NONLINEAR MODELLING AND CONTROL OF A THYRISTOR-CONTROLLED
SERIES CAPACITOR FOR POWER FLOW ENHANCEMENT
By
Anele Amos Onyedikachi
Submitted in partial fulfillment of the requirement for the degree:
MAGISTER TECHNOLOGIAE (M-Tech): Electrical Engineering
In the Department of Electrical Engineering/F’SATI
Faculty of Engineering and the Built Environment
Tshwane University of Technology
Supervisor: Professor John Terhile AGEE
Co-Supervisor: Professor Adisa Abdul-Ganiyu JIMOH
2012
Declaration and Copyright
“I hereby declare that this dissertation submitted for the degree of Masters of Technology:
Engineering: Electrical, at the Tshwane University of Technology, is my own original work and
has not previously been submitted to any other institution of higher education. I further declare
that all sources cited or quoted are indicated and acknowledged by means of a comprehensive list
of references”.
_______________________
__________________
ANELE AO
DATE
i
Dedication
This dissertation is dedicated to the ALMIGHTY GOD, my family and loved ones.
ii
Acknowledgement
Firstly, with a heart full of joy, I give thanks to JEHOVAH the MOST HIGH GOD for HIS love,
grace, strength, wisdom, knowledge and understanding. I thank HIM for always being by my
side. Without HIM, the completion of this research study would not have been possible.
My deepest gratitude also goes to my supervisors Prof. John Terhile AGEE and Prof. Adisa
Abdul-Ganiyu JIMOH. I appreciate them for building me up in the field of research and for
contributing immensely towards the completion of my research study. In addition, I seize this
opportunity to specially thank Prof John Terhile AGEE for the efforts he invested and the
amount of time he created for me during the course of my research study.
I would like to appreciate my beloved parents, Mr Lawrence ANELE and Mrs Christiana
ANELE for their love, prayers and encouragement. Also, a big thank-you goes to my uncle,
pastor Gospel AZUATALAM and his wife, Pastor Mabel AZUATALAM for their care, support
and encouragement.
With a sincerity of heart, I would like to thank the Tshwane University of Technology (TUT)
and the French South African Institute of Technology (F’SATI) for their financial support.
iii
Abstract
Efficient supply of electric power to various customer load demands is challenging due to power
system instabilities. Nevertheless, with the help of flexible alternating current transmission
system, power flow in transmission systems can be enhanced. As a result, this dissertation
presents nonlinear modelling and control of a single-machine dynamic load (SMDL) system with
a thyristor-controlled series capacitor (TCSC) for power system stability enhancement. Firstly, to
achieve the objective of this study, a mathematical model describing the internal state of TCSC
is formulated using firing angles of the TCSC thyristors as the inputs. Combining the states
model and the TCSC reactance model reported in the literature, the steady state behaviour of
TCSC under varying capacitance and inductance is investigated. The simulations obtained
showed that the developed model is not only capable of describing the internal state of TCSC but
also gives better understanding on how TCSC operates. Also, the simulations obtained showed
that the steady state behaviour of TCSC is negatively influenced when inappropriate values are
chosen for its capacitance and inductance. To study the behaviour of the SMDL system under
load reactive power demand variation, nonlinear models of the system with and without a TCSC
are formulated based on recent and relevant advances reported in the literature. The simulations
obtained via voltage collapse, bifurcation diagram, time-domain and phase portrait showed that
without the TCSC, the system presents a saddle-node bifurcation (SNB) which is associated with
voltage collapse and dynamic instability in power systems. With the connection of TCSC, the
loadability of the system increased. In addition, although SNB is prevented, the system becomes
unstable. Lastly, to improve the stability of the SMDL system with the TCSC, a linear quadratic
regulator (LQR) optimal control technique is used. MATLAB/Simulink application software is
used for this study.
iv
Table of Contents
Abstract .................................................................................................................................... iv
CHAPTER 1
INTRODUCTION .............................................................................................1
1.0.
Background/Justification ..................................................................................................1
1.1.
Problem Statement ...........................................................................................................4
1.1.1.
Sub-Problem 1 ...........................................................................................................4
1.1.2.
Sub-Problem 2 ...........................................................................................................4
1.1.3.
Sub-Problem 3 ...........................................................................................................5
1.1.4.
Sub-Problem 4 ...........................................................................................................5
1.1.5.
Sub-Problem 5 ...........................................................................................................5
1.2.
Research Methodology .....................................................................................................5
1.3.
Delimitations ....................................................................................................................6
1.4.
Significance of the Study ..................................................................................................6
1.5.
Contributions ....................................................................................................................7
1.6.
Outline of Dissertation......................................................................................................8
CHAPTER 2
LITERATURE REVIEW...................................................................................9
2.1.
Literature Review on FACTS ...........................................................................................9
2.2.
Motivation for TCSC Device .......................................................................................... 12
2.3.
Literature Review on TCSC Modelling ........................................................................... 12
2.4.
Literature Review on TCSC Steady State Behaviour....................................................... 14
v
2.5.
Literature Review on Bifurcation Analysis in Power Systems ......................................... 17
2.6.
Literature Review on Control Strategies for Power System Stability ............................... 20
2.7.
Motivation for LQR Optimal Controller Technique ........................................................ 26
2.8.
Conclusion ..................................................................................................................... 27
CHAPTER 3
MATHEMATICAL DESIGN AND METHODOLOGY .................................. 28
3.1.
Description of the Internal State of TCSC ....................................................................... 28
3.2.
Nonlinear Dynamical Model of SMDL System without TCSC ....................................... 33
3.3.
Nonlinear Dynamical Model of SMDL System with TCSC ............................................ 36
3.4.
LQR Optimal Controller Design Procedure .................................................................... 39
3.4.1.
LQR Optimal Controller Algorithm ............................................................................ 39
3.4.2.
Weight Matrix Selection ............................................................................................. 40
3.5.
LQR Optimal Controller Design for the SMDL System with TCSC................................ 41
3.6.
Conclusion ..................................................................................................................... 43
CHAPTER 4
ANALYSIS AND DISCUSSION OF RESULTS ............................................. 44
4.1.
Description of the Internal State of TCSC ....................................................................... 44
4.2.
Investigating the Steady State Behaviour of TCSC ......................................................... 46
4.3.
Voltage Collapse Simulation .......................................................................................... 51
4.4.
Bifurcation Analysis ....................................................................................................... 53
4.5.
Time Domain Simulation................................................................................................ 56
4.6.
Phase Portrait Simulation................................................................................................ 59
vi
4.7.
LQR Optimal Controller Design Simulation ................................................................... 60
4.8.
Conclusion ..................................................................................................................... 65
CHAPTER 5
CONCLUSION ............................................................................................... 67
5.1.
Introduction .................................................................................................................... 67
5.2.
Objectives and Research Process .................................................................................... 67
5.3.
Summary of Research Results ........................................................................................ 68
5.4.
Recommendations for Further Study............................................................................... 70
5.5.
Final Conclusion ............................................................................................................ 71
References ................................................................................................................................ 72
vii
List of Tables
Table 1: Data used for the description of the internal state of TCSC .......................................... 44
Table 2: Data used for investigating the steady state behaviour of TCSC ................................... 47
Table 3: Data used for voltage collapse simulation .................................................................... 51
Table 4: Data used for the SMDL system without the TCSC ..................................................... 53
Table 5: Data used for bifurcation diagrams .............................................................................. 53
Table 6: Data used for the SMDL system with TCSC ................................................................ 54
viii
List of Figures
Figure 3.1: TCSC circuit diagram .............................................................................................. 28
Figure 3.2: One-line diagram of SMDL system without TCSC .................................................. 33
Figure 3.3: One-line diagram of SMDL system with TCSC ....................................................... 36
Figure 4.1: Description of the internal state of TCSC for
C  254 .8  F and L  7 .89 mH
Figure 4.2: Description of the internal state of TCSC for
C  247  F and L  10 mH
Figure 4.3: Description of the internal state of TCSC for
C  254 .8  F and L  7 .89 mH
................. 45
....................... 45
................. 46
Figure 4.4: Single-TCSC resonance points for   3 .................................................................. 47
Figure 4.5: Multiple-TCSC resonance point for   3 ............................................................... 48
Figure 4.6: Multiple-TCSC resonant point for   3 .................................................................. 48
Figure 4.7: Description of the internal state of TCSC for   3 .................................................. 49
Figure 4.8: Description of the internal state TCSC for   3 ..................................................... 49
Figure 4.9: Description of the internal state of TCSC for   3 ................................................. 50
Figure 4.10: SMDL system without the TCSC: voltage collapse simulation .............................. 51
Figure 4.11: SMDL system with TCSC: voltage collapse simulation ......................................... 52
Figure 4.12: SMDL system without the TCSC – plot of eigenvalues ......................................... 55
Figure 4.13: SMDL system with TCSC – plot of eigenvalues .................................................... 55
Figure 4.14: PQ load simulink model for SMDL system without the TCSC............................... 57
Figure 4.15: PQ load simulink model for SMDL system with TCSC ......................................... 57
Figure 4.16: Time-domain simulation for the SMDL system without TCSC .............................. 58
Figure 4.17: SMDL system with TCSC – Time domain simulation ........................................... 58
Figure 4.18: SMDL system without the TCSC – phase portrait simulation ................................ 59
Figure 4.19: SMDL system with TCSC – phase portrait simulation ........................................... 60
ix
Figure 4.20: Optimal control input for the SMDL system with TCSC........................................ 63
Figure 4.21: Optimal feedback gains for the SMDL system with TCSC .................................... 63
Figure 4.22: Time response of the state variables ...................................................................... 64
Figure 4.23: Output eigenvalues of the closed-loop system........................................................ 64
x
CHAPTER 1
1.0.
INTRODUCTION
Background/Justification
Enhancing power flow in transmission systems is important for efficient power supply because a
developed society demands a large amount of electrical energy for domestic, commercial and
industrial purposes (Machowski, Bialek & Bumby, 2008). Efficient supply of electric power to
various customer load demands is challenging because the electric power utilities are confronted
with many challenges due to the ever increasing complexity in the system’s operation and
structure. Recently, one of the challenges that got wide attention is the power system instabilities
(Singh, Mathew & Chatterji, 2008). Electric power system is subjected to instabilities due to lack
of new generation and transmission facilities as well as the over exploitation of existing
facilities. However, with the help of flexible alternating current transmission system (FACTS),
power flow can be enhanced by improving the voltage profile of the transmission lines,
improving the power system stability, reducing the power system losses, optimizing power flow
between parallel lines and increasing the transmission line capability (Hingorani, 2000;
Hingorani, Gyugyi & El-Hawary, 2000).
Flexible alternating current transmission system (FACTS) is a modern technology that answers
to the demand of enhancing power flow in transmission networks. Unlike the load tap changers
that are slow to compensate for the changes in voltage, FACTS technology can play a very
important role by providing a flexible means to rapidly prevent oscillations, absorb sudden
changes in load, correct voltage profile at the load buses with rapid reactive power control and
allows the generators to find balance with the load at their slow speed (Hingorani, 2000). FACTS
1
devices such as thyristor controlled series capacitor (TCSC), static synchronous series
compensator (SSSC), static synchronous compensator (STATCOM), interline power flow
controller (IPFC), thyristor controlled phase shifting transformer (TCPST) and unified power
flow controller (UPFC) can provide services related to stability, voltage control and network
loading control. However, TCSC is used to achieve the aim of this research study. TCSC is a
capacitive reactance compensator which consists of a series capacitor bank shunted by an
inductor which is in series with a bi-directional thyristors. In this research study, TCSC is
connected in series with a single-machine dynamic load (SMDL) system in order to offset the
inductive reactance of the transmission line. Unlike the complexity in the structure of other
FACTS devices such as SSSC, STATCOM, IPFC, TCPST and UPFC, TCSC does not require
any form of interface such as dc link, storage device, converters and high voltage transformers.
As a result, since this research work focuses on the aspect of improving the stability of a SMDL
system, TCSC becomes the most economical and cost-effective solution compared to these
FACTS devices.
Furthermore, although power systems are normally operated near a stable equilibrium point,
variation of the system load parameters can move the system slowly from one equilibrium point
to another until it reaches collapse point (Chiang, 2004). Lately, power utilities have reported the
difficulties in maintaining the network stability of their systems due to the occurrence of system
perturbations such as contingencies and load disturbances which make certain power system
parameters to vary (Grillo et al., 2010; Jing et al., 2003; Kucukefe & Kaypmaz, 2008). As a
result, bifurcation analysis is used to study the behaviour of the SMDL system with and without
a TCSC under load reactive power demand variation. Bifurcation analysis is a useful tool for
characterizing the nature and stability of an equilibrium point. This nonlinear analytical
2
technique helps to show the regions where changes from stable to unstable, from stationary to
oscillatory or from order to chaos may occur in power systems. Studying several power system
models, the power systems community has recognized local bifurcations such as saddle-node and
Hopf bifurcations as the type of bifurcations that are most likely to be encountered in power
systems (Ajjarapu & Lee, 1992; Chiang, 2004; Grillo et al., 2010). Local bifurcations are known
by observing the eigenvalues of the current operating point and, as certain parameters in the
system change slowly, the system eventually turns unstable, either due to one of the eigenvalues
becoming zero (saddle-node bifurcation) or due to a pair of complex conjugate eigenvalues
crossing the imaginary axis of the complex plane (Hopf bifurcation). Therefore, based on the
nonlinear models formulated for the SMDL system with and without a TCSC, simulations such
as voltage collapse, bifurcation diagram, time-domain and phase portrait are used to analyse the
behaviour of the system under load reactive power demand variation.
Lastly, by nature, power systems continually experience disturbances such as load variations etc.
As a result, power systems are planned and operated to withstand the occurrence of certain
credible disturbances (Chiang, 2004). A major activity in power utility system planning and
operations is to study the impact of a set of credible disturbances on power system dynamical
behaviours such as stability and to develop counter-measures. A suitable counter-measure for
power system instabilities is to apply a control technique (Abido, 2008; Bevrani, Ledwich &
Ford, 2009).
3
1.1.
Problem Statement
Efficient supply of electric power to various customer load demands can be achieved by
enhancing power flow in transmission systems. However, with the help of FACTS devices, more
electric power can be supplied to various customer load demands with a minimum impact on the
environment and at lower investment cost. As a result, the problem statement of this research
study is to enhance power flow in transmission system.
To successfully solve this task, among the various means of enhancing power flow in a
transmission system with the help of FACTS devices, this research work focuses on the aspect of
improving the stability of power systems. As a result, this dissertation presents nonlinear
modelling and control of a SMDL system with TCSC for power system stability enhancement.
SMDL system is used because with its formulated nonlinear models, its stability analysis with
and without a TCSC can be easily investigated.
To achieve the objective of this research work, the following sub-problems are addressed:
1.1.1. Sub-Problem 1
According to (Geng et al., 2002), it was stated that developing a TCSC model is challenging
because of its variant topology and nonlinear circuit. As a result, in this dissertation, an
additional effort in understanding the internal state and operation of TCSC is studied.
1.1.2. Sub-Problem 2
According to (Hu et al., 2004; Meikandasivam, Nema & Jain, 2008), it was stated that the steady
state behaviour of TCSC is negatively influenced when inappropriate values for its capacitance
and inductance are chosen. As a result, this research work also investigates the steady state
4
behaviour of TCSC based on the model formulated in sub-problem 1 and the TCSC reactance
model reported by (Meikandasivam et al., 2008).
1.1.3. Sub-Problem 3
According to (Chiang, 2004), it was stated that to study the nature and stability of a nonlinear
dynamic system, it is important to formulate the system’s nonlinear models. As a result, to study
the behaviour of the SMDL system under load reactive power demand variation, nonlinear
models for the SMDL system with and without a TCSC are formulated based on recent and
relevant advances reported by (Canizares, 2002; Gu et al., 2007).
1.1.4. Sub-Problem 4
Based on the nonlinear models formulated in sub-problem 3, the stability of the SMDL system
with and without a TCSC is analysed via simulations such as voltage collapse, bifurcation
diagram, time domain and phase portrait.
1.1.5. Sub-Problem 5
According to (Abido, 2008; Bevrani et al., 2009; Philips & T., 1995 ), it was stated that a
suitable counter-measure for power system instabilities is to apply a control technique. As a
result, to improve the stability of the SMDL system with TCSC, a linear quadratic regulator
(LQR) optimal control technique is used.
1.2.
Research Methodology
The following are the study methods and design employed in achieving this task:

To develop a mathematical model for the description of the internal state of TCSC
5

To investigate the steady state behaviour of TCSC under varying capacitance and inductance

To formulate nonlinear dynamical models for the SMDL system with and without a TCSC

To investigate the stability of the SMDL system with and without a TCSC under load
reactive power demand variation via simulations such as voltage collapse, bifurcation
diagram, time domain and phase portrait.

To design a LQR optimal control technique for the stability enhancement of the SMDL
system with TCSC.
1.3.
Delimitations
This dissertation does not consider other FACTS devices except TCSC. In addition, in this
dissertation, only a single-machine dynamic load (SMDL) system is used. Lastly, this
dissertation does not implement a physical prototype.
1.4.
Significance of the Study
This research study is worth doing because of the vital information it renders to future power
system engineers and researchers. Energy is the basic necessity for the economic development of
a country. Many functions necessary to present-day living grind to halt when the supply of
energy stops. Energy exists in different forms in nature but the most important form is the
electrical energy. Therefore, enhancing power flow in transmission systems is important for
efficient power supply because a modern-day society requires a large amount of electrical energy
for domestic, commercial and industrial purposes. Among the various means of enhancing power
flow in transmission systems with the help of FACTS devices, this research work focussed on
the aspect of improving the power system stability. Therefore, efficient supply of electric power
6
to various customer load demands can be achieved when the future power system engineers and
researchers carry out nonlinear modelling, bifurcation analysis and control of their system with
FACTS devices such as TCSC for power system stability enhancement.
1.5.
Contributions
The contributions made in this research study are established based on the information presented
in sub-problems 1.1.1 to 1.1.5. Listed below are the tangible research outputs made in this work:

Amos O. Anele, John. T. Agee and Adisa A. Jimoh, “Description of the Internal State of
TCSC”, (literature accepted and presented at the IASTED conference on Power and Energy
Systems, held June 22-24, 2011, in Crete, Greece).

Amos O. Anele, John. T. Agee and Adisa A. Jimoh, “Stability Analysis of a Single-Machine
Dynamic Load System with TCSC”, (literature accepted and presented at the 5th IASTED
Asian conference on Power and Energy Systems, held April 2-4, 2012, in Phuket, Thailand).

A. O. Anele, J. T. Agee and A. A. Jimoh, “Investigating the Steady State Behaviour of
Thyristor Controlled Series Capacitor”, (literature accepted and presented at the IEEE
Energy Tech conference, held May 29-31, 2012, in the Case Western Reserve University,
Cleveland, Ohio, USA).

A. O. Anele, M. O. Ajayi, J. T. Agee and A. A. Jimoh, “Stability Analysis and Control of a
Single-Machine Dynamic Load System with TCSC (literature accepted for presentation at
the IEEE Power & Energy Society Conference and Exposition in Africa, to be held in July 913, 2012, in the University of the Witwatersrand, Johannesburg, South Africa).
7
1.6.
Outline of Dissertation
The remainder of this dissertation is organized as follows:
Chapter 2: This chapter covers literature review on flexible alternating current transmission
systems (FACTS), thyristor controlled series capacitor (TCSC) mathematical modelling, steady
state behaviour of TCSC, bifurcation analysis in power systems and TCSC control strategies for
power system stability enhancement.
Chapter 3: This chapter covers the development of a mathematical model for the description of
the internal state of thyristor controlled series capacitor (TCSC), the formulation of nonlinear
models for the single-machine dynamic load (SMDL) system with and without a TCSC and the
linear quadratic regulator (LQR) optimal controller design.
Chapter 4: This chapter covers the research findings, analysis and discussions for the
description of the internal state of thyristor controlled series capacitor (TCSC), investigating the
steady state behaviour of TCSC under varying capacitance and inductance, voltage collapse
simulation, bifurcation diagram, time domain simulation, phase portrait simulation for the singlemachine dynamic load (SMDL) system with and without a TCSC and lastly the linear quadratic
regulator (LQR) optimal controller design simulation for the SMDL system with TCSC.
Chapter 5: This chapter presents the concluding aspect of this dissertation.
8
CHAPTER 2
LITERATURE REVIEW
This chapter covers literature review on flexible alternating current transmission systems
(FACTS), thyristor controlled series capacitor (TCSC) mathematical modelling, steady state
behaviour of TCSC, bifurcation analysis in power systems and lastly, TCSC control strategies
for power system stability enhancement.
2.1.
Literature Review on FACTS
Energy is the basic necessity for the economic development of a country. Many functions
necessary to present-day living grind to halt when the supply of energy stops (Mehta & Mehta,
2005). Energy exists in different forms in nature but the most important form is the electrical
energy. Thus, enhancing power flow in transmission systems is important for efficient supply of
electric power for domestic, commercial and industrial purposes (Machowski et al., 2008).
Efficient supply of electric power to various customer load demands is challenging because of
power system instabilities. Nevertheless, with the help of flexible alternating current
transmission system (FACTS), power flow in transmission systems can be enhanced by
improving the voltage profile of the transmission lines, improving the power system stability,
reducing the power system losses, optimizing power flow between parallel lines and increasing
the transmission line capability (Hingorani, 2000; Hingorani et al., 2000).
In the late 1980s, the Electric Power Research Institute (EPRI) formulated the vision of FACTS
in which various power-electronics based controllers regulate power flow and transmission
voltage and mitigate dynamic disturbances (Abido, 2008). FACTS is defined by the Institute of
Electrical and Electronics Engineers (IEEE) as “ac transmission systems incorporating power
electronics-based and other static controllers to enhance controllability and increase power
9
transfer capability” (Habur & O’Leary, 2001; Hingorani, 2000; Hingorani et al., 2000).
Similarly, a FACTS controller is defined as “a power electronics-based system or other static
equipment that provides control of one or more ac transmission parameters” (Hingorani, 2000).
Unlike the conventional control actions in power systems such as mechanically switched series
capacitors and reactors, phase shifting transformers, under-load tap-changers, automatic
generation controls and governor controls that are slow to compensate for the changes in voltage,
FACTS technology offers a flexible means to rapidly prevent oscillations, absorb sudden
changes in load, correct voltage profile at the load buses with rapid reactive power control and
allows the generators to find balance with the load at their slow speed (Hingorani, 2000).
The power industry term FACTS covers a number of technologies that enhance the security,
capacity and flexibility of power transmission systems. FACTS solutions enable power grid
owners to increase existing transmission network capacity while maintaining or improving the
operating margins necessary for grid stability. As a result, more power can reach consumers with
a minimum impact on the environment, after substantially shorter project implementation times,
and at lower investment costs - all compared to the alternative of building new transmission lines
or power generation facilities (Shimpi et al., 2010). FACTS controllers are incorporated in
electric power systems to raise the system’s dynamic stability limits and provide better power
flow control. These are achieved when FACTS devices are used for the dynamic control of
voltage, impedance and phase angle of high voltage ac transmission lines.
In general, FACTS controllers can be divided into the following four categories (Abido, 2008;
Glanzmann & Hochspannungstechnik, 2005; Shimpi et al., 2010; Varma, 2010):

Series FACTS controllers: FACTS devices such as thyristor switched series capacitor
(TSSC), thyristor controlled series capacitor (TCSC), thyristor controlled series reactor
10
(TCSR), thyristor switched series reactor (TSSR), static synchronous series compensator
(SSSC) and phase angle regulator (PAR) are examples of series FACTS controllers. These
devices inject voltage in series with the ac transmission line.

Shunt FACTS controllers: FACTS devices such as static var compensator (SVC) and static
synchronous compensator (STATCOM) are examples of shunt FACTS controllers. These
devices inject current into the system at the point of connection. They absorb or supply
current from or into the transmission line.

Combined series-shunt controllers: These set of controllers consist of separate series and
shunt controllers and are controlled in a coordinated manner. FACTS devices such as unified
power flow controller (UPFC) and thyristor controlled phase shifting transformer (TCPST)
are examples of series-shunt FACTS controllers. The series-shunt controllers inject current
into the system with the shunt part of the controller and inject voltage into the system with
the series part of the controller.

Combined series-series FACTS controllers: These set of controllers consist of separate series
controllers and are controlled in a coordinated manner. FACTS devices such as interline
power flow controller (IPFC) and thyristor controlled phase angle reactor (TCPAR) belong to
this category of controllers. These separate series controllers provide independent series
reactive compensation for each transmission line and also transfer real power along the
transmission line via the power link.
11
2.2.
Motivation for TCSC Device
Among other FACTS devices such as SSSC, STATCOM, IPFC, TCPAR, TCPST and UPFC,
thyristor controlled series capacitor (TCSC) is used to achieve the aim of this research work. The
following are the tangible reasons why TCSC is chosen:

Besides the fact that TCSC does not require any form of interface such as dc link, storage
device, converters and high voltage transformers, since this dissertation focuses on the aspect
of improving the stability of a single-machine dynamic load (SMDL) system, TCSC becomes
the most economical and cost-effective solution (Jiang & Lei, 2000).

In this research study, TCSC is connected in series with the SMDL system in order to offset
the inductive reactance of the transmission line which tends to improve electromechanical
and voltage stability and also limit voltage dips at network nodes (Machowski et al., 2008).

