TRANSMISSION LINE MODELING FOR
REAL-TIME SIMULATIONS
Maria Isabel Silva Lafaia Simões
Dissertation submitted to obtain the degree of Master in
Electrical and Computers Engineering
Committee Members
President
Prof. Paulo José da Costa Branco (DEEC, IST)
Supervisor
Profa Maria Teresa Nunes Padilha de Castro Correia de Barros (DEEC, IST)
Member
Prof. Jean Mahseredjian (École Polytechnique de Montréal)
Member
Prof. José António Marinho Brandão Faria (DEEC, IST)
November 2012
Agradecimentos
Agradeço em primeiro lugar à minha família ao meu pai, à minha mãe, ao meu irmão e à minha
madrinha, pelo amor constante e incondicional que me dedicam.
Agradeço à professora Teresa Correia de Barros pela sua orientação e, sobretudo, pela conança que
deposita em mim.
Agradeço ao meu colega Pedro Cruz pela sua amizade, conselhos e palavras de apoio. Agradeço ainda
ao colega e amigo Miguel Fragoso pelo seu exemplo de dedicação e camaradagem.
A todos os que me inspiraram ao longo dos meus estudos no Instituto Superior Técnico Obrigada!
i
Resumo
A simulação em tempo-real de sistemas de energia eléctrica é uma ferramenta importante sempre que
é necessário incluir um componente físico no sistema em estudo, em vez do seu modelo matemático.
O tempo-real é difícil de atingir em simulação digital, pois a um aumento de exactidão corresponde,
geralmente, um aumento do tempo de processamento. Torna-se assim necessário combinar arquitecturas
de processamento paralelo com a utilização de modelos ecientes. As linhas de transmissão permitem o
processamento paralelo ao dividir uma grande rede em pequenas sub-redes independentes.
Uma representação exacta da linha exige que se considere a dependência na frequência dos seus
parâmetros, o que coloca um desao na denição de um modelo adequado. O objectivo desta dissertação
é estabelecer os procedimentos para uma aproximação dos parâmetros de propagação em modelos de
linha adequada para simulações em tempo-real.
O estudo dos modelos existentes constitui uma base para o desenvolvimento do RT_WB Line, que
é uma reformulação do modelo WB Line do EMTP-RV, em linha com o objectivo de tempo-real. Para
atingir uma exactidão superior com recursos reduzidos, consideram-se duas optimizações relativas à identicação dos atrasos modais e à distribuição dos pólos pelos modos.
O RT_WB Line é validado através de simulações no domínio da frequência e do tempo, considerando
soluções exactas ou o WB Line como referência da exactidão pretendida. Os testes conrmam que o
modelo desenvolvido permitirá, no tipo de aplicações nas quais é relevante o tempo-real, reduzir tempos
de processamento, por redução do número de operações requeridas, sem prejuízo da exactidão das soluções
obtidas.
Palavras-chave: Simulação em tempo-real, transitórios electromagnéticos, parâmetros dependentes
da frequência, RT_WB Line, identicação optimizada dos atrasos modais, distribuição optimizada dos
pólos pelos modos.
ii
Abstract
Real-time simulation of power systems transients is an important tool when there is a need to include
physical elements in the system under study, rather than their mathematical models. However, real-time
is hard to achieve in digital simulations, where accuracy runs oppositely to processing speed. It is therefore necessary to combine parallel processing with ecient numerical techniques for model computation.
Transmission lines allow parallel processing in power systems studies, by dividing large networks into
smaller independent subnetworks.
Accurate line representation requires the use of its frequency dependent parameters. This poses a
challenge on the denition of an adequate line model. The goal of this dissertation is to establish adequate
numerical techniques for approximating the propagation parameters for transmission line modeling, allowing real-time simulations.
The study of existing line models provides the basis for the development of the RT_WB Line, which
is a reformulation of the EMTP-RV model WB Line (based on the Universal Model [3]), in-line with the
real-time simulation target. To ensure additional accuracy with reduced tting resources, two optimizations are suggested, concerning the computation of the modal delays and the assignment of the modal
poles.
The RT_WB Line performance is validated through frequency and time domain tests, considering
the exact solutions or the WB Line as a reference of accuracy. The tests conrm that the RT_WB Line
allows, for the applications in which the real-time is important, a reduction of the processing time, by
reducing the required computations, without prejudice to the accuracy of the solutions.
Keywords: Real-time simulations, electromagnetic transients, transmission line modeling, frequency
dependent parameters, RT_WB Line, optimized modal delay computation, optimized modal poles assignment.
iii
Contents
Agradecimentos
i
Resumo
ii
Abstract
iii
List of tables
vi
List of gures
vii
Symbols and abbreviations
x
1 Introduction
1
1.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Objective of the present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Organization of the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Fundamentals on multiphase transmission line theory a brief review
3
5
2.1
Phase domain solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Modal domain solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
The propagation function of a transmission line . . . . . . . . . . . . . . . . . . . . . . . .
8
EMTP-RV
and transmission line modeling
11
3.1
Background on line modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2
Numerical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3
3.2.1
Major challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.2
Recursive calculation of convolution integrals . . . . . . . . . . . . . . . . . . . . . 14
3.2.3
Rational approximation of transmission line functions . . . . . . . . . . . . . . . . 16
3.2.3.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.3.2
Asymptotic Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.3.3
Vector Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.3.4
Asymptotic Fitting versus Vector Fitting . . . . . . . . . . . . . . . . . . 19
Transmission line models provided by the EMTP-RV 2.3
iv
. . . . . . . . . . . . . . . . . . 19
3.4
3.5
3.3.1
CP Line constant parameters line model . . . . . . . . . . . . . . . . . . . . . . 19
3.3.2
FD Line frequency dependent line model . . . . . . . . . . . . . . . . . . . . . . 20
3.3.3
WB Line wide-band line model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Model testing and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1
Frequency response short-circuited and open-ended line . . . . . . . . . . . . . . 21
3.4.2
Line energization and single-phase short-circuit . . . . . . . . . . . . . . . . . . . . 24
3.4.3
Current induced by phase coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.4
Model eciency
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Wide-band model for real-time simulations 31
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2
Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3
Optimized tting of the propagation function . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4
4.5
5
RT_WB Line
4.3.1
Optimal modal delay identication . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.2
Optimal modal poles assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Computer program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.1
Main program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.2
Propagation parameters computation
4.4.3
Ȳc tting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4.4
H̄ tting
4.4.5
Output generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
. . . . . . . . . . . . . . . . . . . . . . . . . 37
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
RT_WB Line
model validation
41
5.1
Validation perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2
Frequency response short-circuited and open-ended line . . . . . . . . . . . . . . . . . . 42
5.3
Line energization and single-phase short-circuit . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4
Current induced by phase coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Conclusions
53
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2
Completion of proposed objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3
Proposals for further improvements in line modeling . . . . . . . . . . . . . . . . . . . . . 55
Bibliography
57
A Transmission line used in model testing
59
v
List of Tables
3.1
Analytical short-circuit frequency response and approximating errors according to the
EMTP-RV 2.3 models, in terms of the magnitude of the current at the sending end of
phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2
Analytical open-end frequency response and approximating errors according to the EMTP-
RV 2.3 models, in terms of the magnitude of the voltage at the receiving end of phase
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3
Number poles used by the FD Line, available on the EMTP-RV 2.3, in the approximation
of the propagation parameters of a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4
Number of poles used by the WB Line, available on the EMTP-RV 2.3, in the approximation of the propagation parameters of a line . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1
Eect of using optimized modal delays average error of approximating propagation
functions according to dierent order applications of the RT_WB Line, in mode and
phase domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2
Average error of the approximation of H̄, according to dierent order applications of the
RT_WB Line. Use of equal (E) or optimized (O) distribution of the modal poles . . . . . 35
5.1
Number of poles used for the approximation of the propagation parameters of a line, for
the applications of tested models WB Line and RT_WB Line
vi
. . . . . . . . . . . . . . 41
List of Figures
2.1
Multi-phase transmission line and convention used to dene the phase currents and voltages.
2.2
Illustration of the physical meaning of the propagation function current source in parallel
with characteristic admittance connected to a short-circuited transmission line. . . . . . .
2.3
9
Illustration of the physical meaning of the propagation function left: current unit impulse
applied to the line; right: short-circuited line response to a current unit impulse. . . . . .
2.4
5
9
Illustration of the physical meaning of the propagation function alternative representation of h(t) through a time function translated to the origin, h0 (t). . . . . . . . . . . . . . 10
3.1
Illustration of the method of recursive convolutions: system represented by its impulse
response h(t). The function g(t) is the response of the system to an input signal f (t). . . 14
3.2
Illustration of the method of recursive convolutions: Input signal f (t − z) as a function of
z inside the interval [−∆t; 0]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3
Inclusion of losses in a CP Line Model, in the form of lumped resistances. . . . . . . . . . 20
3.4
Circuits used to study the short-circuit and open-end frequency responses according to
the line models available on the EMTP-RV 2.3 . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5
Short-circuit frequency response according to the EMTP-RV 2.3 line models, in terms of
the current at the sending end of phase 1 (CP Line thin, FD Line dotted, WB Line
bold). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6
Open-end frequency response according to the EMTP-RV 2.3 line models, in terms of the
voltage at the receiving end of phase 1 (CP Line thin, FD Line dotted, WB Line bold). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7
Circuit used to study the time response to energization followed by single-phase shortcircuit, according to the line models available on the EMTP-RV 2.3. . . . . . . . . . . . . 24
3.8
Response to line energization at t = 20 ms according to the CP Line thin, and to the
WB Line bold, in terms of the voltage at the receiving end of phase 1.
3.9
. . . . . . . . . 25
Response to a short-circuit at the receiving end of phase 3 at t = 180 ms according to CP
Line thin, and to the WB Line bold, in terms of the voltage at the receiving end of
phase 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.10 Circuit used to study the phenomena of phase coupling according to the line models
available on the EMTP-RV 2.3, in terms of the current induced in phase 3 by energization
of phase 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
vii
3.11 Current induced by energization of phase 1 time evolution of the current at the sending
end of phase 3 during the rst 20 miliseconds of the transient according to the EMTP-RV
2.3 line models (CP Line thin, FD Line dashed, WB Line bold). . . . . . . . . . . 27
3.12 Current induced by energization of phase 1 time evolution of the current at the sending
end of phase 3 during the rst second of the transient according to the EMTP-RV 2.3 line
models (CP Line thin, FD Line dashed, WB Line bold). . . . . . . . . . . . . . . . 27
4.1
General structure of the program developed to compute applications of the RT_WB Line
model in-line with the real-time simulation target, with respective input and output data.
5.1
36
Circuits used to study the short-circuit and open-end frequency responses according to
the WB Line and RT_WB Line models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2
Short-circuit frequency response (0.1 Hz - 1 MHz) analytical (bold), WB Line (thin)
and RT_WB Line (dashed). Current at the sending end of phase 1. . . . . . . . . . . . . 43
5.3
Relative error of the short-circuit frequency response (0.1 Hz - 1 MHz) WB Line (bold)
and RT_WB Line (thin). Current at the sending end of phase 1. . . . . . . . . . . . . . . 43
5.4
Detailed relative error of the short-circuit frequency response (700 Hz - 10 kHz) WB
Line (bold) and RT_WB Line (thin). Current at the sending end of phase 1. . . . . . . . 44
5.5
Open-end frequency response (100 Hz - 1 MHz) analytical(bold), WB Line (thin) and
RT_WB Line (dashed). Voltage at the receiving end of phase 1. . . . . . . . . . . . . . . 45
5.6
Relative error of the open-end frequency response (100 Hz - 1 MHz) WB Line (bold)
and RT_WB Line (thin). Voltage at the receiving end of phase 1. . . . . . . . . . . . . . . 45
5.7
Detailed relative error of the open-end frequency response (700 Hz - 10 kHz) WB Line
(bold) and RT_WB Line (thin). Voltage at the receiving end of phase 1. . . . . . . . . . 46
5.8
Circuit used to study the response to line energization followed by single-phase shortcircuit, according to the WB Line and to the RT_WB Line applications. . . . . . . . . . 46
5.9
Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)
and RT_WB Line (thin). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.10 Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)
and RT_WB Line (thin). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.11 Circuit used to study the phenomena of phase coupling according to the WB Line and to
the RT_WB Line applications, in terms of the current induced in phase 3 by energization
of phase 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.12 Current at the sending end of phase 3, induced by energization of phase 1, at the rst 20
miliseconds, according to the WB Line (bold), and to the RT_WB Line (thin). . . . . . . 48
5.13 Current at the sending end of phase 3, induced by energization of phase 1, at the rst
second of simulation, according to the WB Line (bold), and to the RT_WB Line (thin). . 49
5.14 Current at the sending end of phase 3, induced by energization of phase 1, at the rst
20 miliseconds, according to the WB Line (bold), and to the RT_WB Line applications
(low order thin; high order dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
viii
5.15 Current at the sending end of phase 3, induced by energization of phase 1, at the rst second of simulation, according to the WB Line (bold), and to the RT_WB Line applications
(low order thin; high order dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.16 Current at the sending end of phase 3, induced by energization of phase 1, at the rst 50
second of simulation (reaching steady-state), according to the WB Line (bold), and to the
RT_WB Line applications (low order thin; high order dashed). . . . . . . . . . . . . . 50
A.1 Spacial conguration of the transmission line used throughout this dissertation. . . . . . . 59
ix
Symbols and abbreviations
• When denoting system variables, upper case letters refer to frequency domain quantities, whereas
lower case letters denote time domain quantities. For example, V for frequency domain voltage
and v for time domain voltage.
