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Robust WAM-based Controller Coordination of FACTS Devices for
Power System Transient Stability Enhancement
by
Sushama Rajaram WAGH
Achieving International Excellence
This thesis is presented for the degree of Doctor of Philosophy of The University of Western Australia
Energy Systems Centre School of Electrical, Electronic and Computer Engineering
2011
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ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my supervisor, Professor Tam Nguyen, for
giving me the opportunity to undertake this research under his guidance, and for the constant
support and encouragement provided by him throughout my PhD candidature at The University
of Western Australia (UWA).
This thesis could not have been completed without the timely support of Prof. Victor Sreeram,
who extended his cooperation at the critical moments of my stay at UWA. I must also thank
Prof. Brett Nener for facilitating my work at UWA. I would like to place on record the support
extended to me by the staff at the Energy Systems Centre and, especial mention is deserved for
the regular technical inputs along with the healthy dose of moral support provided by Dr. Van
Liem Nguyen.
Off course, all of this could not have been possible without the Ad hoc scholarship awarded by
the Energy Systems Centre, and The University of Western Australia.
I am highly obliged to the Board of Directors of Veermata Jijabai Technological Institute
(VJTI) who has supported me in my endeavour for higher learning. I must take this opportunity
to thank, Prof. H. A. Mangalvedekar, who has been my foremost well-wisher and guide in the
academic career. Special thanks go to Prof. N. M. Singh for his support which helped me to
remain focused throughout my research work.
I am incredibly grateful to my family. My mother and sister Vijaya, brother-in-law, Rajesh,
gave me their unconditional support during all these years. I also appreciate the contribution of
Sunitha and Robert Mendonca who have been a pillar of support for my family. It would have
been difficult to continue to work at UWA, without Sudhir Bhil and Asha Sharma taking care of
things back home during my absence.
To my parents, I owe my love of knowledge and desire to excel. I would like to dedicate this
thesis to my mother, Mrs. Yamuna Rajaram Wagh, who took every effort, in providing
education to her children despite the adverse conditions she had to face after the sudden loss of
our father.
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ABSTRACT This thesis is devoted to the development of new schemes for transient stability
enhancement using FACTS (flexible alternating current transmission system) devices,
and the control coordination of power systems with FACTS devices in transient state of
post-fault scenario. The key objectives of the research reported in the thesis are, through
online control coordination based on the models of power systems having FACTS
devices, those of maximising and restoring system transient stability following a
disturbance or contingency.
The new schemes are developed in two steps with increasing complexity in modeling
and in terms of considering the size of power system. In the first part of the thesis,
explaining detailed dynamic modeling of power system components with a
comprehensive literature review of controllers used in power system, a new model
predictive control based TCSC controller is designed, developed, and implemented for
SMIB. The key contribution of this new MPC-based TCSC controller is the application
of MPC methodology to power systems which is represented by detailed dynamic
modelling and coordinated with power system primary controllers i.e. exciter and
prime-movers. The simulation study is carried out with a single-machine-infinite-bus
system and the effectiveness of the new controller is validated.
Having established the foundation provided by the comprehensive models developed
for representing power systems with FACTS devices, including the TCSC, the research
in the second part focuses on real-time control coordination of power system
controllers, with the main purpose of restoring power system stability following a
disturbance or contingency.
In the second part of the thesis, a detailed literature review is presented to highlight the
need of real-time controllers in WAN (wide-area network), which is monitored and
operated based on information provided by WAM (wide-area measurement systems).
The practical problems in real-time controller requirements are discussed with review
which gives motivation to the new innovative scheme developed in the second part. The
second part of the thesis develops a novel scheme to predict future system status by a
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look-ahead approach and tries to maintain system stability by introducing proper
compensation. The key contribution of this proposed scheme is not only in improving
system stability but it also gives a brief account of various times required for executing
every module of controller in generating required control law.
Although the developed online control coordination scheme is studied and analysed
using only series compensation, the same concept can be further extended for shunt
compensations and improving system performance in different dominations. Therefore,
the model developed in this research can be considered as general.
Another notable contribution with this developed scheme is it is validated using
standard large size power system represented with a detailed dynamic model of every
power system component. The TCSCs used for providing necessary compensation have
been represented by its detailed dynamic models and smooth controller coordination
between power system and TCSCs is achieved.
The key contribution in developing new online control coordination is that of practical
considerations given to various time delays. This thesis develops a general feasible
approach considering the requirements of real-time control while taking into account
various time delays. The communication channel delays and computation delays which
are significant in a large system which is geographically wide-spread are taken into
account for deriving a feasible control coordination scheme. It also considers and
discusses the computation system requirements of a real-time controller, given the large
amount of data to be handled within given time constraints while maintaining stability
in a post-fault region.
Drawing on the constrained optimization based on search technique together with the
new developed control coordination scheme, the method has been validated for a
multimachine power system network of New England. The search technique developed
reduces the computing time significantly as compared to standard optimization methods
such as Newton’s methods.
The third part of the thesis develops the strategy for an estimation of the internal state
variables of a synchronous generator based on available measurements. At present, as
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the cost of phasor measurement units (PMUs) and wide-area communication network is
on the decrease, the research proposes and develops a new estimation strategy to
investigate the actual values of internal state variables where direct measurements are
not available and only available limited sets of measurements can be used. The new
estimation procedure eliminates the assumption of systems at steady state and using set
of nonlinear equations, processed through iterative algorithm, finds the exact state
values. The advantage of this new estimation algorithm results in accurate comment on
further power system studies which will be based on accurate values rather than
assumed values.
The research can be further extended to combine the new controller schemes proposed,
with estimation algorithms developed in third part to improve the effectiveness of
controller coordination schemes in enhancing power system stability in given time
bounds.
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LIST OF PRINCIPAL SYMBOLS SYMBOLS USED IN CHAPTER 2
d-axis direct-axis (machine base)
q-axis quadrature-axis (machine base)
p Derivative operator (d/dt)
is Vector of stator current of synchronous machine
id Direct-axis rotor current (machine base)
iq Quadrature-axis rotor current (machine base)
υr Vector of rotor voltage of synchronous machine
υd Direct-axis rotor voltage (machine base)
υq Quadrature-axis rotor voltage (machine base)
Ψr Vector of rotor flux linkages of synchronous machine
ψfd Main field winding rotor flux
Ψkd Direct-axis damper winding flux
ψfd quadrature-axis damper winding flux
Efd Synchronous machine field voltage
Am, Fm, Km Matrices depending on synchronous machine parameters
ωr Rotor angular frequency of synchronous machine
ωref synchronous speed (reference speed)
δr Rotor angle of synchronous machine
TM mechanical torque input to synchronous machine
Te electromagnetic torque output of synchronous machine
L,G, R, Matrices of synchronous machine parameters
Lss, Lrr, Lsr, Lrs,
Gss, Gsr, Rss, Rrr
Sub-matrices of L,G,R based on synchronous machine
parameters
i vector of stator and rotor currents
υ Vector of stator and rotor voltages of synchronous
machine
Pm, Zm Matrices depending on synchronous machine parameters
and rotor angular velocity
Vref Voltage reference for excitation control system
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Vpss Supplementary signal from power system stabiliser (PSS)
XExe State vector of excitation system
VR automatic voltage regulator output
VRmin , VRmax Minimum and maximum voltage limit for AVR
Rf rate feedback
AExc , BExc Matrices depending on gains and time constants of
excitation system controller
XGov State vector for prime-mover controller
AGov , BGov , CGov ,
DGov
Matrices depending on gains and time constants of
prime-mover controller
PC Initial mechanical power
H Synchronous machine inertia constant (sec.kW/KVA)
M Synchronous machine inertia constant (pu)
VD Real part of synchronous machine voltage (system base)
VQ Imaginary part of synchronous machine voltage (system
base)
ID Real part of synchronous machine current (system base)
IQ Imaginary part of synchronous machine current (system
base)
ngen Total number of generators in power system network
nnode Total number of nodes in power system
nload Total number of load nodes in power system
Ii Current injected at node i
IBus Matrix of node currents
VBus Matrix of node voltages
Ig , Vg The current and voltage at the generator node
IL , VL the current and voltages at the loads node
PL , QL Active and reactive power load
Pr active power received
Qr reactive power received
G1 the load conductance
B1 load susptance
Vr Receiving end voltage
XSVC State vector of SVC main controller
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ASVC , BSVC , CSVC Matrices depending on gains and time constants of SVC
XC , XL Fixed series capacitor and reactor or TCR
α Thyristor firing angle
Xbypass Bypass reactance of TCSC
Xtcscmin minimum (capacitive) limit of TCSC reactance
Xtcscmax maximum (inductive) limits of TCSC reactance
X1, X2 Intermediate state variables of TCSC model
XSDC Supplementary signal from supplementary damping
controller
Pe Electrical power
Xtcsc State vector for TCSC main controller
Atcsc, Btcsc Matrices depending on gains and time constants of TCSC
controller
SYMBOLS USED IN CHAPTERS 4 AND 5
k The current time instant
N Predicted horizon
NC Control horizon
R, Q Weighting matrices (tuning parameters )
Hp prediction horizon
x Vector of state variables
u Vector of control variables
y Vector of output variables
yref Reference trajectory for output variables
∆t time step
SYMBOLS USED IN CHAPTERS 7, 8 and 9
δi1 Relative rotor angle with respect to reference bus
Tk kth time instant
u Control variable
umax and umin Upper and lower limits of control vector respectively
T Control time window size
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T1 First control window
TD1 and TD2 Communication channel delays
TC Computation delay
M Number of control periods
NC Number of polynomial coefficients
NP Number of polynomials
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GLOSSARY
AVR Automatic Voltage Regulator
CTW Control Time window
DC Direct Current
EEAC Extended Equal Area Criterion
FACTS Flexible Alternating Current Transmission System
MPC Model Predictive Control
NR Newton-Raphson
PID Proportional-Integral-Differential (controller)
PMU Phasor Measurement Unit
RHC Receding Horizon Control
SCADA Supervisory Control and Data Acquisition
SSSC Static Synchronous Series Compensator
STATCOM Static Synchronous Compensator
SVC Static VAr Compensator
SVR Secondary Voltage Control
TCR Thyristor Controlled Reactor
TCSC Thyristor Controlled Series compensation
TSC Thyristor Switched Capacitor
UPFC Unified Power Flow Controller
WAN Wide-Area Network
WAMS Wide-Area Measurement System
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LIST OF FIGURES Figure 2.1 Individual synchronous generator connected to external network
Figure 2.2 Angular relationships between external power system network and
individual machine reference axis
Figure 2.3 Transforming individual machine quantities to system frame of
reference for voltage
Figure 2.4 Typical SVC structure with main components
Figure 2.5 SVC dynamic model block diagram
Figure 2.6 STATCOM schematic diagram
Figure 2.7 Phasor diagram of STATCOM operating principle
Figure 2.8 Basic voltage sourced converter structure using GTO thyristors
Figure 2.9 Dynamic model of STATCOM
Figure 2.10 Simplified operating circuit of thyristor controlled series
compensation
Figure 2.11 Typical layout of practical TCSC structure
Figure 2.12 Typical V-I capability characteristics for a single-module TCSC
Figure 2.13 Practical operating range of TCSC for inductive and capacitive
compensation
Figure 2.14 Multi-module TCSC
Figure 2.15 TCSC schematic block diagram with SDC
Figure2.16 Simplified schematic of supplementary damping controller
Figure2.17 Multi-machine power system network having multiple FACTS devices
Figure 3.1 Typical wave specifications expressed in terms of front and tail of
wave
Figure 3.2 Time frame of various transient phenomena
Figure 3. 3 Classification of power system stability studies based on physical
nature of the phenomena
Figure 4.1
Finite horizon and Infinite horizons with no disturbance and perfect
model respectively
Figure 5.1 Proposed strategy for RHC-based TCSC controller
Figure 5.2 Flowchart of RHC implementation algorithm
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Figure 5.3 Single-machine-infinite-bus system
Figure 5.4 Generator rotor angle response without RHC controller
Figure 5.5 TCSC reactance reference input using RHC controller
Figure 5.6 Synchronous generator rotor angle response using RHC based TCSC
controller
Figure 6.1 Power flow and information flow network in wide-area network
Figure 7.1 Control scheme timing diagram: Option 1
Figure 7.2 Control scheme timing diagram: Option 2
Figure 7.3 Flowchart of control coordination scheme
Figure 8.1 Control coordination block diagram
Figure 8.2 10-Generator 39-Node New England test system
Figure 8.3 Relative rotor angle transients without online control coordination of
TCSCs
Figure 8.4 TCSC input references for case 1
Figure 8.5 Relative rotor angle transients with online control coordination of
TCSCs for case 1
Figure 8.6 TCSC input references for case 2
Figure 8.7 Relative rotor angle transients with online control coordination of
TCSCs for case 2
Figure 8.8 TCSC input reference responses Option 2
Figure 8.9 Relative rotor angle transients Option 2
Figure 8.10 TCSC input reference responses Approximate control
Figure 8.11 Relative rotor angle transients Approximate control
Figure 9.1 Comparison between actual and estimated rotor angle
Figure 9.2 Comparison between actual and estimated main field flux
Figure 9.3 Comparison between actual and estimated d-axis damper winding flux
Figure 9.4 Comparison between actual and estimated q-axis damper winding flux
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TABLE OF CONTENTS
Chapter 1 Introduction ………………………………………………….......... 1
1.1 BACKGROUND AND SCOPE OF THE RESEARCH………........ 1
1.2 OBJECTIVES………………………………………………………. 2
1.3 OUTLINE OF THE THESIS……………………………….............. 5
1.4 CONTRIBUTIONS OF THE THESIS……………………………... 9
Chapter 2 Dynamic Modelling of Power System Components……………... 11
2.1 INTRODUCTION…………………………………………………. 11
2.2 SYNCHRONOUS GENERATOR DYNAMIC MODEL…………. 12
2.2.1 Rotor flux dynamics (Generator electrical axis)……………………. 12
2.2.2 Equation of motion (Turbine/Generator mechanical axis)…………. 13
2.2.3 Relation between generator current and voltage……………………. 14
2.2.4 Excitation and automatic voltage regulator (AVR)………………… 14
2.2.5 Prime-mover and governor system…………………………………. 15
2.2.6 Power system stabiliser (PSS)……………………………………… 16
2.3 MULTIMACHINE DYNAMIC MODELLING WITH MACHINE
REFERENCE………………………………………………………..
16
2.4 EFFECT OF MACHINE MODEL USED ON STABILITY
STUDY……………………………………………………………...
19
2.5 MULTI-MACHINE MODELLING WITH SYSTEM
REFERENCE......................................................................................
20
2.5.1 Connection of individual generator to power grid………………….. 20
2.5.2 Transferring quantities from machine reference to system reference
axis …………………………………………………………………
22
2.6 INTRODUCTION TO FACTS DEVICES……………………......... 23
2.7 STATIC VAR COMPENSATOR (SVC)…………………………... 24
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2.8 STATIC SYNCHRONOUS COMPENSATOR (STATCOM)…….. 26
2.8.1 Working principle of STATCOM…………………………………... 27
2.8.2 Dynamic model of STATCOM…………………………………….. 29
2.9 THYRISTOR CONTROLLED SERIES COMPENSATION
(TCSC)………………………………………………………………
29
2.9.1 Structure and operation of TCSC…………………………………… 30
2.9.2 Dynamic model of TCSC with SDC………………………………... 33
2.10 SUPPLEMENTARY DAMPING CONTROLLER (SDC)………… 34
2.11 POWER SYSTEM NETWORK MODELLING…………………… 35
2.12 LOAD MODELLING……………………………………………… 36
2.12.1 Static load modelling……………………………………………….. 38
2.12.2 Dynamic load modelling……………………………………………. 38
2.12.3 Impact of load modelling on power system transient stability……... 39
2.13 CONCLUSIONS……………………………………………………. 40
Chapter 3 Overview of Power System Issues and Solutions………………... 42
3.1 INTRODUCTION TO POWER GRID OPERATION…………….. 42
3.2 POWER SYSTEM ISSUES: CAUSES AND SOLUTIONS………. 43
3.2.1 Power system reliability and security………………………………. 43
3.2.2 Classification of power system transients…………………………... 44
3.2.3 Disturbances/causes of power system problems……………………. 46
3.2.4 Power system stability issues……………………………………….. 47
3.2.5 Solution for power system stability: Need of FACTS devices…....... 51
3.3 TCSC USED FOR POWER SYSTEM PERFORMANCE
ENHANCEMENT…………………………………………………..
52
3.3.1 TCSC implementation history…………………………………........ 53
3.3.2 TCSC placement and its benefits…………………………………… 54
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3.4 REVIEW OF PREDICTIVE CONTROL BASED METHODS…… 55
3.4.1 Voltage control and small-signal stability applications…………….. 56
3.4.2 Predictive control based controllers for transient stability
improvement .……………………………………………………….
57
3.5 CONCLUSION…..…………………………………………………. 59
Chapter 4 Theory and Overview of Predictive Controllers………………. 61
4.1 INTRODUCTION…………………………………………………. 61
4.2 PREDICTIVE CONTROL METHODOLOGY……………………. 62
4.2.1 Model predictive control (MPC)……………………………………. 62
4.2.2 Receding horizon control (RHC)…………………………………… 63
4.2.3 General MPC problem formulation………………………………… 64
4.2.4 Comparison of PID and MPC ……………………………………… 65
4.2.5 Strengths of predictive controlled based methods………………….. 66
4.2.6 Predictive control developments in literature………………………. 67
4.3 MAJOR DRAWBACKS OF PREDICTIVE CONTROL…………. 69
4.3.1 Stability issue……………………………………………………….. 70
4.3.2 Choice of horizon…………………………………………………… 71
4.4 LIMITATIONS OF MPC APPLICATION TO POWER SYSTEM 73
4.4.1 Linear model for nonlinear system issue…………………………… 75
4.4.2 Use of single machine equivalent…………………………………... 76
4.5 CONCLUSIONS…………………………………………………… 77
Chapter 5 MPC-Based TCSC Controller for Power System Transient
Stability Improvement…………………………………………...
78
5.1 INTRODUCTION AND OBJECTIVES………………………….. 78
5.2 AIM OF PROPOSED METHOD…………………………………... 81
5.3 POWER SYSTEM MODELING………………………………… 82
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5.4 RHC ALGORITHM……………………………………………….. 83
5.5 LINEARIZATION AND OBJECTIVE FUNCTION……………… 84
5.6 RHC ALGORITHM FLOW CHART……………………………… 86
5.7 SIMULATION RESULTS…………………………………………. 87
5.8 CONCLUSIONS……………………………………………………. 89
Chapter 6 Overview of Real Time Controllers for Power System Stability
Improvement……………………………………………………...
91
6.1 INTRODUCTION………………………………………………...... 91
6.2 WIDE AREA NETWORK OPERATION………………………….. 92
6.3 ROLE OF WAM IN MAINTAINING WAN OPERATIONS……... 95
6.3.1 Advanced technology used in WAMs……………………………… 97
6.3.2 Role of communication network and various time delays………….. 99
6.4 REVIEW OF POWER SYSTEM CONTROLLERS…………......... 102
6.4.1 Controllers for power system performance enhancement………….. 102
6.4.2 Controllers for transient stability improvement…………………….. 103
6.4.3 Review of real time controllers……………………………………... 105
6.4.4 Communication delay consideration in controller design………….. 107
6.4.5 Computation requirements in controller design…………………….. 111
6.5 CONCLUSION…………………………………………………….. 113
Chapter 7 Online Control Coordination of FACTS Devices for Power
System Transient Stability: Control Method Derivation………..
115
7.1 INTRODUCTION………………………………………………….. 115
7.2 BACKGROUND OF PROPOSED SCHEME……………………… 116
7.3 TRANSIENT STABILITY CONTROL PRINCIPLE……………… 118
7.3.1 Power system model used…………………………………………... 118
7.3.2 Time-domain transient stability simulation………………………… 119
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7.3.3 Nonlinear relationship between maximum relative rotor angles and
control variables……………………………………………………..
120
7.3.4 Transient stability control concept………………………………...... 121
7.4 TRANSIENT STABILITY CONTROL SCHEME………………… 123
7.4.1 Option 1…………………………………………………………….. 123
7.4.2 Option 2…………………………………………………………….. 128
7.5 FORMING TRANSIENT STABILITY INDICES………………… 130
7.6 CONTROL COORDINATION FLOWCHART…………………… 132
7.7 CONCLUSIONS……………………………………………………. 134
Chapter 8 Online Control Coordination of FACTS Devices for Power
System Transient Stability: Computing Time Requirement
Analysis and Case-Study…………………………………………..
135
8.1 INTRODUCTION………………………………………………….. 135
8.2 COMPUTING TIME REQUIRMENTS……………………………. 136
8.3 ANALYSIS OF COMPUTING TIME REQUIRMENT…………… 137
8.3.1 Control coordination structure……………………………………… 137
8.3.2 Computing time…………………………………………………….. 138
8.4 CASE-STUDY SIMULATION RESULTS ………………………. 141
8.5 REPRESENTATIVE POWER SYSTEM………………………….. 142
8.6 SYSTEM RESPONSE WITHOUT ONLINE CONTROL
COORDINATION OF TCSCS……………………………...………
143
8.7 OUTLINE OF TCSCS CONTROL COORDINATION STUDY….. 144
8.8 TIME-DOMAIN SIMULATION COMPUTING TIME
REQUIREMENTS…………………………………………………..
145
8.9 COMPUTING TIME REQUIREMENTS FOR NONLINEAR
FUNCTION SYNTHESES…………………………………………
145
8.10 COMPUTING TIME FOR CONSTRAINED OPTIMIZATION….. 146
8.11 CONTROL COORDINATION STUDY RESULTS……………..... 146
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8.11.1 Control Option 1……………………………………………………. 146
8.11.2 Control Option 2…………………………………………………..... 151
8.11.3 Approximate control...……………………………………………… 152
8.12 CONCLUSION……..……………………………………………… 154
Chapter 9 Dynamic Modelling Application for Estimating Internal States
of a Synchronous Generator in Transient Operating Mode
from External Measurements………………..…………………...
155
9.1 INTRODUCTION…………………………………………………. 155
9.2 BACKGROUND THEORY……………………………………….. 156
9.3 DEVELOPMENT OF ESTIMATION PROCEDURE……………... 159
9.3.1 Continuous-time nonlinear dynamical system and estimation
requirement………………………………………………………….
159
9.3.2 Discrete-time domain system model………………………………... 160
9.3.3 Estimation problem formulation…………………………………..... 161
9.3.4 Solution method…………………………………………………….. 162
9.3.5 Estimation process for subsequent time instants…………………… 165
9.4 APPLICATION TO SYNCHRONOUS GENERATOR………...…. 166
9.4.1 Generator dynamic model………………………………………….. 166
9.4.2 Procedure for generator internal state estimation…………………... 167
9.4.3 Measurement requirements…………………………………………. 168
9.4.4 Discussion…………………………………………………………... 169
9.5 REPRESENTATIVE CASE STUDY……………………………… 171
9.5.1 Results……………………………………………………………… 172
9.5.2 Computing time…………………………………………………….. 173
9.6 CONCLUSIONS……………………………………………………. 174
Chapter 10 Conclusions and Future Work…………………………………… 175
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10.1 CONCLUSIONS…………………………………………………… 175
10.2 FUTURE WORK…………………………………………………… 177
10.2.1 MPC based controller for transient stability using multiple FACTS
devices ………………………………………………………………
177
10.2.2 Online control coordination scheme for multiple FACTS devices…. 178
10.2.3 Control coordination for power system stability improvements in
case of loss of communication signal information/ data…………….
178
10.2.4 Estimation of internal state variables for synchronous generator
with incomplete/inaccurate measurements …………………………
179
10.2.5 Estimation of internal state variables for exciter, turbine and
governor system……………………………………………………..
179
Bibliography …………………………………………………….. 181
Appendices……………………………………………………….. 192
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1
Chapter 1 Introduction
1.1 BACKGROUND AND SCOPE OF THE RESEARCH
The economic growth of any country depends on the growth of advanced modern
technological developments, and advanced technology is impossible without reliable
continuous electric power networks. In terms of reliability and continuity, the cost of
‘power that is not supplied’ can be huge when it comes to the manufacturing/production
industry. In this context, the rate of increasing power demand, development of
technology, need of maintaining system reliability and continuity along with quality
will be the decision factors for future trends in power systems. In developed countries
like Australia, to fulfil this ever increasing demand along with increasing concern about
the environment the focus of the power sector has shifted from the creation of additional
capacity to more effective management and efficient utilisation of existing network
capacity. This is equally applicable to developing countries like India [1, 2].
One of the best and most economical options for maintaining continuity and reliability
of power supply is having ties with neighbouring networks by forming an
interconnected grid, which can allow power exchanging/sharing in peak and slack
periods. This is also known as power pooling where the spinning reserve is reduced or
need of having any spare units for planned or unplanned outages is minimised.
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Such interconnected ties, however, may sometimes be a source of catastrophic
situations because of the characteristics of electrical power networks. As the power
networks connected in parallel, the system fault level keeps increasing. Unlike other
products, electricity cannot be stored and is generated and consumed at the same time.
With unpredictable load changes and system variations, maintaining a power balance,
which is essential for constant frequency operation, becomes difficult. In addition, any
sudden disturbance like a short circuit in any part of the network can affect other
healthy parts in very short duration of time without giving much margin for power
system operators to take any corrective action. Thus, the interconnected grid formed to
maintain continuity can, during cases of disturbances become the cause of cascade
tripping or even a blackout, because of its complexity and power balance problem.
With the above background, it can be seen that the performance of power systems
decreases with increase in the size, the loading and the complexity of the network.
Because of their geographically dispersed networks, power systems require a
functionally complex monitoring and control system to fulfil the immediate and near
future priorities of higher availability and efficiency maintenance. This is possible only
with good information technology based services in energy management systems.
The major motivation of this research is, therefore, to maintain transient stability of an
interconnected grid via the real-time and optimal adjustment of flexible AC
transmission systems (FACTS) devices controller input references in post-fault
conditions. The control coordination law is derived based on real-time requirements, in
terms of time as well as computation capabilities to have a feasible solution.
1.2 OBJECTIVES
As the transient stability is a critical issue in power system reliability, the objective of
this thesis is to develop a control strategy which will introduce necessary compensation
using effective control of FACTS devices input references to maintain system stability
during post-fault conditions. The total thesis work is divided in four major parts. The
first part discusses the detailed modelling of various power system components,
including synchronous machine, exciter, prime-mover, governor, and various FACTS
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devices. After modelling, various power system issues and commonly used methods are
discussed highlighting the need of present research topic of transient stability
improvement using compensating devices.
In the second part, a receding horizon control methodology is developed to control the
TCSC input reference for improving transient stability. The effective control action is
confirmed with MATLAB simulation on a single-machine-infinite-bus system. In the
third part, a real-time controller-coordination based on time-domain simulations and
constrained optimisation is considered to propose optimal adjustment of FACTS
devices controller. The developed control coordination scheme is combined with
practical feasible time constraints to satisfy real-time requirements. The control
algorithm is implemented in two possible options based on how frequently the system
model is updated. Both the control strategies have proven effectiveness in improving
the transient stability of wide-area network validated with a standard New England
system of 10 generators.
The fourth part of the thesis is devoted to the estimation of internal state variables of
synchronous generators using the available measurements, as it is necessary to know the
initial state of the internal state variables which cannot be measured directly. Drawing
on the availability of synchronous generator terminal voltage and current measurements
together with those for rotor speed and/or field voltage, a procedure is derived for
estimating, in transient condition, the generator’s internal operating states, which
include rotor angle and flux linkages associated with field winding and damper
windings. The procedure is based on the fifth-order generator dynamic model formed in
terms of differential and algebraic equations. By applying the numerical integration
formula based on the trapezoidal rule, the generator model is described by a set of
algebraic equations in a recursive form in the discrete time-domain. Combining the
results of external measurements in successive sampling time instants with the set of
recursive equations, leads to a system of nonlinear equations in which the unknown
variables are those representing the internal operating states of the generator. A
minimisation technique based on the sequential quadratic programming method is then
applied to solve the variables in the nonlinear equation system. The estimation
procedure can be applied to any time instants including those in the transient operating
conditions. The effectiveness and accuracy of the procedure developed is tested by
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4
simulation using a representative multimachine power system operating in the transient
mode.
Given the context of the research described above the thesis will have the following
objectives:
(a) Power system modelling for transient stability. The models are to fulfil the
analysis requirements, and at the same time, should be able to give better insight in
internal controller parameters to keep track of the required performance. In this
context, detailed dynamic modelling of synchronous machine, exciter, various
FACTS devices and other power system components is derived and discussed. The
effects of dynamic modelling on transient stability study are discussed in relation to
models considered.
(b) RHC based TCSC controller. A receding horizon based TCSC controller is
developed for power system transient stability improvement for SMIB. The key
feature of the developed controller is that unlike previous literature where classical
models were used for implementation of predictive control methodology, the
detailed power system dynamic model is used for deriving optimal FACTS input
references.
(c) Real-time controller with control coordination schemes for transient stability
enhancement. A real-time control coordination scheme based on time-domain
simulation and constrained optimisation is derived, developed and implemented for
improving multimachine power system transient stability performance using
multiple FACTS devices. To build the most effective control strategy, two different
options are investigated. This is to account for frequent updates in the power system
model and system scenarios due to fault and/or fault clearing occurrences.
(d) Consideration of communication channel delays. Considering the geographically
wide-spread span of wide-area power network, and the necessity for good
information technology, the limitations of communication channel delays are taken
into account while developing control schemes mentioned in (c). Taking the
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5
communication delays into consideration will give a more realistic prediction of
system stability in the future, which is rarely covered in the control designs in
available literature.
(e) Consideration of Computation requirements. Based on the transient stability
control algorithm developed, a comprehensive analysis of the computing time
requirement in implementing the above control coordination algorithm is
performed. The analysis outcome is a set of feasible constraints which are to be
satisfied by computer systems used for online control coordination of FACTS
devices with the aim of enhancing or maintaining power system transient stability
following disturbances. With reference to a representative multimachine power
system having TCSCs, it is demonstrated that even with the available current
technology, the control coordination algorithm can be implemented in real time,
using a cluster of processors operating in parallel. Offline simulation, using realistic
timing parameters for the control scheme, confirms its effectiveness in maintaining
transient stability following a fault disturbance.
(f) Development of internal state estimation algorithm. Drawing on the availability
of synchronous generator terminal voltage and current measurements together with
those for rotor speed and/or field voltage, a procedure is derived for estimating in
transient conditions the generator internal operating states which include rotor angle
and flux linkages associated with field winding and damper windings. The
procedure is based on the fifth-order generator dynamic model formed in terms of
differential and algebraic equations. A minimisation technique based on a
sequential quadratic programming method is then applied for solving the variables
in the nonlinear equation system. The estimation procedure proposed can be applied
to any time instants including those in the transient operating conditions.
1.3 OUTLINE OF THE THESIS
Chapter 1 of the thesis gives the background and scope of research, its objectives and
contribution of the thesis.
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6
Chapter 2 is devoted to dynamic modelling of various power system components,
including the synchronous generator which is the key element to maintain synchronism.
Considering the major drawback of using classical model or third order model in
stability studies, special consideration is given to detailed modelling of synchronous
generator using fifth-order model along with detailed modelling of excitation, prime-
mover and governor system. The single machine modelling is extended to multimachine
dynamic modelling and their connection to the external grid is explained taking into
consideration the individual machine reference and system reference. The chapter
further presents modelling of various shunt and series FACTS devices, load modelling
and supplementary damping controllers. It also gives importance to, and the effects of,
modelling on transient stability study results.
