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A NOVEL DIFFERENTIAL EVOLUTION

ALGORITHMIC APPROACH TO

TRANSMISSION EXPANSION PLANNING

A thesis submitted for the degree of Doctor of Philosophy

by

Thanathip Sum-Im

Department of Electronic and Computer Engineering, Brunel University

March 2009

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ABSTRACT

Nowadays modern electric power systems consist of large-scale and highly complex

interconnected transmission systems, thus transmission expansion planning (TEP) is

now a significant power system optimisation problem. The TEP problem is a large-

scale, complex and nonlinear combinatorial problem of mixed integer nature where the

number of candidate solutions to be evaluated increases exponentially with system size.

The accurate solution of the TEP problem is essential in order to plan power systems in

both an economic and efficient manner. Therefore, applied optimisation methods

should be sufficiently efficient when solving such problems. In recent years a number

of computational techniques have been proposed to solve this efficiency issue. Such

methods include algorithms inspired by observations of natural phenomena for solving

complex combinatorial optimisation problems. These algorithms have been

successfully applied to a wide variety of electrical power system optimisation

problems. In recent years differential evolution algorithm (DEA) procedures have been

attracting significant attention from the researchers as such procedures have been

found to be extremely effective in solving power system optimisation problems.

The aim of this research is to develop and apply a novel DEA procedure

directly to a DC power flow based model in order to efficiently solve the TEP problem.

In this thesis, the TEP problem has been investigated in both static and dynamic form.

In addition, two cases of the static TEP problem, with and without generation resizing,

have also been investigated. The proposed method has achieved solutions with good

accuracy, stable convergence characteristics, simple implementation and satisfactory

computation time. The analyses have been performed within the mathematical

programming environment of MATLAB using both DEA and conventional genetic

algorithm (CGA) procedures and a detailed comparison has also been presented.

Finally, the sensitivity of DEA control parameters has also been investigated.

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ACKNOWLEDGEMENTS

I would like to express my deep gratitude to Dr. Gareth A. Taylor for initiating this

exciting research topic and for his invaluable guidance and encouragement throughout

the duration of my research. I am also grateful to Professor Malcolm R. Irving and

Professor Yong H. Song for their valuable comments and suggestions throughout my

research.

My deep appreciation is to Dr. Thanawat Nakawiro, Dr. Pathomthat Chiradeja

and Dr. Namkhun Srisanit for their valuable discussions, suggestions and great help

during my PhD study. I would like to thank Dr. Jeremy Daniel for his technical support

on computer hardware and software during my research. I am also thankful to the

Brunel Institute of Power Systems at Brunel University for providing me the

opportunity and resources to carry out this research work.

I gratefully acknowledge financial support from the Royal Thai Government

and Srinakharinwirot University.

Finally, I would like to thank my family, especially my mother, my father and

my lovely wife, for their love, support, patience and understanding.

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DECLARATION

The work described in this thesis has not been previously submitted for a degree in this

or any other university and unless otherwise referenced it is the author‟s own work.

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STATEMENT OF COPYRIGHT

The copyright of this thesis rests with the author.

No parts from it should be published without his prior written consent,

and information derived from it should be acknowledged.

©COPYRIGHT BY THANATHIP SUM-IM 2009

All Right Reserved

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TABLE OF CONTENTS

ABSTRACT………………………………………………………………………........ii

ACKNOWLEDGMENTS……………………………………………………………iii

DECLARATION……………………………………………………………………...iv

STATEMENT OF COPYRIGHT…………………………………………………….v

TABLE OF CONTENTS……………………………………………………………..vi

LIST OF TABLES…………………………………………………………………..xiii

LIST OF FIGURES………………………………………………………………….xv

LIST OF ABBREVIATIONS……………………………………………………...xviii

LIST OF NOMENCLATURE………………………………………………………xx

CHAPTER 1 INTRODUCTION……………………………………………………..1

1.1 Research Motivation……………………………………………………………....1

1.2 Problem Statement and Rationale…………………………………………………2

1.3 Contributions of the Thesis………………………….…………………………….3

1.4 List of Publications………………………………………………………………..5

1.4.1 Referred Journal Paper: Accepted…………………….………………......5

1.4.2 Referred Book Chapter: Submitted……………………….………………5

1.4.3 Referred Conference Papers: Published………………….…………….....5

1.5 Thesis Outline…………………………………………………………………....6

CHAPTER 2 FUNDAMENTALS OF TRANSMISSION EXPANSION

PLANNING PROBLEM……………………………………………………………...7

2.1 Introduction……………………………………………………………………..…7

2.2 Treatment of the Transmission Expansion Planning Horizon…….…………….....8

2.3 DC Power Flow…………………………………………………………………....8

2.4 Overview of the Static Transmission Expansion Planning………………………10

2.4.1 Problem Statement………………………………………………………10

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2.4.2 The Objective Function………………………………………………….10

2.4.3 Problem Constraints……………………………………………………..11

2.4.3.1 DC Power Flow Node Balance Constraint……………………11

2.4.3.2 Power Flow Limit on Transmission Lines Constraint…….......11

2.4.3.3 Power Generation Limit Constraint…………………………...11

2.4.3.4 Right-of-way Constraint……………………………………....12

2.4.3.5 Bus Voltage Phase Angle Limit Constraint…………………....12

2.5 Overview of the Dynamic Transmission Expansion Planning……………………12

2.5.1 Problem Statement………………………………………………………12

2.5.2 The Objective Function………………………………………………….13

2.5.3 Problem Constraints……………………………………………………..13

2.6 Review of Solution Methods for Transmission Expansion Planning…………..…14

2.6.1 Mathematical Optimisation Methods…………………………………....14

2.6.1.1 Linear Programming…………………………………………..15

2.6.1.2 Nonlinear Programming……………………………………….15

2.6.1.3 Dynamic Programming………………………………………..15

2.6.1.4 Integer and Mixed-Integer Programming…………………..…16

2.6.1.5 Branch and Bound……………………………………………..16

2.6.1.6 Benders and Hierarchical Decomposition…………………….16

2.6.2 Heuristic and Meta-heuristic Methods……………………………….….17

2.6.2.1 Heuristic Algorithms…………………………………………..17

2.6.2.2 Tabu Search…………………………………………………....18

2.6.2.3 Simulated Annealing………………………………………..…18

2.6.2.4 Expert Systems……………………………………………...…19

2.6.2.5 Evolutionary Algorithms……………………………………....19

2.6.2.6 Genetic Algorithms…………………………………………....20

2.6.2.7 Ant Colony System Algorithm…………………………….…..21

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2.6.2.8 Particle Swarm………………………………………………...21

2.6.2.9 Hybrid Artificial Intelligent Techniques……………………....22

2.7 Conclusions………………………………………………………………………..22

CHAPTER 3 FUNDAMENTALS OF DIFFERENTIAL EVOLUTION

ALGORITHM AND GENETIC ALGORITHMS…………………………………23

3.1 Introduction………………………………………………………………………..23

3.2 Genetic Algorithms………………………………………………………………..24

3.2.1 Background and Literature Review…………………………………..…24

3.2.2 Basis of Genetic Algorithms and Optimisation Process………………...27

3.3 Differential Evolution Algorithm………………………………………………….28

3.3.1 Background and Literature Review…………………………………..…28

3.3.2 Basis of Differential Evolution Algorithm……………………………....32

3.3.3 Differential Evolution Algorithm Optimisation Process……………...…33

3.3.3.1 Initialisation…………………………………………………...33

3.3.3.2 Mutation……………………………………………………….33

3.3.3.3 Crossover…………………………………………………...…34

3.3.3.4 Selection……………………………………………………….34

3.3.3.5 DEA Strategies………………………………………...………36

3.3.3.6 The Example of DEA Optimisation Process………………..…38

3.3.4 Constraint Handling Techniques……………………………………...…40

3.4 Conclusions………………………………………………………………………..41

CHAPTER 4 DESIGN AND TESTING OF DIFFERENTIAL EVOLUTION

ALGORITHM PROGRAM………………………………………………………....43

4.1 Introduction……………………………………………………….……………….43

4.2 Design of the Differential Evolution Algorithm Optimisation Program………….43

4.3 Numerical Benchmark Test Functions…………………………………………….45

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4.4 Experimental Setup and Control Parameters Setting……………………………...47