• Bold letters distinguish between matrix or vector quantities and scalar quantities. For example, a
vector of scalar voltages is written as v = [v1 , v2 , · · · , vn ]T , where T stands for transposition and
n is the vector dimension.
• A bar-hatted letter denotes a complex quantity, with real and imaginary parts. For example,
γ̄ = α + jβ .
• System variables:
R longitudinal/series resistance
L longitudinal/series inductance
G transversal/shunt conductance
C transversal/shunt capacitance
Z̄ longitudinal impedance function
Ȳ transversal admittance function
Γ̄ propagation factor
α attenuation factor
β phase shift factor
τ propagation delay
H̄ propagation function
Ȳc characteristic admittance
Z̄c = Ȳc−1 characteristic impedance
i phase current
I phase current phasor
v phase voltage
V phase voltage phasor
T̄i current transformation matrix
T̄v voltage transformation matrix
x
• Parameters:
t time
∆t time step
ω angular frequency
s complex frequency
√
j −1
d length of line
x distance from the sending end of the line
• Operators:
F Fourier transformation
F −1 inverse Fourier transformation
Re real part of complex quantity
Im imaginary part of complex quantity
∗ convolution
• Subscripts:
k sending end of a line
m receiving end of a line
s source quantity
f it approximating function
1, 2, · · · , n line phase
short short-circuit condition
open open-end condition
• Superscripts:
k (superscript) relative to the k th line mode
m (superscript) modal domain
• Abbreviations:
DC direct current
RMS root-mean-square value
CP Line constant parameters line model
FD Line frequency dependent line model
WB Line wide band line model
EMTP-RV Electromagnetic Transients Program Revised Version
xi
Chapter 1
Introduction
1.1
Overview
Although power systems are in steady-state most of the time, they must be able to withstand the worst
possible stresses to which they may be subjected, which usually occur during transient conditions of the
power system. Therefore, the size and cost of the equipment in a power system is largely determined by
transient conditions, rather than by its steady-state behavior.
It is of the utmost importance to accurately predict the behavior of the system. For instance, the
eectiveness of protective strategies in moderating transient conditions is only properly assessed based on
accurate data. Also, specially for high voltage power systems, any tolerance on equipment specications
may represent a considerable increase of costs with no guarantee of optimum operation.
Two ways of studying transients in a power system are:
• Analogical simulation: the power system is represented by a transient network analyzer (TNA's),
which is a physical down-scaled reproduction of the power system components;
• Digital computer simulation: the power system components are represented through mathematical
models implemented computationally.
Transient network analyzers require physical facilities (space and equipment) and trained personnel.
The simulation of large networks using this method is nancially very demanding. Furthermore, TNA's
have limited ability to represent real physical systems, namely, the distributed and frequency dependent
character of any component parameters.
On the other hand, the digital simulation has low requirements on space and equipment and, therefore, involves lower costs. It is more exible than TNA's since any new component may be simulated
with reduced or null additional costs, provided an adequate model is known. Finally, the development
1
of computer processing capacity allows very rigorous simulations, with few simplications.
Anytime there is a need to include physical elements in the system under study, rather than their
mathematical models, the simulation must be performed in real-time, that is, the quantities of interest
must have their values predicted correctly and within a prescribed period of time. This is usually the
case when studying the interaction between a power system and protection/relaying equipment, control/command systems and power electronic devices.
The real-time is intrinsic to TNA's, but it is not so easy to achieve in digital simulations, where
accuracy runs oppositely to processing speed. The design of a real-time digital simulator has mainly two
areas of development:
• Processor architecture: parallel processing allows to distribute the eort by several processing
units, working simultaneously. To do so, it is necessary to identify the operations that may be
taken independently, and not in a sequential manner;
• Implementation algorithms: eciency of numerical model computation and optimization of the
couple accuracy/complexity of the model.
Transmission lines play an important role on the denition of parallel processing levels within a power
system network: every time the propagation time of a line is suciently larger than the simulation time
step, the subnetworks connected through that line may be considered independently.
An accurate representation of a transmission line (for example, accurately representing distortion)
requires a ne representation of the distributed and frequency dependent character of its parameters.
This poses a challenge on the denition of an adequate transmission line model.
1.2
Objective of the present work
The main goal of this dissertation is to establish adequate numerical techniques for approximating the
propagation parameters for transmission line modeling, allowing real-time simulations. This requires an
ecient use of reduced modeling resources, namely, the introduction of optimization procedures that
ensure additional accuracy.
To accomplish this task, it is rst necessary to take insight into the "state of the art" of line modeling,
specically, the main challenges and its evolution. The study of the characteristics and performance of
the most used line models allows to dened the basic formulation to construct an accurate and ecient
model.
After dening the structure of the model and introducing the adequate numerical techniques, it is
necessary to create a program that computes the developed model applications, using a pre-dened order
2
for the approximating line functions. The validation process consists of a set of tests in frequency and time
domain conditions and must use an application of the developed model which order of approximations
is adequate for real-time performances. A line model provided by the EMTP-RV 2.3 is included in the
test and taken as a reference of accuracy.
1.3
Organization of the text
The work presented in this dissertation is divided into 6 chapters and 1 appendix, summarized as follows:
Chapter 1: Introduction
The introductory chapter gives an overview of the problem of transmission line modeling for realtime simulations. It also presents the objective of this work, which regards the establishment of adequate
numerical techniques for approximating the propagation parameters for transmission line modeling, allowing real-time simulations. The chapter ends with a summary of the organization of the text in this
dissertation.
Chapter 2: Fundamentals on multiphase transmission line theory a brief review
This chapter presents a brief review of the theory necessary to understand the construction of a
mathematical model to represent a multiphase transmission line in transient studies. The equations
representing the line behavior in transient conditions are rst formulated in phase domain. The modal
domain is then introduced as an alternative for studying multiphase lines. The chapter concludes by
providing insight to the meaning of the propagation function for the simple case of a single-phase line.
Chapter 3:
EMTP-RV
and transmission tine modeling
EMTP-RV stands for Electromagnetic Transients Program. It is a widely used software, useful to
study transmission systems. The line models available in the EMTP-RV 2.3 provide a summary of the
evolution of line modeling.
The chapter starts by presenting the program and the main aspects that characterize line models,
namely, which functions are used to characterize the line, how the frequency dependence of line parameters is taken into account and whether the solution to line equations is computed in phase or modal
domain.
Some insight is also given to some of the most important techniques for model eciency. The rst,
concerns the time domain equations that describe the behavior of the line in transient conditions, which
have to be computed at every simulation step and contain convolution integrals. A recursive calculation
[1] of those integrals is an alternative that tackles the high memory and processing time required for a
numerical evaluation.
3
The second technique concerns the representation of the line functions. Instead of using the actual
values of those functions for a large number of frequency samples, they can be approximated by analytical expressions in the form of rational functions of frequency. This representation is not only a basic
requirement for the use of recursive convolutions. It also allows a very ecient representation of the
line functions and a direct analytical transformation to time domain. Two techniques used for rational approximation of line functions are: Asymptotic Fitting [2], based on the magnitude of the original
function, and Vector Fitting [4, 5, 6], which allows to t a set of functions using the same basic terms.
After that, the line models available on the EMTP-RV 2.3 are described, tested and compared. These
line models give a good insight to the evolution of line modeling, starting from the most basic constant
parameters model, called CP Line, to the most accurate WB Line, which takes the modal information
into account to t the phase domain line functions.
Chapter 4: Wide-band model for real-time simulations RT_WB Line
The objective of this dissertation is to establish adequate numerical techniques for approximating
the propagation parameters for transmission line modeling, allowing real-time simulations. This chapter
presents the RT_WB Line, which is a reformulation of the EMTP-RV model WB Line. In order to
ensure additional accuracy, it is necessary to introduce some optimization procedures, each of which is
presented and illustrated by a numerical example.
Chapter 5:
RT_WB Line
model validation
This chapter presents a set of tests that validate the developed transmission line model, RT_WB
Line. This is done through simulations in the EMTP-RV 2.3 environment. The tests analyze the
frequency and time domain behavior of a transmission line, according to an application of the RT_WB
Line, which uses an order for the approximating line functions in-line with the examples in literature
regarding real-time transmission line modeling. The WB Line, generated by the EMTP-RV, is taken as
a reference of accuracy.
Chapter 6: Conclusions
This chapter presents the nal considerations on the performance of the RT_WB Line and on the fulllment of the proposed objectives. A set of ideas are presented for further improvements in transmission
line modeling for real-time simulations.
Appendix: Transmission line used in model testing
The appendix presents a description of the spacial and electromagnetic characteristics of the transmission line used throughout this work for model testing.
4
Chapter 2
Fundamentals on multiphase
transmission line theory a brief
review
2.1
Phase domain solution
Consider an n-phase transmission line of length d, as illustrated in gure 2.1. As it is well known,
penetration of the electromagnetic eld in unperfect conductors introduces the frequency dependence of
the longitudinal transmission line parameters.
Figure 2.1: Multi-phase transmission line and convention used to dene the phase currents and voltages.
Therefore, the multiphase transmission line is characterized by its longitudinal impedance matrix
Z̄ = R(ω) + jωL(ω) and transversal admittance matrix Ȳ = G + jωC and described in frequency
5
domain by a couple of matrix dierential functions:
d2
V̄(ω, x) = Z̄(ω)Ȳ(ω)V̄(ω, x)
dx2
d2
Ī(ω, x) = Ȳ(ω)Z̄(ω)Ī(ω, x)
dx2
(2.1)
(2.2)
where V̄ and Ī are vectors containing the phase voltages and currents of the line. It is possible to deduce
a solution to equations (2.1) and (2.2) which relates V̄ and Ī at the two line terminals as:
Ȳc V̄k − Īk = H̄ Ȳc V̄m − Īm
Ȳc V̄m + Īm = H̄ Ȳc V̄k + Īk
(2.3)
(2.4)
The auxiliary line functions introduced are:
• the characteristic admittance matrix
Ȳc (ω) = Z̄(ω)−1
q
Z̄(ω)Ȳ(ω)
(2.5)
• the propagation matrix (matrix exponential)
H̄(ω) = e−Γ̄(ω)d
(2.6)
• the matrix of the propagation factors
Γ̄(ω) =
q
Ȳ(ω)Z̄(ω)
(2.7)
The transformation of the line equations (2.3) and (2.4) to the time domain must take into account
the frequency dependence of all the line functions, resulting:
(yc (t) ∗ vk (t) − ik (t)) = h(t) ∗ (yc (t) ∗ vm (t) − im (t))
(2.8)
(yc (t) ∗ vm (t) + im (t)) = h(t) ∗ (yc (t) ∗ vk (t) + ik (t))
(2.9)
Due to coupling between the n line phases, the corresponding voltages and currents are interdependent. Therefore, line matrices Z̄ and Ȳ, and consequently Ȳc and H̄, are non-diagonal matrices, and
the total number of convolutions needed to compute the equations (2.8) and (2.9) is proportional to n2 .
2.2
Modal domain solution
The study of a transmission line in terms of the voltages and currents of the n phases is complicated
by coupling phenomena. However, for ordinary multi-conductor transmission line congurations, there
are n independent propagation modes so, alternatively, the line may be studied in terms of the electric
quantities associated to its modes.
The conversion between phase and mode quantities is performed in frequency domain using a voltage
transformation matrix and a current transformation matrix as:
V̄ = T̄v V̄m
(2.10)
Ī = T̄i Īm
(2.11)
6
Substituting these relations into the line dierential equations (2.1) and (2.2) results:
d2 m
V̄ (ω, x) = Λ̄(ω)V̄m (ω, x)
(2.12)
dx2
d2 m
Ī (ω, x) = Λ̄(ω)Īm (ω, x)
(2.13)
dx2
−1
where m stands for modal domain and Λ̄(ω) = T̄−1
v Z̄Ȳ T̄v = T̄i Ȳ Z̄T̄i is a diagonal matrix.
Therefore, T̄v and T̄i are the matrices that diagonalize the products Z̄Ȳ and ȲZ̄, respectively. This
means the columns of T̄v and T̄i are equal to the eigenvectors of the corresponding products.
Generally, Z̄Ȳ and ȲZ̄ are dierent and frequency dependent, and so will be T̄v and T̄i 1 . However,
it is possible to relate them to each other through:
T̄i = T̄tv
−1
(2.14)
where t stands for transposition. It is therefore sucient to compute only one of them.
