- "
NONLINEAR CONTROL OF CENTER-NODE UPFC AND
VSC-BASED FACTS CONTROLLERS
Bin Lu
B. Eng. Tsinghua University, P. R. China
M. Eng. McGill University, Canada
A thesis submitted to McGill University in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
Department of Electrical and Computer Engineering
McGill University
Montreal, Quebec, Canada
August, 2003
© Bin Lu 2003
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ABSTRACT
Voltage-Source Converters form the basic modules of a class of power electronic
controllers of Flexible AC Transmission Systems (F ACTS) which include the STATic
COMpensator (STATCOM), the Static Synchronous Series Compensator (SSSC) and the
Unified Power Flow Controller (UPFC). The mathematical equations which model a
Voltage-Source Converter (VSC) are nonlinear (bilinear) because the system inputs are
multiplied by a state-variable. The high performance characteristics for their operation
must be designed in the face of the nonlinearity. This thesis applies a Nonlinear Control
Method which makes use of a nonlinear transformation to obtain a system of linear
equations. Then linear state feedback is used to move the eigenvalues of the linear system
to achieve fast, stable response.
The Nonlinear Control Method has been applied successfully to 3 F ACTS controllers:
(1) the single VSC SSSC (system order N=3); (2) the 2-VSC UPFC (N=5); and (3) the 3-
VSC Center-no de Unified Power Flow Controller (C-UPFC, N=5). The key to success is
in finding the nonlinear transformation equations which is an art as in aIl integration
efforts and which cannot be taught as in differentiation. Having found the nonlinear
transformation equations for the 3 F ACTS controllers, they can be extended to the entire
family of Voltage-Source Converter based F ACTS controllers.
A Simplified Nonlinear Control Method, which does not sacrifice mathematical ri gour,
is proposed. As the Simplified Nonlinear Control Method does not require knowledge in
advanced control theory, it facilitates adoption of the Nonlinear Control Method by the
power electronics community.
The thesis also covers in-depth research on the Center-node Unified Power Flow
Controller (C-UPFC), an innovative F ACTS controller. The research shows that it has aIl
the capabilities of Lazslo Gyugyi's Unified Power Flow Controller. In addition, as a
controller conceived to operate at the mid-point of a radial transmission line, it can double
the power transmissibility of the line.
RÉSUMÉ
Les convertisseurs commutateurs de tension (CCT) constituent les modules de base
des systèmes de transport à courant alternatif flexibles (sigle anglais F ACTS), une classe
de contrôleurs qui inclut le compensateur statique (STATCOM), le compensateur
synchrone série statique (SSSC) et le variateur de charge universel (UPFC). Leurs
équations mathématiques sont nonlinéaires car elles multiplient leurs entrées avec une
variable d'état. On devra nécessairement intégrer cette nonlinéarité dans notre
modélisation si on veut atteindre un niveau d'exploitation performant. Cette thèse décrit
une méthodologie de commande qui, en un premier temps, transforme les équations du
convertisseur en un système linéaire. Les techniques d'asservissement par variables
d'état permettent ensuite de placer les valeurs propres du système pour assurer une
réponse rapide et stable.
La méthodologie de commande nonlinéaire a été testé avec succès sur trois contrôleurs
FACTS: (1) SSSC à un CCT (d'ordre N=3); (2) UPFC à 2 CCT (N=5); et (3) UPFC à
nœud central à 3 CCT (C-UPFC, N=5). Comme dans tout effort d'intégration, le noeud
de la recherche consiste à trouver la transformation nonlinéaire appropriée; cela relève de
l'art plutôt que de connaissances acquises comme par exemple la différentiation. À partir
des transformations des trois contrôleurs analysés, nous pouvons étendre l'étude à la
famille entière des F ACTS à convertisseurs commutateurs de tension.
Une méthodologie de commande simplifiée est proposée. Celle-ci ne sacrifie rien en
rigueur mathématique mais ne requiert pas une connaissance de la théorie de la
commande avancée, ce qui pourrait faciliter son adoption dans la communauté technique
de l'électronique de puissance.
Cette thèse fait un bilan des recherches courantes sur un contrôleur innovateur, le
variateur de charge universel à nœud central (C-UPFC). Ces recherches démontrent qu'il
reproduit toute la fonctionalité du variateur de charge universel de Lazslo Gyugyi. De
plus, étant conçu pour opérer au point médian d'une ligne de transport radiale, il peut
doubler la capacité de transfert de cette ligne.
11
ACKNOWLEDGEMENTS
My most sincere gratitude firstly and surely cornes to Professor Ooi, my thesis
supervisor, for his exceptional supervision, most valuable guidance, continuous
encouragement and warmest friendship throughout this research. 1 would also thank him
so much for the financial arrangement of my studies, for his careful, kind and considerate
arrangement for the completion ofmy thesis.
1 am very grateful to Professor F. D. Galiana and Professor P. Kabal, my Ph.D.
committee members, for their valuable suggestions, discussions and guidance.
1 would like to thank Professor F. D. Galiana and Professor G. Joos for the use of the
computer facilities in the Power Engineering Laboratory, which enabled me to
accomplish digital simulations, and the preparation of documents and publications.
1 am indebted to Dr. Z. Wolanski for the many discussions about the nonlinear control
topic, and his very useful suggestions and friendship as weIl. Many thanks to Professor D.
McGiIlis for his great friendship and encouragement.
1 am grateful to my friends and colleagues in the power group. 1 leamed a lot from the
discussions with Dr. B. Mwinyiwiwa, Dr. Y. Chen, Dr. W. Lu, Dr. L. Tang, Ms. X.
Huang and Mr. J. Hu. 1 also enjoyed the great friendship and support of Dr. J. Cheng, Dr.
G. Atanackovic, Mr. S. Jia, Ms. E. Radinskaia, Mr. Y. Ren, Mr. F. Zhou, Dr. L. Jiao, Ms.
1. Kockar, Mr. F.A. Rahman AI-Jowder, Mr. F. Bouffard, Mr. C. Abbey, Mr. W. Li, Mr.
C. Luo, Mr. B. Shen and Mr. M. Zou. 1 especially want to thank Dr. L. Tang, Mr. Y. Ren
and Mr. F. Zhou for their kind assistance for aIl the works involved with submission of
this thesis.
III
l am thoroughly grateful to Dr. M. Huneault for his kind help to use his precious time
to do an expert French translation of the Abstract.
l would like to extend my sincere thanks to aU the supporting staffs of the Electrical
and Computer Engineering Department for their continuously kind assistance and
support.
My special thanks to my wife Xuemei for her encouragement, support and sacrifice, to
my parents for their priceless advice and support, and to an other relatives as weIl.
l would like to thank each and every pers on whose name has not been mentioned, but
in one way or another, has contributed to the successful completion of this research work.
IV
TABLE OF CONTENTS
ABSTRACT
RÉSUMÉ 11
ACKNOWLEDGEMENTS III
TABLE OF CONTENTS v
LIST OF FIGURES x
LIST OF TABLES XIV
LIST OF SYMBOLS xv
LIST OF ACRONYMS XVlll
Chapter 1 Introduction
l.1 INTRODUCTION 1
1.1.1 Background of Thesis 1
1.1.2 Brief History on Flexible AC Transmission Systems (F ACTS) 2
1.1.3 Control Research in Power Electronics 7
l.2 OBJECTIVES 10
l.3 ORGANIZA TION OF THE SIS 10
1.4 CONTRIBUTIONS 12
Chapter 2 Center-Node Unified Power Flow Controller (C-UPFC) 14
2.1 INTRODUCTION 14
2.2 OPERATION REQUIRING C-UPFC 16
2.2.1 Constraints-Phase-Shifter Operation 16
2.2.2 Current Continuity 18
2.2.3 Center-Node Voltage Vu 18
2.2.4 Voltage Gap 19
2.2.5 Voltage Bridge 20
2.3 DESCRIPTION OF C-UPFC 20
2.3.1 C-UPFC in Radial Transmission Line 20
2.3.2 Voltage-Source Converters 20
v
2.3.3 Shunt Converter
2.3.4 Series Converters
2.4 MULTI-CONVERTER CONTROL
2.4.1 Estimation of Complex Power Settings and Voltage
Injections of Series Converters
2.4.2 Proportional-Integral Feedbacks
2.5 DIGITAL SIMULATIONS
2.5.1 Simulation Software
2.5.2 Conditions of Tests on C-UPFC
2.6 CONCLUSION
APPENDIX 2-A PROPORTIONAL AND INTEGRAL GAINS OF
FEEDBACK CONTROL
APPENDIX 2-B ACTIVE AC POWER BALANCE IN
SERIES CONVERTERS
Chapter 3 Voltage-Source Converter Modeling and Nonlinear Control
3.1 INTRODUCTION
3.2 MODELING OF A VOLTAGE-SOURCE CONVERTER
3.2.1 Ideal Current Source Equivalent Circuit
3.2.2 Ideal Voltage Sources
3.2.3 Physical Reason for System Nonlinearity
3.2.4 Modeling in a-b-c frame
3.2.5 Modeling in d-q frame
3.3 PRINCIPLE OF NONLINEAR CONTROL
3.3.1 Preliminaries
3.3.2 Mathematical Preliminaries
3.3.2.1 Relative Degree
3.3.2.2 8180 Nonlinear Example
3.3.2.3 Multi-Input Systems
3.3.2.4 Conditions for Feedback Linearization
3.3.2.5 Lie Product or Lie Bracket
3.3.2.6 Examples of Lie Brackets
21
22
23
23
24
25
25
25
29
30
30
32
32
33
34
35
36
37
38
41
41
43
44
44
51
55
56
57
vi
3.3.2.7 Involutive Property
3.3.2.8 Reasonfor using Lie Brackets
3.3.2.9 Synthesizing hJ(J)
3.3.2.10 Synthesizing h2(J)
58
58
60
60
3.3.3 Generalization to m-input n-order Nonlinear System 62
3.3.4 Operating Nonlinear System
3.4 CONCLUSION
Chapter 4 Nonlinear Control of Voltage Source Converter Based
F ACTS Controllers
4.1 INTRODUCTION
4.2 NONLINEAR CONTROL OF SSSC
4.2.1 Modeling ofSSSC
4.3
4.2.2 Nonlinear Control of SSSC
4.2.3 Inverse Transformation
4.2.4 Simulation Results
UPFC NONLINEAR CONTROL
4.3.1 Modeling ofUPFC
4.3.1.1 Shunt Converter
4.3.1.2 Series Converter
4.3.1.3 DC Link Equation
4.3.2 Nonlinear Control ofUPFC
4.3.3 Reference Settings
4.3.4 Simulation Results
4.4 CONCLUSION
APPENDIX 4 SIMULATION P ARAMETERS AND SETTINGS
Chapter 5
5.1
5.2
C-UPFC Nonlinear Control
INTRODUCTION
C-UPFC NONLINEAR CONTROL
5.2.1 Modeling ofC-UPFC
5.2.2 Nonlinear Control ofC-UPFC
5.2.3 Simulation Results
67
68
70
70
71
72
74
77
78
80
81
82
82
82
82
91
92
99
99
101
101
102
102
107
114
VII
5.3 CONCLUSION 129
APPENDIX 5 SIMULATION PARAMETERS AND SETTINGS 130
Chapter 6 Simplified Nonlinear Control of
Voltage Source Converter Based F ACTS controllers 131
6.1 INTRODUCTION 131
6.2 OVERVIEW OF NONLINEAR CONTROL 133
6.2.1 Transformation of,! to ~ 134
6.2.2 Nonlinear Set 134
6.2.3 Linear Set 135
6.2.4 Identity Transformation 135
6.2.5 Linear System of~ 136
6.2.6 Inverse Transformation of w to Y: 137
6.3 BILINEAR EQUATIONS OF PWM-CONVERTER 137
6.3.1 STATCOM Equations 137
6.3.2 Removing Bilinear Terms 139
6.3.3 Transformation of,! to ~ 140
6.3.4 Inverse Transformation of w to Y: 141
6.3.4.1 Solutionfrom Decoupled Equation 142
6.3.4.2 Solutionfrom Coupled Equation 142
6.3.5 Solving the Gain Matrix [E} 143
6.3.6 Steady-state Operating States 144
6.4 TWO-CONVERTER SYSTEMS 144
6.4.1 Unified Power Flow Controller (UPFC) 145
6.4.2 Transformation of,! to ~ 146
6.4.3 Transforming w to Y: 147
6.4.3.1 Solution of Decoupled Equations 147
6.4.3.2 Solution ofCoupled Equations 148
6.4.4 MATLAB Solution of [E} 149
6.4.5 Complex Power Regulators 150
6.4.6 Digital Simulation Results 150
6.5 C-UPFC SYSTEM 152
VIlI
6.5.1 C7-[JjJl'C7
6.5.2 Transformation of~ to ~
6.5.3 Transforming w to Y:.
6.5.3.1 Solution of Decoupled Equations
6.5.3.2 Solution ofC7oupled Equations
6.6 CONCLUSION
152
152
153
153
155
156
APPENDIX 6 SIMULATION P ARAMETERS AND SETTINGS 157
Chapter 7
7.1
Further Development ofNonlinear Control
INTRODUCTION
7.2 ESTIMATION OF SYSTEM VOLTAGES BY WAY OF LOCAL
MEASUREMENTS
7.2.1 Estimation of System Voltages
158
158
159
159
7.2.2 Treatment of Time-varying voltages (VSd, vSq) and (VRd, VRq) 160
7.2.3 Digital Simulations 162
7.3 SIMULATION OF SYSTEM WITH VOLTAGE SWING 165
7.4 CONCLUSION 170
APPENDIX 7 SIMULATION P ARAMETERS AND SETTINGS 170
Chapter 8
8.1
8.2
Conclusions
CONCLUSION
8.1.1 Summary
8.1.2 C7onclusions
SUGGESTIONS FOR FUTURE WORK
REFERENCES
172
172
172
174
178
179
ix
Chapter 2 Fig. 2-1 Fig. 2-2
Fig. 2-3 Fig. 2-4
Fig. 2-5 Fig. 2-6 Fig. 2-7 Fig. 2-8
Chapter 3 Fig. 3-1
Fig. 3-2 Fig. 3-3
Chapter 4 Fig. 4-1 Fig. 4-2 Fig. 4-3 Fig. 4-4 Fig. 4-5 Fig. 4-6
Fig. 4-7
LIST OF FIGURES
Equivalent circuit of C-UPFC Phasor diagram (a) C-UPFC (b) Phasors of series converters Single line diagram of C-UPFC Voltage-source converter (a) Single line diagram representation (b) 3-phase bridge with transformer (c) Equivalent circuit System response for a step change in Psref Operating range for real power P Operating range for reactive power Qs System response for a step change in 8 s
17 17
19 21
26 27 28 28
Voltage-source converter (VSC) with transformer 34 (a) Single line symbolic representation (b) Detail 3-phase bridge of VSC (c) Equivalent circuit of a-phase Equivalent circuit for a voltage-source converter in a-b-c frame 35 Nonlinear Control in VSC applications 41
Single line diagram of SSSC Equivalent circuit of SSSC Step change in Iqo Single line diagram ofUPFC Control diagram ofUPFC Step Changes in complex power P and QR ( 8 =25 0
)
(a) Ps; (b) Qs; (c) QR; (d) ac voltage and cUITent; (e) dc link voltage; (f) modulation inputs. Step Changes in complex power P and QR ( 6 =8 0
)
(a) Ps; (b) Qs; (c) QR; (d) ac voltage and CUITent; (e) dc link voltage; (f) modulation inputs.
71 72 79 80 91 94
95
x
Fig. 4-8 Power ReversaIs (5=25°) 97 (a) Ps; (h) Qs; (c) QR; (d) ac voltage and cUITent; (e) de link voltage; (f) modulation inputs.
Fig. 4-9 Power ReversaIs (5=8°) 98 (a) Ps; (h) Qs; (c) QR; (d) ae voltage and cUITent; (e) dc link voltage; (f) modulation inputs.
Chapter 5 Fig. 5-1 Equivalent circuit ofC-UPFC with three eonverters sharing
one common dc eapaeitor link 103 Fig. 5-2 Control diagram of C-UPFC 108 Fig. 5-3 Real power reversaI 116
(a) V sa, isa (h) VRa, iRa (e) Voa, ioa (d) Ps (e) Qs (f) QR
Fig. 5-4 Real power reversaI 117 (a) Vdc (h) Udl, Uql (e) Ud2, Uq2 (d) Ud3, Uq3
Fig. 5-5 Real power step change 119 (a) Vsa, isa (h) VRa, iRa (e) Voa, ioa (d) Ps (e) Qs (f) QR
Fig. 5-6 Real power step change 120 (a) Vdc (h) Udl, Uql (c) Ud2, Uq2 (d) Ud3, Uq3
Fig. 5-7 Reactive power reversaI at sending end 122 (a) Vsa, isa (h) VRa, iRa (e) Voa, ioa
Xl
(d) Ps (e) Qs (f) QR
Fig. 5-8 Reactive power reversaI at sending end 123 (a) Vdc (b) Udl, Uql (c) Ud2, Uq2 (d) Ud3, Uq3
Fig. 5-9 Real power step change (capacitive compensation) 126 (a) Vsa, isa (b) VRa, iRa (C) Voa, ioa (d) Ps (e) Qs (f) QR
Fig. 5-10 Real power step change (capacitive compensation) 127 (a) Vdc (b) Udl, Uql (c) Ud2, Uq2 (d) Ud3, Uq3
Fig. 5-11 Real power step change (capacitive compensation) 128 (a) VIa, isa (b) V2a, iRa (C) V3a, ioa (d) PI (e) P2 (f) P3
Chapter 6 Fig. 6-1 Diagram of control 133 Fig. 6-2 Single line diagram of STATCOM 138 Fig. 6-3 Power ReversaIs of UPFC 151
(a) Ps; (b) Qs; (c) QR; (d) ac voltage and cUITent; (e) dc link voltage; (f) modulation inputs.
Chapter 7 Fig. 7-1 Equivalent circuit of C-UPFC with three converters
sharing one common dc capacitor link 160 Fig. 7-2 Real power reversaI of C-UPFC 163
(a) Vsa(estimated), isa (b) V Ra( estimated), iRa (C) Voa, ioa (d) Ps
XII
Fig. 7-3
Fig. 7-4
Fig. 7-5
Fig. 7-6
Fig. 7-7
(e) Qs (f) QR Real power reversaI of C-UPFC (a) Vdc
(b) Udl, Uql (c) Ud2, Uq2 (d) Ud3, Uq3 Real power reversaI with oscillation at Vs (a) Vsiestimated), isa (b) V Ri estimated), iRa (C) Voa, ioa (d) Ps (e) Qs (f) QR Real power reversaI with oscillation at Vs (a) Vdc
(b) Phase angle of Vs (c) Vds, Vqs (d) Udl, Uql (e) Ud2, Uq2
(f) Ud3, Uq3 Real power reversaI with oscillation at Vs and V R (a) Vsiestimated), isa (b) V Ri estimated), iRa (C) Voa, ioa (d) Ps (e) Qs (f) QR Real power reversaI with oscillation at Vs and V R (a) Vdc
(b) Phase angle of Vs (c) Phase angle ofVR (d) Udl, Uql (e) Ud2, Uq2
(f) Ud3, Uq3
164
166
167
168
169
xiii
Chapter 7 Table 7-1
Chapter 8
LIST OF TABLES
System parameters adopted in the simulation 162
Table 8-1 List of r=2 output function hl (~) for VSC-based F ACTS controllers 175
XIV
adrg
C
[C], [D]
dh(~)
< dh(~), f(~»
èlE
E
E
[E]
ea, eb, and ee
ed, eq
~(t)
f
f(~)
[G], [H]
gi(~)
ia, ib, and ie
Ide
Id, Iq
l,
Iqo
1(t)
L
LIST OF SYMBOLS
Lie products of f(~) and g(~)
Capacitor
Time invariant matrices
Differential or gradient row vector
Inner product of dh(~) and f(~)
Perturbation magnitude of the injected voltage of a VSC
Magnitude of the injected voltage ofa VSC
Ph as or of the injected voltage of a voltage-source converter (VSC)
State feedback matrix
VSC converter AC side voltage (3-phase)
d-q components of the VSC converter AC side voltage
Converter AC side voltage vector
System frequency
State variable function vector
Time invariant matrices
Control input function vector
A distribution
Output function vector
Current
Current phasor
AC phase currents
DC current
d-q components of the AC phase currents
The magnitude of the system line current
Steady-state operating value of iq
AC current vector
Modulation index
Inductor
xv
Lth(~)
[M], [N], [<D]
P
Q
r
R
S
[T]
V,v
v vit), Vb(t), vc(t)
Vdc
x Xo
y
y(t)Ckl
Zo
Lie derivative or derivative h along f
Time invariance matrices transforming input!! in x-system to w in z-system
Active power
Reactive power
Relative degree
Resistor
Complex power
Power invariant coordinate transformation
Control inputs vector in x-system
Modulating input signaIs ofa VSC (a-b-c frame)
Control inputs in d-q frame
Voltage
Voltage phasor
3-phase system voltage at the terminaIs of a VSC
DC voltage across capacitor C
d-q components of the VSC system voltage
The magnitude of system voltage (line-line)
Receiving-end voltage phasor
Sending-end voltage phasor
System voltage vector at the terminaIs of a VSC
Control inputs vector in z-system
State variables vector in x-system
Line reactance
Equilibrium state in x-system
Output vector in x-system
kth time derivative of output function in time domain
State variables vector in z-system
Equilibrium state in z-system
Phase angle
Perturbation phase angle of the injected voltage of a VSC
XVI
8 Phase angle of the injected voltage of a VSC
Eigenvalues
Angular frequency (= 2nf)
XYIl
AC, ac
AEP
BTB
CCF
C-UPFC
DC,dc
DSP
FACTS
GTO
HVDC
IEEE
IGBT
IGCT
IPC
IPFC
MIMO
M-UPFC
MVA
P-I
P.U.
PWM
SISO
SPWM
SSR
SSSC
STATCOM
SVC
TCSC
LIST OF ACRONYMS
Altemating CUITent
American Electric Power
Back-to-Back
Canonical Controllable Form
Center-no de Unified Power Flow Controller
Direct CUITent
Digital Signal Processor
Flexible AC Transmission Systems
Gate-Tum-Off Thyristor
High Voltage Direct CUITent
Institute of Electrical and Electronics Engineers
Insulated Gate Bipolar Transistor
Insulated Gate Controlled Thyristor
Inter-Phase Controller
Interline Power Flow Controller
Multiple-input Multiple-output
Multi-Terminal UPFC
Mega Volt Ampere
Proportional-Integral
Per Unit
Pulse Width Modulation
Single-input single-output
Sinusoidal Pulse Width Modulation
Subsynchronous Resonance
Static Synchronous Series Compensator
Static Compensator
Static Var Compensator
Thyristor Controlled Series Capacitor
XV III
THD
UPFC
VAR, var
VSC
Total Harmonie Distortion
Unified Power Flow Controller
Voltage Ampere Reactive
Voltage-Source Converter
XIX
Chapter 1
Introduction
1.1 INTRODUCTION
1.1.1 Background of Thesis
The course of research can take a tortuous path and it has been the case for the
research ofthis thesis. Originally, the research topic was the "Center-node Unified Power
Flow Controller" or the C-UPFC, which was to be the title of this thesis. The C-UPFC is
an innovative power electronic controller, which is intended to be a new member in the
family of controllers of Flexible AC Transmission Systems (F ACTS). In the preliminary
stage of research on new circuit topologies, the simple proportional-integral (P-I)
feedback is used as a matter of course in simulation studies with the hope that it would be
adequate. Should the dynamic performance need improvement, one has to apply more
sophisticated control methods at a later stage. As it tumed out, except for the phase-shifter
mode of operation, it has been impossible to stabilize the C-UPFC using proportional
integral (P-I) feedback so that it can operate in the other operating modes predicted for it.
The instability of the C-UPFC is not unexpected. This is because the C-UPFC consists
of 3 independent Voltage-Source Converters (VSCs). Each VSC has 2 independent
control inputs. With as many as 6 control inputs which can be in conflict if they are not
coordinated, it is necessary to find a systematic design method to operate the C-UPFC.
For graduate students outside the control research area in McGill University, systematic
design means linear control with techniques su ch as pole-placement, which are among the
graduate courses taken by power students. But the system equations of the C-UPFC are
nonlinear. While nonlinearity can be dealt with by small perturbation linearization, it will
always raise questions as to whether the region of validity has been exceeded in the
operation of the C-UPFC. The decision not to use small perturbation linearization,
however, requires several books of the Nonlinear Control Method [1-3] to be self-studied.
