer. In such
systems, wells for pumping water are equipped with steel casing pipes cemented
with the surrounding rock [6], which reliably protects groundwater from pollution
by geothermal objects [7]. But these measures significantly increase energy costs,
capital investments for the construction of an injection well and additional costs for
its operation.
Re-injection of waste water is also necessary to maintain reservoir pressure in the
aquifer, which otherwise can lead to a decrease in plant productivity and possible
ground subsidence in the area of the geothermal field.
Environmental damage could be high consumption of fresh water, which is used
by geothermal power plants for cooling and re-injection. In most cases, when using
a closed circulation system, not the entire volume of water pumped out of the underground horizon can be returned back due to the fact that part of the water is lost in
the form of steam. To maintain reservoir pressure, it is necessary to use water from
outside. The required amount of water depends on the capacity of the station and
the technology used. Since there are no strict requirements for the composition of
the injected water, clean water is not always used for this purpose. For example, in
geothermal power plants in California, in the USA, on Geyser Square, non-potable
treated wastewater is pumped.
Environmental Aspects of Geothermal Energy
403
Fresh ground water in geothermal power plants is also used for cooling and
condensation. In the USA, all geothermal power plants use wet recirculation technology with cooling towers [8]. The cooling water consumption of geothermal power
plants per kWh of electricity produced is 4–5 times that of thermal power plants due
to lower efficiency.
Atmosphere. The volume of emissions of harmful gases into the atmosphere
in geothermal power plants is much less than in thermal power plants. In terms
of chemical composition, they differ from emissions from fossil fuel stations. The
steam produced at geothermal stations is 80% water. Gas impurities consist mainly of
carbon dioxide, a small part of methane, hydrogen, nitrogen, ammonia and hydrogen
sulfide. The most dangerous and harmful is hydrogen sulfide (0.0225%). Once in the
atmosphere, hydrogen sulfide turns into sulfur dioxide (SO2 ) and when combined
with water, causes acid rain, which causes great damage to nature, and causes heart
and lung disease in humans and animals. But it should be noted that SO2 emissions
from geothermal power plants are about 30 times lower per 1 MWh than from coalfired power plants, which are the largest sources of SO2 . CO2 emissions per 1 MWh
of generated energy at a geothermal plant are 0.45 kg, while at a thermal power plant
operating on natural gas—464 kg, on fuel oil—720 kg, on coal—819 kg (thirteen)
[9].
The amount of air emissions from geothermal plants depends on whether an open
or closed fluid circuit is used. In systems with a closed loop, gases from the liquid
practically do not enter the atmosphere, because. After use, they are pumped back
into the aquifer and therefore emissions to the atmosphere are minimal.
In open loop systems, air emissions are reduced by filter and scrubber technologies. But this produces a sludge consisting of trapped substances, which include
sulfur, vanadium, silica compounds, chlorides, arsenic, mercury, nickel and other
heavy metals. This toxic sludge must be disposed of in hazardous waste landfills
[10].
Land use. The size of the land area required to accommodate a geothermal power
plant depends on the properties of the underground collector, the capacity of the
station, the energy conversion technology used, the type of cooling system, the
layout of pipelines, and the area of auxiliary buildings. For example, one of the
world’s largest geothermal stations Geysers, USA, has a capacity of 1517 MW, the
station area is about 78 square kilometers. Large geothermal power plants are mainly
located in fault zones, in zones of modern volcanism, in places with high geothermal
gradient where seismic instability and earthquakes are observed. Earthquakes can
occur when drilling deep wells, when hydraulically stimulating rocks to create additional fractures and increase the heat exchange surface for the coolant, as well as in
the development of petrothermal systems, when high-pressure water is pumped into
the underground formation of hot rocks to create fractures in the formation, similar
to technology hydraulic fracturing of natural gas reservoir. These phenomena have
been observed in different parts of the world. Seismic activity in this case is usually
minor, but can lead to damage to buildings, injury and even death. For example,
in 2006 a geothermal exploration project in Basel, Switzerland was charged with
causing a series of earthquakes measuring up to 3.4 on the Richter scale. In 2011,
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A. Pidruchna and Y. Shurchkova
scientists established a definite relationship between geothermal exploration and
seismic activity. On November 15, 2017, a powerful earthquake of magnitude 5.5 on
the Richter scale struck Pohang, South Korea, injuring 135 people and leaving 1700
homeless [11].
Certain problems arise when drilling deep wells during the construction of
GeoPES and GeoTPP, when hydrogen, the reserves of which are quite large at depths
of 2–3 km, is released, as a result of the combination of hydrogen with atmospheric
oxygen, vacuum-type explosions can occur. Dozens of such cases are known in
Russia and, according to unofficial data, happened in Ukraine.
A serious environmental problem in the use of geothermal energy is also the
potential instability of the surface of the geothermal field. This is because when water
and steam are extracted from underground collectors, the ground above them may
slowly sink over time. This risk is significantly reduced when using closed circulation
systems, when the spent coolant is pumped into the aquifer and the formation pressure
is maintained constant.
Duration of operation of geothermal power plants. In world practice, it is believed
that geothermal resources can be used for 20–30 years, although many of them work
longer. After these periods, the volume of energy production decreases and their
further operation becomes unprofitable. Geothermal resources can be exhausted even
before certain deadlines if the rate of heat extraction exceeds the rate of its natural
replenishment. The service life largely depends on the power of the heat source and
the technologies for its use. For example, a geothermal power plant in Larderello,
Italy, has been generating energy since the early 1900s, and at Geysers, USA, since
1960.The problem of reducing the decline in productivity was solved by drilling new
wells and additional injection of treated wastewater into the aquifer [12].
Direct use of geothermal energy. The most common form of use of geothermal
energy is its direct use without transformation—it is space heating and cooling,
including district heating; balneology; pools for swimming and bathing; Agriculture;
greenhouse heating; drying, etc.
More than 80% of the total global geothermal energy capacity is used in heating
and hot water supply systems. At the end of 2019, the total installed capacity of
thermal geothermal plants in the world was 107,727 MW. According to WGC2015,
the increase in capacity for the period from 2010 to 2015 was 52.0, or 8.7% per year.
For district heating, geothermal energy is used in 28 countries. The leaders in
district heating in terms of annual energy consumption are: China, Iceland, France
and Germany. Individual heating is developed mainly in Turkey, USA, Italy, Slovakia
and Russia. There are more than 5000 district heating systems in Europe, and the
district heating market share is about 10% of the total heating market. In Iceland,
more than 90% of the heat supply is based on geothermal heat. In Reykjavik, 99% of
the needs are provided by geothermal heat. In France, the installed capacity of thermal
geothermal plants, including geothermal heat pumps, is 2.3 thousand tons.MW, which
reduces CO2 emissions by about 1.8 million tons. Around Paris, 33 geothermal plants
heat 170,000 homes, saving the equivalent of 144.4 million m3 of natural gas. It is
planned that geothermal thermal stations should provide 60% of the heat demand in
Paris and its environs. The use of geothermal energy for heating needs has a number
Environmental Aspects of Geothermal Energy
405
of advantages compared to both fossil and renewable types of energy: year-round
and 24 h availability, no fuel depots, no labor for loading fuel into boilers, no low
chimneys, etc.
Problems, which arise from the use of underground thermal waters in direct use,
are similar to those that occur during the construction and operation of geothermal
power plants, although to a lesser extent.
Thermal waters for direct use tend to have a lower potential than geothermal
power plants and require medium to shallow wells to extract them. In this regard, the
concentration of impurities in the waters is much lower. But this does not exclude
the need for re-injection of waste water into an underground aquifer. This makes it
possible to protect surface natural water systems from pollution, from waterlogging
of the area and soil salinization. In the case of systems with an open circuit, emissions
of gases and heat into the atmosphere or surface water are possible.
The use of low-grade water in combination with heat pumps in closed circulation
systems can significantly reduce the risks of negative environmental impact.
In systems using surface heat in combination with heat pumps, with shallow
wells (up to 300 m) or heat exchangers, greenhouse gas emissions are practically
absent and the environmental impact is minimized. In such systems, small changes
in the temperature of groundwater or surrounding rocks are possible. The temperature around vertical wells may rise or fall slightly depending on the time of year
and operating conditions. But with a balanced heating or cooling load, the ground
temperature will remain stable.
The size of the land areas required for the placement of thermal geothermal
stations is determined by the type of hydraulic scheme used for the circulation of
thermal waters in the ground complex; the number of production and absorbing
wells; distances between production wells and the geothermal plant, between the
thermal water intake circuit and the injection circuit; placement of peak boilers and
geothermal installations, auxiliary facilities. Usually, station nodes fit into existing
heating systems with boilers and do not occupy large areas.
Near-surface geothermal systems also occupy relatively small areas. For example,
according to the description, in Klamath Falls, Oregon, USA, (Klamath Falls
(Oregon)—Wikipedia) a geothermal thermal plant that provides residential heating,
district heating, a snowmelt system in the city center, heat supply to local industrial
enterprises, has about 600 geothermal wells 100 m deep, almost invisible in the city.
The lifetimes of geothermal thermal plants, if properly managed and operated
properly, can be quite long. For example, the Reykjavik district heating system has
been operating since the early 1930s with little change in performance, while the
Oregon Institute of Technology geothermal heating system has been operating since
the 1950s with no change in performance.
Life cycle assessment studies for geothermal energy production are few and often
conflicts depend on many factors, such as the specific characteristics of geothermal
fields, the uncertainty of the terms of operation, the imperfection of the technologies
used.
Life cycle analysis of geothermal technologies includes the following main steps:
[13]:
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A. Pidruchna and Y. Shurchkova
. characteristics of wells: fluid temperature, depth and size of wells, number of
exploration, injection and production wells, type and amount of materials for
well construction;
. characteristics of the power plant: plant capacity, type and quantity of materials
for the construction of the ground part of the station, geothermal field capacity
and net energy production;
. operational characteristics: characteristics of the working fluid, requirements for
make-up water;
. comparison with the characteristics of other energy production systems.
Environmental impact assessment is carried out taking into account from 1 to 18
indicators.
According to the IPCC, 2011 IPCC Special Report on Renewable Energy and
Climate Change Mitigation, greenhouse gas emissions from near-surface open-loop
geothermal systems during operation are approximately 0.1 pounds (0.0454 kg) of
carbon dioxide equivalent per 1 kWh in closed loop systems, the gases are not
vented to the atmosphere. But in both cases, there are emissions associated with the
construction of stations and service infrastructure.
In geothermal systems, which include deep wells that require energy to drill and
pump water into underground reservoirs to create a developed infrastructure, life
cycle greenhouse gas emissions are approximately 0.2 pounds (0.091 kg) of carbon
dioxide equivalent per kW hr.
For comparison, data are provided to estimate life cycle greenhouse gas emissions
for electricity generated from natural gas—from 0.6 to 2 pounds (0.2722–0.9072 kg)
of carbon dioxide equivalent per 1 kWh, and for electricity, produced on coal—from
1.4 to 3.6 pounds (0.6350–1.6329 kg) of carbon dioxide equivalent per 1 kWh.
However, in most countries, little attention is paid to the analysis of the life cycle
of geothermal systems, while it allows for a deep analysis of the environmental
impact of each stage of the life cycle and targeted management of geothermal energy
production.
Economic and environmental assessment of the use of geothermal energy.
The economic assessment of the environmental benefits of developing geothermal
resources is based on an assessment of the degree of interchangeability of traditional
and geothermal energy sources.
The economic effect of the use of geothermal energy is defined “as the prevented
damage from the negative impact of the extraction of fossil fuels and the production
of heat or electricity on natural resources and the environment” [14]. Mathematical
dependence for assessing the economic efficiency of environmental benefits includes
economic damage from the extraction and use of fossil fuels for the production of
heat or electricity; economic damage from the generation of heat or electricity based
on geothermal resources; economic effect from the additional environmental and
social benefits of a geothermal energy source; unrealized income from the use of
substituted conventional fuel for other purposes; expenses for the elimination of
possible accidents and their consequences at power generating stations:
Environmental Aspects of Geothermal Energy
E g = U t − U g + E g + Dt + Ua ,
407
(1)
where U t —economic damage from the extraction and use of fossil fuels for the
production of heat or electricity; U g —economic damage from the generation of heat
or electricity based on geothermal resources; E g —economic effect of additional
environmental and social benefits of a geothermal energy source; Dt —unrealized
income from use of substituted traditional fuel for other purposes; U a —expenses for
the elimination of possible accidents and their consequences at energy-producing
stations.
When designing geothermal plants, taking into account the amount of prevented
economic damage can significantly affect the reduction of their payback periods.
The design and technological parameters of geothermal systems are influenced
by a large number of factors, such as the geological and geothermal conditions of the
energy source, on the one hand, and a wide range of thermal loads and temperature
conditions of consumers, on the other. Therefore, the assessment of the effectiveness
and feasibility of creating each particular geothermal facility is possible only when
determining its optimal parameters and indicators. The solution of such a problem is
expedient with the use of economic and mathematical modeling of all stages of the
creation of stations [15].
Geothermal energy in Ukraine. The main consumers of thermal energy in Ukraine
are housing and communal services and the population (about 70%). More than 31
thousand boiler houses are operated in the country, 24% of which are equipped with
boiler units that have been in operation for more than 20 years and have an efficiency
below 82%. The total number of installed boilers is 75.8 thousand units. Among them
are a significant number of small boiler houses with a heat output of up to 70 GJ/h, in
which low-quality coals are burned, which leads to air pollution of cities and towns
with a large amount of ash, dust and soot. Most small boiler houses operate without
flue gas cleaning systems and ash collectors, since this increases the cost of heat
generation by 10–25%. Given these circumstances, the development of geothermal
heat can help the housing and communal services sector to get out of the energy
conservation in terms of replacing traditional fuels and reducing the burden on the
environment.