TCSC plays vital roles in the operation and control of power systems. These roles include
current control, damping oscillations, transient and dynamic stability, voltage stability, fault
current limiting etc (Zhou & Liang, 1999).
2.3.
Literature Review on TCSC Modelling
TCSC modelling is not without its challenges because it exhibits a variant topology and
nonlinear circuit (Dongxia et al., 1998; Geng et al., 2002; Hak-Guhn & Jong-Keun, 1998;
Helbing & Karady, 1994; Jovcic & Pillai, 2005; Li et al., 2000; Li et al., 1998; Vuorenpää,
Järventausta & Lavapuro, 2008; Zhizhong, 2010; Zhongdong et al., 1998). A detailed analysis of
the TCSC circuit dynamics was stated in (Helbing & Karady, 1994). The model was formulated
through mathematical equations by separating the thyristor off and on equivalent circuits.
According to (Zhongdong et al., 1998), it was stated that the kind of synchronizing signal
12
applied for the circuit analysis of TCSC influences its transient characteristics. However, the
model was developed by choosing line current as the synchronizing signal. The results obtained
proved not to have over-shoot in inter-zone step. It was also demonstrated in (Li et al., 1998) that
to assess the advantages of TCSC in power systems or to ascertain its suitable control strategy,
an accurate model is required to analyze its dynamic behaviour. It was also stated that TCSC
modelling is difficult because it is composed of both continuous dynamics which is related to the
voltages on capacitors and the currents on inductors and discrete events which are related to the
switching of thyristors. However, the TCSC was modelled as advanced variable impedance
based on the characteristics of its transition period. Analysis of transient and steady state
characteristics using state space analysis was stated in (Hak-Guhn & Jong-Keun, 1998). The
result obtained showed that in the transient state, the zero-crossing of the line current is not fixed
because the conduction angle of the thyristor is asymmetrical. Thus, it is considered to obtain the
accurate simulation results. A TCSC model describing its transient behaviour was stated in
(Dongxia et al., 1998). The aim of the literature was achieved using topology analysis method.
The modelling of TCSC circuit as a fast switch between two equivalent circuits, corresponding
to the thyristor blocking or conduction state was stated in (Li et al., 2000). A TCSC analytical
model which is a first-order difference equation when the line current is used as synchronizing
signal and a second-order difference equation when the capacitor voltage is employed was stated
in (Geng et al., 2002). Analytical modelling of TCSC dynamics was stated in (Jovcic & Pillai,
2005). The aim of the literature was achieved by proposing a simplified fundamental frequency
response model. New modelling techniques for TCSC was proposed by (Vuorenpää et al., 2008).
It was stated in the literature that depending on the firing angles of the thyristors, both inductive
and capacitive could be used as the main control mode. However, the modelling was achieved
13
using the capacitive control mode. According to (Zhizhong, 2010), it was stated that in order to
control the power flow on the line flexibly, there is an increasing need to develop a suitable
model for TCSC such that a controller with good performance can be designed for TCSC
according to the model. To achieve this, the delay angle of thyristor valves was used as the input
signal and the inductor current was chosen as the output signal. Theoretical analysis and
simulation studies showed that TCSC is a nonlinear system and its parameters vary with the
operating point.
Based on this information, this dissertation also formulates a mathematical model describing the
internal state of TCSC. This model is derived using firing angles of the TCSC thyristors as the
inputs. The mathematical analysis is formulated using calculus method, Laplace and inverse
Laplace transform. MATLAB/M-file environment is used for the study.
2.4.
Literature Review on TCSC Steady State Behaviour
It was stated according to (Hu et al., 2004; Joshi & Mohan, 2006; Meikandasivam et al., 2008)
that choosing inappropriate values for the capacitance and inductance of TCSC negatively
influence its steady state behaviour. The relation between the fundamental frequency equivalent
impedance of the TCSC and the basic frequency reactance of the thyristor controlled reactor
(TCR) was stated in (Hu et al., 2004). In this literature, it was revealed that the equivalent
capacitor of TCSC circuit is not a fixed value but varies with control angle  . Also, the effects
of the fundamental frequency reactance and TCR on the resonance point and the controlled range
was also investigated. The results obtained showed that the impact on the waveforms and values
of current and voltage in the thyristor circuit and the capacitance circuit are negatively influenced
when inappropriate values are chosen for the TCSC component parameters. The use of TCSC to
14
solve problems of interconnection of wind turbines to grid was proposed by (Joshi & Mohan,
2006). Two applications of TCSC were considered in this literature. One for fault current
limitation and other unbalance voltage compensation. The results showed that TCSC is quite
effective in both cases. However, the major problem found was designing of inductor and
capacitor values for TCSC. The basics of TCSC device, the analysis of its impedance
characteristics and its associated single and multi resonance conditions were stated in
(Meikandasivam et al., 2008). The impedance characteristics curve is drawn for different values
of inductance in MATLAB using M-files. This study was also helpful in estimating the
appropriate inductance and capacitance values which have influence on multi-resonance point in
TCSC device.
The fundamental effective TCSC reactance X TCSC with respect to the firing angle  of the
thyristor is given as (Meikandasivam et al., 2008):
 4X 2
 X  XL 
X TCSC     X C   C
2     sin2      LC
 X L
 


 2
 cos     tan      tan   


2.1
Where
X LC 
XC XL
XC  XL
X L is the inductive reactance of TCSC in ohms, X C is the capacitive reactance of TCSC in
ohms,  is the parameter which determines the operating performance of TCSC and is given as
 
X C o

XL

(2.2)
15
 is the network frequency while o is the resonant frequency which occurs in a high power
electronic circuit such as TCSC when the inductive reactance X L  L and the capacitive
reactance X C 
o 
1
are of equal magnitude. It can also be expressed as
C
1
(2.3)
LC
Based on equation (2.1), (Geng et al., 2002; Hu et al., 2004; Meikandasivam et al., 2008)
showed that the effective TCSC reactance X TCSC   would be infinity when

      2m 1 ; (m  1,2,3..)
2
or
2m  1 
Crit   
2
(2.4)
Based on equation (2.4), (Meikandasivam et al., 2008) showed that between 90 0 to 180 0 of
firing angle  , a multiple-TCSC resonance point may occur. Therefore, since only a singleTCSC resonance point is allowable for a proper operating performance of TCSC, (Geng et al.,
2002; Hu et al., 2004; Meikandasivam et al., 2008) proposed that one evident way to achieve a
single TCSC resonance point is to restrict the value of factor by
 
X C o

3
XL

(2.5)
Therefore, to confine the value of  to be less than three, appropriate values for the TCSC
capacitance and inductance must be chosen (Hu et al., 2004; Meikandasivam et al., 2008; Souza
et al., 2003).
16
Based on this information, this dissertation also investigates the steady state behaviour of TCSC
under varying capacitance and inductance values. This aim is achieved via the mathematical
model developed for the description of the internal state of TCSC and the fundamental effective
TCSC reactance model reported in (Meikandasivam et al., 2008).
2.5.
Literature Review on Bifurcation Analysis in Power Systems
Electrical power system is a large, complex and dynamic system capable of generating,
transmitting and distributing electrical power over a large geographical area (Machowski et al.,
2008). However, studying its nonlinear dynamical behaviour becomes difficult due to the ever
increasing complexity in its structure and operation. Although power systems are normally
operated near a stable equilibrium point, variation of the system load parameters can move the
system slowly from one equilibrium point to another until it reaches collapse point (Chiang,
2004). Therefore, it is important to study the nonlinear behaviour of power systems under system
load-parameter variation. Bifurcation theory is the mathematical study of changes in the
qualitative or topological structure of a given family such as the solutions of a family of
differential equations (Blanchard & Devaney, 2006 ). It is most commonly applied to the
mathematical study of nonlinear dynamical systems. Bifurcations occur when a small smooth
change made to the bifurcation parameter of a system causes a sudden qualitative or topological
change in its behaviour. Bifurcation analysis is a useful tool for characterizing the nature and
stability of an equilibrium point. It helps to show the regions where changes from stable to
unstable, from stationary to oscillatory or from order to chaos may occur in nonlinear dynamic
systems.
17
Based on the pioneer work of (Tavora & Smith, 1972), the theory of nonlinear dynamics has
become a field of great interest to researchers and engineers in the power system community. A
review of bifurcation and chaos researches in power systems was in (Yusheng, Haiqiang &
Xiaorong, 2002). Bifurcation and chaos analysis of a three-bus power system was presented in
(Chiang et al., 1993), (Abed et al., 1992; Ajjarapu & Lee, 1992; Chiang, Conneen & Flueck,
1994) conducted a numerical bifurcation analysis of a simplified model of a nine-bus power
system and thirty-nine bus power system, (Rosehart & Cañizares, 1999) presented bifurcation
analysis of various power system model, (Canizares, 2002) presented voltage stability
assessment of various power system models and (Gu et al., 2007) presented a literature on Hopf
bifurcation induced by static var compensator. Lately, power utilities have reported the
difficulties in maintaining the network stability of their systems due to the occurrence of system
perturbations such as load disturbances or contingencies which causes certain power system
parameters to vary (Grillo et al., 2010; Gu et al., 2007; Jing et al., 2003; Kucukefe & Kaypmaz,
2008; Pulgar-Painemal & Sauer, 2009). Studying several power system models, the power
systems community has recognized saddle-node, limit-induced, pitchfork, Hopf, cyclic fold,
period doubling, monoclinic chaos and torus as the various types of bifurcations that are most
likely to be encountered in power systems (Ajjarapu & Lee, 1992; Chiang, 2004; Grillo et al.,
2010). As a result, in power systems, bifurcation analysis is applied to investigate various power
system instabilities such as voltage collapse and low frequency electro-mechanical oscillations.
Furthermore, an operational power system can lose its stability when bifurcations emerge. These
types of bifurcations can be divided into two principal classes namely:

Local Bifurcations: such as saddle-node bifurcations (SNBs), Hopf bifurcations (HBs),
limit-induced (or trans-critical) bifurcations and pitchfork bifurcations. These types of
18
bifurcations can be analyzed entirely through changes in the local stability properties of
equilibria, periodic orbits or other invariant sets as a parameter cross through critical
thresholds. A local bifurcation occurs when a parameter change causes the stability of an
equilibrium (or fixed point) to change.

Global Bifurcations: such as cyclic fold bifurcation, period-doubling, torus bifurcation etc.
These types of bifurcations often occur when larger invariant sets of the system collide with
each other, or with equilibria of the system. They cannot be detected purely by a stability
analysis of the equilibria (or fixed points). These bifurcations may emerge in power systems
because of the limit cycles resulting from a Hopf bifurcation.
This study focuses only on saddle-node bifurcation (SNB) and Hopf bifurcations because they
are the most commonly bifurcations that are induced by FACTS device (Gu et al., 2007).

Saddle-Node Bifurcation: in power systems, SNB is associated with voltage collapse and this
usually occurs at maximum loading condition. Local bifurcation such as SNB is known by
observing the eigenvalues of the current operating point and, as certain parameters in the
system change slowly, the system eventually turns unstable due to one of the eigenvalues
becoming zero. Therefore, due to loss of operating equilibrium, the system state changes
dynamically and as a result, system voltages fall dynamically.

