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Modelling and Simulation in Power Networks

Enrique Acha
University of Glasgow, UK

Claudio R. Fuerte-Esquivel
Universidad Michoacana, MEXICO
Hugo Ambriz-Perez

Comision Federal de Electricidad, MEXICO
Cesar Angeles-Camacho
University of Glasgow, UK
Copyright # 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
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Preface xiii

1 Introduction 1
1.1 Background 1
1.2 Flexible Alternating Current Transmission Systems 2
1.3 Inherent Limitations of Transmission Systems 3
1.4 FACTS Controllers 3
1.5 Steady-state Power System Analysis 6
References 6

2 Modelling of FACTS Controllers 9
2.1 Introduction 9
2.2 Modelling Philosophy 11
2.3 Controllers Based on Conventional Thyristors 11
2.3.1 The Thyristor-controlled Reactor 12
2.3.2 The Static VAR Compensator 16
2.3.3 The Thyristor-controlled Series Compensator 18 Thyristor-controlled Series Capacitor Equivalent Circuit 19 Steady-state Current and Voltage Equations 20 Thyristor-controlled Series Capacitor Fundamental
Frequency Impedance 24
2.4 Power Electronic Controllers Based on Fully Controlled Semiconductor Devices 28
2.4.1 The Voltage Source Converter 29 Pulse-width Modulation Control 30 Principles of Voltage Source Converter Operation 33
2.4.2 The Static Compensator 34
2.4.3 The Solid State Series Compensator 35
2.4.4 The Uni¬ed Power Flow Controller 36
2.4.5 The High-voltage Direct-current Based on Voltage Source Converters 38
2.5 Control Capabilities of Controllers Based on Voltage Source Converters 40
2.6 Summary 41
References 41

3 Modelling of Conventional Power Plant 43
3.1 Introduction 43
3.2 Transmission Line Modelling 44
3.2.1 The Voltage-drop Equation 45 Calculation of Lumped RLC Parameters 45 Shunt Admittances 47 Internal Impedances 47 Ground Return Impedances 48
3.2.2 Ground Wires 50
3.2.3 Bundle Conductors 51

3.2.4 Double Circuit Transmission Lines 54
3.2.5 The Per-unit System 55
3.2.6 Transmission-line Program: Basic Parameters 56
3.2.7 Numerical Example of Transmission Line Parameter Calculation 59
3.2.8 Long Line Effects 60
3.2.9 Transmission Line Transpositions 62
3.2.10 Transmission Line Program: Distributed Parameters 63
3.2.11 Numerical Example of Long Line Parameter Calculation 66
3.2.12 Symmetrical Components and Sequence Domain Parameters 67
3.2.13 Transmission Line Program: Sequence Parameters 69
3.2.14 Numerical Example of Sequence Parameter Calculation 69
3.3 Power Transformer Modelling 70
3.3.1 Single-phase Transformers 70
3.3.2 Simple Tap-changing Transformer 72
3.3.3 Advanced Tap-changing Transformer 73
3.3.4 Three-phase Transformers 75 Star“Star Connection 76 Delta“Delta Connection 78 Star“Delta Connection 78
3.3.5 Sequence Domain Parameters 79
3.4 Rotating Machinery Modelling 82
3.4.1 Machine Voltage Equation 83
3.5 System Load 86
3.6 Summary 89
References 90

4 Conventional Power Flow 93
4.1 Introduction 93
4.2 General Power Flow Concepts 93
4.2.1 Basic Formulation 94
4.2.2 Variables and Bus Classi¬cation 97
4.3 Power Flow Solution Methods 97
4.3.1 Early Power Flow Algorithms 97
4.3.2 The Newton“Raphson Algorithm 98
4.3.3 State Variable Initialisation 101
4.3.4 Generator Reactive Power Limits 101
4.3.5 Linearised Frame of Reference 102
4.3.6 Newton“Raphson Computer Program in Matlab1 Code 104
4.3.7 The Fast Decoupled Algorithm 111
4.3.8 Fast Decoupled Computer Program in Matlab1 Code 112
4.3.9 A Benchmark Numerical Example 115
4.4 Constrained Power Flow Solutions 119
4.4.1 Load Tap-changing Transformers 119 State Variable Initialisation and Limit Checking 121 Load Tap Changer Computer Program in Matlab1 Code 122 Test Case of Voltage Magnitude Control with Load
Tap-changing 127 Combined Voltage Magnitude Control by Means of Generators
and Load Tap Changers 130 Control Coordination between One Load Tap Changer and
One Generator 130
4.4.2 Phase-shifting Transformer 132 State Variable Initialisation and Limit Checking 134 Phase-shifter Computer Program in Matlab1 Code 135 Test Cases for Phase-shifting Transformers 140

4.5 Further Concepts in Power Flows 144
4.5.1 Sparsity-oriented Solutions 144
4.5.2 Truncated Adjustments 145 Test Case of Truncated Adjustments Involving Three
Load Tap-changing Transformers 145
4.5.3 Special Load Tap Changer Con¬gurations 147 Test Case of Sensitivity Factors in Parallel Load
Tap-changing Operation 148
4.6 Summary 149
References 150

5 Power Flow Including FACTS Controllers 153
5.1 Introduction 153
5.2 Power Flow Solutions Including FACTS Controllers 154
5.3 Static VAR Compensator 155
5.3.1 Conventional Power Flow Models 155
5.3.2 Shunt Variable Susceptance Model 158
5.3.3 Static VAR Compensator Computer Program in Matlab1 Code 159
5.3.4 Firing-angle Model 162
5.3.5 Static VAR Compensator Firing-angle Computer Program in
Matlab1 Code 162
5.3.6 Integrated Transformer Firing-angle Model 166
5.3.7 Nodal Voltage Magnitude Control using Static VAR Compensators 167
5.3.8 Control Coordination between Reactive Sources 168
5.3.9 Numerical Example of Voltage Magnitude Control using One
Static VAR Compensator 168
5.4 Thyristor-controlled Series Compensator 171
5.4.1 Variable Series Impedance Power Flow Model 171
5.4.2 Thyristor-controlled Series Compensator Computer Program in
Matlab1 Code 173
5.4.3 Numerical Example of Active Power Flow Control using One
Thyristor-controlled Series Compensator: Variable Series
Compensator Model 178
5.4.4 Firing-angle Power Flow Model 180
5.4.5 Thyristor-controlled Series Compensator Firing-angle Computer
Program in Matlab1 Code 182
5.4.6 Numerical Example of Active Power Flow Control using One
Thyristor-controlled Series Compensator: Firing-angle Model 187
5.4.7 Numerical Properties of the Thyristor-controlled Series
Compensator Power Flow Model 189
5.5 Static Synchronous Compensator 191
5.5.1 Power Flow Model 191
5.5.2 Static Compensator Computer Program in Matlab1 Code 192
5.5.3 Numerical Example of Voltage Magnitude Control using One
Static Compensator 198
5.6 Uni¬ed Power Flow Controller 200
5.6.1 Power Flow Model 201
5.6.2 Uni¬ed Power Flow Controller Computer Program in Matlab1 Code 203
5.6.3 Numerical Example of Power Flow Control using One Uni¬ed
Power Flow Controller 213
5.7 High-voltage Direct-current-based Voltage Source Converter 216
5.7.1 Power Equations 217
5.7.2 High-voltage Direct-current-based Voltage Source Converter
Computer Program in Matlab1 Code 218
5.7.3 Numerical Example of Power Flow Control using One
x CONTENTS HVDC-VSC Back-to-back Model 225 HVDC-VSC Full Model 225
5.8 Effective Initialisation of FACTS Controllers 227
5.8.1 Controllers Represented by Shunt Synchronous Voltage Sources 227
5.8.2 Controllers Represented by Shunt Admittances 227
5.8.3 Controllers Represented by Series Reactances 227
5.8.4 Controllers Represented by Series Synchronous Voltage Sources 228
5.9 Summary 228
References 229