To write the line equations in modal domain, it is still necessary to convert the line functions Z̄ and
Ȳ to modal equivalents through:
Z̄m = T̄−1
v Z̄ T̄i
(2.15)
Ȳm = T̄−1
Ȳ T̄v
i
(2.16)
where Z̄m and Ȳm are diagonal matrices. These are used to compute the auxiliary functions:
• the matrix of modal characteristic admittances:
Ȳcm = (Z̄m )−1
p
Z̄m Ȳm
(2.17)
• the matrix of modal propagation functions:
√
H̄m = e−
Ȳ m Z̄m d
(2.18)
The solution in frequency domain to equations (2.12) and (2.13) is then similar to the phase equations
(2.3) and (2.4), as long as all quantities are considered in modal domain. The time domain solution is
then simply:
m
m
m
m
(ycm (t) ∗ vkm (t) − im
k (t)) = h (t) ∗ (yc (t) ∗ vm (t) − im (t))
(2.19)
m
m
m
m
m
(ycm (t) ∗ vm
(t) + im
m (t)) = h (t) ∗ (yc (t) ∗ vk (t) + ik (t))
(2.20)
where ycm (t) and hm (t) are the inverse Fourier transformation of the matrices Ȳcm (ω) and H̄m (ω).
vm (t) and im (t) are vectors the modal voltages and currents in time.
1 An
eigenvector can be arbitrarily scaled, thus T̄v and T̄i are not uniquely dened. The ambiguity in their calculation
∗ +
can be removed by normalizing the matrices columns to vectors of unitary euclidean length, that is, by requiring Tv1 Tv1
∗ + ... = 1 with T ∗ =conjugate complex of the i-th column of T . However, there is still ambiguity in the sense that
Tv2 Tv2
v
vi
each column can be multiplied with a rotation constant ejα and still have unitary vector length.
7
Due to the independence of the line modes, the characteristic admittance and propagation matrices
become diagonal in modal domain. So, each of equations (2.19) and (2.20) in fact represents n independent scalar equations, and each convolution represents only n scalar convolutions. In phase domain
equations (2.8) and (2.9), they represented n2 scalar convolutions.
This means that each mode can be studied as a single-phase line. However, the advantage is not as
good as may seem, since an additional set of convolutions must be taken in order to convert the modal
voltages and currents to the natural domain of phases through:
v(t) = tv (t) ∗ vm (t)
(2.21)
i(t) = ti (t) ∗ im (t)
(2.22)
Notice that, since the transformation matrices are frequency dependent, the last step implies calculating the inverse Fourier of T̄v (ω) and T̄i (ω), thus increasing the complexity of analysis.
2.3
The propagation function of a transmission line
It is worthwhile to analyze the meaning of the propagation function of a transmission line, which is easier
to understand for the case of a single-phase line. For this simple case, the propagation function is scalar
and dened as:
H̄ = e−γ̄(ω)d = e−α(ω)d . e−jβ(ω)d
(2.23)
with γ̄ = α + jωβ , H̄ contains an attenuation factor e−αd as well as a phase shift factor e−jβd , both
functions of frequency.
To go deeper into the meaning of H̄ , consider a current source I¯s in parallel with an admittance
equal to the characteristic admittance of the line (to avoid reections), connected to the sending end, k ,
of a line having the receiving end, m, short-circuited, as illustrated in gure 2.2. In that case, we have
Ȳc V̄k + I¯k = I¯s and V̄m = 0. From equation (2.4):
I¯m = H̄ I¯s
(2.24)
That is, the propagation function H̄ is the ratio (receiving end current)/(source current) of a shortcircuited line fed through a matching admittance Ȳc to avoid reections at the sending end k .
If I¯s = 1 at all frequencies, then its time domain transformation is a unit impulse is (t) = δ(t) (innitely high spike which is innitely narrow with an area of 1). Setting I¯s = 1 in equation (2.24) shows
that H̄(ω) transformed to time domain must be the impulse that arrives at the receiving end m if the
source is a unit impulse. According to (2.23), this response to the unit impulse will be attenuated (no
longer innitely high) and distorted (no longer innitely narrow) as illustrated in gure 2.3 for a typical
single-phase line.
8
If is (t) is an arbitrary function of time, equation (2.24) transforms to the time domain as:
Z +∞
h(τ )is (t − τ )dτ
im = h(t) ∗ is (t) =
(2.25)
τmin
The convolution integral starts in τmin since h(t) = 0 for t < τmin , as illustrated in gure 2.3. This
expression shows that im (t) is constructed as the sum of the samples of is (t) taken τ units of time ago
and weighted according to the value of h(τ ).
Figure 2.4 shows that h(t) can also be expressed as a similar function translated in time to the origin.
In that case:
h(t) = h0 (t − τmin )
(2.26)
H̄(ω) = H̄ 0 (ω)e−jωτmin
(2.27)
which transforms to frequency domain as
that is, a time delay in the time domain becomes a phase shift in the frequency domain.
Figure 2.2: Illustration of the physical meaning of the propagation function current source in parallel
with characteristic admittance connected to a short-circuited transmission line.
Figure 2.3: Illustration of the physical meaning of the propagation function left: current unit impulse
applied to the line; right: short-circuited line response to a current unit impulse.
9
Figure 2.4: Illustration of the physical meaning of the propagation function alternative representation
of h(t) through a time function translated to the origin, h0 (t).
10
Chapter 3
EMTP-RV
and transmission line
modeling
3.1
Background on line modeling
The EMTP-RV 2.3 is a specialized software for the simulation of electromagnetic, electromechanical
and control systems transients in multiphase power systems. The software is used worldwide by many
utilities, companies and consultants. Its main applications include projects, design and engineering or
the solution of problems and unexpected failures.
Specically, the program is useful to study transmission systems, including insulation coordination
and switching design. The transmission line models available in the EMTP-RV 2.3 provide a summary
of the evolution of line modeling, from the simplest constant parameters model, to the more complex
frequency dependent models, which approximate the line functions by analytical expressions in the form
of rational functions of frequency.
Generally, transmission line models represent the line as a multi-port system, that is, the study of the
line behavior is described in terms of the currents and voltages at the two line terminals. Nevertheless,
there are several aspects that distinguish the line models, namely:
• Characterization of the line: a transmission line is characterized by two functions, based on
the parameters per unit length of the line R, L, G and C. The rst alternative is using the
characteristic admittance Ȳc and the propagation function H̄. In this case, the line is analyzed in
terms of reected and incident current waves (Ȳc V̄ ± Ī) at the two terminals k and m. The line
equations in frequency domain are:
(Ȳc V̄k − Īk ) = H̄(Ȳc V̄m − Īm )
(3.1)
(Ȳc V̄m + Īm ) = H̄(Ȳc V̄k + Īk )
(3.2)
11
The other alternative is using the characteristic impedance Z̄c = Ȳc−1 and the propagation function
H̄. In this case, the line is analyzed in terms of reected and incident voltage waves (V̄ ± Z̄c Ī)
at the two terminals k and m. The line equations in frequency domain are:
(V̄k − Z̄c Īk ) = H̄(V̄m − Z̄c Īm )
(3.3)
(V̄m + Z̄c Īm ) = H̄(V̄k + Z̄c Īk )
(3.4)
• Accounting for frequency dependence of line functions: for the case of a constant parameters model, the approximating line functions are constant in frequency. Therefore, the transformation of those functions to time domain is immediate and the line equations have no convolution
integrals. On the other hand, frequency dependent models approximate, within a frequency range
of interest, each line function by a sum of rational terms, which transforms to time domain as a
sum of exponential terms. The time domain equations contain convolution integrals which may be
computed recursively [1], as described in section 3.2.2. Frequency dependent models use dierent
techniques to compute rational approximations of the line functions. Section 3.2.3 describes two
examples of these techniques: Asymptotic Fitting and Vector Fitting.
• Solution domain: a line model can be computed in phase domain, through a set of coupled
equations, or in modal domain, using independent equations for each mode. All EMTP-RV 2.3
line models make use of some information from the modes and all use a constant real transformation
matrix. The optimal frequency at which this matrix is evaluated may be automatically computed
by the EMTP-RV 2.3, or specied by the user1 .
Given the use of an approximating transformation matrix, the models in modal domain are based
on approximated modes, which represents a source of inaccuracy in relation to the phase domain
models. Furthermore, due to the interaction of the line with the outside system (which is modeled
in phase domain) it is necessary to convert the computed modal variables to phase domain at each
simulation step, increasing the model processing time.
These are the basic characteristics that distinguish the several EMTP-RV 2.3 models. Other line
models may have dierent characteristics. For example, tting the line functions in z-domain, instead of
s-domain [11], or considering lumped instead of distributed parameters.
In order to characterize the model eciency, it is also necessary to analyze how it is implemented.
From the chapter on line theory, it was clear that the computation of the line variables implies (1)
computing the inverse Fourier transformation of the line propagation parameters and (2) computing the
line equations which contain convolution integrals. Thus, the accuracy and eciency of the model is
1 The
optimum frequency determination procedure selects an optimum value of frequency for the range of switching
transients. This value is based on asymptotic conditions for the particular line under consideration. Typical values range
from 500 Hz to 5 kHz with a average around 1 kHz. The selection of an optimum value is based on the constancy of the
transformation matrix within the typical frequency range for switching transients. For studies involving other frequency
ranges (lightning, for example) the frequency should be supplied by the user.
12
greatly inuenced by the way the line functions are represented and the techniques for computing the
convolution integrals.
Section 3.2, in this chapter, introduces two numerical techniques which are crucial to line models
eciency. The rst technique involves a recursive computation of the convolution integrals in line
equations. The second consists of representing the line propagation parameters through approximating
analytical expressions in the form of rational functions of frequency. The advantages of both techniques
are claried. After that, section 3.3 presents the line models available in EMTP-RV 2.3. These models
are then tested and compared in section 3.4.
3.2
Numerical techniques
3.2.1
Major challenges
The simulation of a transmission line implies the computation for each time step of the matrix equations
(2.8) and (2.9), or (2.19) and (2.20) for a modal domain analysis2 . This creates two major challenges:
• The calculation of the inverse Fourier transformation of the line functions H̄ and Ȳc , known in
frequency domain. This represents a preprocessing routine;
• The calculation at each time step of the convolution integrals in line equations (2.8) and (2.9).
The most direct approach is to execute these steps using the exact line functions H̄ and Ȳc evaluated
at each frequency sample. The computation of the Fourier transformation of these functions results into
an equal number of time samples. The convolution integrals computed at each time step must then consider the complete range of samples. This procedure is not only highly demanding in terms of memory
and processing time, but also vulnerable to integration errors.
Alternatively, currently used line models approximate the elements of matrices H̄ and Ȳc with
analytical expressions in the form of rational functions of frequency. The advantages of this approach
are:
• Direct calculation of h(t) and yc (t): the inverse Fourier transformation of a rational function of
frequency has a well known analytical form;
• Memory saving: instead of saving a high number of samples of H̄ and Ȳc , it is only necessary to
keep the parameters of their approximating functions;
2 For
a modal domain approach, it is necessary to convert at each time step between phase and modal quantities through
equations vphase (t) = tv (t) ∗ vmodal (t) and iphase (t) = ti (t) ∗ imodal (t), where tv (t) and ti (t) are the inverse Fourier
transformation of the voltage and current transformation matrices. Many line models, however, consider real constant
transformation matrices, turning this equations into simple matrix products with reduced impact on the model eciency.
Thus, this step will be omitted along this chapter.
13
• Possibility of computing the convolution integrals recursively [1], greatly reducing the processing
time for each time step.
The use of this strategy implies an additional eort to compute the functions which approximate the
elements of H̄ and Ȳc . This aditional eort must be included in the preprocessing routine and it does not
interfere with the requirements of real-time simulations. However, the complexity of the approximating
functions will have a direct impact on the model eciency. It is therefore mandatory to obtain adequate
approximations: an optimized reduced order model.
Section 3.2.2 illustrates the technique of recursive convolutions [1]. A general view of the techniques
for rational approximation of line functions is given in section 3.2.3.
3.2.2
Recursive calculation of convolution integrals
The digital simulation of a transmission line implies the calculation, at each time step, of a set of equations involving convolution integrals, in terms of the time domain counterparts of the line functions and
the voltages and currents at the two line terminals.
A numerical solution is prohibitively time and memory consuming. A much more ecient approach is
the recursive solution of the convolutions integrals [1]. This technique consists of dividing the convolution
integral in two smaller integrals: one computed from the beginning of the simulation until the previous
time step; the second computed over the present time step.
The use of this technique reduces considerably the processing time for each simulation step, since it
is only necessary to compute the second part of the integral (the rst comes from the previous iteration).
It also reduces the memory requirements since it is only necessary to keep track of a few past time
steps, to account for the delay of propagation across the line. Nevertheless, the application of recursive
convolutions, for the purpose of transmission line modeling, requires the line functions h(t) and yc (t) to
be represented as a sum of exponentials.
To illustrate the basics of recursive convolution, consider gure 3.1, where a given system is represented by its impulse response h(t). g(t) is the system response to an input signal f (t), computed
through a convolution integral:
Figure 3.1: Illustration of the method of recursive convolutions: system represented by its impulse
response h(t). The function g(t) is the response of the system to an input signal f (t).