Hopf Bifurcation: is known as the appearance or the disappearance of a periodic orbit
through a local change in the stability properties of a steady point. It occurs when a fixed
point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the
linearization around the equilibrium point crosses the imaginary axis of the complex plane.
19
The emergence of Hopf bifurcation in power systems can lead to a variety of low frequency
electro-mechanical oscillations.
Based on this information, the stability analysis of a single-machine dynamic load (SMDL)
system with and without a thyristor controlled series capacitor (TCSC) is also presented in this
dissertation. This aim is achieved by studying the results obtained for the voltage collapse
simulation, bifurcation diagram, time domain simulation and phase portrait simulation based on
the formulated nonlinear models of the SMDL system with and without a TCSC.
2.6.
Literature Review on Control Strategies for Power System Stability
Research on control strategies for power system stability enhancement can be dated back to 1966
when (Kimbark, 1969) analyzed the improvement of transient stability of a two-circuit ac
transmission link using switched series capacitors. The application of switched series capacitors
for the improvement of stability against a permanent fault on one ac circuit and a permanent fault
on a dc line operating in parallel with the ac line was illustrated using equal-area diagrams. With
the application of the simple control, the angular swing caused by a disturbance was decreased
and this successively decreases the fluctuation of load voltages. An optimal control: bang-bang
control of the transmission line reactance for the improvement of power system transient stability
was presented in (Ramarao & Reitan, 1970). The aim of the literature was achieved using
Pontryagin’s maximum principle. However, advances in high-power and high-efficiency power
electronic devices have led to the development of flexible alternating current transmission
systems (FACTS) devices such as thyristor controlled series capacitor (TCSC) in power systems.
Over the past years, many different control techniques have been reported in the literature
pertaining to investigating the effect of TCSC on power system stability (Chang & Chow, 1997;
20
Chen et al., 1995; Lie, Shrestha & Ghosh, 1995; Lu et al., 1996; Rajkumar & Mohler, 1994;
Rouco & Pagola, 1997; Wang et al., 1992; Zhao & Jiang, 1998). Variable-structure FACTS
controllers for power system transient stability was proposed in (Wang et al., 1992). In the
literature, nonlinear variable structure control theory was employed for series capacitor control
and braking resistor control to improve the transient stability of a single-machine infinite bus
(SMIB) system. The results obtained showed that the variable-structure control of the series
capacitor and braking resistor was effective for the enhancement of power system steady state
performance and transient stability. Bilinear generalized predictive control using the TCSC was
proposed for power systems subjected to large faults (Rajkumar & Mohler, 1994). Simulation
results showed that bilinear self-tuning control using TCSC is capable of increasing the region of
stability and providing good damping to a SMIB power system. The design and application of a
fuzzy logic control scheme for transient stability enhancement in power systems was proposed in
(Lie et al., 1995). The fuzzy logic controller was designed to implement variable series capacitor
compensation in the transmission network of interconnected power systems. The effectiveness of
the fuzzy logic controller was performed on a SMIB system. The results obtained showed that
the controller can contribute tremendously to the enhancement of power system transient
stability during disturbances. The application of controlled series compensators (CSCs) to
improve the stability margin of a multi-machine power system was presented in (Chen et al.,
1995). The major contributions of the literature were (a) design of state feedback CSC controller
for a multi-machine power system using a linearised system model and (b) development of a
procedure for selecting the most effective CSC locations and their coordination. Pole placement
technique was used to calculate the feedback controller gains. The results obtained showed that
the most effective locations for CSCs can be obtained by analyzing the “degree of
21
controllability” of oscillation modes owing to the action of controllers placed at different
locations in the system. However, besides the fact that the robustness of the proposed CSC
controller under changing network conditions was not addressed, the applicability of the
proposed controller is reduced because it requires all system states. Decentralized nonlinear
optimal excitation control was proposed in (Lu et al., 1996). A design method laying emphasis
on differential geometric approach for decentralized nonlinear optimal excitation control of
multi-machine systems was suggested in the literature. The control law achieved was
implemented via purely local measurements and it was independent of the parameters of power
networks. Based on the simulation performed on a six-machine system, it was shown that the
nonlinear optimal excitation control could adapt to the conditions under large disturbances.
Besides, it was verified in the literature that the presented nonlinear control law was of
optimality in the sense of quasi-quadratic performance index. The design of a time-optimal
control for a controllable series capacitor to damp transient power swings due to inter-area
modes in interconnected power systems was proposed by (Chang & Chow, 1997). The timeoptimal control design was illustrated using bang-bang control of a series capacitor in a sixmachine system with three coherent groups of machines. However, one major limitation of the
control approach implemented in the literature lies in the fact that only the classical models are
used for the generators and no excitation system effects were considered. An eigenvalue
sensitivity approach to location and controller design of controllable series capacitors for
damping power system oscillations was proposed in (Rouco & Pagola, 1997). Two issues (i.e.,
location and controller design) encountered in the application of CSC for damping
electromechanical oscillations were addressed in the literature. These were achieved using smallsignal models of the power system and the corresponding eigenvalue sensitivities. A robust
22
TCSC controller design using the H  optimization technique to improve the system damping to
the inter-area modes was presented in (Zhao & Jiang, 1998). Based on the small signal
eigenvalue analysis and the full scale nonlinear time domain simulations obtained, it was shown
that the robust controller is effective in providing additional damping to the system under load
variations.
Furthermore, several literatures pertaining to nonlinear control schemes for power system
stability problems have been addressed in (Barkhordary et al., 2006; Cong & Wang, 2001; Jiang
& Lei, 2000; Wang, Tan & Guo, 2002). A nonlinear TCSC control strategy for power system
stability enhancement was proposed in (Jiang & Lei, 2000). In the literature, a nonlinear control
scheme for the TCSC to dampen power oscillations and improve transient stability of power
system was presented. A nonlinear mathematical model was established and was proven as an
affine nonlinear system. With the help of the feedback linearization technique, the affine
nonlinear model was exactly transferred to a linear model, and then, the control scheme is
designed for the TCSC based on the global linearization, where the input signal uses local
measurements only. The effectiveness and robustness of the proposed nonlinear control scheme
were demonstrated with one-machine test system, where the TCSC modelling and power system
simulations were performed by using the program system NETOMAC. In comparison with a
conventional control scheme, significant improvements of dynamical performance in the test
power system were achieved by the proposed nonlinear control strategy for the TCSC. A
coordinated control approach for FACTS and generator excitation system was presented in
(Cong & Wang, 2001) to improve the dynamic stability of power systems. A robust control
approach was proposed in the literature for the coordinated control of geographically distributed
FACTS devices and generator excitation systems to improve the dynamic stability of power
23
systems. Firstly, a nonlinear feedback law which linearizes the system model was found. Then,
the interaction between FACTS and excitation system was treated as parameter uncertainty and
solved by robust control theory. The proposed coordinated controller is designed based on local
measurements. The effectiveness of the proposed approach was implemented on a SMIB power
system with TCSC. The control objective was to achieve both voltage regulation and system
stability enhancement. Results obtained from real-time simulations showed that both voltage
regulation and transient stability enhancement can be achieved regardless of the system operating
point, and compared to other kinds of controllers, the proposed controller gave better dynamic
performance and robustness. A robust feedback linearization controller design for thyristor
control series capacitor to damp electromechanical oscillations was proposed in (Barkhordary et
al., 2006). The literature presented a procedure to design adaptive feedback linearization control
of TCSC for damping one-machine infinite-bus power system oscillations. The effectiveness of
the proposed adaptive feedback linearization method in stabilizing the power system was well
shown. A robust nonlinear coordinated control for power systems excitation and TCSC is
proposed in (Wang et al., 2002) to enhance the transient stability of power systems. A nonlinear
feedback law for the generator which linearizes and decouples the power system model was first
found. Robust nonlinear control theory was then employed to design the nonlinear coordinated
controller that consisted of two controllers, the generator excitation and the TCSC controllers.
The proposed coordinated controller is designed, based on local measurements, and the design of
the resulting controllers is independent of the operating point. Simulation results showed that the
proposed controller can enhance transient stability of the power system under a large sudden
fault, which may occur at the generator bus-bar terminal and even in the case where a fault
occurs at the generator bus-bar terminal.
24
Nevertheless, controller design can also be done through mathematical optimization techniques.
The linear quadratic regulator (LQR) optimal feedback controller technique is one of many tools
to improve the stability of an interconnected power system (Kirk, 2004). Using LQR theory, it
has been established that for a controllable linear time-invariant system, a set of power system
plant optimal feedback gains may be found which minimizes a quadratic index and makes the
closed-loop system stable. An LQR and pole placement controller design for static synchronous
compensator was presented by (Ali & Amin, 2007). In this literature, two kinds of feedback
controllers (pole assignment and LQR) were designed. The results obtained showed that LQR
method is preferred to pole assignment because the determination of the state feedback gains are
easily obtained using LQR method. Application and comparison of LQR and robust sliding mode
controllers to improve power system stability was presented by (Shamshirgar et al., 2000).
Fundamental theory of regular power system stabilizer (PSS), LQR based state feedback,
observer, LQR based PSS, and eventually sliding mode based PSS were presented in the
literature. The results obtained showed that LQR based PSS yielded better stabilizing parameters
than the regular PSS with a smaller control effort. Moreover, after applying a heavy fault, the
regular PSS got unstable while the LQR based PSS was able to stabilize the power system but
with an increased control effort. In conclusion, this literature showed that a designer can apply
LQR to design a power system stabilizer with good gain and phase margins without any worry
about control signals since LQR can compromise between values of state variables and control
efforts.
25
2.7.
Motivation for LQR Optimal Controller Technique
Linear quadratic regulator (LQR) optimal control technique is used to stabilize the singlemachine dynamic load with thyristor controlled series capacitor. Quadratic optimal regulator
system is a linear quadratic regulator (LQR) optimal control technique which is generally
formulated for time-varying systems (Kirk, 2004). Unlike the classical design techniques such as
frequency-response and the root locus as well as the modern design technique such as poleassignment, LQR optimal control technique yields the best control system. This technique is an
optimal design technique, and assumes that we can write a mathematical function which is called
the cost function. The LQR optimal controller design procedure determines the optimal feedback
gain matrix that minimizes the cost function in order to achieve some compromise between the
use of control effort, the magnitude and the speed of response that will guarantee a stable system.
Quadratic cost function is employed because its mathematical function is logical and with it, the
development of the optimal controller design is simple. LQR optimal control technique is used
because of the following remarks (Philips & T., 1995 ): (a) It can be considered for time varying
matrices, (b) It can be extended in several ways to nonlinear systems, (c) It assumes full
knowledge of the state, (d) It can be considered for final times and (e) Unlike the pole-placement
technique, LQR optimal control technique provides a systematic way of computing the state
feedback control gain matrix.
26
2.8.
Conclusion
Chapter two of this dissertation dealt with literature review on flexible alternating current
transmission systems (FACTS), thyristor controlled series capacitor (TCSC) mathematical
modelling, steady state behaviour of TCSC, bifurcation analysis in power systems and TCSC
control strategies for power system stability enhancement. In this chapter, the motivation for
choosing TCSC among other FACTS devices such as SSSC, STATCOM, IPFC, TCPAR,
TCPST and UPFC are also presented. Therefore, based on the useful information obtained from
this chapter, the following study methods and design will be presented in the next chapter:

Development of a mathematical model for the description of the internal state of TCSC.

Formulation of nonlinear dynamical models for the SMDL system with and without a TCSC.

Design of a LQR optimal control technique for the stability enhancement of the SMDL
system with TCSC.
27
CHAPTER 3
MATHEMATICAL DESIGN AND METHODOLOGY
This chapter covers the development of a mathematical model for the description of the internal
state of thyristor controlled series capacitor (TCSC), the formulation of nonlinear models for the
single-machine dynamic load (SMDL) system with and without a TCSC and lastly, the linear
quadratic regulator (LQR) optimal controller design for the SMDL system with TCSC
3.1.
Description of the Internal State of TCSC
The circuit diagram shown in Fig. 3.1 is used for the development of a mathematical model for
describing the internal state of TCSC. All calculations assume steady-state; this is because it
ignores the effects of metal-oxide varistor as well as the reactor’s and the thyristors’ resistance
(Geng et al., 2002).
Figure 3.1: TCSC circuit diagram
28
The mathematical model is developed in the following four phases:
Phase 1: for a firing angle range between 0   t   , the thyristors do not conduct. As a result,
only the capacitor conducts. Therefore, applying Kirchhoff’s current law (KCL) to Fig. 3.1
yields
I Line (t )  iC (t )
(3.1)
Phase 2: for a firing angle range between   t   , one of the thyristors conducts in the
positive half cycle. Therefore, applying KCL to Fig. 3.1 yields
I Line (t )  iC (t )  iTH (t )
(3.2)
Phase 3: for a firing angle range between   t     , the thyristors do not conduct. As a
result, only the capacitor conducts. Therefore, applying KCL to Fig. 3.1 yields
I Line (t )  iC (t )
(3.3)
Phase 4: for a firing angle range between     t  2 , one of the thyristors conducts in the
negative half cycle. Therefore, applying KCL to Fig. 3.1 yields
I Line (t )  iC (t )  iTH (t )
(3.4)
From phase 1,
I O sin  t  
CdV C ( t )
dt
Where I Line(t)  I o sint  and i C t  
(3.5)
CdV C ( t )
dt
I Linet  : Line current, iC (t) : Capacitor current, I o : Peak current and VC (t ) : Capacitor voltage
29
From equation (3.5), the capacitor voltage can be given as
V C  t  
1
1
I o sin t dt  
I cos t 

C
C o
(3.6)
Therefore, since the capacitive reactance is given as X C 
VC (t )  IO X C cost 
1
, equation (3.6) becomes
C
(3.7)
In phase 2, one of the thyristors conducts in the positive half cycle. Therefore, equation (3.2)
becomes
I O sin  t   iT H (t ) 
CdV C ( t )
dt
(3.8)
From Fig. 3.1, it is shown that the capacitor voltage equals to the voltage across the inductor. As
a result, the capacitor voltage can be given as
VC ( t ) 
LdiTH (t )
dt
(3.9)
Therefore, substituting equation (3.9) into (3.8) yields
LCd 2 iTH (t )
I O sin t   iT H (t ) 
dt 2
(3.10)
The Laplace Transform of equation (3.10) is given as
  
IO  2
 I T H (s)[1  LCs2 ]  LCsiTH (0)  LCi'TH (0)
2 
 s  
30
(3.11)
Where iTH 0  0 , is the initial value of the thyristor current just before the thyristor was fired
and i'TH 0  0 is the first derivative of the initial thyristor current. As a result, equation (3.11)
becomes
  
IO  2
 I T H ( s)[1  LCs 2 ]
2 
 s  
(3.12)
Therefore, substituting equation (2.3) into equation (3.12) yields
2
s 2  o
  
IO  2

I
(
s
)[
]
TH
2 
o 2
 s  
(3.13)
Making IT H (s) the subject of equation (3.13) yields
2
   o
I TH ( s)  I O  2
[
]
2 
2
 s    s 2  o
(3.14)
The inverse Laplace transform of equation (3.14) is given as
2
 o 2

o

iTH (t )  I O  2
sin t  
sin  o t 
2
2
2
( o   )  o
 ( o   )

(3.15)
o 2
, equation (3.15) becomes
Letting P 
2
(o   2 )



iTH (t )  I O  P sint   P sino t 
o


(3.16)
31
Therefore, substituting equation (2.2) into equation (3.16) yields the formulation for the thyristor
current as
1


iTH (t )  I O P sin t   sin t 



(3.17)
In addition, substituting equation (3.17) into equation (3.9) yields the capacitor voltage as
VC t   I O PX L cost   cost   I O cos 
(3.18)
Following the same mathematical analysis for phase 3 and phase 4, the complete mathematical
model developed for the description of the internal state of TCSC is given as follows:

Capacitor current
i C t   I Line t 
0  t  
i C t   I Line t   i TH t 
  t  
i C t   I Line t 
  t    
i C t   I Line t   i TH t 
     t  2

(3.19)
Thyristor current
I TH t   0
0  t  
1


I TH t   I O P sin t   sin  t 



  t  
I TH t   0
  t    
1


I TH t    I O P sin  t   sin  t 



    t  2
(3.20)
32

Capacitor voltage
VC (t )  I O X C cost 
3.21
VC (t )  I O PX L cost   cost  - I O X C cos( )
VC (t )  I O X C cost   I O PX L cos   cos  - I O X C cos( )
VC (t )  I O PX L cost   cost   I O X C cos     I O PX L cos   cos   I O X C cos( )
3.2.
Nonlinear Dynamical Model of SMDL System without TCSC
As shown in Fig. 3.2 is a single-machine dynamic load (SMDL) system without the thyristor
controlled series capacitor (TCSC).
PG  jQG
jX
PL  jQL
E
V0
Figure 3.2: One-line diagram of SMDL system without TCSC
Where PG is the generated active power of the generator in per unit, QG is the generated
reactive power of the generator in per unit, PL is the load active power demand in per unit, QL is
the load reactive power demand in per unit, X is the transmission line reactance in per unit, E
is the terminal voltage of the generator in per unit and V is the bus voltage of the dynamic load
system in per unit.
33
The nonlinear models for the SMDL system without the TCSC are based on (Canizares, 2002;
Gu et al., 2007) and are formulated based on the following assumptions (Machowski et al.,
2008):

A round-rotor synchronous generator (i.e., turbo-generator) is employed. This is considered
in this dissertation because round-rotor synchronous generators are normally used for turbo
units driven by high-speed steam or gas. As a result, they have the ability to withstand high
centrifugal forces.

