OPERATING MODES AND THEIR REGULATIONS OF VOLTAGE-SOURCED CONVERTER …chowj/Xia Jiang PhD...
Transcript of OPERATING MODES AND THEIR REGULATIONS OF VOLTAGE-SOURCED CONVERTER …chowj/Xia Jiang PhD...
OPERATING MODES AND THEIR REGULATIONSOF VOLTAGE-SOURCED CONVERTER BASED
FACTS CONTROLLERS
By
Xia Jiang
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subject: Electrical Engineering
Approved by theExamining Committee:
Joe H. Chow, Thesis Adviser
Sheppard J. Salon, Member
Jian Sun, Member
Murat Arcak, Member
Behruz Fardanesh, Member
Abdel-Aty Edris, Member
Rensselaer Polytechnic InstituteTroy, New York
March 2007(For Graduation May 2007)
OPERATING MODES AND THEIR REGULATIONSOF VOLTAGE-SOURCED CONVERTER BASED
FACTS CONTROLLERS
By
Xia Jiang
An Abstract of a Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subject: Electrical Engineering
The original of the complete thesis is on filein the Rensselaer Polytechnic Institute Library
Examining Committee:
Joe H. Chow, Thesis Adviser
Sheppard J. Salon, Member
Jian Sun, Member
Murat Arcak, Member
Behruz Fardanesh, Member
Abdel-Aty Edris, Member
Rensselaer Polytechnic InstituteTroy, New York
March 2007(For Graduation May 2007)
c© Copyright 2007
by
Xia Jiang
All Rights Reserved
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CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Description of Shunt and Series Voltage-Sourced Converters . . . . . 5
2.2 Summary on Voltage-Sourced Converter Operating Modes . . . . . . 6
2.2.1 Shunt VSC Operating Modes . . . . . . . . . . . . . . . . . . 6
2.2.2 Series VSC Operating Modes . . . . . . . . . . . . . . . . . . 6
2.3 Overview of VSC-Based FACTS Controller Loadflow Models . . . . . 8
2.3.1 Decoupled FACTS Controller Loadflow Model . . . . . . . . . 8
2.3.2 Power Injection Model . . . . . . . . . . . . . . . . . . . . . . 10
2.3.3 Voltage Source Model . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 14
2.4 Overview of VSC-Based FACTS Controllers in Time-Domain Simu-lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 17
3. FACTS CONTROLLER STEADY-STATE DISPATCH . . . . . . . . . . . 18
3.1 VSC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Dispatch Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Shunt VSC Operating Modes . . . . . . . . . . . . . . . . . . 21
3.2.2 Series VSC Operating Modes . . . . . . . . . . . . . . . . . . 22
3.2.2.1 Standalone or “Slave” Operation . . . . . . . . . . . 22
3.2.2.2 Coupled Operation . . . . . . . . . . . . . . . . . . . 23
3.3 Newton-Raphson Algorithm . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Rated-Capacity Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Operating Limits . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.2 Dispatch Strategies . . . . . . . . . . . . . . . . . . . . . . . . 27
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3.4.2.1 Standalone Operation . . . . . . . . . . . . . . . . . 27
3.4.2.2 Coupled Operating Mode . . . . . . . . . . . . . . . 28
3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.1 Voltage Stability Improvement by the SSSC . . . . . . . . . . 31
3.5.2 Operator Training Simulator (OTS) for NYPA’s CSC . . . . . 34
3.5.3 Maximum Dispatchbility for the UPFC and IPFC . . . . . . . 38
3.5.3.1 Maximum UPFC Dispatchability . . . . . . . . . . . 39
3.5.3.2 Maximum IPFC Dispatchability . . . . . . . . . . . . 40
3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 43
4. FACTS CONTROLLER DYNAMIC MODELS AND SETPOINT CON-TROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 VSC Dynamic Modeling and Control . . . . . . . . . . . . . . . . . . 45
4.1.1 VSC Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.2 VSC Setpoint Controller Models . . . . . . . . . . . . . . . . . 48
4.1.2.1 Shunt VSC Model . . . . . . . . . . . . . . . . . . . 48
4.1.2.2 Standalone Series VSC Model . . . . . . . . . . . . . 49
4.1.2.3 Coupled Series VSC Model . . . . . . . . . . . . . . 52
4.1.2.4 The IPFC Model . . . . . . . . . . . . . . . . . . . . 53
4.1.3 DC Link Capacitor Dynamics . . . . . . . . . . . . . . . . . . 56
4.2 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Nonlinear Dynamic Models . . . . . . . . . . . . . . . . . . . 58
4.2.1.1 Shunt Operating Modes . . . . . . . . . . . . . . . . 59
4.2.1.2 Standalone Series Dispatch Modes . . . . . . . . . . 60
4.2.1.3 Coupled Series Dispatch Modes . . . . . . . . . . . . 61
4.2.1.4 IPFC Operating Modes . . . . . . . . . . . . . . . . 62
4.2.2 Network Equations . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.4 Integration Method . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 FACTS Controller Dynamic Simulations . . . . . . . . . . . . 71
4.3.1.1 STATCOM Dynamics . . . . . . . . . . . . . . . . . 71
4.3.1.2 SSSC Dynamics . . . . . . . . . . . . . . . . . . . . . 72
4.3.1.3 UPFC Dynamics . . . . . . . . . . . . . . . . . . . . 72
4.3.1.4 IPFC Dynamics . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Transient Power Transfer Capability Analysis Example . . . . 73
4.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 75
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5. LINEARIZED MODELS AND MODAL DECOMPOSITION OF MULTI-MACHINE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 Small-Signal Linearization . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 System Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3 Multi-Machine Modal Decomposition Approach . . . . . . . . . . . . 87
5.4 Application: A 20-Bus System Study . . . . . . . . . . . . . . . . . . 90
5.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 92
6. DAMPING CONTROLLER DESIGN . . . . . . . . . . . . . . . . . . . . . 95
6.1 Damping Controller Block Diagram . . . . . . . . . . . . . . . . . . . 96
6.2 Input Signal Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Design for the STATCOM . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4 Design for the SSSC . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.5 Design for the UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5.1 UPFC Series Regulator in Vd,Vq Mode . . . . . . . . . . . . . 108
6.5.2 Impact of the Series P ,Q Regulators . . . . . . . . . . . . . . 109
6.5.3 Dynamic Simulation of the UPFC Damping Controllers . . . . 111
6.6 Design for the IPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 117
7. MAIN CONTRIBUTIONS AND FUTURE WORK RECOMMENDATIONS120
7.1 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Future Research Recommendations . . . . . . . . . . . . . . . . . . . 121
LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
APPENDICES
A. DATA FILE OF A 22-BUS POWER SYSTEM . . . . . . . . . . . . . . . 131
B. DATA FILE OF A 20-BUS POWER SYSTEM . . . . . . . . . . . . . . . 137
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LIST OF TABLES
3.1 Transmission Line Data of the 4-Bus Radial Test System . . . . . . . . 32
3.2 Operating Modes of a Reconfigurable VSC-Based FACTS Controller . . 36
3.3 Dispatch Computation of an Operator Training Scenerio . . . . . . . . . 38
4.1 Operating Conditions of the STATCOM in Var Control Mode . . . . . . 71
4.2 Operating Conditions of the SSSC in Vm Control Mode . . . . . . . . . 72
4.3 Operating Conditions of the UPFC in V ,Vd,Vq Control Mode . . . . . . 72
4.4 Operating Conditions of the IPFC in Inverter Voltage Control Mode . . 73
4.5 Comparison of Transient Power Transfer Capability Analysis withoutand with Various FACTS Controllers . . . . . . . . . . . . . . . . . . . 74
5.1 State Modes of the 20-Bus System . . . . . . . . . . . . . . . . . . . . . 94
6.1 MDI Indices for the UPFC Series Vd,Vq Mode v.s. the STATCOM . . . 110
6.2 MDI Index Values for Measured Signals to IPFC Regulators . . . . . . 115
6.3 Controllability and Observability Gain Product Index . . . . . . . . . . 116
6.4 Designed Damping Controllers for the IPFC . . . . . . . . . . . . . . . 116
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LIST OF FIGURES
2.1 Single-Line Diagrams of VSC-Based FACTS Controllers . . . . . . . . 5
2.2 Operating Mode Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Decoupled Loadflow Model of the UPFC . . . . . . . . . . . . . . . . . 9
2.4 Power Injection Model of the UPFC . . . . . . . . . . . . . . . . . . . 10
2.5 Decoupled Power Injection Model of the UPFC . . . . . . . . . . . . . 12
2.6 Shunt Admittance Model of the UPFC . . . . . . . . . . . . . . . . . . 13
3.1 Injected Voltage-Sourced Model of a Shunt VSC . . . . . . . . . . . . . 20
3.2 Injected Voltage-Sourced Model of a Series VSC . . . . . . . . . . . . . 20
3.3 Injected Series Voltage Modification in the Master VSC . . . . . . . . . 29
3.4 Injected Series Voltage Modification When the Slave VSC Cannot Sup-port Enough Real Power . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 4-Bus System with a Series VSC . . . . . . . . . . . . . . . . . . . . . . 32
3.6 PV Characteristics of the SSSC . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Series VSC Injected Voltage . . . . . . . . . . . . . . . . . . . . . . . . 34
3.8 The CSC Connection Scheme . . . . . . . . . . . . . . . . . . . . . . . 35
3.9 UPFC Control Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Injected Series Voltage Reference . . . . . . . . . . . . . . . . . . . . . 39
3.11 UPFC Series Line Incremental P -Q Curves . . . . . . . . . . . . . . . . 40
3.12 Incremental P -Q Curves of IPFC Lines . . . . . . . . . . . . . . . . . . 41
3.13 Injected Series Voltage of IPFC Slave VSC . . . . . . . . . . . . . . . . 42
3.14 Incremental P -Q Curves of IPFC Lines . . . . . . . . . . . . . . . . . . 42
3.15 Injected Series Voltage of the IPFC . . . . . . . . . . . . . . . . . . . . 43
4.1 Voltage-Sourced Converters Showing DC Capacitors . . . . . . . . . . 46
4.2 Voltage-Sourced Converter Models . . . . . . . . . . . . . . . . . . . . 47
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4.3 Setpoint Control Schemes of a Shunt VSC . . . . . . . . . . . . . . . . 49
4.4 Setpoint Control Schemes of a Standalone or “Slave” Series VSC in LineActive Power Regulation Mode . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Setpoint Control Schemes of a Standalone or “Slave” Series VSC inFixed Injected Voltage Mode . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Setpoint Control Schemes of Coupled Series VSC . . . . . . . . . . . . . 53
4.7 Setpoint Control Schemes of an IPFC in the Line Power Regulation Mode 54
4.8 Setpoint Control Schemes of an IPFC in the Fixed Injected Voltage Mode 55
4.9 DC Link Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.10 Block Realization of the PI regulator and LP filter . . . . . . . . . . . 58
4.11 22-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.12 STATCOM Var Control Mode Simulation . . . . . . . . . . . . . . . . . 77
4.13 SSSC Inverter Voltage Magnitude Control Mode Simulation . . . . . . . 78
4.14 UPFC V ,Vd,Vq Control Mode Simulation . . . . . . . . . . . . . . . . . 79
4.15 IPFC Inverter Voltage Control Mode Simulation – I . . . . . . . . . . . 80
4.16 IPFC Inverter Voltage Control Mode Simulation – II . . . . . . . . . . . 81
4.17 Comparison of No FACTS and UPFC in V ,Vd,Vq Mode When PL2=3235MW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1 Modal Decomposition of a Linearized Multi-Machine System with aNetwork Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 20-Bus Test System Single-Line Diagram and Flows . . . . . . . . . . . 91
5.3 Compass Plots for the Four Swing Modes . . . . . . . . . . . . . . . . 92
6.1 Damping Controller Block Diagram . . . . . . . . . . . . . . . . . . . . 96
6.2 Washout Loop Block Realization . . . . . . . . . . . . . . . . . . . . . 97
6.3 Phase Compensator Block Realization . . . . . . . . . . . . . . . . . . 98
6.4 Low Pass Filter Block Realization . . . . . . . . . . . . . . . . . . . . . 99
6.5 STATCOM MDI Index Plots Varying Kp . . . . . . . . . . . . . . . . . 102
6.6 STATCOM MDI Index Plots Varying Ki . . . . . . . . . . . . . . . . . 103
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6.7 STATCOM Damping Controller Signal . . . . . . . . . . . . . . . . . . 104
6.8 Dynamic Simulation with a STATCOM Damping Controller – I . . . . 105
6.9 Dynamic Simulation with a STATCOM Damping Controller – II . . . . 106
6.10 SSSC MDI Index Plots Varying Regulation Control Gains . . . . . . . . 107
6.11 SSSC Damping Controller Signal . . . . . . . . . . . . . . . . . . . . . . 107
6.12 Dynamic Simulation with an SSSC Damping Controller – I . . . . . . . 108
6.13 Dynamic Simulation with an SSSC Damping Controller – II . . . . . . . 109
6.14 MDI Index of the UPFC in V ar,P ,Q Mode . . . . . . . . . . . . . . . 111
6.15 UPFC Root-Locus Plots of the Four Swing Modes . . . . . . . . . . . . 112
6.16 UPFC Damping Controller Signal . . . . . . . . . . . . . . . . . . . . . 113
6.17 Dynamic Simulation with a UPFC Damping Controller – I . . . . . . . 113
6.18 Dynamic Simulation with a UPFC Damping Controller – II . . . . . . . 114
6.19 Bus 4 Voltage without and with Damping Controllers . . . . . . . . . . 117
6.20 DC Capacitor Voltage without and with Damping Controllers . . . . . 117
6.21 IPFC Line Flows without and with Damping Controllers . . . . . . . . 119
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ACKNOWLEDGMENT
I would like to thank my advisor, Prof. Joe H. Chow, for the invaluable guidance
and inspiration that he has provided during the course of this work.
Special thanks are given to Dr. Abdel-Aty Edris (EPRI), Dr. Bruce Fardanesh
(NYPA), Dr. Sheppard J. Salon, Dr. Robert C. Degeneff, Dr. Murat Arcak, and
Dr. Jian Sun for their interacts of this work. I thank Ms. Edvina Uzunovic, Ms.
Jane (Jiyun) Sun, Ms. Liana Hopkins, Mr. Bruce Fardanesh, Mr. Mike Parisi, and
Mr. Mark Graham at NYPA for their great cooperation in the Operator Training
Simulator (OTS) project. I am also grateful to Dr. Graham Rogers of Cherry Tree
Scientific Software for providing the Power System Toolbox (PST).
I would like to thank my colleagues Dr. Xuan Wei, Mr. Xinghao Fang, and
Mr. Luigi Vanfretti for always being there to offer help.
I would also like to thank my husband Hui, parents Zhenghua and Juzhen,
sister Feng, and brother Liang, who have always been supportive in different stages
of my life.
This research is supported in part by Electric Power Research institute (EPRI)
and New York Power Authority (NYPA), and in part by National Science Founda-
tion (NSF).
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ABSTRACT
Voltage-sourced converter (VSC) based FACTS controllers are capable of providing
fast voltage support and active power flow control to improve the power transfer
capability over congested transmission paths. In most published literature, a shunt
VSC such as a Static Synchronous Compensator (STATCOM) is set to control the
bus voltage and a series VSC such as a Static Synchronous Series Compensator
(SSSC) is set to control the line power flow. In practical operations, however, there
are other control modes that are more appropriate, such as fixed reactive power
setpoint control for a shunt converter and fixed injected voltage control for a series
converter.
In this research work, we aim to investigate the modeling, simulation, and
control of various operating modes and their regulations of VSC-based FACTS con-
trollers embedded in transmission networks. The first major task of this research
work is to study the impact of these FACTS controllers in both normal operation and
rated-capacity operation. The second major task is to develop dynamic models so
that the regulations of the various control modes can be properly investigated. The
third major task is to design damping controllers supplemental to the regulations
to improve small-signal stability.
In this thesis work, an efficient control mode implementation has been pro-
posed to implement steady-state dispatch of various operating modes of FACTS
controllers, using an approach of separate models for a shunt VSC and for a series
VSC. If the DC buses of the two converters are coupled, then an appropriate active
power circulation constraint can be added to the VSC operating constraints. With
this implementation, we only need to select and combine the appropriate equations
of the shunt VSC, the series VSC, and the DC link coupling to form the specified
FACTS controller and to operate it in the desired operating mode. Because the
maximum dispatch benefit of an FACTS controller often occurs when it operates at
its rated capacity, efficient dispatch strategies to optimize line power flow transfer
have also been proposed when one or both VSCs of the FACTS controller are loaded
xi
to their rated capacity.
Following the steady-state dispatch, dynamic regulator models of FACTS con-
trollers, which take into consideration the dynamics of DC Links, have been devel-
oped and implemented to evaluate their impact on transient stability during system
faults and lightly damped inter-area oscillations. Based on the dynamic models,
linearized models of FACTS controllers in multi-machine systems have been derived
using small-signal perturbations.
In addition to their capability of regulating power flow transfer, FACTS con-
trollers can be utilized to improve small-signal stability by providing damping con-
trol supplemental to their regulation controls. To study damping control effects of
FACTS controllers, a new modal decomposition approach, which fully decouples all
the modes in the system and considers the interaction of the other system modes
to the mode of interest, has been proposed to quantify levels of controllability, ob-
servability, and inner-loop gains of the linearized models. A comprehensive process
that examines the controller gain limitation, the selection of damping controller in-
put signals, and modal damping selectivity signal selection have been developed to
design damping controllers.
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CHAPTER 1
INTRODUCTION
Flexible AC Transmission Systems (FACTS) based on power electronics offer an
opportunity to enhance controllability, stability, and power transfer capability of
AC transmission systems [1]. In recent years, different FACTS controllers, which are
defined as power electronics-based systems or other static equipments that provide
control of one or more AC transmission parameters, have performed a wide variety
of compensating functions.
In general, FACTS controllers can be classified into two distinct generations.
The earlier generation is based on the line-commutated thyristor devices with only
gate turn-on but no gate turn-off capability. Application of this technology started
with the Static Var Compensator (SVC) in the 1970s [2], followed by the Thyristor-
Controlled Series Capacitor (TCSC) [3]. The newer generation of FACTS Con-
trollers is based on the self-commutated voltage-sourced converters (VSC), which
utilize thyristors/transistors with gate turn-off capability, such as GTOs, MTOs,
IGCTs, and IGBTs1. The voltage-sourced converter (VSC) technology has been
successfully applied in a number of installations world-wide of Static Synchronous
Compensators (STATCOM) [4]-[12], Back-To-Back STATCOM [13], Unified Power
Flow Controllers (UPFC) [14], and Convertible Series Compensators (CSC) [15].
The family of VSC-based FACTS controllers also includes the Synchronous Series
Compensator (SSSC) [16], the Interline Power Flow Controller (IPFC) [17], and the
Generalized Unified Power Flow Controller (GUPFC) [18]. Among all these types
of VSC-based FACTS controllers, the CSC is the most versatile FACTS device con-
ceived, which can be operated as STATCOM, SSSC, UPFC, or IPFC in 11 different
configurations.
The VSC-based FACTS controllers offer several significant advantages over
the thyristor-based configurations such as SVCs and TCSCs. First of all, they are
usually quite compact and most likely fit into existing substations, thus avoiding
1GTO: Gate Turn-off Thyristor, MTO: MOS Turn-Off Thyristor, IGCT: Integrated Gate-Commutated Thyristor, and IGBT: Insulated Gate Bipolar Transistor.
1
2
the need for land acquisition and associated environmental concerns. Second, faster
control responses and lower output distortion can be achieved with suitable internal
controls. Third, they can improve dispatch flexibility by circulating active power
between their AC and DC terminals if there is a suitable power source or energy
storage connected to the DC terminals, or there are more than one VSCs coupled
together. Moreover, the combined VSC-based FACTS controllers, such as the UPFC
and CSC, provide complete controllability for controlling not only bus voltages but
also line flows. In this research, we focus our study on VSC-based FACTS controllers.
This research work focuses on steady-state and dynamic modeling, simula-
tion, and control of various operating modes and their regulations of VSC-based
FACTS controllers embedded in transmission networks. The steady-state modeling
and dispatch in part builds on Dr. Xuan Wei’s research work [19]-[20] of studying the
modeling, dispatch, and control of various VSC-based FACTS Controllers. In par-
ticular, she investigated dispatch strategies of these FACTS Controllers to optimize
the voltage profile and power transfer for both normal operation and rated capacity
operation conditions [21]-[23]. In her work, the FACTS controllers are classified by
type, which means for a new configuration or even for a new operating mode, a
complete set of codes for the model needs to be included.
In continuing the steady-state dispatch work in [19], three major improvements
are provided in this thesis work. First, an efficient control mode implementation has
been developed by advocating separate modeling for a shunt VSC and for a series
VSC. If the DC buses of two converters are coupled, then an appropriate active power
circulation constraint can be added to the VSC models. With this implementation,
we only need to select and combine the appropriate shunt VSC, series VSC, and
DC link coupling equations to form the specified FACTS controller and to operate
it in the desired operating mode. Second, in addition to the shunt voltage setpoint
control mode and the line power flow regulation mode, the reactive power setpoint
control mode and the reactive power reserve mode for the shunt VSC and the fixed
injected voltage control mode for the series VSC and their rated-capacity dispatch
have been incorporated into the control mode implementation. Third, line active
power priority rule is applied to re-calculate series injected voltage setpoints when a
3
coupled series VSC is operated at rated capacity, which does not require additional
optimization programs to specify the setpoints.
Besides the steady-state dispatch, this thesis work also extends the control
mode implementation to transient stability analysis for various FACTS controllers.
A comprehensive set of regulator models of FACTS controllers, which take into ac-
count the DC link capacitor dynamics, are proposed to evaluate their impact on
transient stability during system faults and lightly damped inter-area oscillations.
The shunt VSC controls and the series VSC controls are modeled as separate reg-
ulators. When a VSC changes its operating mode, only the input signals of the
corresponding regulator need to be adjusted. The comprehensive set of regulator
models are readily incorporated into a positive-sequence time-domain simulation
program.
Following the development of dynamic models in various control modes, lin-
earized models of FACTS controllers in multi-machine systems are derived based on
the dynamic models using small-signal perturbations. In this approach, dynamic
simulation and small-signal analysis are able to share a common set of codes for
FACTS controllers.
Furthermore, damping control design using VSC-based FACTS controllers for
damping inter-area modes are investigated in this thesis work. A new modal de-
composition approach, which fully decouples all the system modes and consider the
interaction of other state modes to the mode of interest, is proposed to to quantify
levels of controllability, observability, and inner-loop gain of the small-signal lin-
earized models of FACTS controllers in multi-machine systems. A comprehensive
process that will examine the controller gain limitation, the selection of damping
controller input signals, and modal damping selectivity signal selection have been
developed to design damping controllers.
This thesis is organized as follows. This chapter gives a short introduction
of VSC-based FACTS controllers and background of this research work. Chapter 2
introduced shunt VSCs and Series VSCs and their operating modes and reviewed
loadflow models and dynamic models for FACTS controllers. Steady-state dispatch
of FACTS controllers in both normal operation and rated capacity is described in
4
Chapter 3. Chapter 4 includes the details for dynamic simulation of FACTS con-
trollers. Chapter 5 presents linearized models and modal decomposition approach
for small-signal stability analysis of multi-machine systems. Damping control design
is discussed in Chapter 6. The main contributions of this thesis work and future
work recommendations are summarized in Chapter 7.
CHAPTER 2
LITERATURE REVIEW
2.1 Description of Shunt and Series Voltage-Sourced Con-
verters
The positive-sequence representation of a shunt connection and a series con-
nection of VSC-based FACTS controllers are shown in Figures 2.1 (a) and (b),
respectively. From a DC input voltage source, provided by the charged capacitor,
each converter produces a set of controllable three-phase output voltages with the
frequency of the AC power system [1].
+ _
Z1 Z2V1V2 V3
Ssh
Vdc
From-bus
~ ~~
+ _
Z4V2Z3
V1V3 V4~ ~ ~ ~
Vdc
From-bus To-busSse
(a) Shunt Connection (b) Series Connection
Figure 2.1: Single-Line Diagrams of VSC-Based FACTS Controllers
The output voltage of the shunt VSC is connected to the corresponding AC
system voltage by a shunt coupling transformer. By varying the amplitude and
phase angle of the output voltage produced by the shunt VSC, the power exchange
Psh and Qsh between the converter and the AC system can be controlled. If the shunt
VSC operates standalone as a STATCOM and is not integrated with other energy
storage systems, the output voltage will be in phase with the from-bus voltage and
thus no active power exchanges between the converter and the AC system.
5
6
The series VSC injects its output voltage into the transmission line via a series
coupling transformer. By regulating the amplitude and phase angle of its output
voltage, it exchanges both reactive and active power with the transmission system.
If the series VSC operates standalone as an SSSC and is not integrated with other
energy storage systems, the output voltage will be in quadrature with the line current
and thus no active power exchanges between the converter and the AC system.
If a VSC is integrated with an energy storage system or coupled with other
VSCs via DC link capacitors, active power will circulate between their AC and DC
terminals.
2.2 Summary on Voltage-Sourced Converter Operating Modes
As discussed in most FACTS Controllers literature, the most common dispatch
mode of a shunt VSC is to regulate the bus voltage and of a series VSC is to regulate
the real power flow on the line. When a shunt VSC and a series VSC are coupled at
their DC bus to form a UPFC, the line reactive power flow can also be controlled.
For practical applications, however, other operational modes should be considered,
as is the case with the NYPA CSC [24], [25].
2.2.1 Shunt VSC Operating Modes
The possible operating modes of a shunt VSC include:
(Sh1) Control the shunt bus voltage with a reference value Vref and a droop α, that
is, V1 = Vref − αIshq, where Ishq is the reactive current injected by the shunt
VSC. The droop function can be turned off by setting α = 0.
(Sh2) Control the Var output of the shunt VSC to a desired value Ishqref .
(Sh3) Operate in the Var reserve mode which is the operating mode (Sh1) with the
Var output of the shunt VSC limited to [Ishqmin, Ishqmax]. The operating V -I
characteristic of mode (Sh3) is shown in Figure 2.2(a).