6 Three-phase Power Flow 231
6.1 Introduction 231
6.2 Power Flow in the Phase Frame of Reference 232
6.2.1 Power Flow Equations 233
6.2.2 Newton“Raphson Power Flow Algorithm 234
6.2.3 Matlab1 Code of a Power Flow Program in the Phase
Frame of Reference 236
6.2.4 Numerical Example of a Three-phase Network 244
6.3 Static VAR Compensator 249
6.3.1 Variable Susceptance Model 250
6.3.2 Firing-angle Model 251
6.3.3 Numerical Example: Static VAR Compensator Voltage
Magnitude Balancing Capability 252
6.4 Thyristor-controlled Series Compensator 253
6.4.1 Variable Susceptance Model 253
6.4.2 Firing-angle Model 255
6.4.3 Numerical Example: Power Flow Control using One
Thyristor-controlled Series Compensator 257
6.5 Static Compensator 257
6.5.1 Static Compensator Three-phase Numerical Example 260
6.6 Uni¬ed Power Flow Controller 261
6.6.1 Numerical Example of Power Flow Control using
One Uni¬ed Power Flow Controller 264
6.7 Summary 264
References 265

7 Optimal Power Flow 267
7.1 Introduction 267
7.2 Optimal Power Flow using Newton™s Method 268
7.2.1 General Formulation 268 Variables 268 Objective Function 269 Equality Constraints 269 Inequality Constraints 270
7.2.2 Application of Newton™s Method to Optimal Power Flow 270
7.2.3 Linearised System Equations 271
7.2.4 Optimality Conditions for Newton™s Method 272
7.2.5 Conventional Power Plant Modelling in Optimal Power Flow 272 Transmission Lines 273 Shunt Elements 274 Synchronous Generators 275
7.2.6 Handling of Inequality Constraints 275 Handling of Inequality Constraints on Variables 275 Handling of Inequality Constraints on Functions 277

7.3 Implementation of Optimal Power Flow using Newton™s Method 278
7.3.1 Initial Conditions in Optimal Power Flow Solutions 279
7.3.2 Active Power Schedule 279
7.3.3 Lagrange Multipliers 280
7.3.4 Penalty Weighting Factors 280
7.3.5 Conjugated Variables 280
7.3.6 An Optimal Power Flow Numerical Example 281
7.4 Power System Controller Representation in Optimal Power Flow Studies 283
7.5 Load Tap-changing Transformer 283
7.5.1 Load Tap-changing Lagrangian Function 283
7.5.2 Linearised System of Equations 284
7.5.3 Load Tap-changing Transformer Test Cases 285
7.6 Phase-shifting Transformer 286
7.6.1 Lagrangian Function 286
7.6.2 Linearised System of Equations 287
7.6.3 Phase-shifting Transformer Test Cases 289 Case 1: No Active Power Flow Regulation 289 Case 2: Active Power Flow Regulation at LakePS 290
7.7 Static VAR Compensator 291
7.7.1 Lagrangian Function 291
7.7.2 Linearised System of Equations 292
7.7.3 Static VAR Compensator Test Cases 293 Firing-angle Model 293 Susceptance Model 295
7.8 Thyristor-controlled Series Compensator 296
7.8.1 Lagrangian Function 296
7.8.2 Linearised System of Equations 297
7.8.3 Thyristor-controlled Series Compensator Test Cases 299
7.9 Uni¬ed Power Flow Controller 301
7.9.1 Uni¬ed Power Flow Controller Lagrangian Function 301
7.9.2 Direct-current Link Lagrangian Function 301
7.9.3 Uni¬ed Power Flow Controller Power Flow Constraints 302
7.9.4 Linearised System of Equations 302
7.9.5 Uni¬ed Power Flow Controller Test Cases 305
7.9.6 Uni¬ed Power Flow Controller Operating Modes 307
7.10 Summary 307
References 308

8 Power Flow Tracing 311
8.1 Introduction 311
8.2 Basic Assumptions 312
8.3 Mathematical Justi¬cation of the Proportional Sharing Principle 313
8.4 Dominions 315
8.4.1 Dominion Contributions to Active Power Flows 317
8.4.2 Dominion Contributions to Reactive Power Flows 319
8.4.3 Dominion Contributions to Loads and Sinks 320
8.5 Tracing Algorithm 321
8.6 Numerical Examples 322
8.6.1 Simple Radial Network 322
8.6.2 Simple Meshed Network: Active Power 324
8.6.3 Meshed Network with FACTS Controllers: Reactive Power 329
8.6.4 Large Network 331
8.6.5 Tracing the Power Output of a Wind Generator 331 The Wind Generator Model 335 Numerical Example 336

8.7 Summary 339
References 340

Appendix A: Jacobian Elements for FACTS Controllers in
Positive Sequence Power Flow 343
A.1 Tap-changing Transformer 343
A.2 Thyristor-controlled Series Compensator 344
A.3 Static Synchronous Compensator 345
A.4 Uni¬ed Power Flow Controller 345
A.5 High-voltage Direct-current-based Voltage Source Converter 347

Appendix B: Gradient and Hessian Elements for Optimal Power
Flow Newton™s Method 349
B.1 First and Second Partial Derivatives for Transmission Lines 349
B.1.1 The Gradient Vector 349
B.1.2 The Matrix W 350
B.2 Phase Shifter Transformer 352
B.3 Static VAR Compensator 355
B.4 Thyristor-controlled Series Compensator 356
B.5 Uni¬ed Power Flow Controller 357

Appendix C: Matlab1 Computer Program for Optimal Power
Flow Solutions using Newton™s Method 365

Index 399

Flexible alternating-current transmission systems (FACTS) is a recent technological
development in electrical power systems. It builds on the great many advances achieved in
high-current, high-power semiconductor device technology, digital control and signals
conditioning. From the power systems engineering perspective, the wealth of experience
gained with the commissioning and operation of high-voltage direct-current (HVDC) links
and static VAR compensator (SVC) systems, over many decades, in many parts of the globe,
may have provided the driving force for searching deeper into the use of emerging power
electronic equipment and techniques, as a means of alleviating long-standing operational
problems in both high-voltage transmission and low-voltage distribution systems. A large
number of researchers have contributed to the rapid advancement of the FACTS technology,
but the names N.G. Hingorani and L. Gyugyi stand out prominently. Their work on FACTS,
synthesised in their book, Understanding FACT “ Concepts and Technology of Flexible AC
Transmission Systems (Institute of Electronic and Electrical Engineers, New York, 2000), is
a source of learning and inspiration.
Following universal acceptance of the FACTS technology and the commissioning of a vast
array of controllers in both high-voltage transmission and low voltage distribution systems,
research attention turned to the steady-state and dynamic interaction of FACTS controllers
with the power network. The research community responded vigorously, lured by the novelty
of the technology, turning out a very healthy volume of advanced models and high-quality
simulations and case studies. Most matters concerning steady-state modelling and
simulations of FACTS controllers are well agreed on, and the goal of our current book:
FACTS: Modelling and Simulation in Power Networks, is to provide a coherent and
systematic treatise of the most popular FACTS models, their interaction with the power
network, and the main steady-state operational characteristics.
The overall aims and objectives of the FACTS philosophy are outlined in Chapter 1. The
inherent limitations exhibited by high-voltage transmission systems, which are in¬‚exible and
overdesigned, are brought to attention as a means of explaining the background against
which the FACTS technology developed and took hold. The most promising FACTS
controllers and their range of steady-state applicability are described in this chapter.
Chapters 2 and 3 provide a thorough grounding on the mathematical representation of the
most popular FACTS controllers and power plant components. The models are derived from
¬rst principles: by encapsulating the main steady-state operational characteristics and
physical structure of the actual equipment, advanced power system models are developed in
phase coordinates. As a by-product, more restrictive models are then derived, which are
suitable for positive sequence power system analysis. Software written in Matlab1 code is
given for the most involved aspects of power plant modelling, such as transmission Line
parameter calculation.