14
Z
+∞
Z
+∞
−∞
(3.5)
Ae−bτ f (t − τ )dτ
h(τ )f (t − τ )dτ =
g(t) = h(t) ∗ f (t) =
τ =0
where h(t) = Ae−bt for t > 0. At an instant t + ∆t, one may write:
Z ∞
Ae−bτ f (t + ∆t − τ )dτ
g(t + ∆t) =
(3.6)
τ =0
Now consider the change of variables:
z = τ − ∆t
dz = dτ
The application of this change of variables in (3.6) leads to:
Z 0
Z
−b∆t
−bz
Ae f (t − z)dz +
g(t + ∆t) = e
=e
z=−∆t
Z 0
−b∆t
∞
Ae
−bz
f (t − z)dz
(3.7)
z=0
Ae−bz f (t − z)dz + g(t)
(3.8)
z=−∆t
The rst integral in (3.8) may be developed if the function f (t − z) is approximated inside the interval
z ∈ [−∆t; 0] by a polynomial function. Consider, for simplicity, a rst order approximation. According
to gure 3.2:
Figure 3.2: Illustration of the method of recursive convolutions: Input signal f (t − z) as a function of z
inside the interval [−∆t; 0].
f (t) − f (t + ∆t)
z + f (t) = k1 z + k2 , for z ∈ [−∆t; 0]
(3.9)
∆t
and k2 = f (t) are determined at each time step. This allows to solve the rst
f (t − z) ≈
where k1 =
f (t)−f (t+∆t)
∆t
integral in (3.8) by parts, resulting into:
Z 0
−b∆t
e
Ae−bz f (t − z)dz = α f (t) + β f (t + ∆t)
(3.10)
z=−∆t
where α and β are constants given by:
1
−b∆t
−b∆t
−e
+
(1 − e
)
b∆t
A
1
β=
1−
(1 − e−b∆t )
b
b∆t
A
α=
b
15
(3.11)
(3.12)
Equation (3.9) may now be rewritten in a recursive manner as:
g(t + ∆t) = α f (t) + β f (t + ∆t) + γ g(t)
(3.13)
with α and β given by (3.11) and (3.12) and γ = e−b∆t . Notice that calculation of (3.13) implies only
the calculation of three products involving scalar quantities (the coecients are constant throughout
the simulation). In the general case of an impulse response approximated by a sum of exponentials
P
h(t) = i Ai e−bi t , thus resulting:
X
g(t) =
gi (t)
(3.14)
i
where gi (t) = hi (t) ∗ f (t). This method has proven to be very accurate and stable, as explained in [1].
Notice that the number of exponentials used to approximate the impulse response h(t) increases the
accuracy of the method, and also its processing time.
3.2.3
Rational approximation of transmission line functions
3.2.3.1 Background
As mentioned before, transmission line models characterize the line through the propagation function
and the characteristic admittance (or, alternatively, the characteristic impedance). Generally, instead of
using the "exact" value of their samples, frequency dependent models approximate those functions with
analytical expressions in the form of rational functions of frequency. Consider a general function of the
complex frequency s = jω , G(s), which could approximate any of the line functions:
QM
G(s) =
(s − zm )
c0 Qm=1
N
n=1 (s − pn )
=
N
X
cn
s
−
pn
n=1
(3.15)
where M and N ≥ M are the number of zeros (zm ) and poles (pn ) used in the approximation of G(s).
c0 is a constant and cn is the residue of G(s) corresponding to pole pn , which is:
cn = Res[G(s)]pn = lim
s→pn
dk−1
[(s − pn )G(s)]
dsk−1
(3.16)
where k if the multiplicity of pole pn . This way of representing the line functions presents several
advantages:
• Direct analytical inverse Fourier transformation of the line functions: the time-domain counterpart
of (3.15) is given by:
g(t) = F −1 [G(ω)] =
N
X
cn epn t
(3.17)
n=1
• Memory saving: instead of keeping a high number of time samples, only the parameters of the
approximating line functions are needed, which identify both their frequency and time-domain
counterparts.
• Processing time saving: thanks to the possibility of using recursive convolutions to compute the
line variables at each time step of a simulation.
16
The computation of those approximating functions represents a preprocessing routine to a frequency
dependent model. The accuracy and eciency of the model is directly related to the complexity of the
approximating functions. Higher order approximations provide more accurate results, but increase the
processing time associated to the model.
Two techniques have been widely used in line modeling for the rational approximation of frequency
responses: Asymptotic Fitting [2] and Vector Fitting [4, 5, 6].
3.2.3.2
Asymptotic Fitting
The Asymptotic Fitting technique has been introduced in line modeling by J. Marti [2] and it is based
on the approximation of the magnitude of the original function.
From zero up to a maximum frequency, at which the original function approaches zero or becomes
constant, the original function is compared to the approximating function. Poles and zeros are assigned to
the tting function as needed, that is, when the dierence between the two functions is above a maximum
accepted error. Thus, the order of the approximation is not established a priori, but determined by the
approximating routine.
For a stable model, all poles lie on the left side of the complex plane. The Asymptotic Fitting uses
only real poles and zeros to avoid ripples or local peaks in the approximating function.
It is important to refer that this technique considers that the characteristic admittance (or impedance)
and the propagation function (after extracting the minimum propagation delay3 ) are approximately
minimum phase shift functions. For this class of functions, the phase of the original function matches
the phase of the corresponding approximating function. For the propagation function, the minimum
propagation delay is usually computed by comparing the phases of approximating and original functions.
The condition of minimum-phase-shift function is achieved by setting all zeros of the approximating
function on the left half of the complex plane.
3.2.3.3
Vector Fitting
The Vector Fitting technique [4, 5, 6] was introduced in line modeling by Gustavsen and Semlyen [14, 15]
and the program that implements this technique is available on the internet [7].
Vector Fitting technique consists of approximating a frequency response (magnitude and phase) in
an iterative manner using a prescribed set of starting poles. To illustrate the method, consider the
approximating function:
f (s) ≈
N
X
cn
+ d + se
s
−
an
n=1
(3.18)
The residues cn and poles an are real or complex quantities. d and e are real quantities allowing
dierent degrees of accuracy. The objective of the method is to t these parameters in order to obtain a
3 This
method has been used in line models computed in mode domain. Thus, each mode has its own propagation delay
and it is possible to represent the modal propagation function as H̄(ω) = H̄ 0 (ω)e−jωτ , where τ is the minimum propagation
delay of the corresponding mode and H̄ 0 (ω) is a minimum-phase-shift function.
17
least squares approximation of f (s) over a given frequency interval. This is a non-linear problem since
the parameters an appear on the denominator. Vector Fitting solves it as a linear problem in two stages,
both with known poles.
The rst stage covers the poles identication. Consider a set of initial poles ān and multiply f (s) by
an unknown function σ(s). An additional equation is introduced for a rational approximation of σ(s),
resulting into a higher order description:
σ(s)f (s) ≈
σ(s) ≈
N
X
cn
+ d + se
s − ān
n=1
(3.19)
N
X
ĉn
+1
s
−
ān
n=1
(3.20)
Multiplying the second row of (3.20) by f (s) yields to:
!
!
N
N
X
X
ĉn
cn
+ d + se ≈
+ 1 f (s)
s − ān
s − ān
n=1
n=1
(3.21)
(3.22)
=⇒ (σf )f it (s) ≈ σf it (s)f (s)
Equation (3.21) shows a linear dependence as regards cn , ĉn , d and e, and it is solved as a linear
least squares problem. A rational approximation for f (s) can now be obtained. This becomes evident
by writing:
QN +1
(σf )f it (s) =
From (3.22):
QN
(s − zn )
e Qn=1
N
n=1 (s − ān )
n=1
σf it (s) = QN
(s − ẑn )
n=1 (s
− ān )
QN +1
(s − zn )
(σf )f it (s)
= e Qn=1
f (s) ≈
N
σf it (s)
n=1 (s − ẑn )
(3.23)
(3.24)
That is, the poles of f (s) are an approximation of the zeros of σ(s). Therefore, by computing the
zeros of σ(s), one gets a good set of poles to t f (s).
The second stage involves computing the residues of f (s). Using the zeros of σ(s) as the new poles,
equation (3.18) becomes linear in terms of cn , d and e, and is solved as a least squares problem.
In order to achieve a good approximation of f (s), it is necessary to repeat these two stages iteratively,
using the computed poles as new starting poles until an acceptable overall error is achieved. Vector Fit-
ting as been improved [5, 6] in order to accelerate convergence of the method.
All the poles are forced to be stable by inverting the sign of their real part when needed. For a fast
convergence, the initial poles should be well distributed over the frequency range of interest.
Instead of approximating a single function, Vector Fitting may be applied to an array of functions,
using the same set of poles for all. This can be very useful in line modeling, for example, for a column-wise
approximation of the characteristic admittance matrix.
18
3.2.3.4
Asymptotic Fitting
versus
Vector Fitting
For real-time simulations, it is important to limit the order of the line model. With Vector Fitting this is
ensured a priori, by dening an initial set of poles for the approximating functions. This can't be done
with Asymptotic Fitting, where the number of poles is dened by the approximating routine.
For the same order of approximation, Vector Fitting generally gives more accurate results. First,
because it ts the real and imaginary part of the original function, and not just its magnitude function
like Asymptotic Fitting 4 . Second, Vector Fitting is not constraint to real poles, thus making it a more
exible technique than Asymptotic Fitting.
Finally, by Vector Fitting it is possible to t a set of functions with the same poles. This can be particularly useful, for example, for a low order tting of the characteristic admittance of a transmission line.
Therefore, Vector Fitting is generally the most indicated technique to use in transmission line modeling for the possibility of predening the order of the model and to achieve a more accurate representation
with lower order approximations.
3.3
3.3.1
Transmission line models provided by the
EMTP-RV 2.3
CP Line constant parameters line model
This line model is based on the work by Dommel [10].
The CP Line model is based on modal analysis and each of the n line modes is characterized in terms
of the corresponding characteristic admittance Ȳc and propagation function H̄ , which are determined
through a real constant transformation matrix.
The model approximates the line as an ideal lossless line, that is, with R = 0 and G = 0. The
inductance L is considered constant and evaluated at the same frequency used for the transformation
matrix. Therefore, each modal function simply becomes:
r
C
Ȳc =
L
H̄ = e−jωτ
(3.25)
(3.26)
√
where C , L and τ = d LC are the capacitance, inductance and propagation delay of the corresponding
mode. Each mode is studied as a single-phase line by using equations:
r
r
C
C
vk (t) − ik (t) =
vm (t − τ ) − im (t − τ )
L
L
r
r
C
C
vm (t) + im (t) =
vk (t − τ ) + ik (t − τ )
L
L
(3.27)
(3.28)
technique of Asymptotic Fitting is based on the assumption that the original function is a minimum phase shift
function, which is generally just an approximation.
4 The
19
where all voltages, currents and parameters correspond to a specic mode. The electric quantities computed from these equations must then be converted into phase domain at each simulation step, by using
the constant real transformation matrix.
This model can include the eect of losses in the form of a constant resistance R. To do so, the ideal
lossless line with distributed parameters is divided in two segments of half the length (that is, half the
propagation delay). The resistance R is then inserted in the form of a lumped parameter in discrete
positions: R/4 at the terminals and R/2 between the two segments. This is illustrated in gure 3.3.
Figure 3.3: Inclusion of losses in a CP Line Model, in the form of lumped resistances.
3.3.2
FD Line frequency dependent line model
The FD Line model is based on the work by J. Marti in [2].
This model is based on modal analysis and characterizes each of the n line modes through the corresponding characteristic impedance Z̄c and propagation function H̄ . These functions are computed
from the line parameters in phase domain, by using a real constant transformation matrix. The line
parameters R and L are considered frequency dependent. A non-zero shunt conductance is included on
the admittance matrix Ȳ of the line (default value 0.2 × 10−9 S/km).
For each line mode, the characteristic impedance and propagation function are approximated by
Asymptotic Fitting [2] in the s-domain, as:
H̄(s) ≈
Nz
X
kx
s
−
px
x=1
!
Nh
X
ky
e−sτmin
s
−
p
y
y=1
Z̄c (s) ≈ k0 +
(3.29)
(3.30)
where Nz and Nh are the number of poles used to approximate the corresponding modal functions and
τmin is the minimum propagation delay of the mode.
3.3.3
WB Line wide-band line model
This model is based on the Universal Line Model, presented in a work by Gustavsen et al. [3].
The WB Line model describes the line in phase domain through matrices Ȳc and H̄. The line parameters R and L are considered as frequency dependent. A non-zero shunt conductance is included on
20
the admittance matrix Ȳ of the line (default value 0.2 × 10−9 S/km).
The admittance matrix Ȳc (ω) is tted column-by-column by using Vector Fitting [4]. The elements
of the propagation matrix H̄(ω) are all approximated with the same poles and delays, dened by the
approximated modes. These approximations are described in equations (3.31) and (3.32), where:
Nj
X
kx
jω − px
x=1
!
Nk
n
X
X
cmkij
H̄ij (ω) ≈
e−jωτk
jω
−
p
mk
m=1
Ȳcij (ω) ≈ k0 +
(3.31)
(3.32)
k=1
where:
• n is the number of line modes,
• Nj is the number of poles used to t the elements of the j th column of Ȳc ,
• Nk is the number of poles used to t the k th modal propagation function and
• τk is the minimum propagation delay associated to the k th mode.
The poles of Ȳc are generally real, whereas those of H̄ may be real or complex. The modal poles and
delays that approximate H̄ are obtained by applying Vector Fitting to each modal propagation function.
The residues cmkij are computed from a set of samples of H̄, by solving a linear least squares problem.
3.4
Model testing and comparison
This section presents a set of tests that analyze and compare the line models provided by the EMTP-RV
2.3, in terms of eciency and accuracy. The tests contemplate frequency and time domain simulations
including the line described in appendix, represented by the models. For this line, the optimal frequency
computed by the EMTP-RV to evaluate the constant real transformation matrix and line parameters is
1.0956 kHz, for all line models.