2.2.2 Series VSC Operating Modes
The possible operating modes of a series VSC operating standalone include:
7
ICshqmax ILshqmax0
V1
Vref
ICshq ILshq
ICshqres ILshqres Vd
VqV1~
Vm~
Iline
(a) Shunt Sh3 (b) Series SeC2
Figure 2.2: Operating Mode Diagrams
(Se1) Control the line active power flow to a desired value Pdes.
(Se2) Fix the injected voltage magnitude, in either a quadrature leading or lagging
direction with respect to the line current.
The standalone series mode also applies to the series VSC operating as the
“slave” in an Interline Power Flow Controller (IPFC) [26].
If the series VSC is coupled to another VSC, then the possible operating modes
include:
(SeC1) Control the line active and reactive power flow to the desired values Pdes and
Qdes.
(SeC2) Fix the magnitude of the d-axis and q-axis injected voltages at Vd and Vq,
determined with respect to the from-bus voltage vector V1 (Figure 2.2(b)).
The (SeC1) mode is the most commonly cited mode of operation of a UPFC in
the published literature, as influenced by the UPFC operation in Inez [14]. Fixing the
line P and Q flow, however, may prevent the line from carrying a higher amount of
8
flow in case of a contingency. The operating mode (SeC2) would allow for additional
line power flow when appropriate.
The operating modes listed here must also respect the VSC operating limits.
When a shunt VSC is at its operating limit, it may not be able to control the bus
voltage to its desired value, and when a series VSC is at its operating limit, it may
not be able to control the line active power to its desired value. Instead, their
injected voltages will be fixed at their maximum values. For coupled VSCs such as
a UPFC, operation at capacity limits can be more involved.
2.3 Overview of VSC-Based FACTS Controller Loadflow
Models
The challenge of loadflow modeling for the series and shunt VSCs arises from
the fact that the traditional loadflow model consists of only shunt injections and
shunt voltage sources. As a result, the decoupled FACTS Controller loadflow model
[27, 28] and the power injection model (PIM) [29]-[31] are proposed to accommodate
a series VSC controller installed on a transmission line. Other researchers use voltage
source model (VSM) where the series VSCs are modeled directly as series voltage
sources [20], [32]-[37].
In this section, we will review and compare different loadflow models of VSC-
based FACTS Controllers, especially from the aspect of their dispatch calculations
in various operating modes and rated-capacity operation.
2.3.1 Decoupled FACTS Controller Loadflow Model
The decoupled FACTS loadflow model [27, 28] decomposes a UPFC and mod-
els its voltage-controlled bus (from-bus) as a generator and the other bus (to-bus)
as a load, as shown in Figure 2.3. Note that the equivalent load at Bus 2 satisfies
P2 = −Pto (2.1a)
Q2 = −Qto (2.1b)
9
Z4V2Z3
V1V3 V4~ ~ ~ ~
PV bus PQ bus
P1, V1 P2, Q2
Figure 2.3: Decoupled Loadflow Model of the UPFC
where Pto and Qto are the line flow of the series branch of the FACTS controller.
Assuming that the VSC operation is lossless, the active power injected from the
generator satisfies
P1 = P2 (2.2)
The reactive power Q1 injected from the generator is the amount required to keep
voltage V1 at its regulated value.
If the shunt VSC is operating in the reactive power setpoint control mode,
we can model the from bus as a PQ load bus with the equivalent load of P1 and
Q1, instead of a PV bus. However, note that here P1 and Q1 do not equal to the
original shunt injections Psh and Qsh by the shunt VSC, so it is not able to obtain
appropriate setpoint for Q1 directly from the setpoint of Qsh.
A standard loadflow can be carried out with the equivalent PV bus and PQ
bus. After the loadflow has converged, the original control variables need to be
calculated from the set of FACTS controller steady-state nonlinear equations, which
requires an iteration process to solve in order to match the phases of V1 and V2 across
a reactance of the series transformer.
Although the decoupled model is capable of modeling the power flow regulation
modes for the series VSC, it is not applicable to the fixed voltage injection mode,
where the setpoints for P2 and Q2 of the equivalent load are not known or specified
in advance. Moreover, the decoupled model is not applicable to the rated-capacity
operating mode. In this model, the original FACTS control variables are calculated
only after the conventional loadflow converges. As a result, during the loadflow
10
iterations it is impossible to check or enforce the limits of the control variables and
other constraints such as MVA ratings and maximum current magnitudes.
2.3.2 Power Injection Model
The power injection model (PIM) [29] represents a FACTS controller as a
set of active and reactive nodal power injections P′1, Q
′1, P
′2, and Q
′2, with a series
reactance Xt2 connecting the from-bus and to-bus as shown in Figure 2.4. The paper
Z4V2Z3
V1V3 V4jXt2
~ ~ ~ ~
P'1 ,Q
'1 P'
2 ,Q'2
Pu, Qu1 Pu, Qu2
Iu~
Figure 2.4: Power Injection Model of the UPFC
[38] gives the expressions of these power injections of the UPFC as
P′1 = Psh + rV 2
1 sin(γ)/Xt2
Q′1 = Qsh + rV 2
1 cos(γ)/Xt2
(2.3)
P′2 = −rV1V2 sin(θ12 + γ)/Xt2
Q′2 = −rV1V2 cos(θ12 + γ)/Xt2
(2.4)
and the expressions of the active and reactive power supplied by the series VSC as
Pse = −rV1V2 sin(θ12 + γ)/Xt2 + rV 21 sin(γ)/Xt2
Qse = −rV1V2 cos(θ12 + γ)/Xt2 + rV 21 cos(γ)/Xt2 + r2V 2
1 /Xt2
(2.5)
where r is the ratio of the series VSC inserted voltage magnitude to the from-bus
voltage magnitude, γ is the angle difference between the series VSC inserted voltage
angle and the from-bus voltage angle, θ12 = θ1 − θ2 is the difference between the
from-bus and to-bus voltage angles, and Psh and Qsh are the real and reactive power
injection by the shunt VSC, respectively.
11
Assuming that the DC link voltage is held constant and the VSC model is
lossless, the active power circulation is balanced between the shunt VSC and the
series VSC, that is
Psh + Pse = 0 (2.6)
Substituting (2.5) and (2.6) into (2.3), the power injections P′1 and Q
′1 can be ex-
pressed as
P′1 = rV1V2 sin(θ12 + γ)/Xt2
Q′1 = Qsh + rV 2
1 cos(γ)/Xt2
(2.7)
Note that here we denote the power injections as P′1, Q
′1, P
′2, and Q
′2 because these
variables are different from the power injections in (2.1) and (2.2) in the decoupled
model.
The power flowing into the UPFC to-bus can be expressed as
Pto = Pu + P′2
Qto = Qu2 + Q′2
(2.8)
Note that in PIM, the UPFC line current Iu and flows Pu, Qu1, and Qu2 solved from
the loadflow do not equal to those on the actual equipment.
In [29], the reactive power injection Q′1 is set to be either zero or the maximum
value allowed by the shunt converter MVA rating, thus no longer regulating the from-
bus voltage regulation via the shunt VSC. The series VSC control variables Vm2 and
α2 are adjusted manually by trial and error in order to achieve a power flow solution
that matches the targeted requirements, which means searching through the feasible
solution space of the control variables to achieve the UPFC setpoints.
Another approach of implementing the voltage and power setpoint regulation
with the PIM is to use the following iterative algorithm:
1. Solve the loadflow with no power injections, and find the uncompensated power
Pu and Qu2 into the UPFC to-bus, as shown in Figure 2.4.
2. Set P′2 = Pdes−Pu, Q
′2 = Qdes−Qu2, P
′1 = −P
′2, and Q
′1 = K(Vref −V1), where
K is a proportional gain to regulate V1 to its reference Vref .
12
3. Solve the loadflow using the power injections P′1, Q
′1, P
′2, and Q
′2. Check
the setpoint mismatch criteria. If the criteria are not achieved, update the
uncompensated power Pu and Qu2 and go back to step 2.
The drawback of this method is that it introduces a Gauss update loop outside the
loadflow, which could lead to poor overall convergence of the solution.
Z4V2Z3
V1V3 V4jXt2
~ ~ ~ ~
P'1, P'
2 ,Q'2Qsh Qsc
_
Pu, Qu1 Pu, Qu2
Iu~
Figure 2.5: Decoupled Power Injection Model of the UPFC
The PIM is further developed by Xiao et al in [30, 31], where the decomposed
power injection model (DPIM) is proposed. As shown in Figure 2.5, the DPIM
separates the reactive power injection Q′1 in the PIM (2.7) at the from-bus into two
components: Qsh and −∆Qsc, where Qsh is the reactive power injected by the shunt
VSC and −∆Qsc is the difference of Q′1 and Qsh. From (2.7), we have
−∆Qsc = rV 21 cos(γ)/Xt2 (2.9)
Another variation of the PIM is to transform the power injections at the UPFC
from-bus and to-bus into equivalent shunt admittances [39] (see Figure 2.6), resulting
in a π section representation of the UPFC. The admittance Ysh, Y1, and Y2 are
functions of the corresponding power injections and bus voltages. A conventional
loadflow is then carried out with the pre-specified equivalent power injections.
In summary, to bypass the direct modeling of the voltage injected by a series
VSC, the PIM adds additional power injections at the FACTS controller from-bus
and to-bus in the conventional loadflow. An external iteration loop is necessary to
enforce the setpoint regulation, which could lead to poor convergence.
13
Z4V2Z3
V1V3 V4jXt2
~ ~ ~ ~
Ysh Y1 Y2
Pu, Qu1 Pu, Qu2
Pu,~
Figure 2.6: Shunt Admittance Model of the UPFC
For the same reasons of the decoupled model, the PIM is not applicable to
model rated-capacity mode for FACTS controllers.
2.3.3 Voltage Source Model
The voltage source model (VSM) [20], [32]-[37] represents the shunt and series
VSCs directly as shunt and series voltage injections, respectively. The VSMs of the
FACTS controllers can be readily incorporated into the Newton-Raphson loadflow
algorithm, with the FACTS control variables as part of the expanded state variables.
The mismatch equations are expanded to include the FACTS Controller setpoint
equations, such that the Jacobian matrix is also expanded.
With the Newton-Raphson solution technique, the VSM is capable of model-
ing various operating modes by substituting appropriate mismatch equations and by
modifying the corresponding Jacobian matrix. Depending on the operation mode,
the FACTS Controller mismatch equations can be either the voltage and power
setpoint mismatches or the fixed control variable setpoint mismatches, or the con-
troller’s active constraints. Furthermore, the FACTS control variables and line
currents and flows on the actual equipment are readily available at each iteration,
which allows the limit constraints of the FACTS Controllers to be checked during
each loadflow iteration. If a limit is violated, the device is fixed at that limit and
one or more regulated setpoints are no longer enforced.
14
The details of the VSM loadflow technique and the dispatch strategies for
FACTS controllers operating at various operating modes and rated capacity will be
described in Chapter 3.
2.3.4 Summary and Conclusions
Although the decoupled model and the power injection model are capable
of modeling the voltage and power flow regulation mode for a FACTS controller,
they both require additional computation effort in the regulation mode, and are not
applicable to the rated-capacity operation mode.
On the other hand, the voltage source model is intuitive and efficient. By
directly modeling the shunt and series VSCs, the VSM is capable of modeling
VSC-based FACTS controllers with any combination of operating modes of the
coupled shunt and series VSCs [40]. The Newton-Raphson algorithm shows good
convergence properties [20, 33, 34] by simultaneously adjusting all voltage variables.
At each loadflow iteration, the limit constraints of the FACTS controllers can be
checked. If a limit is violated, it is fixed at that limit and one or more regulated
setpoints will no longer be enforced. A drawback of the VSC model is that it re-
quires substantial effort to implement in a legacy loadflow programs, because of the
need to expand the solution vector and the Jacobian. However, the implementation
of VSM in Power System Toolbox (PST) is relatively straightforward.
Most important, the VSM represents a common modeling framework, where
the VSC variables from the loadflow solution can be used to directly initialize elec-
tromagnetic transient programs and dynamic simulation programs. In particular,
the sensitivity analysis of the FACTS control variables using network equations
consisting of shunt and series voltage injections can be readily obtained in such a
framework [20].
2.4 Overview of VSC-Based FACTS Controllers in Time-
Domain Simulation
Time-domain simulations are predominant in studying transient stability en-
hancement using FACTS controllers. There are two possible ways to carry out the
15
time-domain studies.
The first one is the three-phase electromagnetic transient simulation approach,
in which electric power systems including FACTS devices have to be modeled in
detail, using some standard software, such as EMTP [41] and PSCAD/EMTDC
[42]. In these studies, the results should represent the time functions of physical
quantities with the fast transients. The system voltages and currents are represented
as sinusoidal functions, which requires a considerable amount of computational time
because of the comprehensive modeling and the short integration step size. As the
electro-mechanical response of a power system is relatively slower, such an approach
for studying transient stability is too computationally intensive.
The second one is the positive-sequence approach, in which an electric power
system is modeled as a balanced three-phase system. Since sinusoidal quantities are
not dealt with, the integration step size may be larger, and the modeling is simpler,
such that the simulation procedure is much faster than in the electromagnetic tran-
sient approach. With proper modeling, the results for transient stability should be
very close to those achieved in the electromagnetic transient approach. For these
reasons this approach is chosen as a basis for our investigation.
In this section, we will review and compare different models of VSC-based
FACTS controllers suitable for positive-sequence time-domain simulation.
A. The Instantaneous Control Model, without DC Link Capacitor Dynamics Involved
The instantaneous control model [29, 43, 44] determines the controllable vari-
ables of FACTS controllers instantaneously by solving algebraic equations in each
integration step. Those algebraic equations are usually solved by using optimization
techniques to the preliminary static models, which assume that the DC link voltage
maintains constant and the VSC models are lossless. Thus the DC link capacitor
dynamics is not involved in this type of model.
The application of the instantaneous control model is limited to those spec-
ified open-loop control strategies which drive the FACTS controllers to operate at
the rated capacity and thus the controllable variables of FACTS controllers are in-
stantaneously available. A new model, which is capable of modeling the closed-loop
setpoint regulation controls of the FACTS controllers in different operating modes,
16
is required.
B. The Regulator Model, without DC Link Capacitor Dynamics Involved
The regulator model [45]-[48] uses regulators to model the controls with feed-
back to determine the controllable variables of FACTS controllers. In this type
of model, the VSC controllers are usually modeled as voltage sources or current
sources. The independent variables of the voltage sources or current sources are
controlled by the regulators. These regulators can be represented as nonlinear dif-
ferential equations and then incorporated into the conventional positive-sequence
time-domain dynamic simulation program.
In [45]-[48], the DC link capacitor voltage is assumed to maintain a constant,
and thus the active power circulation equals zero. This active power balance equa-
tion is used to determine the dependent variables of the FACTS controllers. How-
ever, during transient stability studies, the DC link capacitor of FACTS controllers
will exchange energy with the system and consequently its voltage varies. Thus for
transient stability studies the active power balance equation would not apply.
C. The Regulator Model, with DC Link Capacitor Dynamics Involved
References [49, 50] include the DC link capacitor dynamics in the regulator
model. The DC link dynamics is expressed as a differential equation associated
with the DC link capacitor voltage. The constant DC link capacitor control for the
UPFC is regulated by controlling the firing angle of the shunt VSC.
[49] and [50] considered only the voltage and power flow regulation modes
for the UPFC. However, there are other control modes that are more appropriate,
such as the fixed reactive power setpoint control mode and the reactive power re-
serve mode for a shunt converter and the fixed injected voltage control for a series
converter.
D. A Comprehensive Set of Regulator Models
A comprehensive set of regulator models of FACTS controllers, which include
the DC link capacitor dynamics and take into account various operating modes, are
proposed in this thesis work. In the modeling, shunt VSC controllers and series VSC
controllers are modeled as controllable voltage sources with equivalent transformer
reactance.
17
To complete this model, a control mode implementation is applied. The shunt
VSC controls and the series VSC controls are modeled as separate regulators. When
a VSC changes its operating mode, only the input signals of the corresponding
regulator need to be adjusted. With this implementation, we only need to select
and combine the logics of the shunt VSC, the series VSC and the DC link coupling
to form the specified FACTS controller and to operate it in the desired operating
mode.
2.5 Summary and Conclusions
This chapter first gives a basic description of FACTS controllers their operating
modes and then reviews various modeling methodologies of VSC-based FACTS Con-
trollers, for both loadflow and dynamic simulations. This comparison between dif-
ferent FACTS Controller models is important. Based on the discussion, we propose
to use the voltage source model (VSM) for stead-state dispatch and a comprehensive
set of regulator models for transient stability analysis. Details of the voltage source
model and the comprehensive set of regulator models will be presented in Chapter
3 and Chapter 4, respectively.
CHAPTER 3
FACTS CONTROLLER STEADY-STATE DISPATCH
Because of their flexible performance, converter-based transmission controllers such
as the UPFC can be very effective in improving power transfer capability over con-
gested transmission paths [1]. In most published literature, a shunt converter such
as a STATCOM is set to control the bus voltage and a series converter such as an
SSSC is to control the line power flow. In practical operations, however, there are
other control modes that are more appropriate, such as fixed reactive power output
for a shunt converter and fixed injected voltage for a series converter. The CSC
installed at the NYPA’s Marcy substation [26], [25] can operate in 11 configurations
of different shunt and series connections. When the converters in each configuration
are allowed to operate in multiple modes, the CSC is capable of operating in a total
of 49 different control modes. It is thus important that a dispatch tool can allow for
all the operating modes and be able to compute the operating conditions efficiently.
To reduce the complexity associated with the many dispatch modes, it is pro-
posed to separate the shunt VSC and the series VSC models. The separation of
models can readily accommodate all VSC configurations, including a GUPFC [18],
[25] which contains more than two VSCs coupled to a common DC bus. In this
approach, the unknown variables of the loadflow solution are always kept the same,
independent of the VSC controller operating mode. In this way, when a VSC con-
troller changes mode, only two equations for each shunt VSC and two equations for
each series VSC need to be adjusted.
The injected shunt and series voltage sources are used to model voltage-sourced
converters (VSC) [34], [20], based on which a Newton-Raphson loadflow solution can
be readily developed. This use of injected voltage sources is consistent with detailed
simulation of a VSC in the electromagnetic transient program (EMTP) [27]. An
advantage of directly including injected voltage sources in the loadflow model is that
the loadflow solution can iterate on the bus voltages and injected voltage sources
simultaneously, without needing an outer loop of adjusting the equivalent injected
18
19
bus power or current. It eliminates the potential hunting of the solution in case of
small VSC series transformer reactance and multiple UPFCs and IPFCs. Another
advantage is that the line current and power are readily computed, allowing direct
enforcement of equipment limits [51]. A further advantage is that if the loadflow
uses injected voltage source models, then all subsequent analysis such as sensitivity
computation, control design, and dynamic simulation, can also make use of the same
modeling framework and directly work with the injected voltage sources.
In this chapter, we will focus on the loadflow formulation for various regulation
modes of FACTS Controllers. The modeling details and setpoint regulation loadflow
equations for shunt and series VSCs are summarized in Sections 3.1 and 3.2. The
Newton-Raphson solution technique is described in Section 3.3. Rated-capacity
dispatch when a VSC reaches a limit is discussed in Section 3.4. Application results
are given in Section 3.5.
3.1 VSC Model
Each VSC in Figure 2.1 can be modeled as an injected voltage source with
transformer reactance. In the shunt configuration (Figure 3.1), Vm1 = Vm1ejα1 is
the complex injection voltage, Vi = Viejθi , i = 1, 2, 3, are the complex bus voltages.
The reactance Xt1 is short-circuit reactance of the shunt transformer. From Figure
3.1, the injected current Ish from the shunt VSC into the system is
Ish =Vm1 − V1
jXt1
(3.1)
such that the injected power Ssh is
Ssh = V1I∗sh = Psh + jQsh (3.2)
In the series configuration (Figure 3.2), Vm2 = Vm2ejα2 is the complex injection
voltage, and Vi = Viejθi , i = 1, ..., 4, are the complex bus voltages. The reactance
Xt2 is the winding reactance on the high-side of the series transformer, which is
20
jXt1
Ish
+_ Vm1
~
Z1 Z2V1V2 V3
~ ~ ~
Ssh,~ ~
Figure 3.1: Injected Voltage-Sourced Model of a Shunt VSC
Z4V2Z3
V1V3 V4jXt2 + _Vm2~
Ise
~ ~ ~ ~
From-bus To-busSse,
S2~
~ ~
Figure 3.2: Injected Voltage-Sourced Model of a Series VSC
typically very small. From Figure 3.2, the line current Ise is given by
Ise =V1 − (Vm2 + V2)
jXt2
(3.3)
such that the power injected by the series VSC is
Sse = (V2 − V1)I∗se = Pse + jQse (3.4)
and the power injected into the to-bus (Bus 2) is
S2 = V2I∗se = Pto + jQto (3.5)
Note that the model of the capacitor on the DC link of the VSC is not included
21
in (3.1) through (3.5). In coupled VSC operations, the DC bus would allow the
coupled VSCs to circulate active power. Additional constraints need to be placed
on (3.1) to (3.5) to represent isolated and coupled VSC operations.
3.2 Dispatch Computation
In this section, we show the use of the injected voltage model for VSCs for
dealing with the different modes of dispatch. In particular, the equations for the
shunt and series converters are separately modeled. For example, for a UPFC to
perform from-bus voltage control with droop and line P ,Q flow control, we select
from the list of options for the shunt VSC to be in the (Sh1) mode and for the
series VSC to be in the (SeC1) mode, with the second equation of the shunt VSC
to facilitate the necessary active power circulation on the coupled DC bus. It is no
longer necessary to provide a model specific to this operating mode.
We provide the equations for formulating the different shunt and series VSC
dispatch modes in Subsections 3.2.1 to 3.2.3.
3.2.1 Shunt VSC Operating Modes
The possible operating modes of a shunt VSC include:
(Sh1) Control the shunt bus voltage to a desired value with the droop α, that is,
V1 = Vref − αIshq (3.6)
where Ishq is the reactive current injected by the shunt VSC. The droop func-
tion can be turned off by setting α = 0.
(Sh2) Control the reactive current of the shunt VSC to a desired value Ishqref ,
Ishq = Ishqref (3.7)
(Sh3) Operate in the Var reserve mode which is the operating mode (Sh1) with
the reactive current of the shunt VSC limited to [ICshqres, ILshqres], that is, if
ICshqres ≤ Ishq ≤ ILshqres, then (3.6) is applicable. Otherwise,
22
Ishq =
ILshqres if Ishq > ILshqres
ICshqres if Ishq < ICshqres
(3.8)
If the shunt VSC is operated standalone, the circulating active power is
Pcirc = 0 (3.9)
which can be equivalently represented by
θ1 − α1 = 0 (3.10)
If the DC bus of the shunt VSC is integrated with an energy supply system, such
as a battery park, then the circulating active power Pcirc, instead of (3.9), is set to
the active power output PES of the energy supply device
Pcirc = PES (3.11)
If the shunt VSC is coupled to other VSCs, then the circulating power Pcirc is equal
to the active power collectively absorbed or generated by the other coupled VSCs.
When a shunt VSC is coupled with a series VSC, such as in a UPFC, the circulating
power is
Pcirc = −Psh = Pse (3.12)
3.2.2 Series VSC Operating Modes
The possible operating modes of a series VSC are separately described under
standalone and coupled operations.
3.2.2.1 Standalone or “Slave” Operation
(Se1) Control the line active power flow Pto to a desired value Pdes,
Pto =V2(Vm2 sin(θ2 − α2) − V1 sin(θ2 − θ1))
Xt2
= Pdes (3.13)
(Se2) Fix the injected voltage magnitude, in either quadrature leading or lagging
23
with respect to the transmission line current
Vm2 = Vmdes (3.14)
For the standalone operation of a series VSC, we also need to set the series
VSC to operate with zero circulating power
Pcirc = 0 (3.15)
which can be expressed as
V1 sin(θ1 − α2) − V2 sin(θ2 − α2) = 0 (3.16)
If the DC bus of the series VSC is integrated with an energy supply system, then
the active power injected into the line will become
Pcirc = PES (3.17)
In cases when the series VSC is operated as the “Slave” VSC in an IPFC,
(3.15) becomes
Pcirc = Pse1 = −Pse2 (3.18)
where Pse1 and Pse2 are the injected active power from VSC 1 and VSC 2, respec-
tively.
3.2.2.2 Coupled Operation
When a series VSC is coupled to another VSC or an energy supply system,
such as the “Master” VSC in a UPFC, it has an additional degree of freedom so
that two variables can be regulated.
(SeC1) Control the line active and reactive power flow Pto and Qto to the desired
values Pdes and Qdes, respectively,
Pto =V2(Vm2 sin(θ2 − α2) − V1 sin(θ2 − θ1))
Xt2
= Pdes (3.19)
24
Qto =−V2(V2 − V1 cos(θ2 − θ1) + Vm2 cos(θ2 − α2))
Xt2
= Qdes (3.20)
(SeC2) Fix the d-axis and q-axis of the injected voltage at Vd and Vq with respect to
the from-bus voltage vector V1
Vm2 =√
V 2d + V 2
q , φ = tan−1(Vq
Vd
) (3.21)
where φ is the angle between the injected voltage vector and the from-bus
voltage vector.
(SeC3) Fix the magnitude of the q-axis injected voltage at Vq, determined with respect
to the from-bus voltage vector V1. Also, satisfy the real power circulation
balance between two VSCs, as shown in (3.18).
.
3.3 Newton-Raphson Algorithm
In an N -bus power network with Ng generators and without any VSCs, the
loadflow equations can be formulated as N − 1 equations for the active power bus
injections/loads P and N − Ng equations of reactive power bus injections/loads Q
fP (v) = P
fQ(v) = Q(3.22)
where
v = [V T θT ]T = [V1 V2 · · · VN θ1 θ2 · · · θN ]T (3.23)
is a 2N − Ng − 1 vector variable of bus voltage magnitudes and angles, with Ng
generator bus voltages removed and the angle of the swing bus set to 0◦. In the
Newton-Raphson algorithm, the Jacobian matrix J
J =
∂fP /∂V ∂fP /∂θ
∂fQ/∂V ∂fQ/∂θ
(3.24)
25
is used to iteratively find the solution of (3.22).
