UNIVERSITY OF NAIROBI
SCHOOL OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING
DE-COUPLED LOAD FLOW STUDY METHOD
PROJECT INDEX: PRJ (71)
BY
KETER SAMSON KIPKIRUI
F17/30052/2009
SUPERVISOR:
EXAMINER:
DR.N.O. ABUNGU
Prof. MBUTHIA
This Project report submitted in partial fulfillment of the
Requirement for the award of the degree
Of
Bachelor of Science in Electrical and Electronic Engineering of the
University Of Nairobi.
SUBMITTED ON: 28TH APRIL 2014
1
DECLARATION OF ORIGINALITY
NAME OF STUDENT: KETER SAMSON KIPKIRUI
REGISTRATION NUMBER: F17/30052/2009
COLLEGE: Architecture and Engineering
FACULTY/SCHOOL/INSTITUTE: Engineering
DEPARTMENT: Electrical and Information Engineering
COURSE NAME: Bachelor of Science in Electrical and Electronic Engineering
TITLE OF WORK: DE-COUPLED LOAD FLOW STUDY METHOD
1) I understand what plagiarism is and I am aware of the university policy in this regard.
2) I declare that this final year project report is my original work and has not been submitted
elsewhere for examination, award of a degree or publication. Where other people’s work or my
own work has been used, this has properly been acknowledged and referenced in accordance
with the University of Nairobi’s requirements.
3) I have not sought or used the services of any professional agencies to produce this work.
4) I have not allowed, and shall not allow anyone to copy my work with the intention of passing
it off as his/her own work.
5) I understand that any false claim in respect of this work shall result in disciplinary action,
in accordance with University anti-plagiarism policy.
Signature: ………………………………………………………………………
Date: …………………………………………………………………………
2
DEDICATION
To my family for the endless support and bringing the best of me at early age
i
ACKNOWLEDGEMENTS
I would like to express my deep gratitude to Dr. N. O. Abungu my project supervisor, for his patient
guidance, enthusiastic encouragement and useful critiques of this research work.
I would like to express my great appreciation to Mr. Peter Musau for his valuable and
constructive suggestion during planning and development of this project work. His willingness
to give his time so generously and keeping my progress on schedule has been very much
appreciated.
Special thanks to the Dean-Faculty of Engineering; Chairman-Department of Electrical and
Information Engineering and all my lecturers at the University of Nairobi for their support
which contributed greatly to the provision of knowledge as well as the completion of this
project
Sincere thanks should be given to all my friends and especially Mark Musembi and Erick
Mbugua for the special insights and valuable ideas that help me in understanding load flow
problem and MATLAB programming, May God’s blessing always be with them.
I thank God for His guidance, faithfulness throughout my life as a student and for giving me
peace throughout my final academic year.
I also extend my appreciation to my parents for their continued support and encouragement
throughout my studies.
ii
DECLARATION AND CERTIFICATION
This BSc. work is my original work and has not been presented for a degree award in this
or any other university.
………………………………………..
KETER SAMSON KIPKIRUI
F17/30052/2009
This report has been submitted to the Department of Electrical and Information Eng.,
University of Nairobi with my approval as supervisor:
………………………………
Dr. Nicodemus Abungu Odero
Date: ………………………
iii
LIST OF ABBREVIATIONS
DLF
Decoupled Load Flow
FDLF
Fast Decoupled Load Flow
GS
Gauss-Siedel Load Flow Method
IEEE
Institute of Electrical and Electronics Engineering
MATLAB
Matrix Laboratory
MVA
Mega Voltage Ampere
MVAR
Reactive Power in Mega watts
MW:
Real power in Mega Watts
NR
Newton Raphson Method
P.U
Per Unit
PSAT
Power System Analysis Toolbox
P-V
Voltage Controlled Bus
P-Q
Load Bus
iv
TABLE OF CONTENT
Contents
DECLARATION OF ORIGINALITY............................................................................... 2
DEDICATION ..................................................................................................................... i
ACKNOWLEDGEMENTS ................................................................................................ ii
DECLARATION AND CERTIFICATION ...................................................................... iii
LIST OF ABBREVIATIONS ............................................................................................ iv
TABLE OF CONTENT ...................................................................................................... v
LIST OF FIGURES .......................................................................................................... vii
LIST OF TABLES ........................................................................................................... viii
ABSTRACT ....................................................................................................................... ix
CHAPTER 1 ...................................................................................................................... 10
INTRODUCTION........................................................................................................... 10
1.1 Load flow studies .............................................................................................. 10
1.2 Constraints on load flow solution....................................................................... 10
1.3 Solution to Load flow ........................................................................................ 11
1.4 survey of earlier work ............................................................................................ 12
1.5 Problem statement ................................................................................................. 13
1.4 Objectives.............................................................................................................. 13
1.6 Organization of the Report..................................................................................... 14
CHAPTER 2 ....................................................................................................................... 15
LITERATURE REVIEW ................................................................................................ 15
2.1 Load flow study ..................................................................................................... 15
2.2 Importance of load flow studies ............................................................................. 15
2.3 Load flow Analysis ................................................................................................ 16
2.4 Methods of load flow analysis ............................................................................... 23
2.5 Load Flow Methods ............................................................................................... 24
2.6 Convergence procedure ......................................................................................... 36
2.7 Acceleration of convergence .................................................................................. 36
2. 8 Algorithm modification when PV Buses are also present ...................................... 37
2.9 Comparison of Load Flow Methods ....................................................................... 38
CHAPTER 3 ...................................................................................................................... 40
METHODOLOGY .......................................................................................................... 40
3.1Computational procedure for decoupled load flow method [1]. ............................... 40
3.2 Design Flow Chart ................................................................................................. 41
3.3 IEEE 14 Bus Test Network .................................................................................... 42
3.4 Load Flow Data ..................................................................................................... 44
3.5 Assembling load flow MATLAB data. .................................................................. 47
CHAPTER 4 ...................................................................................................................... 49
RESULTS, ANALYSIS AND DISCUSSION ............................................................. 49
4.1 Results Analysis, Discussion and Validation .......................................................... 49
4.2Performance Analysis ............................................................................................. 51
4.3 Comparative Results .............................................................................................. 52
4.4 Charts and Graphs ................................................................................................. 55
CHAPTER 5 ...................................................................................................................... 60
CONCLUSION AND RECOMMENDATION ................................................................ 60
5.1 Conclusion ............................................................................................................ 60
5.2 Recommendations for Future Work ....................................................................... 61
v
REFERENCES ............................................................................................................... 62
APPENDIX..................................................................................................................... 64
PROGRAM LISTING ................................................................................................. 64
vi
LIST OF FIGURES
FIGURE 2.1: Π LINE FLOW REPRESENTATION .......................................................................... 27
FIGURE 3.1: DECOUPLED LOAD FLOW CHART-[1] .................................................................. 41
FIGURE 3.2: IEEE 14 BUS
SYSTEM [7]. .................................................................................
43
FIGURE 3.3: DIAGRAM OF A TWO-WINDING TRANSFORMER CIRCUIT [16]. ............................... 46
FIGURE 3.4:ONE
LINE DIAGRAM FOR 14 BUS TEST SYSTEM- ...................................................
48
FIGURE 4.1: NEWTON RAPHSON VOLTAGE PROFILE ............................................................... 55
FIGURE 4.2: DECOUPLED LOAD FLOW VOLTAGE PROFILE........................................................
56
FIGURE 4.3: ANGLE PROFILE FOR DLF AND NR .................................................................... 56
FIGURE 4.4: DLF REAL AND REACTIVE POWER FLOW............................................................ 57
FIGURE 4.5: DLF AND NR POWER FLOW .............................................................................. 57
FIGURE 4.6: DLF LINE LOSSES. ............................................................................................ 58
FIGURE 4.5:S LINE LOSSES .................................................................................................... 58
vii
LIST OF TABLES
TABLE 2.1: SUMMARY OF BUS VARIABLES ............................................................................ 21
TABLE 3.1: BUS DATA ......................................................................................................... 44
TABLE 3.2: LINE DATA ......................................................................................................... 45
TABLE 4.1: BUS VOLTAGES, POWER GENERATED AND LOAD AFTER CONVERGENCE OF
DECOUPLED LOAD FLOW. ...............................................................................................
49
TABLE 4.2: REAL AND REACTIVE POWER FLOW OVER DIFFERENT LINES AND LOSSES ............. 50
TABLE 4.3: VOLTAGE, ANGLE, GENERATION AND LOAD POWER COMPARISON BETWEEN DLF
AND NR ........................................................................................................................
53
TABLE 4.4: REAL AND COMPLEX BUS POWER COMPARISON FOR THE DLF AND NR METHOD.. 54
TABLE4.5: PSAT SIMULATED RESULTS ................................................................................ 55
TABLE 4.6: DATA USED TO SHOW RELATIVE ACCURACY OF THE RESULTS OF EACH METHOD .. 58
viii
ABSTRACT
Load flow study is the analysis of a network under steady state operation subjected to inequality
constraints in which the system operates. Load flow analysis is the backbone of power system
analysis and design. They are necessary for planning, operation, economic scheduling and
exchange of power between utilities. The principal information of power flow analysis is to
find the magnitude and phase angle of voltage at each bus and the real and reactive power flows
in each transmission lines. Therefore, load flow analysis is an importance tool involving
numerical analysis applied to a power system.
In this analysis, iterative techniques are used because there are no known analytical method to
solve the load flow problem. This iterative techniques includes; Gauss Siedel, Newton
Raphson, Decoupled method and Fast Decoupled method. Load flow analysis is difficult and
time consuming to perform by hand. The Decoupled load flow method in detail; Formulation
of static load flow equations and computational algorithm is clearly discussed.
The objective of this project is to develop a load flow program based on Decoupled method
that will ease the analysis of load flow problem. MATLAB software was used as a
programming platform. The program was run on an IEEE 14-bus system test network and the
results compared with those from other methods, i.e. Newton Raphson method and finally,
validated by simulated results from Power System Analysis (PSAT) simulation software.
The load flow results obtained were analyzed and discussed. Both the decoupled load flow and
Newton-Raphson methods gave similar results. However, the decoupled method converged
faster than the Newton-Raphson method. The bus voltage magnitudes, angles of each bus along
with power generated and consumed at each bus has been tabulated in Table 4.1 and 4.2. It is
seen from this tables that the total power generated is 273.590 MW whereas the total power
consumed is 259 MW. This indicates that there is a line loss of about 14.590 MW for all the
lines put together. For Newton Raphson method, the total power generated were 272.593MW
whereas the power demand were 259 MW thus a loss of 13.593MW. The power loss as
obtained from PSAT was 29.4125MW. The results indeed, compares very well. Therefore the
decoupled load flow method was verified to be effective and reliable method of obtaining
optimum solution for a load flow problem.
ix
CHAPTER 1
INTRODUCTION
1.1 Load flow studies
Load flow solution is a solution of the network under steady state operation subjected to certain
inequality constraints under which the system operates. Load flow studies are important in
planning and designing future expansion of power systems. The study gives steady state
solutions of the voltages at all the buses, for a particular load condition. Different steady state
solutions can be obtained, for different operating conditions, to help in planning, design and
operation of the power system [1],[9].
Generally, load flow studies are limited to the transmission system, which involves bulk power
transmission. The load at the buses is assumed to be known. Load flow studies throw light on
some of the important aspects of the system operation, such as: violation of voltage magnitudes
at the buses, overloading of lines, overloading of generators, stability margin reduction,
indicated by power angle differences between buses linked by a line, effect of contingencies
like line voltages, emergency shutdown of generators, etc. Load flow studies are required for
deciding the economic operation of the power system. They are also required in transient
stability studies. Hence, load flow studies play a vital role in power system studies. Thus the
load flow problem consists of finding the power flows (real and reactive) and voltages of a
network for given bus conditions. At each bus, there are four quantities of interest to be known
for further analysis: the real and reactive power, the voltage magnitude and its phase angle.
Because of the nonlinearity of the algebraic equations, describing the given power system, their
solutions are obviously, based on the iterative methods only.
1.2 Constraints on load flow solution
The constraints placed on the load flow solutions could be: The Kirchhoff’s relations holding
well, Capability limits of reactive power sources, Tap-setting range of tap-changing
transformers, Specified power interchange between interconnected systems, Selection of initial
values, acceleration factor and convergence limit.
In load flow analysis, an electrical power system network consists of hundreds of buses and
branches with impedances specified in per unit on a common MVA base. Performance of
power system network both in normal operating conditions and under fault should be
continuously analyzed [3]. For optimal operation of an electrical power system requires that;
10
Generation must supply the load plus losses, The bus voltage magnitudes must remain close to
rated values, generators must operate within specified real and reactive power limits and that
transmission lines and transformers should not be overloaded for long periods. [2];
Load flow study covers a wide range of time constants which include
steady state and
transient conditions. The symmetrical steady state operation of an electrical power system is
the most important mode of operation since it ensures supply of real and reactive power
demanded by various loads, the frequency and bus voltages being maintained within specified
tolerances and with optimum economy [4].
Load flow deals with the flow of electrical power from one or more sources to loads consuming
energy through available paths as commonly shown in a one line diagram [3]. Electric energy
flow in a network divides among branches according to their respective impedances until a
voltage balance is reached in accordance to Kirchhoff’s Laws [5]. The flow will shift anytime
the circuit configuration is changed or modified, generation is shifted or load requirements
changes.
1.3 Solution to Load flow
Load flow study is the determination of steady-state conditions of a power system for a
specified power generation and load demand. The load flow problem is the computation of
voltage magnitude and phase angle at each bus and also active and reactive flows in a power
system.
Load flow analysis is performed extensively both for system planning purposes, to analyze
alternative plans of future systems operation and to evaluate different operating conditions of
existing systems. In static contingency analysis, load flow study is used to assess the effect of
branch or generator outages. In transfer capability analysis, repetitive power flow analysis is
performed to calculate the power transfer limits.
In load flow analysis, it is normal to assume that the system is balanced and that the network
is composed of constant, linear, lumped-parameter branches. In the most basic form of the
power flow, transformer taps are assumed to be fixed [1]. This assumption is relaxed in
commercial load flow. Therefore, nodal analysis is generally used to describe the network.
However, because the injection and demand at bus bars is generally specified in terms of real
and reactive power, the overall problem is nonlinear. Accordingly, the load flow problem is a
set of simultaneous nonlinear algebraic equations. Numerical techniques are required to solve
this set of equations [2].
11
Traditional solutions of the load-flow problems follow an iterative process by assigning
estimated values to the unknown bus voltages and angles and calculating a new value for each
bus voltage and angle from the estimated values at the other buses, the real power specified,
and the specified reactive power or voltage magnitude given in some buses. A new set of values
for voltage and angle are thus obtained for each bus and still used to calculate another set of
bus voltages and angles in a sequential algorithm. The iterative process is repeated until the
changes at each bus are less than the specified tolerance value, (0.00001<ε<0.0001).
Load flow analysis has become in recent years one of the major areas of research in electrical
engineering. However load flow study is a difficult task. First, the load distribution network is
a complex system and exhibits lots computational procedure hence time consuming. Secondly,
there are losses in electrical network distribution hence quantification and minimization of
losses is important because it will determine the economic operation of the power system.
1.4 Survey of Earlier Work
Over the years, the direction of research has shifted, replacing old approaches with newer and
more efficient ones. Apparently due to their limited success, a number of old approaches seem
to no longer in use. These include such methods as Runge-Kutta, Iwamoto, and Ward and Hale
methods load-flow study methods. There is also considerably less emphasis on methods such
as AC and DC Decoupled methods, Gauss-Seidel load-flow study. The rapidly increasing
power of the personal computer is making it possible to apply more complicated solution
techniques methods based on few and faster iterations technique such as Newton-Raphson
(NR), Decoupled load flow and Fast Decoupled Load Flow methods [6, 10]. For large scale
power transmission system, decoupled load flow has been found to be an alternative strategy
for improving the computational efficiency and reducing computer storage requirements. This
method uses an approximate version of NR procedure. The DLF requires more iterations than
NR method, but, requires considerably less time per iterations and thus power flow solution is
obtain rapidly. This technique is very useful tool in contingency analysis where numerous
outages are to be simulated or when a power flow solution is required for line control.
Fast Decoupled load flow method is a variation on Newton-Raphson that exploits the
approximate decoupling of active and reactive flows in well-behaved power networks, and
additionally fixes the value of the Jacobean during the iteration in order to avoid costly matrix
decompositions. It is achieved by only inverting the Jacobean matrix once within its algorithm.
It has 3 assumptions. First, the conductance between the buses is zero. Second, the magnitude
12
of the bus voltage is one per unit. Thirdly, the sine of phases between buses is zero, whereas
the cosine of phases is 1. In reality the decoupled load flow can return the answer within
seconds, whereas a Newton Raphson method takes much longer to return an answer. The
decoupled load flow is a computer-driven method, in the sense that it is not necessary for the
researcher to calculate manually each and every computational procedures in order to arrive at
the final solution of the load flow problem instead a computer based algorithm can be used to
solve large and complex load flow problems with an ease.
1.5 Problem Statement
The purpose of this work is to understand the theory of load flow analysis and to develop a
reliable and effective program based on Decoupled Load Flow study method. MATLAB
7.6(r2013b) software was used as a programming platform and PSAT (Power System Analysis
Toolbox) as a validating tool. The proposed Decoupled load flow method should accurately
calculate and analyze a well-conditioned load flow study with minimal losses on the buses,
branches and the minimal number of iteration required for convergence. The effectiveness of
the decoupled program is tested on IEEE 14 bus test network to give reliable results. To achieve
this relevant input variables are to be identified, formulated and gathered from the load flow
data.
1.4 Objectives
The objective of the research can be stated as follows:

To understand Decoupled load flow study method and use it to find the optimal
solution for load flow of a 14 bus test network.

To develop a decoupled load flow program using MATLAB as programming
platform
13
1.6 Organization of the Report
This project report has been arranged into five chapters.
In chapter 1, general introduction to load flow is made, it also addresses the load flow
constraints, statement of problem and objectives. Finally, organization of the report is also
presented in this chapter
In Chapter 2, a literature review of electrical power flow study has been conducted followed
by the load flow study methods which are the Gauss Siedel method, Newton-Raphson method,
Decoupled load flow and finally, fast decoupled load flow. Each subject has been
independently broken down and addressed separately in detail. Further, the Decoupled load
flow was expounded on how it is used to solve load flow problem.
In Chapter 3, the algorithm and the flow chart of decoupled load flow was discussed. The
IEEE test network and data of the 14 bus network was featured in this chapter. The Validating
tool, Power Analysis Tool Box (PSAT) is also featured. Data from the field as well as other
sources are introduced, analyzed, interpreted and validated. The data has been plotted using
MATLAB for easier analysis and validated by (PSAT. The selection of the data was as a result
of research n determining the most suitable input variables for the decoupled load flow method.
The step by step process of calculation and simulation of decoupled load flow study.
In Chapter 4, the results are discussed and analyzed giving brief explanations of what can
be drawn from the output of the Decoupled load flow method which includes number of
iteration required for solution to converge and its level of accuracy in making a load flow
analysis.
In Chapter 5, conclusion and recommendation of the report. Recommendation of the report
by giving a review of the study in the preceding chapters and identifies some problems for
future work in this area.
14
CHAPTER 2
LITERATURE REVIEW
2.1 Load flow study
In power engineering, the power flow study, also known as load-flow study, is an analysis of
the voltages, currents and power flows in a power system under the steady conditions. Load
flow is an important tool involving numerical analysis applied to a power system [9]. It usually
uses simplified notation such as a one-line diagram and per-unit system, and focuses on various
components of Alternating Current AC power i.e.: voltages, voltage angles, real power and
reactive power. The study is based on normal operation of a power system and operating under
balanced conditions [11]. Conducting a load flow study helps ensure that the power system is
adequately designed to satisfy the required performance criteria. A properly designed system
helps contain initial capital investment and future operating costs. It also helps develop
equipment specification guidelines, optimize circuit usage, minimize KW and KVAR losses
and identify transformer tap settings. . The principal information obtain from a power flow
study is the magnitude and the phase angle of the voltage at each bus and the real and reactive
power flowing in each line and line loses [9]
This information obtained is important for the continuous monitoring of the current state of the
system and for analyzing the effectiveness of alternative plans for future system expansion to
meet increased load demand [1], [6]. When the current state of the system is monitored and
found to be unsatisfactory e.g. if voltage at bus is too low, then a control action is taken to
correct the voltage e.g. put HVDC or use compensation. The load continues to increase and
hence the system needs to be expanded regularly. For the continuously increasing load demand
plans has to be made to match with generation facility.
2.2 Importance of load flow studies
Load flow studies are performed in major areas of power system development and operation
because of the following rationale;
1. Planning: Necessary for planning, economic scheduling, and control of an existing
system as well as planning its future expansion. This is the future development of a
system in which load flows are used to study the effects and feasibility of changes in
network configuration such as the removal or addition of lines, new generation units,
or increased loads due to a growing consumer demand. Load flow is central to the
15
stability analysis performed on the proposed system. System security is also determined
and multiple load flows are performed -to evaluate contingencies.
2. Operation and Control, the configuration of the network changes due to loss of
generation units or transmission circuits, or –the change in demand of consumer load.
Load flow studies are used to evaluate these changes and compensate for high or low
bus voltages by the addition or removal of static capacitors, the altering of ratios of
transformers or by changing the reactive power of synchronous condensers or generator
units. Stability analysis and system security Studies are also performed.
3. The Economic Operation: As loads change throughout the day there is a need to
determine the best generating pattern to minimize costs of operation and provide the
best voltage regulation. Load flow is used to obtain the optimum settings of transformer
taps, shunt capacitance and unit generation; subject to the operational constraints of
equipment in the system
4. Load-flow studies are performed to determine the steady-state operation of an electric
power system. It calculates the voltage drop on each feeder, the voltage at each bus,
and the power flow in all branch and feeder circuits.
5. Determine if system voltages remain within specified limits under various contingency
conditions, and whether equipment such as transformers and conductors are
overloaded.
6. Load-flow studies are often used to identify the need for additional generation,
capacitive, or inductive VAR support, or the placement of capacitors and/or reactors to
maintain system voltages within specified limits.
7. Losses in each branch and total system power losses are also calculated.
2.3 Load flow Analysis
The goal of load flow analysis is to obtain complete voltage angle and magnitude information
for each bus in a power system for specified load and generator real power and voltage
conditions. Once this information is known, real and reactive power flow on each branch as
well as generator reactive power output can be analytically determined. Due to the nonlinear
nature of this problem, numerical methods are employed to obtain a solution that is within an
acceptable tolerance. Load flow uses a mathematical algorithm of successive approximation
by iteration, or the repeated application of calculation steps on the non-linear load flow
equations [4]. These steps represent a process of trial and error that starts with assuming one
array of numbers for the entire system, comparing the relationships among the numbers to the
16
laws of power flow equations, and then repeatedly adjusting the numbers until the entire array
is consistent with both physical law and the conditions stipulated by the user. In practice, this
is a computer program to which the operator gives certain input information about the power
system, and which then provides output that completes the picture of what is happening in the
system. There are variations on what types of information are chosen as input and output.
Typically the input data is divided into: Line data, Bus data, Generator data, Transformer data
and Load data. This data is included with every load flow output file in order to document the
system, load configuration that the solution applies for. The load flow study have a predefined
set of criteria that the system evaluated must meet. These criteria are not exception of:

Voltage criteria in which bus voltages must be within their limits.

Power flows on cables and transformers must be within equipment ratings.