Unlike the salient-pole synchronous generator, since a round-rotor synchronous generator
(i.e., turbo-generator) is considered, the effect of rotor saliency is neglected.

A two-pole round-rotor synchronous generator is employed.

The mechanical power input to the machine, Pm is assumed constant.

For the sake of simplicity but without loss of generality, the resistances of stator winding of
machine and transmission lines are neglected.

Lastly, it is assumed that in the dynamic model of the SMDL system with and without the
TCSC, the load is assumed to be at steady state.
Therefore, based on the assumptions, the following sets of nonlinear differential equations for
the SMDL system without the TCSC can be formulated as

The swing equation of the synchronous generator is given as (Gu et al., 2007):
  
 
1
M
(3.22)
EV


 Pm  X sin   D 


(3.23)
34

Based on the last assumption, the PQ load model is given as (Canizares, 2002):

1  EV cos V 2
V  

 QL 
 X
X

(3.24)
EV
EV cos  V 2
P

D


P




sin

,
and
Q

V



D
G
Where G
X
X
X
Where  is the rotor angle of the generator in radian,   is the speed deviation of the
generator in electrical radians per second, M is the inertia constant of the generator in seconds,
Pm is the mechanical input power of the generator in per unit,  is the voltage time constant of
the load system in seconds, PD is the damping power in per unit and D is the damping
coefficient.
From equation (3.24), the behaviour of the bus-voltage versus the load reactive power demand
can be studied. This is achieved by expressing the PQ load model in quadratic form as
V 2  E cos V  QL X  0
(3.25)
Therefore, applying quadratic formula to equation (3.25) results to
V
E cos   E 2 cos 2    4Q L X
(3.26)
2
From equation (3.26), the load reactive power demand is considered to be the bifurcation
parameter.
35
Also, from equations (3.22-3.24), the Jacobian matrix of the SMDL system without the TCSC is
given as


0
1
  EV cos 
D
A

MX
M

  EV sin  
0

X

3.3.


0

  E sin    
 MX  
E cos   2V 

X

(3.27)
Nonlinear Dynamical Model of SMDL System with TCSC
As shown in Fig. 3.3 is a single-machine dynamic load (SMDL) system with thyristor controlled
series capacitor (TCSC).
PG  jQG
jX
2
jX
2
XC
PL  jQL
E
XL
V 0
T1
T2
Figure 3.3: One-line diagram of SMDL system with TCSC
Also, the nonlinear models for the SMDL system with TCSC are based on (Canizares, 2002; Gu
et al., 2007) and are formulated based on the following assumptions (Machowski et al., 2008):
36

TCSC device is normally located either at the line terminals or at the middle of the
transmission line. It is located in the middle of the transmission line because fault currents
are lower and transmission line protection is easier. Also, it is located at the line terminal for
easy access necessary for maintenance, control and monitoring of its device. Although
dissertation does study a situation with fault occurrence, TCSC is located at the middle of the
transmission line.

Plus the assumption stated in section 3.2.
Therefore, based on the assumptions, the following sets of nonlinear differential equations for
the SMDL system with TCSC can be formulated as
  
 
1
M
(3.28)


EV sin 
 D 
 Pm 
 X  X t csc  


(3.29)
EV sin 
Where PG     X  X  
t csc

1  EV cos 
V2
V  

 QL 
   X  X t csc    X  X t csc  

(3.30)
EV cos 
V2
Q


G
Where
 X  X t csc    X  X t csc  
The dynamic model of TCSC is given as (Barkhordary et al., 2006; Cong & Wang, 2001)
1
X t csc     X t csc    X t csc 0  k t u t 
Tc
(3.31)
37
Where
X TCSC     X C  C1 2     sin2     C2 cos2     tan      tan   
X LC
XC XL
 X  XL

, C1   C
XC  XL


 4X 2

LC
C

,
 2 

X L






X t csc   is the equivalent output TCSC reactance in per unit,  is the firing angle of TCSC in
radian, Tc is the time constant of TCSC in seconds, k t is the gain of TCSC regulator and ut is
the control input of TCSC regulator, X L is the inductive reactance of TCSC in ohms, X C is the
capacitive reactance of TCSC in ohms and  is the parameter which determines the operating
performance of TCSC.
From equation (3.30), the behaviour of the bus-voltage versus the load reactive power demand is
also studied. This is carried out in order to examine the effects of TCSC in the PQ load model.
V
E cos   E 2 cos 2    4QL  X  X t csc  
2
(3.32)
From equation (3.32), the load reactive power demand is also considered to be the bifurcation
parameter. Also, from equations (3.28-3.31), the Jacobian matrix of the SMDL system with
TCSC is given as
0
1

   EV cos  
D

 
M

X

X




M

t csc


A     EV sin  
0
   X  X t csc   


0
0

0
0

  E sin     EV sin   


 
2 
 M  X  X t csc     M  X  X t csc    

 E cos   2V  1  EV cos   V 2  




2
  X  X t csc       X  X t csc    

1

0


Tc
38
(3.33)
3.4.
LQR Optimal Controller Design Procedure
The linear quadratic regulator (LQR) optimal controller design procedure is guaranteed to
produce an optimal feedback gain matrix that stabilizes the system as long as the following holds
(Kirk, 2004):

Selecting the system matrices A , B and C which can be easily obtained from the state space
model of the nonlinear dynamic system.

Selecting the control input U to be optimized

Selecting the design parameter matrices Q and R . According to (Kirk, 2004), it is stated that
large Q penalizes transients of the state variable X while large R penalizes the usage of
control actions

Solving the algebraic Riccati equation for the positive-definite matrix P

Determining the optimal feedback gain matrix K

Determining the eigenvlaues of the output closed-loop system
3.4.1. LQR Optimal Controller Algorithm
For a given continuous-time linear system described by
x  Ax  Bu
(3.34)
y  Cx
(3.35)
determine the optimal feedback gain matrix K of the LQR vector
u   Kx
(3.36)
in order to minimize the quadratic cost function
39
J
N
 x
0
T

Qx  u T Ru dt
(3.37)
subject to the constraints
Q  0, R0
Where the optimal feedback gain matrix K can be computed by
K  R 1 B T P
(3.38)
in which the positive definite matrix P is found by solving the finite continuous time algebraic
Riccati equation:
AT P  PA  PBR 1 BT P  Q  0
(3.39)
Therefore, by substituting equation (3.36) into (3.34) yields the output closed loop system as
x  Ax  BKx   A  BK x
(3.40)
Where equation (3.40) is the output closed loop system and is a linear, time-varying, full-state
feedback regulator that minimizes the quadratic cost function.
3.4.2. Weight Matrix Selection
From equation (3.37), Q (an n x n matrix) and R (an m x m matrix) are the weight matrices
which are to be selected by the design engineer. Depending on how these parameters are
selected, the closed loop system will exhibit a different response. The quadratic cost function is
well-defined if Q is a positive definite or semi-definite symmetry matrix and R is a positive
definite symmetry matrix. This implies that the scalar quantity x T Qx is always positive or zero
40
at each time t for all functions xt  and the scalar quantity u T Ru is always positive at each time t
for all values of ut  . A positive definite or semi-definite symmetry matrix can be achieved by
selecting Q and R to be diagonal matrix. In addition, according to (Philips & T., 1995 ), Q can
be obtained by multiplying the transpose of the output matrix with the output matrix (i.e.,
C T  C ) while the value of R can be large so as to lessen the effort of the control action.
3.5.
LQR Optimal Controller Design for the SMDL System with TCSC
The linearized state space model of the single-machine dynamic load (SMDL) system with a
thyristor-controlled series capacitor is obtained based on equations (3.34-3.35) and is given as
0
1

D
  EV cos

 M X  X




M
t csc

x  
 EV sin 

0
   X  X t csc  


0
0

1

0
y
0

0
0
 E sin 
M  X  X t csc  
E cos  2V
  X  X t csc  
0
0
 EV sin 

0


  EV sin   P  Q  R  S 





M  X  X t csc  2  
MT
 


2

    P  Q  R  S  EV cos   V 2
EV cos   V
 V
 
T

  X  X t csc  2  
 P  Q  R  U 
  X t csc   
1


Tc

Tc


0 0 0 x1 
 
1 0 0 x2 
0 1 0 x3 
 
0 0 1 x4 

3.41
3.42
Where
The control input is given as u t   
 x1   

 x    

The state variables is given as  2   
x3   V

  

 x 4   X t csc  









and

 x 1   

 x  
 2     

 x 3   V

  
 x 4   X t csc  
41
The Jacobian matrix of the state variables is given as
0
1

 EV cos 
D


 M X  X
M
t csc  

 EV sin 
A

0
   X  X t csc  


0
0

0
 E sin 
M  X  X t csc  
E cos   2V
  X  X t csc  
0
0
 EV sin 



M  X  X t csc  2 
2 
EV cos   V

  X  X t csc  2 

1


Tc

The Jacobian matrix of the control inputs is given as
0

  EV sin   P  Q  R  S 

MT

2
B    P  Q  R  S  EV cos   V

T

 P  Q  R  U 

Tc












Where
2 
 4X
LC

 2 cos    2 cos 2    2 
0
0
 X L 


P 
2 cos  0   2
2 
 4X
LC
 sin 2   2   tan    
Q  
0
0
 X L 




2 
 4X
LC

 2 cos    2
0
 X L 
2 
 4X

LC

 2 cos    2 sin 2    2  2  
0
0
 X L 
2

R  
cos  0   2
 X  XL 
 X  XL 
S  2 c
 cos 2  0  2    2  c







2 
  X  X  4X
L 
LC  cos 2   2    2  X c  X L 
U   2 c



0

X L  

 




 X  XL 
 X  XL 
T  2 X c  2 c
    0    c
 sin 2    0  








2 
 4X
LC

 cos 2     tan      
0
0
 X L 


42
2 

 4X
LC

 cos 2     tan     
0
0
 X L 




2
3.6.
Conclusion
Chapter three of this dissertation presented the study methods and design employed in achieving
the objective of this research work. These are established by presenting the mathematical design
for the description of the internal state of thyristor controlled series capacitor (TCSC), the
formulation of nonlinear models for the single-machine dynamic load (SMDL) system with and
without a TCSC and linear quadratic regulator (LQR) optimal controller design for the SMDL
system with TCSC. Therefore, the research findings, analysis and discussion of results obtained
from these designs are presented in the next chapter.
43
CHAPTER 4
ANALYSIS AND DISCUSSION OF RESULTS
This chapter covers the research findings, analysis and discussions for the description of the
internal state of thyristor controlled series capacitor (TCSC), investigating the steady state
behaviour of TCSC under varying capacitance and inductance, voltage collapse simulation,
bifurcation diagram, time domain simulation and phase portrait simulation for the singlemachine dynamic load (SMDL) system with and without a TCSC and the linear quadratic
regulator (LQR) optimal controller design simulation for the SMDL system with TCSC.
4.1.
Description of the Internal State of TCSC
Using the mathematical model developed as given in equations (3.19-3.21), the waveforms
shown in Figs. 4.1-4.3 are obtained with the aid of MATLAB/M-file. The results are obtained
based on the data given in Table 1 (Meikandasivam et al., 2008):
Table 1: Data used for the description of the internal state of TCSC

Inductance
Capacitive
Inductive
F 
mH
Reactance  
Reactance  
254.80
7.89
12.49
2.48
2.24
247.00
10.00
12.89
2.39
2.32
Capacitance
44
I L( A )
1
0
-1
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
IT (A )
5
0
-5
IC (A )
2
0
-2
V C(V )
20
0
-20
Firing Angle of the Thyristor (Radian)
Figure 4.1: Description of the internal state of TCSC for
C  254 .8  F and L  7 .89 mH
C a p a c ito r.C u rr(A ), T h y ris to r.C u rr(A ) & C a p a c i to rV o l t(V )
80
60
40
20
0
-20
-40
Capacitor Current (A)
Thyristor Current (A)
Capacitor Voltage (V)
-60
-80
0
10
20
30
40
50
60
70
Firing Angle of the Thyristor (Radian)
Figure 4.2: Description of the internal state of TCSC for
45
C  247  F and L  10 mH
80
C a p a c ito r.C u rr(A ), T h yristo r.C u rr(A ) & C a p a cito rV o lt(V )
80
60
40
20
0
-20
-40
Capacitor Current (A)
Thyristor Current (A)
Capacitor Voltage (V)
-60
-80
0
10
20
30
40
50
60
70
80
Firing Angle of the Thyristor (Radian)
Figure 4.3: Description of the internal state of TCSC for
C  254 .8  F and L  7 .89 mH
Shown in Figs. 4.1-4.3 are the simulations obtained for the description of the internal state of
TCSC. The results obtained show that the mathematical model developed is capable of
describing the internal state of TCSC and also gives better understanding on how TCSC
operates.
4.2.
Investigating the Steady State Behaviour of TCSC
Using the mathematical model developed for the description of the internal state of TCSC as
given in equations (3.19-3.21) and the fundamental effective TCSC reactance given in equation
(2.1), the waveforms shown in Figs. 4.4-4.9 are obtained using MATLAB/M-file. These results
are obtained based on the data given in Table 2 (Habur & O’Leary, 2001; Shimpi et al., 2010;
Varma, 2010).
46
Table 2: Data used for investigating the steady state behaviour of TCSC