If an N -bus power network also includes M VSCs, then the loadflow equation
(3.22) will expand by 2M equations, resulting in
fP (v) = P
fQ(v) = Q
fVSC(v) = R
(3.25)
where
v = [V T θT V Tm αT ]T (3.26)
= [V1 · · · VN θ1 · · · θN Vm1 · · · VmM α1 · · · αM ]T
(3.27)
is a 2(N +M)−Ng−1 vector variable of bus voltage magnitudes and angles, and the
last (third) equation in (3.25) is determined by the VSC operating modes, where R is
a vector of the VSC-based controller setpoints or reference values. In our approach,
the P and Q (the first and second) equations in (3.25) will remain unchanged for
all operating modes. The VSC equations (3.6) to (3.21) constitute the formulation
of the third equation in (3.25).
To apply the Newton-Raphson algorithm to the augmented system (3.25), the
Jacobian matrix J becomes
J =
∂fP /∂V ∂fP /∂θ ∂fP /∂Vm ∂fP /∂α
∂fQ/∂V ∂fQ/∂θ ∂fQ/∂Vm ∂fQ/∂α
∂fVSC/∂V ∂fVSC/∂θ ∂fVSC/∂Vm ∂fVSC/∂α
(3.28)
Note that the first 2 × 2 blocks of J (namely, the (1,1), (1,2), (2,1), and (2,2)
entries) are identical to the Jacobian J in (3.24), except for the additional terms
due to the shunt and series VSC transformer reactance and injection terms. Thus
an attractive feature of this Newton-Raphson algorithm for solving loadflow is that
the formulation of the third equation of (3.25) can be readily built into an existing
conventional Newton-Raphson algorithm. For large data sets, sparse factorization
26
techniques can be used to achieve an efficient solution.
For a FACTS Controller changing to a different control mode, only the equa-
tions constituting the third equation of (3.25) need to be changed, and consequently,
only the last row of the Jacobian matrix J (3.28) need to be changed. Although it
is possible in the case of the shunt VSC controlling the bus voltage to eliminate the
load bus voltage magnitude as an unknown variable, doing so would require changes
to all the loadflow equations and its Jacobian matrix, and thus require extensive
coding. Our approach of changing only the third equation of (3.25) is an efficient
way to handle the various operating modes of a FACTS Controller.
3.4 Rated-Capacity Dispatch
3.4.1 Operating Limits
A number of operating limits are imposed on both the shunt and series VSCs:
designed physical limitations, overload protection, as well as the limitations of bus
voltages [20],[51],[52],[53]. These operating limits need to be considered in the load-
flow solution process when the operating limits of VSCs are exceeded. This is
important in assessing the impact of the VSCs on maximum transfer capability.
For a shunt VSC, the operating limits are listed as follows, where the subscripts
max and min denote maximum and minimum, respectively.
1. Shunt VSC current limit: |Ish| ≤ Ishmax
2. Shunt VSC injected voltage magnitude limit: Vm1 ≤ Vm1max
3. Shunt VSC MVA rating: |Ssh| ≤ Sshmax
4. If the shunt VSC is coupled with an energy supply system or another VSC,
the active power transfer Psh is bound by |Psh| ≤ Pcirc max.
For a series VSC, the operating limits include:
1. Series VSC line current limit: |Ise| ≤ Isemax
2. Series VSC injected voltage magnitude limit: Vm2 ≤ Vm2max
3. Series VSC MVA rating: |Sse| ≤ Ssemax
27
4. The voltage magnitude limits at adjacent buses: Vmin ≤ |V1|, |V2| ≤ Vmax
5. If the series VSC is coupled to an energy supply system or another VSC, the
active power transfer Pse is bound by |Pse| ≤ Pcirc max.
If any of these limits are violated, the voltage or flow setpoints of a VSC can no
longer be enforced. In such cases, the dispatch strategies to achieve a rated-capacity
loadflow solution have been derived.
3.4.2 Dispatch Strategies
Rated-capacity operation is important for VSCs because they are often dis-
patched to their limits to achieve maximum benefit. Most likely, a shunt VSC is
loaded to its current limit and MVA rating, and a series VSC is loaded to its in-
serted voltage limit. As discussed below, standalone VSC rated-capacity operation
strategies are relatively straightforward but coupled VSC operation strategies can
be quite complex.
3.4.2.1 Standalone Operation
i. Shunt VSC
• Reactive current control – Reset the reference current values if they are outside
the limits:
– If Ishref > Ishmax, enforce Ishref = Ishmax;
– If Ishref < Ishmin, enforce Ishref = Ishmin.
• Voltage control – If in reaching the reference voltage values, a current limit is
violated, then switch to the current control mode in the following manner:
– If Ish > Ishmax, enforce Ishref = Ishmax;
– If Ish < Ishmin, enforce Ishref = Ishmin.
ii. Series VSC
28
• Line active power control – If in controlling the desired line power flow ref-
erence results in Vm > Vmmax, switch to fixed injected voltage control and
enforce Vmref = Vmmax.
• Fixed injected voltage control: If Vmref > Vmmax, enforce Vmref = Vmmax.
3.4.2.2 Coupled Operating Mode
When the DC buses of N VSCs are coupled, N − 1 VSCs (called the master
VSCs) will be able to control two variables, and the Nth VSC (called the slave VSC)
will control one variable and provide the appropriate power circulation. The general
strategy in rated-capacity operation for the master VSC is to maintain its active
power flow (real power priority strategy [54]) and for the slave VSC is to provide
the active power circulation dictated by the master VSC.
i. The Master VSC
• Line active and reactive power (P ,Q) control mode – If enforcing the desired
line P and Q values results in the series VSC insertion voltage magnitude
greater than its limit, adjustments need to be applied on the control strategy
– the line active power will be either enforced to its desired value or maximized
while the reactive power will deviate from the desired value. To accommodate
the modification, the control mode is switched to the Vd,Vq setpoint control
mode. Because Vq affects strongly the line active power and the Vd affects
the line reactive power, in the real power priority strategy Vq should be either
kept constant or maximized and Vd is modified to satisfy the voltage magnitude
constraints, as described below:
– If series√
V 2d + V 2
q > Vmmax and Vq ≥ Vmmax, enforce Vqref = Vmmax and
Vdref = 0 (Figure 3.3(a)).
– If series√
V 2d + V 2
q > Vmmax and Vq < Vmmax, enforce Vqref = Vq and
Vdref =√
V 2mmax − V 2
q (Figure 3.3(b)).
• Vd,Vq setpoint control – If√
V 2dref + V 2
qref > Vmmax, then scale them back pro-
29
portionally (Figure 3.3(c)):
V newdref =
Vmmax
V 2dref + V 2
qref
Vdref , V newqref =
Vmmax
V 2dref + V 2
qref
Vqref (3.29)
D
Q
dV
qV refqV
mV
D
Q
mV
refqV
dV
refdVmaxmV
(a) (b)
refmV
refqV
refdV
maxmV
newrefqV
newrefdV
D
Q
(c)
Figure 3.3: Injected Series Voltage Modification in the Master VSC
ii. The Slave VSC
The slave VSC needs to support the DC bus active power circulation to the
master VSC. So in rated-capacity operation, we need to consider whether the slave
VSC can still support the active power to the master VSC or VSCs. Thus the
control strategy has two possibilities.
30
• The slave VSC can provide the required active power circulation – In this case,
we can modify the setpoint of the slave VSC to the limit and satisfy the active
power balance between the two VSCs at the same time.
– Series VSC: switch to Vm control mode and enforce
Vm = Vmmax, Pse = −Pcirc (3.30)
– Shunt VSC: switch to reactive current control and enforce
Ishqref =Sshmax√P 2
sh + Q2sh
Ishq, Psh = −Pcirc (3.31)
• The slave VSC cannot provide the required active power circulation – This
case often occurs when the master VSC is required to significantly impact the
line flow, which requires more active power circulation that can be provided by
the slave VSC. In such cases, the master VSC has to scale back its setpoints to
achieve active power balance in the coupled link, which means that it cannot
both regulate the line active power and reactive power [55]. Based on the real
power priority rule, we release the reactive power control for the master line,
so that
Vqref = Vq, Pse = Pcirc (3.32)
as shown in Figure 3.4(a). Simultaneously, the dispatch of the slave VSC
should also be modified to provide the maximum active power circulation to
the master VSC.
– IPFC configuration (Figure 3.4(b)): The slave VSC is switched to the
(SeC3) control mode to satisfy
Vm = Vmmax, Pse2 = Pse2max (3.33)
– UPFC configuration: The shunt VSC is switched to reactive current con-
trol to enforce
Ishqref = 0, Psh = Pshmax (3.34)
31
D
Q
LineCurrent
Solutionwith Limit
Solutionwithout Limit
D
QLine
Current
Solutionwith Limit
Solutionwithout Limit
(a) Master Line (b) Slave Line
Figure 3.4: Injected Series Voltage Modification When the Slave VSCCannot Support Enough Real Power
3.5 Applications
The control mode implementation for the steady-state dispatch of FACTS
controllers is applied to a 4-bus test system and a 1673-bus test system used in the
Operator Training Simulator (OTS) [40]. Rated-capacity dispatch is enforced when
a VSC reaches its limit.
3.5.1 Voltage Stability Improvement by the SSSC
Consider the four-bus radial system in Figure 3.5, where a 100 MVA series
VSC is located between Buses 2 and 4 to enhance the power transfer capability
from the generator on Bus 1 to the load on Bus 3. The maximum series line current
and injected voltage of the series VSC are 10 pu and 0.1 pu on the system base of 100
MVA, respectively. The maximum and minimum voltage magnitudes at adjacent
buses are 1.5 pu and 0.5 pu, respectively. The maximum power transfer between
the converter and the energy storage system is 10 MW. The series VSC transformer
leakage reactance is Xt = 0.002 pu. The line parameters, given on the system base,
are listeded in Table 3.1.
32
4
2 31
generator
B
Sse
EnergyStorage
S1
S2
L3
Figure 3.5: 4-Bus System with a Series VSC
Table 3.1: Transmission Line Data of the 4-Bus Radial Test System
Line Resistance Reactance Charging(pu) (pu) (pu)
1-2 0.00163 0.03877 0.788002-3 0 0.08154 0.394003-4 0 0.07954 0.39400
Note that by closing the Switch B the SSSC is bypassed, which is referred to
as the uncompensated system (base case). The SSSC is in service if Switch B is
open. By also closing the Switches S1 and S2, the SSSC is integrated with an energy
storage system.
Because the VSC is a reactive power source, the objective is to show the impact
of the VSC on the system voltage stability as the power transfer on the transmission
lines is increased. In a base case, Switch B is closed so that the VSC is not deployed.
Figure 3.6 shows the variation of the voltage V3 at Bus 3 when the load L3 on Bus
3 is increased from the base value of 400 MW. In particular, V3 drops to 0.95 pu
when L3 reaches about 565 MW. Next, Switch B is opened with Switches S1 and
S2 open and the SSSC is inserted and is set to carry 62% of the load L3. As shown
in Figure 3.6, V3 drops to 0.95 pu when L3 reaches about 615 MW, showing a 50
MW increase in the power transfer. Figure 3.7 shows the magnitude of the injected
33
400 450 500 550 600 650 700 7500.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
Pload
(MW)
V3 (
pu)
No FACTSSSSCSSSC ES Pc = −10 MWSSSC ES Pc = 0 MWSSSC ES Pc = 10 MW
Figure 3.6: PV Characteristics of the SSSC with and without EnergyStorage
voltage as a function of the power dispatch. Note that the injected voltage Vm
reaches the maximum value of 0.1 pu (on the system base) at 510 MW and remains
at the maximum value for higher values of power transfer.
Then Swtiches S1 and S2 are closed to form a SSSC integrated with the energy
storage system. The load L3 on Bus 3 is again increased from the base load of 400
MW, with 62% carried on the series VSC. The dispatch results are shown in Figures
3.6. Without any power circulation, that is, the active power flowing out of the
energy storage system into the series VSC is zero, the dispatching is exactly the
same with the SSSC without the energy storage system. By circulating 10 MW
from the energy storage system to the series VSC, the power transfer is improved
by another 30 MW to 645 MW at V3 = 0.95 pu. On the other hand, if active
power circulates from the series VSC to recharge the energy storage system, the
power transfer is decreased. The magnitude of the injected series voltage source
is shown in Figure 3.7, depicting the saturation of the inserted voltage magnitude.
34
400 450 500 550 600 650 700 7500.075
0.08
0.085
0.09
0.095
0.1
0.105
Pload
(MW)
Vm
(pu
)
SSSCSSSC ES Pc = −10 MWSSSC ES Pc = 0 MWSSSC ES Pc = 10 MW
Figure 3.7: Series VSC Injected Voltage
There is one corner point at each curve. For example, the corner point of the curve
with squares (SSSC ES Pc = 10 MW) is at a load of 490 MW, which means that
the SSSC switches from line active power regulation mode to fixed series injected
voltage source magnitude mode at that point.
3.5.2 Operator Training Simulator (OTS) for NYPA’s CSC
To illustrate the potential modes of operation of a reconfigurable VSC-based
controller, we consider the Convertible Static Compensator (CSC) installed at the
NYPA’s Marcy 345 kV substation, which has been fully operational since June 2004
[25]. The station connection scheme is shown in Figure 3.8.
The CSC, consisting of two 100-MVA voltage-sourced converters, enables volt-
age control at the Marcy bus as well as power flow control on two 345 kV lines (the
Marcy-New Scotland (UNS) line and the Marcy-Coopers Corners (UCC) line) ex-
iting the Marcy substation. There is one shunt step-down transformer with two
secondary windings and two series transformers. The controller can be connected
35
INV1100 MVA
INV2100 MVA
DC Bus 1 DC Bus 2
Marcy 345 kV
TR-SH200MVA
DC-SW
LVB
HSB
HSB
LVB
TR-SE2100 MVA
TR-SE1100 MVA
TBS2TBS1
NewScotland(UNS)
CoopersCorners(UCC)
Figure 3.8: The CSC Connection Scheme
as a shunt controller (STATCOM) or as a series controller, inserting controllable
voltages in series in the two 345 kV lines (SSSC), or can function as a combination
of shunt and series controllers. In addition to STATCOM and SSSC, the CSC can
operate as a UPFC, or an IPFC.
The CSC is capable of operating in 11 different configurations, as shown in
Table 3.2 [25]. For each configuration, such as “STATCOM100-1”, there can be sev-
eral modes of operation, such as voltage control with droop or Var reference control.
Table 3.2 lists the number of possible modes of operations for each configuration,
with a total of 49. It becomes obvious that for studying the dispatch of the CSC, one
cannot treat each of the operating modes as a special case and provide customized
code for it. Instead, each operating mode should be considered as a combination of
appropriate shunt and series modes, as discussed in Section 3.1.
A CSC operation dispatch tool is being developed with this approach as the
36
Table 3.2: Operating Modes of a Reconfigurable VSC-Based FACTSController
Configuration VSC 1 VSC 2 Total Numbermodes modes of modes
1. STATCOM100-1 3 0 32. STATCOM100-2 0 3 33. STATCOM200 3 3 34. SSSC100-UCC 2 0 25. SSSC100-UNS 0 2 26. SSSC100-UCC 2 2 4
SSSC100-UNS7. STATCOM100-1 3 2 6
SSSC100-UNS8. SSSC100-UCC 2 3 6
STATCOM100-29. UPFC100/100-UNS 3 2 610. UPFC100/100-UCC 2 3 611. IPFC100-UCC/100- 2 (Master) 2 (Slave) 4
UNS 2 (Slave) 2 (Master) 4
basis of a CSC Operator Training Simulator. The Training Simulator will allow
an operator to adjust the VSC-based controller using the manufacturer’s control
screens (see Figure 3.9 for a screen shot of Configuration 9) and see the impact of
the controller reflected on the station one-line diagrams. Such a training simulator,
which can be used to dispatch the CSC in different configurations and in different
modes, will provide NYPA system operators with an off-line tool to gain experience
in operating the CSC, which cannot be adjusted for training when it is in operation.
We now use the UPFC in Figure 3.9 to illustrate the versatility of this tool
to study power dispatch with the computation results shown in Table 3.3. The
simulator uses a 1673-bus power system, with VSC 1 in the shunt connection and
VSC 2 in the series connection on the UNS line. To start, the system operation is
solved without the UPFC. The Marcy bus voltage and the flows on the Marcy to
New Scotland (UNS) line are shown in the first row of Table 3.3. Then the UPFC
is inserted with the shunt voltage setpoint at the pre-insertion Marcy voltage and
with the series VSC voltage set to zero. The second row of Table 3.3 shows that the
UNS line power flow drops slightly because of the effect of the series transformer
37
Figure 3.9: UPFC Control Screen
reactance. Next the line flow on the UNS line is dispatched to P = 638 MW and
Q = −1 MVar, with the shunt bus voltage set to Vref = 1.0327 pu. The dispatch
result is shown in the third row of Table 3.3. From the P ,Q dispatch, we use the
series insertion quadrature voltages Vd and Vq as the setpoint to switch to the fixed
voltage insertion mode for the series VSC, as shown in the fourth row of Table 3.3.
In doing so, we expect the fixed inserted voltage mode dispatch to be identical to
the P ,Q mode dispatch. Then the ENS line, which is parallel to the UNS line and
carrying about 500 MW, is tripped. With the fixed inserted voltage dispatch, the
UNS line is able to carry an additional 180 MW. On the other hand, if the series
VSC is in the P ,Q mode, the rest of the network, in particular, the lower kV system,
needs to transport this 180 MW, which can cause low voltages in some portions of
the network.
38
Table 3.3: Dispatch Computation of an Operator Training Scenerio
Config. Setpoints Loadflow ResultsShunt Series Shunt Series Line Line BusVSC VSC VSC VSC UNS UCC Marcy
Vf (pu) P (pu) Psh(pu) Pse(pu) P (pu) P (pu) V (pu)α Q(pu) Qsh(pu) Qse(pu) Q(pu) Q(pu)
Qsh(pu) Vd(pu) Vinj(pu)Qres(pu) Vq(pu)
No 0 0 5.548 3.60 1.0327CSC - - 0 0 0.035 −0.539
0UPFC Vf : 1.0327 Vd : 0.0 0.0 0.0 5.577 3.585 1.0327Vd,Vq α: 0.03 Vq : 0.0 0.001 −0.01 −0.011 −0.544Mode 0.033UPFC Vf : 1.0327 P :6.38 0.055 −0.055 6.38 3.48 1.0336P,Q α: 0.03 Q : −0.01 −0.03 0.278 −0.01 −0.551
Mode 0.824UPFC Vf : 1.0327 Vd : −0.197 0.055 −0.055 6.38 3.48 1.0336Vd,Vq α: 0.03 Vq : 0.839 −0.03 0.278 −0.01 −0.551Mode 0.824UPFC Vf : 1.0327 Vd : −0.197 0.089 −0.089 8.183 4.073 1.0297
Vd,Vq & α: 0.03 Vq : 0.839 0.1 0.35 0.394 −0.046Trip ENS 0.814
3.5.3 Maximum Dispatchbility for the UPFC and IPFC
The steady-state dispatch at rated capacity of shunt and series VSCs is il-
lustrated using the CSC OTS described in Subsection 3.5.3. In the following, the
UPFC and IPFC configurations are being investigated.
The parameters of the VSCs are specified as follows. The maximum shunt and
series injected voltages are 1.5 pu and 0.056 pu on 100 MVA system base, respec-
tively. The maximum shunt injected current and series line current are 1.0 pu and
18.1 pu, respectively. The maximum and minimum voltage magnitudes at the adja-
cent buses are 1.5 pu and 0.5 pu. The maximum active power circulation between
the converters is 50 MW. Note that in the following discussions the maximum series
injected voltage is scaled to 1.0 pu on the converter base.
39
3.5.3.1 Maximum UPFC Dispatchability
The UPFC configuration consists of the first VSC in the shunt configuration
and the second VSC in the series configuration. To study the maximum dispatcha-
bility of the UPFC, the series VSC is loaded to its maximum voltage insertion. As
shown in Figure 3.10(a), 12 values of Vd,Vq uniformly spaced on the unit circle are
used. For each set of Vd,Vq, the shunt VSC is set at three different Mvar reference
settings, namely, 50% capacitive (0.5 pu), neutral (0 pu), and 50% inductive (0.5
pu).
D
Q 1
23 4 5
6
7
8
91011
12
1.0
1.0
D
Q 12
34
5 6 7891011
12
1.0
1.0
Cases
0.50.5
13141516
1718
Case 1,2,43
(a) UPFC (b) IPFC Cases
Figure 3.10: Injected Series Voltage Reference
The resulting incremental P ,Q flows, denoted by ∆P and ∆Q with respect to
the uncompensated base case, on the series compensated line are shown in Figure
3.11, where the points correspond to those in Figure 3.10(a). Note that dispatch
traces are elliptically shaped ∆P -∆Q curves, which are not strongly dependent on
the shunt VSC reactive power injections. The regions contained in the ∆P -∆Q
curves are the feasible dispatchable flow of the UPFC, given the ratings of the
VSCs. The controllable line real power incremental flow ranges from −102 MW to
107 MW, and line reactive power incremental flow ranges from −112 Mvar to 120
Mvar.
40
-150
-100
-50
0
50
100
150
-150 -100 -50 0 50 100 150Line ∆P (MW)
Qsh=0.5 CapacitiveQsh=0.0Qsh=0.5 Inductive
1
2
345
6
7
8
9 1011
12
Figure 3.11: UPFC Series Line Incremental P -Q Curves
3.5.3.2 Maximum IPFC Dispatchability
In the IPFC configuration, the VSCs are inserted in series on two lines in
different paths of the same transfer interface of the system. The line compensated
by VSC 2 is normally heavily loaded in the nominal system without any VSCs. Here
the line flow regulation of the IPFC is demonstrated.
A. VSC 1 as the Master and VSC 2 as the Slave
In the first set of dispatch computation, VSC 1 is set as master VSC such that
its injected voltage magnitude reference is kept constant at 1.0 pu while its angle
varies with step variations of 20◦ for a set of 18 values, as shown in Figure 3.10(b).
The dispatch is computed for two injected voltage reference settings of the slave
VSC (VSC 2) (Figure 3.10(b)):
• (Case 1) Vqref = 0.23 pu,
41
• (Case 2) Vqref = 0.8 pu.
The resulting dispatch of the master and slave VSCs is shown in Figure 3.12.
Both the master and slave VSC ∆P -∆Q curves are shaped like ellipses, with the
slave ∆P -∆Q ellipses being more narrow and the points corresponding to those in
Figure 3.10(b). Note also that Case 2, which has a higher Vqref , results in about 60
MW and 20 Mvar more power flow on the slave SVC line and about 10 MW less
on the master line than Case 1. The master SVC line reactive power flows for both
cases are very close.
-100
-50
0
50
100
-100 -50 0 50 100Master Line ∆P (MW)
Case 1Case 2
-100
-50
0
50
100
-50 0 50 100 150Slave Line ∆P (MW)
Case 1Case 2
1
5
14
189
1
5
9
14
181
18
14
9
5
Figure 3.12: Incremental P -Q Curves of IPFC Lines
Figure 3.13 shows the d-axis and q-axis components of the injected voltage of
the slave VSC. When the slave VSC reference value is high as in Case 2, with the
master VSC simultaneously requiring large active power circulation (|Vd| > 0.6 pu),
the slave VSC voltage insertion will exceed its limit. Based on the power circulation
priority rule in Section 3.4.2.2.ii, the slave VSC Vq cannot keep its reference value
any more, and it will be reduced to ensure the slave VSC voltage satisfies its limit.
B. VSC 2 as the Master and VSC 1 as the Slave
In the second set of dispatch, VSC 2 is set as the master such that its magnitude
of the injected voltage reference is kept constant while its angle varies, as shown in
Figure 3.10(b). The dispatch is computed for two settings:
42
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1Slave Vq (pu)
Case1Case2
1
5
9
14
181
9
5
14
18
Figure 3.13: Injected Series Voltage of IPFC Slave VSC
• (Case 3) Master VSC Vmref = 0.5 pu,
• (Case 4) Master VSC Vmref = 1.0 pu.
The slave inverter VSC 1 reference is set at Vqref = 0.1 pu.
-100
-50
0
50
100
-100 -50 0 50Slave Line ∆P (MW)
Case 3Case 4
-100
-50
0
50
100
-150 -100 -50 0 50 100 150Master Line ∆P (MW)
Case 3Case 4
1
18
5
9
14
A1
A2
118
5
9
14
1
2
10
11
Figure 3.14: Incremental P -Q Curves of IPFC Lines
The resulting incremental P -Q curves of the master and slave VSCs are shown
in Figure 3.14. Case 4, which has higher injected voltage magnitude of the master
43
VSC, has a larger line flow dispatch region than Case 3. However, compared to Case
3, in Case 4 the top and bottom of the near-elliptical ∆P -∆Q curves are clipped
because of the limits of the slave VSC. Note that two additional reference points
A1 and A2 in Case 4 are added to the set of 18 values for a clearer illustration of
this limitation. Figure 3.15 shows the d-axis and q-axis components of the injected
voltages of the IPFC. When the master VSC reference Vdref is too high (|Vdref | > 0.6
pu), that is, the master VSC requires larger active power circulation, even though
the slave Vd is set to its limit, it is still unable to support the power circulation.
Based on the power circulation priority rule, the master VSC Vd will be reduced to
allow the slave VSC to provide enough active power circulation.
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1Slave Vq (pu)
Case 3Case 4
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1Master Vq (pu)
Case 3Case 4
1(10)1(10)
56
(6)5
14
14
1 1
5
5 6
6
10 10
(15)
14
1415
15
15
Figure 3.15: Injected Series Voltage of the IPFC
3.6 Summary and Conclusions
In this chapter, we have presented a novel computation approach required for
dispatching the many control modes associated with multi-functional VSC-based
FACTS controllers. The shunt or series VSCs are separately modeled and then
functionally coupled by the circulating active power between them. This approach
can be adopted in all dispatch computation tools involving converter-based con-
trollers.
44
Rated-capacity operation strategies have also been implemented, such that
maximum dispatchability of VSCs can be studied. This feature may be used as
a tool for both the siting and sizing of converter-based transmission controllers.