Chapter 3 explains power grid formation, its problems, causes and issues associated in
maintaining reliable, continuity and quality of supply. For improvement in power
system performance, the need for FACTS devices and their applications for improving
stability are discussed and the effectiveness of TCSC is presented in this chapter.
Considering the need of predicting system scenarios in advance, based on a look-ahead
approach, a detailed review of the controllers based on prediction control methodology
is made which gives the foundation for further development of RHC based controller in
Chapter 5.
Chapter 4 gives an overview of receding horizon control methodology, as developed in
the field of control systems and its limitations. It also highlights the issues of concern
for application of predictive control strategies to power system issues. To get familiar
with the development of prediction control methodology, a brief review is presented to
explain the principle of RHC/MPC, its applications in other fields like control system or
chemical industry where it was well developed and well-implemented. Knowing the
practical limitations of MPC applications, a detailed review is presented of MPC
applications to power system specific problems focusing on stability related issues.
Chapter 5 develops a RHC based prediction scheme for investigating the required
TCSC reference inputs for enhancing power system transient stability. The power
system model considered is a single-machine-infinite-bus (SMIB) system, represented
by a detailed dynamic model of synchronous generator including a detailed dynamic
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7
model of an exciter and governor systems. The TCSC is modelled in variable reactance
form and based on the output of a MPC controller. Input is given to TCSC which will
insert the necessary series compensation in post-fault conditions. The method is
validated with case study results of the SMIB system taking into consideration a 3-
phase short circuit where the fault is cleared after critical clearing time. The case study
presented shows that without a controller the system will lose synchronism because of
high acceleration, while with the controller it is proved to regain stable operating
conditions in a short time.
Chapter 6 gives a detailed literature review of operations, maintenance and technologies
used in wide-area network and the role of wide-area measurement systems. It also
highlights the importance of good communication networks for fast and huge
information exchange, giving a brief account of various time delays in wide-area
network and its effect on stability studies. The chapter explains about the structure of
wide-area network and their limitations in terms of communication due to their wide-
spread nature in stability problems. A detailed literature review of methods and schemes
addressing the problems of communication and computation delays for real-time
applications is presented.
A new method is developed in Chapter 7 for real-time transient stability control in a
power system with FACTS devices. Central to the method is the control, in successive
time periods, of a synchronous generator is relative rotor angles to satisfy the nominated
transient stability criterion via the real-time and optimal adjustment of FACTS devices
controllers input references. In each period, the dependencies of maximum relative rotor
angles on input references are expressed as nonlinear functions which are synthesised
from the results of time-domain transient stability simulations, using the prevailing
power system models and conditions. The constrained optimisation problem is then
formed from the synthesised functions, and solved for the optimal input references.
Practical issues related to computing time and communication channel time delays are
considered in the control methodology.
In the context of the online transient stability control method developed in chapter 7,
based on time-domain simulations and constrained optimisation for real-time and
optimal adjustment of FACTS devices controllers’ input references, Chapter 8 has a
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8
focus on the analysis of computing time requirements in implementing the method, and
presents results of study cases. The computing time components of the individual steps
for executing the control algorithm are identified, determined, and then combined to
form overall feasibility constraints to then be satisfied by computer systems used in
control method implementation. Computational tasks which are independent of one
another are identified so that they can be performed using parallel computing systems.
The effectiveness of the control coordination in maintaining transient stability is
verified by a simulation study, with the control scheme having realistic timing
parameters applied to a representative multimachine power system.
Chapter 9 explains and derives new methodologies for internal state variable estimation.
Drawing on the availability of synchronous generator terminal voltage and currents,
rotor speed and field winding voltage measurements, a procedure is derived for
estimating in transient condition the generators internal operating states which are the
rotor angle and flux linkages associated with field winding and damper windings. The
procedure is based on the fifth-order generator dynamic model. By applying the
numerical integration formula based on the trapezoidal rule, the generator model is
described by a set of algebraic equations of a recursive form in the discrete time-
domain. With external measurements, the unknown variables in the equations are those
representing the generator’s internal operating states. The nonlinear equations derived
for successive time instants are solved by applying the Newton-Raphson method.
However, if the number of equations is greater than that of variables, a minimisation
technique based on sequential quadratic programming method is then applied to solve
the nonlinear equation system. The estimation procedure can be applied to any time
instants including those in the transient operating conditions, without the requirement
for specifying the steady-state condition. The effectiveness and accuracy of the
procedure developed are verified by simulation using a representative multimachine
power system operating in the transient mode.
Chapter 10 gives the overall conclusion for the thesis and the future scope of the
ongoing research in relation to further development of the schemes proposed in
Chapters 5, 7, 8 and 9.
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9
Appendices are provided for detailed derivations of major formulation which are used in
the modelling of power system components, developing MPC algorithms and estimation
problem formulation. They also give the complete input data files for the power system
case-studies used in the thesis for validation of results.
1.4 CONTRIBUTIONS OF THE THESIS
The thesis has made five original contributions described below:
(a) Development of an RHC-based TCSC controller for power system transient
stability enhancement considering a detailed dynamic model of a power system.
(b) Development of robust online controller coordination schemes for improvement of
power system transient stability in multi-machine, multiple FACTS devices using
two different strategies.
(c) This controller is further developed to take into consideration the size of power
system networks spread over wide geographical areas causing various time-delays
in WAN communication channels.
(d) Special consideration is given to the computation time and computation burden
forced on computing system requirements in developing this real-time control
coordination scheme for multi-machine, multi-TCSC hybrid network considering
detailed nonlinear dynamic models.
(e) New method is derived to find out values of internal state variables of a
synchronous generator based on practical feasible measurements for real time
application.
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10
The thesis is supported by two published international conference papers and two
journal papers submitted which are under review at present. The list of publications is as
follows:
1) Nguyen, T.T., and Wagh, S.R., “Model Predictive Control of FACTS Devices
for Power System Transient Stability,” Proceedings of the IEEE Transmission
and Distribution Conference, Seoul, Korea, October, 2009.
2) Nguyen, T.T., and Wagh, S.R., “Predictive Control-Based FACTS Devices for
Power System Transient Stability Improvement,” Proceedings of the 8th IET
International Conference on Advances in Power System Control, Operation and
Management, APSCOM 2009, Hong Kong, November, 2009.
3) Nguyen, T.T., and S. R. Wagh, “Application of Dynamic Modelling for
Estimating Internal States of a Synchronous Generator in Transient Operating
Mode from External Measurements”, submitted to IEEE Trans. Power Systems,
2011 (under review).
4) Nguyen, T.T., and S. R. Wagh, “Online Control Coordination of TCSCs for
Power System Transient Stability”, submitted to IET Generation, Transmission
and Distribution, December 2011 (under review).
For reference, copies of the above four publications are given in Appendix H.
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11
Chapter 2 Dynamic Modelling of Power System
Components
2.1 INTRODUCTION
The first step for power system stability study is dynamic modelling of various power
system components. The power system elements include synchronous generators,
transformers, transmission lines, various compensating devices such as that of FACTS
devices which are modelled in forming the complete network model for stability study.
For steady-state power-flow analysis, power system can be well represented using nodal
variables including node voltages and node currents. For dynamic stability studies of
multimachine power system network, first dynamic models of individual items should
be formed and then their interaction with external power system can be considered.
With reference to the transient stability study of power systems after a large disturbance,
the system dynamic performance is decided by rotating machines dynamics and power
system controller including FACTS devices responses which are part of interconnected
network. Hence, the entire power system dynamics can be divided in two major parts,
first, the individual synchronous machine which is working on its own individual
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12
machine axis reference, and second, its connection to the external power system
network which is working on a common system reference axis. To understand the
dynamic behaviour of power systems, first, it is necessary to focus on the detailed
dynamic model of synchronous machines which consist of synchronous machine along
with its exciter and prime mover. In the present chapter the dynamic modelling of
synchronous machine is discussed along with exciter and prime mover which are further
extended and developed for multi-machine system. The latter part of the chapter will
focus on the dynamic modelling of various FACTS devices and loads. The chapter also
discusses the effects of choosing appropriate dynamic models on the power system
performance parameters.
2.2 SYNCHRONOUS GENERATOR DYNAMIC MODEL
Due to the various limitations with classical and reduced order models, in this thesis, the
synchronous machine is represented by the fifth-order model having d-q axis as the
rotor reference frame. To develop basic dynamic equations for balanced , symmetrical,
three-phase synchronous machine, a generator with one field winding on direct axis and
two damper windings, one on d-axis and one on q-axis, is considered [3].
In the two-axis theory of synchronous machines, the three phases of the armature
winding are replaced by fictitious direct-axis and quadrature-axis. Hence the various
self and mutual inductances between stator and rotor circuits can be summarized as
given in [4], with d-q axis terminology in terms of shaft angle θ. From the expression
(A1) of voltage as given in Appendix A, the equations of rotor flux and stator voltages
can be derived in terms of d-q axis terminology which is useful in deriving rotor
dynamic model in the next sections.
2.2.1 Rotor flux dynamics (Generator electrical axis)
As shown in the Appendix A, the advantage can often be taken, in practical stability
studies based on the synchronously rotating reference frame, of reducing the computing
time expanded in analysis by eliminating the terms of stator-voltage transients
corresponding to the rate of change of stator flux linkages with respect to time. These
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13
terms contribute very little dynamic response analyses that are dominated by the inertial
characteristics of rotating machines and they may be readily eliminated. In this research,
the synchronous machine is represented by the fifth-order model of the d-q axis having
a rotor frame of reference. Having one main field winding and two damper windings,
the rotor flux equation can be given by[4, 5] :
rsmrmr viFAp ++= ψψ (2. 1)
which can be expanded for three rotor fluxes separately as:
fdmqmdmkqmkdmfdmfd EKiFiFAAA 111211131211 +++++=•
ψψψψ (2. 2)
qmdmkqmkdmfdmkd iFiFAAA 2221232221 ++++=•
ψψψψ (2. 3)
qmdmkqmkdmfdmkq iFiFAAA 3231333231 ++++=•
ψψψψ (2. 4)
where constants Am and Fm are dependent on machine parameters as derived in
Appendix A. Flux linkages fdψ , kdψ and kqψ constitute the rotor flux linkages vector
while di and qi are stator current components along d- and q-axis respectively.
It should be noted that, physically, exciter output voltage/current and generator field
voltage/current are the same, but distinction is made only in their per unit values to
allow independent selection of the per unit systems for modelling excitation systems
and synchronous machines. Hence, the constant Km11 is used for proper interfacing
between excitation system and synchronous machine field circuits when modelled in per
unit system and is calculated based on machine parameters as explained in [6] .
2.2.2 Equation of motion (Turbine/Generator mechanical axis)
The equation of motion which is also known as the swing equation is expressed as
shown in (2.5) and (2.6). It gives the relationship between the rotor angle and rotor
angular frequency and is represented in rotor mechanical axis.
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14
refr ωωδ −=•
(2. 5)
eM TTM −=•ω (2. 6)
where, rδ is rotor angle, ωref is synchronous speed while ω is rotor angular frequency.
M is calculated as 2H/ωref, where H is the machine inertia constant. TM is the
mechanical torque input to the synchronous generator, while Te is the electromagnetic
torque output of the generator. The electrical torque Te can be expressed in terms of
generator currents as given in (2.7)
BBIIAAIT tss
tse refref ωω += (2.7)
where AA and BB are matrices dependent on machine parameters and are derived in
Appendix A (as A31).
2.2.3 Relation between generator current and voltage
Having discarded stator flux transients which are very fast compared to other dynamics,
the stator voltage vector can be represented in terms of stator current by an algebraic
equation as shown in (2.8) which is derived in Appendix A.
smrms izpv −= ψ (2.8)
where, Pm and Zm matrices are dependent on machine parameters and rotor angular
frequency and can be calculated as shown in Appendix A (given by A13).
2.2.4 Excitation and automatic voltage regulator (AVR)
In transient stability study it is customary to consider excitation circuit model in detail
to obtain an accurate system response. While many excitation models for stability
studies are specified in [7], they generally consist of five major blocks including the
main exciter; regulator; terminal voltage transducer and load compensator; power
system stabiliser; limiters and protective circuits as mentioned in [6] and shown in Fig.
B1 in Appendix B.
In this research, the IEEE Type I excitation model is considered and its schematic block
diagram is shown in Fig. B2. As derived in Appendix B, the exciter model can be given
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15
by a set of equations representing the entire excitation system including an automatic
voltage regulator with its limits. However, the set of equations can be expressed in a
general form as:
)9.2(ExcExcExcExcExc VBXAX +=•
where, XExc is state vector of the excitation system, VExc is the synchronous machine
terminal voltage, VPSS is the supplementary signal from PSS and Vref is the voltage
reference, while AExc and BExc are the constant matrices dependent on the gains and
time constants of the controller. It should be also be noted that (2.9) provides a link
between the exciter controller and the synchronous generator terminal output voltage
through matrix VExc as shown in the detailed model in Appendix B.
2.2.5 Prime-mover and governor system
It is well known that the frequency of the ac voltage at the terminals of a synchronous
generator is determined by its shaft speed and the number of magnetic poles of the
machine. The steady-state speed of a synchronous machine is determined by the speed
of the prime mover that drives the shaft. The prime-movers can be of steam type, gas
turbine or hydro turbines. In [8, 9] various dynamic models for steam and hydro
turbines are explained. A nonreheat type of steam turbine chosen as the prime mover.
Fig. B3 shows the block diagram representation of turbine and governor combined
model. In this present study, steam chest dynamics and the effects of steam valve
positions (PSV) on the synchronous machine torque (TM) are modelled as shown in [10].
Equations (B7)-(B8) give a complete model of turbine and governor systems which can
be represented in compact notations as shown in (2.10) where XGov is the state vector of
the prime-mover controller and AGov, BGov, CGov, and DGov are matrices dependent on
gain and time constants of the controller.
CGovGovGovGovGovGov PDCBXAX +++=•
ωω ref (2.10)
It can be seen from the detailed derivation in Appendix B, that (2.10) provides a link of
interconnection of the prime-mover output to the synchronous generator power.
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16
2.2.6 Power system stabiliser (PSS)
In addition to the above discussed elements, the power system stabiliser (PSS) is
another common stabiliser used to introduce modulating signals through the excitation
system to contribute to rotor oscillation damping.
A block diagram for PSS is shown in Fig. B4, which consists of a gain block, a
washout, a lead-lag block and a limiter. A washout is necessary to guarantee that PSS
responds only for the disturbances and not for any steady-state condition when speed or
power is used as input. The output of PSS is added to the exciter error signal and can be
used as a supplementary signal.
The state equations derived from the given block diagram are explained in detail in
Appendix B and can be summarised in compact form as :
XBXAX PSSPSSPSSPSS += (2.11)
where XPSS is the vector of state variables of PSS as described in Appendix B and, APSS,
BPSS are matrices elements depending on the gain and time constant of the PSS
controller while X is the speed variation.
2.3 MULTIMACHINE DYNAMIC MODELLING WITH MACHINE
REFERENCE
Having modelled the main components of synchronous machines, the overall generator
dynamics for single machine systems can be summarized as a set of differential
equations collected and represented together as follows:
rsmrmr viFA ++=•
ψψ (2.1)
refr ωωδ −=•
(2.5)
eM TTM −=•ω (2.6)
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17
ExcExcExcExcExc VBXAX +=•
(2.9)
CGovGovGovGovGovGov PDCBXAX +++=•
ωω ref (2.10)
For interconnected networks with a number of synchronous machines, a generalised
equivalent machine model can be formed by extending the same modelling concept as
explained in Section 2.2. Using the same terminology for a multimachine system having
N number of machines, the rotor flux dynamics on its own generator electrical axis will
be given by:
rNsNmNrNNmrN viFAp ++= ψψ (2.12)
which can be expanded on the same line as that of a single machine for extracting
individual rotor fluxes for field winding, as well as both damper windings as:
fdNNmqNNmdNNm
kqNNmkdNNmfdNNmfdN
EKiFiF
AAA
111211
131211
++
+++=•
ψψψψ (2.13)
qNNmdNNm
kqNNmkdNNmfdNNmkdN
iFiF
AAA
2221
232221
+
+++=•
ψψψψ
(2.14)
qNNmdNNm
kqNNmkdNNmfdNNmkqN
iFiF
AAA
3231
333231
+
+++=•
ψψψψ (2.15)
where constants AmN and FmN are dependent on machine parameters as explained in
Appendix B. Flux linkages 𝜓𝑓𝑑𝑁, 𝜓𝑘𝑑𝑁 and 𝜓𝑘𝑞𝑁 constitute the rotor flux linkages
vector, while idN and iqN are stator current components along d- and q-axes respectively.
The equation of motion based on the turbine/generator mechanical axis for all given N
synchronous generators can be generalised as:
refNNrN ωωδ −=•
(2.16)
eNMNNN TTM −=•
ω (2.17)
where rNδ is the vector of the rotor angle of all individual machines, ωNref is the
synchronous speed and ωN is the vector of the rotor angular frequency. MN is the
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18
machine inertia constant calculated from the per unit inertia constant H, as 2HN/ωNref,
TMN is the mechanical torque input to the synchronous generator, while TeN is the
electromagnetic torque output of the generator. The mechanical torque in (2.17) will be
as an input to the synchronous generator which is available as an output of turbine and
governor system, while the output expressed as power or electrical torque TeN can be
expressed as:
NtsNsNN
tNseN BBIIAAIT refNrefN ωω += (2.18)
For a transient stability study of multimachine system, the exciter and automatic voltage
regulators can be generally represented in compact notations as:
ExcNExcNExcNExcNExcN VBXAX +=•
(2.19)
The XExc is the vector of excitation system variables and the constants AExc and BExc are
dependent of given excitation system model chosen. The details of the notations used in
(2.19) are as given in the list of symbols.
Similar to the multimachine model formation of synchronous machine or exciter, even
the prime-mover and governor can be generalised for multimachine system. Equation
(2.20) gives a complete model of turbine and governor system which can be represented
in compact notations for multi-machine system having N number of machines, where
XGovN is state vector of prime-mover controller and AGovN, BGovN, CGovN, and DGovN are
matrices dependent on gain and time constants of the controller.
CNGovNNGovNGovNGovNGovNGovN PDCBXAX +++=•
ωω refN (2.20)
The overall generator dynamic model for the transient stability study of multimachine
can be summarized by the following sets of differential equations
rNsNmNrNmNrN viFA ++=•
ψψ (2.12)
refNNrN ωωδ −=•
(2.16)
eNMNNN TTM −=•
ω (2.17)
ExcNExcNExcNExcNExcN VBXAX +=•
(2.19)
CNGovNNGovNGovNGovNGovNGovN PDCBXAX +++=•
ωω refN (2.20)
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19
2.4 EFFECT OF MACHINE MODEL USED ON STABILITY STUDY
Synchronous machines may be modelled in as much detail as possible in the study of
most categories of power system stability. This includes appropriate representation
(subject to the available data) of the dynamics of the field circuit, excitation system, and
rotor damper circuits. With today’s computing tools, it is no longer a necessity to use
the simplified model. Experience has shown that critical problems may be masked by
the use of simplified models often perceived to be acceptable for a particular type of
study.
It is particularly important to represent the dynamics of the field circuit, as it has
significant influence on the effectiveness of excitation system in enhancing large-
disturbance rotor-angle stability. In current literature, generally a classical model for
study is used with only a rotor swing equation. Assuming that the variation in rotor
flux, exciter parameters and governor action is very small the other machine dynamic
equations are ignored and the computational burden and complexity in handling
dynamic equations are reduced. Assumptions for considering use of a classical model
are justified as follows:
(i) As magnetic energy is dissipated in the field winding resistance, flux decrement
effects will cause the rotor emf to decrease with time. If the fault clearing time is very
short then this flux decrement effect can be neglected for transient stability
considerations [11].
(ii) Similarly, the governor being a mechanical system has a large time constant so
compared to the fast action of transients, the effect of variation in governor parameters
is very small on transient performance.
However, if the fault-clearing time is long enough then the decay of this rotor emf will
have a considerable effect [11]. The result of considering flux decrement effects is in
reducing deceleration for given acceleration which will deteriorate the transient
stability. Similarly, most modern generators are equipped with an automatic voltage
regulator (AVR) which increases the synchronising torque component at steady state
[6]. AVR action may reduce the damping of rotor swing following a small disturbance
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20
or large disturbance. When fault occur the generator terminal voltage drops and the
large regulation error forces AVR to increase the generator field current. The effect of
AVR is to increase the field current leading to an increase in the transient emf so a
strong action of AVR may prevent a loss of synchronism after a large disturbance.
Although the fast acting AVR reduces the first rotor swing, it can increase the second
and following swings depending on the system parameters, the dynamic properties of
AVR and the time constant of the field winding. Consequently the use of a classical
model may lead to an optimistic assessment of the critical clearing time although it
reduces complexity and computational burden.
Considering all of the above problems a compromised approach is accepted in most of
the previous literature [12-14] or the sake of simplicity and classical modeling
consisting of rotor swing equations is used for transient stability studies. However, these
devices maintain the terminal voltage of the generator at a specified value and, in the
process, modulate the field voltage, and hence, the field current, thus supplying the
required reactive power to the load.
2.5 MULTI-MACHINE MODELLING WITH SYSTEM REFERENCE
The first part of this chapter has discussed the synchronous machine modelling along
with exciter and governor. However, this modelling was based on its own individual
machine axis and for the study of multimachine networks special care should be taken
to convert all parameters to form a dynamic model on one common base. This section
will describe in detail the connection and interaction of individual machines with the
external system network.
2.5.1 Connection of individual generator to power grid
In general, the dynamics of the post-fault power system after any disturbance can be
described by a set of nonlinear differential-algebraic equations. Typically, the states can
be associated with each machine’s armature and rotor currents, rotor dynamics, AVR,
and turbine-governor dynamics [6, 10, 15]. The overall dynamics can be best
summarized in terms of Fig. 2.1 in the form of block a diagram adopted from [6] and
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21
modified further to explain the structure of differential-algebraic equation model
formation for individual generators (having its own d-q reference frame) connected to
the external power system network (having a D-Q/Re-Im reference frame of
interconnected grid).
Transmission network equations
Including static loads (AE)
Excitation system (Field Excitation) Efd
Prime mover and governor system (mechanical power input) PM, ω
Generator rotor circuit Rotor flux and emf equations
Main field winding circuitd-axis damper winding circuits q-axis damper winding circuits
(DE)
Acceleration or swing equation Rotor angle
Speed(DE)
Machine rotor base reference frame: d-q
Other generators
Motors
Other dynamic
devices e.g. SVC
Generator voltages Reactive power generated
Generator frequency Active power generated
Stator circuit equations Machine stator base: D-Q
(AE)
Stator
Rotor
Generator connected to outside power system network common system base : Re-Im
Individual machine
Generator outputP, Q, |V|, δ
Fig.2.1 Individual synchronous generator connected to external network [6]
It can be seen from Fig. 2.1 that the transient stability study of power system networks
can be divided into two major parts, the individual synchronous machine and its
connection to the external power system network. While developing the mathematical
models of such an interconnected network, the first stage is the selection of the frame of
reference for the electrical quantities. The equations for each machine are expressed
with reference to pairs of individual axis (d-q) which rotate in synchronism with the
rotors of the machines; while the algebraic power flow equations of the power system
network are with system reference axes (D-Q/Re-Im).
Figure 2.2 shows the reference of frame for two individual machines as d1-q1, d2-q2. It
can be seen that machine 1 reference axes d1-q1 is displaced by δ1 with respect to D-Q
axis while for machine 2 this displacement angle is δ2. In steady state all these axes will
rotate at the same speed, but in transient conditions the angles will vary as the machine
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22
speed varies. Therefore, it is necessary to obtain a relationship between the deviations of
the current and voltage variables of individual machines from their steady-state
equilibrium values after disturbance.
D
Q
d1
d2q1
q2
δ1
δ2
Fig. 2.2 Angular relationships between the external power system network and the
individual machine reference axes
2.5.2 Transferring quantities from machine reference to system reference axis
With reference to the representative voltage vector shown in Fig. 2.3, (2.21) and (2.22)
can be used to transfer quantities from system reference to individual machine reference
while, (2.23) and (2.24) can be used for transforming from individual machine axes to
the main reference. The detailed derivation of these standard transformations can be
found in literature [16] .
δsinVδcosVV QDd += (2.21)
sinδVcosδVV DQq −= (2.22)
sinδVcosδVV qdD −= (2.23)
cosδVsinδVV qdQ += (2.24)
______________________________________________________________________
23
D
Q
d1
Vs
q1
δ
θ
VdVq
VQ
VD
θ- δ
Fig.2.3 Transforming individual machine quantities to system frame of reference for
voltage
In general, while developing a power system model for multimachine system, each
machine quantity will be transformed with their respective dq-DQ displacement using
given transformation.
2.6 INTRODUCTION TO FACTS DEVICES
Having modelled the basic synchronous generator, exciter and prime-mover systems,
the next part of this chapter will focus on dynamic modelling of FACTS devices and
main power system components which plays important role in maintaining power
system performance in normal as well as abnormal conditions.
With the recent developments in power electronics and especially thyristor sizing and
switching technology, the FACTS family has many devices developed for power system
performance improvement depending on targeted issues. FACTS devices consist of
power electronics components and conventional equipment which can be combined in
different configurations. It is therefore relatively easy to develop new devices to meet
extended system requirements. The FACTS family members include static VAr
______________________________________________________________________
24
compensators (SVC), static synchronous compensator (STATCOM), unified power
flow controller (UPFC), and thyristor controlled series compensation (TCSC) which
play different roles in steady-state as well dynamic stability improvement of
interconnected power systems. The SVC, STATCOM have prove their effectiveness in
voltage stability applications while series compensations like TCSC are always useful in
improving power transfer under normal steady state operation as well as after major
disturbances. The SVCs and STATCOM provide fast voltage control, reactive power
control and power oscillation damping features. As reported in an article on power
transmission and distribution published by Siemens, Germany, as an option, SVCs can
control unbalanced system voltages while fixed series compensation is widely used to
improve the stability and transmission capacity in long distance transmissions. In [17-
19] benefits of various FACTS devices in power system performance enhancement are
discussed. This chapter will present a brief dynamic modelling of the most commonly
used FACTS devices followed by dynamic modelling of other power system
components including supplementary damping controller (SDC) and load modelling
which are necessary to consider for system operation. The chapter also highlights on the
effect of modelling these components on performance calculation showing the
importance of using the correct models for study of transient stability.
2.7 STATIC VAR COMPENSATOR (SVC)
Static var compensators (SVCs) are shunt-connected static generators and/or absorbers
whose outputs are varied to control specific parameters of the electric power system.
The term static var system (SVS) is an aggregation of SVCs and mechanically switched
capacitors (MSCs) or reactors (MSRs) whose outputs are coordinated. There are many
basic reactive power control elements as listed in [6] which form all or part of any static
var system, and can be combined to form different types of SVS configurations. In
general, the main components and structure of a typical SVC can be represented as
shown in Fig. 2.4.
Almost always the SVC is connected to the transmission network via a coupling step-up
transformer. At the low-voltage side node of the transformer there are, in general, three
types of elements employed: thyristor controlled reactor (TCR), thyristor switched
capacitors (TSC) and fixed harmonic filters [20].
______________________________________________________________________
25
(i) Thyristor controlled reactor (TCR): A schematic diagram of the single-phase
TCR consisting of the antiparallel thyristor pair and the linear reactor is shown in
typical SVC structure in Fig. 2.4. The controlled switching of the thyristors combined
with the linear reactor response enables the effective supply-frequency reactance of the
TCR, which is a function of the thyristor firing angle. For continuous variation of
reactance the firing angle can be varied smoothly from thyristor fully conducting to
fully non-conducting range (blocking).
High-voltage node
Low-voltage node
Coupling transformer
Filters
TCRTSC
↑QSVC
Fig. 2.4 Typical SVC structure with main components [19]
(ii) Thyristor switched capacitors (TSCs): Thyristors in this model have the function
of switching capacitors on or off as required so as to combine with the TCR to provide a
continuous range covering both inductive and capacitive compensation.
(iii) Fixed harmonic filters: The filters provide low-impedance paths for harmonic
currents generated from the TCR operation. Moreover, the filters also provide
capacitive compensation at the fundamental frequency.
______________________________________________________________________
26
In general, by changing the thyristor firing delay angle, the effective reactance of the
TCR varies which in turn changes the effective reactance of the SVC. By doing so, the
SVC can supply or consume reactive-power from a transmission system. From an
operational point of view, the SVC can be considered as a shunt-connected variable
reactance, which either generates or absorbs reactive-power in order to regulate the
voltage magnitude at the point of connection to the transmission network.
The steady-state control objective of the SVC is that of voltage control function which
is expressed in terms of a V-I characteristic decided by the operating limits of TCR and
TSC under steady-state conditions. However, for a transient stability study, the dynamic
model of SVC is represented as shown in Fig. 2.5 [21, 22] and is simplified as derived
in Appendix C to represent in compact form as:
refSVCtSVCSDCSVCSVCSVCSVC VDVCXBXAX ⋅+⋅+⋅+⋅= (2.25)
In (2.25) SVCX , is a vector of SVC state variables, while ASVC, XSVC, BSVC, CSVC, and
DSVC are matrices whose elements are dependent on the gains and time constants of the
controllers. |Vt| is the magnitude of terminal voltage and Vref is the reference voltage
setting.
Ksvc
Bsvc(min)
svc
svcsTsT
2
111++
svcsT+11 X1svc
Bsvc(max)
Bsvc∑
XSDC
Vref
|Vt|
-
-+
Fig. 2.5 SVC dynamic model block diagram
2.8 STATIC SYNCHRONOUS COMPENSATOR (STATCOM)
The STATCOM performs the same function as that of SVCs. However at voltages
lower than the normal voltage regulation range, the STATCOM can generate more
reactive power than the SVC. This is due to the fact that the maximum capacitive power
______________________________________________________________________
27
generated by a SVC is proportional to the square of the system voltage (constant
susceptance) while the maximum capacitive power generated by a STATCOM
decreases linearly with voltage (constant current). This ability to provide more
capacitive reactive power during a fault is one important advantage of the STATCOM
over the SVC. In addition, the STATCOM will normally exhibit a faster response than
the SVC because with the VSC, the STATCOM has no delay associated with the
thyristor firing (in the order of 4 ms for a SVC) [17].
2.8.1 Working principle of STATCOM
The basic principle of the STATCOM is to use a voltage sourced converter (VSC)
technology based on gate turn-off (GTO) thyristor or insulated gate bi-polar transistor
(IGBT)) that have the capability to interrupt current flow in response to a gating
command. This allows the STATCOM to generate an AC voltage source at the
converter terminal at the desired fundamental frequency with controllable magnitude. A
block diagram of a STATCOM is given in Fig. 2.6 [23].
ø
αjTT eVV =
CqCpC jIII +=
φjdcqCpCC eVkjVVV =+=
XC
Vdc
Fig.2.6 STATCOM schematic diagram
The voltage difference across the coupling transformer reactance produces active-power
and reactive-power exchanges between the network and the STATCOM. As shown in
Fig. 2.7, the phase reference for STATCOM voltage VC is the terminal voltage VT which
in phase with p-axis will decide the direction of active power flow while the exchange
______________________________________________________________________
28
of reactive-power with the network is obtained by controlling the magnitude of the
voltage source. The active-power exchange is only used to control the DC voltage of the
capacitor. In steady-state conditions where the capacitor voltage is constant, the active-
power exchange is, therefore, zero if the VSC losses are discounted.
VCq
D
p
VC
VT
VCpøα
Fig. 2.7 Phasor diagram of STATCOM operating principle
There are several VSC structures currently used in actual power system operation. Fig.
