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Load flow study is the analysis of a network under steady state operation subjected to inequalityconstraints in which the system operates. Load flow analysis is the backbone of power systemanalysis and design. They are necessary for planning, operation, economic scheduling andexchange of power between utilities. The principal information of power flow analysis is tofind the magnitude and phase angle of voltage at each bus and the real and reactive power flowsin each transmission lines. Therefore, load flow analysis is an importance tool involvingnumerical analysis applied to a power system.
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1 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: DR.N.O. ABUNGU EXAMINER: 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: 28 TH APRIL 2014
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1

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: DR.N.O. ABUNGU EXAMINER: 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

2

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: …………………………………………………………………………

i

DEDICATION

To my family for the endless support and bringing the best of me at early age

ii

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.

iii

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: ………………………

iv

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

v

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

vi

REFERENCES ............................................................................................................... 62

APPENDIX ..................................................................................................................... 64

PROGRAM LISTING ................................................................................................. 64

vii

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

viii

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

ix

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.

10

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;

11

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].

12

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

13

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

14

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.

15

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

16

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

17

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

18

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].

�� = �� + ��� = ����∗ � = 1, 2, … . � (2.1)

Where � �the voltage at the ��ℎ bus and with respect to ground and �� is the source current

injected into the bus. The load flow problem is handled more conveniently by use of �� rather

than��∗. 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(��� + �� − ��) ; � = 1, 2, … � (2.4)

��(�������� �����) = −|��|� |��|

���

|���|sin(��� + �� − ��) ; � = 1, 2, … � (2.5)

19

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

20

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 generation

.

=Total

generation.

−Totalload

.

�� = 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].

21

Table 2.1: Summary of bus variables

Bus type Specified variables Unknown variables

Slack or reference bus | �� | , �� �� , ��

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), ��� or |��|

Non-control variables ��� and ���

State variables |�� | 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)

22

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

in Esq. (2.4) and (2.5) ��� ≈ 90° and��� ≈ −90°.

ii. (�� − ��) 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

�� = |��|� |��|

���

|���| (�� − ��) ; � = 1, 2, … � (2.13)

�� = −|��|� |��|

������

|���|cos(�� − ��) + |��|� |���|; � = 1, 2, … � (2.14)

Since |��|s are specified, Eq. (2.13) represents a set of linear algebraic equations in ��s which

are (n-1) in number as�� is specified at the slack bus (�� = 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��, ��, … , ��, which when

substituted in Eq. (2.14), yields ��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

23

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 Gauss-

Seidel, 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

24

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

25

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)

�� =1

���

⎣⎢⎢⎡�� − � �����

������ ⎦

⎥⎥⎤

(2.16)

Substituting for �� 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.

26

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, … . . , � (2.19)

��� =���

��� � = 2, 3, … . . , �; (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,

�∆��(���)

� = ���(���)

− ��(�)

� < �;� = 2, 3, … . , � (2.22)

4 Computation of slack bus power: substitution of all bus voltages computed in

step 3 along with V� yieldsS�∗ = P� − jQ�.

5 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)

27

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(��� − �� + ��)

Bus � ��� Bus k ����

����

���� ���

� ����

��� ��� ���

��� �� ��

28

−�|��|� |���|sin��� − |��||��||���|sin(��� − �� + ��) − |��|������ � (2.34)

∴ ��� = −|��|� |���|cos���

+ |��||��||���|cos(��� − �� + ��) (2.35)

��� = |��|� |���|sin��� − |��||��||���|sin(��� − �� + ��) − |��|�����

� � (2.36)

Similarly power flows from k to � can be written as:

��� = −|��|� |���|cos��� + |��||��||���|cos(��� − �� + ��) (2.37)

��� = |��|� |���|sin��� − |��||��||���|sin(��� − �� + ��) − |��|������ � (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{��� + (�� −

��)}]

= −(|��|� + |��|�)|���|cos��� + 2|��||��||���|cos��� cos(�� − ��)

∴ �������= [2|��||��|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(��� − �� + ��) + sin(��� − �� + ��)]