The developed dispatch software has been implemented in an Operator Training
Simulator (OTS), which is customized to the CSC installed at NYPA’s Marcy 345
kV Substation.
CHAPTER 4
FACTS CONTROLLER DYNAMIC MODELS AND
SETPOINT CONTROL
A comprehensive set of regulator models of FACTS controllers, which include the
DC link capacitor dynamics and are applicable to various operating modes, are
proposed in this thesis work. In our approach, shunt VSC controllers and series VSC
controllers are modeled as controllable voltage sources with equivalent transformer
reactance. In the control model implementation, the shunt VSC controls and the
series VSC controls are modeled as separate regulators. When a VSC changes its
operating mode, only the input signals of the corresponding regulator need to be
adjusted. With this implementation, we only need to select and combine the proper
functionalities of the shunt VSC, the series VSC and the DC link coupling to form
the specified type of a FACTS controller and to operate it in the desired operating
mode.
In this chapter, we will focus on the formulation of the regulation model for
multi-functional FACTS Controllers. The modeling and setpoint control for the
shunt VSC, the series VSC, and the DC link capacitor are summarized in Sec-
tions 4.1. The nonlinear differential equation formulation for the different operating
modes, the algebraic equations of the network solution, and the numerical simulation
are included in Section 4.2. Application results are given in Section 4.3.
4.1 VSC Dynamic Modeling and Control
4.1.1 VSC Dynamic Model
Figure 4.1 shows the schematic diagrams for a shunt VSC and a series VSC,
where γsh and γse are modulation ratio signals to control the shunt and series con-
verter voltage magnitudes, respectively, and αsh and αse are firing angles of the shunt
VSC and series VSC, respectively. Note that αsh and αse in the dynamic models are
measured with respect to the angle of the from-bus voltage V1, while α1 and α2 in
the loadflow models in Chapter 3 are measured with respect to the swing bus angle.
45
46
+
_
Z1V1V2
~ ~ ~
Vdc
Ssh
C
Idc1
γsh αsh
Z2V3
From-bus
+
_
Z4V2Z3
V1V3 V4~ ~ ~ ~
Vdc
From-bus To-bus
Sse
C
Idc2
γse αse
(a) Shunt VSC (b) Series VSC
Figure 4.1: Voltage-Sourced Converters Showing DC Capacitors
In the time scale of transient stability, in which the VSC switching dynamics
are neglected, the model of a VSC with modulation ratio γ and firing angle α can
be represented as a voltage source
Vm = kVdcεjα (4.1)
where k is a factor which relates the inverter DC-side voltage to its AC-side terminal
voltage. Note that k is dependent on the modulation ratio γ.
The dynamic balanced positive-sequence model of a shunt VSC is shown in
Figure 4.2 (a). The shunt VSC is modeled as a controllable injected voltage source
Vm1 behind an equivalent transformer reactance Xt1, where Vm1 can be expressed as
Vm1 = k1Vdcεj(αsh+θ1) = Vm1ε
j(αsh+θ1) (4.2)
where k1 is the factor between the DC-side voltage Vdc and the AC-side voltage
magnitude Vm1 of the shunt VSC and θ1 is the angle of the from-bus voltage V1.
The injected current Ish and the injected power Ssh from the shunt VSC into the
system are the same as given in (3.1) and (3.2) for the steady-state shunt VSC
model, respectively.
As shown in Figure 4.2 (b), the series VSC is modeled as a controllable injected
47
jXt1
Ish
+_ Vm1
~
Z1 Z2V1V2 V3
~ ~ ~
Ssh,~ ~
From-bus
Z4V2Z3
V1V3 V4jXt2 + _Vm2~
Ise
~ ~ ~ ~From-bus To-bus
Sse,
S2~
~ ~
(a) Shunt VSC (b) Series VSC
Figure 4.2: Voltage-Sourced Converter Models
voltage source Vm2 behind an equivalent transformer reactance Xt2, where Vm2 can
be expressed as
Vm2 = k2Vdcεj(αse+θ1) = Vm2ε
j(αse+θ1) (4.3)
where k2 is the factor between the DC-side voltage Vdc and the AC-side voltage
magnitude Vm2 of the series VSC and θ1 is the angle of the from-bus voltage V1. The
line current Ise, the power injected by the series VSC Sse, and the power injected into
the to-bus (Bus 2) S2 are the same as given in (3.3), (3.4), and (3.5) for the steady-
state series VSC model, respectively. The series injected voltage Vm2 can be split
into two components: Vd is a component in phase with the from-bus voltage which
mainly affects the reactive power of the compensated line and Vq is a component in
quadrature with the from-bus voltage which mainly affect the active power of the
compensated line.
During transient studies, the DC link capacitor of FACTS controllers will ex-
change energy with the system and consequently its voltage will vary. The variation
of the DC capacitor voltage is dependent on its current inflow, which can be modeled
as
CdVdc
dt= Idc (4.4)
where Idc is the current flowing into the DC capacitor C from the VSC. In steady-
state operations when the power transfer is balanced, Idc = 0, and hence, dVdc/dt is
zero.
48
The FACTS dynamic models will be interfaced with the other dynamic com-
ponents in a power system, such as synchronous machines and excitation systems,
through the algebraic network equations. In using injected voltage sources for the
VSCs in the loadflow formulation, this transition to dynamic modes will be seamless,
because Vm1 and Vm2 are operational states in the loadflow models.
4.1.2 VSC Setpoint Controller Models
The same separation of shunt and series control modes for loadflow calculation
can be implemented dynamically also. The following subsections show separately
the Proportional-Integral (PI) regulators for each of the setpoint control modes of
the shunt VSC and the series VSC. Each of the regulators allows for setpoint control
by creating an error signal between the desired setpoint value and the actual value.
4.1.2.1 Shunt VSC Model
The shunt VSC can be operated either in voltage control mode or Var control
mode. The block diagrams for these two operating modes are shown in Figure 4.3
(a) and (b), respectively. The magnitude Vm1 and angle α1 of the inverter voltage
are generated by the control systems.
In the magnitude control of the shunt VSC, the factor k1 between its AC-side
voltage Vm1 and DC-side voltage Vdc is set to a constant. Thus the changes in the
magnitude of the inverter output voltage are achieved by charging or discharging
the DC bus capacitor to a different voltage.
For the angle control of the shunt VSC, an outer voltage regulation loop and
an inner current regulation loop are built to regulate the from-bus voltage V1 in the
voltage control mode, whereas in the Var control mode the shunt reactive current
Ishq is directly controlled to its reference value without the outer voltage regulation
loop.
The outer voltage regulation loop in the voltage control mode, which consists of
an integral controller Kv/s and a feedback droop α, is used to regulate the from-bus
voltage V1 towards its setpoint Vref . This loop produces a reactive current reference
I∗shq for the inner current loop. The shunt reactive current Ishq is controlled to I∗
shq
by the inner current loop, which consists an PI controller Kp + Ki/s and an LP
49
V1
Vref +_
+sKv
sKiKp+
αDroop
I*shq +
Ishq
θ1
-αsh
Vm1
α111+Ts
k1Vdc
_
_
_
(a) Voltage Regulation Mode
+_s
KiKp+Ishqref +
Ishq_
θ1
-αsh
Vm1
α111+Ts
k1Vdc
(b) Var Control Mode
Figure 4.3: Setpoint Control Schemes of a Shunt VSC
filter 1/(1 + Ts). The output of the inner current loop is the minus shunt inverter
voltage angle −αsh. The inverter voltage angle α1 can then be obtained with the
information of the from-bus voltage angle θ1.
In the var control mode, the shunt reactive current setpoint Ishqref is directly
specified in the operator screens. The shunt reactive current Ishq is controlled to
I∗shq by an PI controller Kp + Ki/s and an LP filter 1/(1 + Ts). This current loop
produces the angle information −αsh.
In steady state the angle αsh is zero, which means that the inverter output
voltage is kept essentially in phase with the from-bus voltage. Small transient posi-
tive or negative deviations in αsh cause nonzero active power to go through the DC
capacitor and thus result in an increase or decrease of the DC bus voltage Vdc.
4.1.2.2 Standalone Series VSC Model
The standalone series VSC can be either in the line active power control mode
or the inverter voltage magnitude control mode. The block diagrams for these two
50
modes are shown in Figures 4.4 and 4.5, respectively. The magnitude Vm2 and angle
α2 of the inverter voltage are generated by the control systems.
P
Pref
+sKiKp++
θl∆αse α2
π2
+11+Ts
Vm2k2Vdc
+-1
_
_
(a) Pref ≥ P0
P
Pref
+sKiKp++
_
θl∆αse α2
π2
+11+Ts
Vm2k2Vdc
+1
_
(b) Pref ≤ P0
Figure 4.4: Setpoint Control Schemes of a Standalone or “Slave” SeriesVSC in Line Active Power Regulation Mode
The standalone series VSC is also operated with a constant k2 between its AC-
side voltage Vm2 and DC-side voltage Vdc, and hence the changes in the magnitude
of the inverter output voltage are achieved by charging or discharging the DC bus
capacitor to a different voltage.
An PI controller Kp + Ki/s and an LP filter 1/(1 + Ts) are applied for the
angle control of the standalone series VSC. The input signal for the line active power
control mode is the difference of the line active power setpoint Pref and its measured
value P , while the input signal for the inverter voltage magnitude control mode is
the difference of the inverter voltage magnitude setpoint |Vm2ref | and k2Vdc. In each
operating mode, the output signal from the LP filter is the angle deviation ∆αse.
In steady state, ∆αse is zero, which means that the inverter output voltage is kept
essentially in quadrature with the current of the compensated line. Small transient
positive or negative deviations in the phase of the inverter voltage cause nonzero
51
|Vm2ref|
+sKiKp++
θl∆αse α2
π2
+11+Ts
Vm2k2Vdc
+-1
k2Vdc
_
_
(a) Vm2ref ≤ 0
|Vm2ref|
+sKiKp++
θl∆αse α2
π2
+11+Ts
Vm2k2Vdc
+1
k2Vdc
_
_
(b) Vm2ref ≥ 0
Figure 4.5: Setpoint Control Schemes of a Standalone or “Slave” SeriesVSC in Fixed Injected Voltage Mode
active power to go through the DC capacitor and thus result an increase or decrease
of the DC bus voltage.
In the line active power control mode, when the line active power setpoint Pref
is larger than the original line active power without compensation P0, the inverter
voltage angle α2 is (Figure 4.4 (a))
α2 = θ� − π
2+ ∆αse (4.5)
such that in steady state the inverter voltage is 90 degree lagging the line current
vector. When Pref ≤ P0, the inverter voltage angle α2 is (Figure 4.4 (b))
α2 = θ� +π
2− ∆αse (4.6)
such that in steady state the inverter voltage is 90 degree leading the line current
vector.
In the inverter voltage magnitude control mode, a polarity is added to the
52
inverter voltage reference Vm2ref to indicate whether leading or lagging voltage in-
jection is required. When Vm2ref ≤ 0, the inverter voltage angle α2 is (Figure 4.5
(a))
α2 = θ� − π
2+ ∆αse (4.7)
Thus in steady state the inverter voltage is 90 degree lagging the line current vector,
which means that it will increase the line active power. When Vm2ref ≥ 0, the inverter
voltage angle α2 is (Figure 4.5 (b))
α2 = θ� +π
2− ∆αse (4.8)
Thus in steady state the inverter voltage is 90 degree leading the line current vector,
which means that it will decrease the line active power.
4.1.2.3 Coupled Series VSC Model
The UPFC shunt VSC is operated in the same way as a STATCOM. For the
UPFC series VSC control, both the DC-to-AC ratio of the inverter and the phase
angle of the inverter output voltage are controlled.
The coupled series VSC can be either in the inverter voltage Vd,Vq control
mode or the line power P ,Q control mode. The block diagrams for both modes are
shown in Figure 4.6 (a) and (b), respectively. The magnitude Vm2 and angle α2 of
the inverter voltage are generated by the control systems.
Because the q-axis output voltage of the series VSC Vq has a strong impact on
the line active power flow P while the d-axis output voltage of the series VSC Vd
has a significant effect on the line reactive power flow Q. Therefore, in the line P ,Q
control mode, line active power P regulation and reactive power Q regulation are
implemented by independently controlling the q-axis and d-axis output voltage of
the series VSC Vq and Vd by using the PI controllers and LP filters as shown in 4.6
(a). Then the magnitude and angle of inverter voltage Vm2 and α2 can be obtained
as
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq
Vd)
(4.9)
53
+
sKiKp+
Pref +
P
θ1
αse α2
11+Ts
+sKiKp+
Qref +
Q
11+Ts
Magnitudeand AngleCalculator
Vm2Vq
Vd
_
_
(a) Line Power Regulation Mode
+θ1
αse α2
+
Magnitudeand AngleCalculator
Vm2Vdref
Vqref
(b) Fixed Injected Voltage Mode
Figure 4.6: Setpoint Control Schemes of a Coupled Series VSC
where θ1 is the angle of the from-bus voltage V1.
In the inverter voltage Vd,Vq control mode, the magnitude and angle of inverter
voltage Vm2 and α2 are instantaneously calculated from the setpoints Vdref and Vqref
as
Vm2 =√
V 2dref + V 2
qref
α2 = θ1 + tan−1(Vqref
Vdref)
(4.10)
4.1.2.4 The IPFC Model
An IPFC can be implemented by having a combination of the standalone and
coupled series regulators discussed above. However we make an exception for the
IPFC model here to include certain special control features associated with the real
hardware. In this IPFC control, both the DC-to-AC ratio of the inverter and the
phase angle of the inverter output voltage are controlled. The DC bus voltage is
held at an essentially constant value by the control action, while the inverter output
voltages can take on any values between zero and the maximum.
The IPFC VSCs can be either in the inverter voltage control mode or the line
power control mode. The block diagrams for these two operating modes are shown
54
in Figure 4.7 and 4.8, respectively. One VSC of an IPFC is operated as the Master
VSC, and the other is operated as the Slave VSC. The magnitude Vm2 m and angle
α2 m of the Master inverter voltage and the magnitude Vm2 s and angle α2 s of the
Slave inverter voltage are generated by the control systems.
VdcVdcref +
++
sKiKp+
KαDroop
∆Vd +
V*d
+
θ1
αse
Vm2_m
α2_m
11+Ts Magnitude
and AngleCalculator
V*q
Vq
Vd
Q
Qref +_
sKiKp+ 1
1+Ts
PPref +
sKiKp+ 1
1+Ts
_
_
_
(a) The Master VSCVdc
Vdcref +_
++
sKiKp+
∆Vd
αse
Vm2_s
α2_s
11+Ts Magnitude
and AngleCalculator
V*q
Vq
Vd
PPref +
sKiKp+ 1
1+Ts
_
(b) The Slave VSC
Figure 4.7: Setpoint Control Schemes of an IPFC in the Line Power Reg-ulation Mode
In these two operating modes, a DC bus voltage regulation loop, which consists
of an PI controller Kp + Ki/s and an LP filter 1/(1 + Ts), is applied for each VSC
of the IPFC. The control difference between the DC bus voltage regulation loops of
the Master and Slave VSCs is that there is a nonzero feedback droop for the Slave
VSC, while there is no such a feedback loop for the Master VSC. Thus the DC
bus voltage is more strictly controlled by the Slave VSC. The output signal of each
DC bus voltage regulator, denoted as ∆Vd, are the error signal to form the d-axis
inverter voltage Vd of the corresponding VSC.
55
VdcVdcref +
_
++
_ sKiKp+
KαDroop
∆Vd +
Vdref
+
θ1
αse
Vm2_m
α2_m
11+Ts Magnitude
and AngleCalculatorVqref
Vd
(a) The Master VSC
Vdc
Vdcref +
_
+
sKiKp+
∆Vd +
Vdref
+
θ1
αse
Vm2_s
α2_s
11+Ts Magnitude
and AngleCalculatorVqref
Vq
Vd
+
(b) The Slave VSC
Figure 4.8: Setpoint Control Schemes of an IPFC in the Fixed InjectedVoltage Mode
In the line power control mode, the Master line active and reactive power
P and Q regulations are implemented by independently controlling the q-axis and
d-axis voltages of the Master VSC V ∗q and V ∗
d by using the PI controllers and LP
filters as shown in 4.7 (a). The Master inverter voltage can then be obtained as
Vd = V ∗d + ∆Vd
Vq = V ∗q
Vm2 m =√
V 2d + V 2
q
αm2 m = θ1 + tan−1(Vq
Vd)
(4.11)
For the Slave VSC, only its line active power P is controlled by the PI controller
and LP filter as shown in 4.7 (b). The regulator output is the q-axis voltage of
the Slave VSC v∗q . The d-axis component v∗
d is not specified in order to meet the
power circulation constraint of the IPFC. Moreover, Vq of the Slave inverter voltage
is limited as a function of Vd to prioritize the transfer of power circulation. The
56
Slave inverter voltage can then be obtained as
Vd = ∆Vd
Vq =
V ∗q if Vm2 s ≤ Vm max
sign(V ∗q ) ·
√V 2
mmax − V 2d if Vm2 s ≥ Vm max
Vm2 s =√
V 2d + V 2
q
αm2 s = θ1 + tan−1(Vq
Vd)
(4.12)
where Vm max is the maximum limit of the Slave inverter voltage.
In the inverter voltage control mode, d-axis and q-axis inverter voltage ref-
erences Vdref and Vdref are specified directly in the operator screens. The Master
inverter voltage can be obtained as
Vd = Vdref + ∆Vd
Vq = Vqref
Vm2 m =√
V 2d + V 2
q
αm2 m = θ1 + tan−1(Vq
Vd)
(4.13)
And the Slave inverter voltage can be obtained as
Vd = Vdref + ∆Vd
Vq =
Vqref if Vm2 s ≤ Vm max
sign(Vqref) ·√
V 2mmax − V 2
d if Vm2 s ≥ Vm max
Vm2 s =√
V 2d + V 2
q
αm2 s = θ1 + tan−1(Vq
Vd)
(4.14)
4.1.3 DC Link Capacitor Dynamics
The AC instantaneous active power injection into the power system by a shunt
VSC is given by
Psh =V1Vm1 sin(αsh)
Xt1
(4.15)
and by a series VSC is given by
Pse = −V1Vm2 sin(αse) − V2Vm2 sin(αse + θ1 − θ2)
Xt2
(4.16)
57
Thus the AC instantaneous active power flowing into a single shunt VSC, such as a
STATCOM, is
Pac = −Psh (4.17)
and into a single series VSC, such as an SSSC, is
Pac = −Pse (4.18)
If the DC bus of a FACTS controller is coupled with M shunt VSCs and N series
VSCs, the AC instantaneous active powers flowing into the VSCs from the system
is given as
Pac = −(M∑i=1
Pshi+
N∑i=1
Psei) (4.19)
Assuming that the VSC model is ideal, the total AC instantaneous active
powers on the AC-side is equal to the DC-side active power, that is
Pac = VdcIdc (4.20)
From (4.4) and (4.20), we have
dVdc
dt=
1
CVdc
Pac (4.21)
Equation (4.21) is, in general, not per-unitized.
The block diagram of the DC link dynamics is shown in Figure 4.9.
1s
Pac VdcCVdc
1
Figure 4.9: DC Link Dynamics
58
4.2 Numerical Computation
The power system dynamic models can be written as a set of differential equa-
tions (4.22) and a set of algebraic equations (4.23) in vector form as
x = f(x, V ) (4.22)
I(x, V ) = Y V (4.23)
where I and V are complex injection currents and voltage vectors of dimension n,
respectively, and x is a state variable vector of dimension m. The number n is equal
to the number of nodes in the system and the number m depends on the number
and the type of the dynamic models used for the actual equipment. For example,
for a generator modeled with subtransient reactance, the state variables are its rotor
angle δ, speed ω, and direct- and quadrature-axis fluxes E′q, ψd, E
′d, and ψq [56].
In the explicit integration approach, (4.22) is used to update the state variables
x and then the algebraic variables V in (4.23) can be solved iteratively by a Newton
method given by (4.62), at every integration step.
4.2.1 Nonlinear Dynamic Models
The FACTS controls are represented as nonlinear differential equations for
transient stability studies.
The block realization of a PI regulator in series with an LP filter is shown in
Figure 4.10. The time-domain state equation is derived as
+ 1sTj
1_
ej zj.
zj
+1s
xj.
xjKij
Kpj+
Figure 4.10: Block Realization of the PI regulator and LP filter
59
xj = Kijej
zj = (Kpjej + xj − zj)/Tj
(4.24)
where xj and zj are the state variables for the jth regulators.
We introduce three additional state variables I∗shq, x1, and z1 for a shunt VSC,
two state variable x2 and z2 for a standalone VSC, four state variables x3, z3, x4,
and z4 for a coupled series VSC, and ten state variables x5, z5, x6, z6, x7, z7, xM ,
zM , xS, and zS for an IPFC into the state variable vector x in (4.22). Also the DC
link dynamic state variable xdc = Vdc will be incorporated.
In this section we provide the equations for formulating the different shunt
and series VSC operating modes.
4.2.1.1 Shunt Operating Modes
(Sh1) Voltage control mode with droop α: the differential equations of state variables
I∗shq, x1, and z1 can be expressed as
I∗shq = Kv(Vref − V1 − αI∗
shq)
x1 = Ki1(I∗shq − Ishq)
z1 = [Kp1(I∗shq − Ishq) + x1 − z1]/T1
(4.25)
where Kv is the gain of the voltage regulator, Kp1 and Ki1 are the proportional
and integral gain coefficients of the PI controller, and T1 is the time constant
of the LP filter (Figure 4.3 (a)). The shunt injected voltage source can be
obtained as
Vm1 = k1Vdc
α1 = θ1 − z1
(4.26)
where k1 is the constant ratio between Vm1 and Vdc and θ1 is the from-bus
voltage angle. Note that α1 is with regard to to the system swing bus angle.
(Sh2) Control the Var output of the shunt VSC to a desired value Ishqref : the dif-
ferential equations of the state variables I∗shq, x1, and z1 can be expressed
60
as
I∗shq = 0
x1 = Ki1(Ishqref − Ishq)
z1 = [Kp1(Ishqref − Ishq) + x1 − z1]/T1
(4.27)
where Kp1 and Ki1 are the proportional and integral gain coefficients of the PI
controllers and T1 is the time constant of the LP filter (Figure 4.3 (b)). The
shunt injected voltage source can be obtained as
Vm1 = k1Vdc
α1 = θ1 − z1
(4.28)
where k1 is the constant ratio between Vm1 and Vdc and θ1 is the from-bus
voltage angle. Note that α1 is with regard to to the system swing bus angle.
4.2.1.2 Standalone Series Dispatch Modes
(Se1) Control the line active power flow P to a desired value Pref : the differential
equations of state variables x2 and z2 can be expressed as
x2 = Ki2(Pref − P )
z2 = [Kp2(Pref − P ) + x2 − z2]/T2
(4.29)
where Kp2 and Ki2 are the proportional and integral gain coefficients of the
PI controller and T2 is the time constant of the LP filter (Figure 4.4). The
series injected voltage source can be obtained as
Vm2 = k2Vdc
α2 =
θl − π/2 + z2 if Pref ≥ P0
θl + π/2 − z2 if Pref ≤ P0
(4.30)
where k2 is the constant ratio between Vm2 and Vdc and θ� is the line current
angle.
(Se2) Fix the injected voltage magnitude, in either the quadrature leading or lag-
ging direction with respect to the transmission line current: the differential
61
equations of state variables z2 and z2 can be expressed as
x2 = Ki2(Vm2ref − Vm2)
z2 = [Kp2(Vm2ref − Vm2) + x2 − z2]/T2
(4.31)
where Kp2 and Ki2 are the proportional and integral gain coefficients of the
PI controller and T2 is the time constant of the LP filter (Figure 4.5). The
series injected voltage source can be obtained as
Vm2 = k2Vdc
α2 =
θl − π/2 + z2 if Vm2ref ≤ 0
θl + π/2 − z2 if Vm2ref ≥ 0
(4.32)
where k2 is the constant ratio between Vm2 and Vdc and θ� is the line current
angle.
4.2.1.3 Coupled Series Dispatch Modes
(SeC1) Control the line active and reactive power flow P and Q to their desired values
Pref and Qref , respectively: the differential equations of state variables x3, z3,
x4, and z4 can be expressed as
x3 = Ki3(Pref − P )
z3 = [Kp3(Pref − P ) + x3 − z3]/T3
x4 = Ki4(Qref − Q)
z4 = [Kp4(Qref − Q) + x4 − z4]/T4
(4.33)
where Kp3, Ki3, Kp4, and Ki4 are the proportional and integral gain coefficients
of the PI controllers and T3 and T4 are the time constants of the LP filters
(Figure 4.6 (a)). The series injected voltage source can be obtained as
Vq = z3
Vd = z4
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq/Vd)
(4.34)
62
(SeC2) Fix the d-axis and q-axis of the injected voltage at Vdref and Vqref with respect
to the from-bus voltage vector V1: the differential equations of state variables
x3, z3, x4, and z4 can be expressed as
x3 = 0
z3 = 0
x4 = 0
z4 = 0
(4.35)
Note that (4.35) is listed here only for completeness. The series injected voltage
source can be obtained as
Vq = Vqref
Vd = Vdref
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq/Vd)
(4.36)
4.2.1.4 IPFC Operating Modes
A. The Master VSC
(SeM1) Control the Master line active and reactive power flow P and Q to their desired
values Pref and Qref , respectively: the differential equations of state variables
x5, z5, x6, z6, xM , and zM can be expressed as
x5 = Ki5(Pref − P )
z5 = [Kp5(Pref − P ) + x5 − z5]/T5
x6 = Ki6(Qref − Q)
z6 = [Kp6(Qref − Q) + x6 − z6]/T6
xM = KiM(Vdcref − Vdc − KαzM)
zM = [KpM(Vdcref − Vdc − KαzM) + xM − zM ]/TM
(4.37)
where Kp5, Ki5, Kp6, Ki6, KpM, and KiM are the proportional and integral gain
coefficients of the PI controllers and T5, T6, and TM are the time constants
of the LP filters (Figure 4.7 (a)). The Master injected voltage source can be
63
obtained as
∆Vd = zM
Vq = z5
Vd = z6 + ∆Vd
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq/Vd)
(4.38)
(SeM2) Fix the d-axis and q-axis of the injected voltage at Vdref and Vqref with respect
to the from-bus voltage vector V1: the differential equations of state variables
x5, z5, x6, z6, xM , and zM can be expressed as
x5 = 0
z5 = 0
x6 = 0
z6 = 0
xM = KiM(Vdcref − Vdc − KαzM)
zM = [KpM(Vdcref − Vdc − KαzM) + xM − zM ]/TM
(4.39)
where KpM and KiM are the proportional and integral gain coefficients of the
PI controllers and TM is the time constant of the LP filter (Figure 4.8 (a)).