3.4.1
Frequency response short-circuited and open-ended line
This test is specially adequate to infer the accuracy of the line models: given the simplicity of the
boundary conditions, it is possible to derive the analytical expression of the frequency response of the
line, and use it as a reference to analyze model accuracy. The two cases - short-circuited and open-ended
line, are illustrated in gure 3.4, where the source feeding the line is a three-phase ideal symmetrical
source of 1VRMS value.
21
Figure 3.4: Circuits used to study the short-circuit and open-end frequency responses according to the
line models available on the EMTP-RV 2.3
The analytical formulas of the frequency response of a line in short-circuit and open-end are:
Īk_short = Ȳc I − H̄2
V̄m_open
−1
(3.33)
I + H̄2 V̄s
−1
= I − H̄2
H̄ 2V̄s
(3.34)
where I is the identity matrix. Īk_short is an array with the phase currents at the sending end of the
short-circuited line. V̄m_open represents the phase voltages at the receiving end of the line in open-end.
Based on equations (3.33) and (3.34) and on line data, it is possible to compute the exact value
of these frequency responses. Tables 3.1 and 3.2 show the exact values of the current/voltage and the
deviation of the values computed according to the three line models. For simplicity, only the frequencies
0.1 Hz, 50 Hz, 1 kHz and 100 kHz are presented.
Table 3.1: Analytical short-circuit frequency response and approximating errors according to the
EMTP-RV 2.3 models, in terms of the magnitude of the current at the sending end of phase 1
Model
0.1 Hz
50 Hz
1 kHz
100 kHz
Analytical
0.059407
0.022864
0.0017535
0.0048819
CP Line
−38.478%
−11.304%
+3.0265%
−28.848%
FD Line
+16.423%
+0.5536%
+3.1092%
−0.3263%
WB Line
+11.668%
−0.6456%
+3.1400%
+0.0139%
Observing table 3.1, which concerns short-circuit results, it is evident that the CP Line is generally
the least accurate model. An exception is veried at 1 kHz, where the model provides the best approximation of the line response. Notice that this frequency is very close to that chosen by EMTP-RV to
compute the transformation matrix for all models and to process the propagation parameters of the CP
Line.
As regards the open-end results in table 3.2, the errors of the frequency response according to all
22
models are smaller than the ones concerning the short-circuit scan. The CP Line is again the least
accurate model, with the exception of very low frequencies.
Table 3.2: Analytical open-end frequency response and approximating errors according to the
EMTP-RV 2.3 models, in terms of the magnitude of the voltage at the receiving end of phase 1
Model
0.1 Hz
50 Hz
1 kHz
100 kHz
Analytical
1
1.006
1.8731
0.96315
CP Line
0%
+0.0696%
−0.8494%
+35.393%
FD Line
−0.0002%
−0.0020%
−0.7763%
−2.0098%
WB Line
−0.0001%
+0.0129%
−0.9332%
−2.1639%
Concerning the FD Line and the WB Line, both give very similar and acceptable results. Actually,
the FD Line is more accurate for many frequency points, both for short-circuit and for open-end conditions. However, in these cases, the dierence between the two models is almost negligible.
A comparison of the EMTP-RV models performance is found in gures 3.5 and 3.6. For the shortcircuit response, the plots represent the sending end current of phase 1 according to the three models,
whereas the open-end response is analyzed regarding the phase 1 receiving end voltage.
These plots show the great dierence between the results generated with the CP Line in relation to
the other two models. Regarding the short-circuit scan, there is a great dierence of results both for
low and high frequencies. For the open-end scan, the higher frequencies are more critical. Concerning
the FD Line and the WB Line, the results seem coincident for most of the frequencies, except for the
short-circuit scan, where the results of the two models diverge for low frequencies.
Figure 3.5: Short-circuit frequency response according to the EMTP-RV 2.3 line models, in terms of
the current at the sending end of phase 1 (CP Line thin, FD Line dotted, WB Line bold).
23
Figure 3.6: Open-end frequency response according to the EMTP-RV 2.3 line models, in terms of the
voltage at the receiving end of phase 1 (CP Line thin, FD Line dotted, WB Line bold).
3.4.2
Line energization and single-phase short-circuit
This test shows how the models represent the behavior of the line under two common transient conditions:
line energization and single-phase short-circuit. Figure 3.7 illustrates the circuit used, where a threephase line is connected through ideal switches to a three-phase ideal symmetrical source of 1V peak
voltage and 50 Hz. The three phases are connected to the source simultaneously at t = 20 milliseconds.
After the transient of line energization, a short-circuit occurs at t = 180 milliseconds in phase 3 of the
line. The voltage at the receiving end of phase 1, vm1 (t), is observed.
Figure 3.7: Circuit used to study the time response to energization followed by single-phase
short-circuit, according to the line models available on the EMTP-RV 2.3.
The results of this test are plotted in gures 3.8 and 3.9, which represent only the CP Line and the
WB Line approximations. This is done for simplicity, given the FD Line generates practically the same
results as the WB Line. As regards the CP Line, the results are noticeably dierent for both transient
24
conditions. Particularly, this model doesn't manage to represent the distortion introduced by the line
in the energization transient, as perceived by the square waves in gure 3.8. Furthermore, the CP Line
also shows a stronger attenuation. On the other hand, the WB Line shows more realistic results, with
a smoother wave, representing the distortion phenomena.
Concerning the transient induced by the short-circuit on phase 3, illustrated in gure 3.9, the dierence between the results of the two models basically resumes to a delay and weaker attenuation in the
response generated by the CP Line.
Figure 3.8: Response to line energization at t = 20 ms according to the CP Line thin, and to the WB
Line bold, in terms of the voltage at the receiving end of phase 1.
Figure 3.9: Response to a short-circuit at the receiving end of phase 3 at t = 180 ms according to CP
Line thin, and to the WB Line bold, in terms of the voltage at the receiving end of phase 1.
An important note is that line energization directly aects all phases, whereas the short-circuit af25
fects phase 1 only through induction. Therefore, the results of the energization transient reect mainly
the accuracy of the models in approximating the diagonal elements of the line matrices (H̄ and Ȳc or
Z̄c ). As for the short-circuit transient only the o-diagonal terms of those matrices are taken into account.
This test shows that for common transient conditions, as the ones observed, both FD Line and WB
Line provide accurate and similar approximating responses. On the other hand, the CP Line provides
a poor representation of the line behavior and, therefore, should be used with care and preferably only
for didactic purposes.
3.4.3
Current induced by phase coupling
The last carried out test is one in which all models show signicantly dierent results. Consider gure
3.10, where the transmission line is short-circuited at all terminals except at the sending end of phase 1,
which is connected to a DC voltage source of 1V, at t = 1 millisecond, by an ideal switch. The current
induced at the sending end of phase 3, ik3 (t), is observed.
Figure 3.10: Circuit used to study the phenomena of phase coupling according to the line models
available on the EMTP-RV 2.3, in terms of the current induced in phase 3 by energization of phase 1.
This test results are plotted in gures 3.11 and 3.12, for the rst 20 miliseconds and 1 second of
the simulation, respectively. Figure 3.11 plots the very initial transient on the current of phase 3. It is
evident that the responses according to the dierent line models become considerably dierent as time
goes by. Again, the results of the CP Line disagree with the responses computed according to the other
two models, which present similar results for the initial period of the transient. Consider, however, a
longer period of the same test, as plotted in gure 3.12. Now, the dierence between the models results
is far evident, given that each model presents a dierent response. However, only the WB Line shows
results physically acceptable.
In fact, due to the source, the circuit will reach a DC steady-state, extinguishing coupling phenomena.
Since phase 3 is grounded, and due to resistivity of line and ground, the current on this phase goes to
zero. The WB Line is the only with a satisfying result in these conditions the current declines to zero.
The inaccuracy in the models response is due to the problem of tting the o-diagonal elements of the
26
line functions, which are related to the coupling between the line phases. Generally, these elements are
very small, compared to the respective diagonal elements. Therefore, if the tting process is based on
absolute errors, their approximation may be less accurate.
This is a critical test, as perceived by the diverse responses obtained from the various line models.
Nevertheless, it allows to verify that the WB Line is more accurate than the FD Line, an thus, can be
use in a wider variety of transient conditions and still provide physically acceptable results.
Figure 3.11: Current induced by energization of phase 1 time evolution of the current at the sending
end of phase 3 during the rst 20 miliseconds of the transient according to the EMTP-RV 2.3 line
models (CP Line thin, FD Line dashed, WB Line bold).
Figure 3.12: Current induced by energization of phase 1 time evolution of the current at the sending
end of phase 3 during the rst second of the transient according to the EMTP-RV 2.3 line models (CP
Line thin, FD Line dashed, WB Line bold).
27
3.4.4
Model eciency
The model eciency is associated both to the processing time required for each step of the simulation
and to the accuracy of the generated results. These two aspects are conditioned by the complexity of the
model. CP Line is by far the fastest model, but this is only possible due to using constant line parameters.
The other two EMTP-RV models consider the frequency dependence of the line functions, which are
tted by rational functions of frequency. The line functions are thus more accurately represented for a
wide range of frequencies, but the eort to compute the convolution integrals on line equations depends
on the order of those approximations. Table 3.3 summarizes the order (that is, the number of poles)
of the approximating functions generated by EMTP-RV for the FD Line. As regards the WB Line,
the same set of poles is used to t all the columns of Ȳc (ω). The order of the approximating functions
generated by EMTP-RV are summarized in table 3.4.
Table 3.3: Number poles used by the FD Line, available on the EMTP-RV 2.3,
in the approximation of the propagation parameters of a line
Function
Mode 1
Mode 2
Mode 3
Total
Z̄c (ω)
17
15
18
51
H̄(ω)
23
23
23
69
Table 3.4: Number of poles used by the WB Line, available on the EMTP-RV 2.3,
in the approximation of the propagation parameters of a line
Function
Mode 1
Mode 2
Mode 3
Total
Ȳc (ω)
11
H̄(ω)
6
7
8
21
The WB Line uses a total of 11 + 21 = 32 poles compared to the 51 + 69 = 120 poles used by the FD
Line. Furthermore, the test results of this chapter have showed that the WB Line is the most accurate
of the line models provided by the EMTP-RV. Therefore, the WB Line presents a great improvement in
eciency, by obtaining better results with less resources. The eciency of the WB Line is reinforced by
the fact of being a phase-domain model there is no need to convert from phase to modal quantities,
and vice-versa, at each simulation step.
3.5
Conclusions
The EMTP-RV 2.3 is a specialized software particularly useful to study transmission systems, including
insulation coordination and switching design. The transmission line models available in the EMTP-RV
provide a summary of the evolution from the constant parameters model, to the more complex frequency
dependent models, that approximate the line functions by analytical expressions in the form of rational
functions of frequency.
28
Generally, transmission line models treat the line as a two-port system, that is, the study of the line
behavior is described in terms of the currents and voltages at the two line terminals. Nevertheless, there
are several aspects that distinguish the line models:
• They may represent the line behavior in terms of incident and reected current waves, which
corresponds to using the line functions H̄ and Ȳc , or in terms of incident and reected voltage
waves, by using H̄ and Z̄c ;
• Another dierentiating aspect of line models is the account for the frequency dependence of line
parameters. Frequency dependent models approximate, within a frequency range of interest, each
line function by a sum of rational terms, which transforms to time domain as a sum of exponential terms. The time domain equations contain convolution integrals which may be computed
recursively [1];
• The line models may also dier on the domain of solution, whether they described the line in terms
of modal or phase quantities. Given the use of a constant real transformation matrix, the modal
domain approach will be based on approximated modes, which represents a source of inaccuracy in
relation to phase domain models. Furthermore, due to the interaction of the line with the outside
system (which is modeled in phase domain) it is necessary to convert the computed modal variables
to phase domain at each simulation step, increasing the model processing time.
In order to characterize the model eciency, it is necessary to analyze how it is implemented, and
specically, which computation techniques are used. Section 3.2.2 gives an introduction to the technique
of recursive convolutions, which allows a very ecient computation of the integrals in the line equations.
As concerns the rational approximation of the line functions, Asymptotic Fitting and Vector Fitting
are described in section 3.2.3 as two alternative techniques. Vector Fitting approximates both real and
imaginary part of the original function, and it allows real or complex conjugate pairs of poles. On the
other hand, Asymptotic Fitting approximates only the magnitude of the original function, which means
considering it as a minimum phase shift function generally, an approximation. Furthermore, Asymp-
totic Fitting allows only for real poles. Generally, Vector Fitting achieves more accurate results with
lower order approximations than Asymptotic Fitting. Furthermore, Vector Fitting allows to pre-establish
the number of approximating poles, and is therefore the preferable technique to be used by line models
in-line with the target of real-time simulations.
After giving an overview on some line modeling issues, section 3.3 presents a description of the line
models available on the EMTP-RV 2.3. The CP Line is a modal domain model which approximates
the line parameters as constant. The model represents the losses on the line as lumped resistances
inserted at particular points of the line. The FD Line is a modal domain frequency dependent line
model, which ts the line functions using the technique of Asymptotic Fitting, thus originating robust
high order approximations. The WB Line is another frequency dependent model. Though this is a
phase domain model, it approximates the elements of the propagation matrix H̄ using the poles and de29
lays dened by the modes, and computed by applying Vector Fitting to the modal propagation functions.