Ukraine has all the prerequisites for the development of geothermal heat energy:
there are significant reserves of geothermal energy of medium potential, spread
almost throughout the territory; has scientific potential; scientific and technical
experience in the design and implementation of geothermal thermal power plants.
In 1996, the Institute of Technical Thermophysics developed the State Target
Program “Environmentally friendly geothermal energy”, approved by the Cabinet of
Ministers of Ukraine №100 on 17.01.1996 different regions of Ukraine. However,
today in Ukraine there are no existing commercial projects to create electricity generation at the GeoPPP or heat supply stations. And this despite the fact that the annual
technically achievable thermal potential of geothermal energy in the country is equivalent to about 90,000 million kWh/year (according to the State Energy Efficiency of
Ukraine), and its use saves about 10 billion cubic meters. m of gas and significantly
reduce emissions.
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A. Pidruchna and Y. Shurchkova
The government periodically takes some measures to develop geothermal energy:
in 2003 the Law of Ukraine “On Alternative Energy Sources” of 20.02 was
adopted.2003 No. 555-1U; in March 2015, Iceland and Ukraine signed a memorandum of cooperation in the development of geothermal energy. As part of the
Memorandum of Understanding in the fields of energy efficiency and renewable
energy between the State Energy Efficiency and the National Energy Administration
of Iceland (Orkustofnun), the parties agreed to implement joint projects to develop
geothermal resources in Ukraine. Of course, the experience of Iceland, which heats
about 93% of residential premises using this type of energy, is very important for
Ukraine. However, this direction was frozen; The Ministry of Environmental Protection and Natural Resources plans to launch a monitoring, reporting and verification
system for greenhouse gas emissions from 2021. (Law of Ukraine “On ambush
monitoring, and verification of greenhouse gas emissions” dated April 29, 2019 No.
0875).
However, according to the State Agency for Energy Efficiency, as of the end
of 2020, geothermal energy is not among the recently commissioned renewable
energy facilities, while the use of geothermal energy for heating, ventilation and air
conditioning can reduce energy consumption by 25–50% comparable to traditional
systems. According to the International Geothermal Agency, the use of geothermal
heat in 2015 saved 52.5 million tons of oil equivalent per year and significantly
reduces the consumption of traditional fuels and reduces environmental pollution.
Studies by the US Department of Energy in the field of geothermal heating have
shown that a geothermal heating system can reduce carbon dioxide emissions by 46
million tons compared to the use of fuel oil.
1 Conclusions
The main advantage of using renewable energy sources in comparison with other
types of energy carriers is their environmental friendliness and minimal impact on
the environment. Comparison of the impact intensity of different energy production
technologies is based on the analysis of their environmental parameters. For this
purpose, life cycle analysis is used, which takes into account the entire life cycle of
the analyzed system from the receipt of materials during construction to operation
and disposal of waste at the end of its life, and helps to determine the key stages that
affect the effectiveness of the selected technology.
At present, it is impossible to talk about approaching zero greenhouse gas emissions when increasing the capacity of power stations using renewable energy sources,
since this requires a simultaneous increase in fossil fuels for the operation of enterprises that produce materials and equipment for the creation and operation of these
stations, which leads to an increase in pollution environment, including the atmosphere. Zero greenhouse gas emissions will be possible only when complete production cycles are created that ensure the production of renewable energy without
the participation of traditional fuels at all stages of the life cycle. When creating
Environmental Aspects of Geothermal Energy
409
geothermal stations, it is necessary to take into account environmental problems, such
as the discharge of waste water and condensate contaminated with chemical impurities; changes in the level of groundwater and soil failures; swamping and salinization
of soils; gas emissions; pollution of groundwater and aquifers; heat emissions to the
atmosphere or surface water; change in temperature fields of underground horizons;
land alienation.
Since the design and technological parameters of geothermal plants are influenced
by a large number of factors, not only geological and geothermal, but also the operating modes of the consumer, the assessment of the effectiveness and feasibility of
creating a particular geothermal facility is possible only when determining its optimal
parameters and indicators. The solution of such a problem must be optimized using
economic and mathematical modeling of all stages of the creation of stations.
In Ukraine, due to the obvious shortage of hot water, the low efficiency of the old
scorching boilers, the high level of pollution of the middle ground with greenhouse
gases, the development of geothermal heat energy can be an alternative to modernizing the living room of the housing and communal state. In Ukraine p all changes
of mind for the development of geothermal heat energy: p significant reserves of
geothermal energy of the average potential, expanding practically throughout the
territory; p scientific potential; scientific and technical report on the design and
implementation of geothermal thermal stations. World experience shows that the
construction of geothermal heat supply stations is economically feasible, and the
replacement of boiler houses operating on traditional fuels with geothermal stations
is beneficial not only from an environmental and economic point of view, but also
has an important social aspect, because burning environmentally dirty fuel causes
serious harm to the environment and human health.
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Environmental and Circular Economic
Indicators
Valerii Havrysh
and Vasyl Hruban
Abstract The country settlements of Ukraine use primarily natural gas for heating.
Last year there was a drastic rise in natural gas prices. It burdens the budget of
village councils. This state forces the local authorities to look for alternative energy
resources. Currently, in Ukraine, large amounts of agricultural residues are left in the
field. They can be used for heat generation. That is why the purpose of this paper is
to make an assessment of economic viability for substitution of natural gas by straw
pellet production and utilization. Renewable energy is a pillar of the circular economy.
The circular economy is an alternative to the linear economy in solving global issues.
This study determines some indicators which are improved by the straw-based heat
supply system in the countryside. We have made a feasibility analysis (economic,
energy, environmental, and sensitivity) for pellet heat supply in the Shevchenkovo
village council (Mykolaiv province, Ukraine). Investment and operating costs were
estimated at a pellet plant capacity of 590.27 t (annual pellet demand). Feedstock
(straw) is the largest component (34.71%) in the production costs structure. The
current energy prices at EUR37.98/GJ for natural gas and EUR11.98/GJ for straw
pellets are favorable for biomass pellets to be competitive. The calculated straw pellet
production cost is EUR172.87/t. The simple payback period is less than one year.
Sensitivity analyses have shown that the project is most sensitive to investment costs
and natural gas prices.
Keywords Energy · Renewable · Biomass · Crop residue · Heating ·
Countryside · Circular economy · Emissions
1 Introduction
Biomass, including agricultural residues, is a low-carbon energy source. Its use is
important for the sustainable development of modern civilization [1]. Agricultural
residues are one of the elements supporting the European Green Deal targets [2].
V. Havrysh (B) · V. Hruban
Mykolayiv National Agrarian University, Mykolaiv, Ukraine
e-mail: havryshvi@mnau.edu.ua
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_25
411
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V. Havrysh and V. Hruban
The main disadvantages of biomass are low energy density and low yield. These
factors result in a high cost of biomass delivery. That is why pellets have valueadded advantages over raw biomass, including straw. Pelletization reduces moisture
content, increases energy density, and enhances combustion efficiency compared to
raw biomass [3]. The bulk density of biomass pellets is almost ten times higher
compared to straw [4]. It makes easier its handling and transport. For this reason,
densification is the widely used method [5, 6]. Therefore, pellets are a more attractive
form of biomass-based energy.
Crop residues as renewable energy may be a pillar for the circular economy.
The circular economy (CE) promotes the responsible and cyclical use of resources.
In the recent decade, the idea of CE has been supported as an effective policy
to stimulate further economic growth with minimum environmental impact [7].
This concept includes lowering material input and minimizing waste generation to
decouple economic growth from natural resource use [8, 9].
In Ukraine, crop residues are the most abundant and cheapest biomass. They are
the primary raw materials for pellet production. Ukraine is ranked first among pellet
producers (934 thousand tonnes in 2016) [10]. The country settlements of Ukraine
use primarily natural gas for heating. Its high price burdens the budget of village
councils. Financial expenses, energy security, global warming, and exhaustibility of
fossil fuels force using of biomass utilization. Biomass is, as a rule, a local energy
resource. It is nearly carbon-neutral, hence its utilization helps to mitigate greenhouse
gas emissions and strengthen energy security. Currently, in Ukraine, large amounts
of agricultural residues are left in the field to rot. They could be used to produce
pellets to be used as a natural gas substitution.
An increase in European wholesale gas prices (Fig. 1) is encouraging utilities to
use more cheap fuels electricity and heat generation. European coal prices have also
resin too [11–13]. In December 2021, natural gas cost USD1488 per 1000 m3 . This
cost included the transportation expenditure to the Ukrainian border. It was 52%
higher compared to November 2021. In December 2020, import natural gas cost
USD258 per 1000 m3 [14]. The high price is a result of the following reasons:
• there are low reserves in European countries underground gas storage facilities;
200
180
160
Price, EUR/MWh
Fig. 1 Natural gas price
history in the European
Union and Ukraine
140
120
100
80
60
40
20
0
26-09-2020
04-01-2021
14-04-2021
23-07-2021
Period
EU
Ukraine
31-10-2021
08-02-2022
Straw Pellets for Heat Supply in the Countryside: Economic, …
413
• there is Nord Stream 2 certification delay (the project does not comply with
European legislation);
• natural gas supply through Yamal gas pipeline has been suspended;
• there has been a decrease in the gas supply from Norway.
In Ukraine, natural gas price correlates with European trends. This European
natural gas crisis is set to gain momentum. According to the international experience,
the best solution is to support consumers to overcome this problem. Many European
countries have taken steps to normalize the situation. Renewable energy may be a
solution too.
Many scientists have explored the use of biomass as an energy source. Economic
and environmental analyzes are important elements for the development of pellet
utilization. These problems are in the spotlight. Thomson and Liddel [15] studied
the feasibility of biomass pellet-based heat supply systems. They paid attention to the
advantages and barriers. The economic performance was analyzed by some scientists [16–19]. An environmental evaluation was done by Hendricks et al. [20–22].
Sunflower husk utilization for combined heat and power supply was studied too [23].
Environmental impacts of electricity from wheat straw pellets were investigated by
Giuntoli et al. [24]. Li et al. [25] carried out a life cycle assessment of straw pellets in
the Canadian Prairies. Kwasniewski and Kubon [26] studied the economic efficiency
of straw pellet production.
The purpose of the paper is to make an assessment of economic viability for
substitution of natural gas by an alternative energy resource, namely of agricultural
residue for pellet production and utilization on the example of Shevchenkovo village
council (Mykolaiv region, Ukraine). To reach the aim, some objectives must be
studied:
• energy resource analysis;
• availability of feedstock;
• estimation of pellet production cost of agricultural biomass (e.g. wheat, barley,
and oat straw) in the Shevchenko village council;
• determination of the optimal location for the pellets plant;
• carbon dioxide emission saving;
• impact on circular economy indicators;
• economic assessment.
The scope of this research is to conduct a techno-economic assessment for developing a straw pellet plant operating for 20 years using wheat, barley and oats straw.
This includes harvesting and collection, handling, storage, transportation, pellet
production, pellet boiler installation.
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V. Havrysh and V. Hruban
2 Methodology
Data collection was carried out for the development of a techno-economic model
of straw pellet production and its utilization. The determination of cost was based
on data taken from the literature, statistical data, websites, personal communication
with equipment suppliers, experts, and author developed data.
System boundaries
In this study, a life cycle analysis was applied. The production chain comprises all the
stages from straw production to heat generation. The system boundaries are presented
in Fig. 2.
Economic indicators
Technical, technological, agricultural, and economic factors were used in the study.
Technical factors were: the efficiency of the boiler, the lower heating value of fuels
or energy resources. The efficiency of the boiler and specific fuel consumption were
used as technological factors. Crop straw yield variations were agricultural factors.
Investment costs, payback period, the energy cost of fuel, the production cost of
alternative fuel or energy resources were used as economic factors.
The energy cost of fuel was determined as
C E = F pr · (Q · ρ)−1 , EUR/GJ,
where Fpr is the price of fuel, EUR/m3 ; Q is the lower heating value of the fuel,
MJ/kg; ρ is the density of the fuel, t/m3 .
Fig. 2 System boundaries
for straw pellet pathway
Cultivating and harvesting of straw
Transport (straw)
Pellet mill
Transport (pellet)
Consumers
Straw Pellets for Heat Supply in the Countryside: Economic, …
415
The efficiency of the boiler depends on a number of factors, including the type
of fuel used. Therefore, it is advisable to determine the energy cost that will be used
for useful heating
C E E = C E · η−1 = F pr · (η · Q · ρ)−1 , EUR/GJ,
where η is the efficient of the boiler.
The above for an electric boiler is equal to
U ECe = 1000 · E pr · (3.6 · ηe )−1 , EUR/GJ,
where ηe is the efficient of the electric boiler; Epr is the price of electricity, EUR/kWh.
The same for a heat pump is equal to
U ECe = 1000 · E pr · (3.6 · C O P)−1 , EUR/GJ,
where COP is the coefficient of performance for a heat pump.
The techno-economic model was developed for a straw pellet plant operating for
20 years. All life cycle costs of the pellet production and utilization were considered (the straw harvesting, transporting to the pellet plant, producing and utilization
pellets). Capital cost, energy cost, employee cost, and consumable cost have been
factored into the calculations. To develop the model, yields of wheat, barley, and
oat straws were considered. The optimum location of the plant was determined by
applying mathematical programming for average, maximum, and minimum biomass
yields.