4.4.1 DEA Control Parameters and Their Effect……………………………...47

4.4.1.1 Population Size (NP)…………………………………………..48

4.4.1.2 Mutation Factor (F)……………………………………………48

4.4.1.3 Crossover Probability (CR)……………………………………48

4.4.1.4 Number of Problem Variables (D)…………………………….49

4.4.1.5 Convergence Criterion ( )……………………………………..49

4.4.2 Control Parameters Setting……………………………………………...49

4.5 Experimental Results and Discussion……………………………………………..50

4.5.1 Sphere Function Test Results…………………………………………....50

4.5.2 Rosenbrock1 Function Test Results……………………………………..52

4.5.3 Rosenbrock2 Function Test Results……………………………………..54

4.5.4 Absolute Function Test Results………………………………………….56

4.5.5 Salomon Function Test Results………………………………………….58

4.5.6 Schwefel Function Test Results………………………………………....60

4.5.7 Rastrigin Function Test Results…………………………………………62

4.6 Overall Analysis and Discussion on Test Results…………………………………64

4.7 Conclusions………………………………………………………………………..65

CHAPTER 5 APPLICATION OF DIFFERENTIAL EVOLUTION

ALGORITHM TO STATIC TRANSMISSION EXPANSION PLANNING…….66

5.1 Introduction………………………………………………………………………..66

5.2 Primal Static Transmission Expansion Planning - Problem Formulation………....67

5.3 Implementation of DEA for Static Transmission Expansion Planning Problem….68

5.3.1 Initialisation Step………………………..………………………………69

5.3.2 Optimisation Step………………………………………………………..69

5.3.3 Control Parameter Settings……………………………………………...69

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5.3.4 DEA Optimisation Program for Static TEP problem – Overall

Procedure………………………………………………………………..70

5.4 Test Systems and Numerical Test Results………………………………………...72

5.4.1 Garver 6-Bus System……………………………………………………72

5.4.1.1 Without Generation Resizing - Garver‟s System……………...73

5.4.1.2 With Generation Resizing - Garver‟s System…………………74

5.4.2 IEEE 25-Bus System…………………………………………………….76

5.4.2.1 Without Generation Resizing - IEEE 25-Bus System…………77

5.4.2.2 With Generation Resizing - IEEE 25-Bus System…………….78

5.4.3 Brazilian 46-Bus System……………………………………………...…80

5.4.3.1 Without Generation Resizing - Brazilian 46-Bus System……..81

5.4.3.2 With Generation Resizing - Brazilian 46-Bus System………...82

5.5 Overall Analysis and Discussion on the Results…………………………………..86

5.6 Conclusions………………………………………………………………………..86

CHAPTER 6 APPLICATION OF DIFFERENTIAL EVOLUTION

ALGORITHM TO DYNAMIC TRANSMISSION EXPANSION PLANNING…88

6.1 Introduction………………………………………………………………………..88

6.2 Primal Dynamic Transmission Expansion Planning - Problem Formulation……..89

6.3 Implementation of DEA for Dynamic Transmission Expansion Planning

Problem……………………………………………………………………………90

6.3.1 Initialisation Step………………………………………………………..91

6.3.2 Optimisation Step………………………………………………………..91

6.3.3 Control Parameter Settings……………………………………………...91

6.3.4 DEA Optimisation Program for Dynamic TEP problem – Overall

Procedure……………………………………………………………….92

6.4 Test System and Numerical Test Results…………………………….……………93

6.5 Discussion on the Results…………………………………………………………97

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6.6 Conclusions………………………………………………………………………..98

CHAPTER 7 INTERPRETATION OF TEST RESULTS IN TRANSMISSION

EXPANSION PLANNING PROBLEM…………………………………………….99

7.1 Introduction………………………………………………………………………..99

7.2 Sensitivity Analysis of DEA Control Parameters on Static Transmission Expansion

Planning………………………………………………………………………..…99

7.2.1 Sensitivity of Population Size (NP)…………………………………….100

7.2.1.1 Static TEP Problem - without Generation Resizing……….…100

7.2.1.2 Static TEP Problem - with Generation Resizing……………..103

7.2.2 Sensitivity of Scaling Mutation Factor (F)………………………….…105

7.2.2.1 Static TEP Problem - without Generation Resizing……….…105

7.2.2.2 Static TEP Problem - with Generation Resizing……………..107

7.2.3 Sensitivity of Crossover Probability (CR)……………………………..110

7.2.3.1 Static TEP Problem - without Generation Resizing………….110

7.2.3.2 Static TEP Problem - with Generation Resizing……………..112

7.3 Sensitivity Analysis of DEA Control Parameters on Dynamic Transmission

Expansion Planning……………………………………………………………....114

7.3.1 Sensitivity of Population Size (NP)………………………………….…115

7.3.2 Sensitivity of Scaling Mutation Factor (F)…………………………….117

7.3.3 Sensitivity of Crossover Probability (CR)……………………………..119

7.4 Overall Discussions……………………………………………………………...121

7.5 Conclusions………………………………………………………………………122

CHAPTER 8 CONCLUSIONS AND FUTURE WORK…………………………124

8.1 Conclusions……………………………………………………………………...124

8.2 Future Work……………………………………………………………………...127

8.2.1 Further Work Concerning in the Modified DEA Procedure…………...127

8.2.2 Further Work Concerning in Power System Problems……………...…127

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REFERENCES……………………………………………………………………...129

APPENDIX A TEST SYSTEMS DATA………………………………………..….139

A1 Garver 6-Bus System…………………………………………………….139

A2 IEEE 25-Bus System……………………………………………………..141

A3 Brazilian 46-Bus System…………………………………………………144

A4 Colombian 93-Bus System……………………………………………….148

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LIST OF TABLES

Tables

4.1 Numerical benchmark test functions……………………………………………...46

4.2 Parameters used in the implementation…………………………………………...50

4.3 Comparison of simulation results for Sphere function (f1)………………………..51

4.4 Comparison of simulation results for Rosenbrock1 function (f2)…………………53

4.5 Comparison of simulation results for Rosenbrock2 function (f3)…………………55

4.6 Comparison of simulation results for Absolute function (f4)……………………...57

4.7 Comparison of simulation results for Salomon function (f5)……………………...59

4.8 Comparison of simulation results for Schwefel function (f6)……………………..61

4.9 Comparison of simulation results for Rastrigin function (f7)……………………...63

5.1 Summary results of Garver 6-bus system without generation resizing case……...74

5.2 Summary results of Garver 6-bus system with generation resizing case………….75

5.3 Summary results of IEEE 25-bus system without generation resizing case………78

5.4 Summary results of IEEE 25-bus system with generation resizing case………….80

5.5 Summary results of Brazilian 46-bus system without generation resizing case…..82

5.6 Summary results of Brazilian 46-bus system with generation resizing case……...84

5.7 Results of static transmission expansion planning problem………………………85

5.8 Computational effort of static transmission expansion planning problem……..…85

6.1 Summary results of Colombian 93-bus system…………………………………...95

6.2 The expansion investment cost calculation of Colombian 93-bus system………..96

6.3 The best results comparison of Colombian 93-bus system………………………..97

A1.1 Generation and load data for Garver 6-bus system…………………………….139

A1.2 Branch data for Garver 6-bus system…………………………………………..139

A2.1 Generation and load data for IEEE 25-bus system…………………………….141

A2.2 Branch data for IEEE 25-bus system…………………………………………..142

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Tables

A3.1 Generation and load data for Brazilian 46-bus system……………………...…144

A3.2 Branch data for Brazilian 46-bus system………………………………………145

A4.1 Generation and load data for Colombian 93-bus system………………………148

A4.2 Branch data for Colombian 93-bus system…………………………………….151

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LIST OF FIGURES

Figures

3.1 The main flowchart of the typical GA optimisation process…………………...…28

3.2 The main flowchart of the typical DEA optimisation process…………………….35

4.1 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 1……………………………………………………………..52

4.2 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 2…………………………………………………………......54

4.3 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 3……………………………………………………………..56

4.4 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 4…………………………………………………………..…58

4.5 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 5…………………………………………………………..…60

4.6 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 6……………………………………………………………..62