The hard work is rewarded because as the research unfolded, it has been found that all the
FACTS controllers based on pulse width modulated Voltage-Source Converters (PWM
VSCs) are amenable to design by the Nonlinear Control Method.
On returning to apply the Nonlinear Control Method to the C-UPFC, it is found that
the research in applying the Nonlinear Control Method to the many VSC-based F ACTS
controllers (such as the Static Synchronous Series Compensator (SSSC), the Unified
Power Flow Controller (UPFC) and the C-UPFC) has far outweighed the research on the
C-UPFC. The thesis takes the present title because of this reason. But because chapters 2
and 5 contain the research on the C-UPFC, it is necessary to explain why they are there
through this background note.
1.1.2 BriefHistory on Flexible AC Transmission Systems (FACTS)
Controllers in the electric power utility system are few. Traditionally, they are the
governor system of the prime movers (steam or hydro turbines), the field exciter system
in the alternators and the transformer tap changers. In recent times, there is a growing
need for more and better controllers to cope with the many problems related to: (i)
extensive ac interconnections; (ii) very long distance transmissions; (iii) congestions in
transmission corridors; (iv) power utility deregulation and restructuring.
Power electronic controllers first entered the picture when thyristor (thyratron before
it) bridges were introduced to rectify ac power to the dc field current of alternators and to
2
use the phase angle delay to control the reactive power output of the alternators [4].
Power electronics became more prominent when mercury-arc based and later thyristor
based High Voltage Direct Current (HVDC) transmission systems [5-7] were used firstly
in undersea cable crossings, th en for very long distance transmission and for
asynchronous ties to interconnect large regional ac systems (whose frequencies differ,
60Hz to 50 Hz, or whose frequencies are the same but there is need of a phase shifter to
span an overly large voltage angle difference). The capability of HVDC stations to
control the flow of real power introduces a new control degree of freedom in the electric
power utility system. But as HVDC stations are very expensive, ac transmission is the
preferred option.
As ac interconnections multiplied and ac transmission lines stretched over distances
approaching sizeable fractions of the wavelength of 60 Hz or 50 Hz, the effects of the
distributed line inductances and capacitances manifest themselves as overvoltages during
light loads and voltage sags during heavy loads [8]. There is a need to regulate the ac
voltages of the transmission line and this is done by var compensation at strategic points,
using a combination of thyristor-switched capacitors and thyristor-controlled reactors
together with thyristor-based Static Var Compensators (SVC) to provide continuous var
control [9-11].
Distant ac transmission line also meets the problem of increasingly large inductive
reactance of the line which reduces the transient stability limit. The transient stability
limit can be raised by series capacitor compensation but in thermal stations where there
are several turbine stages, the mechanical torsional resonance of the multi-inertia
torsional spring shaft system can interact adversely with the L-C electrical system to give
rise to subsynchronous resonance (SSR) instability [12]. SSR instability is not an issue
3
when HVDC is used for long distance transmission. But as already mentioned, the
converter stations of HVDC are expensive. Recently, there has been a come-back by ac
transmission because SSR oscillations across the compensating series capacitors can be
suppressed by electronic control using back-to-back thyristors connected in parallel
across the series capacitors, a scheme which has been given the name Thyristor
Controlled Series Capacitor (TCSC) [13]. The ac power electronic controllers, the SVC
and the TCSC, now come under the name of F ACTS (Flexible AC Transmission System)
controllers [14-16].
Up to this point, the technology has been based on the thyristor (and the mercury-arc
rectifier before it). The thyristor is an imperfect solid state switch because although it can
be tumed "on" electronically by a gating signal, it has to be tumed "off" by reversing the
voltage across the anode and cathode. In ac circuit, it is tumed "off' during the negative
half cycle of the ac voltage. This is called "line-commutation".
Since the eighties of 20th century, the Gate-Tum-Off Thyristor (GTO) [17] has entered
the field. As its name suggests, the GTO has gate-tum-off capability which makes it a
more perfect switch. GTO technology promises that the advantages of pulse width
modulation techniques (PWM) [18], which have been well appreciated in motor drive [19,
20] and uninterrupted power supply [20, 21] applications, can be extended to the high
power environment of electric utilities. Unlike line-commutation which has to wait for the
negative half cycle voltage to appear, the sampling rate of PWM-GTO is many times
higher. One immediate consequence in the increase in frequency bandwidth is that it
allows the standards on the Total Harmonic Distortion (THD) factor in voltage and
CUITent to be satisfied by smaller and therefore more economic filters.
4
Since the late 1980s, the thyristor based power electronic controllers have found new
embodiments as GTO controllers:
• Static Var Controllers as Static Compensators (STATCOMs) [22-26];
• Thyristor Controlled Series Capacitor (TCSC) as Static Synchronous Series
Compensator (SSSC) [14, 23, 27];
• Thyristor HVDC as Voltage-Source Converter HVDC (VSC-HVDC) [28-34]
The STATCOM, the SSSC and the converter stations ofVSC-HVDC are aIl based on
the same basic converter configuration, the GTO Voltage-Source Converter (VSC). When
operated under pulse width modulation, each VSC can be considered as 3 voltage
amplifiers, one for each phase amplifying the modulating signal of the phase [20]. The
VSC is the building block from which new controllers can be realized.
Unlike the STATCOM and the SSSC which, in the main, replicate the functions of the
SVC and the TCSC with improvements, the VSC-HVDC offers new functions. In
addition to controlling the real power, the VSC-HVDC converter stations can control the
V AR outputs on their ac sides, a feature which thyristor HVDC does not have.
As already menti one d, one application of the HVDC is as a phase shifter. The back-to
back HVDC link forming an asynchronous connection has a 3600 phase shift range.
Lazslo Gyugyi pointed out that there are many situations where the phase shift required is
only a small angle and the costly HVDC link cannot be justified since each of the 2 VSC
converters has to be rated at the full ac voltage and the full ac current. U sing the building
block concept, he proposed rearranging the 2 VSCs, one connected in shunt (almost as a
STATCOM) and the other in series (almost as an SSSC) but with their dc terminaIs
connected back-to-back to enable real power exchange between the shunt VSC and the
5
senes VSC. The shunt VSC does not have to carry the full ac CUITent. The senes
converter does not have to bear the full ac voltage. Therefore, their combined MV A is
less than that of the HVDC link. Furthennore, because the shunt VSC and the series VSC
are connected back-to-back at their dc tenninals, they do not have to operate strictly as a
STATCOM or as a SSSC. They can admit and output real power by way of the dc link.
This greater freedom allows the VSCs to control the vars on their ac sides. The
combination exercises 3 degrees of control: Ci) the power through the combination and (ii)
(iii) the vars of the sending end and the receiving end of the transmission line. Lazslo
Gyugyi rightly claimed it to be the ultimate ac controller, giving it the name Unified
Power Flow Controller (UPFC) [23]. A 160 MVA prototype UPFC has been in service in
the Inez area in the south central part of the AEP (American Electric Power) system [35,
36]. Besides the var controlling capability, the UPFC can be used in the following 3
applications (which the C-UPFC of the thesis must be able to equal):
• Phase shifting application
• SSSC (from series converter) application -- Series capacitor compensation
• Reversing power application (operating against nonnal direction of power
flow)
The opportunities offered by configuring VSC building blocks have not stopped with
the UPFC. Mc Gill University proposed a Multi-Tenninal UPFC (M-UPFC) [37], which
was conceived to facilitate energy trading in the deregulated energy market. Lazslo
Gyugyi soon followed his UPFC with another invention, the Interline Power Flow
Controller (IPFC) [38].
6
The C-UPFC (Center-no de UPFC) ofthis thesis is the sequel of an earlier publication,
which pointed out that the mid-point of the transmission line is the optimal location of
any VSC-based F ACTS controller [39]. The mid-point should also be optimal for the
UPFC. The research objective is to find out whether the functions of the UPFC can be
better realized by a new configuration based on 3 VSCs, with 2 VSCs in series (one on
either side of the center no de ) and the third VSC in shunt, operating as a ST A TCOM. The
research, therefore, should find out if the C-UPFC can equal the UPFC in operating in the
3 application modes. But very early in the research [40], it is found that it has been
impossible to stabilize the C-UPFC using simple P-I feedback apart from the phase
shifter mode.
In summary, power electronic research has followed technology-push and market-pull.
The technology-push cornes from the availability of Gate-tum-off thyristors (GTOs) and
related solid-state switches such as the Insulated Gate Bipolar Transistors (IGBTs) [41,
42], which are now available at high voltage and high current ratings. The continuing
challenge is to realize GTO or IGBT controllers which solve market-pull related problems:
• Relieving transmission congestion. The F ACTS controllers are conceived to
overcome the transient stability limit so that existing lines can be operated up to
their thermallimits.
• Facilitating energy trading in a deregulated market place by ensunng that
contractual power is transferred through designated routes with minimal loop
flows.
1.1.3 Control Research in Power Electronics
As most researchers in power electronics are hardware orientated, they value the
familiarity of classical control methods such as the proven proportional and integral
7
feedback. Apart from conservatism, the adherence to P-I type of feedback control
stemmed from the fact that hardware implementation, until the era of digital signal
processors (DSPs), was analogue. The control functions of analogue computing are
limited to multipliers, adders and simple single-input single-output nonlinear blocks.
With ever increasing computing power ofDSPs, more of the methods of modem linear
control theory can be applied to real world, real-time control problems [43]. Not
surprisingly, the most sophisticated control methods have been applied to systems with
very long time constants such as chemical processes. In power electronics, the early
successes of digital controls were in motor drives where the rotor shaft inertia systems
have relatively long time constants. Vector control of ac motor drives [19,44] represents
early achievement. However, when the "plant" in question is none other th an the Voltage
Source Converter itself, the time constant is shorter by an order of magnitude. Digital
control over the F ACTS controllers therefore requires greater computing speeds still.
Fortunately, not only are DSPs becoming faster year after year, but many have features
which allow them to be paralleled.
In the early 1990s, the Power Electronics Research Group of McGill University
initiated a research program on paralleling DSPs for real-time power electronics control.
The first prototype successfully paralleled 3 DSPs, the Texas Instrument TMS320C25
[45]. This computing platform was used to stabilize a PWM Voltage-Source Converter
using pole-placement technique [46] and to stabilize a CUITent-Source Converter system
[47].
The second prototype was based on 5 DSPs, the more powerful Texas Instrument
TMS320C30 [48]. At that time, the computing power from this second prototype
8
exceeded the control requirement of any power electronic experiment. It was applied as a
real-time digital simulator of three turbo-generators.
The lesson leamed from this research pro gram is: not only are faster DSPs coming into
the market year after year, but if their individual computing power is insufficient, more
power will be available by paralleling. This assurance allows the Nonlinear Control
Method to be seriously considered as a research topic.
The Nonlinear Control Method was brought to McGill University by Dr. Zbigniew
Wolanski, who applied it successfully to the STATCOM [49]. At that time, it was a minor
breakthrough. Unfortunately, his paper was not archived in any IEEE Transaction and the
possible reasons are noted here because they are relevant to the research of this thesis also.
In general, research on applying new control methods belongs to a "no-man's-land". To
reviewers in the Transaction on Control, any control method, which can find engineering
applications, cannot be new. In general, unfamiliar notations and new mathematical ideas
are not welcome by reviewers of the Transaction on Power Electronics.
The status of research in applying the Nonlinear Control Method to power electronic
systems is that it has been successfully applied to system order, n=3 (the order of the
STATCOM) [49-51]. This thesis increases the system order to n=5, the system orders of
the UPFC and the C-UPFC analysed in this thesis.
A few words must be added to explain that the increase from n=3 to n=5 is not
insignificant. This is because the Nonlinear Control Method depends on integrating
partial differential equations. As is well known, differentiation can be taught but
integration is an art. The success in applying the Nonlinear Control Method in the thesis
is due to the insights by which the output functions hi(~), h= 1, 2, ... , mare synthesized.
This aspect of the method has resemblance to synthesizing a Lyapunov function.
9
In the nonlinear control area, state space transformation was first studied by Krener
[52]. The approach of transforming a nonlinear system via state feedback has been
adopted for specific applications [53 - 57]. [53] also proposed and treated the feedback
linearization problem. A general philosophy on state feedbacks can be found in [58-60].
The problem of global feedback linearization was studied in [61-65]. Before the
STATCOM, the Nonlinear Control Method has been applied to dc-dc switched power
converters and includes a Lyapunov function-based control [66] and the method of exact
linearization [67-69].
To the best of the candidate' s knowledge, this thesis is the first to apply systematically
the Nonlinear Control Method to the entire family of PWM-VSC-based F ACTS
controllers.
1.2 OBJECTIVES
The objectives of the thesis are: (1) to study the C-UPFC, an innovative PWM-VSC
F ACTS controller, (2) to attain high performance features with the help of a suitable
feedback control algorithm, (3) to seek a systematic control method applicable to aIl
PWM-VSC FACTS controllers.
1.3 ORGANIZATION OF THESIS
Chapter 2 describes the Center-node Unified Power Flow Controller (C-UPFC) and
shows through digital simulations that it operates satisfactorily as a phase shifter. The
control has been based on trial and error, using individual P-I feedback over each of the 3
separate Voltage-Source Converters (VSCs) which form the C-UPFC. However, it has
not been possible to operate in the other application modes predicted for it. The
10
conclusion is that a more systematic approach should be taken. Noting that the system
equations are inherently nonlinear, the Nonlinear Control Method of[1-3] is chosen.
Chapter 3 introduces the mathematical model of the Voltage-Source Converter (VSC),
which forms the building block of the C-UPFC and the entire family ofPWM-VSC-based
F ACTS controllers. As the order is n=3, it is sufficiently small and yet structured to
illustrate how the Nonlinear Control Method is applied.
This chapter is also in part a tutorial in explaining the "short-hand" notations used in
Nonlinear Control literature and, more important, the rudimentary ideas behind the
method. The emphasis is more on how the method is applied to F ACTS controllers and
the precautions which have to be observed. It is assumed that readers interested in the
theory can find it in the references [1-3].
Chapter 4 A step-by-step approach, based on beginning with a low order system before
advancing to a higher order system, has been followed throughout the research. Since the
STATCOM has already been worked on, the Nonlinear Control Method is first applied to
the SSSC, which is still a single VSC controller, n=3. With confidence and experience
with the SSSC, one advances to a 2-VSC system, in this case the UPFC, n=5.
Chapter 5 applies of the Nonlinear Control Method on the C-UPFC. The attack on C
UPFC has been deferred until now because it is a 3-VSC system. However, on closer
examination, it is found that its system order is still n=5. The research of this chapter
completes the research on the C-UPFC by showing that it operates in aIl the 3 application
modes, predicted for it.
Chapter 6 The experiences from applying the Nonlinear Control Method to the SSSC,
the UPFC and the C-UPFC have yielded a combination ofphysical and analytical insights
by which a Simplified Nonlinear Control Method is formulated. "Simplified" is used to
Il
mean "simple to present and understand". The Simplified Method yields identical
simulation results so that there is no "short changing" the rigorous approach of chapters 3,
4 and 5. However, the Simplified Method is restricted to the PWM-VSC family of
controllers. Its advantage is that the Nonlinear Control Method is more accessible to
power electronic designers because the Simplified Method does not require a background
knowledge of the reference texts [1-3]. Chapter 6 applies the Simplified Method to the
STATCOM, the UPFC and the C-UPFC.
Chapter 7 addresses: (i) sensitivity to parameter variations and (ii) remote terminal
voltages which enter into the system equations, the time variations of which need to be
estimated from local measurements. These topics are likely to be the material for another
thesis because these practical issues have to be solved before the Nonlinear Control
Method will be used practically in the field. This brief chapter is in the nature of a
reconnoitre, to find out from a few simulations whether the Nonlinear Control Method
has sufficient robustness to pursue further research.
Chapter 8 contains the conclusions and suggestions for further study.
1.4 CONTRIBUTIONS
To the best of the knowledge of the author, the contributions of the thesis are:
Nonlinear Control Method
(1) The successful application of the Nonlinear Control Method to the following
members of the family of F ACTS controllers which are based on Pulse Width
Modulated, Voltage-Source Converters (PWM-VSC):
(i) Static Synchronous Series Compensator (SSSC)-- see chapter 4
12
(ii) Unified Power Flow Controller (UPFC) -- see chapter 4
(iii) Center-Node Power Flow Controller (C-UPFC) -- see chapter 5.
From physical insights on the F ACTS controllers and analytical insights on the
Nonlinear Control Method, the method can be extended to other members of the
same family:
(iv) Multi-Terminal Unified Power Flow Controller (M-UPFC)
(v) Interline Power Flow Controller (IPFC)
(vi) STATic COMpensator (STATCOM)
(vii) Back-to-Back, Voltage-Source HVDC (BTB-VSC-HVDC)
(2) The presentation of the Simplified Nonlinear Control Method for PWM-VSC
F ACTS controllers.
Center-Node Unified Power Flow Controller (C-UPFC)
(3) Proving the viability of the C-UPFC, which has performance capabilities which
not only rival the Unified Power Flow Controller (UPFC) but can also double the
transmissibility of power because it conceived to be located at the mid-point of a
radial transmission line. -- see chapter 2 and 5
(4) Demonstrating that there is a new method ofmaintaining power balance in the dc
link. In the UPFC, it is by a shunt VSC and a series VSC. In the IPFC, it is by
series VSCs of different radiallines. In the C-UPFC, the dc power balance is by 2
series VSCs in the same transmission line. One series VSC is between the
sending-end and the center-no de and the other VSC is between the receiving-end
and the center-node. -- see chapter 2 and 5
13
Chapter 2
Center-Node Unified Power Flow Controller (C-UPFC)
2.1 INTRODUCTION
Controllers of Flexible AC Transmission Systems (F ACTS) have the potential to
increase the transmission capacity of existing electric utility network and to enhance its
dynamic performance. To date, the F ACTS controllers, from which a selection can be
made, may be categorized as: (1) transformer based---Inter-Phase Controllers (IPC) [70],
(2) thyristor based---Thyristor Controlled Series Capacitor (TCSC) [13], (3) Force
Commutation based---STATic COMpensator (STATCOM) [22-26], Static Synchronous
Series Compensator (SSSC) [14, 23, 27, 71-73], the Unified Power Flow Controller
(UPFC) [23, 35, 36, 74] and the Interline Power Flow Controller [38].
This chapter presents the Center-node UPFC (C-UPFC) to increase the repertoire of
solutions for selection. It is a variant of the UPFC invented by Laszlo Gyugyi [23,35,36]
which consists of a series converter and a shunt converter connected to a common dc bus.
The UPFC has been claimed to be the ultimate F ACTS controller, as it has three
independent control degrees of freedom - one degree for the active power through the
radial line and two degrees for the reactive powers at both ends of the line. Not only has
the originality of the UPFC concept drawn many workers to F ACTS research, but
because of interest from industry, a 160 MY A prototype UPFC is already in operation in
the Inez area in the south central part of the AEP (American Electric Power) system [35,
36].
14
The C-UPFC consists of three converters: two series converters on either side of a
center-node, to which a shunt converter is also connected. The two series converters share
the same dc bus so that the active power, which enters one series converter, exits through
the other series converter.
The shunt converter, which tentatively in this chapter has its separate dc bus, operates
independently as an Advanced SYC or STATCOM. Together with switched capacitors in
parallel with it, the function of the shunt converter is to provide Y AR support to main tain
the AC voltage of the center-no de at regulated rated voltage.
From the center-node, whose AC voltage is regulated by the ST A TCOM, the series
converters inject their ac voltages. By adjusting their magnitudes and phase angles, the
active power through the transmission line and the reactive powers at both ends are
controlled, thus fulfilling the requirements of the UPFC.
The first difference of the C-UPFC from the UPFC hes in the shunt converter. The
shunt converter of the UPFC [35, 36, 74] has twofunctions: (1) to provide reactive power
compensation on one end of the transmission hne or to provide voltage support at the
node at which the UPFC is located; (2) to provide the retum path for the active power
rectified or inverted by the series converter. The shunt converter of the C-UPFC has the
sole function of providing voltage support at the node. The intent is to give the C-UPFC
greater freedom to be located at positions remote from the sending-end bus or the
receiving-end bus, and/or in locations where voltage support is needed. In a previous
paper [39], it has been pointed out that by locating a F ACTS controller at the mid-point of
the transmission hne, the transmitted active power can be doubled. The C-UPFC has been
conceived to exploit such gain in power transmissibihty. Although the mid-point is the
15
optimal position, the C-UPFC still functions when off-centered. This conclusion follows
from theoretical considerations and has been demonstrated in digital simulations.
The second difference lies in the management of the active power rectified (or
inverted) by the series converter(s), as the ac currents are not in time quadrature with the
injected voltages. In the UPFC, the shunt converter plays the role of the "power slack" to
the series converter so that the algebraic sum of active powers injected into the dc bus is
zero. In the C-UPFC, one series converter plays the role of the "power slack" of the other
series converter.
For clarity in the presentation of this chapter, line resistances and converter losses are
assumed to be negligible. This assumption is made only for the purpose of simplifying the
phasor diagrams and the presentation of the underlying concepts in this chapter. The
operation of the C-UPFC and the analysis do not depend on these assumptions.
2.2 OPERATION REQUIRING C-UPFC
Fig. 2-1 shows the single-line diagram of a radial transmission line joining the
sending-end voltage, Vs, to the receiving-end voltage, V R. The sending-end and
receiving-end voltages are given, in per unit values, as Vs = 1.0 L 8 and VR=1.0L.O, 8
being the voltage angle between them.
2.2.1 Constraints-Phase-Shifler Operation
In the illustrative example chosen, as depicted in the phasor diagram of Fig. 2-2, the
voltage angle, 8, is a large angle, requiring the C-UPFC to operate as a combination of a
STATCOM and an electronic Phase-Shifter. This application choice has been taken for
16
LmES LmER
v 0' center-node
jXs : :Es~ - - -/j~;: jXR 1-----1--------1 rv rv f---r-----'
L ____________ ~
C - UPFC
Fig. 2-1 Equivalent circuit of C-UPFC
"Es Vo E" -' _____________ --------____ R "
(a)
Fig. 2-2 Phasor diagrarn (a) C-UPFC (b) Phasors of series converters
(b)
clarity in the diagrarn. (For another application, in which 8 is a srnall angle and the line
17
reactance is very large, and the C-UPFC is required to serve as a combination of a
STATCOM and a Series Capacitor Compensator, the resulting Phasor Diagram is very
c1uttered and crammed. The Phasor Diagram of Fig. 2-2 is still valid for this application,
although it would have a different appearance.)
It is desired to send complex power Ss=Ps+jQs and recelve complex power
-SR=PR+jQR at both ends of the transmission line. From the lossless assumption, the
active power sent and received are the same, Ps=PR. In general, it is not possible to
specify Qs and QR independently. In order for this to be possible, the radial transmission
line is broken at the center-no de into two lengths, LINE S and LINE R as illustrated in
Fig. 2-1 and Fig. 2-3. Designating the currents through LINE Sand LINE R to be
respectively Is and IR, the CUITent phasors Is and IR can be calculated from Is = (Ss / Vs )*
and IR = - (SR / V R)*, where * denotes the complex conjugate operation.
2.2.2 Current Continuity
From the CUITent phasors depicted in Fig. 2-2, it is evident that the C-UPFC, whose
schematic is shown in Fig. 2-3, must have a shunt path at the center-node. For Kirchhoffs
Current Law to be satisfied, the required shunt CUITent is:
(2-1)
2.2.3 Center-Node Voltage Vo
Fig. 2-2 illustrates the construction of the CUITent phasor, 10= Is-IR. When the current 10
flows through the switched capacitors and the Shunt Converter (which is made to operate
as a Capacitive Reactance), capacitive VAR is provided to support a voltage Vo at the
center-node. Vo must lag 10 by 90°. With negative feedback control of Eo, the voltage
18
injected by the Shunt Converter, the center-node voltage, Vo, is regulated at 1.0 p.u.
voltage. Thus Vs, V Rand Vo alliie on the circle with unit radius as shown in Fig. 2-2.
2.2.4 Voltage Gap
The sides of the two hatched triangles in Fig. 2-2 display the voltage phasor
summations of (Vs - jXs1s) and (V R + jXRIR) of UNE S and UNE R, whose line
reactances are assumed to be jXs and jXR respectively. Clearly, there is a voltage gap
between the hatched triangles, which the C-UPFC must bridge, in order that the specified
complex powers, Ss and (-SR), are delivered at the sending-end and the receiving-end
respectively.