Sensitivity analysis
Sensitivity analysis investigates the impact of changes in project variables on the
base indicators. As a rule, only adverse changes are assessed. The primary aim of
sensitivity analysis is to identify the variables which have the greatest impact on
the project performance. This analysis must be carried out systematically. We acted
under the following recommendations [27]:
• the identification of the key variables;
• the calculation of the effect on the base project indicator (simple payback period);
• the analysis of the direction and scale of changes in the project indicator for each
key variable.
The sensitivity variables are as follows: field costs, investment costs, employee
costs, natural gas price, electricity price, and lifetime. For each variable, the base
value was increased or decreased by 50% [28].
Available crop residues
Crop residue potential was estimated for 30 years. We used Ukrainian official statistical reports. For this study, we selected three widespread crops: wheat, barley,
and oats. We took into account crop harvest and a Residue-to-Crop Ratio (RCR)
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V. Havrysh and V. Hruban
Table 1 Residue-to-crop
ratios and calorific value of
selected crops
Crop
Residue-to-crop ratio
Lower heating value of
straw, MJ/kg
Wheat
0.8–1.8
15.0–18.1
Oats
1.0–2.0
15.0–18.1
Barley
0.9–1.8
15.0–18.1
to calculate the crop residue quantity
MR =
n
Σ
(Moi · RC Ri ), t,
i=1
where Moi is the average annual production of ith crop, t; RCPi is the Residue-to-Crop
Ratio of ith crop; i is the crop number; n is the number of crops.
Residue to crop ratios and calorific values for selected crops are shown in Table
1 [29–32].
Carbon dioxide emissions
Lifecycle carbon dioxide emissions include several factors such as fuel combustion
and well-to-tank emissions. Moreover, we took into account the emissions associated
with electricity generation and associated with straw production. The carbon dioxide
emission factor for electricity generation in Ukraine is equal to 323 g/kWh [33].
Natural gas has WTT carbon dioxide emissions of 56.38 gCO2 /MJ
(0.203 kgCO2 /kWh) [34]. Well-to-wheel (WTW) emissions are equal to:
W T W = BN G
)
(
11
+ W T TN G , kg,
· CC N G ·
3
where CC NG is the carbon content in natural gas, CC NG = 0.75 kg/kg; WTT NG is the
well-to-tank carbon dioxide emissions of natural gas, kg CO2 /kg; BNG is the natural
gas combustion.
Power generation results in the following carbon dioxide emissions [23]:
C D E G = W · E Fe, kgCO2 ,
where EFe is the emission factor, kg CO2 /kWh; W is the electricity consumption,
kWh.
Optimal location of the pellet plant
The optimal location of the pellet plant is the destination, when the pellet
transportation work is minimized [35]. The objective function is
||
||
S = ||di j · Gp j || → min,
Straw Pellets for Heat Supply in the Countryside: Economic, …
417
Table 2 Indicators for monitoring the circular economy transition
Classification
Indicator
Footprint
Renewable energy share
Material and waste
Annual total waste generation
Aggregated GHG emissions (CO2 equivalents)
Circular material use rate
Socio-economic impact
Investment costs related to circular economy sectors
Jobs related to circular economy sectors
Number of new circular business created to implement the circular
economy initiative
where d ij is the distance from ith destination to jth destination, km; Gpj is the annual
consumption of pellets by jth destination, t.
And for our case, the optimal location is
Si =
6
Σ
Si, j , i ∗ = arg min Si ,
i=1
where i* is the optimal destination for the pellet plant location.
Circular economy indicators
Circular Economy is a major topic, especially in the European Union. For monitoring
the Circular Economy transition, we selected the following indicators (Table 2) [36–
38].
The following sections demonstrate the application of this methodology of technical and economic assessment and optimization for agricultural pellet production
in the Shevchenko village council (Mykolaiv province, Ukraine).
3 Initial Data
In this study, we used prices which were set on February 2022. Electricity price in
Mykolaiv oblast in 2021 (Fig. 3) [39]. From February 1, 2022, the price of natural
gas for commercial consumers was set at UAH40500 per thousand m3 or EUR1276
per thousand m3 (EUR127.6/MWh) [7].
The Shevchenkovo village council has area of 296.81 km2 . Its farmers cultivate
20.5 thousand ha of arable land. We have determined the annual natural gas consumption of public buildings such as schools, kindergartens, and cultural institutions.
According to our analysis, their annual consumption is 226.58 thousand m3 (Table 3).
The village of Shevchenkovo has the highest gas consumption of 45.867 thousand m3 .
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V. Havrysh and V. Hruban
120
Fig. 3 Electricity price in
Mykolaiv province in 2021
Price, EUR/MWh
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
11
12
Month
Table 3 Annual natural gas consumption, m3
Settlement
Schools
Kindergartens
Cultural institutions
Sum
Poligon
23,500
8128
6796
38,424
Kotlyareve
35,000
8672
0
43,672
Shevchenkovo
29,000
5702
11,165
45,867
Zarya
0
0
0
0
Luch
17,600
5702
0
23,302
Myrne
41,900
0
0
41,900
Zelenyy Hay
28,000
5417
0
33,417
Total
175,000
33,621
17,961
226,582
Public buildings are equipped by gas boilers. Their maximum power ranges from
16.96 to 99.17 kW (Table 4). Capacity of boilers were determined by the maximum
gas consumption in the coldest month of the year. The school in the village of Myrne
is equipped with a boiler with the highest capacity (99.17 kW). The kindergarten of
the village of Zelenyy Hay has a boiler with the lowest capacity of 15.81 kW.
Table 4 Maximum power of
boilers, kW
Settlement
Schools
Kindergartens
Cultural
institutions
Poligon
64
28.23
16.96
Kotlyarovo
87.5
33.88
0
Shevchenkovo
81
0
28.33
Zarya
0
0
0
Luch
46.67
22.59
0
Zelenyy Hay
72.33
15.81
0
Myrne
99.17
0
0
Straw Pellets for Heat Supply in the Countryside: Economic, …
419
Table 5 Distances between locations, km
Settlement
Poligon
Kotlyareve
Shevchen-kovo
Zarya
Luch
Zelenyy
Hay
Myrne
Poligon
0
16.5
19
22
22.8
16.2
29.7
Kotlyarovo
17.3
0
4.6
5.5
10.3
19.4
17.1
Shevchenkovo
19.7
3.3
0
8.6
12.7
21.9
19.6
Zarya
22.7
5.5
8.6
0
15.7
8.2
22.6
Luch
23.6
10.3
12.8
15.8
0
25.7
10
Zelenyy Hay
15.1
19.9
22.3
8.2
26.1
0
33
Myrne
29.7
17.1
19.6
22.6
10
33
0
To determine the optimal location of the pellet plant, it is necessary to minimize
transport work. We used data on the distances between settlements to solve the
optimization problem. The distance between settlements is presented in Table 5.
4 Alternatives
In our case, there is possibility for some alternatives:
•
•
•
•
natural gas boiler;
electric boiler;
heat pump;
solid biofuel boiler (pellet boilers).
Each alternative has its advantages and disadvantages (Table 6). Small-scale pellet
combustion has been identified as one of the significant sources of particulate matter.
Fine particles have an adverse effect on the environment. One way to solve this
problem is to increase the efficiency of pellet boilers. There are a lot of investigations
devoted to ensuring an energy-efficient of these boilers [40–42].
5 Straw Availability
The annual potential volume of straw can be assessed. The actual amount depends
on many factors (biomass species, biomass yield, location, climate, and technology).
The yield of residue is an important parameter for a project. It affects the production
cost of pellets. The lifespan of a typical bioenergy facility is 20–30 years. It requires
a continuous and constant supply of feedstock. This is particularly true for facilities
that depend on annual crop production.
The total average yield of wheat, barley, and oats over the last 30 years (1990–
2020) has been 2941; 2239; and 1586 kg per ha respectively (Fig. 4) [43, 44].
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V. Havrysh and V. Hruban
Table 6 Advantages and disadvantages of fuels (energy resources)
Fuel (energy resource)
Advantages
Disadvantages
Natural gas
Controllable
Existing infrastructure
High price
Instability of price
Exhaustibility
Electric boiler
Controllable
Ecologically clean
More expensive as compared to
natural gas
Additional investment costs
Heat pump
Low electricity consumption
Controllable
Ecologically clean
High investment costs
High operating costs
Solid biofuel (pellets)
Renewable
Ecologically clean
Additional investment costs in new
boilers
Additional investment costs in solid
biofuel production
Investment costs in warehouses
Smoke control need
Expensive transportation
4000
Fig. 4 Evolution of crop
yields
3500
Yield, kg/ha
3000
2500
2000
1500
1000
500
0
1990
1995
2000
2005
2010
2015
2020
Period, year
wheat
barley
oat
The available straw production volumes are typically determined by applying
straw to grain mass ratios. After an analysis of technical charts and data, the ratios
adopted in this study for estimating crop residue for wheat, barley, and oats are
1.1; 0.8 and 1.1, respectively. To determine the net yield of straw, additional factors
have been taken into consideration: retained straw for soil conservations; organic
fertilizer; some straw is left on the field in accordance with the efficiency of the
combine harvesters; mulching; lost through handling, transport, and storage. The
quantity of straw also depends on its moisture content. After the literature analysis,
0.75 t/ha was allocated to soil conservation. In this study, we assumed:
• the moisture content of the straw—14%;
• the harvest loss—30% [45–47];
Straw Pellets for Heat Supply in the Countryside: Economic, …
421
2250
Fig. 5 The evolution of
straw yield for selected crops
2000
1750
Yield, kg/ha
1500
1250
1000
750
500
250
0
1990
1995
2000
2005
2010
2015
2020
Period, year
wheat
barley
oat
• the storage and transportation loss—15% [46, 47].
The average yields of wheat, barley, and oats are 1480; 620 and 599 tons per ha,
respectively in the Mykolaiv province (Fig. 5).
A wide variability was observed in the net yields of straw over the years. To
develop our techno-economic model, we have considered three cases: the average
yield, the maximum yield, minimum yield, fuel and residue properties.
The area needed for straw production can be calculated by the following formula
Fs =
ξ·
Mp
(1 − 0.01 · W p)
, ha,
·
i=1 (Ui · εi ) (1 − 0.01 · W s)
Σn
where Mp is the annual pellet production, t; ξ is the arable area to total area ratio,
ξ = 0,691; εi is the ith crop share; U i is the ith crop yield, t/ha; Wp is the moisture
content of pellets, %; Ws is the moisture content of straw, %.
We assumed the circular arrangement of the fields. In this case the radius is
/
R=
0.01 ·
Fs
, km.
π
The average distance of transportation is determined as
Rt =
2
· R, km.
3
Values for radiuses of fields and distances of transportation for three scenarios are
shown in Table 7.
The pellet demand was calculated taking into account its lower heating value of
14.51 MJ/kg and pellet boiler efficiency—80%. It constitutes 590.27 tons per year
(Table 8). To endow that mass of the pellet, it is necessary to have the corresponding
crop area (Table 9).
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V. Havrysh and V. Hruban
Table 7 Radiuses of fields
and distances of
transportation
Scenario
Radiuses of
fields, km
Distances of
transportation,
km
556.43
2.45
1.63
Minimum
yield
1528.36
4.06
2.71
Maximum
yield
350.59
1.95
1.30
Average
yield
Arable area
needed, ha
Table 8 Annual pellet demand, tons
Kindergartens
Cultural institutions
Sum, t
Poligon
61.22
21.17
17.70
100.10
Kotlyareve
91.18
22.59
0.00
113.77
Shevchenkovo
75.55
14.85
29.09
119.49
0.00
0.00
0.00
0.00
Settlement
Schools
Zarya
Luch
Zelenyy Hay
Myrne
45.85
14.85
0.00
60.70
109.15
0.00
0.00
109.15
72.94
14.11
0.00
87.05
Total
Table 9 Needed crop area,
ha
590.27
Crop species
Scenarios
Average yield
Minimum yield
Maximum
yield
wheat straw
422.72
996.59
292.18
barley straw
1008.66
4818.37
518.46
oat straw
1055.93
2909.71
605.07
The educational and cultural institutions of the village council own 200 ha of
arable land. Thus, to ensure the need for biomass, it is necessary to use agricultural
residues of the nearest farms.
6 Pellet Production Cost
The production of pellets from agricultural residue involves harvesting, handling,
storage, transportation and pellet production. Total production cost can be divided
into four main components:
• field cost (straw);
Straw Pellets for Heat Supply in the Countryside: Economic, …
423
7
Specific freighting cost, cent/(t*km)
Fig. 6 Freight cost
6
5
4
3
y = 7.4564e-0.047x
R² = 0.9557
2
1
0
0
5
10
15
20
Carrying capacity, t
Actual freight cost
Trend
• cost of transportation from field to pellet plant;
• depreciation and maintenance;
• salary of employees.
Field cost was evaluated as follows. The estimated price of biomass can vary
from a producer and crop species [48]. In our case, the field cost of agricultural
residue was assumed from market prices of EUR37.5/t [49]. It includes storage
cost. Transportation cost has two components. The fixed component of the cost is
the cost of loading and unloading cost. The variable component includes wages,
fuel, and maintenance. These variable costs are proportional to the distance traveled.
Specific transportation cost depends on the carrying capacity of a truck (Fig. 6) [50].
When transporting straw, transportation costs increase by about 50%. The typical
loading and unloading cost for truck transportation is around USD5.45/t [51, 52].
The transportation distance is proportional to the square root of the crop area needed
for the pellets plant Fig. 8.
The minimum yield scenario is based on yields obtained in the drought years.
The straw-pellet plant has a capacity of 590.3 t/year. Pellet production cost is
EUR172.87/t. In the European countries, bulk pellet prices are in the range of
EUR150/t to EUR321/t. And the average prices ranged from EUR250/t to EUR270/t
[53]. Therefore, the results of our calculations correspond to the European trends.