4.7 Convergence curves of DEA strategies and CGA procedure on mathematical

benchmark function 7……………………………………………………………..64

5.1 Convergence curves of DEA1and CGA for Garver 6-bus system without

generation resizing case…………………………………………………………...74

5.2 Convergence curves of DEA3and CGA for Garver 6-bus system with generation

resizing case……………………………………………………………………….76

5.3 Convergence curves of DEA3 and CGA for IEEE 25-bus system without

generation resizing case…………………………………………………………...78

5.4 Convergence curves of DEA3 and CGA for IEEE 25-bus system with generation

resizing case…………………………………………………………………….…80

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Figures

5.5 Convergence curves of DEA3 and CGA for Brazilian 46-bus system without

generation resizing case…………………………………………………………..82

5.6 Convergence curves of DEA3 and CGA for Brazilian 46-bus system with

generation resizing case…………………………………………………………..84

7.1 Comparison of various population sizes obtained from DEA1-DEA10 for static

TEP problem without generation resizing on Garver 6-bus system…………….102

7.2 Comparison of various population sizes obtained from DEA1-DEA10 for static

TEP problem with generation resizing on Garver 6-bus system………………...104

7.3 Comparison of various scaling mutation factors (F) obtained from DEA1-DEA10

for static TEP problem without generation resizing on Garver 6-bus system…..107

7.4 Comparison of various scaling mutation factors (F) obtained from DEA1-DEA10

for static TEP problem with generation resizing on Graver 6-bus system……...109

7.5 Comparison of various crossover probabilities (CR) obtained from DEA1-DEA10

for static TEP problem without generation resizing on Graver 6-bus system…...111

7.6 Comparison of various crossover probabilities (CR) obtained from DEA1-DEA10

for static TEP problem with generation resizing on Graver 6-bus system………113

7.7 Comparison of various population sizes obtained from DEA1-DEA10 for dynamic

TEP problem without generation resizing on Colombian 93-bus system……….116

7.8 Comparison of various scaling mutation factors (F) obtained from DEA1-DEA10

for dynamic TEP problem without generation resizing on Colombian 93-bus

system……………………………………………………………………………118

7.9 Comparison of various crossover probabilities (CR) obtained from DEA1-DEA10

for dynamic TEP problem without generation resizing on Colombian 93-bus

system…………………………………………………………………………...121

8.1 Chromosome structure of self-adaptive DEA method…………………………...127

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Figures

A1 Garver 6-Bus System………………………………………………………….....140

A2 IEEE 25-Bus System……………………………………………………………..143

A3 Brazilian 46-Bus System…………………………………………………………147

A4 Colombian 93-Bus System……………………………………………………….153

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LIST OF ABBREVIATIONS

TEP : Transmission Expansion Planning

DEA : Differential Evolution Algorithm

GA : Genetic Algorithm

CGA : Conventional Genetic Algorithm

ILP : Integer Linear Programming

CHA : Constructive Heuristic Algorithm

TM : Transportation Model

TS : Tabu Search

SA : Simulated Annealing

STEP : Short Term Expansion Planning

FDLF : Fast Decoupled Load Flow

IGA : Improved Genetic Algorithm

TNEP : Transmission Network Expansion Planning

ACS : Ant Colony Search

PSO : Particle Swarm Optimisation

AI : Artificial Intelligence

EAs : Evolutionary Algorithms

EP : Evolutionary Programming

ESs : Evolution Strategies

AEC-GA : Advanced Engineered-Conditioning Genetic Algorithm

UC : Unit Commitment

PGA : Parallel Genetic Algorithm

CPF : Continuation Power Flow

SQP : Sequential Quadratic Programming

HDE : Hybrid Differential Evolution

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VSHDE : Variable Scaling Hybrid Differential Evolution

MDE : Modified Differential Evolution

DES : Differential Evolution Strategy

ORPF : Optimal Reactive Power Flow

TSCOPF : Transient Stability Constrained Optimal Power Flow

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LIST OF NOMENCLATURE

Pi : Real power of bus i

Qi : Reactive power of bus i

Vi : Voltage magnitude of bus i

i : Voltage phase angle of bus i

Vk : Voltage magnitude at bus k.

Gik and Bik

: Real and imaginary parts of element (i,k) of bus admittance matrix

N : Total number of buses in the system

v : Transmission investment cost

cij : Cost of a circuit which is a candidate for addition to the branch i-j

nij : Number of circuits added to the branch i-j

: Set of all candidate branches for expansion

g : Real power generation vector in the existing power plants

d : Real load demand vector in all network nodes

B : Susceptance matrix of the existing and added lines in the network

: Bus voltage phase angle vector.

fij : Total branch power flow in the branch i-j

fijmax : Maximum branch power flow in the branch i-j

nij0 : Number of circuits in the original base system

xij : Reactance in the branch i-j

i and j : Voltage phase angle of the terminal buses i and j

gi : Real power generation at node i

gimin : Lower real power generation limits at node i

gimax : Upper real power generation limits at node i

nij

: Total integer number of circuits added to the branch i-j

nijmax : Maximum number of added circuits in the branch i-j

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ctij : The cost of a candidate circuit added to branch i-j at stage t

ntij : Number of circuits added to branch i-j at stage t.

tinv : The discount factor used to calculate the present value of

an investment cost at stage t

Xi : A candidate solution, which is a D-dimensional vector, containing as

many integer-valued parameters as the problem decision parameters D

xj,i(G=0) : The initial value (G = 0) of the jth parameter of the ith individual vector.

xjmin : Lower bounds of the jth decision parameter

xjmax : Upper bounds of the jth decision parameter

NP : Population size

F : A scaling mutation factor

CR : A crossover constant or crossover probability

Ui : The trial vectors

Vi : The mutant vectors

: Convergence criterion

D : Number of problem variables

Gmax : Maximum number of iterations or generations

gbest : The best fitness value of the current iteration

pbest : The best fitness value of the previous iteration

Fs(X) : The fitness functions of the static TEP problem

Os(X) : The objective functions of the static TEP problem

P1(X) : The equality constraint penalty functions

P2(X) : The inequality constraint penalty functions

X : The individual vector of decision variables

1and

2 : Penalty weighting factors that are set to “0.5” in this research

l : The penalty coefficient of the lth inequality constraint

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c : An inequality constraint constant that is used when an individual

violates the inequality constraint.

nb : The number of buses in the transmission system

nc : The number of considered inequality constraints in TEP problem

FD(X) : The fitness functions of the dynamic TEP problem

OD(X) : The objective functions of the dynamic TEP problem

T : A horizon of time stage planning used in dynamic TEP problem

I : An annual interest rate value used in dynamic TEP problem

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CHAPTER 1

INTRODUCTION

1.1 Research Motivation

Electric energy is the most popular form of energy because it can be transported

easily at high efficiency and reasonable cost. Nowadays the real-world electric power

systems are large-scale and highly complex interconnected transmission systems. An

electric power system can be subdivided into four major parts that are generation,

transmission, distribution and load. The purpose of a transmission system is to

transfer electric energy from generating units at various locations to the distribution

systems that ultimately supply the load. Transmission lines that also interconnect

neighbouring utilities permit economic power dispatch across regions during normal

conditions as well as the transfer of power between regions during emergency.

Over the past few decades, the amount of electric power energy to be

transferred from generation sites to major load areas has been growing dramatically.

Due to increasing costs and the essential need for reliable electric power systems,

suitable and optimal design methods for different sections of the power system are

required. Transmission systems are a major part of any power system therefore they

have to be accurately and efficiently planned. In this research, electric power

transmission systems are studied with regard to optimising the transmission

expansion planning (TEP) problem.

Electric power transmission lines are initially built to link remote generating

power plants to load centres, thus allowing power plants to be located in regions that

are more economical and environmentally suitable. As systems grew, meshed

networks of transmission lines have emerged, providing alternative paths for power

flows from generators to loads that enhance the reliability of continuous supply. In

regions where generation resources or load patterns are imbalanced, transmission

interconnection eases the requirement for additional generation. Additional

transmission capability is justified whenever there is a need to connect cheaper

generation to meet growing load demand or enhance system reliability or both.

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1.2 Problem Statement and Rationale

Transmission expansion planning has always been a rather complicated task

especially for large-scale real-world transmission networks. First of all, electricity

demand changes across both area and time. The change in demand is met by the

appropriate dispatching of generation resources. As an electric power system must

obey physical laws, the effect of any change in one part of network (e.g. changing the

load at a node, raising the output of a generator, switching on/off a transmission line

or a transformer) will spread instantaneously to other parts of the interconnected

network, hence altering the loading conditions on all transmission lines. The ensuing

consequences may be more marked on some transmission lines than others,

depending on electrical characteristics of the lines and interconnection.

The electric transmission expansion planning problem involves determining

the least investment cost of the power system expansion and operation through the

timely addition of electric transmission facilities in order to guarantee that the

constraints of the transmission system are satisfied over the defined planning horizon.

