optimal placement of statcom for voltage stability ... · optimal placement of statcom for voltage...
TRANSCRIPT
Optimal Placement of STATCOM for voltage stabilityenhancement using particle swarm optimization
Luıs Miguel de Granja Silva e Sousa Rocha
Dissertacao para obtencao do Grau de Mestre em
Engenharia Electrotecnica e de Computadores
JuriPresidente: Prof. Paulo Jose da Costa BrancoOrientador: Prof. Rui Manuel Gameiro de CastroCo-Orientador: Prof. Jose Manuel Dias Ferreira de JesusVogal: Prof. Pedro Manuel Santos de Carvalho
Junho 2012
Take the risk of thinking for yourself. Much more happiness, truth, beauty and wisdom will come to youthat way.
Christopher Hitchens
Agradecimentos
Ao Professor Rui Castro, pela confianca depositada em mim quando eu primeiro lhe comuniquei
a minha vontade de o ter como orientador da minha tese de mestrado e pela disponibilidade sempre
demonstrada nas inumeras vezes que bati a sua porta, nao so em relacao a duvidas relacionadas
com esta tese mas tambem em outras ocasioes.
Ao Professor Ferreira de Jesus, tambem pela total disponibilidade e empenhos demonstrados ao
longo desta tese e de todo o mestrado para resolver as minhas muitas duvidas que tive ao longo
deste, e sempre de boa disposicao.
A toda a minha familia, em particular aos meus pais, que sempre me apoiaram em todos os mo-
mentos da minha vida, e que sao os principais responsaveis pela minha formacao nao so academica
mas tambem pessoal. Um especial agradecimento a eles e a toda a minha familia pela estabilidade
e ajuda que me proporcionaram ao longo de toda a minha vida.
Por ultimo, mas nao em grau de importancia, aos meus amigos. Aqueles que mantive desde os
tempos da escola, que sempre me apoiaram e cujas inumeras conversas se tornaram quase inde-
spensaveis no dia a dia, e aqueles que ganhei durante a minha formacao academica no IST e cujo
companheirismo e bom ambiente fez com que estes ultimos anos da minha formacao no IST sejam
irrepetiveis. Destes ultimos, um especial agradecimento ao Bruno Pereira, Luıs Lourenco e ao Joao
Neto, que me ajudaram na correcao deste documento.
A todos, um grande obrigado.
iii
Abstract
The major of concern of power utilities is to maintain, in all situations, the supply of electrical power
to all its customers without any failure. However, due to recent phenomenon such as demand power
increase, insufficient power generation and other economical and environmental factors, most power
system utilities operate with its equipment very close to their limits. With this situation, occurrences
such as voltage instability or even voltage collapse become likely to occur.
The Static Synchronous Compensator (STATCOM), a shunt connected Flexible AC Transmission
Systems (FACTS) device which is capable of regulating the voltage level through the generation or
absorption of reactive power, is an important tool in order to prevent these occurrences from happen-
ing.
In this thesis, a deterministic approach to the problem of where these devices have to be inserted
and what size should they have in order to enhance the voltage stability margin to a specified level
is introduced in an algorithmic form. This algorithm is constructed by using the Continuation Power
Flow (CPF) method to calculate the point of maximum loadability of the power system and using the
Particle Swarm Optimization (PSO) concept to find the optimal combination of locations and sizes of
a group of STATCOM units.
The results from the application of the CPF, PSO methods and the new algorithm proposed in this
thesis in different situations are examined, and it is confirmed that the new algorithm is able, with
the proposed design, to generate optimal locations and sizes of a group of STATCOMs in several
networks with different sizes.
Keywords
Voltage Stability, Voltage Collapse, Continuation Power Flow, Particle Swarm Optimization, Static
Synchronous Compensator (STATCOM).
v
Resumo
A principal preocupacao de um sistema de energia electrica e manter o fornecimento de ener-
gia a todos os seus clientes, sem qualquer falha. No entanto, devido a fenomenos recentes como
o aumento do consumo de energia, geracao insuficiente desta e a outros factores economicos e
ambientais, a maioria destes sistemas opera com o seu equipamento proximo dos limites. Neste
paradigma, instabilidade de tensao ou mesmo colapso de tensao tornam-se fenomenos possıveis de
ocorrer.
O Compensador Sıncrono Estatico (denominado por STATCOM), um dispositivo membro da famılia
FACTS que e capaz de regular a tensao atraves do transito de potencia reactiva, e uma importante
ferramenta na prevencao destas ocorrencias.
Nesta tese, uma abordagem determinıstica para calcular as localizacoes e dimensoes destes
dispositivos de modo a aumentar a margem de estabilidade de tensao para um valor especıfico e
introduzida na forma de um algoritmo. Este e construıdo usando o metodo Continuation Power Flow
(CPF) para calcular o ponto de colapso e o conceito de Particle Swarm Optimization (PSO) para
calcular a combinacao optima de localizacoes e dimensoes do grupo de STATCOMs.
Os resultados da aplicacao dos metodos CPF, PSO e do novo algoritmo desenvolvido nesta tese
sao estudados e e confirmado que o novo algoritmo e capaz de gerar localizacoes e dimensoes
optimas de um grupo de STATCOMs em diferentes cenarios.
Palavras Chave
Estabilidade de Tensao, Colapso de Tensao, Continuation Power Flow, Optimizacao por Enxame
de Particulas, Compensador Sıncrono Estatico.
vii
Contents
1 Introduction 1
1.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Continuation Power Flow 5
2.1 Voltage Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Reactive Power Limits of Synchronous Generators . . . . . . . . . . . . . . . . . 7
2.1.3 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Voltage Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Newton-Raphson Method Singularity . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Continuation Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Locally Parametrized Continuation Technique . . . . . . . . . . . . . . . . . . . . 15
2.3.4 Reformulation of the Power Flow Equations . . . . . . . . . . . . . . . . . . . . . 16
2.3.5 Predictor Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.6 Corrector Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.7 Choosing the Continuation Parameter . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.8 Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.9 Generator Limit Integration in the CPF method . . . . . . . . . . . . . . . . . . . 21
2.3.10 Transmission Limit vs Reactive Power Generation Limit . . . . . . . . . . . . . . 22
2.3.11 Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.1 2 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 5 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.3 IEEE New England 39 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . 31
ix
3 Particle Swarm Optimization 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 PSO Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Original Version with Inertia Weight . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.2 Constriction Factor Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.3 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.4 Discrete Variable Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Rosenbrock Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.2 Schaffer F6 function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.3 Discrete De Jong F1 function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Optimal Placement of STATCOM 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Reactive Power Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Capacitor Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.2 Synchronous Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 Static Synchronous Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 PSO Algorithm Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2 Problem’s Hyperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.3 Speed Up Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.1 5 Bus Network with 1 STATCOM Unit . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.2 Porto Santo Island 12 Bus Network with 2 STATCOMs . . . . . . . . . . . . . . . 73
4.5.3 IEEE Midwest 57 Bus Network with 4 STATCOMs . . . . . . . . . . . . . . . . . 76
5 Conclusions 79
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Bibliography 83
Appendix A Data of the 5 Bus Network A-1
A.1 Bus Base Case Power Flow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
A.2 Generator’s Reactive Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
A.3 Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
x
Appendix B Data of the IEEE New England 39 Bus Network B-1
B.1 Bus Base Case Power Flow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B.2 Generator’s Reactive Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
B.3 Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
B.4 Transformer Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4
B.5 Bus Data at Point of Collapse without Swing Bus Reactive Limits . . . . . . . . . . . . . B-5
Appendix C Data of the Porto Santo Island 12 Bus Network C-1
C.1 Bus Base Case Power Flow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
C.2 Generator’s Reactive Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
C.3 Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
C.4 Transformer Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
Appendix D Data of the IEEE Midwest 57 Bus Network D-1
D.1 Bus Base Case Power Flow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2
D.2 Generator’s Reactive Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3
D.3 Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-4
D.4 Transformer Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5
xi
List of Figures
2.1 Single Line Diagram of a Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Synchronous Generator Operating Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 PI-Equivalent Circuit of a Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 2 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Plot of V2 as a function of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Plot of V2 as a function of Pd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 CPF Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Flow Chart of the CPF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9 2 Bus Network with data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.10 Plot of V2 as a function of λ as a result of the CPF method . . . . . . . . . . . . . . . . . 27
2.11 Plot of V2 as a function of Pd as a result of the CPF method . . . . . . . . . . . . . . . . 27
2.12 Single Line Diagram of the 5 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.13 PV Curves of the 5 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.14 PV Curves of the 5 Bus Network without Swing Bus Reactive Limits . . . . . . . . . . . 31
2.15 Single Line Diagram of the New England 39 Bus Network ([1]) . . . . . . . . . . . . . . 32
2.16 Pdi vs PdTotal for the New England 39 Bus Network . . . . . . . . . . . . . . . . . . . . 33
2.17 Pdi vs PdTotal for the New England 39 Bus Network from [2] . . . . . . . . . . . . . . . . 33
2.18 Pgi vs PdTotal for the New England 39 Bus Network . . . . . . . . . . . . . . . . . . . . . 34
2.19 Pgi vs PdTotal for the New England 39 Bus Network from [2] . . . . . . . . . . . . . . . . 34
2.20 Qgi vs PdTotal for the New England 39 Bus Network . . . . . . . . . . . . . . . . . . . . 35
2.21 Qgi vs PdTotal for the New England 39 Bus Network from [2] . . . . . . . . . . . . . . . . 35
2.22 Vi vs λ for the New England 39 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . 36
2.23 Vi vs λ for the New England 39 Bus Network from [2] . . . . . . . . . . . . . . . . . . . . 36
2.24 Vi vs PdTotal for the New England 39 Bus Network . . . . . . . . . . . . . . . . . . . . . 37
2.25 Vi vs PdTotal for the New England 39 Bus Network from [2] . . . . . . . . . . . . . . . . 37
3.1 Movement of a PSO particle in a two dimensional plane . . . . . . . . . . . . . . . . . . 43
3.2 Flow Chart of PSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Plot of the Rosenbrock function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Contour Plot of the Rosenbrock function . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Plot of the Schaffer F6 function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xiii
3.6 Plot of the Schaffer F6 function with x2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.7 Plot of the De Jong f1 function with n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1 Single Line Network with FACTS devices . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Detailed Scheme of the STATCOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Memory Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Single Line Diagram of the 5 Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Plot of the Objective Function, using the configuration proposed in [3] with the first set
of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 2D Representation of the Objective Function, using the configuration proposed in [3]
with the first set of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Plot of the Objective Function, using the configuration proposed in [3] with the second
set of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.8 2D Representation of the Objective Function, using the configuration proposed in [3]
with the second set of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.9 Plot of the Objective Function, using the configuration proposed in [3] with the third set
of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.10 2D Representation of the Objective Function, using the configuration proposed in [3]
with the third set of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.11 Plot of the Objective Function, using the configuration proposed in this thesis . . . . . . 73
4.12 2D Representation of the Objective Function, using the configuration proposed in this
thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.13 Single Line Diagram of the Porto Santo Island Network . . . . . . . . . . . . . . . . . . 74
4.14 Single Line Diagram of the IEEE Midwest 57 Bus Network ([1]) . . . . . . . . . . . . . . 76
xiv
List of Tables
2.1 Points of Voltage Collapse of Figures 2.5 and 2.6 . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Base Case Power Flow Data of the 2 Bus Network for tan(φ) = 0.4 . . . . . . . . . . . . 25
2.3 Comparison between the Analytical Method and the CPF Method . . . . . . . . . . . . . 28
2.4 CPF Method Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Bus Data at the Point of Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Bus Data at Point of Collapse without Swing Bus Reactive Limits . . . . . . . . . . . . . 30
2.7 Kd values of load buses scheduled to increase . . . . . . . . . . . . . . . . . . . . . . . 32
2.8 Kg values of generation buses scheduled to increase . . . . . . . . . . . . . . . . . . . 33
3.1 PSO Parameters for the Rosenbrock Function Optimization . . . . . . . . . . . . . . . . 49
3.2 Number of Iterations for the Optimization of the Rosenbrock Function . . . . . . . . . . 50
3.3 PSO Parameters for the Schaffer F6 Function Optimization without the improvement-
based criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Success Rate of each Series of Simulations for the Schaffer F6 Function without the
improvement-based criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Success Rate of each Series of Simulations for the Schaffer F6 Function with the
improvement-based criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 PSO Parameters for the De Jong F1 Function Optimization with n = 2 . . . . . . . . . . 53
3.7 Number of Iterations for the Optimization of the Discrete De Jong F1 Function with n = 2 53
3.8 Number of Iterations for the Optimization of the Continuous De Jong F1 Function with
n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.9 PSO Parameters for the De Jong F1 Function Optimization with n = 30 . . . . . . . . . 54
3.10 Number of Iterations for the Optimization of the Discrete De Jong F1 Function with n = 10 54
4.1 PSO Parameters for the 5 Bus with 1 STATCOM Optimization . . . . . . . . . . . . . . . 68
4.2 Location of the Reactive Generation Limit and Transmission Limit of the Porto Santo
Island Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Location of the Reactive Generation Limit and Transmission Limit of the Porto Santo
Island Network with STATCOMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 PSO Parameters for the IEEE Midwest 57 Bus Network with 4 STATCOMs Optimization 77
4.5 Optimal Placement and Sizing of four STATCOMs in the IEEE Midwest 57 Bus Network 78
xv
Abbreviations
emf Electromotive Force
pu Per Unit
CPF Continuation Power Flow
PSO Particle Swarm Optimization
STATCOM Static Synchronous Compensator
FACTS Flexible AC Transmission Systems
SSSC Static Synchronous Series Compensator
UPFC Unified Power Flow Compensator
VSC Voltage Source Converter
DFACTS Distribution Network Flexible AC Transmission Systems
xvii
List of Symbols
E Synchronous Generators Emf Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Xs Synchronous Reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
VSG Synchronous Generator Terminal Voltage Phasor . . . . . . . . . . . . . . . . . . . . 7
Pg Active Power Generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Qg Reactive Power Generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Zl Line Longitudinal Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
YT Line Transversal Admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Rl Line Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Xl Line Reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Bl Line Susceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I Line Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Pd Active Power Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Qd Reactive Power Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
α Complex Power Increase Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
S0d Base Case Complex Power Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
P 0d Base Case Active Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Q0d Base Case Reactive Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
nPQ Number of Load Buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
nPV Number of Voltage Buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Pi Injected Active Power at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Pgi Active Power Generated at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Pdi Active Power Demanded at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Vi Voltage Amplitude at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Vj Voltage Amplitude at Bus j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Gij Element (i, j) of the Nodal Conductance Matrix . . . . . . . . . . . . . . . . . . . . . 13
θi Voltage Angle at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
θj Voltage Angle at Bus j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Bij Element (i, j) of the Nodal Susceptance Matrix . . . . . . . . . . . . . . . . . . . . . 13
Qi Injected Reactive Power at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Qgi Reactive Power Generated at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Qdi Reactive Power Demanded at Bus i . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
xix
λ Load Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
nP Number of Active Power Balance Equations . . . . . . . . . . . . . . . . . . . . . . . 16
nQ Number of Reactive Power Balance Equations . . . . . . . . . . . . . . . . . . . . . 16
Kdi Constant defining the rate of change of demanded power in bus i as λ variates . . . 17
Kgi Constant defining the rate of change in active power generation in bus i as λ variates 17
Fθ Partial Derivatives Matrix of F with respect to the set of voltage angles used as state
variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
FV Partial Derivatives Matrix of F with respect to the set of voltage amplitudes used as
state variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Fλ Partial Derivatives Column Vector of F with respect to λ . . . . . . . . . . . . . . . . 18
σ CPF Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
xcpf Vector of all state variables including the load parameter . . . . . . . . . . . . . . . . 19
ξ Value of the continuation parameter during the continuation process . . . . . . . . . 19
pbesti Personal Best Solution of Particle i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
gbest Global Best Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
xi(t) Position Vector of Particle i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
vi(t) Velocity Vector of Particle i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
w(t) Inertia Weight at Iteration t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
c1 Cognitive Acceleration Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
c2 Social Acceleration Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
|vmax| Maximum limit for a particle’s velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
wmax Maximum Value for the Inertia Weight . . . . . . . . . . . . . . . . . . . . . . . . . . 45
wmin Minimum Value for the Inertia Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
tmax Maximum Number of Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
K Constriction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
ϕ Sum of the Cognitive and Social Acceleration Constants . . . . . . . . . . . . . . . . 45
T Threshold Value for the Improvement-based Criterion of PSO . . . . . . . . . . . . . 46
I Maximum Number of Iterations without Improvement greater than T . . . . . . . . . . 46
|x|max Domain limit within which the particles can move . . . . . . . . . . . . . . . . . . . . 49
ω Angular Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
C Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
m Number of Functions in the Objective Function . . . . . . . . . . . . . . . . . . . . . 62
NB Number of Buses in the Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
NL Number of Lines in the Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
PLi Active power losses in the line i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
QiST Reactive Power Range of the STATCOM Unit i . . . . . . . . . . . . . . . . . . . . . 63
λ∗max Specified Voltage Stability Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
NS Number of STATCOM Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
xx
1Introduction
Contents1.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1
In this chapter the subject of this thesis is introduced and contextualized, along with the presen-
tation of the motivations for this study and its objectives. By the end of this introduction chapter, the
original contributions of this thesis and its outline are described.
1.1 Framework
Since the introduction of the electrical power network, in the end of the 19th century, the human
society has used electrical energy that is provided by the network in a variety of ways. In the early
days, the electric power generation was made in its DC form, which meant that, since there were no
power electronics available at the time and since transformers need time-varying currents to perform
their function, the voltage level could not be changed between the generation and the loads which
required different values for the voltage. Due to this flaw of the DC transmission system, the genera-
tion units on these first generation networks needed to be close to their loads in order to prevent big
voltage drops between the generation and load. This was especially true in networks that used low
voltage levels and that hence the power transmission was made using high levels of electrical current.
To combat these inconveniences, by the end of the 1800s, Nikola Tesla, introduced a new ap-
proach to the electric power system paradigm using AC polyphase generators and transformers. By
using AC voltage combined with the use of the transformer, the voltage level could now be changed in
what way necessary on the route from the generation units to the electrical loads. This meant that it
was no longer necessary to have the generation units close to their loads, which in turn signified that
different generation units in different cities could be interconnected over a wide area and that the use
of remote and low cost energy sources such as hydroelectric power sources could be utilized.
The proliferation of this approach to the concept of electric power transmission eventually led to
the power systems that are in place today. Nowadays, due to several different phenomenon such
as the increase of load power being demanded, transmission expansion and other economical and
environmental restrictions and also due to the characteristics of the AC transmission system, voltage
drops through the lines have once again become a growing concern. As a consequence of the
reactive power nature of most electrical loads, the reactive power flowing through the lines causes
the voltage to drop between the generation unit and the load. If these decreases in the value for the
voltage amplitude are not controlled via voltage-reactive power control, it may send the system to a
state of voltage instability or even voltage collapse. This concern is aggravated by the fact that, in the
last century, electrical energy transmission has become completely essential to the human society.
1.2 Motivation and Objectives
It has already been discussed that due to a variety of factors, many power systems today operate
with their equipment very close to their limits, and hence are prone to have voltage instability occur-
rences or even voltage collapse. Thus, the main motivation for this thesis is to find a way to, during
2
the normal operation of a power system, prevent these occurrences from happening. This can be
done by enhancing the steady state voltage stability margin of the network that is being analyzed, so
that during the everyday variations the system does not enter in a state of voltage instability, or even
worse, of voltage collapse.
In this thesis, the increase in the voltage stability margin is done through the insertion in the net-
work of one or several devices known as the Static Synchronous Compensator (STATCOM), which
are capable of generating or absorbing reactive power in order to control the voltage amplitude of
the bus where they are connected. However, due to the great number of buses in the majority of the
electric networks and to the variety of sizes for these devices, the insertion of these devices in the
network would have better results if it were to be done using a deterministic approach to the problem.
In order to achieve this deterministic approach to the STATCOM placement and sizing problem, in
this thesis it is constructed an algorithm using Particle Swarm Optimization (PSO) in order to find the
placement and sizing of a specified number of STATCOMs so that it enhances the voltage stability
margin to a specified level, i.e., the optimal solution. However, due to the nature of the methods used
to perform power flow studies, the location of the point of voltage collapse of a certain network cannot
be calculated using these methods. With the objective of computing the point of collapse, and hence,
the voltage stability margin of a specific network, the concept of Continuation Power Flow (CPF) is
investigated.
Therefore, the main objectives of this thesis are to present a theoretical explanation and formula-
tion of the concepts of CPF and PSO, and use these methods to construct an algorithm that computes
the optimal placement and sizing of a specified number of STATCOM units in order to enhance the
voltage stability margin.
1.3 Original Contributions
In other studies, the problem of optimal placement and sizing of STATCOM units has been already
approached ([3], [4]). In [4], this optimization problem is solved with the objective of improving the
voltage deviation in all the buses and all the active power losses in all the lines in the network during
its normal operation. However, in this work and in [3], the optimization is set to improve these param-
eter plus the voltage stability margin.