The main components of pellet production cost are agricultural residue costs and
salary costs (Fig. 7).
7 Economical Efficiency of the Project
Existing natural gas boilers should be replaced by pellet boilers or dual fuel boilers.
It needs EUR57,020.31 (Table 10). The above investment costs may be covered
during 0.84 years (Table 11). In the previous study [54], the payback periods of
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V. Havrysh and V. Hruban
Fig. 7 Production cost
structure, %
depreciation&re
pair
19%
others
17%
energy
7%
salary
22%
straw
35%
depreciation&repair
Table 10 Investment costs
for replacement of boilers
energy
straw
salary
others
Item
Price, EUR
Number
Sum, EUR
Boiler Kalvis (100 kW)
3437.50
4
13,750.00
Boiler Kalvis (70 kW)
2656.25
1
2656.25
Boiler Kalvis (50 kW)
1792.19
1
1792.19
Boiler Kalvis (40 kW)
1300.00
3
3900.00
Boiler Kalvis (25 kW)
1078.13
3
3234.38
Fuel supply system
390.63
12
4687.50
1562.50
12
18,750.00
Installation
156.25
12
1875.00
pellet warehouse
312.50
12
3750.00
transportation
156.25
12
1875.00
Other expenses
62.50
12
750.00
Project (design)
Total
57,020.31
similar projects were in the range of 0.57–5.2 years. In our study, the benefit of
the straw project is the revenue from pellet production and utilization for a natural
gas substitution in heat supply systems. Pellets need to compete with conventional
sources of energy used for heating. The rise in natural gas prices makes these projects
highly profitable (Fig. 8).
8 Sensitivity Analysis
The sensitivity analysis was carried out for the average yield case by changing the
values for different costs and technical factors from − 50% to + 50% in steps of
10% for each case. Cost factors (field, investment, employee, energy) and lifetime
were included in the analysis. Figure 9 shows the results of the sensitivity analysis.
Straw Pellets for Heat Supply in the Countryside: Economic, …
Table 11 Simple payback
period
425
Unit
Item
Value
Annual natural gas consumption
Thousand
Annual pellet consumption
t
m3
226.58
590.27
m3
Natural gas price
EUR/1000
Pellet production cost
EUR/t
172.87
Total investment costs
Thousand EUR
156.42
Annual cost of natural gas
Thousand EUR
289.12
Annual cost of pellets
Thousand EUR
102.04
Return
EUR
187.08
Simple payback period
Years
0.84
Fig. 8 Energy cost
1276.00
Energy cost, EUR/GJ
40
30
20
10
0
Natural gas
Electricity
(electric
boiler)
Electricity
(heat pump)
Pellet
It can be seen that the cost of pellet production is more sensitive to a decrease in
natural gas price and an increase in investment costs. Lifetime and employee costs
have a negligible impact on the project’s payback period.
250
Relative simple payback period, %
Fig. 9 Sensitivity analysis:
relative simple payback
period versus variables
225
200
175
150
125
100
75
50
-50
-40
-30
-20
-10
0
10
20
30
Change, %
Field cost
Employee costs
electricity price
Investment costs
Natural gas price
Lifetime
40
50
426
V. Havrysh and V. Hruban
9 Carbon Dioxide Emissions
Specific carbon dioxide emissions are a function of energy inputs (natural gas, electricity), the fuel carbon contents, the emission factors, well-to-tank emissions, and
carbon dioxide emission associated with straw formation. For pellet production,
specific carbon dioxide emission is equal to
SC D E p
=
V ngp ·
( 11
3
)
· CCng + W T T ng + EC · E Fc + Ms · C DS F
, kgCO2 /MJ,
Ms · L H V s
where Vngp is the natural gas consumption for pellet production, m3 ; CC is the natural
gas carbon content, CC = 0.75; WTTng is the well-to-tank carbon dioxide emissions
for natural gas, WTTng = 2.03 kgCO2 /m3 ; EC is the electricity consumption, kWh;
EFe is the emission factor for electricity, EFe = 0.365 kgCO2 /kWh; Ms is the annual
straw consumption, kg; CDSF is the carbon dioxide emissions associated with straw
production, kgCO2 /t; LHVs is the lower heating value of straw, MJ/kg.
The same indicator for natural gas-based heat supply system is equal to
SC D Eng =
11
3
· CCng + W T T ng
, kgCO2 /MJ,
L H V ng
The same indicator for electric heat supply system is calculated by the equation
SC D Ee =
E Fe
, kgCO2 /MJ.
3.6
The use of straw pellets emits the least carbon dioxide. Straw pellets have the
lowest specific carbon dioxide emission for heat supply systems (Fig. 10). The carbon
dioxide emissions in our study are comparable with available studies [24, 55, 56].
Therefore, the use of straw pellets to substitute natural gas can achieve high carbon
dioxide emission savings.
10 Circular Economic Indicators
Our civilization is extracting fossil resources to meet its requirements in energy and
materials. Since 2017, the total annual extraction exceeded 100 billion tons [57]. A
linear (traditional) economy is based on the use of natural resources. It generates a
lot of waste and pollution [58]. The concept of the circular economy eliminates the
disadvantages of the linear economy. The circular economy includes crop residue
recycling too. Majeed and Luni underlined that renewable energy is an important
pillar of the circular economy because it does not generate waste, reduces the use
Straw Pellets for Heat Supply in the Countryside: Economic, …
120
Specific carbon dioxide emissions, kgCO2/MJ
Fig. 10 Specific carbon
dioxide emissions
427
100
80
60
40
20
0
Natural gas
Electricity
Heat pump
Pellet
Type of fuel
Table 12 Circular economy indicators
Classification
Footprint
Material and waste
Indicator
Value
Renewable energy per capita
MJ per capita
726.86
Carbon dioxide emissions saving
kg per capita
82.51
Annual total waste generation
t
21,763
Circular material use rate (straw)
Socio-economic impact
Unit
Investment costs related to circular
economy sectors
t
590.27
%
2.7
Thousand EUR
156.42
Jobs related to circular economy
sectors
3
Number of new circular business
created to implement the circular
economy initiative (pellet
production)
1
of exhaustible resources and carbon dioxide emissions [59]. Table 12 presents the
impact of a pellet-based heat supply system on the circular economy indicators.
11 Conclusions
A techno-economic model was developed to estimate the pellet production cost and
determine the optimum location of the pellet plant. Agricultural residues (wheat,
barley and oat straw) were considered for average, maximum and minimum yield
cases. The total cost was calculated from the harvest of straw to pellet production.
428
V. Havrysh and V. Hruban
The techno-economic model was applied to Shevchenko village council, Mykolaiv
region, Ukraine.
The use of crop residues for heat supply systems in the countryside has become
more attractive due to drastically emerge of natural gas price. Currently, pellet energy
cost is equal to around 41% of natural gas energy costs. Only heat pump systems can
provide cheaper heat. However, they need higher investment costs.
An investment project includes costs suck as a pellet mill and pellet boilers for
consumers. The last item exceeds 50% of the total costs. Under current conditions,
the payback period is less than one year. A decrease in natural gas price has the
strongest impact on the project profitability.
The sensitivity analysis has shown that the pellet production cost is more sensitive
to the natural gas price decrease, and an increase in field costs.
The use of straw pellet drastically reduces WTW carbon dioxide emissions. They
are 20-fold less compared to natural gas-based heat supply systems. Electrical and
heat pump heat supply systems are characterized by a large emission of carbon
dioxide as well.
Renewable energy for heat supply improve circular economy indicators. In this
study, renewable energy and the circular economy were investigated. Crop residues
as renewable energy may be the primary pillar of the circular economy. This study
measured some indicators of the CE: carbon dioxide emissions savings, renewable
energy per capita, circular material use rate (straw), investment costs in renewable
energy, job creation, and a number of new circular businesses. Promotion of the
circular economy, the use of waste biomass, and renewable energy are suggested to
local and state authorities. Unlike fossil fuels, renewable energy does not harm the
environment.
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Comparative Analysis
of Energy-Economic Indicators
of Renewable Technologies in Market
Conditions and Fixed Pricing
on the Example of the Power System
of Ukraine
Mykhailo Kulyk , Tetiana Nechaieva , Oleksandr Zgurovets ,
Sergii Shulzhenko , and Natalia Maistrenko
Abstract World energy is currently experiencing a period of rapid development of
wind (WPP) and solar (SPP) power plants in the structure of generating capacity
of power systems. Back in 2016, the European Union (EU) recommended that EU
member states in their energy policy on the development and use of WPP and SPP
move to purely market relations. However, not all EU member states have taken
advantage of these recommendations and continue to work in this area on the principles of fixed pricing with preferences for WPP and SPP owners. In Ukraine, such
preferences are among the highest in Europe. This paper analyzes in detail and determines the factors and amounts of financial losses incurred by the IPS of Ukraine
represented by NEC “Ukrenergo” and consumers of electricity generated by WPPs
and SPPs. The consequences of such activities are projected both for NPC “Ukrenergo” and for the country’s economy and society. It is shown that such energy policy
leads (paradoxically) to a significant deterioration of the environmental situation
in the country. Recommendations have been developed for ways out of the critical
state of the country’s energy in connection with the hypertrophied development of
WPPs and SPPs in the structure of its power system. The obtained results and experience of the authors can be useful for specialists in countries with natural conditions
comparable to Ukraine, and who carry out measures to decarbonize their own energy.
Keywords Wind power plant · Solar power plant · Reserve power plant ·
Electricity production · Greenhouse gases · Electricity cost · Revenues · Profits
1 Introduction
World energy is currently experiencing a period of rapid development and use of
renewable energy sources (primarily wind (WPP) and solar (SPP) power plants) in
M. Kulyk · T. Nechaieva (B) · O. Zgurovets · S. Shulzhenko · N. Maistrenko
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: nechaieva.tan@gmail.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_26
433
434
M. Kulyk et al.
the generating capacity structure of power systems. This process has been developing
for a long time without taking into account two extremely important factors, namely:
(1) WPPs and SPPs are sources with zero guaranteed capacity; (2) due to their
technological nature, WPPs and SPPs cannot ensure the normalized stability of the
frequency and power of electricity which is supplied to the power system. At the same
time, the relative share of WPP and SPP capacities in the initial period of their use
in power systems were, firstly, insignificant, and, secondly, the required volumes of
regulating capacities were forcibly attracted from primary and secondary regulation
reserves provided in each power system in accordance with regulatory requirements
for stabilization of normal and emergency modes of its (power system) operation.
That is, to ensure the stable operation of wind and solar power plants as part of
integrated power systems involved high-speed reserve capacity, designed for other
purposes. This approach did not create problems in the power systems as long as
the capacity of WPPs and SPPs (renewable energy sources—RES) was low. As
their capacity increased significantly due to green tariff laws in many countries,
severe systemic accidents began, all the way to blackouts (South Australia) and the
disconnection of large regions with a total capacity of several thousand megawatts
(Germany and other countries). At the same time, the rapid growth of the use of
wind and solar power plants in integrated power systems (IPS) has been and is being
carried out almost without proper scientific support, by trial and error. The IPS of
Ukraine is no exception. As of October 2019, about 4000 MW of SPPs capacity
and about 750 MW of WPPs capacity were commissioned. Two years later, the
capacity of the SPPs was already about 6500 and the WPPs 1500 MW, that is, the
total capacity of RES in the IPS of Ukraine has almost doubled. Additional highspeed capacities designed to stabilize the modes of operation of the IPS of Ukraine
when using significant capacity of WPPs and SPPs, since the signing of the laws “On
Alternative Energy Sources”, “On the Electricity Market” (hereinafter—the laws on
“green tariff”) was not entered.
Research by the Institute of General Energy (IGE) of the National Academy of
Sciences of Ukraine has established that to ensure normalized frequency stability in a
integrated power system with powerful RES requires high-speed electrical regulators
such as batteries (AB), the total capacity of which must be not less than 30% of the
operating capacity of RES. In addition, to ensure the continuous operation of the
power system during weather shutdowns of RES, the reserve capacity of traditional
power plants is required, which capacity practically coincides with the capacity of
RES. Laws on “green” tariff exempt RES owners from construction in the power
system adequate to them in terms of regulatory and reserve capacities. This provision
of the law on “green” tariff grossly violates the principles of a market economy, which
provides equal rights for market participants. All power plants with conventional
technologies, according to the current regulations, must ensure the stability of the
frequency and power of the electricity which is supplied to the power system. In
addition to this unjustified and very significant benefit, the owners of WPPS and
SPPs according to the mentioned laws till 2018 inclusive had even greater preferences
for electricity sales tariffs, which are 2–3 times higher than the electricity tariffs of
traditional power plants.
Comparative Analysis of Energy-Economic Indicators of Renewable …
435
In the current laws on the “green” tariff, the relations between the owners of
WPPS and SPPs and NPC “Ukrenergo” are governed by the principle of “take or
pay”. This is the most burdensome preference for the consumer, which these laws
provide to the owner of WPP/SPP. According to this principle, with the functional
ability of WPPs or SPPs, the dispatch control (DC) of the IPS of Ukraine is obliged
to put them into operation regardless of the conditions in the power system. In case
the DC is forced to restrictions these RES due to any factors (in particular, due to
lack of current demand), the electricity market is obliged to pay the owners of WPPs
and SPPs compensation for lost profits in the amount of electricity sold at a reduced
rate. It is clear that the introduction of the declared SPP and WPP capacity in the
structure of generating capacities of IPS of Ukraine will lead to its (power system)
technological incapacity (blackout due to unacceptable frequency deviations), or to
economic inability of the energy market to settle with WPP and SPP owners, since
under such conditions all its total profit may be less than the total claims of the
owners of WPP and SPP to compensate for their commercial benefits arising from
the provisions of these laws.
