optimal location of facts to optimize power system security · in academical and companionship...
TRANSCRIPT
Optimal Location of FACTS to Optimize Power SystemSecurity
Madalena Coelho de Oliveira Ferreira da Trindade
Thesis to obtain the Master of Science Degree in
Electrical and Computer Engineering
Examination Committee
Chairperson: Prof. Dr. Maria Eduarda de Sampaio Pinto de Almeida PedroSupervisor: Prof. Dr. José Manuel Dias Ferreira de Jesus
Member of the Committee: Prof. Dr. Pedro Alexandre Flores Correia
July 2013
ii
To my parents.
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iv
Acknowledgments
I will start by acknowledging my supervisor Professor Jose Ferreira de Jesus for always being avail-
able to discuss the thesis with me and for his wise advice.
Also Professor Mangalore Pai who is an author of the main references that I follow along the thesis
and Professor Anil Kulkarni helped me to decode some necessary data for the final results. I really
appreciate their availability. Marco Seabra is another student currently working on his own thesis having
as a basis the same MATLABTM
program. He was always available to help me with any issues that I
came across with.
An affectionate word is here written to my grandmother who was the person I lived with during my
college years. Also my aunts and uncles were important for me while being away from home. Still
thanking my family, a word is here written for my brother and sister for meaning what they mean to me.
A sincere and respectful thanks is destined to my parents for giving me all their unconditional support
when I needed it, for all their love and for never letting me down. Their advice and affection along my
academic journey were greatly responsible for the opportunity of writing these words right now in a
master’s thesis. I am forever grateful.
In academical and companionship terms, I really appreciate the help from Joao Pedro Alvito for
always being my group partner, Ines do O for the tireless days of studying and Lıgia Fernandes for every
single year of my academic path. I have always felt very fortunate to have you as friends.
Finally, to Pedro Lima a huge and fond thanks for helping me whenever I needed it academically
and especially for always being by my side emotionally. You have never let me down in tough times for
always being patient and loving.
IST has the resources for students to feel at home and unfortunately it is still one of the few universi-
ties with these characteristics in Portugal. Definitely being part of the basketball team during my student
years helped me a lot with keeping my mind away from hard work.
To all fellow engineering students and mainly to the ones that will graduate in my home university,
remember that ”pain is temporary, pride is forever”.
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Resumo
A estabilidade dos sistemas de energia electrica (SEE) e um dos topicos mais importantes ao nıvel
de transmissao. Os SEE estao sujeitos a varios tipos de perturbacoes que induzem oscilacoes e even-
tualmente instabilidade. Um metodo usual para detectar instabilidades e analisar os valores proprios
da matriz que contem todos os modelos dinamicos dos componentes do sistema. Lidando com pe-
quenas oscilacoes, e possıvel linearizar os modelos dinamicos do sistema em torno de um ponto de
equilıbrio. Assim, com uma abordagem baseada em valores proprios, e possıvel afirmar se o sistema
e estavel ou nao. O objectivo da tese e lidar com estabilidade de pequenas perturbacoes para melho-
rar a seguranca de operacao das redes do sistema. Alem disto, os sistemas de transmissao flexıvel
em corrente alternada (FACTS) serao introduzidos para compreender o seu impacto na estabilidade.
Tambem as consequencias de diferentes tipos de carga serao tidos em conta assim como outro tipo de
perturbacoes induzidas diferente do aumento incremental da potencia activa de uma das cargas.
Os FACTS serao inicialmente implementados no modelo de equacoes diferenciais e algebricas para
se compreender como eles afectam o comportamento de todo o sistema. Esta e a analise dinamica.
Posteriormente, de modo a ter-se um modelo mais completo e realista, os FACTS serao introduzidos
no power flow que precede a analise dinamica. Ambos os tipos de FACTS em serie e paralelo serao
abordados e explicados ao longo do texto assim como a sua localizacao. No final, varios sistemas do
IEEE foram testados e as conclusoes foram que a bifurcacao de Hopf pode ser modificada introduzindo
os modelos dinamicos dos FACTS no sistema e a singularidade do Jacobiano so se modifica se os
FACTS forem introduzidos logo no power flow.
Os resultados do modelo dinamico confirmam que os FACTS melhoram a estabilidade do sistema
atrasando a bifurcacao de Hopf quando uma perturbacao ocorre. Verificou-se tambem que o TCSC e
bastante mais eficiente que o SVC nao so por ser independente da localizacao na linha mas tambem
porque amortece as oscilacoes de modo mais eficiente. De facto, para combater o pouco amorteci-
mento que o SVC fornece, um controlador auxiliar foi adicionado ao SVC, o que permitiu uma melhor
actuacao em relacao a estabilidade. Tambem se confirmou que a compensacao em paralelo tem mel-
hores resultados quando colocada a meio de uma linha em vez de num barramento ja que as perdas
de tensao se reduzem ao longo da linha. Em contraste, a compensacao em serie foi colocada junto ao
barramento problematico numa localizacao previamente determinada.
As cargas foram modeladas como dependentes da tensao, o que se revelou ter um grande impacto
na localizacao da bifurcacao de Hopf. Para alem disto, se uma perturbacao mantem a relacao entre
potencias activa e reactiva da carga que esta a ser aumentada (tanφ de carga constante), o sistema
fica instavel mais rapidamente.
O barramento que contem a perturbacao induzida foi escolhido com base numa analise estatica
(que tem como base o power flow) para avaliar o pior cenario de instabilidade possıvel. Esta analise foi
baseada em valores proprios e factores de participacao. Apos saber o barramento onde se incrementa
a carga, as linhas adjacentes foram avaliadas em termos da relacao entre potencia activa e reactiva. A
linha menos explorada de acordo com esta relacao foi aquela na qual se introduziram os FACTS.
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Palavras-chave: FACTS, bifurcacao de Hopf, valores proprios, estabilidade de pequenas
oscilacoes, power flow
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Abstract
Power System Stability is one of the most important topics at transmission level. Systems are sub-
jected to various types of disturbance that will induce oscillations and eventually instability. A typical way
to detect instabilities is to analyze the eigenvalues of the system matrix containing all dynamic models
of the system components. By dealing with only small oscillations, it is possible to linearize the dynamic
models of the system around an equilibrium point. Then, with an eigenvalue approach it is possible to
determine whether the system is stable or not. The purpose of the thesis is to deal with small signal
stability for the enhancement of system security. Moreover, FACTS devices (flexible AC transmission
systems) will be introduced in the systems in order to understand their impact on stability. Also the
consequences of different types of loads will be taken into account as well as other type of induced
instabilities other than a progressive increase of active power load at a chosen bus.
FACTS will be at first implemented on the differential-algebraic equations model to realize how their
behavior affects the whole system. This is the dynamic analysis. Later, in order to have a more complete
and realistic model, FACTS are introduced as well in the load-flow analysis that precedes the dynamic
one. Both types of FACTS shunt and series will be mentioned and explained along the text and also
the location of these devices will be studied. In the end, several IEEE bus systems were tested and the
conclusions taken were that the Hopf bifurcation can be modified introducing FACTS dynamic models in
the system and the Jacobian singularity can only be modified if FACTS are introduced in the load-flow.
The dynamic model results confirmed that FACTS improve the system stability delaying the Hopf
bifurcation when a disturbance occurs. It was also noticeable that the thyristor-controlled series capacitor
is much more efficient than the static var compensator, not only because it is independent of the line
location but it also damps oscillations in a more effective way. In fact, in order to counter the small
damping provided by the static var compensator, it was equipped later with an auxiliary controller that
allowed a better performance in terms of stability. It was also confirmed that the shunt compensation
had better results when placed in the middle of a line instead of at a bus since voltage losses are further
reduced along the line. In contrast to this, the series compensation was placed near the problematic bus
on a location previously determined.
The loads were modeled as voltage-dependent which revealed an impact on the location of the Hopf
bifurcation. Furthermore, if the disturbance maintains the relation between active and reactive power of
the increasing load, the system becomes unstable more rapidly.
The bus containing the induced disturbance was chosen based on a static analysis (load-flow basis)
in order to evaluate the worst case scenario for the system instability. This analysis was based on the
system eigenvalues and participation factors. After knowing the bus on which the load is to be increased,
the adjacent lines to that bus were evaluated in terms of the rate of active and reactive power. The less
exploited line according to this rate was the one to introduce a FACTS device.
Keywords: FACTS, Hopf bifurcation, eigenvalues, small signal stability, load-flow
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Small Signal Stability 9
2.1 Small Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Model Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Synchronous Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1.1 GENROE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1.2 GENRED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1.3 GENSAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Generator Exciters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2.1 IEEET1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2.2 ST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Generator Governors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3.1 TGOV1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3.2 HYGOV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3.3 GAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
xi
2.3.1 Type of Loads Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Perturbation maintaining tanφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 FACTS 23
3.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Static Var Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Thyristor Controlled Series Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Inclusion of FACTS in Small Signal Stability 41
4.1 Steady-State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Power Flow with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Power Flow with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Transient Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Inclusion of the SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2.1 Model of the SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2.2 Model inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2.3 SVC integration in DAE model . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.3 Inclusion of the TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3.1 Model of the TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3.2 Model inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.3.3 TCSC integration in DAE model . . . . . . . . . . . . . . . . . . . . . . . 58
5 Model Validation 61
5.1 IEEE 9 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 9 bus for different types of loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.2 9 bus with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.3 9 bus with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.4 9 bus with both SVC and TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 IEEE 5 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.1 5 bus without FACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.2 5 bus with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.3 5 bus with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Results and Further Assumptions 73
6.1 IEEE 9 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.1 9 bus with SVC - located in the middle of a line vs at a bus . . . . . . . . . . . . . 73
6.1.2 9 bus with SVC equipped with auxiliary controller . . . . . . . . . . . . . . . . . . . 74
xii
6.1.3 9 bus with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.4 Influence of the type of perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Influence of FACTS dynamic parameters on stability . . . . . . . . . . . . . . . . . . . . . 76
6.2.1 SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2.2 SVC with auxiliary controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.3 TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Determination of Locations for FACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3.1 Results from Locations for FACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4 IEEE 14 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.5 IEEE 30 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.5.1 30 bus with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5.2 30 bus with SVCac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5.3 30 bus with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7 Conclusions 85
7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A Dynamic models 89
A.1 GENROE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.2 GENSAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.3 IEEET1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.4 TGOV1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A.5 HYGOV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.6 GAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B Modifications in algorithms from [9] 94
C Results 96
C.1 IEEE 9 bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.2 IEEE 9 bus data with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
C.3 IEEE 9 bus data with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C.4 IEEE 5 bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.5 IEEE 5 bus data with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.6 IEEE 5 bus data with TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.7 Program interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
D Data Files 107
D.1 SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.2 SVC with auxiliary controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.3 TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
xiii
Bibliography 116
xiv
List of Tables
3.1 Power electronics at transmission level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 Eigenvalues of case PL05 = 1.5pu,QL05 = 0.5pu with constant power load (type 0) . . . . 62
5.2 Eigenvalues of case PL05 = 1.5pu,QL05 = 0.5pu with constant current load (type 1) . . . . 62
5.3 Eigenvalues of case PL05 = 1.5pu,QL05 = 0.5pu with constant impedance load (type 2) . 63
5.4 Eigenvalues of case PL05 = 4.5pu,QL05 = 0.5pu with constant power load (type 0) . . . . 63
5.5 Eigenvalues of case PL05 = 4.5pu,QL05 = 0.5pu with constant current load (type 1) . . . . 64
5.6 Eigenvalues of case PL05 = 4.5pu,QL05 = 0.5pu with constant impedance load (type 2) . 64
5.7 Eigenvalues of nominal case P6 = 1.25pu, V6 = 0.996pu without SVC . . . . . . . . . . . . 65
5.8 Eigenvalues of nominal case P6 = 1.25pu, V6 = 0.996pu with SVC . . . . . . . . . . . . . . 66
5.9 Eigenvalues at Hopf bifurcation case P6 = 4.69pu, V6 = 0.8499pu without SVC . . . . . . . 66
5.10 Eigenvalues at Hopf bifurcation P6 = 4.77pu, V6 = 0.8418pu with SVC . . . . . . . . . . . . 67
5.11 Eigenvalues of nominal case P6 = 1.25pu,Ki = 1 with TCSC . . . . . . . . . . . . . . . . 69
5.12 Eigenvalues at Hopf bifurcation case P6 = 5.03pu,Ki = 1.25 with TCSC . . . . . . . . . . 69
5.13 Eigenvalues at Hopf bifurcation case P6 = 5.07pu,Ki = 1.25 with SVC and TCSC . . . . . 70
5.14 IEEE 5 bus power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.15 IEEE 5 bus power flow with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.16 IEEE 5 bus power flow with SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.1 SVC in the middle of a line vs SVC at a bus . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Eigenvalues and their damping for P6 = 1.25pu . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3 Eigenvalues and their damping for P6 = 1.25pu . . . . . . . . . . . . . . . . . . . . . . . . 75
C.1 Branches data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C.2 Bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C.3 Generator data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C.4 Exciter data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C.5 Branches data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
C.6 Bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.7 Generator data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.8 Exciter data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.9 SVC data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
xv
C.10 Bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C.11 SVC auxiliary controller data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C.12 Branches data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.13 Bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.14 TCSC data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.15 TCSC data with a different Ki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.16 Branches data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.17 Bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.18 SVC data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.19 Branches data of modified network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.20 Bus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.21 TCSC data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
xvi
List of Figures
1.1 Hopf Bifurcation - reprinted from [66] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Static Exciter (ST) - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Voltage as a function of active power load (P-V curve) - reprinted from [55] . . . . . . . . 21
3.1 Investment on FACTS - reprinted from [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Static var compensator - courtesy of [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Uncompensated line - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Uncompensated line vector diagram - reprinted from [18] . . . . . . . . . . . . . . . . . . 26
3.5 Voltage vs. transmission distance - reprinted from [18] . . . . . . . . . . . . . . . . . . . . 27
3.6 Shunt compensation - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Shunt compensation vector diagram - reprinted from [18] . . . . . . . . . . . . . . . . . . 27
3.8 Shunt compensation transmission line - reprinted from [18] . . . . . . . . . . . . . . . . . 28
3.9 Impact of shunt compensation in stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.10 Characteristics of FC and TCR - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . 28
3.11 Characteristic of the FC-TCR - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . 28
3.12 Characteristics of TSC and TCR - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . 29
3.13 Characteristic of the TSC-TCR - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . 29
3.14 Required regulation in a transmission system - reprinted from [18] . . . . . . . . . . . . . 29
3.15 Ability of FACTS for regulation of the voltage - reprinted from [18] . . . . . . . . . . . . . . 29
3.16 System Thevenin equivalent - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . 29
3.17 Droop control - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.18 Line diagram of a TCR - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.19 TCR waveforms - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.20 Schematic of the FC-TCR - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . 32
3.21 Line diagram of the TSC - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.22 Voltages of the TSC - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.23 Waveforms for TSC - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.24 nTSC-TCR characteristic - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.25 Thyristor controlled series capacitor - courtesy of [2] . . . . . . . . . . . . . . . . . . . . . 34
3.26 Series compensation - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
xvii
3.27 Series compensation vector diagram - reprinted from [18] . . . . . . . . . . . . . . . . . . 35
3.28 TCSC line diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.29 Characteristic of TCSC reactance vs. firing angle α - reprinted from [18] . . . . . . . . . . 37
3.30 TCSC voltage reversal line diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.31 TCSC voltage Vc shifted up - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . 38
3.32 TCSC waveforms - reprinted from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Augmented Jacobian - reprinted from [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 SVC power flow model - reprinted from [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Shunt Variable Susceptance Model - reprinted from [9] . . . . . . . . . . . . . . . . . . . . 43
4.4 Series Variable Impedance Model - reprinted from [9] . . . . . . . . . . . . . . . . . . . . 44
4.5 9 bus network - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Operating ranges with droop control - reprinted from [53] . . . . . . . . . . . . . . . . . . . 47
4.7 SVC block diagram - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.8 SVC block diagram with auxiliary controller - reprinted from [53] . . . . . . . . . . . . . . . 49
4.9 TCSC V-I characteristics - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.10 TCSC line diagram - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.11 TCSC block diagram - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 Eigenvalues trajectory of the IEEE 9 bus network with SVC . . . . . . . . . . . . . . . . . 67
6.1 π-model of a transmission line - reprinted from [56] . . . . . . . . . . . . . . . . . . . . . . 78
6.2 SVC best location eigenvalues, nominal case . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3 SVC best location eigenvalues plotted, nominal case . . . . . . . . . . . . . . . . . . . . . 80
6.4 TCSC best location eigenvalues, nominal case . . . . . . . . . . . . . . . . . . . . . . . . 81
6.5 TCSC best location eigenvalues plotted, nominal case . . . . . . . . . . . . . . . . . . . . 81
6.6 P-V curve for 14 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.7 P-V curve for 30 bus network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.1 GENROE - reprinted from [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.2 GENSAL - reprinted from [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.3 IEEET1 - reprinted from [69] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.4 TGOV1 - reprinted from [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A.5 HYGOV - reprinted from [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.6 GAST - reprinted from [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
C.1 9 bus network - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.2 9 bus network with SVC - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . . 98
C.3 9 bus network with TCSC - reprinted from [53] . . . . . . . . . . . . . . . . . . . . . . . . 100
C.4 5 bus network - reprinted from [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.5 5 bus network with SVC - reprinted from [9] . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xviii
C.6 5 bus network with TCSC - reprinted from [9] . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.7 Type of Load menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.8 9 bus network eigenvalues, static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.9 9 bus network bus participation factors, static analysis . . . . . . . . . . . . . . . . . . . . 105
C.10 SelectingP
Qratio in print/plot menu, static analysis . . . . . . . . . . . . . . . . . . . . . . 106
C.11P
Qratio, static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
D.1 Data for SVC in .raw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.2 Data for SVC in .dyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.3 Data for SVC with auxiliary controller in .raw . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.4 Data for SVC with auxiliary controller in .dyr . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D.5 Data for TCSC in .raw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
D.6 Data for TCSC in .dyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xix
xx
Nomenclature
α Firing angle
α2 Firing angle for the second thyristor
αs Extinction angle
αik Angle of the ikth entrance of the admitance matrix
cosφ Power factor
δ Difference between angles of voltages at the sending end and at the receiving end
δi Rotor angle
λ Complex Eigenvalue
ω Angular velocity
ωλ Eigenvalue imaginary part
ωi Rotor angular speed
Ψ1d Amortisseur flux linkage in d-axis
Ψ2d Amortisseur flux linkage in q-axis
σ Conduction angle
σλ Eigenvalue real part
θ Voltages angle
ϕ Angle between voltage and current
ξ Eigenvalue damping
Asys System matrix
B Susceptance
BC Susceptance of capacitor
Bσ Susceptance of transformer
xxi
Bmax Maximum susceptance
Bmin Minimum susceptance
BrefSV CReference susceptance of SVC
BSV C Susceptance of SVC
BTCR Susceptance of TCR
D Speed Damping
E System’s voltage
E′
d Electromotive force due to flux linkage in d-axis
E′
q Electromotive force due to flux linkage in q-axis
Efd Steady-state induced electromotive force
f Frequency of oscillation
G Conductance
H Rotor inertia
Hm Tranfer function
I Current
i Grid current
ic Current through capacitor
Id d-axis stator current component
iL Current through reactor
Iq q-axis stator current component
Icap Capacitive current
Iind Inductive current
ISV C Current of SVC
JAE Algebraic-equations Jacobian
JLF Load-flow Jacobian
KA Amplifier gain
KB Auxiliary controller gain
KE Exciter gain
xxii
KF Feedback gain
Ki Integral gain
KP Proportional gain
KR Regulator gain
M Quocient between two times the inertia and the rotor angular speed
m Number of generators
n Number of buses
npi Load indice of active power
nqi Load indice of reactive power
P Active power
PL Active power of a load
Ploss Power losses
Q Reactive power
QL Reactive power of a load
Qlim Limit value of reactive power
QSV C Reactive power injected by the SVC
R Resistance
Rf Field resistance
Rs Armature resistance
RnF Power system controller state variables
S Apparent power
SE Saturation function
T Voltage period
TA Amplifier time constant
Tb Transport delay
TE Exciter time constant
TF Feedback time constant
TM Mechanical torque
xxiii
Tm Tranfer function’s time constant
TR Regulator time constant
T′′
d0 Initial condition of the open-circuit subtransient time constant in d-axis
T′
d0 Initial condition of the open-circuit transient time constant in d-axis
T′′
q0 Initial condition of the open-circuit subtransient time constant in q-axis
T′
q0 Initial condition of the open-circuit transient time constant in q-axis
u Inputs
V Voltage
vc Voltage of capacitor
Vg Voltage of generators
Vl Voltage of loads
vL Voltage of reactor
Vm Voltage at the middle of a line
VR Voltage regulator
Vr Voltage at the receiving end
Vs Voltage at the sending end
Veq Equilibrium voltage
Vref Reference voltage
vSW Voltage of switch
VTCSC Voltage of TCSC
X Reactance
x State variables
XC Reactance of capacitor
Xd Synchronous reactance in d-axis
X′′
d Subtransient reactance in d-axis
X′
d Transient reactance in d-axis
XL Reactance of a line
XL Reactance of reactor
xxiv
Xq Synchronous reactance in q-axis
X′′
q Subtransient reactance in q-axis
X′
q Transient reactance in q-axis
XFC Reactance of fixed capacitor
Xls Leakage reactance of the rotor windings
XnAC AC network state variables
XSV C Reactance of SVC
XTCSC Reactance of TCSC
Y Admitance
y Injected currents and bus voltages
ya Algebraic variables
yb Load-flow variables
Z Impedance
Zeq Equilibrium impedance
xxv
xxvi
Acronyms and Abbreviations
AC Alternated Current
DAE Differential Algebraic Equations
DC Direct Current
FACTS Flexible AC Transmission Systems
FC Fixed Capacitor
GAST Gas-Turbine Governor
GENRED Round Rotor Synchronous Generator with no
saturation
GENROE Round Rotor Synchronous Generator with Ex-
ponential saturation
GENSAL Salient pole Synchronous Generator
HVDC High Voltage DC
HYGOV Hydro-Turbine Governor
IEEET1 IEEE Type I exciter
IEEE Institute of Electrical and Electronics Engineers
LHP Left Half Plane
PI Proportional Integral
RHP Right Half Plane
SSR Subsyncronous Resonance
ST Static Exciter
SVCac Static Var Compensator with auxiliary controller
SVC Static Var Compensator
TCR Thyristor-Controlled Reactor
TCSC Thyristor-Controlled Series Capacitor
TGOV1 Steam-Turbine Governor
TSC Thyristor-Switched Capacitor
xxvii
xxviii
Chapter 1
Introduction
1.1 Motivation
Initially, electricity was used in a direct current (DC) form. The development of the transformer revo-
lutionized the electric world and as a consequence since this device needed alternated current energy
became transmitted as alternated current (AC). This was the starting point to transmit power at long dis-
tances with lower losses. Nevertheless, DC benefits are still exploited in transmission power systems.