Generator reactive outputs must be within the limits defined by the generator capability
curve
2.3.1 Types of Variables
Basically load-flow analysis deals with known real and reactive power flows at each bus, and
those voltage magnitudes that are explicitly known, and from this information calculating the
remaining voltage magnitudes and all the voltage angles is made possible [3], [4], and [9].
Owing to the nonlinear nature of the load-flow problem, it may be impossible to find one
unique solution because more than one answer is mathematically consistent with the given
configuration. However, it is usually straightforward in such cases to identify the “true”
solution among the mathematical possibilities based on physical plausibility and common
sense. Conversely, there may be no solution at all because the given information was
hypothetical and does not correspond to any situation that is physically possible. Still, it is true
in principle and most important for a general conceptual understanding that two variables per
node are needed to determine everything that is happening in the system. In practice, current is
not known at all; the currents through the various circuit branches turn out to be the last thing
that are calculated once the load-flow analysis has been completed. Voltage is known explicitly
for some buses but not for others. More typically, what is known is the amount of power going
into or out of a bus.
Load-flow analysis consists of taking all the known real and reactive power flows at each bus,
and those voltage magnitudes that are explicitly known, and from this information calculating
the remaining voltage magnitudes and all the voltage angles, this is the hard part. The easy part
17
is to calculate the current magnitudes and angles from the voltages, knowing how to calculate
real and reactive power from voltage and current, power is basically the product of voltage and
current, and the relative phase angle between voltage and current determines the respective
contributions of real and reactive power.
Conversely, one can deduce voltage or current magnitude and angle if real and reactive power
is given, but it is far more difficult to work out mathematically in this direction.
This is because each value of real and reactive power would be consistent with many different
possible combinations of voltages and currents. In order to choose the correct ones, one has to
check each node in relation to its neighboring nodes in the circuit and find a set of voltages and
currents that are consistent all the way around the system.
2.3.2 Load flow problem formulation
The complex power injected by the source into the ℎ bus of a power system is [1, 9].
=
Where
+
∗
=
= 1, 2, … .
(2.1)
ℎ bus and with respect to ground and
the voltage at the
is the source current
injected into the bus. The load flow problem is handled more conveniently by use of
than
∗
rather
. Therefore, taking the complex conjugate of Eq. (2.1), hence
−
=
∗
∑
;
= 1, 2, …
(2.2b)
Equating real and imaginary parts
(
∗
)=
(
(2.3 )
∗
)=−
(2.3 )
In polar form
=| |
=|
|
Real and reactive powers can be expressed as
(
(
)=| |
| ||
) = −| |
| cos(
| ||
+
| sin(
−
+
);
−
= 1, 2, …
) ;
= 1, 2, …
(2.4)
(2.5)
18
Equations (2.4) and (2.5) represent 2n power flow equations at n buses of a power system (n
real power flow equations and n reactive power flow equations). Each bus is characterized by
four variables: , , | | and . Resulting in a total of 4n variables. Equations (2.4) and (2.5)
can be solved for 2n variables if are specified. Practical considerations allow a power system
analyst to fix a priori two variables at each bus. The solution for the remaining 2n bus variables
is rendered difficult by the fact that Equations. (2.4) and (2.5) are non-linear algebraic equations
(buses voltages are involved in product form and sine and cosine terms are present) and
therefore, explicit solution is not possible. Solution can only be obtained by iterative numerical
techniques. Depending upon which two variables are specified a priori, the buses are classified
into three categories.
2.3.3 Types of Buses
Load flow analysis buses are represented as nodes, but there are many types of buses (typically
3) which should be known for better understanding. [1].The three main types of buses are [9,
5]:
1. Load buses. This is a bus without any generators connected to it, both Power generated and
reactive power generated are zero and the real power
; and reactive power
drawn from
the system by the load (negative inputs into the system) are known from historical record, load
forecast, or measurement. Quite often in practice only the real power is known and the reactive
power is based on an assumed power factor such as 0.85 or higher.
2. Voltage-controlled buses (P-V). Any bus of the system at which the voltage magnitude is kept
constant is said to be voltage controlled. At each bus to which there is a generator connected,
the megawatt generation can be controlled by adjusting the prime mover, and the voltage
magnitude can be controlled by adjusting the generator excitation. Therefore, at each generator
bus the power generated and voltage magnitude are specified. With Power demand of the bus
also known, we can define mismatch .Generator reactive power
required to support the
scheduled voltage magnitude cannot be known in advance, and so reactive power mismatch is
not defined. Therefore, at a generator bus ∆ voltage angle
is the unknown quantity to be
determined [9].
3. Slack bus. The bus is also known as swing or reference bus. The known qualities are the
voltage magnitude at the bus | | and the voltage angle . The voltage of the slack bus serves
as reference for the angles of all the other bus voltages where the usual practice is set. The
unknown quantities are the active and reactive power
and
at this bus and mismatches are
19
therefore not defined for the slack bus [9], [1].The slack bus is usually designated as bus 1 and
there is only one type of this bus in a power system.
2.3.4 Need for a slack bus
Unlike the other two buses which represent physical systems conditions, this bus is more a
mathematical requirement. It is needed to provide a ‘reference’ angle to which all the other
angles are referred [2]. Also in a load flow study active and reactive power cannot be fixed a
priori at all the buses as the net complex power flow into the network is not known in advance.
This is because the system power loses are unknown till the load flow solution is completed
[9]. In order that the variations in real and reactive power at the slack bus during the interactive
process are a small percentage of the generating capacity, the slack bus is normally selected as
the bus connected with the largest generating station [1].
Real power .
Total
Total.
=
−
generation. load
generation
=I R
=
(2.6 )
−
( 2.6 )
Real power losses are loses in the transmission lines and transformers of the network.
Individual currents in various transmission lines of the network cannot be calculated until after
the voltage magnitude and angle are known at all the buses of the system and hence
is
initially unknown.
∑
Accounts for the combined MVARS associated with line charging, shunt capacitors
and reactors at buses and the
losses in the series reactance of the transmission lines. It is
given by the difference between the total MVARS supplied by the generator at the buses and
the MVARS received by the loads [9].
After load flow problem has been solved the difference (slack) between the total specified
power going into the system at lay the other buses and the total output power plus losses are
assigned to the slack bus [9].
20
Table 2.1: Summary of bus variables
Bus type
Specified variables
|
Slack or reference bus
Unknown variables
|,
Generator or PV bus
,|
Load or PQ bus
,
,
|
,
|
|,
2.3.5 Variable types and Limits
2.3.5.1 Variable types

Control variables
(excepting slack bus),

Non-control variables

State variables |
or | |
and
| and _
2.3.5.2 Variable limits
(i) Voltage magnitude | | must satisfy the inequality
| |
≤| |≤| |
(2.7)
The power system equipment is designed to operate at fixed voltages with allowable variations
of ±(5 − 10)% of the rated values [7].
(ii) Certain of the
(state variables) must satisfy the inequality constraint of Power angle.
|
−
|
≤|
−
|≤|
−
|
(2.8)
This constraint limits the maximum permissible power angle of transmission line connecting
buses and
and is imposed by considerations of system stability [1].
(iii)Owing to physical limitations of P and Q generation sources,
and
are
Constrained as follows Power limits:
,
,
≤
≤
,
(2.9)
≤
≤
,
(2.10)
2.3.6 Power Balance Equations
Power balance equations it is, of course obvious that the total generation of real and reactive
power must equal the total load demand plus losses, i.e. [4, 10].
=
=
+
+
(2.11)
(2.12)
21
Where
and
are system real and reactive power loss, respectively. This leads to optimal
sharing of active and reactive power generation between sources.
Once
and V are known, the voltage angle and magnitude at every bus, it can be easy to find
the current through every transmission link; it becomes a simple matter of applying law to each
individual link. (In fact, these currents have to be found simultaneously in order to compute
the line losses, so that by the time the program announces
’s and V’s, all the hard work is
done.) Depending on how the output of a load-flow program is formatted, it may state only the
basic output variables, as in it may explicitly state the currents for all transmission links in
amperes, or it may express the flow on each transmission link in terms of an amount of real
and reactive power owing, in megawatts (MW) and (MVAR).
2.3.7 Static load flow solution
The following assumptions and approximations are made in the load flow Esq. (2.4) and (2.5).
i.
Line resistances being small are neglected (shunt conductance of overhead lines
is always negligible), i.e. PL, the active power loss of the system is zero. Thus
≈ 90° and
in Esq. (2.4) and (2.5)
ii.
(
−
≈ −90° .
) Is small (< /6) so that sin(
−
) ≈(
−
). This is justified
from considerations of stability.
iii.
All buses other than the slack bus (numbered as bus 1) are PV buses, i.e. voltage
magnitudes at all the buses including the slack bus are specified.
Equations (2.4) and (2.5) then reduce to
=| |
= −| |
|(
| ||
| ||
−
);
| cos(
−
= 1, 2, …
(2.13)
) + | | | |;
= 1, 2, …
(2.14)
Since | |s are specified, Eq. (2.13) represents a set of linear algebraic equations in
are (n-1) in number as
is specified at the slack bus (
s which
= 0). Nth equation corresponding to
slack bus (n=1) is redundant as the real power injected at this bus is now fully specified as
Equations (2.13) can be solved explicitly (non-iteratively) for ,
substituted in Eq. (2.14), yields
,…,
, which when
s, the reactive power bus injecting. It may be noted that the
assumptions have decoupled Esq. (2.13) and (2.14) so that these need not be solved
simultaneously but can be solved sequentially [solution of Eq. (2.14) follows immediately upon
22
simultaneous solution of Eq. (2.14)]. Since the solution is non-iterative and the dimension is
reduced to (n-1) from 2n, it is computational highly economical.
2.3.8 General building rules of YBUS
1 Self-admittance of node ,
equals the algebraic sum of all the admittances
connected to node .
2
Mutual admittance between nodes
and ,
,
equals the negative sum of all
admittances connecting nodes and k.
3
=
Characteristics of YBUS
1. It is symmetric
2. It is very sparse (>90% for more than 100 buses)
2.4 Methods of load flow analysis
The numerical analysis involving the solution of algebraic simultaneous equations forms the
basis for solution of the performance equations in computer aided electrical power system
analyses, such as during linear graph analysis, load flow analysis (nonlinear equations),
transient stability studies (differential equations), etc. Hence, it is necessary to review the
general forms of the various solution methods with respect to all forms of equations as under.
There are various methods in which the load flows can be done. Some of them include GaussSeidel, Newton Raphson, Decoupled load flow Fast decoupled load flow and various other
novel methods are being proposed. In this project we made use of the decoupled load flow
method which is one of the most basic methods of load flow analysis introduced in power
system analysis. The more successful methods of load flow solution are based on the
admittance matrix [y] representation of a system. The advantages gained are ease of problem
and data preparation and changes made to the system do not involve the recalculation of all
network elements. The admittance matrix is sparse for a practical power system, i.e. it has only
a few non-zero elements for large systems. By contrast the impedance matrix [Z] of a system
(which is the inverse of the admittance matrix) is full, and changes in system configuration
affect the whole of the matrix.
The first practical digital solution methods for load flow were the Y matrix--iterative methods,
these were suitable because of the low storage requirements, but had the disadvantage of
converging slimly or not at all. Z matrix methods were developed which overcame the
reliability problem but a sacrifice was made of storage and speed with large systems. The
Newton-Raphson method was developed this time and was found to have very strong
23
convergence. The current problems faced in the development of load flow are: an ever
increasing size of systems to be solved, on-line applications for automatic control, and system
optimization. Newer and modified methods of load flow have been developed to overcome
these problems.
2.4.1 Properties of load flow solution method.

High computational speed. This is especially important when dealing with large
systems, real time applications (on-line), multiple case load flows such as in system
security assessment, and also in interactive applications.

Low computer storage. This is important for large systems and in the use of computers
with small core storage availability, e.g. mini-computers for on-line application.

Reliability of solution. It is necessary that a solution be obtained for ill-conditioned
problems, in outage studies and for real time applications.

Versatility. An ability on the part of load flow to handle conventional and special
features (e.g. the adjustment of tap ratios on transformers; different representations of
power system apparatus), and its suitability for incorporation into more complicated
processes.