2 Formation of Normative and Legal Legislation
on the RES Operation as Part of the Integrated Power
System of Ukraine
The first “green” tariffs in Ukraine were introduced in 2008 [1]. The value of the
“green” tariff was set annually at twice the weighted average electricity tariff for
energy generating companies operating in the wholesale electricity market of Ukraine
on price bids for the year preceding the year of tariff setting. Such incentives for RES
producers were to last for 10 years from the date of its provision.
In 2009, the conditions for providing “green” tariffs were significantly changed
[2] with a breakdown by type of RES with a fixation in euros and lasting until
January 1, 2030. The value of the “green” tariff was set at the level of the retail
tariff for consumers of the second voltage class in January 2009, fixed in euros,
multiplied by the “green” tariff coefficient determined for each type of alternative
energy. As of January 2009, the retail price of electricity for second-class consumers
was 58.46 kopecks/kWh [3], which was 5.4 eurocents/kWh in euros at the official
exchange rate of the National Bank of Ukraine. For producers of electricity from
solar radiation and hydropower in calculating the size of the “green” tariff used an
additional multiplier—the increasing coefficient of peak load [3, 4]. The green tariff
rate for terrestrial solar power plants was 46.5 eurocents/kWh, for wind farms from
2 MW to 11.3 eurocents/kWh.
At the end of 2012, the coefficients of green tariffs were revised downwards
from April 1, 2013 [5]. For power plants that were commissioned or significantly
upgraded after 2014, 2020 and 2024, this coefficient was reduced by 10%, 20% and
30% relative to the 2013–2014 coefficients.
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M. Kulyk et al.
Stimulating the introduction of renewable generation in Ukraine at the expense
of “green” tariffs were important at the initial stage of their development. In particular, “green” tariffs were introduced to cover capital investment in RES, to promote
investment in Ukrainian industry. The rates of the “green” tariff at the time of the
adoption of the Law were focused on the cost of equipment at that time and the
actual costs of implementing RES projects. The majority of the cost of RES projects
is equipment. Thus, according to IRENA, 64% of the cost of a wind power plant
are wind turbines. At the same time, most wind generation projects in Ukraine are
implemented through wind turbines manufactured abroad. The situation is similar
with solar panels. Despite the significant development of RES generation in recent
years, most of the investments attracted have been used to support the economies of
other countries by supplying imported equipment. At the same time, the experience
of other countries shows that with “green” tariffs it is possible to develop industry
and mechanical engineering. Thus, Germany and Denmark, through the production
of wind turbines, ensured the development of the machine-building industry. China
has taken a similar approach by localizing the production of RES equipment, in
particular solar panels.
According to NPC “Ukrenergo” [6], as of the end of 2020, 5360 MW of SPPs and
1110 MW of WPPs were installed in the IPS of Ukraine, which was more than 12%
of the IPS total installed capacity. In fact, more than twice as much solar generation
has been built, and wind generation is half as much as planned by the National
Renewable Energy Action Plan [7], which envisages the introduction of 2280 MW
of wind generation and 2300 MW of solar generation by the end of 2020. This rapid
growth in solar energy is due to high tariffs for solar power plants, which has made
them more attractive for investment than wind generation.
Throughout the period of validity of the “green” tariffs, the issue of changing the
amount of the “green” tariffs has been raised repeatedly. Thus, in September 2014,
the National Commission for State Regulation of Energy and Utilities (NCRECP)
stopped reviewing “green” tariffs, and in February and March 2015 it reduced “green”
tariffs for RES producers. However, these reductions did not comply with the provisions of the Law “On Electricity” and were challenged in the courts by RES producers
[8]. By the end of 2015, RES producers received compensation for unjustified revision and reduction of tariffs. This situation has contributed to a change in the order
of revision of “green” tariffs and a partial reduction in tariffs for future projects. In
June 2015, green tariff coefficients were revised downwards [9] to bring green tariff
levels closer to the world average and to eliminate over-incentives for solar power
plants. Also, for the SPPs, the coefficient for the peak period of time was excluded
from the calculation of the value of the “green” tariff.
The high level of the “green” tariff in Ukraine, especially for solar power plants,
created an excessive price burden for consumers of electricity in Ukraine, which
began to grow rapidly with the commissioning of new power plants.
The transition from the “green” tariff to auctions announced in 2018 has intensified the activities of companies designing and commissioning new RES facilities,
achieving the highest growth rates of installed capacity of the RES sector in 2019. In
Comparative Analysis of Energy-Economic Indicators of Renewable …
437
2019, 3.2 times more new RES capacity was built in Ukraine than in all the previous
10 years of the “green” tariff.
With the launch of the new electricity market in 2019, a public service obligations
mechanism (PSO) was introduced, which aims to provide affordable electricity to the
households and pay a “green” tariff. Responsibility for the implementation of PSO
rests with the Guaranteed Buyer, who is obliged to purchase all electricity generated
from RES at a fixed “green” tariff. Part of the special responsibilities is assigned to
the Transmission System Operator (Ukrenergo), which is obliged to send the funds
received from the transmission tariff to the Guaranteed Buyer to pay the “green”
tariff.
The proposed PSO model created a payment crisis in the first months of the new
electricity market, one of the reasons being the low transmission tariff, which led
to Ukrenergo’s inability to meet its obligations. Guaranteed Buyer in 2020 almost
completely stopped paying for electricity produced at the “green” tariff. Therefore, to
resolve the situation on the electricity market, the Cabinet of Ministers, the European
Energy Agency and the Ukrainian Wind Energy Association signed a “Memorandum
of Understanding on June 10, 2020” (hereinafter—the Memorandum), in which
producers voluntarily agreed to reduce the “green” tariff by 15% for existing SPPs
and by 7.5% for existing WPPs. In addition, liability was introduced in the form of
fines for imbalances in the deviation of the actual RES electricity production schedule
from projected. In turn, the state has committed itself to resolving the payment of
existing debts and ensuring the operation of the newly introduced auction model of
RES support. The main provisions of the Memorandum were subsequently enshrined
in law [10] by taking into account in the Law “On Alternative Energy Sources” [11]
the peculiarities of the “green” tariff in the period from August 1, 2020 by introducing
reduction factors. The current rates of “green tariffs” for WPP and SPP are given in
Table 1.
3 Problem Statement
In the period up to 2019, RES tariffs for energy in Ukraine were many times higher
than market prices for electricity using traditional technologies. As proved in [12],
this factor was one of the main factors that led to the loss of the Ukrainian electricity
market and threatened it with bankruptcy.
Since 2019, there have been radical changes in the tariff formation for electricity
produced by both wind and solar power plants. The fixed tariffs established by law
for wind farms are currently close to the tariffs of the Ukrainian electricity market.
The current electricity tariffs of SPPs are even lower than market prices (Table 1).
Annex 5 of the official document [13] defines the forecast estimates of the installed
capacity of WPPs and SPPs in the IPS of Ukraine for the period up to 2030. In combination with the data in Table 1, this provides an opportunity to make a detailed analysis of the energy economic situation projected in the IPS of Ukraine and its energy
1.04.2021
9.41
16.96
WPP from 2000 kW
SPP from 10 MW
a From
01.01.2015 –30.06.2015
Date of commissioning
14.42
9.41
01.07.2015 –31.12.2015
13.60
9.41
2016
12.77
9.41
2017–2019
Table 1 Current rates of “green” tariffs, euro cents per kilowatt-hour in accordance with [11]
10.97
8.82
2020
8.82
4.20
8.82
2022
7.61/4.35a
2021
4.05
8.82
2023–2024
3.90
7.72
2025–2029
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M. Kulyk et al.
Comparative Analysis of Energy-Economic Indicators of Renewable …
439
market at 2030. This task is relevant both today and in the long run, as extremely negative forecasts made in [12] are already confirmed in the current state of Ukraine’s
electricity sector in general, its power system and electricity market in particular.
This is already manifested in the irrational use of existing generating capacity, especially high-economy nuclear generation, unreasonably high electricity prices in the
domestic market, the associated significant imports of electricity with large excess
capacity of its own generation and a number of other negative phenomena. Therefore,
there is an opportunity and urgent need to develop reasonable, objective forecasts of
the energy situation in the electricity sector of a country with a high level of RES
development, its power system and electricity market, as well as to develop appropriate conclusions. This problem is relevant not only for the electricity industry of
Ukraine, it is equally important for the energy complexes of most industrialized countries, which are moving to the principles of low-carbon development. The purpose
and content of this publication are the development of directions and basic measures
for solving the main tasks caused by this problem.
At once it is necessary to note the main difficulty in the decision of problems
associated with this problem. Availability of zero guaranteed power in RES leads to
the need to use additional specific equipment in the structure of the IPS, which ensures
the stability of the frequency and power supplied by RES to the system. In order to
formulate technological requirements for this equipment, it is necessary to have the
tools to analyze its operation within the IPS. It was necessary to develop specific
mathematical models of frequency and power control in the IPS, and their (models)
should have included mathematical blocks that reflect not only the characteristics
(primarily frequency) of RES and traditional technologies, but also the characteristics
of this additional technological equipment and interconnections between all IPS
equipment, including RES, additional technological and traditional. An additional
complication in such models is the synthesis of mathematical blocks that reflect the
behavior of wind and solar radiation as a working fluid.
Numerous specialized literature on RES usually examines the relationship and
behavior between individual RES and additional equipment for this purpose. A fairly
detailed analysis of these publications is given in [14]. Consideration of the problems
of analysis of the functioning of RES in the IPS among the publications known to
the authors was not found.
Currently, a large number of studies on the functioning of RES as part of the
IPS are conducted at the Institute of General Energy of the National Academy of
Sciences of Ukraine. This uses a set of several mathematical models with different
functionality. A model and software package for the study of the joint functioning
of wind farms, solar power plants, hydroelectric power plants (HPPs) and battery
energy storage (AB) in the IPS of Ukraine [14–16], which have passed various tests
and applications on real data.
A modification of the model and software package for forecasting the long-term
development of power systems with wind and solar power plants using statistical
information to increase the flexibility of the power system [17, 18]. Using the developed models and software complex, the role and mechanism of the influence of
440
M. Kulyk et al.
derivatives from control capacities on frequency stability in power systems with
wind power plants have been studied [19].
For estimates the economic efficiency of joint operation of a RES power plant, a
battery energy storage system and a conventional reserve power plant in conditions
of ensuring a stable level of power developed an appropriate life cycle model of such
a system [20].
An important result of the problem under study is the creation and study of an
adaptive frequency and power control system in power systems with wind farms
[21].
For forecasting the long-term development of the structure of generating capacity
of the power system, taking into account the commissioning and decommissioning
dynamics of capacities and changes in their technical and economic indicators during
the forecast period, developed a partial integer mathematical model [22].
Due to the results presented, in particular, in publications [14–22], researchers
have a reasonable opportunity to choose the types and power of regulators that provide
the necessary frequency stability of IPS, in the structure of which operate RES of
one nature or another. If, for example, high-capacity wind farms operate in the IPS,
then frequency stabilization in it can be provided only by AB or high-power HPPs.
Stable operation of IPS, in which mainly SPPs operates, can be ensured even by
plain HPPs. However, neither in the first nor in the second case can thermal power
plants of any physical nature be used to stabilize the frequency in the IPS. Taking
into account, in particular, this information, energy and economic indicators of wind
and solar power plants in the structure of the IPS of Ukraine at the level of 2030 were
developed.
4 Energy and Economic Indicators of SPPs Operation
in the IPS of Ukraine at the Level of 2030
According to the source [11] and Table 1, the electricity tariff of SPPs in Ukraine is
legally established for 2030 in the amount of 3.9 eurocents per 1 kWh. This tariff is
significantly lower than current prices on the Ukrainian electricity market [23]. This
circumstance gives rise to possible estimates and claims that starting from 2023, solar
energy will no longer have such a devastating impact on the state of the country’s
energy complex, which is described in [12] and whose manifestations are observed
in reality today. This situation and the above goal prompted the authors to conduct
this study.
Initial data for the study of energy efficiency indicators of SPP at the level of
2030
Solar power plants
Installed SPP capacity—9947 MW;
operating life of SPP—25 years;
Comparative Analysis of Energy-Economic Indicators of Renewable …
441
specific investments of SPP—$1000/kW;
annual capacity factor of SPP—0.17;
SPP electricity tariff (2029)—3.9 eurocents/kWh.
Reserve power plants (modernized coal-fired thermal power plants)
Installed reserve TPP capacity—9947 MW;
specific investments—$400/kW;
annual capacity factor (estimated)—0.63;
specific fuel consumption—345 g sc/kWh;
operating life—35 years;
CO2 emissions payment—$3/ton.
Energy-economic calculations of SPP indicators by processing large amounts of
information (Appendix 1) according to known dependencies and algorithms. The
exception is the value of the annual capacity factor of 0.63 instead of 0.83 due to the
presence of sufficient intensity of solar radiation on average for 11 h a day.
The results of calculations of energy-economic indicators of SPP in the structure
of IPS of Ukraine at the level of 2030 are provided in two forms: Table 2, which shows
the main indicators that determine the main conclusions and recommendations, and
Annex 1, which contains all necessary basic and intermediate information in the form
of numerical data and algorithms used to determine the required energy-economic
indicators.
Table 2 shows the energy-economic indicators of the SPP + reserve TPP complex
for comparison with similar indicators of the alternative TPP (modernized coal-fired).