Finally, section 3.4 tests and compares the line models CP Line, FD Line and WB Line through a set
of frequency and time domain simulations. The rst of the tests regards the short-circuit and open-end
frequency responses of the line represented in appendix, according to the three models. The CP Line
shows high relative errors for both conditions, whereas the FD Line and the WB Line provide acceptable and similar results in any conditions. The second test simulates line energization and single-phase
short-circuit conditions. Again, the FD Line and the WB Line generate practically the same results,
whereas the CP Line provides unrealistic time responses, namely in what concerns representing distortion phenomena. The last test concerns the study of coupling phenomena between line phases. For this
test, all line models provide dierent responses. Nevertheless, only the WB Line generates physically
acceptable results, as regards approximating the steady-state condition of the line.
The described tests show that the CP Line should be avoided for its inaccurate results in most
of the cases observed. As regards the FD Line, it provides very accurate results for several typical
transient conditions. However, it should be avoided for simulating more complex transient conditions,
specially those concerning coupling phenomena. The WB Line proved to be the most accurate of the
tested models, even in approximating the coupling phenomena between the line phases. Furthermore,
the WB Line uses lower order approximations than the FD Line, and is therefore the most ecient of
the EMTP-RV line models.
30
Chapter 4
Wide-band model for real-time
simulations 4.1
RT_WB Line
Introduction
The goal of this work is to establish adequate numerical techniques for approximating the propagation
parameters for transmission line modeling, allowing real-time simulations. This requires an ecient use
of reduced modeling resources. In order to ensure additional accuracy, it is necessary to introduce some
optimization procedures. The resulting model is called RT_WB Line, as it is a reformulation of the
EMTP-RV model WB Line, in-line with the real-time simulation target.
The real-time requirement means that during the digital simulation, each time step should have a
processing time never greater than the period represented. This possibility depends on the order of the
model and on the specic computer processing capacity a faster computer can perform real-time simulations with higher order models. To accomplish its function in any conditions, that is, in any computer,
it is assumed that the correct order of the model for real-time performance, that is, the number of poles
to use in the approximation of the line functions, is pre-dened.
In order to test the developed model, it must be able to interface with the EMTP-RV. This is ensured
by writing the model data into an output le using the same template of the WB Line. This allows
testing the applications of the developed model in the EMTP-RV environment as if they had been computed by the software itself.
The following text covers all the steps taken for the denition and implementation of the RT_WB
Line : section 4.2 starts by introducing the theoretical formulation of the model; then, section 4.3 presents
the optimization procedures introduced to ensure increased accuracy with reduced order approximations;
nally, section 4.4 presents the conceptual structure of the routine developed to compute the applications
of the RT_WB Line, with a summary of its subroutines, main variables, input and output data.
31
4.2
Model formulation
The developed line model, called RT_WB Line, uses an approach similar to the WB Line provided by
the EMTP-RV 2.3. It is a phase domain model based on the rational approximations of the propagation
matrix H̄ and characteristic admittance Ȳc , which are computed in frequency domain using the line
characteristic parameters:
√
H̄ = e(−
Ȳc = Z̄−1
Ȳ Z̄)d
(4.1)
p
(4.2)
Z̄Ȳ
The elements of the characteristic admittance matrix are all tted by the same set of poles, through
Vector Fitting. For the ij th element of Ȳc :
Ȳcij (ω) ≈ y0ij +
Ny
X
ynij
jω
− pn
n=1
(4.3)
where Ny is the number of poles used to t the characteristic admittance matrix. Generally, all the
tting parameters in (4.3) are real quantities.
The elements of the propagation matrix H̄ are tted by the poles and delays dened by the approximated modes, obtained through a constant real transformation matrix evaluated at 1 kHz1 . For the ij th
element of H̄:
H̄ij (ω) ≈
n
X
k=1
Nk
X
cmkij
jω
− pmk
m=1
!
e−jωτk
(4.4)
where the poles pmk and residues cmkij are real quantities or come as complex conjugate pairs.
The rst step to t the elements of the propagation matrix is to gather the modal data, that is, the
delays and poles dened by the approximation of the modal propagation functions. Section 4.3.1 presents
the method used to compute the propagation delays τk . After extracting a constant propagation delay
to each of the modes, the correspondent modal propagation function becomes approximately a minimum
phase shift function, being approximated through Vector Fitting, so as to obtain the modal poles pmk .
After obtaining the modal poles and delays, these are used to compute the residues of each element
of H̄. To do so, it is necessary to write (4.4) for several frequencies, so as to obtain an overdetermined
linear matrix equation of the form A X = B, where X contains the unknown residues. Each row in A
and B corresponds to a frequency point, and each column in X and B corresponds to an element of H̄.
The equation A X = B is solved as a linear least squares problem.
1 The
modal poles dier slightly from the accurate ones. However, this has little impact on the nal approximation (4.4)
since a small displacement of the poles will be compensated by a small displacement of the corresponding residues.
32
4.3
Optimized tting of the propagation function
The RT_WB Line model is based on the rational approximations of the line matrices H̄ and Ȳc . The
elements of Ȳc are generally smooth functions of frequency and can easily be tted by low order functions. The tting of H̄ is a more challenging task, due to the contribution of the various line modes, all
with a dierent frequency dependent propagation delay. Furthermore, it must respect the limited order
of the model, pre-dened by real-time requirements.
This section presents a set of optimization procedures used on the tting process of H̄ in order to
ensure increased accuracy of approximation for the pre-dened model order.
4.3.1
Optimal modal delay identication
This optimization process regards the computation of the modal propagation delays, necessary for the
process of approximating the phase domain propagation matrix H̄. Consider the propagation function
of the k th line mode:
(4.5)
H̄ k (ω) = e−(αk (ω)d+jωτk (ω))
where τk (ω) is the propagation delay of the k th mode. Each function H̄ k (ω) may be approximated
0
by a rational function and a constant time delay factor e−jωτk :
k
0k
H̄ (ω) = H̄ (ω)e
−jωτk0
≈
Nk
X
cn
jω − pn
n=1
!
0
e−jωτk
(4.6)
where Nk is the number of poles used to t the k th mode. The extraction of a constant propagation
delay ensures a more accurate tting of H̄ k (ω), using the same number of poles.
The constant delay τk0 may be computed so that H̄ 0k (ω) becomes approximately a minimum phase
shift function. According to [12], this is done through:
1 π d ln|H̄ k (ω1 )| τk0 ≈ τk (ω) +
ω1 =ω
ω 2 d ln ω1
(4.7)
computed for a frequency ω such that |H̄ k (ω)| = 0.1.
However, according to [12], the computed delay may not correspond to the most accurate tting of
H̄ (ω). Therefore, the process of optimization suggested regards nding the modal delays leading to the
k
most accurate rational approximation obtained with (4.6). Several tests involving dierent lines have
showed that a good estimation for τk0 can be found in the interval [ 0.9 τk0 ; 1.1 τk0 ], where τk0 is given by
(4.7). Tests have further showed that generally it is enough to search with an iteration of 1% of the base
modal delay.
Though the accuracy in the approximating modes is not directly related to the phase domain results
[12], this optimization routine has a positive impact on the accuracy of the approximating propagation
matrix H̄, in phase domain. This is conrmed in the following numerical example.
33
Numerical Example: Consider the line described in appendix. Table 4.1 shows the average error
of the approximating propagation functions, in modal and phase domain, using a simple modal delay
computed through (4.7) or using an optimized modal delay.
Table 4.1: Eect of using optimized modal delays average error of approximating propagation
functions according to dierent order applications of the RT_WB Line, in mode and phase domain.
Modal Poles
Modal Delay
Mode domain error
Phase domain error
1
Not optimized
3.8524 × 10−2
1.7446 × 10−2
Optimized
3.5934 × 10−2
1.5711 × 10−2
Not optimized
1.0959 × 10−3
3.7699 × 10−4
Optimized
1.0875 × 10−3
3.6292 × 10−4
Not Optimized
1.1505 × 10−3
4.2359 × 10−4
Optimized
9.6221 × 10−4
3.3057 × 10−4
9
13
The table shows that, for certain orders of the approximating functions, the use of optimized modal
delays leads to increased accuracy both in modal and phase domain, when compared to the approximating
functions obtained by simply using the lossless modal delays. Nevertheless, it must be noted that the
improvements introduced by this process vary with the particular line and with the order of the model.
4.3.2
Optimal modal poles assignment
The RT_WB Line model approximates the propagation matrix of a line using the poles and delays dened by the modes, as expressed in (4.4). In order to respect the pre-dened order of the model, the sum
of the poles assigned to each mode, Nk , must be equal to the maximum number of poles allowed for H̄,
P
that is,
Nk = Nmax . The simplest would be to assign an equal number of poles to each mode. However, practical tests have showed that this choice generally does not lead to the most accurate tting of H̄.
Therefore, it is advantageous to optimize the number of poles assigned to each modal propagation
function. This is done by trying all the possible distributions of the available number of poles among
the modes, with the requirement that the total number of poles must respect the pre-dened order of
the model. The following numerical example illustrates and justies this procedure.
Numerical example: Consider the line described in appendix. Table 4.2 shows the average error of
the approximating propagation matrix considering an equal or optimized distribution of modal poles, for
several applications of the RT_WB Line, which dier the order of the approximations.
As table 4.2 shows, the optimized assignment of modal poles has a positive impact on the accuracy
of the approximating propagation matrix, with a reduction of the approximation error of at least 6 % in
relation to the equal distribution of modal poles, for all the tested conditions.
34
Table 4.2: Average error of the approximation of H̄, according to dierent order applications of the
RT_WB Line. Use of equal (E) or optimized (O) distribution of the modal poles
Total poles
Distribution of poles
Phase domain error
6
E = [2 − 2 − 2]
7.5693 × 10−3
O = [2 − 3 − 1]
5.4463 × 10−3
9
E = [3 − 3 − 3]
3.8117 × 10
−3
O = [4 − 4 − 1]
3.5990 × 10−3
12
E = [4 − 4 − 4]
2.5625 × 10
−3
O = [5 − 0 − 7]
1.1446 × 10−3
15
E = [5 − 5 − 5]
1.4659 × 10
−3
O = [5 − 1 − 9]
8.3372 × 10−4
18
E = [6 − 6 − 6]
7.9743 × 10
−4
O = [8 − 1 − 9]
6.7828 × 10−4
Dierence (%)
−28%
−6%
−55%
−43%
−15%
Another interesting aspect showed in table 4.2 is that, for a low number of approximating poles, the
optimized distribution tends to assign more poles to modes 1 and 2, whereas for a higher approximating
order, modes 1 and 3 are preferred. Nevertheless, only in exceptional cases one mode is neglected, being
assigned zero tting poles, as showed in table 4.2 for a total of 12 poles. Therefore, there is not an
explicit tendency of optimal pole distribution that allows to dene a single strategy that works both for
low and high orders of approximation. For example, if one mode is not very signicant for the phase
tting, how can it be dened whether it should be assign 1 or 0 poles, without checking the phase error
obtained in the two cases?
Therefore, in order to achieve the optimal approximation of the propagation matrix H̄, the optimization procedure tries all the possible assignements of modal poles. This is not the most ecient process,
but it certainly reaches the most accurate result, based on the average error of approximating H̄.
4.4
Computer program
This section presents the structure of the program aimed at transmission line modeling for real-time
simulation. It is a MATLAB routine which is original, except for the use of an external function called
vectt3.m. This is a free access MATLAB routine available on the Internet [7], which computes a rational
expression to approximate a function of frequency using the technique of Vector Fitting [4, 5, 6].
The RT_WB Line program is formed by several subroutines which represent the main steps of a
transmission line modeling process. The structure of the program, as well as its input and output, are
illustrated in gure 4.1. The expected input to the program are the location of the le containing the
information about the specic transmission line to be modeled and the prescribed order of the model,
35
that is, the number of poles allowed to t the characteristic admittance Ȳc and the number of poles to
t the propagation matrix H̄ of the line.
Figure 4.1: General structure of the program developed to compute applications of the RT_WB Line
model in-line with the real-time simulation target, with respective input and output data.
Sections 4.4.1 to 4.4.5 provide a description of the main program and each of its subroutines, namely
its objective and expected input and output.
4.4.1
Main program
• Objective:
Computing an application of the RT_WB Line using pre-dened orders for approximating
the line propagation parameters Ȳc and H̄.
• Input Data:
Line_data_rv.lig location of the EMTP-RV le containing the number and value of the
frequency samples to consider, the line parameters per unit length Z̄ and Ȳ computed for
those samples, and the constant real transformation matrix.
Ny order of the approximating characteristic admittance matrix of the line, that is, the total
number of poles used to t the matrix Ȳc .
Nh order of the approximating propagation matrix of the line, that is, the total number of
poles used to t the matrix H̄.
36
• Output Data:
model.dat location of the generated le, containing the description of the RT_WB Line
application computed. The template of this le follows that of an EMTP-RV WB Line
model.
4.4.2
Propagation parameters computation
• Objective:
The computation of the frequency samples of the propagation parameters of the line to model,
namely, the characteristic admittance Ȳc , the propagation matrix H̄, the modal propagation
functions H̄ k and the corresponding propagation delays and τ k , for k = 1, ..., n (n is the
number of line modes).