The transmission system planner is entrusted with ensuring the above-stated goals

are best met whilst utilising all the available resources. Therefore the purpose of

transmission system planning is to determine the timing and type of new

transmission facilities. The facilities are required in order to provide adequate

transmission capacity to cope with future additional generation and power flow

requirements. The transmission plans may require the introduction of higher voltage

levels, the installation of new transmission elements and new substations.

Transmission system planners tend to use many techniques to solve the transmission

expansion planning problem. Planners utilise automatic expansion models to

determine an optimum expansion system by minimising the mathematical objective

function subject to a number of constraints.

In general, transmission expansion planning can be categorised as static or

dynamic according to the treatment of the study period [1]. In static planning; the

planner considers only one planning horizon and determines the number of suitable

circuits that should be installed to each branch of the transmission network system.

Investment is carried out at the beginning of the planning horizon time. In dynamic

or multistage planning; the planner considers not only the optimal number and

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location of added lines and type of investments but also the most appropriate times to

carry out such expansion investments. Therefore the continuing growth of the

demand and generation is always assimilated by the system in an optimised way. The

planning horizon is divided into various stages and the transmission lines must be

installed at each stage of the planning horizon.

Many optimisation methods have been applied when solving the transmission

expansion planning problem. The techniques range from expert engineering

judgements to powerful mathematical programming methods. The engineering

judgements depend upon human expertise and knowledge of the system. The most

applied approaches in the transmission expansion planning problem can be classified

into three groups that are mathematical optimisation methods (linear programming,

nonlinear programming, dynamic programming, integer and mixed integer

programming, benders decomposition and branch and bound, etc.), heuristic methods

(mostly constructive heuristics) and meta-heuristic methods (genetic algorithms, tabu

search, simulated annealing, particle swarm, evolutionary algorithms, etc).

Over the past decade, algorithms inspired by the observation of natural

phenomena when solving complex combinatorial problems have been gaining

increasing interest because they have been shown to have good performance and

efficiency when solving optimisation problems [2]. Such algorithms have

successfully applied to many power system problems [3, 4], for example power

system scheduling, power system planning and power system control. In this

research, a differential evolution algorithm (DEA) and genetic algorithm (GA) will

be proposed and developed to solve both static and dynamic transmission expansion

planning problems by direct application to the DC power flow based model.

1.3 Contributions of the Thesis

The major contribution of this thesis is the research and development of a novel DEA

procedure and the investigation of the applicability of DEA method when applied to

both static and dynamic TEP problems. In addition a detailed comparison of various

DEA strategies used for solving these two electrical power system optimisation

problems is presented. The most significant original contributions presented and

investigated in this thesis are outlined as follows:

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Firstly, this thesis proposes the methodology where a novel DEA procedure is

developed and improved by applying several DEA mutation strategies. In order

to validate its searching capability and reliability, the proposed methodology has

been tested with some selected mathematical benchmark test functions that are

as follows: Sphere, Rosenbrock1, Rosenbrock2, Absolute, Salomon, Schwefel

and Rastrigin functions, respectively. Regarding the obtained results, the

proposed method performs effectively and gives better solutions in all cases

when compared with a conventional genetic algorithm (CGA) procedure.

Regarding the effectiveness of DEA method as tested on several numerical

benchmark test functions. The proposed methodology has been successfully

implemented to solve a real-world optimisation problem that is the static TEP

problem. For this research, two different scenarios of the static TEP problem,

with and without generation resizing, have been investigated and reported in this

thesis. In addition, a heuristic search method has been adopted in order to deal

with the static TEP when considering the DC power flow based model

constraints.

In addition, this research utilises the proposed effective methodology to deal

with the dynamic or multistage TEP problem, which is more complex and

difficult when compared with the static TEP problem. In this thesis, the dynamic

TEP problem considering the DC power flow based model constraints has been

analysed and considered in the separation of the planning horizon into multiple

stages, which is an especially difficult task with regard to large-scale real-world

transmission systems. A novel DEA method as applied to solve the dynamic TEP

problem is tested on a realistically complex transmission system the Colombian

93-bus system.

Finally, the influence of control parameter variation on the novel DEA method

when applied to static and dynamic TEP problems has been investigated in this

thesis. The simulation results clearly illustrate that the proposed algorithm

provides higher robustness and reliability of approaching optimal solutions in

both applications when compared to the CGA procedure.

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1.4 List of Publications

Arising from this research project, a journal paper and a book chapter have been

submitted. In addition, three conference papers have been presented and published in

conference proceedings. The papers are listed as follows:

1.4.1 Refereed Journal Paper: Accepted

T. Sum-Im, G. A. Taylor, M. R. Irving and Y. H. Song, “Differential evolution

algorithm for static and multistage transmission expansion planning,” IET Proc.

Gener. Transm. Distrib., (Accepted 2009).

1.4.2 Refereed Book Chapter: Submitted

T. Sum-Im, G. A. Taylor, M. R. Irving and Y. H. Song, “Differential evolution

algorithm for transmission expansion planning,” in Intelligent techniques for

power system transmission, G. K. Venayagamoorthy, R. Harley and N. G

Hingorani, Ed., Wiley, (Submitted 2008).

1.4.3 Refereed Conference Papers: Published

T. Sum-Im, G. A. Taylor, M. R. Irving and Y. H. Song, “A comparative study of

state-of-the-art transmission expansion planning tools,” Proc. the 41st

International Universities Power Engineering Conference (UPEC 2006),

Newcastle upon Tyne, United Kingdom, pp. 267-271, 6th-8th Sep. 2006.

T. Sum-Im, G. A. Taylor, M. R. Irving and Y. H. Song, “A differential evolution

algorithm for multistage transmission expansion planning,” Proc. the 42nd

International Universities Power Engineering Conference (UPEC 2007),

Brighton, United Kingdom, pp. 357-364, 4th-6th Sep. 2007.

T. Sum-Im, G. A. Taylor, M. R. Irving and Y. H. Song, “Transmission expansion

planning using the DC model and a differential evolution algorithm,” Proc. the

1st School of Engineering and Design Research Student Conference (RESCon

2008), Brunel University, United Kingdom, pp. 43-44, 25th-26th Jun. 2008.

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1.5 Thesis Outline

Chapter 1 provides an introduction to the transmission expansion planning problem.

In addition, the research contributions of applying the novel differential evolution

algorithm to transmission expansion planning problems are presented.

Chapter 2 presents an overview of static and dynamic transmission expansion

planning problems including problem formulation, treatment of the planning horizon

and available literature.

Chapter 3 provides a review of DEA and genetic algorithms. The optimisation

process and constraint handing techniques of the proposed algorithm are presented.

Chapter 4 presents the DEA optimisation procedure and program, which is tested on

various numerical benchmark functions. The numerical test results and discussion are

explained in this chapter.

Chapter 5 provides the implementation and development of the novel differential

evolution algorithm for solving the static transmission expansion planning problem.

Moreover the experimental results and comments are discussed in this chapter.

Chapter 6 presents the implementation of the novel DEA for solving the dynamic

transmission expansion planning problem. In addition, the numerical test results for

realistic transmission systems and comments are included in this chapter.

Chapter 7 gives the interpretations of results from chapter 5 and 6 with regard to

sensitivity and convergence analysis of the DEA on static and dynamic transmission

expansion planning problems.

Chapter 8 presents the overall conclusions of the research reported in this thesis and

indicates further possible research directions.

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CHAPTER 2

FUNDAMENTALS OF TRANSMISSION

EXPANSION PLANNING PROBLEM

2.1 Introduction

In general, the objective of electric transmission expansion planning (TEP) is to

specify addition of transmission facilities that provide adequate capacity and in the

mean time maintain operating performance of electric transmission system [5]. To

achieve effective plan, exact location, capacity, timing and type of new transmission

equipment must be thoroughly determined to meet demand growth, generation

addition and increased power flow. However, cost-effective transmission expansion

planning becomes one of the major challenges in power system optimisation due to

the nature of the problem that is complex, large-scale, difficult and nonlinear.

Meanwhile, mixed integer nature of TEP results in an exponentially increased

number of possible solutions when system size is enlarged.

To find an optimal solution of TEP over a planning horizon, extensive

parameters are required; for instance topology of the base year, candidate circuits,

electricity demand and generation forecast, investment constraints, etc. This would

consequently impose more complexity to solving TEP problem. Given the above

information, in–depth knowledge on problem formulation and computation

techniques for TEP is crucial and therefore, this chapter aims essentially at presenting

fundamental information of these issues.