The Master injected voltage source can be obtained as
∆Vd = zM
Vq = Vqref
Vd = Vdref + ∆Vd
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq/Vd)
(4.40)
B. The Slave VSC
(SeS1) Control the Slave line active power flow P to its desired value Pref : the differ-
64
ential equations of state variables x7, z7, xS, and zS can be expressed as
x7 = Ki7(Pref − P )
z7 = [Kp5(Pref − P ) + x7 − z7]/T7
xS = KiS(Vdcref − Vdc)
zS = [KpS(Vdcref − Vdc + xS − zS]/TS
(4.41)
where Kp7, Ki7, KpS, and KiS are the proportional and integral gain coefficients
of the PI controllers and T7 and TS are the time constants of the LP filters
(Figure 4.7 (b)). The Slave injected voltage source can be obtained as
∆Vd = zS
Vq = z7
Vd = ∆Vd
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq/Vd)
(4.42)
(SeS2) Fix the q-axis of the injected voltage at and Vqref with respect to the from-bus
voltage vector V1: the differential equations of state variables x7, z7, xS, and
zS can be expressed as
x7 = 0
z7 = 0
xS = KiS(Vdcref − Vdc)
zS = [KpS(Vdcref − Vdc + xS − zS]/TS
(4.43)
where KpS and KiS are the proportional and integral gain coefficients of the
PI controllers and TS is the time constant of the LP filter (Figure 4.8 (b)).
65
The Slave injected voltage source can be obtained as
∆Vd = zS
Vq = Vqref
Vd = Vdref + ∆Vd
Vm2 =√
V 2d + V 2
q
α2 = θ1 + tan−1(Vq/Vd)
(4.44)
4.2.2 Network Equations
The bus admittance matrix equation of a power system without FACTS con-
trollers and non-conforming loads can be written as follows
Ygg Ygl
Ylg Yll
Vg
Vl
=
Ig
0
(4.45)
where Vg is the generator bus voltage vector and Vl is the bus voltage vector for all
the load buses.
If a shunt VSC is connected to Bus f of the power system, the bus admittance
equation is expanded to
Ygg Ygf Ygl 0
Yfg Yff + 1jXt1
Yfl − 1jXt1
Ylg Ylf Yll 0
Vg
Vf
Vl
Vm1
=
Ig
0
0
(4.46)
Rearranging (4.46) by moving Vm1 to the right hand side, we obtain
Ygg Ygf Ygl
Yfg Yff + 1jXt1
Yfl
Ylg Ylf Yll
Vg
Vf
Vl
=
Ig
Vm1/jXt1
0
(4.47)
If a series VSC is inserted into the line with from-bus f and to-bus t of the
66
power system, the bus admittance equation is expressed as follows
Ygg Ygf Ygt Ygl 0
Yfg Yff + 1jXt2
Yft − 1jXt2
Yfl − 1jXt2
Ytg Ytf − 1jXt2
Ytt + 1jXt2
Ytl1
jXt2
Ylg Ylf Ylt Yll 0
Vg
Vf
Vt
Vl
Vm2
=
Ig
0
0
0
(4.48)
Rearrange (4.48) by moving Vm2 to the right hand side, we obtain
Ygg Ygf Ygt Ygl
Yfg Yff + 1jXt2
Yft − 1jXt2
Yfl
Ytg Ytf − 1jXt2
Ytt + 1jXt2
Ytl
Ylg Ylf Ylt Yll
Vg
Vf
Vt
Vl
=
Ig
Vm2/jXt2
−Vm2/jXt2
0
(4.49)
If a VSC has the same from-bus or to-bus with some other shunt or series
VSCs, the effect of all these VSCs on the bus admittance matrix equation can be
added together.
Suppose the total number of distinct from-buses of FACTS controllers is L,
and the total number of distinct to-buses of FACTS Controllers is R in a specific
power system and let the from-bus fi have Nf i shunt VSCs and Mf i series VSCs
connected to it, for i = 1, . . . , L, and the to-bus tk have Mtk series VSCs connected
to it, for k = 1, . . . , R. The bus admittance matrix equation can be expressed as
Ygg YgF YgF Ygl
YFg YFF YFT YFl
YTg YTF YTT YTl
Ylg YlF YlT Yll
Vg
VF
VT
Vl
=
Ig
IF
IT
0
(4.50)
67
where
YFF =
Y′f1f1
. . . 0
Y′fifi
0. . .
Y′fLfL
L×L
(4.51)
YTT =
Y′t1t1
. . . 0
Y′tktk
0. . .
Y′tRtR
R×R
(4.52)
YFT = Y TFT =
Y′f1t1
· · · Y′f1tk
· · · Y′f1tR
.... . .
......
Y′fit1
· · · Y′fitk
· · · Y′fitR
......
. . ....
Y′fLt1
· · · Y′fLtk
· · · Y′fLtR
L×R
(4.53)
Y′fifi
= Yfifi+
Nf i∑i=1
1
jXt1i
+Mf i∑i=1
1
jXt2i
, i = 1, . . . , L (4.54)
Y′tktk
= Ytktk +Mtk∑k=1
1
jXt2k
, k = 1, . . . , R (4.55)
Y′fitk
=
Yfitk , no series VSCs in Line fitk
Yfitk −M�s∑s=1
1
jXt2s
, M�s series VSCs in Line fitk(4.56)
i = 1, . . . , L; k = 1, . . . , R
Ifi=
Nf i∑i=1
Vm1i
jXt1i
+Mf i∑i=1
Vm2i
jXt2i
=Nf i∑i=1
Vm1iej(α1i+θ1i)
jXt1i
+Mf i∑i=1
Vm2iej(α2i+θ1i)
jXt2i
68
i = 1, . . . , L (4.57)
Itk = −Ntk∑i=1
Vm2i
jXt2i
= −Ntk∑i=1
Vm2iej(α2i+θ1i)
jXt2i
k = 1, . . . , R (4.58)
where Yfifi, Ytktk , and Yfitk are the nodal admittances and mutual admittances at the
bus fi and bus tk of the system without considering FACTS Controllers, respectively.
Next, we reduce the bus admittance matrix to the generator internal buses
and the FACTS controllers’ from-buses and to-buses. The corresponding reduced
bus admittance matrix equation takes the form
YGG YGF YGT
YFG YFF YFT
YTG YTF YTT
E′′
Vf
Vt
=
Ig
IF
IT
(4.59)
where E′′
is the generator internal voltage vector behind the transient or subtransient
reactance.
It is clear that in (4.57) and (4.58), Vm1, α1, Vm2, and α2 are known from
the control outputs Vm1, α1, Vd, and Vq in Section 4-1. The only unknown is the
from-bus angle θ1, which can be obtained from the network solution. An iterative
process can be applied to obtain the solutions of Vf , θ1, and Vt.
4.2.3 Newton’s Method
Rearranging the second and third equations of (4.59), we have
YFF YFT
YTF YTT
VF
VT
=
IF − YFGE
′′
IT − YTGE′′
(4.60)
Define the functions ∆F1 and ∆F2 as
∆F1
∆F2
=
YFFVF + YFTVT + YFGE
′′ − IF
YTFVF + YTTVT + YTGE′′ − IT
(4.61)
69
Use the Newton’s method to solve for the variables VFre , VFim, VTre , and VTim
itera-
tively as
VFre
new = VFre
old + ∆VFre
VFim
new = VFim
old + ∆VFim
VTre
new = VTre
old + ∆VTre
VTim
new = VTim
old + ∆VTim
(4.62)
where the updates are computed as
∆VFre
∆VFim
∆VTre
∆VTim
=
∂∆F1re
∂VFre
∂∆F1re
∂VFim
∂∆F1re
∂VTre
∂∆F1re
∂VTim
∂∆F1im
∂VFre
∂∆F1im
∂VFim
∂∆F1im
∂VTre
∂∆F1im
∂VTim
∂∆F2re
∂VFre
∂∆F2re
∂VFim
∂∆F2re
∂VTre
∂∆F2re
∂VTim
∂∆F2im
∂VFre
∂∆F2im
∂VFim
∂∆F2im
∂VTre
∂∆F2im
∂VTim
−1
∆F1re
∆F1im
∆F2re
∆F2im
(4.63)
Then we get VF = VFre + jVFimand VT = VTre + jVTim
. Substituting VF and
VT into the first equation of (4.59) gives the current injections Ig into the generator
internal buses
Ig = YGGE′′
+ YGFVF + YGTVT (4.64)
4.2.4 Integration Method
The predictor-corrector scheme [56] is used to solve the problem of (4.22). It
consists of two main steps, a predictor step:
xk+1 = xk + f(xk, tk)∆t (4.65)
and a corrector step:
xk+1 = xk +[f(xk, tk) + f(xk+1, tk+1)]
2∆t (4.66)
This multi-step scheme will result in a second-order accuracy of the solution, that
is, the local error of the method, which is the difference between the approximate
solution xk obtained by using this method and the exact solution x∗k of the differential
equation, is O((∆t)3) as ∆t → 0.
70
4.3 Simulation Results
The regulator models of the VSC-based FACTS controllers are simulated in a
22-bus test system as shown in Figure 4.11, which has 6 equivalent generators and
3 equivalent loads. The loads of the test system are concentrated in the southeast
part of the system, while the generations are mainly in the northwest area. The
arrows indicate the direction of the active power flows. A 100 MVA shunt VSC can
be connected to Bus 4 by closing its switch and two 100 MVA series VSCs can be
inserted into Line 4-11 and Line 4-12, which are the two major paths between the
generations and the loads, by opening their bypass switches, respectively. Note that
the system base is 100 MVA.
4
2 3
7
9
6
8
10
1 5
11
1213
14
15
17
19
20
21
1618
VSC 2
Load 1Load 2
VSC 1
22
G 1
G 2
G 3
G 4
G 5
G 6
VSC 3
Figure 4.11: 22-Bus Test System
71
In the dynamic simulations, each generator are modeled with a subtransient
reactance, controlled by a simple voltage regulator. The loads are modeled as con-
stant impedances.
4.3.1 FACTS Controller Dynamic Simulations
By manipulating the switches, four configurations of FACTS controllers, named
a STATCOM, an SSSC, a UPFC, and an IPFC, can be simulated to study their
dynamic effects in the 22-bus test system. The simulation results for one operating
mode in each configuration are displayed in the following subsections.
4.3.1.1 STATCOM Dynamics
The STATCOM configuration is formed by connecting the shunt VSC 1 to
Bus 4 and leaving the bypass switches of the series VSC 2 and VSC 3 closed.
Table 4.1: Operating Conditions of the STATCOM in Var Control Mode
Control Original DisturbanceGains Setpoint Event
Kp=0.01 At t=0.2 s, the reactive current injectionKi=0.1 Ishqset = −1.0 pu reference has a step change from full
T=0.02 s inductive (−1 pu) to full capacitive (1 pu).
Figure 4.12 shows the simulation results of the STATCOM in var control mode
under the operating conditions in Table 4.1. A positive value of reactive power in-
jection reference Ishqset implies capacitive shunt reactive power compensation, while
a negative value implies inductive compensation.
When the reactive current reference changes from inductive to capacitive, the
shunt reactive power injected into the from bus by the VSC will respond to the
change, and thus will cause the from-bus voltage increase. Both the DC capacitor
voltage and the inverter voltage increase but the ratio between them is kept constant.
As shown in Figure 4.12, when the shunt reactive power compensation changes
from full inductive to full capacitive, the DC capacitor voltage increases from about
−20% below to 20% above nominal.
Note that the fast oscillations in the system voltage are due to the FACTS
controller and generator automatic voltage controllers, and the slower oscillations
72
are the effect of the superposition of the impact of all machine swing modes. This
also explains the voltage oscillations for all the following cases.
4.3.1.2 SSSC Dynamics
The SSSC configuration is formed by opening the bypass switch of the series
VSC 2 to insert the series VSC 2 into Line 4-11, while leaving the switch of the
shunt VSC 1 open and the bypass switch of the series VSC 3 closed.
Table 4.2: Operating Conditions of the SSSC in Vm Control Mode
Control Original DisturbanceGains Setpoint EventKp=20 At t=0.01 s, Vmref has a rampKi=200 Vmref = −0.05 pu increase from −0.05 pu to 0.05T=0.02 pu in 10 s.
Figure 4.13 shows the simulation results of the SSSC in inverter voltage magni-
tude control mode under the operating conditions in Table 4.2. The polarity of Vmref
indicates that the insertion of the SSSC is either inductive when it is positive or is
capacitive when it is negative. We observe that the DC capacitor voltage decreases
from over 10 kV to zero and then goes back up, while the from-bus voltage keeps
decreasing from 1.0122 pu to 1.0058 pu. The line active power decreases about 160
MW with the SSSC control from 0.05 pu capacitive to 0.05 pu inductive.
4.3.1.3 UPFC Dynamics
The UPFC configuration is formed by connecting the shunt VSC 1 to Bus 4
and inserting the series VSC 2 into Line 4-11, while leaving the bypass switch of the
series VSC 3 closed.
Table 4.3: Operating Conditions of the UPFC in V ,Vd,Vq Control Mode
Shunt Series Original DisturbancesGains Gains Setpoint Event 1 Event 2 Event 3
Kv=500 Kp=0.01 Vset=1.025 pu At t=1 s, At t=7 s, At t=13 s,Kp=0.01 Ki = 0.1 α=0.03 shunt Vset has series Vdref has series Vqref hasKi = 0.1 T=0.02 Vdref=0.01 pu a step increase a step increase a step decreaseT=0.02 Vqref=−0.02 pu of 0.01 pu. of 0.02 pu. of 0.02 pu.
73
Figure 4.14 shows the simulation results of the UPFC in V ,Vd,Vq control mode
under the operating conditions in Table 4.3. We observe that the from-bus voltage,
the series VSC voltage Vd, and the series VSC voltage Vq are independently con-
trolled to their reference values. The change of Vq mainly affects line active power,
while the change of Vd mainly affects line reactive power.
4.3.1.4 IPFC Dynamics
The IPFC configuration is formed by inserting the series VSC 2 into Line 4-11
as the Master VSC and inserting the series VSC 3 into Line 4-12 as the Slave VSC,
while leaving the switch of the shunt VSC 1 open.
Table 4.4: Operating Conditions of the IPFC in Inverter Voltage ControlMode
Master Slave Original DisturbancesGains Gains Setpoint Event 1 Event 2 Event 3
Kα=100 Kp=.1 Master At t=1 s, At t=7 s, At t=13 s,Kp=.1 Ki = 1 Vdref=0.02 pu Master Vqref has Slave Vqref has Master Vdref hasKi = 1 T=0.02 Vqref=0.0 pu; a step decrease a step decrease a step decreaseT=0.02 Slave of 0.01 pu. of 0.01 pu. of 0.02 pu.
Vqref=−0.03 pu
Figure 4.15 and 4.16 show the simulation results of the IPFC in inverter voltage
control mode under the operating conditions in Table 4.4. We observe that the
Master VSC Vd, the Master VSC Vq, and the Slave VSC Vq are independently
controlled to their reference values. Note that the DC capacitor voltage is also
controlled to its reference value. The step change (-0.01 pu) of the Master line Vdref
causes a 0.009 pu (3 kV in nominal) increase of the to-bus voltage of the Master
line and the same amount of decrease of the to-bus voltage of the Slave line at the
same time.
4.3.2 Transient Power Transfer Capability Analysis Example
In order to evaluate maximum power transfer capability of the critical paths
in the 22-bus system, we stress the system by gradually increasing the active powers
of Load L2 on Bus 17 and Generators G1, G2, and G3. The original active power of
Load L2 is 2500 MW. At time t=0.1 s, a three-phase line-to-ground fault is applied
74
on the Bus 3 side of Line 3-13, which is a line paralleled with Line 4-11. Its near end
is cleared at t=0.15 s and remote end is cleared at t=0.17 s. The maximum load
on Bus 17 that the system can stand during the fault and the corresponding power
transfers on Line 4-11 and Line 4-12 are displayed in Table 4.5 for seven different
system configurations. The setpoints for these configurations are simply specified
on their rated capacity.
Table 4.5: Comparison of Transient Power Transfer Capability Analysiswithout and with Various FACTS Controllers
Max Load Line PowerConfiguration Setpoints on Bus 17 Transfer (MW)
(MW) Line 4-11 Line 4-121. No FACTS - 3235 1390 904
2. STATCOM 100 MVA Ishqref = 1.0 pu 3268 1403 912Var Control
3. STATCOM 200 MVA Ishqref = 2.0 pu 3300 1415 920Var Control
4. STATCOM 200 MVA Vref = 0.9121 pu, 3298 1421 924Voltage Control α = 0.03
5. SSSC 100 MVA L4-11 Vmref = −0.055 pu 3275 1460 897Inverter Vm Control
6. UPFC (Sh) Ishqref = 1.0 pu,100/100 MVA L4-11 (Se) Vdref = 0.0 pu, 3343 1469 904Var,Vd,Vq Control Vqref = −0.055 pu
7. UPFC (Sh) Vref = 0.91 pu,100/100 MVA L4-11 α = 0.03, 3340 1473 905
V ,Vd,Vq Control (Se) Vdref = 0.0 pu,Vqref = −0.055 pu
8. IPFC 100/100 MVA (M) Vdref = 0.0 pu,L4-11(M)/Line4-12(S) Vqref = −0.055 pu, 3298 1443 952Inverter Vd, Vq Control (S) Vqref = −0.055 pu9. IPFC 100/100 MVA (M) Vdref = 0.0 pu,L4-12(M)/Line4-11(S) Vqref = −0.055 pu, 3300 1459 935Inverter Vd, Vq Control (S) Vqref = −0.055 pu
As shown in Table 4.5, a STATCOM with 200 MVA rating can support over
60 MW more load on Bus 17 than the configuration without a FACTS controller
(Config. 1). The corresponding power flows on Line 4-11 and Line 4-12 increase
about 30 MW and 20 MW, respectively. Compared with Config. 1, the system with
75
an 100 MVA SSSC in Line 4-11 (Config. 5) stands 40 MW more load on Bus 17.
The corresponding power flow on Line 4-11 increases 70 MW while that on Line
4-12 decreases 7 MW.
Considering the system with a UPFC, which consists a 100 MVA shunt VSC
on Bus 4 coupled with a 100 MVA series VSC in Line 4-11, if the series VSC of
the UPFC is operated in the line P ,Q control mode, the system will crash during
the fault because the system can not transmit enough power from the Northwest
generators to the Southeast loads with Line 3-13 tripped and the P setpoint of Line
4-11 fixed. So the series VSC of the UPFC should be operated in the inverter Vd,Vq
control mode as in Config. 6 and Config. 7. Each of these two configurations can
support over 100 MW more load on Bus 17 than Config. 1 and transmit about 80
MW more power flow on Line 4-11 while keep that on Line 4-12 unaffected.
Config. 8 and Config. 9 are configurations with an IPFC, which consists of a
100 MVA series VSC in Line 4-11 coupled with a 100 MVA series VSC in Line 4-12.
Both configurations can stand over 60 MW more load on Bus 17 than Config. 1.
Each carries over 100 MW more the total power flows of Lines 4-11 and 4-12.
Figure 4.17 shows the dynamic simulations of Config. 1 and Config. 7 in the
same loading conditions where the active power of Load L2 on Bus 17 is 3235 MW.
The setpoints of Config. 7 is the same as shown in Table 4.5. We observe that the
UPFC reduces transient oscillations in voltages on Bus 3 and Bus 4 and power flows
on the critical paths Line 4-11 and Line 4-12 during the fault, in addition to its
capability on increasing post-fault bus voltages and power transfers.
To summarize, FACTS controllers can substantially improve the transient
power transfer capability of a transmission system during a fault.
4.4 Summary and Conclusions
In this chapter, we have discussed a comprehensive set of regulator models and
their efficient control mode implementation for time-domain dynamic simulation of
various operating modes associated with VSC-based FACTS controllers.
We have incorporated special control features of the actual hardware into the
dynamic models, so that the effect of exercising these features can be illustrated to
76
the operators and equipment engineers, without performing the experiments on the
real hardware.
77
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51
1.005
1.01
1.015
1.02
1.025
1.03
1.035
1.04
1.045
time in seconds
AC
vol
tage
per
uni
t
STATCOM from−bus voltage magnitude
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7x 10
4
time in seconds
DC
vol
tage
vol
ts
STATCOM DC capacitor voltage
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5
−1
−0.5
0
0.5
1
1.5
2
time in seconds
Rea
ctiv
e cu
rren
t per
uni
t
STATCOM reactive current injection
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1
−0.5
0
0.5
1
1.5
2
time in seconds
Rea
ctiv
e cu
rren
t inc
rem
enta
l per
uni
t
STATCOM ∆Ishq
= I*shq
−Ishq
I*shq
Ishq
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.8
0.9
1
1.1
1.2
1.3
1.4
1.5
time in seconds
AC
vol
tage
per
uni
t
Shunt VSC voltage magnitude
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−12
−10
−8
−6
−4
−2
0
2
4x 10
−3
time in seconds
angl
e ra
d
Shunt VSC voltage angle w.r.t. from−bus voltage angle
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
time in seconds
Rea
l pow
er p
er u
nit
STATCOM active power injection Psh
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5
−1
−0.5
0
0.5
1
1.5
2
time in seconds
Rea
ctiv
e po
wer
per
uni
t
STATCOM reactive power injection Qsh
Figure 4.12: STATCOM Var Control Mode Simulation
78
0 1 2 3 4 5 6 7 8 9 101.004
1.006
1.008
1.01
1.012
1.014
time in seconds
AC
vol
tage
per
uni
t
SSSC from−bus voltage magnitude
0 2 4 6 8 100
5000
10000
15000
time in seconds
DC
vol
tage
vol
ts
SSSC DC capacitor voltage
0 1 2 3 4 5 6 7 8 9 100
0.01
0.02
0.03
0.04
0.05
0.06
time in seconds
AC
vol
tage
per
uni
t
Series VSC voltage magnitude
0 2 4 6 8 10−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
time in seconds
angl
e ra
d
Series VSC voltage angle w.r.t. from−bus voltage angle
0 1 2 3 4 5 6 7 8 9 106.5
7
7.5
8
8.5
time in seconds
pow
er fl
ow p
er u
nit
SSSC line active power into the to−bus
0 2 4 6 8 10−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
time in seconds
pow
er fl
ow p
er u
nit
SSSC line reactive power into the to−bus
Figure 4.13: SSSC Inverter Voltage Magnitude Control Mode Simulation
79
0 2 4 6 8 10 12 14 16 18 201.01
1.015
1.02
1.025
1.03
1.035
1.04
1.045
1.05
1.055
time in seconds
AC
vol
tage
per
uni
t
UPFC from−bus voltage magnitude
0 5 10 15 201.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9x 10
4
time in seconds
DC
vol
tage
vol
ts
UPFC DC capacitor voltage
0 2 4 6 8 10 12 14 16 18 207.6
7.7
7.8
7.9
8
8.1
8.2
time in seconds
pow
er fl
ow p
er u
nit
UPFC line active power into the to−bus
0 5 10 15 20−0.65
−0.6
−0.55
−0.5
−0.45
−0.4
−0.35
−0.3
−0.25
time in seconds
pow
er fl
ow p
er u
nit
UPFC line reactive power into the to−bus
0 2 4 6 8 10 12 14 16 18 201
1.1
1.2
1.3
1.4
1.5
1.6
1.7
time in seconds
AC
vol
tage
per
uni
t
Shunt VSC inserted voltage magnitude
0 5 10 15 200.005
0.01
0.015
0.02
0.025
0.03
0.035
time in seconds
angl
e ra
d
Shunt VSC injected voltage angle w.r.t. from−bus voltage angle
0 2 4 6 8 10 12 14 16 18 200.005
0.01
0.015
0.02
0.025
0.03
0.035
time in seconds
AC
vol
tage
per
uni
t
Series VSC inserted voltage Vd
0 5 10 15 20−0.045
−0.04
−0.035
−0.03
−0.025
−0.02
−0.015
time in seconds
angl
e ra
d
Series VSC inserted voltage Vq
Figure 4.14: UPFC V ,Vd,Vq Control Mode Simulation
80
0 2 4 6 8 10 12 14 16 18 201.0225
1.023
1.0235
1.024
1.0245
1.025
time in seconds
AC
vol
tage
per
uni
t
IPFC from−bus voltage magnitude
0 5 10 15 200.95
1
1.05
1.1
1.15
1.2
1.25x 10
4
time in seconds
DC
vol
tage
vol
ts
IPFC DC capacitor voltage
0 2 4 6 8 10 12 14 16 18 201.002
1.004
1.006
1.008
1.01
1.012
1.014
1.016
time in seconds
pow
er fl
ow p
er u
nit
IPFC master to−bus voltage magnitude
0 5 10 15 201.032
1.034
1.036
1.038
1.04
1.042
1.044
1.046
1.048
1.05
time in seconds
pow
er fl
ow p
er u
nit
IPFC slave to−bus voltage magnitude
0 2 4 6 8 10 12 14 16 18 207.4
7.45
7.5
7.55
7.6
7.65
time in seconds
pow
er fl
ow p
er u
nit
IPFC master line active power into the to−bus
0 5 10 15 20−0.65
−0.6
−0.55
−0.5
−0.45
−0.4
−0.35
−0.3
time in seconds
pow
er fl
ow p
er u
nit
IPFC master line reactive power into the to−bus
0 2 4 6 8 10 12 14 16 18 207.74
7.76
7.78
7.8
7.82
7.84
7.86
7.88
time in seconds
pow
er fl
ow p
er u
nit
IPFC slave line active power into the to−bus
0 5 10 15 200.8
0.85
0.9
0.95
1
1.05
time in seconds
pow
er fl
ow p
er u
nit
IPFC slave line reactive power into the to−bus
Figure 4.15: IPFC Inverter Voltage Control Mode Simulation – I
81
0 2 4 6 8 10 12 14 16 18 200.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
time in seconds
AC
vol
tage
per
uni
t
Master VSC inserted voltage Vd
0 5 10 15 20−12
−10
−8
−6
−4
−2
0
2x 10
−3
time in seconds
angl
e ra
d
Master VSC inserted voltage Vq
0 2 4 6 8 10 12 14 16 18 20−0.024
−0.022
−0.02
−0.018
−0.016
−0.014
−0.012
−0.01
−0.008
time in seconds
AC
vol
tage
per
uni
t
Slave VSC inserted voltage Vd
0 5 10 15 20−0.042
−0.04
−0.038
−0.036
−0.034
−0.032
−0.03
−0.028
time in seconds
angl
e ra
d
Slave VSC inserted voltage Vq
Figure 4.16: IPFC Inverter Voltage Control Mode Simulation – II
82
0 1 2 3 4 50.5
0.6
0.7
0.8
0.9
1
time in seconds
volta
ge in
pu
Bus 3 Voltage Magnitude
0 1 2 3 4 50.5
0.6
0.7
0.8
0.9
1
time in seconds
volta
ge in
pu
Bus 4 Voltage Magnitude
0 1 2 3 4 58
10
12
14
16
time in seconds
pow
er fl
ow in
MW
Line 4−11 Active Power Flow
0 1 2 3 4 56
7
8
9
10
time in seconds
pow
er fl
ow in
MW
Line 4−12 Active Power Flow
No FACTSUPFC VVdVq
Figure 4.17: Comparison of No FACTS and UPFC in V ,Vd,Vq Mode WhenPL2=3235 MW
CHAPTER 5
LINEARIZED MODELS AND MODAL
DECOMPOSITION OF MULTI-MACHINE SYSTEMS
Following the development of dynamic models with various operating modes, we
generate linearized models of multi-machine systems with VSC-based FACTS con-
trollers for small-signal stability analysis and further study on the control effect
of transmission controllers to inter-area modes. We use the modal decomposition
technique to analyze controllability, observability, and inner-loop gains for shunt,
series, and coupled VSCs in multi-machine systems [57], [58], [59]. For the domi-
nant inter-area modes, the network power flow and voltage sensitivities will be used
to generate the modal quantities. This can be accomplished by using the coherency
idea [60] such that the modal torque is aggregated from the torques on the individ-
ual machines and the modal observability is obtained from a simultaneous, weighted
perturbation of the machine angles.