In [3], the voltage stability margin is set to be maximized, however, in this thesis, for reasons that
are going to be explained in chapter 4, it is proposed that the objective function of the optimization
should be such that the optimal solution does not maximizes the voltage stability margin but puts it at
a specified level.
3
Apart from this improvement in the design of the objective function used by the algorithm, in this
work, a different solution, from the one presented in [4], to the management of unfeasible solutions is
presented and also some speed up measures that take advantage from the environment where the
algorithm is run (MATLABTM
) and from the techniques used during the construction of the algorithm
(CPF and PSO) are proposed.
1.4 Thesis Outline
As explained earlier, the three main objectives of this thesis are to study and present the CPF and
PSO concepts and to use these methods to construct an effective algorithm that is able to find the
optimal location and sizing of multiple STATCOM devices in a power network, in order to optimize a
set of parameters. Each one of these main objectives has a dedicated chapter on this thesis. Closing
this document is a chapter dedicated to the explanations of the conclusions made in this thesis.
A brief description of each chapter is given in the next few paragraphs:
• Chapter 2, named Continuation Power Flow, begins with the presentation of the voltage stability
concept and of the voltage collapse phenomenon, followed by the theoretical explanation and
formulation of the CPF algorithm. Closing this chapter is the presentation, discussion and in
some cases comparison with other sources, of the results of the application of the CPF method
to three different networks.
• Chapter 3, has the purpose of giving an abstract presentation of the PSO algorithm, thus named
Particle Swarm Optimization. It begins by giving a theoretical presentation of the formulation of
two different approaches to the PSO paradigm followed by the illustration and discussion of the
results from the use of the PSO algorithm to optimize several different mathematical functions
that are especially built to test the effectiveness of this kind of algorithms in various situations.
• Chapter 4 begins by addressing the question of what is a STATCOM unit and where is this
device inserted in the power network’s paradigm. It then merges the two concepts described in
chapters 2 and 3 in order to construct an algorithm that finds the optimal locations and sizes of a
certain number of STATCOM units in order to enhance the voltage stability margin of a network.
The chapter then closes by giving sets of recommendations about the optimal placements and
sizings of STATCOM unit for three different networks.
• Chapter 5 is dedicated to review the conclusions made during this thesis and to propose possi-
ble further work on this matter.
4
2Continuation Power Flow
Contents2.1 Voltage Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Voltage Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Continuation Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5
This chapter is dedicated to give a full description of the CPF method. Since the concept of CPF
is strongly entwined to voltage stability and voltage collapse phenomena, sections 2.1 and 2.2 will
be dedicated to explain these fundamental concepts. After this introduction stage, section 2.3 will be
focused on the actual presentation and explanation of the CPF method followed in section 2.4 by the
application of this method in several different scenarios.
2.1 Voltage Stability
Power System Stability, according to [5], can be generally defined as the ability of a power system
to remain in the state of equilibrium under normal operating conditions and to achieve an acceptable
state to equilibrium after being subjected to a disturbance. These disturbances can have an impact in
mainly two kinds of stabilities:
Rotor Angle Stability: The ability of interconnected synchronous generators within a power system
to stay in synchronism. A fundamental factor in this problem is the manner in which the power
outputs of each synchronous generator vary as their rotors oscillate;
Voltage Stability: The ability of a power system to maintain acceptable voltage at all buses under
normal operating conditions and after being subjected to a disturbance.
Although both have great importance, the objective of this thesis is to enhance the voltage stability
margin in a power system and thus the following sections will reveal more information about this
subject.
2.1.1 Introduction
A power system enters a state of voltage instability when a disturbance (i.e., an increase of load
demand), or change in the system (e.g., large generating unit out of service) causes a progressive and
uncontrollable voltage drop. This is mainly due to the inability of the power system to meet the reactive
power demand and consequently, voltage stability can be attained by an effective Voltage-Reactive
Power Control. This control, as stated by [6], should satisfy the following objectives:
• Maintain the voltage at all buses within acceptable limits;
• Minimize reactive power transmission, in order to reduce active and reactive power losses in
lines, hence maximizing active power transmission capabilities.
Although during normal operating conditions the voltage-reactive power control ensures voltage
stability, when the system is heavily loaded the voltage instability problem can arise. This is mainly
due to the reactive power generation limits within the power system.
6
2.1.2 Reactive Power Limits of Synchronous Generators
Synchronous machines can either absorb or generate reactive power, depending on their excita-
tion currents (DC current on the rotor field winding). This characteristic can be illustrated by consid-
ering the diagram of Figure 2.1. This diagram is a monophasic scheme of a synchronous generator
where E is the Electromotive Force (emf) of the synchronous generator, Xs is the synchronous reac-
tance, VSG is the voltage at the generator’s terminals and I is the current in the circuit.
Figure 2.1: Single Line Diagram of a Synchronous Generator
Using the voltage VSG as reference (VSG = VSG 0), the active and reactive powers generated by
the synchronous machine can be written as shown in equations 2.1 and 2.2 ([6]), where δ is the angle
between the reference voltage VSG and E.
Pg =EVSGXs
sin(δ) (2.1)
Qg =VSGXs
(E cos(δ)− VSG) (2.2)
Analyzing equation 2.1 it is concluded that the maximum value for Pg is achieved when δ = π/2.
The generated reactive power on the other hand, by interpreting equation 2.2, depends on the factor
∆, depicted in equation 2.3.
∆ = E cos(δ)− VSG (2.3)
Admitting that the generator is connected to a large network and therefore the voltage amplitude
at its terminals VSG remains constant, the generated reactive power is then only controllable by the
emf E which is controllable by the excitation current. Normal excitation occurs when ∆ = 0 ([6]). If
the current is increased, ∆ becomes positive and the machine becomes over-excited and generates
reactive power. In contrast if the current decreases, ∆ becomes negative and the machine becomes
under-excited and absorbs reactive power.
However, there are limits for the amount of generated or absorbed reactive power. Figure 2.2
illustrates the operating limits of a synchronous generator ([6]). Limit 1, when the generator is heavily
over-excited, is imposed by a over-excitation limiter in order to avoid damage in the field winding
([6],[7]). Limit 2 is the locus of the maximum stator current, limited to avoid damage in the stator
7
winding ([6]). Limit 3, when the generator is heavily under-excited, the excitation current reaches
its minimum which is imposed to maintain the synchronous motion in the machine ([6]). It can also
be inferred that, due to the maximum and minimum values allowed for the excitation current, the
maximum reactive power which can be absorbed is less than the maximum reactive power that can
be generated.
1
2
3
P(pu)
Q(pu)
Figure 2.2: Synchronous Generator Operating Limits
2.1.3 Transmission Lines
Despite the reactive generation limits imposed by synchronous generators, another important fac-
tor that contributes to the demand of reactive power in the system, and hence to voltage stability, is
the loadability of its transmission lines. Transmission lines throughout this thesis are represented by
its PI-Equivalent circuit, represented in Figure 2.3.
Rl
jXl
jBl/2jBl/2
Figure 2.3: PI-Equivalent Circuit of a Transmission Line
In the PI-equivalent circuit, the line is characterized by a line or longitudinal impedance Zl and by
a transversal admittance YT.
Zl = Rl + jXl YT = jBl (2.4)
Since these two elements either absorb (Longitudinal Impedance) or generate (Transversal Ad-
8
mittance) reactive power it is deduced that, depending on the value of the parameters of the cable,
the length of the line and its loading level, the line will either absorb or generate reactive power.
The most common transmission lines in a large power system are overhead lines ([6]). This kind of
lines rely on their loading level to define their reactive power behavior. They generate reactive power
under light load since their production due to the transversal admittance exceeds the reactive power
requirement due to the longitudinal impedance. In contrast, under heavy load, the reactive power
being absorbed exceeds the reactive power being generated ([7]).
Underground lines on the other hand don’t have this twofold result. Independently of the loading
level, due to their high shunt capacitance, underground lines always generate reactive power since
the reactive power requirement due to the longitudinal impedance never exceeds the reactive power
being generated. As a result, a small amount of the reactive power demand is met by the reactive
power generated by underground lines.
2.2 Voltage Collapse
2.2.1 Introduction
In recent years, several phenomenon such as demand increase, transmission expansion and eco-
nomical restrictions have led power utilities to operate power networks close to their transmission
limits ([8]). As the power demand increases during the day, voltage magnitudes in buses throughout
the network slowly decline due to the voltage drops in transmission lines. These decreases can be
controlled, as referred in section 2.1.1, via voltage-reactive power control. Nevertheless, continuous
increases lead to greater voltage drops through lines and can cause some generators to reach their
reactive power generation limits. These eventually drive the system to an unstable state where volt-
age magnitudes decrease rapidly ([8]). This phenomena is known as voltage collapse.
Voltage collapse can be demonstrated using a simple 2 Bus network like the one illustrated in
Figure 2.4.
V1 V2
I
Pd+jQd
Figure 2.4: 2 Bus Network
Using the voltage from Bus 1, V1, as reference (V1 = V1 0) and neglecting the transversal admit-
9
tance in the line, the current I can be calculated as:
I =S∗dV∗2
=Pd − jQdV2 θ
(2.5)
Using equation 2.5, one can formulate:
V1 = V2 + ZlI = V2 −θ + (Rl + jXl)Pd − jQdV2 θ
(2.6)
Multiplying both sides by V∗2:
V1V2 θ = V 22 + (Rl + jXl)(Pd − jQd) (2.7)
Decomposing the equation 2.7 into real and imaginary parts:
V1V2 cos(θ) = V 22 +RlPd +XlQd (2.8)
V1V2 sin(θ) = XlPd −RlQd (2.9)
Finally, squaring and adding the real and imaginary parts of equation 2.7, results in the following
equation:
V 42 + [2(RlPd +XlQd)− V 2
1 ]V 22 + (R2
l +X2l )(P 2
d +Q2d) = 0 (2.10)
Equation 2.10 is a biquadratic equation, whose roots are:
V2 = ±
√−b±
√b2 − 4ac
2a(2.11)
Where:
a = 1 b = 2(RlPd +XlQd)− V 21 c = (R2
l +X2l )(P 2
d +Q2d)
These four solutions characterize V2, the voltage magnitude in Bus 2, since V1 and the line pa-
rameters Rl and Xl are constant, as a function of active and reactive power demanded in Bus 2.
However, two of the solutions describe V2 as being negative, which have no significance in a power
system’s view of the equations. This leaves only the two feasible solutions in equation 2.12.
V2 =
√−b±
√b2 − 4ac
2a(2.12)
As explained earlier, voltage collapse occurs when a continuous increase in the load level of
the network leads to unstable voltage drops through the lines. With the purpose of modeling this
continuous increase in the load level, the complex power in Bus 2, Sd, can be expressed as a function
of a complex power increase factor, α, as in equation 2.13.
Sd(α) = S0d(1 + α) = (P 0
d + jQ0d)(1 + α) (2.13)
Where S0d, P 0
d and Q0d are respectively the base case complex, active and reactive powers. Using
this formulation the voltage magnitude V2 can be plotted as a function of α, and thereby, using the
solutions in equation 2.12, as a function of the demand power. In Figures 2.5 and 2.6 it is represented
the voltage magnitude in Bus 2, in Per Unit (pu) as a function of α and Pd respectively, with Rl =
0.004 pu, Xl = 0.03 pu, S0d = 1 + 1× tan(φ)j [pu] and for tan(φ) = 0 and tan(φ) = 0.4.
10
Figure 2.5: Plot of V2(α)
Figure 2.6: Plot of V2(Pd)
The curves depicted in Figures 2.5 and 2.6 are known as PV curves or nose curves due to its
characteristic shape. In these examples it is observed the phenomena of voltage collapse. Consid-
ering that the generator in Bus 1 as no reactive power generation limits, and therefore, it maintains
V1 constant, V2 suffers a decrease due to the voltage drop on the line as Sd increases. If preventive
measures are not undertaken, further increases in the power demand causes the system to go into
a unstable state where V2 decreases rapidly, hence reaching the point of voltage collapse. It is also
important to remark that, as pointed out by [9], for loads under the maximum loading condition, there
are two solutions for the voltage amplitude V2, the one provided by the upper curve is physically ac-
ceptable, the one provided by the lower curve has no physical interest. In the examples of Figures 2.5
11
and 2.6, voltage collapse occurs at the points (α, V2) and (Pd, V2) presented on Table 2.1.
Table 2.1: Points of Voltage Collapse of Figures 2.5 and 2.6
Figure 2.5 Figure 2.6
tan(φ) = 0 (13.592, 0.6646) (14.592, 0.6646)tan(φ) = 0.4 (9.289, 0.5791) (10.289, 0.5791)
In this example, the problem of determining the point of collapse was solved using a solution that
was obtained analytically. However, as pointed out by [10], for larger networks this problem involves
the solution of a system of non-linear equations, and therefore, all the methods utilized to solve this
problem are classified as iterative.
2.2.2 Newton-Raphson Method Singularity
There are several methods that are used to perform the power flow study of a power system. The
Newton-Raphson method is the most widely used due to its superior convergence velocity when com-
pared to other methods ([6],[10]). It is named after Isaac Newton and Joseph Raphson and it is an
iterative method used in numerical analysis to solve real-valued non-linear equations. In the following
paragraphs a conceptual description of this method will be provided followed by its adaptation to the
power flow problem.
Considering the following system of n equations and n variables represented in matrix form:f1(x)...
fn(x)
=
y1...yn
(2.14)
If all the functions are differentiable, the values of the components in the solution vector [x] at
iteration k + 1 of the Newton-Raphson method can be computed through the equation 2.15, where
[Jk] is the jacobian matrix of fk[x] defined by 2.16.
[xk+1] = [xk] + [Jk]−1(
[y]− fk[x])
(2.15)
[Jk] =
∂fk
1
∂x1. . .
∂fk1
∂xn
.... . .
...∂fk
n
∂x1. . .
∂fkn
∂xn
(2.16)
As demonstrated by equation 2.15, one iteration’s solution depends directly upon its previous value
and therefore the initial conditions of the problem [x0] have to be provided with reasonable values. The
algorithm then runs until the difference |[y]− fk[x]| is lower than a specified tolerance.
The objective of the power flow study is to compute the amplitudes and phases of the voltages
in each bus. Thus, since the Newton-Raphson Method solves real value equations, these must be
formulated so that the solution vector has the value of the amplitudes and phases of the voltages.
Another factor that contributes to the formulation of the problem is that of the existence of various
types of buses within a power system:
12
Swing Bus (SW): Bus where the amplitude and phase of the voltage are controlled and therefore
known;
Voltage Bus (PV): Bus where the amplitude of the voltage is controlled within the generators limits;
Load Bus (PQ): Bus where neither the amplitude or phase of the bus is known;
Since some of the buses in power systems have specified values for their voltages, it is concluded
that the problem must be formulated initially with 2nPQ + nPV equations and variables, where nPQ
and nPV represents respectively the number of PQ and PV buses in the network. The equations used
to formulate the problem are those that translate the power balances in each node, described in this
work by equations 2.17 and 2.18.
Pi = Pgi − Pdi =
n∑j=1
ViVj
[Gij cos(θi − θj) +Bij sin(θi − θj)
]i = 1, . . . , n (2.17)
Qi = Qgi −Qdi =
n∑j=1
ViVj
[Gij sin(θi − θj)−Bij cos(θi − θj)
]i = 1, . . . , n (2.18)
Another adaptation that has to be made to the conceptual algorithm is the verification of the gen-
erators limits in PV buses. After every iteration the generated reactive power in each of the PV buses
has to be computed. If the computed power is off the limits of the generator, that bus is reclassified
as a false PQ bus. If in the next iteration, the computed generated reactive power is within the limits
of the generator, that bus goes back to being a PV bus. This accounts for the fact that generators can
only hold the amplitude of the voltage in the bus where they are connected while the injected reactive
power is within specified limits.
Although this method works perfectly in most power flow studies, in the first part of this work, the
objective is to encounter a way to determine the point of collapse of a specific network. By using this
exact formulation of the Newton-Raphson power flow method one can construct a simple algorithm
that conceptually could compute the point of collapse of a network through successive power flow
calculations for continuous increasing loads until the point of collapse is reached.
Using the same 2 Bus network example that was used in section 2.2.1, the value of V2 θ2 in each
load level could be computed by applying the Newton-Raphson method to equations 2.19 and 2.20.
P2 = −Pd = V2V1
[G21 cos(θ2 − θ1) +B21 sin(θ2 − θ1)
]+ V 2
2 G22 (2.19)
Q2 = −Qd = V2V1
[G21 sin(θ2 − θ1)−B21 cos(θ2 − θ1)
]− V 2
2 B22 (2.20)
The problem of obtaining the point of collapse could then be done by calculating V2 for each value
of Sd or α:
Sd = S0d(1 + α) (2.21)
Although this formulation of the problem behaves well for values of Sd far from the point of col-
lapse, as the algorithm approaches the point of collapse the determinant of the Newton-Raphson
13
Jacobian begins to decrease. As a consequence, attempts of obtaining power flow solutions near the
point of collapse are prone to divergence and error. In fact, obtaining a power flow solution in the point
of collapse results in a singular Jacobian matrix ([11]).
The following section is dedicated to present a method that avoids this singularity by slightly re-
formulating the power flow equations and by applying a locally parametrized continuation technique.
This method is called Continuation Power Flow.
2.3 Continuation Power Flow
2.3.1 State of the Art
In order to overcome the singularity problem encountered in the Newton-Raphson method, ac-
cording to [12], in early 1990’s researchers proposed a series of new methods to replace it.
In 1991, a first attempt was made in [13] to obtain not only the critical loading point but also the
whole PV curve through an homotopy continuation method based on the Newton-Raphson power flow
method. Although the author in [13] claimed that this method can overcome the numerical difficulties
related to the singularity of the Jacobian matrix, according to [12], the problem wasn’t fully solved and
this method still faces singularity.
Later, in 1992, the method used in this work was developed in [11]. The authors proposed a
method for finding a continuum of power flow solutions starting at a base load and leading to the
voltage stability limit, i.e. point of collapse, by a slight modification of the power flow equation and by
applying a locally parametrized continuation technique in order to avoid the singularity problem. The
research made in [11] resulted in a computational research tool called UWPFLOW which is provided
by the University of Waterloo. In 1995, in [14], it was proposed a computer package called CPFLOW
that improved the computational performance in comparison with the UWPFLOW package.
Another variation of the CPF method was proposed in [15], which was published in 1993, combin-
ing the continuation method developed in [11], and the point of collapse method developed in [16].
According to [12], the method proposed by [15] runs under the continuation method until it comes to
the singularity where it uses the point of collapse method. The disadvantage with this proposal is the
need to run under two different methods and therefore needs more computational effort.
2.3.2 Introduction
In section 2.2.2 it was concluded that the Newton-Raphson method, usually used in power flow
studies, is not appropriate for obtaining the power flow solution at the point of collapse due to its
ill-conditioning at this point. In contrast, the CPF technique, presented in this section, was created to,
according to [11], find a continuum of power flow solutions starting at some base load leading to the
14
steady state voltage stability limit (Point of Collapse) of the system. This means that the main feature
of this new method must be its ability to remain well-conditioned throughout the whole experience
and specifically at and near the point of collapse of the system. In Figure 2.7, it is depicted the main
principle of this method.
Load
V(pu)
Corrector
Predictor
Point of
Collapse
Figure 2.7: CPF Principle
For each solution of the PV curve, the CPF method uses a predictor-corrector scheme to a set of
power flow equations that are reformulated to include the load parameter λ. This parameter is similar
to the complex power increase factor α used in section 2.2.
From a base case power flow solution, the algorithm uses a tangent vector as a predictor of the
subsequent solution corresponding to a different value of the load parameter. The estimated solution
is then corrected using a Newton-Raphson method with a slightly modified formulation. This process
is then repeated until the point of collapse of the system that is being studied is reached.
2.3.3 Locally Parametrized Continuation Technique
In section 2.3.2, it was roughly explained the scheme used by the CPF algorithm to obtain all the
solutions between the base case solution and the voltage stability limit solution. However, this scheme
is not possible without local parameterization. According to [2], parameterization is a mathematical
means of identifying each solution on the branch, a kind of measure along a branch. In other words,
parameterization provides an identification for each solution in a branch.
In the Newton-Raphson power flow method the problem is formulated using a set of nonlinear
equations, F ([x]) = 0, that describe the power balance in each node and where consequently [x]
describes a set of state variables composed by voltage amplitudes and angles. However in the CPF
method, the algorithm has to compute the solution vector [x] over a branch described by the load
parameter. This means that a specific solution in the CPF technique is described by the original set
of state variables [x] plus the load parameter, resulting in a formulation described by equation 2.22.
F ([x], λ) = 0 (2.22)
15
This results in a problem, the number of equations has remained the same, although a new state
variable was added. Consequently in each continuation step, the variation of one of the state variables
has to be specified. For example, in the Newton-Raphson based process proposed in section 2.2.2,
the formulation of the problem would be similar and required at each iteration a constant increment of
α.