DC main advantages over AC are described:
• each conductor can support more power which results in minor costs in overhead lines;
• less losses;
• there is the possibility of ground return;
• skin effect1 does not exist, dielectric losses are more reduced and for underground cables since
there is no reactive power production the cable length is not a constraint;
• the reactance is also a non-constraint factor meaning that compensation is not necessary;
• possible asynchronous connection between networks (there is no instability);
• active power directly controllable from both sending and receiving ends;
• the converters do not contribute to the short-circuit current.
On the other hand, DC faces some drawbacks when compared to AC:
• converters are needed which cause many harmonics that are eliminated by expensive filters;
• the converters have a very limited overloading capability requiring extremely fast control and pro-
tection systems;
• circuit breakers are the main drawback of DC since current does not pass through zero in order to
cut it.1Skin effect results from the repulsion between electromagnetic current lines creating the trend of flowing at the conductor
surface. It is proportional to the frequency and to both electrical and magnetic properties of the conductor. It is then responsiblefor the apparent resistance increase due to the decrease of the effective area of the conductor.
1
With these information it is possible to present some examples of DC applications used:
• long distance transmissions in overhead lines (typically over 600km);
• in transmission underground cables with lengths over 30km;
• asynchronous interconnections;
• networks interconnection without increasing the short-circuit power;
• distribution in urban towers of great extension where underground cables prevail.
Another interesting comparison besides DC vs. AC is low voltage against high voltage2. The higher
the voltage the lower the losses. This is due to the fact that losses are proportional to the square of the
current across the transmission line and so one is interested in lowering substantially the current, equa-
tion (1.1), since the line resistance is something that cannot be modified, it is a physical characteristic.
Ploss = RI2 (1.1)
Hence, in order to keep the same power flowing as before lowering the current, the voltage has to
increase (equation (1.2)). This leads to high voltage transmission lines.
P = V I (1.2)
This is why the transformer is such a revolutionary device, it allows the electrical transmissions to
be performed in high voltages and then it lowers the voltage for domestic consumption3. Concluding, it
would be more efficient to have High Voltage DC (HVDC) transmission lines. Yet most transmission lines
were conceived AC. Here is where devices called Flexible AC Transmission Systems (FACTS) emerge.
Despite being such expensive devices, as explained further on, it would become much more expensive
to change AC lines to DC.
This background is important to the main subject related to this thesis: stability of power systems.
Power System Stability is an important topic that is dealt with in every transmission system. This is due
to the fact that the systems are always being subjected to various types of oscillations. If these are not
well damped they can lead to instability. Apart from that they can even induce stress in the mechanical
shaft. Therefore, it is really important to know how to deal with oscillations and to reduce them.
The oscillations considered in the thesis are related to small signal stability since only small distur-
bances are taken into account. One way to analyze them can be done by defining the dynamic models
of the system (including generators, machines, FACTS, loads, etc), linearize them around an equilib-
rium point and with the aid of the resultant matrix compute its eigenvalues which determine whether the
power system is stable or not.
Furthermore, the FACTS technology is becoming more and more important these days. This hap-
pens because one cannot afford to have considerable losses in a transmission line and not use it up to
2Considered above 35KV.3Domestic consumption uses low voltage because of security issues and having smaller components/devices.
2
its thermal limit. FACTS are devices that improve the amount of active power transmitted by compen-
sating reactive power using reactive components such as capacitors and inductors. These components
will modify the reactance of the line and so less reactive power will be transmitted. As a consequence,
the line will be able to transmit more active power.
Besides increasing the active power flow, FACTS also improve voltage stability and power oscillation
damping. This thesis should be able to clarify and demonstrate this with numerical results.
Finally, a reasonable location of the previous devices will also be taken into account since it will affect
the stability of the system and the larger the stability margin the safer the system is.
This thesis comes after previous work in the main topic ”Small Signal Stability” by adding FACTS into
the system and also taking into account the type of load (constant power, constant current and constant
impedance).
1.2 State-of-the-art
FACTS are alternating current transmission
systems incorporating power-electronics
based and other static controllers to enhance
controllability and power transfer capability.
IEEE 4
FACTS can be defined in the following way ”FACTS (Flexible Alternating Current Transmission Sys-
tems) is a power industry term for technologies that enhance the security, capacity and flexibility of
power transmission networks. FACTS solutions help power companies increase transmission capacity
over existing AC power lines, providing fast voltage regulation, active power control and load flow con-
trol in meshed power systems. The main purpose is to minimize bottlenecks in existing transmission
systems, and improve the availability, reliability, stability and quality of the power supply.” (in [3]).
The related work mentioned in this subsection is directly related to this thesis since it is the continua-
tion of previous investigation research and utilizes the programs and methodologies developed in them.
Mainly three theses can be enunciated. They were written by students of Instituto Superior Tecnico
(IST), Lisbon, Portugal, namely Pedro Araujo, Carlos Anjos and Marco Seabra. Although those theses
are the main ones related to this thesis, they are integrated in a continuous research work developed in
IST for the past few years that complements them.
The first one is based on a MATLABTM
program that computes the power flow calculations in real-
sized networks, [17]. The power flow computes the normal steady-state operation of a system net-
work. In order to do so, since its equations are nonlinear, numerical methods are used to get a solution
within an acceptable tolerance. The most robust one (and the one that is actually implemented) is the
Newton-Raphson method due to its quick convergence5. The information given to the algorithm are the
4Institute of Electrical and Electronics Engineers.5Three to five iterations.
3
generators voltages and real power as well as the specified loads and the lines characteristics (resis-
tance, reactance and susceptance). The information obtained on the other hand are the voltages (both
magnitudes and angles) at each bus, active and reactive power flowing in each line and reactive power
of generators. Power flow is performed since in the real world one can only measure the information
given (enumerated before) in order to know the quantities obtained from it. An arbitrary bus containing
a generator is used as a Slack bus and so its voltage angle is zero, i.e. it is the reference bus. All other
generator buses are called PV (since active power, P , and magnitude voltage, V , are known) and the
remaining ones PQ (if there is a load, both active, PL, and reactive powers, QL, are known, otherwise
they are null). The number of unknowns for N buses and R generators is given by 2(N − 1) − (R − 1).
A very short resume of Newton-Raphson method is enunciated (for further literature, refer to [56] or [17]):
initialize variables with initial unknowns guess;
while algorithm does not converge do
solve power balance equations;
linearize system equations around equilibrium voltages;
solve equations for change in V ;
update | V | and ∠V ;
check for stopping conditions;
if there is no convergence then
go to step 2;
end
endAlgorithm 1: Newton-Raphson Power Flow
Furthermore, many dynamic models of network devices were introduced in [17]. The models previ-
ously implemented from synchronous generators are round rotor and salient pole, IEEE type 1 for the
excitation controller and the governor system TGOV1. Besides these models, new models had to be
introduced since the goal was to represent realistic networks not only in terms of size but also in terms
of existing devices. Thus, the speed regulation system composed by a gas turbine-governor and the
excitation control system Type DC1A were developed.
On the other hand, the author of the second thesis [16] developed a program also in MATLABTM
that studies small signal stability. The first step for this program is the calculation of power flow utilizing
[17]. Then, two distinct ways of determining whether the system is stable or not were taken: a static
analysis and a dynamic analysis. From a more complete point of view, the last is the one that the author
presents results with. In fact, static analysis was followed during many years in the past. It was based on
the PV-curve and stated that when the determinant of the Jacobian changed sign, meaning that it was
singularity-induced, only then, the system became unstable. As time passed by, researchers found that
the system could become unstable much before the change of sign of the Jacobian determinant which
can be represented with a Hopf bifurcation. This is why the small signal stability analysis was necessary
4
to perform and is related to dynamic stability.
In [16] the key factors that lead to instability are identified by computing the eigenvalues of the system
matrix that is described ahead in the thesis. Also for the computation of system matrices related to
dynamic models, the loads were considered of constant power type. Afterwards these matrices were
used to obtain the system matrix that defines the system and its eigenvalues gave information about
stability. This approach was possible since one only deals with small signals and thus the dynamic
models can be linearized around an equilibrium point. The eigenvalues of the system determine the
damping of the oscillations which is crucial for system security.
Finally, the third thesis ([67] in development) improves the program by correcting some aspects
related to the dynamic models on the thesis aforementioned, [16], and even introducing new models,
namely the Hydro-Turbine Governor (HYGOV) and the Gas-Turbine Governor (GAST). Although these
models were not important for this thesis to show results, they are present for the user to have more
choices when testing networks.
The main bibliography used in small signal stability related to the thesis is [66] since the dynamic
analysis follows the methodology presented in the book. These authors had a great impact when it
comes to small signal analysis. Another book that will be followed along the thesis is [53] since it follows
the previous one and introduces the dynamic models of FACTS.
Regarding FACTS location, one of the most used methods is based on the sensitivities of the line
components and on both matrices of modal analysis and Jacobian. The bigger the sensitivities the more
plausible the location since weak places are detected. The main references are [22] and [23]. Another
simpler method is to find the losses in all network lines and to use FACTS in the lines with higher losses
which is explained in [41].
One of the goals for implementing so many dynamic models in a simple program from a user point of
view is to create an alternative for PSS/ETM
, a program developed by SiemensTM
but quite complex. The
new program is intended to be a simple and practical alternative in terms of files to use, documentation,
etc. This allows the user to study networks stability in a simpler way.
1.2.1 Hopf bifurcation
A bifurcation is a qualitative change in the response of a dynamic system due to variations in con-
trol parameters (usually denominated as µ). The Hopf bifurcation6 states that the system is stable for
some controller parameter values until it reaches some point from which the system becomes unstable,
[26]. The eigenvalues are complex and its location determines if the system is (un)stable. This type of
bifurcation is used in several study areas such as small signal analysis.
A P-V curve example for a bus is represented in Figure 1.1. The static analysis gives information
regarding the change of sign of the Jacobian determinant, point B. On the other hand, the dynamic
analysis stated that the system could become unstable before the change of sign of the Jacobian de-
terminant. In fact, point A (Hopf bifurcation) represents when two complex eigenvalues split into real
ones that move in opposite directions in the real axis and this makes the system unstable (since the6Also known as Poincare-Andronov-Hopf bifurcation since these three researchers have contributed to its study.
5
eigenvalues are in the right half plane - RHP). Then, increasing further the active power, the system
reaches B that can be seen as one of the eigenvalues reaching again the left half plane via +∞ and
finally C. This last point will be stable again since both eigenvalues are again in the left half plane (LHP).
Figure 1.1: Hopf Bifurcation - reprinted from [66]
1.3 Goals
The main goals of the thesis are to introduce the dynamic models of FACTS namely the static var
compensator (SVC) and the thyristor-controlled series capacitor (TCSC) in order to determine their
impact on power systems stability. Furthermore, their location will also be studied so that their utilization
is optimized and the transmission lines are used up to their thermal limits therefore being possible to
transmit more active power.
1.4 Outline
In order to understand FACTS inclusion in the small signal analysis, it is necessary to clarify the back-
ground of this decomposition. That is what Chapter 2 deals with, it explains what is already implemented
in the program and how.
Chapter 3 is dedicated to the explanation of FACTS devices before their introduction in the main
program. It will only focus on the SVC and the TCSC and so they will be extensively explained.
The fourth Chapter is the sequence of 2. It is only written in this part of the thesis since an explanation
of FACTS is required to understand the meaning of their inclusion in the analysis (Chapter 3). This
chapter details where exactly on the DAE model (deduced in Chapter 2) FACTS are introduced.
FACTS model validation is placed in Chapter 5. These models were explained in Chapters 2 and 4
and are compared to references [66], [53] and [9].
The approach to the determination of locations for FACTS is explained in Chapter 6. Not only the
concepts but also how to use them on the program are accounted. Further results from small signal
6
analysis are also presented and explained. This chapter details the perturbation maintaining the ratio
between the load active and reactive power constant, FACTS dynamic models and their location in a
network as well as their comparison with each other.
The last chapter, Chapter 7, enunciates what future work could be done starting from this thesis by
explaining the conclusions of the whole work based on the results.
7
8
Chapter 2
Small Signal Stability
2.1 Small Signal Analysis
The approach chosen to evaluate instability is described. When a network is analyzed, firstly power
flow results are obtained. From here, voltages, reactive powers from generators and power flowing
in the transmission lines are known, as explained in Section 1.2. To measure the system stability,
a perturbation is induced. What is commonly done (as in references [66] and [53] for instance) is
increasing the active power of an arbitrary load: the system is evaluated for the nominal case, then an
active power load is increased1 and the system is re-evaluated. This is done until there is an instability
on the system. When the power flow diverges it means that the system cannot tolerate such higher
active power load on the chosen bus. This is part of the static analysis commonly used in the past which
was studied in [16]. The information to retain is that the system was described mathematically by a
system of equations based exclusively on the power flow and that when power flow diverged, the load-
flow Jacobian2 became singular. However, more recent studies such as [66] state that the system can
become unstable much sooner than the power flow divergence. Not only the load-flow Jacobian is part
of the mathematical formulation but also the system dynamic components3 were crucial for the system
stability analysis. The analysis becomes dynamic and so mathematically the network is described by
a system of equations that contains not only the load-flow Jacobian but also the components dynamic
models. Creating a system describing matrix it was possible to compute its eigenvalues and to conclude
if the system is stable (eigenvalues in the LHP) or unstable (eigenvalues in the RHP). This type of
instability is called Hopf bifurcation.
For a complete background please refer to [66]. This section only intends to do a brief review on
small signal analysis. This theme was extensively explained in [16] and thus only the crucial points
for understanding FACTS inclusion will be mentioned here. Also the main concepts of the theme are
referred in Chapter 1.
A dynamic study is required to analyze a power system. In fact, as referred in Chapter 1, the static
1In this thesis increments of 1MW were used.2Introduced in Section 1.2.3Dynamics of electrical machines namely generators associated with governors and exciters are described in this thesis.
9
analysis is less robust than the dynamic one. As aforementioned, the dynamic analysis consists in
evaluating the system matrix eigenvalues as the systems suffer a progressive perturbation, in this case
an increasing active power load at a bus. The system becomes unstable as the eigenvalues cross the
imaginary axis into the RHP direction. The network will be described as a dynamic system in the next
subsections.
2.1.1 Model Linearization
The way to deal with nonlinear control involves linearization techniques. Here the electrical machine
equations are linearized and together with the stator algebraic equations and network equations form a
differential-algebraic equations (DAE) model. The nonlinear models that are dealt with in this thesis are
defined by equations (2.1) (differential equations) and (2.2) (algebraic equations).
x = f(x, y, u) (2.1)
0 = g(x, y) (2.2)
where x are the state variables, y includes Id−q and V from the machines models (type of generator,
exciter, governor, etc) and u are the inputs.
If y is arranged according (2.3), where ya corresponds to the algebraic variables and yb to the load-
flow variables, the system can be linearized around an equilibrium point, (2.4) and (2.5), and give matri-
cially (2.6).
y = [ITd−q θ1 V1 · · · Vm | θ2 · · · θn Vm+1 · · · Vn]T = [yTa | yTb ]T (2.3)
∆x =∂f
∂x∆x+
∂f
∂y∆y +
∂f
∂u∆u (2.4)
0 =∂g
∂x∆x+
∂g
∂y∆y (2.5)
ddt∆x
0
0
=
A B
CD11 D11
D21 JLF
∆x
∆ya
∆yb
+ E[∆u] (2.6)
where JLF is the load-flow Jacobian.
To find the system matrix Asys, both ∆ya and ∆yb are eliminated and so (2.7) leads to (2.8) where
JAE is composed by sub-matrices D11, D12, D21 and JLF .
∆x = Asys∆x (2.7)
Asys = (A−BJ−1AEC) (2.8)
This formulation is now applied to a specific problem widely discussed in this thesis, namely at
10
FACTS inclusion and final results. This will clarify the approach to use in order to implement the theory
discussed. Consider a three machine case all being GENRED with IEEET1 type of exciter (see Section
2.2 to understand these models).
The differential equations are found, see Subsections 2.2.1.2 and 2.2.2.1. They are therefore seven,
meaning that if m is the number of machines, in the end one has 7m state variables x.
The static algebraic equations are
E′di − Vi sin(δi − θi)−RsiIdi +X ′qiIqi = 0 (2.9)
E′qi − Vi cos(δi − θi)−RsiIqi +X ′diIdi = 0 (2.10)
where i = 1, ...,m (generator buses).
Plus, considering n the total number of buses, the network equations are
IdiVi sin(δi − θi) + IqiVi cos(δi − θi) + PLi(Vi)−n∑k=1
ViVkYik cos(θi − θk − αik) = 0 (2.11)
IdiVi cos(δi − θi) + IqiVi sin(δi − θi) +QLi(Vi)−n∑k=1
ViVkYik sin(θi − θk − αik) = 0 (2.12)
for i = 1, ...,m and
PLi(Vi)−n∑k=1
ViVkYik cos(θi − θk − αik) = 0 (2.13)
QLi(Vi)−n∑k=1
ViVkYik sin(θi − θk − αik) = 0 (2.14)
for i = m+ 1, ..., n (non-generator buses).
Linearizing the model differential equations according to (2.4) and (2.5) and putting into a matrix
formulation, (2.15) can be written.
11
∆δi
∆ωi
∆ ˙E′qi
∆ ˙E′di
∆ ˙E′fdi
∆ ˙VRi
∆ ˙RFi
=
0 1 0 0 0 0 0
0 −Di
Mi− Iqi0Mi
− Idi0Mi0 0 0
0 0 − 1T ′
di00 1
T ′di0
0 0
0 0 0 − 1T ′
qi00 0 0
0 0 0 0 fsi(Efdi0) 1TEi
0
0 0 0 0 −KAiKFi
TAiTTi− 1TAi
KAi
TAi
0 0 0 0 KFi
T 2Fi
0 − 1TFi
∆δi
∆ωi
∆E′qi
∆E′di
∆E′fdi
∆VRi
∆RFi
+
0 0
Iqi0(X′di−X′
qi)−E′di0
Mi
Idi0(X′di−X′
qi)−E′qi0
Mi
−Xdi−X′di
T ′di0
0
0Xqi−X′
qi
T ′qi0
0 0
0 0
0 0
∆Idi
∆Iqi
+
0 0
0 0
0 0
0 0
0 0
0 −KAi
TAi
0 0
∆θi
∆Vi
+
0 0
1Mi
0
0 0
0 0
0 0
0 KAi
TAi
0 0
∆TMi
∆Vrefi
, i = 1, ...,m
(2.15)
Denoting,
∆Idi
∆Iqi
= ∆Igi,
∆θi
∆Vi
= ∆Vgi and
∆TMi
∆Vrefi
= ∆ui, (2.15) becomes for the
m-machine system (2.16).
∆x = A1∆x+B1∆Ig +B2∆Vg + E∆u (2.16)
Next are the stator algebraic equations to be linearized and written in a matrix formulation, (2.17).
12
−Vi0 cos(δi0 − θi0) 0 0 1 0 0 0
0 Vi0 sin(δi0 − θi0) 0 1 0 0 0
∆δi
∆ωi
∆E′qi
∆E′di
∆E′fdi
∆VRi
∆RFi
+
−Rsi X ′qi
−X ′di −Rsi
∆Idi
∆Iqi
+
Vi0 cos(δi0 − θi0) − sin(δi0 − θi0)
−Vi0 sin(δi0 − θi0) − cos(δi0 − θi0)
∆θi
∆Vi
, i = 1, ...,m
(2.17)
(2.17) can be written as
0 = C1∆x+D1∆Ig +D2∆Vg. (2.18)
Still for the generators, the algebraic network equations are
0 =
C21
. . .
C2m
∆x1
...
∆xm
+
D31
. . .
C3m
∆Ig1...
∆Igm
+
D41,1 · · · D41,m
......
...
D4m,1 · · · D4m,m
∆Vg1...
∆Vgm
+
D51,m+1 · · · D51,n
......
...
D5m,m+1 · · · D5m,n
∆Vlm+1
...
∆Vln
(2.19)
which can be written in a more compact form
0 = C2∆x+D3∆Ig +D4∆Vg +D5∆Vl , (2.20)
where ∆Vli represents the same as ∆Vgi but for non-generator buses.
Finally, for the non-generator buses the algebraic network equations become
0 =
D6m+1,1 · · · D6m+1,m
......
...
D6n,1 · · · D6n,m
∆Vg1...
∆Vgm
+
D7m+1,m+1 · · · D7m+1,n
......