Analytical controllability and observability conditions of a dynamic system
mostly give a yes or no answer [62]. For practical controller design, controllabil-
ity, and observability are rarely a binary issue and controller interactions may be
a significant consideration. To this end, the modal decomposition technique was
applied to SVC [57] and TCSC [58] to quantify levels of controllability and ob-
servability, which is an extension of the Heffron and Phillips model [63], [64] for a
single-machine infinite-bus system to a power network with multiple swing modes.
But given a mode λi of interest, [57] and [58] require additional effort to relate the
modal angle ∆δmi and speed ∆ωmi to λi. To simply identify the inter-area mode of
interest and corresponding state variables, [59] proposed a complete modal decompo-
sition method by performing canonical state transformation, which fully decouples
all the state modes. The complete multi-modal decomposition can represent all the
swing modes individually such that controllability and observability are defined for
each swing mode. However, [59] ignored the interaction of other swing modes to
the mode of interest. The inner-loop sensitivity KIL(s) are relatively constant for
83
84
all the swing modes in the sense that the inner loop does not include the interac-
tion of them and the dominant component is the network effect. In this thesis, a
modal decomposition approach is proposed based on the method in [59] by using
the canonical state transformation to fully decouple all the state modes. But in
the new approach, the interaction of other swing modes to the mode of interest, is
included in the inner-loop sensitivity KIL(s) to provide a better representation of
the linearized multi-machine systems.
In this chapter, small-signal linearization of a multi-machine system is dis-
cussed in Section 5.1. System modal analysis methods to study the linearized system
and to identify the inter-area mode are described in Section 5.2. The new multi-
machine modal decomposition approach is presented in Section 5.3. An application
example is given in Section 5.4.
5.1 Small-Signal Linearization
The stability of a power system operating point subject to small disturbances
is termed small-signal stability. To test for small-signal stability the system dynamic
equations (4.22-4.23) are linearized about the steady-state operating point to get a
linear set of state equations.
In some programs for small-signal stability, the state matrices are calculated
using the analytical Jacobian of the non-linear state equations. In this thesis work,
on the other hand, linearization is performed by calculating the Jacobian numeri-
cally. This has the advantage of using identical dynamic model codes for transient
and small-signal stability computation. However, there is some loss of accuracy,
particularly in the zero eigenvalue which is characteristic of most inter-connected
power systems.
Starting from the states determined from the model initialization, a small
perturbation is applied to each state in turn. The resulting deviation in the rates of
change of each state divided by the magnitude of the perturbation gives a column
of the state matrix corresponding to the perturbed state. Following each rate of
change calculation, the perturbed state is returned to its equilibrium value and the
intermediate dependent variable values are reset to their initial values. Each step in
85
this process is a single step in the dynamic simulation program. The input matrix
B, the output matrix C, and the feedforward matrix D can be determined in a
similar manner.
The linearized model of a multi-machine system can be expressed in the state
space form
x = Ax + Bu, y = Cx + Du (5.1)
where u and y are the vectors of control and measurement variables, respectively,
and x is the vector of state variables, which can be arranged as
x = [∆δ1 ∆δ2 . . . ∆δn ∆ωg1 ∆ωg2 . . . ∆ωgn zT ]T (5.2)
where the ∆δ’s and the ∆ωg’s represent the perturbed generator angles and speeds,
respectively, and z is the vector of all the other state variables.
5.2 System Modal Analysis
Eigenvalues and eigenvectors, participation factors, and compass plots are uti-
lized to study the small-signal stability of the linearized system (5.1) and to identify
its inter-area modes.
A. Eigenvalue and Eigenvector
The eigenvalues show the damping ratios and frequencies of system modes. A
linear system whose eigenvalues all have negative real parts is stable. The nature of
each mode may be identified from the corresponding eigenvector.
Given an eigenvalue λi, the right eigenvector Ui of the state matrix A satisfies
AUi = λiUi (5.3)
The right eigenvector gives the mode shape, i.e., the relative activity of the state
variables when a particular mode is excited. For example, the degree of activity
of the state variable xj in the ith mode is given by the element uj,i of the right
eigenvector Ui. The magnitudes of the elements of Ui give the extents of the activities
of the n state variables in the ith mode, and the angles of the elements give phase
86
displacements of the state variables with regard to the mode. For a n-mode system,
the right eigenvectors for all the eigenvalues form a n×n matrix U = [U1 U2 . . . Un].
A left eigenvector of the state matrix A is defined as a row vector Vi satisfying
ViA = λiVi (5.4)
The left eigenvector Vi identifies which combination of the original state variables
displays only the ith mode. Thus the jth element of the right eigenvector Ui mea-
sures the activity of the variable xj in the ith mode, and the jth element of the
left eigenvector Vi weighs the contribution of this activity to the ith mode. For
a n-mode system, the left eigenvalues for all the eigenvalues form a n × n matrix
V = [V T1 V T
2 . . . V Tn ]T .
B. Participation factor
Participation factors are nondimensional scalars that measure the interaction
between the modes and the state variables of a linear system. Participation factors
give the sensitivity of an eigenvalue to a change in the diagonal elements of the state
matrix.
Participation factors for an n-mode system are defined as [65] and can be
formed as
pk,i = uj,ivi,j (5.5)
where j = 1, 2, . . . , n, i = 1, 2, . . . , n, uj,i is the element on the jth row and ith
column of the right eigenvector matrix U , and vi,j is the element on the ith row and
jth column of the left eigenvector matrix V of the state matrix A.
The participation factors computed from the eigenvectors associated with the
critical mode provide information in improving voltage stability.
C. Compass Plot
A compass plot shows the rotor angle state terms of the eigenvector of a
complex eigenvalue of interest. The eigenvector associated with a mode indicates
the relative motions of the states which could be observed when that mode is excited.
It enables us to confirm that a particular mode is an inter-area mode, if a group of
generators are oscillating against another group. Besides, the largest components
87
of the eigenvector mean that the inter-area mode may be most readily observed
by monitoring those states. It does not necessarily mean that these states are
necessarily good for controlling the inter-are mode.
5.3 Multi-Machine Modal Decomposition Approach
In addition to its capability of regulating power flow transfer, a FACTS con-
troller can be utilized to improve small-signal stability by providing supplemental
damping control, in addition to its regulation control. To address important de-
sign issues, such as feedback signal selection and regulator selection, multi-machine
modal decomposition is a logical approach to provide valuable insights. The ap-
proach is derived from the state space format in (5.1) as shown below.
Assuming that state matrix A is diagonalizable, there exists a state coordinate
transformation T relating the state vector x in (5.2) to a new state variable vector
xc [59]
xc = Tx (5.6)
such that
x = [∆δm1 ∆δm2 . . . ∆δmn ∆ωmg1 ∆ωmg2 . . . ∆ωmgn zTm]T (5.7)
where the ∆δm’s and the ∆ωm’s represent perturbed modal generator angles and
speeds, respectively. This transformation produces a canonical state-space realiza-
tion of (5.1)
xc = Amxc + Bmu, y = Cmxc + Dmu (5.8)
where the real eigenvalues appear on the diagonal of the state matrix Am and the
88
complex eigenvalues appear in 2-by-2 blocks on the diagonal of Am, that is,
Am = TAT−1 =
σ1 ω1 . . . 0 0 0
−ω1 σ1 0 0 0...
. . ....
...
0 0 σn ωn 0
0 0 . . . −ωn σn 0
0 0 . . . 0 0 Az
(5.9)
and
Bm = TB, Cm = CT−1, Dm = D (5.10)
where σi and ωi, i = 1, . . . , n, are the real and imaginary parts of system swing
modes λi, respectively, and Az is the state matrix where all the other modes appear
on its diagonal.
For a swing mode λi = σi + jωi corresponding to the inter-area mode of
interest, the state variables in xc can be rearranged such that the modal angle ∆δmi
and speed ∆ωmi correspond to become the first and second state variables, resulting
in the system representation
∆δmi
∆ωmi
zr
=
σi ωi 0
−ωi σi 0
0 0 Ar
∆δmi
∆ωmi
zr
+
bmi1
bmi2
Br
u (5.11)
y =[
cm1 cm2 Cr
]
∆δmi
∆ωmi
zr
+ Dmu (5.12)
where zr is the vector of all the other state variables except the inter-area mode
state variables ∆δmi and ∆ωmi, and Ar, Br, and Cr are state matrices associated
with zr.
The main advantage of this system representation is that the state equations
for ∆δmi and ∆ωmi are decoupled from those of the other state variables. As a
89
result, we have
∆δmi
∆ωmi
=
σi ωi
−ωi σi
∆δmi
∆ωmi
+
bmi1
bmi2
(5.13)
ymi =[
cmi1 cmi2
] ∆δmi
∆ωmi
+ Dmiu (5.14)
The transfer function from u to ymi can be expressed as
Tmi(s) =[
cmi1 cmi2
] s − σi −ωi
ωi s − σi
−1
bmi1
bmi2
=
βi1s + βi2
s2 + αi1s + αi2
(5.15)
where αi1 = −2σi, αi2 = σ2i +ω2
i , βi1 = bmi1cmi1 + bmi2cmi2, and βi2 = −σibmi1cmi1 +
ωibmi2cmi1 − σibmi2cmi2 − ωibmi1cmi2.
The numerator of Tmi(s) consists of the product of the controllability Kci(s)
and the observability Koi(s) of the inter-area mode of interest, that is
Tmi(s) = Kci(s)Koi(s) = βi1s + βi2 (5.16)
We construct a block diagram of (5.11)-(5.12) to represent the system using the
transfer functions, as shown in Figure 5.1, where KPSDC(s) is the transfer function
of the power system damping controller. Note that KIL(s), denoted as the inner-
loop transfer function, is the effect from the control u to the measured variable y
excluding the inter-area mode. It consists of KILg(s) and KILn(s), where KILg(s)
is the inner-loop gain due to all the other swing modes and KILn(s) consists of the
effects of the network and all the other modes. The dominant part of KILn(s) is the
effect on the network variables imposed by the FACTS controller.
The transfer functions Kci(s), Koi(s), KPSDC(s), and KIL(s) when evaluated at
s = jω, are complex, providing both gain and phase information. This information
can be used to design damping control, as stated in the subsequent chapter.
90
Koi(s)
KIL(s)
Inner LoopFeedback
KPSDC(s)u
Power System Damping Controller
y
Kci(s)
++
Inter-Area Mode of Interest
s1
s1
αi1
αi2
_
_
Inner Loop
Figure 5.1: Modal Decomposition of a Linearized Multi-Machine Systemwith a Network Controller
5.4 Application: A 20-Bus System Study
Figure 5.2 shows the one-line diagram of the 20-bus test system. The active
power values of generators, loads, and line flows are also displayed in the diagram.
The PST data of the test system is listed in Appendix B.
The system is linearized to obtain state-space matrices, and then eigenvalues,
participation factors, and compass plots are studied. The number of dynamic states
in this model is 40, with 6 generator states and 2 exciter states for each generator.
Table 5.1 shows all the modes for the test system. Note that Mode 1 is effectively
the zero eigenvalue.
By reviewing the participation factors, we identify that −0.29 ± j1.79 (mode
6,7), −0.22 ± j5.81 (mode 10,11), −0.35 ± j6.72 (mode 12,13), and −0.47 ± j9.32
(mode 16,17) are the four swing modes for the five generators.
Figure 5.3 shows the compass plots of the rotor angle state terms of the swing
91
4
23
7
9
6
8
10
1
5
11
1213
14
15
17
19
20
18
16
VSC 2
Load 1Load 2
VSC 1
G 1
G 2
G 3
G 4
G 5
VSC 3
522
1433
786
628
1177
15262700
59
Figure 5.2: 20-Bus Test System Single-Line Diagram and Flows
mode eigenvectors. We observe that eigenvalues 10 and 11 correspond to the inter-
area mode between the northwest and southeast areas, the damping ratio of which
is less than 0.05.
Based on above analysis, we identify the inter-area mode of interest, which has
a damping ratio of δ = −0.22 and a frequency of ω = 5.81 rad/sec (Mode 10,11).
It is mainly due to Generators 1, 2, and 3 in the northwest area oscillating against
Generator 5 in the southeast area.
92
0.2
0.4
0.6
0.8
30
210
60
240
90
270
120
300
150
330
180 0
G4G3G2G1G5
0.02
0.04
0.06
0.08
30
210
60
240
90
270
120
300
150
330
180 0
G2G3G5G4G1
(a) Swing Mode 6,7 (b) Swing Mode 10,11
0.05
0.1
0.15
30
210
60
240
90
270
120
300
150
330
180 0
G2G3G5G4G1
0.05
0.1
0.15
0.2
0.25
30
210
60
240
90
270
120
300
150
330
180 0
G3G2G5G4G1
(c) Swing Mode 12,13 (d) Swing Mode 16,17
Figure 5.3: Compass Plots for the Four Swing Modes
5.5 Summary and Conclusions
In this chapter, linearized models using small-signal perturbations based on
nonlinear system dynamic models are described. This method allows the lineariza-
tion and nonlinear dynamic models to share the same codes for generators, exciters,
and transmission controllers. Eigenvalue analysis, participation factors, and com-
pass plots are good tools to study the linearized system to identify the inter-area
mode of interest.
93
The new multi-machine modal decomposition is an approach to quantify the
inner-loop transfer function and the product of the controllability and observability
transfer functions for a multi-machine systems with FACTS controllers. In next
chapter, we will study damping control design based on the modal decomposition
approach.
94
Table 5.1: State Modes of the 20-Bus System
Mode # Eigenvalues Damping Ratio Frequency (Hz)1 4.92 × 10−5
2 −0.203 −0.554 −0.665 −0.78
6,7 −0.29 ± j1.79 0.160 0.2848 −1.989 −3.07
10,11 −0.22 ± j5.81 0.038 0.92512,13 −0.35 ± j6.72 0.051 1.07014 −6.7915 −8.94
16,17 −0.47 ± j9.32 0.050 1.48318,19 −6.98 ± j9.43 0.595 1.50120,21 −8.78 ± j8.72 0.710 1.38722,23 −5.97 ± j12.99 0.417 2.06724,25 −7.67 ± j12.57 0.521 2.00126,27 −7.30 ± j16.28 0.409 2.59128 −20.7329 −22.7430 −24.5631 −27.9632 −30.1533 −33.2134 −37.1535 −38.7536 −100.4937 −100.6238 −100.8939 −102.3840 −103.41
CHAPTER 6
DAMPING CONTROLLER DESIGN
In [57] and [58] influence factors and control effectiveness were established to select
appropriate controller input signals for SVC and TCSC, respectively. Using these
techniques it has been demonstrated in several example power systems that for
effective damping control, shunt controllers (like SVC) should use flow (such as
line current magnitude) measurements and series controllers (like TCSC) should
use nodal (bus voltage) measurements [66], [67]. This duality in controllability
and observability is perhaps not surprising because intuitively a series controller
regulating line flows would need an orthogonal nodal signal to get the damping
information. The converse is true for a shunt controller.
Reference [68] presented the control strategies of the UPFC, controllable series
capacitor, and quadrature boosting transformer (QBT) for damping of electrome-
chanical power system oscillations based on the Control Lyapunov Functions (CLF).
The authors in [68] used Lyapunov function candidates in feedback design itself by
making the Lyapunove derivative negative when choosing the control. However,
considering the main functions of FACT controllers are to regulate system voltages
and power flows while damping power system oscillations is a supplemental function,
we aim to design the damping controllers based on their regulation controllers.
In this chapter, we use the new modal decomposition technique described in
Chapter 5 to analyze controllability, observability, and inner-loop gains for shunt,
series, and coupled VSCs in multi-machine systems to investigate the design of VSC-
based damping controllers supplemental to their regulation controllers for inter-area
modes. Although we will be interested in all FACTS Controllers, the design for
stand-alone shunt VSCs (STATCOMs) follows a similar line as in SVC [57]. The
focus here is on series VSCs, and series SVCs coupled to other VSCs (such as UPFCs
and IPFCs). Several papers [49], [70] on damping control design for UPFCs have
been published, but most of these papers only discussed the design mechanism.
Here we will pursue a comprehensive approach that will examine the selection of
95
96
damping controller input signals, the controller gain limitation, and modal damping
selectivity.
This chapter is organized as follows. Section 6.1 gives the block diagram
of a damping controller to be designed. For damping input signal selection, three
quantifiable indices based on the new multi-machine modal-decomposition approach
are described in Section 6.2. Design examples for the STATCOM, SSSC, UPFC,
and IPFC are discussed in Sections 6.3-6.5.
6.1 Damping Controller Block Diagram
Figure 6.1 shows a damping controller supplemental to a regulator of FACTS
controllers. The damping controller KPSDC(s) is designed to consist of a washout
loop Gw(s), a phase compensator Gp(s), a low pass (LP) filter Gf (s), a constant
gain k, and saturation limits [umin, umax]. The damping signal u, which is added
with the error signal and sent to a regulator of FACTS controllers, can be expressed
as
u =
umax, if u ≥ umax
KPSDC(s)y, if umin < u < umax
umin, if u ≤ umin
(6.1)
where
KPSDC = kGf (s)Gp(s)Gw(s) (6.2)
Error signal+
11+Tf s 1+Tds
1+Tns1+Tws
Tws Measuredsignal y
k
+Dampingsignal u
FACTSRegulator
PowerSystem
umax
uminGf (s) Gp(s) Gw(s)
KPSDC(s)
Figure 6.1: Damping Controller Block Diagram
97
The washout transfer function is
Gw(s) =Tws
1 + Tws(6.3)
where Tw is the time constant. Its time-domain state equation is derived as
xw = (y − xw)/Tw
Fo1 = y − xw
(6.4)
where xw is the state variable. Its realization in block diagram is shown in Figure
6.2.
+
1Tw s
1
Measuredsignal y
_
Fo1
xw. xw
Figure 6.2: Washout Loop Block Realization
The phase compensation design is
Gp(s) =1 + Tns
1 + Tds(6.5)
where Tn and Td are constant coefficients. Its time-domain state equation is derived
as
xp = (Fo1 − Fo2)/Td
Fo2 = xp + Fo1Tn/Td
(6.6)
where xp is the state variable. Its realization in block diagram is shown in Figure
6.3.
Coefficients Tn and Td are designed based on phase compensation to generate
a proper damping signal. If phase-lead compensation is needed, we have
Gp(s) = αlds + zld
s + pld
(6.7)
98
+1Tds
1_
Fo2
xp.xp
Td
Tn
Fo1+
+
Figure 6.3: Phase Compensator Block Realization
under the conditions of√
zldpld = ω (6.8)
pld = αldzld, αld > 1 (6.9)
where ω is the frequency of the mode of interest to be compensated and αld deter-
mines the amount of the phase-lead compensation. The larger the αld, the higher
the phase-lead compensation is, although the relationship is not linear. In phase
compensation design, the values of αld are varied to find an optimal choice.
For a given try value of αld, the coefficients Tn and Td of a phase-lead com-
pensator can be calculated based on (6.7)-(6.9) as
Tn =√
αldω, Td =1√αldω
(6.10)
For phase-lag compensation, we have
Gp(s) =1
αlg
s + zlg
s + plg
(6.11)
under the conditions of√
zlgplg = ω (6.12)
pld =zlg
αlg
, αld > 1 (6.13)
The larger the αlg, the higher the phase-lag compensation is. Similarly, the coeffi-
cients Tn and Td for a phase-lag compensator can be calculated based on (6.11)-(6.13)
99
as follows
Tn =1√αlgω
, Td =√
αlgω (6.14)
The low pass filter has a transfer function
Gf (s) =1
1 + Tfs(6.15)
where Tf is the time constant. Its time-domain state equation is derived as
xf = (Fo2 − xf )/Tf
Fo3 = xf
(6.16)
where xf is the state variable. Its realization in block diagram is shown in Figure
6.4.
+1Tfs
1_
Fo3 xf.
xf Fo2
Figure 6.4: Low Pass Filter Block Realization
State equations (6.4), (6.6), and (6.16) are incorporated into the dynamic
simulation program, together with other state equations for FACTS controllers in
Chapter 4. Then all the dynamic equations are linearized by performing small
perturbations.
Given a choice of shunt and series VSC configurations, two important issues
in damping control design are: which regulator configuration should be used and
which measured signal should be used as the damping input signal y. For a specified
regulator and measured signal, the damping controller as shown in Figure 6.1 can
be readily designed based on root-locus plot and bode plot techniques.
100
6.2 Input Signal Selection
With the tools available to analyze observability and inner-loop gain restric-
tions, we evaluate the use of local signals for inter-area mode damping enhancement
for each FACTS Controller. The local signals include bus frequencies, bus voltages,
and line currents and power flows. Specifically, bus voltage magnitudes, active pow-
ers of the compensated lines, and line current magnitudes are considered because
those variables normally have positive quantities and thus do not change signs. The
objectives are to investigate the suitability of the signals for shunt VSC controllers
and series VSC controllers, and to develop some guidelines for using these signals.
Based on the block diagram in Figure 5.1, the effective control action Kei(s)
for a power system damping controller to be designed can be expressed as
Kei(s) = Kci(s)KPSDC(s)
1 − KPSDC(s)KIL(s)Koi(s) (6.17)
which describes the impact of a given damping controller KPSDC(s) on the ith swing
mode.
By analyzing the transfer functions and the effective control action at the
frequency of the inter-area mode, we create two useful indices which provide insights
to the performance of a damping controller with the given measurements.
A. Maximum Damping Influence (MDI) Index
By assuming that |KPSDC(s)KIL(s)| >> 1, the gain of the effective control
action in (6.17) can be simplified as
MDI =
∣∣∣∣∣Kci(s)Koi(s)
KIL(s)
∣∣∣∣∣ (6.18)
which is denoted as the maximum damping influence (MDI) index [58]. The MDI
index is a measure of the eigenvalue shift per unit control gain achievable based on
the assumption. The value of the MDI index is that it indicates the effectiveness of
measurements having high observability gain and low inner-loop gain. The MDI is
a useful index to exclude those damping signal candidates who are not suitable as
input signals.
There is also the gain margin consideration that the control gain of KPSDC(s)
101
is limited, such that the assumption |KPSDC(s)KIL(s)| >> 1 is not applicable by
itself. Thus, it is necessary to create other indices to evaluate the candidates chosen
by the MDI index.
B. Controllability and Observability Gain Product Index
A KPSDC(s) with a smaller control gain is preferable because a smaller gain
would provide a higher gain margin. Considering that the candidates chosen by the
MDI index and inner-loop gain index usually have small inner-loop gain, we can
assume that |KPSDC(s)KIL(s)| << 1. The first derivative of the effective control
action gain to the gain k of KPSDC(s) = kGc(s) is expressed as
∣∣∣∣∣dKei(s)
dk
∣∣∣∣∣ =
∣∣∣∣∣ Kci(s)Gc(s)Koi(s)
(1 − kGc(s)KIL(s))2
∣∣∣∣∣ (6.19)
Under the assumption of |KPSDC(s)KIL(s)| << 1, (6.19) can be simplified as
∣∣∣∣∣dKei(s)
dk
∣∣∣∣∣ = |Kci(s)Koi(s)| · |Gc(s)| (6.20)
which indicates that as k increases from zero, the larger the product of controlla-
bility and observability gains is, the faster the control effect changes. This means
a relatively small k could achieve the desired damping improvement. Thus a large
value of the product of controllability and observability gains is preferred. Based
on this discussion, we select candidate signals with high values of the product of
controllability and observability gains from those ranked by the MDI index. They
become candidates for damping controller design for testing by dynamic simulations.
The candidate that achieves the best dynamic performance will be finalized as the
selected signal for damping controller design.
6.3 Design for the STATCOM
Considering designing a damping controller for a STATCOM, which has only a
single damping signal input option, the issue left is to select a damping input signal
from a list of candidate signals.