In the CPF method, the process is slightly different. Choosing the load parameter as the state
variable that receives a specified variation throughout the experiment, although initially a solution to
2.22 can be found for each value of λ, causes problems to arise when a solution does not exist for a
maximum value of λ (Point of Collapse). To solve this problem, the parameter that receives a speci-
fied variation is chosen locally at each continuation step and thus resulting in a process that is locally
parameterized. This parameter is called continuation parameter and the method of choosing this pa-
rameter will be presented in section 2.3.7. As stated by [2], in summary, local parameterization allows
not only the added load parameter λ, but also the original state variables to be used as continuation
parameters.
2.3.4 Reformulation of the Power Flow Equations
In order to execute the CPF method, the set of equations must be reformulated to accommodate
the load parameter but first this parameter must be properly defined. The load parameter λ is a
dimensionless scalar that serves as global indicator of the load level within a power system. At this
point, it is essential to explain that, although one can easily infer that the load parameter affects the
demanded powers in all buses, it must also affect the active power generated in PV buses. This fact
has to do mainly to the original construction of the original problem in the Newton-Raphson power
flow method. In section 2.2.2 it was explained that:
• The number of active power balance equations (nP ) in the system to be solved equals the sum
of the number of PQ (nPQ) and PV (nPV ) buses;
• The number of reactive power balance equations (nQ) in that same system equals the number
of PQ (nPQ) buses.
Hence, assuming that each bus with dynamic power generation, i.e., that can regulate its power
outlet to suit the network’s needs, is either a swing bus or a PV bus, the generating active powers that
could appear in the formulation of the problem belong to PV buses and therefore, if the generation of
active power is to accompany the increase of the load power, the load parameter must also affect the
active power generated in PV buses.
16
In section 2.2.2, the power flow equations used for the Newton-Raphson method were defined in
the format expressed in equation 2.23.
f1([x])...
fn([x])
=
y1...yn
⇔
...Pi([x])
...Qi([x])
...
=
...Pgi − Pdi
...Qgi −Qdi
...
(2.23)
However, due to the appearance of a new state variable, the load parameter, the generated and
demanded powers are no longer constant and depend of λ resulting in a new set of equations de-
scribed by 2.24.
f1([x], λ)...
fn([x], λ)
= 0⇔
...Pi([x]) + Pdi(λ)− Pgi(λ)
...Qi([x]) +Qdi(λ)−Qgi
...
= 0 (2.24)
It can be seen in equation 2.24, that there is no need to define the generated reactive powers to
depend upon λ. This is due to the fact that the reactive power balance equations either belong to
PQ buses, which do not have dynamic generation, or to false PQ buses, which have generation, but
at a constant value. Despite this, there is a need to construct a loading model that helps define the
dependence of the demanded powers and generated active powers upon λ in the equations.
As indicated by [9], there are several ways of integrating this parameter in the power flow equa-
tions which only vary on the manner of how the power being demanded and the active power being
generated change as the load parameter varies. In this work, the loading model used is the same
used by [11] and it is presented in equations 2.25, 2.26 and 2.27.
Pdi(λ) = P 0di(1 + λKdi) (2.25)
Qdi(λ) = Q0di(1 + λKdi) (2.26)
Pgi(λ) = P 0gi(1 + λKgi) (2.27)
It has been already defined that the load parameter λ is a global indicator of the load level within
a power system and that it affects not only the demanded power in the system but also the generated
active power in PV buses. However, this model defines the need for constants Kdi that, since the
load in each bus may increase at a different rate than the rest, define the load increase in bus i as
λ increases. The same happens in the case of the active power generation in each PV bus, as λ
increases, the constants Kgi define the increase in active power generation in bus i. In summary, the
constants Kdi and Kgi define the specific variation of load and active power generation in each bus
as λ changes.
17
If the loading model presented above is inserted on the power flow equations, it results in the
equations 2.28 and 2.29.
Pi([x]) + P 0di(1 + λKdi)− P 0
gi(1 + λKgi) = 0 i = 1, . . . , n (2.28)
Qi([x]) +Q0di(1 + λKgi)−Qgi = 0 i = 1, . . . , n (2.29)
This concludes the reformulation of the set of equations, necessary to apply the predictor-corrector
process described in the following sections.
2.3.5 Predictor Process
In section 2.3.2 it was referred that, before initiating the CPF method, first a base case power flow
solution has to be found. This solution can be found by applying the Newton-Raphson power flow
method to the set of equations of the system when λ = 0.
Once the base case solution has been found, the CPF method initiates by making a prediction of
the subsequent solution. This prediction is made by taking a sized step in a tangent direction which is
calculated by obtaining the tangent vector. This vector is obtained by taking the derivative on the both
sides of equation 2.22. Remembering that the variable [x] represents [[θ], [V ]] where [θ] represents
the set of voltage angles and [V ] represents the set of voltage amplitudes used as state variables and
that F is used to denote the whole set of equations, the derivative of equation 2.22 takes the form of
equation 2.30.
dF ([x], λ) = dF ([θ], [V ], λ) = Fθ[dθ] + FV [dV ] + Fλdλ = 0 (2.30)
Factorizing equation 2.30, results in the matrix equation 2.31.
[Fθ FV Fλ
] [dθ][dV ]dλ
= 0 (2.31)
Analyzing the equation 2.31, it can be seen that on the left side of the equation there is a matrix
of the partial derivatives multiplied by a vector composed by the differentials. The matrix of partial
derivatives can be seen as the original power flow Jacobian, composed by the matrices Fθ and FV ,
augmented with one column Fλ, representing the partial derivatives of the set of equations F with
respect to the added state variable λ. The vector of differentials is the so called tangent vector that is
being sought.
t =[[dθ] [dV ] dλ
]T(2.32)
However to calculate the tangent vector, the problem of having a set of 2nPQ + nPV equations to
2nPQ + nPV + 1 state variables must be overcome. This problem is solved through the process of
local parameterization presented in section 2.3.3. As stated in section 2.3.3, to solve this problem,
one state variable must have a specified variation in each continuation step. Which is equivalent to
say, using the notion of the tangent vector, that the continuation parameter must have a specified non-
zero magnitude in its component of the tangent vector and therefore resulting in adding an equation to
the matrix equation 2.31 that states that the continuation parameter component in the tangent vector
18
is equal to a non-zero value. In this work, in accordance with [2] and [11], this specified value is ±1,
depending on the previous direction of the vector. This reformulation leads to the matrix equation
2.33. [Fθ FV Fλ
ek
]t =
[0±1
](2.33)
In equation 2.33, ek is an appropriately dimensioned row vector with all elements equal to zero
except the element corresponding to the continuation parameter, which equals one. The tangent
vector t can now be calculated by inversing the augmented jacobian and thus resulting in equation
2.34.
t =[[dθ] [dV ] dλ
]T=
[Fθ FV Fλ
ek
]−1 [0±1
](2.34)
After the tangent vector is obtained, the predicted value of the subsequent solution can be found
from equation 2.35. [θ][V ]λ
k+1
Pred
=
[θ][V ]λ
k + σ
[dθ][dV ]dλ
k (2.35)
Where the symbol σ stands for the step size and k is the number of the iteration. The value of
this step is of great importance to the algorithm and it is a subject that will be picked up later in this
work. This is also a factor that withdraws any importance from the specified magnitude attributed to
the continuation parameter component in the tangent vector. To the predicted solution is then applied
the corrector process described in the following section.
2.3.6 Corrector Process
In section 2.3.5 it was explained how to obtain a prediction of the subsequent solution as a part of
the CPF method. However, this prediction may result in incorrect results, and therefore, a method of
correcting the predicted solution is needed.
Correcting this solution involves using a modified Newton-Raphson power flow method. As said
in section 2.2.2, the Newton-Raphson method is a means of solving a set o real value equations. In
power flow analysis this method is applied to a set of power balance equations with a format expressed
by equation 2.23. In this case the Newton-Raphson is applied to the set of equations expressed by the
matrix equation 2.24 plus an added equation needed to perform local parameterization. The objective
of this added equation is to make the value of the continuation parameter constant throughout the
corrector process, thus meaning that the value of the continuation parameter obtained in the predictor
process is unaltered. Let xcpf be the set of all state variables, including the load parameter.
[xcpf ] =
[θ][V ]λ
xcpf ∈ R2nPQ+nPV +1 (2.36)
Considering the definition of xcpf , then the new set of equations used in the corrector process
would be described by equation 2.37. F ([xcpf ])
xcpfk − ξ
= 0 (2.37)
19
Where xcpfk specifies the continuation parameter and ξ the value of its predicted solution.
In summary, the corrector process finds the set of values of the state variables that, in the vicinity
of the predicted solution, respect all the power balance equations included in the problem and at the
same time maintains the value of the continuation parameter.
2.3.7 Choosing the Continuation Parameter
Throughout the description of the predictor-corrector process, in sections 2.3.5 and 2.3.6, the no-
tion of the continuation parameter is referred several times. However a method of choosing which
state variable serves as continuation parameter has yet to be described. This section is dedicated to
that process.
According to [11], the continuation parameter must correspond to the state variable that has the
greatest rate of change near the given solution. Using the notion of the tangent vector, the continua-
tion parameter corresponds to the state variable that has the largest, in absolute value, tangent vector
component.
As referred in section 2.3.3, when starting the process from the base solution, the load parameter
is a good choice for continuation parameter. This is especially true if the base case is characterized
by normal or light loading ([11]), i.e., far from the point of collapse. In these conditions, voltage am-
plitudes and angles within the power system have a slow decrease under load change and therefore
the variation in load change is expected to be larger than the variation in the voltage values.
Once the algorithm approaches the point of collapse, the variation in voltage values becomes
progressively significant, this makes the choice of the load parameter for continuation parameter less
valid since it may have a small variation in comparison to the other state variables. For these reasons,
as is presented in section 2.3.3, the choice of which state variable is the continuation parameter has
to be made at each iteration of the algorithm.
In this thesis, the choice of which state variable is used as the continuation parameter is made
by analyzing which state variable had the greater absolute variation in the previous iteration. In
section 2.3.5 it is described that the value of the tangent vector component that corresponds to the
continuation parameter is either +1 or −1. This depends on the component previous value signal. If
the continuation parameter, in the previous iteration of the algorithm, was decreasing (e.g., a voltage
amplitude is expected to decrease during the algorithm), then the value attributed to its component in
the tangent vector of the current iteration is −1. On the other hand, if the continuation parameter was
increasing (e.g., the load parameter is expected to increase), the value attributed to its component in
the tangent vector of the current iteration is +1. This differentiation happens so that when choosing
the state variable that serves as continuation parameter, the previous direction of the component that
20
corresponds to that state variable remains unaltered.
2.3.8 Stopping Criterion
Until this point, the predictor-corrector process that is the basis of the CPF method has been de-
fined, however, a stopping criterion for this method has yet to be defined. In this work, the CPF method
was constructed to compute the point of maximum loadability, i.e., the point of voltage collapse, thus,
it follows that the CPF method can end immediately after reaching the point of collapse.
For the algorithm to end after reaching the point of collapse, it must have the means of identifying
this particular point. The point of voltage collapse is characterized as the point where the loading
within a power system reaches a maximum, and hence the maximum point of the load parameter λ.
After this point, as shown by Figures 2.5 and 2.6, the loading level begins to decrease. For this reason
the point of collapse can be easily identifiable by checking for a change in the signal of the tangent
vector component dλ, i.e., the component that corresponds to the load parameter. If this component
has a positive value, it means that the point of collapse has not been passed. On the other hand, if it
has a negative value, it means that the point of collapse has been passed and that the algorithm must
end.
2.3.9 Generator Limit Integration in the CPF method
It has already been established, in earlier sections, that the CPF method basis itself on increasing
the load level within a power system until the point of collapse is reached. Therefore, it is presumable
that on the way to the point of collapse, during the algorithm, some generators reactive outlet may
reach their maximum limits.
In the Newton-Raphson power flow method, as referred in section 2.2.2, these occurrences are
managed by transforming the PV buses whose limit has been reached to false PQ buses, therefore
adding the subsequent equations that will define the new state variables (i.e., the voltage amplitudes
of those buses) to the set of power flow equations of the problem. In this work, the same approach is
used within the CPF method to manage buses whose limits have been reached. In the end of every
iteration of the CPF method it is checked if there are PV buses that have reached their reactive power
limits, and if so, those buses are redefined as false PQ buses.
With this limit management defined, since it only manages the limits of PV buses, one can ad-
vocate if the swing bus of a particular power system should not also have generated reactive power
limits. This could be the case if the swing bus represents a generation unit with similar capacities
of the other generation units in the system. On the other hand, if the swing bus represents a larger
power system and not actually a generation unit, these limits may not be considered, since the reac-
tive outlet of that bus is not a result of a physical generation unit and is rather a result of the reactive
supply capacities of a greater power system.
21
2.3.10 Transmission Limit vs Reactive Power Generation Limit
Picking up on the 2 Bus example of section 2.3.2, it was assumed that the generation unit in Bus
1 could supply any amount of active and reactive power. However, if this hypothesis is removed, it
could be the case that during the increase of load power in Bus 2 from the base case value to the
point of collapse caused by the voltage drop in the transmission line, there can be a value for α for
which the reactive power generated in Bus 1 is the maximum one. Since the system has no other
reactive power source, the load power cannot be increased any further. If this proposition is true, the
point of collapse of the network is not caused by the voltage drop in the transmission line, i.e., the
transmission limit, but by the reactive power generation limit.
Extrapolating this assertion to larger power systems, there can only be a reactive power generation
limit for the whole network, if every generation unit in it has reactive generation limits, which is usually
the case when the reactive generation limits of the swing bus are considered. In order to identify the
correct location of the point of collapse, the CPF algorithm constructed in this work has the ability of
detecting the reactive power generation limit of a network if it occurs before the transmission limit and
if so, it stops the computation and establishes the point of collapse by being characterized by the load
level where the reactive power generation limit occurs.
2.3.11 Step Size
In section 2.3.5 it is mentioned a need of the algorithm for a step value σ. This factor is of great
importance to the algorithm. At each iteration of the CPF method, the value of the step size σ has
to be such that the resulting predicted solution is within the radius of convergence of the corrector
method. This can be achieved by two different solutions, either applying a small constant step that
guarantees convergence or using a dynamic step scheme as suggested in [2] and [11]. The dynamic
step scheme can shorten the number of iterations needed to reach the point of collapse but adds a
degree of complexity to the method which is inherited from the computation of the step value in each
iteration. On the other hand, using a small constant step, as said earlier, guarantees convergence
throughout the algorithm but the number o iterations is larger.
In this work, the CPF method is used to calculate the value of a fitness function within a optimiza-
tion algorithm that will be explained in the following chapters. Although the dynamic step scheme can
be used as a speed up measure within the whole algorithm, it was preferred to use a very small con-
stant step that prevents divergence of the algorithm and use speed up measures that take advantage
of the characteristics of both the CPF method and the optimization algorithm that is used in this work.
22
Start
Calculate base case
power flow solution
Compute tangent
vector
Calculate predicted
solution
Check for the point of
collapse End
Correct predicted
solution
Choose Continuation
Parameter
Yes
No
Check Reactive Limits
Figure 2.8: Flow Chart of the CPF method
This section concludes the presentation of the CPF method. In Figure 2.8 is presented the flow
chart that summarizes the whole process.
23
2.4 Results
In the earlier sections the process of Continuation Power Flow was described. The objective of
this section is to present and interpret the data resulting from applying the CPF method constructed in
this work to several different systems. These simulations are performed with three different networks.
Firstly the 2 Bus network used in 2.2.1 will be tested, followed by a 5 Bus network and a 39 Bus
network.
2.4.1 2 Bus Network
Since the point of collapse of the 2 Bus network used in this work has already been found in sec-
tion 2.2.1, it serves as an optimum initial result comparison, in this case between the CPF method
and the analytical method used in the initial computation. For the sake of readability, the data of this
network is displayed again in Figure 2.9.
V1=1 pu V2
Zl=0.004+0.03j
Sd=1+tan(ϕ)j0
Figure 2.9: 2 Bus Network with data
As was considered in section 2.2.1, in this simulation it is assumed that the generator in Bus
1 is capable of maintaining V1 at 1 pu as the load in Bus 2 increases, i.e., generation limits are not
considered. Hence the equations that define the problem are the two power balance equations of Bus
2. Using the formulation suggested in section 2.3.4, these equations can be written as the following:
f1 = V2V1
[G21 cos(θ2 − θ1) +B21 sin(θ2 − θ1)
]+ V 2
2 G22 + P 0d (1 + λKd2) = 0 (2.38)
f2 = V2V1
[G21 sin(θ2 − θ1)−B21 cos(θ2 − θ1)
]− V 2
2 B22 +Q0d(1 + λKd2) = 0 (2.39)
Where:
V1 = V1 θ1 = 1.0 0 [pu] P 0d = 1.0 pu Q0
d = P 0d tan(φ) pu
In order to construct a good comparison between the results achieved in section 2.2.1 and the
results achieved by the CPF algorithm, the problem must be set in the same way. While solving the
problem analytically, the load power in Bus 2 was set to increase with the complex power increase
factor α in the following manner:
Sd = S0d(1 + α) = (P 0
d + jQ0d)(1 + α) (2.40)
24
In order to the problem be the same in the two computations, the constant Kd2 must be equal
to 1, resulting in a direct comparison between the parameters α and λ. In this case, the problem of
dimensioning the constant Kd2 can be addressed in another way. By remembering that the objective
of this constant is to define the local increase in power as λ increases and since there is no other load
besides the one in Bus 2, it can be asserted that the purpose for this constant no longer exists.
By observing equations 2.38 and 2.39 and as explained in section 2.3, it is concluded that the
state variables calculated along this simulation are the voltage amplitude of Bus 2 V2, the voltage
angle of Bus 2 θ2 and the load parameter λ. Utilizing the nomenclature used in section 2.3.6 during
the generic description of the CPF process, [xcpf ] can be defined as:
[xcpf ] =
θ2V2λ
One of the advantages of using this simple 2 Bus example is that the application of the CPF
method to this network is much simpler than with larger networks, this is also the case with the pre-
sentation of the results since there is only one bus where the voltage amplitude will decay. Taking
this into account, the next few paragraphs will describe in detail the first iteration of the CPF method
for tan(φ) = 0.4, followed by the presentation of the results regarding the simulations for both cases
analyzed in section 2.2.1, tan(φ) = 0.4 and tan(φ) = 0.
As explained earlier, the CPF method must start with an initial point, this point being the base
case power flow solution, i.e., the solution of the equation system formed by equations 2.38 and 2.39
when λ = 0. Using the Newton-Raphson power flow method, the power flow data in Table 2.2 can be
presented.
Table 2.2: Base Case Power Flow Data of the 2 Bus Network for tan(φ) = 0.4
|V| [pu] θ [rad] Pg [pu] Qg [pu] Pd [pu] Qd [pu]
Bus 1 1.0 0.0 1.004799 0.435991 0.0 0.0Bus 2 0.98331 −0.028886 0.0 0.0 1.0 0.4
Now that the base case power flow solution has been calculated, the predictor-corrector scheme
that is the basis of the CPF algorithm can start. The first step to compute the subsequent predicted
solution is to choose which state variable represents the continuation parameter in this first iteration.
As stated in section 2.3.7, for the first iteration of the algorithm, the load parameter λ is the logical
choice for continuation parameter.
After this decision has been made, the following action, as described in section 2.3.5, is to compute
the tangent vector that will make the prediction. This vector is calculated using equation 2.34 which is
25
rewritten in equation 2.41 using the values of this example.
t =
dθ2dV2dλ
=
∂f1∂θ2
∂f1∂V2
∂f1∂λ
∂f2∂θ2
∂f2∂V2
∂f2∂λ
0 0 1
−1 0
0+1
=
32.067 3.2769 1.0
−5.222 31.7977 0.4
0 0 1.0000
−1 0
0+1
=
=
−0.029406−0.017409
1.0000
(2.41)
As expected, the component of the continuation parameter (in this case λ) in the tangent vector has
a value 1.0 that is fixed by the algorithm while the other components are computed through equation
2.41. With this tangent vector components calculated, the predicted solution can be determined using
equation 2.42. θ2V2λ
1
Pred
=
θ2V2λ
0
+ σ
dθ2dV2dλ
0
︸ ︷︷ ︸Tangent Vector
(2.42)
Using the values presented in Table 2.2 and in equation 2.41, it results in the values of equation
2.43. θ2V2λ
1
Pred
=
−0.0288860.98331
0
0
+ 0.5
−0.029406−0.017409
1.0000
0
=
−0.0435890.9746
0.5
(2.43)
The step value used in this case was 0.5. The objective of this chosen value is to inflict a significant
error, so that the corrector step has visible results to the reader. However, if the CPF method were to
use 0.5 as a constant step value throughout the whole process, the algorithm would not achieve its
purpose due to divergence in the corrector step.
With the predicted values of the state variables determined, it is time to use the corrector step
to erase the error inherited by the prediction. This error can be shown by substituting the predicted
values on equations 2.38 and 2.39.
