tesis_ operating modes and their regulations of voltage sourced converters based facts controllers

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

ii

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

iii

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


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


iv

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


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


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

v

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

vi

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

vii

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

viii

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

ix

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).

x

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.

xii

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


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


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


(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


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


(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


(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


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


(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


(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.


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


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


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


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


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


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


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


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


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


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


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


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


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

];

tesis_ operating modes and their regulations of voltage sourced converters based facts controllers

Documents

summaryandconclusions

linearized models

main contributions

dispatchstrategies 27iii3