The organisation of this chapter is as follows: section 2.2 presents the

overview of treatment of the transmission expansion planning horizon, while in

section 2.3 the overview and formulation of DC power flow model is introduced.

Section 2.4 and 2.5 present the problem formulation and the mathematical model of

static and dynamic transmission expansion planning, respectively. Section 2.6

presents the review of solution methods for transmission expansion planning found

in the international technical literature. Finally, a summary of this chapter is made in

section 2.7.

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2.2 Treatment of the Transmission Expansion Planning Horizon

Based on the treatment of planning horizon, transmission expansion planning can be

traditionally classified into two categories, namely static (single-stage) and dynamic

(multi-stage) planning. In static planning, only a single time period is considered as a

planning horizon. In contrast, dynamic planning considers the planning horizon by

separating the period of study into multiple stages [1].

For static planning, the planner searches for an appropriate number of new

circuits that should be added into each branch of the transmission system and in this

case, the planner is not interested in scheduling when the new lines should be

constructed and the total expansion investment is carried out at the beginning of the

planning horizon [6]. Many research works regarding the static TEP are presented in

[5, 8, 11, 14, 15, 19, 21, 22, 25, 67, 68, 74] that are solved using a variety of the

optimisation techniques.

In contrast, time-phased or various stages are considered in dynamic planning

while an optimal expansion schedule or strategy is considered for the entire planning

period. Thus, multi-stage transmission expansion planning is a larger-scale and more

complex problem as it deals with not only the optimal quantity, placement and type

of transmission expansion investments but also the most suitable times to carry out

such investments. Therefore, the dynamic transmission expansion planning

inevitably considers a great number of variables and constraints that consequently

require enormous computational effort to achieve an optimal solution, especially for

large-scale real-world transmission systems. Many research works regarding the

dynamic TEP [6, 12, 13, 19, 68, 73] are presented some of the dynamic models that

have been developed.

2.3 DC Power Flow

For a long-term TEP study, some assumptions are invented and introduced for

solving such planning problem, for example, a consideration of the reactive power

allocation is neglected in the first moment of the planning. In this stage, the main

concern is to identify the principal power corridors that probably will become part of

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the expanded system. There are several types of the mathematical model employed

for representing the transmission network in the TEP study; AC power flow model,

DC power flow model, transportation model, hybrid model, and disjunctive model

[8].

Basically, the DC power flow model is widely employed to the TEP problem

and it is frequently considered as a reference because in general, networks

synthesized by this model satisfy the basic conditions stated by operation planning

studies. The planning results found in this phase will be further investigated by

operation planning tools such as AC power flow analysis, transient and dynamic

stability analysis and short-circuit analysis [3]. In the simulation of this research, the

DC power flow model is considered as it is widely used in transmission expansion

planning [5, 8, 25, 66, 67].

The formulation of DC power flow is obtained from the modification of a

general representation of AC power flow, which can be illustrated by the following

equations.

1

[ cos() sin()]

N

iik ikik ikik

k

PVVGB

(2.1)

1

[ sin() cos( )]

N

iik ikik ikik

k

QVVGB

(2.2)

where Pi and Qi are real and reactive power of bus i respectively. Vi and

i

are voltage magnitude and voltage phase angle of bus i respectively. Vk is voltage

magnitude at bus k. Gik and Bik are real and imaginary parts of element (i,k) of bus

admittance matrix respectively. N is total number of buses in the system.

To modify AC power flow model to the DC power flow based model, the

following assumptions are normally considered [7]:

Bus voltage magnitude at each bus bar is approximate one per unit ( Vi = 1

p.u. for all i buses);

Line conductance at each path is neglected (Gik = 0), or on the other hand

only line susceptance (Bik) is considered in the DC model;

Some trigonometric terms of AC model in equations (2.1) and (2.2) can be

approximated as following terms:

sin ( i - k) i - k and cos ( i - k) 1

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10

Given these assumptions, the AC power flow equation in (2.1) is therefore

simplified to yield the DC power flow equation as follows:

1

(), 1,...,

N

i ikik

k

PBiN (2.3)

where Bik is the line susceptance between bus i and k.

2.4 Overview of the Static Transmission Expansion Planning

In this section, the static transmission expansion planning is formulated as a

mathematical problem. The objective of solving this problem is typically to fulfil the

required planning function in terms of investment and operation restriction. The

detailed discussion is as follows.

2.4.1 Problem Statement

In general, transmission expansion planning problem can be mathematically

formulated by using DC power flow model, which is a nonlinear mixed-integer

problem with high complexity, especially for large-scale real-world transmission

networks. There are several alternatives to the DC model such as the transportation,

hybrid and disjunctive models. Detailed reviews of the main mathematical models

for transmission expansion planning were presented in [8, 9].

2.4.2 The Objective Function

The objective of transmission expansion planning is to minimise investment cost

while satisfying operational and economic constraints. In this research, the classical

DC power flow model is applied to solve the TEP problem. Mathematically, the

problem can be formulated as follows.

( , )

i j

min

ij ij

c n

v

(2.4)

where v, cij and nij represent, respectively, transmission investment cost, cost

of a candidate circuit for addition to the branch i-j and the number of circuits added

to the branch i-j. Here is the set of all candidate branches for expansion.

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2.4.3 Problem Constraints

The objective function (2.4) represents the capital cost of the newly installed

transmission lines, which has some restrictions. These constraints must be included

into mathematical model to ensure that the optimal solution satisfies transmission

planning requirements. These constraints are described as following:

2.4.3.1 DC Power Flow Node Balance Constraint

This linear equality constraint represents the conservation of power at each node.

gdB (2.5)

where g, d and B is real power generation vector in existing power plants, real

load demand vector in all network nodes, and susceptance matrix of the existing and

added lines in the network, respectively. Here is the bus voltage phase angle vector.

2.4.3.2 Power Flow Limit on Transmission Lines Constraint

The following inequality constraint is applied to transmission expansion planning in

order to limit the power flow for each path.

0

ij

max

ij

()

ijij

fnnf

(2.6)

In DC power flow model, each element of the branch power flow in

constraint (2.6) can be calculated by using equation (2.7):

0

ij

()

()

ij

ijij

ij

nn

f

x

(2.7)

where fij, fijmax, nij, nij0 and xij represents, respectively, total branch power flow

in branch i-j, maximum branch power flow in branch i-j, number of circuits added to

branch i-j, number of circuits in original base system, and reactance of the branch i-j.

Here i and j is voltage phase angle of the terminal buses i and j respectively.

2.4.3.3 Power Generation Limit Constraint

In transmission expansion planning problem, power generation limit must be

included into the problem constraints. This can be mathematically represented as

follows:

minmax

iii

ggg

(2.8)

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where gi, gimin and gimax is real power generation at node i, the lower and

upper real power generation limits at node i, respectively.

2.4.3.4 Right-of-way Constraint

It is significant for an accurate transmission expansion planning that planners need to

know the exact location and capacity of the newly required circuits. Therefore this

constraint must be included into the consideration of planning problem.

Mathematically, this constraint defines the new circuit location and the maximum

number of circuits that can be installed in a specified location. It can be represented

as follows.

max

ij

n

0

ij

n

(2.9)

where nij and nijmax represents the total integer number of circuits added to the

branch i-j and the maximum number of added circuits in the branch i-j, respectively.

2.4.3.5 Bus Voltage Phase Angle Limit Constraint

The bus voltage magnitude is not a factor in this analysis since a DC power flow

model is used for transmission planning. The voltage phase angle is included as a

transmission expansion planning constraint and the calculated phase angle ( ijcal)

should be less than the predefined maximum phase angle ( ijmax). This can be

represented as the following mathematical expression.

cal

ij

max

ij

(2.10)

2.5 Overview of the Dynamic Transmission Expansion Planning

In this section, a mathematical representation of the dynamic transmission expansion

planning problem is discussed as following details.

2.5.1 Problem Statement

The purpose of dynamic transmission expansion planning is to minimise the present

value of investment cost for transmission expansion over an entire planning periods.

Normally, the problem of dynamic TEP requires a huge computational effort to

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search for an optimal solution. The DC power flow model was applied to the static

TEP problem in the previous section and it can be extended to more complex

dynamic transmission expansion planning as well. The dynamic planning problem is

a mixed integer nonlinear programming problem that is difficult for solving

especially medium-scale and large-scale transmission systems.