For the purpose of objective comparison, the electricity production at the alternative
TPP (ATPP) coincides with its production at the SPP (paragraph 3, Table 2). This
makes it possible to compare the CO2 emissions from WPP + TPP complex (item
7) with emissions from ATPP (item 11). It can be seen that the CO2 emissions of the
SPP + TPP complex are almost 4 times higher than the emissions made by ATPP.
At the same time, the total costs of this complex are 14 times higher than the costs of
ATPP. A similar pattern is maintained in the ratio of the cost of electricity production
(for the consumer) by the SPP + TPP complex (item 9) to the cost of APEC (item
13, Table 2), which is equal to 3.
5 Energy and Economic Indicators of WPPs Operation
in the IPS of Ukraine at the Level of 2030
The method of constructing this section is similar to that used in the previous section.
According to the source [13] the WPP capacity at the level of 2030 is determined at
5033 MW. The reserve power plant identified a modernized coal-fired thermal power
plant with a similar installed capacity.
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M. Kulyk et al.
Table 2 Basic energy-economic indicators of solar power plants in the power system of Ukraine
at the level of 2030
№
Indicator
Unit
Value
1
Installed SPP capacity
MW
9947
2
Electricity generation, total
kWh
55.77 × 109
3
of the SPP
kWh
11.85 × 109
4
at the reserve TPP
kWh
43.916 × 109
5
SPP owner costs (1 year of operation),
total
$
457.72 × 109
6
Payback period of the SPPs owner’s
capital
Year
8.87
7
CO2 emissions from the reserve TPP
Ton
55.55 × 106
8
Consumer costs for electricity generated $
by the SPP + TPP complex, total
9.411 × 109
9
Cost of electricity generated at the SPP
+ TPP complex (for the consumer)
$/kWh
0.169
Alternative TPPs (modernized coal-fired)
10
Installed power
kW
1.691 × 106
11
CO2 emissions alt. TPP
Ton
15 × 106
12
Total costs for alt. TPP (1 year of
operation)
$
650.05 × 106
13
The cost of energy produced on alt. TPP $/kWh
0.0549
Initial data for the study
Wind power plants
Installed capacity—5033 MW;
operating life—25 years;
specific investments—$1400/kW;
annual capacity factor—0.35;
WPP electricity tariff—7.72 eurocents/kWh.
Reserve power plants
Installed capacity—5033 MW;
specific investments—$400/kW;
annual capacity factor—0.65;
specific fuel consumption—345 g sc/kWh;
operating life—35 years;
CO2 emissions payment—$3/ton.
Calculated basic energy efficiency indicators of wind farms in the IPS of Ukraine
are given in Table 3, and their detailed description—in Annex 2.
Comparative Analysis of Energy-Economic Indicators of Renewable …
443
Table 3 Basic energy-economic indicators of wind power plants in the power system of Ukraine
at the level of 2030
№
Indicator
Unit
Value
1
Installed WPP capacity
MW
5033
2
Electricity generation, total
kWh
35.27 × 109
3
of WPP
kWh
12.344 × 109
4
at the reserve TPP
kWh
22.926 × 109
5
WPP owner costs (1 year of operation), $
total
323.52 × 106
6
Payback period of the WPP owner’s
capital
Year
0.535
7
CO2 emissions from the reserve TPP
Ton
29 × 106
8
Consumer costs for electricity
generated by WPP + TPP
$
8.2475 × 109
9
Cost of electricity generated at the
$/kWh
WPP + TPP complex (in relation to the
consumer)
0.234
Alternative TPP (modernized coal-fired)
10
Installed power
kW
1.761 × 106
11
CO2 emissions alt. TPP
ton
15.61 × 106
12
Total costs for alt. TPP
$
677.06 × 106
13
The cost of electricity alt. TPP
$/kWh
54.85 × 10−3
Table 3 and Annex 2 provide an opportunity to compare the main energy-economic
indicators of the WPP + reserve TPP complex with similar indicators of the alternative coal-fired TPP, which produces the same amount of energy as the WPP (12,344 ×
109 kWh) in 1 year. As in the previous case, the alternative TPP provides much better
energy-economic indicators than the WPP + TPP complex. Carbon dioxide emissions (the main indicator that entitles to a number of large preferences of WPP) of
this complex is almost twice the emissions of alternative thermal TPP. At the same
time, the total costs of the WPP + TPP complex exceed the costs of the alternative
TPP by more than 12 times, and the ratio of the cost of electricity production by the
complex (item 9, Table 3) to the cost of alternative TPP (item 13) is 4.27.
6 Conclusions and Recommendations
Of all the available power sources suitable for frequency balancing in IPS with WPP
and SPP capacities, rechargeable batteries are the most efficient. To perform these
functions, powerful HPPs can be used no less effectively, which Ukraine does not
have and cannot have due to its natural conditions. It is advisable to use thermal
(modernized coal-fired) TPPs as reserve capacities when working in complex with
444
M. Kulyk et al.
WPP (SPP) in Ukraine, as IPS of Ukraine has a surplus of sources of this type, and
others either do not have the necessary maneuverability (NPP) or are too expensive to operate (gas reciprocating units) due to the use of large volumes of natural
gas. Currently, the frequency stability in Ukraine’s UES is ensured by importing
energy from Russia of powerful Volga Cascade HPPs on very favorable for Ukraine
conditions of daily zero balance, when Ukraine imports very expensive high-speed
hydropower energy and even has the right to pay for nuclear power electricity.
However, such conditions threaten the country’s energy security, as Russia could
cut off (and for a long time) its energy supply at any time, and then there will be a
collapse of Ukraine’s IPS, which can be eliminated only by restriction of all WPP
and SPP powers.
Even if the import of regulatory powers from Russia is maintained, other arguments about the feasibility of using WPPs and WPPs in the conditions of Ukraine’s
IPS remain irrefutable. The main political argument, which is primarily used to justify
the need to use these technologies, is to reduce greenhouse gas emissions. However,
as irrefutably proved above, in reality the opposite situation is true, namely, greenhouse gas emissions increase by 2–4 times compared to the emissions of alternative
TPP, which produces the same amount of energy as WPP or SPP. It is also very
important that the total consumer costs for the SPP + TPP complex are 14 times
higher than its costs for the alternative TPP. For the WPP + TPP complex, these
costs are 12 times higher.
From this analysis it is irrefutably expedient in the conditions of Ukraine to refuse
further new construction of WPP and SPP, freezing their installed capacity at the level
of 2021.
As shown in Sect. 4, further reductions in SPP electricity tariffs may bring their
owners to the brink of bankruptcy, but the consumer’s costs remain sky-high, as
the consumer reimburses the costs of frequency stabilization and redundancy. This
situation will be observed with further reduction of tariffs for WPPs The requirement
of the Law of Ukraine “On the Electricity Market”, which in practice is formed in the
form of “Take or Pay”, is unfounded. It contradicts the Transmission System Code,
which is a normative document of the National Commission of Ukraine for Energy
Regulation and Public Utilities, which is not covered by the laws of the Verkhovna
Rada of Ukraine in the field of energy. In addition, the application of this principle to
the operation of wind farms and wind farms leads to additional losses to consumers
and worsens the environmental situation in Ukraine. Therefore, this principle needs
to be urgently removed from this law.
The obtained results and experience of the authors can also be useful for specialists
from countries with natural conditions comparable to Ukraine, and who carry out
measures to decarbonize their own energy.
Comparative Analysis of Energy-Economic Indicators of Renewable …
445
Appendix 1
Energy-economic indicators of solar power plants as part of the energy system
of Ukraine at the level of 2030
№
Indicator
Unit
Value
1
Installed SPP capacity
MW
9947
2
Operating capacity of SPP and reserve MW
TPP (p.1 × 0.8)
3
Electricity generation:
3.1
of the SPP (p.2 × 0.17 × 8.76 ×
103 )
kWh
11.85 × 109
3.2
at the reserve TPP (p.2 × 0.63 ×
8.76 × 103 )
kWh
43.916 × 109
4
The cost of electricity:
7957.6
4.1
of SPP (p.3.1 × 0.039E)
$
522.2 × 106
4.2
at the reserve TPP (p.3.1 × 2.717₴)
$
4.143 × 109
5
SPP owner costs (for 1 year of
operation)
5.1
Capital costs, taking into account the
operating life (p.1 × 103 /25 × 1.112)
$
442.44 × 106
5.2
Staff salaries with accruals (820
persons)
$
7.321 × 106
5.3
Other costs (materials, etc.) (2% of
item p.5.1)
$
7.958 × 106
5.4
Total SPP owner’s gross costs (p.5.1
+ p.5.2 + p.5.3)
$
457.72 × 106
6
Gross revenue of the SPP owner
(p.4.1)
$
522.2 × 106
7
Gross profit of the SPP owner (p.6 −
p.5.4)
$
64.48 × 106
8
Net profit of the SPP owner (p.7 ×
0.8)
$
51.58 × 106
9
Payback period of the owner’s capital
(p.5.4/p.8)
year
8.87
10
CO2 emissions from the reserve TPP
(p.3.2 × 0.345 × 10−3 × 44/12)
ton
55.55 × 106
11
Fee for CO2 emissions of the reserve
TPP (p.10 × 3$/t)
$
166.66 × 106
12
Consumer costs for electricity
generated by the SPP + TPP complex
12.1
The cost of electricity generated by
SPPs itself (p.4.1)
$
522.2 × 106
12.2
Purchase of electricity for SPP reserve $
2.893 × 109
(continued)
446
M. Kulyk et al.
(continued)
№
Indicator
12.3
Purchase of HPP electricity to
$
stabilize the frequency (p.3.1 × 0.3 ×
1.5$)
5.332 × 109
12.4
Fee for CO2 emissions of the reserve
TPP (p.11)
166.66 × 106
12.5
Total consumer costs (p.12.1 + p.12.2 $
+ p.12.3 + p.12.4)
9.411 × 109
13
Total produced electricity (SPP +
TPP) (p.3.1 + p.3.2)
55.77 × 109
14
The cost of electricity generated at the $/kWh
SPP + TPP complex (p.12.5/p.13)
15
Alternative TPP (modernized
coal-fired)
15.1
Installed power (p.3.1/0.8(18.76 ×
103 ))
kW
1.691 × 106
15.2
Capital expenditures for 1 year,
including construction
$
21.49 × 106
15.3
Staff salaries with accruals (250
persons)
$
2.232 × 106
15.4
Other costs (materials, etc.) (2% of
item p.15.2)
$
0.4298 × 106
15.5
Fuel consumption (p.3.1 × 0.345 ×
10−3 )
ton
5.11 × 106
15.6
Fuel costs (p.15.5 × 3.274 × 103 ₴)
$
580.9 × 106
15.7
CO2 emissions alt. TPP (p.3.1 ×
0.345 × 10−3 × 44/12)
ton
15 × 106
15.8
Fee for CO2 emissions of alt. TPP
(p.15.7 × 3)
$
45 × 106
15.9
Total costs for alt. TPP (p.15.2 +
p.15.3 + p.15.4 + p.15.6 + p.15.8)
$
650.05 × 106
15.10
Cost of electricity generated at alt.
TPP (p.15.9/p.3.1)
$/kWh
0.0549
Unit
$
kWh
Value
0.169
Appendix 2
Energy-economic indicators of wind power plants as part of the energy system
of Ukraine at the level of 2030
Comparative Analysis of Energy-Economic Indicators of Renewable …
№
Indicator
447
Unit
Value
1
Installed WPP capacity
MW
5033
2
Working power (p.1 × 0.8)
MW
4026.4
3
Electricity generation
kWh
12.344 × 109
kWh
22.926 × 109
3.1
of WPP (p.2 × 106 × 8.76 × 103 × 0.35)
3.2
at the reserve TPP (p.2 ×
4
106
× 8.76 ×
103
× 0.65)
The cost of electricity
4.1
of WPP (p.3.1 × 7.72 × 10−2 E)
$
1.0795 × 109
4.2
at the reserve TPP (p.3.2 × 2.717₴)
$
2.163 × 109
5
WPP owner costs (for 1 year of operation)
5.1
Capital expenditures taking into account construction
$
313.41 × 106
5.2
Staff salaries with accruals (430 persons)
$
3.839 × 106
5.3
Other costs (materials, etc.) (2% from p.5.1)
$
6.268 × 106
5.4
Total owner’s gross costs (p.5.1 + p.5.2 + p.5.3)
$
323.52 × 106
6
Gross revenue of the owner (p.4.1)
$
1.0795 × 109
7
Gross profit of the owner (p.4.1 − p.5.4)
$
755.98 × 106
8
Net profit of the owner (p.7 × 0.8)
$
604.78 × 106
9
Payback period of the owner’s costs (p.5.4/p.8)
year
0.535
ton
29 × 106
$
87 × 106
10−3
10
CO2 emissions from the reserve TPP (p.3.2 × 0.345 ×
× 44/12)
11
Fee for CO2 emissions of the reserve TPP (p.10 × 3$/t)
12
Consumer costs for electricity generated by WPP + TPP
complex
12.1
Purchase of electricity generated by WPP (p.4.1)
$
1.0795 × 109
12.2
Purchase of electricity to reserve WPP
$
1.526 × 109
12.3
Purchase of HPP electricity to stabilize the frequency (p.3.1
× 0.3 × 1.5$)
$
5.555:109
12.4
Fee for CO2 emissions of the reserve TPP (p.11)
$
87 × 106
12.5
Total consumer costs (p.12.1 + p.12.2 + p.12.3 + p.12.4)
$
8.2475 × 109
13
Total electricity generated by the WPP + TPP complex (p.3.1 kWh
+ p.3.2)
35.27 × 109
14
The cost of electricity generated at the WPP + TPP complex
(p.12.5/p.1.3)
$/kWh
0.234
15
Alternative TPP (modernized coal-fired)
15.1
Production of electricity on alt. TPP (p.3.1)
kWh
12.344 × 109
15.2
Installed power (p.3.1/0.8(18.76 ×
kW
1.761 × 106
15.3
Capital investment for 1 year, including construction
$
22.38 × 106
15.4
Staff salaries with accruals (250 persons)
$
2.232 × 106
15.5
Other costs (materials, etc.) (2% from p.15.3)
$
0.4476 × 106
15.6
Coal consumption (p.15.1 × 0.345 × 10−3 )
ton
5.323 × 106
103 ))
(continued)