...
D7n,m+1 · · · D7n,n
∆Vlm+1
...
∆Vln
(2.21)
or
13
0 = D6∆Vg +D7∆Vl. (2.22)
Rewriting (2.16), (2.18), (2.20) and (2.22) together the final DAE model is presented:
∆x = A1∆x+B1∆Ig +B2∆Vg + E∆u
0 = C1∆x+D1∆Ig +D2∆Vg
0 = C2∆x+D3∆Ig +D4∆Vg +D5∆Vl
0 = D6∆Vg +D7∆Vl
(2.23)
The DAE model can however be simplified further since one is not interested in evaluating ∆Ig, only
the voltages (due to voltage stability) and so using (2.18), ∆Ig is substituted in the remaining system
equations.
∆Ig = −D−11 C1∆x−D−1
1 D2∆Vg (2.24)
Let K1 = D4 −D3D−11 D2 and K2 = C2 −D3D
−11 C1. The system becomes then
∆x = (A1 −B1D−11 C1)∆x+ (B2 −B1D
−11 D2)∆Vg + E∆u
0 = K2∆x+K1∆Vg +D5∆Vl
0 = D6∆Vg +D7∆Vl
(2.25)
Defining the voltage vector ∆Vp = [∆yTc ∆yTb ]T = [∆θ1∆V1 · · ·∆Vm | ∆θ2 · · ·∆θn∆Vm+1 · · ·∆Vn]T ,
the system relates to a single system matrix Asys.
∆x
0
0
=
A′ B′1 B′2
C ′1 D′11 D′12
C ′2 D′21 D′22
∆x
∆yc
∆yb
+
E
0
0
[∆u] (2.26)
where the new sub-matrices are related with the reordering of the voltage vector. The system matrix can
then be defined as
Asys = A′ − [B′1B′2][J ′AE ]
C ′1
C ′2
, (2.27)
where J ′AE is the algebraic Jacobian
J ′AE =
D′11 D′12
D′21 D′22
, (2.28)
to give the DAE model
∆x = Asys∆x+ E∆u . (2.29)
Stability is thus evaluated by using Asys eigenvalues placement from both the dynamic models and
14
the Jacobian and therefore this method is more powerful than static analysis which is only based on the
Jacobian.
2.2 Dynamic Models
In this section the used models are illustrated. Some of them were already implemented, see refer-
ences [17] and [16], while others were not.
2.2.1 Synchronous Machines
Although the thesis [17] explains several generator models, only two were really implemented in [16].
They were in fact the only necessary ones that the author used to validate results.
In this thesis no other developed models were needed. However, in [67] new models were developed
and part of them were incorporated in this one. Although they will not be explained here, a brief summary
will be presented so that the reader can visualize them.
2.2.1.1 GENROE
GENROE is a round rotor synchronous generator model with exponential saturation in d-q axis4. It
is displayed the same way as in PSS/ETM [71] because of modeling validation that was used in past
theses.
First of all, it is necessary to get the differential model equations from the block diagram, Figure A.1.
These equations are on Appendix A.1.
Then, they are linearized to be incorporated in the DAE model defined in Section 2.1. Since they
were not used for result validation in this thesis, the reader may know more about the linearization in
[66]. Apart from the differential equations there are also algebraic equations related to the stator.
Vi sin (δi − θi)−X
′qi−X
′′qi
X′qi−Xlsi
E′
di +X
′qi−X
′′qi
X′qi−Xlsi
Ψ2qi +RSiIdi −X′′
qiIqi = 0
Vi cos (δi − θi)− X′′di−Xlsi
X′di−Xlsi
E′
qi +X
′di−X
′′di
X′di−Xlsi
Ψ1di +RSiIqi −X′′
diIdi = 0(2.30)
where Vi is the voltage absolute value at the bus containing the generator and θi is the voltage angle at
the bus containing the generator.
Finally, both systems of equations (A.1) and (2.30) are incorporated in matrices A1i, B1i, B2i, Ei,
C1i, D1i and D2i from DAE model (as illustrated in Subsection 2.1.1).
4Direct–quadrature–zero (dq0) transformation is a mathematical transformation used to simplify the analysis of three-phasecircuits. In case of balanced three-phase circuits, the application of the dq0-transformation reduces the three AC quantities to twoDC quantities.
15
2.2.1.2 GENRED
The GENRED model was actually the most used model. As a consequence, this subsection de-
scribes how its model was incorporated in the DAE model of the system. This type of generator is
a particular case of GENROE but neglecting both stator and network transients as well as saturation.
Therefore, the model becomes much simpler.
dδidt
= ωi − ωS (2.31)
dωidt
=ωS2H
[TMi −Di(ωi − ωS)− (E′
qi −X′
diIdi)Iqi − (E′
di +X′
qiIdi)Idi] (2.32)
T′
d0i
dE′
qi
dt= −E
′
qi − (Xdi −X′
di)Idi + Efdi (2.33)
T′
q0i
dE′
di
dt= −E
′
di + (Xqi −X′
qi)Iqi (2.34)
Also the algebraic equations simplify.
E′
di − Vi sin(δi − θi)−RsiIdi +X′
qiIqi = 0 (2.35)
E′
qi − Vi cos(δi − θi)−RsiIqi +X′
diIdi = 0 (2.36)
2.2.1.3 GENSAL
GENSAL is a salient pole synchronous machine with quadratic saturation in d-axis. Since it was not
used for result validation, only its block diagram is presented to give a general idea to the reader, Figure
A.2.
2.2.2 Generator Exciters
In order to compare results between bibliographic references and the developed program, it became
necessary to implement a new type of exciter and to use one that was already implemented.
The excitation system provides direct current to the synchronous machine field winding. Further-
more, it performs control and protective functions so that the capability limits of the machine are not
exceeded: field voltage and current are controlled. The following subsections focus mainly on the dy-
namic modeling of the exciters.
There are three types of exciters: DC (direct current), AC (alternating current) and ST (static). DC
exciters use DC generators as sources of excitation power and can be either self-excited or separately
16
excited.
The AC excitation systems use AC machines as sources of the main generator excitation power.
Their rectifiers can be either stationary or rotating. No AC exciter is used in this thesis.
Static exciters are called this way since all their components are static or stationary. The power
supply to the rectifiers comes from the main generator through a transformer to step down the voltage
to the needed level.
2.2.2.1 IEEET1
The IEEET1 type of exciter was already implemented. For further information about it the reader can
consult [17], [16] and PSS/ETM bibliography, namely [69] and [71].
IEEET1 is a DC type of exciter that when self-excited, the exciter gain KE is selected so that initially
the regulator voltage output VR is null. The generator voltage goes through a terminal voltage transducer
and is compared with reference voltage. Then a fast response is obtained with lead-lag compensation
and feedback. Finally, the excitation function provides saturation to the control system.
The IEEE type I exciter is related to the system of equations (A.2) deducted from its block diagram,
Figure A.3 found in Appendix A.3 (see Section 2.1 for further information about IEEET1).
2.2.2.2 ST
To test the inclusion of FACTS devices in the program, it became necessary to develop another type
of exciter. This was useful to compare the eigenvalues FACTS results with reference [53]. It is in fact
a much simpler model than IEEET1 since only a PI controller is needed to provide excitation to the
generator. The static exciter developed model is quite simple and is represented below.
Figure 2.1: Static Exciter (ST) - reprinted from [53]
The ST dynamic model is constituted by only one differential-algebraic equation, (2.37), that comes
directly from the block diagram shown in Figure 2.1. Moreover, it does not have any saturation function.
It is integrated in the DAE model from Section 2.1.
TAdEfddt
= −Efd +KA(Vref − V + VS) (2.37)
According to the DAE model derived in Section 2.1, equation (2.37) will only be part of matrices Ai,
B2i and Ei. This can be verified if one linearizes the exciter equation according to equations (2.4) and
17
(2.5). Moreover, the matrices include the type of generator used to result verification, GENRED (in the
example provided). Also for the GENROE the ST was implemented. Since there was no particular need
for getting results with governors, the ST is only implemented for the synchronous machines without any
governor associated.
d∆Efdidt
= − 1
TA∆Efd −
KA
TA∆Vi +
KA
TA∆Vref (2.38)
(2.38) leads to the system matrices
Ai =
0 1 0 0 0
0 −Di
Mi− Iqi0Mi
− Idi0Mi0
0 0 − 1T
′di0
1T
′di0
0 0 0 − 1T
′qi0
0
0 0 0 0 − 1TAi
, B2i =
0 0
0 0
0 0
0 0
0 −KAi
TAi
and Ei =
0 0
1Mi
0
0 0
0 0
0 KAi
TAi
. (2.39)
2.2.3 Generator Governors
Generator governors provide power and frequency control when load demand changes. Due to their
time constants their response is slow which may influence system stability. They were not necessary to
test if the program was producing correct results but it is part of the dynamic models implemented. The
three implemented governor models are illustrated below. The thesis did not focus much on these and
so if the reader would like to know more about them refer to [71] and [16] (TGOV1). Once more, HYGOV
and GAST are presented in [67] and are also part of this thesis despite they were not used to test the
networks considered for the results.
2.2.3.1 TGOV1
TGOV1 is a PSS/ETM steam turbine-governor model representing governor action and the reheater
time constant effect for a steam turbine. As it was mentioned in previous sections, from the block diagram
(Figure A.4) differential equations can be written in order to be incorporated in the DAE model.
2.2.3.2 HYGOV
HYGOV (hydraulic turbine and governor), Figure A.5, represents a straightforward hydro electric
plant governor, with a simple hydraulic representation of the penstock with unrestricted head race and
tail race, and no surge tank.
2.2.3.3 GAST
GAST (gast turbine-governor), Figure A.6, represents the main dynamic characteristics of industrial
18
gas turbines driving generators connected to electric power systems.
2.3 Loads
This section is destined to explain the effect of type of load in the stability of power systems. The
load modeling is in fact voltage-dependent, (2.40) and (2.41).
PLi = PLi0
(ViVi0
)npi
(2.40)
QLi = QLi0
(ViVi0
)nqi
(2.41)
where i represents any bus, i.e. i=1,...,n, PLi0 and QLi0 are the nominal real and reactive powers
respectively, Vi0 is the nominal voltage and npi and nqi are the load indices.
The load indices are related to the type of load considered. There are three different types: constant
power (np = nq = 0), constant current (np = nq = 1) and constant impedance (np = nq = 2). Constant
power type was the only one implemented previously in the MATLABTM developed program from thesis
[16]. The remaining two were considered interesting to evaluate the power system behavior.
2.3.1 Type of Loads Inclusion
The place in the DAE model where the type of load has influence is on (2.20) and (2.22)5, more
specifically in matrices D4 and D7. It is then necessary to derive both expressions (2.40) and (2.41) and
afterwards substitute np and nq by 0, 1 or 2 and the remaining variables by the operational point.
∂PLi∂Vi
=PLi0
Vnpi
i0
npiVnpi−1i (2.42)
∂QLi∂Vi
=QLi0
Vnqi
i0
nqiVnqi−1i (2.43)
For constant power type (0) there is no need for modifications in D4 or D7 as one can see in demon-
stration equations (2.44).
∂PLi∂Vi
= 0
∂QLi∂Vi
= 0 (2.44)
For the remaining types the load will in fact modify matrices D4 and D7 because the derivatives are
no longer zero. Type 1 is in (2.45) while type 2 is in (2.46).
5If there is a load on these buses, they can be either PV or PQ.
19
∂PLi∂Vi
=PLi0Vi0
∂QLi∂Vi
=QLi0Vi0
(2.45)
∂PLi∂Vi
= 2PLi0Vi0
∂QLi∂Vi
= 2QLi0Vi0
(2.46)
At this point, the partial derivatives∂PLi∂Vi
and∂QLi∂Vi
should be added to some cells of matrices D4
and D7. In the matrices the lines refer to either active or reactive power and the columns to the voltage
angle or magnitude. The cells to modify are the ones related to the ith bus, both at active and reactive
powers but only on the voltage magnitude. They are in bold below for the first load bus example (bus
m+1). For this example matrix D4 is not represented since it will not be modified, this is not a PV bus but
a PQ. The fact that only the ith bus has its load modeled is a consideration following [66]. Nevertheless,
in the program it was left but not used the part of the code that allows all loads to be modeled if one is
interested in evaluating these results.
D7 =
∂Pm+1
∂θm+1· · · ∂Pm+1
∂θn
∂Pm+1
∂Vm+1· · · ∂Pm+1
∂Vn...
. . ....
.... . .
...∂Pn∂θm+1
· · · ∂Pn∂θn
∂Pn∂Vm+1
· · · ∂Pn∂Vn
∂Qm+1
∂θm+1· · · ∂Qm+1
∂θn
∂Qm+1
∂Vm+1· · · ∂Qm+1
∂Vn...
. . ....
.... . .
...∂Qn∂θm+1
· · · ∂Qn∂θn
∂Qn∂Vm+1
· · · ∂Qn∂Vn
(2.47)
From the load models it is expected and demonstrated in Chapter 6 that voltage stability will be
modified since Hopf bifurcation location will change. This is due to matrices D4 and D7 influencing
system matrix Asys.
2.4 Perturbation maintaining tanφ
As explained before and considered in the program, the small perturbation induced is a progressive
increase of active power load at an arbitrary bus (containing a load). Hence, the program starts with
a nominal value and in each simulation this value, PLi, is increased (typically 1MW) until the Jacobian
20
becomes singularity-induced. It is in fact the perturbation assumed for the bibliography considered in
[66] and [53].
The ratio between active and reactive power load is called tanφ [55]. Directly from (2.48) it is possible
to notice that tanφ increases when QL increases or PL decreases. Due to the proportional relation
between tanφ and QL, stability is questioned as tanφ increases, see Figure 2.2. This means that the
perturbation assumed is actually improving the system in one way, only when PL increases a lot one
can see the system stability getting worse because of the network response.
tanφ =QLPL
(2.48)
Figure 2.2: Voltage as a function of active power load (P-V curve) - reprinted from [55]
To discard this inherent improvement of the system allied to the perturbation, it is possible to increase
the reactive power load in the same ratio as the increment of the active power. Then, when the PL
increment is performed in the program, one should also increment QL according to (2.49) where tanφ
is constant and calculated according to (2.48) for the nominal case.
QL = PL tanφ (2.49)
This thesis was done according to the PL increment, i.e. what is typically done in these type of
studies. PL and QL increments were performed simultaneously to show that an increase only in PL is
optimistic meaning that the stability margin is larger. In Chapter 6 these perturbations are illustrated.
21
22
Chapter 3
FACTS
3.1 General Description
The expansion of networks and their demands has been a major target field in research work in order
to optimize the cost-utility factor in power systems. Not only the need of feeding loads has taken place
but also some new technologies such as renewable energies force power systems to be efficient. As a
consequence, many devices have been developed for the past few decades in order to achieve reliability
in a more economical way.
Power systems can be subdivided into two different levels namely the transmission level and the
distribution level. At distribution level, custom power devices mainly improve quality in voltage. This is
very important for factories that work with much sensitive and expensive equipment that if not fed by
the grid with quality can lead to the loss of millions of euros. These devices are able to reduce flicker,
perform active filtering, mitigate voltage dips and even interruptions by having other sources of supply
besides the main grid. On the other hand, at transmission level it is known today that the transmission
of power in direct current is much more efficient than in AC (explained in Chapter 1). However, there are
still many more AC lines than DC. As it is much more expensive to modify from AC to DC, engineers
came up with developing devices that maximize this AC power transmission. These devices are called
Flexible AC Transmission Systems or, in short, FACTS. Figure 3.1 gives an idea of the investment costs
where the solid line represents the total investment costs and the dashed line the equipment costs.
Furthermore, the transmission power losses can reach for example 5% and the voltage across the line
6% which results in a great waste of money.
23
Figure 3.1: Investment on FACTS - reprinted from [19]
FACTS were introduced in transmission lines with the main goal of operating them up to their thermal
limits. Without FACTS, the lines still transmit lots of power but with a great percentage of reactive power.
Reactive power is the main issue that does not allow active power flow to increase. Thus, less active
power can flow across the line since a great amount of reactive power is also flowing which disables the
line efficiency since its temperature rises and just a small portion of active power can be transmitted1.
In addition, FACTS can also improve considerably the stability of the system, voltage control and
power oscillation damping. This is why it is so important to study the stability of power systems having
in consideration these devices.
FACTS can be divided into two subgroups: shunt compensation and series compensation. To do so,
reactive elements compose FACTS to compensate for the excess of reactive power that does not allow
the active power to increase. What FACTS do is to modify the line reactance in order to achieve this
goal.
In shunt compensation it is intended that the device changes the amplitude of the voltage but not the
phase. In fact, both the amplitude and phase modify the amount of reactive power in a line, equation
(3.1), however the phase difference of terminal voltages is only used up to a certain limit ( 35o) ensuring
that power flow is well below the static stability limit (P-V curve) so that the system is capable of handling
transients. Voltage regulation is mainly used to change reactive power. In this type of compensation, the
location of the device is crucial: it is much more efficient when it is located in the electrical middle point
of the transmission line as it is explained in Section 3.2.
Qs =Vs(Vs − Vr cos δ)
XL(3.1)
where s stands for sending end, r for receiving end and L for line.
1This can be better understood with an example. Consider a beer. As one knows, it can have more or less foam but the goal isto have less so that more beer is in the glass. If the beer is well poured, it should come with little foam and more liquid so that onetakes more advantage of the glass. In transmission systems, active power is the liquid whilst reactive power is the foam.
24
In series compensation, the line length is apparently shortened since the line reactance is reduced
and therefore active power flow is increased. It improves angular and voltage stability and prevents
systems from falling out of synchronism at disturbances. This type of compensation is more efficient
than shunt since for the same amount of active power less reactive power is needed (see Section 3.4).
Also, not always power electronics are needed as in shunt.
For a FACTS to change its reactance it must have power electronics as well as reactive components.
Power electronics (thyristors in this case) will be controlled so that the equivalent reactance can be
regulated. This will be better explained along this section. What is important to understand is that the
dynamic models will be related to these controllers.
Many FACTS devices have already been developed but not all of them have vindicated because of
their extremely expensive price or inefficiency. Table 3.1 represents the devices split according to their
characteristics at transmission level and main roles. In this thesis only SVC and TCSC are considered
since they are the most popular FACTS in their categories, although STATCOM2 is a promising device
because of its flexibility on being a solution for all type of issues including working as a power custom
device when equipped with energy storage.
Transmission Level CompensationShunt Series
regulation of bus voltage increase of active power flow
Table 3.1: Power electronics at transmission level
The following sections analyze both the SVC and the TCSC. However, the modeling and control
of these devices will not be extensively explained. If the reader would like to know more about these
aspects, refer to [53]. Only a direct word for the controllers will be given in Chapter 4 since those models
are applied directly to the program.
3.2 Static Var Compensator
Figure 3.2: Static var compensator - courtesy of [3]2Static Synchronous Compensator.
25
The SVC is an electrical device that provides fast-acting reactive power on high voltage electric trans-
mission networks. Yet, the compensation is done with only capacitor banks in most cases. Nevertheless
the SVC is much quicker and more reliable than these capacitors. A bank capacitor provides reac-
tive power compensation with power factor correction, decrease of line losses and voltage stability. On
the other hand, an SVC has a near-instantaneous response, has a higher capacity, is faster and more
reliable but much more expensive.
The benefits of equipping a grid with an SVC are the increase of stability limits dictated by the
system voltage and a more controllable voltage profile. This results in a more robust grid with enhanced
capacity, flexibility and predictability. The SVC main control features can be resumed as voltage control,
reactive power control, damping of power oscillations and unbalance control [6]. Typical voltage levels
vary within the range of 33 < kV < 800 and its overall ratings are within the interval 40 < Mvar < 800 [2]
, meaning that these are extremely powerful devices and therefore expensive. It should be noticed that
a transformer is usually placed between the high voltage grid and the SVC so that a lower voltage allows
a reduced size and number of SVC components although larger conductors/thyristors are needed due
to higher currents associated with lower voltages.
The largest companies in the world manufacturing SVCs are ABBTM
from Sweden and SiemensTM
from Germany. They have been producing these devices for countries all over the world. Some examples
follow: Australia, Denmark, USA, Germany, Sweden, Brazil, South Africa, Mexico, Columbia, Indonesia,
Tanzania, Ghana, UK, Libya, Saudi Arabia, New Zealand, etc.
3.2.1 Theoretical Description
The SVC is now described based on its line diagram and its V-I characteristic.
Another drawback of shunt compensation comparing to series (besides that for the same amount of
active power variation less reactive power injection is needed) is that in shunt the location of the device
in a line is crucial regarding its effectiveness. In fact, all shunt compensators should be located at the
exact electrical midpoint of the line. The midpoint is characterized by having an angle of voltage that
is half of the difference between those that are in both ends of the line. It happens that it is impossible
to know exactly this location. It should be in the middle of the transmission line but it is not since the
reactances on the left and right are not the same at every time instant.
Figure 3.3: Uncompensated line - reprinted from
[18]Figure 3.4: Uncompensated line vector diagram -
reprinted from [18]
26
Without compensation, the active power is given by (3.2).
P =VsVrXL
sin δ =V 2
XLsin δ . (3.2)
SVC function is to increase the voltage magnitude in its location to the same value in the sending
and receiving ends so that less power losses are induced (see Figure 3.5 where Po is the characteristic
power). Ideally, a line would behave like a naturally loaded line if it had an infinite number of SVCs.
Figure 3.5: Voltage vs. transmission distance - reprinted from [18]
In (3.3) is the modification in active power flow when equipped with an SVC.
Figure 3.6: Shunt compensation - reprinted from
[18]Figure 3.7: Shunt compensation vector diagram -
reprinted from [18]
As Vm = Vs = Vr, the maximum of active power is increased for the double (considering unitary sin).