Simplicity. The ease of coding a computer program of the load flow algorithm

The type of solution required from a load flow also determines the method used:
accurate or approximate unadjusted or adjusted offline or on—line single case or
multiple cases
2.5 Load Flow Methods
2.5.1 Gauss-Seidel Method
The Gauss-Siedel (GS) method is an iterative algorithm for solving a set of non-linear algebraic
equations [6, 1]. To start with, a solution vector is assumed, based on guidance from practical
experience in a physical situation. One of the equations is the used to obtain the revised value
of a particular variable by substituting in it the present values of the remaining values. The
solution vector is immediately updated in respect of these variables. The process is then
repeated for all the variables thereby completing one iteration. The iterative process is repeated
till the solution vector converges within prescribed accuracy. The convergence is quite
sensitive to the starting values assumed. Fortunately, in load flow study a starting vector close
to the final solution can be easily identified with previous experience
24
To explain how the GS method is applied to obtain the load flow solution, let it be assumed
that all the buses other than the slack bus are PQ buses. We shall see later that the method can
be easily adopted to include PV buses as well. The slack bus voltage being specified, there are
(n-1) bus voltage starting values of whose magnitudes and angles are assumed. These values
are then updated through an iterative process. During the course of iteration, the revised voltage
at the ℎ bus is obtained as follows:
=
(
−
)
(2.15)
∗
From equation (2.27)
=
Substituting for
1⎡
⎢ −
⎢
⎣
⎤
⎥
⎥
⎦
(2.16)
from equation (2.38) into (2.39)
=
1⎡
⎢
⎢
⎣
−
∗
−
⎤
⎥ ; = 2, 3, … … ,
⎥
⎦
(2.17)
The voltages substituted in the right hand side of Eq. (2.17) are the most recently calculated
(updated) values for the corresponding buses. During each iteration voltages at buses =1, 2,
3… n are sequentially updated through use of Eq. (2.17). V1, the slack bus voltage being fixed
is not required to be updated. Iterations are repeated till no bus voltage magnitude changes by
more than a prescribed value during iteration. The computation process is then said to converge
to a solution. If instead of updating voltages at every step of iteration updating is carried out at
the end of a complete iteration, the process is known as the Gauss iterative method. It is much
slower to converge and may sometimes fail to do so.
2.5.2 Algorithm for load flow solution
Presently we shall continue to consider the case where all the buses other than the slack are PQ
buses. The steps of a computational algorithm are given below:
1.
With the load profile known at each bus i.e.P , Q
are known, allocate e P and Q
to all generating stations.
While active and reactive generations are allocated to the slack bus, these are permitted
to vary during iterative computation. This is necessary as voltage magnitude and angle
are specified at this bus (only two variables can be specified at any bus)
With this step, bus injections (P + jQ ) are known at all buses other than the slack bus.
25
2.
Assembly of bus admittance matrix YBUS: with the line and shunt admittance data
stored in the computer, YBUS is assembled by using the rule for self and mutual
admittances. Alternatively YBUS is assembled using Eq. (2.4), where input data is the
form of primitive matrix Y and singular connection matrix A.
3.
Iterative computation of bus voltages (V ; i = 2, 3 … . , n): to start the iterations a set of
initial voltage values is assumed. Since, in a power system the voltage is not too wide,
it is normal practices to use a flat voltage start, i.e., initially all voltages are set to (1 +
j0) except the voltage of the slack bus which is fixed. It should be noted that (n − 1)
equation (2.40) complex numbers are to be solved iteratively for finding(n − 1)
complex voltagesV , V , … . , V . If complex number operation are not available in
computer , Equation (2.40) can be converted into 2(n − 1) equations in real unknowns
(e , f or |V |, δ ) by writing
V = e + jf = |V |e
(2.18)
A significant reduction in the computer time can be achieved by performing in advance all the
arithmetic operations that do not change with iterations.
Define
−
=
= 2, 3, … . . ,
=
= 2, 3, … . . , ;
(2.19)
(2.20)
= 2, 3, … . . , ;
≠
Now for the ( + 1) ℎ iteration, the voltage Eq. (2.17) becomes
(
)
=
(
−
)
( )
= 2, 3, … . ,
(2.21)
(
The iterative process is continued till the change in magnitude of bus voltage,
( ) ∗
)
∆
4
5
(
)
=
−
(
)
−
( )
< ; = 2, 3, … . ,
(2.22)
Computation of slack bus power: substitution of all bus voltages computed in
step 3 along with V yieldsS ∗ = P − jQ .
Computation of line flows and line losses: this is the last step in the load flow
analysis wherein the power flows on the various lines of the network are
computed [1, 4]. Consider the lines connecting buses and k. The line and the
transformers at each end can be represented by a circuit with series admittance
y and two shunt admittances y
and y
as shown in Fig (2)
26
Bus k
Bus
Figure 2.1: π line flow representation
The current field fed by bus into the line can be expressed as
=
+
=(
(2.23)
−
)
(2.24)
=
(2.25)
From Eqns. (2.23), (2.24) and (2.25), we get,
=(
)
−
(2.26)
+
The power fed into the line from bus is:
=
+
∴
(2.27)
+
∗
=
(2.28)
Using Eqns. (2.26) and (2.28), we get
+
=
[(
)
−
∴
+
=
(
∴
−
=
∗
∴
−
=| |
(
∗
+
∗) ∗
−
)
−
∗
+
+
∗
−
)]∗
(
(
)∗
∗
+| |
( 2.29)
Similarly, power fed into the line from bus k is
=| |
−
∗
−
+| |
(2.30)
=−
Now
∴
(2.32)
=−
From Eqns. (2.30) and (2.32), we get
=|
+
∗
=| |<
,
= −| |
−
|<
,
+| |
∗
(2.33)
=| |<−
=
∴
−
= [−| | |
| cos
+ | || ||
| cos(
−
+
)
27
−| | |
| sin
− | || ||
= −| | |
∴
| sin(
−
)−| |
+
(2.34)
| cos
+ | || ||
=| | |
| cos(
| sin
−
− | ||
)
+
||
(2.35)
| sin(
−
)−| |
+
(2.36)
Similarly power flows from k to can be written as:
= −| | |
=| | |
| cos
| sin
+ | || ||
− | ||
||
| cos(
| sin(
−
−
)
(2.37)
)−| |
(2.38)
+
+
Now real power loss in the line ( → ) is the sum of the real power flows determined from Eqn. (2.35)
and (2.37)
∴
=
+
= −| | |
∴
+ | ||
||
| cos
+ | || ||
| cos(
= (| | + | | )|
−
−
) –| | |
+
| cos
)
+
| cos
| cos(
+ | || ||
|[cos{
−(
)} + cos{
−
+(
−
)}]
= −(| | + | | )|
= [2| ||
∴
| cos
| cos(
+ 2| || ||
−
| cos
cos(
) − | | − | | ]|
−
)
| cos
(2.39)
Let
=
+
=|
| cos
=|
| sin
=[
∴
2| || | cos(
−
)−| | −| | ]
(2.40)
Reactive power loss in the line ( → ) is the sum of the reactive power flows determined
from Equations (2.36) and (2.38), i.e.
∴
|
∴
=
+
=| | |
| sin
| |
| sin
− | || ||
− | || ||
= (| | + | | )
− | |
∴
= (| | + | | )
∴
=
| sin(
− | ||
| sin(
−
+
−
+
)−|
||
|[sin(
− 2| || |
cos(
−
+
)−| |
+
|
) + sin(
−
+
)]
+| |
[| | + | | − 2| || |
cos(
−
−
)− | |
)] − | |
+|
|
+| |
(2.41)
The power loss in the ( → ) ℎ line is the sum of the power flows determined from equation (2.40)
and (2.41). Total transmission loss can be computed by summing all the line flows (
+
) for all
28
, . it may be noted that the slack bus power can also be found by summing the flows on the lines
terminating at the slack bus.
2.5.3 Newton-Raphson Method
Newton-Raphson is an iterative method which approximates the set of non-linear
simultaneous equations to set of linear equations using Taylor’s series expansion and the
terms are restricted to first order approximation [6, 1].
Given a set of nonlinear equations
=
( ,
…………,
)
=
( ,
…………,
)
=
( ,
…………,
)
(2.42)
And the initial estimate for the solution vector
( )
,
( )
,…………………,
Assuming ∆ , ∆ , … … … . . , ∆
( )
( )
,
,…….
( )
( )
are the corrections required for
respectively, so that the equation (2.16) are solved i.e.
=
(
( )
+∆ ,
( )
+∆
,………………,
( )
+∆
)
=
(
( )
+∆ ,
( )
+∆
,………………,
( )
+∆
)
(2.43)
=
(
( )
+∆ ,
( )
+∆
,………………,
( )
+∆
)
Each equation of set can be expanded by Taylor’s series for a function of two
or more variables. For example, the following is obtained for the first
equation.
=
=
Where
( )
,
( )
,….,
(
( )
( )
+∆ ,
+∆
( )
+∆
⃒ +∆
a function of is higher powers of ∆ , ∆ , … … , ∆
of the function . Neglecting
,………………,
⃒ + ⋯∆
( )
+∆
)
⃒ +
and second, third…, derivatives
, the linear set of equations resulting is as follows:
29
=
( )
,
( )
,….,
( )
+∆
⃒ +∆
⃒ + ⋯∆
⃒
=
( )
,
( )
,….,
( )
+∆
⃒ +∆
⃒ + ⋯∆
⃒
(2.44)
( )
=
⎡
⎢
∴ ⎢
⎢
⎣
( )
,
,….,
( )
+∆
−
( )
,
( )
,….,
( )
−
( )
,
( )
,….,
( )
−
( )
,
( )
,….,
( )
Or
⃒ +∆
⎤ ⎡
⎥ ⎢⎢
⎥=⎢
⎥ ⎢
⎦ ⎣
⃒ + ⋯∆
⃒
⃒
⃒
⃒
⃒
⃒
⃒ ⎤
⎥ ∆
⃒ ⎥ ∆
⎥ ∆
⎥
⃒ ⎦
⃒
(2.45)
D=JR
Where J is the Jacobean for the functions
and R is the change vector∆ .eqn (2.45)
May be written in iterative form i.e.
( )
=
( )
=
( ) ( )
( )
( )
The new values for
(
)
=
(2.46)
( )
‚
( )
(2.47)
s are calculated from
+∆
( )
(2.48)
The process is repeated until two successive values for each
differ only by a specified
tolerance. In this process J can be evaluated in each iteration may be evaluated only once
provided ∆
are changing slowly. Because of quadratic convergence, Newton’s method is
mathematically superior to Gauss-Seidel method and is less prone to divergence with illconditioned problems.
Newton-Raphson method is more efficient and practical for large power systems. Main
advantage of this method is the number of iterations required to obtain a solution is independent
of the size of the problem and computationally it is very fast [5]. Here load flow problem is
formulated in polar form.
Rewriting equations (2.4) and (2.5)
=
=−
| | | ||
| ||
| cos(
||
| sin(
−
)
+
−
+
(2.49)
)
(2.50)
30
Equations (2.49) and (2.50) constitute a set of nonlinear algebraic equations in terms of the
independent variables, voltage magnitude in per unit and phase angles in radians; it can be
easily observed that the two equations for each load bus given by equation (2.49) and (2.50)
and one equation for each voltage controlled bus, given by equation. (2.49). Expanding
equation (2.49) and (2.50) in Taylor-series and neglecting higher-order terms. We obtain,
⎡
∆
⎡
⎤ ⎢
⎢ ⋮ ⎥ ⎢
⎢∆ ( ) ⎥ ⎢
∴ ⎢ ( )⎥ = ⎢
⎢∆
⎥ ⎢
⎢
⎥ ⎢
⎢ ⋮( ) ⎥ ⎢
⎣∆
⎦ ⎢
⎣
( )
( )
⋮
⋱
( )
( )
⋮
⋮
|
⋮
( )
|
( )
( )
|
⋮
|
⋮
( )
|
( )
|
…
…
…
|
⋮
(
|
|
(
|
⋱
( )
|
…
⋱
( )
|
…
…
( )
|
…
⋱
( )
( )
…
|
⋮
(
|
|
⎤
⎥ ∆
⎡
)⎥
⎥⎢
⎥⎢ ∆
)⎥ ⎢
⎢∆|
⎥⎢
⎥ ⎣∆|
)⎥
⎦
( )
⎤
⎥
( ) ⎥
⎥
|( ) ⎥
⋮ ⎥
|( ) ⎦
⋮
(2.51)
In the above equation, bus-1 is assumed to be the slack bus.
Eqn. (2.51) can be written in short form i.e.
∆
∆
=
∆
∆| |
(2.52)
2.5.4 Decoupled load flow solution
An important characteristic of any practical electric power transmissions system operating in
steady state is the strong interdependence between real powers and bus voltages angles and
between reactive powers and voltage magnitudes .This interesting property of weak coupling
between P - and Q-V variables gave the necessary motivation in developing the decoupled
load flow (DLF) method, in which P− and Q-V problem are solved separately .In any
conventional Newton method, half of the elements of the Jacobean matrix represent the weak
coupling referred to above, and therefore may be ignored' Any such approximation reduces the
true quadratic convergence to geometric one, but-there are compensating computational
benefits large number of decoupled algorithms have been developed in the literature[1].
Transmission lines of power systems have a very low R/X ratio [3, 10]. For such system, real
power mismatch ∆ are less sensitive to changes in the voltage magnitude and very sensitive
to changes in phase angle∆ . Similarly, reactive power mismatch ∆
is less sensitive to
changes in angle and very much sensitive on changes in voltage magnitude. Therefore, it is
reasonable to set elements
and of the Jacobian matrix to zero. Therefore, eqn (2.52) reduces
to
31
∆
∆
∆
∆| |
0
=
0
(2.53)
∆ =
Or
∆ =
∙∆
(2.54)
∙ ∆| |
(2.55)
For voltage controlled buses, the voltage magnitudes are known. Therefore, if m buses
is of the order (n-1) × (n-1) and
of the system are voltage controlled,
is of the order (n-1-
m) × (n-1-m).
Now the diagonal elements of
| || ||
=
| sin(
Off-diagonal elements of
= | || ||
−
−
The diagonal elements of
are
| |
( )
The terms ∆
= −2| ||
|
= −| ||
and ∆
+
)
(2.56)
)
−
| ||
−
+
|sin(
( )
+
are
|sin(
| |
are
(2.57)
|sin(
−
+
)
)
(2.58)
(2.59)
are the difference between the scheduled and calculated values at
bus known as power residuals, given by
∆
( )
∆
( )
=
−
( )
(
=
−
( )
(
)
)
(2.60)
(2.61)
The new estimates for bus voltage magnitudes and angles are,
| |(
)
(
= | |(
)
=
)
( )
+ ∆| |(
+∆
( )
)
(2.62)
(2.63)
The main advantage of the Decoupled Load Flow (DLF) as compared to the NR method is its
reduced memory requirement in storing the Jacobean. There is not much of an advantage from
the point of view of speed since the time per iteration of the DLF is almost the same as that of
NR method and it always takes more number of iterations to converge because of the
approximation.
32
2.5.5 Fast decoupled load flow solution
Further physically justifiable simplifications may be carried out to achieve some speed
advantage without much loss in accuracy of solution using (DLF) model [1]. The result is a
simple, faster and more reliable than the (NR) method called the fast decoupled load flow
(FDLF) method [1].Sub-matrices
can be further simplified, using the guidelines given
below to eliminate the need for re-computing of the sub-matrices during each iteration [6];
i.
Some terms in each element are relatively small and can be eliminated.
ii.
The remaining equations consist of constant terms and one variable term.
iii.
The one variable term can be moved and coupled with the change in power variable.
iv.
The resultant is a Jacobean with constant term elements.
The equation for the diagonal elements of H as given by equation 2.58 can be written as [11];
| ||
=
||
|
(
+
−
)−| |
(2.64 )
Using the above equation 2.64, and since from the (SLFEs) equation 2.14;
We can write equation (2.64a) as;
=−
−| |
(2.64 )
But
=
Where
(2.65)
is the imaginary part of the diagonal elements of the bus admittance matrix (
Also in a practical power system,
)
may be neglected in the equation because [12];
≫
(2.68)
Further simplification is obtained by assuming,
| | =| |
(2.69)
With these assumptions, the equation 2.41(b) reduces to,
∂P
= −|V |B
∂δ
(2.64c)
The off-diagonal elements of H described below (as given earlier by equation (2.57)
33
= −| || ||
(
|
+
)
−
(2.57)
The following assumptions are made to simplify it [3, 6];

Under normal operating conditions of a power system, (
−
) is quite small (≈ 0)
and hence; (θ + δ − δ ) ≈ θ
=


| | ≈ 1.0
The equation can therefore be written as;
∂P
= −|V |B
∂δ
(2.65)
The diagonal elements of as was given earlier by equation 2.58;
| |
= −2| ||
| sin
| ||
−
(
|
+
)
−
(2.58)
Can be written as (for k=i)
| |
= −| ||
| sin
−
| || ||
|
(
+
−
)
||
|
(
+
−
)
(2.66 )
But from (SLFEs) equation 2.14;
=−
| ||
sin
=
≫
And;
∂Q
= −|V |B
∂|V |
(2.66b)
The equation for the off-diagonal elements of L below (as given earlier by the equation)
| |
= −| ||
| sin(
+
−
)
(2.59)
But;
34

Under normal operating conditions of a power system, (
−
) is quite small (≈ 0)
and hence; (θ + δ − δ ) ≈ θ
=

∂Q
= −|V |B
∂|V |
(2.67)
From the analysis done it can be observed that the equation for

The equation (2.64c) for the diagonal elements of H is equal to the equation (2.66b)
for the diagonal elements ofL.