2.8 shows the basic structure of a three-phase, full-wave converter having six switches
with each consisting of a GTO thyristor connected antiparallel with a diode. With the
aim of producing an output voltage waveform as near to a sinusoidal waveform as
possible, the switching of individual GTO thyristors in the VSC are controlled by the
switching control module designed to minimise the harmonics generated in the VSC
operation and requirement for harmonic filters.
+
-
Va
Vb
Vc
Output
Fig.2.8. Basic voltage sourced converter structure using GTO thyristors
______________________________________________________________________
29
2.8.2 Dynamic model of STATCOM
The dc voltage represented in Fig. 2.6 can be combined with the transfer functions
shown in Fig. 2.9 to form a complete dynamic model for a STATCOM main controller
as shown in (2.26) - (2.30).
φstadc VAV = (2.26)
SDCstaCqstaTstaTrefsta XEIDVCVBV +++= (2.27)
φstaSDCstaTstaTrefstadcstastaC VLXKVJVHVGVFX +++++= (2.28)
φNXMφ staCsta += (2.29)
where, sinφVV Tφ = (2.30)
and Asta, Bsta, Csta, Dsta, Esta, Fsta, Gsta, Hsta, Jsta, Ksta, Lsta, Msta, and Nsta are matrices
elements dependent on STATCOM and its controller parameters are as derived in
Appendix C.
∑+VTref
-+
XSDC
|VT|
-
droop ICq
sK sta1 ÷
Ksta
|V|+
Vdc
stasT+11XC ø
max
min
∑( )
sta
stastasT
TK
2
22 1++
Fig.2.9 Dynamic model of STATCOM
2.9 THYRISTOR CONTROLLED SERIES COMPENSATION (TCSC)
The series counterpart of a shunt-connected SVC is a TCSC, which is connected in
series with a transmission line to provide improved stability of interconnected power
systems, increasing power transfer and directing power flows in desired transmission
______________________________________________________________________
30
paths [24, 25]. A typical TCSC module in one phase is given in Fig. 2.10 [24]
consisting of a series capacitor in parallel with a TCR.
XFC
XP
Xc
XL Xtcsc
Fig.2.10 Simplified operating circuit of thyristor controlled series compensation
The operating constraints of a TCSC are associated with its reactance and can be
expressed in terms of the following inequalities:
tcscmaxtcsctcscmin XXX ≤≤ (2.31)
In (2.31), Xtcscmin and Xtcscmax are the minimum (capacitive) and maximum (inductive)
limits of TCSC reactance. They are dynamic limits which depend on the transmission
line current and can be determined using the capability characteristics. If the TCSC
reactance hits a limit, the specified control function will no longer be applicable, and the
TCSC will behave like a fixed reactance corresponding to the respective limit value.
2.9.1 Structure and operation of TCSC
Based on the above simplified operation circuit, it can be seen that a TCSC module can
operate in two extreme modes, i.e. thyristor path totally blocked, using it like a
conventional capacitor XC or continuously gated where it appears as a small inductance
with net reactance of Xbypass. In other words as mentioned in [26, 27], TCSC is often
treated as a variable reactance of a transmission line.
However, in practice, the practical layout of TCSC structure is as shown in Fig. 2.11
[28] in which an indispensable component is a highly nonlinear metal oxidized varistor
(MOV) which operates in both steady state and transient process of the power system.
______________________________________________________________________
31
breaker
Varistor
TCSC Reactor
breaker
Capacitor bank
Fig.2.11 Typical layout of practical TCSC structure
During transient operation, the MOV bypasses excessive currents and limits the voltage
across the capacitor banks. This produces distortion of the TCSC voltage waveform and
consequently, sharply changes the fundamental frequency reactance of the TCSC.
Effective design and accurate evaluation of the TCSC control strategy depend on the
simulation accuracy of this process.
The operation of a TCSC module is constrained by an overvoltage protection provided
by the MOV which imposes the voltage limit on the TCSC operation in the capacitive
zone, and by harmonic and thyristor current ratings which constrain the TCSC operation
in the inductive zone. Fig. 2.12 [23] shows a typical TCSC capability characteristic for a
single module in terms of voltage versus the line current. Considerations of the thyristor
delay angle limits, voltage limits for the safe operation of the series capacitor and
thyristor current limits lead to the operating boundaries of the form in the voltage-
current plane in Fig. 2.12 within which the TCSC operation is allowed. The response
time periods are considered in the construction of the operating boundaries which are
applicable for short-term transient, long-term transient and continuous (steady-state)
operations, respectively.
______________________________________________________________________
32
With reference to the above discussion of practical operating limits for a single module,
there is a gap as shown in Fig. 2.13 in the control range between blocked reactance, XC,
and bypassed reactance, Xbypass, for which no thyristor firing angle exists. This restricts
the application of TCSC in the transmission system where a smooth variation in the
combined reactance of the TCSC and transmission line is often required.
In order to eliminate this gap, the TCSC is split into multiple modules as shown in Fig.
2.14 which operate independently in the inductive and capacitive modes. By doing so, a
continuous transition from the capacitive to the inductive domain becomes feasible.
The larger the number of the modules into which the TCSC is divided, the narrower is
the gap between the capacitive and inductive regions. With a sufficient number of
modules, the TCSC reactance can vary continuously from capacitive value to inductive
value, and the reactance limits approach a closed locus within which TCSC operation is
feasible.
Vol
tage
(pu)
Indu
ctiv
eC
apac
itive
Maximum Thyristor Current
-2
0
-2
1 2
MOV protection Level
Max
imum
Firing
Adv
ance
Maximum Firing Delay
Harmonic Heating Limit
No Thyristor Current (blocked, slope= XC)
Full Thyristor Conduction(slope=Xbypass)
Line current (pu)
Fig.2.12 Typical V-I capability characteristics for a single-module TCSC
______________________________________________________________________
33
δ (d
eg)
Xbypass XC
900
0 2-2 -1 1 300
1800
Maximum Firing Delay
Maximum Firing advance
Xnet(pu)
Unavailable
Fig. 2.13 Practical operating range of TCSC for inductive and capacitive compensation
MOV MOV MOV
Conventional Series Capacitor
VC
ILine
Multi-module TCSC
Fig. 2.14 Multi-module TCSC
2.9.2 Dynamic model of TCSC with SDC
For a transient stability study, the schematic block diagram of TCSC is shown in Fig.
2.15 in which TCSC dynamics are represented by one first-order block along with the
supplementary damping controller (SDC) block consisting of washout and two lead-lag
______________________________________________________________________
34
compensators. The time lag used with TCSC is associated with the firing controls and
natural response of the TCSC and is represented by single time constant Ttcsc.
∑csc
csc1 t
tsT
K+
Pe
Xref
-
+
XSDC
Xtcsc
Xtcsc(max)
Xtcsc(min)XSDC(max)
XSDC(min)
SDC4
SDC311
sTsT
+
+ X2SDC
SD C2
SD C111
sTsT
+
+ X1SDC
SDC
SDC1 sT
sT+ SDC1K
Fig. 2.15 TCSC schematic block diagram with SDC
The reactance limits of the TCSC must be considered for static modelling as well as
dynamic. These limits are relatively complex and time dependent. The TCSC dynamic
model is derived from the above block diagram and represented in compact notations
as:
ettttT PBXAX••
+= cscTcscTcscTcsc (2.32)
In (2.32), ATtcsc and BTtcsc matrix constants are dependent on controller design (i.e. gains
and time constants) and are derived as shown in Appendix C.
Although TCSC control parameters can be tuned with changing system status and
controller output as discussed in [29], for the purpose of this study the tuning
parameters are fixed throughout to avoid complexity in focusing on transient stability
improvement for a multimachine system, considering its nonlinear dynamics.
2.10 SUPPLEMENTARY DAMPING CONTROLLER (SDC)
While using TCSCs for transient stability improvement focusing on first-swing stability,
there are chances of system oscillations in the next successive cycles because of slow
action of AVR. Hence, after taking care of the first-swing the next successive
oscillations can be taken care by using supplementary damping controllers (SDC) which
work almost the same as power system stabilisers.
______________________________________________________________________
35
It has a washout filter and dynamic compensator as two main components followed by a
limiter. The simplified structure of SDC is represented as shown in Fig. 2.16. As the
SDC is expected to respond only for transient variations and avoid responding to any dc
offset of input signal, this washout filter acts like a high pass filter allowing only
frequency of interest. This can be achieved by selecting proper time constants to control
local modes or inter area modes. A dynamic compensator is a two-stage lead-lag
network and a limiter is used to keep the controller response within a specified tolerance
limit restricting it from large deviations.
PeSDC
SDC1 sT
sT+ ( )sT
( )minSDCX
( )maxSDCX
Washout Dynamic compensator
Fig.2.16 Simplified schematic of supplementary damping controller
Though there are many parameters which can be used as input signals to this SDC,
including line current, active power, reactive power and bus voltage magnitude in
present research active power input is used as an input signal to SDC. The two main
reasons for using active power as an input signal are, firstly, it is available with local
measurements and secondly, this will provide correct control action in maintaining
power transfer over lines when severe fault occurs. In [29], a qualitative analysis is
presented with different pros and cons in selecting a particular parameter as input signal.
2.11 POWER SYSTEM NETWORK MODELLING
Having formed the dynamic models of key compensating devices in power system, to
estimate the power flow over given transmission lines or voltages and phase angles at
every node, it is necessary to conduct power flow study. Hence, a proper representation
of power system network is necessary. The general power system network can be
represented as shown in Fig.2.17 which constitutes generator nodes, load nodes
including multiple FACTS devices inserted at some nodes for compensation.
______________________________________________________________________
36
Representing total generator nodes by ngen and nsvc as the number of SVC nodes, nsta as
the total of STATCOM nodes and ntcsc as the nodes at which TCSCs are inserted, the
system has in all nnode nodes including load nodes.
Following the notations of Vk as the nodal voltage vector and Ybus as the general
admittance matrix based on a power system structure as shown in Fig. 2.17, the vector
of the current injected at every node including generator and load nodes, can be
represented in general form as:
bus
n
1kbusbus YVI
node
∑=
= (2.33)
where,
Ibus=
noden
i
1
I
I
I
; Vbus=
noden
i
1
V
V
V
; Ybus =
nodenodenode
node
nn1n
ii
2221
1n1k1211
YYY
YYYYYY
(2.34)
The general active and reactive power flow equations at any given node are represented
as given in (2.35):
= ∑
*
Rek
kikii VYVP and
= ∑
*
Imk
kikii VYVQ
(2.35)
where, i=2,3,. . ., nnode and Pi and Qi, are active and reactive powers injected at ith node
respectively.
2.12 LOAD MODELLING
As explained in [30], a load model is a mathematical representation of the relationship
between a bus voltage (magnitude and frequency) and the power (active and reactive) or
current flowing into the bus load. The term ‘load model’ may refer to the equations
themselves or the equations plus specific values for the parameters (e.g. coefficients,
exponents) of the equations. Depending on the computational implementation of these
equations in a specific program, the load power or current may not be included
explicitly, but it is useful to think of the model in these terms.
______________________________________________________________________
37
It is difficult to quantify the benefits of improved load representation. However, several
studies, reported in literature, have demonstrated the impact that different load models
can have on the different types of study results. In some cases the impact can be
significant. A common philosophy, in the absence of accurate data on load
characteristics, is to assume what is believed to be a pessimistic representation, in order
to provide some safety margin in the system design and operating limits.
1 (slack)
2
ngen
gen
SVC1 ngen+1
SVCn_svc ngen+nsvc
STATCOM1 ngen+nsvc+1
STATCOMn_sta ngen+nsvc+nsta
TCSC1
TCSCn_tcsc
ngen+nsvc+nsta+1
ngen+nsvc+nsta+ntcsc
ngen+nsvc+nsta+2*ntcsc+1
n_svc1
n_svcn_svc
n_sta1
n_stan_sta
ngen+nsvc+nsta+ntcsc+1
ngen+nsvc+nsta+2*ntcsc
n_node
svc
sta
TCSC
nodeload
Fig.2.17 Multi-machine power system network having multiple FACTS devices
______________________________________________________________________
38
2.12.1 Static load modelling
Static load modeling can be defined as in [30] as model that expresses the active and
reactive power at any instant of time as a function of the bus voltage magnitude and
frequency at the same time. Static load models are used both for essentially static load
components, e.g. resistive and lighting load and as an approximation for dynamic load
components e.g. motor-driven loads.
In this research, the static loads are modelled as equivalent admittances. The required
data for calculations of these admittances is obtained from load flow studies. Thus for
an active load of PL and reactive load of QL, at any load bus having voltage as VL, the
equivalent load admittance at a given bus can be calculated as:
2L
L2
L
LL
V
QjV
PY −= (2.36)
Although the static load model of the constant admittance form is most popular one,
other static models such as those based on constant current, constant power and
exponential functions have also been proposed and reported in [30-32] .
2.12.2 Dynamic load modelling
The dynamic load model as defined by [30], is a model that expresses the active and
reactive powers at any instant of time as a function of voltage magnitude and frequency
at any past instant of time and, usually including the present instant. Difference or
differential equations can be used to represent such models.
Even though power system load has gain more attention in literature, it is still
considered as one of the most uncertain and difficult components to model due to the
large number of diverse load components, its high distribution, variable compositions
and also because of lack of precise information on the composition of the load [33]. The
induction motor type of loads connected to the power system, maintain their stability,
when unbalances due to voltage changes arise by shifting their driving points slightly.
______________________________________________________________________
39
The induction motor is a typical load under constant torque operation. It compensates
for a shortage of power by increasing the slip and for a surplus of power by decreasing
its slip, resulting in balanced operation. Such a mechanism is called self-controllability.
As mentioned in [33], with various developments in power system, it’s not only
induction motors which should be considered as dynamic loads, but tap-changers or
spontaneous load variations should also be considered for load modelling. Navarro in
[33], has discussed various types of load and their static as well as dynamic load
models such as, frequency load models, induction load models and exponential dynamic
load models.
On the time scale of importance for transient stability properties, generators and motor
loads are the primary source of angular dynamics following a network fault. However,
in common with [34, 35] and [13] systems are considered with no motor loads. This
assumption is proposed for three reasons: an accurate representation of load behaviour
is difficult to obtain; measurement of motor load dynamics is difficult; and the constant
impedance load assumption simplifies the control design process. Though the effect of
dynamic loading in stability study is discussed in previous literature, in present research
only static load models are considered. However, the control schemes developed in this
thesis for RHC based TCSC controllers or online control coordination schemes can be
equally applicable for dynamic load consideration by modifying existing power system
models with additional equations of dynamic load models.
2.12.3 Impact of load modelling on power system transient stability study
The load modelling is important from a first-swing stability study point of view or
transient stability point of view. A first-swing problem exhibits large and rapid voltage
excursions during the initiating fault and slower voltage excursions during the first
power-angle swing, which lasts one second or less. Load response to these voltages is
important. System voltages are normally depressed during the first angular swing
following the fault. The power consumed by the loads during this period will affect the
generation-load power imbalance and thereby affect the magnitude of the angular
excursion and the first swing stability of the system. As mentioned in [30], in case of a
constant impedance load model, the power consumption would vary with the square of
______________________________________________________________________
40
the voltage and, therefore, would be lower than the actual load during the depressed
voltage period. For loads near accelerating machines, this will give pessimistic results,
since the generation-load imbalance will be increased. However, for loads remote from
accelerating machines (near to decelerating machines), there will be an optimistic
impact on the results. On the other hand, a constant MVA load model would have the
opposite impact on the results since it would hold the load power at higher value during
the depressed voltage. It is therefore difficult to select a model that is guaranteed to be
conservative for all parts of the system and for various disturbances.
There is also a brief frequency excursion during the power-angle swing, so frequency
characteristics of loads close to accelerating or decelerating generators can also be
important. Chandrashekhar and authors in [36] have discussed the roles played by
structure and load models in direct stability assessment. One of the major drawbacks of
using classical modelling with constant-impedance loads is difficulty in accounting
rigorously for transfer conductance between the machine voltage nodes and the
dependency of stability properties on the structural features of the network getting
masked. These disadvantages are essentially due to the process of assuming impedance
models for loads and then using a reduced network model. The resistive part of the
loads leads to transfer conductances and the network structure is lost in the reduction.
However, with present advanced computer technology the network reduction can be
avoided and system structure can be preserved for better accuracy.
2.13 CONCLUSIONS
The state-of-the-art models for the synchronous generator in dynamic condition have
been explained. The dynamic modelling of exciter, prime-mover and PSS are explained
for transient stability study. The importance and effect of using accurate models of
synchronous machine, exciter and governor are discussed. The interaction of exciter
and prime-mover with synchronous generator is explained and how to incorporate these
individual machine models with external power system networks for transient stability
study is shown by proper transformation of reference axes.
The chapter has discussed the dynamic modelling of various key FACTS devices
including SVCs, STATCOMs and TCSCs. Using simplified schematic representative
______________________________________________________________________
41
dynamic models, the effect of modelling these equipments on transient stability is
discussed in brief. The power system network equations are considered using multiple
FACTS devices. Based on the various aspects of dynamic modelling of key power
system elements for transient stability, a comprehensive model will be formed and used
for deriving various control schemes which are proposed in the next part of this thesis.
______________________________________________________________________
42
Chapter 3 Overview of Power System Issues and
Solutions
3.1 INTRODUCTION TO POWER GRID OPERATION
In recent years, as a consequence of the deregulation of the electric power industry,
power sources and consumers geographically dispersed, resulting in bulk power
exchanges over long distances. Though the supply should be increased to cope with
increasing demand, the development of generation is much slower than the rate of load
increment. As mentioned in [37] , with reference to report [38] from the Electric Power
Research Institute, the generation capacity margin has consistently decreased in the past
20 years. For example, the generation capacity margin in 2000 was only one third of a
half of the increase of electricity demand. In short, the demand is ever increasing and
the new transmission infrastructure to cope with this increasing demand is restricted
because of several factors like economy and environment. This forces the existing
power system network to work at its maximum possible permissible limits, making it
vulnerable to becoming unstable in case of any sudden disturbance. In such
circumstances to maintain, continuity and reliability, interconnection of various areas
becomes essential. Forming interconnected grid helps in many ways including less
______________________________________________________________________
43
installed capacity, less spinning reserve required, efficient loading which results in
better transmission efficiency, maintaining continuity, giving breathing time for some
outages and maintenance work. However, interconnected systems operating close to
their transient stability limit are vulnerable to any major sudden disturbances, where
cascade tripping problem and total or partial blackouts can occur if necessary corrective
action is not taken in time.
3.2 POWER SYSTEM ISSUES: CAUSES AND SOLUTIONS
3.2.1 Power system reliability and security
As explained in [39] an electrical power system consists of numerous components
connected together to form a large, complex system which is generating, transmitting
and distributing electrical power. Electric power systems and additional preventive
control schemes are designed in such a way, that the system should be able to withstand
any single contingency, that is, outage of any single component without loss of stability
and all system variables kept within predefined ranges [6]. Not all possible
disturbances, however, can be foreseen at the planning stage and these may result in
instability leading eventually to collapse or islanding of the system. Furthermore,
because of environmental constraints on the extension of the transmission capacity,
increased electricity consumption and new economic constraints imposed by the
liberalised power market, power systems are operated closer and closer to their stability
limits. During normal operation, the focus is on economic optimisation of system
operation, while during more challenging network conditions (alert or emergency
situations) the focus shifts to stability consideration rather than economy [40]. The
ultimate objective is keeping as much as possible of the network and generators
connected to the grid intact, as breakdown will normally result in more severe problems.
Hence, the main concern in the emergency state is the system security. Security is an
online operational characteristic which describes the ability of power systems to
withstand different contingencies without service interruptions. Security is closely
related to reliability: an unreliable system cannot be secure. The security level of the
power system desired is to be high enough to enable robust operation changes
dynamically as the power system operation changes and depends on the factors outside
the control of power system operators such as weather [41] . With this background, it is
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44
vital to give top priority to dynamic security and stability before pursuing other targets,
like economical operation, optimal load flow and fair deregulation in power market
[42]. To understand the dynamic behaviour of system, it is necessary to know the
classification of power system transients and its time frames of response.
3.2.2 Classification of power system transients
Power systems comprise of many components including generators, loads, transmission
lines, which interact with each other in various ways and on various time scales (i.e
dynamic speed ranging from milliseconds up to years). Dynamics present in power
systems may be very nonlinear and have a hybrid nature having both continuous and
discrete state variables [43] . At every instant power system networks experience some
sort of disturbances that may be small or large in amplitude and varying over different
time duration. The severity of transients is classified based on the rate at which the wave
rises from 10% of its value to 90%, i.e. value of front of waveform and the time for
which it stays on system i.e. fall time measured from peak to 50% of its value, which is
also known as the tail of wave as shown in Fig.3.1. The sharp front can stress the
system pushing suddenly away from normal operating conditions, while the long tail
may keep the system oscillating for a longer time.
0.90
0.10
TailFront
Time
0.50
Fig.3.1 Typical wave specifications expressed in terms of front and tail of wave
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45
The time frame for some important disturbances is shown in Fig. 3.2 [44, 45]
classifying the transients based on their behaviour. The class A type of transients are
known as ultrafast transients and are also categorized as surge phenomena, having a
very steep wave front. The class B type of transients are medium-fast and are due to
short-circuit phenomena, while the class C represents slow transients which comes
under the category of transient stability varying in time range of few seconds to few
minutes.
10-7 10-5 10-3 10-1 101 103 10-5
Lightning
Switching
Subsynchronous resonance
Transient stability
Long term dynamics
Tie-line regulation
Daily load following
SVC,TCSC etc
Generator control
Protection
Prime mover control
LFC
Operator actions
1 cycle 1 second 1 minute 1 hour 1 day
Pow
er sy
stem
phe
nom
ena
Pow
er sy
stem
con
trol
s
Timescale (seconds)Class AClass B
Class C
Fig: 3.2 Time frame of various transient phenomena
The modelling of power system equipment and their responses are also compared in
Fig.3.2 for these same time frame disturbances and it can be seen that no power system
control can really respond to class A types of transients. However, with fast switching
thyristor controls, FACTS devices can be used to enhance power system performance
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46
for medium-fast or class B types of transients. It can be observed that prime mover and
generator controls are active to respond during class C transients which are transient
stability responses indicating that assumption or simplified modelling of these elements
may lead to pessimistic results. With reference to Fig.3.2 and as explained in Chapter 2
it can be seen that any accuracies in dynamic modelling of power system elements will
result in inaccurate prediction of system stability in transient stability conditions.
3.2.3 Disturbances/causes of power system problems
Even knowing the classification of transients on power systems, the response of various
power system elements to these transients and having most accurate designed protection
schemes, power systems are always exposed to various serious disturbances which can
lead to the interruption of power supply to consumers. Even the best planned system
cannot predict all possible contingencies, and any unpredictable events can stress the
system beyond planned limits. Some of the more well-known reasons as well as those
mentioned in [41], why completely reliable system operation is not achievable [46]:
(a) Globalisation/liberalisation: Deregulation and privatisation [18] has opened the
market for independent suppliers and transmission companies giving rise to different
load patterns than those for which it was previously designed.
(b) Transmission congestion and stressed conditions: problems of uncontrolled loop
flows, over loading and excessive short-circuit levels gives rise to instabilities and
outages which are bottlenecks in transmission. This gives rise to an infinite number of
possible operating contingencies in modern interconnected power system networks. The
evolving nature of power systems generating unpredictable changes, giving power
system operation a totally different scenario from the expectations of the system
designers, particularly during emergency. In addition, a combination of unusual and
undesired events such as human error combined with heavy weather and scheduled or
unscheduled maintenance outages of the important system element. For example,
deregulation provided financial motivation to transfer power from generation e.g.
independent power producers to remote loads. As existing power systems were not
designed for those transfers, the system has to bear additional stress.
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47
(c) Weak connections: Extension of interconnected systems is one of the solutions for
an economical, efficient and reliable power system. However, the increased power
exchange among the long interconnected links can give rise to weak link connections
which can put the system in a condition of cascade tripping.
(d) Unexpected events, stability threats or hidden failures in protection systems
The development of transmission systems follows closely the increasing demand on
electrical energy. However, the performance of the power systems decreases with
increasing size and complexity of the networks. This is related to problems with load
flow, power oscillations and voltage quality. Such problems have been highlighted by
the deregulation of the electrical power markets, worldwide. Contractual transactions
now result in power flows that are much different from those of the original network
design criteria, and the operational constraints of the existing network, especially where
the connecting AC links are weak.
The interconnection of power systems offers numerous benefits for power transmission,
including pooling of various energy resources, reduction of reserve capacity in the
systems and increasing the transmission efficiency. However, if the size of the system is
too large, dynamic problems can occur which can jeopardize the reliability and
availability of the synchronous operation of the interconnected grids. By using the
advanced solutions, based on modern power electronics, the performance of
transmission systems can be improved.
3.2.4 Power system stability issues
The severity and response time for power system stability will vary depending on the
type of transients arising in power systems as explained in Section 3.2.2. Deregulation
has resulted in more competition in the power market so that it is becoming necessary to
allow the operating points of the generators to vary quickly to meet the needs according
to the power purchase agreement between the seller and buyer. Conventional control
strategies based on approximate linearised models have proven to be very valuable to
ensure satisfactory transient performance due to the inherent nonlinear nature of the
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48
power system. In fact, the nature of a power system is not only highly nonlinear but it
exhibits uncertainty somewhat as their operating condition keeps changing frequently.
Particularly after a large disturbance, the power system topology can change drastically
due to fault clearing which may be by removal of any of the transmission lines [47]. In
such circumstances, it is necessary to know whether the system will regain any stable
post-fault operation or collapse. With reference to this, various stability definitions are
given in literature.
As mentioned in [48, 49] , stability can be defined as, the ability of power system for a
given initial operating condition, to regain a state of operating equilibrium after being
subjected to a physical disturbance, with most system variables bounded so that
practically the entire system remains intact. During normal operating conditions an
interconnected grid will help for power exchange over tie-lines, showing its usefulness.
However, during any sudden disturbances it also affects the entire network and
“stiffness” of the system decides the system stability in such conditions. In such
circumstances, the stability of the power system is defined by its ability to restore to
normal operation after any disturbance.
The transient stability is defined as the ability of the power system to come back to
stable operating conditions, subjected to any sudden disturbance. Following a
disturbance, protective relays will sense the type of disturbance and will give the
tripping command to the corresponding circuit breaker, to isolate the faulty part from
healthy network. However, system stability cannot be guaranteed even after fault
clearance, as total fault clearing time will decide the system behaviour in its post-fault
region. For guaranteed system recovery after such disturbances it is necessary to
understand the behaviour and response of various transients which arise in power
systems.
Though power system stability is a single problem, because of its high dimensionality
and complexity, to analyse specific types of problems using an appropriate degree of
details of system representation, simplifying assumptions can be used. Analysis of
stability, including key factors that contribute to instability and devising methods of
improving stable operation, is greatly facilitated by the classification of stability into
appropriate categories. Classification, therefore, is essential for meaningful practical
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49
analysis and resolution of power system problems which is shown in Fig.3.3.
Phenomena that create wide-area power system disturbances are divided into the
following major categories: angular stability, voltage stability, overloads, power system
cascading, etc. using variety of protective relaying and emergency control measures are
employed to combat them [41].
The angular instability or loss of synchronism condition occurs when generators in one
part of the network accelerate while other generators in some other area decelerate
thereby creating a situation where the system is likely to get separated into two parts.
The most common predictive scheme to combat loss of synchronism is the equal area
criteria and its variations. This method assumes that a power system behaves like a two-
machine model where one area oscillates against the rest of the system. Whenever the
underlying assumption holds true, the method has potential for fast detection.
Often the causes of angular instability can be overloading or short circuiting. Overloads
frequently occur during wide-area disturbances due to the increasingly high utilisation
of equipment capability. These overloads may result in faults or equipment damage if
overload protection is not provided. Outage of one or more power system elements due
to overload may result in overload of other elements in the system. If the overload is not
alleviated in time, the process of power system cascading may start, leading to power
system separation. Uncontrolled separation often occurs as a result of a transmission
line short-circuits protection system interpreting power swing as a short-circuit. When a
power system separates, islands with an imbalance between generation and load are
formed with a consequence of frequency deviation from nominal value. If the imbalance
cannot be handled by generators, load or generation, shedding is necessary. A quick,
simple and reliable way to re-establish active power balance is to shed load by
underfrequency relays.
While the system frequency is a final result of the power deficiency, the rate of change
of frequency is an instantaneous indicator of power deficiency and can enable incipient
recognition of the power imbalance. However, change of the machine speeds is
oscillatory by nature, due to the interaction among generators. The main cause of
system oscillations is smaller system inertia which gives rise to larger peak-to-peak
value for rate of change of frequency oscillations [41].
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50
Power system Stability
Rotor-angle stability
Frequency stability
Large-disturbance angle stability /
transient stability
Voltage stability
Small-disturbance angle stability /
small-signal stability
Large-disturbance voltage stability
Small-disturbance voltage stability
Slow and small amplitude
disturbance
Different modes of oscillations
Sudden large amplitude
disturbance
First swing and oscillations
problem
3-5s following disturbance is
period of interest
Few seconds to several minutes
following disturbance is the period of
interest
Fig.3.3 Classification of power system stability studies based on physical nature of the
phenomena
The rotor-angle stability also known as simply angle stability is concerned with the
ability of interconnected synchronous machines of a power system to remain in
synchronism under normal operating conditions and after being subjected to a
disturbance [50]. A fundamental factor in this aspect of stability is the manner in which
torque or power outputs of the synchronous machines vary as their rotors oscillate. The
mechanism by which the synchronous machine maintains synchronism with one another
is through the development of restoring torques whenever there are forces tending to
accelerate or decelerate the machine with respect to each other. When system and
synchronous generators to remain in synchronism and to give desired performance in
post-disturbance condition, use of some compensating devices becomes essential to
maintain system performance.
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3.2.5 Solution for power system stability: Need of FACTS devices
With the formation of wide-area network (WAN), if the power should be transmitted
through an interconnected system over longer distances, transmission needs to be
supported, which is possible with the help of recent developments in FACTS devices.
One of the major key consequences of deregulation as mentioned in [18] is that the
power transfer across the system is nowadays much more wide spread and fluctuating
than initially designed by the system planners. The system elements are going to be
loaded up to their limits, with the risk of losing that (n-1) safety criterion. The major
blackouts as reported in [18] show that, once the cascading sequence is started, it is
difficult or even impossible to stop it, unless the direct causes are eliminated.
This motivates researchers to find some solution for improving transient stability by
some controller, compensations, fast enough to help in the recovery of system in a post-
fault region. With recent developments in power electronics and especially thyristor
technology, the FACTS family is taking a leading role in power system performance
improvement. These FACTS devices are used not only in transient stability
improvement but also used in normal healthy operations to improve power transfer
capability, voltage stability etc.
In order to study the effects of FACTS devices installed in power systems, proper
models of power systems with FACTS elements need to be established which is already
discussed and derived in Chapter 2. However, Table 3.1 shows selected FACTS devices
with their basic scheme and comparative performance in terms of their impact on
various power system studies (load flow and transient stability) based on the report
[51].
For most applications in AC transmission systems and for network interactions, SVC,
FSC and TCSC are fully capable to match the essential requirements of the grid. Out of
all FACTS devices, the only device which is most commonly inserted in series, for
power system performance is TCSC. Thyristor-controlled series compensation can
provide improved stability for interconnected power systems allowing higher power
transfer levels and directing flows on desired transmission paths. Knowing the necessity
of FACTS controllers for power system performance, the next section of this chapter
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will give a brief review of various power system controllers used for stability
enhancement.