V2V1
[G21 cos(θ2 − θ1) +B21 sin(θ2 − θ1)
]+ V 2
2 G22 + P 0d (1 + λKd2) = 0.00508 6= 0
V2V1
[G21 sin(θ2 − θ1)−B21 cos(θ2 − θ1)
]− V 2
2 B22 +Q0d(1 + λKd2) = 0.00523 6= 0
To erase this error, the corrector step is applied. As explained in section 2.3.6, the corrector step
uses a Newton-Raphson power flow method variant with the predicted values as an initial point to
find in the vicinity of this point, a solution that respects the set of equations of the problem. Using this
method to the set of equations described by 2.38 and 2.39 and using the predicted solution calculated
in 2.43 as an initial point, results in the following solution.θ2V2λ
1
=
−0.04373250.9744137
0.5
(2.44)
Since there are no reactive power generation limits, this concludes the first iteration of CPF
method. It should be noted that the choice of λ as the initial continuation parameter was confirmed by
26
the values of computed tangent vector. The continuation parameter component has a greater abso-
lute value than the other two components thereby confirming that, in this case, in the initial iterations
of CPF method, the load parameter varies more rapidly than the other state variables.
With the first iteration example completed, the results from the application of the CPF method to
the 2-Bus network will be presented in the following paragraphs. As referred earlier, using 0.5 as the
value for the constant step of the process would result in divergence, thus the step value used in the
following simulations was chosen to be 0.001.
Figure 2.10: Plot of V2(λ)
Figure 2.11: Plot of V2(Pd)
27
In Figures 2.10 and 2.11 the results of both simulations (tan(φ) = 0 and tan(φ) = 0.4) are pre-
sented in form of plots representing, in Figure 2.10, the voltage magnitude in Bus 2 V2 as a function
of λ, and in Figure 2.11, V2 as a function of the demanded active power in Bus 2 Pd.
By comparing these PV curves with the ones achieved in section 2.2.1, at first glance, one dif-
ference is clear. When using the CPF method constructed in this work, the algorithm only finds the
solutions in the upper curve. The reason for this difference is, as explained in section 2.3.8, the stop-
ping criterion used in this work for the algorithm, that obligates the algorithm to stop when it reaches
the point of collapse of a specific network.
Apart from this distinction, the point of collapse achieved by the CPF algorithm for this network is
identical to the point calculated in section 2.2.1. These similarities are shown in Tables 2.3 and 2.4.
Table 2.3: Comparison between the Analytical Method and the CPF Method
Analytical Method CPF MethodV2(λ) V2(Pd) V2(λ) V2(Pd)
tan(φ) = 0 (13.592, 0.6646) (14.592, 0.6646) (13.592, 0.6641) (14.592, 0.6641)tan(φ) = 0.4 (9.289, 0.5791) (10.289, 0.5791) (9.289, 0.5789) (10.289, 0.5789)
Table 2.4: CPF Method Error
CPF ErrorV2(λ) V2(Pd)
tan(φ) = 0 (0%, 0.075%) (0%, 0.075%)tan(φ) = 0.4 (0%, 0.035%) (0%, 0.035%)
Table 2.4 illustrates the error of the results achieved by the CPF method. Despite having no error
in computing the load level where the point of collapse occurs, the voltage amplitude at the point of
collapse suffers a very small error due to the iterative nature of the modified Newton-Raphson power
flow method used in the corrector step.
2.4.2 5 Bus Network
After testing the CPF algorithm in a simple 2 Bus example, this section dedicates itself to the
application of the CPF method to a 5 Bus network presented in Figure 4.4, used in [7], which consists
on a 5 buses, 2 generators and 6 lines network where Bus 1 was chosen to be the swing bus. The
network data from this network is presented in Appendix A and since there is a need to perform a
power flow study before initiating the CPF method, the data from this network is presented with the
base case power flow results.
28
1 2
35
4
G1
G2
Figure 2.12: Single Line Diagram of the 5 Bus Network
With a careful reading of the network data, it can be seen that the two generation units have
reactive power limits, including the one present in the swing bus, thus it should be expected that,
during the CPF algorithm, at least one of these limits is expected to be reached. Also, in this study,
since there is no clear picture of how the buses should have their loads increased and likewise how
the increases of active power would be managed by the two generation units, it was decided to use
Kg = 1 and Kd = 1 for all generation units and loads respectively. In the case of the Kd constants,
in this way, there is a complete correlation between the value of λ and the increase in each load power.
Performing the CPF method to this network with a constant step of 0.001 results on the PV curves
of Figure 2.13 and the respective bus data at the point of collapse in Table 2.5.
Table 2.5: Bus Data at the Point of Collapse
BUS DATA at λ = 0.288 :
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu] Qmax [pu] Qmin [pu]1 Swing 1.039 0 0.8372 0.3864 3.03 1.5 1.5 −0.12 PQ 0.9537 −0.1553 1.4812 0.7728 0 0 0 0
3 PV 1.02 −0.0939 0.9016 0.5152 2.318 1.5 1.5 −0.14 PQ 0.8733 −0.2613 0.9016 0.3864 0 0 0 0
5 PQ 0.9411 −0.1454 1.0948 0.5152 0 0 0 0
As expected, swing and PV buses such as Bus 1 and 3, can regulate their reactive power outlet
to maintain a constant voltage amplitude until their maximum limits are reached. In fact, during this
simulation, the maximum value of generated reactive power of Bus 1, which is 1.5 pu, is reached at
λ = 0.284 and as a result, during the rest of the simulation, the voltage amplitude in Bus 1 suffers a
slight decrease. As for the generator in Bus 3, it reaches its maximum limit, also 1.5 pu, at λ = 0.288
or Pmax = 521.64 MW. Since there are only 2 generation units, the generation reactive power limit
is reached and hence this limit yields the point of collapse. Also, since it was decided earlier to use
Kd = 1 for all the buses, the value of lambda represents the increase in load power in each bus, in
relation with the base case value, and hence, it can be said that the point of collapse is reached when
29
every load in the network grows by 28.8%.
Figure 2.13: PV Curves of the 5 Bus Network
As a second scenario, the same network without reactive limits in its swing bus can be tested. This
scenario not only serves as an extra different network to apply the CPF method, but it also serves as
an example to interpret the influence on to the network of one generation unit reaching its reactive
maximum limit.
With this hypothesis established, the point of collapse of the 5 Bus network changes to λ = 0.6462
and the application of the CPF method results in the PV curves depicted in Figure 2.14 and the
respective bus data at the point of collapse displayed in Table 2.6.
Table 2.6: Bus Data at Point of Collapse without Swing Bus Reactive Limits
BUS DATA at λ = 0.6462 :
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu] Qmax [pu] Qmin [pu]1 Swing 1.04 0 1.07 0.4939 4.126 3.847 +∞ −∞2 PQ 0.8074 −0.2365 1.8931 0.9877 0 0 0 0
3 PV 0.8435 −0.1076 1.1523 0.6585 2.963 1.5 1.5 −0.14 PQ 0.5719 −0.4977 1.1523 0.4939 0 0 0 0
5 PQ 0.7802 −0.2180 1.3993 0.6585 0 0 0 0
In the first scenario, the swing bus generation unit hit its maximum reactive power limit at λ = 0.284,
and because the other generation unit in the network (Bus 3) only reached its maximum limit after this
point, the decrease in the voltage magnitudes in this second scenario is the same as in the first until
λ = 0.284. After this point, since there is no limits to the amount of reactive power made available
by the generation unit in Bus 1 and therefore it keeps supplying reactive power after λ = 0.284, the
generation unit in Bus 3 only hits its limit in λ = 0.293 when before it reached its maximum at λ = 0.288.
30
Figure 2.14: PV Curves of the 5 Bus Network without Swing Bus Reactive Limits
After this stage, voltage magnitudes within the network begin to decrease more rapidly. This is a
result of all the reactive power needs being supplied by a single generation unit, leading to more re-
active power flowing through the lines and consequently augmenting the voltage drop associated with
them. By increasing the loading level even further causes the system to reach its power transmission
limit and thus reaching its point of collapse at λ = 0.6462 or Pmax = 666.69. Once again, since it was
established earlier to use Kd = 1, it is concluded that the point of collapse occurs when every load in
the network grows by 64.62%.
2.4.3 IEEE New England 39 Bus Network
This section presents the results from the application of the CPF method to the IEEE 39 Bus net-
work from the New England region of the United States of America. The single line diagram of this
network is illustrated in Figure 2.15 and the node and branch data is presented in Appendix B. The
purpose of this application is to test the CPF method constructed in this work with a real-life network
and compare the results with the ones exhibited in [2]. In order to achieve an effective comparison,
the same scenario has to be tested and with the same values.
Only 9 of the 18 load buses are scheduled to increase, respectively buses 7, 8, 15, 16, 18, 20, 21
and 23, their loads are increased proportionally to their initial load powers, or using the nomenclature
used in this work, their Kd are constant throughout the algorithm and lastly all the constants Kd
concerning each bus have the same value. These values, according to [2] are calculated using
equation 2.45 and are presented in Table 2.7 together with the base values of the loads scheduled to
increase.
Kd =P 0dTotal
P 0d7 + P 0
d8 + P 0d15 + P 0
d16 + P 0d18 + P 0
d20 + P 0d21 + P 0
d23
= 2.221326 (2.45)
31
Figure 2.15: Single Line Diagram of the New England 39 Bus Network ([1])
Table 2.7: Kd values of load buses scheduled to increase
Bus Number P0d [pu] Q0
d [pu] Kd
7 2.338 0.84 2.2213268 5.22 1.76 2.22132615 3.20 1.53 2.22132616 3.294 1.323 2.22132618 1.58 0.30 2.22132620 6.80 1.03 2.22132621 2.74 1.15 2.22132623 2.475 0.846 2.221326
In regard to the generation units, 9 out of the 10 existent are scheduled to accompany the in-
creases of active load power, respectively buses 30, 31, 32, 33, 34, 35, 36, 37 and 38. The active
power output concerning these buses, except the swing bus 31, are also increased proportionally
to their initial active power output and all the constants Kg regarding each generation bus schedule
to increase have the same value. Once again, these values, according to [2], are calculated using
equation 2.46 and are presented in Table 2.8 together with the base values of the active power being
generated in said buses. In this scenario, the swing bus reactive limits are not considered.
Kg =P 0gTotal
P 0g30 + P 0
g31 + P 0g31 + P 0
g32 + P 0g33 + P 0
g34 + P 0g35 + P 0
g36 + P 0g37 + P 0
g38
= 1.19294 (2.46)
Using the data presented, the CPF method was performed using once again a constant step value
of 0.001. The bus data at the point of collapse is presented in Appendix B. Figure 2.16 shows the
increase of each individual bus load that was scheduled to increase. As referred earlier, since each
load was programmed to increase proportionally to its initial load level, buses that have a greater initial
32
load level in relation to the rest also suffer a bigger increase in their load during the algorithm. These
results can be confirmed by observation of a similar plot in [2] presented here in Figure 2.17.
Table 2.8: Kg values of generation buses scheduled to increase
Bus Number P0g [pu] Kg
30 2.30 1.1929431 7.229 –32 6.30 1.1929433 6.12 1.1929434 4.88 1.1929435 6.30 1.1929436 5.40 1.1929437 5.20 1.1929438 8.10 1.19294
Figure 2.16: Pdi vs PdTotal
Figure 2.17: Pdi vs PdTotal in [2]
33
In Figure 2.18 is presented the evolution of the active power being generated by the generation
units scheduled to accompany the evolution in load during the algorithm. Once more, as in the
latter paragraph, since each generation unit was scheduled to increase its output in active power
proportionally to its initial output, generation units that have a greater initial active output suffer a
relatively bigger increase in their active power output during the algorithm, with one exception, the
swing bus. As the name implies, the swing bus has to close the power balance in the network and
therefore the generated active power is computed for this purpose. In [2], the author achieved similar
results as can be seen in Figure 2.19.
Figure 2.18: Pgi vs PdTotal
Figure 2.19: Pgi vs PdTotal in [2]
With the evolution of active power being demanded and generated in the system covered, the
evolution of reactive power being generated in the system will now be analyzed. In Figure 2.20 it is
34
exhibited the variation of the reactive power output by all generation units. By observing Figure 2.20
and consulting the reactive power limits form each bus in Appendix B, it is concluded that several
generation units reach their limits during the algorithm, namely the generation units from buses 30,
32, 33, 34, 35, 36. Also, as expected, as some generation units reach their limits, the effort of ac-
companying the increase of reactive power demand in the system is transfered to other generators.
This consequence is particularly visible in buses 38, 39 and the swing bus 31 which in this scenario
created by [2] had its reactive power limits ignored and thus surpassing by a large amount the maxi-
mum reactive power output limit suggested by the data in Apprendix B, 600 MVAR. In Figure 2.21, it
is represented the plot achieved in [2] that suggests the same behavior depicted in this work.
Figure 2.20: Qgi vs PdTotal
Figure 2.21: Qgi vs PdTotal in [2]
Since it would be unpractical to present the PV curves of every bus in this network, it was chosen
to present the PV curves of the same buses that were chosen in [2] in order to perform a correct
35
comparison. The PV curves of buses 19, 21, 22 and 23 are represented in Figure 2.22 with respect
to λ and in Figure 2.24 with respect to PdTotal. When λ = 0, voltage amplitudes in these buses equal
those of the base case power flow solution, and as the load level increases within the network, these
voltage amplitudes suffer a decrease due to the limitations in the power transportation and also due
to the limitations in the generation of reactive power in some generation units in the network. They
eventually reach the point of collapse at λ = 0.4228 or PdTotal = 8737.6 MW.
Figure 2.22: Vi vs λ
Figure 2.23: Vi vs λ in [2]
The PV curves achieved by [2] are exhibited in Figures 2.23 and 2.25. Although the results of
[2] point to the same location of the point of collapse, the level of voltage amplitudes throughout the
experience in [2] seem divergent from the ones achieved in this work. The base case power flow
solution achieved in this work and in [2] suggest the same values to these voltage amplitudes when
36
λ = 0 (PdTotal = 6147.4 MW), V19 = 1.0432, V21 = 1.0122 pu, V22 = 1.0387 pu and V23 = 1.0322 pu.
These values are in agreement with Figures 2.22 and 2.24. However in Figures 2.23 and 2.25, the
voltage amplitudes at λ = 0 (PdTotal = 6147.4 MW) are all under 1.04 pu. Thus suggesting that for
reasons unknown, the values displayed in these figures suffered a negative offset.
Figure 2.24: Vi vs PdTotal
Figure 2.25: Vi vs PdTotal in [2]
37
38
3Particle Swarm Optimization
Contents3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 PSO Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
39
This work basis itself on two fundamental concepts in two rather different scientific dimensions.
The first fundamental concept is the CPF method, presented in chapter 2, which is inserted in the
power engineering dimension of this work. On the other hand, the concept introduced in this chapter,
PSO, on itself has nothing to do with power engineering, since it is a mathematical optimization
method. This chapter starts off by giving a brief introduction to the PSO concept in section 3.1
followed by the actual detailed description of the algorithm utilized in this work. Closing this chapter
is the presentation of the results of applying this method to several mathematical functions.
3.1 Introduction
The concept of the optimization of non-linear functions using a particle swarm methodology was
first introduced in [17] by social psychologist James Kennedy and electrical engineer Russel Eber-
hart. This particular strange mixture of fields of science reveals part of the nature of the optimization
method presented in this chapter. PSO is a population based stochastic optimization technique and
it is inspired by the social behavior of certain groups of animal species, in fact, as stated in [17] and
[4], PSO is strongly related to evolutionary computation and particularly genetic algorithms.
As several species of animals evolved in the planet through Darwinian principles of evolution,
some developed social behaviors which were optimal to both the species and their surroundings by
maximizing the survival of the species. PSO takes advantage of these optimal social behaviors such
as the ones observed during fish schooling and bird flocking to optimize non-linear functions. Fish
schooling and bird flocking are the means which by some fish and bird species move in their habitat
in a coordinated manner. This observation, as referred in [18], leads to the assumption that per-
sonal and social information is shared within the group. This social behavior can also be observed
in humans. According to [18] and [19], the behavior of each human individual has its roots on the
behavior patterns constant within social groups and on the individual’s experience. PSO simulates
the movement of a group of individuals with the behavior described above to find the optimum solution
to a mathematical problem, as a flock of birds uses its social behavior to find locations of food sources.
The main principle of PSO is the use of particles, these represent individuals within a social group
and each particle represents a potential mathematical solution to the problem. Each particle moves
through the problem’s hyperspace and during its ”flight” adjusts its position according to its own ex-
perience (self awareness) and the experience of all the particles in the group (social awareness). To
do so, each particle keeps track of the coordinates in the problem’s hyperspace which are associated
with the best solution that it has achieved so far and with the best solution achieved by all the particles
until that moment.
Although this technique has its uses ranging from engineering problems ([20]) to simulating social
behavior ([19]), recently it has shown great promise in power system optimization problems ([4], [18],
40
[21] and [3]). This is mainly due to its capability to handle both continuous and discrete variables with
little effort and to integrate optimization constraints, required in some power system’s problems, that
are difficult to handle by mathematical propositions.
3.2 State of the Art
The PSO concept was first developed in [17] during the year of 1995. The objective of this initial
research was to take advantage of natural optimal processes in order to construct a simple and robust
concept and paradigm which could be successful in optimizing a wide range of functions. Although
the objective of this paper was fulfilled, it was admitted by the author that much further research was
needed to develop this technique.
Later, in 1998, the inclusion of a inertia weight into the particle’s laws of movement was proposed
by [22] to balance the local and global search characteristics of PSO in order to improve its perfor-
mance. In [23], this idea was further enhanced to include a linear decreasing inertia weight throughout
the algorithm with the objective of favoring global search in the beginning of the algorithm and local
search in the end of the algorithm.
Another modification of the original laws of movement attributed to the swarm of particles was
the introduction, in [24], of the constriction factor that according to [25], improves the PSO’s ability
to constraint and control the velocities that particles take in the hyperspace. In [26], the author con-
cludes that this approach combined with a upper limit for the velocities significantly improved the PSO
performance when compared with the inertia weight inclusion.
3.3 PSO Formulation
3.3.1 Original Version with Inertia Weight
The main purpose of a standard continuous optimization technique is to find the best of all feasible
solutions to a optimization problem, usually, minimizing or maximizing a continuous function with
respect to several constraints. In the case of minimization, mathematically, such a problem can be
stated as:
minimizex
f(x) , f(x) : Rn → R
subject to gi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p
Where f(x) is called objective or fitness function and gi(x) and hi(x) respectively define the in-
equality and equality constraints. To solve these problems, PSO proposes a new approach by mim-
icking the movement behavior of some social groups encountered in nature with PSO particles acting
as individuals in such a group.
41
As described before, these particles represent feasible solutions to the problem and move through
the problems hyperspace according to previous social and personal behavior. However, before de-
scribing how these laws of movement of the particles are formulated, the problems of how these
particles should be initially generated into the hyperspace and how many should be generated have
to be addressed. The former can be addressed by randomly generating particles within defined limits
within the problem’s hyperspace where the particles are allowed to move so that computational effort
is not wasted locating mathematical solutions that are not feasible to the problem. The latter relates
to the problem’s complexity (number of dimensions, number of local maxima and minima, constraints,
etc) as stated in [27]. According to [27], the number typically ranges from 20 to 40 particles, depend-
ing on the problem’s complexity and on the balance between number of calculations in each iteration
of the algorithm and the number of iterations needed for the algorithm to converge.
As explained in section 3.1, particles move in the hyperspace according to two ”best” solutions.
The first one is the personal best solution achieved by each particle in the hyperspace until a partic-
ular moment. The set of coordinates values associated to this solution is named in this work pbesti
for particle i. The second one is the best solution achieved by all particles moving in the hyperspace.
Consequently the value is shared by all particles. The set of coordinates values associated to this
global best solution is called gbest. Defining these criteria enables the laws of movement of the set of
particles in the hyperspace to be constructed.
Each particle i has an associated position vector xi(t) and velocity vector vi(t) in the hyperspace.