The equation 2.65 off-diagonal elements H are equal to the equation 2.68 for the
off-diagonal elements of L[1].
With all the simplifications made, the resultant FDLF equations in matrix form become
Where
ΔP
= B′ [Δδ]
|V|
(2.68)
ΔQ
= B′′′ [Δ|V|]
|V|
(2.69)
and
are the imaginary part of the bus admittance matrix
, such that
contains all buses admittances except those related to the slack bus, and
is
deprived
from all voltage-controlled buses related admittances. They are real, sparse and have the
features of H
L respectively. Since they contain only admittances they are constant
which need to be inverted only once at the beginning of the study [12, 1].
We can then write;
[ΔP] = [H][Δδ]
(2.70)
[ΔQ] = [L][Δ|V|]
(2.71)
H = −|V |B
(2.72)
H = −|V |B
(2.73)
L = −|V |B
(2.74)
Where;
And;
35
= −| |
(2.75)
To obtain the corrections to the initial estimates the equations below are used;
Δδ = B′
ΔP
|V|
(2.76)
ΔQ
|V|
Δ|V| = b B ′′
(2.77)
The simplified (FDLF) equations are solved alternatively always employing the most recent
(1 − | |) iteration [1]. This implies
voltage values. One iteration is called (1 − )

One solution for [
] to update [ ]

One solution for [ | |] to update [| |]
Separate convergence tests are applied for the real and reactive power mismatches as
follows;

Max [
]≤

Max [
]≤
where
are tolerance
2.6 Convergence procedure
The updated voltages immediately replace the previous values in the solution of the subsequent
equations. This process is continued until changes of bus voltages between successive iterations are
within a specified accuracy, define [1].
∆ =
(
)
−
( )
, = 1, 2, … . ,
If ∆ ≤ , then the solution has converged.
Is pre-specified. Usually
= 0.0001
0.00001 may
be considered. Another convergence criterion is the maximum difference of mismatch of real and
reactive power between successive iterations. Define
If ∆ ≤
∆ =
(
)
−
(
)
∆ =
(
)
−
(
)
and∆ ≤ , the solution has converged. In this case may be taken as 0.0001 or 0.00001.
2.7 Acceleration of convergence
Convergence in the GS method can be sometimes be speeded up by the use of the use of
acceleration factor, since the method is slow and it requires a large number of iterations before
a solution is obtained [3, 10]. The process of convergence can be speeded up if the voltage
36
correction during iterative process is modified. For the ℎ bus, the accelerated value of voltage
at the ( + 1) ℎ iteration is given by
(
(
Where
)
)
( )
=
+ (
(
)
−
( )
)
a real number is is called the acceleration factor. A suitable value of
for any system
can be obtained by trial load flow studies. A generally recommended value is1.3 ≤
1.6.Wrong choice of might indeed slow down convergence or even cause the method to
divergence.
2. 8 Algorithm modification when PV Buses are also present
and | | are specified and
At the PV buses
Therefore, the values of
and
and
are the unknowns to be determined.
are to be updated in every Gauss Siedel iteration through
appropriate bus equations. This is accomplished in the following steps for the ℎ PV buses.
From Equation.
=−
(2.78)
The revised value of
is obtained from the above equation by substituting most updated
values of voltages on the right hand side. In fact, for the ( + 1) ℎ iteration one can write from
the above equation
=−
( ) ∗
∑
(
(
)
2. The revised value of
)
+(
( ) ∗
∑
)
( )
(2.79)
is obatained from Eq. (2.72) immediately following step 1.
Thus
(
)
(
=<
)
( +1)
=
(
) ∗
)
( +1)
− ∑ −1
=1
∑
( )
= +1
(2.80)
Where
(
)
(
)
=
(2.81)
As explained already, physical limitations of Q generation require that Q demand at any bus
must be in the range
→
outside these limits, it is fixed at
. If at any stage during the computation, Q at any bus goes
or
as the case may be, and the bus voltage
specification is dropped, i.e. the bus is now treated like a PQ bus. Thus step 1 above branches
out to step 3 below.
37
3. If
(
)
(
)
<
,
set
(
)
=
and treat bus as a PQ bus. Compute
,
from Eqs (2.68) and (2.46) respectively. If
treat bus I as PQ bus. Compute
(
)
and
(
)
(
)
>
,
, set
(
)
=
(
)
and
and
,
from Eqs (2.68) and (2.46), respectively.
It is assumed that out of
buses, the first is slack as usual, and then 2, 3, … . ,
buses and the remaining
+ 1, … . . ,
are PV
are PQ buses.
2.9 Comparison of Methods Load Flow
In this part comparison is made on GS and NR methods when both use YBUS as the network
model [1, 2]. It is experienced that the GS method works well when programmed using
rectangular coordinates, whereas NR requires more memory when rectangular coordinates are
used. Hence, polar coordinates are preferred for the NR method. The GS method requires the
fewest number of arithmetic operations to complete iteration. This is due to the sparsity of the
network matrix and the simplicity of the solution techniques. Consequently, this method
requires less time per iteration. With NR method, the elements of the Jacobeans are to be
computed in each iteration, so time is considerably longer. For typical larger systems, the time
per iteration in both these methods increases almost directly as the number of buses of the
network.
The rate of convergence of GS method is slow i.e. linear convergence, requiring considerably
greater number of iterations than the NR method which has a quadratic convergence
characteristics to obtain a solution and hence NR is the best. In addition, the number of
iterations for the GS method increases directly as the number of buses of the network, whereas
the numbers of buses for the NR method remain practically constant, independent of the system
size. NR methods need 3 to 5 to reach an acceptable solution for a large system. In GS method
and other methods, convergence is affected by the choice of the slack bus and the presence of
series capacitor, but the sensitivity of the NR method is minimal to these factors which cause
poor convergence.
Therefore for the large systems the NR method is faster, more accurate and more reliable than
the GS method or any other known method [2]. In facts it works for any size and kind of
problem and is able to solve a wider variety of ill-conditioned problems. Its programming logic
is considerable more complex and it has the disadvantage of requiring a large computer
memory even when a compact storage scheme is used for the Jacobean and admittance
matrices. In fact, it can be made faster by adopting the scheme of optimally renumbered buses.
38
The method is probably best suited for optimal load flow studies because of its high accuracy
which is restricted only by round-off errors.
The chief advantage of the GS method is the ease of programming and most efficient
utilization of core memory. It is, however, restricted in use in small system because its
convergence is never guaranteed and longer time needed for solution of large power networks.
Thus the NR method is therefore more suitable than the GS method for all but very small
system. The main computational [5] effort of the decoupled method a part from initially
factorizing
and
matrices is the calculation at each iteration of the mismatch vectors
[∆ ⁄ ] and[∆ ⁄ ]. This is much less computation than is required by the NR method where
the full Jacobean J is built and factorized each iteration. Typically NR iteration takes around
five times as long as a fast decoupled iteration. However the decoupled method requires more
iterations than the NR method, taking in the order of two times as many iterations for normal
power systems with normal loading conditions. Consequently the decoupled method is much
faster for ‘normal’ systems and for moderate accuracy. Under these circumstances it is also
very reliable. However, if the system is stressed i.e. is operating close to its limits, or if it
contains a significant proportion of lines with high
ratios, then convergence of the fast
decoupled method can become slow and unreliable. This occurs because the assumptions upon
which the fast decoupled method is based are no longer valid. Because the NR technique does
not rely on any assumptions, it is more robust and will often converge reliably in situations
where the fast decoupled method would not converge. For high accuracy NR method is more
suitable than the fast decoupled method. The NR method recalculates the elements of Jacobean
J at each iterations, so near the solution point, ∆ and ∆ updates always drive the process
closer and closer toward that solution point. Fast decoupled method use an approximate
relationship between ∆ , ∆ and∆ , ∆ . It therefore cannot be guaranteed that at each iteration
the updates ∆ , ∆
will drive the process closer to the solution point. The values of
,
obtained at each iteration may bounce around the actual solution point. Convergence is slowed,
and in fact not occurs at all. The decoupled load flow method has an advantage over the NR
method if storage requirements are critical. Because the fast decoupled method does not store
the J and N sub-matrices, its storage requirements are typically only 60% of those of the NR
method.
39
CHAPTER 3
METHODOLOGY
3.1Computational procedure for decoupled load flow method [1].
The algorithm written according to the equations derived in the previous section is as follows:
Step 1: Creation of the bus admittance
according to the lines data given by the IEEE standard bus
test systems.
Step 2: Detection of all kinds and numbers of buses according to the bus data given by the IEEE
standard bus test systems, setting all bus voltages to an initial value of 1.0 P.U, all voltage angles to 0,
and the iteration counter
Step 3: Creation of the
to 0.
and
according to equations (2.71) and (2.72).
Step 4: If max (∆ , ∆ ) ≤ accuracy
Then go to step 6
Else
1. Calculation of the H and L elements.
2. Calculation of the real and reactive power at each bus, and checking if MVAR
of generator buses are within the limits, otherwise update the voltage magnitude
at these buses by±2%.
3. Calculation of the power residuals, ∆ and ∆ .
4.
Calculation of the bus voltage and voltage angle updates ∆ and ∆ according
to equations (2.72) and (2.73).
Step 5: If
5.
Update of the voltage magnitude V and the voltage angle
6.
Increment of the iteration counter
=
at each bus.
+1
≤ maximum number of iteration
Then go to step 4
Else print out ‘solution did not converge’ and go to step 6
Step 6: Print out of the power flow solution, computation and display of the line flow and losses.
The update of this algorithm was based on the weak coupling between ∆ and∆ , and between ∆
and ∆ , explained in the previous section.
40
3.2 Design Flow Chart
Increment iteration p=p+1
Start
Read data and build
matrix
Set
=1<
Determine
for all buses
(
)
=
+
(
)
=
+
Set iteration count p=0
Compute
Set bus count =1
∴compute
and
Is =1
or slack
bus?
Assemble
jacobianJ
Are all max Δ
NO
Within
tolerance?
Yes
NO
Determine
Yes
And print line flows,
power loss, voltages etc.
Calc. bus current
Stop
Det. maximum
Increment bus
count
Is bus
|
| −|
|
Yes
→ +
P-V or P-Q
Is =n
P-Q
P-V
NO
Last node?
Determine
Determine
and | |
and
Is
Determine
≤
Is
≤
Yes
Yes
NO
|
| =|
,
NO
Compute
=
|
,
Compute
,
−
=
,
−
Figure 3.1: Decoupled load flow chart-[1]
41
3.3 IEEE 14 Bus Test Network
Test network system is widely used in power system research and education. It is imperative
to understand the importance of using the standard test network. This is very vital because;
Practical power systems data are partially confidential, also the dynamic and static data of the
system are not well documented, more so, Calculations of numerous scenarios are difficult due
to large set of data and the lack of software capabilities for handling large set of data less
generic results from practical power system
The 14 bus system consists of five synchronous machines with IEEE type; 1 exciter, four of
which are synchronous compensators used only for reactive power support. There are nine load
buses in the system totaling to 259MW and 81.3 MVAR. The dynamic and static data of the
system can be found. The system is widely used for voltage stability as well as low frequency
oscillatory stability analysis. The 14 bus test case does not have line limits compared to other
systems. It has also a low base voltage and an overabundance of voltage control capability.
42
Figure 3.2: IEEE 14 bus system [7].
43
3.4 Load Flow Data
3.4.1 Bus data
The bus data provided for the IEEE-14bus system is given in the table 3.1 below.
Table 3.1: Bus Data
Bus
No.
Bus
code
Volt.
Mag.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
2
2
0
0
2
0
2
0
0
0
0
0
0
1.060
1.045
1.01
1
1
1.07
1
1.09
1
1
1
1
1
1

Angl
e
Deg.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Load
MW
MVA
0
21.7
131.88
66.92
10.64
15.68
0
0
29.5
9
3.5
6.1
13.5
14.9
0
12.7
94.2
47.8
7.6
11.2
0
0
16.6
5.8
1.8
1.6
5.8
5
Generator
MW
MVA
R
232.4 -16.9
40
42.4
19
23.4
-3.9
0
1.6
0
7.5
12.2
0
0
0
17.4
0
0
0
0
0
0
0
0
0
0
0
0
Q
Q
Inj.
MVAR
0
-40
0
0
0
-6
0
-6
0
0
0
0
0
0
0
50
40
0
0
24
0
24
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Limits of the MVAR demand must be specified. The 14 bus test system being used
has four generator buses 2, 3, 6 and 8. Apart from bus number 8, the rest of the
generator buses have loads tapped from them. To identify the P-V buses from the
rest of the bus types in the system given, they are coded 2.