Table: 3.1 Overview and functions of most popular FACTS devices
Principle Devices Scheme Impact on system performance Load
flow Stability Voltage
quality Variation of line impedance : Series compensation
FSC TPSC TCSC
Small Small Medium
Strong Strong Strong
Small Small Small
Voltage control: Shunt compensation
SVC STATCOM
No/ Low No/ Low
Medium Medium
Strong Strong
FSC: Fixed series compensation
TPSC: Thyristor protected series compensation
TCSC: Thyristor controlled series compensation
SVC: Static VAr compensation
STATCOM: Static synchronous compensator
3.3 TCSC USED FOR POWER SYSTEM PERFORMANCE ENHANCEMENT
The power system transient structure forms firstly, with continuous dynamics which is
to be controlled and secondly with the discrete-event controller. To trigger the action of
the controller and link these two, some trajectory reference is set with suitable
parameters. The sensitive parameter is used to activate the controller which can be an
error in the given state variables. For example the reference load angle delta or speed
can be specified. In an occurrence of any disturbance the system dynamics start
evolving and deviate from steady conditions introducing errors in actual system
parameters and reference values set, which give rise to triggered action of the controller.
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53
In [52] , broad classification of various emergency schemes, such as event-based and
response-based is given. The event-based controller works based on predefined
disturbances. For example, look-up tables while response-based will react based on
getting triggered for a particular situation. As a power system is nonlinear and non-
stationary, the response-based controller will be adaptive and flexible to provide a best
control action to assure system stability, compared to event-based type control. Event-
based control schemes require no intervention of system operator and have a special
protection system as a defence scheme which responds automatically when the
contingency is detected [53].
Though there are many controllers developed for various power system applications
including voltage control, power system oscillations damping or transient stability
improvement using various FACTS devices, the focus of this research is using TCSC
for improvement of transient stability. The next section will provide the implementation
history of TCSC to discuss its practical performance and explain its benefits.
3.3.1 TCSC implementation history
Authors in [24], have reported the world’s first 3-phase, 2x165 MVAr TCSC
installations in 1992, in a 230kV transmission line at the Kayenta Substation in Arizona
and the Slatt Substation in Oregon. The aim of these installations was to increase power
transfer up to the thermal limits of line and to evaluate the effect of TCSC in controlling
power flow to damp electromechanical power oscillation. However, soon it was realized
that in addition to increase in power transfer by 30%, TCSC provides effective means
for damping electromechanical power oscillations. The on-site observations also
showed that it can provide series compensation without causing risk of sub-synchronous
resonance (SSR) as in case of a fixed series capacitor. With this experience, ABB
installed, the world’s first TCSC for sub-synchronous resonance in Stode, Sweden in
1998. As mentioned in [54], a thyristor-switched series capacitor (TSSC) controller
was experimented for compensating the 345kV transmission line at the Kanawa River
Substation in West Virginia. A TCSC controller is also installed on a 500kV
transmission line in Fengtun, China. Kirschner et.al. in [18] report that the rating of
shunt connected FACTS controllers is up to 800 MVAr while, series FACTS devices
are implemented on high voltage levels up to 765 kV, this can increase the line
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transmission capacity up to several GW. Also [18] found that the interconnection of the
1000 km long, North-South Brazilian system operating at 500kV AC is unstable without
the damping function of TCSC. If only one TCSC is in operation, the interconnection
becomes stable, and with both devices working, the inter-area oscillations are well
damped. It is also reported that, under increased load conditions, the TCSC damping
function is activated up to several hundred times per day, thus saving power
transmission and keeping the return investments constantly ‘running’. To find out the
optimised placement of TCSC and the effect of load increase and generation
rescheduling, a method based on trajectory sensitivity has been given by [55-57].
3.3.2 TCSC placement and its benefits
The authors in [24] have given some fundamental and concise study results of TCSC
behaviour. There are two strategies in the use of TCSC, firstly, the ‘constant line power
strategy’ where the power flow over a particular line is maintained constant with TCSC
series compensation and secondly by the ‘constant angle strategy’. While, in general,
the system stability condition improves with the placement of a TCSC in a multi-
machine system, the placement of the TCSC in some of the lines may have a
detrimental effect on system stability. Therefore, as mentioned in [57], it is extremely
important to identify the lines that give maximum benefit, and the lines that adversely
affect the system stability.
Zhou and Liang [58] give a detailed survey of various control schemes for enhancing
power system stability performance using TCSC. TCSC is not only useful in large
disturbances but it also improves the performance of systems under normal conditions.
Further benefits include scheduling of power flow over line under normal operating
condition, small-signal stability, damping oscillations, limiting short-circuit currents,
and many more the explained in [59] and [17]. However, for transient stability
problems, TCSC is considered to be most effective solution as compared to other shunt
types of compensators and FACTS devices. The authors in [47] have presented a
nonlinear control scheme for the TCSC to improve transient stability and to dampen
power system oscillations. However, an established nonlinear mathematical model
which has proven to be an affine nonlinear system is based on a single-machine-infinite-
bus (SMIB) system. On the use of the SMIB approach is given in Chapter 4. In a study
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of TCSC controller design for power system stability improvement, [29] provides a
detailed analysis of some of the fundamental aspects of proper TCSC controller design.
The limitations of using linear control techniques for controller design detail with large
disturbance in a realistic power system network are discussed in. A detailed analysis of
TCSC control performance for improving system stability with different input signals is
discussed and the need for proper input signal selection and coordination of the different
control levels are highlighted. Author, Sun in [60], has discussed the dynamic response
of TCSC through time domain digital simulation of an example TCSC circuit and then
proposed an improved closed loop reactance control method for the application of
TCSC.
Following a disturbance, protective relays will act to clear the fault by tripping the
faulted component. However, the power system is not guaranteed to be stable after a
clearing fault. In order to improve the transient stability performance, early detection of
fault, and fast fault clearance is most important. Along with fast controllers, the
prediction algorithms can help to predict the system conditions in advance so that
necessary corrective action can be implemented accurately well before the system
moves in critical stages. Knowing the potential of various prediction algorithms, the
next section of the chapter is devoted to reviewing of predictive control based methods
which are used for power system performance enhancement.
3.4 REVIEW OF PREDICTIVE CONTROL BASED METHODS USED FOR
POWER SYSTEM APPLICATIONS
Predictive controllers are well-developed in control systems and have proved their
effectiveness in chemical industry applications [61, 62]. However, Power systems
exhibit several features of complex systems, such as hybrid nature (mixed and
continuous dynamics), nonlinear dynamics and very large size. Such complex features
of power systems, offer challenges to predictive controls which require reasonable time
in optimisation computation. Several authors have addressed difficulties in applying
predictive methodology for power systems because of its complexity, large size, and
hybrid nature. The power system can be modelled as a hybrid system incorporating
nonlinear dynamics, discrete events and discrete manipulated variables. The nonlinear
behaviour of the system calls for methods based on a dynamic model in order to account
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for changes of operating point and the network state. This is complicated by the fact that
most control moves are inherently discrete-valued, for example switching on of
capacitor banks. However with recent advances in computation, communication and
power system instrumentation technology, and more specifically phasor measurement
units and wide-area measurements, the coordination and model based approach have
become more tractable. To deal with these challenges, Marek Zima [43] tried to use
MPC for power system applications using a trajectory sensitivity concept which claimed
that trajectory sensitivities allow an accurate reproduction of the nonlinear system
behaviour using a considerably reduced computational burden as compared to full
nonlinear integration of the system trajectories. Unless employed control changes
significantly, trajectory sensitivity reproduces the system behaviour quite accurately,
even considering nonlinear dynamics. However, it is very difficult to bind a region, in
which the changes can be considered ‘reasonably small’. Another possible source of
errors may be certain types of discrete events. A discrepancy between the model and the
actual system response can be corrected in the receding horizon manner. With this
background, the next section of this chapter will discuss various predictive control
applications for power system stability enhancement.
3.4.1 Voltage control and small-signal stability applications
There are a number of attempts of applying predictive control methodology for voltage
stability and small-signal stability. In [63, 64], Zima has used trajectory sensitivities
based model predictive control (MPC) for emergency control of voltage stability
problems. It has used simplified MPC concepts in which only one control input set is
computed and tracking of certain reference values are disregarded. Using the same
trajectory sensitivities in order to reduce the modelling complexity and computational
burden [43] has presented closed loop MPC used in both emergency and normal
operation conditions in power system. Geyer and Larsson [39] used mixed logic
dynamics framework in connection with MPC for dealing with the hybrid nature of
power systems to predict and prevent voltage collapse in a power system.
In [65], for predictive frequency stability control based on wide-area phasor
measurements, a method is proposed in terms for a predictive control strategy. First a
single-machine equivalent model of the system is formed based on the collected
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measurements, which is then used to estimate the active power imbalance in the system
and subsequently a predicted steady-state frequency. The same model is then used to
find out the amount of load shedding required to keep the frequency above some target
value. Finally, the calculated amount is allocated to different feeders using a simple
iterative method considering the actual load on feeders. To maintain frequency stability,
load shedding is an easy and quick solution; however, it is not a preferred or
recommended solution when it comes to quality and continuity of supply.
3.4.2 Predictive control based controllers for transient stability improvement
The interconnection of power systems offers numerous benefits for power transmission,
such as pooling of various energy resources, reduction of reserve capacity in the
systems and increasing the transmission efficiency. However, if the size of the system is
too large, dynamic problems can occur which could and can jeopardize the reliability
and availability of the synchronous operation of the interconnected grids [51] .
Following a disturbance, protective relays will act to clear the fault by tripping the
faulted component. However, the power system is not guaranteed to be stable after a
clearing fault. In order to improve the transient stability performance, early detection of
fault, and fast fault clearance is most important so a neural network-based system was
proposed in [52]. For making accurate predictions of transient stability status of power
systems, in [52], training examples were added continuously to reflect the most recent
operating conditions.
In the emergency control given in [65], a prediction method is used for estimating
future frequency decline after disturbance and necessary corrective action to maintain
wide-area frequency stability problem. However, the prediction is based on
approximation. Authors in [66] have proposed a response-based emergency control
scheme to prevent transient instability with the help of a hybrid model developed with
continuous dynamics and discrete events interacted. The problem of deciding
emergency control action is embedded into the MPC framework. For response-based
emergency schemes, an essential but challenging step is to make real-time predictions
of the transient stability status of a power system within a short time interval, say about
15ms after clearing the fault. In such critical conditions, a compromise has to be made
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between prediction accuracy and speed. Though many event-based and response-based
emergency control schemes are reported in literature, the response-based emergency
control leads over event-based because of its adaptability and flexibility to changing
operating conditions and disturbance.
It has been found in [67] that, when the transmission line impedance is controlled by
FACTS equipments like TCSC, the control action is multiplicative. This necessitates
accurate nonlinear models to make use of the most effective, economical use of
available control resources. In this context, variable structure control [68] and nonlinear
model-based self-tuning control [69] have been proposed for a class of transient
stability problems. In [67], a nonlinear time series, model-based, generalised predictive
controller for a simplified power system, using rotor angles as the measured output and
a TCSC controller is designed and developed. Generalised predictive controllers offer
advantages of being easy to implement in real-time and allowing systematic methods of
handling input constraints.
Ford et al. [13] have tried to develop an efficient and robust model predictive control for
first swing transient stability using FACTS devices. However, it is based on simplified
low order model and unrealistic fault duration times and is more focused on the power
transfer capacity of transmission lines and several other assumptions.
To study the effectiveness of various control methods applied to power systems, [67]
compared generalised predictive control (GPC), feedback-linearisation and LQR
methods. The authors in [67] observed that the nonlinear GPC provides the best
damping out of the three within the shortest possible time and using the least amount of
control. Rajkumar et al [69] presented generalised predictive control schemes for TCSC
to raise the transient stability limits and provide rapid damping to the power system
oscillation. However, for nonlinear systems, the online optimisation numerical
computation burden is huge and the demand of real-time control may not be satisfied.
To address this issue, [70] designed a TCSC controller with closed-form analytical
solution control law based on nonlinear optimal predictive control theory. The authors
claim that this controller does not require online optimisation and, hence, will satisfy the
demand of real-time control. A major advantage of the nonlinear GPC over the other
two controllers is that it allows a systematic way of handling control constraints albeit
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numerically intensive, which may restrict the look-ahead horizon for real-time
implementation.
Feedback-linearising controllers demand excessive control efforts, compared with
LQR’s due to the extended control efforts of feedback-linearising the nonlinear
dynamics. When the control saturates, the nonlinear dynamics are not cancelled entirely,
and the system is left with residual nonlinear dynamics with properties that are different
from the original nonlinear power system. Feedback-linearising controllers need a
perfect reference model and measurements to provide the exact cancellation.
Uncertainties in the reference model can lead to deteriorated robustness of the
controller. There exists the danger of destabilisation, by imperfect cancellation power
system dynamics which possess an unstable equilibrium, e.g. dynamically unstable
situations. The presence of dynamic uncertainties, such as time-varying infinite-bus
voltage, can lead to non-robust performance of the feedback linearising controller. LQ
regulators perform worst than the nonlinear GPC, but better than the feedback
linearisers, in identical conditions. Although the performance of LQR on nonlinear
systems is sub-optimal, it is seen to have good robustness properties.
In view of the many attempts of predictive control algorithms to power system
applications, the next chapter will develop the theory and scheme for transient stability
enhancement using MPC based TCSC controllers.
3.5 CONCLUSION
The chapter has reviewed the problems, causes and power system issues. Knowing the
limitations of limited infrastructure and maximum utilisation of transmission networks,
the chapter has highlighted the role of FACTS devices in enhancing power system
performance and has given emphasis on the effectiveness of series compensation in
maintaining transient stability. With a brief review of power system controller
requirements the chapter has discussed various prediction based methods its limitations
and applications related to power systems. In view of the many attempts of applying
predictive control algorithms to power system applications, the first part of the research,
developing a receding horizon control based TCSC controller for implementation and
validation single machine infinite bus sample system. With reference to this the next
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chapter will discuss the problems associated with application of existing predictive
control schemes to power systems and will prepare a background theory which is used
for developing the RHC-based controller in subsequent chapters.
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61
Chapter 4 Theory and Overview of Predictive
Controllers
4.1 INTRODUCTION
Having reviewed various predictive control-based controllers used for power system
stability improvement in Chapter 3, this chapter will develop the necessary background
theory for a new proposed predictive control-based TCSC controller for transient
stability enhancement. The chapter will first explain predictive control theories
developed basically in control and chemical fields in general, along with their pros and
cons. While the second part of the chapter will cover necessary background theory for
the application of predictive control strategy in power systems. While there are many
predictive control schemes developed and reported in literature [71-73], in this research
a FACTS controller is developed based on the receding horizon principle (RHC).
The three control strategies which had been investigated independently, RHC is one
type of predictive control while the other two well-known predictive controls are
generalised predictive control (GPC) and model predictive control (MPC). GPC is
based on single-input and single-output models such as the auto regressive moving
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average, or the controlled auto-regressive integrated moving average models, which
have been used for most adaptive controls. The basic idea of GPC is to calculate a
sequence of future control signals in such a way that it minimises a multistage cost
function defined over a prediction horizon. The index to be optimised is the expectation
of a quadratic function measuring the distance between the predicted system output and
some predicted reference sequence over the horizon plus a quadratic function measuring
the control efforts [73] .
The term ‘predictive’ is used in GPC since the minimum variance is given in predicted
values on the finite future time. The term ‘predictive’ is used in MPC since the
performance is given in predicted values on the finite future time which can be
computed by using the model. The performance for RHC is the same as the one for
MPC. Thus, the term ‘predictive’ can be incorporated in RHC as the receding horizon
predictive control (RHPC) or simply RHC.
4.2 PREDICTIVE CONTROL METHODOLOGY
4.2.1 Model predictive control (MPC)
MPC has been developed on a model basis process industry area as an alternative
algorithm to the conventional proportional integrate derivative (PID) control that does
not utilise the model. The purpose of MPC is to achieve online accurate tracking of the
trajectory delivered by the dynamic real-time optimiser. The MPC solves a constrained
optimisation problem online and determines an optimal control input over a fixed future
time horizon, based on the predictive future behaviour of the process. Although more
than one control move is generally calculated, only the first one is implemented. At the
next sampling time, the optimisation problem is reformulated and solved with new
measurements obtained from the system. The optimal reference trajectories for the
manipulated and controlled variables are produced by the dynamic real-time trajectory
optimiser and passed to the MPC.
The status of system variables (including control variables), updates of the model
parameters and estimates of disturbance signals are assumed to be available online and
able to be imputed to the MPC. Given factors such as the initial system status,
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information about the type of disturbances, and the reference trajectories, the optimiser
produces the manipulated variables such that the input and output trajectories follow the
reference trajectories as closely as possible subject to the constraints imposed in the
optimisation.
The commercially available, model predictive controllers vary in many details but they
all are based on finite time horizon optimisation problems, based on one linear model at
a time. The characteristic feature of MPC is that the control strategy is determined by
the optimisation of a performance function on a finite time interval. This interval
stretches from the current time to a time instant, which is a fixed time slot ahead. The
optimal control is calculated and implemented only until new measurements are
available. Based on new measurements, an update of the control strategy is determined
by repeating the optimisation of the performance function at the next time step. In this
way, the control strategy depends on the measurements and could therefore be known as
feedback type.
MPC can be described in short as a control methodology, which allows explicit
integration/inclusion of the constraints (imposed on the controller system and/or
employed controls) and explicit expression of the control quality criteria in the control
objective [71, 74]. The main features of MPC are:
(a) It can be used to control a great variety of systems, including those with non-
minimum phase, long time delay or open-loop unstable characteristics;
(b) It can deal with multivariable, multi-input-multi-output as well as single-input-
single-output systems; and
(c) Its system constraints can be readily treated within the optimisation process
4.2.2 Receding horizon control (RHC)
RHC, which is based on state-space framework, has been developed in academia as an
alternative control to the LQ controls. The basic concept of RHC is that: at the current
time, the optimal control is obtained,(either closed-loop type, or open-loop type), on a
finite fixed horizon, from the current time k, say [k, k+N]. Among the optimal control
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on the entire fixed horizon [k, k+N], only the first one is adopted as the current control
law. The procedure is then repeated the next time, say [k+1, k+1+N]. The term ‘receding
horizon’ is introduced, since the horizon recedes as time proceeds.
4.2.3 General MPC problem formulation
One of the reasons for the fruitful achievements of MPC algorithms is the intuitive way
it addresses the control problem. In comparison with conventional control, which often
uses a pre-computed state or output feedback control law, predictive control uses a
discrete-time model of the system to obtain an estimate (prediction) of its future
behaviour. This is done by applying a set of input sequences to a model, with the
measured state/output as the initial condition, while taking into account constraints. An
optimisation problem built around a performance oriented cost function is then solved
to choose an optimal sequence of controls from all feasible sequences. The feedback
control law is then obtained in a receding horizon manner by applying to the system
only the first element of the computed sequence of optimal controls, and repeating the
whole procedure at the next discrete-time step.
In short MPC is built around the following key principles:
(a) The explicit use of a process model for circulating predicting of future plant
behaviour.
(b) Optimisation of an objective function, subject to constraints which yields an optimal
sequence of control.
(c) A receding horizon strategy, so that at each instant the horizon is moved towards the
future, which involves the application of the first control signal of the sequence
calculated at each step.
MPC intends to force the controlled system, which are expressed by the system state, to
follow the desired trajectory. The trajectory applies an optimal sequence of manipulated
control inputs in the time instant/samples within the specified time horizon. Both the
system state values, as well as control inputs, can be subjected to the inequality and/or
equality constraints. To capture the system dynamics and predict the system response to
the control inputs, the model of the system is needed; which is accomplished here by
introducing the equality constraints. The various MPC algorithms propose different cost
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functions for obtaining control law as given in [43, 63, 75] and shown in Appendix D.
However, the general aim is that the future output (y) on the considered horizon should
follow a determined reference signal yref and at the same time, the control efforts (∆u)
necessary for doing so should be penalised. The general expression for such an
objective function will be [71]:
2
1
2
1
221 ])1([)]()([),,( ∑∑
==−+∆++−+=
uN
j
N
Nju jtjttjtNNNJ uRyyQ ref (4.1)
The parameters N1 and N2 are the minimum and maximum prediction horizons and Nu is
the control horizon, which does not necessarily have to coincide with the maximum
horizon. The N1 and N2 mark the limits of the instants in which it is desirable for the
output to follow the reference. The term yref, which is also known as a reference
trajectory provides an advantage in predictive control. If the future evolution of the
reference is known a priori, the system can react before the change has effectively been
made, thus avoiding the effects of delay in the system response. However, in
minimisation the majority of methods usually use a reference trajectory which does not
necessarily have to coincide with the real reference as is the case in most of the power
system scenarios.
The weight matrices/vectors Q and R express the importance of the close tracking of the
reference for various states. The weight matrix R can be used to define the control
efforts. The overall controller performance is tuned using Q and R (e.g. accuracy,
aggressiveness).
4.2.4 Comparison of PID and MPC
The PID controllers by far are the most dominating form of feedback in use for more
than 90% industry applications used for wide range of problems, such as process
control, motor drives etc. However, the performance of conventional PID controllers
can be severely degraded if a process has a relatively large time delay compared to the
dominant time constant, and PID controllers cab only be detuned to retain closed loop
stability resulting in sluggish performance [76]. The widespread use and success of
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MPC applications attests to the improved performance of MPC compared to PID for
control of difficult process dynamics for more advanced controls. In study carried out in
[77], it has been reported that, the MPC controller was capable to maintain the variation
of the controlled variables much closer to the set points than the classical PID
controllers. In addition, it was claimed that using MPC, it is not only possible to save
equipment and energy cost, but the plant can also be exploited at its maximum capacity.
As MPC finds an edge over traditional PID controllers, next section will describe in
detail about various strengths of the predictive controlled based methods.
4.2.5 Strengths of predictive controlled based methods
As mentioned in [14], an issue with power systems is the control of large complex
nonlinear systems. MPC has been shown to be successful in addressing many large
scale nonlinear control problems and therefore is worth considering for stabilisation of
power systems. While MPC is suitable for almost any kind of problem, it displays its
main strength when applied to problems with;
(a) a large number of manipulated and controlled variables;
(b) constraints imposed on both the manipulated and controlled variables;
(c) changing control objectives and /or equipment failure (sensor/actuator); and
(d) time delays.
The strengths of MPC that are relevant to the task of power system stabilisation are the
explicit handling of constraints. Predictive controls based on the state space model can
be dealt with in terms of RHC instead of MPC although MPC based on the state-space
model is same as RHC. The advantages of RHC control as given in [73]:
(a) Applicability to a broad class of systems. The optimisation problem over the
finite horizon, on which RHC is based, can be applied to a broad class of systems,
including nonlinear systems and time-delayed systems. Analytical or numerical
solutions often exist for such systems.
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(b) Systematic approach to obtain closed-loop control. While optimal controls for
linear systems with input and output constraints or nonlinear systems are usually open-
loop controls, RHCs always provide closed-loop controls due to the repeated
computation and the implementation of only first control.
(c) Constraint handling capability. For linear systems with the input and state
constraints that are common in industrial problems, RHC can be easily and efficiently
computed by using mathematical programming, such as quadratic programming (QP)
and semidefinite programming (SDP). Even for nonlinear systems, RHC can handle
input and state constraints numerically in many cases due to optimisation over finite
horizon.
(d) Good tracking performance. RHC presents good tracking performance by
utilising the future reference signal for a finite horizon that can be known in many cases.
In infinite horizon tracking control, all future reference signals are needed for the
tracking performance. However, they are not always available in real applications and
the computation over the infinite horizon is almost impossible. In PID control, which
has been most widely used in the industrial applications, only the current reference
signal is used even when the future reference signals are available on a finite horizon.
This PID control might be too short-sighted for the tracking performance and thus has a
lower performance than RHC, which makes the best of all future reference signals.
There are many such advantages listed in [73] such as adaption to changing parameters
which is very important from a power system application point of view where power
system topology may change after disturbance.
4.2.6 Predictive control developments in literature
MPC was popularised in the 1970s for control of petroleum refinery operations, which
often operate at constraints on manipulated variables or controlled variables. Since then,
MPC has become the benchmark for complex constrained multivariable control
problem in the process industries. A good literature review of past, present and future of
model predictive control is presented in [78-82] . MPC and RHC are forms of control in
which current control action is obtained by online solving at each sampling instant using
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the current state of the plant as the initial state. The sequence and the first control in
this sequence is applied to the system [83]. Linear MPC refers to a family of MPC
schemes in which linear models are used to predict the system dynamics even though
the dynamics of the closed-loop system is nonlinear due to the presence of various
constraints. The linear MPC approach has found successful applications, especially in
the process industries as reported in [84] .
The authors in [85] proposed a predictive control approach for the optimal control of
nonlinear systems, in which it has been claimed that the main features of an explicitly
analytical form of the optimal predictive control are - online optimisation is not
required; stability of the control-loop system is guaranteed, the whole design procedure
is transparent to designers; and the resultant controller is easy to implement. By
establishing the relationship between the design parameters and time-domain transients
it is shown that the design of an optimal generalised predictive controller to achieve
desired time-domain specifications for nonlinear systems can be performed by looking
up tables.
In a survey paper of theory and practice in MPC [82], authors have addressed the
important issues for any control system and then reviewed a number of design
techniques emanating from MPC, namely Dynamic Matrix Control, Model Algorithmic
Control, Inferential Control and Internal Model Control. These are put in perspective
with respect to each other and the relation to more traditional methods like Linear
Quadratic Control is examined.
It has been observed that nonlinear model predictive control is an attractive strategy for
controlling complex systems as it offers good dynamic performance while ensuring
operation within certain physical limits. This feature enables the system operator to run
the system near constraint boundaries, which can increase productivity without
sacrificing quality [86]. In general, MPC computes an optimal sequence of manipulated
inputs, which minimises a tracking error, i.e. the difference between the desired
reference output and its real value, subject to constraints on inputs and outputs. As
explained in [61, 62], this can be formulated in the continuous time domain for a
general case, applicable to nonlinear systems.
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Nonlinear generalised predictive control has been observed to provide a powerful means
of stabilising power systems following large faults. The controller is obtained by
optimising a quadratic criterion over a fixed horizon, using nonlinear reference models
of the power system. Normally, the feedback loop is closed with the first control, and
the computations are repeated at the next sample. This scheme has the disadvantage of
requiring large horizons to assure stability. In [87], it is mentioned that appropriate
selection of the reference trajectory permits the use of short prediction horizons for the
controller to conduct the power system states to a neighbourhood of the post-fault
equilibrium. In this region, local asymptotic stabilisation is provided by the linear
controller.
Though the predictive control method is well established in chemical and control
engineering it has some major issues which will be discussed in next the two sections of
this chapter. The first section will discuss the control strategy related issues while the
second section will highlight the issues focused on application of predictive control to a
power system point of view.
4.3 MAJOR DRAWBACKS OF PREDICTIVE CONTROL
The two major drawbacks of predictive control are: firstly by, its open control loop
nature, and secondly by the computational burden associated with the solution of the
optimisation problem. The first obstacle can be overcome by introducing an implicit
feedback in the form of repetitive computation of control laws in a receding horizon
manner which has been proven for infinite horizon control. The second obstacle has
restricted the application of MPC, mostly to control slower processes with the
dynamics, in an order of minutes (such as in the chemical industry). This is also
probably the factor limiting a wider spreading of MPC in power systems up to now
[43]. Getting an appropriate model of the system is also one of the significant limitation
as mentioned in [74].
In order to solve the predictive control problem, there must be a way of computing the
predicted values of the control variables. This must cover from the best estimation of
the current state and the assumed future inputs as well as the latest input and the
assumed future changes. The way in which the predictions are made has great effect on
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the performance of the closed-loop system running under predictive control. So the
choice of prediction strategy is another ‘tuning parameter’ for predictive control, just as
are choice of horizon and cost functions. Furthermore, the prediction strategy follows in
a rather systematic way from assumptions made about disturbances acting on the system
measurements errors such as noise. Hence, it can be said that rather than choosing a
prediction strategy, a model of the environment is specified.
4.3.1 Stability issue
Predictive control, using the receding horizon idea, is a feedback control policy. There
is therefore a risk that the resulting closed loop might be unstable. Even though the
performance of the plant is being optimised over the prediction horizon, and even
though the optimisation keeps being repeated, each optimisation ‘does not care’ about
what happens beyond the prediction horizon, and so could be putting the system into
such a state that it will eventually be impossible to stabilise. This is particularly likely to
occur when there are constraints on the possible control input signals. The problem
arises because the prediction horizon is too short-sighted, and it turns out that stability
can usually be ensured by making the prediction horizon long enough, or even infinite.
Another way of ensuring stability is to have any length of horizon, but to add a terminal
constraint, which forces the state to take a particular value at the end of the prediction
horizon.
Stability has been one of the main problems in MPC, ever since early MPCs for linear
systems were criticised for their loss of stability [84, 85]. This problem has been solved
for linear systems in various ways such as infinite horizon predictive control, terminal
constraints and the fake algebraic Riccati equations (FARE). Though there are some
good results shown in literature for nonlinear system for addressing stability issues [88],
from a computational point of view solving a nonlinear dynamic optimisation problem
with equality constraints is highly computationally intensive, and in many cases is
impossible to perform within a limited time. In [88], Chen et al, developed a practical
MPC with guaranteed stability for general nonlinear systems. It tried to address several
issues in the implementation of nonlinear MPC, including computational delay, loss of
optimality in the optimisation procedure and stability, but computational burden was not
considered.
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The key feature of all model based predictive control methods is open loop optimal
control rather than closed-loop control in a moving horizon. Application of the MPC
concept to nonlinear systems like power systems leads, in general, to involved nonlinear
programming problems. In general, the optimisation problem is nonconvex and leads to
many difficulties impacting on implementation of MPC. These difficulties are related to
feasibility and optimality, computation and stability aspects. The important distinction
in nonlinear is not linear versus nonlinear, but rather convex versus nonconvex. If the
resulting nonlinear optimisation problem is convex, there exist methods which ensure
convergence to a global minimum, which is unique if the performance criterion is
strictly convex.
On the other hand, if the system to be controlled is nonlinear, even if the cost function
and constraint sets are convex, the control problem will be, in general, a nonconvex
nonlinear optimisation problem. Therefore, finding a global optimum can be a difficult
and computationally very demanding task. In other words, non-convexity makes the
solution of the nonlinear programming uncertain [89].
To overcome the limitations of nonlinear and linear controllers, a combination of both
linear as well as nonlinear controllers is suggested in [87]. When the faults concerned
are large, the proposed nonlinear predictive controller based on TCSC returns the power
system state to a small neighbourhood of the post-fault equilibrium. In this region,
linear controllers are designed to provide effective damping to the origin. The
coordination of linear and nonlinear controller offers powerful means of extending the
stability limit of power systems.
4.3.2 Choice of horizon
One of the main issues in optimal control is whether or not the closed-loop system
under derived optimal control law is stable. As discussed in [72] the first obvious option
can be an infinite horizon technique to minimise the performance objective determined
by cost. However, the open-loop optimal control problem, that must be solved online, is
often formulated in finite horizon. The input is parameterised finitely in order to allow a
real-time numerical solution of the nonlinear open-loop control problem. It is obvious
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that the shorter the horizon the less costly the online optimisation solution. Thus, from a
computational point of view it is desirable to implement a predictive control with a
short horizon. However, when a finite prediction horizon is used, the actual closed-loop
input and trajectories will differ from the predicted open-loop trajectories, even if no
model mismatch and no disturbances are present.