Line R --1
S . R
enes Converter S Series Converter R
transformer
ER +'lf~I~1 transformer
~----~'------1~+ L--______ -+-___ ~ - V deI
Center node
switched [ capacitor
banks l [f[ III
transformer
Shunt -converter
Fig. 2-3 Single line diagram of C-UPFC
19
2.2.5 Voltage Bridge
Using Vo as a center-no de voltage support, the voltage phasors Es and (-ER) are used
to span over the voltage gaps between the hatched triangles in Fig. 2-2. The phasors Es
and ER are from the injected voltages of Series Converter S and Series Converter R of the
C-UPFC respectively. Kirchhoffs Voltage Law applied to UNE S and UNE R yields:
Vs = jXsIs + Es + Vo
Vo=jXRIR-ER +VR
(2-2)
(2-3)
As the operation of C-UPFC depends active power exchange between the two series
converters, it is necessary to be assured that there is active power balance. A proof of
active power balance is given in Appendix 2-B.
2.3 DESCRIPTION OF C-UPFC
2.3.1 C-UPFC in Radial Transmission Line
Fig. 2-3 shows the single line diagram of the C-UPFC in which the bridging voltages
Es and ER are injected by Series Converter S and Series Converter R. From the center
node, the shunt current, 10, flows through a Shunt Converter in parallel with the Switched
Capacitors.
2.3.2 Voltage-Source Converters
Each of the three converters in Fig. 2-3 is a 3-phase, voltage-source converter whose
detail is shown in Fig. 2-4. In Fig. 2-4 (b), each of the six symbols, consisting of the 'V'
within the rectangular box, represents a GTO, IGBT or IGCT switch. The details of the
operation of the voltage-source converter will not be discussed here and interested readers
are referred to [75]. In addition, chapter 3 will describe the modeling of a voltage-source
20
converter. Here, it suffices to say that each converter produces balanced 3-phase
sinusoidal voltages at line frequency, which are injected by way of the transformers into
the transmission system. The single-line equivalent circuit of each converter, consisting
of the voltage phasor, E, behind the reactance, jXe, is shown in Fig. 2-4( c). In order to
avoid complicated algebraic expressions, Xe of Series Converter S and Series Converter R
are lumped into Xs and XR. It is within the art in High Power Electronics to control the
magnitude and the phase angle of the voltage phasor, E.
2.3.3 Shunt Converter
The total shunt cUITent, 10, provides the CUITent "slack" so that Kirchhoff s CUITent
Law at the center-no de can be satisfied. As 10 flows across the switched capacitors and
the Shunt Converter, the ac voltage, Vo, is supported and regulated to 1.0 p.U. at the
center- node.
A
+ .. _~D EI~E
transformer
N converter
(a)
-E-+ -A~N~' ~ . ~
transformer
converter
(b)
Fig. 2-4 Voltage-source converter (a) Single line diagram representation (b) 3-phase bridge with transformer (c) Equivalent circuit
(c)
21
The switched capacitors provide coarse but cheaper VAR support. The more costly
shunt converter provides continuous VAR support and close regulation of the ac voltage,
Vo. Tentatively in this chapter, the shunt converter has its separate dc bus, whose
regulated dc voltage V dc2 projects the ac voltage phasor, Eo, at its ac terminaIs. From the
magnitude and phase angle of Eo, there are two control degrees of freedom. As the
operation of shunt converter as a STATCOM is well-known, nothing needs to be added.
2.3.4 Series Converters
From the center-node, the voltages Es and ER of the Series Converters R and S, are
inserted by the series transformers to UNE S and UNE R respectively. As each voltage
phasor has two degrees offreedom (magnitude and phase angle), there are altogether four
control degrees of freedom to specify the complex powers, Ss=Ps+ jQs and SR=PR+jQR at
both ends of the transmission line.
The series converters share a common dc bus whose dc voltage is regulated at V de 1.
The dc bus provides the channel by which active power rectified by one series converter
is inverted out of the dc link by the other series converter so as to satisfy the real power
balance requirement Re (Es1s*-ERIR*) = O.
One of the series converters is assigned the dut y of a dc voltage regulator, which
automatically assumes the role of a "power slack" to take the active power rectified or
inverted by the other series converter. The voltage regulator employs the phase angle of
its injected ac voltage to control the active power admitted into the dc bus to null the dc
voltage error between the measured value OfVdcl and the reference setting.
Having used up one control degree of freedom, the series converters are left with three
degrees of freedom: the two voltage magnitudes for Qs and QR, and the remaining phase
angle to control PS=PR.
22
2.4 MULTI-CONVERTER CONTROL
The automatic control of the C-UPFC consists of two parts: (1) Estimation of the
Complex Power Settings and Voltage Injections of the Series Converters, (2)
Proportional-Integral Feedback.
2.4.1 Estimation of Co mp lex Power Settings and Voltage Injections of Series Converters
The estimation of reference settings is found to increase the speed of response. The
measurements taken at the terminaIs of the C-UPFC are the hne CUITent measurements, Isi
and I RI, and the voltage measurements, V csl
, at the terminaIs between UNE S and Series
Converter S and V CRI, between UNE R and Series Converter R. Since jXs and jXR are
known, from the measured values of V csl, V CRI, ISI and hl, the estimations of the remote
voltages V Si and V RI can be made from V SI=(V csl + jXsISI) and V RI=(V CRI - jXRIR
I).
At each sample interval, the new required line cUITents, ISII and IRII , can be computed
from the latest estimations of the voltages V Si and V RI and the specified or updated
complex powers SSI and SRI.
From (2-1), 1011 is computed. Voll is 90° behind and has a magnitude of 1.0 p.u.
Knowing jXs, jXR accurately, and having the estimations of V Si, V RI and VOII, the injected
voltage phasors ESI (=EsILSsl) is computed from ESI = VCSI-VOII and ERI (=ER1LSR
1) is
computed from ERI = Voll -Vcl The computed values ESI and E R
I are sent to Series
Converter S and Series Converter R as their "open loop" control voltage settings. The
complex powers, (V cs' IslI*) and (V CRI IRII*), are assigned as "closed Ioop" reference
settings of the feedback controis of the Series Converter S and Series Converter R.
23
2.4.2 Proportional-Integral Feedbacks
As the first iteration in research on the C-UPFC, proportional-integral feedback is
considered sufficient. Further refinement using more sophisticated control will be used, if
needed, in later iterations. This section shows how local closed loop feedback have been
implemented using the estimated settings from 2.4.1.
The complex powers to the C-UPFC at the terminaIs of the Series Converter S and the
Series Converter Rare measured and compared with their complex power reference
settings, (V csl ISII*) and (V CRI IRII*). The Real parts and the Imaginary parts of the
complex errors, after passing through Proportional and Integral Transfer Function Blocks,
are negatively fedback to control the perturbation magnitudes, ~Es and ~ER, and the
perturbation phase angles, ~es and ~eR, of the complex voltages
Es=(Esl+~Es)L(esl+~es) and ER=(ERI+~ER) L(eRl+~eR). The perturbation variables, ~es
and ~eR, are used to null the error of the active power of one Series Converter and the
error of the dc voltage of the other Series Converter, which has been designated to operate
as the DC Voltage Regulator. The remaining perturbation variables, ~Es and ~ER, are
used to null the errors of the Reactive Powers. Altogether, there are four Proportional
Gains and four Integral Gains to fine-tune for fast response.
In addition, there are two Proportional Gains and two Integral Gains of the magnitude
and phase angle controls of the Shunt Converter to be fine-tuned for fast operation as a
STATCOM.
24
2.5 DIGITAL SIMULATIONS
2.5.1 Simulation Software
A well-known, digital simulation software package in which industry-users have
confidence, was initially used to validate the concepts of the C-UPFC. This was not
successful, as no provisions had been made in the software package to model the two
Series Converters. As a result, the digital simulations of the research are based on
software written in MA TLAB.
2.5.2 Conditions of Tests on C-UPFC
The uncompensated line reactance is Xs+ XR=0.3 p.u. As a point of reference, the
transmitted power across the uncompensated line is taken as 1.0 p.u wh en operating at
8=17.4°.
In the example chosen, it is assumed that the buses at both ends of the line make an
angle of 8=60° so that the C-UPFC is required for phase-shifting, in addition to reactive
power compensation.
Successful multiple-variable control of the three Voltage-Source Converters is critical
to the realization of the C-UPFC. For this reason, the tests have been designed to show
that this is possible. Appendix 2-A lists the Proportional and Integral Feedback gains.
Fig. 2-5 shows the response to a "step" demand of the active power, Ps. The fast, 3-
cycle long response in which the transients are relatively small is made possible by tuning
of the P-l gains and by using a gentle incline in the "step" change. The magnitude of Voa,
Qs and QR are constant except during the brief transient. Zero Qs and QR have been
specified because it is easy to see in Fig. 2-5 that the sending-end and receiving-end
25
currents of the a-phase (isa, ira) are in phase with their corresponding voltages (vsa, vra).
The voltages and currents are represented in light and heavy lines respectively. The shunt
converter current ioa leads the voltage Voa by 90°, confirrning that STATCOM operation
has been achieved. Since Fig. 2-5 is obtained from digital simulations, the C-UPFC is
stable un der the condition of the test. Otherwise, the simulations would not have
converged to the steady-state equilibrium solutions displayed.
A number of simulations similar to Fig. 2-5 have been conducted in order to show that
the stable, operating range is extensive. Fig. 2-6 presents the operating points, marked by
x, which have been shown to be stable in the range of the active power, Pref. The other
control settings are QSreFQRreFO.O and VOreF 1.0 p.U. Fig. 2-7 is for the range of the
receiving-end reactive power, QRref. The other control settings are QSreFO.O, VOreF 1.0 p.U.
and PSreFO.9 p.u.
(p.u.)
::: -H .. ; ci u -~
Ira -1 _un .... -----------'---_________________ 1
2' ---~---. -.---~----,-. -T--------
-~ t- ____ ~_~ ____ ~----'__ ._ L ____ .______ ,._ .. _______ , __
Ps ~~[;==-~====~/_~------~----~----~ Qs 0g ~ ____ -_--____ -y--_-_---_-______ _
-0.5~--~------------~------~--------~------~
QR O.g ~-- -, -------..M._' =-_~_--.--r--_____ _ -0.51_._ ._, __ . ___ .~ __ ._.L. __
1.08 1.12 1.16 1.2 1.24 1.28
Fig. 2-5 System response for a step change in Psref
(- voltage - cUITent)
time (5)
26
The operating condition of the test of Fig. 2-8 is one in which the steady-state
equilibrium solutions have been reached un der constant control settings of VOref, PSref,
QSref and QRref. Then a step-change is introduced in the voltage angle from 8=600 to 8=900•
Such a test can only be performed by simulation because in the field the voltage angle
cannot change so fast. The simulations show that Vo, Ps, Qs and QR are held unchanged
by the local feedback controls alone. The test indicates that when 8 oscillates during
inertial hunting, the quantities Vo, Ps, Qs and QR will be held constant also.
The results of the test of Fig. 2-8 can be better appreciated, when one is reminded that
the feedback signaIs to the controls of the C-UPFC are local, i.e. they are taken at the
terminaIs of the C-UPFC. The regulated reactive powers Qs and QR are delivered at the
ends of the transmission line. The step-change in the voltage angle, 8, also occurs at the
ends of the transmission line. Thus the C-UPFC manages the control ofremote quantities
from local information.
P (p.u.)
2.5
2
1.5
1
0.5 /X//
X/
/X/ X/
/
/X/
/X/ /X/
/X/
/X/
xP /X///
o 0.5 1 1.5 2 2.5
Fig. 2-6 Operating range for real power P Simulation data X
Pref(P·U.)
27
-1
QR (p.U.) 1
0.5
0.5 1 Q (p.u.) Rref
Fig. 2-7 Operating range for reactive power Qs Simulation data X
(degge~) l{g' _--..:_-...;..-___ ....J---'~~_-_'_'_--_---~~~~~~~~~____'I Ysa 6 1
lsa -1
~ Y:aa _~ : 1
~ Yoa 6 '1 '-" loa -1
Ps 1.51-------~Ir_· --------'--~---- 1 0.5 ~----1 _ ______' _ ______' _ ______' _ ____'_ _ ____'_ _ ____'_ _ ____'_ ___ ~
Qs _~:~ L-E_L--_-_-_--'--r ___ -'-------1-
T
------,--_-'----------1-' _-L-------1n 1
QR_~:~ E~__ --~-~=_~_L--' '---~_-I 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28
time (5)
Fig. 2-8 System response for a step change in 0 s (- voltage - cUITent)
28
2.6 CONCLUSION
This chapter has given a detailed description of the Center-Node Unified Power Flow
Controller (C-UPFC) which has four independent control degrees of freedom. The C
UPFC has a novel structure which consists of three converters: two in series and one in
shunt. Active ac power, which enters one series converter, exits through the second series
converter. This is in contrast to the UPFC of L. Gyugyi in which the active ac power
which enters (or exits) from the series converter finds its exit (or entry) by the shunt
converter. The two series converters of the C-UPFC implement the original three degrees
of control of the UPFC: control of the active power through a radial line and the reactive
powers at both ends of the line. The shunt converter of the C-UPFC operates exclusively
as a STATic COMpensator (STATCOM) whose function is to regulate the AC voltage of
the center-no de and this constitutes the 4th independent degree of control freedom.
The digital simulations have shown that the C-UPFC is stable and operates with fast
response under P-I control for Phase-Shifter Operation. Unfortunately, it has not been
able to stabilize the C-UPFC using the same P-I control for two other applications: (2)
Capacitive Reactance Compensation and (3) Power Flow in Reversed Direction. Since the
C-UPFC consists of three Voltage-Voltage Converters and the P-I feedbacks operate
separately on each converter, in retrospect, there is no reason to hope that the y will act
cooperatively. Therefore, it is decided that a systematic control method, which considers
the C-UPFC as a single entity, should be employed. Furthermore, to avoid another
disappointment along the way, the systematic control method sought should not depend
on simplifying assumptions such as small perturbation linearization. Therefore, the
29
decision was made to learn the Nonlinear Control Method and to apply it to stabilize the
C-UPFC.
APPENDIX 2-A
PROPORTIONAL AND INTEGRAL GAINS OF FEEDBACK CONTROL
Series Converter S
~Es Kp=0.004
~8s Kp=0.0017
Series Converter R
~ER Kp=0.04
~8R Kp=0.4072
Shunt Converter
Kp=OA
Kp=4.2
Kr=5.0
Kr=OA363
K[=10.0
Kr=42.0
K[=1O.0
K[=80.0
APPENDIX 2-B
ACTIVE AC POWER BALANCE IN SERIES CONVERTERS
Fig. 2-2 (b) extracts the voltage and CUITent phasors (Es, Is) and (ER, IR) from Fig. 2-2
(a). As the voltages and CUITent phasors are not 90° apart, active ac powers are absorbed
or generated by the injected voltages. The C-UPFC, as a F ACTS Controller, cannot be a
source or sink of active power. As the Shunt Converter is operated exclusively as a
30
Capacitive Reactance, it is necessary that Series Converter S be a source or sink of the
active AC power from Series Converter R, with the dc bus serving as a conduit of the
power transfer. It is necessary to show mathematically that the total active AC powers of
the Series Converter S and the Series Converter R have an algebraic sum equal to zero.
Proof
Multiply (2-2) by Is*
Vs Is*= jXsIsIs *+ EsIs*+ Vols*
Multiply (2-3) by IR*
Vo IR *= jXRh IR* - ER IR* + VRIR*
Adding (A2-1) and (A2-2)
(Vs Is* - VRIR*)= (jXsIsIs *+ jXRIR IR*)
+ (EsIs* - ERIR*) +Vo(ls-IR)*
(A2-1)
(A2-2)
(A2-3)
Since PS=PR, that is the active power sent and received are the same, Re (Vs Is* -
VRIR*) = O. For the reactance components, Re (jXsIs Is*)= 0 and Re (jXRIR IR*)= O.
Finally, Re {Vo (Is- IR)*}= 0, since Vo is 90° apart from 10= Is-IRo It follows that by
taking the real part of (A2-3), the active power balance of the series converters is satisfied
as:
Re (EsIs* - ER IR*) = 0
End ofProof
(A2-4)
31
Chapter 3
Voltage-Source Converter Modeling and N onlinear Control
3.1 INTRODUCTION
In chapter 2, it has been found that with simple P-I control, the C-UPFC is able to
operate in only one of the 3 operation modes. This thesis proposes to apply the Nonlinear
Control Method, associated with the books of A.Isidori [1] and H.Nijmeijer and A.J.van
der Shaft [2], to the C-UPFC. This is because it is a systematic approach to design fast
stable control for nonlinear systems such as the C-UPFC. Apart from researchers of
Control Research, the rest of the Graduate School in the Department of Electrical and
Computer Engineering are unfamiliar with the Nonlinear Control Method. For this
reason, it is necessary to use this chapter to introduce the notations employed and to
sketch an outline of the ideas behind the method. The emphasis is to show the steps,
which have to be taken, and to state the rules, which have to be respected, so that its
applications to the SSSC and the UPFC in chapter 4 and to the C-UPFC in chapter 5 can
be followed. This chapter can be criticized for its lack of mathematical rigor. In defense,
it may be said that mathematicians and control engineers have to rely on mathematical
rigor and proofs to advance from lemma to lemma and from theorem and theorem. Power
electronic engineers can only be thankful for their wonderful breakthroughs. They pick up
32
from where the control theorists have left off. They have digital simulations (and at a later
stage laboratory experiments) to verify the theory. The criticism, which will be
considered fair, is whether the Nonlinear Control Method has been applied correctly.
What th en are the contributions of the application researcher? The Nonlinear Control
Method involves solving partial differential equations, the solutions of which, as in aIl
integration, fall in the realm of art. It is through intimate knowledge of the system
equations of the Voltage-Source Converter that the solutions have been found.
In introducing the Nonlinear Control Method, the first step is to develop the
mathematical model of the Voltage-Source Converter in Section 3.2. It is small and
structured enough (system order n=3, number of inputs m=2) to serve as an illustrative
example on which the Nonlinear Control Method is applied. The principles of the
Nonlinear Control Method are introduced as a tutorial in Section 3.3. After a firrn grasp in
the specifies of a small system, the method is generalized so that it can be applied in
chapters 4 and 5.
3.2 MODELING OF A VOLTAGE-SOURCE CONVERTER
Fig. 3-1 presents the diagrammatic representations of a voltage-source converter
(VSC). Fig. 3-1 (a) is the single line diagram while its equivalent circuit is shown in Fig.
3-1 (c). Fig. 3-1 (b) shows the detail circuit of the voltage-source converter (VSC). In Fig.
3-1 (b), each of the six symbols, consisting of a 'V' within a rectangular box, represents a
33
solid state switch, which may be a GTO, an IGBT or a IGCT. Each ac phase has an upper
switch and a lower switch, which are tumed on and off according to a pulse width
modulation (PWM) sequence.
3.2.1 Ideal Current Source Equivalent Circuit
For the Voltage-Source Converter configuration, a capacitor C (not shown) is al ways
connected between D and E on the dc side. Fig. 3-2 shows that on the dc side, the
converter is represented by an ideal current source ide. The current ide originates from the
currents ia, ib, and ie from the ac side. The ac currents pass through the switches or their
antiparallel diodes in the form of width-modulated current pulses which charge the
capacitor to a voltage Vde.
A
transformer transformer
N converter
converter
(a) (h) (c)
Fig. 3-1 Voltage-source converter (VSC) with transformer (a) Single line symbolic representation (b) Detail3-phase bridge ofVSC (c) Equivalent circuit of a-phase
34
3.2.2 Ideal Voltage Sources
The switching of the solid state switches produces a train of positive and negative
width-modulated voltage pulses with the magnitude of v de at the ac terminal of each of the
three phases, which are represented as ideal voltage sources ea, eb, and ee. When the
Sinusoidal Pulse Width Modulation (SPWM) is applied, the fundamental frequency
component of a train of pulses is the linearly amplified voltage of the sinusoidal
modulating signal of its phase, when the dc voltage v de is constant. For the purpose of this
thesis, it is assumed that the switching frequency is sufficiently high [76] so that the
carrier frequencies and the sidebands of the SPWM switching can be economically
removed by tuned L-C filters (not shown).
R jX ea + ,-------"'Y-v'------1I0\----------,
+
N
Fig. 3-2 Equivalent circuit for a voltage-source converter in a-b-c frame
1
35
3.2.3 Physical Reasonfor System Nonlinearity
The system nonlinearity, because of which the Nonlinear Control Method has to be
applied, is due to the fact that in general the dc voltage Vde is not held constant. Thus the
ideal voltage sources ea, eb, and ee cease to be linear amplifiers of the modulating input
signaIs Ua, Ub, and Ue respectively.
In this thesis, the following assumptions have been adopted:
(a) The ac voltages at the converter terminaIs form the 3-phase balanced system,
described by the voltage vector y(t) = [vit), Vb(t), Ve(t)]T.
(b) AlI inductive parameters of the transformer and the filtering reactor are
represented by L (=X/(2TTf)), and aIl resistive losses are represented by R.
(c) The PWM switching frequency is sufficiently high, therefore the discontinuous
converter model can be represented by a continuous time-averaged state space
model.
On the basis of these assumptions, the ac side instantaneous voltages y(t) and ~(t) can
be expressed in the following forms:
cos(wt + i\)
y(t) = HV1 cos(wt + Dv - 231t
)
21t cos(wt + Dv +-)
3
(3 -1)
36
where
2n cos(wt + 8e --)
3 2n
cos(wt + 8e +-) 3
VI - the magnitude of system voltage (line-line)
w= 2nf and fis the system frequency
Cv - system voltage phase angle
k Mod - modulation index
(3 - 2)
As it can be seen, the magnitude of converter ac side voltage ~ is controlled by the
modulation index kMod and is dependent on the capacitor voltage v de as well, while the
phase is controlled by the angle Ce with respect to the reference.
3.2.4 Modeling in a-b-c frame
The continuous-time averaged converter model is thus obtained as the following:
2::(t) = L :t i(t) + Ri(t) + ~(t) (3 - 3)
(3 - 4)
(3 - 5)
37
cos(œt + DJ
= ·fi.!, cos(œt + Di -~TC) 3
2 cos(œt + Di + -TC)
3
where Il is the magnitude of the system line current.
(3 - 6)
Equation (3-4) is the converter power balance equation. The ac power on the left side
of the equation is transferred to the dc side in the form of v dcidc,
The equivalent circuit for a voltage-source converter in a-b-c frame is shown in Fig. 3-
2.
3.2.5 Modeling in d-q frame
The model described in (3-3), (3-4) and (3-5) can be transformed into a rotating
reference d-q frame via the well-known orthogonal, power invariant coordinate
transformation:
1 1 1 l cos(wt +e) sin(wt + e) 01 2 2
cos(œt + 8) 0 H J3 J3 [T] = -Sin(~ + 8) 0 (3 -7)
2 2 o 1 1 1 1
J2 J2 J2
Applying (3-7) to (3-1), (3-2) and (3-6):
(3 -8)
38
[T]~(t) =
Hk MOd V deCOS(Ùe - 8)
HkModVdeSin(Ùe -8)
o
(3 -9)
(3 -10)
The zero sequence components (vo, eo, io) in the transformed voltage and CUITent items
are aU zeroes and the corresponding equation can be removed. Thus, by matrix pre-
multiplying [T] on both sides of (3-3), (3-4), and (3-5) and dropping out the 0 elements,
the transformed converter model in d-q coordinates can be obtained as the foUowing:
(3 -11)
d . R. . 1 1 -1 =--1 -an --v u +-v dt q L q 1 d L de q L q
(3-12)
(3 -13)
where CUi = CU + d8 . In this thesis, d8 = 0 is assumed. Therefore, CUi = CU • dt dt
By selecting id, iq and Vdc as the state variables, and Ud and Uq as the control inputs,
which are:
39
The equations (3-11) - (3-13) can be rewritten in following standard forrn:
~ = f(~) + gl (~)UI + g2 (~)U2
where
- -
R 1 --XI + (ùj x 2 +-Yd L L
R 1 --x -(ù·X +-Y L 2 IlL q
-!r c
1 --x L 3
o 1
--x L 3
1 -x C 2
(3 -14)
(3 -15)
(3 -16)
(3 -17)
40
It is clear that the converter model is nonlinear, or more precisely bilinear, since the
control input.!! is multiplied to the state ~ in the state equation, as shown in (3-14) - (3-
17).