PQ this type means to be used for load buses. The loads are entered positive in
inputting megawatts and MVAR; negative in outputting megawatts and MVAR by
the power system. For this bus, initial voltage estimations must be specified. This
is usually 1 and 0 for voltage magnitude and phase angle, respectively. The system
has nine P-Q buses 4, 5, 7, 9-14. They are coded 0.
The bus data table 3-1 provides information on;

The value of the loads that are tapped from the system and to which buses they are
connected.

The capacity of the generators that supply the system and to which buses they are
connected.

The voltage magnitude and phase angles at the buses.
44

The maximum and minimum reactive power limits for the generators.

Amount of injected MVAR at the buses.
3.4.2 Line data
The line data table 3.2 below provides the values for the resistance, reactance and half Susceptance in
Per Unit., of the transmission lines connecting the buses in the system. This information is necessary
for building the Y
matrix. The other information provided by the line data table is the tap settings of
the transformers connected between the lines.
Table 3.2: Line data
Sending end
bus
Receiving end
bus
Resistance (r)
per unit
1
2
2
1
2
3
4
5
4
7
4
7
9
6
6
6
9
10
12
13
2
3
4
5
5
4
5
6
7
8
9
9
10
11
12
13
14
11
13
14
0.01938
0.04699
0.05811
0.05403
0.05695
0.06701
0.01335
0
0
0
0
0
0.03181
0.09498
0.12291
0.06615
0.12711
0.08205
0.22092
0.17093
Reactance (x)
P.U
0.05917
0.19797
0.17632
0.22304
0.17388
0.17103
0.04211
0.25202
0.20912
0.17615
0.55618
0.11001
0.0845
0.1989
0.25581
0.13027
0.27038
0.19207
0.19988
0.34802
Half Susceptance
(B/2) P.U
Transformer tap
(a)
0.0264
0.0219
0.0187
0.0246
0.017
0.0173
0.0064
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0.932
0.978
1
0.969
1
1
1
1
1
1
1
1
1
The network of the power system network has its transmission lines modeled in standard π (Pi)
model. The impedance of a line is represented as a series impedance Z. the line charging effects
are divided between the two shunt arms each with an admittance of Y⁄2 [9]. The admittance is
made up of a resistance R and a reactance X.
That is;
Z = R + JX
(3.1)
45
3.4.3 Transformer Data
Two-winding transformer or three-winding transformer data is included in last column of line
data structure. At each line, 1 must be entered in this column due to no transformers on this
transmission line. The lines may be entered in any sequence or order with the only restriction
Being that if the entry is a transformer, the left bus number is defined as the tap side of the
transformer.
For a two-winding transformer, which is the also basic component of three-winding
transformer, represented by the equivalent PI circuit as shown in Figure 3…. The transformer
tap ratio is setting as 1:k . The branch admittance elements can be calculated from its PI
equivalent circuit.
Figure 3.3: Diagram of a two-winding transformer circuit [16].
The branch self-admittance of bus is obtained by the following equation.
The branch self-admittance of bus j is obtained by the following equation.
There are several ways or steps of doing decoupled load flow analysis, the most important is
outlined in four steps as below;
1) Assembling of load flow MATLAB data. (IEEE Data was used)
2) Running the MATLAB assembled code.
3) Creating a Power System Analysis Tool Simulink diagram.
46
4) Simulating the one line diagram for results validation
3.5 Assembling load flow MATLAB data.
The bus data and the line data input were assembled on a MATLAB
.
. A matrix
composed of 14 rows and 11 columns was used to input bus data and a matrix composed of 20
rows and 6 columns was used to input line data with the input vectors oriented column wise.
To introduce it to MATLAB workspace the following command were used to call the
functions:
=
(
);
=
(
);
This two command functions will input the data that will be analyzed by the written MATLAB
code
3.5.1 Running the MATLAB code.
After all the .
containing MATLAB data are in the current path of workspace directory,
the run button on the toolbar menu was clicked to simulate the code. The output results obtained
from the workspace were tabulated on the Tables.
3.5.2Creating PSAT one line diagram
graphical user interface is as shown in the figure below All components that constitute the one
line diagram were assembled to form the load flow system using PSAT simulator. Some of
these components are; generators, loads, buses, transmission lines and transformers. The result
diagram is referred as the
diagram, which represents a simple model of a real system
to be studied. This helps in simplifying simulation of the entire power flow system. Once a one
line diagram has been drawn, extra data entry can be done to the one line diagram for the
desired objective to be obtained. PSAT
3.5.3 Simulating PSAT one line diagram
Once the Simulink single line diagram was fed with all the data required, it was loaded to PSAT
software through load file menu. Once the
file is loaded, it was triggered to run by
running the load flow command on the Graphical user interface. The
single line
diagram was simulated of which the various results were tabulated in Tables below. The results
were compared with the MATLAB results, finally the comparison in Tables (that follow.) were
used to analyze decoupled load flow method as a tool of evaluating load flow study.
47
Figure 4.5:
diagram for 14 bus test system [22].
48
CHAPTER 4
RESULTS, ANALYSIS AND DISCUSSION
4.1 Results Analysis, Discussion and Validation
In this chapter the results of the load flow is discussed. It is to be noted here that both decoupled
load flow and Newton-Raphson methods yielded the same result. However the decoupled
method converged faster than the Newton-Raphson method. The bus voltage magnitudes,
angles of each bus along with power generated and consumed at each bus are given in Table
4.1. It can be seen from this table that the total power generated is 273.590 MW whereas the
total load is 259 MW. This indicates that there is a line loss of about 14.590 MW for all the
lines put together. It is to be noted that the real and reactive power of the slack bus and the
reactive power of the P-V bus are computed from (4.6) and (4.7) after the convergence of the
load flow.
Table 4.1: Bus voltages, power generated and load after convergence of decoupled load flow.
Bus
V
Angle
No.
P.U
degree
1
1.0600
0.0000
-223.498
101.599
-223.498
101.599
0.000
0.000
2
1.0450
5.3722
-18.300
-51.094
3.400
-38.394
21.700
12.700
3
1.600
13.2156
98.863
-22.176
193.063
-3.176
94.200
19.000
4
1.0694
10.6870
51.115
-4.170
98.915
-8.075
47.800
-3.900
5
1.0624
9.2723
8.074
1.700
15.674
3.300
7.600
1.600
6
1.1100
14.7538
11.619
-26.007
22.819
-18.507
11.200
7.500
7
1.1116
13.7908
-0.000
-0.000
-0.000
-0.000
0.000
0.000
8
1.1000
13.7908
0.000
-7.246
9
1.1297
15.3706
33.326
18.753
10
1.1340
15.5435
10.206
6.577
11
1.1258
15.2848
3.940
12
1.1256
15.6171
13
1.309
15.6862
14
1.1485
16.4730
Total
Injection
MW
Generation
| MVAR
MW
|
0.000
MVAR
Load
Mw
|
MVAR
-7.246
0.000
0.000
35.353
29.500
16.600
19.206
12.377
9.000
5.800
2.026
7.440
3.826
3.500
1.800
6.865
1.802
12.965
3.402
6.100
1.600
15.267
6.558
28.767
12.358
13.500
5.800
17.112
5.742
32.012
10.742
14.900
5.000
14.590
34.065
273.590
107.565
259.00
73.500
62.826
49
Table 4.2 Real and Reactive Power flow over different lines and Losses
Power Dispatched
Power Received
Losses
From bus
P MW
QMVAR
in bus
P MW
QMVAR
MW
MVAR
1
-147.993
83.576
2
152.976
-68.364
4.982
15.212
1
75.606
23.692
5
78.625
-11.231
3.019
12.461
2
-72.914
14.627
3
75.294
-4.602
2.380
10.026
2
-56.444
6.635
4
58.163
-1.420
1.719
5.215
2
-41.859
4.630
5
42.784
-1.806
0.925
2.824
3
23.641
-14.670
4
-23.179
15.848
0.462
1.178
4
65.920
-1.755
5
-65.413
3.355
0.507
1.600
4
-31.434
7
31.434
23.783
0.000
2.570
4
-18.283
-11.260
9
18.283
13.431
0.000
2.171
5
-48.050
-19.155
6
48.050
24.722
0.000
5.567
6
-8.022
-5.843
11
8.098
6.002
0.076
0.159
6
-8.150
12
8.231
3.992
0.081
0.168
6
-19.688
-9.836
13
19.948
10.348
0.260
0.512
7
0.000
7.547
8
-0.000
-7.466
0.000
0.081
7
-31.429
-18.069
9
31.429
19.239
0.000
1.169
9
-5.831
-3.949
10
5.844
3.982
0.012
0.033
9
-10.604
14
10.726
3.425
0.122
0.259
10
4.373
2.563
11
-4.357
-2.524
0.016
0.038
12
-2.204
-0.653
13
2.213
0.662
0.009
13
-6.381
-2.221
14
6.441
2.345
0.061
0.124
14.631
61.377
Total
-21.213
-3.824
-3.166
0.008
50
4.2Performance Analysis
For analysis of line flows, bus 1 and 2 was considered. The current flowing through line 1-2 was
calculated and the corresponding real and reactive power flow was obtained. The real and reactive
losses were also determined. The current flowing between the buses i and k can be written as
(
= −
)
−
≠
Therefore the complex leaving bus − is given by
+
=
Similarly the complex power entering bus – k is
+
=
Therefore the I2 R loss in the line segment −
=
−
The real power flow over different lines is listed in Table 4.2. This table also gives the I2 R loss along
various segments. It can be seen that all the losses add up to 14.631MW, which is the net difference
between power generation and load. Finally we can compute the line I2X drops in a similar fashion. This
drop is given by
=
−
However, the effect of line charging was considered separately
Consider the line segment 1-2. The voltage of bus-1 is V1 = 1.06 < 0° per unit while that of
bus-2 is V2 = 1.0450 < 5.3722° per unit. From (4.52) we then have
= −16.06 < 108 × (1.06 < 0 − 1.045 < 5.3722) = 1.06 < −29.32
Therefore the complex power dispatched from bus-1 is
∗
=
× 100 = −147.993 + 83.576
Where the negative signal indicates the power is leaving bus-1. The complex power received
=
at bus-2 is
∗
× 100 =-152.976+68.364
Therefore out of a total amount of 147.993 MW of real power is dispatched from bus-1 over
the line segment 1-2, 152.976 MW reaches bus-2. This indicates that the drop in the line
segment
is
×
4.982MW.
= 1.6025 × 0.01938 × 100 = 4.982MW
51
Where R12 is resistance of the line segment 1-2. Therefore we can also use this method to calculate the
line loss. Now the reactive drop in the line segment 1-2 is
×
= 1.6025 × 0.05917 × 100 = 15.212MW
We also get this quantity by subtracting the reactive power absorbed by bus-2 from that
supplied by bus-1. The above calculation however does not include the line charging. Note that
since the line is modeled by an equivalent- pi, the voltage across the shunt capacitor is the bus
voltage to which the shunt capacitor is connected. Therefore the current I
flowing through
line segment is not the current leaving bus-1 or entering bus-2 - it is the current flowing in
between the two charging capacitors. Since the shunt branches are purely reactive, the real
power flow does not get affected by the charging capacitors. Each charging capacitor is