P =VsVrXL
2
sinδ
2=
2V 2
XLsin
δ
2. (3.3)
Directly from Figure 3.7 it is intuitive that what happens is represented in Figure 3.8.
27
Figure 3.8: Shunt compensation transmission line - reprinted from [18]
The impact of shunt compensation will reflect mainly in voltage angle stability. Also the maximum
active power that can be transmitted doubles. From equal-area criterion, Figure 3.9 can be drawn.
Figure 3.9: Impact of shunt compensation in stability
There are two types of SVCs: fixed capacitor - thyristor-controlled reactor (FC-TCR) and thyristor
switched capacitor - thyristor-controlled reactor (TSC-TCR). They both have two components because
one of them provides controllability in the capacitive range (FC and TSC) while the other does it in the
inductive range (TCR). This means that FC and TSC are mainly constituted by a capacitor while TCR is
by an inductor (or reactor). Both V-I characteristics are displayed in Figures 3.11 and 3.13.
Figure 3.10: Characteristics of FC and TCR -
reprinted from [18]
Figure 3.11: Characteristic of the FC-TCR -
reprinted from [18]
28
Figure 3.12: Characteristics of TSC and TCR -
reprinted from [18]Figure 3.13: Characteristic of the TSC-TCR -
reprinted from [18]
They look similar although the working mechanisms differ from one another in terms of reactive power
supply. One can see that for negative current (capacitive range) only FC and TSC have a V-I charac-
teristic while for positive current only the TCR has it. In fact, what is needed in a transmission line is
control in both ranges as illustrated in Figure 3.14 in order to maintain the same voltage despite it is
being modified (Figure 3.15). This figure is associated with the system since it can be represented by
its Thevenin equivalent (Figure 3.16).
Figure 3.14: Required regulation in a transmission system - reprinted from [18]
Figure 3.15: Ability of FACTS for regulation of the
voltage - reprinted from [18]
Figure 3.16: System Thevenin equivalent - reprinted
from [18]
However, SVC control is not really as Figure 3.15. This provides ideal control which means that the
29
voltage is always regulated to the same value, for instance 1 p.u.. What actually happens is that the
voltage can have a little variation. Typically 2 − 5% is acceptable, i.e in the limit |Vr||Vs| ≤ 0.95. Despite
having a slightly worse voltage regulation, less current injection is needed in both ranges (except near
the inductive limit case). Furthermore, it can be seen in Figure 3.17 that controllability increases in the
inductive range. In order to understand this, one can draw multiple system lines as represented in Figure
3.14 along the SVC characteristic. This type of control is called droop control.
Figure 3.17: Droop control - reprinted from [18]
First of all, the TCR which is common to both FACTS is explained. Its structure is represented in
Figure 3.18, two thyristors and one reactor. One thyristor is fired according to the needs (firing angle α)
and then the other one is subjected also to the first one, it will be fired 180o after (see (3.5) where α2 is
the firing angle of the second thyristor). It is impossible to both thyristors to be conducting at the same
time since if one is conducting the other is reversely biased, meaning that even if it is given an impulse
to start conducting it will not be able to do so. The thyristors conduct during σ that is the conduction
angle while αs is the extinction angle. Due to the restriction of the firing angles and since one does not
want the TCR to work as a rectifier (that is what would happen if only one thyristor was correctly fired), α
will be confined to an interval from 0o to 90o. This limit angle (90o) is the impedance angle of the circuit,
in this case, it is only an inductor.
Moreover, the phase angle modulation switched by the thyristors makes the TCR to be variably
switched into the circuit and so it provides Mvar injection or absorption to the electrical network.
σ = π − 2α , (3.4)
α2 = α+ π (3.5)
and
αs = σ + α (3.6)
make
30
αs < α2 ⇒ α > 0 (3.7)
and
σ > 0⇒ α <π
2. (3.8)
Figure 3.18: Line diagram of a TCR - reprinted from [18]
The resultant waveforms are drawn in Figure 3.19 and by observing them, one understands that the
circuit reactance will be affected which was what one desired in the first place in order to control reactive
power flow. The equivalent reactance is therefore dependent on alpha.
ILF (α) =V
ωL
(1− 2
πα− 1
πsin (2α)
)(3.9)
Figure 3.19: TCR waveforms - reprinted from [18]
The FC-TCR is the simplest SVC built due to not having power electronics associated with the fixed
capacitor. However, this is a drawback for the device since one cannot turn-off the capacitor while in
the TSC-TCR the TSC can be turned-off. This drawback will result in great losses at zero current, i.e.
moving from one range to another. This is due to the current being trapped in the loop (Figure 3.20).
One advantage when compared to the TSC is that the lack of power electronics avoids a control system
and reduces the size of the device. The fixed capacitor supplies reactive power that the TCR needs
31
to consume. Moreover, when the system reactive load is capacitive (leading), the SVC uses the TCR
to consume VARs from the system to decrease the voltage and when it is inductive (lagging), the FC
provides VARs to increase the system voltage.
Figure 3.20: Schematic of the FC-TCR - reprinted from [18]
On the other hand, although the TSC-TCR has the same V-I characteristic as FC-TCR, it has many
fewer losses due to the TSC being based on power electronics, i.e. it can be completely turned-off. TSC
structure is represented in Figure 3.21.
Figure 3.21: Line diagram of the TSC - reprinted from [18]
It is composed by a controllable static switch, a capacitor and a small reactor. The inductor is needed
in order to limit the current during switching operations. The main drawback of this design is the size of
the thyristors. They have to be oversized since the capacitor can never be discharged (see Figure 3.23
where the current stops flowing) and so the voltage at the terminal of the switches is twice what it would
be without the capacitor.
32
Figure 3.22: Voltages of the TSC - reprinted from [18]
The size of the inductor leads to (3.10).
vsw ≈ v − vc (3.10)
Figure 3.23: Waveforms for TSC - reprinted from [18]
To solve this oversized problem, the switching is done when the difference between the voltage at the
capacitor terminals and the voltage at the TSC terminals is minimum so that the switching voltage is also
minimum. This will induce a transient period after switching that is minimized according to the strategy
taken for minimizing the switching voltage vsw. Another drawback is due to the presence of an inductor
with a capacitor and so one has a tuned LC circuit which will induce oscillations. Nevertheless, the TSC
provides a more efficient solution than the FC because it can be turned-off. Here, the firing angle is
either 0 or π/2 and nothing in between meaning that the TSC can only be fully on or off. This way the
controllability in the capacitive range is yet not very efficient. To work around this issue, it is common
to use more than one TSC (nTSC-TCR) and play with their on and off function to get the desired result
(Figure 3.24). Although TSC-TCR gives a much smoother control, FC-TCR was adopted for this thesis
since the main reference was [53].
33
Figure 3.24: nTSC-TCR characteristic - reprinted from [18]
By controlling the amount of reactive power exchanged with the grid regulating its voltage, the SVC
maximizes system security and stability.
3.3 Thyristor Controlled Series Capacitor
Figure 3.25: Thyristor controlled series capacitor - courtesy of [2]
The TCSC is an electrical device that works as a controllable voltage source. It is used on long lines
to shorten them indirectly by changing the impedance of the line. Moreover, it provides damping of low
frequency electromechanical oscillations. Typical ratings of the TCSC are hundreds of kV and MVar.
The main features of the TCSC are to reduce the line voltage drop, to limit load dependent voltage
drops, it influences the load flow in parallel transmission lines (controls current), increases power transfer
capability, reduces the transmission angle and increases the system stability [7].
A proper design of a TCSC enables inherent immunity against subsyncronous resonance (SSR), i.e.
34
SSR mitigation and protection3. Also, the cost-effectiveness of long transmission lines is limited by this
SSR.
The largest companies in the world manufacturing TCSCs are once again ABBTM
and SiemensTM
.
Examples of countries where TCSCs are placed are Brazil, USA, China, India, Tanzania, Mexico, Chile,
etc. due to the use of very long lines.
3.3.1 Theoretical Description
The TCSC can be seen as a reactance placed in series with the line. With this added reactance, the
voltage across the line is elevated to reduce losses. This is shown using again the voltage at the middle
of the line although it is recalled here that the TCSC location is independent of its efficiency, this is just
a way to compare active and reactive powers with shunt compensation.
Figure 3.26: Series compensation - reprinted from
[18]Figure 3.27: Series compensation vector diagram -
reprinted from [18]
The line current is now given by (3.11) and the middle voltage is (3.12).
I =2V
(1− k)Xsin
(δ
2
)(3.11)
with k = Xc
X where Xc stands for compensation reactance and X the line reactance.
Vm = V cos
(δ
2
)(3.12)
This gives (3.13) for active power and (3.14) for reactive power.
P = VmI =V 2
(1− k)Xsin δ (3.13)
Qc = I2Xc =2V 2
X
k
1− k(1− cos δ) (3.14)
The basic scheme of a TCSC consists of a TCR that operates according to what was explained
3SSR are resonant frequencies below the fundamental that result from interaction between series capacitors and nearbyturbine-generators. They can even induce shaft failures due to vibrations that will grow exponentially.
35
in Section 3.2 and a capacitor in parallel. Logically, the TCR reactance is the only one capable of
being modified and is ruled by (3.15). Its dependence on a firing angle makes TCSC reactance also
being dependent on α, (3.16). Providing a rapid dynamic modulation of the inserted reaction, the TCSC
reduces transmission line inductance, meaning that the line is shortened. Thus, the transmission angle
is reduced and so the active power transfer is increased.
Figure 3.28: TCSC line diagram
XL(α) = XLπ
π − 2α− sin(2α), XL ≤ XL(α) ≤ ∞ (3.15)
XTCSC(α) = Xc//XL(α) (3.16)
Seeing the TCSC as the parallel of a variable reactor and a capacitor, the equivalent impedance can
be written as
Zeq =1
Bc +BL= −j 1
ωC − 1ωL
(3.17)
If ωL > 1ωC the overall reactance is capacitive. Naturally, if ωL < 1
ωC the overall reactance is
inductive. However, ωL = 1ωC has to be avoided in order not to have resonance. As a consequence,
no smooth transition between ranges is possible, see Figure 3.29 on which αr is the critical angle. If
resonance occurred, the impedance inside the loop would become zero (capacitor in series with TCR)
and therefore the equivalent impedance would become infinite.
36
Figure 3.29: Characteristic of TCSC reactance vs. firing angle α - reprinted from [18]
In the small circuit diagram included in Figure 3.29, one can see the loop composed by the capacitor,
the inductor and the switch. At the instant of closing the switch, the line current will continue (dis)charging
the capacitor while at the same time the charge of the capacitor will be reversed during the resonant
half-cycle of the LC circuit formed by the switch closing. This resonant charge reversal also known as
TCSC voltage reversal produces a DC offset for the next positive half-cycle of the capacitor voltage.
By maintaining the same α the DC offset is reversed and thus the voltage becomes symmetrical in
each half-cycle. The waveforms for TCSC voltage and current are explained in detail below. The first
assumption to take is that the grid works as a constant current source, i.e. it is strong enough not to be
affected by the changing of the TCSC reactance. As one can see in the waveforms in Figure 3.31, the
voltage is a bit different from a perfect sinusoid and therefore this assumption is not 100% true. Another
assumption is that all currents are positive from left to right (illustrated in Figure 3.30) and the same for
Vc.
Figure 3.30: TCSC voltage reversal line diagram
Imagine that the capacitor voltage has the opposite direction indicated in Figure 3.30 and i starts
37
negatively according to the second assumption. Furthermore, Vc0 will be in quadrature with the current
i because of the capacitor. If Vc is negative initially it means iL is also negative when the upper thyristor
is fired. The current inside the loop is now traveling clockwise (ic = i − iL) and so it passes through
the reactor and will charge the capacitor becoming Vc positive. This is the TCSC voltage reversal in
the capacitive range. The voltage Vc is then shifted up since the zero-crossing of Vc is anticipated, see
Figure 3.31.
Figure 3.31: TCSC voltage Vc shifted up - reprinted from [18]
Both capacitive and inductive range are represented below by their waveforms. As one can see,
one of the particularities of the TCSC is that it is understandable by the waveforms on which range it is
operating.
(a) Capacitive range (b) Inductive Range
Figure 3.32: TCSC waveforms - reprinted from [18]
3.4 Summary
Summarizing what was explained in Sections 3.2 and 3.3, both devices are associated with many
38
harmonics illustrated in Figures 3.19 and 3.32. As a consequence, the devices have to be equipped in
association with huge filters. This is one of the reasons why the dimension of FACTS is so large.
The fact that the SVC has to be implemented in the middle of a transmission line is a major drawback
when comparing to the TCSC. Transmission lines have components that cannot be measured physically
but they have to be measured electrically, which is the case of the electrical middle of a line. It is
impossible to know exactly this location at every time instant which results in lower efficiencies for the
SVC. What is usually done is to place the SVC in the length middle of the power line.
The TCSC has a resonance region that has to be avoided so that the device works correctly. This is
due to the LC loop circuit in series with the line. If resonance occurred, no current would flow out of the
loop.
Both devices are thyristor controlled-based and thus the firing angles of those electronic components
are crucial. While the SVC injects/consumes reactive power (in shunt) according to the line needs, the
TCSC shortens the line in order to avoid having great amounts of reactive power. Both of them allow
more active power flow lowering the reactive power.
A sequence of thoughts is now demonstrated to understand how the SVC is less efficient than the
TCSC. For the same amount of active power less reactive power is needed. For series, voltage from
(3.18)
Vm = V cosδ
2(3.18)
and current from (3.19)
I =2V
Xsin
δ
2(3.19)
lead to (3.20)
P =2V 2
Xsin
δ
2cos
δ
2. (3.20)
The variation of active power is related to the variation of the line reactance as well as the variation
of reactive power.
∆P =dP
dX∆X = −2V 2
X2sin
δ
2cos
δ
2∆X (3.21)
∆Qseries = I2 ·∆X =4V 2
X2sin2 δ
2·∆X . (3.22)
This results in (3.23) that gives the variation of reactive power for a fixed variation in active power.
∣∣∣ ∆P
∆Qseries
∣∣∣ =cos δ22 sin δ
2
=1
2 tan δ2
. (3.23)
On the other hand, it is more common to modify the line susceptance instead of reactance in shunt
compensation. Therefore, with a similar approach, the active power becomes
39
P = I · Vm =2V 2
Xsin
δ
2cos
δ
2(3.24)
The variations of active (related to the midpoint voltage as indicated in table 3.1) and reactive powers
become
∆P =dP
dVm∆Vm =
2V
Xsin
δ
2∆Vm (3.25)
∆Qshunt = V 2m ·∆B = V 2 cos2 δ
2·∆B , (3.26)
where
∆Vm =∆I
2
X
2=Vm ·∆B ·X
4. (3.27)
This results in a ratio between variations of power of
∣∣∣ ∆P
∆Qshunt
∣∣∣ =2VX sin δ
2V cos δ2∆B·X
4
V 2cos2 δ2∆B
=12 sin δ
2
cos δ2=
1
2tan
δ
2. (3.28)
From (3.23) and (3.28),
∆P∆Qshunt
∆P∆Qseries
= tan2 δ
2, (3.29)
which is less than one for the angle that is indeed used (no more than 35o).
40
Chapter 4
Inclusion of FACTS in Small Signal
Stability
4.1 Steady-State Model
FACTS have been introduced both in power flow and transient programs. The introduction of FACTS
in power flow is related to their steady-state behavior. Transient behavior is accounted for in the transient
program. In the power flow program, instead of controlling reactive power as in PV buses, FACTS have
reactance control, i.e. their reactance cannot go beyond an interval. This interval is illustrated in the V-I
characteristics; if a reactance goes outside this interval no more control is provided.
FACTS power flow was implemented in the developed program but it was not necessary to use it for
result comparison with [53]. In fact, [53] considered all FACTS buses PQ since their implemented control
assumed that due to its time constants being small both SVC and TCSC can change their reactance
extremely fast. The authors were concerned with the Hopf bifurcation that is dependent on the dynamic
models of the system components. Therefore, system power flow considering PQ buses produces
results as if the devices control was off. Then at dynamic analysis the control input is what comes
from the power flow and is compared with a reference so that in the end both devices compensate this
difference with a simple reactance.
Power flow for FACTS was added to the program to evaluate results considering these devices. This
section shows its implementation which is the same as [9]1.
First of all it is based on the Newton-Raphson algorithm. In fact, in large scale systems the algorithm
is the most successful one due to its strong convergence characteristics. Its essence is based on Taylor
series expansion. Introducing FACTS to the algorithm is done in a very modular way. Also, the algorithms
are extremely detailed and ready to use (except for some issues that are referred in Appendix B) in [9].
It is also worth mentioning that the power flow works for several FACTS devices but only if they are of
the same type.
1[9] had already a well-prepared algorithm for FACTS ready to use with several types of control such as power flow, voltage,firing angle, etc.
41
In order to introduce FACTS, a unified approach is adopted. It consists of blending the AC network
and power system controller state variables in a single simultaneous equations system.
f(XnAC , RnF ) = 0
g(XnAC , RnF ) = 0,(4.1)
where XnAC stands for the AC network state variables and RnF for the power system controller state
variables. Thus, the Jacobian will have its columns and rows increased since the number of new vari-
ables is proportional to the number of FACTS controllers. Figure 4.1 illustrates the augmented Jacobian
that the power flow has to deal with.
Figure 4.1: Augmented Jacobian - reprinted from [9]
4.1.1 Power Flow with SVC
As mentioned in Section 3.2, droop control is the most useful and realistic one. It can be represented
by connecting the SVC to an auxiliary bus coupled to the high-voltage bus by an inductive reactance
consisting of the transformer reactance and the SVC slope. This auxiliary bus is represented as PV
while the high-voltage bus is PQ, meaning that two buses are needed. However, this model is only valid
for operations inside reactance (or susceptance) limits.
Figure 4.2: SVC power flow model - reprinted from [9]
For operations outside limits, the SVC has a fixed susceptance related to reactive power and there-
42
fore only one bus represents the model.
Bsvc = − QlimV 2SV C
(4.2)
where VSV C is the new voltage due to Qlim being exceeded.
The main drawback of the combined generator-susceptance representation is that the SVC needs
two models that use a different number of buses. As a consequence, the network Jacobian is always
being re-dimensioned and reordered as verifications are made to check whether the SVC is inside or
outside limits.
The shunt variable susceptance model can be adjusted by two means: firing-angle limits or reactance
limits. The model implemented is the last one since it can be more intuitive and simpler than the first.
Using the firing-angle requires indeed harder computational efforts because it is calculated from SVC
suceptance and these two quantities are nonlinearly related.
Figure 4.3: Shunt Variable Susceptance Model - reprinted from [9]
What the power flow needs to incorporate are the SVC linearized equations and then update the
shunt susceptance at all iterations. From Figure 4.3 comes straight forward
Isvc = jBSV CVk, (4.3)
which gives reactive power injected at bus k
Qsvc = −V 2k BSV C . (4.4)
Since the device is represented by a pure susceptance, only reactive power is injected. Along itera-
tions it has to be updated.
∆Pk
∆Qk
(i)
=
0 0
0 Qk
(i) ∆θk
∆BSV C
BSV C
(i)
(4.5)
43
At the end of each iteration, BSV C is updated according to (4.6). This way, the desired voltage at the
bus containing the SVC is always satisfied.
B(i)SV C = B
(i−1)SV C +
(∆BSV CBSV C
)(i)
B(i−1)SV C (4.6)
4.1.2 Power Flow with TCSC
Similarly to the SVC, the TCSC can be modeled based on a simple reactance or a firing-angle. Once
again the firing-angle model relates two quantities in a nonlinear way whereby the implemented model
was the reactance model. Its representation is in Figure 4.4 and according to the operating range the
reactance is an inductor (4.4(a)) or a capacitor (4.4(b)).
(a) Inductive range (b) Capacitive range
Figure 4.4: Series Variable Impedance Model - reprinted from [9]
Unlike the SVC, the TCSC is related to two buses instead of only one. Therefore the model equation
becomes (4.7).
Ik
Im
=
jBkk jBkm
jBmk jBmm
Vk
Vm
, (4.7)
where for inductive range
Bkk = Bmm = − 1XTCSC
Bkm = Bmk = 1XTCSC
(4.8)
and for capacitive range the signs are reversed.
Injected active and reactive powers at bus k are therefore given by (4.9) while at bus m the signs
must be changed.
Pk = VkVmBkm sin(θk − θm)
Qk = −V 2k Bkk − VkVmBkm cos(θk − θm)
(4.9)
From here the Newton-Raphson linearized subsystem to implement is given in (4.10).
44
∆Pk
∆Pm
∆Qk
∆Qm
∆PXTCSC
km
(i)
=
∂Pk∂θk
∂Pk∂θm
∂Pk∂Vk
Vk∂Pk∂Vm
Vm∂Pk
∂XTCSCXTCSC
∂Pm∂θk
∂Pm∂θm
∂Pm∂Vk
Vk∂Pm∂Vm
Vm∂Pm
∂XTCSCXTCSC
∂Qk∂θk
∂Qk∂θm
∂Qk∂Vk
Vk∂Qk∂Vm
Vm∂Qk
∂XTCSCXTCSC
∂Qm∂θk
∂Qm∂θm
∂Qm∂Vk
Vk∂Qm∂Vm
Vm∂Qm
∂XTCSCXTCSC
∂PXTCSC
km
∂θk
∂PXTCSC
km
∂θm
∂PXTCSC
km
∂VkVk
∂PXTCSC
km
∂VmVm
∂PXTCSC
km
∂XTCSCXTCSC
(i)
∆θk
∆θm
∆Vk
Vk
∆Vm
Vm
∆XTCSC
XTCSC
(i)
,
(4.10)
where
∆PXTCSC
km = P regkm − PXTCSC,cal
km (4.11)
and P regkm is the amount of active power flowing from bus k to bus m that series reactance regulates
and PXTCSC,cal
km is the calculated power between these two buses, i.e. (4.11) is the active power flow
mismatch.