3.3 PRINCIPLE OF NONLINEAR CONTROL
3.3.1 Preliminaries
The nonlinear method presented III this thesis is based on the exact linearization
method of A.Isidori [1] and H.Nijmeijer and A.J.van der Shaft [2]. The block diagram of
Fig.3-3 illustrates the steps taken when applied to VSC of Fig.3-2. The nonlinear
equations of (3-14), inside the VSC block, is viewed as an m=2 input, n=3 order system
of the form:
r-;;;- V de - 5 Nonlinear E
I:! ~ f---- d-q ~ transformation ~ i(t) Id
y(t) Iq
Transformation
Ux)
VSC Converters
.!! ,-----------,~
Inverse ~ ~-----1 Transformation ~:?;_-'
w
,-------1 1 1
l 1
1 1 1 1 z ZO
: 1- Linear -1 : Feedbaek
1 1
1 1 1 1
~"-l "-l "-1 Linearized Model
1 1
'- _______ 1
Fig. 3-3 Nonlinear Control in VSC applications
41
• 2
~ = f(~) + IgJ~)Ui (3 -18) i=l
In the Linear Feedback block are state equations of the form:
(3 -19)
where the dimensions of the state vector ~ and the input vector w are chosen to be
compatible with n=3 and m=2 and the time invariant matrices [C] and [D] matrices are
dimensioned 3x3 and 3x2 respectively.
Since linear system theory is better known, it is easier to introduce the Nonlinear
Control Method by first recalling the weIl known state-feedback technique to improve the
system dynamics of the linear system ~: =[C]~ +[D]w. In state feedback, the state vector
~ is used as the output which is fed back to the input w after passing through a linear
transformation, i.e. w=[E]~ where [E] is a 2x3 matrix. Substituting w=[E]~, the dynamic
equations are ~: ={[C]+[D][E]}~. There are now software from control toolboxes which
solves for the matrix [E] for any desired specifications of the eigenvalues of the matrix
{[C]+[D][E]} .
Although the above paragraph describes what is inside the Linear Feedback block in
Fig.3.3, the intent of the paragraph is also to bring out the fact that the Linear Feedback
block itself plays the role of feedback matrix [E] in the bigger context of the nonlinear
42
system of the Vsc. In w=[E].?;, the input w is activated by the output, which in state-
feedback is the state vector .?;.
Retuming to the nonlinear system, in order to improve its system dynamics its input 1!
must in the same way be derived from an equivalent the [E] matrix which takes
information from the output vector y of the nonlinear system. However, up to this point
no output vector y has been defined. This leaves the opportunity for output y=h(~(t)) to - -
be synthesized. The output y=h(~(t)) is used as the basis for transforming to the .?;-based
linear system. The transformation from the nonlinear ~-system to the linear .?;-system is
represented by the Nonlinear Transformation block in Fig.3-3.
In the linear .?;-system, linear state feedback is applied to improve the system dynamics
by pole-placement. The retum from the linear .?;-system to the nonlinear ~-system is
equivalent to closing the feedback loop. The Inverse Transformation block shows the path
of closure. This block derives the input vector 1! of the nonlinear ~-system from the input
vector w of the linear .?;-system. It is not necessary to make an inverse transformation
from.?; to~.
3.3.2 Mathematical Preliminaries
Readers preferring a complete description of the Nonlinear Control Method should
consult the references [1, 2] which are books on the subject. This section is, in part, a
brief tutorial explaining the notations which are used and sorne of the essential
mathematical background of [1, 2] which is required in order to follow the method, when
applied narrowly to the VSC-based controllers of this thesis. The objective is to give an
43
idea of what the method is about and no mathematical rigour is claimed. Chapter 6 will
present an altemate formulation which by-passes the mathematical background of [1, 2]
altogether.
Firstly, it is necessary to introduce the notion of relative degree and sorne
mathematical notations.
3.3.2.1 Relative Degree
The notion of relative degree is best explained using a linear, single-input single-
output (SISO) system. In the linear SISO case, f(~)=[ A ]~, g(~) = [B] and the output
y=h=[C]~. The transfer function relating the output to input is H(s)=[C]{s[I]-[A]r1[B].
The relative degree is the integer r which is the difference between the degree of the
denominator polynomial and the degree of the numerator polynomial in H(s).
3.3.2.2 SISO Nonlinear Example
Retuming to the nonlinear system of (3-18) but still keeping to the single input single
output case, differentiating the output y(t) with respect to time once and using y(t)(l) to
denote dy( t) dt
n 8h(x) dx n 8h(x) y(t)(I) = I-----J =I---{fj(~)+gj(~)u}
j=1 ax j dt j=1 ax j (3 - 20)
Because the summation operation is used repeatedly, the following notation called the
Lie derivative or the derivative ofh alongfis defined and used through out the text as:
44
(3 - 21)
An alternative formulation of Lfh(~) is based on the difJerential or gradient row vector
defined as:
(3 - 22)
Thus, the Lie derivative may be expressed as a vector product
(3 - 23)
or as an inner product
(3 - 24)
The operations can be extended. For example, the difJerential or gradient of Lth(~)
by applying (3-22) is
(3 - 25)
Likewise, the derivative of h along g is defined as:
(3 - 26)
45
or
(3 - 27)
Retuming to the first time derivative ofy(t), it can be written more briefly as:
(3 - 28)
When Lgh(~{t)) is not zero, the relative degree r=1.
However, when Lgh(~(t))=O, then y(t)(l)=L f h(~(t)). The second time derivative of y(t)
consists ofreplacing h(~(t)) with Lth(~(t)) in (3-20) and is
(3 - 29)
which can be rewritten as
(3 - 30)
Defining
(3 - 31)
or
46
(3 - 32)
the second time derivative ofy(t) is rewritten as
(3 - 33)
When LgLth(2i(t))u(t):;t:O, the relative degree r=2.
If the degree is higher than 2, the differentiation continues. Thus the relative degree is
the number of times by which the output is differentiated before the input u(t) appears.
The kth derivative, y(t)(k), contains the Lie derivative of the form L~h(2i(t)).
Beginning an output h(2i) whose relative degree is r, one can define a coordinate
transformation as follows:
(3 - 34)
Then since, d;l = L f h(2i(t)) + Lgh(2i(t))u(t) and since Lgh(2i(t))u(t)=O, one can define
(3 - 35)
or
(3 - 36)
One can keep on defining
47
where
and
dZr
_1 --=z
dt r
until u(t) appears in the rth terrn
which can be written as
dz _r = b(z) + a(z)u(t) dt - -
where
(3 - 37)
(3 - 38)
(3 - 39)
(3 - 40)
(3 - 41)
(3 - 42)
48
In this system whose single output has a relative degree of r, u(t) appears after the rth
differentiation because it is the way relative degree is defined. In (3-42) ~=<l>-l(?;) is the
inverse of the nonlinear transformation ?;=<l>(~).
Another simple example of a nonlinear transformation given here is the case when the
relative degree r is equal to the system dimension n (i.e. r=n). Thus the n functions h(~(t)),
Lfh(~(t)), L~h(lf(t)), .... L';lh(lf(t)) are used as the elements in the local coordinate
transformation.
In the z-system, the system equations appear in the form:
dZ 1
dt dZ 2
Zz
dz dt Z3
= = (3 - 43) dt
dZ n' l zn dt b(?;) + a(?;)u
dZ n
dt
One can make the new system of (3-43) linear and controllable by rewriting it as:
Z2
dz Z3
= (3 -44) dt
zn
w
The input of this linear system is w which is formed the last row of (3-44)
49
byequatingw=b(?;)+a(?;)u (3-45)
The linear equations of (3-44) in the matrix form is given for the case n=3 by (3-46)
below:
(3 - 46)
where
la 1 0] [C]= 001
000
and
Provided a(?;) is not zero, the closed loop feedback of Fig.3-3 has the input u which
can be obtained from u=(w-b(?;»/a(?;).
It should be pointed out that the Nonlinear Control Method only glves the
requirements of the output function y(t)=h(~(t» but not what the formula of it. One is
reminded that in the case when the relative degree is r, it means that h(~(t» must
simultaneously satisfy r equations, Lgh(~(t»=O, LgLth(~(t»=O, Lgefh(~(t» = 0, ...
LgL(;-')h(~(t» = O. As (3-21) and (3-31) serve as reminders, they are partial differential
50
equations of increasingly high orders. It has been found that for the systems analysed in
this thesis, the relative degree is at most r=2. Thus only Lgh(~(t))=O has to be satisfied.
3.3.2.3 Mufti-Input Systems
As exemplified by (3.14) to (3-17) of the Voltage-Source Converter, it is necessary to
deal with multi-input nonlinear systems. In this case, there are m=2 inputs for the n=3
order system. The Nonlinear Control Method has treatment for rn-input, rn-output
(MIMO) nonlinear system, so that m=2 output functions, hl(~) and h2(~) of relative
degree rI and r2 respectively can be assigned. There is a requirement that the relative
degrees of the rn-outputs must satisfy the equality rl+ r2+ ... rm=n. In the VSC case, this
requirement can be satisfied by letting hl (~) to have rI =2 and h2(~) to have r2= 1.
It is allowed in the nonlinear transformation to consider the MIMO system as an
aggregate of m independent SISO linear channels.
Substituting Yl=hl(~), one has
(3 - 47)
Since rl=2, it is required that Lglhl(~(t))=O and Lg2hl(~(t))=O so that the inputs
UI(t) and U2(t) do not appear. This leaves
(3 - 48)
51
The second tirne derivative is:
(3 -49)
which can be rewritten as:
(3 - 50)
u](t) and U2(t) appear in this equation. As in the SISO case, the ab ove derivations can be
used as part of the nonlinear transformation.
The z-systern state-space equations can be formatted as:
dZ 2 -=w dt 1
Substituting Y3=h2(~), one has
(3 - 51)
52
Using the above derivation to complete the nonlinear transformation, one defines
Completing the dynamic state-space equations in the z-frame,
In summary, the nonlinear transformation equation is:
(3 - 52)
These are the equations in ~(2f) in the Nonlinear Transformation block ofFig.3-3.
Collecting the z-state-variables, the state-equation is
(3 - 53)
where
lo 1 0J [C]= 000 ,
000
(3-53-a)
53
(3-53-b)
Tests using [C] and [D] show that the ~-system is controllable. Thus, it is possible to
design a 2x3 [E] matrix in the state-feedback w=[E]~ in which the eigenvalues of
{[C]+[D][E]}of the resultant state equation dz ={[C]+[D][E]}z are placed in desired dt -
locations. These are the equations in the Linear Feedback block in Fig.3-3.
The inputs w of the z-frame is related tO!! of the x-frame by the matrix equations:
(3 - 54)
Collecting the terms already derived above, one has:
(3 - 55)
The feedback acts on the nonlinear system through the inputs
54
inverse transformation block in Fig.3-3.
3.3.2.4 Conditions for Feedback Linearization
In the derivations above, it has been assumed that two conditions have been met:
(i) When h1(2f) has relative degree r1=2, it has been assumed from (3-47) that
(3 - 56)
(ii) For the inputs w and!! to be coupled, from (3-50) and (3-51) it is assumed that:
This means that the 2x2 matrix
[
L g, Lfh[ (~(t))
Lg, h 2 (~(t))
L g2 Lfh[ (~(t))] Lg, h 2 (~(t))
to be nonsingular to be invertible.
(3 - 57a)
(3 - 57b)
(3 - 58)
55
In order to avoid dealing with high order partial differential equations in Lg, Lfh\ (~(t))
and L g2 Lfh\ (~(t)), conditions (i) and (ii) are usually stated in terms of the rank and the
involutiveness of nested sets Go ç G\ ç G2 ••• which consist of vector fields of g\ (~),
g2 (~) and new vectors defined as adrg\, adrg2, which are respectively the Lie products of
3.3.2.5 Lie Product or Lie Bracket
The zero order Lie bracket is defined as:
(3 - 59)
The first order Lie product or Lie bracket is defined as:
(3 - 60)
8g(~) The expreSSIOns and
8x 8f(~) are the Jacobian matrices of g and f. The
8x
quantities adrg], adrg2 are also vectors.
Similarly, the second order Lie product or Lie bracket is defined as:
8(ad g ) 8f ad 2g.= f , (x)f(x)--=(x)ad g.(x)
f 1 8x - - - 8x - f 1 -(3-61a)
And the /h order Lie product or Lie bracket is defined as:
(3 - 61 b)
56
3.3.2.6 Examples of Lie Brackets
For example, from gl (~), g2 (~) and f(~) of (3-14) ~ (3-17), one has the following Lie
products.
R 1 -- ()). 0 o 0-- L 1
1 --x L 3 L R
= 0 0 0 f(~) - - ())i - LOO
~O 0 0 0 0 C
R l --x +-
L2 3 LC
())i = --x L 3
R ())i l --x +-x +-y
LC 1 C 2 LC ct
1 -x C 1
000 R
-- ()). 0 LlO
1 R = 0 0 - - f(x) - - ()). - - 0 L - - 1 L
O~ 0 0 0 0 C
()). _IX L 3
R l = --x +-
L2 3 LC
())i R l --x --x +-y
C 1 LC 2 LC q
1 --x L 3
1 -x C 2
57
3.3.2.7 Involutive Pro pert y
A distribution is involutive if the Lie bracket adrg of any pair of vectors f and g
belongs to the same vector field of f and g . Having the same rank is a standard test of
fulfilling the involutive property, thus:
rank [f,~] = rank [f,~, adfg] (3 - 62)
Roughly, involutiveness has to do with spanning the same vector space. Thus adrg is a
linear combination of f and g.
3.3.2.8 Reasonfor using Lie Brackets
From (2-6) on page 10 of 1 st edition (1985) [1] Isidori has listed the following property
which explains why the Lie bracket is used:
(3 - 63)
B(L h (x)) B(L h (x)) B(L h (x)) . vector dl h (x) = [ f 1 - f 1 - f 1 - ] lS coupled to g (x) and g (x)
f 1- BK' BK ' BK ---.l-~-1 2 3
so that
(3 - 64)
58
As a second partial differentiation operation is required in dLthl(~), it is preferred to
sufficient to show that
(3 - 65)
Thus the requirement is that dhl(~) spans the vectors adrgl and adrg2.
According to Theorem 6.3 of [2], the conditions of whether the nonlinear system is
feedback linearizable is related to:
Go = span { g l , g 2 }
G2 = span{gl' g2 ,adrgl, adrg2, adfgi' adfg2} - -
It can be verified that:
Go = span {gl , g2 } has constant rank 2 and is involutive;
G1 = span {gl , g2 , adfg 1 , adf g2 } has constant rank 3 and is involutive;
As G1 has full rank of n=3 and Go ç G1 ç G2 , G2 will have full rank of n=3 as weIl.
For this reason, the rank of G2 does not need to be checked and fortunately, the
calculations of adfgl and adfg2 are not necessary, thus saving lots ofwork.
As a result, the nonlinear system is feedback linearizable. From section 3.3.2.3 as n=3
(system order) and m=2 (inputs), the two output functions hl(~) and h2(~) must have
59
relative degrees of 2 and 1, respectively. Equation (3-56) is used as a guide in
synthesizing hl(~). Choosing h2(~) is more difficult still. The requirement that (3-58) is
non-singular is another guide. However, (3-58) includes terms relating to hl(~), which
presumes that the correct hl(~) has been found which satisfies (3-56).
3.3.2.9 Synthesizing hJ(J)
From (3-15), (3-16) and (3-17), for Go=span{ad~g\, ad~g2}= span{fu' g2 }, which
is involutive and of constant rank 2. It follows from the requirements L g, h 1 (~) = 0 and
(3 - 66)
It can be verified that hl = L X l2 + L X 2
2 + C x/ satisfies (3-66). This function is the 2 2 2
expression of the total energy stored in the converter. A similar total storage energy is
used throughout the thesis as an output function. In fact it is an important key which
enables the Nonlinear Control Method to be successfully applied to the VSC-based
F ACTS controllers in the thesis.
3.3.2.10 Synthesizing h2(J)
The requirement placed on h2(~) is that (3-58) is not singular. This corresponds to
satisfying (3-57a) or (3-65), and (3-57b) simultaneously. Both requirements are stated as:
60
In order to satisfy the inequality, the deterrninant of the matrix [tft] should be non-zero:
(3 - 67)
By ea1culation, the inner produets are:
1 8h 2 1 8h 2 <dh g > =--x -+-x -2' 1 L 3 ax C 1 ax
1 3
1 8h 2 1 8h 2 <dh g > =--x -+-x -2' 2 L 3 ax C 2 ax
2 3
The state variables Xl, X2 and X3 are possible ehoiee for h2. In ehoosing h2 = CX3, (3-67)
beeomes
2R Il] (--x X +-x +-v X ) L 2 3 C 2 L q 3 :;t:0
x 2
In the VSC, the de voltage is positive so that X3 > O. The only other requirement is
- V dX2 + V qX 1 :;t: O. This is satisfied wh en the reaetive power is non-zero.
61
W ' h hl' f h L 2 L 2 C 2 d h C h N l' It t e se ectlOn 0 1 =-X1 +-x 2 +-x 3 an 2= X3, t e on mear 2 2 2
Transformation equations are:
2 2 Z 2 = L f hl = - R( XI + X 2 ) + V ct X 1 - Ix 3 (3 - 68)
Z3 = h 2 = CX 3
With the following nonsingular state feedback transformation of
the linear state space model becomes
(3 - 69)
where [C] and [D] are time invariant matrices of (3-53-a) and (3-53-b) and w is the
input vector.
3,3.3 Generalizatian ta rn-input n-arder Nanlinear System
The method can now be generalized for m inputs u l, U2 '" um, and the nonlinear
system is now of the form:
• m
1f = f(~J + Lfu. (1f)u i (3 - 70) i=1
62
It is a matter of choosing m output functions hl(~), h2(~), ... hm(~) where h(x) E 91n
and is smooth. The MIMO system is said to have a vector relative degree ri with respect
to the i-th output hi(~) at a point ~o, if [1][49]
variables here), and for aIl ~ in the neighborhood of~o,
(b) The (m x m) decoupling matrix
. . 1 0 IS nonsmgu ar at ~ = ~ .
Lg", q-:'h, (~) :
Lgm Cr'h m (~)
m
(3 - 71)
If the relative degrees ri are such that 2:>i = n , then an exact linear state equivalent of ;=,
the system exists and can be obtained through sorne coordinate transformations and a
state feedback (aIl possibly nonlinear).
Therefore, the controlIer design is simplified by transforrning the original nonlinear
model to a linear model with outputs Yi and their Lie derivatives Cr'Yi being the new
state variables. Before proceeding to solve the output functions hl(~), h2(~), '" hm(~), it
needs to check if the nonlinear system of (3-70) is feedback linearizable. The following
two conditions must be fulfilled in order for (3-70) to be feedback linearizable:
63
(i) G{ = span{adigj: 1::::; i ::::; m, 0::::; j::::; i}, 0::::; i ::::; n - 2 is involutive and of constant rank;
as described in section 3.3.2.5, adigj is the Lie bracket in local coordinates ~ and it is
defined as:
8(ad g.) 8f ad;gj= 8~ 1 (~)f(~) - 8~ (~)adfgj (~), and so on.
- -
(ii) Rank Gn-I = n.
It is worth pointing out that for all the VSC F ACTS controllers which are studied in
this thesis, the number of inputs m is always m=n-l, where n is the system order.
Therefore, wh en rank Go=m=n-l and rank GI=n have already been ascertained, the
system is feedback linearizable. This is because rank G2, ... Gn-I will all equal to n due to
the fact that Go ç G1 ç G2 ... ç Gn- 1 • In this application, only first order Lie Brackets need
to be calculated. Thus one does not need to compute the second and higher order Lie
Brackets.
Once the nonlinear system of (3-70) is justified to be feedback linearizable, then it
needs to solve the m output functions hl (~), h2(~), ... hm(~), which satisfy:
(3 -72)
and the matrix
64
l dh d rl-I
< l' ~ f gl >
dh d r -1 < m,a t gl>
dh drl-I l ... < l' ~ f gm>
< dh drm-Ig > m' a f m
(3 - 73)
is nonsingular at ?Seo.
Here the operations < dh j , Grj
_2 > and < dh j , adtlgj > refer to the Lie derivatives of
the output function h j with respect to Grj2 , adtlgj respectively.
As the number of inputs m=n-l for aIl the nonlinear systems studied in this thesis, the
relative degree rI will be 2 and the remaining relative degrees (r2, ... rm) will aIl equal to 1.
The fourth stage is to form the new state space coordinates ?;(?Se), where
hl
VI-Ih f 1
z= (3 -74)
hm
Vm-Ih f m
The nonsingular state feedback transformation will be calculated as the following:
L VI-Ih J gm f 1
: u
L Vm-Ih gm f m
(3 - 75)
With (3-74) and (3-75), a new linear model is thus obtained as shown in (3-76). As has
just been pointed out, for aIl the F ACTS systems considered in this thesis, the maximum
65
value of the relative degrees is just 2. This enables a "simplified" nonlinear control
solution to be derived, as will be shown in chapter 6.
(3 - 76)
where [C] and [D] are shown in (3-77), (3-78) respectively.
fi fm ~~
010 ... 00 ... 000 ... 00
001 ... 00 ... 000 ... 00
000 ... 10 ... 000 ... 00
000 ... 00 ... 000 ... 00
[c]= (3 -77)
000 ... 00 ... 010 ... 00
000 ... 00 ... 001 ... 00
000 ... 00 ... 000 ... 10
000 ... 00 ... 000 ... 00
66
fi ~
0 ... 0
0 ... 0 rI
0 ... 0
1...0
[D]= . (3 - 78)
0 ... 0
0 ... 0
rm 0 ... 0
0 ... 1
It can be verified that a system described in (3-76) is controllable. Therefore, any
conventional control method for eigenvalue-placement by state feedback can be applied
to the system. Software for moving the eigenvalues is available, as will be employed in
the later chapters ofthis thesis.
The final step is to retum from the linear system to the nonlinear system. This can be
done by the following inverse transformation of (3-79):
[
L L'I.-lh gl t 1
u= :
L L'm-1h gl f m
(3 - 79)
3.3.4 Operating Nonlinear System
Since !! has already been committed in c10sing the feedback loop, it is necessary to
show how the nonlinear system is made to fulfi1 its operating functions. The operating
67
function is described by the equilibrium state Xo. For example, the Voltage-Source
Converter can be called upon to operate from one equilibrium state Xo 1 and then to make
a transition to another operating state X02 . By the Nonlinear Transformation,
corresponding to each Xo, there exists Zoo Having chosen the feedback gain matrix [E] so
that w=[E]?; in the state-feedback, the dynamic equations of the linear system is:
(3-80)
Since the eigenvalues of {[C]+[D][E]} have been chosen to converge rapidly to zero, it
means that ?; converges quickly to ZOo In the nonlinear system, ~ converges on Xo in the
same time.
3.4 CONCLUSION
The model of a Voltage-Source Converter in the d-q frame (derived from its a-b-c
model) has been developed, and it is a nonlinear (or more exactly, bilinear) system. The
principle of the Nonlinear Control Method has been introduced in a tutorial form. The
n=3 order, m=2 input, Voltage-Source Converter model has been used as an example to
illustrate the principles and how they are applied. From this example, the method has
been generalized for application to the higher order systems in chapter 4 and chapter 5.
68
The Nonlinear Control Method is not easy to apply because it requires output functions
hi(~), i=l, 2 ... m, to be synthesized under complicated rules. For the case of the Voltage
Source Converter, two output functions hl(~), h2(~) have to be synthesized. It has been
found that hl(~) should be the sum of the storage energy terms on the ac-side and the dc
side of the Voltage-Source Converter. In the tutorial example, it has also been found that
h2(~)=CX3 is suitable. In fact, h2(~) can be any one of the state-variables of the nonlinear
system, i.e. Xl, X2 or X3.
These findings are significant because together with the insights into the system
equations of the other F ACTS controllers, it is possible to apply the Nonlinear Control
Method to them in chapters 4 and 5.