− �|��|������ � + |��|�����

� ��

∴ �������= (|��|� + |��|�)��� − 2|��||��|��� cos(�� − ��) − �|��|�����

� � + |��|������ ��

∴ �������= ���[|��|

� + |��|� − 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

29

�, �. 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 ∆��, ∆��, … … , ∆�� and second, third…, derivatives

of the function��. Neglecting��, the linear set of equations resulting is as follows:

30

�� = �� ���(�)

, ��(�)

, … . , ��(�)

� + ∆�� ���

���⃒� + ∆��

���

���⃒� + ⋯ ∆��

���

���⃒�

�� = �� ���(�)

, ��(�)

, … . , ��(�)

� + ∆�� ���

���⃒� + ∆��

���

���⃒� + ⋯ ∆��

���

���⃒�

(2.44)

�� = �� ���(�)

, ��(�)

, … . , ��(�)

� + ∆�� ���

���⃒� + ∆��

���

���⃒� + ⋯ ∆��

���

���⃒�

⎣⎢⎢⎢⎡�� − �� ���

(�), ��

(�), … . , ��

(�)�

�� − �� ���(�)

, ��(�)

, … . , ��(�)

�� − �� ���(�)

, ��(�)

, … . , ��(�)

�⎦⎥⎥⎥⎤

=

⎣⎢⎢⎢⎢⎡

���

���⃒�

���

���⃒�

���

���⃒�

���

���⃒�

���

���⃒�

���

���⃒�

���

���⃒�

���

���⃒�

���

���⃒�⎦

⎥⎥⎥⎥⎤

�∆��

∆��

∆��

� (2.45)

Or 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.

�(�) = �(�)�(�) (2.46)

� (�) = ��(�)��(�)

�(�) (2.47)

The new values for ��‚ 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 ill-

conditioned 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(��� − �� + ��) (2.49)

�� = − � |��|

���

|��||���|sin(��� − �� + ��) (2.50)

31

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

32

�∆�∆�

� = �� 00 �

� �∆�

∆|�|� (2.53)

Or ∆� = � ∙ ∆� (2.54)

∆� = � ∙ ∆|�| (2.55)

For voltage controlled buses, the voltage magnitudes are known. Therefore, if m buses

of the system are voltage controlled, � is of the order (n-1) × (n-1) and � is of the order (n-1-

m) × (n-1-m).

Now the diagonal elements of � are

���

���= � |��||��||���|sin(��� − �� + ��) (2.56)

������

Off-diagonal elements of � are

���

���= |��||��||���|sin (��� − �� + ��)��� (2.57)

The diagonal elements of � are

���

�|��|= −2|��||���|������ − � |��||���|sin (��� − �� + ��)

������

(2.58)

���

�|��|= −|��||���|sin (��� − �� + ��)��� (2.59)

The terms ∆��(�)

and ∆��(�)

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.

33

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

������ ��� = ��� (2.65)

Where ��� 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�

∂δ�= −|V�|B�� (2.64c)

The off-diagonal elements of H described below (as given earlier by equation (2.57)

34

���

���= −|��||��||���|���(��� + �� − ��) (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�

∂δ�= −|V�|B�� (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�|= −|V�|B�� (2.66b)

The equation for the off-diagonal elements of L below (as given earlier by the equation)

���

�|��|= −|��||���|sin(��� + �� − ��) (2.59)

But;

35

Under normal operating conditions of a power system, (�� − ��) is quite small (≈ 0)

and hence; (θ�� + δ� − δ�) ≈ θ��

������ ��� = ���

∂Q �

∂|V�|= −|V�|B�� (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

�ΔP

|V|� = �B′�[Δδ] (2.68)

�ΔQ

|V|� = �B′′′�[Δ|V|] (2.69)

Where �� 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)

Where;

H�� = −|V�|B�� (2.72)

H�� = −|V�|B�� (2.73)

And;

L�� = −|V�|B�� (2.74)

36

��� = −|��|��� (2.75)

To obtain the corrections to the initial estimates the equations below are used;

Δδ = �B′��� ΔP

|V| (2.76)

Δ|V|= b�B′′��� ΔQ

|V| (2.77)

The simplified (FDLF) equations are solved alternatively always employing the most recent

voltage values. One iteration is called (1 − �) ��� (1 − |�|) iteration [1]. This implies

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 [��] ≤ �� where �� ��� ��are tolerance

Max [��] ≤ ��

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

37

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

At the PV buses �and |�| are specified and �and � are the unknowns to be determined.