The power ¬‚ow method is the most basic system analysis tool with which to assess the
steady-state operation of a power system. It has been in existence for almost half a century,
having reached quite a sophisticated level of development, in terms of both computational
ef¬ciency and modelling ¬‚exibility. The Newton“Raphson method is the de facto standard
for solving the nonlinear power equations, which describe the power systems, owing to its
reliability towards convergence. Chapter 4 covers the theory of positive sequence power ¬‚ow
in depth, and makes extensions to incorporate cases of adjusted solutions using two
conventional power system controllers. This serves as a preamble to the material presented in
Chapter 5, where a wide range of positive sequence power ¬‚ow models of FACTS controllers
are developed. Test cases and software written in Matlab1 code is provided for each
controller to enable the reader to gain ample experience with the various models provided.
Furthermore, suitable coding of the Jacobian elements given in Appendix A enables more
general FACTS power ¬‚ow computer programs than those given in Chapter 5.
The concepts used in the study of positive sequence power ¬‚ow in Chapters 4 and 5 are
extended in Chapter 6 to address the more involved topic of three-phase power ¬‚ow. The ¬rst
part deals with the Newton“Raphson in-phase coordinates using simpli¬ed representations of
conventional power plant components. Software written in Matlab1 code is provided to
enable the solution of small and medium-size three-phase power systems. Advanced models
of conventional power plants are not included in the Matlab1 function given in this chapter
but their incorporation is a straightforward programming exercise. The second half of
Chapter 6 addresses the modelling of three-phase controllers within the context of the power
¬‚ow Newton“Raphson method, where the voltage and power ¬‚ow balancing capabilities of
shunt and series FACTS controllers, respectively, are discussed.
The topic of optimal power ¬‚ow is covered in depth in Chapter 7. Building on the ground
covered in Chapters 4 and 5, the theory of positive sequence power ¬‚ow is blended with
advanced optimisation techniques to incorporate economic and security aspects of power
system operation. The optimisation method studied in this chapter is Newton™s method,
which exhibits strong convergence and ¬ts in well with the modelling philosophy developed
throughout the book. Both conventional plant equipment and FACTS controller representa-
tions are accommodated with ease within the frame of reference provided by Newton™s
method. To facilitate the extension of a conventional optimal power ¬‚ow computer program
to include FACTS representation, Appendix B gives the Hessian and gradient elements for all
the FACTS controllers presented in Chapter 7. Software written in Matlab1 code is provided
in Appendix C to carry out non-FACTS optimal power ¬‚ow solutions of small and medium-
size power systems. The timely issue of power ¬‚ow tracing is presented in Chapter 8. The
method is based on the principle of proportional sharing and yields unambiguous information
on the contribution of each generator to each transmission Line power ¬‚ow and load in the
system. Several application examples are presented in the chapter.


The preparation of the book is a testimony to international collaboration, overcoming ¬xed
work commitments, continental distances, and widely differing time zones to bring the
project to fruition: we would like to thank our respective families for the time that we were
kindly spared throughout the duration of the project. In this tenor, we would also like to
acknowledge the unstinting support of our colleagues in the Power Engineering Group at the

University of Glasgow. The research work underpinning most of the new modelling
concepts and methods presented in this book were carried out at the University of Glasgow
by the authors over a period of more than 10 years. We would like to thank Mr Colin Tan
Soon Guan and Dr Jesus Rico Melgoza for their early contribution to the research project. It
is fair to say that the dream was only made possible by the generous support of the role
model for all research councils in the world, the Consejo Nacional de Ciencia y Tecnolog±a
(CONACYT), Mexico. The dream goes on . . . thanks CONACYT.
We are grateful to the staff of John Wiley & Sons, particularly Kathryn Sharples, Simone
Taylor, and Susan Barclay for their patience and continuous encouragement throughout the
preparation of the manuscript.


The electricity supply industry is undergoing a profound transformation worldwide. Market
forces, scarcer natural resources, and an everincreasing demand for electricity are some of
the drivers responsible for such an unprecedented change. Against this background of rapid
evolution, the expansion programmes of many utilities are being thwarted by a variety of
well-founded, environmental, land-use, and regulatory pressures that prevent the licensing
and building of new transmission lines and electricity generating plants. An in-depth
analysis of the options available for maximising existing transmission assets, with high
levels of reliability and stability, has pointed in the direction of power electronics. There is
general agreement that novel power electronics equipment and techniques are potential
substitutes for conventional solutions, which are normally based on electromechanical
technologies that have slow response times and high maintenance costs (Hingorani and
Gyugyi, 2000; Song and Johns, 1999).
An electrical power system can be seen as the interconnection of generating sources and
customer loads through a network of transmission lines, transformers, and ancillary
equipment. Its structure has many variations that are the result of a legacy of economic,
political, engineering, and environmental decisions. Based on their structure, power systems
can be broadly classi¬ed into meshed and longitudinal systems. Meshed systems can be
found in regions with a high population density and where it is possible to build power
stations close to load demand centres. Longitudinal systems are found in regions where
large amounts of power have to be transmitted over long distances from power stations to
load demand centres.
Independent of the structure of a power system, the power ¬‚ows throughout the network
are largely distributed as a function of transmission line impedance; a transmission line with
low impedance enables larger power ¬‚ows through it than does a transmission line with high
impedance. This is not always the most desirable outcome because quite often it gives rise
to a myriad of operational problems; the job of the system operator is to intervene to try to
achieve power ¬‚ow redistribution, but with limited success. Examples of operating problems
to which unregulated active and reactive power ¬‚ows may give rise are: loss of system
stability, power ¬‚ow loops, high transmission losses, voltage limit violations, an inability to
utilise transmission line capability up to the thermal limit, and cascade tripping.

FACTS: Modelling and Simulation in Power Networks.
´ ´
Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho
# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2

In the long term, such problems have traditionally been solved by building new power plants
and transmission lines, a solution that is costly to implement and that involves long
construction times and opposition from pressure groups. It is envisaged that a new solution
to such operational problems will rely on the upgrading of existing transmission corridors
by using the latest power electronic equipment and methods, a new technological thinking
that comes under the generic title of FACTS “ an acronym for ¬‚exible alternating current
transmission systems.