69
Chapter 4
Nonlinear Control of Voltage Source Converter Based
FACTS Controllers
4.1 INTRODUCTION
In chapter 3, the nonlinear control method was introduced and, as a simple example,
feedback linearization was applied to the nonlinear equations of a voltage source converter
(VSC) mode!. Although stabilizing the C-UPFC of Chapter 2 is the motivation for
reaching out to the nonlinear control method [1-3], the research proceeds by applying the
nonlinear control method cautiously to FACTS (Flexible AC Transmission Systems)
controllers with increasing complexity, as measured by the dimension of the systems. AlI
the FACTS controllers are based on VSC modules. The first test case considered is the
SSSC (Static Synchronous Series Compensator) [14, 23, 27, 71-73], which has only one
VSC module and has the dimension N=3. Then advancing in complexity the second test
example is the UPFC (Unified Power Flow Controller) [23, 35, 36, 74], which has two
VSC modules and dimension N=5. Digital simulations will be presented for verification
purpose. After mastery over the less complicated examples, only then will the nonlinear
method be applied to the C-UPFC, which has 3 VSC modules. This is the subject of the
next chapter.
70
It should be emphasized that since the controls of the SSSC and the UPFC have never
been designed systematically with their nonlinearities taken into account, this chapter
makes advancement in the control methodology of the VSC family ofFACTS controllers.
4.2 NONLINEAR CONTROL OF SSSC
As its name suggests the Static Synchronous Series Compensator (SSSC) is a VSC
which is inserted by a transformer into a transmission line (see single-line diagram
Fig.4-I) so that it injects an equivalent series capacitive voltage (Vo=Is/jmC) to
compensate the large line inductive reactance jmLIs voltage. The power transmissibility of
Vs ~----+----.. ~,------,------'I\/UYV\--.tv R -- .-- transformer
Is
I __ @ + -1- Vdc
Converter
sssc
Fig. 4-1 Single line diagram of SSSC
71
the line is increased from V s V Rsin(8s-8R)/coL, without compensation, to
Vs V Rsin(8s-8R)/( coL-l/coC), with capacitive reactance compensation. Fig.4-2 shows the
per phase equivalent circuit in the radial transmission line with the SSSC, which is
modeled by the voltage ph as or Vo. The sending-end and receiving-end voltage phasors are
Vs and V R respectively.
4.2.1 Modeling of SSSC
The formulation of the dynamic equations reqmres transformation of the 3-phase
voltages and currents from the a-b-c frame to the synchronously rotating o-d-q frame. The
0- or zero sequence is neglected throughout because faults and unbalanced operation are
outside the scope of this thesis. Since the transformed equations of the VSC are similar to
(3-11)~(3-13) in chapter 3, it is necessary only to relate the phasor quantities in Fig.4-2 to
the d-q quantities which are used:
Vo R L + r-~~~~~---\ ~ )-----___ !\I\JVY\ _____ ~I
Fig. 4-2 Equivalent circuit of SSSC
72
Sending-end voltages Vs VSd, vSq
Receiving-end voltages V R VRd, VRq
Line-Currents Is Id, Iq
SSSC injected voltages Vo VdcUd, VdcUq
As in chapter 3, Vdc is the voltage across the dc bus of the VSC and Ud, uq are the control
inputs, which in the VSC are o-d-q transformations of the SPWM modulation signaIs.
It can be easily verified that the system equations in d-q frame are:
(4 -1)
(4 - 2)
(4 - 3)
Rand L are the line resistance and inductance.
Defining ~T=[Xl, X2, X3] as the state variables of id, iq and Vdc, and gT=[Ul, U2] as the
control inputs Ud and uq,:
equations (4-1) - (4-3) can be rewritten in following standard form:
~ = f(~) + gl (~)Ul + g2 (~)U2 (4 - 4) - -
73
where
gl (~) =
R 1 1 - - XI + mi x 2 + - V Sd - - V Rd
L L L R 1 1
--x -m·x +-v --v L 2 IlL Sq L Rq
o
1 --x L 3
0
1 -x C 1
o 1
--x L 3
1 -x C 2
4.2.2 Nonlinear Control ofSSSC
(4 - 5)
(4 - 6)
(4 -7)
The SSSC model is similar to the converter model which has been developed in Chapter
3. Therefore, sorne of the steps which, have been developed in detail previously, will be
passed over quickly.
It can be verified that the system is feedback linearizable. The system has n=3 (order),
m=2 (inputs) and the relative degrees with respect to the two output functions hl and h2
are: rl=2, r2=1.
74
As a result, the two functions hl and h2 have to fuI fi 11 the fo11owing requirements:
(4 - 8)
where Go=span { g 1 CS), g 2 C~) }.
(4 -9)
It can be verified that the choice of hl = LXI 2 + L x 2 2 + C X 3 2 satisfies (4-8). 2 2 2
If h2=x2 is selected, (4-9) becomes
1 ~ X 3 *- 0 and Xl*- - (v Sd - V Rd)
2R
The above requirements can be easily satisfied. Therefore, the requirements in (4-9) is
fuIfi11ed.
The new state space coordinates are to be selected as fo11ows:
75
2 2 Z 2 = L f h 1 = - R( XI + X 2 ) + (V Sd - V Rd )X 1 + (V Sq - V Rq )X 2 (4 -10)
Z3 = h2 = X 2
As for the nonsingular state feedback transformation, it takes the form of (4-11):
(4-11)
where
From (4-10) and (4-11), the following new state space model can be derived:
76
dZ I -=Zl dt
dZ l -=w dt 1
dZ 3 -=Wl dt
which is linear and controllable.
In fact, by the use of the following linear state feedback:
o w 1 =-Â1Âl(ZI -Zj )+(Âj +Âl )Z2
o w 2 = Â3 (Z3 - Z3 )
(4-12) becomes
~-l::-Z'~-Â3 Z-Z 3 3
(4 -12)
(4-13)
and (4-13) is a CCF (Canonical Controllable Form) with Al, A2, A3 as its three
eigenvalues and (Zlo, 0, Z30), the system equilibrium point.
4.2.3 Inverse Transformation
From (4-11), the inverse transformation will be in the following form:
(4-14)
With this transformation, the control inputs of the linear equivalent are inversely
transformed back to that of the nonlinear system.
77
4.2.4 Simulation Results
The simulation results are shown in Fig. 4-3. As the FACTS controller is a SSSC, the
test has been planned to show its ability to provide fast capacitive compensation. The test
consists of making two step changes in Iqo, the steady-state operating value of iq. The step
changes in Iqo, cause Id, P, and Q to change accordingly. The simulation shows that the
control is very fast (the system transients are completed within 2 cycles). The parameters
for this simulation are listed in APPENDIX 4.
78
1
0.5 id
0
-0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0.5
0 iq
-0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
1.05
"de
1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
1
0.5 P s
0
-0.50 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0.2 Qs
0
-02 . 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 1
0
-1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
time(s)
Fig. 4-3 Step change in Iqo
79
4.3 UPFC NONLlNEAR CONTROL
Fig. 4-4 shows the single line diagram of the Unified Power Flow Controller (upFC).
The radial transmission line is modeled by resistance R2 and inductive reactance X2=coL2
between sending-end voltage Vs and receiving-end voltage V R. The UPFC is situated at
the sending-end. It consists of a shunt VSC, represented by voltage phasor Et with
resistance RI and inductance LI in the circuit, and a series VSC, represented by voltage
phasor E2. The two VSCs are connected back-to-back across their dc terminaIs so that
they exchange real power through the dc link, i.e. Real(E212*)+Real(Etl t*)=O.
,------------~ISI +E2_ ,----I---.-----{ ~ J_--+ __ I
L ___________ _
UPFC
Fig. 4-4 Single line diagram ofUPFC
80
The inventor, L. Gyugyi, c1aims that his UPFC is the ultimate power controller
because it can not only control independently the real power across the radial
transmission line but the reactive powers Qs at the sending-end and QR at the
receiving-end.
4.3.1 Modeling of UPFC
Using the same a-b-c to O-d-q transformations as (3-7), the results of transformation
are:
Vs <=> Vds, Vqs
V R <=> VdR, VqR
Il <=> IdI, IqI
12 <=> Ict2, Iq2
Defining the dc link voltage to be Vdc, the d-q equivalent of the modulation signaIs of
VSCI to be (Uctt, UqI) and those of VSC2 to be (Ud2, Uq2), then the transformation of the
VSC ac voltages are:
The equations which mode1 the UPFC in the d-q frame are:
81
_.&. w 0 0 0 Vdc + 1 V Id!
LI Id!
- r::;- U dl 1:;- ds
Iql -W-.&.O
LI 0 0
Iql VdC +! V -r::;- U ql 1:;- qs
d Id2 = 0 O-~ w 0 Id2 + - ~~ U d2 + ~2 (V ds - V dR) (4 -15) dt L2 Iq2 0 O-W -~O L2
Iq2 - ~~ U q2 + ~2 (Vqs - VqR )
Vde ~~ Ud2 Uq20 V de 0 CCC C
4.3.1.1 Shunt Converter
The first and second rows of (4-15) are the dynamic equations of the shunt converter.
The resistance and inductance of the shunt converter circuit are RI and LI and the voltages
across them are the sending-end voltages (V dS, V qS) and the shunt converter voltages
(lidIVdc, l!qIVdc). The shunt converter is controlled by the modulation variables (UdI, UqI).
4.3.1.2 Series Converter
The third and fourth rows of (4-15) are the dynamic equations of the series converter.
The resistance and inductance of the series converter and the transmission line are R2 and
L2. The series converter voltages are (Ud2Vdc, l!q2Vdc). The receiving-end voltages are
(VdR, VqR). The series converter is controlled by the modulation variables (Ud2, l!q2).
4.3.1.3 DC Link Equation
The fifth row of (4-15) models the charging of the voltage Vdc ~f the dc capacitor, C,
by the dc currents of the shunt and the series converters.
4.3.2 Nonlinear Control ofUPFC
Defining the 5-tuple state-variable vector K and the 4-tuple input vector.!! as:
82
(4-15) can be re-fonned as
. 2f = f(2f) + g) (2f)u) + g2 (2f)u2 + g3 (2f)u3 + g4 (2f)u 4 (4 -16)
where
(4 -17)
_..Lx LI 5
0
g) (2f) = 0 (4 -18)
0 ) c- x )
0
_..Lx LI 5
g2 (2f) = 0 (4 -19)
0 )
C- X 2
0
0
g3 (2f) = ---Lx L 2 5
(4 - 20)
0 )
C- X 3
83
0
0
g4(~) = 0 (4-21)
---Lx L2 5
1 C X4
Is this system feedback linearizable?
It can be verified that:
Go = span {gl, gz, g3, ~} has constant rank 4 and is involutive;
G1 = span{gl, gz, g3, ~, adrgl, adrgz, adrg3, ad®i} has constant rank 5 and lS
involutive;
As having already been shown in chapter 3, the Lie brackets are define as:
Therefore, the system is feedback linearizable.
As the system has n=5 (order), m=4 (inputs), the relative degrees with respect to the
four output functions hl, hz, 14 and 14 are: rl=2, rz=l, r3=1, r4=1.
Therefore, it needs to find the four functions hl, hz, h3, 14 which satisfy:
< dh j , Gr'_2 > = 0 for j ~ i J
84
< dh2, Gr2
- 2 > = < dh2, G_! > = 0
< dh3, Gr3 -2 > = < dh3, G_! > = 0
<dh4,Grç2 > = <dh4 ,G_! > =0
From (4-22),
_J..- xs ah! +~x! ah! = 0 L! &! C &s
1 ah! 1 ah! --xs --+-x2 --= 0
L! &2 C &s
1 ah! 1 ah! --XS--+-x3-=0
L2 &3 C &s
1 ah! 1 ah! --xs --+-x4 --= 0
L2 &4 C &s
(4-22)
) not useful.
(4-23)
Furthermore, another requirement is that the following matrix needs to he nonsingular:
< dh!, adfg! > < dh p adfg 2 > < dh!, adfg3 > < dh!, adfg4 >
<P= <dh 2,g!> < dh2,g2 > < dhz, g3 > <dhZ ,g4 >
<dh3,gt> < dh3,g2 > <dh3,g3 > < dh3, g4 >
<dh4,gt> < dh4,gz > <dh4,g3 > < dh4,g4 >
The requirement is ohvious as the inverse form of matrix <t> will he used to inversely
transform w, the control inputs of the linear equivalent, to g, the control inputs of the
nonlinear system. Upon the choice of
85
(4-24)
and through further ca1culation, the new state space coordinates can be obtained as
follows:
(4-25)
The next step is to find the nonsingular state feedback transfonnation
w= [M]+[N]u (4-26)
where
V'h f 1 L~hl V2 h L f h 2 [M]= f 2 (4 - 27) = V3 h Lrh 3 f 3
V4 h f 4 L f h 4
86
L Vi-Ih gt f 1 L Vt-Ih g2 f 1
L Vt-Ih g3 f 1 L Vt-Ih g4 f 1
L V2-lh L V2-lh L V2-lh L V2-lh [N]=
gt f 2 g2 f 2 g3 f 2 g4 f 2
L V3-lh L V3-lh L V3-lh L V3-lh gt f 3 g2 f 3 g3 f 3 g4 f 3
L V4-lh gt f 4 L V4-lh g2 f 4 L V4-lh gl f 4 L V4-lh g4 f 4
LgtLfhl Lg2 Lfhl Lg3 Lfhl Lg4 Lfh l
Lgt h2 Lg2 h2 Lgl h2 Lg4 h2 =<1> (4- 28) =
Lgt h3 Lg2h3 Lg3 h3 Lg4 h3
Lgt h4 Lg2h4 Lg3 h4 Lg4 h4
and
87
X s L h =--gl 2 L
1
L h =0 g2 2
L h =0 g3 2
L h =0 g4 2
L h =0 gl 3
L h =0 g2 3
X s L h =--g3 3 L
2
L h =0 g4 3
L h =0 gl 4
L h =0 g2 4
88
L h =-~ g4 4 L
2
By taking the inverse transformation form: u=[Nr\w-[M]), the time derivatives of z
are in the forms of the following:
•
Z3 = [M] + [N]u (4-29)
Therefore, with this nonsingular state feedback transformation, the non-linear
equations can be transformed into the following linear ones:
. z=[C]z+[D]w (4 -30)
where
89
01000
00000
[C]= 00000 ;
00000
00000
0000
1000
[D]= 0100
0010
0001
The ab ove system is a linear system and controllable. As mentioned in previous
section, linear state feedback control method can be applied to place the system
eigenvalues of the linear system ta where as far as from the origin in the negative axis to
get a fast system response. That is to say, with the linear state feedback w=[E]z, (4-30)
becomes,
• z = ([C] + [D][E])z (4-31)
so that the eigenvalues out of ([C]+[D][E]) can be placed in a pre-determined values.
The last step is to transform the control actions of the linear system equivalent back to
that of the nonlinear system, and this is done by the inverse transformation
u=[Nr1(w-[M]).
The control diagram ofUPFC system is thereby summarized in Fig. 4-5.
90
4.3.3 Reference Settings
Up to this point, no mention has been made as to how Reference Settings are
introduced in the Nonlinear Control. The box "Reference Generator" in Fig.4-5 computes
from the complex power assignment of the UPFC P, QSrefand QRref the operating states Xo
of (4-15). From (4-25) the linearized operating states Zo of (4-29) are computed from Xo.
In the d-q frame the steady-state dXo/dt=O so that dZo/dt=O. Since Z;-Zo is a solution to
(4-31), it follows that dz/dt=([C]+[D] [E])(Z;-Zo). Because the eigenvalues have been
chosen so that Z; converges to Zo quickly, it follows that X converges to Xo within the
same time.
r------, , , - vdc J!l , , , , ~ Nonlinear , , Q) Transformation ,
, 1 Reference 1 ... , , l Generator ::1 d-q gj Z(x) , Q) ~ transformation
, ~ i(t) Idl
, , , , y(t) Iql , ~ ~o ''''',
'1' 1
, , , , Id2 , Lmear , , , , Feedback , Linearized M
'---Iq2 , , , ~ ,
- L ___ ___ ..J
~ !! UPFC Inverse F-
Transformation ~
w
Fig. 4-5 Control diagram ofUPFC
-p
QSref
QRref
odel
91
4.3.4 Simulation Resu/ts
The capability of Nonlinear Control to implement fast response, stable operation is
demonstrated by digital simulations of the UPFC system of Fig. 4-5. In this chapter, the
sending-end voltage Vs and the receiving-end voltage VR are assumed to be constant and
therefore the tests are performed under the same condition. The simulation tests consist of
two different cases: (a) the power angle between the sending-end voltage Vs and the
receiving-end voltage V R is ù=ùs-ùR=25° and (b) ù=ùs-ùR=8° . The tests show that the
responses to step changes are very fast transients, which are completed within two 50 Hz
cycles. The simulation results point to the effectiveness of Nonlinear Control in
controlling the UPFC.
In field operation, the sending-end voltage Vs and the receiving-end voltage V Rare not
he Id constant. In particular, the voltage angle Ù=ÙS-ÙR have low frequency oscillation
because the inertias on the sending-end side swing with respect to the inertias on the
receiving-end side. Chapter 7 shows that Nonlinear Control can be made to handle this.
For the present, the simulation results are the properties of the UPFC alone under
Nonlinear Control.
A. Step Increase in Complex Powers
Because the responses to step changes are very fast, each simulation can accommodate
2 step changes whichare completed within 0.12s. Fig. 4-6 is for the case of ù=25 ° and
Fig. 4-7 is for ù=8 0 •
92
Fig. 4-6 (a), (b) and (c) show the transients of P, Qs and QR associated with two
successive step changes. In the first step change, QR is increased from -0.243 p.u. to 0.0,
while keeping constant Qs=0.243 p.U. and P=0.97 p.u. After the transients have subsided,
P is stepped from 0.998 to 1.97 p.U., while keeping constant Qs=0.243 and QR=O.O. In
Fig. 4-6 (d), the CUITent iRa and voltage VRa have phase angle between them when
QR=-0.243 and they are in phase when QR=O.O. The dc link voltage Vdc is kept at 1.0 by
the component in the reference setting Zoo It is only slightly affected during the step
change of QR. but there is a capacitor charge depletion which has caused it to dip to about
0.975 p.u. during the increase of P from 0.998 to 1.97 p.U. It should be noted that the
variation of Vdc is the source of the nonlinearity. Fig. 4-6 (f) display the inputs in the
vector.l:! =[Ul, U2, U3, U4]=[UdI UqI lid2 Uq2]T to show that at no point are they beyond the 1.0
p.U. limit, beyond which the modulating signaIs would be saturated.
Fig. 4-7 are the simulation results for the case 8=8 0 • In the first step change, QR is
increased from -0.07 p.U. to 0.0, while keeping constant Qs=0.07 p.U. and P=0.998 p.U.
After the transients have subsided, P is stepped from 0.998 p.u. to 1.46 p.u., while
keeping constant Qs=0.07 p.U. and QR=O.O p.u. For a small voltage of 8=8 0 ,the power
increase is made possible by the series converter operating as a capacitive reactance to
compensate the line inductive reactance.
93
3
2 (a) Ps
1
00 0.02 0.04 0.06 0.08 0.1 0.12 0.5
(b) 0 Qs V/~~
-0.50 0.02 0.04 0.06 0.08 0.1 0.12
0.5
(c) 0 QR
-0 5 . 0 0.02 0.04 0.06 0.08 0.1 0.12
5
(d) 0
-5 0 0.02 0.04 0.06 0.08 0.1 0.12
1.05
(e) 1 vdc
0.95 '------'----'--------'----'--------'------'
(f)
o 1
0.5
o -0.5
0
0.02 0.04 0.06 0.08
U4 Ut
u2 U3 1 \
1" ...
0.02 0.04 0.06 0.08
Fig. 4-6 Step Changes in complex power P and QR ( 0 =25 0
)
(a) Ps;
(b) Qs;
(c) QR;
(d) ac voltage and current;
(e) dc link voltage;
(f) modulation inputs.
0.1 0.12
o. 1 0.12
94
2
(a) 1.5 Pi 10 0.02 0.04 0.06 0.08 0.1 0.12
0.5
0
(b) -0.5
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
0.1
(c) 0
~ -0 1 . 0 0.02 0.04 0.06 0.08 0.1 0.12
5
(d) 0
-5 0 0.02 0.04 0.06 0.08 0.1 0.12
1.02
(e) 1 vdc
0.980 0.02 0.04 0.06 0.08 0.1 0.12
1
0.5 u1
(f) 0 u 2 '--a. U3 "- U4 "-
~ /' ~
----0 5 . 0 0.02 0.04 0.06 0.08 0.1 0.12
Fig. 4-7 Step Changes in complex power P and QR ( 0 =8 0 )
(a) Ps;
(b) Qs;
(c) QR;
(d) ac voltage and current;
(e) de Hnk voltage;
(f) modulation inputs. 95
B. Step Power ReversaIs
K.K. Sen pointed to the unique capability of the UPFC for fast power reversaI [77].
Fig. 4-8 shows two successive step changes with P reversing from +0.97 to -0.97 and
then from -0.97 to +0.97 when the power angle between the receiving-end voltages is 25
degree (ù=25° ). The reactive powers are kept constant: QR=-0.243 and Qs=0.243.
Fig. 4-9 shows again two successive step changes with P reversing from +0.998 to
-0.998 and then from -0.998 to +0.998 when the power angle between the receiving-end
voltages is 8 degree (ù=8° ). The reactive powers are kept constant: QR=-0.07 and
Qs=0.07.
In the 4 sets of simulation results of FigA-6 to 4-9, the 50 Hz cycles of the ac voltage
VRa and CUITent iRa have been recorded in (d) to emphasize how quickly the transients are
completed. From an engineering viewpoint, one is concemed as to whether the very fast
response is obtained by exceeding practicallimits. For this reason, the de link voltage Vdc
is monitored in (e) to ensure that at no time instant is the capacitor discharged and is there
a transient over-voltage exceeding 5%. Also the modulation index of the controls of the
Voltage-Source Converter should not exceed 1.0. For this reason, the magnitudes of UI,
U2, U3 and 14 have been monitored and are found to remain within bounds in (f).
96
(a) o
-1~----~----~~==~======~----~----~ o 0.02 0.04 0.06 0.08 0.1 0.12
0.4~----~------~----~------~----~----~
(b) 0.2
00 0.02 0.04 0.06 0.08 0.1 0.12 -0.23 .--------------,----------,-----------,--------,----------,-------------,
(c) -0.24 QR
-025~----~---=~------~----~----~----~
. 0 0.02 0.04 0.06 0.08 0.1 0.12
2~----~------~----~------~----~----~
(d) 0
(e)
(f)
-2~----~------~----~------~----~----~ o 0.02 0.04 0.06 0.08 O. 1 0.12
1. 02 ....--------r------.-------r----.-------r-----,
o 1
0.5
o
-05 . 0
U2
0.02 0.04 0.06
Ut
U3~ If
0.02 0.04 0.06
Fig. 4-8 Power ReversaIs ( 0 =25 0
)
(a) Ps;
(b) Qs;
(c) QR;
(d) ac voltage and current;
(e) dc link voltage;
(f) modulation inputs.
0.08 0.1 0.12
\--CU4
V 0.08 0.1 0.12
97
2
1
(a) 0
-1 0
0.2
(b) 0.1
00
-0.065
(c) -0.07
-0.075 0
2
(d) 0
-2 0
1.015
(e) 1.01
1.005
10
1
0.5
(f) 0
-05 . 0
Ps ( 0.02 0.04 0.06 0.08 0.1 0.12
0.02 0.04 0.06 0.08 0.1 0.12
QR
0.02 0.04 0.06 0.08 0.1 0.12
0.02 0.04 0.06 0.08 0.1 0.12
vdc
0.02 0.04 0.06 0.08 0.1 0.12
U1
u2 ----.. U3--"'~ \ ~u
4 r 0.02 0.04 0.06 0.08 0.1 0.12
time(s)
Fig. 4-9 Power ReversaIs ( 6 =8 0
)
(a) Ps;
(h) Qs;
(c) QR;
(d) ac voltage and CUITent;
(e) de link voltage;
(f) modulation inputs. 98
4.4 CONCLUSION
The nonlinear control method has been successfully applied to two F ACTS controllers:
the SSSC, which has one converter, and the UPFC, which has two converters. The digital
simulation results show that good system dynamics and performance are achieved by
applying the nonlinear control method.
APPENDIX 4 SIMULATION P ARAMETERS AND SETTINGS
SSSC
Vs: 1.0 p.u.
VR: 1.0 p.u.
ô: 25°
R:X=1:27.5
À 1 = -190
À 2 = -220
À 3 =-210
UPFC
Vs: 1.0 p.u.