Therefore, the values of � and � 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

�� = −�� �(��(�)

)∗ ∑ �����(���)���

��� + (��(�)

)∗ ∑ �����(�)�

��� � (2.79)

2. The revised value of �� 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���� → ����. If at any stage during the computation, Q at any bus goes

outside these limits, it is fixed at ���� 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.

38

3. If ��(���) < ��,��� set ��

(���) = ��,��� and treat bus � as a PQ bus. Compute ��(���)

and

��(���)

from Eqs (2.68) and (2.46) respectively. If ��(���) > ��,��� , set ��

(���) = ��,��� and

treat bus I as PQ bus. Compute ��(���)

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, … . , � are PV

buses and the remaining � + 1, … . . , � 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.

39

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.

40

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 ���� to 0.

Step 3: Creation of the �� 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).

5. Update of the voltage magnitude V and the voltage angle � at each bus.

6. Increment of the iteration counter ���� = ���� + 1

Step 5: If ���� ≤ 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.

41

3.2 Design Flow Chart

Figure 3.1: Decoupled load flow chart-[1]

Determine

��(���)

= ���

+ ����

��(���)

= ���

+ ����

Read data and build ����

matrix

Set ��=1<��for all buses

Set iteration count p=0

Set bus count �=1

Calc. bus current ��

Is bus �

P-V or P-Q

Determine

���

and |��|�

Determine

���

and ���

Compute

����

= ��,��� − ���

Compute

����

= ��,��� − ���

Is

���

≤ �����

Is ���

�����

Determine

�|���

|� = |��,���|� − |���

|�

Is �=n

Last node?

Increment bus

count

→ � + �

Det. maximum

���,��� ��� �|��|�

Are all max Δ

Within

tolerance?

Assemble

jacobianJ

Compute ��� ∴compute

����

and ����

Increment iteration p=p+1

Determine

And print line flows,

power loss, voltages etc.

Is �=1

or slack

bus?

NO NO

NO

NO NO

Yes

Yes

Yes

Yes Yes

P-V P-Q

Stop

Start

42

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.

43

Figure 3.2: IEEE 14 bus system [7].

44

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.

Angle Deg.

Load Generator Inj. MVAR MW MVA MW MVA

R Q ��� Q ���

1 1 1.060 0 0 0 232.4 -16.9 0 0 0 2 2 1.045 0 21.7 12.7 40 42.4 -40 50 0 3 2 1.01 0 131.88 94.2 19 23.4 0 40 0 4 0 1 0 66.92 47.8 -3.9 0 0 0 0 5 0 1 0 10.64 7.6 1.6 0 0 0 0 6 2 1.07 0 15.68 11.2 7.5 12.2 -6 24 0 7 0 1 0 0 0 0 0 0 0 0 8 2 1.09 0 0 0 0 17.4 -6 24 0 9 0 1 0 29.5 16.6 0 0 0 0 0 10 0 1 0 9 5.8 0 0 0 0 0 11 0 1 0 3.5 1.8 0 0 0 0 0 12 0 1 0 6.1 1.6 0 0 0 0 0 13 0 1 0 13.5 5.8 0 0 0 0 0 14 0 1 0 14.9 5 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.