As referred earlier, the position update of each particle should be done by taking into account coor-
dinates pbesti and gbest. This is done by calculating the signed distance between these coordinates
and the current position of each particle at iteration t and integrating these computations into the
calculation of the updated velocity vector. The velocity of particle i at iteration t is determined, as
suggested in equation 3.1, by taking into account:
• the previous velocity of the particle i, vi(t − 1) weighed by a inertia factor w(t) suggested by
[22]. This component models the tendency of the particle to continue in the same direction it
has been traveling ([4]);
• the signed distance between the coordinates associated with the personal best value, i.e.,
pbesti, and the current position of the particle, weighed by R1, a random value within the range
[0, 1] generated by a uniform distribution at each iteration, and by c1, a positive valued constant
named the cognitive acceleration constant, with a suggested value of 2.0 by [17]. This second
component represents the linear attraction towards the best position ever found by particle i
pbesti and therefore is named ”memory” or ”self-knowledge” ([4]);
• the signed distance between the coordinates associated with the global best value, i.e., gbest,
and the current position of the particle, weighed by R2, a random value within the range [0, 1]
generated by a uniform distribution at each iteration, and by c2, a positive valued constant named
42
the social acceleration constant, with a suggested value of 2.0 by [17]. This last component is
described by [4] as the linear attraction towards the best position ever found by any particle in
the group, gbest, thus receiving the name ”cooperation” or ”social knowledge”.
vi(t) = w(t)vi(t− 1) + c1R1
[pbesti − xi(t− 1)
]+ c2R2
[gbest− xi(t− 1)
](3.1)
The position of each particle, at iteration t is then determined by the sum of the previous position
vector xi(t − 1) and the updated velocity vector vi(t) computed by 3.1. Interpreting the relationship
between a position and a velocity demonstrated in equation 3.2 leads to conclusion that each iteration
of the algorithm represents one unit of time in physical terms.
xi(t) = xi(t− 1) + vi(t) (3.2)
Figure 3.1 illustrates the movement of particle i in a two dimensional plane during one iteration of
the PSO algorithm according to the laws of movement defined above.
pbesti
gbest
vi(t-1)
xi(t-1)
xi(t)
vi(t)
Figure 3.1: Movement of a PSO particle in a two dimensional plane
In this example, since the particle moves in a two dimensional space, the vectors xi, vi, pbesti and
gbest would be defined as:
xi, vi, pbesti, gbest ∈ R2 f(x) : R2 → R (3.3)
Although the laws of movement defined by equations 3.1 and 3.2 describe the movement of the
particles throughout almost the whole process, in the first iteration of the PSO algorithm not only the
position of all particles must be randomly generated, as was described earlier, but also the velocities
on which they enter the problem’s hyperspace must be randomly generated.
During the algorithm constructed in this work, the particles velocities are limited by a maximum
value, |vmax|, suggested in [3] and [4]. This maximum limit assures the local exploration of the prob-
lem space and the realistic simulation of changes in animal behavior. Moreover, if |vmax| is too high
particles will achieve velocities which may cause them to move erratically in the hyperspace ([4]) and
may ”fly” past viable solutions ([3]). If |vmax| is too low the particle’s movement becomes limited and
may cause particles to not explore sufficiently beyond local solutions ([3]).
43
Although velocities have been limited, it is expected that, during the algorithm, some particles
might transpose the feasible domain limits. In [4], it is suggested that, particles that exceed this do-
main are re-randomized into the feasible domain. This re-randomizing includes the computation of
a random position which will be filled by such particle and of a random velocity within the defined
limits. However, this option causes optimal solutions near the boundaries to be harder to reach since
particles exploring a region near a boundary keep being re-randomized into the feasible domain.
In order to keep all the particles within the problem’s boundaries, the PSO algorithm constructed
in this work obligates that if a particle i hits a boundary with velocity vi(t), it bounces off with velocity
−vi(t). This way, not only the particle stays within the boundaries but also continues to explore the
same region in the hyperspace that it was exploring before hitting the boundary.
The inertia factor in equation 3.1 is not part of the original formulation of the PSO technique. It was
first introduced in [22] as a way of balancing the influence of the three components represented in
equation 3.1. Without this factor, the velocity at iteration t of particle i would be described by equation
3.4.
vi(t) = vi(t− 1) + c1R1
[pbest− xi(t− 1)
]+ c2R2
[gbest− xi(t− 1)
](3.4)
The right side of this equation can still be divided in three parts, as was earlier. The first one is
the previous velocity of the particle and the second and third parts, which are now the only ones to
contribute to the change of velocity, are described, as earlier, as the self and social knowledge compo-
nents. Without these last two components, the PSO technique would be pointless since each particle
would keep moving at the current speed in the same direction until they eventually hit a boundary.
However the influence of the first component is less apparent. As investigated in [22], without the first
component and assuming at the initial iterations of the algorithm particle i has the best global posi-
tion, then it would have velocity 0 and it would remain still until another particle encounters a better
position. At the same time, other particles are moving based on gbest and pbest. With c1 = 2 and
c2 = 2 and without the first component, [22] concludes that the algorithm would unroll in the following
manner. Since, statistically, pbest exerts the same influence as gbest, the swarm of particles would
contract to the current best position until another particle finds a better position during this contraction,
from which time all particles begin statistically contracting to the new global best position. With this
proposition, only when the global optimum is within the initial search domain is there a chance for
PSO to find it.
With the insertion of the first component comes the ability to expand the search space, i.e., the
swarm of particles gains a global search ability. The inertia weight w(t) brings a trade-off between
the local search ability and the global search ability intrinsic to equation 3.4. According to [22], when
using c1 = 2 and c2 = 2, the range [0.8, 1.2] is a good area to choose w(t) from. However, further
research in [23] reveals that a PSO algorithm with a linearly decreasing inertia weight throughout the
algorithm is a better solution than the use of a constant value. One option to describe this decrease
44
in the inertia value is offered in [3] and it is described in equation 3.5 where wmax and wmin are the
boundaries of the range in which the inertia weight operates and tmax is the maximum iteration value.
w(t) = wmax −wmax − wmin
tmax× t (3.5)
In the beginning, this allows the algorithm to scatter the particles in the hyperspace with the aim
of favoring global search and as the algorithm begins to approach the end, favoring local search by
using a small inertia weight value.
3.3.2 Constriction Factor Approach
Another approach to the PSO paradigm is the use of a constriction factor, which was first proposed
in [24]. This approach derived from analyzing the system’s inner dynamics, in particular the movement
of one particle when it is subjected to the original PSO formulation, through eigen values evaluation.
These analyses led to the introduction of the constriction factor in the particle’s velocity equation in
order, according to [24], to control the system’s convergence tendencies. The new expression for the
particle’s velocity is shown in equation 3.6.
vi(t) = K(vi(t− 1) + c1R1
[pbest− xi(t− 1)
]+ c2R2
[gbest− xi(t− 1)
])(3.6)
In this equation, the constriction factor, represented by K is defined as:
K =2∣∣∣2− ϕ−√ϕ2 − 4ϕ
∣∣∣ ϕ = c1 + c2, ϕ > 4 (3.7)
By analyzing this new formulation, it is concluded that when c1 = c2 = 2.0, which is suggested in
the original formulation, the constriction factor has value 1. Thus the original formulation of PSO can
be considered as a particular case of the constriction factor approach of PSO.
The convergence characteristic of the system is controlled in this new approach by the parame-
ter ϕ. In [24] it was concluded that ϕ must be greater than 4.0 to guarantee stability. However, as
referred in [28], as ϕ increases, the constriction factor and the velocity both suffer a decrease and
consequently, the diversification is reduced. Typically, when using this approach, ϕ is set to 4.1 with
c1 = c2 = 2.05, giving the self and social knowledge components the same influence ([26],[28]).
When comparing the performance of PSO using an inertia weight while using a constriction factor,
in [26], it is concluded that this new approach achieves better results than the insertion of the inertia
weight.
3.3.3 Stopping Criteria
In sections 3.3.1 and 3.3.2 the operation of the PSO algorithm with an inertia weight and with a
constriction factor was described. In either variant of PSO, the main objective is to find the global
optimum through the convergence of the swarm of particles to a particular point in the problem’s
45
hyperspace. However, to efficiently construct this algorithm, stopping criteria have to be properly de-
fined.
When defining the purpose of the inertia weight, equation 3.5 defined a maximum limit for the
number of iterations, tmax. If no other stopping criteria are defined, the algorithm will only stop when
the maximum number of iterations is reached whether if the global optimum has been reached or not.
This is the only stopping criterion used in [21] and [3].
However, if the value of tmax is not properly dimensioned, it can lead to undesirable consequences.
If the defined value is too small, it may result in error when obtaining the global optimum. On the other
hand, when the value is too high, it leads to time being wasted in processing iterations when there is
no need for it. This is especially true if calculating the fitness value of each particle takes a reasonable
amount of time. In summary, the algorithm will always end without knowing if its job has been fulfilled,
which seems a rather inelegant solution.
With this in mind, an extra criterion, suggested in [29], has been integrated in the PSO algorithm
constructed in this work. The objective of this new criterion is to monitor the improvement of the
fitness value of gbest. At the beginning of the algorithm, this value is expected to suffer extreme
variations, however, after the the global optimum is nearly reached, this value is expected either to
remain constant or to suffer minor variations. So if the improvement of the fitness value of gbest falls
below a given threshold value T for a number of iterations I, the optimization run is terminated.
3.3.4 Discrete Variable Integration
One of the advantages of the PSO technique is the ability of easily integrating discrete variables in
the optimization problem, as was investigated by [30], although in its original version PSO was only to
be utilized with continuous variables. Moreover, according to [30], the performance of PSO in discrete
problems is shown to be better than other evolutionary computational methods. By observing the
basic formulation of the PSO algorithm, it is concluded that the method basis itself on how the velocity
of each particle is calculated at every iteration. To integrate discrete variables the method does not
need a reformulation of this principle.
In the case of non-linear optimization problems with one or more discrete variables, which is the
case of mixed integer non-linear optimization problems ([18]), the PSO technique can be slightly mod-
ified by rounding off the particle’s position to the nearest discrete value. This modification maintains
the validity of equations 3.1, 3.2 and 3.6 but once the updated position of a particle has been found,
the algorithm proceeds to rounding the updated position to the nearest discrete value.
This concludes the theoretical presentation of the PSO technique. In Figure 3.2 is presented the
flow chart of the whole PSO process.
46
Start
Initialize Particles:
Positions, Velocities
Update global best
solution gbest
All particles checked?
Compute fitness
value
YesNo
Update pbest
Update particle’s
velocity
Check for velocity
limit
Update particle’s
position
Compute inertia
weight
Check feasibility
limits
EndCheck Stopping
Criteria
No
Yes
Figure 3.2: Flow Chart of PSO
3.4 Results
The following sections are intended to display the results from the application of PSO to three
different mathematical functions with the objective of interpreting the influence of some of the PSO
parameters and test the performance of the PSO technique in different optimization problems.
47
3.4.1 Rosenbrock Function
The first non-linear function used is the Rosenbrock three dimensional function, also known as
the banana function due to its parabolic shaped valley. This function has been extensively utilized as
an objective function in minimization problems due to the difficulty in finding its global minimum in the
parabolic valley. The function is expressed by equation 3.8 and it is illustrated in Figure 3.3.
f(x1, x2) = (1− x1)2 + 100(x2 − x21)2 x1, x2 ∈ R (3.8)
Figure 3.3: Plot of the Rosenbrock function
The contour plot of f(x1, x2), illustrated in Figure 3.4, demonstrates the difficulty in computing the
global minimum of the Rosenbrock function due to the low gradient inside the parabolic valley.
As explained in section 3.3.1, the objective of an optimization problem can be either obtaining the
global maximum or the global minimum of a fitness function with respect to several constraints. In
this example it was preferred to define the problem as an minimization with no constraints, since the
global minimum is well known to be in x1 = 1 and x2 = 1 with fitness value f(1, 1) = 0. Three series
of 50 simulations were performed with respectively 20, 25 and 30 particles with both versions of the
PSO technique presented in this chapter, with an inertia weight and with a constriction factor. The
parameters used in these simulations are in Table 3.1. For the sake of readability, the definition of
these parameters is once again presented:
tmax: Maximum number of iterations;
wmax: Upper bondary for the range where the inertia weight operates;
wmin: Lower bondary for the range where the inertia weight operates;
T: Threshold value for the improvement-based stopping criterion;
48
I: Maximum number of iterations without improvement greater than T ;
|x|max: Domain limit within which the particles can move;
|v|max: Maximum limit for a particle’s velocity.
In regards to the values for the cognitive and social acceleration constants, for the inertia weight
version of the algorithm, the values used were c1 = 2.0 and c2 = 2.0, however, for the constriction
factor version it was used c1 = 2.05 and c2 = 2.05.
Figure 3.4: Contour Plot of the Rosenbrock function
Table 3.1: PSO Parameters for the Rosenbrock Function Optimization
tmax wmax wmin T I |x|max |v|max
10000 0.9 0.6 0 100 30 30
Note that for T = 0 and I = 100, the improvement based criterion is fulfilled when the value of
gbest doesn’t change for 100 iterations. All the simulation runs ended with the correct result for the
global optimum. Table 3.2 presents the mean, maximum and minimum of the number of iterations for
each set of simulations.
The values for the standard deviation in each set of simulations presented in Table 3.2 bring to
light the randomness of the process inherited from the initial scattering of particles and from equation
3.4. As a result, the consequence of any modification of the PSO parameters or even of its algorithm
can only be asserted statistically. By observing the mean values, two conclusions can be inferred.
49
The first, as expected, with an increase in the number of particles moving in the two dimensional
plane comes, statistically, a decrease in the number of iterations. The second, as was concluded in
[26], the constriction factor approach decreases substantially the time consumed by each simulation.
Table 3.2: Number of Iterations for the Optimization of the Rosenbrock Function
With Constriction With Inertia Weight20 25 30 20 25 30
Success Rate 100% 100% 100% 100% 100% 100%Mean Value 1220.8 1144.74 1099.6 1805.54 1689.22 1600.5Best Value 1050 944 964 1341 1689.22 1223Worst Value 1467 1293 1280 2320 2117 1910Standard Deviation 95.79 74.74 71.9 192.52 183.9 154.73
3.4.2 Schaffer F6 function
Another non-linear function that is extensively used as an optimization problem for the application
of the PSO algorithm is the Schaffer F6 function described in equation 3.9 and in Figure 3.5. This
function provides an immense variety of local maxima and minima where the swarm of particles can
mistakenly converge. Hence, this function consists in a great test for the improvement based criterion
used to stop the algorithm. Due to the location of the global minimum (x1 = 0, x2 = 0 with f(0, 0) = 0),
if the dimensioning of the constants T and I is not correct, the algorithm may end with the wrong result.
f(x1, x2) = 0.5 +
[sin(
√x21 + x22)
]2− 0.5[
1.0− 0.001(x21 + x22)]2 x1, x2 ∈ R (3.9)
The extreme variety in this function of local optima is due to its sinusoidal pattern illustrated in
Figure 3.6 and its cylindrical symmetry shown in Figure 3.5. In Figure 3.6 it can also be observed the
location of the global minimum at x1 = 0 and x2 = 0.
In this section, the objective is to interpret the influence of the improvement based criterion on the
accuracy of the algorithm. With this purpose, and taking into account the conclusions achieved in the
last section, all of the simulations performed in this section used the constriction factor approach with
c1 = 2.05 and c2 = 2.05. Four series of 100 simulations, with respectively 20, 30, 40 and 50 particles,
with the purpose of optimizing this function were run without this stopping criterion, which is the same
as stating that the simulations were run with T = 0 and I = tmax. The parameters used in this first
set are presented in Table 3.3.
Table 3.3: PSO Parameters for the Schaffer F6 Function Optimization without the improvement-based criterion
tmax T I |x|max |v|max
10000 0 10000 70 70
50
Figure 3.5: Plot of the Schaffer F6 function
Figure 3.6: Plot of the Schaffer F6 function with x2 = 0
Due to the number of local minima present in the problem’s domain, not all the the simulations
achieved the correct result. The success rate of each series of simulations is presented in Table 3.4.
Table 3.4: Accuracy of each Series of Simulations for the Schaffer F6 Function without the improvement-basedcriterion
No. of Particles20 30 40 50
Success Rate 69% 79% 84% 95%
Evaluating the values in Table 3.4, it is clear that the number of particles plays an important role
not only in decreasing the number of iterations, as was concluded in section 3.4.1, but also in the
success rate of the simulations. In order to evaluate the influence of the improvement based stopping
criterion in the success rate of algorithm, another set of four series of 100 simulations were performed,
51
this time with T = 0 and I = 1500. The success rate of each of the series is illustrated in Table 3.5.
Table 3.5: Accuracy of each Series of Simulations for the Schaffer f6 Function with the improvement-basedcriterion
No. of Particles20 30 40 50
Success Rate 60% 73% 81% 86%
These results show that in some situations, the integration of the improvement based criterion
diminishes the success rate of the algorithm. In this optimization problem, due to the variety of local
minima, the global best position gbest can stay constant for a large number of iterations. Because the
new stopping criterion states that if the improvement of the fitness value of gbest falls below T for I
iterations the algorithm must stop, the inclusion of the this criterion causes the algorithm to end before
the global minimum is encountered.
3.4.3 Discrete De Jong F1 function
The last function used in this chapter is the De Jong F1 function described in equation 3.10,
also known as the Sphere function. In this section, there are two optimization problems based on
this function. The first is used to test the performance of PSO with constriction on discrete valued
problems and the second tests it on n-dimensional problems with n > 2.
f(x) =
n∑i=1
x2i x = [x1, . . . , xi, . . . , xn] xi ∈ Z (3.10)
Figure 3.7: Plot of the De Jong f1 function with n = 2
The first optimization problem considered in this section is the minimization of a discrete De Jong
F1 function with n = 2, also known as a paraboloid. This particular function is described by equation
52
3.11 and is plotted in Figure 3.7.
f(x) = x21 + x22 x1, x2 ∈ Z (3.11)
Once again, as was with the Schaffer’s f6 function, the global minimum is well known to be in
x1 = 0 and x2 = 0 with fitness value f(0, 0) = 0. Three series of 50 simulations were performed with
different swarm sizes, respectively, 5, 15, and 30 particles and with the constriction factor approach
with c1 = 2.05 and c2 = 2.05. The parameters used in this first optimization problem are displayed
in Table 3.6. The smaller values of the proposed swarm sizes and of tmax and I are due to the less
complex nature of the problem when comparing it with the optimization problems proposed in sections
3.4.1 and 3.4.2.
Table 3.6: PSO Parameters for the De Jong F1 Function Optimization with n = 2
tmax T I |x|max |v|max
1000 0 50 100 100
All 150 simulations obtained the correct value for the global optimum. Table 3.7 presents the
values for the number of iterations needed for each of the series of simulations.
Table 3.7: Number of Iterations for the Optimization of the Discrete De Jong F1 Function with n = 2
No. of Particles5 15 30
Success Ratio 100% 100% 100%Mean Value 74.1 73.34 68.36Best Value 53 53 54Worst Value 98 88 79Standard Deviation 14.67 8.26 6.49
As expected, due to the simple nature of the problem, the number of iterations needed to compute
the global minimum of this function is much smaller when compared with the number of iterations
needed in section 3.4.1. However, another reason for this extreme decrease in the number of itera-
tions is that the function is only defined for a discrete set of values, in this case, the set of integers. For
the sake of comparison, Table 3.8 presents the values of the number of iterations needed to perform
this optimization when x1 ∈ R and x2 ∈ R with the same PSO parameters.
Table 3.8: Number of Iterations for the Optimization of the Continuous De Jong F1 Function with n = 2
No. of Particles5 15 30
Success Rate 100% 100% 100%Mean Value 118.3 113.7 110.5Best Value 53 54 54Worst Value 163 136 127Standard Deviation 32.48 21.24 13.52
This decrease from a continuous variable problem to a discrete variable problem is due to the
variety of positions that a particle can occupy. With a continuous variable problem the positions that a
53
particle can occupy are infinite, however, using discrete variables, a particle can only occupy the po-
sitions where the fitness function is defined. In this example, since x1 ∈ Z, x2 ∈ Z and |xmax| = 100,
the number of feasible positions is limited to 40000.
With the purpose of testing the performance of the PSO method constructed in this work with multi-
dimensional functions, the second optimization problem considered in this section is the minimization
of a continuous De Jong F1 function with n = 10, described in the equation 3.12.
f(x) =
10∑i=1
x2i x = [x1, . . . , xi, . . . , x30] xi ∈ R (3.12)
By increasing the number of dimensions of the problem, its complexity escalates, comparing it
with the same problem with n = 2, due to the increase in the area where each particle can move.
Acknowledging this fact leads to the modification of some of the PSO parameters to accommodate
this increase. Hence, the size of the swarm size in this problem was increased, in all three series of
50 simulations, to 20, 30 and 40 particles. All the other parameters are presented in Table 3.9.
Table 3.9: PSO Parameters for the De Jong f1 Function Optimization with n = 30
tmax T I |x|max |v|max
10000 0 100 100 100
Although the number of dimensions was increased from 2 to 10, the global minimum remains in
the same place, i.e., the origin. All 150 simulations were executed and all obtained the correct value
for the global minimum. The values of the number of iterations in the three series of simulations are
presented in Table 3.10.
Table 3.10: Number of Iterations for the Optimization of the Discrete De Jong F1 Function with n = 10
No. of Particles20 30 40
Success Rate 100% 100% 100%Mean Value 204.18 191.08 184.14Best Value 169 167 166Worst Value 264 235 237Standard Deviation 16.93 13.20 12.32
As explained earlier, the increase in the number of dimensions of the problem results in a increase
of complexity and consequently, despite some modifications applied to the PSO parameters, the
number of iterations needed for the optimization suffers an increase when comparing with the same
problem with n = 2.
3.4.4 Discussion
In section 3.4.1 it was concluded, through a variety of tests, that the constriction factor approach
achieved the same results as the inertia weight approach while using less iterations of the algorithm.