Finally,
∆XTCSC = X(i)TCSC −X
(i−1)TCSC (4.12)
which means that the updated impedance is given by (4.13).
X(i)TCSC = X
(i−1)TCSC +
(∆XTCSC
XTCSC
)(i)
X(i−1)TCSC (4.13)
4.2 Transient Model
4.2.1 General Description
The methodology adopted in this chapter follows [53]. The model derived is therefore the same as
the 9 bus case presented in [66] which can be represented by the system matrix (4.14). With this starting
point, FACTS are included in a modular way.
45
Figure 4.5: 9 bus network - reprinted from [53]
∆xg1
∆xg2
∆xg3
0
0
0
0
0
=
A11 0 0
0 A22 0
0 0 A33
A2 0 A4 0 0
B1 B2 B3 B4 B5 0
C1 C2 C3 C4 C5 C6
D1 D2 D3 D4 D5 D6
0 0 E3 E4 E5 E6
F1 F2 F3 F4 F5 F6
∆xg1
∆xg2
∆xg3
∆Id−q
∆θ1
∆Vg
∆θLF
∆VLF
(4.14)
4.2.2 Inclusion of the SVC
Introducing an SVC in the system improves system stability. Its working mechanism was described
in Section 3.2. The reactive power associated to the device is treated as an injection into the bus where
it is located. Although two different models were explained in this thesis, only the FC-TCR is modeled
since it was the one assumed by [53]2.
4.2.2.1 Model of the SVC
As aforementioned, the reactance of the device is a function of the angle α which is in turn modified
via a PI-controller. As a consequence, it is possible for the SVC to maintain the voltage after it is
disturbed for several reasons indicated in Chapter 1. Droop control is used since it is the most realistic
and practical type of control. As one can see in Figure 4.6, if the voltage of the system VSV C = ESV C −
jXSV CISV C is lower than the reference voltage Vref , the operating range is the capacitive whilst when
the system voltage is bigger the inductive range is the operating one.
2The FC-TCR is much simpler than the TCS-TCR in terms of control and since the main goal of this thesis was to follow themethodology in [53], the FC-TCR became the SVC implemented.
46
Figure 4.6: Operating ranges with droop control - reprinted from [53]
The block diagram used is
Figure 4.7: SVC block diagram - reprinted from [53]
for which the state equations are
x1 = 1Tb
(VSV C(1 +Kx3)− x1)
x2 = Ki(Vref,SV C − x1)
x3 = 1Tc
(x2 +KP (Vref,SV C − x1)− x3)
QSV C = V 2SV Cx3 .
(4.15)
These equations are now linearized
47
∆x1 = 1Tb
(VSV C(1 +Kxeq3 ) + V eqSV CK∆x3 −∆x1)
∆x2 = Ki(∆Vref,SV C −∆x1)
∆x3 = 1Tc
(∆x2 +KP (∆Vref,SV C −∆x1)−∆x3)
∆QSV C = 2V eqSV Cxeq3 ∆VSV C + V eq2SV C∆x3 .
(4.16)
4.2.2.2 Model inclusion
According to the system of equations (4.16), the input is ∆VSV C while the output is ∆QSV C . In
addition, the state space model uses the following state variables:
∆xSV C = [∆x1 ∆x2 ∆x3]T . (4.17)
Hence, the system model with the SVC becomes
∆xg1
∆xg2
∆xg3
∆xSVC
0
0
0
0
0
=
A11 0 0 0
0 A22 0 0
0 0 A33 0
0 0 0 ASVC
A2 0 A4 0 A6
B1 B2 B3 B4 B5 0
C1 C2 C3 C4 C5 C6
D1 D2 D3 D4 D5 D6
E1 0 E3 E4 E5 E6
F1 F2 F3 F4 F5 F6
∆xg1
∆xg2
∆xg3
∆xSVC
∆Id−q
∆θ1
∆Vg
∆θLF
∆VLF
(4.18)
where the bold items are the ones that must be changed comparing with (4.14). Subsection 4.2.2.3
presents the system model that is actually included in the DAE model with explained matrices; equation
4.18 is just illustrative.
In order to damp oscillations the SVC may have an auxiliary controller. In fact, the device by itself
already sets some damping but not as efficiently as an auxiliary controller does. This type of SVC
controller is explained below.
48
Figure 4.8: SVC block diagram with auxiliary controller - reprinted from [53]
Figure 4.8 illustrates the block diagram of the SVC equipped with the auxiliary controller. The con-
troller is commanded by an uc signal that can be any of the signals derived from the system, typically
the frequency or reactive power flow into the adjacent lines of the SVC bus. Here, the second choice is
taken. In addition, G(s) is a simple lead-lag controller with a gain, meaning that one extra variable must
be added to the system.
G(s) = KB1 + sT1
1 + sT2(4.19)
Signal uc, according to Figure C.2 and considering it placed at bus 4, is given by
uc = V4V6Y46 sin (θ4 − θ6 − α46) + V4V8Y48 sin (θ4 − θ8 − α48) , (4.20)
where
Yij = Yij∠αij . (4.21)
Hence, the state equations become
49
x1 = 1Tb
(VSV C(1 +Kx3)− x1)
x2 = Ki(Vref,SV C − x1 + V1)
x3 = 1Tc
(x2 +KP (Vref,SV C − x1 + V1)− x3)
x4 = 1T2
[(KB −KB
T1
T2
)uc − x4
]
QSV C = V 2SV Cx3 .
(4.22)
which linearized give
∆x1 = 1Tb
(VSV C(1 +Kxeq3 ) + V eqSV CK∆x3 −∆x1)
∆x2 = Ki
(∆Vref,SV C −∆x1 + ∆x4 +KB
T1
T2∆uc
)
∆x3 = 1Tc
[∆x2 +KP
(∆Vref,SV C −∆x1 + ∆x4 +KB
T1
T2∆uc
)−∆x3
]
∆x4 = 1T2
[(KB −KB
T1
T2
)∆uc −∆x4
]
∆QSV C = 2V eqSV Cxeq3 ∆VSV C + V eq2SV C∆x3 .
(4.23)
Additionally, uc must also be linearized.
∆uc = [V6Y46 sin (θ4 − θ6 − α46)V8Y48 sin (θ4 − θ8 − α48)]∆V4+
V4Y48 sin (θ4 − θ8 − α48)∆V8 + V4Y46 sin (θ4 − θ6 − α46)∆V6+
[V4V6Y46 cos (θ4 − θ6 − α46) + V4V8Y48 cos (θ4 − θ8 − α48)]∆θ4−
V4V8Y48 cos (θ4 − θ8 − α48)∆θ8 − V4V6Y46 cos (θ4 − θ6 − α46)∆θ6 .
(4.24)
Lastly, the system model is modified to
50
∆xg1
∆xg2
∆xg3
∆xSVC
0
0
0
0
0
=
A11 0 0 0
0 A22 0 0
0 0 A33 0
0 0 0 ASVC
A2 0 A4 A5 A6
B1 B2 B3 B4 B5 0
C1 C2 C3 C4 C5 C6
D1 D2 D3 D4 D5 D6
E1 0 E3 E4 E5 E6
F1 F2 F3 F4 F5 F6
∆xg1
∆xg2
∆xg3
∆xSVC
∆Id−q
∆θ1
∆Vg
∆θLF
∆VLF
(4.25)
4.2.2.3 SVC integration in DAE model
At this point, by having a clear idea of how the SVC controller is modeled, it is possible to define
specifically its inclusion on the MATLABTM program that has been developed for the past few theses
[17], [16] and [67]. This allows the reader to know exactly the matrices from [66] that should be modified
and also the ones that should be added. Having the system of equations (4.26) as a starting reference
(also presented in Chapter 2) and equations (4.16), one can modify the system in order to get the new
DAE model (4.27).
∆x = A1∆x+B1∆Ig +B2∆Vg + E1∆u
0 = C1∆x+D1∆Ig +D2∆Vg
0 = C2∆x+D3∆Ig +D4∆Vg +D5∆Vl
0 = D6∆Vg +D7∆Vl
(4.26)
Then the final system of equation takes form
∆x = A1∆x+B1∆Ig +B2∆Vg +B3∆Vl + E1∆u
0 = C1∆x+D1∆Ig +D2∆Vg
0 = C2∆x+D3∆Ig +D4∆Vg +D5∆Vl
0 = C3∆x+D6∆Vg +D7∆Vl
(4.27)
This demonstration is intended for the reader to compare the eigenvalues of the system matrix Asys
to previous examples and as a consequence one can neglect the input matrix E1 and only focus on the
others. Following this line of thought, the matrices explained are A1, B3, C3 and D7.
Matrix A1 will have its dimensions increased since the state variables will also increase. It is a result
51
from the differential equations in (4.16). Every SVC can be represented by
Asvc =
− 1
Tb0
V eqsvcK
Tb
−Ki 0 0
−Kp
Tc
1
Tc− 1
Tc
. (4.28)
Then, as the SVC is independent of the other state variables, the remaining matrix entries are null,
meaning that Asvc is added to A1 diagonally. The final matrix has the dimensions [df × df ] being df the
number of differential variables.
A1svc =
A1 0
0 Asvc
(4.29)
On the same equation, a new matrix B3 must be added in order to have into account the coefficient
that multiplies by the voltage of the bus containing the SVC (first equation in (4.16)): ∆Vsvc. Its dimen-
sions are [df × 2(n−m)] (remember that n is the number of buses and m the number of generators so
n−m gives the non-generator buses). Apart from this entry, all others are zero.
B3 =
0 0 · · · 0 0
......
......
...
0 · · · 1Tb
(1 +Kxeq3 ) · · · 0
0 0 · · · 0 0
0 0 · · · 0 0
(4.30)
The SVC algebraic equation leads to a modification in the algebraic equation for non-generator
buses. D7 is just modified in the entry regarding reactive power in the SVC bus, 2V eqsvcxeq3 is added
to this entry. On the other hand, C3 is created to have in consideration the multiplicative coefficient of
∆x3. Its dimensions are [2(n−m)× df ].
C3 =
0 · · · 0
.... . . V eqsvc
2
0 · · · 0
(4.31)
Once again, ∆Ig is not relevant for this problem and so it can be eliminated.
∆Ig = −D−11 C1∆x−D−1
1 D2∆Vg (4.32)
(4.32) is substituted in (4.27) resulting in a new system of differential equations.
52
∆x = (A1svc −B1D−11 C1)∆x+ (B2 −B1D
−11 D2)∆Vg +B3∆Vl
0 = (C2 −D3D−11 C1)∆x+ (D4 −D3D
−11 D2)∆Vg +D5∆Vl
0 = C3∆x+D6∆Vg +D7∆Vl(4.33)
As in [66], one can create two new matrices just for simplification, K1 = D4 − D3D−11 D2 and
K2 = C2 −D3D−11 C1. Finally (4.34) is obtained and ready to be implemented.
∆x
0
0
=
A1svc −B1D
−11 C1 B2 −B1D
−11 D2 B3
K2 K1 D5
C3 D6 D7
∆x
∆Vg
∆Vl
(4.34)
Finally, the equilibrium points x1, x2 and x3 have to be determined. By setting the derivatives from
the differential equations in (4.15) to zero, one gets
x1 = Vref,svc
x2 =Vref,svc − Vsvc
KVsvc
x3 =Vref,svc − Vsvc
KVsvc
, (4.35)
where Vref,svc is the voltage at which the SVC bus should be and Vsvc comes from the power flow (often
represented as V eqsvc since it is an equilibrium point). These equilibrium values, namely xeq3 and V eqsvc are
really important for being part of the system matrices. Nevertheless x3, SVC susceptance, is not usually
calculated this way, it is either zero (considering that there is no reactive power injection initially3) or a
value coming from power flow (when considering FACTS implemented there).
The same is done for the SVC with the auxiliary controller. Asvc becomes
Asvc =
− 1
Tb0
V eqsvcK
Tb0
−Ki 0 0 Ki
−Kp
Tc
1
Tc− 1
Tc
Kp
Tc
0 0 0 − 1
T2
. (4.36)
B3 is formed in the same way as without the auxiliary controller but with more non-zero entrances
since there are now coefficients that multiply by both angle and magnitude voltages of several buses
due to uc. Furthermore, D7 is modified in the same way as before and so is C3. The only relevant initial
conditions (used further ahead) and excepting x3 for the same reasons stated before are given by
3[53] does not consider in the power flow that an SVC is implemented.
53
x1 = Vsvc
x4 = uc
(KB −KB
T1
T2
)(4.37)
At this point, it is possible to simulate the system in order to evaluate its eigenvalues and observe the
impact of the SVC in power oscillations damping. Later on, it shall be attainable to estimate the location
where to place the device.
4.2.3 Inclusion of the TCSC
The TCSC will have an impact on the system similar to the SVC since they both improve stability.
However this device works in a different way which was explained in Section 3.3. Due to the fast switch-
ing provided by the thyristors, power flow in a line is considered to change almost instantaneously. Also
damping is introduced with the device since it influences the whole system.
4.2.3.1 Model of the TCSC
Two types of control can be used, constant current control and constant angle control. As one will
see further on, they do not differ significantly since only a variable is changed. In the thesis the chosen
approach was the constant current control. The differences between types of control are represented in
Figure 4.9, constant current regulates power flow since the current is kept constant while constant angle
control maintains the voltage angle. Having constant current means that the firing angle from the TCSC
reactance is modified via a controller.
(a) Constant current control (b) Constant angle control
Figure 4.9: TCSC V-I characteristics - reprinted from [53]
To introduce the controller, considerations on the TCSC model are firstly enunciated. The adopted
diagram of the TCSC is shown in Figure 4.10. [53] considered that at normal state operating conditions
the TCSC is a capacitor, meaning that its reactance is always a negative number. Furthermore, inside
control range the TCSC is able to keep constant current while outside the range it delivers either the
maximum or minimum series compensation.
54
Figure 4.10: TCSC line diagram - reprinted from [53]
The block diagram used is represented in Figure 4.11. The droop is only used for constant angle
control and therefore Sk takes the value − 1
XL. On the other hand, for impedance constant control
Sk = 0. Tt and Ts are time constants with a small value since FACTS must act really fast. Finally, Ki
block is an integral controller.
Figure 4.11: TCSC block diagram - reprinted from [53]
From the block diagram it is possible to state the relations in (4.38).
Im =1
Tt(ITCSC − SkVTCSC − Im)
x1 = Ki(Iref − Im)
xTCSC =1
Ts(x1 − xTCSC)
(4.38)
where
VTCSC =| Vi − Vj |=√V 2i + V 2
j − 2ViVj cos(θi − θj) (4.39)
and
55
ITCSC =| Vi − Vj || XTCSC |
=
√V 2i + V 2
j − 2ViVj cos(θi − θj)
| XTCSC |. (4.40)
Substituting (4.39) and (4.40) in (4.38) gives (4.41).
Im =1
Tt
[√V 2i + V 2
j − 2ViVj cos(θi − θj)(
1
| XTCSC |− Sk
)− Im
]
x1 = Ki(Iref − Im)
xTCSC =1
Ts(x1 − xTCSC)
(4.41)
Equations (4.41) can then be linearized. For simplification, a new variable is defined which represents
VTCSC at equilibrium, i.e. from load flow (equation (4.42)).
Heq =√V eqi
2+ V eqj
2 − 2V eqi V eqj cos(θeqi − θeqj ) (4.42)
Which leads to:
∆Im =1
Tt
[Heq
XeqTCSC
2 ∆XTCSC+
V eqi −V
eqj cos(θeqi −θ
eqj )
Heq
(1
|XeqTCSC |
− Sk)
∆Vi+
V eqj −V
eqi cos(θeqi −θ
eqj )
Heq
(1
|XeqTCSC |
− Sk)
∆Vj+
V eqi V eq
j sin(θeqi −θeqj )
Heq
(1
|XeqTCSC |
− Sk)
∆θi+
−V eqi V eq
j sin(θeqi −θeqj )
Heq
(1
|XeqTCSC |
− Sk)
∆θj
]
∆x1 = Ki(∆Iref −∆Im)
∆xTCSC =1
Ts(∆x1 −∆XTCSC).
(4.43)
The conventional series capacitor is approximately 30% of XL (transmission line reactance) while
the control range from the FC-TCR module is 10%XL−40%XL and thus the equilibrium point for XTCSC
is usually 40%XL − 70%XL.
4.2.3.2 Model inclusion
Once again it is necessary to start from the simple model (4.14) and get to a model with the device
included. Although two buses will be affected by the TCSC and not just one as in the SVC, the principle
for including the thyristor controlled series capacitance is the same. According to equations (4.43), the
input is ∆Iref (variable that will be compared) whilst the output is ∆XTCSC . Furthermore, the state
56
space model state variables are
∆xTCSC = [∆Im ∆x1 ∆xTCSC ]T . (4.44)
Hence, the system model with the TCSC becomes
∆xg1
∆xg2
∆xg3
∆xTCSC
0
0
0
0
0
=
A11 0 0 0
0 A22 0 0
0 0 A33 0
0 0 0 ATCSC
A2 0 A4 A5 A6
B1 B2 B3 B4 B5 0
C1 C2 C3 C4 C5 C6
D1 D2 D3 D4 D5 D6
E1 0 E3 E4 E5 E6
F1 F2 F3 F4 F5 F6
∆xg1
∆xg2
∆xg3
∆xTCSC
∆Id−q
∆θ1
∆Vg
∆θLF
∆VLF
(4.45)
where the bold items are the ones that must be changed comparing with (4.14). Because of the two
affected buses, not only more sub-matrices are modified but a new one is created, A5.
In addition, the TCSC will also have some algebraic equations modified due to its reactance being
between two buses and it is now a dynamic variable. AsXTCSC will affect the basic power flow equations
between buses i and j (where the TCSC is placed), the following holds true:
Gii = Gij = 0 (4.46)
Bii = −Bij = − 1
XTCSC(4.47)
The power flow equations from bus i to j are given by (4.48) and (4.49).
Pij = V 2i Gii + ViVj [Gij cos(θi − θj) +Bij sin(θi − θj)] (4.48)
Qij = −V 2i Bii + ViVj [Gij sin(θi − θj)−Bij cos(θi − θj)] (4.49)
Substituting (4.46) and (4.47) on the power flow equations leads to
Pij = ViVj sin(θi − θj)1
XTCSC(4.50)
Qij = [V 2i − ViVj cos(θi − θj)]
1
XTCSC(4.51)
Now equations (4.52) and (4.53) must be linearized to be incorporated in the DAE model. Here only
57
the partial derivative in order to XTCSC is taken since the remaining derivatives are already considered
in the model.
∆Pij = −V eqi V eqj sin(θeqi − θ
eqj )
XeqTCSC
2 ∆XTCSC (4.52)
∆Qij = −V eqi
2 − V eqi V eqj cos(θeqi − θeqj )
XeqTCSC
2 ∆XTCSC (4.53)
These equations will be added to active and reactive power equations at buses i and j. That is why E1
and F1 are modified in the DAE model (4.45).
4.2.3.3 TCSC integration in DAE model
The steps needed for implementation in the program are now detailed. Starting with the final DAE
model one can understand that it is also needed to create/modify matrices as in the SVC. The DAE
model has thus a similar shape as in Subsection 4.2.2.3.
∆x = A1∆x+B1∆Ig +B2∆Vg +B3∆Vl + E1∆u
0 = C1∆x+D1∆Ig +D2∆Vg
0 = C2∆x+D3∆Ig +D4∆Vg +D5∆Vl
0 = C3∆x+D6∆Vg +D7∆Vl
(4.54)
Once more E1 is neglected for eigenvalue analysis. Matrix A1 will have the parameters in 4.55.
Atcsc =
− 1
Tt0
1
Tt
Heq
XeqTCSC
2
−Ki 0 0
01
Ts− 1
Ts
(4.55)
Then, as in the SVC, Atcsc is added to A1 diagonally which results in dimensions [df × df ].
A1tcsc =
A1 0
0 Atcsc
(4.56)
B3 is a new matrix that needs to be created to account for PQ buses voltage magnitudes (last two
non-zero entries) and angles (first two non-zero entries) that are related to both terminal buses of the
TCSC. Its dimensions are also [df × 2(n−m)] and all other entries are zero. Variables a, b, c and d are
just separated from B3 since they would not fit in visualization.
B3 =
· · · a · · · b · · · c · · · d · · ·...
......
......
......
......
0 0 0 0 0 0 0 0 0
, (4.57)
58
a =1
Tt
V eqi − Veqj cos(θeqi − θ
eqj )
Heq
( 1
| XeqTCSC |
− Sk),
b =1
Tt
V eqj − Veqi cos(θeqi − θ
eqj )
Heq
( 1
| XeqTCSC |
− Sk),
c =1
Tt
V eqi V eqj sin(θeqi − θeqj )
Heq
( 1
| XeqTCSC |
− Sk),
d = − 1
Tt
V eqi V eqj sin(θeqi − θeqj )
Heq
( 1
| XeqTCSC |
− Sk).
(4.58)
The TCSC algebraic equations are already incorporated in matrix D7 since the data given in .raw file
indicate all lines reactances. Apart from that and similarly to the static var compensator, a new matrix
C3 [2(n −m) × df ] is created to incorporate the active and reactive power portions that multiply by the
state variables. All other entries are null.
C3 =
0 0 0 0 0 0...
......
......
...
· · · −Veqi V eq
j sin(θeqi −θeqj )
XeqTCSC
2 · · · −Veqi
2−V eqi V eq
j cos(θeqi −θeqj )
XeqTCSC
2 · · · 0
......