Fig.4.1 (a) and (b) shows the basic difference in finite and infinite receding horizon
formulation of predictive control. With reference to Fig.4.1, at time k, a particular
trajectory is optimal over the prediction horizon Hp at time (k+1) with no disturbance
and no model mismatch. The system will be in same state as it was predicted at the
previous time step. With this initial observation, it is expected that the optimal trajectory
over prediction horizon from time (k + 1) to (k +1+Hp) should coincide with previously
computed optimal trajectory. But in previous optimisation the interval between (k + Hp)
to (k +1+ Hp) was not considered. As a result when a new time interval enters it may
give a totally different optimal trajectory than what was computed in earlier steps.
(a)
(b)
Fig.4.1. (a) and (b) Finite horizon and Infinite horizons with no disturbance and perfect
model respectively
On the other hand, with second case, at time k, an optimal trajectory is determined for
the entire infinite horizon implicitly. As a result, at time (k+1), no new information
enters in the optimisation problem and optimal trajectory continues as the tail of the
previously computed trajectory. The Bellman’s principle of optimality states that the tail
of any optimal trajectory is itself the optimal trajectory from its starting point which is
thus applicable to the infinite horizon problem, while, the finite horizon it does not
apply because at every step new optimisation problems arise.
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With reference to above discussion, it can be concluded that short prediction horizon
gives rise to too short-sighted control and may also give rise to stability issues while
stability can be ensured using longer horizons. However, as the horizon length increases
the computational burden increases making solution costlier. In addition, too long a
prediction horizon can degrade the performance as errors in prediction are also large
[90]. In such a scenario, the stability can be ensured by three ways-first by imposing
terminal constraints, second using infinite horizons as explained above and third, the
FARE approach.
4.4 LIMITATIONS OF MPC APPLICATION TO POWER SYSTEM
There are many difficulties that limit the use of this kind of model such as:
(a) computational load in applying MPC to large systems with fast time constants;
(b) lack of identification techniques for nonlinear processes;
(c) requirement of an appropriate system model.
The general tools for nonlinear MPC are not necessarily well developed for the specific
nonlinearities of the power system. The major disadvantage associated with RHC, is
longer computation time compared with conventional nonoptimal controls. In [75],
Zima and Anderson have discussed various techniques in relation to minimisation of
computation efforts. The first option given is, the use of full nonlinear mode and
computations which is most accurate but most time consuming also. This type of
approach can be applied for small sized systems with slow dynamics. The second
frequently used method is to linearise the system equations around the present operating
(equilibrium) point and apply linear MPC. Discrepancy between the linearised model
and the actual system behaviour is then compensated in the next controller step.
However, this involves a risk that the system may undergo large excursions from the
optimal trajectory and even violate imposed constraints. The third approach is based on
approximation of the expected trajectory (Euler prediction) of the system if control
inputs would remain unchanged and the numerical computation of sensitivities of
control inputs impact on this trajectory.
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Camponogara, in [90] has discussed distributed model predictive control from a power
system point of view. Typically, MPC is implemented in a centralised fashion. The
complete system is modelled, and all the control inputs are computed in one
optimisation problem. This is possible for small size plants such as in the chemical
industry where it is possible to get all measurements. In large-scale applications, such as
power systems, it is useful (sometimes necessary) to have distributed or decentralised
control schemes where local control inputs are computed using local measurements and
the reduced-order models of the local dynamics. With this background, [90] has tried to
achieve some degree of coordination among agents that are solving MPC problems with
locally relevant variables, costs and constraints, but without solving centralised MPC
problem. Such coordination schemes are considered to be useful when local
optimisation problems are much smaller than a centralised problem, such as new
deregulated power markets. A power system with a two area case is studied in [90] for
load-frequency problems using the distributed MPC technique considering
communication network and coordination problems in power systems. However, the
power system model used is very much simplified.
As concluded in [14] that centralised or global MPC may face some difficulties such as
lack of knowledge from system and computation costs because of complex and highly
nonlinear structure of power systems, the authors have developed a new approach to the
control angle difference of multi-machine systems using a combination of MPC and
energy function. However, the power system model used is again a simplified low order
dynamic model. Centralised MPC is not well suited for control of large-scale,
geographically expansive systems such as power systems. However, the performance
benefits obtained with centralised MPC can be realized through distributed MPC
strategies. Such strategies rely on decomposition of the overall system into
interconnected subsystems, and iterative exchange of information between these
subsystems [91]. As TCSC devices are mainly installed in long distance, high voltage
AC transmission system, [70] has considered simplified models of interconnected
power systems for developing nonlinear optimal predictive controller for TCSC and
then extended it for multi-machine power systems with some approximations and
assumptions.
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4.4.1 Linear model for nonlinear system issue
For computational reasons, MPC applications largely have been limited to linear models
i.e. those in which the dynamics of the system models are linear. Such models often do
not capture the dynamics of the system adequately, especially in regions that are not
close to the target state. In these cases, nonlinear models are necessary to describe
accurately the behaviour of physical systems. From an algorithmic point of view,
nonlinear model predictive control (NMPC) requires the repeated solution of a
nonlinear optimal control problem.
A fundamental difficulty with the NMPC approach is that the implementation platform
must be capable of solving a constrained optimisation problem within a specified time
limit. This time decreases as the speed of the dynamics to be controlled increases. As a
result, the implementation of NMPC has, to date, been generally limited to plants with
slow or otherwise very simple dynamics so that the time constraints in computing a
solution are relaxed. Surmounting this difficulty of computational overhead to achieve
the benefits of MPC for linear systems has attracted research attention [86].
The relationship between the linear model given in Appendix D by (D1-D3) and the real
system need careful consideration of predictive control. In most control methodologies
the linear model is used offline, as an aid to analysis and design. In predictive control it
is used as part of the control algorithm, and the resulting signals are applied directly to
the system. Therefore careful attention must be paid to appropriate treatment of
measurements before using them in the control computation algorithm, and the
computed control signal.
The two major problems of this computational burden are first, large computational
delay and second, achieving global and sometimes even local minimum in a given time
limit of each optimisation cycle. To tackle the online computational issue, one solution
is proper choice of prediction horizon. However, it may be difficult to predict system
output for a longer horizon and a smaller horizon may not guarantee required close-loop
stability [72]. Though there have been many efforts to extend MPC from linear systems
to nonlinear systems the two major obstacles to the extension of MPC from linear to
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nonlinear systems like power systems are: stability and the online computational burden
[88].
In [87], Rajkumar has proposed a high performance nonlinear predictive controller
using TCSC for the stabilisation and damping of multi-machine power systems. The
controller is designed using the classical model of power systems and effective means
of extending the stability limit is achieved by such linear and nonlinear controller
coordination. After a large disturbance, the nonlinear predictive controller based on
TCSC, brings the system to a small neighbourhood of the post-fault equilibrium. Linear
controllers will then provide effective damping to the origin.
4.4.2 Use of single machine equivalent
It has been observed that, to overcome computational requirements and avoid
complexity of power systems, the control strategies are developed based on a single
machine equivalent model. The methods relying on a one-machine infinite bus (OMIB)
equivalent are based on the observation that the loss of synchronism of a multimachine
power system originates from the irrevocable separation of its machines into two groups
that they successively replace by a two-machine equivalent and further by a one-
machine infinite bus equivalent. Thus an OMIB may be viewed as a transformation of
the multidimensional multimachine dynamic equations into a single dynamic equation.
Depending on power system modelling and the assumed behaviour of the machines
within each group, OIMB can be distinguished in three types as explained in [92, 93] as
time-invariant, time varying and generalised. The time-invariant OMIB is based on
assumption of simplified power system model and coherency of the machine within
each one of the two groups, while time-invariant is based on simplified model but no
coherency in the group. Using detailed power system model instead of simplified one is
categorised as generalised OMIB.
With reference to various advantages and disadvantages [92], generalised OMIB is
found much better for quantifying the severity of instability and also deciding required
compensation to make such a system stable. To restore transient stability [94] has
proposed a well-behaved optimal power flow model with embedded transient stability
constraints which can be used for both dispatching and re-dispatching. The transient
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stability constraints are formulated by reducing the initial multi-machine model to a
one-machine infinite-bus equivalent FACTS controller input signal derivation.
4.5 CONCLUSIONS
Starting from a basic introduction, the chapter has discussed various strengths and
limitations of predictive control strategies to develop its application. After critically
reviewing the problems associated with predictive control based methods to power
systems, first an attempt will be made to develop an RHC-based TCSC controller for a
single-machine-infinite-bus system for improving transient stability improvement.
It has been seen that insertion of thyristor controlled series compensation (TCSC) can
lead to increased power transfer and in turn transient stability. Following the
disturbance, the entire system dynamics change and many parameters undergo variation
which need to be controlled, but the relative rotor angle of synchronous machine is the
key parameter which will decided the overall dynamics of the perturbed system. As per
the first swing stability criteria, the relative rotor angle swing will be an ideal indicator
of system stability. The issues related to the prediction scheme will be selected to
maximise the performance and assure system stability in post-fault conditions.
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Chapter 5 MPC-based TCSC Controller for
Power System Transient Stability
Improvement
5.1 INTRODUCTION AND OBJECTIVES
The previous chapter has given a basic introduction to predictive control strategies and
discussed their strengths and limitations with application to power systems. This chapter
focuses on developing an RHC-based TCSC controller, to address the power system
stability related problem, using MPC controller.
It has been seen that insertion of TCSC can lead to increased power transfer and in turn
transient stability. Following a disturbance, the entire system dynamics change and
many parameters undergo variations which need to be controlled. The relative rotor
angle of synchronous machine is the key parameter which will decided the overall
dynamics of a perturbed system. As per the first swing stability criteria, the relative
rotor angle swing will be an ideal indicator of system stability. The issues related to the
prediction scheme will be selected to maximise the performance and assure system
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stability in post-fault conditions. In current literature where RHC controllers are
developed for power system, in most cases, the model used is classical. This has
limitations which does not give insight in machine dynamics which plays important role
in system stability. Knowing the various pros and cons of RHC related methods and
their applications in power system, an attempt has been made here to apply RHC
controller to power system transient stability in order to maintain its detailed fifth order
rotor dynamic model, having 10 variables including exciter and prime-mover systems.
In order to improve the transient stability performance, early detection of faults, and fast
fault clearance is most important. In an initial attempts reported in [95], a bang-bang
type control of switched series capacitors was proposed but the technique for
determining the required switching instant was not given. In addition, the use of fixed
series capacitors can give rise to the risk of sub-synchronous resonance (SSR) [24].
However, with recent advanced developments in power electronics, the FACTS devices
can be used for power system stability enhancement in healthy or post-fault conditions,
as explained in [47, 96].
FACTS devices of both the shunt and series form can contribute to stability
enhancement. However, it is accepted in general that series compensation is more
effective in improving or maintaining system transient stability. Among the FACTS
devices-based series compensators, the TCSC is the most popular, and is extensively
used in power systems, particularly where long-distance transmission interconnections
are required. A TCSC is directly connected in series with the transmission line for
which it provides the compensation, without the use of a coupling transformer.
At present, the reference input to a TCSC, either active-power reference or reactance
reference, is determined, based mainly on a steady-state operating condition and offline
calculation. Small-disturbance stability enhancement is achieved by a supplementary
damping controller (SDC) in conjunction with the TCSC main controller. There has
been extensive research on the design of SDCs, including the online and adaptive tuning
of their parameters [97]. However, research on the real-time control of TCSC for
enhancing or maintaining power system transient stability following a large disturbance
has been very limited, where a simplified power system is adopted for forming the
control law [13]. It is acknowledged that the real-time and optimal control of the
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TCSC reactance reference is the most effective means of transient stability enhancement
[12-14] .
In this chapter an attempt has been made to derive the control law based on the power
system detailed dynamic model in the prevailing operating condition. Based on the
concept of receding horizon control (RHC) [12, 13] the overall control is subdivided
into a series of time horizons, and the power system dynamic model is used in a
predictive mode in each time horizon. At the start of each time horizon, for which
current system dynamic response is available via wide-area measurements (WAMs),
subsequent system responses within the time horizon are predicted using the system
dynamic model, and are optimised, subject to FACTS devices operating limits, to obtain
the optimal reference inputs to FACTS devices controllers. The objective function in the
optimisation represents the system dynamic performance index (DPI) expressed in
terms of relative rotor angles. The relationship, required in forming the objective
function, between the DPI and the FACTS controllers input references, is derived
through the linearisation of the power system model around the current operating point.
The variables in the relationship derived are the FACTS controllers input references,
and their optimal values obtained from the optimisation are used for setting the input
references to the FACTS devices controllers. The RHC is applied repeatedly for
successive time horizons in each of which an optimal set of FACTS devices input
references is derived and implemented.
The control algorithm proposed is implemented in software, and then tested by
simulation with a single-machine-infinite-bus (SMIB) power system having a TCSC.
The results presented in the chapter indicate the effectiveness of the control
methodology proposed. There is definitely a need for predicting in advance where the
system goes before the control actions are deployed [46] i.e. a look-ahead approach. A
simulation environment is proposed using fresh real-time information from the EMS, in
addition to historical data recordings that can be fed into the online dynamic simulation.
The accuracy of the system state prediction simulation environment depends on how
reliable our schedules and limits are in the short-term time frame. Under catastrophic
conditions a window of approximately few minutes ahead may help to reduce the risk of
system operation.
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The fast state prediction simulation and, a change of simulation scope and region of
interest may help. The state prediction could be triggered automatically (or on demand)
to assess control movements and reconfiguration plans under disturbances. A lot of
potential problems can be caught after control deployment. For emergency conditions,
models including only the measurements that can be delivered in the quickest time to
the control centre are required. These measurements may include items such as:
important generator outputs, frequency and voltage measurements, important flow
measurements, most important high transmission backbone lines internal to a specific
control area; and important new measurements such as angle measurements. The goal is
to estimate where the system is heading and what the security level is going to be.
The system recovery depends very much on new changed system topology. The online
dynamic simulation will recognise and characterise situations, it will predict collapse
and unstable behaviour and it will recommend re-adjustment to prevent, or corrections
to recover, from failures. The need for high frequent solutions of a quadratic program
and repeated linearisation of the nonlinear model determines the main computational
load for MPC.
5.2 AIM OF PROPOSED METHOD
The SMIB system is represented by two mechanical axes or swing equations, three rotor
flux equations, three differential equations describing excitation dynamics and two
differential equations for prime-mover and governor dynamics. The controller
coordination between FACTS devices, power system dynamics and proposed MPC
controller is shown in Fig.5.1.
The output of the controller will decide the input reference for FACTS device for which
this predictive controller needs some reference trajectory to follow which is represented
by Yref in Fig.5.1 above. The reference output Yref, may be pre-selected on the basis of
contingency studies. Keeping the reference output fixed at the post-fault equilibrium
may impose large computation burdens on the controller, since a large horizon may
have to be selected to assure a solution to the minimisation problem. Different options
for the selection of the reference output, such as e.g. keeping it fixed at some value
close to the post-fault equilibrium, can be helpful in realising the control objectives with
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shorter horizon. Though, in the present study, reference output is kept the same as that
of pre-fault system status, it can be changed as per the changed power system network
or any other value based on selected criteria. The system performance is observed in
every stage of pre-fault, during-fault and post-fault. In post-fault the system stability is
observed when fault clearing time is greater than critical clearing time.
Model Predictive ControllerOriginal Nonlinear system model
Linearized Model for optimization
Quadratic programming QP problem
Cost function and Constraints
𝛥x
Y
YrefY
FACTS
Power System
Fig.5.1 Proposed strategy for RHC-based TCSC controller
5.3 POWER SYSTEM MODELLING
The power system which incorporates both continuous dynamic and discrete events can
be divided into the two parts. Firstly, the continuous dynamical system which is
modelled as a differential algebraic equation (DAE) system which is formed from two
ordinary differential equations (ODE) of the load and the algebraic equations (equality
constraints). The ODE as well as the model of the algebraic equations is nonlinear.
Additionally, the saturation of the internal Automatic Voltage Regulator (AVR) of
generators can be included as part of the continuous dynamics.
As proved in various studies [98] that rotor angle trajectory is the dominating factor in
determining stability, in the present proposed scheme, rotor angle is considered as
stability performance index. Similarly, for the choice of FACTS devices which can
provide effective control in power system stability improvement, TCSC is preferred
over others. One of the reasons why TCSC is preferred is because of its distinctive
quality of extremely simple main circuit topology. The capacitor inserted directly in
series with the transmission line and thyristor controlled inductor mounted directly in
parallel with this capacitor, thus requires no interfacing equipments such as high voltage
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transformers. This makes TCSC more economical than other competing FACTS
devices. There are various advantages of TCSC as compared to other FACTS devices
which are addressed well in literature [18, 24, 99].
In the first part of the thesis, as the focus is to use a nonlinear dynamic model and
coordinate it with the MPC controller with FACTS devices for first swing control, a
simplified single machine system is considered. To keep the complexity within the
limited range, TCSC is represented as a simple variable reactance. In the second part of
the thesis, multimachine systems and detailed dynamic models of TCSC will be
developed and considered.
5.4 RHC ALGORITHM
The RHC module solves online a constrained optimisation problem and determines an
optimal control input over a fixed future time-horizon, based on the predicted future
behaviour of the system and on the desired reference trajectory. As by now, linear MPC
theory is quiet mature and important issues such as online computation, the interplay
between modelling/identification and control and issues like stability are well addressed
[84]. In this proposed method, the predicted future system behaviour is represented as
the sum of a nonlinear prediction component and a component based on linear time-
varying models defined along the reference trajectory, which needs to be tracked. The
first component constitutes a future output prediction using non-linear simulation
models, given initial system inputs and disturbance history. The second component uses
linearised models for prediction of future process output as required for calculation of
optimum future system manipulation.
The constrained optimisation problem leads to a quadratic programming problem which
is a convex optimisation problem. The status of real-time optimisation is transferred to
the RHC to ensure feasibility and proper functionality of all system components. A need
for high frequent solutions of a quadratic program and the repeated linearisation of the
nonlinear model determine the main computational load for RHC. As explained in [71,
73] the RHC concept is well suited for finding control laws in an optimal way for
hybrid systems like power networks.
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The concept of RHC [72, 100] in the discrete time-domain is that of subdividing the
control into a number of durations each of which is referred to as a time horizon with a
nominated number of time steps. The aim of RHC is to derive a set of control variables
for individual time steps within a time horizon. The control variables derivation is based
on the optimisation of the system responses in a given time horizon. If a particular time
horizon to be considered at the present time, tp, is [ ]tNttttt ppp ∆+∆+∆+ ,,2, where
tt p ∆+ is the start of the horizon, t∆ the time step, and N the number of time steps in
the horizon, then the optimal set of values for the control variables, following the
optimisation, are represented as ( ) ( ) ( ){ }tNttttt ppp ∆+∆+∆+ uuu ,,2, . One option is
to implement these control variable values at time instant
[ ]tNttttt ppp ∆+∆+∆+ ,,2, respectively and then to move on to the next time
horizon starting at time ( )( )tNt p ∆++ 1 . However, this option might present a problem
if within the horizon ( ) ( ) ( ){ }tNttttt ppp ∆+∆+∆+ ,,2, , there are events occurring after
pt , which have not been represented in the system model at time pt , used for response
evaluations and optimisation. The second option is to implement the control value
( )tt p ∆+u at time tt p ∆+ , and move on to the next horizon which starts at time
tt p ∆+ 2 , i.e. the time horizon to be considered is ( )[ ]tNttt pp ∆++∆+ 1,,2 . The
system model is then updated for time tt p ∆+ , and the RHC procedure is then repeated
for the horizon starting from tt p ∆+ 2 , which will lead to the optimal value for the
control vector at time tt p ∆+ 2 to be implemented.
5.5 LINEARISATION AND OBJECTIVE FUNCTION
In principle, the nonlinear power system model as developed in Chapter 2 can be
represented by the following sets of differential equations and algebraic equations as:
), uyh(x,x =•
(5.1)
( ) 0uy,x,g = (5.2)
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In (5.1) and (5.2), h and g are, in general, nonlinear vector functions; x is the vector of
state variables arising from modelling synchronous generators together with their
controllers, FACTS devices and dynamic loads; y is the vector of network nodal voltage
variables, and u is the vector of control variables to be optimised in real time for
maximising transient stability margins. In the present work, the control variables
represent the input references to the main controllers of FACTS devices which have
been selected for participating in the control.
Equation (5.1) and (5.2) can be applied, through a numerical integration, for predicting
the system responses required in individual time horizons. However, it is, particularly in
terms of computing time, difficult, if not impossible, to carry out the optimisation of the
system responses which are implicit nonlinear functions of the control variables. To
avoid this difficulty, the MPC method is based on the linearisation of the nonlinear
model in (5.1) and (5.2) by which the state variables describing power system dynamic
responses are expressed explicitly in terms of a linear vector function of control variable
vector u. This allows the prediction of the system responses within each time horizon in
terms of the variation of the control variables, which provides the basis for forming the
objective function to be minimised. A representative objective function often quoted in
control theory literature is
( )( ) ( )( )∑∑∑∑
====∆+∆+−∆+
N
jpiij
L
i
N
jrefkpkkj
M
ktjtuWZtjtZW
1
2
11
2
1 (5.3)
In (5.3), kZ ’s (for k= 1, 2, . . . , M) denotes the power system responses selected for
optimisation which are formed approximately using the responses at time tp and the
deviation predicted by the linearised system model; refkZ represents the values of the
system responses in the pre-disturbance condition, and the second summation in (3) has
the purpose of minimising the deviation of the control vector from that at time tp. The
objective function in (5.3) is a quadratic function in ( )tjtu pi ∆+∆ (for i= 1, 2, . . . , L
and j= 1,2, . . . , N). Constrained optimisation based on the quasi-Newton method is
directly applicable for minimising the objective function subject to bounds imposed on
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the control variables, to identify the optimal values for ( )tjtu pi ∆+∆ from which the
total values of the control variables are formed and implemented. If the second option
referred to in Section 5.4 is adopted, the control vector to be implemented at time
tt p ∆+ is ( ) ( )ttt pp ∆+∆+ uu , and the RHC procedure will be repeated for the next
horizon, starting at time tt p ∆+ 2 as described in the algorithm of Section 5.4.
5.6 RHC ALGORITHM FLOW CHART
Fig.5.2 shows the complete block diagram for implementation of a RHC controller
applied to the SMIB system for transient stability improvement. At the start of any RHC
time horizon, the power system configuration together with its operating state is
available via WAMs and current statuses of the circuit breakers and isolators. This in
conjunction with the system database allows the construction of the nonlinear power
system model relevant to transient stability analysis and simulation. With the RHC
algorithm adopted, linearisation is then required to form the objective function for
minimisation. The outcome of the minimisation is the updated value of the control
variable, which in this case is the TCSC reactance reference. The updated control
variable is sent to the TCSC main controller, for adjusting its input reference as shown
in Fig.5.2. The RHC sequence is repeatedly applied for successive time horizons as
indicated in the loop shown in Fig.5.2.
The solution for output of the optimisation problem is a required TCSC reactance
setting value which will be used to set the input reference for given FACTS devices, i.e.
in this case TCSC. The TCSC input reference is adjusted as per the output of the
controller which will be inserted in series with the transmission line to modify its
reactance to improve system performance in post-fault conditions. The updated system
variables are again used to form a power system model for next time instant and entire
process will be repeated for every time step to update control law, FACTS devices input
references as per changed system conditions by using principle of receding horizon.
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Power systemWAMs
Circuit breakers/isolators statuses Database
Nonlinear power system model
Linearisation and prediction
Objective function formulation and
Constrained minimization
Updating the FACTS-devices control variables
FACTS Devices
New Xtcsc optimized value
Fig.5.2 Flowchart of RHC implementation algorithm
5.7 SIMULATION RESULTS
The general RHC algorithm developed in Section 5.2 is applied to the single-machine-
infinite-bus power system with a TCSC as shown in Fig.5.3. The system data is given in
[101]. For dynamic simulation of the power system and the RHC, the fifth-order
generator model [10, 101] is used. The excitation controller is based on IEEE Type-
ST1, the prime-mover and governor is as per [9] and the TCSC, represented in terms of
variable reactance associated with reactance control mode, is adopted from [55].
Fig.5.4 shows the variation of TCSC reactance reference derived from the output of the
RHC. The effectiveness of the control applied to the TCSC input reference is confirmed
in the rotor angle response of Fig.5.5 which indicates that the system transient stability
is maintained throughout the transient operating period, and a new steady-state
condition is reached after about three seconds subsequent to the fault onset.
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Fig.5.3 Single-machine-infinite-bus system
The TCSC operates in the reactance control mode. Based on this, the control variable in
the study is that which represents the TCSC reactance reference input. For transient
stability control, the system response selected for forming the objective function in (5.3)
is the generator rotor angle, with the infinite bus as the reference. The system details
and data are given in Appendix E.
The simulation is carried and analysed with two cases. A three-phase-to-earth fault is
applied at bus 3, with the fault clearing time of 200 ms. With reference to the time
origin of Fig.5.4, the fault starts at time 0.5s. The time step length adopted in the study
is 10 ms. The first case study is that when there is no transient stability control by RHC
where the TCSC reference input remains fixed at the pre-fault value. The rotor angle
response in Fig.5.4 indicates that the system transient stability is lost after fault and fault
clearance. The second case study represents the RHC for optimally adjusting the TCSC
reactance reference input signal, based on the control sequence in Fig.5.5.
Fig.5.4 Generator rotor angle response without RHC controller
0 1 2 3 4 50
500
1000
1500
2000
Time (s)
Del
ta (D
eg)
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Fig.5.5 TCSC reactance reference input using RHC controller
Fig.5.6 Synchronous generator rotor angle response using RHC based TCSC controller
5.8 CONCLUSIONS
The chapter has developed a comprehensive, flexible, and systematic method for the
formulation of RHC-based TCSC control law, together with its software
implementation which has been validated with many case-studies on SMIB of which
representative results are presented in this chapter.
0 1 2 3 4 5
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Time (s)
TCSC
Rea
ctan
ce (p
u)
0 1 2 3 4 50
50
100
150
200
Time (s)
Del
ta (D
eg)
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The simulation is carried out in two parts, first without a RHC controller, which shows
continuous increase of rotor angle with rapid rate, leading systems to a ‘run away’
situation because of acceleration. The second part of the simulation is carried out with a
RHC controller switched on after a clearing fault, where the fault-clearing time is again
the same as that of case one, which is greater than critical clearing time. It has been
observed that as the RHC-based TCSC controller provides the required series
compensation the system regains its stability even if the fault is cleared after the critical
clearing time.
The research and results presented in the chapter have confirmed that it is possible to
form control law which is adaptive to power system operating condition, and effective
in improving or maintaining its transient stability. This is achieved by directly deriving
the relationship between the relative rotor angle and the control variable through
linearisation in individual time horizons, which leads to the objective function to be
minimised for forming successive optimal values for the control variable.
The further development in this research is to extend this strategy for multi-machine
systems with large numbers of generators. The major concerns in this development are
first, due to computation burden because of scale of operations and requirements, the
optimisation problem to be solved online are large. Secondly, the solution must be
obtained in a limited amount of time because it is implemented in a receding horizon
fashion. The third aim is to attempt to develop this RHC algorithm with capabilities for
control of different system dynamics and for the various constraints handling along with
good computational efficiency in optimisation to allow online RHC application.
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Chapter 6 Overview of Real Time Controllers
for Power System Stability
Improvement
6.1 INTRODUCTION
The first part of this thesis (Chapter 2 to Chapter 5) has covered detailed dynamic
modelling of various power system components along with FACTS devices as
compensating devices. A new predictive control based FACTS controller is developed
for enhancement of transient stability performance of power systems. Although the
proposed predictive scheme for FACTS controller has shown its effectiveness by
applying linear models for a nonlinear power system and validated it for a single-
machine infinite bus, in reality, power systems are inherently nonlinear. Together with
higher quality specifications increasing demand, tighter environment regulations and
demanding economic considerations are pushing power systems closer to the
boundaries of the admissible operating regions. In such circumstances, linear models are
often inadequate to describe the system dynamics, and nonlinear models must be used.
In addition, the simplified dynamic models of components which are used to overcome
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the problem of computation requirement may not reflect correct system status which
might lead to wrong prediction and in turn system stability. This has motivated the
development of the second part of the thesis in which an online control coordination
strategy is developed considering nonlinear models for a real multimachine system with
real time requirements.
To address these limitations, the second part of thesis will consider the requirements for
real multimachine systems, including their detailed dynamic models for accuracy. It will
also consider computation time requirements and computation burden that computing
systems should have for real-time application of controllers. With reference to this,
given that recent advances have been made in computer technology and wide-area
measurement systems, the second part of the thesis is devoted to real-time controllers
and their requirements. Starting with the review of real-time controllers this chapter will
discuss the requirements of real-time power system controllers, including those of
FACTS devices, for achieving power system stability. With the background of reviews
presented in this chapter, the next chapter will develop new online control coordination
of FACTS devices for transient stability improvement for a realistic multimachine
power system network.
6.2 WIDE AREA NETWORK OPERATION
The growing trend towards restructuring the power industry and the ever increasing
demand for power exchange calls for the employment of WAMS for near to real-time
measurements to maintain or improve the stability of the system. To achieve technical
and economical advantages, power systems have been extended by interconnections to
the neighbouring systems. Regional systems have been built-up towards national grids
and later to interconnected systems with the neighbouring countries. Such large systems
came into existence, covering parts of or even whole continents, to gain the following
well known advantages:
(a) Reduction of reserve capacity in the systems (i.e. less spinning reserve);
(b) Utilisation of the most efficient energy resources;
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(c) Parallel operation to share power in peak and off peak demand periods making
power generation economical as well as efficient;
(d) Reliability and continuity along with the quality of supply
With the above advantages of interconnection, generators and loads that are over
thousands of miles apart are connected to form a large system. As a result, the general
configuration of a modern power system consisting of generation, transmission and
distribution is often geographically dispersed.
The general trend in power system planning utilises tight operating margins, with less
redundancy because of new constraints placed by economical and environmental
factors. At the same time, factors such as - addition of non-utility generators and
independent power producers, an interchange increase; an increasingly competitive
environmental; and the introduction of FACTS devices make the power system more
and more complex to operate and to control, and thus more vulnerable to a disturbance.
The interesting example of an actual blackout that occurred in Italy can be reviewed
here to explain power system problems, their cause and the complexity in controlling
the severity of these problems. As mentioned in [18], the Italian blackout was initiated
by a line trip in Switzerland. Reconnection of the line after the fault was unsuccessful
because of too large phase angle difference which was about 60 degrees, leading to
blockage of the synchro-check devices. Twenty minutes later a second line tripped,
followed by a fast trip sequence of all interconnecting lines to Italy due to overload.
As such, this example shows that interconnected system have high fault levels and are
prone to becoming unstable, leading to total blackout due to its complexities and power
balance problems when faced with large and severe disturbances. Hence, imminent if
not immediate priorities must be the higher availability and maintaining of efficiency
which is possible only with good information technology based services in power
system management. Conventional SCADA and Energy management system (EMS)
stability control systems currently do not provide efficient solutions in occurrences of
cascaded outages, through any coordinated or optimised stabilising actions. [41]
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In modern interconnected networks, a fast developing emergency may involve a wide
area. Since operator response may be too slow and inconsistent, fast automatic actions
are implemented to minimise the impact of the disturbance. These automatic actions
may use local or centralised intelligence or a combination of both. In [40], the author
has discussed the trends in wide area protection of power system. The normal automatic
control can provide either preventive or corrective action. During normal operation, the
focus is on economic optimisation of system operation, while during more challenging
network conditions, such as alert state, or emergency situations, the focus of control
shifts towards stability considerations. The ultimate objective is keeping the maximum
of possible networks intact and the generators connected to the grid. The breakdown
normally results in one or more severe problems in the power system. The main concern
in the emergency state is the system security. System protection schemes form a last
line of defence in the case of severe disturbances.