In its most general expression, the FACTS concept is based on the substantial incorporation
of power electronic devices and methods into the high-voltage side of the network, to make
it electronically controllable (IEEE/CIGRE, 1995).
Many of the ideas upon which the foundation of FACTS rests evolved over a period of
many decades. Nevertheless, FACTS, an integrated philosophy, is a novel concept that was
brought to fruition during the 1980s at the Electric Power Research Institute (EPRI), the
utility arm of North American utilities (Hingorani and Gyugyi, 2000). FACTS looks at ways
of capitalising on the many breakthroughs taking place in the area of high-voltage and high-
current power electronics, aiming at increasing the control of power ¬‚ows in the high-
voltage side of the network during both steady-state and transient conditions. The new
reality of making the power network electronically controllable has started to alter the way
power plant equipment is designed and built as well as the thinking and procedures that go
into the planning and operation of transmission and distribution networks. These
developments may also affect the way energy transactions are conducted, as high-speed
control of the path of the energy ¬‚ow is now feasible. Owing to the many economical and
technical bene¬ts it promised, FACTS received the uninstinctive support of electrical
equipment manufacturers, utilities, and research organisations around the world (Song and
Johns, 1999).
Several kinds of FACTS controllers have been commissioned in various parts of the
world. The most popular are: load tap changers, phase-angle regulators, static VAR compen-
sators, thyristor-controlled series compensators, interphase power controllers, static
compensators, and uni¬ed power ¬‚ow controllers (IEEE/CIGRE, 1995).
It was recognised quite early on the development programme of the FACTS technology
that, in order to determine the effectiveness of such controllers; on a networkwide basis, it
would be necessary to upgrade most of the system analysis tools with which power engineers
plan and operate their systems. Some of the tools that have received research attention and,
to a greater or lesser extent, have reached a high degree of modelling sophistication are:

 positive sequence power ¬‚ow;
 three-phase power ¬‚ow;
 optimal power ¬‚ow;
 state estimation;
 transient stability;
 dynamic stability;
 electromagnetic transients;
 power quality.

This book covers in breadth and depth the modelling and simulation methods required for
a thorough study of the steady-state operation of electrical power systems with FACTS
controllers. The ¬rst three application areas, which are clearly de¬ned within the realm of
steady-state operation, are addressed in the book. The area of FACTS state estimation is still
under research and no de¬nitive models or simulation methods have emerged, as yet. A great
deal of research progress has been made on the modelling and simulation of FACTS con-
trollers for transient and dynamic stability, electromagnetic transients, and power quality,
but the simulation tools required to conduct studies in such application areas are not really
suited to conduct steady-state power systems analysis, and they are not covered in this book.


The characteristics of a given power system evolve with time, as load grows and generation
is added. If the transmission facilities are not upgraded suf¬ciently the power system
becomes vulnerable to steady-state and transient stability problems, as stability margins
become narrower (Hingorani and Gyugyi, 2000).
The ability of the transmission system to transmit power becomes impaired by one or
more of the following steady-state and dynamic limitations (Song and Johns, 1999):
 angular stability;
 voltage magnitude;
 thermal limits;
 transient stability;
 dynamic stability.
These limits de¬ne the maximum electrical power to be transmitted without causing damage
to transmission lines and electric equipment. In principle, limitations on power transfer can
always be relieved by the addition of new transmission and generation facilities. Alternati-
vely, FACTS controllers can enable the same objectives to be met with no major alterations
to system layout. The potential bene¬ts brought about by FACTS controllers include
reduction of operation and transmission investment cost, increased system security and
reliability, increased power transfer capabilities, and an overall enhancement of the quality
of the electric energy delivered to customers (IEEE/CIGRE, 1995).


Power ¬‚ow control has traditionally relied on generator control, voltage regulation by means
of tap-changing and phase-shifting transformers, and reactive power plant compensation
switching. Phase-shifting transformers have been used for the purpose of regulating active
power in alternating current (AC) transmission networks. In practice, some of them
are permanently operated with ¬xed angles, but in most cases their variable tapping facilities
are actually made use of.
Series reactors are used to reduce power ¬‚ow and short-circuit levels at designated
locations of the network. Conversely, series capacitors are used to shorten the electrical
length of lines, hence increasing the power ¬‚ow. In general, series compensation is switched
on and off according to load and voltage conditions. For instance, in longitudinal power

systems, series capacitive compensation is bypassed during minimum loading in order to
avoid transmission line overvoltages due to excessive capacitive effects in the system. Con-
versely, series capacitive compensation is fully utilised during maximum loading, aiming at
increasing the transfer of power without subjecting transmission lines to overloads.
Until recently, these solutions served well the needs of the electricity supply industry.
However, deregulation of the industry and dif¬culties in securing new ˜rights of way™ have
created the momentum for adopting new, radical technological developments based on high-
voltage, high-current solid-state controllers (Hingorani and Gyugyi, 2000). A few years ago,
in partnership with manufacturers and research organisations, the supply industry embarked
on an ambitious programme to develop a new generation of power electronic-based plant
components (Song and Johns, 1999). The impact of such developments has already made
inroads in all three areas of the business, namely, generation, transmission, and distribution.
Early developments of the FACTS technology were in power electronic versions of the
phase-shifting and tap-changing transformers. These controllers together with the electronic
series compensator can be considered to belong to the ¬rst generation of FACTS equipment.
The uni¬ed power ¬‚ow controller, the static compensator, and the interphase power
controller are more recent developments. Their control capabilities and intended function
are more sophisticated than those of the ¬rst wave of FACTS controllers. They may be
considered to belong to a second generation of FACTS equipment. Shunt-connected
thyristor-switched capacitors and thyristor-controlled reactors, as well as high-voltage
direct-current (DC) power converters, have been in existence for many years, although their
operational characteristics resemble those of FACTS controllers.
A number of FACTS controllers have been commissioned. Most of them perform a useful
role during both steady-state and transient operation, but some are speci¬cally designed to
operate only under transient conditions, for instance, Hingorani™s subsynchronous resonance
(SSR) damper.
FACTS controllers intended for steady-state operation are as follows (IEEE/CIGRE,

 Thyristor-controlled phase shifter (PS): this controller is an electronic phase-shifting
transformer adjusted by thyristor switches to provide a rapidly varying phase angle.
 Load tap changer (LTC): this may be considered to be a FACTS controller if the tap
changes are controlled by thyristor switches.
 Thyristor-controlled reactor (TCR): this is a shunt-connected, thyristor-controlled reactor,
the effective reactance of which is varied in a continuous manner by partial conduction
control of the thyristor valve.
 Thyristor-controlled series capacitor (TCSC): this controller consists of a series capacitor
paralleled by a thyristor-controlled reactor in order to provide smooth variable series
 Interphase power controller (IPC): this is a series-connected controller comprising two
parallel branches, one inductive and one capacitive, subjected to separate phase-shifted
voltage magnitudes. Active power control is set by independent or coordinated adjust-
ment of the two phase-shifting sources and the two variable reactances. Reactive power
control is independent of active power.
 Static compensator (STATCOM): this is a solid-state synchronous condenser connected in
shunt with the AC system. The output current is adjusted to control either the nodal
voltage magnitude or the reactive power injected at the bus.

 Solid-state series controller (SSSC): this controller is similar to the STATCOM but it is
connected in series with the AC system. The output current is adjusted to control either the
nodal voltage magnitude or the reactive power injected at one of the terminals of the
series-connected transformer.
 Uni¬ed power ¬‚ow controller (UPFC): this consists of a static synchronous series
compensator (sssc) and a STATCOM, connected in such a way that they share a common
DC capacitor. The UPFC, by means of an angularly unconstrained, series voltage injection,
is able to control, concurrently or selectively, the transmission line impedance, the nodal
voltage magnitude, and the active and reactive power ¬‚ow through it. It may also provide
independently controllable shunt reactive compensation.