......
......
0 · · · −Veqi V eq
j sin(θeqj −θeqi )
XeqTCSC
2 · · · −Veqj
2−V eqi V eq
j cos(θeqi −θeqj )
XeqTCSC
2 · · ·...
......
......
...
0 0 0 0 0 0
(4.59)
Eliminating ∆Ig one gets (4.60).
∆x = (A1tcsc −B1D−11 C1)∆x+ (B2 −B1D
−11 D2)∆Vg +B3∆Vl
0 = (C2 −D3D−11 C1)∆x+ (D4 −D3D
−11 D2)∆Vg +D5∆Vl
0 = C3∆x+D6∆Vg +D7∆Vl(4.60)
Finally, matricially the system becomes:
∆x
0
0
=
A1tcsc −B1D
−11 C1 B2 −B1D
−11 D2 B3
K2 K1 D5
C3 D6 D7
∆x
∆Vg
∆Vl
(4.61)
The only relevant equilibrium point for the model is XeqTCSC which is the same as the line reactance
between the two buses that bound the TCSC: XeqTCSC = Xij .
At this point it is possible to perform a TCSC simulation and get conclusions from it.
59
60
Chapter 5
Model Validation
This chapter is dedicated to compare results with references [66] (type of load), [53] (inclusion of
FACTS) and [9] (power flow for FACTS). Not only the data are provided but also how the user should
perform according to what he/she wants to simulate is explained. The networks are represented and
parts of the program are illustrated (referring to the Appendix) to familiarize the user with it.
5.1 IEEE 9 bus network
IEEE 9 bus network is commonly used in bibliography concerning small signal analysis and enhance-
ment of system stability. By using it, it is easier to compare results from the developed program in the
thesis to other references. The main references taken into account for the following subsections are [66]
and [53]. The system data are presented as well as the network diagram for each result comparison.
The data are referred as p.u., the nominal values can be found in the mentioned references or the files
used for simulations.
Subsection 5.1.1 uses model I according to [53] while Subsections 5.1.2, 6.1.3 and 5.1.4 deal with
model II. The only difference is that model I uses the IEEET1 type of exciter [66] while model II uses a
Static Exciter (introduced in Section 2.2.2).
5.1.1 9 bus for different types of loads
In Section 2.3 it became clear that the type of load, i.e. constant power, constant current or constant
impedance, has influence on the system stability. In order to visualize this fact, the following results
and their comparison with [66] illustrate it. Except for one result in Table 5.5 that produces a huge error
(almost 30%) all others are according to [66]. This can be a printing error in [66] since the immediate
previous value of the imaginary part of the concerned eigenvalue is exactly the same.
The results are from the 9 bus network which is represented in Figure C.1.
The data for the network were taken from [53] and are represented in Appendix C.1. Moreover these
data were used to make the files 9bus.raw and 9busGENRED+IEEET1+D.dyr. (The D refers to damping
that was not on 9busGENRED+IEEET1.dyr.)
61
Running the program with the mentioned files for a stable case one gets the results on Tables 5.1,
5.2 and 5.3. Only bus 5 was modeled as a different type of load. In Figure C.7 it is shown how the user
indicates options type of load and operating point.
Table 5.1: Eigenvalues of case PL05 = 1.5pu,QL05 = 0.5pu with constant power load (type 0)
MATLABTM program [66] Re error εr(%) Im error εi(%)
−0.7920± j12.7628 −0.7927± j12.7660 0.088 0.025
−0.2848± j8.3661 −0.2849± j8.3675 0.035 0.017
−5.5187± j7.9509 −5.5187± j7.9508 0.000 0.001
−5.3325± j7.9240 −5.3325± j7.9240 0.000 0.000
−5.2238± j7.8156 −5.2238± j7.8156 0.000 0.000
−5.2035 −5.2019 0.031 -
−3.4041 −3.4040 0.003 -
−0.4426± j1.2243 −0.4427± j1.2241 0.023 0.016
−0.4404± j0.7413 −0.4404± j0.7413 0.000 0.000
0.0000 −0.0000 0.000 -
−0.1975 −0.1975 0.000 -
−0.4274± j0.4979 −0.4276± j0.4980 0.047 0.020
−3.2258 −3.2258 0.000 -
Table 5.2: Eigenvalues of case PL05 = 1.5pu,QL05 = 0.5pu with constant current load (type 1)
MATLABTM program [66] Re error εr(%) Im error εi(%)
−0.7897± j12.7654 −0.7904± j12.7686 0.089 0.025
−0.2767± j8.3432 −0.2768± j8.3447 0.036 0.018
−5.5214± j7.9516 −5.5214± j7.9516 0.000 0.000
−5.3335± j7.9247 −5.3335± j7.9247 0.000 0.000
−5.2273± j7.8259 −5.2273± j7.8259 0.000 0.000
−5.2046 −5.2030 0.031 -
−3.4464 −3.4462 0.006 -
−0.4536± j1.1824 −0.4537± j1.1822 0.022 0.017
−0.4411± j0.7416 −0.4412± j0.7416 0.022 0.000
0.0000 −0.0000 0.000 -
−0.1974 −0.1974 0.000 -
−0.4275± j0.4979 −0.4276± j0.4980 0.023 0.020
−3.2258 −3.2258 0.000 -
62
Table 5.3: Eigenvalues of case PL05 = 1.5pu,QL05 = 0.5pu with constant impedance load (type 2)
MATLABTM program [66] Re error εr(%) Im error εi(%)
−0.7879± j12.7674 −0.7887± j12.7706 0.101 0.025
−0.2702± j8.3256 −0.2703± j8.3271 0.037 0.018
−5.5236± j7.9524 −5.5236± j7.9523 0.000 0.001
−5.3344± j7.9254 −5.3344± j7.9253 0.000 0.001
−5.2302± j7.8337 −5.2301± j7.8337 0.002 0.000
−5.2054 −5.2039 0.029 -
−3.4805 −3.4801 0.011 -
−0.4616± j1.1490 −0.4617± j1.1489 0.022 0.009
−0.4419± j0.7418 −0.4419± j0.7418 0.000 0.000
0.0000 −0.0000 0.000 -
−0.1973 −0.1973 0.000 -
−0.4275± j0.4979 −0.4277± j0.4980 0.047 0.020
−3.2258 −3.2258 0.000 -
One unstable case for type 0 load, PL05 = 4.5pu and QL05 = 0.5pu, is represented on Tables 5.4,
5.5 and 5.6. It is noticeable that despite the fact that the system becomes unstable at P5 = 4.5 for
constant power load, it is stable for both constant current and constant impedance load. The reason
comes directly from the mathematical formulation of loads.
Table 5.4: Eigenvalues of case PL05 = 4.5pu,QL05 = 0.5pu with constant power load (type 0)
MATLABTM program [66] Re error εr(%) Im error εi(%)
−0.7748± j12.7336 −0.7751± j12.7373 0.039 0.029
−0.2843± j8.0707 −0.2845± j8.0723 0.070 0.020
−6.7303± j7.8879 −6.7291± j7.8883 0.018 0.005
−5.6033± j7.9237 −5.6034± j7.9238 0.002 0.001
−5.2941± j7.6426 −5.2935± j7.6433 0.011 0.009
−5.2551 −5.2541 0.019 -
0.1316± j2.2834 0.1268± j2.2798 3.785 0.158
−2.5503 −2.5529 0.102 -
−0.4858± j0.7475 −0.4858± j0.7475 0.000 0.000
−0.0000 −0.0000 0.000 -
−0.5341± j0.5305 −0.5341± j0.5306 0.000 0.019
−0.1976 −0.1976 0.047 0.020
−3.2258 −3.2258 0.000 -
63
Table 5.5: Eigenvalues of case PL05 = 4.5pu,QL05 = 0.5pu with constant current load (type 1)
MATLABTM program [66] Re error εr(%) Im error εi(%)
−0.7331± j12.7807 −0.7335± j12.7842 0.054 0.027
−0.2496± j8.0630 −0.2497± j8.0650 0.040 0.025
−6.7680± j7.9729 −6.7669± j7.9730 0.016 0.001
−5.6287± j7.9557 −5.6287± j7.9557 0.000 0.000
−5.2813± j7.8419 −5.2812± j7.8419 0.002 0.000
−5.2725 −5.2715 0.019 -
−3.5295 −3.5296 0.002 -
−0.5020± j1.2534 −0.5020± j1.2531 0.000 0.024
−0.0000 −0.0000 0.000 0.000
−0.4910± j0.7561 −0.4910± j0.7561 0.000 0.000
−0.5360± j0.5324 −0.5360± j0.7561 0.000 29.586
−0.1972 −0.1972 0.000 -
−3.2258 −3.2258 0.000 -
Table 5.6: Eigenvalues of case PL05 = 4.5pu,QL05 = 0.5pu with constant impedance load (type 2)
MATLABTM program [66] Re error εr(%) Im error εi(%)
−0.7281± j12.7901 −0.7285± j12.7936 0.055 0.027
−0.2444± j8.0639 −0.2444± j8.0659 0.000 0.025
−6.7772± j7.9894 −6.7760± j7.9895 0.018 0.001
−5.6338± j7.9639 −5.6338± j7.9639 0.000 0.000
−5.2939± j7.8712 −5.2938± j7.8712 0.002 0.000
−5.2799 −5.2790 0.017 -
−3.8110 −3.8105 0.013 -
−0.5304± j1.0434 −0.5303± j1.0434 0.019 0.000
−0.4950± j0.7653 −0.4950± j0.7653 0.000 0.000
−0.5371± j0.5336 −0.5371± j0.05336 0.000 0.000
−0.0000 −0.0000 0.000 -
−0.1970 −0.1970 0.000 -
−3.2258 −3.2258 0.000 -
5.1.2 9 bus with SVC
As aforementioned, the SVC produces better outcomes when placed in the (electrical) middle of a
line. To simplify the network, a new bus is added in the middle of the line 5-7 for the purpose of installing
64
the SVC there. The representation of the modified network is in Figure C.2. Also, renumbering and
changing of data is necessary. The new bus 4 is treated as PQ since the power flow is run the same
way as before and it produces an absolute value for bus 4 voltage. V4 will then be compared with the
SVC reference that the user has set in the first place (controller that was explained in Chapter 4).
The data presented in Appendix C.2 were used for making both 9bus svc+modelB.raw and 9bus-
GENRED+ST svc.dyr files. Furthermore, for the case with no SVC (SVC off) the data files were named
10bus modelB.raw and 9busGENRED+ST.dyr.
Reference [53] assumes the injected reactive power of the SVC to be null (Qsvc = 0) and therefore
the equilibrium point for the device susceptance is also null (Beqsvc = xeq3 = 0) and not calculated as a
regular case. This is the same as assuming the SVC not to be modeled in the power flow.
After running the developed program with the indicated files, it is possible to compare results with
[53]. As one can see, the program gives really accurate results inasmuch as they are equivalent at least
until the fourth decimal place. The increasing load power is at bus 6 and the loads are assumed as
constant power type.
First, one has to evaluate the case where no SVC is set in order to compare later if the device really
puts off the Hopf bifurcation a little farther from the nominal case (Table 5.7).
Table 5.7: Eigenvalues of nominal case P6 = 1.25pu, V6 = 0.996pu without SVC
MATLABTM program [53] Associated States Error ε(%)
−0.8492± j12.7672 −0.8492± j12.7672 ω3, δ3 0.0
−0.2512± j8.3648 −0.2512± j8.3648 ω2, δ2 0.0
−2.2421± j3.0195 −2.2421± j3.0195 Efd1, E′
q1 0.0
−4.6654± j1.3830 −4.6654± j1.3830 Efd3, Efd2, E′
d3, E′
d2 0.0
−3.4855± j1.0014 −3.4855± j1.0014 Efd2, E′
q2, E′
fd1 0.0
−3.2258 −3.2258 E′
d1 0.0
−2.2613 −2.2613 E′
d2, Efd2 0.0
0.0000 −0.0000 δ1 0.0
−0.8882 −0.8882 E′
q3 0.0
−0.1365 −0.1365 ω1 0.0
The nominal case with SVC is illustrated on Table 5.8 so that the reader realizes where the SVC
eigenvalues are placed. This, as one will see, is very relevant for this specific device. Furthermore, it is
noticeable that the SVC affects the voltage stability modes.
65
Table 5.8: Eigenvalues of nominal case P6 = 1.25pu, V6 = 0.996pu with SVC
MATLABTM program [53] Associated States Error ε(%)
−0.8432± j12.7698 −0.8432± j12.7698 ω3, δ3 0.0
−0.2677± j8.4245 −0.2677± j8.4245 ω2, δ2 0.0
−2.6818± j2.0672 −2.6818± j2.0672 Efd1, E′
q1 0.0
−4.6981± j1.3196 −4.6981± j1.3196 Efd3, Efd2, E′
d3, E′
d2 0.0
−3.8082± j1.5021 −3.8082± j1.5021 Efd2, E′
q2, E′
fd1 0.0
−3.2258 −3.2258 E′
d1 0.0
−1.7352 −1.7352 E′
d2, Efd2 0.0
0.0000 −0.0000 δ1 0.0
−0.8871 −0.8871 E′
q3 0.0
−0.1365 −0.1365 ω1 0.0
−10.2417± j26.2143 −10.2417± j26.2143 x2, x1, x3 0.0
−78.4325 −78.4325 x1, x3 0.0
Table 5.9 shows that the inducement of the instability is due to an exciter mode. This way what is
expected for the SVC to be able to do is to put off the Hopf bifurcation farther from the nominal case so
that the stability of the system is improved.
Table 5.9: Eigenvalues at Hopf bifurcation case P6 = 4.69pu, V6 = 0.8499pu without SVC
MATLABTM program [53] Associated States Error ε(%)
−0.8800± j12.6855 −0.8800± j12.6855 ω3, δ3 0.0
−0.2805± j8.0166 −0.2805± j8.0166 ω2, δ2 0.0
0.0824± j6.5781 0.0824± j6.5781 E′
q1, Efd1, E′
q2, Efd2 0.0
−4.4888± j1.2500 −4.4888± j1.2500 Efd3, E′
d3 0.0
−2.7595± j1.2951 −2.7595± j1.2951 Efd1, E′
q1 0.0
−3.2258 −3.2258 E′
d1 0.0
−4.4044 −4.4044 E′
d2 0.0
0.0000 −0.0000 δ1 0.0
−1.3028 −1.3028 E′
q3 0.0
−0.1361 −0.1361 ω1 0.0
Finally, 9 bus results with SVC at the Hopf bifurcation are shown on Table 5.10. As expected, the
device delays the system to become unstable at higher loads (for bus 6 in this specific case). Besides,
it is not at the exciter modes that it occurs but at the SVC modes. Figure 5.1 represents the eigenvalues
trajectory where the marked eigenvalues are the SVC initial eigenvalues. With the represented trajectory
it is possible to see the SVC eigenvalues crossing the imaginary axis.
66
Table 5.10: Eigenvalues at Hopf bifurcation P6 = 4.77pu, V6 = 0.8418pu with SVC
MATLABTM program [53] Associated States Error ε(%)
−0.7566± j12.7553 −0.7566± j12.7553 ω3, δ3 0.0
−0.2440± j7.9110 −0.2440± j7.9110 ω2, δ2 0.0
−2.7050± j3.2341 −2.7050± j3.2341 Efd1, E′
q1 0.0
−4.4521± j1.1671 −4.4521± j1.1671 Efd3, Efd2, E′
d3, E′
d2 0.0
−2.9925± j1.1198 −2.9925± j1.1198 Efd2, E′
q2, E′
fd1 0.0
−3.2258 −3.2258 E′
d1 0.0
−3.9913 −3.9913 E′
d2, Efd2 0.0
0.0000 −0.0000 δ1 0.0
−1.3249 −1.3249 E′
q3 0.0
−0.1364 −0.1364 ω1 0.0
0.1994± j43.7892 0.1994± j43.7892 x2, x1, x3 0.0
−93.4680 −93.4680 x1, x3 0.0
Figure 5.1: Eigenvalues trajectory of the IEEE 9 bus network with SVC
From these results it becomes clear that the static var compensator enhances the security of the
system. In fact, because of voltage control, Bsvc allows the system to work closer to its thermal limits
67
since the reactive power at the SVC transmission line is considerably reduced.
5.1.3 9 bus with TCSC
Unlike the SVC, the TCSC is placed in series with the transmission line that is compensated. This
is indeed a great advantage of series compensation over shunt compensation as mentioned in Chapter
3. To simulate this device, once more the 9 bus network was modified. As a starting network, [53]
includes the SVC bus (with the SVC turned off) and then includes another bus numbered 11. Figure C.3
illustrates the new network. Line 4-6 was split in two so that it is compensated. The new line 4-11 is
just like the others, however line 11-6 is supposed to simulate just a TCSC and therefore is composed
only by a reactance Xtcsc. Usually it is within the range of 40%XL − 70%XL for compensation (in this
case it was chosen 40%). The data for this system are the same as in Subsection 5.1.2 with only some
changes indicated on Tables C.12, C.13 and C.14. Once more, buses 4 and 11 are treated as PQ.
The data presented were used for making both 9bus tcsc+modelB.raw and 9busGENRED+ST tcsc.dyr
files. The comparison with the case of no TCSC can be once more illustrated with Tables 5.7 and 5.9.
The equilibrium point of xtcsc is the reactance of line 11-6 used for determining the power flow. From
here, the other relevant equilibrium points (Im in this case) are computed. It is considered that the TCSC
reactance is always negative for assuming it to be a capacitor. Also, Sk = 0 since it is assumed for the
current across the line to be constant. Regarding Ki, it is changed during the results demonstration
since [53] had different eigenvalues for the nominal case if Ki = 1.25.
Once more there is result comparison between the developed program and [53]. The increasing load
power is at bus 6 again and the loads are assumed as constant power type.
The TCSC nominal case is illustrated on Table 5.11. Once again, as expected, the device affects the
voltage stability modes.
68
Table 5.11: Eigenvalues of nominal case P6 = 1.25pu,Ki = 1 with TCSC
MATLABTM program [53] Associated States Error ε(%)
−0.8462± j12.7752 −0.8462± j12.7752 ω3, δ3 0.0
−0.3111± j8.5292 −0.3111± j8.5292 ω2, δ2 0.0
−2.2614± j2.9860 −2.2614± j2.9860 Efd1, E′
q1 0.0
−4.6805± j1.3669 −4.6805± j1.3669 Efd3, E′
d2 0.0
−3.6411± j1.0263 −3.6411± j1.0263 Efd1, E′
fd2 0.0
−3.2258 −3.2258 E′
d1 0.0
−1.9956 −1.9956 E′
d2, Efd2 0.0
0.0000 −0.0000 δ1 0.0
−0.8800 −0.8800 E′
q3 0.0
−0.1365 −0.1365 ω1 0.0
−9.3897± j29.7347 −9.3897± j29.7347 x1, Im, xtcsc 0.0
−81.1618 −81.1618 Im, xtcsc 0.0
The Hopf bifurcation is now at P6 = 5.03pu. Here a change in Ki is needed just for having the exact
same results as [53]. It should be the same value however this was done in order to get the same results
as [53]. Despite the fact that for Ki = 1 and Ki = 1.25 the Hopf Bifurcation is the same, the eigenvalues
differ. Assume the data from Table C.15 to present results on Table 5.12. Unlike the SVC, the instability
occurs due to the same exciter modes as in the network with no TCSC.
Table 5.12: Eigenvalues at Hopf bifurcation case P6 = 5.03pu,Ki = 1.25 with TCSC
MATLABTM program [53] Associated States Error ε(%)
−0.8265± j12.7169 −0.8265± j12.7169 ω3, δ3 0.0
−0.6967± j7.9918 −0.6967± j7.9918 ω2, δ2 0.0
0.0650± j6.3502 0.0650± j6.3502 Efd1, E′
q1 0.0
−4.4976± j1.2171 −4.4976± j1.2171 Efd3, E′
d3 0.0
−2.7598± j1.1326 −2.7598± j1.1326 Efd1, E′
q1 0.0
−3.2258 −3.2258 E′
d1 0.0
−4.4837 −4.4837 E′
d2 0.0
0.0000 −0.0000 δ1 0.0
−1.3125 −1.3125 E′
q3 0.0
−0.1363 −0.1363 ω1 0.0
−1.5385± j47.4112 −1.5385± j47.4112 x1, Im, xtcsc 0.0
−97.4498 −97.4498 Im, xtcsc 0.0
Concluding, also the TCSC puts off the Hopf bifurcation farther from the nominal case. Its advantage
69
over the SVC is the independence of the location. This leads to better enhancement of stability since
instabilities appear for higher power loads. The system is now operating closer to its limits since less
reactive power is transmitted.
5.1.4 9 bus with both SVC and TCSC
One more test with FACTS is taken for the 9 bus 3-machine system with Qsvc = 0, increasing load at
bus 6 and loads of constant power type. The data used are from Tables C.7, C.8, C.9, C.12, C.13 and
C.15 and are part of files 9bus svc+tcsc+modelB.raw and 9busGENRED+ST svc+tcsc.dyr. Again the
instability is induced by the SVC associated states x1svc, x2svc and x3svc. Having both devices turned
on, the Hopf bifurcation is now at P6 = 5.07pu having considerably increased in comparison with the
initial 9 bus network, Figure C.1.