Inductance
Capacitive
Inductive
F 
mH
Reactance  
Reactance  
247.00
7.89
12.89
2.45
<3
212.20
9.55
15.00
3.00
>3
450.00
1.10
7.07
0.35
>3
In d u c tiv e R e g io n (O h m s )
2500
C a p a c itiv e R e g io n (O h m s )
Capacitance
0
2000
1500
1000
500
-500
-1000
-1500
-2000
90
100
110
120
130
140
150
Firing Angle of Thyristor (Radian)
Figure 4.4: Single-TCSC resonance points for   3
47
160
170
180
Inductive Region (O hm s)
C apacitive Region (O hm s)
300
200
100
0
-100
-200
-300
-400
-500
-600
90
100
110
120
130
140
150
160
170
180
160
170
180
Firing Angle of Thyristor (Radian)
Capacitive Region (Ohms)
Inductive Region (Ohms)
Figure 4.5: Multiple-TCSC resonance point for   3
150
100
50
0
-50
-100
-150
-200
-250
-300
90
100
110
120
130
140
150
Firing Angle of Thyristor (Radian)
Figure 4.6: Multiple-TCSC resonant point for   3
48
Lin e C urrent (A), Th yristo r C urren t (A), Ca pacitor Current (A) and C apacitor Vo ltage (Volts)
Plot of Line Current (A), Capacitor Current (A), Thyristor Current (A) and Capacitor Volatge (Volts) versus Firing Angle (Radian)
80
60
40
20
0
-20
-40
Line Current
Thyristor Current
Capacitor Current
Capacitor Voltage
-60
-80
0
10
20
30
40
50
Firing Angle Range of Thyristor (Radian)
60
70
80
Line Current (A), Capacitor Current (A),ThyristorCurrent (A) &CapacitorVolatge(Volts)
Figure 4.7: Description of the internal state of TCSC for   3
Plot of Line Current (A), Capacitor Current (A), Thyristor Current (A) and Capacitor Volatge (Volts) versus Firing Angle (Radian)
80
Line Current
Thyristor Current
Capacitor Current
Capacitor Voltage
60
40
20
0
-20
-40
-60
-80
0
10
20
30
40
50
Firing Angle Range of Thyristor (Radian)
Figure 4.8: Description of the internal state TCSC for   3
49
60
70
80
Line Current (A), Capac itorCurrent(A),Thy ristorCurrent (A),Capac itorVoltage(Volts)
Plot of Line Current (A), Capacitor Current (A), Thyristor Current (A), Capacitor Voltage (Volts) versus Firing Angle (Radian)
60
40
20
0
-20
Line Current
Thyristor Current
Capacitor Current
Capacitor Voltage
-40
-60
0
10
20
30
40
50
Firing Angle Range of Thyristor (Radian)
60
70
80
Figure 4.9: Description of the internal state of TCSC for   3
Shown in Figs. 4.4-4.9 are the waveforms obtained for studying the steady state behaviour of
TCSC. The simulation obtained in Fig. 4.4 shows that a single-TCSC resonance point occurs. As
a result, the steady state behaviour of TCSC (see Fig. 4.7) is not negatively affected because an
allowable single TCSC resonance point is obtained. A single resonance point is obtained because
appropriate values for the TCSC capacitance and inductance are chosen. These appropriate
values are obtained because the parameter which describes the TCSC operating performance is
less than three (i.e,   3 ). The simulations obtained in Figs. 4.5-4.6 show that the steady state
behaviour of TCSC (see Figs. 4.8-4.9) is negatively affected because an unacceptable multiple
TCSC resonance points occur simply because the parameter which describes the TCSC operating
performance is greater than three (i.e.,   3 ). This implies that inappropriate values for the
TCSC component parameters are chosen. In addition, it is also observed as shown in Figs. 4.54.6 and Figs. 4.8-4.9 that due to the presence of multiple resonance points, the following occur:
50

Current spikes in the capacitor and thyristor controlled reactor circuit

Capacitor voltage overshoot and oscillations
4.3.
Voltage Collapse Simulation
The voltage collapse simulations for the SMDL system with and without a TCSC are obtained
using equations (3.26) and (3.32). By varying the load reactive power demand and applying the
data given in Table 3 (Gu et al., 2007) to equations (3.26) and (3.32), the waveforms shown in
Figs. 4.10-4.11 are obtained with the aid of MATLAB/M-file.
Table 3: Data used for voltage collapse simulation
X  p.u 
E p.u
Pm  p.u 
 rad
X t csc   p.u 
0.5
1
1
0
0.25
1
0.9
0.8
Bus-Voltage (p.u)
0.7
0.6
SNB
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Load Reactive Power Demand (p.u)
0.4
0.45
Figure 4.10: SMDL system without the TCSC: voltage collapse simulation
51
0.5
1
0.9
0.8
Bus-Voltage (p.u)
0.7
0.6
QL max
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Load Reactive Power Demand (p.u)
0.8
0.9
1
Figure 4.11: SMDL system with TCSC: voltage collapse simulation
Shown in Figs. 4.10 and 4.11 are the voltage collapse simulations for the SMDL system with and
without a TCSC. The result obtained in Fig. 4.10 shows that at load reactive power demand of
0.5 per-units, a saddle-node bifurcation (SNB) occur. In power systems, SNB is associated with
voltage collapse and dynamic instability and this usually occurs at maximum loading condition
(Gu et al., 2007). This implies that power systems generally have a maximum loading condition
which is associated with a SNB. With the connection of TCSC, simulation obtained shows that
an additional load reactive power demand of 0.5 per-units is injected. This implies that TCSC is
capable of improving the system’s loadability.
52
4.4.
Bifurcation Analysis
The bifurcation diagrams shown in Figs. 4.12 and 4.13 are got by applying the data given in
Tables 4, 5 and 6 to the Jacobian matrices given in equations (3.27) and (3.33). It is important to
note that the data given in Tables 4 and 6 are not chosen arbitrarily but are obtained from the
voltage collapse simulation.
Table 4: Data used for the SMDL system without the TCSC
S/N
V  p.u
 rad 
QL  p.u
1st
1.0000
0.5236
0.0000
2nd
0.9472
0.5561
0.1000
3rd
0.8873
0.5986
0.2000
4th
0.8163
0.6592
0.3000
5th
0.7236
0.7629
0.4000
6th
0.7000
0.7956
0.4200
7th
0.6700
0.8424
0.4400
8th
0.6414
0.8939
0.4600
9th
0.6000
0.9851
0.4800
10th
0.5707
1.0677
0.4900
11th
0.5000
1.5708
0.5000
Table 5: Data used for bifurcation diagrams
 secs
Msecs
E p.u
X  p.u 
Tc sec s 
0.001
0.1
1
0.5
0.015
53
Table 6: Data used for the SMDL system with TCSC
S/N
V  p.u  rad 
X t csc  p.u
QL  p.u
25-70%
Compensation
1st
1.0000
0.3844
0.1250
0.0000
2nd
0.9743
0.3785
0.1400
0.1000
3rd
0.9472
0.3728
0.1550
0.2000
4th
0.9183
0.3676
0.1700
0.3000
5th
0.8873
0.3629
0.1850
0.4000
6th
0.8536
0.3591
0.2000
0.5000
7th
0.8162
0.3567
0.2150
0.6000
8th
0.7739
0.3564
0.2300
0.7000
9th
0.7236
0.3550
0.2450
0.8000
10th
0.6581
0.3530
0.2600
0.9000
11th
0.6414
0.3500
0.2750
0.9200
12th
0.6225
0.3441
0.2900
0.9400
13th
0.6000
0.3310
0.3050
0.9600
14th
0.5707
0.3209
0.3200
0.9800
15th
0.5500
0.3150
0.3350
0.9900
16th
0.5000
0.3047
0.3500
1.0000
54
0.1
Stable Equilibrium Point
Unstable Equilibrium Point
Stable Equilibrium Point
0.08
0.06
Im aginary-ax is
0.04
SNB
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-2.5
-2
-1.5
-1
Real-axis
-0.5
0
0.5
Figure 4.12: SMDL system without the TCSC – plot of eigenvalues
0.5
0.4
0.3
Stable Equilibrium Point
Unstable Equilibrium Point
Stable Equilibrium Point
Stable Equilibrium Point
Almost
approaching SNB
at Q=0.99 per unit
Im aginary -ax is
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-3.5
-3
-2.5
-2
-1.5
Real-axis
Figure 4.13: SMDL system with TCSC – plot of eigenvalues
55
-1
-0.5
0
0.5
Shown in Figs. 4.12 and 4.13 are the bifurcation diagrams for the SMDL system with and
without a TCSC. The result obtained in Fig. 4.12 shows that without the connection of TCSC,
the SMDL system presents a saddle-node bifurcation (SNB) at a load reactive power demand of
0.5 per unit. Local bifurcation such as SNB is known by observing the eigenvalues of the current
operating point and, as certain parameters in the system change slowly, the system eventually
turns unstable due to one of the eigenvalues becoming zero. Therefore, due to loss of operating
equilibrium, the system state changes dynamically and as a result, system voltages fall
dynamically (see Fig. 4.10). With the connection of TCSC, the result obtained (see Fig. 4.13)
shows that at a load reactive power demand of 0.99 per-units; the SMDL system almost
approached SNB. Compared to the result obtained in Fig. 4.12, this implies that an additional
load reactive power demand of 0.49 per-units is injected by the TCSC. In addition, it is also
observed that at a load reactive power demand of 1.0 per-unit, the SMDL system becomes
unstable because of the presence of a real and positive eigenvalue.
4.5.
Time Domain Simulation
To verify if the results obtained in Figs. 4.12 and 4.13 are valid, time domain simulations for the
SMDL system with and without a TCSC are obtained by designing the PQ load model given in
equations (3.26) and (3.32) with the aid of MATLAB/Simulink. Time domain simulations are
obtained by applying the value of QL that corresponds to the bifurcation point to the PQ
Simulink model as shown in Figs. 4.14 and 4.15. That is, at QL  0.5 per - units, the system
without TCSC presents a saddle-node bifurcation (SNB) while at QL  1.0 per - units, the system
becomes unstable. As a result, the time domain simulations as shown in Figs. 4.16 and 4.17 are
obtained.
56
1
t
Clock
Constant
To Workspace
E
V
1.5708
Constant 2
0.5
Constant 3
d
fcn
1
xo s
V1
X
0.5
Scope
Integrator
Constant 4
0.001
Constant 5
Q
t
V
Embedded
MATLAB Function
To Workspace 1
0.5
v
Figure 4.14: PQ load simulink model for SMDL system without the TCSC
t
1
Clock
To Workspace
Constant
E
V
0.3047
Constant 2
0 .5
Constant 3
d
X
fcn
1
xo s
V1
Q
1
Scope
Integrator
t
Constant 4
V
X_c
0 .001
Constant 5
To Workspace 1
Embedded
MATLAB Function
0.5
0.35
v
Constant 1
Figure 4.15: PQ load simulink model for SMDL system with TCSC
57
0.5
0.4
0.3
Bus-Voltage (p.u)
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.5
1
1.5
2
Time (seconds)
2.5
3
3.5
Figure 4.16: Time-domain simulation for the SMDL system without TCSC
0.85
0.8
Bus-Voltage (p.u)
0.75
0.7
0.65
0.6
0.55
0.5
0
0.005
0.01
0.015
0.02
0.025
Time (seconds)
0.03
Figure 4.17: SMDL system with TCSC – Time domain simulation
58
0.035
0.04
0.045
0.05
Shown in Figs. 4.16 and 4.17 are the time domain simulations for the SMDL system with and
without a TCSC. The result obtained in Fig. 4.16 shows that at t  3.15sec s , the system without
TCSC undergoes a voltage collapse which is induced by the SNB and this occurs at maximum
loading condition (see Figs. 4.10 and 4.12). With the connection of TCSC, the simulations
obtained in Fig. 4.17 shows that although the SMDL system resulted to instability, the system’s
loadability increased (see Figs. 4.11 and 4.13).
4.6.
Phase Portrait Simulation
Phase portrait is the geometric representation of the trajectories of nonlinear dynamic systems.
The waveforms shown in Figs. 4.18 and 4.19 are the phase portraits of the SMDL system with
and without a TCSC. These results are obtained by writing MATLAB scripts for the nonlinear
models of the system with and without a TCSC.
Figure 4.18: SMDL system without the TCSC – phase portrait simulation
59
Figure 4.19: SMDL system with TCSC – phase portrait simulation
As shown in Figs. 4.18 and 4.19 are the phase portrait simulations for the SMDL system with
and without a TCSC. The result obtained in Fig. 4.18 shows that without the TCSC, the system is
far from the point of stability and with the connection of TCSC, the result obtained in Fig. 4.19
shows that the system approaches stability.
4.7.
LQR Optimal Controller Design Simulation
The results obtained from the stability analysis of SMDL system with TCSC showed that that
except for the presence of an unstable equilibrium point, the SMDL system remains generally in
its stable state. Therefore, to enhance the stability of the SMDL system with the TCSC, a linear
quadratic regulator (LQR) optimal control technique is used.
60
Applying the data given in Table 1 and the 16 th bifurcation point given in Table 6 to the
linearized state space model given in equations (3.41-3.42) yields
0
0.001
0
0
 x1  
  x1   0 
 x    0.0191
0
 0.0304  0.0653  x 2   0.25 
 2  


 x 3   0.9798
0
 2.6506 3.5185   x3   1.14 
  
  