The computation of these functions is based on the longitudinal impedance matrix Z̄, on the
transversal admittance matrix Ȳ, as well as on the transformation matrix, as explained on
chapter 2, concerning line theory. These matrices are provided by the input le Line_data_rv.lig.
• Input Data:
Line_data_rv.lig location of the EMTP-RV le containing the number and value of the
frequency samples to consider, the line parameters per unit length Z̄ and Ȳ computed for
those samples, and the constant real transformation matrix.
• Output Data:
f the vector containing the set of sampling frequencies considered.
Ȳc the characteristic admittance matrix of the line, evaluated at all the sampling frequencies.
H̄ the propagation matrix of the line, evaluated at all the sampling frequencies.
H̄ k , for k = 1, · · · , n the set of n modal propagation functions of the line, evaluated at all
the sampling frequencies.
τ k , for k = 1, · · · , n the set of n modal propagation delays of the line, computed through
equation (4.7).
4.4.3
Ȳc tting
• Objective:
The computation of the approximating rational functions of frequency that t the elements
of the characteristic admittance matrix of the line using a pre-dened number of poles.
As explained earlier on this chapter the elements of Ȳc are all tted together by the same set
of poles, using the technique of Vector Fitting. The tting parameters are the poles used to
t the whole matrix, and for each element of the matrix, the set of residues corresponding to
those poles (see equation (4.3)).
37
• Input Data:
f the set of sampling frequencies considered.
Ny order of the rational approximation of the elements of the characteristic admittance
matrix of the line, that is, the total number of poles used to t the matrix Ȳc .
Ȳc the characteristic admittance matrix of the line, evaluated at all the sampling frequencies.
• Output Data:
pn , for n = 1, · · · , Ny the set of poles used to t the matrix Ȳc .
y0ij the value used to t the ij th elements of Ȳc when s → ∞.
ynij for n = 1, · · · , Ny the residues used to t each of the ij th elements of Ȳc .
These tting parameters are all included in a structure called Yc_data .
4.4.4
H̄ tting
• Objective:
The computation of approximating rational functions of frequency that t the elements of the
propagation matrix of the line using a pre-dened number of poles.
The elements of H̄ are tted by the poles and delays dened by the modes through the solution
of a least squares problem (see equation (4.4)). The modal delays are optimized so as to allow
the most accurate modal representation, as described in section 4.3.1. After extracting the
optimal modal delay, each modal propagation function is approximated by Vector Fitting,
dening the modal poles. The number of poles assigned to each mode is that leading to the
most accurate tting of the propagation matrix H̄, as described in section 4.3.2.
• Input Data:
f the set of sampling frequencies considered.
Nh order of the rational approximation of the elements of the propagation matrix of the
line, that is, the total number of poles used to t the matrix H̄.
H̄ the characteristic admittance matrix of the line, evaluated for all the sampling frequencies.
H̄ k , for k = 1, · · · , n the set of n modal propagation functions of the line, evaluated for all
the sampling frequencies.
τ k , for k = 1, · · · , n the set of n modal propagation delays of the line, computed through
equation (4.7), which are a base for the corresponding optimization process.
• Output Data:
(τ k )opt for k = 1, · · · , n the optimized propagation delay of each modal propagation function;
38
pmk for m = 1, · · · , Nk the set of poles used to t the kth modal propagation function, where
Nk is the optimal number of poles assigned to the k th mode.
cmkij the residues corresponding to the poles pmk used to t each of the ij th elements of
the propagation matrix H̄.
These tting parameters are all included in a structure called H_data .
4.4.5
Output generation
• Objective:
Write the parameters of the generated application of the RT_WB Line model into a le,
using the same template of the WB Line model, generated by the EMTP-RV.
The tting parameters to write are those included in the structures Yc_data and H_data .
• Input Data:
Yc_data the tting parameters used to approximate the elements of the characteristic admittance matrix of the line to model.
H_data the tting parameters used to approximate the elements of the propagation matrix
of the line to model.
• Output Data:
model.dat location of the generated le, containing the description of the developed model
computed, following the template of a WB Line le, as computed by the EMTP-RV.
4.5
Conclusions
The goal of this work is to establish adequate numerical techniques for approximating the propagation
parameters for transmission line modeling, allowing real-time simulations. This requires an ecient use
of reduced modeling resources. In order to ensure additional accuracy, it is necessary to introduce some
optimization procedures. The resulting model is called RT_WB Line, as it is a reformulation of the
EMTP-RV 2.3 model WB Line, in-line with the real-time simulation target.
The RT_WB Line is a phase domain model based on the rational approximations of the characteristic admittance and propagation matrix. The elements of the characteristic admittance matrix are all
tted by the same set of poles, whereas the elements of the propagation matrix are tted by the poles
and delays dened by the approximated modes.
The elements of Ȳc are generally smooth functions of frequency and can easily be tted by low order
functions. The tting of H̄ is a more challenging task, due to the contribution of the various line modes,
39
each with dierent frequency dependent propagation delays. Furthermore, it must respect a limited
order of the model pre-dened by real-time requirements.
In order to ensure increased accuracy of the approximating propagation function for the pre-dened
order of the model, the RT_WB Line takes a set of optimization procedures regarding the computation
of the modal delays and the assignment of the poles to the line modes. The rst optimization process
is based on the fact that the extraction of the lossless delays from the modal propagation functions may
not lead to the most accurate modal tting. Therefore, it is necessary to search within a given interval
around the lossless delay for an ideal value. The second optimization procedure concerns the number of
poles assigned to each mode tests have showed that, generally, assigning the same number of poles to
all modes does not lead to the most accurate approximation of the elements of H̄. The tests have also
showed that there is not an explicit logic that allows to dene a strategy to decide which modes should be
preferred and which should be neglected, or whether a mode with little inuence on the phase quantities
should be assign zero or one pole, in order to achieve the most accurate phase tting. Therefore, the
optimization process consists of searching within all the possible assignments of poles to the modes for
the one leading to the most accurate tting of H̄.
40
Chapter 5
RT_WB Line
5.1
model validation
Validation perspective
The purpose of the present chapter is the validation of the RT_WB Line, which is a reformulation of
the EMTP-RV model WB Line, in-line with the real-time simulation target. The formulation of the
developed model, as well as the optimizations introduced to ensure additional accuracy with low order
approximations, are presented in chapter 4.
The validation process consists of frequency and time domain simulations in the EMTP-RV 2.3
environment, using an application of the RT_WB Line, which performance is compared to that of the WB
Line, computed by the EMTP-RV and taken as a reference of accuracy. The order of the approximating
line functions used for the two models, that is, the number of poles used to t the propagation matrix
H̄ and the characteristic admittance matrix Ȳc , is presented in table 5.1. As already mentioned, the
elements of H̄ are tted by the delays and poles dened by the approximated modes, whereas the poles
used to t the characteristic admittance matrix Ȳc are the same for all of its elements.
Table 5.1: Number of poles used for the approximation of the propagation parameters of a line, for the
applications of tested models WB Line and RT_WB Line
Model
H̄ Mode 1
H̄ Mode 2
H̄ Mode 3
Ȳc
Total
WB Line
6
7
8
11
32
RT_WB Line
4
4
1
9
18
The application of the RT_WB Line assigns 9 poles to each line function. The distribution of modal
poles to t H̄ is optimized as described in section 4.3. The total number of poles used by this application is in-line with the examples in real-time line modeling literature, namely [8, 9]. It is demonstrated
throughout this chapter that it is possible to achieve very accurate simulation results even using this
order for the approximating line functions.
41
The next sections, dedicated to the tests, are organized in the following way: section 5.2 presents a
study of the frequency response in short-circuit and open-end conditions, according to the two models;
then, section 5.3 concerns two typical conditions in transmission line transients studies: line energization
and single-phase short-circuit; nally, section 5.4 refers a critical test concerning the current induced by
phase coupling, for which line models usually show substantially dierent results. Finally, section 5.5
presents a summary of the analysis of the tests results.
5.2
Frequency response short-circuited and open-ended line
This test evaluates the accuracy of the line models by comparing their frequency responses with the
expected behavior, according to analytical expressions computed with the exact line functions, which are
rewritten, respectively, as:
Īk_short = Ȳc I − H̄2
V̄m_open
−1
I + H̄2 V̄s
−1
= I − H̄2
H̄ 2V̄s
(5.1)
(5.2)
For the short-circuit condition, the quantity observed is the current at the sending end of the line.
In open-end, the line is studied in terms of the receiving end voltage. Figure 5.1 illustrates these two
situations. In both cases, the line is connected to a symmetrical sinusoidal voltage source of 1VRMS
voltage. Given the symmetry of the problem, it is sucient to analyze one phase of the line (phase 1
was chosen).
Figure 5.1: Circuits used to study the short-circuit and open-end frequency responses according to the
WB Line and RT_WB Line models.
Figure 5.2 plots the approximating short-circuit responses according to the WB Line and to the
RT_WB Line, as well as the expected value for those functions of frequency, according to expression
(5.1).
42
Figure 5.2: Short-circuit frequency response (0.1 Hz - 1 MHz) analytical (bold), WB Line (thin) and
RT_WB Line (dashed). Current at the sending end of phase 1.
The approximating frequency responses are specially innacurate for range of frequencies up to 50 Hz,
for which the developed model is particularly bad. However, for higher frequencies, including the range
from 100 Hz to 1 kHz (relevant for switching transients studies) the approximations of both models tend
to be very accurate.
Figure 5.3: Relative error of the short-circuit frequency response (0.1 Hz - 1 MHz) WB Line (bold)
and RT_WB Line (thin). Current at the sending end of phase 1.
43
Figure 5.4: Detailed relative error of the short-circuit frequency response (700 Hz - 10 kHz) WB Line
(bold) and RT_WB Line (thin). Current at the sending end of phase 1.
For a closer analysis of the models performance, gure 5.3 shows the relative error of the approximating short-circuit responses, according to the two models. It is now clear that the RT_WB Line
is inadequate for transients studies involving low frequencies, specially under the 50 Hz, for which the
approximating errors go over the 10 %. The WB Line is also not very adequate for this range of frequencies, though the correspondent errors are far lower. On the other hand, for higher frequencies, both
models provide accurate approximations, except when approximating the various peaks in the expected
frequency response.
Figure 5.4 shows a zoom of the relative errors of the approximating frequency responses from 700
Hz to 10 kHz, which includes the switching transients frequencies. For this range, the RT_WB Line is
more accurate than the WB Line, despite using lower order approximations.
The open-end response according to the developed model and to the WB Line, as well as its
expected value, are plotted in gure 5.5. At rst sight, both models seem very accurate, even for the low
frequencies.
Taking a closer look at the relative errors of the approximating frequency responses, as gure 5.6
shows, it is clear that both the WB Line and the RT_WB Line are very accurate for low frequencies.
However, their approximating errors tend to be higher for the frequency points correspondent to the
voltage peaks in the open-end response. The errors of the RT_WB Line approximation tend to increase
with frequency. Nevertheless, apart from the peaks of the open-end response, the developed model is
very accurate for the range of the switching transients frequencies.
44
Figure 5.7 shows a zoom into the relative errors of the approximating open-end responses, from 700
Hz to 10 kHz. As observed in the short-circuit scan, the RT_WB Line is more accurate than the WB
Line for this range of frequencies. Except for the rst peak observed, the relative approximation error
of the developed model is always below the 2 %, whereas the WB Line may reach a correspondent value
of 10 %.
Figure 5.5: Open-end frequency response (100 Hz - 1 MHz) analytical(bold), WB Line (thin) and
RT_WB Line (dashed). Voltage at the receiving end of phase 1.
Figure 5.6: Relative error of the open-end frequency response (100 Hz - 1 MHz) WB Line (bold) and
RT_WB Line (thin). Voltage at the receiving end of phase 1.
45
Figure 5.7: Detailed relative error of the open-end frequency response (700 Hz - 10 kHz) WB Line
(bold) and RT_WB Line (thin). Voltage at the receiving end of phase 1.
5.3
Line energization and single-phase short-circuit
This test evaluates how the models represent the behavior of the line under typical transient conditions.
Figure 5.8 illustrates the circuit used, where the line is connected by ideal switches to a three-phase
symmetrical source of 1V peak voltage and 50 Hz. The line energization occurs at t = 20 milliseconds,
with the closure of the switches connecting the line to the source. After reaching steady-state, a new
transient is originated, at t = 180 milliseconds, by closing the receiving end switch (short-circuit on phase
3). The voltage at the receiving end of phase 1 is observed.
Figure 5.8: Circuit used to study the response to line energization followed by single-phase
short-circuit, according to the WB Line and to the RT_WB Line applications.
The results of this test are plotted in gures 5.9 and 5.10, regarding the energization transient and
the short-circuit of phase 3, respectively.
46
Both gures show very good agreement between the two models performance. A dierence is perceived
only in the energization condition, where the approximating line response according to the RT_WB Line
denotes a slightly weaker attenuation of the voltage at the receiving end of phase 1.
Figure 5.9: Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)
and RT_WB Line (thin).
Figure 5.10: Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)
and RT_WB Line (thin).