448
M. Kulyk et al.
(continued)
№
Indicator
Unit
Value
15.7
Fuel costs
$
605.16 × 106
15.8
CO2 emissions alt. TPP (p.15.1 × 0.345 × 10−3 × 44/12)
ton
15.61 × 106
15.9
Fee for CO2 emissions alt. TPP (p.15.8 × 3)
$
46.84 × 106
15.10
Total costs for alt. TPP (p.15.3 + p.15.4 + p.15.5 + p.15.7 + $
p.15.9)
677.06 × 106
15.11
Cost of electricity generated at alt. TPP (p.15.10/p.15.1)
54.85 × 10−3
$/kWh
References
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shchodo vstanovlennia «zelenoho» taryfu. https://zakon.rada.gov.ua/laws/show/601-17/ed2
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Ukrainian)
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shchodo zabezpechennia konkurentnykh umov vyrobnytstva elektroenerhii z alternatyvnykh
dzherel enerhii. https://zakon.rada.gov.ua/laws/show/514-19/ed20150604#Text (in Ukrainian)
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shchodo udoskonalennia umov pidtrymky vyrobnytstva elektrychnoi enerhii z alternatyvnykh
dzherel enerhii. https://zakon.rada.gov.ua/laws/show/810-20/ed20200721#Text (in Ukrainian)
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555-15?lang=en#Text
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storage. In: Babak, V., Isaienko, V., Zaporozhets, A. (eds.) Systems, Decision and Control in
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Prospects and Energy-Economic
Indicators of Heat Energy Production
Through Direct Use of Electricity
from Renewable Sources in Modern Heat
Generators
Volodymyr Derii , Oleksandr Teslenko , Eugene Lenchevsky ,
Viktor Denisov , and Natalia Maistrenko
Abstract The chapter presents the results of research on the use of electric heat
generators in district heating systems as consumers—regulators who can provide
ancillary services to the Integrated Power System of Ukraine to regulate its load. Electric heat generators can use excess electricity from wind and solar power plants in the
daytime and during the night “failure” of the daily schedule of electrical loads. Modes
of their operation for balancing the power system are determined by dispatchers of
the Ukrainian IPS. Calculations have shown the competitiveness of electric heat
generators compared to gas boilers. Implementation of electric heat generators with
aggregated capacity of about 2500 MW to regulate the power system load will make
the structure of the Ukrainian IPS generation more effective by increasing the level
of NPP basic generation, reduce the natural gas consumption by 604.5 million m3
per year and TPP coal used by 296.7 thousand tons per year, reduce greenhouse
gas emissions by 1691.3 thousand tons per year. Also, the Ukrainian IPS become
more resistant to load changes, which will increase the Ukraine energy security and
independence.
Keywords District heating system · Ukrainian integrated power system · Electric
heat generators · Wind power plant · Solar power plant · Electric boiler · Thermal
networks
1 Introduction
Currently, the Integrated Power System (IPS) of Ukraine operates as part of the
energy systems unification of Russia, Ukraine, Belarus and the Baltic states. The
stability of this unification is ensured by Russian hydroelectric power plants, which
provide system services to Ukraine. One of the significant problems of Ukrainian
IPS is the lack of its own maneuverability to ensure the stability of the energy
V. Derii (B) · O. Teslenko · E. Lenchevsky · V. Denisov · N. Maistrenko
General Energy Institute of NAS of Ukraine, Kyiv, Ukraine
e-mail: derii.volodymyr@gmail.com
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. Zaporozhets (ed.), Systems, Decision and Control in Energy IV, Studies in Systems,
Decision and Control 454, https://doi.org/10.1007/978-3-031-22464-5_27
451
452
V. Derii et al.
system. Due to the objective factors, that have developed in the economy of Ukraine
(the structure production change, increasing the population electricity consumption,
etc.), this problem will only worsen [1]. Particularly significant is the lack of shunting
(regulating) power during the night “failure” hours (23 p.m.–7 a.m.) of the daily electrical loads schedule (DSEL). During this time period, forced shutdowns of coal-fired
power units 150, 200 and 300 MW of thermal power plants (TPP) are used to regulate
the production-consumption ratio. According to the national operator NEC Ukrenergo report on conformity assessment of generating capacities [2], from 7 to 10 power
units of coal-fired thermal power plants are disconnected daily, which leads to accelerated depletion of their resource, excess fuel consumption, increased maintenance
and repair costs. In addition, due to a number of political, economic, technological and other factors, Russia may at any time stop providing system services, or
significantly increase their cost and jeopardize Ukraine’s energy security. Therefore, research aimed at full or partial solution of the own maneuverability of the of
Ukrainian IPS problem increasing, is appropriate and relevant today.
2 Basic Material
To estimate the maximum value of the required regulating power during the nighttime
“failure” of the electricity load schedule of Ukrainian IPS, the indicator “depth of
nighttime failure” proposed, determined by formula (1)
Δ P = P23 − Pmin ,
(1)
where
Δ P depth of nighttime “failure” of DSEL (MW);
Pmin minimum electrical load of DSEL (between 4 and 5 a.m.) (MW);
P23 electrical load of the DSEL at 11 p.m. of the previous day (MW).
The statistical study results of the DSEL nighttime “failure” depth for the period
2014–2021 are shown in Fig. 1.
As can be seen from Fig. 1, the DSEL depths of night “failures” (both maximum
and average values) during 2014–2018 tended to decrease. This indicates that the
processes of changing the load at night and the mode of operation of consumers
established. But in the period 2018–2021 depth of night “failures” began to increase
due to the influence of wind (WPP) and solar (SPP) power plants, which generating
power depends on weather conditions and is stochastic. According to the legislation
of Ukraine, the purchase of the entire amount of electricity from SPP and WPP is
guaranteed, which determines their operation at the basic level. In the spring–summer
day periods, when the generation volumes from SPP and WPP are large, the system
operator uses all available offers of TPP and HPP manufacturers for unloading within
the balancing market. Further, in order to maintain the stability of the Ukrainian IPS,
the system operator is forced to switch Hydro Power Pumping Station (HPPS) to
Prospects and Energy-Economic Indicators of Heat Energy Production …
453
Fig. 1 DSEL nighttime “failure” depth for 2014–2021
pump operation, which reduces their capacity to operate during the night “failure”
of DSEL, as shown in Fig. 2 for 22.07.2021.
If these measures are not enough, the SPP and WPP power limits of are applied,
as shown in Table 1.
Table 1 shows that the maximum total power limitation in 2020 reached
2178.86 MW (07.06.2020) due to a significant increase WPP and SPP power. The first
restriction of renewable energy sources (RES) power in 2021 was already on March
11 at 12 p.m., when the RES generation power reached 4.46 GW. Given surplus power
and exhausted unloading reserves to ensure the IPS operational safety, this necessitated 400 MW the SPP and WPP power total unloading. RES electricity generation
restrictions in 2021 also occurred on March 28, 1, 3, 10 and 11 April (last two days
the total unloading power of about 1.5 GW each day) [3].
Fig. 2 Power generation of renewable energy sources and electricity consumption of Hydro Power
Pumping Station in the period 09 a.m–7 p.m. 22.07.2021
454
V. Derii et al.
Table 1 Power limits of WPP and SPP in 2020 [3]
#
Date
Total electric power limitation, MW
Type of generation
1
07.01
929.5
WPP, SPP
2
10.03
510
WPP
3
14.03
282.5
WPP
4
15.03
400
WPP, SPP
5
26.03
407
WPP, SPP
6
28.03
409
WPP, SPP
7
02.04
390.4
WPP, SPP
8
03.04
597.6
WPP, SPP
9
04.04
1363.4
WPP, SPP
10
05.04
1656.7
WPP, SPP
11
09.04
400
SPP
12
11.04
958.46
WPP, SPP
13
12.04
644.27
SPP
14
19.04
380
SPP
15
10.05
910
WPP, SPP
16
07.06
2178.86
WPP, SPP
17
13.08
350
WPP, SPP
18
28.08
300
WPP, SPP
19
09.09
498.1
WPP, SPP
20
04.10
1113.16
WPP, SPP
Due to the lack of market pricing mechanisms, the possibility of adjusting the rates
of the “green” tariff and reducing price of the SPP and WPP electricity, in recent years
there has been a rapid increase in their number and installed power. At the beginning
of 2021, the installed power SES was 6.87 GW and WPP—1.31 GW, and by 2030
it is planned to increase their installed power to 10.5 GW and 5.0 GW, respectively
[4]. Such plans cannot be implemented without providing the IPS of Ukraine with
sufficient own maneuverability, which is currently insufficient. To provide IPS of
Ukraine with balancing power, the system operator NEK Ukrenergo plans to introduce highly maneuverable gas power plants with a total capacity of 2 GW and battery
energy storage systems with a total capacity of 700 MW [3].
Problem definition: significant imbalance of the power system is forecasted due to the
increase in the share of RES generation with non-guaranteed electricity generation
in the absence of the necessary shunting power of the Ukrainian IPS.
The purpose of research: to determine the prospects and energy efficiency of thermal
energy production through the direct use of electricity from renewable sources in
modern heat generators.
Prospects and Energy-Economic Indicators of Heat Energy Production …
455
A much economically feasible solution to this problem may be the transformation
of electricity into thermal energy using controlled electric heat generators (EHG):
electric boilers, compression heat pumps with electric motors, dynamic cavitation
heat generators and more. These EHGs are operated as consumers-regulators on
the instructions of the IPS dispatcher with daily accumulation of thermal energy.
The essence of the method is that the excess electrical energy in the power system
is converted into thermal energy using EHG. In case of electricity shortage in the
power system, the EHG is turned off by order of the IPS dispatcher. The main
conditions for the use of this method are the presence of consumers of thermal energy
from EHG and the possibility of its partial or complete accumulation. Consumption
of all thermal energy from EHG should occur every day during the year. District
heating systems (DHS) are best suited for the implementation of this method. DHSs
have consumers of hot water (year-round consumption) and main networks, which
allow the accumulation of thermal energy by current regulations of Ukraine [5]. To
determine the needs for EPG control capacities, daily information on the power/load
of the Ukrainian IPS for 2021 was collected and analyzed [6]. As a result of the
statistical analysis, the probability (P) dependence of the load deficit coverage (Pl)
of the Ukrainian IPS on the power of the electric load of the EHG was constructed,
as shown in Fig. 3.
As can be seen from Fig. 3, ENG with an electric load of about 3000 MW is able
to eliminate the deficit of shunting power of the Ukrainian IPS with a high level of
probability (P = 0.9). Thus, the need for regulatory capacity of EHG to cover the
night “failure” of the Ukrainian IPS as of 2021 was 3000 MW. Ukraine is one of the
countries with a high level of centralization of heating systems. The share of DHS in
the total heat supply of urban settlements in Ukraine is about 52%. The main equipment of the DHS of Ukraine is physically worn out and technologically obsolete,
which caused a number of significant problems for both consumers and heat supply
companies. Today, the DHS of Ukraine needs to be renewed through mass reconstruction and modernization. When planning the reconstruction and modernization
Fig. 3 The probability (P) dependence of the load deficit coverage (Pl) of the Ukrainian IPS on
the power of the electric load EHG
456
V. Derii et al.
of the DHS, it is necessary to provide for the introduction of EHG for heat production
and ancillary services to power systems, which will solve one of the urgent problems of the Ukrainian IPS—reducing the deficit of shunting power. To assess the
possibility of the DHS to provide ancillary services to the Ukrainian IPS through
the use of EHG, the heat supply systems of large cities with a population of over
100,000 inhabitants with boilers with a power of more than 20 Gcal/h (23.25 MW)
were studied. Such DHSs have well-developed main networks and supply hot water
to consumers all year round. The analysis of these DHSs determined that the average
heat load for domestic hot water supply systems (DHWS) is 3012 MW. In fact, these
are potential shunting capacities of EHG, which are able to replace the power of
existing boilers in the mode of hot water. To determine the prospects for the use of
EHG in DHS, the influencing factors were identified and the forecast changes in the
load of hot water systems was built. The biggest influencing factors on the hot water
system load are the population of Ukraine reduction and the DHS decentralization
processes. The demographic scenario of the Institute of Economics and Forecasting
of the National Academy of Sciences of Ukraine was used for the analysis. It predicts
the average rate of change in birth rates, life expectancy and net migration in Ukraine
[7]. Data on the processes of heat supply systems decentralization have been used
from [8]. Decentralization processes are primarily due to the low quality of DHWS
services and their high prices. Consumers massively refuse these services and use
household electric water heaters [9]. When constructing the DHWS load forecast,
it was assumed that in 2020–2035 large-scale reconstruction and modernization of
DHS will be carried out, restored DHWS systems and their services will be cheaper
than the use of domestic boilers and decentralization of heating systems will stop.