45

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 Reactance (x)

P.U

Half Susceptance (B/2) P.U

Transformer tap (a)

1 2 0.01938 0.05917 0.0264 1 2 3 0.04699 0.19797 0.0219 1 2 4 0.05811 0.17632 0.0187 1 1 5 0.05403 0.22304 0.0246 1 2 5 0.05695 0.17388 0.017 1 3 4 0.06701 0.17103 0.0173 1 4 5 0.01335 0.04211 0.0064 1 5 6 0 0.25202 0 0.932 4 7 0 0.20912 0 0.978 7 8 0 0.17615 0 1 4 9 0 0.55618 0 0.969 7 9 0 0.11001 0 1 9 10 0.03181 0.0845 0 1 6 11 0.09498 0.1989 0 1 6 12 0.12291 0.25581 0 1 6 13 0.06615 0.13027 0 1 9 14 0.12711 0.27038 0 1 10 11 0.08205 0.19207 0 1 12 13 0.22092 0.19988 0 1 13 14 0.17093 0.34802 0 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)

46

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.

47

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.

48

Figure 4.5: �������� diagram for 14 bus test system [22].

49

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

No.

V

P.U

Angle

degree

Injection

MW | MVAR

Generation

MW | MVAR

Load

Mw | MVAR

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 0.000 -7.246 0.000 0.000

9 1.1297 15.3706 33.326 18.753 62.826 35.353 29.500 16.600

10 1.1340 15.5435 10.206 6.577 19.206 12.377 9.000 5.800

11 1.1258 15.2848 3.940 2.026 7.440 3.826 3.500 1.800

12 1.1256 15.6171 6.865 1.802 12.965 3.402 6.100 1.600

13 1.309 15.6862 15.267 6.558 28.767 12.358 13.500 5.800

14 1.1485 16.4730 17.112 5.742 32.012 10.742 14.900 5.000

Total 14.590 34.065 273.590 107.565 259.00 73.500

50

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 -21.213 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 -3.824 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 -3.166 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 0.008

13 -6.381 -2.221 14 6.441 2.345 0.061 0.124

Total 14.631 61.377

51

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

52

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

53

Table 4.3: Voltage, Angle, Generation and Load Power Comparison between DLF and NR

Bus

No.

Analysis

Techniques

Voltage

Magnitude V

(P.U)

Angle

Generation

MW MVAR

Load

MW MVAR

Bus-1

DLF Value 1.0600 0.0000 -223.498 101.599 0.000 0.000

NR Value 1.0600 0.0000 232.593 -15.233 0.000 0.000

DFF 0.000 0.000 0.3677 11.755 0.000 0.000

Bus-2

DLF Value 1.0450 5.3722 3.400 -38.394 21.700 12.700

NR Value 1.0450 -4.9891 40 47.928 21.7 12.7

DFF 0.000 0.3831 0.000 5.045 0.000 0.000

Bus-3

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

DFF 0.590 0.4666 0.000 7.956 0.000 0.000

Bus-4

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

Bus-5

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

Bus-6

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

Bus-7

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

Bus-8

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

Bus-9

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

Bus-10

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

Bus-11 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

Bus-12 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

Bus-13 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

Bus-14 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

54

Table 4.4: Real and Complex Bus Power comparison for the DLF and NR method

POWER DISPATCHED POWER RECEIVED ANALYSIS TECHNIQUE ANALYSIS TECHNIQUE LINE LOSSES

DLF METHOD NR METHOD DLF METHOD NR METHOD NR METHOD DLF METHOD

FROM BUS

P MW Q MVAR P MW Q MVAR IN BUS P MW QMVA

R

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 78.625 -11.231 -72.740 3.464 2.773 11.445 3.019 12.461

2 -72.914 14.627 73.396 5.936 3 75.294 -4.602 -71.063 3.894 2.333 9.830 2.380 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 41.733 4.738 5 42.784 -1.806 -40.813 -1.929 0.920 2.809 0.925 2.824

3 23.641 -14.670 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 -59.585 11.574 5 -65.413 3.355 60.064 -10.063 0.479 1.511 0.507 1.600

4 -31.434 -21.213 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 -3.824 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 -3.166 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 0.008

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

TOTAL LOSSES 13.593 56.910 14.631 61.377

55

Table4.5: PSAT Simulated Results

Bus � �. � Angle (�)