54
However, in section 3.4.2, it was discovered that, in some cases, due to the variety of local optima,
the PSO technique does not achieve the correct results 100% of the time (also observed in [26] and
[25]), which begs the question of how does an user of PSO knows that the correct result has been
computed in a real-life optimization problem.
In order to get around this lack of accuracy of the PSO algorithm, it is suggested in this thesis, as
in [3], [4] and [20], to execute several runs of the PSO algorithm for the same optimization problem.
If several different solutions emerge, the user should then choose the best solution according to the
function value of each solution and/or to other relevant criteria to the problem.
55
56
4Optimal Placement of STATCOM
Contents4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Reactive Power Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 PSO Algorithm Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
57
4.1 Introduction
In chapter 2, a tool to determine the voltage stability margin of a specific power system was pre-
sented in the form of the CPF technique. This presentation was followed in chapter 3 by the PSO
algorithm, which in recent times as shown great promise in power system’s optimization problems.
The following chapter proposes a novelty approach for the optimal placement and sizing of mul-
tiple STATCOMs in power systems with the main objective of enhancing the voltage stability margin
of such systems. This approach merges the methods described in the last two chapters since it shall
be based on a PSO formulation using the CPF technique to compute the voltage stability margin of
different network configurations.
This chapter begins by giving a description of several devices used to perform reactive power
compensation, where the STATCOM is inserted, followed by the presentation of the PSO formulation
used in this thesis to perform the optimization. This chapter ends with the examination of the results
of the application of the technique described in this chapter to various networks.
4.2 Reactive Power Compensation
4.2.1 Capacitor Bank
One way to perform reactive power compensation in a power system is through capacitor banks.
These can be used to improve the power factor of a particular bus by generating the necessary
reactive power and therefore reducing the flow of reactive power in the lines to that bus and thus
reducing the voltage drop in those lines. The reactive power generated by these elements is given by
equation 4.1, where ω is the angular frequency of the network where the capacitor bank is inserted,
C is the capacitance of the group of capacitors and Vc is the voltage amplitude of the bus.
Qc = ωCV 2c (4.1)
Observing this equation, it is concluded that the reactive power generated by the capacitor bank
depends heavily of the voltage amplitude in the bus where the device is inserted. This constitutes
a problem since, during the day, when the load peak is expected to occur, the voltage amplitudes in
buses begins to decline, bringing the power system closer to the point of collapse. In these situations,
the reactive power outlet of capacitor banks, instead of ideally suffering an increase to compensate
the voltage drops in the lines, it begins to decrease and further diminishes the voltage amplitudes.
4.2.2 Synchronous Compensator
The synchronous machine has many uses in power systems engineering. Apart from its general
use in massive electrical energy production that stands as the base of any big power system, it can
also be used as an electrical motor, which is the case in pumped-storage hydroelectric power plants
58
and in some electric vehicles and, lastly, as reactive power compensation device. This third mode of
operation of this device is achieved when the power angle δ of the synchronous machine is zero.
The active and reactive power generated by a synchronous machine are given by equations 4.2
and 4.3.
Pg =EVSGXs
sin δ (4.2)
Qg =VSGXs
(E cos δ − VSG) (4.3)
In these equations, E represents the machine’s emf, VSG is the voltage at the generator’s ter-
minals, Xs is the synchronous reactance and δ is the angle between the phasors VSG (used as
reference) and E. From these equations it is observed that when δ = 0, the active power generated
by the machine, Pg, equals 0. However, the generated reactive power, Qg, can be either positive or
negative, depending on the value of the emf E and consequently of the DC excitation current. This
mode of operation is named as synchronous compensator. In this mode, the synchronous machine
acts as a motor whose shaft spins freely and enables the continuous variation, through the excitation
current, of the reactive power generated (synchronous capacitor) or absorbed (synchronous reac-
tance). Which means that, unlike the capacitor bank, the reactive power outlet of the machine can
be adjusted, within the limits described in section 2.1.2, in every situation. However this method of
reactive power compensation requires the use of synchronous generator unit that could be used to
generate active and reactive power.
4.2.3 Static Synchronous Compensator
The static synchronous compensator, also known as STATCOM, is inserted on a new power trans-
mission concept known as Flexible AC Transmission Systems (FACTS). This new concept is defined
in [31] as a ”power electronic based system and other static equipment that provide control of one
or more AC transmission system parameters to enhance controllability and increase power transfer
capability”. It is important to add to this definition that, as stated in [32], the concept of flexibility is to
be intended as the capability to vary the system’s parameters in a quick and continuous way.
The objective of the use of FACTS devices can be further explained, as is done in [32], by using
the network in Figure 4.1. Using this network, the active power that flows through the line can be
described by equation 4.4.
P12 =V1V2XL
sin(θ1 − θ2) (4.4)
Interpreting this expression it is inferred that the flow of active power in the line is controlled by the
voltage amplitudes in the buses (V1, V2), the angles of their phasors (θ1, θ2) and the line impedance
(XL). The FACTS devices, like the ones illustrated in Figure 4.1, are designed to act upon one or
more of these parameters in order to influence the flow of active and reactive power. In Figure 4.1
there are displayed three kinds of FACTS devices:
59
STATCOM: Connected in shunt with the power system, it is designed to control the voltage level in
the bus where it is connected through the generation or consumption of reactive power;
SSSC: The Static Synchronous Series Compensator (SSSC) is, as the name implies, connected in
series with the line in order to control the line reactance and therefore influence the flow of both
active and reactive power;
UPFC: The Unified Power Flow Compensator (UPFC) is the combination of STATCOM and a SSSC
in the same device and therefore is connected both in series and in shunt with the network
and has the capability of controlling the voltage in the bus where it is shunt connected and the
reactance of the line where it is connected in series.
V1 V2
XL
STATCOM SSSC
UPFC
Figure 4.1: Single Line Network with FACTS devices
The FACTS device where this thesis has its focus is the STATCOM, illustrated with more detail in
Figure 4.2. This device, as described earlier, has the objective of either supplying reactive power to
the bus where it is connected or absorbing reactive power from that same bus in order to control the
bus’s voltage amplitude.
V1
AC ↔ DC
Controller
Figure 4.2: Detailed Scheme of the STATCOM
60
To do this, the STATCOM is based on a Voltage Source Converter (VSC) which is constituted by
an inverter with controlled power electronic semiconductors (ex.: IGBT for voltage controlled or GTO
for current controlled) and a DC voltage source in the form of capacitors. To cope with the different
voltage levels in both the network and the VSC, the STATCOM also has a transformer. By controlling
the semiconductors on the inverter, the STATCOM can use the VSC to generate currents lagged +90◦
from the network in the case of the need for reactive power in the bus or lagged −90◦ if the bus needs
reactive power to be absorbed. The operation of this device can then be described in the following
manner:
• If the voltage amplitude in the bus is smaller than a specified voltage Vref , the STATCOM sup-
plies the bus with reactive power;
• On the other hand, if the voltage amplitude in the bus is greater than the specified voltage Vref ,
the STATCOM absorbs reactive power.
Hence the reason for its name, since it has the same behavior as a synchronous compensator but
unlike this machine, it has no moving parts. The application of the STATCOM ranges from the use as
a voltage stability margin enhancer ([3]) to the use in helping restore acceptable voltage levels more
efficiently after the occurrence of faults in the network ([32]). The sizes of this device can range from
1 MVA ([32]) to 200 MVA like the one installed recently in Austin, Texas, whose picture is in the cover
of this thesis.
4.3 State of the Art
The PSO algorithm has been used to solve engineering problems almost since its conception
([20]). It has recently shown great promise in power engineering. These implementations range from
the optimal placement of STATCOMs, in [3] and [4], and other devices such as Distribution Network
Flexible AC Transmission Systems (DFACTS) in [21] to optimal economic dispatch of generation units,
in [28].
In regards with the optimal placement of STATCOM, in [4], in 2006, the concept of this optimization
problem was first introduced using the PSO algorithm. However, the optimal placement and sizing of
the STATCOM units took only in consideration the power flow solution of the network.
In 2007, [3] introduced another criteria from which to judge the optimal placement and sizing of
the STATCOM units in the form of the voltage stability margin. In this article, the optimal placement
and sizing was to be found by achieving a balance between maximizing the voltage stability margin
and minimizing the size of the STATCOM units involved. In this thesis, however, another approach
was designed and compared with the one presented in [3].
61
4.4 PSO Algorithm Implementation
As was stated in section 4.1, the placement and sizing of the STATCOMs is to be made with
a correct implementation of the PSO algorithm. However, to achieve this correct implementation,
several aspects of the PSO have to be designed to this particular problem. First a proper objective
function that evaluates the performance of each particle has to be defined. Then, taking in account
the definition of the objective function, the particle’s position vector has to be defined, i.e., how many
dimensions does the problem have and how should this dimensions be defined, followed by the char-
acterization of unfeasible solutions.
In order to present this implementation in an enlightened way, it is important to state that the PSO
algorithm shall be used, for this implementation, in its constriction factor variant. Closing this section
is the presentation of several speed up measures used to minimize the computational effort.
4.4.1 Objective Function
The main aspect in designing an optimization problem is the definition of its objective function.
Before describing the objective function used, it is important to state that this optimization problem
is defined in this thesis, as a minimization. In this particular optimization problem there exist several
aspects of the network that should be evaluated. In order to do this, the objective function F (x)
designed in this thesis, in accordance with [3] and [4], is defined by the sum of functions Fi(x) that
define each parameter that influences the value of the objective function and the respective weight ωi
that defines the influence of one parameter against another.
F (x) = ω1F1(x) + . . .+ ωmFm(x) (4.5)
In equation 4.5, m is the number of functions. Although it has been stated before in this thesis that
the main objective is to enhance the voltage stability margin, the base case power flow solution of the
network must also have its weight when computing the optimal placement of STATCOMs, since it is
in the neighborhood of this state where the network is going to operate most of the time. In section
4.2.3, it was explained that the STATCOM controls the voltage level of the bus where it is connected,
and so, the first function to integrate is the sum of the voltage deviation in all the buses, in relation
to the nominal voltage (1.0 pu). This function is presented in equation 4.6, where NB stands for the
number of buses in the network and Vi is the amplitude of voltage of bus i.
F1 =
√√√√NB∑i=1
(Vi − 1)2 (4.6)
In this way the objective function evaluates the improvement in the voltage levels throughout the
network, caused by the insertion of the STATCOM(s).
Another aspect of the base case power flow solution is that the insertion of STATCOMs is expected
to improve the active power losses in the lines, again, through the improvement of the voltage ampli-
62
tudes in certain buses. Hence, the second function that composes the objective function constructed
in this section is, as described in equation 4.7, the sum of all the active power losses in all the lines of
the network. In equation 4.7, NL stands for the number of lines in the network and PLi is the active
power losses in the line i.
F2 =
NL∑i=1
PLi (4.7)
At this point, there is a need for the insertion of a function that enables the enhancement of the
voltage stability margin. This margin is defined as the load interval between the base case power flow
solution and the point of collapse. Since, in the base case power flow solution, the load parameter of
the CPF method equals 0, the voltage stability margin can be represented, using the load parameter
by λmax, which stands for the point where the load level is maximum, i.e., the point of collapse. In [3],
this parameter is integrated into the objective function in the manner described in equation 4.8. In this
way, the algorithm will compute the location(s) and size(s) that maximize λmax in order to minimize
F3.
F3 =1
λmax(4.8)
However, using the objective function as the weighted sum of these three functions, the optimal
solution will always be one where the sizes of the STATCOM(s) are maximum, since it is only with the
maximum size that λmax is also maximum. With the objective of avoiding this, in [3], a fourth function
is proposed in order to minimize the size(s) of the STATCOM(s).
F4 = α
NS∑i=1
QiST (4.9)
In equation 4.9, the parameter α is an adimensional constant in order that all functions have val-
ues which are comparable in magnitude and QiST is the reactive power range of STATCOM unit i.
This configuration of the objective function leads to a problem, since the F3 and F4 are contradicting
each other, i.e., in order to minimize one, the other is maximized. This is dealt with in [3] by choosing
the right values of α, ω3 and ω4. However, if these parameters are ill-dimensioned, the algorithm will
either converge on a solution that has its STATCOM sizes all 0, or in a solution that has its STATCOM
sizes all with the maximum value. The problem is aggravated with the fact that there is no reason
for choosing a set of values for these parameters against one other different set, resulting in a global
optimum whose STATCOM sizes are directly influenced by the choices of α, ω3 and ω4.
Instead of using the F3 and F4 functions, in this thesis is proposed to use only one function to
integrate the voltage stability margin into the objective function. The objective of this optimization
is to increase λmax but not maximize it, since that would lead to the optimum STATCOM(s) size(s)
being equal to the maximum value. Thus it follows that, when designing the optimization problem, it
should be specified how much λmax should increase. With this value specified, the algorithm then
encounters the optimal solution that increases λmax by the value specified and minimizes both the
voltage level deviation (equation 4.6) and the active power losses in the lines (equation 4.7). This
optimization scheme can be easily done by integrating the function described in equation 4.10 into
63
the objective function.
F ′3 = |λmax − λ∗max| (4.10)
This not only adds a degree of specification to the problem, by specifying the voltage stability
margin wanted, λ∗max, but it also avoids the ill-conditioning of the design proposed by [3], since if
the optimal solution contains in its component(s) dedicated to the STATCOM(s) size(s) the value 0,
it means that the specified voltage stability margin is smaller or similar to the one already existent.
On the other hand if the optimal solution has the STATCOM(s) size(s) all with the maximum value, it
means that, with the number of STATCOM(s) specified, either the specified voltage stability margin
can only be achieved with this solution or that this margin cannot be reached.
4.4.2 Problem’s Hyperspace
In a optimization problem, the optimal solution is defined by the set of coordinates that minimize
or maximize a particular function. In this particular optimization problem, the objective is to find
the location and the size of one or multiple STATCOM unit(s) that optimizes a certain number of
criteria. Therefore, it follows that the optimal solution should have the location(s) and size(s) of the
STATCOM unit(s) and that the problem’s hyperspace has 2NS dimensions, where NS is the number
of STATCOMs. Taking into account that a particle in the PSO paradigm is considered as a possible
solution, the position vector of a particle i should be defined as described in equation 4.11, where Lj
and Sj stand for the location and size of the STATCOM j.
xi = [L1, . . . , Lj , . . . , LNS, S1, . . . , Sj , . . . , SNS
] (4.11)
The location of a particular STATCOM unit is described by the bus number where that unit is
connected. Hence, the particles in the swarm can only occupy positions in the hyperspace which the
first NS coordinates (which are the ones that describe the locations) are integers and are between
the number 1 and the number of buses in the network. It is also trivial to conclude that, in the case
of NS > 1, two or more STATCOM units cannot have the same location and that the STATCOM(s)
should be inserted in buses where the voltage is not already controlled by generation units. This
restrictions are presented mathematically in equation 4.12, where NB stands for the number of buses
in the network and PQ stands for a bus where the voltage is not controlled.
L1, . . . , LNS∈ Θ Θ =
{Li, Lj ∈ N : Li 6= Lj , type{Li} = PQ , 1 ≤ Li ≤ NB
}(4.12)
With respect to the size(s) of the STATCOM unit(s), they describe the reactive power range of the
unit. In [4] it is considered that the variety of STATCOM sizes available equals the set of integers in the
interval [0, QMaxST ], where QMax
ST is the maximum size available for the problem. However, in reality, this
extreme variety of sizes does not exist and considering otherwise can lead to optimal solutions that
despite having a mathematical significance, cannot be applicable in reality based problems where the
sizes of STATCOM available is more limited.
64
Another restriction, necessary to maintain the number of STATCOMs, to the coordinates that rep-
resent the sizes of STATCOM, is that, for optimization problems with NS > 1, one or more STATCOM
sizes cannot have size 0. However, it is necessary to consider positions in the hyperspace with
all STATCOM sizes with size 0 as possible optimal solutions of the problem in the case of an ill-
dimensioning of the objective function.
Taking in consideration these restrictions to the sizes of the STATCOMs, it follows that the PSO
particles can only occupy positions in the hyperspace that have in their coordinates Si values that
belong to the set of STATCOM sizes available and that have all the Si components of the position
vector greater than zero or all zero. Thus eliminating positions that either have STATCOM sizes that
do not belong to the set of sizes available or have some but no all the STATCOM sizes equal to zero.
These restrictions can be expressed mathematically as expressed in equation 4.13, where Ψ1 is the
set of numbers that describe the available STATCOM sizes.
S1, . . . , SNS∈ Ψ1 ∩Ψ2 Ψ2 =
{Si ∈ N0 : Si 6= 0 ∪
( NS∑j=1
Sj = 0)}
(4.13)
The outer boundaries of the hyperspace are integrated in the PSO algorithm in the manner pre-
sented in section 3.3.1, i.e., if a particle i approaches one of the outer boundaries with velocity vi
it bounces of the boundary with velocity −vi. However, in order to completely integrate this restric-
tions to the PSO algorithm, applying the discrete variable integration described in section 3.3.4 is not
enough, since the particle can occupy a discrete position in the hyperspace and still be unfeasible.
For these specific positions, it is suggested in [4] that these particles should be re-randomized in to
the hyperspace. However, it was concluded during this work, that while computing the optimal place-
ment and sizing of multiple STATCOMs, particles assume too often unfeasible positions that if it was
to be assumed that these particles should have been re-randomized, the dynamic of the whole group
would have been damaged as a result. Hence, in this thesis, another approach is used to solve this
problem. Particles that, during the process, assume unfeasible positions receive automatically a fit-
ness value of 500, which is a value higher than any fitness value achieved by any particle. In this way,
a particle that occupies an unfeasible position is not considered for the global best, computational
effort is not spent in calculating its fitness value, and also maintains its movement in the next iteration
and consequently does not damage the algorithm’s inner dynamics.
The velocity limits applied in this optimization problem equal the limits of the hyperspace. In other
words, for the velocity vector components that correspond to the STATCOM locations, |v|max is equal
to the number of buses in the network NB , for the velocity components that correspond to the STAT-
COM sizes, |v|max is equal to the maximum size available, QMaxST .
With the objective function and the respective restrictions specified, a formal mathematical expres-
65
sion of the optimization problem at hand is presented in equation 4.14.
minimizex
ω1
√√√√NB∑i=1
(Vi − 1)2 + ω2
NL∑i=1
PLi + ω3|λmax − λ∗max|
subject to L1, . . . , LNS∈ Θ
S1, . . . , SNS∈ Ψ1 ∩Ψ2
(4.14)
Where:xi = [L1, . . . , Lj , . . . , LNS
, S1, . . . , Sj , . . . , SNS]
Θ ={Li, Lj ∈ N : Li 6= Lj , type{Li} = PQ , 1 ≤ Li ≤ NB
}Ψ1 := Set of available STATCOM sizes
Ψ2 ={Si ∈ N0 : Si 6= 0 ∪
( NS∑j=1
Sj = 0)} (4.15)
4.4.3 Speed Up Measures
The PSO process on itself is not a very time consuming computation, however, in this case, for
each particle, in order to compute its fitness value, the algorithm has to do one CPF calculation to
achieve the respective point of collapse λmax. Since the swarm size typically ranges from 20 to 40
particles and taking in account the possible number of iterations needed to achieve the global opti-
mum, it is concluded that the optimal placement and sizing of STATCOM devices can be a very time
consuming process. With the objective of reducing the time needed for this computation, some speed
up measures were integrated in the process.
In order to speed up the process, the most logical procedure is to find a way to utilize the whole
computational power of the machine where the algorithm is running. The algorithm was constructed
in the MATLABTM
environment which is single core program. Nowadays, the majority of all personal
computers have more than one processor core, in fact, this algorithm was run in a quad-core ma-
chine. In order to distribute work to all the processor cores in the computer, MATLABTM
provides the
Parallel Computing ToolboxTM
which makes use of the function matlabpool to start a pool of MATLABTM
executables, each one dedicated to one of the processor’s cores. The work distribution can then be
done by using the parfor computational loop, which is similar to the for loop but where each iteration
is done in parallel in the processor’s cores with other iterations.
In the PSO algorithm, there are many fitness value computations being made in every iteration. It
follows that, since in this case, each fitness value computation requires some computational effort, it
would be useful to save the results from the fitness value computation of each particle in order to use
them if another particle reaches the same position. This is the second speed up measure integrated in
the optimal STATCOM placement and sizing process. After each particle’s fitness value computation,
the position vector components values, the fitness value and the λmax achieved by that particle are
stored in a table, like the one in Figure 4.3, that is kept in memory throughout the whole process.
66
L1=1 L2=3 S1=10 S2=20 F=0.0638 λmax=0.6
L1=1 L2=4 S1=30 S2=20 F=0.0608 λmax=0.65
L1=3 L2=4 S1=20 S2=10 F=0.0628 λmax=0.62
L1=2 L2=3 S1=10 S2=10 F=0.075 λmax=0.56
L1=2 L2=7 S1=30 S2=20 F=0.055 λmax=0.7
L1=1 L2=5 S1=10 S2=30 F=0.059 λmax=0.67
L1=4 L2=6 S1=10 S2=20 F=0.0655 λmax=0.64
Figure 4.3: Memory Table
Before trying to calculate the fitness value of a particle, the algorithm first searches, using the
information in the position vector, in the table for the fitness value. If this search is successful, the
fitness value is obtained through the table and not resorting to the CPF method.