Table 5.13: Eigenvalues at Hopf bifurcation case P6 = 5.07pu,Ki = 1.25 with SVC and TCSC
MATLABTM program [53] Associated States Error ε(%)
−0.7422± j12.7668 −0.7422± j12.7668 ω3, δ3 0.0
−0.2382± j8.1122 −0.2382± j8.1122 ω2, δ2 0.0
−2.7193± j3.0691 −2.7193± j3.0691 Efd1, E′
q1 0.0
−4.4630± j1.1475 −4.4630± j1.1475 Efd3, E′
d3 0.0
−2.9148± j1.0296 −2.9148± j1.0296 Efd1, E′
q1 0.0
−3.2258 −3.2258 E′
d1 0.0
−4.1631 −4.1631 E′
d2 0.0
0.0000 −0.0000 δ1 0.0
−1.3223 −1.3223 E′
q3 0.0
−0.1365 −0.1365 ω1 0.0
0.0736± j39.1361 0.0736± j39.1361 x1svc, x2svc, x3svc 0.0
−92.4878 −92.4878 x1svc, x3svc 0.0
−5.2713± j46.1114 −5.2713± j46.1114 x1, Im, xtcsc 0.0
−92.9532 −92.9532 Im, xtcsc 0.0
5.2 IEEE 5 bus network
The IEEE 5 bus network was used to verify that power flow for FACTS is working properly. It is
recalled here that this analysis was not used for result comparison with [53]. When the user needs to
use it in the future, it is possible to start from this power flow instead of considering only FACTS in the
dynamic analysis. Result comparison is between the MATLABTM developed program and [9].
70
5.2.1 5 bus without FACTS
In order to easier apply FACTS over a Newton-Raphson Power Flow, it was decided that a new
algorithm had to be implemented [9]. This subsection illustrates an example used for power flow without
FACTS.
Both network and its data are presented in Appendix C.4. The network data were taken from [5] and
were placed in 5bus.raw. The .dyr file is 5bus.dyr but it was just created so that the program could run
(remember that it always asks for two files), it does not mean anything in terms of generators or exciters.
The way to get these results is to set a breakpoint immediately after power flow runs in file Analise DynVS.m
and write ”main” in MATLABTM command line. This is just for proving that power flow results are correct
which means that it is not necessary to run the whole program, just the power flow part. That is why only
part of one iteration in dynamic analysis is taken.
Power flow results for IEEE 5 bus system are the same as in [9], see Table 5.14. The error is
determined taking into account only the decimal places of both voltage magnitudes and angles that [9]
shows.
Table 5.14: IEEE 5 bus power flow
MATLABTM program [9]V error εv(%) θ error εθ(%)
V θ V θ
1.0600 0.0000 1.060 0.00 0.0 0.00
1.0000 -2.0612 1.000 -2.06 0.0 0.00
0.9872 -4.6367 0.987 -4.64 0.0 0.00
0.9841 -4.9570 0.984 -4.96 0.0 0.00
0.9717 -5.7649 0.972 -5.77 0.0 0.17
With a working simple power flow it was possible to introduce in a modular way FACTS devices. In
this case they were the SVC and the TCSC.
5.2.2 5 bus with SVC
To simulate a static var compensator, the system’s only further information was the SVC’s. As men-
tioned in Section 4.1, this device can be modeled as a simple susceptance set in any bus. In the following
example, the SVC was placed in the bus called ”Lake”. The algorithm requires the SVC location, its de-
sired voltage, its initial susceptance and both inferior and superior susceptance limits. All values are in
p.u..
The files needed to perform the simulation are 5bus svc.raw and 5bus svc.dyr. The simulation is
done in the same way as for the previous case (simple power flow). Once more the .dyr file was
created just for the program to run. If the program was run until the end no meaningful results would be
accomplished. This is due to the fact that the SVC control was already done at the power flow and not
71
with the elapse of the dynamic simulation.
As expected, the SVC sets bus Lake’s voltage to 1pu and by doing so modifies its susceptance. Bsvc
was the same as in [9], i.e. 0.2047pu.
Table 5.15: IEEE 5 bus power flow with SVC
MATLABTM program [9]V error εv(%) θ error εθ(%)
V θ V θ
1.0600 0.0000 1.060 0.00 0.0 0.000
1.0000 -2.0533 1.000 -2.05 0.0 0.000
1.0000 -4.8379 1.000 -4.83 0.0 0.002
0.9944 -5.1073 0.994 -5.11 0.0 0.000
0.9752 -5.7975 0.975 -5.80 0.0 0.000
5.2.3 5 bus with TCSC
To simplify the TCSC inclusion a new bus was created, ”Lakefa”, to compensate the transmission line
connected between Lake and Main. The purpose of this compensation was to maintain active power
flow from Lakefa to Main at 21MW. Similarly to the SVC, the TCSC data include its location (sending
and receiving ends) and TCSC reactance initial value and its limits. The new network data and design
are illustrated below. All data are present in 5bus tcsc.raw and the dynamic file to use is 5bus tcsc.dyr.
By compensating a transmission line, the TCSC modifies its reactance: Xtcsc = −0.0216pu (same
value as in [9]). [9] only shows the voltages of the buses that were already there so Lakefa will not have
its results compared.
Table 5.16: IEEE 5 bus power flow with SVC
MATLABTM program [9]V error εv(%) θ error εθ(%)
V θ V θ
1.0600 0.0000 1.060 0.00 0.0 0.000
1.0000 -2.0380 1.000 -2.04 0.0 0.000
0.9870 -4.7274 0.987 -4.72 0.0 0.002
0.9876 -4.4605 0.988 -4.46 0.0 0.000
0.9844 -4.8113 0.984 -4.81 0.0 0.000
0.9718 -5.7009 - - - -
72
Chapter 6
Results and Further Assumptions
The results obtained from previous sections are represented in this chapter. With validated models
from Chapter 5 it is possible to consider different network scenarios in order to take conclusions from
there. Also different tests with the SVC in particular were performed as well as its comparison with the
TCSC in terms of oscillation damping.
6.1 IEEE 9 bus network
6.1.1 9 bus with SVC - located in the middle of a line vs at a bus
In Chapters 3 and 4 it was mentioned that an SVC was more efficient when located in the middle
of a line instead of at a specific bus. For the same network in Subsection 5.1.1 and data from Tables
C.1, C.7, C.8 and C.9 the SVC was placed at bus 5, Table C.10. These data were placed on files
9bus svc atbus+modelB.raw and 9busGENRED+ST svc atbus.dyr.
The results obtained for constant power loads and an increasing real power at bus 5 prove that the
SVC eigenvalues become unstable earlier than in the case of Subsection 5.1.2.
Table 6.1: SVC in the middle of a line vs SVC at a bus
Hopf bifurcation
Line - PL6[MW ] Bus - PL5[MW ]
477 475
Although the difference is not very significant for this specific network, when an electrical energy
transmission company buys an SVC it expects to recover the investment as soon as possible exploiting
the most of it. Therefore, the SVC must be as efficient as attainable. Furthermore, it is reminded here
that this is the worst case that the system faces regarding instability.
73
6.1.2 9 bus with SVC equipped with auxiliary controller
Another interesting test performed was to implement an SVC with auxiliary controller on the IEEE 9
bus network and to compare with the simple SVC. The same data in Subsection 5.1.2 were used along
with the data provided on Table C.11 from [53] related to the auxiliary controller. The files to use are
9bus svc auxc+modelB.raw and 9busGENRED+ST svc auxc.dyr.
After running the program, several conclusions could be taken. The SVC with auxiliary controller
increases the damping in several imaginary eigenvalues comparing to the original 9 bus network. How-
ever, the SVC by itself already provides damping and the SVC with auxiliary controller does not improve
these dampings considerably. Yet, it improves the SVC eigenvalues damping and also the stability mar-
gin. Table 6.2 shows the results for the eigenvalues at the nominal case for the 9 bus network without
SVC, with a simple SVC and with an SVC with auxiliary controller (the last two are referred to the 9 bus
network with a new bus where the SVC is placed as in Subsection 5.1.2).
The damping is defined as (6.3). Equations (6.1) and (6.2) represent a complex eigenvalue and its
frequency of oscillation, respectively. To more information refer to [16].
λ = σλ ± jωλ (6.1)
where σλ and ωλ are the complex eigenvalue real and imaginary part, respectively.
f =ωλ2π
(6.2)
ξ = − σλ√σ2λ + ω2
λ
(6.3)
74
Table 6.2: Eigenvalues and their damping for P6 = 1.25pu
Without SVC With SVC With SVC and auxiliary controller
eigenvalues damping eigenvalues damping eigenvalues damping
−0.8485± j12.7640 0.066 −0.8432± j12.7698 0.066 −0.8475± j12.7686 0.066
−0.2507± j8.3597 0.030 −0.2677± j8.4245 0.032 −0.2610± j8.3872 0.031
−2.2412± j3.0203 0.596 −2.6818± j2.0672 0.792 −2.4335± j2.6923 0.671
−4.6659± j1.3824 0.959 −4.6981± j1.3196 0.963 −4.6714± j1.3675 0.960
−3.4832± j0.9992 0.961 −3.8082± j1.5021 0.930 −3.5678± j1.1487 0.952
−3.2258 1.000 −3.2258 1.000 −3.2258 1.000
−2.2668 1.000 −1.7352 1.000 −2.0717 1.000
0.0000 − 0.0000 − 0.0000 −
−0.8880 1.000 −0.8871 1.000 −0.8879 1.000
−0.1365 1.000 −0.1365 1.000 −0.1365 1.000
− − −10.2417± j26.2143 0.364 −15.4222± j11.3618 0.805
− − −78.4325 1.000 −63.1559 1.000
6.1.3 9 bus with TCSC
Also, the TCSC has a better capability of providing damping to the system than the SVC either with
or without auxiliary controller. In order to compare the eigenvalue damping, Table 6.2 is represented
again along with the TCSC case, Table 6.3.
Table 6.3: Eigenvalues and their damping for P6 = 1.25pu
Without FACTS With SVC With SVC and auxiliary controller With TCSC
eig damp eig damp eig damp eig damp
−0.8485± j12.7640 0.066 −0.8432± j12.7698 0.066 −0.8475± j12.7686 0.066 −0.8434± j12.7824 0.066
−0.2507± j8.3597 0.030 −0.2677± j8.4245 0.032 −0.2610± j8.3872 0.031 −0.3276± j8.6436 0.038
−2.2412± j3.0203 0.596 −2.6818± j2.0672 0.792 −2.4335± j2.6923 0.671 −2.2617± j2.9757 0.605
−4.6659± j1.3824 0.959 −4.6981± j1.3196 0.963 −4.6714± j1.3675 0.960 −4.6907± j1.3439 0.961
−3.4832± j0.9992 0.961 −3.8082± j1.5021 0.930 −3.5678± j1.1487 0.952 −3.6936± j1.0287 0.963
−3.2258 1.000 −3.2258 1.000 −3.2258 1.000 −3.2258 1.000
−2.2668 1.000 −1.7352 1.000 −2.0717 1.000 −1.9119 1.000
0.0000 − 0.0000 − 0.0000 − 0.0000 −
−0.8880 1.000 −0.8871 1.000 −0.8879 1.000 −0.8715 1.000
−0.1365 1.000 −0.1365 1.000 −0.1365 1.000 −0.1365 1.000
− − −10.2417± j26.2143 0.364 −15.4222± j11.3618 0.805 −14.2046± j16.1163 0.661
− − −78.4325 1.000 −63.1559 1.000 −71.5103 1.000
75
6.1.4 Influence of the type of perturbation
Another type of perturbation was explained in Section 2.4. It was adopted for demonstrating that in
fact the single increase of PL at a bus influences both Hopf bifurcation and Jacobian singularity. Bus 5
was the bus for increasing the load in the 9 bus network without FACTS once again.
Maintaining the ratio between QL5 and PL5, i.e. tanφ5, the Hopf bifurcation was pulled back to
PL5 = 372MW
QL5 = 148.8MW
(6.4)
and the Jacobian singularity, i.e. when power flow does not converge, was also pulled back to
PL5 = 413MW
QL5 = 165.2MW
. (6.5)
6.2 Influence of FACTS dynamic parameters on stability
Although there is information about FACTS dynamic parameters for the 9 bus network in [53], it
is important to understand how the system responds to different data. This is done not only to get
a deeper knowledge about the control parameters as well as how one should perform if the network
is larger (more buses) and therefore more demanding in terms of system stability. These tests were
performed increasing/decreasing a parameter at a time and comparing to the original 9 bus network
with the corresponding FACTS. The comparison is based on the Hopf bifurcation.
6.2.1 SVC
The controller of static var compensator is composed by five parameters. Each one of them has a
different influence on the overall system dynamic stability. The limit values referred are dependent on
the system and so no specific amounts are given.
• K: stability is (slightly) improved when it takes lower values;
• Tc: stability is improved when it takes lower values. However, time constants in a controller cannot
take extremely low values since SVCs do not respond as quickly in reality;
• Tb: the same observation as Tc is done for this time constant;
• Kp: stability increases as Kp rises; however, if Kp is increased further than a limit value, stability
does not become any better;
• Ki: stability is improved when Ki diminishes until a certain limit value.
76
6.2.2 SVC with auxiliary controller
The SVC parameters influences were explained in Subsection 6.2.1. It is now time to look at the
SVC equipped with an auxiliary controller.
• KB : until a certain limit value, as | KB | increases, stability is improved;
• T1: stability is improved as T1 increases until a limit value;
• T2: stability improves as T2 decreases.
6.2.3 TCSC
• SK : since constant current model was adopted for the TCSC, Sk is always null;
• Tt: maintaining the same value as [53] or decreasing it stability remains unaffected;
• Ts: the same happens as Tt;
• K: stability is (slightly) improved as K decreases.
6.3 Determination of Locations for FACTS
One of the main goals of this research work was to achieve a plausible efficient location for FACTS.
As explained before, the small perturbation used for determining the system stability is a progressive
increase of active power load. Therefore, the first thing to do is to determine which load to increase.
It was assumed to be the load that firstly led the system to instability. The explanation for this is quite
straightforward. If one wants to evaluate system stability, the interesting result is a stability margin. This
margin should always be the worst case to guarantee all cases are stable considering it. In this way it is
possible to state that the system is stable from the nominal value of an active power load to another one
farther away from it.
The static analysis developed in [16] is an efficient and sufficient way to find the bus where to increase
the active power load. It is sufficient because the first step is to evaluate where the eigenvalues of the
initial network are placed. The choice of the bus for increasing the load in the static analysis is not
very relevant for this purpose since the eigenvalues for the initial conditions vary little from one bus to
another. As a consequence, one can choose any bus for the load increasing and then evaluate the
eigenvalues for the nominal case of PLi. The closer to zero the lower the stability margin. This means
that the eigenvalue to consider is the one that stands out in terms of being closer to instability.
Starting from the chosen eigenvalue, it is now time to look at the participation factors. They will give
information about the buses that most influence each eigenvalue, namely the chosen one in the first
step. The bigger the participation factor the greater influence it has on that eigenvalue. This bus is the
one that leads to the worst case of instability and thus the chosen one to apply the small perturbation.
77
Since the chosen bus is the main responsible for instability, a FACTS device should be placed in an
adjacent line to it. The new step is now to choose which one. A simple implementation was to evaluate
the efficiency of the line measuring both active and reactive powers that are transmitted. These powers
are computed according to (6.6) and then a ratio between active and reactive powers is computed.
Afterwards, an average for the terminal buses is taken betweenPijQij
andPjiQji
and called ratio PQ (6.7).
This ratio represents the active power flow in a line comparatively to the reactive power flow. It is
therefore a representative measure of taking advantage of the line that goes deeply into the definition
of FACTS: they lower the reactive power in order to increase the active one. One is thus interested in
compensating the lines that have a low active/reactive power ratio.
Skij =
(1
Z∗Lk
+Y∗Tk
2
)V 2i −
1
Z∗Lk
ViV∗j (6.6)
where i stands for the sending end while j the receiving end and ZLk= Rk + jXk and YTk
= jBk are
characteristics of each k line according to π-model, Figure 6.1.
Figure 6.1: π-model of a transmission line - reprinted from [56]
ratioPQ =
Pij
Qij+
Pji
Qji
2(6.7)
This also could be compared with a measurement of line efficiency, the power factor (6.8). In fact,
power factor relates real power flowing to the load with the apparent power of the circuit (6.9). The closer
to one the more efficient a line is.
cosϕ =P
S(6.8)
where ϕ is the phase angle between current and voltage and the apparent power is given by
S =√P 2 +Q2 (6.9)
being P and Q the active and reactive powers respectively.
78
6.3.1 Results from Locations for FACTS
Once again the 9 bus network is taken to perform the analysis of a good FACTS location. What
this illustrates are the results for the approach taken in Section 6.3. To do so, some new files had
to be created to introduce the SVC and the TCSC in the line that corresponded to the best one for
compensation. The devices were introduced in the same way as in Sections 5.1.2 and 6.1.3, i.e. the
SVC was placed in the middle of a line and the TCSC was placed near a bus giving 40% compensation
to the line.
As referred in Section 6.3, the first thing to look for is where to increase the load, meaning that it will be
the location for small perturbations. It starts with running the program with the simple 9 bus network files,
9bus+ST.raw and 9busGENRED+ST.dyr, and by setting Static Analysis → Run and selecting on ”print”
Eigenvalue Analysis and Bus Participation Factors. Now, one must look at the eigenvalue analysis.
Remember that for the Static Analysis, the eigenvalues are positive and the farther from zero the more
stable the system is. Therefore, by looking at Figure C.8, it is possible to indicate that the most troubled
eigenvalue will be the fourth on the list. It is reminded that one must always look at the initial values
(first columns in eigenvalue analysis and bus participation factors) for these tests because one is only
interested on the initial conditions of the network.
Now one shall look at bus participation factors. For the fourth eigenvalue, the bus that has the bigger
participation factor is bus number 5 which has a load. Thus, this is the bus which one should select to
increase the load (worst case scenario for instability).
It has now ended the increasing load bus choice. Since bus 5 is the most troubled and the one that
will lead to instability in the first place, a FACTS device should be introduced near it. Just one more step
is needed, the choice of which line that should be compensated. There are only two adjacent lines to
bus 5. It is now time to look at the average ratioPijQij
. Line 5-4 has a lower ratio than line 5-7. This
means that line 5-4 is less exploited. Since a FACTS will raise the transmitted active power lowering the
reactive power, this line is a suitable place (and the best one) for setting one FACTS device to get the
most of the power system.
Firstly, the SVC was placed at line 5-4 instead of 5-7 which was done in Subsection 5.1.2. The
new files are called 9bus svc+modelB+loc.raw and 9busGENRED+ST svc+loc.dyr. Then the program
was run selecting Dynamic Analysis and bus 5 as the one to increase the power load. It is noticeable
that the system becomes unstable for a higher active power load at bus 5 than in Subsection 5.1.2,
PL5 = 4.79pu vs PL5 = 4.69pu, which means that this is in fact a better location for the SVC. The
nominal case eigenvalues are represented in Figures 6.2 and 6.3. It is possible to see that they got
farther away from the RHP.
79
Figure 6.2: SVC best location eigenvalues, nominal case
Figure 6.3: SVC best location eigenvalues plotted, nominal case
The same thing was done for the TCSC in order to compare with Subsection 6.1.3. The files to
use are 9bus tcsc+modelB+loc.raw and 9busGENRED+ST tcsc+loc.dyr. From the unstable case at
PL5 = 5.03pu one got PL5 = 6.79pu, a significant improvement to the older network.
80
Figure 6.4: TCSC best location eigenvalues, nominal case
Figure 6.5: TCSC best location eigenvalues plotted, nominal case
6.4 IEEE 14 bus network
Another power system network was subjected to stability tests. The data for the IEEE 14 bus network
81
were taken from [1] and placed in files 14bus.raw and 14bus.dyr. After performing the static analysis,
the load at bus 14 was the chosen one to be increased and line 14-9 was selected to be equipped with
FACTS. The devices used were the SVC with auxiliary controller1 and the TCSC2. No power flow for
FACTS was simulated and the loads were assumed as constant power type.
For the simple 14 bus system, instabilities appeared at PL14 = 162.9MW and the Jacobian became
singular at PL14 = 169.9MW . For the SVC with auxiliary controller instability was postponed until
PL14 = 164.9MW , a slightly improvement, and the Jacobian became singular at the same loading
PL14 = 169.9MW since power flow was not affected by the introduction of the device.
On the other hand, when the TCSC was placed, the system suffered a much more accentuated
improvement: instabilities occurred only at PL14 = 178.9MW and singularities of the Jacobian at PL14 =
187.9MW . One can see that power flow was affected due to the initial conditions of the TCSC reactance
since it was placed in the line 14-9.
Figure 6.6: P-V curve for 14 bus network
6.5 IEEE 30 bus network
A last sequence of tests was performed in order to summarize and gather all the developed work
described along the thesis, namely in Chapter 6.
The IEEE 30 bus network data can be found in [4] or in files 30bus.raw and 30bus.dyr. Adopting
the method that has been described in this chapter, the load at bus 30 should be increased. Performing
a dynamic analysis and considering the loads as constant power type, it is noticeable that the system
does not become unstable before power flow diverges, i.e. before the Jacobian becomes singular (at
1Files 14bus svcac.raw and 14bus svcac.dyr.2Files 14bus tcsc.raw and 14bus tcsc.dyr.
82
PL30 = 58.6MW ). As a consequence, FACTS do not improve the system in terms of dynamic stability.
It became then necessary to use the FACTS power flow. In fact, this is what happens in reality because
one cannot ignore the physical reactance of these devices. Neglecting it was an assumption described
in [53] and in most results along the thesis3. The authors from [53] were in fact more concerned about
the Hopf bifurcation that is the first instability point in the system and less concerned with the singularity
of the Jacobian.
The line to be compensated was 30 - 10 according to the approach for placing FACTS considered in
this chapter.
6.5.1 30 bus with SVC
Using the same dynamic data for the static var compensator as in Section 5.1.2, the SVC modes
became unstable at PL30 = 72.6MW 4. To take more advantage of the SVC the system needed a
better control data for the device. According to Subsection 6.2.1, the time constants were decreased,
Tc = Tb = 0.002s. With the adapted data, the system became unstable only at PL30 = 91.6MW .