0
0
0
 0.7333  x 4   0.15
 x 4  
1
0
y
0

0
0 0 0  x1 
1 0 0  x 2 
0 1 0  x3 
 
0 0 1  x 4 
Where
0.001
0
0 
 0
  0.0191
0
 0.0304  0.0653

A
,
 0.9798
0
 2.6506 3.5185 


0
0
 0.7333
 0
 0 
 0.25 

B
 1.14 


 0.15
and
1
0
C
0

0
0
1
0
0
0
0
1
0
0
0
0

1
u     Kx
Where the control input u is the firing angle of the TCSC thyristor  . Therefore, to determine
the optimal feedback gain matrix K of the LQR vector in order to minimize the quadratic cost
function


20 
J    x1
0



x2
x3
1
0
x 4 
0

0

0 0 0  x1 




1 0 0  x 2 

  T 20 dt
0 1 0  x 3 

 

0 0 1  x 4 

61
Subject to the constraints
R  0 and is chosen to [20]
1
0
T
Q  0 and is chosen to be Q  C C  
0

0
0
1
0
0
0
0
1
0
0
0
0

1
T
 0 


1  0.25 
The optimal feedback gain matrix K is given as K  [20]
P
 1.14 


  0.15
in which the positive definite matrix P is found by solving the finite continuous time algebraic
Riccati equation
T
T
0
0.001
0
0
0
0.001
0
0




 0 
 0 
1









0
 0.0304  0.0653
 0.0191
0
 0.0304  0.0653
0.25 
0.25 
0
  0.0191
P  P
 P
[ 20]1 
P
  0.9798
 0.9798
 1.14 
 1.14 
0
0
 2.6506 3.5185 
0
 2.6506 3.5185 









0
0
0
 0.7333
0
0
0
 0.7333


 0.15
 0.15
0
0 0 0

1 0 0
0
0 1 0

0 0 1
Therefore, by the substituting the control input into the linearized state space model x , yields the
output closed loop system as
 0
0.001
0
0   0    x1 


 0.0191
0
 0.0304  0.0653  0.25    x2 


x 

K
  0.9798
0
 2.6506 3.5185   1.14    x3 

 
  
0
0
 0.7333  0.15   x4 
  0
With the aid of MATLAB/M-file, the following results as shown in Figs. 4.20-4.23 are obtained:
62
0.15
Optimal Control Input u(t)
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0
2
4
6
8
10
time (t)
12
14
16
18
20
Figure 4.20: Optimal control input for the SMDL system with TCSC
0.3
K1
K2
K3
K4
0.2
F eedbac k G ain K (k )
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
time (t)
12
14
Figure 4.21: Optimal feedback gains for the SMDL system with TCSC
63
16
18
20
1
Rotor speed: x2(t)
Rotor angle: x1(t)
1
0.5
0
-0.5
0
5
10
15
time (sec)
TCSC reactance: x4(t)
Bus-voltage: x3(t)
0.5
0
-0.5
0
5
10
15
time (sec)
0
-0.5
20
1
0.5
0
5
10
15
time (sec)
20
0
5
10
15
time (sec)
20
1
0.5
0
-0.5
20
-1
Figure 4.22: Time response of the state variables
0.06
0.04
Imaginary-axis
0.02
0
-0.02
-0.04
-0.06
-0.9
-0.8
-0.7
-0.6
-0.5
Real-axis
-0.4
Figure 4.23: Output eigenvalues of the closed-loop system
64
-0.3
-0.2
-0.1
Shown in Figs. 4.20 to 4.23 are the results obtained for the optimal control input, optimal
feedback gain matrix, time response of the state variables and the eigenvalues of the output
closed loop system. The firing angle of the TCSC is the control input required to be optimized
(see Fig. 4.20). The optimal control input is achieved because of the results obtained for the
optimal feedback gain matrix (see Fig. 4.21). As a result, as the time increases, the state variables
stabilized (see Fig. 4.22). The stability of the system is guaranteed because a real and negative
eigenvalues are obtained for the output closed-loop system (see Fig. 4.23).
4.8.
Conclusion
Chapter four of this dissertation presented the research findings, analysis and discussions of
results for the description of the internal state of thyristor controlled series capacitor (TCSC),
investigating the steady state behaviour of TCSC under varying capacitance and inductance,
voltage collapse simulation, bifurcation diagram, time domain simulation and phase portrait
simulation for the single-machine dynamic load (SMDL) system with and without a TCSC and
the linear quadratic regulator (LQR) optimal controller design simulation for the SMDL system
with TCSC.
The simulations obtained for the formulated mathematical model of TCSC showed that the
developed model is not only capable of describing the internal state of TCSC but also gives
better understanding on how TCSC operates (see Figs. 4.1-4.3 and Fig. 4.7). In addition, the
simulations obtained based on the mathematical model formulated and the TCSC reactance
model stated in (Meikandasivam et al., 2008) showed that the steady state behaviour of TCSC is
negatively influenced when inappropriate values are chosen for its capacitance and inductance
(see Figs. 4.5-4.6 and 4.8-4.9). Therefore, to benefit from TCSC control, it is important to restrict
65
the value of the parameter  which determines the TCSC operating performance between 2.2
and 2.7 (Hu et al., 2004). Futhermore, the simulations obtained via voltage collapse, bifurcation
diagram, time-domain and phase portrait showed that without the TCSC, the system presents a
saddle-node bifurcation (SNB) at a load reactive power demand of 0.5 per-units (see Figs. 4.10,
4.12, 4.16 and 4.18). In power systems, SNB is associated with voltage collapse and dynamic
instability and this usually occurs at maximum loading condition (Gu et al., 2007). With the
connection of TCSC, simulation obtained showed that an additional load reactive power demand
of 0.5 per-units was injected (see Fig. 4.11). This implies that TCSC is capable of improving the
system’s loadability. In addition, it was clearly seen that SNB was prevented but at a load
reactive power demand of 1.0 per unit, the system becomes unstable (see Figs. 4.13, 4.17 and
4.19). As a result, LQR optimal controller technique was designed to improve the stability of the
system. The simulations obtained for the optimal control input, time response of the state
variables, optimal feedback gains and eigenvalues of the closed-loop system showed that LQR
optimal controller is capable of stabilizing the SMDL system with TCSC (see Figs. 4.20-4.23).
66
CHAPTER 5
5.1.
CONCLUSION
Introduction
Enhancing power flow in transmission systems is important for efficient power supply because a
developed society demands a large amount of electrical energy for domestic, commercial and
industrial purposes (Machowski et al., 2008). Efficient supply of electric power to various
customer load demands is challenging because of power system instabilities. Nevertheless, with
the help of flexible alternating current transmission system (FACTS), power flow can be
enhanced by improving the voltage profile of the transmission lines, improving the power system
stability, reducing the power system losses, optimizing power flow between parallel lines and
increasing the transmission line capability (Hingorani, 2000; Hingorani et al., 2000). As a result,
the problem statement of this research study is to enhance power flow in transmission system. To
successfully solve this task, among the various means of enhancing power flow in a transmission
system with the help of FACTS devices, this research work focussed on the aspect of improving
the stability of power systems. Based on this information, the objective of this dissertation is to
carryout nonlinear modelling and control of a single-machine dynamic load (SMDL) system with
a thyristor-controlled series capacitor (TCSC) for power system stability enhancement.
5.2.
Objectives and Research Process
Firstly, to achieve the objective of this research work, a mathematical model describing the
internal state of TCSC was formulated using firing angles of the TCSC thyristors as the inputs.
Combining the states model and the TCSC reactance model reported in (Meikandasivam et al.,
2008), the steady state behaviour of TCSC under varying capacitance and inductance was
67
investigated. Furthermore, to study the behaviour of the SMDL system under load reactive
power demand variation, nonlinear models of the system with and without a TCSC are
formulated based on recent and relevant advances reported by (Canizares, 2002; Gu et al., 2007).
Then based on the nonlinear models formulated, the stability of the system was analyzed. Lastly,
to improve the stability of the SMDL system with the TCSC, a linear quadratic regulator (LQR)
optimal control technique was employed. The following are the study methods and design
employed in achieving this task via MATLAB/Simulink application software:

Development of a mathematical model for the description of the internal state of TCSC

Investigating the steady state behaviour of TCSC under varying capacitance and inductance

Formulation of nonlinear dynamical models for the SMDL system with and without a TCSC

Stability analysis of the SMDL system with and without a TCSC under load reactive power
demand variation via simulations such as voltage collapse, bifurcation diagram, time domain
and phase portrait

Design of a LQR optimal control technique for the stability enhancement of the SMDL
system with TCSC
5.3.
Summary of Research Results
Firstly, according to (Geng et al., 2002), it was stated that developing a TCSC model is
challenging because of its variant topology and nonlinear circuit. As a result, in this research
work, an additional effort in understanding the internal state and operation of TCSC is studied.
Secondly, according to (Hu et al., 2004; Meikandasivam et al., 2008), it was stated that the
steady state behaviour of TCSC is negatively influenced when inappropriate values for its
capacitance and inductance are chosen. As a result, this research work also investigates the
68
steady state behaviour of TCSC based on the model formulated and the TCSC reactance model
reported by (Meikandasivam et al., 2008). Thirdly, according to (Chiang, 2004), it was stated
that to study the nature and stability of a nonlinear dynamic system, it is important to formulate
the system’s nonlinear models. As a result, to study the behaviour of the SMDL system under
load reactive power demand variation, nonlinear models for the SMDL system with and without
a TCSC are formulated based on recent and relevant advances reported by (Canizares, 2002; Gu
et al., 2007) and based on the nonlinear models formulated, the stability of the SMDL system
with and without a TCSC is analysed via simulations such as voltage collapse, bifurcation
diagram, time domain and phase portrait. Lastly, according to (Abido, 2008; Bevrani et al., 2009;
Philips & T., 1995 ), it was stated that a suitable counter-measure for power system instabilities
is to apply a control technique. As a result, to improve the stability of the SMDL system with
TCSC, a linear quadratic regulator (LQR) optimal control technique is used. The simulations
obtained for the formulated mathematical model of TCSC showed that the developed model is
not only capable of describing the internal state of TCSC but also gives better understanding on
how TCSC operates (see Figs. 4.1-4.3 and Fig. 4.7). In addition, the simulations obtained based
on the mathematical model formulated and the TCSC reactance model stated in (Meikandasivam
et al., 2008) showed that the steady state behaviour of TCSC is negatively influenced when
inappropriate values are chosen for its capacitance and inductance (see Figs. 4.5-4.6 and 4.8-4.9).
Therefore, to benefit from TCSC control, it is important to restrict the value of the parameter 
which determines the TCSC operating performance between 2.2 and 2.7 (Hu et al., 2004).
Futhermore, the simulations obtained via voltage collapse, bifurcation diagram, time-domain and
phase portrait showed that without the TCSC, the system presents a saddle-node bifurcation
(SNB) at a load reactive power demand of 0.5 per-units (see Figs. 4.10, 4.12, 4.16 and 4.18). In
69
power systems, SNB is associated with voltage collapse and dynamic instability and this usually
occurs at maximum loading condition (Gu et al., 2007). With the connection of TCSC,
simulation obtained showed that an additional load reactive power demand of 0.5 per-units was
injected (see Fig. 4.11). This implies that TCSC is capable of improving the system’s loadability.
In addition, it was clearly seen that SNB was prevented but at a load reactive power demand of
1.0 per unit, the system becomes unstable (see Figs. 4.13, 4.17 and 4.19). As a result, LQR
optimal controller technique was designed to improve the stability of the system. The simulations
obtained for the optimal control input, time response of the state variables, optimal feedback
gains and eigenvalues of the closed-loop system showed that LQR optimal controller is capable
of stabilizing the SMDL system with TCSC (see Figs. 4.20-4.23).
5.4.
Recommendations for Further Study
An electrical power system is a large, complex and dynamic system capable of generating,
transmitting and distributing electrical power over a large geographical area (Machowski et al.,
2008). However, studying its nonlinear dynamical behaviour becomes difficult because of
instabilities which occur due to disturbances such as contingencies and load disturbances.
Although power systems are normally operated near a stable equilibrium point, the variation of
the system load parameters can move the system slowly from one equilibrium point to another
until it reaches collapse point (Chiang, 2004). As a result, this dissertation presents nonlinear
modelling, bifurcation analysis and control of a single-machine dynamic load (SMDL) system
with a thyristor-controlled series capacitor (TCSC) for power system stability enhancement. For
further research studies, the following topics could be investigated:

Stability analysis and control of a multi-machine power system with any FACTS device
70