VR: 1.0 p.u.
99
ô: 25° or 8°
Rl:Xl=I:27.5
R2:X2=1:55
À 1=-190
À 2 = -220
À 3 =-210
À 4 =-210
À 5 = -150
100
Chapter 5
C-UPFC N onlinear Control
5.1 INTRODUCTION
In chapter 4, the nonlinear control method has been applied successfully to VSC-based
nonlinear systems of increasing dimensionality, the SSSC (single VSC controller, n=3)
and the UPFC (two VSCs, n=5). This chapter retums to the C-UPFC of chapter 2, which
promises to be more complex still because it is made up of 3 VSCs, two of which are in
series, the third being in shunt. It should be remembered that in chapter 2 the C-UPFC
succeeded in operating only in the "phase-shifter" mode, which is to bridge the large
voltage angle ù between the sending-end voltage and the receiving-end voltage. There are
the other two application modes in which the C-UPFC has been expected to excel: the
"capacitor compensation" mode in which the series VSCs would off-set the large
transmission line reactance thus effectively "shortening" the transmission distance; and
the "power reversaI mode" in which the sending-end voltage continues to transfer power
to the receiving-end voltage even though the voltage angle ù is negative. Unfortunately, it
was found that the conventional PID control method could not stabilize the C-UPFC apart
from "phase-shifter" mode. It is because of the failure of PID control that nonlinear
control method has been sought as a systematic way to stabilize the C-UPFC for all the
101
three modes of operation. As chapters 3 and 4 have shown, the nonlinear control
method tackles the problem of the nonlinearity inherent in the VSC while allowing the
eigenvalues of the linearized system to be assigned.
Closer examination shows that the system dimension ofthe three VSCs in the C-UPFC
is 5 and not 7. This is because the currents of the shunt VSC are not independent state
variables since the Kirchhoffs Current Law at the central nodes require them to be
related to the currents of the series VSCs. Although there are 6 control voltage levers in
the 3 VSCs, only 4 of them can be used independently. After showing how the nonlinear
control method is applied to C-UPFC, the objective of this chapter is to display the
capability of the C-UPFC in the "capacitor compensation" mode and the "power reversaI"
mode.
5.2 C-UPFC NONLINEAR CONTROL
5.2.1 Modeling ofC-UPFC
In the model of C-UPFC in Chapter 2, the shunt converter does not have the common
dc bus link with the two series converters. This chapter will adopt the C-UPFC model
with all the three converters sharing just one corumon dc capacitor link. In this case, the
equivalent circuit ofthe C-UPFC is shown in Fig. 5-1.
In the d-q frame, Kirchhoffs Voltage Laws applied to the shunt converter are:
102
LINE 1
-+1 1
LINE2 V 0' center-node
jXj :~E-j~-A;~: jX2 R2 ~ f\;
1
131
L ____________ -1
C - UPFC
Fig. 5-1 Equivalent circuit of C-UPFC with three converters
sharing one common dc capacitor link
L did3 R· . 3 --+ 31d3 - (j) Iq3 = V do - U d3 V de
dt (5 -1)
(5 -2)
where L3, R3 (not shown in Fig. 5-1) are the shunt circuit inductance and resistance and
V do, V qo are the voltages of the mid-point between the sending-end and receiving-end
buses.
Kirchhoff's Voltage Laws applied to the first series converter are:
103
(5 -3)
(5 -4)
where LI, RI are the transmission line inductance and resistance between the
sending-end bus and the mid-point and v dS , V qS are the voltages at the sending-end
buses. The inductance and resistance associated with this series converter have been
combined into LI and RI. respectively.
Kirchhoffs Voltage Laws applied to the second series converter are:
L did2 R' . 2 --+ 21d2 - OJ lq2 = V do - V dR + u d2 V de
dt (5 - 5)
(5 - 6)
where L2, R2 are the transmission line inductance and resistance between the mid-point
and receiving-end bus and v dR' V qR are the voltages at the receiving-end buses. The
inductance and resistance associated with this series converter have been combined into
L2 and R2, respectively.
Kirchhoffs CUITent Law on the DC side is:
(5 -7)
Kirchhoffs CUITent Law on the AC side are:
104
· . . Id3 = Idl -1 d2 (5 - 8)
(5 -9)
Furthermore, as explained in Chapter 2 section 2.3.3, the shunt converter operates as a
STATCOM and regulates the magnitude of the center-node voltage at 1 p.u., which can
be expressed in the following two equations:
(5-10)
(5-11)
At the first glance of above equations (5-1) ~ (5-11), the system has 7 state variables
(idl , iqI, id2 , iq2, id3 , iq3 , Vdc), and six control inputs (Udl, Uql, Ud2, Uq2, Ud3, Uq3). Therefore,
one would presume that it is a i h order system. However, because of (5-8) and (5-9)
which are reminders that only 2 of the 3 branch currents at one node can be independent,
it is actually a 5th order system. In fact, after re-arrangement of (5-1) ~ (5-9), a set of
following equations can be obtained:
I dl ail a l2 a l3 0 0 Idl
d lql a 21 a 22 0 a 24 0 I q1
1d2 = a 31 0 a 33 a 34 0 Id2 +B (5 -12) dt
lq2 0 a 42 a 43 a 44 0 lq2
V de 0 0 0 0 0 V de
105
where
m(LJL2 + LJL3 + L3L2) aJ2 =
-R2LJ -R2L3 -R3LJ a 33 =
106
[B] =
L 2 +L3 L3 L 2 L 2 +L3 L3 - VdcUdl +-Vdc Ud2 --Vdc Ud3 + VdS --VdR
al al al al al
L2 +L3 L3 L 2 L2 +L3 L3 - VdcU ql +-Vdc Uq2 --Vdc Uq3 + VqS --VqR
al al al al al
L3 LI +L3 LI L3 LI +L3 --VdcUdI + vdc u d2 +-vdc u d3 +-vdS - VdR
al al al al al
L3 LI + L3 LI L3 LI + L3 --VdcUql + Vdc Uq2 +-Vdc Uq3 +-vqS - VqR
al al al al al
!(idIUdl +iqlU ql -id2 u d2 -id2 u d2 + (i dl -i d2 )u d3 + (i ql -i q2 )u q3 ) C
As this system has 5 state variables, the system dimension is n=5. From the [B] matrix
in (5-12), the system has 6 control inputs (lldl, Uql, Ud2, Uq2, Ud3, Uq3). The fact that the
coefficients of Ud3 and Uq3 in the fifth row of [B], namely (id1 -id2) and (iq1 -iq2), are not
independent has served as a guide not to use Ud3 and Uq3 as inputs in the feedback control.
Thus Ud3 and Uq3 are used only in open loop control, and their values will change only
when the C-UPFC system changes its steady state operating point, which is the case for a
step change of active power and/or reactive power, etc. Therefore, Udl, Uql, Ud2, Uq2 are
considered to be the four control inputs. From chapter 2, the relative degrees of the SSSC
and the UPFC is {2, 1} and {2, 1, 1, 1} respectively. Applying this experience to the
C-UPFC system, 4 output functions (hl, h2, h3, 14) will be required for the 4 inputs (Udl,
5.2.2 Nonlinear Control ofC-UPFC
The control diagram of C-UPFC system is summarized in Fig. 5-2.
107
14], (5-12) becomes
(5 -13)
where
L2 +L3 L3 L2 allx I + a12 x2 + a13 x3 + v dS --v dR --XSu d3 al al al
L2 +L3 L3 L2 a 21 xI +a22 x2 +a24 x4 + vqS --vqR --xsu q3 al al al
L3 LI +L3 LI a 31 xI +a33 x3 +a34 x4 +-vdS - vdR +-XSu d3
al al al
L3 LI + L3 LI a 42 x 2 + a 43 x 3 + a 44 x 4 + - V qS - V qR + - X sU q3
al al al
,-------, , , r-- vdc 2l , , , ,
Iii Non!inear ,
li E Transformation
, Reference 1 !:! ::> __ d-q ,
, 1 Generator ~ Z(x) , , .,
transformation . ::s i(t) ~dI , ,
r- , , y(t) lqI , ~ ~o
~, '1 L' 1
Id2 , mear , " , Feedback , Linearized M
lq2 , , -
, ~ , ode!
l L ___ ----'
~ 11 C-UPFC Inverse ...=---
Transformation ~
w
Fig. 5-2 Control diagram of C-UPFC
108
o _ L3 -x a 5
1
o
_ L3 -x a 5 1
L3 -x a 5
1
109
g4(!) = 0
LI +L3 xs
al
1 --x C 4
It can be verified that Go and G1 have constant rank 4 and 5 respectively, and they are
all involutive. Therefore, the system is feedback linearizable.
As the system has n=5 (order), m=4 (inputs), the relative degrees with respect to the
Thus the four functions hl, h2, h3, 114 need to be found and they have to fulfill the
following requirements:
<dhz,Gr2 -z > = <dhz,G_I > =0 }
< dh3 , Gr3
-z > = < dh3 , G_I > = 0
< dh 4 , Gr4 -z > = < dh 4 , G_I > = 0
which is not useful.
From (5-14), the following equation can be obtained.
(5-14)
110
(5 -15)
is one solution to (5 -15).
Furthermore, the following matrix needs to be nonsingular as weil.
< dht, adfg t > < dhl' adfg2 > < dht, adfg3 > < dhl' adfg 4 >
<1>= <dh2,gt> <dh2,g2> < dh2,g3 > < dh2, g4 >
<dh3,gt> <dh3,g2 > < dh3,g3 > < dh3, g4 >
<dh4,gt> < dh4,g2 > <dh4,g3 > < dh4,g4 >
In choosing
(5 -16)
111
the new state space coordinates will be:
Z2 = Lfh t , details for this equation are not given here as it contains many items.
With the following nonsingular state feedback transformation:
w = [M]+[N]y (5-17)
where
Vth f t efh t
V2 h Lfh 2 [M]= f 2 (5 -18) = V3 h Lfh 3 f 3
V4 h f 4 L f h 4
112
L Vt-lh gt f 1 L Vt-lh g2 f 1
L Vt-lh g3 f 1 L Vt-lh g4 f 1
L V2-l h L V2-l h L V2-lh L V2-lh [N]=
gt f 2 g2 f 2 g3 f 2 g4 f 2
L V3-l h L V3-l h L V3-lh L V3-lh gt f 3 g2 f 3 g3 f 3 g4 f 3
L V4-lh gt f 4 L V4-lh g2 f 4 L V4-lh g3 f 4 L V4-lh g4 f 4
LgtLfhl Lg2 Lfhl Lg3 Lfhl Lg4 L f hl
Lgt h2 Lg2 h2 Lg3 h2 Lg4 h2 (5 -19) =
Lgt h3 Lg2h3 Lg3 h3 Lg4 h3
Lgt h4 Lg2h4 Lg3 h4 Lg4 h4
From (5-17), the inverse transformation form is: y=[Nr1(w-[M)).
With the choice of new coordinates, the time derivatives of ~ can be calculated as
follows:
Z3 = [M] + [N]g (5 - 20)
Therefore, with this nonsingular state feedback transformation (5-17), the non-linear
equations can be transformed into the following linear ones:
113
· ~ = [C]~ + [D]w
where
01000
00000
[C]= 00000 ;
00000
00000
0000
1000
[D]= 0100
0010
0001
(5-21)
It can be verified that the above system is a linear system and controllable. As a result,
linear state feedback control method can be applied. The procedures to be followed are
exactly the same as that in section 4.3.2 and therefore are not repeated here.
5.2.3 Simulation Results
The simulations, which have been conducted, serve two objectives:
(1) to demonstrate that the nonlinear control method has been successfully
implemented;
114
(2) to show that the C-UPFC has the desirable operating features of a F ACTS
controller.
Four cases are presented here: real power reversaI, real power step change, reactive
power reversaI and real power step change (capacitive compensation). The simulation
results show that the system has a very fast response, which can be clearly seen in
transients which are completed within two 60 Hz cycles.
A. Real Power ReversaI
In this case, the reference value for the real power is step reversed from +0.998 p.u. to
-0.998 p.u. Fig. 5-3 (a), (b) and (c) show the transients of phase A voltage and CUITent
waveform at sending end, receiving end and middle point while Fig. 5-3 (d), (e) and (f)
show the transients of Ps, Qs and QR. Fig. 5-4 (a), (b), (c) and (d) show the dc link
voltage, the control inputs of the sending-end side series converter, the control inputs of
the receiving-end si de series converter and the control inputs of the shunt converter.
This simulation result shows the success of the nonlinear control method. For
comparison, in Chapter 2 the system could not be stabilized when a unit real power
reversaI command was given. Here the simulation shows that the system is very robust
and its transients associated with a unit real power reversaI are completed within two
cycles.
115
1
(a) 0
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
1
(b) 0
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(c) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(d) 0 Ps
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
0.2
(e) 0 Qs
-02 . 0 0.02 0.04 0.06 0.08 0.1 0.12 -0.06
(f) -0.07
-0.08
-009 . 0 0.02 0.04 0.06 0.08 0.1 0.12 (s)
Fig. 5-3 Real power reversaI
(a) Vsa, isa
(b) VRa, iRa
(c) Voa, ioa
(d) Ps (e) Qs
(f) QR 116
1.015
1.01
(a) 1.005
1 Vdc
0.995 0 0.02 0.04 0.06 0.08 0.1 0.12
0.8
0.6 ud1 0.4 uq1
(b) 0.2
0
-0 2 . 0 0.02 0.04 0.06 0.08 0.1 0.12
1
0.5 ud2 (c)
0 uq2
-0.50 0.02 0.04 0.06 0.08 0.1 0.12
0.8
0.6 \Ud3 0.4
(d) 0.2
0 uq3
-02 . 0 0.02 0.04 0.06 0.08 0.1 0.12(s)
Fig. 5-4 Real power reversaI
(a) Vdc
(h) Udb Uql
(c) Ud2, Uq2
(d) Ud3, Uq3
117
B. Real Power Step Change
In this case, the real power reference has a step change from +0.8 p.u. to + 1.995 p.u.
Fig. 5-5 (a), (b) and (c) show the transients of phase A voltage and CUITent waveform at
sending end, receiving end and mid-point while Fig. 5-5 (d), (e) and (f) show the
transients of Ps, Qs and QR. Fig. 5-6 (a), (b), (c) and (d) show the dc link voltage, the
control inputs of the sending-end si de series converter, the control inputs of the
receiving-end side series converter and the control inputs of the shunt converter. As
explained in section 5.2.1, the control inputs of the shunt converter, Ud3 and Uq3, are
normally kept constant and are changed only when the C-UPFC system changes its steady
state operating point. And this is shown in Fig. 5-4 (d), Fig. 5-5 (d), Fig. 5-6 (d) and Fig.
5.8.
118
2
(a) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(b) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(c) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
3
2 (d)
1 Ps
00 0.02 0.04 0.06 0.08 0.1 0.12 0.5
0 Qs
(e)
-0.5 ~ 10-3 0.02 0.04 0.06 0.08 0.1 0.12
10
5 (f)
0 QR
-5 0 0.02 0.04 0.06 0.08 0.1 0.12 (s)
Fig. 5-5 Real power step change
(a) V sa, isa
(b) VRa, iRa
(c) Voa, ioa
(d) Ps (e) Qs
(f) QR 119
1.04~----~----~------~----~----~------'
1.02 (a)
Vdc 11-------------J
0.2
o
o
(b) -0.2
-0.4
(c)
(d)
-0.60
0.4
0.2
o -0.2
-0.4
-0.60
1
0.5
o
0.02 0.04 0.06
Uq1 r/ ud1
0.02 0.04 0.06
Uq2
t~~
0.02 0.04 0.06
1
ud3
uq3
0.02 0.04 0.06
Fig. 5-6 Real power step change
(a) V dc
(b) Udh Uql
(c) Ud2, Uq2
(d) Ud3, Uq3
0.08 0.1 0.12
-----
0.08 0.1 0.12
0.08 0.1 0.12
0.08 0.1 0.12(8)
120
C. Reactive Power ReversaI
In this case, the reference value for the reactive power is step reversed from +0.07 p.u.
to -0.07 p.u. Fig. 5-7 (a), (b) and (c) show the transients of phase A voltage and CUITent
WaVefOlTI1 at sending end, receiving end and middle point while Fig. 5-7 (d), (e) and (f)
show the transients of Ps, Qs and QR. Fig. 5-8 (a), (b), (c) and (d) show the dc link
voltage, the control inputs of the sending-end si de series converter, the control inputs of
the receiving-end side series converter and the control inputs of the shunt converter. The
control inputs of the shunt converter, Ud3 and Uq3, are norrnally kept constant and are
changed only when the C-UPFC system changes its steady state operating point
121
1
(a) 0
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
1
(b) 0
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
2
1 (c)
0
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
1.04
(d) 1.02 Ps
10 0.02 0.04 0.06 0.08 0.1 0.12 0.2
(e) 0
-0.20 0.02 0.04 0.06 0.08 0.1 0.12
1
0 (f)
-1
-2 0 0.02 0.04 0.06 0.08 0.1 0.12(s)
Fig. 5-7 Reactive power reversaI at sending end
(a) Vsa, isa
(b) VRa, iRa
(c) Voa, ioa
(d) Ps
(e) Qs
(f) QR 122
1.001 ,--------,----------,--------,---,-------,-------,
1 f-----------,
0.999
(a)0.998
0.997
0.9960
0.15
0.1
(b) 0.05
o
-005 . 0
0.1
0.05
(c) 0
-0.05
0.8
0.6
(d) 0.4
0.2
0
-0 2 . 0
0.02 0.04 0.06 0.08
Uq1
V ud1
0.02 0.04 0.06 0.08
Ud2
uq2
~
0.02 0.04 0.06 0.08
0.02 0.04 0.06 0.08
Fig. 5-8 Reactive power reversaI at sending end
(a) V dc
(b) Udh Uql
(c) Ud2, Uq2
(d) Ud3, Uq3
0.1 0.12
0.1 0.12
0.1 0.12
0.1 0.12(5)
123
D. Real Power Step Change (Capacitive Compensation)
In this case, the real power reference has a step change from +0.8 p.u. to + 1.995 p.U.
As the C-UPFC operates in pure inductive or capacitive compensation mode, the phase
angle between the voltage Vs and current Is at the sending end, and the phase angle
between the voltage V R and current IR at the receiving end both have to remain constant;
therefore, a step change in the real power reference me ans an associated step change in
the reactive power references at the sending end and receiving end accordingly. Fig. 5-9
(a), (b) and (c) show the transients of phase A voltage and current waveform at sending
end, receiving end and middle point while Fig. 5-9 (d), (e) and (f) show the transients of
Ps, Qs and QR. Fig. 5-10 (a), (b), (c) and (d) show the dc link voltage, the control inputs
of the sending-end si de series converter, the control inputs ofthe receiving-end si de series
converter and the control inputs of the shunt converter. Fig. 5-11 (a) shows the phase A
voltage injected by the sending-end si de series converter and the phase A current at the
sending end; Fig. 5-11 (b) shows the phase A voltage injected by the receiving-end si de
series converter and the phase A CUITent at the receiving end; Fig. 5-11 (c) shows the
phase A voltage injected by the shunt converter and the phase A shunt CUITent at the
middle point; Fig. 5-11 (d), (e) and (f) show the transients ofthe real powers flowing in or
out of the sending-end side series converter, the receiving-end si de series converter and
the shunt converter. The simulation results show that the C-UPFC operates in an
inductive compensation mode before the step change of the real power is initiated, and
this is c1early seen by the 90 degree phase angle leading of the injected phase A voltages
124
(VIa, V2a) ofthe two series converters with respect to the associated currents (isa, iRa). After
the step change, the C-UPFC changes to capacitive compensation mode of operation as
the two just-mentioned phase angle differences move to 90 degree lagging from 90
degree leading. The above fact is again proven in Fig. 5-11 (d), (e) and (f) as the real
power coming in or out of the three VSCs is zero at steady state.
125
2
(a) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(b) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(c) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
1.5 (d)
1 Ps 0.5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.5
(e) 0
-0.50 0.02 0.04 0.06 0.08 0.1 0.12
0
(f) OR
-0.2
-0.40 0.02 0.04 0.06 0.08 0.1 0.12
Fig. 5-9 Real power step change (capacitive compensation)
(a) Vsa, isa
(b) VRa, iRa
(c) Voa, ioa
(d) Ps
(e) Qs
(f) QR 126
1.02
Vdc (a) 1 f--------,
0.2
o (b) -0.2
-0.4
-0.60
0.02
ud1
0.02
0.04
0.04
0.06 0.08 0.1 0.12
Uq, ~
0.06 0.08 0.1 0.12
0.2~----~----~----~------~----~-----'
(c) -o.~ f-----------------.U
,-----d2-Uq-z-f= --0.4
-0.60 0.02 0.04 0.06 0.08 0.1 0.12
0.8 r----------,--~I==============]
0.6
(d) 0.4 0.2
uq3 O!-------~--~----------------------~
0.02 0.04 0.06 0.08 0.1
Fig. 5-10 Real power step change (capacitive compensation)
(a) V dc
(h) Udb Uql
(c) Ud2, Uq2
(d) Ud3, Uq3
0.12
127
2
(a) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(b) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(c)
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
0
(d) -0.5
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
0 t I(~~ (e) -0.5
P2-
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
0.05
P3 (f) 0
-0.050 0.02 0.04 0.06 0.08 0.1 0.12
Fig. 5-11 Real power step change (capacitive compensation)
(a) Via, isa
(h) V2a, iRa
(c) V3a, ioa
(d) PI
(e) P2
(f) P3 128
5.3 CONCLUSION
This chapter has successfully applied the nonlinear control method to the C-UPFC.
Once again very fast response and stable operation have been achieved. Thus the
nonlinear control method has been extended to a 3-VSC FACTS controller.
After applying the nonlinear control method to the SSSC, the UPFC and now the
C-UPFC, it is c1ear that the same procedure can be used for: the back-to-back
VSC-HVDC system [28-34], the Interline Power Flow Controller (IPFC) [38] and the
Multi-Terminal Unified Power Flow Controller (M-UPFC) [37].
At the level of research of innovative topologies for F ACTS controllers, this chapter
and chapter 2 have shown that the C-UPFC has aIl the capabilities of Lazslo Gyugyi's
Unified Power Flow ControIler, which are: independent control over the real power
through the radial line and the reactive powers at the sending-end and receiving-end. On
top of these capabilities, the C-UPFC has been conceived to be sited at the mid-point of
the transmission line so that it can double the power transmissibility. The innovative
aspect consists of showing that the real power entering a series VSC does not have to pass
through a shunt VSC. There is an alternative route which is through another series VSC in
the same radialline, but in the line section on the other si de of the center-node.
In chapter 2, the C-UPFC has already operated as a "phase shifter". This chapter
shows that with nonlinear control, the C-UPFC is capable of fast power reversaI. Power
transmission, when the voltage angle Ô is negative, is equivalent to "pumping uphill" in
129
hydraulic engineering. Finally, the 2 series VSCs can operate as 2 SSSC's, to "shorten the
transmission distances" by capacitive reactance compensation.
APPENDIX 5 SIMULATION P ARAMETERS AND SETTINGS
vs: 1.0 p.u.
VR: 1.0 p.u.
0: 50°
Rl:Xl=1:55
R2:X2=1:55
R3:X3=1:27.5
À 1 = -190
À 2 = -220
À 3 =-210
À 4 =-210
À 5 =-150
130
Chapter 6
Simplified N onlinear Control of
Voltage Source Converter Based F ACTS controllers
6.1 INTRODUCTION
The Nonlinear Control Method, associated with A.Isidori [1], H.Nijmeijer and Al.van
der Shaft [2], has so far been applied to the low order STATCOM (N=3) [49-51], and the
system dimensionality has stagnated at N=3 in power electronic applications. There are
two reasons for the lack of progress: (1) synthesizing the output vector 11(20 is not a
science but an art; (2) unfamiliarity with the advanced mathematical language used by
A.Isidori, H.Nijmeijer and Al.van der Shaft. Although chapters 4 and 5 have shown how
the Nonlinear Control Method can be applied to the UPFC and the C-UPFC, thus raising
the order to N=5, the two aforesaid reasons remain valid.