Besides the load at which the system became unstable, the load at which power flow diverged was
also interesting to analyze. Since the SVC was incorporated into the power flow analysis, the Jaco-
bian became singular only at PL30 = 104.6MW . In order to have these results, the initial value of the
reactance was 0.01pu and the limits ±3.0pu. These limits are in fact unrealistic, they are supposed to
simulate ±∞ due to the fact that there are no limits in the dynamic analysis having as a reference [53].
6.5.2 30 bus with SVCac
As expected, power flow diverged also at PL30 = 104.6MW 5. To corroborate that the SVC with
auxiliary controller had indeed a better improvement than the SVC itself, the same data as Section 6.1.2
were used and the result was an instability at PL30 = 92.6MW .
6.5.3 30 bus with TCSC
For the TCSC case, instabilities occurred at PL30 = 61.6MW and power flow diverged at PL30 =
63.6MW 6. The fact that these results are worse than for the SVC and SVC with auxiliary controller is
due to the choice of parameters that the power flow needs a priori7. Nevertheless, one can see once
more that both locations of Hopf bifurcation and Jacobian singularity were moved farther away with the
implementation of the device (Figure 6.7 represents a sketch for all tested devices on which the SVC is
represented for data including Tc = Tb = 0.02s).
3The main goal was to implement the models in [53].4Files 30bus svc.raw and 30bus svc.dyr.5Files 30bus svcac.raw and 30bus svcac.dyr.6Files 30bus tcsc.raw and 30bus tcsc.dyr.7TCSC reactance, TCSC reactance limits and power flow regulation on the compensated line.
83
Figure 6.7: P-V curve for 30 bus network
84
Chapter 7
Conclusions
An expanding world and the emergence of new technologies demand a higher efficiency in power
transmissions. As most transmission lines are AC, the most efficient way, HVDC, is out of the question
to be used since the conversion from AC to DC would be extremely expensive. The only choice left is
to exploit better the existent AC lines. Devices called FACTS have been studied for decades in order to
improve power transmission. The investment is really expensive but rewarding. Besides, not only better
quality power is transmitted but also the security of the system is improved.
A way to study small perturbations of the network to conclude about its stability is small signal anal-
ysis. The idea is to define dynamic models of the system components such as generators, governors,
exciters and FACTS to define a DAE model. From here a system is written having as a basis the DAE
model. The evaluation of the system matrix eigenvalues dictates whether the network is stable or not.
The small perturbation typically chosen is a progressive increase of active power load of a bus. However,
this perturbation can improve the system by itself since for the same amount of reactive power one has
more active power. Tests were performed maintaining the relation between these two powers in order
to confirm it. Yet, all other tests were performed increasing only the active power sticking to the main
references [66] and [53].
Voltage-dependent loads were found to have a great impact on the Hopf bifurcation. In fact, different
models stated that the system was stable while others stated that the system was not stable for the
same value of active power load.
For the IEEE 9 bus system tests were performed with and without FACTS, namely the SVC and
the TCSC. It was concluded that FACTS are a stabilizer factor since the Hopf bifurcation was put off
farther away from the nominal case. One could also conclude that the TCSC is more efficient regarding
oscillation damping and stability margin. Furthermore, the SVC was more efficient when placed in the
middle of a line than at a bus and when equipped with an auxiliary controller.
FACTS location had better results when placed at the less exploited line in terms of relation ac-
tive/reactive power near to the bus where the perturbation was occurring.
By defining the control parameters of the devices one could get different stability results. However,
the parameters must be chosen wisely in order to get realistic scenarios.
85
For bigger bus systems (IEEE 30 bus) the worst case scenarios for instability can make it difficult
for FACTS to enhance the security of the system. To get more realistic results it became necessary to
consider the devices in the power flow. This way FACTS modified the location of the Jacobian singularity
and also the Hopf bifurcation with the dynamic analysis.
7.1 Contributions
• Development of the dynamic model of the static exciter (it works along GENROE and GENRED
without governors). This allowed result comparison with reference [53];
• development of the voltage-dependent loads to take conclusions and to compare with [66];
• generalization of the program, in fact some functions were only prepared for the IEEE 9 bus system;
• the program was set to operate with as many generators as one wants. Only six generators
maximum were possible until now which is a major limitation for large networks;
• the program was set to operate with different types of generators, exciters and governors. The
program was working only with the same type since only identical models had the same number
of variables. This was in fact a starting point to do the same for FACTS;
• development of a power flow algorithm in order to implement FACTS more easily. From here SVC
and TCSC were developed;
• development of dynamic models of the SVC and the TCSC to include in the DAE model derived in
Chapter 2;
• to choose a good location for implementing FACTS;
• creation of several input files for the program. This includes:
– simple larger bus networks;
– inclusion of FACTS parameters for dynamic analysis in .raw and .dyr files;
– inclusion of FACTS parameters for power flow in .raw files;
– extension of networks for inclusion of FACTS (creation of new buses and transmission lines).
7.2 Future Work
Many things can be done having the developed work in this thesis as a starting point. This allows to
educate more people about FACTS and to have a simple program to use in order to evaluate the stability
of a system. Some ideas follow:
• more dynamic models of other FACTS can be developed in order to have a more complete system
in terms of choice of the devices;
86
• to develop another SVC considering a model based on nTSC-TCR to compare to the existing one;
• to modify the power flow for FACTS so that different types of devices can be simulated;
• in the existing models, it would be more realistic if reactance/susceptance limits were implemented,
on both power flow and dynamic analysis;
• to gather and analyze realistic information regarding the losses with and without FACTS;
• to perform an economic balance to conclude whether it is economically viable to invest on FACTS,
i.e. to calculate the money losses due to the under exploitation of the transmission lines and the
saved and invested money on FACTS.
87
88
Appendix A
Dynamic models
A.1 GENROE
Figure A.1: GENROE - reprinted from [71]
89
dδidt = ωi − ωS
dωi
dt = ωS
2Hi
[TMi −Di(ωi − ωS)−
(X
′′di−Xlsi
X′di−Xlsi
)E
′
qi0Iqi −
(X
′di−X
′′di
X′di−Xlsi
)Ψ1dIqi −
(X
′′qi−Xlsi
X′di−Xlsi
)E
′
diIdi
+
(X
′qi−X
′′qi
X′qi−Xlsi
)Ψ2qiIdi + (X
′′
qi −X′′
di)IdiIqi
]
dE′di
dt = 1T
′q0i
[− E′
di + (Xqi −X′
qi)
(Iqi −
X′qi−X
′′qi
(X′qi−Xlsi)2
(Ψ2qi + (X′
qi −Xlsi)Iqi + S(2)di )
)]
dE′qi
dt = 1T
′d0i
[− E′
qi + (Xdi −X′
di)
(Idi − X
′di−X
′′di
(X′di−Xlsi)2
(Ψ1di + (X′
di −Xlsi)Idi − E′
qi)
)+ Efdi − S(2)
qi
]
dΨ1di
dt = 1T
′′d0i
[−Ψ1di + E′
qi − (X′
di −Xlsi)Idi
dΨ2qi
dt = 1T
′′q0i
[−Ψ2qi − E′
di − (X′
qi −Xlsi)Iqi
(A.1)
δ is the rotor angle
ω is the rotor angular speed
H is the rotor inertia
D is the speed damping
E′
d is the electromotive force due to flux linkage in d-axis
E′
q is the electromotive force due to flux linkage in q-axis
Ψ1d is the amortisseur flux linkage in d-axis
Ψ2d is the amortisseur flux linkage in q-axis
Id is the d-axis stator current component
Iq is the q-axis stator current component
Xls is the leakage reactance of the rotor windings
Xd, X′
d, X′′
d are the synchronous, transient and subtransient reactances in d-axis, respectively
Xq, X′
q, X′′
q are the synchronous, transient and subtransient reactances in q-axis, respectively
T′
d0, T′′
d0 are the open-circuit transient and subtransient time constants in d-axis, respectively
T′
q0, T′′
q0 are the open-circuit transient and subtransient time constants in q-axis, respectively
The 0 indices mean initial conditions.
90
A.2 GENSAL
Figure A.2: GENSAL - reprinted from [71]
A.3 IEEET1
Figure A.3: IEEET1 - reprinted from [69]
91
TEdEfd
dt = −(KE + SE(Efd))Efd + VR
TFdRf
dt = −Rf + TF
KFEfd
TAdVR
dt = −VR +KARf−KAKF
TFEfd +KA(Vref − V + VS)
(A.2)
KA is the amplifier gain
TA is the amplifier time constant
KE is the exciter gain
TE is the exciter time constant
KF is the feedback gain
TF is the feedback time constant
Efd is the field voltage output from the exciter (steady-state)
Rf is the field resistance
SE is the saturation function
VR is the regulator voltage output
A.4 TGOV1
Figure A.4: TGOV1 - reprinted from [71]
92
A.5 HYGOV
Figure A.5: HYGOV - reprinted from [71]
A.6 GAST
Figure A.6: GAST - reprinted from [71]
93
Appendix B
Modifications in algorithms from [9]
To introduce the algorithm exactly how it appears in [9], the developed program just created variables
with the same name as it appears in the algorithm so that implementation was easier. However, some
bugs prevented the program from running. The modifications needed are indicated below. Furthermore,
the transformers are assumed to have a unitary relation m = 1, however since the admittance matrix Y
was calculated before for the power flow algorithm that was already implemented, it was not necessary
to compute another one.
In function SVCNewtonRaphson, where function PowerMismatches and NewtonRaphsonJacobian
are called, there should be one more input, ”nmax”:
[DPQ,DP,DQ,flag] = PowerMismatches(nmax,nbb,tol,bustype,flag,PNET,QNET,PCAL,QCAL)
[JAC] = NewtonRaphsonJacobian(nmax,nbb,bustype,PCAL,QCAL,VM,VA,YR,YI)
On the same function, where StateVariablesUpdates is called no variable called ”it” should be intro-
duced because of how StateVariablesUpdates was created.
[VA,VM] = StateVariablesUpdates(nbb,D,VA,VM)
In function main regarding TCSC where TCSCNewtonRaphson is called it misses variable ”nmax”.
[VM,VA,it,X] = TCSCNewtonRaphson(nmax, tol, itmax, ngn, nld, nbb, bustype, genbus, load-
bus, PGEN, QGEN, QMAX, QMIN, PLOAD, QLOAD, YR, YI, VM, VA, NTCSC, TCSCsend, TCSCrec,
X, XLo, XHi, Flow, Psp, PSta)
In function TCSCNewtonRaphson the same variable (”nmax”) misses in the function definition be-
cause it is needed later on.
94
function [VM,VA,it,X] = TCSCNewtonRaphson(nmax, tol, itmax, ngn, nld, nbb, bustype, gen-
bus, loadbus, PGEN, QGEN, QMAX, QMIN, PLOAD, QLOAD, YR, YI, VM, VA, NTCSC, TCSCsend,
TCSCrec, X, XLo, XHi, Flow, Psp, PSta)
Also in TCSCNewtonRaphson, ”nmax” is missing where PowerMismatches is called.
[DPQ,DP,DQ,flag]=PowerMismatches(nmax,nbb,tol,bustype,flag,PNET,QNET,PCAL,QCAL)
Still in TCSCNewtonRaphson, ”it” is not necessary where StateVariablesUpdating is called.
[VA,VM]=StateVariablesUpdates(nbb,D,VA,VM)
In function TCSCPQflows where Qtcsc is computed it is missing a ”V” to indicate the variable ”VM”.
Qtcsc(ii,kk) = - VM(TCSCsend(ii))2*Bmm - VM(TCSCsend(ii))*VM(TCSCrec(ii))*Bmk*cos(A)
Finally, some further instructions were written to have the information of convergence in the main
program. The convergence flag was added to functions NewtonRaphson, SVCNewtonRaphson and
TCSCNewtonRaphson.
95
Appendix C
Results
Here the data used for results are presented as well as some parts of the program interface.
C.1 IEEE 9 bus data
Figure C.1: 9 bus network - reprinted from [53]
96
Table C.1: Branches data
Sending end Receiving end R X Bc
4 5 0.0100 0.0850 0.176
4 6 0.0170 0.0920 0.158
5 7 0.0320 0.1610 0.306
6 9 0.0390 0.1700 0.358
7 8 0.0085 0.0720 0.149
8 9 0.0119 0.1008 0.209
Table C.2: Bus data
Bus Type P Q V
1 slack - - 1.04 ∠ 0o
2 PV 1.63 - 1.025
3 PV 0.85 - 1.025
4 PQ 0.00 0.00 -
5 PQ 0.00 0.00 -
6 PQ 1.25 0.50 -
7 PQ 0.90 0.30 -
8 PQ 0.00 0.00 -
9 PQ 1.00 0.35 -
Table C.3: Generator data
Sending end Receiving end X xd x′
d xq x′
q T′
do T′
qo H D
1 4 0.0576 0.1460 0.0608 0.0969 0.0969 8.96 0.310 23.64 0.02540
2 7 0.0625 0.8958 0.1198 0.8645 0.1969 6.00 0.535 6.40 0.00660
3 9 0.0586 1.3125 0.1813 1.2578 0.2500 5.89 0.600 3.01 0.00260
Table C.4: Exciter data
Bus KA TA KE TE KF TF
1 20 0.2 1.0 0.314 0.063 0.35
2 20 0.2 1.0 0.314 0.063 0.35
3 20 0.2 1.0 0.314 0.063 0.35
97
C.2 IEEE 9 bus data with SVC
Figure C.2: 9 bus network with SVC - reprinted from [53]
Table C.5: Branches data
Sending end Receiving end R X Bc
5 7 0.0170 0.0920 0.158
10 7 0.0390 0.1700 0.358
10 9 0.0119 0.1008 0.209
8 9 0.0085 0.0720 0.149
8 4 0.0160 0.0805 0.153
4 6 0.0160 0.0805 0.153
6 5 0.0100 0.0850 0.176
98
Table C.6: Bus data
Bus Type P Q V
1 slack - - 1.04 ∠ 0o
2 PV 1.63 - 1.0253
3 PV 0.85 - 1.0253
4 PQ 0.00 0.00 1.0150
5 PQ 0.00 0.00 -
6 PQ 1.25 0.50 -
7 PQ 0.90 0.30 -
8 PQ 0.00 0.00 -
9 PQ 1.00 0.35 -
10 PQ 0.00 0.00 -
Table C.7: Generator data
Sending end Receiving end X xd x′
d xq x′
q T′
do T′
qo H D
1 5 0.0576 0.1460 0.0608 0.0969 0.0969 8.96 0.310 23.64 0.01254
2 8 0.0625 0.8958 0.1198 0.8645 0.1969 6.00 0.535 6.40 0.00680
3 10 0.0586 1.3125 0.1813 1.2578 0.2500 5.89 0.600 3.01 0.00480
Table C.8: Exciter data
Bus KA TA
1 20 0.2
2 20 0.2
3 20 0.2
Table C.9: SVC data
K Tc Tb Kp Ki
0.1 0.02 0.02 0.0 100
99
Table C.10: Bus data
Bus Type P Q V
1 slack - - 1.04 ∠ 0o
2 PV 1.63 - 1.025
3 PV 0.85 - 1.025
4 PQ 0.00 0.00 -
5 PQ 0.00 0.00 1.015
6 PQ 1.25 0.50 -
7 PQ 0.90 0.30 -
8 PQ 0.00 0.00 -
9 PQ 1.00 0.35 -
Table C.11: SVC auxiliary controller data
KB T1 T2
-0.035 0.044 0.02
C.3 IEEE 9 bus data with TCSC
Figure C.3: 9 bus network with TCSC - reprinted from [53]
100
Table C.12: Branches data
Sending end Receiving end R X Bc
5 7 0.0170 0.0920 0.158
10 7 0.0390 0.1700 0.358
10 9 0.0119 0.1008 0.209
8 9 0.0085 0.0720 0.149
8 4 0.0160 0.0805 0.153
4 11 0.0160 0.0805 0.153
11 6 0.0000 -0.0644 0.000
6 5 0.0100 0.0850 0.176
Table C.13: Bus data
Bus Type P Q V
1 slack - - 1.04 ∠ 0o
2 PV 1.63 - 1.0253
3 PV 0.85 - 1.0253
4 PQ 0.00 0.00 1.0150
5 PQ 0.00 0.00 -
6 PQ 1.25 0.50 -
7 PQ 0.90 0.30 -
8 PQ 0.00 0.00 -
9 PQ 1.00 0.35 -
10 PQ 0.00 0.00 -
11 PQ 0.00 0.00 -
Table C.14: TCSC data
Xtcsc Tt Ts Ki Sk
-0.0644 0.02 0.02 1.0 0.0
Table C.15: TCSC data with a different Ki
Xtcsc Tt Ts Ki Sk
-0.0644 0.02 0.02 1.25 0.0
101
C.4 IEEE 5 bus data
Figure C.4: 5 bus network - reprinted from [9]
Table C.16: Branches data
Sending end Receiving end R X Bc
1 2 0.0200 0.0600 0.060
1 3 0.0800 0.2400 0.050
2 3 0.0600 0.1800 0.040
2 4 0.0600 0.1800 0.040
2 5 0.0400 0.1200 0.030
3 4 0.0100 0.0300 0.020
4 5 0.0800 0.2400 0.050
Table C.17: Bus data
Bus Type PG QG PL QL V
1 slack 0.00 0.00 0.00 0.00 1.06 ∠ 0o
2 PV 0.40 0.00 0.20 0.10 1.00
3 PQ - - 0.45 0.15 1.00
4 PQ - - 0.40 0.05 1.00
5 PQ - - 0.60 0.10 1.00
102
C.5 IEEE 5 bus data with SVC
Figure C.5: 5 bus network with SVC - reprinted from [9]
Table C.18: SVC data
Bus Vset Binit Binf Bsup
3 1.00 0.1 -0.25 0.25
C.6 IEEE 5 bus data with TCSC
Figure C.6: 5 bus network with TCSC - reprinted from [9]
103
Table C.19: Branches data of modified network
Sending end Receiving end R X Bc
1 2 0.0200 0.0600 0.060
1 3 0.0800 0.2400 0.050
2 3 0.0600 0.1800 0.040
2 4 0.0600 0.1800 0.040
2 5 0.0400 0.1200 0.030
4 5 0.0800 0.2400 0.050
6 4 0.0100 0.0300 0.020
Table C.20: Bus data
Bus Type PG QG PL QL V
1 slack 0.00 0.00 0.00 0.00 1.06 ∠ 0o
2 PV 0.40 0.00 0.20 0.10 1.00
3 PQ - - 0.45 0.15 1.00
4 PQ - - 0.40 0.05 1.00
5 PQ - - 0.60 0.10 1.00
6 PQ - - 0.00 0.00 1.00
Table C.21: TCSC data
Sending end Receiving end Xinit Xinf Xsup
3 6 -0.015 -0.05 0.05
104
C.7 Program interface
Figure C.7: Type of Load menu
Figure C.8: 9 bus network eigenvalues, static analysis
Figure C.9: 9 bus network bus participation factors, static analysis
105
Figure C.10: SelectingP
Qratio in print/plot menu, static analysis
Figure C.11:P
Qratio, static analysis
106
Appendix D
Data Files
The creation of data files is important for those who will follow up the developed work. Some ex-
amples of data files containing FACTS data are provided. In the .raw files FACTS data begin after
transformers data and in .dyr files after all data. The value of ID FACTS that appears in .raw files
depends on the type of device: 1 for SVC, 2 for TCSC and 3 for SVC with auxiliary controller.
D.1 SVC
In the .raw file the data regarding the SVC are provided, namely the sending end (I), the receiving
end (J), the voltage reference (V set), type of FACTS (ID FACTS), initial susceptance value (init),
lower limit for susceptance value (Lo) and higher limit for susceptance value (Hi), ordered from left to
right considering Figure D.1. The receiving end is always zero for the SVC since it is a shunt device.
The values related to the susceptance are only used when power flow for FACTS is calculated.
Figure D.1: Data for SVC in .raw
The .dyr file has the appearance of Figure D.2. From left to right, top to bottom are the data compo-
nents for the SVC. For simplicity, the SVC and the SVC with auxiliary controller have the same number
of fields in the file but the last three components (related to the auxiliary controller) are represented as
zero for the SVC. Nevertheless, if they take other values other than zero the program itself neglects this
107
information. On the first line are represented by the same order announced the sending end bus, the
name of FACTS (”SVC” in this case), the ordered number of SVCs in the system, K, Tc, Tb and Kp. On
the last line only the first field is relevant which is Ki.
Figure D.2: Data for SVC in .dyr
D.2 SVC with auxiliary controller
The SVC with auxiliary controller files have a similar outlook as the SVC files. Only ID FACTS is
now 3 and the name of the device is ”SVCac”. Moreover, the last three components of the second line
in the .dyr file are now different from zero and represent respectively KB , T1 and T2.
Figure D.3: Data for SVC with auxiliary controller in .raw
Figure D.4: Data for SVC with auxiliary controller in .dyr
D.3 TCSC
Similarly to the SVC, the TCSC components in the .raw file are the sending end (I), the receiving
end (J), the voltage reference (V set), type of FACTS (ID FACTS), initial reactance value (init), lower
limit for reactance value (Lo) and higher limit for reactance value (Hi), Figure D.5. The values related to
the reactance are only used when power flow for FACTS is calculated.
108
Figure D.5: Data for TCSC in .raw
The .dyr file has the appearance of Figure D.2. The data components for the TCSC are respectively
the sending end bus, the name of FACTS (”TCSC” in this case), the ordered number of TCSCs in the
system, SK , Tt, Ts and K.
Figure D.6: Data for TCSC in .dyr
109
110
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