Bifurcation analysis and control of various power system load models
5.5.
Final Conclusion
The objective of this research work is to carry out nonlinear modelling and control of a singlemachine dynamic load (SMDL) system with a thyristor-controlled series capacitor (TCSC). The
results obtained via voltage collapse, bifurcation diagram, time domain and phase portraits
showed that without the TCSC, the system presents saddle-node bifurcation (SNB) at a load
reactive power demand of 0.5 per-units. This kind of bifurcation is associated with voltage
collapse and dynamic instability in power systems. With the connection of TCSC, simulations
obtained showed that an additional load reactive power demand of 0.5 per-units was injected.
This implies that TCSC is capable of improving the system’s loadability. In addition, unlike the
situation where static var compensator (SVC) induced Hopf bifurcation after increasing the
system’s loadability (Gu et al., 2007), it was clearly seen that with the connection of TCSC, SNB
was prevented but at a load reactive power demand of 1.0 per unit, the system becomes unstable.
As a result, LQR optimal controller technique was designed to improve the stability of the
system.
71
References
ABED, E. H., ALEXANDER, J. C., WANG, H., HAMDAN, A. M. A. & LEE, H. C. 1992.
Dynamic bifurcations in a power system model exhibiting voltage collapse. Circuits and
Systems, 1992. ISCAS '92. Proceedings., 1992 IEEE International Symposium on., San Diego,
USA. , Vol. 5:2509-2512
ABIDO, M. 2008. Power system stability enhancement using FACTS controllers: A review.
The Arabian Journal for Science and Engineering, Vol. 34:153-172.
AJJARAPU, V. & LEE, B. 1992. Bifurcation theory and its application to nonlinear dynamical
phenomena in an electrical power system. Power Systems, IEEE Transactions on, 7(1):424-431.
ALI, N. M. & AMIN, V. 2007, July 26 2007-June 31 2007. An LQR/Pole Placement Controller
Design for STATCOM. Paper presented at the Control Conference, 2007. CCC 2007. China.
BARKHORDARY, M., NABAVI, S., MOTLAGH, M. R. J. & KAZEMI, A. 2006. A Robust
Feedback Linearization Controller Design for Thyristor Control Series Capacitor to Damp
Electromechanical Oscillations [Electronic Version]. 2587-2592.
BEVRANI, H., LEDWICH, G. & FORD, J. J. 2009. On the use of df/dt in power system
emergency control. Power Systems Conference and Exposition, 2009. PSCE '09. IEEE/PES,
Seattle, WA:1-6.
72
BLANCHARD, P. & DEVANEY, R. L. 2006 Differential Equations. London: Hall, G. R.,
Thompson.
CANIZARES, C. A. 2002. Voltage Stability Assessment: concepts, practices and tools. (Final
document, available at http://www.power.uwaterloo.ca Aug, 2002).
CHANG, J. & CHOW, J. H. 1997. Time-optimal series capacitor control for damping interarea
modes in interconnected power systems. Power Systems, IEEE Transactions on, 12(1):215-221.
CHEN, X. R., PAHALAWATHTHA, N. C., ANNAKKAGE, U. D. & KUMBLE, C. S. 1995.
Controlled series compensation for improving the stability of multi-machine power systems.
Generation, Transmission and Distribution, IEE Proceedings., New Delhi, Vol. 142(4):361-366.
CHIANG, H. D. 2004. Bifurcation Control. Application of bifurcation analysis to power
systems., Lecture notes in Control and Information Sciences., Beijing, China, 293:1-28.
CHIANG, H. D., CONNEEN, T. P. & FLUECK, A. J. 1994. Bifurcations and chaos in electric
power systems: Numerical studies. Journal of the Franklin Institute, 331(6):1001-1036.
CHIANG, H. D., LIU, C. W., VARAIYA, P. P., WU, F. F. & LAUBY, M. G. 1993. Chaos in a
simple power system. Power Systems, IEEE Transactions on, 8(4):1407-1417.
73
CONG, L. & WANG, Y. 2001. A coordinated control approach for FACTS and generator
excitation system. Transmission and Distribution Conference and Exposition, 2001 IEEE/PES,
Atlanta, GA, 1:195-200.
DONGXIA, Z., LUYUAN, T., ZHONGDONG, Y. & ZHONGHONG, W. 1998. An analytical
mathematical model for describing the dynamic behavior of the thyristor controlled series
compensator.
Power System Technology, 1998. Proceedings. POWERCON '98. 1998
International Conference on., Beijing, China., Vol.1:420-424
GENG, J., TONG, L., GE, J. & WANG, Z.
2002.
Mathematical model for describing
characteristics of TCSC. Power System Technology, 2002. Proceedings. POWERCON 2002.
International Conference on, Beijing, China., Vol. 3:1498-1502.
GLANZMANN, G. & HOCHSPANNUNGSTECHNIK, I. F. E. E. U. 2005. FACTS: flexible
alternating current transmission systems. ETH, Eidgenössische Technische Hochschule Zürich,
EEH Power Systems Laboratory.
GRILLO, S., MASSUCCO, S., MORINI, A., PITTO, A. & SILVESTRO, F. 2010. Bifurcation
analysis and Chaos detection in power systems. International Journal of Innovations In Energy
Systems and Power., Padova:1-6.
GU, W., MILANO, F., JIANG, P. & TANG, G. 2007. Hopf bifurcations induced by SVC
Controllers: A didactic example. Electric power systems research, 77(3-4):234-240.
74
HABUR, K. & O’LEARY, D. 2001. FACTS: For Cost Effective and Reliable Transmission of
Electrical Energy. Siemens Power Transmission and Distribution Group.
HAK-GUHN, H. & JONG-KEUN, P. 1998. Analysis of operating modes of thyristor controlled
series compensators using state space analysis. Power Electronics Specialists Conference, 1998.
PESC 98 Record. 29th Annual IEEE., Fukuoka, 1:829-834.
HELBING, S. G. & KARADY, G.
1994.
Investigations of an advanced form of series
compensation. Power Delivery, IEEE Transactions on, 9(2):939-947.
HINGORANI, N. G. 2000. Role of FACTS in a Deregulated Market, Power Engineering
Society Summer Meeting, Seattle, WA. Vol. 3:1463-1467.
HINGORANI, N. G., GYUGYI, L. & EL-HAWARY, M.
2000.
Understanding FACTS:
concepts and technology of flexible AC transmission systems. IEEE press New York.
HU, G., CHENG, M., CAI, G. & 2004. Relation between Fundamental Frequency Equivalent
Impedance and Resonant Point for Thyristor Controlled Series Compensation. . 30th Annual
Conference IEEE Industrial Electronics Society., Busan, Korea, Vol. 2:1128-1132.
75
JIANG, D. & LEI, X. 2000. A nonlinear TCSC control strategy for power system stability
enhancement.
Advances in Power System Control, Operation and Management, 2000.
APSCOM-00. 2000 International Conference on, Hong Kong, Vol. 2:576-581
JING, Z., XU, D., CHANG, Y. & CHEN, L. 2003. Bifurcations, chaos, and system collapse in
a three node power system. International Journal of Electrical Power & Energy Systems,
25(6):443-461.
JOSHI, N. N. & MOHAN, N.
2006.
Application of TCSC in Wind Farm Application.
SPEEDAM International Symposium on Power Electronics, Electrical Drives, Automation and
Motion, Taormina:1196-1200.
JOVCIC, D. & PILLAI, G. N. 2005. Analytical modeling of TCSC dynamics. Power Delivery,
IEEE Transactions on, 20(2):1097-1104.
KIMBARK, E. W. 1969. Improvement of power system stability by changes in the network.
Power Apparatus and Systems, IEEE Transactions on(5Part-I):773-781.
KIRK, D. E. 2004. Optimal control theory: an introduction. Dover Pubns.
KUCUKEFE, Y. & KAYPMAZ, A.
2008.
HOPF Bifurcation in the IEEE Second
BENCHMARK Model for SSR Studies, 16th PSSC, Glasgow, Scotland [Electronic Version].
http://www.ijesp.com/vol3No2/IJESP3-4kucukefe.pdf: 1-7. Retrieved July 14-15.
76
LI, B. H., WU, Q. H., TURNER, D. R., WANG, P. Y. & ZHOU, X. X. 2000. Modelling of
TCSC dynamics for control and analysis of power system stability. International Journal of
Electrical Power and Energy Systems, Vol. 22(1):43-49.
LI, X., DUAN, X., YU, J., DING, H. & HE, Y. 1998. Fundamental frequency model of
thyristor-controlled series capacitor for transient stability studies. Power System Technology,
1998. Proceedings. POWERCON '98. 1998 International Conference on., Beijing, China, Vol.
1:333-337
LIE, T., SHRESTHA, G. & GHOSH, A. 1995. Design and application of a fuzzy logic control
scheme for transient stability enhancement in power systems. Electric power systems research,
33(1):17-23.
LU, Q., SUN, Y., XU, Z. & MOCHIZUKI, T. 1996. Decentralized nonlinear optimal excitation
control. Power Systems, IEEE Transactions on, 11(4):1957-1962.
MACHOWSKI, J., BIALEK, J. W. & BUMBY, J. R. (Eds.). 2008. Power system Dynamics:
Stability and Control. 2 ed. West Sussex, United Kingdom: John Wiley & Sons, Ltd.
MEHTA, V. K. & MEHTA, R. 2005. Principles of Power Systems. first ed. New Delhi: S.
Chand & Company LTD.
77
MEIKANDASIVAM, S., NEMA, R. K. & JAIN, S. 2008. Behavioral Study of TCSC Device A MATLAB/Simulink Implementation.
World Academy of Science, Engineering and
Technology., Munich, Germany:694-699.
PHILIPS, C. L. & T., N. H. 1995 Digital Control System Analysis and Design. Third ed. New
Jersey: Prentence Hall, Englewood Cliffs.
PULGAR-PAINEMAL, H. A. & SAUER, P. W. 2009, June 28 2009-July 2 2009. Bifurcations
and loadability issues in power systems. Paper presented at the PowerTech, 2009 IEEE
Bucharest.
RAJKUMAR, V. & MOHLER, R. 1994. Bilinear generalized predictive control using the
thyristor-controlled series-capacitor. Power Systems, IEEE Transactions on, 9(4):1987-1993.
RAMARAO, N. & REITAN, D. K. 1970. Improvement of power system transient stability
using optimal control: bang-bang control of reactance. Power Apparatus and Systems, IEEE
Transactions on(5):975-984.
ROSEHART, W. D. & CAÑIZARES, C. A. 1999. Bifurcation analysis of various power
system models. International Journal of Electrical Power & Energy Systems, 21(3):171-182.
78
ROUCO, L. & PAGOLA, F.
1997.
An eigenvalue sensitivity approach to location and
controller design of controllable series capacitors for damping power system oscillations. Power
Systems, IEEE Transactions on, 12(4):1660-1666.
SHAMSHIRGAR, S. S., GOLKHAH, M., RAHMATI, H. & NEKOUI, M. 2000. Application
and Comparison of LQR and Robust Sliding Mode Controllers to Improve Power System
Stability [Electronic Version]. 1-6.
SHIMPI, R. J., DESALE, R. P., PATIL, K. S., RAJPUT, J. L. & CHAVAN, S. B. 2010.
Flexible AC Transmission System.
International Journal of Computer Applications IJCA,
1(15):61-64.
SINGH, P., MATHEW, M. L. & CHATTERJI, S.
2008, March 29.
MATLAB Based
Simulation of TCSC FACTS Controller. Paper presented at the Proceedings of 2nd National
Conference on Challenges & Opportunities in Information Technology (COIT-2008)
RIMT-IET, Mandi Gobindgarh.
SOUZA, L. F. W. D., WATANABE, E. H., ALVES, J. E. R. & PILOTTO, L. A. S. 2003.
Thyristor and Gate Controlled Series Capacitors Comparison of Components Rating. Power
Engineering Society General Meeting, 2003, IEEE, Vol. 4:2542-2547.
TAVORA, C. J. & SMITH, O. J. M. 1972. Equilibrium analysis of power systems. Power
Apparatus and Systems, IEEE Transactions on(3):1131-1137.
79
VARMA, R. K.
2010.
Elements of FACTS Controllers.
Transmission and Distribution
Conference and Exposition, 2010 IEEE PES; New Orleans, LA, USA:1-6.
VUORENPÄÄ, P., JÄRVENTAUSTA, P. & LAVAPURO, J. 2008. Dynamic modeling of
thyristor controlled series capacitor in PSCAD and RTDS environments. NORPIE/2008, Nordic
Workshop on Power and Industrial Electronics, June 9-11, 2008:1-6.
WANG, Y., MOHLER, R., SPEE, R. & MITTELSTADT, W. 1992. Variable-structure FACTS
controllers for power system transient stability.
Power Systems, IEEE Transactions on,
7(1):307-313.
WANG, Y., TAN, Y. L. & GUO, G. 2002. Robust nonlinear co-ordinated excitation and TCSC
control for power systems. IET Journals & Magazines, Vol. 149(3):367-372.
YUSHENG, X., HAIQIANG, Z. & XIAORONG, G. 2002. A REVIEW OF BIFURCATION
AND CHAOS RESEARCHES IN POWER SYSTEMS [J].
Automation of Electric Power
Systems, 16.
ZHAO, Q. & JIANG, J. 1998. A TCSC damping controller design using robust control theory.
International Journal of Electrical Power & Energy Systems, 20(1):25-33.
80
ZHIZHONG, M. 2010. A new modeling and control scheme for thyristor-controlled series
capacitor. Control Theory Application 81-86.
ZHONGDONG, Y., LUYUAN, T., YONGTIN, C., DONGXIA, Z., CHUNLIN, G. &
ZHONGHONG, W. 1998. A study on the characteristics of TCSC based on digital simulations
and physical experiments. Power System Technology, 1998. Proceedings. POWERCON '98.
1998 International Conference on., Beijing, China, Vol. 1:328-332.
ZHOU, X. & LIANG, J. 1999. Overview of control schemes for TCSC to enhance the stability
of power systems. IET Journals & Magazines, Vol. 146(2):125-130.
81