�� MW

�� MVAR

�� MW

�� MVAR

�� MW

�� MVAR

1 1.06 0 352.013 -28.20 352.013 -28.20 0 -28.20 2 1.045 -7.7738 9.6195 77.08 39.39 94.86 30.38 94.86 3 1.01 -15.149 -131.88 33.14 -0.00023 59.74 131.88 59.74 4 0.9978 -13.0033 -66.92 -5.6 -7e-05 1e-05 66.92 1e-05 5 1.0029 -21.1744 -10.64 -2.4 -4e-05 -1e-05 10.64 -1e-05

6 1.07 -19.445 -15.68 33.93 -1e-05 44.43 15.68 44.43 7 1.036 -21.7195 0 0 0 0 0 8 1.09 -22.0282 0 33.40 1e-05 33.40 0 33.40 9 1.0129 -21.7633 -41.3 -23.24 0 0 41.3 0 10 1.0122 -22.3792 -12.6 -8.12 -3e-05 0 12.6 0 11 1.0357 -22.4298 -4.9 -2.52 0 0 4.9 0 12 1.0462 -22.3792 -8.54 -2.24 1e-05 0 8.54 0 13 1.0366 -22.4298 -18.9 -8.12 1e-05 0 18.9 0 14 0.99695 -23.5234 -20.86 -7 0 0 20.86 0

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.

Figure 4.1: Newton Raphson voltage profile

0

0.5

1

1.5

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Vo

ltag

e (

p.u

)

Buses

Voltage Profile p.u

Voltage Profile p.u

56

Figure 4.2 Decoupled load flow voltage profile

Figure 4.3: Angle profile for DLF and NR

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Vo

ltag

e p

.u

Buses

Voltage Magnitude

Voltage Magnitude

-20

-15

-10

-5

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Bus Number

Final values of voltage angle(deg.)at@bus

Angle DLF

Angle NR

57

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.

Figure 4.4: DLF Real and Reactive power flow

Figure 4.5: DLF and NR power flow

-200

-150

-100

-50

0

50

100

150

200

Real and Reactive power flow

Dispatched P MW dispatched QMVAR Received P MW Received QMVAR

-200

-150

-100

-50

0

50

100

150

200

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

BRANCH SEQUENCE NUMBER

MW ,MVAR FLOWS

NR METHOD Q MVAR

NR METHOD P MW

DLF METHOD Q mvar

DLF METHOD P MW

58

Figure 4.6: DLF Line losses.

Figure 4.7: Line losses

Table 4.6: Data used to show Relative Accuracy of the results of each method

Analysis technique Total MW Loss Total MVAR Loss DLF 14.631 61.377 NR 13.593 56.910

PSAT 29.4125 90.11

0

5

10

15

20

25

Real and Reactive line losses

Losses MW losses MVAR

0

5

10

15

20

25

30

35

40

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

Linelosses

Series4

Series3

Series2

Series1

59

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.

60

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.

61

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.

62

REFERENCES

[1] Prof.D.P.Kothari, Modern Power Systems, 2003.

[2] I. A. Hisken, Power Flow Analysis, Nov.6.2003.

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of the Australian Universities Power Engineering Conference(AUPEC'94), pp. 24-27, 27.Sept.

2000.

[4] D. S. S. a. B. R. Reddy, Power SYstem Analysis, New Delhi: Laxmi , 1997.

[5] D. L. a. T. Blackburn, "Modified Algorithm of Load Flow Simulation for Loss Minimization in

Power System".

[6] D. Das, "Electrical Power System," Department of Electrical Engineering Indian Institute of

Technology, pp. 147-184.

[7] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, New York:

Springer, 1994.

[8] p. F. Milano, "Continuous Newton’s Method for Power Flow Analysis,," IEEE Transaction on

Power System , vol. 24, Feb 2009.

[9] J. J. G. a. W. D.StevensonJr, Power systems analysis’, New Delhi: Mc Graw Hill, 1994.

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and Iwatto Method with Facts Devices," IEEE Transaction Power Application.

[11] D. Das, Electrical Power Systems, New DELHI: New Age International (P) Ltd, 2006.

[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

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[14] P. B. H. Chowdhury, " Load-Flow Analysis in Power Systems," Electrical & Computer Engineering

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[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 .