Power electronic and control technology have been applied to electric power systems for
several decades. HVDC links and static VAR compensators are mature pieces of technology:

 Static VAR compensator (SVC): this is a shunt-connected static source or sink of reactive
 High-voltage direct-current (HVDC) link: this is a controller comprising a recti¬er
station and an inverter station, joined either back-to-back or through a DC cable. The
converters can use either conventional thyristors or the new generation of semiconductor
devices such as gate turn-off thyristors (GTOs) or insulated gate bipolar transistors

The application of FACTS controllers to the solution of steady-state operating problems is
outlined in Table 1.1.

Table 1.1 The role of FACTS (¬‚exible alternating current transmission systems) controllers in power
system operation
Operating problem Corrective action FACTS controller
Voltage limits:
Low voltage at heavy load Supply reactive power STATCOM, SVC,
High voltage at low load Absorb reactive power STATCOM, SVC, TCR
High voltage following Absorb reactive power; STATCOM, SVC, TCR
an outage prevent overload
Low voltage following Supply reactive power; STATCOM, SVC
an outage prevent overload
Thermal limits:
Transmission circuit overload Reduce overload TCSC, SSSC, UPFC, IPC, PS
Tripping of parallel circuits Limit circuit loading TCSC, SSSC, UPFC, IPC, PS
Loop ¬‚ows:
Parallel line load sharing Adjust series reactance IPC, SSSC, UPFC, TCSC, PS
Postfault power ¬‚ow sharing Rearrange network or use IPC, TCSC, SSSC, UPFC, PS
thermal limit actions
Power ¬‚ow direction reversal Adjust phase angle IPC, SSSC, UPFC, PS


In order to assist power system engineers to assess the impact of FACTS equipment on
transmission system performance, it has become necessary to write new power system
software or to upgrade existing software (Ambriz-Perez, 1998; Fuerte-Esquivel, 1997).
This has called for the development of a new generation of mathematical models for
transmission systems and FACTS controllers, which had to be blended together, coded, and
extensively veri¬ed. This has been an area of intense research activity, which has given rise
to a copious volume of publications. Many aspects of FACTS modelling and simulation
have reached maturity, and we believe that the time is ripe for such an important and large
volume of information to be put together in a coherent and systematic fashion. This book
aims to achieve such a role in the area of steady-state operation of FACTS-upgraded power
From the operational point of view, FACTS technology is concerned with the ability to
control, in an adaptive fashion, the path of the power ¬‚ows throughout the network, where
before the advent of FACTS, high-speed control was very restricted. The ability to control
the line impedance and the nodal voltage magnitudes and phase angles at both the sending
and the receiving ends of key transmission lines, with almost no delay, has signi¬cantly
increased the transmission capabilities of the network while considerably enhancing the
security of the system. In this context, power ¬‚ow computer programs with FACTS
controller modelling capability have been very useful tools for system planners and system
operators to evaluate the technical and economical bene¬ts of a wide range of alternative
solutions offered by the FACTS technology.
Arguably, power ¬‚ow analysis “ also termed load ¬‚ow analysis in the parlance of power
systems engineers “ is the most popular analysis tool used by planning and operation
engineers today for the purpose of steady-state power system assessment. The reliable
solution of real-life transmission and distribution networks is not a trivial matter, and
Newton“Raphson-type methods, with their strong convergence characteristics, have proved
most successful (Fuerte-Esquivel, 1997). Extensive research has been carried out over the
past 10 years in order to implement FACTS models into Newton“Raphson-type power
¬‚ow programs. This book offers a thorough grounding on the theory and practice of
positive sequence power ¬‚ow and three-phase power ¬‚ow. In many practical situations, it is
desirable to include economical and operational considerations into the power ¬‚ow
formulation, so that optimal solutions, within constrained solution spaces, can be obtained.
This is the object of optimal power ¬‚ow algorithms (Ambriz-Perez, 1998), a topic also
covered in the book.

Ambriz-Perez, H., 1998, Flexible AC Transmission Systems Modelling in Optimal Power Flows Using
Newton™s Method, PhD thesis, Department of Electronics and Electrical Engineering, University of
Glasgow, Glasgow.
Fuerte-Esquivel, C.R., 1997, Steady State Modelling and Analysis of Flexible AC Transmission Systems,
PhD thesis, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow.

Hingorani, N.G., Gyugyi, L., 2000, Understanding FACTS: Concepts and Technology of Flexible AC
Transmission Systems, Institute of Electrical and Electronic Engineers, New York.
IEEE/CIGRE (Institute of Electrical and Electronic Engineers/Conseil International des Grands
Reseaux Electriques), 1995, FACTS Overview, special issue, 95TP108, IEEE Service Centre,
Piscata-way, NJ.
Song, Y.H., Johns, A.T., 1999, Flexible AC Transmission Systems (FACTS), Institution of Electrical
Engineers, London.
Modelling of FACTS


Two kinds of emerging power electronics applications in power systems are already well
de¬ned: (1) bulk active and reactive power control and (2) power quality improvement
(Hingorani and Gyugyi, 2000). The ¬rst application area is know as FACTS, where the latest
power electronic devices and methods are used to control the high-voltage side of the
network electronically (Hingorani, 1993). The second application area is custom power,
which focuses on low-voltage distribution and is a technology created in response to reports
of poor power quality and reliability of supply affecting factories, of¬ces, and homes. It is
expected that when widespread deployment of the technology takes place, the end-user will
see tighter voltage regulation, minimum power interruptions, low harmonic voltages, and
acceptance of rapidly ¬‚uctuating and other nonlinear loads in the vicinity (Hingorani, 1995).
The one-line diagram shown in Figure 2.1 illustrates the connection of power plants in an
interconnected transmission system, where the boundary between the high-voltage
transmission and the low-voltage distribution is emphasised. The former bene¬ts from the
installation of FACTS equipment whereas the latter bene¬ts from the installation of custom
power equipment.
To a greater or lesser extent, high-voltage transmission systems are highly meshed. For
many decades the trend has been towards interconnection, linking generators and loads into
large integrated systems. The motivation has been to take advantage of load diversity,
enabling a better utilisation of primary energy resources.
From the outset, interconnection was aided by breakthroughs in high-current, high-power
semiconductor valve technology (Arrillaga, 1998). Thyristor-based high-voltage direct-
current (HVDC) converter installations provided a means for interconnecting power systems
with different operating frequencies “ e.g. 50/60 Hz, for interconnecting power systems
separated by the sea and for interconnecting weak and strong power systems (Hingorani,
1996). The most recent development in HVDC technology is the HVDC system based on
solid-state voltage source converters, which enables independent, fast control of active and
reactive powers (McMurray, 1987).