5.4
Current induced by phase coupling
The last test presented is a "hard" test, in the sense that generally all model applications generate
considerably dierent results. Figure 5.11 illustrates the circuit used, where the line is short-circuited at
all terminals, except at the sending end of phase 1, connected to a 1VDC voltage source by an ideal
switch, 1 millisecond after the simulation start. The current at the sending end of phase 3 is observed.
47
Figure 5.11: Circuit used to study the phenomena of phase coupling according to the WB Line and to
the RT_WB Line applications, in terms of the current induced in phase 3 by energization of phase 1.
The energization of phase 1 introduces a transient condition on the system. Since phase 2 and phase
3 are grounded, they present a current which is induced by the time varying electric quantities in phase
1. After reaching a steady-state condition, the whole system must be in DC, to be in accordance with
the voltage source. Therefore, the coupling phenomena is extinguished, and the current on phase 2 and
phase 3 decline to zero. The simulation results for these conditions are plotted in gures 5.12 and 5.13.
As gures show, the time responses according to the two models agree only for the very initial period
of the transient. Furthermore, the response according to the RT_WB Line presents two undesired
peaks, as plotted in gure 5.13. The induced current in steady-state is another important aspect, which
approximation is more accurately computed using the WB Line. Therefore, this test is an example of
transient conditions for which the use of the developed RT_WB Line is not particularly adequate.
Figure 5.12: Current at the sending end of phase 3, induced by energization of phase 1, at the rst 20
miliseconds, according to the WB Line (bold), and to the RT_WB Line (thin).
48
Figure 5.13: Current at the sending end of phase 3, induced by energization of phase 1, at the rst
second of simulation, according to the WB Line (bold), and to the RT_WB Line (thin).
Though this test is not relevant for switching transients studies, it must be noted that the inaccuracy
of the results is a consequence of the low order used for the approximating line functions of the RT_WB
Line application. To demonstrate this, consider including in the test another application of the developed
model, which order of the approximating functions H̄ and Ȳc is the same as that used by the WB Line
application, generated by the EMTP-RV. The new results are plotted in gures 5.14, 5.15 and 5.16.
Figure 5.14: Current at the sending end of phase 3, induced by energization of phase 1, at the rst 20
miliseconds, according to the WB Line (bold), and to the RT_WB Line applications (low order thin;
high order dashed).
49
Figure 5.15: Current at the sending end of phase 3, induced by energization of phase 1, at the rst
second of simulation, according to the WB Line (bold), and to the RT_WB Line applications (low
order thin; high order dashed).
Figure 5.16: Current at the sending end of phase 3, induced by energization of phase 1, at the rst 50
second of simulation (reaching steady-state), according to the WB Line (bold), and to the RT_WB
Line applications (low order thin; high order dashed).
Figure 5.14 shows that it is possible to reach a good agreement with the WB Line time responses, by
using the higher order application of the developed model. Furthermore, the accentuated peaks observed
in the approximating response of the low order RT_WB Line application are practically eliminated when
using the higher order approach.
A very important aspect in this test is the correct approximation of the steady-state condition, where
50
the induced current must decline to zero. Figure 5.16 shows the test results after a long simulation
period, and it is possible to observe that the best approximation is the one computed using the higher
order application of the RT_WB Line, though it uses the same order for the approximating line functions
as the WB Line application.
5.5
Conclusions
The purpose of the present chapter is the validation of the model RT_WB Line, which is a reformulation
of the EMTP-RV model WB Line, in-line with the real-time simulation target. The formulation of the
developed model, as well as the optimizations introduced to ensure additional accuracy with low order
approximations, are presented in chapter 4.
The validation process consists of frequency and time domain simulations in the EMTP-RV 2.3 environment, using an application of the RT_WB Line, which performance is compared to that of the
WB Line, computed by the EMTP-RV and taken as a reference of accuracy. The WB Line application
uses 21 + 11 = 32 poles to t the propagation matrix H̄ and the characteristic admittance matrix Ȳc ,
whereas the RT_WB Line uses 9 + 9 = 18, respectively. The total number of poles used by this developed model application is in-line with the examples in real-time line modeling literature, namely [8, 9].
Section 5.2 presents the rst test, concerning the approximation of the line frequency response under short-circuit and open-end conditions. As regards the short-circuit frequency response both
models generated inaccurate approximations for the range of frequencies up to 50 Hz, particularly the
RT_WB Line application. However, for higher frequencies, both models provide reasonable approximations. Specically for the range from 700 Hz to 10 kHz (including the range of switching transients),
the RT_WB Line provides the most accurate results. Regarding the open-end frequency response,
both models provide good results for low frequencies. The approximating errors are more pronounced
for the voltage peaks of the analytical open-end response and for very high frequencies, which are not
relevant for switching transient studies. Once again, the RT_WB Line generates the most accurate
approximations for the range from 700 Hz to 10 kHz.
Section 5.3 concerns two typical transmission line transients: line energization and single-phase shortcircuit. In both cases, there is a good agreement on the performance of two models. A dierence is
perceived only in the energization condition, where the approximating line response according to the
RT_WB Line denotes a slightly weaker attenuation of the voltage at the receiving end of phase 1.
Finally, section 5.4 refers a critical test concerning the current induced by phase coupling, for which
line models usually show substantially dierent results. The response provided by the RT_WB Line is
inaccurate both for the initial period of the transient and to the steady-state condition, for which the
induced current is expected to decline to zero. Though this test is not relevant for switching transients
51
studies, it must be noted that this result is a consequence of the low order used by the approximating
functions of the RT_WB Line application. This is observed by including in the test another application
of the developed model, which order of the approximating functions H̄ and Ȳc is the same as that used
by the WB Line application, generated by the EMTP-RV. The new results show a good agreement
between the new approach and the WB Line. Furthermore, the high order RT_WB Line application
provides the most accurate approximation of the induced current in steady-state, even though it uses
the same number of poles as the WB Line.
52
Chapter 6
Conclusions
6.1
Introduction
This chapter presents the conclusions of the work in this dissertation, whose objective is to establish
adequate numerical techniques for approximating the propagation parameters for transmission line modeling, allowing real-time simulations, which requires an ecient use of reduced modeling resources. In
order to ensure additional accuracy, it is necessary to introduce some optimization procedures. The resulting model is called RT_WB Line, as it is a reformulation of the EMTP-RV model WB Line, in-line
with the real-time simulation target.
The applications of the developed model are computed by a MATLAB program specically created
in this work. The real-time requirement imposes a limited order for the model, which varies according
to the processor. Therefore, the order of approximations to use in computed applications is assumed
as a pre-dened input to the program. The validation of the developed model consists of testing in the
EMTP-RV 2.3 environment a set of applications of the RT_WB Line, covering both steady-state and
transient conditions.
EMTP-RV is not a real-time simulator, so it is not possible to test the speed of the models applications using this program. Therefore, the analysis is made mainly from the point of view of accuracy.
The real-time requirement has been enforced by using orders for the model that are usually adequate for
this type of simulation (see examples in [8, 9]).
This chapter presents a summary of the conclusions, concerning the performance of the developed
modeling program, and as well it presents a set of proposals for future improvements on transmission
line modeling in-line with the real-time simulation target.
53
6.2
Completion of proposed objectives
The accurate representation of a transmission line requires the use of its frequency dependent parameters. This poses a challenge on the denition of an adequate line model. The goal of this work is to
establish adequate numerical techniques for approximating the propagation parameters for transmission
line modeling, allowing real-time simulations.
The study of existing line models, namely those provided by the EMTP-RV 2.3, leads to the conclusion that the WB Line, that is, the Universal Model [3] approach, presents an increased eciency when
compared to modal domain frequency dependent models, by obtaining better results with less resources.
This eciency is reinforced by the fact that the WB Line is a phase-domain model there is no need to
convert from phase to modal quantities, and vice-versa, at each simulation step.
The line model developed in this work, called RT_WB Line, is a reformulation of the WB Line, inline with the real-time simulation target. Therefore, the RT_WB Line is a phase domain model which
ts the propagation matrix H̄ using the poles and delays dened by the modes. To ensure additional
accuracy with reduced tting resources, two optimizations are introduced, regarding the computation of
the modal delays and the assignment of the modal poles.
The applications of the RT_WB Line are computed by a MATLAB program, specically built for
this dissertation, which enforces the real-time requirement by assuming the order of the approximating
line functions as a pre-dened input. The developed program must additionally receive the location of
the le generated by the EMTP-RV 2.3, containing the line characteristic parameters Z̄ and Ȳ, computed for a set of frequency samples. These parameters are used to compute the original line functions
to be tted H̄ and Ȳc . The output of the program is a le containing the modeling parameters of the
computed application.
The validation of the developed model consists of frequency and time domain simulations in the
EMTP-RV environment, using an application of the RT_WB Line, which performance is compared to
that of the WB Line, computed by the EMTP-RV and taken as a reference of accuracy. The total
number of poles used by the developed model application is in-line with the examples in real-time line
modeling literature, namely [8, 9].
The developed model presents some weaknesses, for example in approximating the low frequencies of
the short-circuit and open-end responses, plotted in gures 5.3 and 5.6, or in approximating the current
induced by phase coupling, as the plots in gures 5.12 and 5.13 show.
These less accurate results are not due to a problem in the model computing routine, but a consequence of the low order used for the approximating functions. This is proved by considering another
54
application of the developed model which approximating functions have the same order as those of the
WB Line application. In fact, this new application provides an approximation of the steady-state induced current which is more accurate than that of the WB Line application, as plotted in gure 5.16.
Furthermore, these tests for which the RT_WB Line is not particularly adequate, are not the applications in which the real-time is more relevant. On the other hand, real-time is of utmost importance
for the study of switching transients, and the tests show that the numerical techniques and optimization
procedures introduced in the RT_WB Line allow to produce low order applications that generate accurate simulation results in switching transient conditions. Examples of this good performance are the
short-circuit and open-end frequency scans for the range of 700 Hz to 10 kHz, plotted in gures 5.4 and
5.7, respectively. The test of line energization followed by single-phase short-circuit, illustrated by the
plots of gures 5.9 and 5.10, is another case of good agreement between the performance of the the WB
Line and RT_WB Line applications, despite the dierence on the order of approximations.
6.3
Proposals for further improvements in line modeling
The rst proposal for further improvements in line modeling is motivated by the test presented in section 5.4, regarding the approximation of the current induced by coupling between phases according to
an application of the RT_WB Line, and taking as a reference the application of the WB Line computed
by the EMTP-RV 2.3. The results of the test show several problems with the developed model performance, namely the existence of accentuated peaks in the current response of the line and the inaccurate
approximation of the steady-state condition.
Section 5.4 demonstrates that the inaccuracy of the RT_WB Line application for this specic test
is a consequence of the low order of its approximations, which is imposed by the real-time target. The
described problems are overcome by using higher order approximations. However, to ensure a real-time
performance alternative strategies must be found.
One of the reasons for the model inaccuracy is that the referred test concerns coupling phenomena.
Therefore, the quantities observed are computed using the o-diagonal elements of the approximating
line matrices H̄ and Ȳc , which generally have a magnitude lower than the correspondent diagonal elements. The RT_WB Line is based on the approximation of these functions using the Universal Model
scheme [3], which computes the tting parameters by solving a least-squares problem. This technique is
based on the absolute deviation between original and approximating functions. Therefore, by enforcing
a given maximum deviation, the relative error is higher for the functions of reduced magnitude, which
is the case for the o-diagonal elements of the line matrices. A possible strategy to tackle this problem
and still use the least-squares technique is to dene a scaling strategy for the elements of H̄ and Ȳc that
assures the relative error is the same for all approximating functions.
55
The other point in the test of section 5.4 is the inaccurate approximation of the steady-state condition,
which is due to a bad approximation of the low frequency samples of the line functions. One possible
solution is to use more samples in the low frequencies. Another alternative is to dene an adequate
weighting scheme that concentrates more eort in approximating the low frequency samples, without
neglecting the frequencies of switching transients.
Another proposal for an improvement in line modeling concerns the optimization presented in section
4.3.2. The procedure regards the approximation of H̄, which uses the poles and delays dened by the
modes. The objective of this optimization is to compute the number of poles to assign to each mode,
given a total number of poles, in order to minimize the error of approximation. Section 4.3.2 provides an
example of several tests performed in order to dene a searching strategy. Given the disparity of results
for dierent orders of approximation tested, it is not possible, within the range of this work, to dene
that strategy. Therefore, the approximation procedure used to compute the RT_WB Line searches all
the possible distributions of modal poles. This motivates a deeper study, including a wider number of
lines, in order to dene a strategy that may represent a major time saving in the pre-processing of the
model tting parameters.
56
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58
Appendix A
Transmission line used in model testing
This dissertation presents several tests using dierent line models. One set of tests compares the line
models provided by the EMTP-RV 2.3. The other compares the EMTP-RV model WB Line with two
applications of the RT_WB Line model, developed for this work.
The line represented in these tests is a three-phase line with a spacial conguration as illustrated in
gure A.1. Other characteristics of this line and of near conductive ground are:
• Length: 100 km
• Number of phases: 3
• Number of ground wires: 0
• DC resistance of each conductor: 0.168228 Ohm/km
• Conductor relative permeability (µr ): 1
• Ground resistivity: 100 Ohm/km
Figure A.1: Spacial conguration of the transmission line used throughout this dissertation.
59