This chapter presents an alternative approach which does not require familiarity with
the notations of A.Isidori, H.Nijmeijer and Al.van der Shaft. The Simplified Nonlinear
Control Method of this chapter is orientated to power electronic systems made up of
several Voltage-Source Converters connected together at their dc buses across a dc
capacitor, examples being the UPFC and the C-UPFC of chapters 2, 4 and 5. The back-to
back VSC-HVDC [28-34] and the Multi-Terminal UPFC [37] are within the range of
Simplified Nonlinear Control Method. It should be emphasized that there is no loss in
mathematical rigour and the term "simplified" is intended to mean "simple to
131
understand". It is based on foUowing every step of the method of A.lsidori, H.Nijmeijer
and A.J.van der Shaft and applying the physical insights in the Voltage-Source
Converters.
The key to the success of the Nonlinear Control Method is in synthesizing a felicitous
output vector h(x). It is not a science but an art. Three physical insights regarding the
Voltage-Source Converter are helpful: (1) Kirchhoffs Voltage Law applied to the ac-side
circuits of the Voltage-Source Converters is an equation rooted on power balance; (2) the
bilinear nonlinearity in the Voltage-Source Converter has its origin in the power balance
equation relating the ac-side power to the dc-side power; (3) the time rate of change of
energy stored in a capacitor C or an inductor L is the power of the capacitor C or the
inductor L. The stumbling block in applying the Nonlinear Control Method is in not
knowing how to synthesize a nonlinear output function zN=h(X). This chapter shows that
the nonlinear function zN=h(X) is always the sum of the energy in the storage circuit
elements immediately on the ac-side and the dc-side of the converter(s).
As the Simplified Nonlinear Control Method is based on systematicaUy manipulating
the system equations, the method is presented by examples. Because the dimensionality
of the STATCOM is the lowest, it is used in this chapter first to explain the essentials of
the method. After that, the UPFC is chosen as the second example to iUustrate how to
cope with increases in system dimension. FinaUy, one retums to the C-UPFC as it is
origin of aU the research effort. In fact, with the procedures outlined in this chapter, there
is no limit to the system size. The algorithm can be applied to back-to-back VSC-HVDC
[28-34] and Multi-Terminal-UPFC [37] without modification.
132
6.2 OVERVIEW OF NONLINEAR CONTROL
The block diagram in Fig. 6-1 presents the same overview of the Nonlinear Control
Method as in Fig.3-2. For greater generality, the block containing UPFC in Fig.3-2 is
replaced by PWM-VSC to indicate that the method is suitable for FACTS controllers
involving combinations of Voltage-Source Converter (VSC) under Pulse-Width
Modulation (PWM) control. The ac-side voltage vectors y(t) and current vectors i(t) after
transformation from the a-b-c frame to become vectors in the d-q frame (Yd, Yq) and (id, iq)
of the Voltage-Source Converter (PWM-VSC). The current vectors (id, iq) together with
Vdc, the de voltage form an N-tuple state-vector x. The control of the PWM-VSC is a (N-
1) tuple input-vector.!!. The dynamic equation of motion is:
r-------, 1 1 1 1 1 1
Yct Non!inear Transformation
,.--. d-q trans- Yq Z(x)
1 1 1 1
1 l , Reference 1 1 l Generator 1
--p fonnation .
1 1 !ct 1 1 iq 1 1; 1;0
~, i(t) 1 l' 1 1 Lmear l "
Vdc y(t) 1 Feedback 1 Linearized M 1 1
1 Measurements 1 1 Yi. 1 Vdc L ___ ----'
!ct !q
~ PWM-VSC !! Inverse F-
Transformation 1;
w
Fig. 6-1 Diagram of control
QSref
QRref
ode!
133
where f(~) is a nonlinear function of~.
In Fig. 6-1, ~ of the PWM-VSC is first transfonned to a N-tuple vector ~, the state-
vector of a linear system. The linear equations are fonnulated as dydt=[G]~+[H]w, where
the (N-1) tuple vector w is the input vector of the linear system. The time-invariant [G]
and [H] matrices are chosen to ensure controllability and to facilitate the transfonnation
ofw back to 11. In the linear domain, using state-feedback so that w=[E]~, where [E] is the
gain matrix, the system equations become dYdt={[G]+[H][E]}~. For specifications ofthe
eigenvalues of the matrix {[G]+[H][E]}, algorithms are available, in MATLAB for
example, by which [E] is solved off-line. Having solved [E], the input vector of the linear
system is known because w=[E]~. What is left to do is to find the input vector 11 from w
by inverse transfonnation.
The problem to be solved in implementing the Nonlinear Control Method consists of:
(1) finding a nonlinear transfonnation by which ~ is transfonned to ~,
(2) ensuring that there is an inverse transfonnation from the input w in the ~-system to
the input 11 of the ~-system.
6.2.1 Transformation of ~ to ~
In the Simplified approach, "Nonlinear Transfonnation" is less fearsome because the
nonlinear transfonnation involves only one variable, the e1ement ZN of the vector ~. The
remaining transfonnation of~ to ~ consists oflinear transfonnations.
6.2.2 Nonlinear Set
The only one nonlinear transfonnation is:
l34
(6-1)
It has to be chosen so that dzN/dt is independent of the input vector g. This choice
allows the state-variable dimension N to fit with the input dimension (N-1) as will be
explained next.
6.2.3 Linear Set
In VSC systems of dimension N, the input vector g has a dimension (N-1). In order to
be dimensionally compatible, ~ state-variable and w the input variable of the linear
system must have dimensions of N and (N-1) respective1y. A convenient way of
accommodating the difference in dimensions between ~ and w is to structure the ~ vector
such that the (N_1)th element of~ shall be defined as:
The element Z(N-I) belongs to the linear set.
6.2.4 Identity Transformation
(6-2)
From the N-tuple state-vectors K and ~, one defines (N-2)-tuple sub-vectors K(N-2) and
~(N-2), where K(N-2)T=[Xr, X2, .... XN-2] and ~(N_2)T=[ZI, Z2, .... ZN-2]. The identity transformation
is applied to relate the two (N-2) tuple vectors:
~(N-2) = K(N-2) (6-3)
From (6-2) and (6-3), it is c1ear that the term "Nonlinear Transformation" is more
fearsome than it sounds as it involves finding only the transformation of (6-1) only.
135
6.2.5 Linear System ofz
The ~-system state variables are ~T=[Zl, Z2, .... ZN-2, ZN]=[~(N_2)T,ZN]. One fonns a linear
system by defining the state equation:
d~/dt=[G]~+[H]w (6-4)
where the [G] and [H] are time-invariant NxN and Nx(N-1) matrices whose fonns are
chosen to facilitate the inverse transfonnation of the inputs (w to 11). Applying state
feedback to the linear system, one relates ~ to w through a yet unspecified (N-1)xN gain
matrix [E]:
w=[E]~ (6-5)
Substituting (6-5) in (6-4), (6-4) becomes
dydt={[G]+[H][E]) ~ (6-6)
Linear theory allows the specification of the eigenvalues of ([G]+[H][E]) to be made
and software packages such as MATLAB can find the solution of the matrix [E] which
will meet the eigenvalue specifications. When the real parts of aIl the eigenvalues are
large negative values, then for any initial value ~(O), the trajectory of ~(t) is damped to Q
very quickly. Since K(t)=Q when ~(t)=Q, it follows that K(t) too will be damped as rapidly
toQ.
It should be noted that (6-4) to (6-6) are for the purpose of theoretical development
only. The only information processing in the Linear Block of Fig. 6-1 is w=[E]?;.
Therefore in real-time control, the gain matrix [E] is first solved off-line by MATLAB (or
136
other software packages) and then placed in the memory of a DSP, which then computes
y(t) for use in controlling the Voltage-Source Converters (VSCs).
6.2.6 Inverse Transformation of w to Y:
It is significant to note that the method does not require an inverse transformation of ~
back to K. Rather, only w needs to be inverted to y. As will be seen later, the identity
transformation of (6-3) and the selection of the form of the [G] and the [H] matrices are
orientated to recovering y from w directly.
Since the Simplified Nonlinear Control Method is orientated to controllers based on
VSCs, from this point on the method will be presented using specific VSC-based
controllers.
6.3 BILINEAR EQUATIONS OF PWM-CONVERTER
6.3.1 STATCOM Equations
Fig. 6-2 shows the single-line diagram of a PWM-Converter connected as a
STATCOM. RI and LI are the line resistance and line inductance on the ac side. After
transformation from the a-b-c frame to the d-q frame, the ac-side currents are (id!' iqI ) and
the terminal ac source voltages are (Vds, Vqs). The converter ac voltages are (UdIVdc,
UqIVdc), where Vdc is the dc voltage across the capacitor C, and (UdI, UqI) are the control
inputs. Kirchhoffs Voltage Laws ofthe ac-side yields the following equations:
(6-7)
(6-8)
137
The power balance equation relating the ac-side power P ae and the dc-side power P de of
the converter is:
(6-9)
By dividing (6-9) throughout by Vde, one obtains ide, the dc current injected by the
converter,
(6-10)
This allows the Kirchhoff s Current Law on the DC side to be written, which for the
STATCOM circuit is:
Cdv dc/dt= id llid 1 +iq 1 Uq 1
. r------------, lql Uql V dc
~ RI LI ~dlVdc r----/V'V---fYY\----t----i ~ }---------,
V dS + ~
V qS -
L.. ___________ _
STATCOM
Fig. 6-2 Single line diagramofSTATCOM
(6-11)
138
Equations (6-7), (6-8) and (6-11) can be rewritten in the first-order standard form of
state equations with ~T=[idI, iqI , Vde]=[XI, X2, X3] and control inputs gT=[lldI, UqI]=[UI, U2].
The system is of order N=3. Because the control inputs (UdI, UqI) are multiplied by the
state-variable Vde in (6-7) and (6-8) and bythe state-variables (idI , iqI ) in (6-11), equations
(6-7), (6-8) and (6-11) cannot be arranged in the first-order standard form of linear
equations. The form of nonlinearity whereby the inputs are multiplied by state variables is
given the name "bilinear".
6.3.2 Removing Bilinear Terms
The following observations are significant to the understanding of the Nonlinear
Control Method:
(1) Since the injected dc current, ide in (6-10), is based on ac si de voltage control (UdI,
uqI), there is no independent controller in the dc link equation, as exemplified by (6-11).
Thus the dimension of the input vector gis (N-l). To be compatible, the dimension ofw
has to be (N-l) also.
(2) Bilinearity has its origin from (6-9), the power balance equation. With this insight,
the input terms which lead to bilinearity can be removed by multiplying (6-7) by idI, (6-8)
by iq 1, (6-11) by v de and summing the three resultant equations which is:
(6-12)
The left-hand side term can be written as dWen/dt, where
(6-13)
139
It is significant to note from the right-hand-side of(6-12) that dWen/dt does not contain
any input terms (Udl, Uql). For this reason, Wen is an excellent candidate for nonlinear
transformation. By equating ZN=Wen, one can obtain dZN/dt=z~-l) to satisfy (6-2). The
function Wen defined in (6-13) is the sum of storage energy in the inductances L and the
capacitor C. Its time derivative from the power equation of (6-12), shows that dWen/dt is
equal to the power from the ac voltage sources less the losses in the line resistances.
6.3.3 Transformation of~ tO;f
Writing the state-vector of the converter as KT =[idl,iql,Vdc]=[Xl, X2, X3], and proposing
the nonlinear function in the nonlinear transformation of (6-1) as h(K)= Wen, then
Since dWen/dt is equal to the left-hand si de term of (6-12), one can define
where from (6-12),
R (. 2 . 2) v· V· Z2=- 1 Idl + Iql + dsIdl+ qsIql
(6-14)
(6-15)
(6-16)
which is independent of the input vector yT=[Udl, Uql]=[UI,U2]. The ability to synthesize
the Nonlinear function ZN(K), such that z~_I)=dzN/dt as in (6-2), ensures that w is a (N-l)
dimension vector which fits (6-42) below, a step which facilitates the inverse
transformation of w to y.
140
Having Z2 and Z3 of the N=3 state-variables of?;, there is Z1 left which, as has been
mentioned in (6-3), will be based on identity transformation. One can choose: either
Z1=X}, Z1=X2 or Z1=X3. For this STATCOM example, the transformation equations are
chosen to be:
(6-17)
6.3.4 Inverse Transformation ofw to y.
For the linear system equations of (6-4), the following [G] and [H] matrices, which
have the same structure as used in the previous chapters, facilitate the transformation of w
tog:
lOOO] [G]= 000
010
(6 -18)
(6 -19)
It can be shown that for the choice of (6-18) and (6-19) for [G] and [H], the system of
(6-4) is controllable.
141
Making the substitution of(6-18) and (6-19) in (6-4), one has
(6-20)
(6-21)
Equation (6-20) and (6-21) respectively belong to the decoupled and coupled portions
in the recovery of.!! from w.
6.3.4.1 Solutionfrom Decoupled Equation
Substituting dXl/dt from (6-7) into (6-22),
where al=(-RlXl+rox2+Vds)
Thus
(6-22)
(6-23)
(6-24)
Because Ul is solved from Wl alone, this solution belongs to the de-coupled set. In
general, .!!(N-2) are solved in the de-coupled manner as exemplified by (6-24).
6.3.4.2 Solutionfrom Coupled Equation
From (6-21)
(6-25)
Substituting (6-16) in (6-25)
142
(6-26)
Substituting dXl/dt= dz1/dt=Wl and dX2/dt from (6-7) in (6-26)
(6-27)
Therefore
Because both Wl and W2 are required to solve for U2, the solution is regarded as
belonging to coupled equations. In general, the last control input e1ement U(N-l) requires
all the elements ofw in its solution, as exemplified by (6-28).
6.3.5 Solving the Gain Matrix [E}
For the [G] and [H] matrices of(6-18) and (6-19),
dzl/dt=w1
(6-29)
one can specify any three eigenvalues Âl, Â2 and Â3 by using the 2x3 [E]-matrix as:
[El=[~ (6-30)
143
As a check, one can show that the characteristic equation of {[G]+[H][E]} is given by
(s-À1)(s-À2)(S-À3)=O.
For systems of higher dimension, [E] IS solved by software packages such as
MATLAB.
6.3.6 Steady-state Operating States
The operation of the STATCOM is determined by the steady-state operating point Xo.
By the transformation of (6-17), Xo has a corresponding ZOo Having chosen eigenvalues
with fast damping, the solution of z; in
d(z;-Zo)/dt= {[G]+[H] [E]} (Z;-Zo) (6-31)
converges quickly to Zo and K converges on Xo in the same time. The control to the
STATCOM is based on transforming w tO!! where
(6-32)
6.4 TWO-CONVERTER SYSTEMS
This section is intended to show how the Simplified Nonlinear Control Method is
extended to two Voltage-Source Converters. The Unified Power Flow Controller (UPFC),
which has been treated in section 4.3 of chapter 4, is reworked here as another example.
The same equations are also applicable to twO Voltage-Source Converters operating as
back-to-back PWM-HVDC.
144
6.4.1 Unified Power Flow Controller (UPFC)
In chapter 4, Fig. 4-4 shows the single-line diagram of the UPFC. In the d-q frame,
Kirchhoff s Voltage Laws applied to the shunt converter are:
(6-33)
(6-34)
LI, RI are the shunt circuit inductance and resistance; V dS, V qS are the voltages of the
sending-end buses.
Kirchhoffs Voltage Laws applied to the series converter are:
(6-35)
(6-36)
L2, R2 are the transmission line inductance and resistance; V dR, V qR are the voltages at
the receiving-end buses.
Kirchhoff s CUITent Law on the DC side is:
(6-37)
The system order is N=5. The 5-tuple state-vector is XT=[ idI, iq!, id2, iq2, Vdc]=[X! ,X2,
X3, X4, X5]. The 4-tuple input vector is !l=[Udt, uq!, lid2, Uq2]=[Ut, U2, U3, U4]. The
dimensions of~ and w are 5 and 4 respectively.
145
6.4.2 Transformation of:J. to ~
Applying the identity transformation of ~(N-2) = X(N-2) of (6-3): Zl=Xl, Z2=X2 and Z3=X3.
From (6-2) Z4= dzs/dt, and from (6-1) the nonlinear transformation function is zs=h(X). By
multiplying (6-33) by id1 , (6-34) by iql, (6-35) by id2, (6-36) by iq2, (6-37) by Vdc and
summing the 5 products, the sum is independent of input vector, which is .!::!7=[Udh Uql, Ud2,
Uq2]. Thus following the example ofthe STATCOM, the Nonlinear function is assumed to
be zs=0.5[L1X12+ L1xl+ L2Xl+ L2xi+Cxl]. In summary, the transformation equation is:
2 2 2 2C 2] zs=0.5[L1Xl + L1X2 + L2X3 + L2X4 + Xs .
Following the pattern of (6-18) and (6-19), [G] and [H] of (6-4) are:
00000
00000
[G]= 00000
00000
00010
1000
0100
[H] = 0010
0001
0000
(6-38)
(6 - 39)
(6 - 40)
146
6.4.3 Transforming w to Y:.
Substituting (6-39) and (6-40) in (6-4), it follows that
(6-41)
6.4.3.1 Solution of Decoupled Equations
Defining (N-2) tuple vectors K(N_2)T=[XI, X2, X3] and W(N-2{=[Wl' W2, W3], since in
(6-38), ~(N-2) = K(N-2), it follow from (6-41) that
d!5.(N-d dt=w (N-2) (6-42)
In order to facilitate symbolic manipulation, one also defines the following (N-2) tuple
vectors:
(6-43)
where
(6-44)
147
One defines [L](N-2) as an (N-2)x(N-2) diagonal matrix having diagonal tenns:
(6-45)
Using the newly defined quantities, (6-43), (6-44) and (6-45), one can rewrite (6-33)
(6-35) as:
[L](N-2) d!i(N-2/dt= ~-2) - Vdc!!(N-2) (6-46)
Substituting (6-42) in (6-46),
[L ](N-2) w (N-2)= ~(N-2) - V dc!!(N-2) (6-47)
Thus
(6-48)
6.4.3.2 Solution of Cou pIed Equations
The remaining 14=Uq2 is solved from dz4/dt=W4 from (6-41). Applying z4=dzs/dt,
(6-49)
Differentiating (6-49)
148
w4=(dXj/dt)(VdS-2RjXj)+(dx2/dt)(Vqs-2RjX2)+(dx)idt)(VdS-VdR
-2R2X3)+(dx4/dt)(V qS-V qR-2R2X4)
Substituting (6-42) and noting from (6-36) that
w 4=Wj (V ds-2Rjxj)+W2(V qs-2RjX2)+W3(V dS-V dR-2R2X3)
+(~- Vdcli4)(VqS-VqR-2R2X4)/L2
Renee
U4=(~/Vdc)-L2{W4-[Wj(VdS-2RjXj)+W2(Vqs-2RjX2)
+W3(VdS-VdR-2R2X3)]}/Vdc(VqS-VqR-2R2X4)
(6-50)
(6-51)
(6-52)
(6-53)
With .!!(N-2) solved from (6-48) and li4 from (6-53), the input veetor.!! T=[.!!(N_2)T, li4].
6.4.4 MATLAB Solution of [E]
For eigenvalues, Îq=-190, Â2=-220, Â3=-210, Â4=-210 Âs=-150 and system parameters:
L j=0.2266 p.u.,L2=0.4226 p.u., Rj=0.0041 p.u.,R2=0.0077 p.u., VdS=1.0 p.u., Vqs=O p.u.,
VdR=0.99 p.u., VqR=0.14 p.u., the MATLAB solution of[E] is:
2.85
o [E]= o
o
0.034 0
o 0.021
o 0
o 0
o o 0.021
o
o o o 0.022
149
6.4.5 Complex Power Regulators
In the power system, HVDC and F ACTS controllers are expected to regulate complex
power SREF=PREF+jQREF. Since the system parameters are known, it is a matter of
computing the steady-state Xo and thereafter Zo for specifications ofPREF and QREF.
6.4.6 Digital Simulation Results
Fig. 6-3 shows the digital simulation results of the UPFC under transients when two
step changes are made to the real power reference PREF from 0.998 to -0.998 p.u. and then
back to 0.998 p.u. again. The reactive powers at the sending- and receiving-end are kept
constant at QSREF=0.07 p.u. and QRREF= -0.07 p.u. The simulation results show that the
simplified nonlinear control method works perfectly weIl. After all, the same equations
as the method developed in Chapter 3, 4 and 5 have been simulated.
150
(a) _~f~i~ps--,--2-L--------J: l o 0.02 0.04 0.06 0.08 0.1 0.12
~)::t~ Qs rs: ~:s;] o 0.02 0.04 0.06 0.08 0.1 0.12
(c~l QR~. == l o 0.02 0.04 0.06 0.08 0.1 0.12
2
~ /\VRa~i~/\ ~ (d):~.VI
o 0.02 0.04 0.06 0.08 0.1 0.12
(eq v·,E: E 1 o 0.02 0.04 0.06 0.08 0.1 0.12
m ~~l u'~+f{ , u, è;; j o 0.02 0.04 0.06 0.08 0.1 0.12
time(s)
Fig. 6-3 Power Reversais ofUPFC (a) Ps; (b) Qs; (c) QR; (d) ac voltage and current; (e) dc link voltage; (f) modulation inputs.
151
6.5 C-UPFC SYSTEM
This section applies the Simplified Nonlinear Control to the C-UPFC system.
6.5.1 C7-[Jj>J?(7
The equivalent circuit the C-UPFC is shown in Fig. 5-1 and the system equations are
the same as (5-1) to (5-9), (5-13) ofchapter 5 section 5.2. The 5-tuple state-vector is KT=[
id!' iqJ , id2, iq2, Vdc]=[Xl ,X2, X3, )4, xs]. The 4-tuple input vector is !/=[Udl, uqJ, Ud2,
Uq2]=[UJ, U2, U3, U4]. The dimensions of Z and w are 5 and 4 respectively. Therefore, the
procedures to follow are the same as for the UPFC in section 6.4. Nevertheless, they are
repeated here to emphasize that the Simplified Nonlinear Control Method can be
generally applied to VSC-based F ACTS controllers.
6.5.2 Transformation of:J. to ~
Applying the identity transformation of Z(N-2) = K(N-2) of (6-3): ZJ=XJ, Z2=X2 and Z3=X3.
From (6-2) Z4= dzs/dt, and from (6-1) the Nonlinear-type function is zs=f(K). Following
the example of the STATCOM, the Nonlinear function is assumed to be zs=O.5[LJx/+
LIX/+L2Xl+L2X/+L3(XJ-X3i+L3(X2-X4i+cXS2]. In summary, the transformation equation
is:
(6-54)
z4=dzs/dt
zs=O.5[LJX12+ LJxl+ L2Xl+ L2X/+L3(XJ-X3i+L3(X2-)4)2+Cxl].
152
Following the pattern of(6-18) and (6-19)
00000
00000
[G]= 00000
00000
00010
1000
0100
[H] = 0010
0001
0000
6.5.3 Transforming w to li
Substituting (6-55) and (6-56) in (6-4), it follows that
6.5.3.1 Solution ofDecoupled Equations
(6-38), ~(N-2) = ~(N-2), it follow from (6-57) that
(6 - 55)
(6 - 56)
(6-57)
153
d?5.(N-2Y dt=w (N-2) (6-58)
In order to facilitate symbolic manipulation, one also defines the following (N-2) tuple
vectors:
1!(N_2)T=[ U}, U2, U3]=[ lldl, Uql, 1ld2]
~_2)T=[a}, a2, a3 ]
where
al=(-R\Xl+CO L\X2+VdS- Vdo)
a2=(-R\X2-CO LIXl+VqS- Vdo)
a3=( -R2X3+CO L2X4+ V do-V dR)
One defines [L](N-2) as an (N-2)x(N-2) diagonal matrix having diagonal terms:
LI 1 (N-2)=L1
L22(N-2)=L1
L33(N-2)=L2
(6-59)
(6-60)
(6-61)
Using the newly defined quantities, (6-59), (6-60) and (6-61), one can rewrite (5-3) -
(5-5) as:
[L](N-2) dXCN-2/dt= ~N-2) - Vdc1!(N-2) (6-62)
Substituting (6-58) in (6-62),
[L ] (N-2) w (N-2)= ~(N-2) - V dc1!(N-2) (6-63)
154
Thus
!!(N-2) = -(lIvdc){[L](N-2) W(N-2) - ~-2)} (6-64)
6.5.3.2 Solution ofCoupled Equations
The remaining U4=Uq2 can be solved from dz4/dt=w4 from (6-57). It is to be recalled
that zs=W, the sum of aIl the storage energy in LI, L2, L3 and C. In fact, W=O.5[LIX12+
Llxl+ L2xl+ ~xl+L3(XI-X3)2+L3(X2-X4i+Cxl]. From power balance, one has:
z4=dzsldt=dW/dt=[VdlXl+VqlX2-Vd2X3-Vq2X4-Rl (Xl 2+X22)-R2( X32+X/)
R3(XI-X3i -R3(X2-X4i]. (6-65)
Therefore on differentiating (6-65) and substituting dz4/dt=W4 from (6-57), one has
W4=[Vdl dx l/dt+Vq 1 dX2/dt+Vd2dx3/dt+Vq2dX4/dt-2Rl (x 1 dXl/dt+X2 dX2/dt)-2R2(X3dX3/dt
+X4dx4/dt)- R3(XI-X3)(dxl/dt-dx3/dt)-R3(X2-X4)(dx2/dt-dx4/dt)]. (6-66)
One sees from (5-3) to (5-6) that the input variables UI,U2,U3 and li4 are contained in the
dxIfdt, dX2/dt, dX3/dt and dxJdt terms respectively in (6-66). Since Ul,U2 and U3 are
elements in the vector !!(N-2) and have already been solved in (6-64), they can be
substituted in (6-66). This leaves U4 as the remaining unknown which can be solved from
(6-66).