The load of hot water systems by years was calculated by the formula
[
p
q ]
hwl
qthwl = q2000
× (1 − ∂t )(1 − ∂t ) ,
(2)
hwl
where qthwl —heat load of domestic hot water systems per year t; q2000
—heat load of
p
d
domestic hot water systems in 2000; ∂t , ∂t —the rate of change in the population of
Ukraine and the rate of decentralization of DHS per year t, respectively, %.
The results of calculations and assumptions are given in Table 2.
Table 2 shows that under the influence of population decline and decentralization
of DHS, the total heat load of domestic hot water systems for the period 2000–2050
Table 2 Forecast of changes in the load of hot water systems
Indicator/Year
2020
2025
2030
2035
2040
2045
2050
The rate of change in the
population of Ukraine, %
0
1.80
1.83
2.34
2.39
2.21
2.51
Rate of decentralization of 0
DHS, %
6
7
5
2
0
0
Heat load of domestic hot
water systems, MW
2685.5
2451.7
2274.7
2175.8
2127.8
2074.5
3012.0
Prospects and Energy-Economic Indicators of Heat Energy Production …
457
Table 3 Possibilities of accumulation of thermal energy in thermal networks, GJ [10]
Appellation
Season period
Heating
Not heated
Maximum power of thermal networks for
the accumulation of thermal energy
Design temperature graph—237,471
Actual temperature graph—147,007
108,560
The need for accumulation of thermal
energy
96,100
will decrease by 31% and reach 2074.5 MW in 2050. For the period up to 2040,
the total electrical load (consumption) of EHG should be chosen about 2500 MW.
One of the conditions for using ETGs is that all the heat energy they produce must
be consumed during the day. This is due to the cyclic mode of operation of the
ETG (during the night “failure” of the DSEL). But during the night “failure” in
the non-heating season, the consumption of hot water is minimal, and the only way
to ensure the operation of the EHG at this time is the accumulation of thermal
energy produced by them. As an option that does not require large investments
in the construction of heat accumulators, is the use of thermal networks for these
purposes. Thus, according to the current regulations of Ukraine [5], the accumulation
of thermal energy is possible only in the main networks. And according to the Law
of Ukraine “On Heat Supply”, the main networks have boilers with a power of at
least 20 Gcal/h (23.25 MW). This fact is an additional limitation on the power of
boilers where ETG can be used. Therefore, it is necessary to investigate the technical
possibilities of thermal networks for the accumulation of thermal energy from EHG
in the absence of its consumption at night. It is necessary to take into account the
fact that part of the main heating networks has exhausted its technical resource. To
increase the reliability of heating networks, heat supply companies were forced to
move from design temperature schedules of 150/70 °C to schedules with lower limit
temperatures (about 120/70 °C). Such a study has already been conducted for the
period of April 14, 2018 in [10], in which the magnitude of the night “failure” of
DSEL was 3002 MW, and the need for thermal energy accumulation—22,953 Gcal
(96,100 GJ). In [10], in addition to the needs for the accumulation of thermal energy
from EHG, the maximum storage power of thermal networks of large cities of Ukraine
was also assessed (Table 3).
As seen from Table 3, the maximum storage power of thermal networks is sufficient for the accumulation of thermal energy in the design and actual temperature
schedule in both the heating and non-heating periods.
3 Research Methodology
When determining the feasibility of using EHG in the DHS, a comparative analysis
of the use of traditional gas boilers (GB) and electric boilers (EB) was conducted.
458
V. Derii et al.
An indicator such as the Levelized weighted average cost of heat (LCOH) is used as
a criterion. To compare the efficiency of different energy generation technologies in
the world, the method of estimating the average cost of energy over the life cycle is
used—LCOE/LCOHC (Levelized cost of energy/levelized cost of heat (cold)). This
method is universal and convenient in the comparative analysis of different types
of energy production technologies (electricity, heat and cold) and is used by many
reputable organizations, including the International Energy Agency.
According to the European Commission’s guidelines for the development of
renewable energy support systems (SWD (2013)) [11], three steps are envisaged
for determination tariffs:
(1) determination of parameters and methodology for calculating direct costs;
(2) forecasting costs and revenues;
(3) conversion of LCOE to the appropriate level of support.
The methodology for estimating the indicator depends on the degree of complexity
of the assumptions (financial, economic and technical). LCOH is defined as the
constant cost of generating one kWh of heat/cold, which is equal to the discounted
costs spent throughout the life cycle [12–15]. The main calculation formula of this
method is:
Σ N (It +Mt +Ft )
LC O H =
(1+r )t
Ht
t=1 (1+r )t
t=1
ΣN
,
(3)
where LCOH—average cost of heat for the life cycle; t—current year of the system
since the beginning of construction (index of component costs); N—the duration of
the project; I t —annual investment; M t —annual conditionally fixed costs for maintenance and repair, Ft—conditionally variable costs (for fuel, electricity, materials,
taxes due to emissions of pollutants and greenhouse gases); H t —annual heat production, r—discount rate (discount), which reflects the rate of decline in investment
capital over the years.
Initial data and results of calculations. Comparative analysis of LCOH was
performed under the following conditions and initial data. The main technical and
economic indicators of boilers are shown in Table 4.
Table 4 The main technical and economic indicators of boilers
Heat
Thermal Electric
Conversion Specific
Operating costs
Resource,
generator power, power
factor
capital
(conditionally-fixed) years
MW
consumption,
expenditures
MW
[16], e/kW
Electric
boiler
50.0
Gas
boiler
58.15
51.02
–
0.98
30.6
0.25 e/MW/year
25
0.93
26.3
3 e/kW/year
20
Prospects and Energy-Economic Indicators of Heat Energy Production …
459
Table 5 Prices for energy resources and emissions taxes
Year
Indicator
2020 2025
Natural gas, e/1000
m3
250
Electricity (night “failure” and day surplus), 50.1
e/MWh
2030 2035
2040
2045
2050
291.1 292
311.1 320.3 329.8 339.5
63.9
63.9
62.5
63.8
63.9
63.3
Payments for CO2 emissions, e/t
0.3
2.1
8
15
22
27
34
Emission payments CO, e/t
3.1
4.6
6.1
7.9
12.6
19.2
25.7
Emission payments NOx , e/t
82.6
111
128
145
165
209
250
Forecasts of prices for electricity and natural gas, ancillary services, taxes on
greenhouse gas emissions and pollutants are given in Table 5.
According to current trends in the theory and practice of financial activities, the
cost of capital of the enterprise is recommended to be calculated based on the use of
the so-called model of weighted average cost of capital WACC (Weighted Average
Cost of Capital) [17]:
W ACC =
E
D
· i d · (1 − t) +
· ie
D+E
D+E
(4)
where D—share of debt capital, accepted 15%; E—share of equity, accepted 85%;
id —cost of debt capital (interest rate on the loan), accepted 6%; ie —cost of equity,
accepted 10%; t—income tax rate accepted 0%.
The discount rate was determined as the weighted average value of equity and
borrowed capital was 6.6%. The share of borrowed funds is 85% of capital expenditures for the implementation of heat generating equipment (the cost of basic
and auxiliary equipment, design, construction, installation and commissioning).
Contingencies are assumed to be equal to 10% of the total project cost.
Modeling of the GB life cycle was conducted for the mode of operation during
the year. At the same time, the initial cost of natural gas was variable—(250, 450,
500, 550, 600) e/1000 m3 . At the same time, the tendencies of changes in natural
gas prices by years (Table 5) remained unchanged.
Modeling the use of EB was performed in the following modes of operation:
. generation of thermal energy during the night “failure” of DSEL power systems;
. generation of thermal energy during the night “failure” of DSEL with the provision
of ancillary services to power systems.
The following assumptions were also made:
. thermal energy from the EB will be partially consumed, and the rest—accumulated
by thermal networks;
. during peak modes of operation of power systems, when the EB will be completely
disconnected, the accumulated thermal energy from thermal networks will be
supplied to consumers.
460
V. Derii et al.
In addition, the cost of connecting the EB to the electricity grid was taken into
account when estimating investment costs. Fees for non-standard connection to the
electricity grid were calculated for 24 large cities of Ukraine according to the methodology of the national regulator in the field of electricity generation and supply [18].
For further modeling, their average value was used, which is 87.4 e/kW of the
installed electric power of the EB.
The cost of ancillary services as of 2020 is assumed to be e9.48/MWh. Variables
in the simulation were the initial cost of electricity: 29; 50.1 and 60 e/MWh. At
the same time, the trends in changes in electricity prices over the years (Table 5)
remained unchanged.
The amount of electricity consumed by the electric boiler and the heat energy
produced by it was determined by the formulas
E E B = k f s · PL E B · tn f · n y ,
(5)
HE B = E E B · η E B ,
(6)
where E EB —amount of electricity consumed by the electric boiler; k f s —coefficient
of filling the schedule of night failure of power system (0.733); PL E B —electric boiler
load; tn f – duration of night failure (8 h); n y —number of days per year; H EB —the
amount of thermal energy produced by the electric boiler; η E B —conversion factor
of electric boiler.
The reduction in natural gas consumption of DH system, which is due to the
replacement of thermal energy produced by the EB, is determined based on the
formula
VG =
E E B ηE B
,
9.42ηG B
(7)
where V G —reduction of natural gas consumption; E EB —electricity consumption by
electric boiler; H EB —conversion factor of electric boiler (accepted 0.98); 9.42—
calorific value of natural gas, MWh/1000 m3 .
The results of modeling the use of an electric boiler with a thermal power of
50 MW showed that:
.
.
.
.
.
.
.
.
annual heat production will be 107 thousand MWh;
annual electricity consumption—109.2 million kWh;
shunting electric load—51.02 MW;
specific investment costs are 120 e per 1 kW of installed power;
annual savings of natural gas—12,622.6 thousand m3 ;
annual reduction of thermal power plant coal consumption—6.06 thousand tons;
annual reduction of greenhouse gas emissions: CO2 —24,528.6 tons;
annual reduction of nitrogen oxide emissions: NOx —26.86 tons.
The results of LCOH modeling are given in Table 6.
Prospects and Energy-Economic Indicators of Heat Energy Production …
461
Table 6 LCOH of gas and electric boilers
Gas boiler 58.15 MW
Gas cost, e/1000 m3
250
450
500
550
600
LCOH, e/MWh
29.8
53.6
59.5
65.4
71.4
29
50.1
60
Electric boiler 50 MW
Electricity cost, e/MWh
LCOH, e/MWh
Without payment for AS
40.3
66.9
79.4
With payment for AS
23.0
49.5
61.9
AS ancillary services
To determine the sustainability of EB implementation projects, their sensitivity
analysis was performed. The key parameter in our case will be LCOH of thermal
energy, which depends on a number of factors, the main of which are the price
of energy resources, the amount of investment costs, discount rate and amount of
energy produced during the year (installed power utilization). In fact, the procedure
for determining sensitivity is nothing more than finding partial derivatives of the
function of many variables. In the practice of preparation of investment projects,
usually the change of the key parameter is determined by a consistent change of
influential factors by 10%. This was done for LCOH of thermal energy, which is
produced using GB and EB. The results of the sensitivity analysis are shown in Table
7.
As seen from Table 7, projects for the introduction of GB in the DH system are
quite resistant to influencing factors. The main influencing factor on the LCOH of
thermal energy is the change in the price of natural gas, and on the EB is the change
in the price of electricity and ancillary services. The significance of this impact is
moderate and acceptable for their implementation projects.
Table 7 The results of the sensitivity analysis of the introduction of gas and electric boilers
Impact factor (change by +
10%)
Change of LCOH, %
Gas boiler with a thermal
power of 58.15 MW
Electric boiler with a thermal
power of 50 MW
Investment costs
0.25
3.1
Discount rate
0.1
2.7
The price of natural gas
9.96
The price of electricity
–
36.6
Cost of ancillary services (−
10%)
–
10.1
The amount of heat produced
per year (− 10%)
0.13
3.2
462
V. Derii et al.
4 Discussion of Research Results
As a result of the analysis of simulation results, it was found that the EB can be used
without the provision of ancillary services to the power system under the following
conditions. With the cost of surplus electricity 29.0 and 50.1 e/MWh, the cost of
natural gas should not be less than 350 and 600 e/1000 m3 , respectively.
When providing ancillary services to the energy system with the help of the EB
(cost of electricity 29.0; 50.1 and 60 e/MWh), the cost of natural gas should not be
less than (250, 450 and 550 e/1000 m3 ), respectively.
The results of modeling the use of EB with a consumed electrical power of
51.02 MW for the provision of ancillary services to power systems allow to determine
the indicators of the use of EB with a total electrical power of 2500 MW:
.
.
.
.
.
the amount of investment costs—e300 million;
electricity consumption—5.35 billion kWh/year;
natural gas savings—618.5 million m3 /year;
reduction of coal consumption—573 thousand tons/year;
reduction of greenhouse gas emissions—1691.3 thousand tons/year.
5 Conclusions
Electric heat generators are used as consumers-regulators when there is an excess
of electricity from non-guaranteed generators (solar and wind power plants) and
during a nighttime “failure” of the daily schedule of electrical loads. The modes of
their operation are determined by the dispatchers of the Ukrainian IPS for balancing
the power system. Calculations have shown the competitiveness of such electric
heat generators in comparison with gas boilers. The introduction of electric heat
generators with a capacity of about 2500 MW to regulate the load of power systems
will make the generation structure of Ukrainian nuclear power plants more efficient by
increasing the level of nuclear power plants basic generation, reduce the consumption
of natural gas by 604.5 million m3 per year and coal used by thermal power plants by
296.7 thousand tons per year, reduce greenhouse gas emissions by 1691.3 thousand
tons per year. Also, the Integrated Power System of Ukraine will be more resistant
to changes in its load, which will increase energy security and energy independence
of Ukraine.
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