FACTS: Modelling and Simulation in Power Networks.
´ ´
Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho
# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2

Power plant

120 “ 765 kV 3 “ 34 kV

400 V

(custom power)

Figure 2.1 A simpli¬ed one-line diagram of a power system. Redrawn, with permission, from N.G.
Hingorani, ˜Introducing Custom Power™, IEEE Spectrum 32(6) 41“48, # 1995 IEEE

Power electronics is a ubiquitous technology that has affected every aspect of electrical
power networks, not just HVDC transmission but also alternating current (AC) transmission,
distribution, and utilisation. Deregulated markets are imposing further demands on
generating plants, increasing their wear and tear and the likelihood of generator instabilities
of various kinds. To help to alleviate such problems, power electronic controllers have
recently been developed to enable generators to operate more reliably in the new
marketplace. The thyristor-controlled series compensator (TCSC) is used to mitigate
subsynchronous resonances (SSRs) and to damp power system oscillations (Larsen et al.,
1992). However, it may be argued that the primary function of the TCSC, like that of its
mechanically controlled counterpart, the series capacitor bank, is to reduce the electrical
length of the compensated transmission line. Hence, the aim is still to increase power
transfers signi¬cantly, but with increased transient stability margins. With reference to the
schematic network of Figure 2.1, the TCSC is deployed on the FACTS side.
For most practical purposes the thyristor-based static VAR compensator (SVC) has made
the rotating synchronous compensator redundant, except where an increase in the short-
circuit level is required along with fast-acting reactive power support (Miller, 1982).
However, as power electronic technology continues to develop further, the replacement of
the SVC by a new breed of static compensators based on the use of voltage source
converters (VSCs) is looming. They are known as STATCOMs (static compensators) and
provide all the functions that the SVC can provide but at a higher speed (IEEE/CIGRE,
1995); it is more compact and requires only a fraction of the land required by an SVC
installation. The STATCOM is essentially a VSC interfaced to the AC system through a
shunt-connected transformer. The VSC is the basic building block of the new generation of

power electronic controllers that have emerged from the FACTS and custom power
initiatives (Hingorani and Gyugyi, 2000). In high-voltage transmission, the most popular
FACTS equipment are: the STATCOM, the uni¬ed power ¬‚ow controller (UPFC) and
the HVDC-VSC. At the low-voltage distribution level, the SVC provides the core of the
following custom power equipment: the distribution STATCOM, the dynamic voltage
restorer, and active ¬lters.


The remit of this book is the study of models and procedures with which to assess the
steady-state operation of electrical power systems at the fundamental frequency. The power
system application tool is termed ˜power ¬‚ows™, and the most popular variants of the tool
are presented in this book; namely, positive sequence power ¬‚ow (Stagg and El-Abiad,
1968), optimal power ¬‚ow (Wood and Wollenberg, 1984), and three-phase power ¬‚ow
(Arrillaga and Arnold, 1990). The ¬rst two applications deal with cases of balanced
operation, for nonoptimal and optimal solutions, respectively. The third application deals
with unbalanced operation induced by imbalances present either in plant components or in
system load. In this book, all three applications incorporate representation of conventional
power plant components and FACTS controllers.
The modelling of FACTS controllers in both the phase domain and the sequence domain
is addressed in this chapter, and Chapter 3 deals with the representation of conventional
power plant components in both domains. All models are developed from ¬rst principles,
with strong reference to the physical structure of the equipment. Such an approach is
amenable to ¬‚exible models useful for assessing the operation of plant components in
network-wide applications, taking due care of equipment design imbalances, which are
naturally present in all power plant equipment. However, if such imbalances are small and
can be neglected in the study, then simpler models of plant components become readily
available, in the form of sequence domain models.
It should be kept in mind that, in this book, the interest is in steady-state analysis at the
fundamental frequency, and the models developed re¬‚ect this fact. They are not suitable for
assessing the periodic steady-state operation of power systems (Acha and Madrigal, 2001)
or their dynamic or transient operation (Kundur, 1994).


Power electronic circuits using conventional thyristors have been widely used in power
transmission applications since the early 1970s (Arrillaga, 1998). The ¬rst applications took
place in the area of HVDC transmission, but shunt reactive power compensation using fast
controllable inductors and capacitors soon gained general acceptance (Miller, 1982). More
recently, fast-acting series compensators using thyristors have been used to vary the
electrical length of key transmission lines, with almost no delay, instead of the classical
series capacitor, which is mechanically controlled. In distribution system applications, solid-
state transfer switches using thyristors are being used to enhance the reliability of supply to
critical customer loads (Anaya-Lara and Acha, 2002).

In this section, the following three thyristor-based controllers receive attention: the
thyristor-controlled reactor (TCR), the SVC and the TCSC. The operational characteristic of
each one of these controllers is studied with particular reference to steady-state operation.

2.3.1 The Thyristor-controlled Reactor

The main components of the basic TCR are shown in Figure 2.2(a). The controllable
element is the antiparallel thyristor pair, Th1 and Th2, which conducts on alternate half-
cycles of the supply frequency. The other key component is the linear (air-core) reactor of
inductance L (Miller, 1982). The thyristor circuit symbol is shown in Figure 2.2(b).

iTCR (t)

Anode (G)
v(t) Th2 Th1


Figure 2.2 Thyristor-based circuit: (a) Basic thyristor-controlled reactor (TCR); (b) thyristor circuit

The overall action of the thyristor controller on the linear reactor is to enable the reactor
to act as a controllable susceptance, in the inductive sense, which is a function of the ¬ring
angle . However, this action is not trouble free, since the TCR achieves its fundamental
frequency steady-state operating point at the expense of generating harmonic distortion,
except for the condition of full conduction.
First, consider the condition when no harmonic distortion is generated by the TCR, which
takes place when the thyristors are gated into conduction, precisely at the peaks of the
supply voltage. The reactor conducts fully, and one could think of the thyristor controller as
being short-circuited. The reactor contains little resistance and the current is essentially
sinusoidal and inductive, lagging the voltage by almost 90 (p/2). This is illustrated in
Figure 2.3(a), where a fundamental frequency period of the voltage and current are shown.
It should be mentioned that this condition corresponds to a ¬ring angle of p/2, which is
the current zero-crossing measured with reference to the voltage zero-crossing. The
relationship between the ¬ring angle and the conduction angle  is given by
 ¼ 2°p À Þ: °2:1Þ
Partial conduction is achieved with ¬ring angles in the range: p=2 < < p, in radians. This
is illustrated in Figures 2.3(b)“2.3(d), where TCR currents, as a function of the ¬ring angle,
i, v


0 90 180 270 360
0 90 180 270 360
Phase (degrees) Phase (degrees)
(a) (b)


0 90 180 270 360 90 180 270 360
Phase (degrees) Phase (degrees)
(c) (d)

Figure 2.3 Current waveforms in the basic thyristor-controlled reactor: (a) ¼ 90 ,  ¼ 180 ;
(b) ¼ 100 ,  ¼ 160 ; (c) ¼ 130 ,  ¼ 100 ; (d) ¼ 150 ,  ¼ 60 ; for convenience, angles are
given in degrees. Note: i, current; v, voltage; , ¬ring angle; , conduction angle. Reproduced by
permission of John Wiley & Sons Inc. from T.J.E Miller, 1982, Reactive Power Control in Electric

are shown. Increasing the value of ¬ring angle above p/2 causes the TCR current waveform
to become nonsinusoidal, with its fundamental frequency component reducing in
magnitude. This, in turn, is equivalent to an increase in the inductance of the reactor,
reducing its ability to draw reactive power from the network at the point of connection.
For the voltage condition shown in Figure 2.2(a), with v°tÞ ¼ 2 V sin !t, the TCR
instantaneous current iTCR °tÞ is given by
1 !t p¬¬¬ 2V
iTCR °tÞ ¼ 2 V sin !t dt ¼ °cos À cos !tÞ °2:2Þ
in the interval !t ° þ Þ, and is zero otherwise. V is the root mean square (rms)
voltage, and ! ¼ 2pf , where f is the operating frequency.
Using Fourier analysis, an expression for the fundamental frequency current, ITCRf1 , is
ITCR f1 ¼ ½2°p À Þ þ sin 2 Š: °2:3Þ

If the ¬ring angles of Th1 and Th2 are balanced, no even harmonics are generated, and the
rms value of the hth odd harmonic current is given by
4 V sin°h þ 1Þ sin h À 1Þ sin h
ITCR h ¼ þ À cos ; °2:4Þ
j!Lp 2°h þ 1Þ 2°h À 1Þ h
where h ¼ 3, 5, 7, 9, 11, 13 . . . .
Power system TCR installations are three-phase and use ¬lters and other harmonic
cancellation arrangements to prevent the harmonic currents from reaching the high-voltage
side of the network. Also, the TCR inductors will have a small resistive component. By way
of example, Figure 2.4 shows a three-phase, delta-connected TCR. This topology uses six
groups of thyristor and is commonly known as a six-pulse TCR.