2.5.2 The Objective Function

A DC model can be applied to the dynamic planning in order to determine the

financial investment for the most economical schedule [6]. The investment plan of

transmission expansion is generally obtained with reference to the base year.

Considering an annual rate I, the present values of the transmission expansion

planning investment costs in the base year t0 with a horizon of T stages are as follows:

where

1020

0

12

1

inv

2

inv

12

( )

c x

(1) ( ) (1

c x

) ( )

... (1)( )

( ) ( ) ...

c x

( )

T

tttt

tt

T

T

inv T

IIc x

Ic x

c xc x

(2.11)

0

1

ttt

t

inv

I

Using the above relations, the dynamic planning for the DC model assumes

the following form:

1( , )

i j

min

T

tt

ij

t

ij inv

t

v c n

(2.12)

where v, ctij, and ntij represents, respectively, the present value of the

expansion investment cost of the added transmission system, the cost of a candidate

circuit added to branch i-j at stage t and the number of circuits added to branch i-j at

stage t. Here is the set of all candidate right-of-ways for expansion. tinv is the

discount factor used to find the present value of an investment at stage t.

2.5.3 Problem Constraints

The objective function (2.12) represents the present value of the dynamic expansion

planning investment costs of the new transmission lines subject to the restrictions as

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described in the previous section. Therefore, these planning constraints must also be

considered in the multi-stage mathematical formulation in order to guarantee that the

achieved solutions satisfy transmission planning requirements. The constraints of

dynamic transmission expansion planning can be formulated in a similar fashion to

those of the static model and are presented as follows:

tttt

dBg (2.13)

0

ij

max

ij

1

()

t

ts

ijij

s

fnnf

(2.14)

0

ij

1

()

()

t

s

ij

ttt

j

s

iji

ij

nn

f

x

(2.15)

,min,max

tt

i

t

ii

ggg

(2.16)

,max

ij

n

0

t

ij

t

n

(2.17)

max

ij

n

1

T

t

ij

t

n

(2.18)

cal

ij

max

ij

(2.19)

The variables of the dynamic transmission expansion planning constraints in

(2.13)-(2.19) are similar to those of static transmission expansion planning except the

addition of the index t, which indicates the specific stage of planning involved.

2.6 Review of Solution Methods for Transmission Expansion

Planning

Over past few decades, many optimisation techniques have been proposed to solve

the transmission expansion planning problem in regulated power systems. These

techniques can be generally classified into mathematical, heuristic and meta-heuristic

optimisation methods. A review of these methods is discussed in this section.

2.6.1 Mathematical Optimisation Methods

Mathematical optimisation methods search for an optimal expansion plan by using a

calculation procedure that solves a mathematical formulation of the planning

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problem. In the problem formulation, the transmission expansion planning is

converted into an optimisation problem with an objective function subject to a set of

constraints. So far, there have been a number of applications of mathematical

optimisation methods to solve the transmission expansion planning problem, for

instance, linear programming [10], nonlinear programming [11] and [12], dynamic

programming [13], branch and bound [14] and [15], mixed-integer programming [16]

and [17] and Benders decomposition [18].

2.6.1.1 Linear Programming

In 1970, Garver proposed a linear programming method to solve the transmission

expansion planning problem [10]. This original method was applied to long-term

planning of electrical power systems and produced a feasible transmission network

with near-minimum circuit miles using as input any existing network plus a load

forecast and generation schedule. Two main steps of the method, in which the

planning problem was formulated as load flow estimation and new circuit selection

could be searched based on the system overloads, were presented in [10]. The linear

programming was used to solve the minimisation problem for the needed power

movements, whereas the result was called “linear flow estimate”. A circuit addition

was selected based on the location of the largest overload in this flow estimate. These

two steps were repeated until no overloads remain in the system.

2.6.1.2 Nonlinear Programming

In 1984, an interactive method was proposed and applied in order to optimise the

transmission expansion planning by Ekwue and Cory [11]. The method was based

upon a single-stage optimisation procedure using sensitivity analysis and the adjoint

network approach to transmit power from a new generating station to a loaded AC

power system. The nonlinear programming technique of gradient projection followed

by a round-off procedure was used for this optimisation method.

2.6.1.3 Dynamic Programming

Discrete dynamic optimising (DDO) was proposed to solve the transmission

planning problem by Dusonchet and El-Abiad [13]. The basic idea of this method

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was to combine deterministic search procedure of dynamic programming with

discrete optimising a probabilistic search coupled with a heuristic stopping criterion.

The proposed method provides a way of dealing with two problems, which are size

and complexity of the procedures for evaluating the performance of alternate

strategies, through the use of a probabilistic search procedure and dynamic

programming. Another advantage of this method is the probability through the

neighbourhood concept to take into account in solution process the planner‟s

experience.

2.6.1.4 Integer and Mixed-Integer Programming

In 2003, Alguacil et al. [17] proposed a mixed-integer linear programming approach

to solve the static transmission expansion planning that includes line losses

consideration. The proposed mixed-integer linear formulation offers accurate optimal

solution. Meanwhile, it is flexible enough to build new networks and to reinforce

existing ones. The proposed technique was tested to Graver‟s 6-bus system, the IEEE

reliability test system and a realistic Brazilian system whereas the results confirm the

accuracy and efficiency of this computation approach.

2.6.1.5 Branch and Bound

Haffner et al. [15] presented a new specialised branch and bound algorithm to solve

the transmission network expansion planning problem. Optimality was obtained at a

cost, however: that was the use of a transportation model for representing the

transmission network. The expansion problem then became an integer linear

programming (ILP) which was solved by the proposed branch and bound method. To

control combinatorial explosion, the branch and bound algorithm was specialised

using specific knowledge about the problem for both the selection of candidate

problems and the selection of the next variable to be used for branching. Special

constraints were also used to reduce the gap between the optimal integer solution

(ILP program) and the solution obtained by relaxing the integrality constraints.

2.6.1.6 Benders and Hierarchical Decomposition

A new Benders decomposition approach was applied to solve the real-world power

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transmission network design problems by Binato et al. [18]. This approach was

characterised by using a mixed linear (0-1) disjunctive model, which ensures the

optimality of the solution found by using additional constraints, iteratively evaluated,

besides the traditional Benders cuts. In [18], the use of Gomory cuts iteratively

evaluated from master sub-problem and the use of Benders cuts evaluated from

relaxed versions of the slave sub-problem. Gomory cuts within Benders

decomposition was used to improve the practical convergence to the optimal solution

of the Benders approach.

2.6.2 Heuristic and Meta-heuristic Methods

In addition to mathematical optimisation methods, heuristic and meta-heuristic

methods become the current alternative to solve the transmission expansion planning

problem. These heuristic and meta-heuristic techniques are efficient algorithms to

optimise the transmission planning problem. There have been many applications of

heuristic and meta-heuristic optimisation methods to solve transmission expansion

planning problem, for example heuristic algorithms [5, 19], tabu search [20],

simulated annealing [21], genetic algorithms [6, 22, 23, 24], artificial neural

networks [25], particle swarm [31] and hybrid artificial intelligent techniques [25].

The detail of these methods is as discussed below.

2.6.2.1 Heuristic Algorithms

Constructive heuristic algorithm (CHA) is the most-widely used heuristic algorithms

in transmission expansion planning. A constructive heuristic algorithm is an iterative

process that searches a good quality solution in a step-by-step process. Romero et al.

[19] presented and analysed heuristic algorithms for the transportation model in static

and multistage transmission expansion planning. A constructive heuristic algorithm

for the transportation model (TM) of Garver‟s work [10] was extensively analysed

and excellent results were obtained in [19]. Furthermore, the Garver algorithm was

extended to accommodate multistage planning, which is especially important to

define financial investment according to the most economical scheduling. The CHA,

which was proposed for the generalised transportation model, reaches quality

topologies for all test systems even though its efficiency decreased as the complexity

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of system increased [19]. In 2005, Romero et al. [5] proposed constructive heuristic

algorithm for the DC model in network transmission expansion planning. A novel

constructive heuristic algorithm worked directly with the DC power flow model in

[5]. This proposed algorithm was developed from Garver‟s works [10] that was

applied to the transportation model. The algorithm presented excellent performance

for systems with low complexity in Garver‟s 6-bus and medium complexity in IEEE

24-bus. The principal advantage of the algorithm was that it worked directly with the

solution given by the DC model with relaxed integer variables.