Although the approach is different from the more general method of A.lsidori,
H.Nijmeijer and A.J.van der Shaft, the Simplified Nonlinear Control Method yie1ds
155
identical digital simulation results as in chapter 5 so that there is no point in repeating
themhere.
6.6 CONCLUSION
The Simplified Nonlinear Control is based rigorously on the more general Nonlinear
Control Method of AIsidori [1], H.Nijmeijer and AJ.van der Shaft [2]. The
"simplification" is possible because the application is restricted to controllers based on
the Voltage-Source Converters. Physical insights into the nature of the nonlinearity of
Voltage-Source Converters and a mathematical understanding of the method of A.lsidori,
H.Nijmeijer and AJ.van der Shaft have enabled a simpler formulation of the method to
be presented without requiring the advanced mathematics of the authors. This chapter has
already applied the Simplified Nonlinear Control Method to the STATCOM, the UPFC
and the C-UPFC. From this application experience, there is no limit to the dimensionality
of VSC-based controller and the controls of many existing multi-converter systems such
as the M-UPFC [37] and the back-to-back VSC-HVDC [28-34] are the next candidates.
The Simplified Nonlinear Control Method will make method of A.lsidori, H.Nijmeijer
and AJ.van der Shaft more accessible to design engineers. Presently the Simplified
Method is restricted to control systems based on VSCs. The future may reveal that the
method has broader applications but this will require further research using the insights
developed in this chapter.
156
APPENDIX 6 SIMULATION PARAMETERS AND SETTINGS
UPFC
Vs: 1.0p.u.
VR: 1.0 p.u.
ô:25°
Rt:Xt=1:27.5
R2:X2=1:55
À, t = -190
À, 2 = -220
À,3=-21O
À,4=-210
À,5=-150
157
Chapter 7
Further Development of N onlinear Control
7.1 INTRODUCTION
The Nonlinear Control Method, in the original fonn as described in chapter 3, or, in its
simplified version as described in chapter 6, has been successfully applied to the SSSC,
the UPFC and the C-UPFC. To complete the thesis, this chapter calls attention to three
practical issues which have to be addressed successfully before the Nonlinear Control
Method will find engineering applications. The objective of this chapter is to present
preliminary results from simulations which show that the issues can be resolved.
However, thorough conclusions will require further research from future theses.
The first issue concems the infonnation of the sending-end voltage Vs and/or
receiving-end voltage VR. Since they are remote1y located from the FACTS controller,
the infonnation (V Sd, V Sq) and (V Rd, V Rq) have to be telemetered to the location of
Nonlinear Controller. This will require reliable and fast communication channels. The
expense of telemetering can be saved if CV Sd, V Sq) and (V Rd, V Rq) can be estimated from
measurements taken locally at the site of the F ACTS controller. This chapter shows that
the Nonlinear Control Method is still successful when (VSd, Vsq) and (VRd, VRq) are
estimated from local measurements taken at the location of the F ACTS controller.
The second issue relates to the d-q frame voltages (V Sd, V Sq) and (V Rd, V Rq) which, in
"
all the examples simulated, have been kept constant. In the field, the angle 8=8s-8R
between the sending-end voltage phasor Vs and/or receiving-end voltage phasor VR may
158
have a low frequency swing oscillation between the inertial systems during transients.
Thus their d-q frame voltages are in fact non time-invariant quantities (VSd, vSq) and (VRd,
VRq).
The third issue relates to the uncertainties of the system parameters, for example the
line parameters RI. XI. R2 and X2. Parameters deviations of 10% have been introduced in
the simulations to give an idea as to how robust the Nonlinear Control Method really is.
Only the C-UPFC will be used in this chapter to illustrate how all the three issues can
be addressed. This is because the C-UPFC is by far the most complicated F ACTS
controller dealt with in the thesis.
7.2 ESTIMATION OF SYSTEM VOLTAGES BY WAY OF LOCAL
MEASUREMENTS
As the C-UPFC system is located in the middle of a long transmission line as shown in
Fig. 7-1, the sending-end voltages (VSd, vSq) and receiving-end voltages (VRd, VRq) are not
available. Only the voltages and currents at the terminal of the three converters (Vdl, VqI.
idI. iqI. Vd2, Vq2, id2, iq2) can be locally measured. Thus, it is necessary to estimate the the
sending-end voltages (VSd, vSq) and receiving-end voltages (VRd, VRq) by making use of the
local measurements VdI. Vql, idl , iql , Vd2, Vq2, id2, iq2 and available transmission line
parameters.
7.2.1 Estimation of System Voltages
As shown in Fig. 7-1, with the knowledge oflocal measurements (VdI. idI. Vql, iql ) and
line 1 parameters (RI, Xl), the estimation of (VSd, vSq) is fairly straight-forward by using
159
equations (5-3), (5-4), (5-5) and (5-6), and the statements of Vdl=Vdo+UdlVdc and
Vql=Vqo+UqlVdc. So is the estimation of (VRd, VRq) at the receiving-end.
7.2.2 Treatment ofTime-varying voltages (VSd. vsq) and (VRd. VRq)
When (VSd, vSq) and (VRd, vRq) are not time-invariant it is necessary to modify the
inverse transformation from the input vector w of the linearized system to !! in the
original nonlinear system equations. In order to make this point with minimum
complication, only the equations of the SSSC in section 4.2 of chapter 4, will be used as
example here. The equations of the SSSC are:
LINE 1 LINE2 Y 0' center-node
:~Ë-l~-A;;: jX2 R2 '----+---1 "v "v r---ir------'
1
Y+ Yl,Ill S "v 1
(measured) 1
---'1 2
Y2,I2 : (measured)
L ____________ .J
C - UPFC
Fig. 7-1 Equivalent circuit ofC-UPFC with three converters sharing one common dc capacitor link
(4-1)
160
The linearized equations are:
dz) -=Z2 dt
dZ 2 -=w dt )
(4-2)
(4-3)
(4-10)
(4-12)
In the inverse transformation, Wl is equated to dz2/dt. In differentiating Z2 in (4-10),
one has to differentiate the sending-end and receiving-end voltages. Since (dVSd/dt,
dvsq/dt) and (dVRd/dt, dvRqldt) are no longer zeroes (as have been assumed in the previous
chapters), the differentiated terms must be retained. As is well known, differentiation
introduces errors because differentiation magnifies high frequency noise. The saving
grace is that because of the large inertias of the power system, (VSd, vSq) and (VRd, VRq) are
161
in the low frequency end of the frequency spectrum. Thus the high frequency noise in the
estimates can be filtered before proceeding to differentiation to obtain (dVSd/dt, dvsq/dt)
and (dVRd/dt, dVRq/dt) in practice. For the simulations, a simple first-order differentiation
method (i.e., desd_dt = (esd-esd_old)/ ilt) is adopted and no filter is used.
7.2.3 Digital Simulations
The next step is to evaluate how weIl the Nonlinear Control Method performs using
the values of (VSd, vSq) and (VRd, vRq) which have been estimated from (Vld, Vlq) and (V2d,
V2q' In the first simulations test, the C-UPFC is connected to time-invariant sending-end
and receiving-end voltages (V Sd, V Sq) and (V Rd, V Rq ) and is given a step power reversaI
ofreference power Ps from + 1.0 p.u. to -1.0 p.u. The reactive powers Qs and QR are held
constant. The simulation results are shown in Fig. 7-2 and continued in Fig. 7-3. In the
simulation, up to 10% deviations with the system parameters RI, XI, R2 and X2 have been
introduced in order to assess how robust the Nonlinear Control Method is. The results
show that with the local measurements, the system response is still comparable to that of
ideal conditions.
The parameters used in the simulation are listed in the following table:
Table 7-1 System parameters adopted in the simulation
RI (p.u.) XI (p.u.) R2 (p.u.) X2 (p.u.)
Under ideal 0.0014282985 0.0785248121 0.0026637767 0.1464487745
conditions
With 10% 0.00128546865 0.07067233089 0.00239739903 0.13180389705 ~ ~ ~ ~
deviation 0.00157112835 0.08637729331 0.00293015437 0.16109365195
162
2
(a) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(b) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
2
(c) 0
-2 0 0.02 0.04 0.06 0.08 0.1 0.12
1
(d) 0 Ps
-1 0 0.02 0.04 0.06 0.08 0.1 0.12
0.2
(e) 0
-0.20 0.02 0.04 0.06 0.08 0.1 0.12
-0.06
-0.07 (f) QR
-0.08
-0.090 0.02 0.04 0.06 0.08 0.1 0.12(s)
Fig. 7-2 Real power reversaI ofC-UPFC (a) Vsa(estimated), isa (b) V Ra( estimated), iRa (c) Voa, ioa (d) Ps (e) Qs (f) QR
163
1.015
1.01
(a)1.005
(b)
(c)
1
0.9950
0.8
0.6
0.4
0.2
0
-0.20
1
0.5
0
-0.50
0.8
0.6
(d) 0.4 0.2
o -0.2
0
vdc
0.02 0.04 0.06 0.08
ud1 uq1
0.02 0.04 0.06 0.08
uq2
0.02 0.04 0.06 0.08
\ ud3
uq3
0.02 0.04 0.06 0.08
Fig. 7-3 Real power reversaI ofC-UPFC (a) Vdc
(b) Ud1, Uq1
(c) Ud2, Uq2
(d) Ud3, Uq3
0.1 0.12
0.1 0.12
0.1 0.12
0.1 0.12(s)
164
7.3 SIMULATION OF SYSTEM WITH VOLTAGE SWING
Further tests are needed to evaluate the effect of non time-invariant voltage sources. In
the next two tests, phase angle swings in the sending-end voltage and the receiving-end
voltage are introduced in the simulations: (1) the sending-end voltage angle is made to
oscillate (results shown in Fig.7-4 and continued in Fig.7-5); (2) the voltage angles of
both sides are made to oscillate (results shown in Fig.7-6 and continued in Fig.7-7). In
both cases, the reference setting of Ps has been given a step change from +1.0 to -1.0
p.u. The reactive powers Qs and QR are held constant throughout.
As depicted in Fig.7-5 (b), the oscillation in the sending-end is given a frequency of
2.5 Hz with a decaying time constant of 0.5 s. The time axis has been extended to 1s so
that there are now many cycles of ac voltages and currents in Fig. 7-4 (a), (b) and (c).
Fig. 7 -7 (b) and (c) show the sending-end oscillation at a frequency of 2.5 Hz with a
decaying time constant of 0.5 s and a receiving-end frequency of 0.5 Hz with decaying
time constant of 1 s. The time axis has been extended to 2s.
The simulation results show that the Nonlinear Control Method under practical
considerations (requiring estimation of both the non time-invariant sending-end and
receiving-end voltages) is quite robust. The system can still achieve very fast response
when there is a step change (real power reversaI). AlI these simulations strengthen the
argument that the Nonlinear Control Method is very promising from the practical point of
vlew.
165
2
(a) 0
-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2
(b) 0
-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2
(c) 0
-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
(d) 0 Ps
-1 \ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5
(e) 0 Qs
-0.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.06
-0.07 >-V OR (f) -0.08
-0.090 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (s)
Fig. 7-4 Real power reversaI with oscillation at Vs (a) VsaC estimated), isa (h) VRa(estimated), iRa (c) Voa, ioa (d) Ps (e) Qs (f) QR
166
(a)
(b)
(c)
(d)
(e)
(t)
1 vdc
0 0.1 35
30
25
200 0.1
1 '----
0.5 /~
00
V
1
0.5
o j--...\ ~
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
phase angle of V s
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
vds
vqs
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
U q l
udl
-0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1r---~--~--~--~--~--~--~--~----~
0.5
0 1><==, ,Ud2 / uq2
-0.5 \... ,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1
0.5 ~ u d3
0 U q3
-0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cs)
Fig. 7-5 Real power reversaI with oscillation at Vs (a) Vdc (b) Phase angle of Vs (c) Vds, Vqs (d) Ud!, uq! ( e) lid2, Uq2
(f) lid3, Uq3
167
2
(a) 0
-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2
(b) 0
-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2
(c) 0
-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1
(d) 0 Ps
-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(e) O.:~J~
-0.50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.06
-0.07 (f) If OR
-0.08
-0.090 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (s)
Fig. 7-6 Real power reversaI with oscillation at Vs and V R
(a) Vsa(estimated), isa (b) VRaC estimated), iRa (c) Voa, ioa (d) Ps (e) Qs (f) QR
168
(a) vdc 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 35
30 (b) 25
20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
35
30 (c) 25
20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1
0.5 ~ uq1
(d) 0 ud1 -----
-0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1
(e) 0 L Ud2 /uq2 ----.. Ji'
-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1
0.5 tl ud3
(f) 0 uq3
-0.50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (s)
Fig. 7-7 Real power reversaI with oscillation at Vs and V R
(a) Vdc
(b) Phase angle of Vs (c) Phase angle ofVR
(d) UdJ, Uql
(e) Ud2, Uq2
(f) Ud3, Uq3
169
7.4 CONCLUSION
From chapters 3 to 6, the Nonlinear Control Method has yielded excellent control
characteristics. The just criticism is that as aIl the simulations have been done under
idealized conditions, there is no way of knowing of whether the excellent control
characteristics will be produced in an engineering environment. Translating a
mathematical method into a practical method requires extensive work.
This chapter has identitied 3 practical issues and has shown through simulations that
they can be resolved and the excellent control characteristics can be realized. The tirst
issue relates to the remote terminal voltages which, without telemetering, are not directly
accessible and have to be estimated from local measurements. The second issue relates to
the remote terminal voltages not being time-invariant in the d-q frame formulation. The
third issue relates to the uncertainties in the parameters of the plant.
More research will be required to uncover other issues and to tind ways to surmount
the difticulties. For the present, the simulations of this chapter have shown that the
Nonlinear Control Method is not without robustness and there is promise that it can be
practically implemented.
APPENDIX 7 SIMULATION PARAMETERS AND SETTINGS
C-UPFC
Vs: 1.0 p.u. (simulated)
VR: 1.0 p.u. (simulated)
ô: 500 (simulated)
170
R1:X1=1:55 with 10% deviation
R2:X2=1 :55 with 10% deviation
R3:X3=1:27.5
À 1 = -190
À 2 = -220
À 3 =-21O
À 4 =-210
À 5 =-150
171
8.1.1 Summary
Chapter 8
Conclusions
8.1 CONCLUSION
Through the six chapters of the thesis, the Nonlinear Control Method has been applied
successfully to the family of F ACTS controllers based on Pulse-Width Modulated
Voltage-Source Converters (PWM -VSCs).
Chapter 2 initiates research on the Center-Node Unified Power Flow Controller (C
UPFC). The C-UPFC is an innovative topology of F ACTS (Flexible AC Transmission
Systems) controllers. It has been conceived to perform better than the Unified Power
Flow Controller (UPFC) which is universally acknowledged as the ultimate F ACTS
controller. The C-UPFC has been configured to operate as a UPFC, but one dedicated to
operate at the center-node of a radial transmission line. At the center-no de node, the
transmission distance is effectively halved so that the power transmissibility is double that
of the UPFC.
As in preliminary stages ofresearch in control topologies, simple proportional-integral
(P-I) feedbacks have been applied to the controls of the C-UPFC. The P-I feedbacks are
adequate for operating the C-UPFC as a phase shifter. But they have not been able to
stabilize the system when operating the C-UPFC as a series capacitor compensator or as a
controller to reverse power flow. Therefore, the instability problem is a source of
frustration. Noting that the Voltage-Source Converters (VSCs) which form the basic
172
modules of the C-UPFC are nonlinear, the Nonlinear Control Method has been brought in
to save the C-UPFC from a still-bom fate.
Chapter 3 is, in part, a tutorial introducing the Nonlinear Control Method. The
mathematical model of the Voltage-Source Converter (VSC) module is derived and used
to illustrate how the princip les are applied.
The method of attack throughout the thesis has been to apply the Nonlinear Control
Method to a small system tirst before proceeding to a higher order system. The order of
the research plan is: single-VSC system tirst, then advance to two-VSC systems before
progressing to the three-VSC C-UPFC.
Chapter 4 applies the Nonlinear Control Method to the single-VSC Static Synchronous
Series Compensator (SSSC) and then to the two-VSC Unitied Power Flow Controller
(UPFC).
Chapter 5 continues with the three-VSC C-UPFC.
The interrupted research on the C-UPFC is resumed. With the Nonlinear Control
Method on board, the C-UPFC is shown to have the performance capabilities which have
been expected. Like the UPFC, the C-UPFC has independent control over the real power
and reactive power and is able to be used as: (i) a phase-shifter, (ii) a capacitive reactance
compensator and (iii) as controller to reverse the direction ofpower Dow.
From the experience of successfully applying the Nonlinear Control Method to the
family of VSC-based F ACTS controllers, the insights gained enables a Simplitied
Nonlinear Control Method to be proposed.
173
Chapter 6 describes the Simplified Nonlinear Control Method and applies it to the
STATCOM, the UPFC and the C-UPFC. The Simplified Method is not a new method.
Instead it is a way of applying the Nonlinear Control Method without requiring advanced
mathematical background.
Chapter 7 is a short chapter which shows through simulations that the Nonlinear Control
Method has indications of sufficient robustness for practical application in the field. The
preliminary robustness tests relate to: (i) small system parameter variations and (ii) low
frequency voltage angle swings of the electric power system.
8.1.2 Conclusions
Nonlinear Control Method
The initial success in applying the Nonlinear Control Method stopped with the
STATCOM (system order n=3) [49-51]. The stumbling block is in synthesizing the
output functions: hl(K), h2(K), ... hm(K). Mathematically, hl(K), h2(K), ... hm(K) are solved by
integration. Unlike differentiation, integration is an art. Thus finding the output functions
will be the challenge in every new application of the method and requires insights of the
equations which model the system.
Relative order r=2 outputfunction h1(x)
In the case ofVSC-based FACTS controllers, m the number of inputs is always less by
1 from n, the order of the system of (3-70). Therefore, m=n-l. This means that the relative
order of one output function must be r=2. An energy function can satisfy the requirement
of the r=2 output function, which has been designated as hl (K) in this thesis. Table 8-1
below lists the output function hl (K) for sorne VSC-based F ACTS controllers.
174
Table 8-1 List of r=2 output function hl (K) for VSC-based F ACTS controllers
Type ofVSC-based Nonlinear transformation function hl (K)
F ACTS controllers
SSSC L 2 L 2 C 2 -- see chapter 4 -x +-x +-x 2 1 2 2 2 3
STATCOM 0.5 {LIX(t+ Llxl+CX3.l} -- see chapter 6
UPFC 0.5[LIXI.l+ LIXil + Lzx/+ Lzx/+Cx/] -- see chapter 4
C-UPFC 0.5[LIXI2+ LIXZ2+ Lzx/+ LZX42+L3(XI-X3)2+L3(xz-X4)2+CXS2]
-- see chapter 5
The research of the thesis has verified aIl the cases of Table 8-1. The output function
of the back-to-back VSC-HVDC station is identical to that of the UPFC. The output
function for the IPFC and the M-UPFC are not listed but they are energy functions similar
to these in Table 8-1.
Relative order r=l outputfunctions h2(x), ... hm(x).
Every row of the x-system equations of (3-70) has one and only one of the m inputs,
ul, Uz, ... um, except the row containing dVdc/dt, which has more than one input. Let Xk be
one of the state variables which is not Vdc, i.e. Xk;.t Vdc, k=I,2, ... n-l. The research has
shown that it is preferable that the output functions hz(x), ... hm(x) be chosen from the
state-variables Xk, k=1,2, ... n-l. The preference is because in solving the input vector!! of
the K-frame from the input vector w of the ?;-frame, (m-l) of the equations are decoupled.
This will facilitate implementation in real-time.
175
Simplified Nonlinear Control Method
The insights from applying the Nonlinear Control Method to several VSC-based
F ACTS Controllers have yie1ded a Simplified Method (chapter 6), which will be
welcomed by power electronic engineers because they will not have to learn the notations
and mathematical background of [1-3]. From familiarity with both the Nonlinear Control
Method and the structure of VSC-based F ACTS Controllers, identical algorithms have
been derived by systematic manipulations of the same equations. The Simplified Method
is identical to the Nonlinear Control Method. The difference is that the steps can be
followed by readers who have only taken linear state-space control courses.
The Simplified Method will enable the Nonlinear Control to gain acceptance more
readily in a community the majority ofwhom are experimentalists.
Practicality of Nonlinear Control Method
Before beginning the research on Nonlinear Control Method a brief survey has given
the assurance that the computation speeds of DSPs or parallel DSPs are capable of
implementing real-time Nonlinear Control algorithms. This research has confirmed that
implementation requires only fast multiplications and additions. There is no requirement
of special functions to be implemented. However, there is a step
[
L V[-lh g[ f 1
u= :
L L'm-1h g[ f fi
(8-1)
which requires the inversion of an m x m matrix. The inversion is not a computation
problem provided the output functions h2(x), ... hm(x) are chosen from the state-variables
176
Xk, k=1,2, ... n-l, as discussed earlier. This is because m-l equations are decoupled. The
decoupled equations are solved first. Then their solutions are back-substituted the mth
coupled equation leaving the remaining unknown to be solved.
Chapter 7 has shown that:
(i) system parameter deviations of the order of 10%;
(ii) low frequency oscillations of the power system
can be tolerated. Chapter 7 is intended to show that the Nonlinear Control Method
deserves further research because it has sorne robustness.
Center-Node Unified Power Flow Controller CC-UPFC)
With the Nonlinear Control Method in charge, the C-UPFC has been shown to the
same capability as the Unified Power Flow Controller of Lazslo Gyugyi [23, 35, 36]. It
consists of independent control over: (i) the real power through the transmission line; (ii)
the reactive power on the sending-end side; and (iii) the reactive power on the receiving
end side. It is also suitable in applications as: (a) a phase-shifter; (b) a capacitive
reactance compensator; or (c) a controller to reverse the direction ofreal power flow.
The C-UPFC is intended for operation at the mid-point of the radial transmission line,
which is the optimal position for a F ACTS controller. This is because with voltage
support at the mid-point, the transmission distance is halved since the spans are between
the sending-end and the mid-point and between the mid-point to the receiving-end. As
the C-UPFC has been specially designed to operate at the mid-point, the series VSC of
the UPFC is conceptually broken into two halves. The research has shown that the two
halves (two separate series VSCs) are capable ofmaintaining power balance in the dc bus.
In the UPFC, dc power balance is maintained by a shunt VSC and a series VSc. The
research ofthe thesis has shown that the new C-UPFC topology is functional.
177
8.2 SUGGESTIONS FOR FUTURE WORK
The suggestions for future works are related to demonstrating that a F ACTS Controller
can be operated under Nonlinear Control.
(1) Following from the promising indications of chapter 7, more simulations studies
are required to confirm that the method has the robustness required for
engineering application. The system voltage estimation method in Chapter 7 is
quite elementary and simple. For improved performance, a more advanced
predictive method may have to be considered.
(2) Software to implement Nonlinear Control algorithms are to be written for
ultra/super fast DSPs or parallel DSPs to control the Voltage-Source Converter
modules which have already been built in the Power Electronics Laboratory.
The hardware research should begin with the STATCOM or the SSSC before
progressing to the UPFC and the C-UPFC.
178
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