Branch 2
Branch 1 Branch 3

Three-phase thyristor-controlled reactor
Figure 2.4

In this arrangement, and under balanced operating conditions, the triplet harmonic
currents generated by the three TCR branches do not reach the external network, only
harmonic orders h ¼ 5, 7, 11, 13, . . . . Moreover, if the TCR is split into two units of equal
rating and connected to the low-voltage side of a transformer having two secondary
windings, one connected in star and the other in delta, then cancellation of harmonic orders
h ¼ 5, and h ¼ 7 is achieved. The alternative arrangement is termed a twelve-pulse TCR.
The lowest harmonic orders reaching the primary winding of the transformer are h ¼ 11,
13, . . . , which are normally removed by using tuned ¬lters (Miller, 1982).
We would assume in the ensuing analysis that suitable harmonic cancellation measures
are in place, as we are concerned only with fundamental frequency operation and
parameters. However, neither balanced operation nor balanced TCR designs will be

assumed a priori. It is not dif¬cult to see from Equation (2.3) that a part of it may
be interpreted as the equivalent susceptances of the basic TCR shown in Figure 2.2, which is
a function of the controllable parameters . Accordingly, Equation (2.3) may be expressed
ITCR ¼ ÀjBTCR V; °2:5Þ
2°p À Þ þ sin 2
BTCR ¼ ; °2:6Þ
and the subscript f1, which indicates fundamental frequency current, has been dropped for
The three-phase nodal admittance representation of a TCR may be obtained by resorting
to linear transformations. For instance, using the result in Equation (2.5), the case of the six-
pulse TCR shown in Figure 2.4 will have the following primitive parameters:
2 32 32 3
ITCR 1 0 0 V1
4 ITCR 2 5 ¼ 4 54 V 2 5;
ÀjBTCR 2 °2:7Þ
0 0
ITCR 3 0 0 V3

and connectivity matrices for phases a, b, c:
23 2 32 3
V1 1 0 Va
°p=6Þ 6
67 76 7
4 V2 5 ¼ p¬¬¬ 4 0 À1 54 Vb 5; °2:8Þ
V3 0 1 Vc
2 3 2 32 3
ITCR a 1 0 ITCR 1
4 ITCR b 5 ¼ °Àp=6Þ 4 À1 0 54 ITCR2 5:
p¬¬¬ °2:9Þ
3 À1
ITCR c 0 1 ITCR3

Substituting Equation (2.8) into Equation (2.7), and the intermediate result into Equa-
tion (2.9), we obtain the following phase domain equivalent circuit for the six-pulse TCR:
2 3 2 32 3
Àj°BTCR 1 þ BTCR 3 Þ
4 ITCR b 5 ¼ 1 4 54 Vb 5:
Àj°BTCR 1 þ BTCR 2 Þ
Àj°BTCR 2 þ BTCR 3 Þ
As as special condition, if all three branches in the TCR have equal equivalent
susceptances (BTCR 1 ¼ BTCR 2 ¼ BTCR 3 ¼ BTCR ), something that is possible to achieve by
careful design, Equation (2.10) simpli¬es to
2 3 2 32 3
4 ITCR b 5 ¼ 1 4 jBTCR 54 Vb 5:
Àj2BTCR °2:11Þ
In this situation, an alternative representation becomes feasible, using the frame of reference
afforded by the concept of symmetrical components. Three sequence components are
associated with three-phase circuits, namely zero (0), positive (1), and negative (2)
sequences. The transformation from phase coordinates to sequence coordinates involves

applying the matrix of symmetrical components TS and its inverse to Equation (2.11),
leading to the following result:
2 32 32 3
0 0 0
4 ITCR °1Þ 5 ¼ 4 0 0 54 V°1Þ 5:
ÀjBTCR °2:12Þ
0 0

The operation required to transform a three-phase quantity into sequence quantities is
explained in detail in Section 3.2.12.
As expected, no zero sequence current can ¬‚ow in this circuit owing to the delta-
connected nature of the TCR. The positive sequence (1) and negative sequence (2) circuits
present equal impedances (susceptances) to their respective current ¬‚ows. Also, it is shown
in Equation (2.12) that no couplings exist between sequences. It should be remarked
that this would not have been the case if symmetrical components had been applied to
Equation (2.10) as opposed to Equation (2.11). The reason is that the admittance matrix
of Equation (2.10) is not necessarily a balanced one, since the condition BTCR 1 6¼ BTCR 2 6¼
BTCR 3 may exist.
Nevertheless, if equal equivalent admittances may be assumed in the six-pulse TCR then
the positive sequence representation becomes

ITCR °1Þ ¼ ÀjBTCR V°1Þ : °2:13Þ

This representation matches the behaviour of the basic (single-phase) TCR shown in
Figure 2.2(a) and given by Equation (2.5).

2.3.2 The Static VAR Compensator

In its simplest form, the SVC consists of a TCR in parallel with a bank of capacitors. From
an operational point of view, the SVC behaves like a shunt-connected variable reactance,
which either generates or absorbs reactive power in order to regulate the voltage magnitude at
the point of connection to the AC network. It is used extensively to provide fast reactive
power and voltage regulation support. The ¬ring angle control of the thyristor enables the
SVC to have almost instantaneous speed of response.
A schematic representation of the SVC is shown in Figure 2.5, where a three-phase, three-
winding transformer is used to interface the SVC to a high-voltage bus. The transformer has
two identical secondary windings: one is used for the delta-connected, six-pulse TCR and
the other for the star-connected, three-phase bank of capacitors, with its star point ¬‚oating.
The three transformer windings are also taken to be star-connected, with their star points
The modelling of one TCR branch has been dealt with in Section 2.3.1, and attention is
now dedicated to a bank of capacitors. The admittances of both branches of the SVC will
then combine quite straightforwardly.
The nodal admittance of the capacitor bank, in phase coordinates, may be expressed with
explicit representation of the star point, which is not grounded. However, it is more
advantageous to perform a Kron reduction to obtain a reduced equivalent, where only the
parameters of phases a, b, and c are represented explicitly.