2.6.2.2 Tabu Search

Tabu search (TS) is an iterative improvement procedure that starts from some initial

feasible solution and attempts to determine a better solution in the manner of a

„greatest descent neighbourhood‟ search algorithm [2]. The basic components of the

TS are the moves, tabu list and aspiration level (criterion). Silva et al. [20] presented

transmission network expansion planning under a tabu search approach. The

implementation of tabu search to cope with long-term transmission network

expansion planning problem was proposed in [20]. Two real-world case studies were

tested and the results obtained by this approach were a robust and promising

technique to be applied to this planning problem. The good quality of results

produced by the intensification phase in both case studies qualifies the strategy used,

i.e. to look for consistent candidate circuits (those that appear in different plans) to

build a consistent transmission expansion plan. The principal improvement of this

approach, comparing with classical methods of optimisation, was related to its ability

in avoiding local optimum solutions, consequently having a greater chance to find

the global optimum solution.

2.6.2.3 Simulated Annealing

Simulated annealing (SA) approach based on thermodynamics was originally

inspired by the formulation of crystals in solids during cooling [2]. Simulated

annealing technique has been successfully applied to a number of engineering

optimisation problems including power system optimisation problems. Romero et al.

[21] proposed a simulated annealing approach for solving the long-term transmission

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system expansion planning problem. The proposed method [21] was compared with

a conventional optimisation method based on mathematical decomposition with a

zero-one implicit enumeration procedure. In [21], two small test systems were used

for tuning the main parameters of the simulated annealing process and then the

proposed technique was applied to a large test system for which no optimal solution

had been known: a number of interesting solutions was obtained with costs about 7%

less than the best solutions known for that particular example system obtained by

optimisation and heuristic methods.

2.6.2.4 Expert Systems

Expert system is a knowledge-based or rule-based system, which uses the knowledge

and interface procedure to solve problems. The state of the field of expert systems

and knowledge engineering in transmission planning was reviewed by Galiana et al.

[26]. The details of that review were the principal elements of transmission planning,

including its aim, the principal activities that constituted transmission planning, the

constraints and prerequisites that must be met by the planner, a general planning

methodology, and a selection to justify the use of expert systems in transmission

planning and to indicate area of potential. Moreover, an expert system approach for

multi-year short-term expansion planning (STEP) was presented in [27] where the

reactive power management issues were addressed in the multi-year STEP to ensure

adequate quality of voltage supply and efficiency of transmission system, which

could be measured by network congestion and percentage losses in the system. An

expert system approach to STEP using enhanced fast decoupled load flow (FDLF)

was proposed to address these reactive power issues.

2.6.2.5 Evolutionary Algorithms

Evolutionary algorithm is based on the Darwin‟s principle of „survival of the fittest

strategy‟. An evolutionary algorithm begins with initialising a population of

candidate solutions to a problem and then new solutions are generated by randomly

varying those of initial population. All solutions are evaluated with respect to how

well they address the task. Finally, a selection operation is applied to eliminate bad

solutions. An evolutionary programming approach for transmission network planning

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in electric power systems was presented in [28]. The proposed evolutionary

programming algorithm was tested in two electric power systems, including Graver

6-bus system and the Mexican electric power system.

2.6.2.6 Genetic Algorithms

Genetic algorithm (GA) is a global search approach based on mechanics of natural

selection and genetics. GA is different from conventional optimisation techniques as

it uses the concept of population genetics to guide the optimisation search. GA

searches from population to population instead of point-to-point search. In 1998,

Gallego et al. [22] presented an extended genetic algorithm for solving the optimal

transmission network expansion planning problem. Two main improvements of GA,

which are an initial population obtained by conventional optimisation based methods

and the mutation approach inspired in the simulated annealing technique, was

introduced in [22].

The application of an improved genetic algorithm (IGA) was also proposed to

solve the transmission network expansion planning problem by Silva et al. [23].

Genetic algorithms (GAs) had demonstrated the ability to deal with non-convex,

nonlinear, integer-mixed optimisation problems, which include transmission network

expansion planning (TNEP) problem, as it generates better performance than a

number of other mathematical methodologies. Some special features had been added

to the basic GAs to improve its performance in solving the TNEP problem for three

real large-scale transmission systems. Results in [23] showed that the proposed

approach was not only suitable but a promising technique for solving such a problem.

In 2001, Gil and Silva presented a reliable approach for solving the

transmission network expansion planning problem using genetic algorithms [24].

The procedure to find the solution was based on the „loss of load limit curve‟ of the

transmission system under study, which was produced utilising unfeasible solutions

found by the GA. A modified procedure made GA more robust to solve the different

large-scale transmission expansion problems and this proposed method was proved

to be efficient for solving in two real large-scale power systems [24].

In 2004, Escobar et al. [6] proposed an efficient genetic algorithm to solve the

multistage and coordinated transmission planning problem, which was a mixed

integer nonlinear programming problem. The proposed GA had a set of specialised

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genetic operators and utilised an efficient form of generation for the initial population

that found high quality suboptimal topologies for large size and high complexity

transmission systems. The achieved results illustrated that an efficient GA was

effectively and efficiently implemented for multistage planning on medium and large

size systems.

2.6.2.7 Ant Colony System Algorithm

Ant colony search (ACS) system was initially introduced by Dorigo in 1992 [32].

ACS technique was originally inspired by the behaviour of real ant colonies and it

was applied to solve function or combinatorial optimisation problems. Gomez et al.

[29] presented ant colony system algorithm for the planning of primary distribution

circuits. The planning problem of electrical power distribution networks, stated as a

mixed nonlinear integer optimisation problem, was solved using the ant colony

system algorithm. In [29], the ant colony system methodology was coupled with a

conventional load flow algorithm for distribution system and adapted to solve the

primary distribution system planning problem. Furthermore, this technique [29] was

very flexible and it could calculate location and the characteristics of the circuits

minimising the investment and operation costs while enforcing the technical

constraints, such as the transmission capabilities, the limits on the voltage

magnitudes, allowing the consideration of a very complete and detailed model for the

electric system.

2.6.2.8 Particle Swarm

Particle swarm optimisation (PSO), using an analogy of swarm behaviour of natural

creatures, was started in the early of the 1990s. Kennedy and Eberhart developed

PSO based on the analogy of swarms of birds and fish schooling [30], which

achieved efficient search by remembrance and feedback mechanisms. By imitating

the behaviours of biome, PSO is highly fit for parallel calculation and good

performance for optimisation problems. A new discrete method for particle swarm

optimisation was applied for transmission network expansion planning (TNEP) in

[31]. The principle of PSO was introduced and an efficient discrete PSO method for

TNEP according to its characters was developed by researchers [31]. Moreover,

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parameter selection, convergence judgment, optimisation fitness function

construction and PSO characters were also analysed in [31]. Numerical results

demonstrated that the proposed discrete method was feasible and efficient for small

test systems.

2.6.2.9 Hybrid Artificial Intelligent Techniques

Al-Saba and El-Amin [25] proposed the application of artificial intelligent (AI) tools,

such as genetic algorithm, tabu search and artificial neural networks (ANNs) with

linear and quadratic programming models, to solve transmission expansion problem.

The effectiveness of these AI methods in dealing with small-scale and large-scale

systems was tested through their applications to the Graver six-bus system, the IEEE-

24 bus network and the Saudi Arabian network [25]. The planning work [25] aimed

to obtain the optimal design using a fast automatic decision-maker. An intelligent

tool started from a random state and it proceeded to allocate the calculated cost

recursively until the stage of the negotiation point was reached.

2.7 Conclusions

This chapter has covered the basis of transmission expansion planning problem,

problem formulation and literature survey on a variety of solution techniques

application to the planning problem. Over several past decades, researchers have

worked on transmission expansion planning and set their interest mostly on static

planning models. Unfortunately, the dynamic and pseudo-dynamic planning models

are still in an undeveloped status as dynamic planning models have some limitations

for their application to real-world transmission systems. The transmission expansion

planning models can be developed and used several different tools, from

spreadsheets to custom-written programs.

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CHAPTER 3

FUNDAMENTALS OF DIFFERENTIAL

EVOLUTION ALGORITHM AND GENETIC

ALGORITHMS

3.1 Introduction

Evolutionary algorithms (EAs) are heuristic and stochastic optimisation techniques

based on the principles of natural evolution theory. The field of investigation,

concerning all EAs, is known as “evolutionary computation”. The origin of

evolutionary computation can be traced back to the late 1950‟s and since then a

variety of EAs have been developed independently by many researchers. The most

popular algorithms are genetic algorithms (GAs), evolutionary programming (EP),

evolution strategies (ESs) and differential evolution algorithm (DEA). These

approaches attempt to search the optimal solution of an optimisation problem via a

simplified model of the evolutionary processes observed in nature and they are based

on the concept of a population of individuals that evolve and improve their fitness

through probabilistic operators via processes of recombination, mutation and

selection. The individuals are evaluated with regard to their fitness and the individual

with superior fitness is selected to compose the population in the next generation.