The authors in [102, 103] have discussed the basic design and special applications of
wide-area monitoring and control systems which complement classical protection
systems and SCADA, also known as Energy Management Systems application.
Currently, the local automatic actions are conservative, act independently from central
control, and the prevailing state of the whole affected area is not considered. Actions
incorporating centralised intelligence are limited to the information anticipated to be
relevant during unforeseen contingencies. There are few schemes that are adaptive to
intelligence gathered from a wide area that respond to unforeseen disturbances or
scenarios.
Historically, only centralised control was able to apply sophisticated analysis because
only at this higher level could computers and communication support be technically and
economically justified. However, with the increased availability of sophisticated
computers communication and measurement technologies, more intelligence can now
be used at a local level. The possibility to close the gap between central and local
decisions and actions will depend on the degree of intelligence put in the local
subsystems. Decentralised subsystems that can make local decisions based on local
measurements and remote information (system wide and emergency control policies)
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and/or send pre-processed information to higher hierarchical levels are an economical
solution to the problem.
Measurements provided by the sensors are usually collected by a host computer in
substations over a local-area communication network. This substation based sensor
network is, in turn, a node of a wide-area network which collects and collates data from
various substations. It then performs various application tasks in order to arrive at
protection and control decisions [104]. These decisions are then communicated to the
substation computers and through them to various actuators in the substations. The
power, communication and computer infrastructures are thus closely interlinked. The
Fig.6.1 shows typical flow of information and communication for various levels of
primary and secondary protections for given disturbances (sensing, computing and
communication infrastructure). Such geographically dispersed system require
functionally complex monitoring and control systems, as the performance of the power
system decreases with the increasing size, loading and complexity of the network.
6.3 ROLE OF WAM IN MAINTAINING WAN OPERATIONS
From Fig.6.1 the power system can be viewed as a large-scale, multi-input, multi-
output, nonlinear system distributed over large geographic areas and needing fast
communication as well as accurate control to maintain reliability. When a major power
system disturbance occurs, protection and control systems have to limit impact, stop the
degradation and restore the system to a normal state by appropriate remedial actions.
WAM and protection systems limit severity of disturbances by early recognition as well
as proposition and execution of coordinated stabilising actions.
There are three approaches to control power system dynamics using wide-area
measurements. First is a control-room operator response to information derived from
WAMS; second is a discontinuous control, such as switching control modes or
protection schemes in response to specific observed dynamic conditions; and third is a
continuous control using wide-area signals as controller input signals.
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System-wide controller and coordinator
Data processing
Data Delivery
PMU 8
PMU 1
PMU 10
PMU 2 PMU
4
PMU 5
PMU 3
PMU 7
PMU 9
PMU 6
Local area control Local area control Local area control
SS 1 SS 2 SS 3Data Acquisition Data Acquisition
Data Acquisition
Power flow direction within local area
Information signal flow (communication links for data flow)
Fig.6.1 Power flow and information flow network in wide-area network
The scheme of WAM alarming during heightened risk of instability with automated
diagnostics and operator responses can help to manage potential dynamics problems.
This is because of their prior knowledge and mitigation procedures, identified in real-
time, for responding unanticipated events. The comparison table of various control
schemes with their advantages and disadvantages is given in [105]. It is argued that the
tools for control-room operators to observe and take action on dynamics problems are a
necessary step before automated systems for security against dynamic issues can be
widely deployed. This stage enables the operator to use discretion in balancing the
increased risk of instability against the cost associated with a dispatch action to reduce
the risk. On the other hand, the advanced measurement and communication technology
in wide-area monitoring and control, FACTS devices can prove better tools to control
the disturbance and a better way to detect and control an emergency.
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In [106], the commercial availability of a wide area monitoring system and its
experiences are described. With the emphasis on higher utilisation of power systems,
monitoring of its dynamics is becoming increasingly important. This requires
information with higher accuracy and update rates that are faster than those usually
provided by traditional protection systems. The introduction of phasor measurement
units as well as advances in communication and computational equipments has made it
technically feasible to monitor the stability of the power systems online, using a wide
area perspective. Power utilities have already deployed PMUs in their grids, mainly for
manual data acquisition and processing [106]. Wide area monitoring systems provide
central data acquisition from already installed and planned PMUs enabling utilities to
utilise phasor information wherever it is needed. The three main goals of WAMS are:
monitoring of the dynamic system behaviour (i.e. stability assessment); monitoring of
transmission corridors (i.e. congestion management); and finally, disturbance analysis
and system extension planning (i.e post-mortem analysis). WAMS includes all types of
measurements that can be useful for system analysis over the wide-area of an
interconnected system. Real-time performance is not required for this type of
application, but is no disadvantage. The main elements are time tags with enough
precision to unambiguously correlate data from multiple sources and the ability to
convert all data to a common format. Accuracy and timely access to data is important as
well. Certainly with its system-wide scope and precise time tags, phasor measurements
are a prime candidate for WAMS.
6.3.1 Advanced technology used in WAMS
As mentioned in the previous section, monitoring, operation and control are the three
major parts of energy management systems and wide-area measurements are the
integral part of power networks which will help in ensuring operation of transmission
networks within their operational limits in the present climate of deregulation.
Emerging techniques in computer technology, communication technology and PMU
technology are being used in WAMS and form the basis for real-time dynamic
monitoring, online security assessment and wide-area stability control of power
systems. In all, WAMS is playing a vital role in interconnected power systems [107] by
providing a wide area system view and increased stability [108].
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In recent years, many attempts have been made to design an integrated controller and
with the wide application of synchronised phasor measurement units in power systems,
WAMS has enabled the use of a combination of measured signals from remote locations
for global control purpose. It is found that if remote signals from one or more distant
locations of the power system are applied to local controller design, system dynamic
performance can be enhanced [109].
Phasor measurement unit (PMU) technology is attractive since it can provide
synchronised, real-time measurements of voltage, incident current phasors, rotor angle
and electrical power at the system buses and lines. Once these real-time signals of the
whole system are available in the form of synchronised phasors, the operators can give
online monitoring of power system operating conditions. Moreover, PMU is also
introduced for the use of stability assessment and wide-area control of the power
system. Due to the large scale and intricate structure of modern power systems, the
demand for structuring WAMS has been increasing. WAMS provides a dynamic
coverage of the wide-area power network and is also able to handle cascaded outage
through coordinated and optimised stabilising actions.
Authors in [110] have proposed a two-level hierarchical structure to optimise the
control of transient swings in multimachine power systems. The control technique
involves a number of independent local controllers communicating with a central
coordinating controller which accounts for nonlinearities and yield global optimal
transient performance. On a similar line, to address voltage regulation and rotor
oscillation problems simultaneously, [111] has proposed a two-level hierarchical
controller based on wide-area measurement for multimachine power systems. The
solution given consists of a local controller for each generator at first level helped by a
multivariable central one at secondary level. The secondary-level controller uses remote
signals from all generators and improves the local controller performance.
Information is now obtained fast and fresh from the synchronised measurement [46].
The power utilities are now placing these devices in selected locations to measure the
voltage and current phasors at the same time. They are transmitted to a central place
where they are compared, analysed and processed. The technology of synchronised
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phasor measurements is well established. It provides an ideal measurement system to
monitor and control power systems, in particular during conditions of stress. The
essential feature of the technique is that it measures positive sequence (and negative
sequence and zero sequence quantities, if needed), voltages and currents of a power
system in real time with precise time synchronisation. This allows accurate comparison
of measurements over widely separated locations as well as potential real-time
measurements based control actions. The synchronisation is achieved through a global
positioning satellite (GPS) system as shown in Fig.6.1.
Starting from WAMS and moving to wide area control systems i.e. WACS is the
challenge of the new century. Authors in [112] have explained about how advances in
digital and optical communication and computation can be exploited to gain the specific
advantage of WACS.
Communication systems are a vital component of a wide area protection system. These
systems distribute and manage the information needed for operation of the wide-area
relay and control system [41]. To meet these difficult requirements, the communications
network will need to be designed for fast, robust and reliable operation.
The introduction of PMU technology can significantly improve the observability of the
power system dynamics, and it can enhance different kinds of wide-area protection and
control [46]. The control actions can either be preventive, or corrective. During normal
operating condition the focus is always on the economics of the system but during
cascading condition the focus is on control shifts towards ensuring power system
security. The objective in that case is to keep, as much as possible, an intact electrical
network with all generators connected to the grid. The energy management systems
EMS applications can improve the security margin using optimisation techniques or
sensitivity routines that, along with time domain simulation, could predict the control
actions to return the system to normal.
6.3.2 Role of communication network and various time delays
Though the above scenario looks very appealing in terms of fast communication, in
reality, there exist various time delays (lags) in power system measurement. A major
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component of system-wide disturbance protection is the ability to receive system-wide
information and commands via the data communication system and to send selected
local information to the SCADA centre. This information should reflect the prevailing
state of the power system [41].
Communication delay plays a crucial role in WAMS by determining the time-lag before
control action is initiated to dampen power system oscillations. As mentioned in [108],
the PMUs measure voltage, current, and frequency phasors using discrete Fourier
transform (DFT) and can detect transients or surges within milliseconds of their
occurrence. In addition to the propagation delay of the particular link, the message
format of the PMU and data rate of the link determine the communication delay in the
system. Furthermore, there is also a processing delay due primarily to the window size
of the DFT. The standard delays calculation associated with various communication
links is given in [108]. It indicates the delays of various communication links when
using PMUs in a WAMS environment and could provide useful delay statics that can be
integrated into simulation and performance analysis of WAMS.
Normally it is assumed that the time frame of the disturbances is shorter than the
response time of the human operators of the power system (less than several minutes).
Thus the scenarios address systems for automatic protection or control as opposed to
manual control. A complete wide-area protection and control system would have the
capability to not only detect incipient disturbance, but also to respond in real time with
effective control action.
The major options for communication used in WAMS are both wireless (micro-wave,
satellites) as well as wired (telephone lines, fibre-optics, power lines) network options
[107]. The basic flow of data and information starts from measurements with the help of
various sensors. These measurements are then sent to signal distributors to send on to
the equipment which use these measurements. This signal is then digitised by signal
converting equipment and then communicated. Traditional short circuit protection
systems measure local signals and respond in four to 40 miliseconds to disturbances in
the local area [104]. Wide-area protection and control systems would gather information
from multiple locations on the system and issue wide area controls as deemed necessary
to respond to disturbances in a somewhat longer time frame.
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The main causes for delay in communication are as mentioned in [108], transducer
delays; delays because of window size of the DFT; processing times; data size of the
PMU output; multiplexing and transition; communication links involved; and data
concentrators which are primarily data collecting centres located at the central
processing unit and are responsible for collecting all the PMU data that is transmitted
over the communication link.
The propagation delay is dependent on the medium characters and the physical route
distance in WAMS. For a local controller, the time delay of feedback signals is very
small (less than 10ms), so the small time delay is often ignored in the controller design.
However, for a wide-area controller, the time delay in an interconnected power system
can vary from ten to several hundred milliseconds or more. An experimental research in
[108] has shown that the time delay caused by different communication links are
different, but all of the delays are more than 100ms. In the case of a satellite link, the
propagation delay could be as high as above 700ms. There could be larger delays when
a large number of signals are to be routed and signals from different areas are waiting
for synchronising. Such large time delays can invalidate many controllers that work
well in with no signal delayed input and even cause disastrous accidents [107]. The
impact of time delay on controller performance has been ignored for a long time in
power systems, but it has significant effect in wide-area control.
Today’s wide area communication topologies such as the synchronous optical network
SONET) are capable of delivering messages from one area of a power system to
multiple nodes on the system in as just 6ms [104]. Assuming decision time of 50ms, a
disturbance on a system could be detected and a corrective response delivered in less
than 200ms. In all, communication systems are a key component of wide area protection
systems. These systems distribute and manage the remote information needed for
operation of the wide area protection and control systems. With rapid advancements in
WAMS technology, the transmission of measured signal to a remote control centre has
become relatively simple.
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6.4 REVIEW OF POWER SYSTEM CONTROLLERS
6.4.1 Controllers for power system performance enhancement
For effective compensation, it is necessary to find the optimal locations for using
FACTS devices depending on the targeted parameter such as voltage stability or
damping oscillations. The given location may be very effective for one type of stability
problem while it may deteriorate the other performance parameters. The authors in
[113] have given a brief literature review of various methods used for finding the best
locations for effective shunt and series compensations for various issues, and have
proposed a method to identify effective locations based on sensitivity analysis for
voltage stability enhancement using series compensation. The authors in [114] have
reported an eigenvalue sensitivity approach for location and controller design of series
compensation for damping power system oscillations.
To tackle with the problem of power system oscillations, Korba in [115] has developed
a model-based approach for monitoring of dominant electromechanical oscillations in
real time. With a similar concept, the authors in [116] have addressed online estimation
of electromechanical oscillatory modes in power systems using dynamic data like
currents, voltages and angle differences measured online across transmission lines.
Authors in [117] have discussed and compared control techniques for damping
undesired inter-area oscillations in power systems, by means of PSS, SVC and
STATCOM. The study on different controllers, their locations and use of various
control signals for effective damping of these oscillations is explained. When multiple
controllers are used in large system, it is necessary to coordinate them properly to
enhance performance. Poor coordination may result in degrading the system
performance. Based on coordination or interactions of multiple FACTS devices, the
authors in [118], have shown interactions between various stabilisers (PSS and/or
FACTS) in multimachine power systems. The issue of coordinated design and
performance of multiple FACTS devices is reported in [119, 120] for power system
oscillation damping, while [121, 122] have used TCSC for damping these inter-area
oscillations. The oscillation problem is analysed from the point of view of Hopf
bifurcations, an extended eigenvalue analysis to study different controllers, their
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locations and the various control signals for the effective damping of these oscillations.
An application of a normalised H-infinity loop shaping technique for design and
simplification of damping controllers in the linear matrix inequalities framework is
illustrated in [123]. Authors in [124] have investigated the use of multiple input signals
for the design of PSS and TCSC controllers for damping power system oscillations. In
[34], a robust damping control is designed for multiple swing modes damping in a
typical power system model using global stabilising signals. A multiple-input-single-
output (MISO) controller is designed for a TCSC to improve the damping of inter-area
nodes.
Using the linearised model of power system, [125] has investigated the enhancement of
damping the power system oscillations via coordinated design of PSS and STATCOM
controllers using an SMIB system. The FACTS devices’ actions may produce additional
synchronisation and/or damping torque improving system capability to absorb
disturbance impacts in the sense of enhancing the capacity of restoring the system
equilibrium and/or making faster the attenuation of the oscillation. To verify this fact, in
[126] the problem of power system stability including effects of TCSC and SVC on
synchronising torque and damping torque are analysed by means of the properties of the
potential part of the transient energy as well as with Lyapunov function time
derivatives. A Lyapunov function was derived for SMIB by a model and it was claimed
that, though some approximations were made by linearisation, the more important
nonlinearities were preserved.
6.4.2 Controllers for transient stability improvement
FACTS controllers like TCSC can be placed in one of the lines of a power system with
a suitable control scheme to improve the transient stability condition of the system
[127]. In [128], it has been shown that the energy of the controlled line can be used to
devise a discrete control scheme for TCSC, with minimum measurements.
In [96, 129] evaluation of transient stability margin is discussed with the help of
multiple FACTS devices, using trajectory sensitivity analysis. To get the information
about each generator, trajectory sensitivity and its corresponding indices are used. It was
shown that the best possible location of the FACTS devices is totally dependent on the
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location of fault. Also STATCOM proved to provide great assistance in the restoration
of the post-fault voltages at different load buses in comparison with the TCSC.
However, this study was carried out using a classical simplified model which may not
be able to reflect true system dynamics.
Athay and authors in [130] have described the development and evaluation of an
analytical method for direct determination of transient stability. It has been claimed that
the study has developed a practical approach, sufficiently accurate and applicable to
realistic problems in power system operation and planning. [131] has given a practical
concern to both transient stability requirements as well as voltage regulation. With these
being the need for two different model requirements, a global controller is proposed to
coordinate the transient stabiliser and voltage regulator. However, the power system is
modelled with a simplified low order model and used for single-machine-infinite-bus
system.
A new fast method for assessing transient stability is proposed in [132] based on the
relationship of transient stability power limits, post-fault static stability power limits and
power impact caused by accelerating power of the failure process. Furthermore, the
method claims that transient stability margins can be assessed efficiently but the power
system model used in the method is a simplified classical model. The authors in [133]
have implemented the transient stability assessment in real-time and investigated the
effectiveness of various available methods as an important tool for energy management.
Various methods have been developed for system assessment considering the
importance of transient stability issues and problems thereafter. However, while, these
tools are capable of capturing the system responses quite accurately, they do not
inherently provide information regarding the degree of stability [134]. In the paper
‘keeping an eye on power system dynamics’ [135], the authors have discussed present
power system scenarios, given some measurements and analysis of events and have
highlighted post-disturbance monitor issues. It gives importance to online
measurements not only in maintaining system stability under disturbance conditions, but
also for post-mortem analysis to prevent (or minimise severity of) the same problems in
future.
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Actually, each of the FACTS devices controls one or more of the network variables
with criteria based on specific control schemes as mentioned in [17, 23, 113]. However,
compared to other FACTS devices such as shunt compensations by SVC or
STATCOM, series compensation by TCSC has proved to be more effective for transient
stability enhancement. Knowing the advantages of TCSC compensation for
performance enhancement in post-fault condition, the focus of the present research is
basically on improvement of transient stability using TCSC. The construction,
modelling and working of TCSC and other FACTS devices is already discussed in the
Chapter 3, so next the section of this chapter will discuss the performance of TCSC and
its applications, highlighting its features.
6.4.3 Review of real time controllers
The introduction of time delay in a feedback loop has a destabilising effect and reduces
the effectiveness of control system damping. In some cases, the system synchronism
may be lost [136] so in order to satisfy the performance specifications for wide-area
control systems, the design of a controller should take into account delay. Moreover, the
controller should tolerate not only the range of operating conditions desired but also the
uncertainty in delay. The impact of time delay on robust controller designs has been
ignored in power systems, but becomes a pertinent topic in recent years with the
proposal of wide-area power system controls.
The advent of real-time phasor measurements and improved communication makes it
attractive to consider new solutions to transient stability. The communication delays
including both propagation and processing delays seem to be approaching values that
are acceptable when considering a transient swing with a period of approximately one
second.
The modern energy system management is supported by SCADA, by numerous power
system analysis tools such as state estimation, power flow, optimal power flow, security
analysis, or transient stability analysis etc in addition to linear and nonlinear
optimisation programs. The available time for running these application programs is the
limiting factor in applying these tools in real time during emergency, and a tradeoff with
accuracy is required. The real time optimisation software and security assessment and
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enhancement software do not include dynamics. Further propagation of a major
disturbance is difficult to incorporate into suitable numerical algorithms, and a heuristic
procedure may be required.
With the increased availability of sophisticated computer, communication and
measurement technologies, more ‘intelligent’ equipment can be used at the local level to
improve the overall emergency response. There seems to be a great potential for wide
area protection and control systems, based on powerful, flexible and reliable system
protection terminals, high speed communication, and GPS synchronisation in
conjunction with careful and skilled engineering by power system analysts and
protection engineers in co-operation [40].
The real-time measurement equipments and associated communication system (i.e.
PMUs and WAMS) can be exploited for developing advanced control techniques and
centralises response-based control architectures. The main idea is that power system
trajectories, acquired in real-time, allow the identification of threats to system security
and degraded dynamic states. If necessary, the control centre can evaluate suitable
corrective control actions and successively transmit the corrective signals to the
actuators. Remedial actions can then be applied through any fast actuator devices such
as FACTS devices [53].
As mentioned in [52] the essential but challenging step in a response-based type
controller is to know the real-time system status using prediction. Online simulations
should be executed continuously to reflect the most current operating condition. A
further, important task is using this system information, deriving the optimal control
action to make the system stable in a short time interval. Counter measures and new
approaches to system security can be based on the adoption of FACTS and HVDC
technologies, dynamic security assessment methodologies, PMU technologies, real-time
measurements and control systems with wide-area measurement systems and wide-area
control systems, automation and control methodology [53].
Power system security can be quantitatively enhanced by developing an online
environment where all control features are implemented on the transient time-scale.
Though Chu and Liu have reported online learning applied to power system transients
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in [52], the attempt was to bring the system back to pre-fault conditions even after
disturbance in post-fault regions. The degree of stability measured by this, the
performance index was hence defined by the deviation of system performance in post-
fault, from its pre-fault values. In reality, there are chances that after disturbance the
system topology is changed due to the loss of one or more transmission networks. In
such cases, giving importance in maintaining system synchronism for keeping
frequency constant, the system operating conditions can be changed with the changed
transmission network. In short, it may not be feasible to bring a system back to the same
pre-fault operating conditions after a disturbance in all given circumstances.
6.4.4 Communication delay consideration in controller design
Traditional stability controlling strategies in power systems only used the local
measuring data. Time delays of the local measuring data were usually very small (<10
ms) [37], and were generally ignored in the past stability study and controller design.
With the rapid development of PMU/WAMS, coordinated stability control strategy has
been paid more and more attention. It uses the remote measuring information from
PMU/WAMS. Since time delay in wide area measurements is usually obvious, it is
important to properly consider its impact on the stability analysis and controller design
in power systems.
Employing phase-measurement units (PMUs) it is possible to deliver the signal at a
speed of as high as 30Hz sampling rate [109]. It is possible to deploy the PMUs at
strategic locations of the grid and obtain a coherent picture of the entire network in real-
time. As mentioned in [137], wide-area can be 10-12 times more effective than local
decentralised control of wide-area oscillations. However, the cost and associated
complexities restrict the use of such sophisticated signal-transmission hardware on a
large scale commercial scale. As a more viable alternative, the existing communication
channels are often used to transmit signals from remote locations. The major problem is
the delay involved between the instant of measurements and that of the signal being
available to the controller. The delay can typically be in the range of a few hundred
milliseconds depending on the distance, protocol of transmission and several other
factors. To consider this effect of time-delay on system stability, [137] has adopted a
predictor approach and discussed its implementation and experimental verification
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using FACTS devices. However, the focus of this time-delay was to investigate its
impact on small-signal stability of power system.
In distributed systems such as protective relay systems, the time delay or latency is
usually less than 10ms [138]. Unlike the small time delay in local control, in wide-area
power systems the time delay can vary from tens to several hundred milliseconds or
more. Fibre optic digital cables are reported to have approximately 38ms for one way,
while considering delays it is over 80ms. Communication systems that entail satellites
may have an even longer delay. The delay of a signal feedback in a wide-area power
system is usually considered to be in the order of 100ms. If routing delay is included,
and if a large number of signals are to be routed, there is a potential of experiencing
long delays and variability.
In [107] the authors have designed a TCSC controller including feedback signal delay,
using a theoretical approach based on time-delay dynamic systems combined with LMI
techniques. The design is discussed and implemented for showing effect of time-delay
and its performance. The analysis of the time delay impact on wide-area system control
is addressed in [138]. In order to eliminate the effect of a power system model’s
nonlinearity and wide-area information’s uncertainty including time delays and
incompleteness, [139] has proposed a nonlinear robust integrated controller.
Optimally tuned conventional controllers, using linear time-invariant (LTI) lead-lag, are
simpler and often provide effective solutions to improve the damping of selected
oscillatory modes. However, they only work within a limited operating range and, in the
case of a changed system configuration with new operating points, can still cause poorly
damped or unstable oscillations due to parameters being tuned to a previous different
setting. To make these types of controllers adaptive for real-time implementation, an
adaptive controller for FACTS has been proposed in [97]. However, while making its
operating range wider than any LTI controllers, it has resulted in high computational
efforts on hardware and large computational time.
The primitive power system controllers such as excitation controllers have been well-
developed and applied to power plants widely. With the use of TCSC equipments, the
applicability of the primitive controllers and their original control strategies under the
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least cost performance index is a considerable problem. The coordination among the
TCSC controller and other controllers is also important. As mentioned in [140], [42]
have given some results about the interactions between TCSC and other controllers.
However, those conclusions are mainly obtained when all the controllers are equipped
with conventional control approaches, such as PID or linear control. When nonlinear
control approaches are employed, the coordination problem should be considered in
more depth.
In fact, multi-machine power systems equipped with TCSCs are characterised by high
nonlinearity and strong coupling, and influenced by exogenous disturbances such as
change of operating points, power system faults, etc. These disturbances have great
impacts on the control design. Although many control design approaches have been
developed for FACTS devices to enhance the power system transient stability, most of
the existing controls are based on the approximately linearised model of the power
system and conventional control principles that are not suitable for the cases with large
disturbances. Though there are many references in literature where TCSC nonlinear
control law is derived, it is based on fixed structure and parameters without considering
the system uncertainty such as loss of transmission line which changes network
topology.
In [140] the authors have developed coordinated nonlinear robust control of TCSC and
excitation for multi-machine systems. Although all the above problems of nonlinear
models, changed system structures, etc. are taken into consideration, the power system
model used to represent multi-machine network is a simplified one.
In the past, studies on time delays in power systems mainly focused on evaluating the
time delay impact on the controller design [37, 107, 109, 138, 141]. In [142] the
influence of time delays on small signal stability regions is investigated and shows that
time delays in power systems can bring both negative and positive influence to the
system’s small signal stability. However, little work has been done to directly analyse
the impact of the time delay on power system stability, especially on transient stability.
In [110] an optimal two level structure for the transient stability problem in
multimachine power systems is proposed. The solution involves a number of
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independent local controllers communicating with a central coordinating controller,
which optimises a global cost function.
At the most basic level of response measurement is the problem of predicting whether
an on-going transient swing is stable or unstable. This is essentially a problem of out-of-
step relaying where there is no control perse but rather a decision to block relay tripping
if the swing is stable and to effect a pre-planned separation if the swing is unstable. This
is a difficult task only with local measurements. In some special situations new
approaches are possible with real-time phasor measurements. If the instability is
modelled as a two-machine system, then the parameters of the two-machine model can
be inferred from the number of measurements in the real world. The equal area-criteria
can be then used to predict whether the swing is stable or not. Unfortunately, when
more than two machine models are required this simple criterion is not useful. Most of
the familiar techniques for investigating transient stability were developed for offline
planning studies and have no direct applicability to real-time problems. The various
energy function techniques can be viewed as a means of avoiding the solution of a large
number of differential equations. In real time, however, nature provides a solution to the
differential equations and the decision about stability or instability must be made on the
basis of measured system variables [104].
Othman et al. [143], proposed a model in d-q axis domain assuming that the line current
is a forcing function to TCSC and therefore an independent quantity. This was a major
drawback of this model, as a change in the TCSC firing angle will cause a change in
line current and hence, it is unreliable in terms of accuracy. The conventional
controllers used for TCSC control utilises a TCSC reactance to firing angle (i.e. Xtcsc
α conversion table) which is generated offline for a given TCSC. This table depends
on the characteristics of devices, so to address all these limitations, in [144], a linearised
discrete-time model of a TCSC-compensated transmission line is presented. The model
derived considers the proper characteristics of TCSC, i.e. the variation in its impedance,
with the firing angler of the TCR. Through the digital computer simulation it is shown
that the eigenvalues of a TCSC-compensated transmission line has two complex
conjugate pairs of poles whose real parts depend only on the line resistance and
reactance. The model thus derived in the d-q axis frames is then linearised around the
nominal operating point and is shown to predict any disturbance very accurately. Three
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different controllers were designed based on this linearised model, which requires the
measurements of local variables only. Two of these controllers were used for regulation,
and were able to reject any major or minor disturbance, whereas the third one was used
for tracking any changes on the power set point.
A lot has been learnt from the offline applications experience, as the first step towards
online applications [46]. Online dynamic analysis could be conducted based on the
most recent wide-area system information PMU data, analogs, statuses and topology
structures. The security of the system could anticipate failure of more than one critical
component and simulations will be available to prevent actions or correct situations.
The interface with short-term simulations will provide a list of devices whose behaviour
could drive the system to instability given the current conditions. The secure region is
constrained by specific limits such as frequency stability and transient stability limits,
whose assessment can be done by full time-domain simulations and approximate
methods [46]. Hiskens, in [145], has established a deterministic nonlinear time-delay
model and incorporated it into systematic hybrid (continuous/discrete) system
representation. It has been shown that time-delay affects the differential-algebraic
model.
A major component of system-wide disturbance protection is the ability to receive
system-wide information and commands via the data communication system and to
send selected local information to the SCADA centre. This information should reflect
the prevailing state of the power system [41].
6.4.5 Computation requirements in controller design
For wide spread geographical power system networks for system monitoring and
operation, factors such as communication, collection of data, interaction and exchange
of information in time bound limits, put lot of pressure on available communication
networks and computation facilities. In addition to communication delays, computation
delays are also playing an important role. Actually, it is not only computational delay,
but even computation burden is one of the major concerns, as the size of power system
network increases. Based on whatever simulation-based methods are used in literature,
their limitations and problems in terms of computation complexity handling are
critically reviewed in [146].
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With the deregulation and constant expansion in power systems, the demand of high
performance computing for power system adequacy and security analysis has been
increased rapidly. High performance plays an important role in ensuring efficient and
reliable communication for power systems operation and control. As explained in [147],
grid computing technology is an infrastructure, which can provide high performance
computation (HPC) and communication mechanism for providing services in these
areas of power systems. In [147], the authors have presented a review on the current
research that has been done in the adoptability of grid computing technology in power
system analysis, operation and trading. It has explained the problems with parallel
processing where the task is divided into a number of subtasks of equal size and allotted
to different workers. For this purpose, it has been pointed out that all machines need to
be dedicated with the same configuration and processing speed, otherwise thus can face
the problem of time difference in giving output results from each one of them. Based on
the various experiments carried out in Western Australian Supercomputer program
(WASP) at The University of Western Australia (UWA), it was observed that, the
proportion of computing speed to size of system do not vary linearly. For very small
systems, parallel processing may not be efficient, while as the size of the system and
amount of computation required increases, the speed up factor of computation time also
increases. However, after a certain threshold, again this may not be efficient as there
may be lot of time consumed in distributing work to different processors and also in
collecting results and compiling it for final output. The overall computing time
requirement is affected by the actual processing speed and time for data transfer
depending on the amount of data transfer and channels available for this transfer. The
parallel processing techniques involve tight coupling of machines and using a
supercomputer is possible only for justifiable size of power system network and
depends on its importance. To address problems of computation, [147] has suggested
grid computing technology by listing various interesting features of grid computing in
terms of sharing resources, for example, taking advantage of time difference of zones
operating on a grid.
The ideal scenario for control of a power system would be to have the capability to
instantly compute optimum operating conditions and keep the system at those operating
conditions using the available controls. This will require knowledge of system topology
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information and the system’s real and reactive load. In this context, instantly or real-
time means fast enough to provide a required bandwidth. In this scenario, considering a
model-based control algorithm, upon occurrence of a system contingency, the change in
system topology will be instantly detected or estimated at a central location. Then as
repeatedly as necessary, the trajectory to the final operating point will be planned
instantly starting from the initial system state, taking into account all constraints , and
the controller will be driven to achieve tracking of the planned trajectory [1].
In order to satisfy specifications for wide-area control systems, the design of a controller
must take into account this delay, to provide a controller robust enough not only for the
range of operating conditions desired, but also for the uncertainty in delay. The impact
of time delay on robust controller designs has been ignored in power systems but has
become a pertinent topic in recent years with the proposal of wide area power system
control [37].
It is found that if a controller is designed for a delay-free system but applied to the
delayed-input system, the close-loop system may lose stability resulting in unacceptable
performance. The aim of a generalised predictive algorithm developed in this research is
to judge the transient stability status of a power system after fault occurrence. The time
interval for prediction should be very short so that it is made in advance before losing
synchronism, leaving enough time for remedial action to be effective. With reference to
above background, the prediction algorithm developed considers the computation time
required for prediction and optimisation to find necessary TCSC reactance. In addition,
it also considers the further communication delay in implementing the corrective
actions, in deciding the effectiveness of controller.