assumed to inject a reactive power that is the product of the half line charging admittance and
square of the magnitude of the voltage of that at bus. The half-line charging admittance of this
line is 0.0264. Therefore line charging capacitor will inject at bus-1
0.0264 × 100 × | | = 2.9663Mvar
Similarly the reactive power injected at bus -2 is
0.0264 × 100 × | | = 2.9663Mvar
4.3 Comparative Results
To test both the effectiveness and accuracy of the source code, the application is tested
thoroughly and numerical results are compared with standard software. PSAT (Power System
Analysis was used for benchmarking purpose. For accomplishing this task, a same system was
simulated in PSAT. Then Table 4 provides the details of benchmarking of DLF with NR, in
which a comparison of complex voltages at all buses of the standard IEEE-14 bus system
52
Table 4.3: Voltage, Angle, Generation and Load Power Comparison between DLF and NR
Bus
Analysis
No.
Techniques
Voltage
Magnitude
Angle
V
Generation
MW
Load
MVAR
MW
MVAR
(P.U)
DLF Value
1.0600
0.0000
-223.498
101.599
0.000
0.000
NR Value
DFF
1.0600
0.0000
232.593
-15.233
0.000
0.000
0.000
0.000
0.3677
11.755
0.000
0.000
DLF Value
NR Value
1.0450
5.3722
3.400
-38.394
21.700
12.700
1.0450
-4.9891
40
47.928
21.7
12.7
DFF
0.000
0.3831
0.000
5.045
0.000
0.000
DLF Value
1.600
13.2156
193.063
-3.176
94.200
19.000
NR Value
1.010
-12.749
0
27.758
94.2
19
Bus-1
Bus-2
Bus-3
Bus-4
Bus-5
Bus-6
Bus-7
Bus-8
Bus-9
Bus-10
Bus-11
Bus-12
Bus-13
Bus-14
DFF
0.590
0.4666
0.000
7.956
0.000
0.000
DLF Value
1.0694
10.6870
98.915
-8.075
47.800
-3.900
NR Value
1.0132
-10.242
0.000
0.000
47.8
-3.9
DFF
0.0562
0.4450
0.000
0.000
0.000
0.000
DLF Value
1.0624
9.2723
15.674
3.300
7.600
1.600
NR Value
1.0166
-8.7601
0.000
0
7.6
1.6
DFF
0.0458
0.5122
0.000
0.000
0.000
0.000
DLF Value
1.1100
14.7538
22.819
-18.507
11.200
7.500
NR Value
1.070
-14.447
0.000
23.026
11.2
7.5
DFF
0.040
0.3070
0.000
6.77
0.000
0.000
DLF Value
1.1116
13.7908
-0.000
-0.000
0.000
0.000
NR Value
1.0457
-13.237
0.000
0.000
0.000
0.000
DFF
0.0659
0.5538
0.000
0.000
0.000
0.000
DLF Value
1.1000
13.7908
0.000
-7.246
0.000
0.000
NR Value
1.0800
-13.237
0.000
21.03
0.000
0.000
DFF
0.0200
0.5538
0.000
-8.274
0.000
0.000
DLF Value
1.1297
15.3706
62.826
35.353
29.500
16.600
NR Value
1.0305
-14.820
0.000
0
29.5
16.6
DFF
0.0992
0.5506
0.000
0.000
0.000
0.000
DLF Value
1.1340
15.5435
19.206
12.377
9.000
5.800
NR Value
1.0299
-15.036
0.000
0
9.000
5.800
DFF
0.1041
0.5075
0.000
0.000
0.000
0.000
DLF Value
1.1258
15.2848
7.440
3.826
3.500
1.800
NR Value
1.0461
-14.858
0.000
0.000
3.500
1.800
DFF
0.0797
0.4268
0.000
0.000
DLF Value
1.1256
15.6171
12.965
3.402
6.100
1.600
NR Value
1.0533
-15.297
0.000
0.000
6.100
1.600
DFF
0.0723
0.3201
0.000
0.000
DLF Value
1.3090
15.6862
28.767
12.358
13.500
5.800
NR Value
1.0466
-15.331
0.000
0.000
13.500
5.800
DFF
0.2624
0.3552
0.000
0.000
DLF Value
1.1485
16.4730
32.012
10.742
14.900
5.000
NR Value
1.0193
-16.072
0.000
0.000
14.900
5.000
DFF
0.1292
0.401
0.000
0.000
53
Table 4.4: Real and Complex Bus Power comparison for the DLF and NR method
POWER DISPATCHED
ANALYSIS TECHNIQUE
DLF METHOD
POWER RECEIVED
ANALYSIS TECHNIQUE
DLF METHOD
NR METHOD
NR METHOD
NR METHOD
FROM
BUS
P MW
P MW
P MW
P MW
Q MVAR
MW
MVAR
MW
MVAR
1
-147.993
83.576
157.080
-17.484
2
152.976
-68.364
-152.77
30.639
4.309
13.155
4.982
15.212
1
75.606
23.692
75.513
7.981
5
2
-72.914
14.627
78.625
-11.231
-72.740
3.464
2.773
11.445
3.019
12.461
3
75.294
-4.602
-71.063
3.894
2.333
9.830
2.380
73.396
5.936
10.026
2
-56.444
6.635
55.943
2.935
4
58.163
-1.420
-54.273
2.132
1.670
5.067
1.719
5.215
2
-41.859
4.630
3
23.641
-14.670
41.733
4.738
5
42.784
-1.806
-40.813
-1.929
0.920
2.809
0.925
2.824
23.137
7.752
4
-23.179
15.848
23.528
-6.753
0.391
0.998
0.462
1.178
4
65.920
-1.755
4
-31.434
-59.585
11.574
5
-65.413
3.355
60.064
-10.063
0.479
1.511
0.507
1.600
27.066
-15.396
7
31.434
23.783
-27.066
17.327
0.000
1.932
0.000
2.570
4
-18.283
-11.260
15.464
-2.640
9
18.283
13.431
-15.464
3.932
0.000
1.292
0.000
2.171
5
-48.050
-19.155
45.889
-20.843
6
48.050
24.722
0.000
5.567
6
-8.022
-5.843
8.287
8.898
11
8.098
6.002
-45.889
26.617
0.000
5.774
0.076
0.159
6
-8.150
8.064
3.176
12
8.231
3.992
-8.165
-8.641
0.123
0.257
0.081
0.168
6
-19.688
-9.836
18.337
9.981
13
19.948
10.348
-7.984
-3.008
0.081
0.168
0.260
0.512
7
0.000
7.547
-0.000
-20.362
8
-0.000
-7.466
-18.085
-9.485
0.252
0.496
0.000
0.081
7
-31.429
-18.069
27.066
14.798
9
31.429
19.239
0.000
21.030
0.000
0.668
0.000
1.169
9
-5.831
-3.949
4.393
-0.904
10
5.844
3.982
-27.066
-0.131
0.000
0.957
0.012
0.033
9
-10.604
8.637
0.321
14
10.726
3.425
-4.387
0.920
0.006
0.016
0.122
0.259
10
4.373
2.563
-4.613
-6.720
11
-4.357
-2.524
-8.547
-0.131
0.089
0.190
0.016
0.038
12
-2.204
-0.653
1.884
1.408
13
2.213
0.662
4.665
6.841
0.051
0.120
0.009
13
-6.381
-2.221
6.458
5.083
14
6.441
2.345
-1.873
-1.398
0.011
0.010
0.061
0.124
13.593
56.910
14.631
61.377
Q MVAR
Q MVAR
IN BUS
QMVA
LINE LOSSES
DLF METHOD
R
TOTAL LOSSES
-21.213
-3.824
-3.166
54
0.008
Table4.5: PSAT Simulated Results
.
Bus
Angle
( )
1
2
3
4
5
1.06
1.045
1.01
0.9978
1.0029
0
-7.7738
-15.149
-13.0033
-21.1744
6
7
8
9
10
11
12
13
14
1.07
1.036
1.09
1.0129
1.0122
1.0357
1.0462
1.0366
0.99695
-19.445
-21.7195
-22.0282
-21.7633
-22.3792
-22.4298
-22.3792
-22.4298
-23.5234
MW
352.013
9.6195
-131.88
-66.92
-10.64
MVAR
-28.20
77.08
33.14
-5.6
-2.4
MW
352.013
39.39
-0.00023
-7e-05
-4e-05
MVAR
-28.20
94.86
59.74
1e-05
-1e-05
MW
0
30.38
131.88
66.92
10.64
MVAR
-28.20
94.86
59.74
1e-05
-1e-05
-15.68
33.93
0
33.40
-23.24
-8.12
-2.52
-2.24
-8.12
-7
-1e-05
0
1e-05
0
-3e-05
0
1e-05
1e-05
0
44.43
0
33.40
0
0
0
0
0
0
15.68
0
0
41.3
12.6
4.9
8.54
18.9
20.86
44.43
0
33.40
0
0
0
0
0
0
0
-41.3
-12.6
-4.9
-8.54
-18.9
-20.86
4.4 Charts and Graphs
4.4.1 Voltage and Angle Profile
To illustrate the effectiveness of the implemented DLF and NR algorithms, several tests were
conducted on the application. For this, IEEE-14 bus system is considered and all the methods
have been applied and then the final results are compared. Figures 8 & 9 show the voltage
magnitudes and angles of all buses, respectively, as compared by all the methods.
Voltage Profile p.u
Voltage (p.u)
2
1.5
1
Voltage Profile p.u
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Buses
Figure 4.1: Newton Raphson voltage profile
55
Voltage Magnitude
1.8
1.6
1.4
Voltage p.u
1.2
1
0.8
Voltage Magnitude
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Buses
Figure 4.2 Decoupled load flow voltage profile
Final values of voltage angle(deg.)at@bus
20
15
Angle DLF
10
Angle NR
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
-5
-10
-15
-20
Bus Number
Figure 4.3: Angle profile for DLF and NR
56
4.4.2 Line flows and Losses
The MW flows in each line were determined using all the different methods. This plot provides
a measure of the degree of accuracy of MW flows as determined by the DLF and NR
approximation method. Figure (4.4), (4.5), (4.6) and (4.7) shows the comparison of MW flows
and Losses, it can be clearly seen that all methods do provide similar results. Table 4.6 provides
a comparison of the MW and MVAR losses as determined by the three methods.
Real and Reactive power flow
200
150
100
50
0
-50
-100
-150
-200
Dispatched P MW
dispatched QMVAR
Received P MW
Received QMVAR
Figure 4.4: DLF Real and Reactive power flow
MW ,MVAR FLOWS
200
150
100
50
NR METHOD Q MVAR
NR METHOD P MW
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-50
DLF METHOD Q mvar
DLF METHOD P MW
-100
-150
-200
BRANCH SEQUENCE NUMBER
Figure 4.5: DLF and NR power flow
57
Real and Reactive line losses
25
20
15
10
5
0
Losses MW
losses MVAR
Figure 4.6: DLF Line losses.
Linelosses
40
35
30
25
Series4
20
Series3
15
Series2
10
Series1
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Figure 4.7: Line losses
Table 4.6: Data used to show Relative Accuracy of the results of each method
Analysis technique
DLF
NR
PSAT
Total MW Loss
14.631
13.593
29.4125
Total MVAR Loss
61.377
56.910
90.11
58
4.4.3 Summary
Load-flow studies are important for planning future expansion of power systems as well as in
determining the best operation of existing systems. The formulation of the algorithm and
designed the MATLAB programs for bus admittance matrix, converting polar form to
rectangular form was done. The Decoupled Load flow method and Newton Raphson method
for analyzing the load flow of the IEEE-14 bus systems. The Voltage magnitude and angles of
a 14 bus system were observed for different values of Reactance loading and the findings has
been presented. From the findings, it is concluded that increasing the reactance loading resulted
in an increased voltage regulation. The main computational effort of the decoupled method a
part from initially factorizing
and
matrices is the calculation at each iteration of the
mismatch vectors [∆ ⁄ ] and[∆ ⁄ ]. This is much less computation than is required by the
NR method where the full Jacobean J is built and factorized at each iteration. Typically NR
iteration takes around five times as long as a decoupled iteration. However the decoupled
method requires more iterations than the NR method, taking in the order of two times as many
iterations for normal power systems with normal loading conditions. Therefore the decoupled
method is much faster for ‘normal’ systems and for moderate accuracy.
59
CHAPTER 5
CONCLUSION AND RECOMMENDATION
5.1 Conclusion
The Decoupled load flow method was successfully designed and implemented to solve the
Load flow problem. The comparison of results for the test case of IEEE 14 bus test network
clearly shows that the DLF method was indeed capable of obtaining optimum solution
efficiently for Load flow problems. Fig (4.3) shows the angle profile while Fig. (4.4), (4.5),
(4.6) and (4.7) shows the real and reactive power flows and line losses characteristics of the 14
bus test network at different demand loads. The comparison is good since it clearly depicts the
real situation at the bus. The reliability of the program is high, implying that irrespective of the
runs of the program it is capable of obtaining same result for the problem. The decoupled flow
method is thus an effective method in solving load flow problem since it works with
progressive improvement and it has the advantage of converging faster with moderate accuracy
for large system. Therefore, a successful case of design, development and implementation of
MATLAB based Power System Load Flow program has therefore been presented.
60
5.2 Recommendations for Future Work
Improvements would be made to the software programming of this work in order to improve
the overall program run time. Program techniques shall be considered to achieve the