After several iterations of the optimisation procedure, the fitness of individuals

should be improved while current individuals explore the solution space for the

optimal value.

In this research, a novel differential evolution algorithm is proposed to be

applied directly to DC power flow based model of transmission expansion planning

problem. In addition, conventional genetic algorithm is employed to compare its

achieved results with that of the proposed method. These two optimisation

techniques are introduced and discussed in this chapter. Moreover, the optimisation

process and constraint handing techniques of the proposed algorithm are also

included in this chapter.

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3.2 Genetic Algorithms

3.2.1 Background and Literature Review

Genetic algorithm (GA) was first introduced in the book “Adaptation in Natural and

Artificial Systems” in 1975 and was mainly developed in the USA by J. H. Holland

[33]. In addition, genetic algorithm was put into practical applications in the late

1980s and it has been continuously used until now.

Genetic algorithm is a mechanism that mimics the process observed in natural

evolution. It is a general-purpose optimisation method that is distinguished from

conventional optimisation techniques by the use of concepts of population genetics to

guide the optimisation search. A population of individuals, representing a potential

candidate solution to a given problem, is maintained through optimisation process. A

fitness value of each individual is assigned according to the fitness function to

indicate the quality of a candidate solution. The individuals then must compete with

others in the population to generate their offspring. The highly fit individuals that are

those with higher fitness value have more opportunities to reproduce through

recombination operation. The offspring inherits genes of their highly fit parents and

will become even fitter, which represent a better solution to the problem concerned.

The lowest fit individuals have few opportunities to reproduce and the trace of their

genes will eventually disappear in the population. Comparison between the newly

generated offspring and their parents, the best individuals are selected regard to their

fitness values to form the population of the next generation. By repeating the GA

optimisation process, the population of individuals will develop into an optimal

solution of the problem.

Over past 20 years, genetic algorithm has been applied to solve various

engineering optimisation problems, especially electrical power system problems such

as economic dispatch [34], unit commitment [35, 36], generator/hydrothermal

scheduling [37, 38], optimal power flow [39], voltage/reactive power control [40],

capacitor placement [41, 42], generation expansion planning [43], transmission

expansion planning [22, 23, 24, 44].

An advanced engineered-conditioning genetic algorithm hybrid (AEC-GA)

with applications in power economic dispatch was proposed by Song and Chou in

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[34]. It was a combined strategy involving local search algorithms and genetic

algorithms. Moreover, several advanced techniques, which enhanced program

efficiency and accuracy such as elite policy, adaptive mutation prediction, non-linear

fitness mapping, different crossover techniques, were also explored in [34]. The

combination of the nonlinear fitness mapping and the sigma truncation scaling was

highly beneficial. Overall, the improved efficiency, accuracy and reliability achieved

by the proposed AEC-GA hybrid demonstrated its advantages in power system

optimisations in [34].

According to [35], a genetic algorithm was applied to solve the unit

commitment problem. It was necessary to enhance a standard GA implementation

with the addition of problem specific operators and the Varying Quality Function

technique in order to obtain satisfactory unit commitment solutions. The proposed

GA-UC was tested in the systems up to 100 units and the obtained results of the

proposed method were compared with Lagrangian relaxation and dynamic

programming in [35].

A genetic algorithm based approach to the scheduling of generators in a

power system was presented in [37]. An enhanced genetic algorithm incorporating a

sequential decomposition logic was employed to provide a faster search mechanism.

The power of the GA presented in [37] relied on the selection and grading of the

penalty functions to allow the fitness function that differentiates between good and

bad solutions. This method guarantees the production of solutions that did not violate

system or unit constraints. The proposed approach demonstrated a good ability to

provide accurate and feasible solutions for a medium-scale power system within

reasonable computational times.

According to [38], the problem of determining the optimal hourly schedule of

power generation in a hydrothermal power system was solved by applying a genetic

algorithm. In [38], a multi-reservoir cascaded hydro-electric system with a nonlinear

relationship between water discharge rate, net head and power generation was

investigated. In addition, the water transport delay between connected reservoirs was

also included in the problem. The proposed method provided a good solution to the

short-term hydrothermal scheduling problem and was able to take into account the

variation in net head and water transport delay factors.

An application of parallel genetic algorithm (PGA) to optimal long-range

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generation expansion planning was presented in [43]. This planning problem was

formulated as a combinatorial optimisation problem that determined the number of

newly generation units at each time interval under different scenarios. The PGA

developed in [43] belonged to the class of coarse-grain PGA in order to achieve the

trade-off between computational speed and hardware cost.

In general, genetic algorithm is a global search method based on the

mechanics of natural selection and genetics. Its characteristics make GA a robust

algorithm to adaptively search the global optimal point of certain class of

engineering problems. There are a number of significant advantages of genetic

algorithm over traditional optimisation techniques have been described in [45].

GA searches the solution from a population of points that is not a single

point. Therefore GA can discover a globally optimal point because each

individual in the population computes independently of each other. GA

has inherent parallel processing nature.

GA evaluates the fitness of each string to guide its search instead of the

optimisation function. GA only needs to evaluate objective function

(fitness) to guide its search. Derivatives or other auxiliary knowledge are

not required by GA. Therefore GA can deal with non-smooth, non-

continuous and non-differentiable functions that are the realistic

optimisation problems.

GA employs the probabilistic transition rules to select generations, which

are not deterministic rules. Therefore GA has the ability to search a

complicated and uncertain area to find the global optimum.

Although GA has many advantages as above explanation, there are also a

number of disadvantages of GA that are as follows:

GA does not always produce an exact global optimum, which may give

the local minima (premature convergence).

GA requires tremendously high computational time since a great number

of complicated fitness evaluations.

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3.2.2 Basis of Genetic Algorithms and Optimisation Process

Genetic algorithms are the most popular form of EAs and belong to the class of

population-based search strategies. They work in a particular way on a population of

strings (chromosomes), in which each string represents a possible candidate solution

to the problem being optimised and each bit (or group of bits) represents a value for a

decision variable of the problem. Firstly, each candidate solution is encoded and each

encoding represents an individual in the GA population. The population is initialised

to random individuals (random chromosomes) at the beginning of the GA

optimisation process and GA then explores the search space of possible

chromosomes for better individuals. The GA search is guided regard to the fitness

value return by an environment, which provides a measure of how well each adapted

individual in term of the problem solving. Therefore, the fitness value of each

individual determines its probability of appearing or surviving in future generations.

Codification is an essential process of GA and binary encoding of the parameters is

traditionally employed. It has been mathematically proven that the cardinality of the

binary alphabet maximises the number of similarity template (schemata) in which

GA operates and hence enhances the search mechanism. The main concept of GA

optimisation process is illustrated in figure 3.1 and a simple GA involves the

following steps:

Encoding: Code parameters of the problem as binary strings of fixed

length;

Initialisation: Randomly generate initial population strings, which evolve

to the next generation by genetic optimisation operators;

Fitness Evaluation: Compute and evaluate each string‟s fitness, which

measures the quality of solutions coded by strings;

Selection: Permit highly-fit strings as parents and produce offsprings

according to their fitness in the next generation;

Crossover: Crossover is the main genetic operator and combines two

selected parents by swapping chromosome parts between their strings,

starting from a randomly selected crossover point. This leads to new

strings inheriting desirable qualities from both chosen parents;

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Mutation: Mutation works as a kind of „life insurance‟ and flips single

bits in a string, which prevents GA from premature convergence by

exploiting new regions in the search space;

Termination: The new strings replace the existing ones and optimisation

process continues until the predetermined termination criterion is

satisfied.

Generate initial population,

Gen = 0

Start

Compute and evaluate the fitness of each

individual

Converged?

End

Form new population

Yes

CrossoverMutation

Gen = Gen + 1

Reproduction

Selection

No

Figure 3.1 The main flowchart of the typical GA optimisation process

3.3 Differential Evolution Algorithm

3.3.1 Background and Literature Review

A differential evolution algorithm (DEA) is an evolutionary computation method that

was originally introduced by Storn and Price in 1995 [46]. Furthermore, they