Figures 6.5 and 6.6 display the MDI indices calculated for a STATCOM on
102
Bus 4 of the same 20-bus test system as shown in Figure 5.2. The STATCOM is
operated in the Var control mode. The list of candidate signals contain three local
signals V4, P4−11, and Im4−12.
52
56
60
64Ki=0.005Ki=0.1Ki=1Ki=5
Ki=0.005Ki=0.1Ki=1Ki=523
24
25
26
5.65
5.7
5.75
5.8
5.85
0 0.05 0.1 0.15 0.2Shunt Kp
Ki=0.005Ki=0.1Ki=1Ki=5
Figure 6.5: STATCOM MDI Index Plots Varying Kp
In Figure 6.5, the four curves in each subplot are corresponding to the integral
gain Ki of the Var regulation loop set to 0.005, 0.1, 1, and 5, respectively. When
the proportional gain Kp of the Var regulation loop varies from 0.0 to 0.2, the MDI
103
52
56
60
64
68Kp=0.005Kp=0.01Kp=0.1Kp=0.5
23
24
25
26
Kp=0.005Kp=0.01Kp=0.1Kp=0.5
5.6
5.7
5.8
5.9
6.0
0.001 0.5 1 1.5 2Shunt Ki
Kp=0.005Kp=0.01Kp=0.1Kp=0.5
Figure 6.6: STATCOM MDI Index Plots Varying Ki
indices for signals Im4−11, P4−11, and V4 converge to 53.0, 25.6, and 5.7, respectively.
For a higher Ki, the MDI indices of a signal are closer to the converged value. When
Ki is high enough, the MDI indices of a signal are not sensitive to the varying of
Kp.
The four curves in Figure 6.6 are corresponding to Kp =0.005, 0.01, 0.1, and
0.5, respectively. When Ki increases from 0.001 to 2.0, the MDI indices for signals
Im4−11, P4−11, and V4 converge to 53.0, 25.6, and 5.7, respectively. For a higher
Kp, the MDI indices of a signal are closer to its converged value. When Kp is high
enough, the MDI indices of a signal are not sensitive to the varying of Ki.
104
For any combination of Kp and Ki in Figures 6.5 and 6.6, we observe that
using Im4−11 as the damping input signal has the highest MDI indices. Thus, Im4−11
is selected as the damping input signal for the STATCOM damping controller to
design. It is consistent with the conclusions in [61], [66], and [67] that flow variables
are more suitable for shunt devices.
We show an example of designing the damping controller on the STATCOM
in the Var control mode with regulation gains Kp = 0.01 and Ki = 0.1. Based on
root-locus plots, the damping controller using input signal Im4−11 is designed as
u = 20 · 0.1s
1 + 0.1s· 1
1 + 0.1s(−0.1 ≤ u ≤ 0.1) (6.21)
The damping signal u(t) in dynamic simulation is shown in Figure 6.7.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
time in seconds
per
unit
damping signal
Figure 6.7: STATCOM Damping Controller Signal
Figures 6.8 and 6.9 show the dynamic simulation results without and with the
designed damping controller on the STATCOM. In the simulation, the STATCOM
Var reference Ishqref has a step increase from 0.0 pu to 0.1 pu at time t = 0.1
s. As shown in Figure 6.8, the damping controller results in substantial damping
improvement on the bus voltage V4, line current Im4−11, and line power flows P4−11
and Q4−11.
Figure 6.9 shows the STATCOM variables including the shunt reactive power
injection Ishq, the DC bus voltage Vdc, and the inverter voltage magnitude Vm1
105
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.031
1.0315
1.032
1.0325
1.033
time in seconds
AC
vol
tage
per
uni
t
from−bus voltage magnitude V4
0 1 2 3 4 57.462
7.464
7.466
7.468
7.47
7.472
time in seconds
curr
ent p
er u
nit
line current magnitude Im4−11
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
7.7
7.705
7.71
7.715
7.72
time in seconds
pow
er fl
ow p
er u
nit
line active power P4−11
0 1 2 3 4 5−0.028
−0.026
−0.024
−0.022
−0.02
−0.018
−0.016
−0.014
time in secondspo
wer
flow
per
uni
t
line reactive power Q4−11
no dmp
dmp input Im4−11
Figure 6.8: Dynamic Simulation with a STATCOM Damping Controller– I
and angle αsh, all of which are affected to have larger oscillations due to injecting
the damping signal into the Var regulator. We also notice that with the damping
controller the voltage V4 and the line power flows P4−11 and Q4−11 have extra drops
between t=0.2 s and t=0.5 s (Figure 6.8). This is the side effect due to the high
injection of the damping signal at that time range. The high injection arises from the
quick drop of the damping input signal at the moment of the setpoint step change.
These are the costs needed to pay for improving system damping by building a
feedback damping controller on the STATCOM’s regulation control.
6.4 Design for the SSSC
An SSSC also has only a single damping signal input option, the issue left is
to select a damping input signal from a list of candidate signals.
To study damping controller design for the SSSC, an SSSC is inserted into Line
4-11 of the same 20-bus system as shown in Figure 5.2. The SSSC is operated in the
106
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.05
0.1
0.15
time in seconds
curr
ent p
er u
nit
shunt reactive current Ishq
0 1 2 3 4 51.235
1.24
1.245
1.25
1.255
1.26
1.265
1.27
1.275x 10
4
time in seconds
DC
vol
tage
vol
ts
DC bus voltage Vdc
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.03
1.035
1.04
1.045
1.05
1.055
1.06
time in seconds
AC
vol
tage
per
uni
t
invter voltage magnitude Vm1
0 1 2 3 4 5−6
−5
−4
−3
−2
−1
0
1
2x 10
−4
time in secondsan
gle
radi
ans
invter voltage angle αsh
no dmpdmp sig I
m4−11
Figure 6.9: Dynamic Simulation with a STATCOM Damping Controller– II
inverter Vm control mode. The MDI indices calculated for the SSSC using signals
V4, P4−11, and Im4−11 as the damping input signal are displayed by the 3 curves
in Figure 6.10, respectively. In this plot, the proportional gain Kp of the SSSC
regulator varies from 0.3 to 25 while the integral gain Ki is kept on Ki = 10Kp. As
Kp increases, the MDI indices for V4, P4−11, and Im4−11 converge to 56.5, 2.1, and
2.8, respectively. When Kp ≥ 1, signal V4 has much higher MDI index than the
other two signals. So V4 is selected to be the damping input signal for the SSSC.
It is consistent with the conclusions in [61], [66], and [67] that nodal variables are
more suitable for series devices.
We show an example of designing a damping controller on the SSSC in the
inverter Vm control mode with regulation gains Kp = 20 and Ki = 200. Based on
root-locus plots, the damping controller using input signal V4 is designed as
u = 10 · 0.1s
1 + 0.1s· 1 + 0.381s
1 + 0.0762s· 1
1 + 0.1s(−0.1 ≤ u ≤ 0.1) (6.22)
107
MDI Index
0
20
40
60
80
5 10 15 20 25Series Kp
Signal V4Signal P4-11Signal Im4-11
1 2
Figure 6.10: SSSC MDI Index Plots Varying Regulation Control Gains
The damping signal u(t) in dynamic simulation is shown in Figure 6.11.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−6
−4
−2
0
2x 10
−3
time in seconds
per
unit
damping signal
Figure 6.11: SSSC Damping Controller Signal
Figures 6.12 and 6.13 show the dynamic simulation results without and with
the designed damping controller for the SSSC. In the simulation, the SSSC inverter
voltage reference Vmref has a step decrease from −0.01 pu to −0.02 pu at time t = 0.1
s. As shown in Figure 6.12, the damping controller results in substantial damping
improvement on the bus voltage V4, line current Im4−11, and line power flows P4−11
and Q4−11.
Figure 6.13 shows the SSSC variables including the DC bus voltage Vdc and the
inverter voltage magnitude Vm2 and angle αse, all of which are affected to have larger
oscillations due to injecting the damping signal into the inverter Vm regulator. We
also notice that with the damping controller the current Im4−11 and the line power
108
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.0295
1.03
1.0305
1.031
time in seconds
AC
vol
tage
per
uni
t
from−bus voltage magnitude V4
0 1 2 3 4 57.05
7.1
7.15
7.2
7.25
7.3
7.35
time in seconds
curr
ent p
er u
nit
line current magnitude Im4−11
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 57.25
7.3
7.35
7.4
7.45
7.5
7.55
time in seconds
pow
er fl
ow p
er u
nit
line active power P4−11
0 1 2 3 4 5
−0.15
−0.1
−0.05
−0.2
time in seconds
pow
er fl
ow p
er u
nit
line reactive power Q4−11
no dmp
dmp input V4
Figure 6.12: Dynamic Simulation with an SSSC Damping Controller – I
flows P4−11 and Q4−11 have extra swells between t=0.27 s and t=0.7 s (Figure 6.12).
This is the side effect due to the high injection of the damping signal at that time
range. These are the costs needed to pay for improving system damping by building
a feedback damping controller on the SSSC’s regulation control.
6.5 Design for the UPFC
Unlike a STATCOM or an SSSC, a UPFC can have multiple shunt and series
regulators. Each regulator can be used to add damping signals to damp the inter-
area mode. More important, the regulators also interact with each other.
6.5.1 UPFC Series Regulator in Vd,Vq Mode
If the series VSC of a UPFC is operated in the inverter Vd,Vq control mode,
it does not have series regulators. Table 6.1 shows the MDI index values for the
UPFC in Vd,Vq mode compared with the STATCOM in same operating conditions
and control gains of their shunt VSCs. The shunt voltage control setpoint is set
109
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52000
3000
4000
5000
6000
time in seconds
DC
vol
tage
vol
ts
DC bus voltage Vdc
no dmpdmp input I
m4−11
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
time in seconds
AC
vol
tage
per
uni
t
series inverter voltage magnitude Vm2
0 1 2 3 4 5
1.48
1.5
1.52
1.54
1.56
1.58
1.6
1.62
time in seconds
angl
e ra
dian
s
invter voltage angle αse
no dmpdmp input V
4
Figure 6.13: Dynamic Simulation with an SSSC Damping Controller – II
to Vref = 1.031 pu and the Var control setpoint is set to Ishqref = 0.0 pu. The
control gains in the shunt voltage control mode are Kv = 500, α = 0.03, Kp = 0.01,
and Ki = 0.1, and the control gains in the shunt Var control mode are Kp = 0.01
and Ki = 0.1. The setpoints of the series VSC are set to Vdref = 0.0199 pu and
Vqref = −0.0373.
Table 6.1 shows that the MDI index values of the UPFC are close to those
of the STATCOM. It implies that the coupled series VSC in Vd,Vq mode has little
impact on the damping control of the shunt regulator. Using Im4−11 as the damping
input signal to the shunt regulator has the highest MDI indices for the UPFC in
Vd,Vq mode and thus Im4−11 is selected as the damping input signal.
6.5.2 Impact of the Series P ,Q Regulators
When the series VSC of a UPFC is operated in the line P ,Q control mode,
it has shunt and series regulators interacting to each other. In order to examine
the impact of the series regulators, we vary the regulation gains Kp of the series P
regulator and Q regulators from 0 to a higher value and keep Ki = 10Kp. When
110
Table 6.1: MDI Indices for the UPFC Series Vd,Vq Mode v.s. the STAT-COM
Local MDI ValuesSignals STATCOM UPFC Series in Vd,Vq Control
Voltage Control Var Control Shunt Voltage Control Shunt Var ControlV4 4.92 5.70 4.89 5.65
P4−11 26.56 25.54 22.90 21.77Im4−11 64.39 53.34 56.96 48.33
Kp = 0, the UPFC is actually degraded into Var,Vd,Vq mode with the two series
regulators disabled. By increasing Kp, the impact of the series regulators increases.
The three plots in Figure 6.14 show the MDI indices for the shunt Var regula-
tor, series P regulator, and series Q regulator of the UPFC in Var,P ,Q control mode,
respectively. The three curves in each plot are corresponding to the MDI indices us-
ing V4, P4−11, and Im4−11 as the damping input signal, respectively, as Kp increases
from 0 to 0.2. In the simulation, the shunt Var setpoint is set to Ishqref = 0.0 pu,
and its control gains are Kp = 0.01 and Ki = 0.1. The series VSC setpoints are set
to Pref = 820 MW and Qref = −2 MVar.
For the shunt regulator, when Kp = 0, which is the case without the impact
of the two series regulators, the signal Im4−11 has the highest MDI indices. As Kp
increases from 0 to 0.2, which means the impact of the series regulators increases,
the MDI indices of Im4−11, P4−11, and V4 drop from 57.4 to 12.3, 20.9 to 14.8, and
5.6 to 4.4, respectively. Based on the above observation, we conclude that the signal
Im4−11, which is a good damping input signal for the STATCOM and the UPFC in
Vd,Vq mode, will be affected significantly by the coupled series P ,Q regulators.
In the second plot the MDI indices of V4, Im4−11, and P4−11 to the series P
regulator decrease from 23.8 to 6.3, 2.9 to 1.2, and 2.1 to 0.04, respectively, as Kp
varies from 0 to 0.2. Among these three signals, V4 has the highest MDI indices.
But the MDI indices of V4 is less than 10 when Kp ≥ 0.03, which is low for a good
damping input signal.
The third plot shows that the MDI indices of Im4−11, P4−11 and V4 to the series
Q regulator decrease from 44.1 to 6.0, 13.6 to 5.8, and 6.6 to 5.0, respectively, as
Kp varies from 0 to 0.2. When Kp ≥ 0.5, all MDI indices of the three signals are
111
MDI Index
0
10
20
30
40
50
60Signal V4Signal P4-11Signal Im4-11
0
5
10
15
20
25Signal V4Signal P4-11Signal Im4-11
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1Series Kp
Signal V4Signal P4-11
Signal Im4-11
Figure 6.14: MDI Index of the UPFC in V ar,P ,Q Mode
less than 10. As a result, none of them is selected as the damping input signal when
Kp ≥ 0.5.
6.5.3 Dynamic Simulation of the UPFC Damping Controllers
In Subsections 6.5.1 and 6.5.2, we showed that the MDI indices of signal Im4−11
for the shunt VSC damping control of a UPFC will not be affected by coupling with
the series VSC in the Vd,Vq control mode, but will be affected largely by the series
112
P ,Q regulators. Thus the UPFC with its series in the Vd,Vq control mode is more
suitable to design a damping controller on its shunt VSC. We use the same 20-bus
system (Figure 5.2) to demonstrate this conclusion. Figures 6.15 (a) and (b) show
the root-locus plots of the four system swing modes in designing a damping controller
on the shunt VSC Var regulator for the UPFC series in the inverter Vd, Vq control
mode and the line P ,Q control mode, respectively. The shunt VSC is operated in
Var control mode with control gains Kp = 0.01 and Ki = 0.1. The series VSC
is inserted into Line 4-12. The control gains of the P ,Q regulators are Kp = 0.1
and Ki = 1. The damping input signal is Im4−11 and is the damping controller is
expressed as
u = k · 0.1s
1 + 0.1s
1
1 + 0.1s− 0.1 ≤ u ≤ 0.1 (6.23)
−2 −1.5 −1 −0.5 01
2
3
4
5
6
7
8
9
10
Real
Imag
Root−Locus Plot
k=30
k=0
−2 −1.5 −1 −0.5 01
2
3
4
5
6
7
8
9
10
Real
Imag
Root−Locus Plot
k=0
k=30
(a) UPFC Series in Vd,Vq Control (b) UPFC Series in P ,Q Control
Figure 6.15: UPFC Root-Locus Plots of the Four Swing Modes
As shown in Figures 6.15 (a) and (b), when the control gain k of the damping
controller increases from 0 to 30, the real part of the inter-area mode is moved from
−0.55 to −1.08 in Vd,Vq control and from −0.27 to −1.59 in P ,Q control, respectively.
Thus, the former one has better damping effect on the inter-area mode.
113
Figures 6.16-6.18 show the dynamic simulation results without and with the
designed damping controller (k = 30) on the shunt Var regulator of the UPFC in
Vd,Vq control. In the simulation, the shunt Var setpoint is set to Ishqref = 0.0 pu and
the series setpoints are set to Vdref = 0.0199 pu and Vqref = −0.0373 pu. A 0.001 pu
step increase of Vdref is applied at time t = 0.1 s.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.04
−0.02
0
0.02
0.04
time in seconds
per
unit
damping signal
Figure 6.16: UPFC Damping Controller Signal
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.034
1.0345
1.035
1.0355
1.036
time in seconds
AC
vol
tage
per
uni
t
from−bus voltage magnitude V4
0 1 2 3 4 58.076
8.077
8.078
8.079
8.08
8.081
8.082
8.083
8.084
time in seconds
curr
ent p
er u
nit
line current magnitude Im4−11
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 58.185
8.19
8.195
8.2
8.205
time in seconds
pow
er fl
ow p
er u
nit
line active power P4−11
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−0.22
−0.215
−0.21
−0.205
−0.2
time in seconds
pow
er fl
ow p
er u
nit
line reactive power Q4−11
no dmp
dmp input Im4−11
Figure 6.17: Dynamic Simulation with a UPFC Damping Controller – I
Figure 6.17 shows that the damping controller results in substantial damping
improvement on the bus voltage V4, line current Im4−11, and line power flows P4−11
114
and Q4−11. Figure 6.18 shows the oscillations on inverter variables including the DC
bus voltage Vdc, the shunt inverter voltage magnitude Vm1 and angle αsh, and the
series d-axis and q-axis voltages Vd and Vq, all of which are affected to have larger
oscillations due to injecting the damping signal into the shunt Var regulator.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.02
0.04
0.06
0.08
0.1
0.12
time in seconds
curr
ent p
er u
nit
shunt reactive current Ishq
0 1 2 3 4 5
1.24
1.245
1.25
1.255
1.26
1.265
1.27x 10
4
time in seconds
DC
vol
tage
vol
ts
DC bus voltage Vdc
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.03
1.035
1.04
1.045
1.05
1.055
1.06
time in seconds
AC
vol
tage
per
uni
t
shunt inverter voltage magnitude Vm1
0 1 2 3 4 50.025
0.0255
0.026
0.0265
time in seconds
angl
e ra
dian
s
shunt inverter voltage angle αsh
no dmp
dmp input Im4−11
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.0198
0.02
0.0202
0.0204
0.0206
0.0208
0.021
time in seconds
AC
vol
tage
per
uni
t
Series inverter d−axis voltage Vd
0 1 2 3 4 5−1.5
−1
−0.5
0
0.5
1
time in seconds
AC
vol
tage
per
uni
t
Series inverter q−axis voltage Vq
no dmp
dmp input Im4−11
Figure 6.18: Dynamic Simulation with a UPFC Damping Controller – II
6.6 Design for the IPFC
We use the 20-bus test system, as shown in Figure 5.2, to illustrate the damping
signal selection process for an IPFC in line power flow control mode. The IPFC is
115
located in Line 4-11 and Line 4-12. The VSC in Line 4-11 is operated as the
Master and the VSC in Line 4-12 is operated as the Slave. The IPFC has 5 possible
regulators: Master capacitor Vdc regulator, Slave capacitor Vdc regulator, Master
line P regulator, Slave line Q regulator, and Slave line P regulator, denoted as R1,
R2, R3, R4, and R5, respectively. The control gains of these regulator are given in
Appendix B.
First, the MDI index is used to exclude inappropriate signals. Table 6.2 shows
the MDI values at the inter-area mode frequency ω = 5.87 rad/s for a list of mea-
sured local signals to each regulator when both VSCs of the IPFC are operated in
power flow control mode. In this example only local measured signals, including bus
voltages, line currents, and line flows, are considered.
Table 6.2: MDI Index Values for Measured Signals to IPFC Regulators
MDI Indexy R1 R2 R3 R4 R5
V4 80.685 44.239 89.687 71.262 91.106P4−11 1.136 1.238 0.343 1.188 14.628Im4−11 0.785 0.834 1.492 0.748 48.15P4−12 1.169 1.046 32.455 1.201 0.890Im4−12 0.597 0.543 10.371 0.623 1.327
As shown in Table 6.2, seven high-MDI elements are highlighted as candidates
for further investigation. We observe that the from-bus voltage magnitude signal V4
is good for all the regulators, the Slave line power signal P4−12 is good for the Master
line P regulator, and the Master line current signal Im4−11 is good for the Slave line
P regulator. This can be explained by the fact that these signals have smaller
sensitivities from the corresponding regulators function, which means smaller inner-
loop gains and thus higher MDI values. Next, the controllability and observability
gain product index is used as an indicator of good damping signals.
Table 6.3 gives the controllability and observability gain product index values
for the four candidate signals selected in the first round. As shown in Table 6.3, the
foure elements in Columns R3 and R5 have high controllability and observability
gain product values and are highlighted as candidates for damping controller design
for testing in dynamic simulations. Figure 6.19 shows the IPFC from-bus voltage
116
Table 6.3: Controllability and Observability Gain Product Index
|Kci(ω)Koi(ω)| Indexy R1 R2 R3 R4 R5
V4 0.00750 0.00797 0.1127 0.02613 0.14589Im4−11 2.0522P4−12 0.63726
magnitude response to a 0.1 pu step change of Master line active power reference at
time t = 0.1 seconds. The five curves represent the cases without damping control
and with damping controllers designed by using the four candidate signals. The
details of the four designed damping controllers are given in Table 6.4. We observe
that the case using measured signal Im4−11 reduced the damping effect in the first 3
circles of oscillation. So this signal is not recommended. Compared the other three
cases with damping controllers, the cases using V4 achieve better damping effect on
the bus voltage. As a result, the two candidates V4, R3 and V4, R5 are selected for
final consideration.
Table 6.4: Designed Damping Controllers for the IPFC
Damping Controller ExpressionV4,R3 u = −300 · 0.1s
1+0.1s· 1+0.341s
1+0.0852s· 1
1+0.1s, −0.1 ≤ u ≤ 0.1
V4,R5 u = −200 · 0.1s1+0.1s
· 1+0.341s1+0.0852s
· 11+0.1s
, −0.1 ≤ u ≤ 0.1
P4−11,R3 u = 40 · 0.1s1+0.1s
· 1+0.341s1+0.0852s
· 11+0.1s
, −0.1 ≤ u ≤ 0.1
Im4−11,R5 u = 3 · 0.1s1+0.1s
· 11+0.1s
, −0.1 ≤ u ≤ 0.1
Figures 6.20 and 6.21 show the line flows and DC capacitor voltage of the
two cases using V4, compared with the case without damping control. The case
V4,R3 damps the inter-area mode by injecting a damping signal to the Master line
P regulator, which results in a 6.1 MW oscillation of the Master line P in the first 2
seconds. The case V4,R5 damps the inter-area mode by injecting a damping signal to
the Slave line P regulator, which results in a 4.4 MW oscillation of the Slave line P in
the first 2 seconds. Considering the operation of an IPFC, the main function of the
Master VSC is to regulate the Master line flow, while the more important function
of the Slave VSC is not to regulate the Slave line flow but to provide real power
circulation to the Master VSC. Thus a smooth Master line active power response is
117
more preferable than a smooth Slave one. As a result, V4,R5 is recommended and
considered as the best damping signal in this example.
Figure 6.19: Bus 4 Voltage without and with Damping Controllers
0 1 2 3 4 51.1995
1.2
1.2005
1.201
1.2015
1.202
1.2025x 10
4
time in seconds
DC
vol
tage
vol
ts
IPFC capacitor voltage Vdc
no dmpdmp:V
4,R
3
dmp:V4,R
5
Figure 6.20: DC Capacitor Voltage without and with Damping Con-trollers
6.7 Summary and Conclusions
In this chapter, two indices are derived from the effective control actions based
on the multi-machine modal decomposition technique and are used in damping input
118
signal selection. The damping controller design processes for the the STATCOM,
SSSC, UPFC, and IPFC are discussed. The designed damping controllers show
substantial improvement of the damping effect on the system inter-area mode.
However, we notice that the damping controllers built on the regulators of
FACTS controllers will cause larger oscillations in the inverter variables. Due to
high injection of the damping signal at the moment of faults or oscillations, a large
disturbance may cause the damping controllers ineffective or even cause severe extra
system oscillations.
The damping controllers built on the basis of the regulator control of FACTS
controllers are a trade-off to the system. And the damping controllers need to be
designed carefully to not cause unexpected problems.
119
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 57.5
7.52
7.54
7.56
7.58
7.6
7.62
time in seconds
pow
er fl
ow p
er u
nit
master line active power P4−11
no dmpdmp:V
4,R
3
dmp:V4,R
5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
6.09
6.1
6.11
6.12
6.13
time in seconds
pow
er fl
ow p
er u
nit
slave line active power P4−12
no dmpdmp:V
4,R
3
dmp:V4,R
5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.56
−0.555
−0.55
−0.545
time in seconds
pow
er fl
ow p
er u
nit
master line reactive power Q4−11
no dmpdmp:V
4,R
3
dmp:V4,R
5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.495
0.5
0.505
0.51
0.515
0.52
time in seconds
pow
er fl
ow p
er u
nit
slave line reactive power Q4−12
no dmpdmp:V
4,R
3
dmp:V4,R
5
Figure 6.21: IPFC Line Flows without and with Damping Controllers
CHAPTER 7
MAIN CONTRIBUTIONS AND FUTURE WORK
RECOMMENDATIONS
In this research work, we focus on the loadflow and dispatch strategies, linearized
models, dynamic simulations, and damping control design for VSC-based FACTS
Controllers in various operating modes.
7.1 Main Contributions
1. An efficient control mode implementation has been developed to reduce the
complexity associated with the many setpoint control modes, by the approach-
ing of advocating separate modelling for a shunt VSC and for a series VSC.
This separation of models can readily accommodate all VSC configurations.
In this approach, the unknown variables of the loadflow solution are always
kept the same, independent of the VSC controller operating mode. In this
way, when a VSC controller changes mode, only two equations for each shunt
VSC and two equations for each series VSC need to be adjusted.
2. Except for shunt voltage setpoint control mode and line power flow regulation
mode, additional reactive power setpoint control mode and reactive power
reserve mode for the shunt VSC and fixed injected voltage control mode for
the series VSC have been incorporated into the control mode implementation
3. Efficient dispatch strategies are developed to optimize power transfer when one
or both VSCs are loaded to their rated capacity, which allows one to study
maximum dispatchability of FACTS controllers in large power systems under
all the operating constraints considered.