At this point is necessary to state that the λmax of a network with STATCOM(s) is larger than the
one achieved with the original network (without STATCOMs), it follows that the λmax achieved by each
particle can be computed in a faster way using the point of collapse achieved by the original network
to compute the first continuation step used in the CPF method. The same logic applies for the λmax
achieved by one particle and the λmax achieved by another particle with the same locations as the
first but with smaller sizes.
Therefore if the initial search turns out to be unsuccessful, the algorithm searches again the table
for a entry that has the same STATCOM locations, if it finds it, the algorithm then checks if all the
STATCOM sizes in the table entry are smaller than the ones in the position vector of the particle. If
they are smaller, the algorithm uses the λmax of the table entry to compute the first step value used in
the CPF method. If not, the algorithm uses the λmax of the original network to compute the first step
value used in the CPF method.
4.5 Results
4.5.1 5 Bus Network with 1 STATCOM Unit
The 5 Bus network used in section 2.4.2 to test the CPF method is a good first example of the
application of both the algorithm designed in this chapter and the one designed in [3]. The power
flow data of this network is displayed in Appendix A. Since it has only 5 buses, in this first example
was chosen to compute only the optimal placement and sizing of one STATCOM unit. The following
optimizations were done considering the reactive power generation limits of the swing bus 1 and with
all the Kdi and Kgi equal to 1, i.e., all the loads and all the generated active powers have the same
67
rate of change during the CPF method.
1 2
35
4
G1
G2
Figure 4.4: Single Line Diagram of the 5 Bus Network
First, the set of STATCOM sizes available for the allocation must be defined. Since, in this section
the objective is to allocate a single STATCOM by applying the two objective function designs described
in section 4.4.1, to then compare the results, it was decided use all the STATCOM sizes from 0 MVA
to 200 MVA in intervals of 10 MVA, and so, the set Ψ1 can be defined in this case as:
Ψ1 ={Si ∈ N0 :
Si10∈ N0, 0 ≤ Si ≤ 200
}[MVA] (4.16)
It also important to present the PSO parameters used in the following simulations. These param-
eters are in Table 4.1.
Table 4.1: PSO Parameters for the 5 Bus with 1 STATCOM Optimization
No. of Particles tmax T I c1 c2
30 3000 0 100 2.05 2.05
The first simulation related to this 5 Bus network, was performed using the objective function
proposed by [3], with the following weights:
ω1 = 0.1 ω2 = 0.1 ω3 = 0.7 α = 0.04 ω4 = 0.1
This configuration of the optimization problem attributes more influence to functions F3 and F4 with
respect with F1 and F2, since the main objective is to increase the voltage stability margin. However,
the values for the weights ω3, α and ω4 were dimensioned so that the objective function doesn’t favors
the value of F3 more than the value of F4 or vice-versa. With this configuration, and with the set of
PSO parameters presented in Table 4.1, the algorithm converges to the global optimum presented on
equation 4.17. The algorithm used 103 iterations with a total time of 21.15 s.
xopt = [5, 130] (4.17)
Which means that, with this configuration, the optimal solution in this network, is to insert a STAT-
COM with a reactive power range of 130 MVA in Bus 5. To confirm these results, the fitness value for
68
every position in the hyperspace was calculated. These values are presented in form of the objective
function plot in Figures 4.5 and 4.6.
Figure 4.5: Plot of the Objective Function, using the configuration proposed in [3] with the first set of weights
Figure 4.6: 2D Representation of the Objective Function, using the configuration proposed in [3] with the firstset of weights
There are several aspects of the objective function plot in Figures 4.5 and 4.6 that need inter-
preting. Firstly, it is confirmed that the algorithm reached the correct global minimum. It can also
be concluded from this figure that this configuration of the objective function generates a parabolic
shaped function where, as expected, for S1 = 0 it has the same value for every value of L1. However,
as S1 increases, the function values for each bus number L1 decrease at different rates, and thus, for
every value L1, there exists a value of S1 where the function has a local minimum.
69
It is important to remember however, that the voltage level is already controlled in the original
network configuration in buses 1 and 3. Observing the function values for these buses in Figures 4.5
and 4.6, it is concluded that inserting a STATCOM unit in one of these buses is not a situation worth
considering, since the local minima achieved with L1 = 1 or L2 = 3 have the highest values amongst
all the local minima.
Figure 4.7: Plot of the Objective Function, using the configuration proposed in [3] with the second set of weights
Figure 4.8: 2D Representation of the Objective Function, using the configuration proposed in [3] with the secondset of weights
It was stated earlier, in section 4.4.1, that the weights ω3, α and ω4 are a direct influence to the
optimum value for the size of the STATCOM. In order to demonstrate this influence, another simulation
70
was executed with this objective function configuration, but with different values for the weights:
ω1 = 0.1 ω2 = 0.1 ω3 = 0.6 α = 0.04 ω4 = 0.2
Using the same PSO parameters, the algorithm converges for a different position in the hyper-
space than before. With these weight values, the global minimum, achieved by the algorithm, is the
one presented in equation 4.18. The algorithm used 106 iterations with a total time of 21.82 s.
xopt = [4, 60] (4.18)
Once again, in order to prove the effectiveness of the algorithm, the function value for every posi-
tion in the hyperspace was calculated and plotted, resulting in the graphics of Figures 4.7 and 4.8.
From Figures 4.7 and 4.8, it is concluded that although the algorithm achieved the correct result
for the global optimum and the characteristics of the objective function plot, described earlier, have all
stayed the same, as a consequence of the slight tinkering of the weight values, the objective function
started to favor smaller sizes for the STATCOM unit, resulting in the global optimum presented in
equation 4.18. If the weight values suffer another alteration in order to favor even more smaller sizes
for the STATCOM, the algorithm will converge to a solution where the STATCOM size is 0. One
example of this ill-conditioning is when the weight values are the following:
ω1 = 0.1 ω2 = 0.1 ω3 = 0.3 α = 0.04 ω4 = 0.5
Figure 4.9: Plot of the Objective Function, using the configuration proposed in [3] with the third set of weights
With these weight values, the plot of the fitness function ceases to have a parabolic shape to have
the shape illustrated in the objective function plot on Figures 4.9 and 4.10.
71
Figure 4.10: 2D Representation of the Objective Function, using the configuration proposed in [3] with the thirdset of weights
Since there is no reason to choose one particular set of values against another, this sort of ill-
conditioning can happen very often. To avoid this, in this thesis is proposed that the objective function
in equation 4.14 is used. In this way, when designing the problem, a value for the voltage stability
margin is specified, and the algorithm finds the solution that minimizes the difference between the
specified λ∗max and the λmax achieved by the STATCOM configuration suggested by each position
in the hyperspace. To test this new objective function, one last simulation was performed with this
network, with the following values for the parameters:
ω1 = 0.1 ω2 = 0.1 ω3 = 0.8 λ∗max = 0.8
Remembering that the voltage stability margin of the original network with the swing bus reactive
power generation limits is λmax = 0.288, in this optimization run, the algorithm will find the optimal
placement and sizing of the single STATCOM unit that puts the voltage stability margin closer to
λmax = 0.8 while still minimizing the voltage level deviation and the active power losses on the base
case power flow solution. With this objective function and with the values for the function’s parameters
presented above, the global minimum, achieved by the algorithm is presented in equation 4.19. In
this case, the algorithm used 104 iterations with a total time of 19.03 s.
xopt = [5, 170] (4.19)
Once again, the objective function was calculated for every position in the hyperspace, which
results on the graphics presented in Figures 4.11 and 4.12. This function, instead of having parabolic
shape, it has a V-shaped form, consequence of the nature of the function F ′3 which calculates the
distance between the specified λ∗max and the λmax achieved by each particle.
72
Figure 4.11: Plot of the Objective Function, using the configuration proposed in this thesis
Figure 4.12: 2D Representation of the Objective Function, using the configuration proposed in this thesis
In this manner, the location of the global minimum is not directly determined by the balance defined
by the values of certain parameters, but by what is the configuration that leads to the voltage stability
enhancement specified.
4.5.2 Porto Santo Island 12 Bus Network with 2 STATCOMs
The Porto Santo island is located in the Madeira archipelago, which is situated in the Atlantic
ocean, 520 kilometres from the African coast line. Its network consists in 12 buses, distributed through
the island, and one diesel generation unit. Apart from this generation unit, the network has connected
to some of its buses, renewable energy sources such as wind farms and solar power plants. The
73
single line diagram of this network is presented in Figure 4.13.
30 kV6.6 kV
1-CNP3
3-CPS3 2-VBL3
5-CPS6.6 6-VBL6.64-CNP6.6
10-WP29-WP111-FP1 12-FP2
7-PTEEM1 8-PTEEM2
5
6
4
CNP CPS VBL
9
11
7 8
12 13
Figure 4.13: Single Line Diagram of the Porto Santo Island Network
Due to the inclusion of renewable power sources in the network, there is the need to address how
should these power sources behave during the CPF process. Since renewable energy sources can-
not control their power outlet due to their intermittent nature, this problem is addressed by making the
power generated by these power sources constant. The base case power flow data of this network is
presented in Appendix C.
Due to the geographical characteristics of the island, the distance between the buses are very
small, and consequently so are the voltage drops in the lines. If the reactive generation limits of
the diesel generator are considered, the point of collapse of the network will be yielded by this limit,
since this is the only generation unit capable of regulating its power outlet. However, in this section,
the objective is to find the optimal placement and sizing of two STATCOM units that will enhance the
voltage stability margin that is defined by the transmission limit, described in section 2.3.10. To do this,
the reactive generation limits of the diesel generation unit will not be considered and consequently it
is assumed, that if the load reaches levels above the reactive generation limit, the power capabilities
of the generation unit are enhanced. In Table 4.2 are presented the load level where the points of
collapse defined by reactive generation limit and the transmission limit are located. These limits were
computed with all the loads with Kd = 1. In this manner, as was referred in chapter 2, there exists a
complete correlation between the value of λ and the increase in each load power.
74
Table 4.2: Location of the Reactive Transmission Limit and Transmission Limit of the Porto Santo Island Network
Reactive Generation Limit Transmission Limit
λmax = 0.961 λmax = 24.9176
The optimal placement and sizing of the two STATCOM units is defined by having the primary ob-
jective of increasing the voltage stability margin caused by the transmission limit. Hence, the objective
function’s weights have the following values:
ω1 = 0.1 ω2 = 0.1 ω3 = 0.8
The set of available STATCOM sizes in this problem is constituted of STATCOM sizes much smaller
than the ones used in section 4.5.1. This is due to the much smaller loads in the base state of the
network, and despite the fact that, the transmission limit of the network is defined for much bigger load
levels, the λmax that defines this limit, is given as a proportionality factor between the base case load
level and the load level at the transmission limit. Thus, if the objective is to give a reasonable increase
to λmax, the available sizes of the STATCOM must be defined taking into account the load level at
the base state of the network. The set of sizes available for this optimization problem is presented in
equation 4.20.
Ψ1 = {0, 1, 5, 10, 20} [MVA] (4.20)
Taking into account the available sizes of the STATCOM units and the value of λmax that defines
the transmission limit, the optimization problem is defined in order to find a solution that enhances the
voltage stability limit due to transmission limit to λ∗max = 27.0.
The PSO parameters used in this problem are equal to those used on the last section (Table 4.1).
With this problem design, the optimal solution achieved by the algorithm is displayed in equation 4.21.
xopt = [2, 4, 10, 5] (4.21)
Using the information on Figure 4.13 to convert the bus number to the bus name, the solution that
increases the voltage stability margin to λ∗max = 27.0 with two STATCOM units, is the insertion in the
CPS6.6 Bus of a 5 MVA STATCOM and in the VBL3 Bus of a 10 MVA. The transmission limit and the
reactive generation limit achieved by this network with the two STATCOMs installed are presented in
the Table 4.3. It is also important to note that the algorithm achieved this solution using 236 iterations,
resulting in a total time of 10m 27s.
Table 4.3: Location of the Reactive Transmission Limit and Transmission Limit of the Porto Santo Island Networkwith STATCOMs
Reactive Generation Limit Transmission Limit
2.512 27.1290
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4.5.3 IEEE Midwest 57 Bus Network with 4 STATCOMs
The last network used in this chapter is the IEEE 57 Bus network ([3]) from the Midwest region
of the United States. This network is depicted in Figure 4.14 and its base case power flow data is
presented on Appendix D.
Figure 4.14: Single Line Diagram of the IEEE Midwest 57 Bus Network ([3])
In this section, the objective is to test the algorithm constructed during this thesis in optimal place-
ment and sizing of four STATCOM devices in a big network such as this one. The voltage stability
margin of this network before the insertion of the STATCOM units is defined by its transmission limit
at λmax = 0.61222 (with all the loads with Kd = 1 and all the generators except the one in the swing
bus with Kg = 1), since according to the data in Appendix D, the swing bus 1 has no reactive power
generation limits.
Like the optimizations done in the last two sections, the objective of the optimal placement and
sizing of the four STATCOMs is, in this section, primarily to enhance the voltage stability margin. Thus,
the objective function’s weights continue to have the same values:
ω1 = 0.1 ω2 = 0.1 ω3 = 0.8
The specified value for the voltage stability margin, defined by λ∗max, is set to 0.7. To do this,
the same logic used in the last section is applied in this network. The set of available STATCOM
sizes must be, preferably, such that the specified value for the voltage stability margin, λ∗max, must
be reached by a combination of locations and sizes. Following this logic, the set of STATCOM sizes,
used in this section, is composed by all the sizes from 0 MVA to 50 MVA in intervals of 10 MVA. This
76
set is mathematically defined in equation 4.22.
Ψ1 ={Si ∈ N0 :
Si10∈ N0, 0 ≤ Si ≤ 50
}[MVA] (4.22)
Due to the number of dimensions of the hyperspace where the particles move, it is suggested to
increase the number of particles used in the PSO. The PSO parameters used to solve this optimization
problem are presented in the Table 4.4.
Table 4.4: PSO Parameters for the IEEE Midwest 57 Bus Network with 4 STATCOM Optimization
No. of Particles tmax T I c1 c2
40 3000 0 100 2.05 2.05
As was discovered in section 3.4.2, the success rate of the PSO algorithm for objective functions
with a great variety of local optima is not 100%. This optimization problem, due to the variety of lo-
cations for the STATCOMs and to the number of STATCOMs, can be considered has having a great
variaty of local minima, thus it is possible that the algorithm could be unsuccessful in achieving the
global minimum in some simulation runs. However, it is possible that there are several local minima
that, due to the difference in the function value to the global minimum, should also be considered
as possible practical solutions. For example, if the algorithm achieves two different sets of locations
and sizes which achieve similar function values but the one that yields the global minimum has bigger
sizes than the other, it can be argued that, for economical reasons, the best solution is the second
solution, which is not the global minima.
In summary, the solutions given by the algorithm constructed in this thesis should always be con-
sidered has theoretical solutions based on a mathematical formulation, and therefore, the inability of
this algorithm to achieve, in some cases, the global minimum 100% of the times can be used to con-
struct a group of possible practical solutions that the user of this algorithm can evaluate and conclude
upon which solution is more appropriate to use.
Taking this into account, to solve the optimization problem proposed in this section, twenty runs
of this algorithm were made. Two solutions emerged from these simulations. These solutions are
presented in equations 4.23 and 4.24 along with the respective fitness values and with the voltage
stability margins achieved by each solution.
x1 = [15, 30, 56, 57, 10, 10, 30, 10] F (x1) = 0.1259 λmax = 0.6994 (4.23)
x2 = [4, 15, 38, 52, 20, 10, 10, 30] F (x2) = 0.1279 λmax = 0.7028 (4.24)
Evaluating the two solutions achieved by the algorithm, taking in account the values displayed
above, it is clear that x1 is a better solution than x2, not only because it achieves a better fitness value
than x2, but also due to the values of the sizes for the STATCOM units. Comparing x1 and x2 leads
to the conclusion that x1 achieves a better fitness value with a smaller set of STATCOM sizes. Nev-
ertheless, it is important to note, that other criteria could be used to compare the achieved solutions,
77
for example, the locations where the STATCOMs would be installed, since in real life problems some
locations are more favorable to the installation of STATCOM units than others.
Table 4.5: Optimal Placement and Sizing of four STATCOMs in the IEEE Midwest 57 Bus Network
STATCOM Unit Bus No. Size [MVA]
1 15 102 30 103 56 304 57 10
78
5Conclusions
Contents5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
79
5.1 Conclusions
From chapter 2, where the CPF method is described is exemplified, some conclusions need to
be presented. Through the application of the CPF method constructed in this thesis to the 2 Bus
and IEEE 39 Bus networks, it was possible to confirm the resulting data and thereby validating the
algorithm. In the case of the 2 Bus network, in section 2.4.1, the results were confirmed, since the
point of collapse could also be achieved analytically, through a comparison with the results that were
computed with the analytical approach, in section 2.2.1. On the other hand, in the case of the IEEE
39 Bus network, the results were confirmed with the comparison with the results achieved in [2].
Through the application of this method to the 5 Bus network, in section 2.4.2, the influence of the
reactive power generation limits of all the buses was studied and consequently the difference of the
point of collapse held by the network’s transmission limit and by the network’s reactive generation limit
was also analyzed.
In chapter 3, which had the purpose of describing the PSO algorithm, the application of this method
to three mathematical functions was also tested. It was concluded, in section 3.4.1 that the constric-
tion factor approach achieves better results than the inertia weight approach and in section 3.4.2 it
was also concluded that, when using the PSO technique to objective functions with a high degree of
complexity, the success rate of the optimization method is not 100%. To get around this lack of accu-
racy it is suggested, in section 3.4.4, to execute several simulation runs of the algorithm and analyze
each solution calculated. The results achieved in this chapter were confirmed by using mathematical
functions with well known global minima.
In the beginning of this thesis, in chapter 1, was established as the major objective of this thesis to
construct a deterministic approach to the problem of where to install STATCOM units in a specific net-
work and what size should these units have. In chapter 4, this approach was constructed by merging
the CPF and PSO concepts. In section 4.4 a new approach to the objective function design was intro-
duced along with new solutions to solve the problem of how to manage unfeasible positions assumed
by particles in the hyperspace. The new design of the objective function was compared with the one
proposed in [3] and, as a result, it was concluded that by swapping the concept of maximization of
the voltage stability margin by the concept of enhancing this margin to a specified level λ∗max, the
objective function’s design has its objectivity increased, since the ill-conditioning of the optimization
ceases to exist. In section 4.5.2, the optimal placement and sizing algorithm was applied to the Porto
Santo 12 Bus network and the optimal locations and sizes of two STATCOM units were found, how-
ever, it is important to remark that since it was concluded that the transmission limit of this network
was extremely far from the state of normal operation, this study is merely hypothetical. Lastly, in this
chapter, the optimal placement and sizing of four STATCOM units were found in relation with the IEEE
57 Bus network, revealing in this case the lack of 100% accuracy of the PSO method mentioned in
section 3.4.4, and how this characteristic of PSO can be used achieve several solutions that can then
80
be evaluated and compared.
Overall, however, the applicability of the PSO algorithm was confirmed due to the easy application
of this technique in multi-dimensional discrete optimization problems. Moreover, this thesis was con-
structed so that other studies regarding the subject of power system’s optimization using PSO or even
regarding the use of the CPF method, can benefit from this document. Hence the structure chosen
for this thesis.
5.2 Future Work
Like every other investigation, there is always space to improve. In this section some suggestions
are made about some of the future work that can be based on this thesis.
The objective of this thesis was to find a deterministic approach to the problem of placement and
sizing of STATCOM units with the objective of enhancing the voltage stability margin. This objective
was accomplished in the last chapter, however, one might argue if there is a even more deterministic
approach to this problem. This argument comes from the fact that, in the algorithm developed in this
thesis, the optimal location(s) and size(s) of the STATCOM(s) are calculated, but the number of STAT-
COM units is still left to the judgment of optimization problem designer. Certainly there exist situations
where the optimal number of STATCOM(s) is one, and in other situation the optimal number of units
is another. Hence further investigation upon this matter should study a way to integrate the number of
STATCOM(s) needed in the optimization problem, so that the optimal solution to the new optimization
problem has the optimal number of STATCOM(s).
Further studies can also investigate if there is a optimal configuration of the PSO algorithm to the
problems of placement and sizing of not only STATCOM unit but of other power network’s equipment.
In this thesis, the configuration used is the one suggested by certain studies that test the performance
of the algorithm with mathematical functions, however, if the performance were to be tested in ob-
jective functions similar to the ones used in the this optimization problem, an optimal configuration
regarding this sort optimization could emerge.