Ia Va
Ib Vb
Ic Vc


I Cb I Ca
I Cc

C1 C2 C3

Branch 2
Branch 1 Branch 3

Figure 2.5 Representation of a three-phase static VAR compensator (SVC) comprising ¬xed
capacitors and thyristor-controlled reactors (TCRs)

In the most general case, when BC 1 6¼ BC 2 6¼ BC 3 , and after having performed Kron™s
reduction, the reduced equivalent model of the bank of capacitors is:
3 2 
2 32 3
B2 1
BC 2 BC 1 BC 3 BC 1 Va
IC a C
6 76 76 7
6 76 76 7
6 IC b 7 ¼ 6 Àj 76 Vb 7;
B2 2
j BC 2 À ÁBC Àj ÁBC °2:14Þ
BC 1 BC 2 BC 3 BC 2
6 76 76 7
4 54  54 5

Àj BC 1 BC 3 Àj BC 2 BC 3 j BC 3 À ÁB3

ÁBC ¼ BC 1 þ BC 2 þ BC 3 ; >
B ¼ !C ;
C1 1
BC 2 ¼ !C2 ; >
BC 3 ¼ !C3 :
Kron™s reduction is a technique used to eliminate mathematically, speci¬c rows and columns
in a matrix equation. It is explained in detail in Section 3.2.3.
If all three branches in the bank of capacitors have equal equivalent susceptances
(BC 1 ¼ BC 2 ¼ BC 3 ¼ BC ), Equation (2.14) simpli¬es to:
2 3 2 32 3
IC a Va
4 IC b 5 ¼ 1 4 ÀjBC j2BC ÀjBC 54 Vb 5: °2:16Þ
IC c Vc

Three-phase models of the SVC in phase coordinates can now be formed with ease. The
most general expression for the six-pulse SVC would be the case when Equations (2.10) and
(2.14) are added together, giving rise to a model where design imbalances in the SVC may
be accounted for.
A more constrained, but still very useful, model is the case when Equations (2.11) and
(2.16) are used as the constituent parts of the SVC model:
2 32 32 3
6 76 76 7
4 ISVC b 5 ¼ 4 IC b 5 þ 4 ITCR b 5
2 32 3
16 76 7
¼ 4 Àj°BC À BTCR Þ j2°BC À BTCR Þ Àj°BC À BTCR Þ 54 Vb 5: °2:17Þ
It is clear that alternative models, of varying functionality, can also be formed. For instance,
combination of Equations (2.10) and (2.16) leads to an SVC model where the three branches
of the TCR may have different equivalent inductances but the three capacitances of the bank
are equal. Use of Equations (2.11) and (2.14) have the opposite functionality effect in the
SVC model.
In any case, only the SVC model given by Equation (2.17) is suitable for deriving a
representation in the frame of reference of symmetrical components. Applying the matrix of
symmetrical components TS and its inverse to Equation (2.17) leads to the following result:
2 32 32 3
0 0 0
4 ISVC °1Þ 5 ¼ 4 0 j°BC À BTCR Þ 54 V°1Þ 5: °2:18Þ
0 0
Similar to the TCR, no zero sequence current can ¬‚ow in the SVC circuit as the star point
of the bank of capacitors is not grounded. The positive sequence and negative sequence
circuits contain equal impedances. However, for cases of balanced operation and balanced
SVC designs only the positive sequence representation is of interest:
ISVC °1Þ ¼ jBSVC V°1Þ ; °2:19Þ
1 XC
½2°p À Þ þ sin 2 Š ; >
p =
XL ¼ !L; >
XC ¼ :
It should be remarked that the positive sequence model of the SVC should also serve the
purpose of representing a single-phase SVC.

2.3.3 The Thyristor-controlled Series Compensator

TCSCs vary the electrical length of the compensated transmission line with little delay. This
characteristic enables the TCSC to be used to provide fast active power ¬‚ow regulation. It

also increases the stability margin of the system and has proved very effective in damping
SSR and power oscillations (Larsen et al., 1992).
In principle, the steady-state response of the TCSC may be calculated by solving the
differential equations that describe its electrical performance, using a suitable numeric
integration method. Alternatively, the TCSC differential equations may be expressed in
algebraic form and then a phasorial method used to solve them. The former approach
involves the integration of the differential equations over many cycles until the transient
response dies out. This solution method is rich in information as the full evolution of the
response is captured, from transient inception to steady-state operation, but it suffers from
excessive computational overheads, particularly when solving lightly damped circuits. Two
different solution ¬‚avours emerge from the phasorial approach. (1) the TCSC steady-state
operation may be determined very ef¬ciently by using fundamental and harmonic frequency
phasors, neatly arranged in the harmonic domain frame of reference (Acha and Madrigal,
2001). The method yields full information for the fundamental and harmonic frequency
TCSC parameters but no transient information is available. (2) Alternatively, a nonlinear
equivalent impedance expression is derived for the TCSC and solved by iteration (Fuerte-
Esquivel, Acha, and Ambriz-Perez, 2000a). The solution method is accurate and converges
very robustly towards the solution, but it only yields information for the fundamental
frequency steady-state solution. This is precisely the approach taken in power ¬‚ow studies,
the application topic covered in this book. Thyristor-controlled series capacitor equivalent circuit

A basic TCSC module consists of a TCR in parallel with a ¬x capacitor. An actual TCSC
comprises one or more modules. Figure 2.6 shows the layout of one phase of the TCSC
installed in the Slatt substation (Kinney, Mittelstadt, and Suhrbier, 1994).

By pass disconnect

Varistor TCSC
(1.99 mF)

(0. 470 mH)
(0 .307 mH)

By pass breaker

Figure 2.6 Physical structure of one phase of a thyristor-controlled series capacitor (TCSC).
Reproduced, with permission, from S.J. Kinney, W.A. Mittelstadt, and R.W. Suhrbier, ˜Test Results
and Initial Operating Experience for the BPA 500 kV Thyristor Controlled Series Capacitor: Design,
Operation, and Fault Test Results, Northcon 95™, in IEEE Technical Conference and Workshops
Northcon 95, Portland, Oregon, USA, October 1995, pp. 268“273, # 1995 IEEE

The TCR achieves its fundamental frequency operating state at the expense of generating
harmonic currents, which are a function of the thyristor conduction angle. Nevertheless,
contrary to the SVC application where the harmonic currents generated by the TCR tend to
escape towards the network, in the TCSC application the TCR harmonic currents are
trapped inside the TCSC because of the low impedance of the capacitor compared with the
network equivalent impedance. This is, at least, the case for a well-designed TCSC
operating in capacitive mode. Measurements conducted in the Slatt and the Kayenta TCSC
systems support this observation. For instance, the Kayenta system generates at its
terminals, a maximum total harmonic distortion (THD) voltage of 1.5 % when operated in
capacitive mode and ¬ring at an angle of 147 (Christl et al., 1992). It should be noted that
there is little incentive for operating the TCSC in inductive mode as this would increase the
electrical length of the compensated transmission line, with adverse consequences on
stability margins, and extra losses.
For the purpose of fundamental frequency power system studies, a complex TCSC
topology, such as the single-phase branch shown in Figure 2.6, may be taken to consist of
one equivalent TCR paralleled by one equivalent capacitor, as illustrated schematically in
Figure 2.7. The surge arrester is not represented as this is a representation intended for
steady-state operation, but the existence of a loop current is emphasised.


I loop



Figure 2.7 Thyristor-controlled series capacitor (TCSC) equivalent circuit. Reproduced with
permission from C.R. Fuerte-Esquivel, E. Acha, and H. Ambriz-Perez, ˜A Thyristor Controlled Series
Compensator Model for the Power Flow Solution of Practical Power Networks™, IEEE Trans. Power
Systems 15(1) 58“64, # 2000 IEEE

This equivalent circuit has an associated equivalent reactance, which is a function of the
thyristor gating signals. Expressions for the various electrical parameters in the TCSC
equivalent circuit are derived in the following two sections. Steady-state current and voltage equations

The TCSC current equations may be obtained with reference to the circuit shown in
Figure 2.8, using Laplace theory. This electric circuit represents, in simple terms, the

i cap
i thy

i line = 1 cos ω t C


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