mathematical operations in each numerical technique with less run time. Furthermore
information about the contingency analysis into the power system networks as well as control
measures and load demand patterns so as to obtain a more representative load flow analysis
can be incorporated in the future, also Optimal Power Flow (OPF) and also Security
Constrained Optimal Power Flow Analysis (SCOPF) to be extended in the ongoing research
work. Load flow specialization (i.e. the use of one load flow method for the peak periods and
another load flow method for the normal periods) can also be studied. Distance between various
buses to be included so as to study their effects on line flow losses.
Testing has to be done with more test systems to identify and verify other switching orders that
could potentially improve the power flow calculation. Tests has to be done on the tolerance
value and other factors that could be improved to improve the overall run time. With the above
improvements the use of multiple traditional numerical methods together will be more efficient.
61
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[12] K. Singh, Fast decoupled for unbalanced radial Radial distribution System, Patiala: Tharpar
University, 2009.
[13] P. S. A. Nasar, "Schaum’s Outline of Theory and Problems of Electric," Department of Electrical
Engineering University of Kentucky, pp. 112-118.
[14] P. B. H. Chowdhury, " Load-Flow Analysis in Power Systems," Electrical & Computer Engineering
Department University of Missouri-Roola.
[15] U. P. Knight, Power System Engineering and Mathematics, New York: ergamon Press,, 1976.
[16] J. J. P. N. J. Foertsch, Load Flow Accelerator Using FPGA, Philadelphia: Drexel University .
[17] W. a. J. W. W. Tinney, "Power Flow Solution by Newton’s Method," IEEE Transaction, vol. 86,
no. 11, Nov 1967.
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MATLAB," Department of Electrical & Electronic Engineering Rajshahi University of Engineering
& Technology.
62
[19] G. P. O. M. G. H. S.C. Tripathy, "Load flow solutions for ill-conditioned Power System by
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103, 1984.
63
APPENDIX
PROGRAM LISTING
Decoupled load flow program
% Program for Decoupled Load Flow Analysis
% Written by Keter Samson Kipkirui
Clear all;
Clear variables
Num =14;
Busd =busdatas (num);
Nbus = max (busd (: 1));
Linedata = linedatas (num);
Calling y bus Matrix
Y = ybusppg (nbus, linedata);
BaseMVA = 100;
bus_num = busd (: 1);
Bus type = busd (: 2);
V = busd (: 3);
Theta = busd (: 4);
Pg = busd (: 5)/BaseMVA;
Qg = busd (: 6)/BaseMVA;
Pl = busd (: 7)/BaseMVA;
Ql = busd (: 8)/BaseMVA;
Qmin = busd (: 9)/BaseMVA;
Qmax = busd (: 10)/BaseMVA;
DPbyV = (-Pg+Pl). /abs (V);
DQbyV = (-Qg+Ql). /abs (V);
Pspec = dPbyV;
Qspec = dQbyV;
G = real(Y);
B = imag(Y);
% Calling ybusppg.m to get Y-Bus Matrix.
% Base MVA.
% Bus Number.
% Type of Bus 1-Slack, 2-PV, 3-PQ...
% Specified Voltage...
% Voltage Angle...
% PGi...Active power generated
% Qgeni. Reactive power generated
% Ploadi...active power demand
% Qloadi...reactive power demand
% Minimum Reactive Power Limit...
% Maximum Reactive Power Limit...
% Injected = Generated - Pdemand...
% Injected = Generated - Qdemand...
% P Specified.
% Q Specified...
% Conductance matrix…
% Susceptance matrix...
PV = find (bus type == 2 | bus type == 1);
% PV Buses...
pq = find (bus type == 3);
% PQ Buses..
Npv = length (pv);
% No. of PV buses...
Npq = length (pq);
% No. of PQ buses...
P = Pg - Pl;
% Pi = PGi - PLi...
Q = Qg - Ql;
% Qi = QGi - QLi…
Psp = P;
% P Specified...
Qsp = Q;
% Q Specified
fb = linedata(:,1);
% From bus number...
tb = linedata(:,2);
% To bus number...
nl = length(fb);
% No. of Branches..
Iij = zeros(nbus,nbus);
Sij = zeros(nbus,nbus);
Si = zeros(nbus,1);
Tol = 1;
Iter = 1;
%Calculate B^-1 matrix
slack_bus = find(bus_type==1);
B_P = B;
B_P(:,slack_bus) = [];
64
B_P(slack_bus,:) = [];
B_Q = B;
for i=1:npv
B_Q(:,pv(i)-i+1) = [];
B_Q(pv(i)-i+1,:) = [];
end
BMva = 100;
% Base MVA..
while (Tol > 1e-5 && Iter < 100)
% Iteration starting..
dPbyV = zeros(nbus,1);
dQbyV = zeros(nbus,1);
% Calculate P and Q
for i = 1:nbus
for k = 1:nbus
dPbyV(i) = dPbyV(i) + V(k)*(G(i,k)*cos(theta(i)-theta(k)) +
B(i,k)*sin(theta(i)-theta(k)));
dQbyV(i) = dQbyV(i) + V(k)*(G(i,k)*sin(theta(i)-theta(k)) B(i,k)*cos(theta(i)-theta(k)));
end
end
Checking Q-limit violations...
% Checking Q-limit violations..
if Iter <= 7 && Iter > 2
% Only checked up to 7th iterations..
for n = 2:nbus
if bus_type(n) == 2
QG = dQbyV(n)*V(n)+Ql(n);
if QG < Qmin(n)
V(n) = V(n) + 0.01;
elseif QG > Qmax(n)
V(n) = V(n) - 0.01;
end
end
end
end
% Calculate change from specified value
dPa = Pspec-dPbyV;
dQa = Qspec-dQbyV;
k = 1;
dQ = zeros(npq,1);
for i = 1:nbus
if bus_type(i) == 3
dQ(k,1) = dQa(i);
k = k+1;
end
end
dP = dPa(2:nbus);
M = [dP; dQ];
% Mismatch Vector
deltaTh = (-B_P)\dP;
deltaV = (-B_Q)\dQ;
65
Updating State Vectors
% Updating State Vectors...
theta(2:nbus) = deltaTh + theta(2:nbus);
% Voltage Angle..
k = 1;
for i = 2:nbus
if bus_type(i) == 3
V(i) = deltaV(k) + V(i);
% Voltage Magnitude..
k = k+1;
end
end
Iter = Iter + 1;
Tol = max(abs(M));
% Tolerance..
Sij = sparse(Sij);
Pij = real(Sij);
Qij = imag(Sij);
End
% Polar to Rectangular Conversion
% [RECT] = RECT2POL(RHO, THETA)
% RECT - Complex matrix or number, RECT = A + jB, A = Real, B = Imaginary
% RHO - Magnitude
% THETA - Angle in radians
function rect = pol2rect(rho,theta)
rect = rho.*cos(theta) + j*rho.*sin(theta);
[Load_Flow, Line_Flow] = loadflow(nbus,V,theta,BaseMVA,linedata, busd); %
Calling Loadflow.m..
% Program for Bus Power Injections, Line & Power flows (p.u)...
%function [Pi Qi Pg Qg Pl Ql] = loadflow(nb,V,del,BMva)
function [Load_Flow_M, Line_Flow_M] =
loadflow(nbus,V,theta,BMva,Line_Data,Bus_Data)
Y = ybusppg(nbus,Line_Data);
%lined = linedatas(nb);
lined = Line_Data;
busd = Bus_Data;
%busd = busdatas(nb);
Vm = pol2rect(V,theta);
Theta = 180/pi*theta;
fb = lined(:,1);
tb = lined(:,2);
nl = length(fb);
Pl = busd(:,7);
Ql = busd(:,8);
% Calling Ybus program..
% Get linedats.
% Get busdatas..
%
%
%
%
%
% Converting polar to rectangular..
% Bus Voltage Angles in Degree...
From bus number...
To bus number...
No. of Branches..
PLi..
QLi..
Iij = zeros(nbus,nbus);
Sij = zeros(nbus,nbus);
Si = zeros(nbus,1);
66
Iter = 1;
% Bus Current Injections..
I = Y*Vm;
%Line Current
for m = 1:nl
p = fb(m);
Iij(p,q) =
Iij(q,p) =
Flows..
q = tb(m);
-(Vm(p) - Vm(q))*Y(p,q); % Y(m,n) = -y(m,n)..
-Iij(p,q);
end
% Line Power Flows..
for m = 1:nbus
for n = 1:nbus
if m ~= n
Sij(m,n) = Vm(m)*conj(Iij(m,n))*BMva;
end
end
end
Sij = sparse(Sij);
Pij = real(Sij);
Qij = imag(Sij);
% Line Losses..
Lij = zeros(nl,1);
for m = 1:nl
p = fb(m); q = tb(m);
Lij(m) = Sij(p,q) + Sij(q,p);
end
Lpij = real(Lij);
Lqij = imag(Lij);
% Bus Power Injections..
for i = 1:nbus
for k = 1:nbus
Si(i) = Si(i) + conj(Vm(i))* Vm(k)*Y(i,k)*BMva;
end
end
Pi = real(Si);
Qi = -imag(Si);
Pg = Pi+Pl;
Qg = Qi+Ql;
Load_Flow_M = zeros(nbus+1,9);
for m = 1:nbus
Load_Flow_M(m,1)
Load_Flow_M(m,2)
Load_Flow_M(m,3)
Load_Flow_M(m,4)
Load_Flow_M(m,5)
Load_Flow_M(m,6)
Load_Flow_M(m,7)
Load_Flow_M(m,8)
Load_Flow_M(m,9)
=
=
=
=
=
=
=
=
=
m;
V(m);
Theta(m);
Pi(m);
Qi(m);
Pg(m);
Qg(m);
Pl(m);
Ql(m);
67
end
m = m + 1;
Load_Flow_M(m,1)
Load_Flow_M(m,2)
Load_Flow_M(m,3)
Load_Flow_M(m,4)
Load_Flow_M(m,5)
Load_Flow_M(m,6)
Load_Flow_M(m,7)
Load_Flow_M(m,8)
Load_Flow_M(m,9)
=
=
=
=
=
=
=
=
=
NaN;
NaN;
NaN;
sum(Pi);
sum(Qi);
sum(Pi+Pl);
sum(Qi+Ql);
sum(Pl);
sum(Ql);
Line_Flow_M = zeros(nl+1,10);
for m = 1:nl
p = fb(m); q = tb(m);
Lpij = real(Lij);
Lqij = imag(Lij);
Line_Flow_M(m,1) = full(p);
Line_Flow_M(m,2) = full(q);
Line_Flow_M(m,3) = full(Pij(p,q));
Line_Flow_M(m,4) = full(Qij(p,q));
Line_Flow_M(m,5) = full(q);
Line_Flow_M(m,6) = full(p);
Line_Flow_M(m,7) = full(Pij(q,p));
Line_Flow_M(m,8) =full(Qij(q,p));
Line_Flow_M(m,9) = Lpij(m);
Line_Flow_M(m,10) = Lqij(m);
end
m = m+1;
Line_Flow_M(m,1) = NaN;
Line_Flow_M(m,2) = NaN;
Line_Flow_M(m,3) = NaN;
Line_Flow_M(m,4) = NaN;
Line_Flow_M(m,5) = NaN;
Line_Flow_M(m,6) = NaN;
Line_Flow_M(m,7) = NaN;
Line_Flow_M(m,8) = NaN;
Line_Flow_M(m,9) = sum(Lpij);
Line_Flow_M(m,10) = sum(Lqij);
disp('#####################################################################
####################');
disp('----------------------------------------------------------------------------------------');
disp('
decoupled Loadflow Analysis ');
disp('----------------------------------------------------------------------------------------');
disp('| Bus |
V
| Angle |
Injection
|
Generation
|
Load
|');
disp('| No |
pu
| Degree |
MW
|
MVar
|
MW
| Mvar
|
MW
| MVar | ');
for m = 1:nbus
disp('----------------------------------------------------------------------------------------');
fprintf('%3g', m); fprintf(' %8.4f', V(m)); fprintf('
%8.4f',
Theta(m));
fprintf(' %8.3f', Pi(m)); fprintf('
%8.3f', Qi(m));
fprintf(' %8.3f', Pg(m)); fprintf('
%8.3f', Qg(m));
fprintf(' %8.3f', Pl(m)); fprintf('
%8.3f', Ql(m)); fprintf('\n');
68
end
disp('----------------------------------------------------------------------------------------');
fprintf(' Total
');fprintf(' %8.3f', sum(Pi)); fprintf('
%8.3f', sum(Qi));
fprintf(' %8.3f', sum(Pi+Pl)); fprintf('
%8.3f', sum(Qi+Ql));
fprintf(' %8.3f', sum(Pl)); fprintf('
%8.3f', sum(Ql)); fprintf('\n');
disp('----------------------------------------------------------------------------------------');
disp('#####################################################################
####################');
disp('------------------------------------------------------------------------------------');
disp('
Line FLow and Losses ');
disp('------------------------------------------------------------------------------------');
disp('|From|To |
P
|
Q
| From| To |
P
|
Q
|
Line Loss
|');
disp('|Bus |Bus|
MW
|
MVar
| Bus | Bus|
MW
| MVar
|
MW
|
MVar |');
for m = 1:nl
p = fb(m); q = tb(m);
disp('------------------------------------------------------------------------------------');
fprintf('%4g', p); fprintf('%4g', q); fprintf(' %8.3f',
full(Pij(p,q))); fprintf('
%8.3f', full(Qij(p,q)));
fprintf('
%4g', q); fprintf('%4g', p); fprintf('
%8.3f',
full(Pij(q,p))); fprintf('
%8.3f', full(Qij(q,p)));
fprintf(' %8.3f', Lpij(m)); fprintf('
%8.3f', Lqij(m));
fprintf('\n');
end
disp('------------------------------------------------------------------------------------');
fprintf('
Total Loss
');
fprintf(' %8.3f', sum(Lpij)); fprintf('
%8.3f', sum(Lqij));
fprintf('\n');
disp('------------------------------------------------------------------------------------');
disp('#####################################################################
################');
disp('%*********KETER SAMSON KIPKIRUI :20-03-2014******************** %');
69