81
82
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85
86
AData of the 5 Bus Network
A-1
A.1 Bus Base Case Power Flow Data
BUS DATA: (Sb = 100 MVA)
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu] εv, ηv
1 Swing 1.04 0 0.65 0.3 2.322 1.096 0
2 PQ 0.9736 −0.1159 1.15 0.6 0 0 0
3 PV 1.02 −0.067 0.7 0.4 1.8 1.005 0
4 PQ 0.9203 −0.1915 0.7 0.3 0 0 0
5 PQ 0.9683 −0.1085 0.85 0.4 0 0 0
A.2 Generator’s Reactive Limits
Reactive Limits:
Bus Qmax [pu] Qmin [pu]1 1.5 −0.13 1.5 −0.1
A.3 Line Data
LINE DATA:
Bus i Bus j Rl [pu] Xl [pu] Bl [pu]1 2 0.0042 0.168 0.041
1 5 0.031 0.126 0.031
2 3 0.031 0.126 0.031
3 4 0.084 0.336 0.082
3 5 0.053 0.21 0.051
4 5 0.063 0.252 0.061
A-2
BData of the IEEE New England 39 Bus
Network
B-1
B.1 Bus Base Case Power Flow Data
BUS DATA: (Sb = 100 MVA)
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu] εv, ηv
1 PQ 1.0436 −0.2339 0 0 0 0 –2 PQ 1.0379 −0.1958 0 0 0 0 –3 PQ 1.0056 −0.2422 3.22 1.224 0 0 0
4 PQ 0.9864 −0.2446 5.0 1.84 0 0 0
5 PQ 0.9924 −0.2137 0 0 0 0 –6 PQ 0.9956 −0.1991 0 0 0 0 –7 PQ 0.9851 −0.2401 2.338 0.84 0 0 0
8 PQ 0.9843 −0.250 5.22 1.76 0 0 0
9 PQ 1.0233 −0.2546 0 0 0 0 –10 PQ 1.006 −0.1644 0 0 0 0 –11 PQ 1.0012 −0.1763 0 0 0 0 –12 PQ 0.9876 −0.1786 0.085 0.880 0 0 0
13 PQ 1.0014 −0.1786 0 0 0 0 –14 PQ 0.9947 −0.2127 0 0 0 0 –15 PQ 0.9909 −0.2328 3.20 1.53 0 0 0
16 PQ 1.0043 −0.2124 3.294 1.323 0 0 0
17 PQ 1.0076 −0.2291 0 0 0 0 –18 PQ 1.0055 −0.2419 1.58 0.30 0 0 0
19 PQ 1.0432 −0.1381 0 0 0 0 –20 PQ 0.9938 −0.1661 6.80 1.03 0 0 0
21 PQ 1.0122 −0.1718 2.74 1.15 0 0 0
22 PQ 1, 0387 −0.0952 0 0 0 0 –23 PQ 1.0322 −0.0988 2.475 0.846 0 0 0
24 PQ 1.0029 −0.2108 3.086 0.922 0 0 0
25 PQ 1.0461 −0.1748 2.24 0.472 0 0 0
26 PQ 1.0299 −0.1989 1.39 0.47 0 0 0
27 PQ 1.0136 −0.2338 2.81 0.755 0 0 0
28 PQ 1.0308 −0.1398 2.06 0.276 0 0 0
29 PQ 1.0318 −0.0914 2.835 1.269 0 0 0
30 PV 1.0475 −0.1566 0 0 2.30 2.068 –31 Swing 0.982 0 0 0 7.229 2.747 –32 PV 0.9831 −0.0276 0 0 6.30 2.540 –33 PV 0.9972 −0.0497 0 0 6.12 1.529 –34 PV 1.0123 −0.0787 0 0 4.88 2.3686 –35 PV 1.0493 −0.0103 0 0 6.30 2.906 –36 PV 1.0635 0.0347 0 0 5.40 1.483 –37 PV 1.0278 −0.0598 0 0 5.20 0.484 –38 PV 1.0265 0.0301 0 0 8.10 1.383 –39 PV 1.03 −0.2563 11.04 2.50 10.0 1.233 0
B-2
B.2 Generator’s Reactive Limits
Reactive Limits:
Bus Qmax [pu] Qmin [pu]30 3.8 −1.031 6.0 −3.032 5.0 −3.033 5.0 −3.034 4.5 −2.535 6.0 −2.536 5.0 −2.237 5.0 −2.238 5.0 −3.039 9.0 −8.0
B.3 Line Data
LINE DATA:
Bus i Bus j Rl [pu] Xl [pu] Bl [pu]1 2 0.0035 0.0411 0.6987
1 39 0.002 0.05 0.375
1 39 0.002 0.05 0.375
2 3 0.0013 0.0151 0.2572
2 25 0.007 0.0086 0.146
3 4 0.0013 0.0213 0.2214
3 18 0.0011 0.0133 0.2138
4 5 0.0008 0.0128 0.1342
4 14 0.0008 0.0129 0.1382
5 6 0.0002 0.0026 0.0434
5 8 0.0008 0.0112 0.1476
6 7 0.0006 0.0092 0.113
6 11 0.0007 0.0082 0.1389
7 8 0.0004 0.0046 0.078
8 9 0.0023 0.0363 0.3804
9 39 0.001 0.025 1.2
10 11 0.0004 0.0043 0.0729
10 13 0.0004 0.0043 0.0729
13 14 0.0009 0.0101 0.1723
14 15 0.0018 0.0217 0.366
15 16 0.0009 0.0094 0.171
16 17 0.0007 0.0089 0.1342
16 19 0.0016 0.0195 0.304
16 21 0.0008 0.0135 0.2548
16 24 0.0003 0.0059 0.068
17 18 0.0007 0.0082 0.1319
17 27 0.0013 0.0173 0.3216
21 22 0.0008 0.014 0.2565
22 23 0.0006 0.0096 0.1846
23 24 0.0022 0.035 0.361
25 26 0.0032 0.0323 0.513
26 27 0.0014 0.0147 0.2396
26 28 0.0043 0.0474 0.7802
26 29 0.0057 0.0625 1.029
28 29 0.0014 0.0151 0.249
B-3
B.4 Transformer Data
TRANSFORMER DATA:
Bus i Bus j Type: Rt [pu] Xt [pu] m
2 30 Fixed Tap 0 0.0181 1.025
6 31 Fixed Tap 0 0.05 1.07
6 31 Fixed Tap 0 0.05 1.07
10 32 Fixed Tap 0 0.02 1.07
12 11 Fixed Tap 0.0016 0.0435 1.006
12 13 Fixed Tap 0.0016 0.0435 1.006
19 20 Fixed Tap 0.0007 0.0138 1.06
19 33 Fixed Tap 0.0007 0.0142 1.07
20 34 Fixed Tap 0.0009 0.018 1.025
22 35 Fixed Tap 0 0.0143 1.025
23 36 Fixed Tap 0.0005 0.0272 1
25 37 Fixed Tap 0.0006 0.0232 1.025
29 38 Fixed Tap 0.0008 0.0156 1.025
B-4
B.5 Bus Data at Point of Collapse without Swing Bus ReactiveLimits
BUS DATA: (Sb = 100 MVA, λ = 0.4228)
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu]1 PQ 1.0162 −0.3931 0 0 0 0
1 PQ 1.0162 −0.3931 0 0 0 0
2 PQ 0.9724 −0.3070 0 0 0 0
3 PQ 0.9042 −0.4108 3.22 1.224 0 0
4 PQ 0.8638 −0.4358 5.0 1.84 0 0
5 PQ 0.8667 −0.4015 0 0 0 0
6 PQ 0.8727 −0.3742 0 0 0 0.
7 PQ 0.8482 −0.4705 8.793 3.159 0 0
8 PQ 0.8478 −0.4917 19.633 6.620 0 0
9 PQ 0.9671 −0.4605 0 0 0 0
10 PQ 0.8870 −0.3065 0 0 0 0
11 PQ 0.8803 −0.3297 0 0 0 0
12 PQ 0.8624 −0.3336 0.085 0.880 0 0
13 PQ 0.8790 −0.3344 0 0 0 0
14 PQ 0.8670 −0.4018 0 0 0 0
15 PQ 0.8506 −0.4929 12.035 5.754 0 0
16 PQ 0.8824 −0.4572 12.389 4.976 0 0
17 PQ 0.8929 −0.4338 0 0 0 0
18 PQ 0.8930 −0.4441 5.943 1.128 0 0
19 PQ 0.9667 −0.3920 0 0 0 0
20 PQ 0.9197 −0.4904 25.575 3.874 0 0
21 PQ 0.8932 −0.4039 10.305 4.325 0 0
22 PQ 0.9510 −0.2666 0 0 0 0
23 PQ 0.9464 −0.2763 9.309 3.182 0 0
24 PQ 0.8841 −0.4491 3.086 0.922 0 0
25 PQ 1.0017 −0.2579 2.240 0.472 0 0
26 PQ 0.9462 −0.2678 1.390 0.470 0 0
27 PQ 0.9108 −0.3677 2.810 0.755 0 0
28 PQ 0.9714 −0.0989 2.060 0.276 0 0
29 PQ 0.9871 −0.0147 2.835 1.269 0 0
30 PV 1.0145 −0.2419 0 0 3.46 3.80
31 SW 0.9820 0 0 0 11.71 8.76
32 PV 0.9122 −0.0532 0 0 9.48 5.0
33 PV 0.9735 −0.2469 0 0 9.21 5.0
34 PV 0.9773 −0.3438 0 0 7.34 4.5
35 PQ 1.0034 −0.1205 0 0 9.48 6.0
36 PV 1.0561 −0.0560 0 0 8.12 5.0
37 PQ 1.0278 −0.0779 0 0 7.82 2.74
38 PQ 1.0265 0.1749 0 0 12.19 4.69
39 PQ 1.0300 −0.4403 11.04 2.5 10.0 4.72
B-5
B-6
CData of the Porto Santo Island 12 Bus
Network
C-1
C.1 Bus Base Case Power Flow Data
BUS DATA: (Sb = 100 MVA)
Number Bus Type |V| [pu] θ [o] Pl [MVA] Ql [MVA] Pg [MVA] Qg [MVA] εv, ηv
1 Swing 1.00 0 0 0 5.6086 4.4950 –2 PQ 0.9992 −0.0128 0 0 0 0 –3 PQ 0.9928 0.1798 0 0 0 0 –4 PQ 0.9993 −0.0753 2.3035 1.2433 0 0 0
5 PQ 0.9983 −0.0765 2.8322 1.5287 0 0 0
6 PQ 0.9917 0.1295 2.379 1.2841 0 0 0
7 PQ 1.0073 0.3197 0 0 0 0 –8 PQ 1.0155 0.2332 0 0 0 0 –9 PQ 1.0269 2.5739 0 0 0.405 −0.251 –10 PQ 1.0258 2.5003 0 0 0.594 −0.3681 –11 PQ 1.0107 0.4658 0 0 0.5 0 –12 PQ 1.0187 0.3770 0 0 0.5 0 –
C.2 Generator’s Reactive Limits
Reactive Limits:
Bus Qmax [MVA] Qmin [MVA]1 8.466 0
C.3 Line Data
LINE DATA:
Number Bus i Bus j Rl [pu] Xl [pu] Bl [pu]4 2 3 0.2778 0.1267 0.0012
5 1 2 0.0302 0.054 0.0006
6 1 2 0.0117 0.021 0.0002
7 5 7 1.8292 1.409 0.0001
8 5 8 3.4867 1.1267 0.0001
9 6 10 4.4913 1.4525 0.0001
11 9 10 0.3546 0.1147 0
12 7 11 0.6708 0.5193 0
13 8 12 0.6708 0.5193 0
C.4 Transformer Data
TRANSFORMER DATA:
Name Bus i Bus j Type: Rt [pu] Xt [pu] m
CNP 1 4 Fixed Tap 0 0.057 1.0
CPS 3 6 Fixed Tap 0 0.06 1.0
VBL 2 5 Fixed Tap 0 0.06 1.0
C-2
DData of the IEEE Midwest 57 Bus
Network
D-1
D.1 Bus Base Case Power Flow Data
BUS DATA: (Sb = 100 MVA)
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu] εv ηv B [pu]1 Swing 1.0400 0 0.55 0.17 4.7879 1.2865 0 0
2 PV 1.0100 −0.0207 0.03 0.88 0.0 −0.00755 0 0
3 PV 0.9850 −0.1045 0.41 0.21 0.40 −0.0131 0 0
4 PQ 0.9808 −0.1281 0.0 0.0 0.0 0 – 0
5 PQ 0.9765 −0.1493 0.13 0.04 0.0 0 0 0
6 PV 0.9800 −0.1516 0.75 0.02 0.0 0.0077 0 0
7 PQ 0.9843 −0.1329 0.0 0.0 0.0 0 – 0
8 PV 1.0050 −0.0786 1.5 0.22 4.5 0.6198 0 0
9 PV 0.9800 −0.1679 1.21 0.26 0.0 0 0 0
10 PQ 0.9863 −0.2001 0.05 0.02 0.0 0 0 0
11 PQ 0.9728 −0.1792 0.0 0.0 0.0 0 – 0
12 PV 1.0150 −0.1830 3.77 0.24 3.1 1.2860 0 0
13 PQ 0.9789 −0.1714 0.18 0.023 0.0 0 0 0
14 PQ 0.9704 −0.1632 0.105 0.053 0.0 0 0 0
15 PQ 0.9882 −0.1255 0.22 0.05 0.0 0 0 0
16 PQ 1.0134 −0.1548 0.43 0.003 0.0 0 0 0
17 PQ 1.0174 −0.0943 0.42 0.08 0.0 0 0 0
18 PQ 1.0008 −0.2046 0.272 0.098 0.0 0 0 0.1
19 PQ 0.9710 −0.2304 0.033 0.006 0.0 0 0 0
20 PQ 0.9651 −0.2339 0.023 0.001 0.0 0 0 0
21 PQ 1.0102 −0.2244 0.0 0.0 0.0 0 – 0
22 PQ 1.0115 −0.2234 0.0 0.0 0.0 0 – 0
23 PQ 1.0101 −0.2245 0.063 0.021 0.0 0 0 0
24 PQ 1.0007 −0.2311 0.0 0.0 0.0 0 – 0
25 PQ 0.9844 −0.3155 0.063 0.032 0.0 0 0 0.059
26 PQ 0.9602 −0.2258 0.0 0.0 0.0 0 – 0
27 PQ 0.9822 −0.2008 0.093 0.005 0.0 0 0 0
28 PQ 0.9971 −0.1829 0.046 0.023 0.0 0 0 0
29 PQ 1.0104 −0.1707 0.17 0.026 0.0 0 0 0
30 PQ 0.9648 −0.3248 0.036 0.018 0.0 0 0 0
31 PQ 0.9386 −0.3360 0.058 0.029 0.0 0 0 0
32 PQ 0.9531 −0.3204 0.016 0.008 0.0 0 0 0
33 PQ 0.9508 −0.3211 0.038 0.019 0.0 0 0 0
34 PQ 0.9625 −0.2439 0.0 0.0 0.0 0 – 0
35 PQ 0.9696 −0.2397 0.06 0.03 0.0 0 0 0
36 PQ 0.9792 −0.2349 0.0 0.0 0.0 0 – 0
37 PQ 0.9894 −0.2316 0.0 0.0 0.0 0 – 0
38 PQ 1.0147 −0.2209 0.14 0.07 0.0 0 0 0
39 PQ 0.9894 −0.2316 0.0 0.0 0.0 0 – 0
40 PQ 0.9749 −0.2355 0.0 0.0 0.0 0 – 0
41 PQ 0.9894 −0.2560 0.063 0.03 0.0 0 0 0
42 PQ 0.9539 −0.2844 0.071 0.044 0.0 0 0 0
D-2
BUS DATA: (Sb = 100 MVA)
Number Bus Type |V| [pu] θ [rad] Pl [pu] Ql [pu] Pg [pu] Qg [pu] εv, ηv B [pu]43 PQ 1.0067 −0.2018 0.02 0.01 0.0 0 0 0
44 PQ 1.0183 −0.2058 0.12 0.018 0.0 0 0 0
45 PQ 1.0366 −0.1612 0.0 0.0 0.0 0 – 0
46 PQ 1.0604 −0.1935 0.0 0.0 0.0 0 – 0
47 PQ 1.0343 −0.2174 0.297 0.116 0.0 0 0 0
48 PQ 1.0286 −0.2191 0.0 0.0 0.0 0 – 0
49 PQ 1.0370 −0.2250 0.18 0.085 0.0 0 0 0
50 PQ 1.0239 −0.2336 0.21 0.105 0.0 0 0 0
51 PQ 1.0524 −0.2188 0.18 0.053 0.0 0 0 0
52 PQ 0.9805 −0.2009 0.049 0.022 0.0 0 0 0
53 PQ 0.9711 −0.2141 0.2 0.1 0.0 0 0 0.063
54 PQ 0.9964 −0.2048 0.041 0.014 0.0 0 0 0
55 PQ 1.0308 −0.1891 0.068 0.034 0.0 0 0 0
56 PQ 0.9503 −0.2965 0.076 0.022 0.0 0 0 0
57 PQ 0.9321 −0.3122 0.067 0.02 0.0 0 0 0
D.2 Generator’s Reactive Limits
Reactive Limits:
Bus Qmax [pu] Qmin [pu]1 +∞ −∞2 0.5 −0.173 0.6 −0.16 0.25 −0.088 2 −1.49 0.09 −0.0312 1.55 −0.5
D-3
D.3 Line Data
LINE DATA:
Bus i Bus j Rl [pu] Xl [pu] Bl [pu] Bus i Bus j Rl [pu] Xl [pu] Bl [pu]1 2 0.0083 0.028 0.129 27 28 0.0618 0.0954 0
2 3 0.0298 0.085 0.0818 28 29 0.0418 0.0587 0
3 4 0.0112 0.0366 0.038 25 30 0.135 0.202 0
4 5 0.0625 0.132 0.0258 30 31 0.326 0.497 0
4 6 0.043 0.148 0.0348 31 32 0.507 0.755 0
6 7 0.02 0.102 0.0276 32 33 0.0392 0.036 0
6 8 0.0339 0.173 0.047 34 35 0.052 0.078 0.0032
8 9 0.0099 0.0505 0.0548 35 36 0.043 0.0537 0.0016
9 10 0.0369 0.1679 0.044 36 37 0.029 0.0366 0
9 11 0.0258 0.0848 0.0218 37 38 0.0651 0.1009 0.002
9 12 0.0648 0.295 0.0772 37 39 0.0239 0.0379 0
9 13 0.0481 0.158 0.0406 36 40 0.03 0.0466 0
13 14 0.0132 0.0434 0.011 22 38 0.0192 0.0295 0
13 15 0.0269 0.0869 0.023 41 42 0.207 0.352 0
1 15 0.0178 0.091 0.0988 41 43 0 0.412 0
1 16 0.0454 0.206 0.0546 38 44 0.0289 0.0585 0.002
1 17 0.0238 0.108 0.0286 46 47 0.023 0.068 0.0032
3 15 0.0162 0.053 0.0544 47 48 0.0182 0.0233 0
5 6 0.0302 0.0641 0.0124 48 49 0.0834 0.129 0.0048
7 8 0.0139 0.0712 0.0194 49 50 0.0801 0.128 0
10 12 0.0277 0.1262 0.0328 50 51 0.1386 0.22 0
11 13 0.0223 0.0732 0.0188 29 52 0.1442 0.187 0
12 13 0.0178 0.058 0.0604 52 53 0.0762 0.0984 0
12 16 0.018 0.0813 0.0216 53 54 0.1878 0.232 0
12 17 0.0397 0.179 0.0476 54 55 0.1732 0.2265 0
14 15 0.0171 0.0547 0.0148 44 45 0.0624 0.1242 0.004
18 19 0.461 0.685 0 56 41 0.553 0.549 0
19 20 0.283 0.434 0 56 42 0.2125 0.354 0
21 22 0.0736 0.117 0 57 56 0.174 0.26 0
22 23 0.0099 0.0152 0 38 49 0.115 0.177 0.003
23 24 0.166 0.256 0.0084 38 48 0.0312 0.0482 0
26 27 0.165 0.254 0
D-4
D.4 Transformer Data
TRANSFORMER DATA:
Bus i Bus j Type: Rt [pu] Xt [pu] m
4 18 Fixed Tap 0 0.555 0.97
4 18 Fixed Tap 0 0.43 0.978
21 20 Fixed Tap 0 0.7767 1.043
24 25 Fixed Tap 0 1.182 1
24 25 Fixed Tap 0 1.23 1
24 26 Fixed Tap 0 0.0473 1.043
7 29 Fixed Tap 0 0.0648 0.967
34 32 Fixed Tap 0 0.953 0.975
11 41 Fixed Tap 0 0.749 0.955
15 45 Fixed Tap 0 0.1042 0.955
14 46 Fixed Tap 0 0.0735 0.9
10 51 Fixed Tap 0 0.0712 0.93
13 49 Fixed Tap 0 0.191 0.895
11 43 Fixed Tap 0 0.153 0.958
40 56 Fixed Tap 0 1.195 0.958
39 57 Fixed Tap 0 1.355 0.98
9 55 Fixed Tap 0 0.1205 0.94
D-5
D-6