6.5 CONCLUSION
The chapter has traced through the evolution of, and advances made in the field of wide-
area measurements and technology use for maintaining wide area network stability. The
comprehensive review has identified two key issues which require further research and
development. The first is that, most of the time, communication delays are ignored in
control law derivations, or if it is included at all at any time, the system used is SMIB
with a classical model. Such approximate or simplified models or control laws without
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consideration of practical issue of communication time delay may not give a correct
picture of system stability. The second key issue which is ignored in previous research
is requirements of computation systems and computation time delays. For development
of real-time controls it is customary to consider these communication and computation
delays and also check for their feasibility as per real-time requirements.
The evolving power system with dispersed generation, power electronics devices
connecting low and medium voltage levels, which are geographically at a far distance,
will cause widening of the interconnected system making it more complex. The new
research should be focused on making this interconnected system safe enough for any
contingency which is possible if propagation of the loss of stability is taken in to
account. Keeping the central theme of tracking in advance the generators which are
expected to fall out of synchronism because of sever disturbances, and system
performance for given emergency situation and control action, a new online controller
coordination scheme is proposed in the next chapter to enhance the transient stability of
an interconnected power system using series compensation.
The total online control coordinated scheme is given in the next two chapters. Chapter 7
will develop the necessary theory for this control coordination scheme, while Chapter 8
will verify the various time requirements for feasibility of a real time controller.
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Chapter 7 Online Control Coordination of
FACTS Devices for Power System
Transient Stability: Control Method
Derivation
7.1 INTRODUCTION
As discussed in the previous chapter with its detailed review of real-time controller
requirements, strong robust communication architecture is essential to meet the features
of accurate detection, decision and reaction times for any system. In the context of real-
time controllers, a new method is developed for real-time transient stability control in a
power system which has FACTS (flexible AC transmission systems) devices. Central to
the method is the control in successive time periods of synchronous generator relative
rotor angles to satisfy the nominated transient stability criterion via the real-time and
optimal adjustment of the input references of the controller FACTS devices. In each
period, the dependencies of maximum relative rotor angles on input references are
expressed as nonlinear functions which are synthesised from the results of time-domain
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transient stability simulations, using the prevailing power system model and condition.
The constrained optimisation problem is then formed from the synthesised functions,
and solved for the optimal input references. Practical issues related to computing time
and communication channel time delays are considered in the control methodology. The
next chapter will examine the computing time requirement, and present the results of the
control coordination study to verify the effectiveness of the proposed control method.
7.2 BACKGROUND OF PROPOSED SCHEME
Transient stability is one of the principal considerations in power systems planning,
design and operation. Following the restructuring and deregulation of the power supply
industry, it is generally acknowledged that the power system transient stability margin is
being reduced. This is a consequence of transmission companies increasing their
competitiveness in market environments. Much research has arisen in addressing this
critical issue [13, 97, 148, 149], drawing on the availability of FACTS (flexible AC
transmission systems) device controllers, and advancements in communications,
measurements and computing. In the area of small-disturbance stability, recent research
[97,137, 149] has applied WAMs based on phasor-measurements units (PMUs), and
self-tuning/adaptive controllers, to coordinate and enhance the dynamic performance of
FACTS device controllers.
There has been very limited development in the area of real-time control for transient
stability enhancement and for large disturbance follow-up maintenance . This is despite
extensive research [94, 98, 150-152] in small-disturbance stability enhancement and
damping of inter-area modes of oscillation. Further where these measures are
formulated and implemented with postulated fault disturbances at time of load dispatch
have not resulted in a workable solution. The key difficulties encountered in the
development of real-time control schemes are those of nonlinearity and high dimension
inherent in power systems, and computing time required for executing the control
algorithms. Notwithstanding the difficulties, there have been some proposals based on
model predictive control (MPC) and extended equal area criterion (EEAC) techniques
[13, 153] for real-time transient stability control. However, the use of simplified
classical generator models only gives approximations and these are limitations in the
application EEAC in MPC as identified in [13]. A number of assumptions have been
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adopted for developing an MPC scheme with cutset EEAC stability measures, which
can lead to inaccuracies near equilibrium conditions and possible compromise in the
control robustness [13].
Against the above background, the objective of this research is to develop a new
control scheme for optimally coordinating the input references to FACTS device main
controllers. This will be implemented in real time, in order to maximise the system
transient stability margin, and maintain transient stability subsequent to a large
disturbance. A key feature of this control scheme is online synthesis with individual
control periods for a set of nonlinear functions that express a generator’s maximum
relative rotor angles in terms of FACTS device controllers input references. These are
referred to as the control variables. As it is difficult, if not impossible, to determine
analytically, in a closed form, these nonlinear functions, a series of time-domain
transient stability simulations are performed to provide the results or data for the
function syntheses. The synthesised functions are then used as system transient stability
indices for deriving the control coordination algorithm. The overall control coordination
is subdivided into a number of control periods. Optimal values of FACTS device
controllers are determined in advance for each control period. These are then
implemented using a constrained optimisation method which minimises the input
reference of the rotor angle that is relatively largest. This is subject to the constraint of
individual rotor angles that are less than the nominated threshold and that control
variables must stay within their bounds. The constrained optimisation problem is
formulated in terms of the synthesised nonlinear functions.
In the present work, nonlinear functions in the form of polynomials are adopted to
represent the nonlinear relationships between relative rotor angles and control variables.
Nonlinear regression analysis is performed, following transient stability simulations
with various values of the control variables (i.e. perturbations of the control variables)
within their ranges, to determine the coefficients in the polynomials. As time-domain
simulations are used, it is straightforward to include detailed dynamic models for
generators, controllers and dynamic loads. The starting values of all of the power
system variables for the transient stability simulations related to each control period are
obtained directly from wide-area measurements, and/or derived from them based on
models for individual items of the generation plant.
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The power system model for simulation is directly constructed from the circuit-breaker
and isolator status data together with generator and controller dynamic models, and the
parameters for individual items of plants are held in the database. There are no
requirements for system model identification. Any control actions are included in the
system model in a straightforward manner, and their effects are represented directly in
the time-domain response evaluations from which transient stability indices, expressed
in terms of maximum relative rotor angles, are formed.
As the control coordination method is based on nonlinear time-domain simulation and
prevailing power system configuration and operating conditions, the following
advantages are achieved:
(a) Inclusion of nonlinear and detailed dynamic models for items of the plant
(b) Being adaptive to variation in power system conditions
(c) High accuracy and robustness in the control
The control coordination process is initiated by circuit-breaker opening operations.
These events include those of faults, and subsequent fault clearance by protection
systems.
The development of the algorithm for the control coordination takes account of practical
issues related to computing time and communications channel time delay. The
feasibility in terms of computing time requirements for implementing the control
coordination proposed using a cluster of high-performance and low-cost processors, will
be investigated in the next chapter.
7.3 TRANSIENT STABILITY CONTROL PRINCIPLE
7.3.1 Power system model used
The starting point of the proposed transient stability control is to construct the model to
represent the power system in the prevailing operating conditions. The control sequence
is initiated by circuit-breaker opening operations. If there have been short-circuit faults
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in the power system, these circuit-breaker operations would be those arising from fault
clearance, subsequent to which, the faults are isolated from the system. On this basis,
the prevailing power system configuration is established directly from the status data of
circuit breakers and isolators received at the control centre. The power system model
relevant to the transient stability phenomena is then constructed from the system
configuration, detailed models adopted for individual items of plant and their
parameters held in the database. The details of forming dynamic models of various
power system components is explained in Chapter 2, which is expressed in symbolic
notations by the following sets of differential equations and algebraic equations:
), uyh(x,x =•
(7.1)
( ) 0uy,x,g = (7.2)
In (7.1) and (7.22), h and g are, in general, nonlinear vector functions; x is the vector of
state variables arising from modelling synchronous generators together with their
controllers, FACTS devices and dynamic loads; y is the vector of network nodal voltage
variables, and u is the vector of control variables to be optimised in real time for
maximising transient stability margins. In the present work, the control variables
represent the input references to the main controllers of FACTS devices which have
been selected for participating in the control.
In terms of representing system loads, composite load models based on admittances
and/or aggregate dynamic load components are used. Their parameters are determined
on the basis of the data obtained from PMUs, pre-determined load compositions at
individual load buses and generic dynamic load models (i.e. individual motor
equivalents) and inertia data [6] .
7.3.2 Time-domain transient stability simulation
Starting from a given set of values for the elements of the state variable vector x at any
instant of time, the sets of equations in (7.1) and (7.2) can be solved simultaneously,
using numerical integration. This gives system responses for any specified future time
periods for control coordination subsequent to the starting time-instant. In principle, this
time-domain simulation process is a straightforward one if there are direct
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measurements of all of the state variables in vector x to be used as the initial condition.
In practice, these direct measurements might be difficult, to achieve. For example,
generator rotor angles, rotor flux variables, and state variables associated with
controllers’ models can be difficult, if not impossible, to measure. However, it would be
possible to obtain the values of the state variables through the available measurements
by synchronised PMUs of network variables such as nodal voltages, nodal currents,
nodal power, branch currents and power flows together with generator rotor speed
and/or field winding voltage measurements. These available measurements combined
with the individual dynamic models for synchronous generators and controllers would
give, through dynamic simulation in the time-domain, solutions to the state variables
associated with the models up until the current time instant. In this way, state variable
vectors would be available for initialising the simulation with the system model
described in (7.1) and (7.2) from any specified time instant (at present or in the past) to
any future time instant as required for control coordination purpose.
7.3.3 Nonlinear relationship between maximum relative rotor angles and control
variables
The key property on which the proposed control coordination draws is that, for any
given period of time subsequent to a disturbance, the relative rotor angle response of
any synchronous generator (except the reference generator) is a nonlinear function of
the control variables (i.e. the input references to the FACTS devices controllers), with a
given power system configuration and parameters. As the focus of the present work is
on transient stability control, individual maximum relative rotor angles, and their
dependence on the control variables are of interest and relevance to the development of
the control algorithm. In any specified time period, the dependence on the control
variables of the maximum (either positive-going or negative-going) relative rotor angle
of each generator is expressed in a compact form as follows, using a functional notation:
( )( ) ( )kiki TfT ,max1 u=δ Ni ,,3,2 = (7.3)
In (7.3), the reference generator is identified by 1; i denotes the ith generator; N is the
total number of generators; ( )( )ki Tmax1δ is the maximum relative rotor angle of the ith
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generator with respect to the reference generator in the time period Tk, and ( )ki Tf ,u is
the nonlinear function of the control vector u, for time period Tk. The kth control period,
Tk, is defined as that from a given time instant tk to the time instant tk+T where T is a
chosen control time window (CTW).
Except for simple or trivial cases, it is not possible to derive analytically in a closed
form the nonlinear function ( )ki Tf ,u referred to in (7.3), for a transient stability control
purpose. The present research proposes a scheme based on a number of time-domain
simulations related to individual control periods Tk to form functions ( )ki Tf ,u . In
Section 7.5 the scheme is developed, based on control vector perturbations and time-
domain simulation, by which fi is synthesised, and expressed in terms of a polynomial
in control variables in vector u.
7.3.4 Transient stability control concept
Subsequent to a disturbance (for example, fault and fault clearance), the control is
subdivided into a number of control periods to be considered sequentially. An optimal
control vector is to be determined in advance for subsequent implementation for each
control period with the objective that all of the relative rotor angle maximum values in
that period are to be within the nominated transient stability threshold, and, for
maximising the transient stability margin, the greatest relative rotor angle is to be
minimised. With this objective, and using the functional notation in (7.3), the
constrained optimisation problem is formulated as follows, for each control period:
Minimise ( )[ ]2
, kj Tf u Mk ,,2,1 = (7.4)
subject to:
( ) thkith Tf δδ ≤≤− ,u Ni ,,3,2 = (7.5)
and
( ) ( )maxmin mmm uuu ≤≤ Lm ,,2,1 = (7.6)
In (7.4), j is the identifier for the synchronous generator having the greatest relative
rotor angle among (N-1) relative rotor angles within the kth control period; M is the
number of control periods; δth is the nominated transient stability threshold (for
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example, 1800) in terms of relative rotor angle; um is the mth element of the control
vector u having upper and lower limits um(max) and um(min) respectively, and L is the
number of control variables. The objective function in (7.4) is formed as the square of
the greatest relative rotor angle, ( )kj Tf ,u , to include the case of its being negative.
The solution of the constrained optimisation problem in (7.4)–(7.6) for each control
period is the optimal values of the control variables (i.e. elements of vector u) which are
used for setting the input references to the participating controllers. In relation to the
frequency at which the controllers’ input references are updated, there are two options:
(a) Option 1: The controller input references are updated, using their optimal values, at
the start (referred to as tk) of the control period, Tk , and then kept constant until the end
of the control period, Tk, which is tk+T. The control sequence will then be repeated for
the next control period Tk+1.
(b) Option 2: Similar to option 1, the controller input references are updated at time
instant tk, using their optimal values. However, the input references, once updated, will
be kept constant only until time instant tk+Tx with Tx < T. The control sequence will
then be repeated for the next control period Tk+1. This implies that, although the
constrained optimisation problem described in (7.4)–(7.6) is solved with respect to the
control period Tk from tk to tk+T, the outcome of the optimisation is used for the period
from tk to tk+Tx (with Tx < T ) only. In this option, the frequency at which the controller
input references are updated is higher, in comparison with that in option1, and there are
overlaps among control periods.
With option 2, it is possible to take account of the change in power system conditions
more closely in the control. However, this has implications in terms of computing time
requirements. In the next section 7.4 are presented the detailed control schemes for both
options1 and 2, taking into account the computing time and communication channel
delay.
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7.4 TRANSIENT STABILITY CONTROL SCHEME
7.4.1 Option 1
Fig. 7.1 shows the control scheme timing diagrams for two successive control cycles
identified by 1 and 2. The timing diagram in Fig.7.1(a) is related to the first control
cycle, following the reception at the control centre of the information that circuit
breakers opening operations were completed at time tc and with reference to the time
origin at which t = 0 in Fig.7.1(a).
There are synchronised measurements of the power system operating states by PMUs at
a given sampling frequency. Time instant tx1 following tc as shown in Fig. 7.1(a) is that
which is closest to tc, and coincides with a time instant when synchronised
measurements are completed. Their results are sent to the control centre which will
receive them at time-instant ty1, after the communication channel time delay of TD1. As
it is not necessary that various measured results would arrive and be received
simultaneously at the control centre, TD1 is set to be the greatest time delay encountered
among the channels, including those for circuit breakers and isolators’ status data. The
calculation related to optimal control coordination for the first control cycle (i.e. k = 1)
is to be carried out, starting at time instant ty1, and completed within the specified time
interval, TC, as indicated in Fig.7.1(a). There are five key calculation steps for the first
control cycle:
(a) Updating the power system model;
(b) Time-domain transient stability simulation;
(c) Nonlinear function synthesis;
(d) Determining optimal values for the control variables; and
(e) Updating the FACTS device controller input references.
In detail, the steps are described as follows:
a) Updating the power system model. Based on the most recently available circuit-
breaker and isolator status data after the circuit-breaker opening operations which is
received at time ty1 the power system model is updated. Parameters and models of
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individual items of plant which are held in the database are used for power system
model construction as described in Section 7.3.1. Also included in this calculation step
is the assembling of the system nodal admittance matrix and its LU factorisation.
b) Time-domain transient stability simulations. This is for the purpose of
generating data for synthesis of nonlinear functions referred to in (7.3) of Section 7.3.3.
At each time step in the simulation, the sets of equations in (7.1) and (7.2) described in
Section 7.3.1 are solved. The set of differential equations in (7.1) is transformed into an
algebraic equation system via a numerical integration formula such as the trapezoidal
rule. This equation system is combined with the network equation set in (7.2) defined
by the nodal admittance matrix formed in step (a).
A series of simulations are performed in this step, using the model from step (a), and
measured initial condition at time tx1 which is received at time ty1. The solution time for
the first simulation is tf1 – tx1 where tf1 is the end of the first control cycle, as indicated in
Fig. 7.1(a).
The first simulation uses the existing values of the control variables (i.e. the input
references to the FACTS device controllers) for the whole time duration from tx1 to tf1.
ty2 tz2 t2 tf2
0T
T
tx2
Timet
(b)
TCTD1 TD2
(a) tx1 ty1 tz1 t1 tf1
0TD1 TD2TC T
Timet tc
Fig.7.1 Control scheme timing diagram: Option 1 (a) The first control cycle: 1=k (b) The second control cycle: 2=k
ty2 = ty1 + T; t2 = tf1
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Each subsequent simulation is carried out for the period from t1 to tf1 which are the start
and end of the first control period respectively, drawing on the initial condition at t1
obtained from the first simulation. For these individual simulations, the FACTS device
controllers input references are perturbed, within their limits, from the existing values.
There is no requirement for the perturbations to be small ones as in linear sensitivity
analysis. From the results of each and every simulation, including the first one, the
maximum value (either positive-going or negative-going) of each relative rotor angle
within t1 and tf1 (i.e. within the control period) is determined, and recorded for
subsequent use in nonlinear function synthesis. In the case of the relative rotor angle
having the maximum value at the start (i.e. t1) of the control period from t1 to tf1, the
value at the end of the period (i.e. tf1) instead of the maximum value is used. This makes
provision for response time delays inherent in the FACTS device controllers and power
systems. It is due to this that the relative rotor angles at the start of each control period
(for example, at t1 for the first control period) would not be influenced by any updated
or revised settings of the controllers input references implemented at the control period
starting time.
The number of simulations depends, in general, on the form of the nonlinear functions
adopted for representing the maximum relative rotor angles. This is specifically in terms
of FACTS device controllers input references, and the number of controllers
participating in the control. Once all of the required simulations are completed, the data
set of maximum relative rotor angles and FACTS device controllers input reference
values which have been used in the simulations from t1 to tf1, is available for the
subsequent syntheses of nonlinear functions expressing the relationships between
maximum relative rotor angles and control variables (i.e. the FACTS device controllers
input references).
c) Nonlinear function synthesis. Each maximum relative rotor angle within the first
control period T1 (period T1 starts at t1 and ends at tf1 = t1+T ) is to be represented by a
nonlinear function, ( )1,Tfi u , (for i = 2, 3, ... , N) as discussed in Section 7.3.3.
In this step, the data set obtained in step (b) is used for the function syntheses. This
involves the postulation of the functional form and determining the parameters or
coefficients in each function, using the data set. The outcome of this step is a set of
nonlinear functions, ( )1,Tfi u ’s, expressed explicitly in terms of the control variables in
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vector u. In Section 7.5 a numerical procedure for forming the nonlinear functions is
presented based on polynomials of control variables and nonlinear regression.
d) Determining optimal values for the control variables. With the nonlinear
functions ( )1,Tfi u , (for i = 2, 3, ... , N) formed in step (c), the constrained optimisation
problem described in (7.4)–(7.6) is formed completely for the first control period with k
= 1. The jth nonlinear function, ( )1,Tf j u , used in forming the objective function in (7.4)
is for the generator with the largest relative rotor angle in period T1, identified from the
results of the simulations performed in step (b).
Solution of the constrained optimisation problem gives the optimal values of the control
variables in vector u. Established methods such as the quasi-Newton method or
sequential quadratic programming [154, 155] can be applied for solving the problem.
As the typical number of FACTS device controllers in a power system is not large, the
dimension, as determined by the number of control variables, of the constrained
optimisation problem encountered in the optimal control coordination is relatively
small. The optimisation is a static one for each control cycle, within which there are no
time-dependent quantities or variables. Time-domain simulations are excluded from the
optimisation loop.
With a relatively small number of FACTS devices controllers, it is possible to solve the
optimisation problem described in (7.4)-(7.6) in Section 7.3.4 by calculating the
objective function and constraint functions for a finite set of specified values of control
vector u with high resolution, and based on the results of the calculation, optimal vector
u is determined. This search method is very suitable for implementation by parallel
computing systems as the function calculations for individual values of control vector
can be performed in parallel and independently of one another. This is an effective
method for exploiting the inherent parallelism in the calculations for solving the
constrained optimisation problem with low computing time, using a computer system
with parallel processing capability.
e) Updating the FACTS device controller input references. The calculations
described in steps (a)–(d) are completed by tz1 (i.e. within the allowed computational
time duration TC). At time instant tz1, the optimal values of the input references,
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determined in step (d), are sent from the control centre to individual FACTS device
controllers. After a time delay TD2 of the communication channels, taking into account
the longest delay encountered in them, the controller input references are updated at
time t1, the start of the first control period. In option 1, the controller input references
are kept constant, following the updating until tf1, the end of the first control period.
Following the completion of the calculations at tz1 related to the first control cycle, the
calculations for the second control cycle (i.e. k=2) are started at time instant ty2 as
shown in the timing diagram of Fig. 7.1(b). With the constraint that the CTW is not less
than the time allowed for calculations, TC, the timing diagram in Fig. 7.1(b) ensures
that, for ty2= ty1+T:
• The calculations for the second control cycle start after the completion of those
for the first control cycle.
• The start, t2, of the second control period coincides with the end, tf1, of the first
control period.
The steps of calculations are similar to those for the first control cycle. If there are no
changes in circuit-breaker and isolator status data, then the power system configuration
to be used would be the same as that for the first control cycle. Load models might have
to be updated, depending on the measured load demands at time tx2. The information on
the measured power system state at tx2 is received, after communication channel time
delay of TD1, at the control centre at time ty2, which provides the initial condition for
time-domain transient stability simulations to be carried out and completed within the
time interval TC between ty2 and tz2 as shown in Fig. 7.1(b). The calculation sequence as
described in steps (b)–(d) for the first control cycle is then repeated. For the first
transient stability simulation in the second control cycle, known values of the control
variables are used throughout the duration from tx2 to tf2 (the end of the second control
period). For the subsequent simulations for the duration from t2 to tf2, perturbed values
of the control variables as with those in the first control cycle are used.
At tz2, (i.e. the end of the calculation period), new optimal values of the control
variables for the second control cycle are sent to the FACTS device controllers for
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updating their input references. Following the updating, they are held constant at the
optimal values until tf2 (the end of the second control period).
The control coordination steps (a)-(e) are then repeated for the subsequent control
cycles (i.e. for k = 3, 4, … , M). The control coordination stopping criterion is as
follows:
• All of the relative rotor angles being within the nominated transient stability
threshold, and
• The amplitudes of relative rotor angle oscillations being within a specified upper
limit.
From t1 (the start of the first control period) to the end of the last control period, the
frequency of updating the FACTS device controllers input references, using option 1, is
1/T. The time delays TD1 and TD2 in the timing diagrams in Fig. 7.1 are settled by the
communication channels themselves. The control coordination designers need to
determine time intervals TC and T, taking into account the computer system processing
capability and the dependence of power system dynamic performance on CTW.
7.4.2 Option 2
Except the possibility that the time intervals TC and T can be different from those of
option 1, the timing diagram for the first control cycle (i.e. k = 1) in option 2 as shown
in Fig. 7.2(a) is the same as that in Fig. 7.1(a) for option 1. However, the control
coordination calculations related to the second control cycle (i.e. k = 2) in option 2 start
at ty2 = ty1+ Tx instead of ty1+T, as indicated in Fig. 7.2(b), with the following constraint:
TTT xC <≤ (7.7)
The constraint in (7.7) that Cx TT ≥ ensures that the calculations for the second control
cycle will start after the completion of those for the first control cycle. With Tx < T, as
indicated in Figs. 7.2(a) and (b), the new optimal values for the control variables
determined for the second control period will be implemented at time t2 which is prior
to tf1, the end of the first control period. The optimal values for the control variables
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determined for the first control period are implemented at time t1, and then kept constant
up until t1+Tx instead of t1+T as in the case of option 1. Similarly, from the control
coordination calculations for the second control period, the FACTS device controller
input references are updated at t2 and kept constant up until t2+Tx which is prior to tf2,
the end of the second control period. The sequence is then repeated for the subsequent
control cycles, with the control coordination stopping criterion being identical with that
of option 1. The calculation steps in Section 7.4.1 (a)–(d) in individual control cycles
are the same for both option 1 and option 2.
In option 2, the optimisation for each control period is carried out for the period T but
the control variable optimal values determined by that optimisation are implemented
only for the period Tx < T before they are updated with the new optimal values
determined for the next control period. With option 2, there are overlaps between
control periods, and the frequency of updating the FACTS device controller input
references is 1/Tx. With the same CTW, the updating frequency for option 2 is higher
than that (i.e. 1/T) for option 1.
The possible benefits offered by option 2 include a closer representation than with
option 1 in the control coordination of any changes in power system operating condition
as detected by measurements. However, the constraint that Tx is to be greater than the
calculation time TC would, in general, lead to the requirement of computer systems with
higher processing capability in comparison with that for option 1.
ty2 tz2t2 tf20
TD1 TD2
tx2
Timet
TTx
Tx(b)
TC
tx1 ty1 tz1 t1 tf10
TD1 TD2TC T
Timet
Tx(a)tc
Fig.7.2 Control scheme timing diagram: Option 2
(a) The first control cycle: 1=k (b) The second control cycle: 2=k
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7.5 FORMING TRANSIENT STABILITY INDICES
As referred to in Section 7.3.3, a key aspect in the control coordination proposed is to
form nonlinear relationships between the maximum relative rotor angles used as
transient stability indices and the control variables, through a series of time-domain
simulations for each control period described in Section 7.4.1 (b) and (c). In principle,
various forms of nonlinear functions can be postulated for expressing the required
relationships. In the present work, polynomial-type functions of the control variables are
adopted in forming the transient stability indices. For the ith generator, the nonlinear
function, fi (u,Tk) , which represents its maximum relative rotor angle within the kth
control period, Tk, is expressed in:
( ) ( ) ( ) ( )
( ) termsorder-higher,
,,,,
1 1 1
1 110
+
+++=
∑ ∑ ∑
∑ ∑∑
=≥=
≥=
=≥==
lnmL
m
L
mnn
L
nll
mnl
nmL
m
L
mnn
mnmL
mmki
uuukia
uukiaukiakiaTf u
(7.8)
In (7.8): i is the ith generator
k is the kth control period
L is number of control variables
( ),,0 kia ( )kiam , , ( )kiamn , , ( )kiamnl , are coefficients of the polynomial function for
the ith generator and kth control period (for m = 1, 2, …, L; n = 1, 2, …, L; l = 1, 2,
…, L).
um, un, ul are the mth, nth and lth elements of control vector u respectively.
The coefficients of the polynomial in (7.8) are to be identified, using the results of the
time-domain transient stability simulations. The identification process is developed in
the following. To achieve more compact notations in the development, the following
vectors are first defined:
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131
( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ),,,,,,
,,,,,,,,,,,,,,,,(,
112111
1211
210
kiakiakiakiakiakia
kiakiakiakiaki
LLL
LL
Lt =α
(7.9)
),,,,,,,,,,,,,1(
211111
211121
LLL
LLLt
uuuuuuuuuuuuuuuuuu=z (7.10)
With the definitions in (7.9) and (7.10), equation (7.8) becomes:
( ) ( )kit Tfki ,, uαz = (7.11)
Various values of the control vector u within its range are specified and used in the
time-domain simulations in period Tk. From each specified control vector u, elements of
vector z in (7.10) are evaluated in a straightforward manner, using their individual
expressions in the RHS of (7.10). Associated with each specified control vector, there is
a set of maximum relative rotor angles within period Tk which are determined from the
time-domain simulation using the specified control vector. On this basis, the following
system of linear equations in terms of coefficient vector ( )ki,α for the ith generator is
formed, using the relation in (7.11):
( ) ( )kifki sp
tp ,, =αz p = 1, 2, …, NP (7.12)
In (7.12):
NP is the number of specified control vectors
zp is Vector z as defined in (7.10) and evaluated using the pth specified control vector
fsp(i,k) is the maximum relative rotor angle of the ith generator determined from the
results of the time-domain simulation for control period Tk with the pth specified control
vector.
Individual linear equations in (7.12) are assembled in a vector/matrix form as follows:
( ) ( )kiki ,, FαA =⋅ (7.13)
in which:
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132
[ ]NPt zzzA 21= (7.14)
and
( ) ( ) ( ) ( )[ ]kifkifkifki sNPsst ,,,, 21 =F (7.15)
The A matrix defined in (7.14) has the dimension of (NP×NC) where NC is the number
of polynomial coefficients (i.e. elements of vector ( )ki,α ). To obtain a unique solution
for vector ( )ki,α , NP is chosen to be greater than, or at least equal to NC. In general, the
inverse of A does not exist (when NP>NC). Vector ( )ki,α is determined by minimising
the error function defined in [156]:
( ) ( )( ) ( ) ( )( )kikikiki t ,,,,E FαAFαA −⋅−⋅= (7.16)
Minimising E in (7.16) based on the solution of ( ) 0
,E
=∂∂
kiα leads to:
( ) ( )kiki ,, FAα += (7.17)
In (7.17): ( ) tt AAAA1−
=+ (7.18)
Matrix A+ in (7.18) is referred to as the pseudo-inverse of matrix A.
Equation (7.17) applies to each and every generator (except the reference generator) for
evaluating the coefficients of individual polynomials expressing the nonlinear
relationships between the maximum relative rotor angles and FACTS devices
controllers input references, for each control period.
7.6 CONTROL COORDINATION FLOWCHART
Applicable to both options 1 and 2, Fig. 7.3 show the flowchart of the overall control
scheme for each control cycle, as described in Section 7.4.1 and 7.4.2., included in the
flowchart is the block representing the communication channels between the wide-area
measurement system and the control coordination system which receives the inputs in
terms of circuit-breakers and isolators statuses, and PMUs’ outputs. Drawing on the
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133
database held in the computer system of the control centre, network data, and dynamic
models of generators and controllers together with their data are available for forming
the network nodal admittance matrix, differential equations and initial values of state
variables for subsequent time-domain transient stability simulations. There are also
communication channels between the output of the control coordination system and the
FACTS devices controllers which receive the optimal input references as referred to in
the flowchart of Fig.7.3.
Wide-area measurement system
Communication channels
Circuit-breaker and isolator statuses
Synchronised PMUs’ outputs:Nodal voltages and currents
Database
Forming network nodal admittance matrix and its LU
factorisation
Forming nodal powers and state
variables
Time-domain transient stability simulations
Nonlinear function syntheses
Solving constrained optimisation problem
Optimal values of control variables
Communication channels
To FACTS device controllers
Fig.7.3 Flowchart of control coordination scheme
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7.7 CONCLUSIONS
Starting from the property that relative rotor angle transients following a disturbance
depend on FACTS device input reference settings, an online control coordination
method has been developed with the objective of maintaining or enhancing power
system transient stability. The method draws on transient stability time-domain
simulations and constrained optimisation for deriving the control algorithm by which
optimal input references for FACTS device controllers are determined to maximise
transient stability margins, taking into account communication channel time delays in
obtaining power system operating state and computing time required in executing the
control algorithm. The dynamic performance of the control method together with its
feasibility in terms of computing time requirement will be investigated and reported in
the next chapter.
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Chapter 8 Online Control Coordination of
FACTS Devices for Power System
Transient Stability: Computing Time
Requirement Analysis and Case-
Study
8.1 INTRODUCTION
The previous chapter has presented online transient stability control method, based on
time-domain simulations and constrained optimisation for real-time and optimal
adjustment of FACTS devices controllers input references. This chapter focuses on the
analysis of required computing time in implementing the method, and presents the
results of study cases. The individual components making up the computing time for
executing the control algorithm are identified, determined, and then combined to form
overall feasibility constraints. The control algorithms and feasibility constraints must
then be satisfied by computer systems used for control method implementation.