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[18] A. A. B. M. Nasiruzzaman, "Astudent friendly toolbox for power system analysis using

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63

[19] G. P. O. M. G. H. S.C. Tripathy, "Load flow solutions for ill-conditioned Power System by

Newton -like Method," IEEE Transaction on Power Apparatus and Syatem.

[20] G. 1. Andersson, Lectures on Modelling and Analysis of Electric Power Systems.

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Coordinares," IEEE transaction on Power system, vol. 20, no. 4, pp. 1667-1674, Nov 2005.

[22] Quick Reference Manual for PSAT version 2.1.2, Power System Analysis Toolbox, June .26.2008.

[23] A. D. a. K. T. Boundary, "Load Flow Solutions," IEEE Transaction on Power System, vol. 19, no. 1,

Feb 2004.

[24] F. F.D. Galiana, "Bound Estimates of the Severity of Line Outages," IEEE Transactions on PAS, p.

103, 1984.

64

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); % Calling ybusppg.m to get Y-Bus Matrix. BaseMVA = 100; % Base MVA. bus_num = busd (: 1); % Bus Number. Bus type = busd (: 2); % Type of Bus 1-Slack, 2-PV, 3-PQ... V = busd (: 3); % Specified Voltage... Theta = busd (: 4); % Voltage Angle... Pg = busd (: 5)/BaseMVA; % PGi...Active power generated Qg = busd (: 6)/BaseMVA; % Qgeni. Reactive power generated Pl = busd (: 7)/BaseMVA; % Ploadi...active power demand Ql = busd (: 8)/BaseMVA; % Qloadi...reactive power demand Qmin = busd (: 9)/BaseMVA; % Minimum Reactive Power Limit... Qmax = busd (: 10)/BaseMVA; % Maximum Reactive Power Limit... DPbyV = (-Pg+Pl). /abs (V); % Injected = Generated - Pdemand... DQbyV = (-Qg+Ql). /abs (V); % Injected = Generated - Qdemand... Pspec = dPbyV; % P Specified. Qspec = dQbyV; % Q Specified... G = real(Y); % Conductance matrix… B = imag(Y); % 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) = [];

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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;

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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); % Calling Ybus program.. %lined = linedatas(nb); % Get linedats. lined = Line_Data; busd = Bus_Data; %busd = busdatas(nb); % Get busdatas.. Vm = pol2rect(V,theta); % Converting polar to rectangular.. Theta = 180/pi*theta; % Bus Voltage Angles in Degree... fb = lined(:,1); % From bus number... tb = lined(:,2); % To bus number... nl = length(fb); % No. of Branches.. Pl = busd(:,7); % PLi.. Ql = busd(:,8); % QLi.. Iij = zeros(nbus,nbus); Sij = zeros(nbus,nbus); Si = zeros(nbus,1);

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Iter = 1; % Bus Current Injections.. I = Y*Vm; %Line Current Flows.. for m = 1:nl p = fb(m); q = tb(m); Iij(p,q) = -(Vm(p) - Vm(q))*Y(p,q); % Y(m,n) = -y(m,n).. Iij(q,p) = -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) = m; Load_Flow_M(m,2) = V(m); Load_Flow_M(m,3) = Theta(m); Load_Flow_M(m,4) = Pi(m); Load_Flow_M(m,5) = Qi(m); Load_Flow_M(m,6) = Pg(m); Load_Flow_M(m,7) = Qg(m); Load_Flow_M(m,8) = Pl(m); Load_Flow_M(m,9) = Ql(m);

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end m = m + 1; Load_Flow_M(m,1) = NaN; Load_Flow_M(m,2) = NaN; Load_Flow_M(m,3) = NaN; Load_Flow_M(m,4) = sum(Pi); Load_Flow_M(m,5) = sum(Qi); Load_Flow_M(m,6) = sum(Pi+Pl); Load_Flow_M(m,7) = sum(Qi+Ql); Load_Flow_M(m,8) = sum(Pl); Load_Flow_M(m,9) = 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');

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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******************** %');


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