4. A versatile regulator model, which includes the DC link capacitor dynamics
and takes into account various operating modes, is proposed. The control
mode implementation is applied to complete this model. The shunt VSC
120
121
controls and the series VSC controls are modeled as separate regulators. When
a VSC changes its operating mode, only the input signals of the corresponding
regulator need to be adjusted. The VSC operating constraints due to various
ratings and operating limits are imposed in the VSC controls. The versatile
regulator model has been incorporated into the positive-sequence transient
stability simulation program to evaluate their impact on transient stability
under oscillations and faults.
5. Small-signal linearized models based on the dynamic models are derived by
using small-signal perturbation. By this approach, dynamic simulation and
small-signal analysis are able to share identical codes for generators, exciters,
FACTS controllers, and so on.
6. A new modal decomposition approach, fully decouples all state modes and
considers the interaction of other state modes to the inter-area mode of in-
terest, is proposed to to quantify levels of controllability, observability, and
inner-loop gains of the linearized models.
7. Damping controllers supplemental to the regulation controls of FACTS con-
trollers are investigated to improve small-signal stability.
7.2 Future Research Recommendations
While the following items for future research are not exhaustive, they are con-
sidered important to improve the operations of the VSC-based FACTS Controllers.
1. Regulation Gain Validation for FACTS Controller Dynamic Models
2. Dynamic Simulation of Large-Scale Systems
3. Multiple FACTS coordination
Further research work is needed on the damping control design of the VSC-
based FACTS Controllers together with other regulation devices at different
122
system operating conditions. We need investigate the interactions of multiple
FACTS Controllers, that is, how one VSC controller affects the effectiveness
of other VSC controllers and how the interactions can be quantified. The
damping control design will be based on potential interactions of nearby or
coupled VSCs. Strategies for best utilizing interacting FACTS controllers for
damping control will then be provided.
4. Real-time simulation
To implement the strategies in real-time, more studies on the automatic control
system are needed.
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APPENDIX A
DATA FILE OF A 22-BUS POWER SYSTEM
22-Bus Power System Test Case
Subtransient Generator Models
Bus Data Format
Bus number, bus voltage magnitude (pu), bus voltage angle (degree), generator
real power (pu), generator reactive power (pu), load real power (pu), load reactive
power (pu), G shunt (pu), B shunt (pu), bus type: 1 for swing bus; 2 for generator
bus (PV bus); 3 for load bus (PQ bus), maximum generator reactive power (pu),
minimum generator reactive power (pu), bus voltage rating (kV), maximum bus
voltage (pu), minimum bus voltage
bus = [
1 1.0646 -5.8094 5.91731 1.19699 0.00 0.00 0 0.0 2 99 -99 345 1.5 0.5;
2 1.0728 0.000 10.6634 1.17505 0.00 0.00 0 0.0 1 99 -99 345 1.5 0.5;
3 1.0267 -14.7321 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
4 1.02665 -14.5892 0.000 0.000 0.00 0.00 0 2.0 3 0.0 0.0 345 1.5 0.5;
5 1.0350 -11.4906 19.61876 2.55597 0.00 0.00 0 0.0 2 99 -99 345 1.5 0.5;
6 1.02074 -39.5822 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
7 1.03356 -33.0712 0.000 0.000 0.593 -0.175 0 2.8 2 3.25 -3 345 1.5 0.5;
8 1.0429 -26.1107 4.87829 -1.4632 0.00 0.00 0 1.35 2 99 -99 345 1.5 0.5;
9 1.0323 -45.4533 0.000 0.000 0.00 0.00 0 2.8 3 0.0 0.0 345 1.5 0.5;
10 1.0403 -39.3350 7.91931 2.000 0.00 0.00 0 1.35 2 99 -99 345 1.5 0.5;
11 1.0267 -15.600 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
12 1.0267 -15.600 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
13 1.02062 -39.6185 0.000 0.000 0.00 0.00 0 2.7 3 0.0 0.0 345 1.5 0.5;
14 1.0377 -48.000 0.000 0.000 0.00 0.00 0 2.7 2 2.7 -3 345 1.5 0.5;
15 1.0375 -49.400 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
16 1.0380 -57.000 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
131
132
17 1.0428 -59.660 0.000 0.000 26.0 1.60 0 2.7 3 0.0 0.0 345 1.5 0.5;
18 1.0429 -53.770 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
19 1.0448 -58.000 5.000 1.000 0.00 0.00 0 0.0 2 0.0 0.0 345 1.5 0.5;
20 1.0367 -58.000 0.000 0.000 18.26 1.00 0 2.7 3 0.0 0.0 345 1.5 0.5;
21 1.0345 -54.710 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
22 1.02953 -33.0626 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
];
Line Data Format
From bus, to bus, line resistance (pu), line reactance (pu), line charging (pu),
tap ratio, tap phase shifter angle (degree), maximum tap, minimum tap, tap size
line = [
9 7 0.00180 0.02640 0.35320 1.0 0.0 0.0 0.0 0.0;
4 12 0.00000 0.00034 0.00000 1.0 0.0 0.0 0.0 0.0;
9 12 0.00389 0.06723 1.17304 1.0 0.0 0.0 0.0 0.0;
22 7 0.00320 0.08290 0.00000 1.0 0.0 0.0 0.0 0.0;
7 8 0.00220 0.02840 0.47760 1.0 0.0 0.0 0.0 0.0;
7 3 0.00220 0.03788 0.66383 1.0 0.0 0.0 0.0 0.0;
7 10 0.00140 0.01830 0.28310 1.0 0.0 0.0 0.0 0.0;
1 4 0.00320 0.06010 0.61305 1.0 0.0 0.0 0.0 0.0;
3 2 0.00351 0.04563 0.59200 1.0 0.0 0.0 0.0 0.0;
3 4 0.00036 0.00067 0.01099 1.0 0.0 0.0 0.0 0.0;
13 6 0.00000 0.00011 0.00000 1.0 0.0 0.0 0.0 0.0;
6 10 0.00140 0.01685 0.27800 1.0 0.0 0.0 0.0 0.0;
4 11 0.00000 0.00034 0.00000 1.0 0.0 0.0 0.0 0.0;
6 11 0.00163 0.03877 0.78800 1.0 0.0 0.0 0.0 0.0;
3 13 0.00410 0.04230 0.68660 1.0 0.0 0.0 0.0 0.0;
5 4 0.00330 0.04787 0.00000 1.0 0.0 0.0 0.0 0.0;
5 4 0.00320 0.04905 0.00000 1.0 0.0 0.0 0.0 0.0;
9 20 0.00130 0.02320 0.39160 1.0 0.0 0.0 0.0 0.0;
9 21 0.00091 0.01624 0.27410 1.0 0.0 0.0 0.0 0.0;
133
13 14 0.00128 0.01320 0.21400 1.0 0.0 0.0 0.0 0.0;
10 14 0.00116 0.01845 0.31498 1.0 0.0 0.0 0.0 0.0;
14 16 0.00194 0.02007 0.32564 1.0 0.0 0.0 0.0 0.0;
14 15 0.00010 0.00053 0.00845 1.0 0.0 0.0 0.0 0.0;
14 18 0.00120 0.01420 0.24519 1.0 0.0 0.0 0.0 0.0;
15 16 0.00194 0.02007 0.32564 1.0 0.0 0.0 0.0 0.0;
21 20 0.00039 0.00696 0.11749 1.0 0.0 0.0 0.0 0.0;
17 19 0.00025 0.00377 1.14099 1.0 0.0 0.0 0.0 0.0;
17 16 0.00021 0.00562 0.11301 1.0 0.0 0.0 0.0 0.0;
17 16 0.00021 0.00562 0.11301 1.0 0.0 0.0 0.0 0.0;
18 19 0.00130 0.01500 0.25970 1.0 0.0 0.0 0.0 0.0;
20 19 0.00070 0.00850 0.14799 1.0 0.0 0.0 0.0 0.0;
];
Machine Data Format
Machine number, bus number, machine base MVA, leakage reactance xl (pu),
resistance ra (pu), d-axis synchronous reactance xd (pu), d-axis transient reactance
x′d (pu), d-axis subtransient reactance x
′′d (pu), d-axis open-circuit time constant T
′do
(sec), d-axis open-circuit subtransient time constant T′′do (sec), q-axis sychronous
reactance xq (pu), q-axis transient reactance x′q (pu), q-axis subtransient reactance
x′′q (pu), q-axis open-circuit time constant T
′qo (sec), q-axis open circuit subtran-
sient time constant T′′qo (sec), inertia constant H (sec), damping coefficient do (pu),
damping coefficient d1(pu), type, saturation factor S(1.0), saturation factor S(1.2)
mac con = [
1 1 900 0.2250 0.00 1.770 0.425 0.3100 6.700 0.0330 1.690 0.607 0.3100
0.401 0.0495 3.500 0.000 0 1 0.1093 0.4655;
2 2 1100 0.2490 0.00 2.033 0.434 0.3020 6.660 0.0500 1.975 1.205 0.3020
1.500 0.2340 4.100 0.000 0 2 0.1200 0.3667;
3 5 2000 0.0928 0.00 0.928 0.350 0.3150 5.000 0.0500 0.625 0.350 0.3150
5.000 0.0500 3.100 0.000 0 5 0.1452 0.6533;
4 8 600 0.2100 0.00 1.740 0.360 0.2750 7.300 0.0333 1.640 0.565 0.2750
134
0.410 0.0570 4.105 0.000 0 8 0.0558 0.2559;
5 10 1050 0.1300 0.00 1.050 0.277 0.1824 12.00 0.0730 0.700 0.277 0.1824
12.00 0.3400 5.580 0.000 0 10 0.1700 0.3600;
6 19 600 0.1400 0.00 1.800 0.245 0.2344 5.400 0.0320 1.480 0.880 0.2344
1.500 0.1500 2.820 0.000 0 19 0.1600 0.4200;
];
Exciter Data Format
Exciter type, machine number, input filter time constant TR, voltage regula-
tor gain KA, voltage regulator time constant TA, voltage regulator time constant
TB, voltage regulator time constant TC , maximum voltage regulator output VRmax,
minimum voltage regulator output VRmin, maximum internal signal VImax, minimum
internal signal VImin, first stage regulator gain KJ , potential circuit gain coefficient
Kp, potential circuit phase angle θp, current circuit gain coefficient KI , potential
source reactance XL, rectifier loading factor KC , maximum field voltage Efdmax,
inner loop feedback constant KG, maximum inner loop voltage feedback VGmax
exc con = [
0 1 0.01 50 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 2 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 3 0.01 400 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 4 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 5 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 6 0.01 400 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
];
FACTS Data Format
• FACTS Power Flow Data
FACTS number, from bus, to bus, FACTS mode, line active power setpoint
(MW), line reactive power setpoint (MVar), bus voltage setpoint (pu), maximum
shunt current (pu), maximum active power transfer (MW), minimum bus voltage
(pu), maximum bus voltage (pu), maximum series current (pu), series reactance
135
(pu), shunt reactance (pu), owner, maximum series inverter voltage (pu), maximum
shunt inverter voltage (pu), series MVA rating, shunt MVA rating, setpoint 1, set-
point 2, series reference code: 1 for bus voltage reference 2 for line current reference,
shunt Var setpoint (pu), voltage droop, shunt mode, series mode
• FACTS Dynamic Data
Shunt Kv, shunt Kp, shunt Ki, shunt T , standalone series Kp, standalone series
Ki, standalone series T , coupled series Kp, coupled series Ki, coupled series T , DC
capacitor voltage (Volts), DC capacitance (µF), maximum DC capacitor voltage
(Volts), minimum DC capacitor voltage (Volts)
facts con = [
1 4 11 1 8.2 -0.2 1.025 1.0 50 0.8 1.2 18.108 0.00034 0.1883
0.056 1.5 100 100 0.01 -0.02 0 0 0.03 1 1
500 .1 .01 .02 1 10 .02 .02 .4 .02 .02 .4 .02
12000 2820 14400 9600;
];
IPFC Data Format
• IPFC Power Flow Data
IPFC number, Master line from bus, Master to bus, Slave line from bus, Slave
line to bus, IPFC mode, Master line active power setpoint (MW), Master line reac-
tive power setpoint (MVar), Slave line active power setpoint (MW), Slave reactive
power setpoint (MVar), maximum Master line current (pu), maximum Slave line
current (pu), Master reactance (pu), Slave reactance (pu), maximum Master in-
verter voltage (pu), maximum Slave inverter voltage (pu), maximum active power
transfer (MW), Master MVA rating, Slave MVA rating, Master d-axis inverter volt-
age setpoint (pu), Master q-axis inverter voltage setpoint (pu), Slave d-axis inverter
voltage setpoint (pu), Slave q-axis inverter voltage setpoint (pu), Master operating
mode, Slave operating mode
• IPFC Dynamic Data
136
Master line active power regulator Kp, Ki, and T , Master line reactive power
regulator Kp, Ki, and T , Master DC bus voltage regulator Kp, Ki, T , and Kα,
Slave DC bus voltage regulator Kp, Ki, and T , DC capacitor voltage (Volts), DC
capacitance (µF), maximum DC capacitor voltage (Volts), minimum DC capacitor
voltage (Volts)
ipfc con = [
1 4 11 4 12 1 7.9 -0.3 7.8 0.0 18.108 18.108 0.00034 0.00034
0.056 0.056 0.5 100 100 0.02 0.0 0.0 -0.03 1 1
.1 2 .02 .1 2 .02 .1 1 .02 100 .1 2 .02 .1 1 .02
12000 2820 14400 9600;
];
Switching File Defines the Simulation Control
• (row 1): col 1 simulation start time (s), cols 2-6 zeros, col 7 initial time step
(s)
• (row 2): col 1 fault application time (s), col 2 bus number at which fault is
applied, col 3 bus number defining far end of faulted line, col 4 zero sequence
impedance in pu on system base, col 5 negative sequence impedance in pu on
system base, col 6 type of fault, col 7 time step for fault period (s)
• (row 3): col 1 finishing time (s), cols 2-7 zeros
sw con = [
0 0 0 0 0 0 0.002; % sets initial time step
0.1 3 13 0 0 6 0.002; % no fault fault at bus 3
20 0 0 0 0 0 0; % end simulation
];
APPENDIX B
DATA FILE OF A 20-BUS POWER SYSTEM
20-Bus Power System Test Case
Subtransient Generator Models
Bus Data Format
Bus number, bus voltage magnitude (pu), bus voltage angle (degree), generator
real power (pu), generator reactive power (pu), load real power (pu), load reactive
power (pu), G shunt (pu), B shunt (pu), bus type: 1 for swing bus; 2 for generator
bus (PV bus); 3 for load bus (PQ bus), maximum generator reactive power (pu),
minimum generator reactive power (pu), bus voltage rating (kV), maximum bus
voltage (pu), minimum bus voltage
bus = [
1 1.0646 -5.8094 13.338 3.19699 0.00 0.00 0 0.0 2 99 -99 345 1.5 0.5;
2 1.0728 0.000 8.8634 1.17505 0.00 0.00 0 0.0 2 99 -99 345 1.5 0.5;
3 1.0267 -14.7321 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
4 1.02665 -14.5892 0.000 0.000 0.00 0.00 0 2.0 3 0.0 0.0 345 1.5 0.5;
5 1.0350 -11.4906 5.21876 2.55597 0.00 0.00 0 0.0 2 99 -99 345 1.5 0.5;
6 1.02074 -39.5822 0.000 0.00 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
7 1.03356 -33.0712 0.000 3.250 0.593 -0.175 0 2.8 2 3.25 -3 345 1.5 0.5;
8 1.0429 -26.1107 6.27829 -1.4632 0.00 0.00 0 1.35 2 99 -99 345 1.5 0.5;
9 1.0323 -45.4533 0.000 0.000 0.00 0.00 0 2.8 3 0.0 0.0 345 1.5 0.5;
10 1.0403 -39.335 0.000 0.000 0.00 0.00 0 1.35 3 99 -99 345 1.5 0.5;
11 1.0267 -15.60 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
1 1.0267 -15.60 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
13 1.02062 -39.6185 0.000 0.000 0.00 0.00 0 2.7 3 0.0 0.0 345 1.5 0.5;
14 1.0377 -48.00 0.000 2.700 0.00 0.00 0 2.7 2 2.7 -3 345 1.5 0.5;
15 1.0375 -49.40 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
16 1.0380 -57.00 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
137
138
17 1.0428 -59.66 0.000 0.000 27.0 1.60 0 2.7 3 0.0 0.0 345 1.5 0.5;
18 1.0345 -54.71 0.000 0.000 0.00 0.00 0 0.0 3 0.0 0.0 345 1.5 0.5;
19 1.0448 -58.00 12.00 1.000 0.00 0.00 0 0.0 1 0.0 0.0 345 1.5 0.5;
20 1.0367 -58.00 0.000 0.000 15.26 1.00 0 2.7 3 0.0 0.0 345 1.5 0.5;
];
Line Data Format
From bus, to bus, line resistance (pu), line reactance (pu), line charging (pu),
tap ratio, tap phase shifter angle (degree), maximum tap, minimum tap, tap size
line = [
9 7 0.00180 0.02640 0.35320 1.0 0.0 0.0 0.0 0.0;
4 12 0.00000 0.00034 0.00000 1.0 0.0 0.0 0.0 0.0;
9 12 0.00389 0.06723 1.17304 1.0 0.0 0.0 0.0 0.0;
7 8 0.01090 0.16990 0.47760 1.0 0.0 0.0 0.0 0.0;
7 3 0.00220 0.03788 0.66383 1.0 0.0 0.0 0.0 0.0;
7 10 0.00140 0.01830 0.28310 1.0 0.0 0.0 0.0 0.0;
1 4 0.00320 0.06010 0.61305 1.0 0.0 0.0 0.0 0.0;
3 2 0.00552 0.05783 0.59200 1.0 0.0 0.0 0.0 0.0;
3 4 0.00036 0.00067 0.01099 1.0 0.0 0.0 0.0 0.0;
13 6 0.00000 0.00011 0.00000 1.0 0.0 0.0 0.0 0.0;
6 10 0.00140 0.01685 0.27800 1.0 0.0 0.0 0.0 0.0;
4 11 0.00000 0.00034 0.00000 1.0 0.0 0.0 0.0 0.0;
6 11 0.00163 0.03877 0.78800 1.0 0.0 0.0 0.0 0.0;
3 13 0.00410 0.04230 0.68660 1.0 0.0 0.0 0.0 0.0;
5 4 0.00114 0.03352 0.08500 1.0 0.0 0.0 0.0 0.0;
5 4 0.00114 0.03352 0.08500 1.0 0.0 0.0 0.0 0.0;
9 18 0.00091 0.01624 0.27410 1.0 0.0 0.0 0.0 0.0;
13 14 0.00128 0.01320 0.21400 1.0 0.0 0.0 0.0 0.0;
10 14 0.00116 0.01845 0.31498 1.0 0.0 0.0 0.0 0.0;
14 16 0.00194 0.02007 0.32564 1.0 0.0 0.0 0.0 0.0;
14 15 0.00010 0.00053 0.00845 1.0 0.0 0.0 0.0 0.0;
139
15 16 0.00194 0.02007 0.32564 1.0 0.0 0.0 0.0 0.0;
18 20 0.00039 0.01696 0.11749 1.0 0.0 0.0 0.0 0.0;
17 19 0.00025 0.00377 0.14099 1.0 0.0 0.0 0.0 0.0;
17 16 0.00021 0.00562 0.11301 1.0 0.0 0.0 0.0 0.0;
17 16 0.00021 0.00562 0.11301 1.0 0.0 0.0 0.0 0.0;
20 19 0.00070 0.01850 0.14799 1.0 0.0 0.0 0.0 0.0;
1 3 0.00251 0.04213 0.59200 1.0 0.0 0.0 0.0 0.0;
];
Machine Data Format
Machine number, bus number, machine base MVA, leakage reactance xl (pu),
resistance ra (pu), d-axis synchronous reactance xd (pu), d-axis transient reactance
x′d (pu), d-axis subtransient reactance x
′′d (pu), d-axis open-circuit time constant T
′do
(sec), d-axis open-circuit subtransient time constant T′′do (sec), q-axis sychronous
reactance xq (pu), q-axis transient reactance x′q (pu), q-axis subtransient reactance
x′′q (pu), q-axis open-circuit time constant T
′qo (sec), q-axis open circuit subtran-
sient time constant T′′qo (sec), inertia constant H (sec), damping coefficient do (pu),
damping coefficient d1(pu), type, saturation factor S(1.0), saturation factor S(1.2)
mac con = [
1 1 2200 0.1490 0.00 1.770 0.425 0.3100 6.700 0.0330 1.690 0.607 0.3100
0.401 0.0495 4.100 5.000 0 1 0.1093 0.4655;
2 2 1500 0.2250 0.00 2.033 0.434 0.3020 6.660 0.0500 1.975 1.205 0.3020
1.500 0.2340 4.000 5.000 0 2 0.1200 0.3667;
3 5 1500 0.0928 0.00 0.928 0.350 0.3150 5.000 0.0500 0.625 0.350 0.3150
5.000 0.0500 3.100 5.000 0 5 0.1452 0.6533;
4 8 3500 0.2100 0.00 1.740 0.360 0.2750 7.300 0.0333 1.640 0.565 0.2750
0.410 0.0570 4.505 5.000 0 8 0.0558 0.2559;
5 19 5000 0.1400 0.00 1.800 0.245 0.2344 5.400 0.0320 1.480 0.880 0.2344
1.500 0.1500 2.820 2.000 0 19 0.1600 0.4200;
];
140
Exciter Data Format
Exciter type, machine number, input filter time constant TR, voltage regula-
tor gain KA, voltage regulator time constant TA, voltage regulator time constant
TB, voltage regulator time constant TC , maximum voltage regulator output VRmax,
minimum voltage regulator output VRmin, maximum internal signal VImax, minimum
internal signal VImin, first stage regulator gain KJ , potential circuit gain coefficient
Kp, potential circuit phase angle θp, current circuit gain coefficient KI , potential
source reactance XL, rectifier loading factor KC , maximum field voltage Efdmax,
inner loop feedback constant KG, maximum inner loop voltage feedback VGmax
exc con = [
0 1 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 2 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 3 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 4 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
0 5 0.01 100 0.05 0 0 5.0 -5.0 0 0 0 0 0 0 0 0 0 0 0;
];
FACTS Data Format
• FACTS Power Flow Data
FACTS number, from bus, to bus, FACTS mode, line active power setpoint
(MW), line reactive power setpoint (MVar), bus voltage setpoint (pu), maximum
shunt current (pu), maximum active power transfer (MW), minimum bus voltage
(pu), maximum bus voltage (pu), maximum series current (pu), series reactance
(pu), shunt reactance (pu), owner, maximum series inverter voltage (pu), maximum
shunt inverter voltage (pu), series MVA rating, shunt MVA rating, setpoint 1, set-
point 2, series reference code: 1 for bus voltage reference 2 for line current reference,
shunt Var setpoint (pu), voltage droop, shunt mode, series mode
• FACTS Dynamic Data
Shunt Kv, shunt Kp, shunt Ki, shunt T , standalone series Kp, standalone series
Ki, standalone series T , coupled series Kp, coupled series Ki, coupled series T , DC
141
capacitor voltage (Volts), DC capacitance (µF), maximum DC capacitor voltage
(Volts), minimum DC capacitor voltage (Volts)
facts con = [
1 4 11 1 8.2 -0.2 1.031 1 50 0.8 1.2 18.108 0.00034 0.1883
0.056 1.5 100 100 0.01 -0.02 0 0 0.03 1 1
500 .1 1 .02 20 200 .02 .1 1 .02 .1 1 .02
12000 2820 14400 9600;
];
IPFC Data Format
• IPFC Power Flow Data
IPFC number, Master line from bus, Master to bus, Slave line from bus, Slave
line to bus, IPFC mode, Master line active power setpoint (MW), Master line reac-
tive power setpoint (MVar), Slave line active power setpoint (MW), Slave reactive
power setpoint (MVar), maximum Master line current (pu), maximum Slave line
current (pu), Master reactance (pu), Slave reactance (pu), maximum Master in-
verter voltage (pu), maximum Slave inverter voltage (pu), maximum active power
transfer (MW), Master MVA rating, Slave MVA rating, Master d-axis inverter volt-
age setpoint (pu), Master q-axis inverter voltage setpoint (pu), Slave d-axis inverter
voltage setpoint (pu), Slave q-axis inverter voltage setpoint (pu), Master operating
mode, Slave operating mode
• IPFC Dynamic Data
Master line active power regulator Kp, Ki, and T , Master line reactive power
regulator Kp, Ki, and T , Master DC bus voltage regulator Kp, Ki, T , and Kα,
Slave DC bus voltage regulator Kp, Ki, and T , DC capacitor voltage (Volts), DC
capacitance (µF), maximum DC capacitor voltage (Volts), minimum DC capacitor
voltage (Volts)
ipfc con = [
1 4 11 4 12 1 7.5 -0.558 6.123 0.0 18.108 18.108 0.00034 0.00034
142
0.056 0.056 50 100 100 0.02 0.0 0.0 -0.03 1 1
1 10 .02 1 10 .02 1 10 .02 10 1 10 .02 1 10 .02
12000 2820 14400 9600;
];
Damping Controller Data Format
Damping controller number, FACTS controller number, damping control gain
k, phase compensator numerator coefficient Tn, phase compensator denominator
coefficient Td, washout block time constant Tw, LP filter time constant Tf , maximum
damping signal, minimum damping signal
dmp con = [
1 1 -3 0.852 0.0341 0.1 0.09 0.6 -0.6;
];
Switching File Defines the Simulation Control
• (row 1): col 1 simulation start time (s), cols 2-6 zeros, col 7 initial time step
(s)
• (row 2): col 1 fault application time (s), col 2 bus number at which fault is
applied, col 3 bus number defining far end of faulted line, col 4 zero sequence
impedance in pu on system base, col 5 negative sequence impedance in pu on
system base, col 6 type of fault, col 7 time step for fault period (s)
• (row 3): col 1 finishing time (s), cols 2-7 zeros
sw con = [
0 0 0 0 0 0 0.002; % sets initial time step
0.1 14 16 0 0 6 0.002; % no fault fault at bus 14
10 0 0 0 0 0 0; % end simulation
];