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136
Computational tasks which are independent of one another are identified so that they
can be performed using parallel computing systems. The effectiveness of the control
coordination in maintaining transient stability is verified by the simulation study, with
the control scheme having realistic timing parameters applied to a representative
multimachine power system.
8.2 COMPUTING TIME REQUIRMENTS
rawing on transient-stability time-domain simulation (TDS) and constrained
optimisation, previous chapter has derived a control coordination method by which
FACTS device input references can be optimally adjusted in real-time to maintain or
enhance transient stability of a power system following a large disturbance.
A key aspect related to the feasibility in implementing the method is that of the
computing time requirements in executing the control algorithm developed in the
previous chapter. The present research performs an analysis of the computing time
requirements which leads to a set of feasibility constraints to be satisfied by a computer
system used for implementing the control method. In the analysis, the computing time
component required for each step of executing the control algorithm is identified and
determined. From this the overall computing time for each control cycle is formed, and
combined with the selected control time window, where relative rotor angles are
controlled to lead to constraints in computing time allocation. To achieve feasibility,
these constraints are to be satisfied by the computer system adopted for the control
method implementation.
Conditions for meeting the feasibility requirements which include ‘faster-than-real-
time’ simulation and the application of parallel computing systems are derived and
discussed in this chapter. As previously developed in Section 7.4, each control cycle
needs several transient stability TDSs. However, they can be performed independently
of one another, using a parallel computing system. In addition, state-of-the-art TDS
technique has reached a high level of maturity, and ‘faster-than-real-time’ simulation is
now feasible [157], with detailed dynamic models for items of plant. Another important
feature of the TDS technique is that the computing time does not increase significantly
with power system size [157]. Based on these state-of-the-art technologies, the research
D
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137
shows that it is feasible to implement, and apply the control coordination method to the
10-generator, 39-node New England test system [158] modified with two thyristor-
controlled series compensators (TCSCs), using a cluster of PCs or high-performance
processors. The feasibility in terms of computing time is achieved with a substantial
margin. Offline simulations of the control scheme with realistic timing parameters
derived from the feasibility constraints are performed to verify the dynamic
performance of the scheme following a large disturbance in the power system.
8.3 ANALYSIS OF COMPUTING TIME REQUIREMENT
8.3.1 Control coordination structure
In Fig. 8.1 is shown the structure in the form of a block diagram for the control
coordination of FACTS devices as developed in the previous chapter. The control
structure applies for each control cycle. To facilitate the subsequent discussion, the time
instants at the inputs and outputs of individual blocks 2 to 8 are indicated in Fig. 8.1, for
the kth control cycle. There are ten functional blocks in the structure of Fig. 8.1, starting
with the wide-area measurement systems (WAMS) identified as block 1. Block 9
represents the FACTS device controllers which receive their input reference signals
formed by the control system comprising blocks 3 to 7. Block 10 represents the power
system to which FACTS devices are connected. Blocks 2 and 8 represent the time
delays in the communication channels described as follows:
Block 2: This represents the time delay of TD1 in sending the results of the
measurements by WAMS to the control centre in which optimal FACTS device input
references are determined for individual control cycles.
Block 8: This represents the time delay TD2 in sending the input reference signals from
the control centre to the FACTS device controllers.
Functional blocks 3 to 7 together with their computing time requirements are discussed
in the following section.
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8.3.2 Computing time
The total computing time requirement for each control period is the sum of individual
computing times for the calculation steps as described before. These computing times
are discussed in the following.
8.3.2.1 Power system modelling (block 3)
This calculation step, being a preparatory one for subsequent time-domain simulations,
is performed only once for each control cycle. The computing time required for this step
depends on the size of the power system for which control coordination is to be carried
out, and the processing speed of the computer system used. To facilitate the subsequent
discussion and development, the computing time required for this step is denoted by Ta.
8.3.2.2 Time domain simulations (block 4)
For each control cycle, there are a number of transient stability TDSs associated with
this functional block. The first simulation in the kth control cycle starts from time instant
txk, the measured system state which is received at the control centre at time instant tyk
after the time delay of TD1. This measured system state provides the initial condition for
the simulation. The duration of the simulation is (TD1+TC+TD2+T) where TC is the
computing time allocation, and T is the control time window. The purposes of the first
simulation are threefold, using the existing values of FACTS device input references.
• To assess the system transient stability with respect to the control stopping criterion
described in Section 7.4.1(e).
• To provide the data related to maximum relative rotor angles within the control time
window T if existing FACTS device input references are used. This data constitutes a
subset of the complete data set used for the syntheses of the polynomial functions
representing the relationships between maximum relative rotor angles and FACTS
device input references.
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139
• To provide the initial condition for subsequent simulations required for forming the
complete data set used for polynomial function syntheses, as described in the following.
As derived in Section 7.5, to obtain the solution for the polynomial coefficient vector
with NC elements, it is required to perform NP (with NP ≥ NC) TDSs to provide NP
subsets of data for maximum relative rotor angles occurring within the control time
window T, with t1k = txk + TD1+TC +TD2 as indicated in the structure of Fig.8.1 (i.e. from
t1k to t1k+T). On this basis, there are, in addition to the first simulation, (NP-1)
simulations which, for each, the solution time is T. In performing these simulations, the
FACTS device input references (i.e. control variables) are perturbed within their ranges.
The perturbed input references form the set of specified control vectors. The initial
condition for starting these simulations is provided by the solution at time instant t1k
obtained in the first simulation.
The computing time for the first simulation is β∙(TD1+TC+TD2+T) where β is a constant
depending on the size of the power system, time step length and processing capability of
the computer system used.
The computing time required for (NP-1) simulations subsequent to the first is β∙(NP-1)T
if they are to be performed sequentially. Therefore, the total computing time required,
Tb, for time-domain simulations related to each control cycle is:
[ ]CDDb TTNPTTT +⋅++= 21β (8.1)
or, if NP simulations each with solution time T are performed in parallel, using a cluster
of processing systems.
[ ]CDDb TTTTT +++= 21β (8.2)
8.3.2.3 Polynomial function syntheses (block 5)
The computing time required for this calculation step for each control cycle depends on
the number of generators, number of specified control vectors, number of control
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140
variables, order of the polynomials adopted in representing the nonlinear functions and
processing capability of the computer system. For convenience in the subsequent
discussion, the computing time required for this step is denoted by Tn.
8.3.2.4 Constrained optimisation (block 6)
With a given processing capability of the computer system used, and constrained
optimisation algorithm, the computing time (denoted by T0) required for this calculation
step in determining the optimal values of the control variables, would depend on the
number of control variables, number of generators and order of the polynomials.
8.3.2.5 Overall requirement
From the individual components identified in 8.3.2.1 to 8.3.2.4, the total computing
time required for the control coordination for each control period is:
0TTTTT nbatotal +++= (8.3)
In (8.3), Tb is given by (8.1) or (8.2), depending on the computer systems used.
8.3.2.6 Feasibility constraints
It is required that, for feasibility, the total computing time formed in (8.3) be less than,
1DTse−
txk tyk
Database
WAMSPower system
modelling.Computing
time:Ta
One TDS from txk to t1k+T.
(NP-1) TDSs from t1k to
t1k+T.Computing
time: Tb
Polynomial function
syntheses. Computing
time: Tn
Constrained optimization. Computing
time:To
FACTS device
controllers
Power system
tyk + Ta+ Tb+Tn
tyk + Ta+ Tb+ Tn+ To
tyk + Ta+ Tbtyk + Ta
mTse− 2DTse−
tzk + TD2 = t1ktzk
(1) (2)(3)
(4)TC
(5) (6)
(7) (8)(9) (10)
Time instants
Fig.8.1. Control coordination block diagram
TDS: time-domain simulation Subscript k identifies the control cycle
s: Laplace transform operator TC: Computing time allocation
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141
or at most equal to the computing time allocation, TC:
Ctotal TT ≤ (8.4)
Block 7 in Fig. 8.1 represents the margin between the completion of the computation
and initiating the transmission of the input references to FACTS device controllers. The
time margin, Tm, shown in block 7 is:
Tm=TC – Ttotal (8.5)
For control option 1 as described in Section 7.4.1 (e), it is also required that:
TTC ≤ (8.6)
However, if control option 2 as referred to in Section 7.4.2 is adopted, the condition in
(8.6) becomes:
xC TT ≤ (8.7)
where Tx is defined in Section 7.3.4 and Fig. 7.2.
For given communication channel time delays, the option adopted for control
coordination, and computer system used, the conditions in (8.4), (8.6) and (8.7) provide
a basis for proper coordination among time intervals T, Tx and TC.
8.4 CASE-STUDY SIMULATION RESULTS
To verify the proposed controller coordination scheme, various case study simulations
were carried out. The first experiment was carried out on smaller power system network
of 4 generators and 12 node system. Being smaller sized power system network, only
one TCSC was introduced and the algorithm was tested for various fault locations and
for different control window sizes. The communication and computation delays were
considered while developing required control law for TCSC. The simulation results
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proved its effectiveness in maintaining first swing rotor angle stability. To verify the
robustness of this coordination scheme, a larger power system network with increase in
number of TCSCs. The selected results of this representative case study are presented in
next section.
8.5 REPRESENTATIVE POWER SYSTEM NETWORK
The 39-bus New England power system having 10 generators as shown in Fig. 8.2 is
adopted in the study for the purpose of illustrating the performance of the proposed
transient stability control coordination scheme, and investigating the feasibility of its
implementation. The system has two thyristor-controlled series compensators (TCSCs) -
the first TCSC in transmission line L11 between nodes N11 and N15, and the second
TCSC in transmission line L8 between nodes N12 and N16.
11
10
8
2
3
4 5
7
6
9
W
N12
N10
N11
N2
N8
N25
N26
N28
N29
N9N24N27N38
N37
N13
N15
N19
N18
N17
N1N16
N14
N31
N3
N20
N32
N33
N34
N35
N36
N21
N39
N30
N4 N5
N7
N23
N22
N6
TCSCL10
L1L2
L3
L4
L5
L6
L7
L8
L9
L11
L12
L13
L14
L15
L16L17
L18L19
L20
L21
L22
L23L24
L25
L26
L27
L28
L29
L30
L31
L32
L34
L33
W W
W
W W
W
W
W
W
W
W
TCS
C
N40
N41
Fig. 8.2. 10-Generator 39-Node New England test system[158]
The actual reactance output limits of TCSC are of a dynamic form which depends on
the TCSC operating current [23]. The TCSC data used in the study is given in Table 8.1.
In the present case-study the transmission lines in which TCSC is to be inserted are
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chosen based on the location and length which will benefit the overall system stability
improvement. With various case-studies carried out with different locations of
placement of TCSC, it is shown that, inserting TCSC in line 11 and line 8 gives the
most effective compensation to favour the entire power system network faster recovery
for transient stability performance.
Table 8.1: Controller parameters
No. Controller Parameters 1
TCSC in line L11
Main Controller: TC= 0.01s; KC=1pu;
pu0.0313Xpu0.05 ref ≤≤− SDC: K1=0.1pu; TW=0.2s; T1=0.2s; T2=0.1s;T3=0.05s;T4=0.2s
pu0.0025Xpu0.0025 SDC ≤≤−
2
TCSC in line L8
Main Controller: TC= 0.01s; KC=1pu;
pu0.0543Xpu0.0868 ref ≤≤− SDC: K1=0.1pu; TW=0.2s; T1=0.2s; T2=0.1s;T3=0.05s;T4=0.2s
pu0.0043Xpu0.0043 SDC ≤≤−
8.6 SYSTEM RESPONSE WITHOUT ONLINE CONTROL COORDINATION
OF TCSCS
The disturbance condition is that of a three-phase-to-earth fault on transmission line L6
near node 37, with a fault clearing time of 160ms. Subsequent to the fault disturbance,
the TCSC reactance input references remain constant at their pre-fault values which
were set at zero. With generator 1 nominated as the reference, the relative rotor angles
of generators 2–10 with respect to the reference generator are shown in Fig. 8.3. The
time step length in the time-domain simulation is 10ms. The fault onset time instant is
200 ms with respect to the origin of the time axis in Fig. 8.3. The responses confirm
that, without online adjustment of TCSCs input references, transient stability is lost
following the fault and fault clearance.
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As the fault happened to be on line 6, near node 37, it can be seen from the relative rotor
angle plots that the entire power system network gets separated in two parts. Generators
4, 5, 6 and 7 form one group or island, while remaining generators (i.e. 2, 3, 8, 9, and
10) form second group with less deviation in their relative rotor angles. It can be
observed from the transient stability response that the machines in the given group
remain in synchronism with each other inside the group. However, the first group of
machines seems to fall out of synchronism with the second group of machines in less
than 1 second’s time. The separation of groups increases as time passes, indicating that
unless corrective action is taken as early as possible, recovering the system stability
may become very difficult or in some cases, even impossible.
Fig.8.3 Relative rotor angle transients without online control coordination of TCSCs
8.7 OUTLINE OF TCSCS CONTROL COORDINATION STUDY
Drawing on the transient stability control schemes developed in Section 7.4 in the
previous chapter, control coordination of TCSCs reactance input references will be
derived and investigated, subject to feasibility constraints in terms of computing time as
discussed in Section 8.3.2.4. The computing time requirements in relation to time-
domain transient stability simulation, nonlinear function syntheses and constrained
0 0.5 1 1.5 2 2.5 3-100
0
100
200
300
400
Time (s)
Rel
ativ
e R
otor
Ang
le (D
eg)
G2 G3 G4 G5 G6 G7 G8 G9 G10
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145
optimisation will be determined. From this the total computing time required for each
control cycle is then calculated, and allocated, taking into account communication
channels delays. The study will also include the investigation of the control
coordination dynamic performance with various control scheme parameters.
8.8 TIME-DOMAIN SIMULATION COMPUTING TIME REQUIREMENTS
Transient stability software systems, particularly those designed for applications in
industry such as Power System Simulator for Engineering (PSS/E), have reached a high
level of maturity. They have been developed and refined over many years, which
implement techniques for speeding up large-scale power system simulation [157].
Those software systems can perform faster-than-real-time transient stability simulation,
even with detailed dynamic models, and, importantly, the computing time does not
significantly increase with power system size [157]. Using a transient stability analysis
function in DIgSILENT PowerFactory implemented on a PC with Intel(R) 2-core CPU
with E6550 processor, a faster-than-real-time transient stability simulation of the power
system in Fig. 8.1 is achieved with factor β ( as defined in Section 8.3.2.2) having the
value of about 0.188. The computing time required for the simulation includes that for
the power system modelling referred to in Section 8.3.2.1. As discussed in Section
8.3.2.2, each control cycle needs multiple transient stability simulations, which means
that a cluster of PCs or processors would be required to perform these individual
simulations in parallel for the control schemes to be feasible, particularly when the
number of separate simulations (i.e. NP as referred to in Section 7.5 in the previous
chapter) required in each control cycle is large.
8.9 COMPUTING TIME REQUIREMENTS FOR NONLINEAR FUNCTION
SYNTHESES
Many studies have been performed in the present work for identifying the nonlinear
relationships between relative rotor angles and TCSCs input references. The results of
the studies indicate that fourth order polynomials can represent with high accuracy the
nonlinear relationships.
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146
With two TCSCs, there are 15 coefficients to be identified for each fourth order
polynomial. The total computing time required in the syntheses of nine polynomials
representing nine relative rotor angles is about 0.014ms, using the algorithm developed
in Section 7.5 with NP = 15, and the PC referred to in Section 8.7.
8.10 COMPUTING TIME FOR CONSTRAINED OPTIMISATION
For the search method described in Section 7.4.1(d), the objective function and nine
constraint functions associated with the relative rotor angles, are to be calculated for
individual values of TCSCs reactance input references specified within their ranges. The
optimal solution is selected from the results of the calculations. For high resolution, a
discrete search space in terms of a 100X100 grid of TCSC reactance reference values,
with uniform spacing, has been used in the calculations. This search method can be
directly implemented by a parallel computing system. For example, the search space in
this case can be subdivided into four equal subspaces, and the search processes for
individual subspaces can be performed in parallel. The global optimal solution is then
obtained from the four optimal solutions for the subspaces. Using the PC as described in
Section 8.7, the computing time for identifying the optimal solution in each subspace is
about 22ms. With a cluster of four processing systems, each with a computing
capability of the PC referred to in Section 8.7, operating in parallel, it is possible to
obtain the global optimal solution in about 22ms. For comparison, the same constrained
optimisation problem has been solved using the quasi-Newton method which achieves
the optimal solution with a computing time of 1.01s. The solution from the search
method with substantially lower computing time is almost identical to that from the
quasi-Newton method.
8.11 CONTROL COORDINATION STUDY RESULTS
8.11.1 Control Option 1
Based on the development presented in Chapter 7, the computing time required for this
calculation step for each control cycle depends on the number of generators, number of
specified control vectors, number of control variables, order of the polynomials adopted
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147
in representing the nonlinear functions and the processing capability of the computer
system. For convenience in the subsequent discussion, the computing time required for
this step is denoted by Tn.
The investigations in Sections 8.7–8.9 have established the value for factor β (0.188)
related to computing time required in transient stability simulation, computing time
requirement for nonlinear function syntheses (0.014ms) and constrained optimisation
(22ms). The computing time component (Ta) for power system modelling referred to in
Section 8.3.2.1 has been included in the estimation of factor β which gives a value on
the conservative side. Based on [41], a communication channel’s time delay of 60ms is
adopted in the study. It remains to select the value of control time window (T) and
computing time allocation (TC) for each control cycle which are subject to:
(i) inequality constraints in (8.4) and (8.6);
(ii) transient stability being maintained following fault and fault clearance.
In terms of the condition in (i) in the above, it is required that, using the values for TD1
(60ms), TD2 (60ms), β (0.188), Tn (0.014ms) and T0 (22ms):
TTT C ≤≤+× .89450.23 (8.8)
The unit for T and TC in (8.8) is ms. In forming (8.8), it is taken that the computing
time for time-domain simulations in each cycle is given by (8.2) where individual
simulations would be performed in parallel as discussed in Section 8.3.2.2, using a
cluster of processing systems. (i.e. NP processing systems in parallel operation). With
the fourth order polynomials and two TCSCs, the requirement is NP=15 as referred
to in Section 8.8.
Various combinations of values of TC and T can satisfy the inequalities in (8.8), to meet
the feasibility requirements related to computing time based on processing systems as
described in Section 8.7. Many case studies have been carried out, using combinations
of TC and T, for investigating the dynamic performance of the control scheme. The
results of two representative cases are presented and discussed in the following.
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8.11.1.1 Case 1: T = 80ms and TC = 80ms
The nominated transient stability threshold in terms of relative rotor angle is 1400. The
control coordination is terminated when all of the relative rotor angles are within the
nominated transient stability threshold, and amplitudes of relative rotor angle
oscillations are less than a specified upper limit of 500. The fault disturbance is the same
as that in Section 8.5. With respect to the origin of the time axis in Fig. 8.4, the fault
onset time instant is 200ms; the fault is cleared at 360ms; and the first control period
starts at 560ms, i.e. the adjustment of TCSCs input references commences 200ms after
the fault clearance. Offline simulations are carried out for successive control cycles to
assess the dynamic performance of the control scheme.
Fig. 8.4 shows the transient responses of TCSCs reactance references (i.e. the control
variables) as determined by the control scheme outputs. Their initial values are 0.00 pu.
Using these controllers’ outputs for adjustments of TCSCs input references leads to the
responses in Fig. 8.5 for nine relative rotor angles. The responses confirm that transient
stability is maintained subsequent to fault and fault clearance. Control coordination
stopping criterion is achieved at 6.24s, i.e. after 71 control cycles as indicated by the
vertical line shown in Fig. 8.5. The transient responses of TCSCs input references are
highly oscillatory between their upper and lower limits during the control coordination
duration of 71 control cycles. Without the control coordination of the TCSCs, transient
stability would be lost as indicated in Section 8.5.
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149
Fig.8.4 TCSC input references for case 1
Fig.8.5 Relative rotor angle transients with online control coordination of TCSCs for case 1
8.11.1.2 Case 2: T = 200ms and TC =110 ms
Subject to the constraints in (8.8), the control time window (T) and computing time
allocation (TC) adopted in study case 1 are minimum possible values. The study in case
2 investigates the sensitivity of the control coordination performance with respect to T
0 1 2 3 4 5 6 7-0.1
-0.05
0
0.05
0.1
Time (s)
Rea
ctan
ce R
efer
ence
(pu)
TCSC1(L11)
TCSC2(L8)
0 1 2 3 4 5 6 7-150
-100
-50
0
50
100
150
Time (s)
Rel
ativ
e R
otor
Ang
le (D
eg)
G2 G3 G4 G5 G6 G7 G8 G9 G10
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150
and TC. Increasing T to 200ms and TC to 110ms leads to TCSCs input references
transient responses being less oscillatory as shown in Fig. 8.6 , in comparison with those
in case 1. However, the maximum relative rotor angles in the first swing are similar in
the two cases, as indicated in Fig. 8.7 for case 2 and Fig. 8.5 for case 1. Although
transient stability is maintained in both cases, the dynamic response of case 1 with
shorter T and TC is marginally better than that of case 2, at least in terms of required
control coordination duration. In case 1, the stopping criterion is achieved at 6.24s,
compared with 6.99s for case 2.
Fig.8.6 TCSC input references for case 2
Fig.8.7. Relative rotor angle transients with online control coordination of TCSCs for case 2
0 1 2 3 4 5 6 7 8-0.1
-0.05
0
0.05
0.1
Time (s)
Rea
ctan
ce R
efer
ence
(pu)
TCSC1(L11)
TCSC2(L8)
0 1 2 3 4 5 6 7 8-150
-100
-50
0
50
100
150
Time (s)
Rel
ativ
e R
otor
Ang
le (D
eg)
G2 G3 G4 G5 G6 G7 G8 G9 G10
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151
8.11.2 Control Option 2
Control coordination in case 2 with option 1 in (a) is modified to form a control
sequence based on option 2 described in Section 7.4.2. The control time window (T)
and computing time allocation (TC) remain at 200ms and 110ms respectively. However,
with option 2, the optimal TCSCs input references obtained from each constrained
optimisation are applied for a time duration Tx < T, instead of T. In the present study, Tx
is chosen to be 120ms. This leads to the TCSC input reference transient responses of
Fig. 8.8. In comparison with control option 1 with the same control time window
(200ms) and computing time allocation (110ms), the TCSCs input reference responses
obtained with control option 2 are more oscillatory as indicated in Figs. 8.6 and 8.8.
However, the relative rotor angle transients in the two control options are similar, as
confirmed in the responses shown in Figs. 8.7 and 8.9. With option 2, the stopping
criterion is achieved at 6.59s which is less than 6.99s for case 2 in option 1.
Fig.8.8 TCSC input reference responses Option 2
0 1 2 3 4 5 6 7-0.1
-0.05
0
0.05
0.1
Time (s)
Rea
ctan
ce R
efer
ence
(pu)
TCSC1(L11)
TCSC2(L8)
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152
Fig.8.9 Relative rotor angle transients Option 2
8.11.3 Approximate control
The TCSCs control schemes in Sections (a) and (b) have been investigated on the basis
of subdividing with high resolution the ranges of TCSCs input references for solving the
constrained optimisation problem. The control schemes are effective in maintaining or
enhancing system transient stability, and feasible in terms of computing time
requirements which can be met by a cluster of PCs or processors operating in parallel. It
is proposed in this section to reduce the computing time requirement by developing an
approximate control scheme in which each TCSC input reference is represented by a
low number of discrete levels. If L is the number of TCSCs, and each TCSC input
reference is represented by ND levels between its lower and upper limits in the control,
then the number of TDSs required in each control cycle for determining the optimal
combination of TCSCs input references for the cycle would be NDL. There would be no
requirements for the syntheses of polynomials and constrained optimisation. As the
number of TCSCs in a power system is relatively small, the required number of TDSs
would be, in general, significantly lower than that in the case of control with a high
resolution, particularly when the number of levels, ND, is low.
0 1 2 3 4 5 6 7-150
-100
-50
0
50
100
150
Time (s)
Rel
ativ
e R
otor
Ang
le (D
eg)
G2 G3 G4 G5 G6 G7 G8 G9 G10
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153
For illustrating the effectiveness of this approximate form of control, the input reference
of each TCSC in the system is represented by two levels: upper and lower limits. With
two TCSCs, each control cycle requires only four TDSs which can be performed by a
cluster of four processors in parallel operation. Optimal values of TCSCs input
references are then selected directly from the comparison among the time-domain
solutions, without the need for polynomial syntheses and constrained optimisation. This
approximate control in the discrete form leads to TCSCs input references in Fig. 8.10
and relative rotor angle transients in Fig. 8.11. The control scheme parameters are the
same as those in case 2, except that only the lower and upper limits of TCSCs input
references are used in the control. The responses in Fig. 8.11 confirm that transient
stability is maintained. The time instant at which the stopping criterion is achieved is
6.99s which is the same as that in case 2. However, the benefit of reducing the
computing system requirement is a substantial one.
Fig.8.10 TCSC input reference responses Approximate control
0 1 2 3 4 5 6 7 8-0.1
-0.05
0
0.05
0.1
Time (s)
Rea
ctan
ce R
efer
ence
(pu)
TCSC1(L11)
TCSC2(L8)
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154
Fig.8.11 Relative rotor angle transients Approximate control
8.12 CONCLUSION
Based on the transient stability control algorithm developed in Chapter 7 the present
chapter has performed a comprehensive analysis of the computing time requirement in
implementing the algorithm. The analysis outcome is a set of feasible constraints which
are to be satisfied by computer systems used for online control coordination of FACTS
devices with the aim of enhancing or maintaining power system transient stability
following disturbances. The various case studies carried out on different power system
configurations, for different fault locations, in various scenarios has confirmed the
robustness of proposed algorithm. With reference to a representative multimachine
power system with TCSCs, the study presented in the chapter indicates that it is
feasible, within the current technology, to implement the control coordination algorithm
in real time, using a cluster of processors operating in parallel. Offline simulation, using
realistic timing parameters for the control scheme, confirms its effectiveness in
maintaining transient stability following a fault disturbance.
0 1 2 3 4 5 6 7 8-150
-100
-50
0
50
100
150
Time (s)
Rel
ativ
e R
otor
Ang
le (D
eg)
G2 G3 G4 G5 G6 G7 G8 G9 G10
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155
Chapter 9 Dynamic Modelling Application for
Estimating Internal States of a
Synchronous Generator in Transient
Operating Mode from External
Measurements
9.1 INTRODUCTION
Drawing on the availability of synchronous generator terminal voltage and current, rotor
speed, and field winding voltage measurements, a procedure is derived for estimating in
transient conditions the generator’s internal operating states. These operating states are
rotor angle and flux linkages associated with field winding and damper windings. The
procedure is based on the fifth-order generator dynamic model. By applying the
numerical integration formula based on the trapezoidal rule, the generator model is
described by a set of algebraic equations of a recursive form in the discrete time-
domain. With external measurements, the unknown variables in the equations are those
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156
representing generator internal operating states. The nonlinear equations derived for
successive time instants are solved by applying the Newton-Raphson method. However,
if the number of equations is greater than that of variables, a minimisation technique
based on sequential quadratic programming method is then applied to solve the
nonlinear equation system. The estimation procedure can be applied to any time instants
including those in the transient operating conditions, without the requirement for
specifying the steady-state condition. The effectiveness and accuracy of the procedure
developed are verified by simulation using a representative multimachine power system
operating in the transient mode.
9.2 BACKGROUND THEORY
Power system stability including transient stability is an issue of increased importance
at present due to reduced stability margins arising from the maximisation of system
utilisation by power companies to increase their competitiveness in market
environments. In an attempt to optimise system dynamic performance, real-time control
methods for enhancing or maintaining system stability have been proposed, developed
and reported in the literature [13, 153, 159]. Central to the methods is the assumed
availability of the internal operating states of synchronous generators in the power
system to which real-time stability control is to be applied. However, the internal
operating states which include generator rotor angles and flux linkages are difficult, if
not impossible, to measure directly for control purpose.
There has been very limited research on real-time estimation of synchronous generator
internal states based on available measurements external to, or at, terminals of a
generator. In [160], a procedure based on a simplified generator model (i.e. single-axis
rotor flux model) in which damper windings and rotor speed transients are discounted
was presented for calculating approximate rotor angle and flux linkage established by
the field winding only. With the simplified model, only algebraic equations are required
in the calculation, using the measurements of generator stator terminal voltages,
currents and power, together with the field winding current. Generator dynamic
responses are not taken into account in the calculation. A method based on artificial
neural networks was proposed in [161] for estimating the rotor angles of synchronous
machines. The method requires offline training of neural networks, the inputs of which
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157
are voltage and current phasors obtained from a phasor measurement unit (PMU) on the
high voltage side of the generator transformer, and the outputs of the trained neural
networks are used for forming the rotor angle. A disadvantage of the method is that the
neural networks need to be updated or retrained when there are changes in network
configuration and/or a combination of generators in operation [161]. In addition, flux
linkages are not estimated in the method.
More recently, an algorithm based on a divide-by-difference filter has been proposed for
estimation of generator rotor angle, using the third-order model of the synchronous
generator [162]. Rotor damper windings are not represented in the estimation. Other
disadvantages of the proposed estimation procedure include the need for specifying a
generator’s steady-state condition for initialising the estimation process, and the
requirement for values of input mechanical torque at individual sampling time instants,
which is difficult, if not impossible, to measure.
Given the above background and state-of-the-art methodology in the estimation of
synchronous generator internal operating states, the present research has the objective of
developing a new procedure for estimating generator flux linkages and rotor angle in
which rotor speed transients and rotor field winding together with damper windings are
represented, requiring only measurements external to the generator. The new procedure
has the following features:
• Requiring only measurements that are practically feasible.
• Elimination of the need for specifying generator steady-state conditions for
initialising the estimation process.
• Representation of generator dynamic responses. The field winding and damper
winding flux linkage transients are included in the estimation process.
• Flexibility of starting the estimation process at any time instant including in the
transient periods.
• Low computing time requirements. This allows real-time estimation, with potential
applications in real-time transient stability control and monitoring.
• High accuracy and numerical stability.
• Robustness. The estimation process is independent of power network configuration
and/or operating conditions.
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158
Based on the standard practice in transient stability analysis and control, the fifth-order
dynamic model of the generator is adopted for deriving the estimation procedure in
which generator dynamic responses are represented. Through the use of the trapezoidal
rule of numerical integration, the relationships among stator terminal voltages, currents
and rotor flux linkages together with rotor angle and speed are transformed into a set of
recursive equations in the discrete time-domain. The time step length to be used in the
numerical integration of the differential equations in the generator model is a typical
one for transient stability analysis, and compatible with the sampling rate of phasor
measurements achieved at present. Combining the generator dynamic model in the
discrete time-domain with the available measurements at successive sampling time
instants of generator terminal voltage and current phasors, together with rotor field
winding voltage and rotor speed, leads to a system of nonlinear equations in which the
unknown variables are the rotor flux linkages and rotor angles at individual sampling
time instants. Values of input mechanical torque are not required in the estimation
procedure.
The estimation procedure can start from any nominated time instant. There is no need
for specifying the steady-state condition for the starting the procedure. The number of
successive sampling time instants of measurements is chosen, subject to the constraints
that the number of nonlinear equations formed at these individual time instants, to be
greater than, or equal to the number of unknown variables. The system of nonlinear
equations where the number of unknown variables is less than that of equations is
solved using an unconstrained minimisation algorithm. In each iteration, an objective
function of a quadratic form in the variables is derived and then minimised by the
Newton’s method. In the particular case where the number of unknown variables is
equal to that of equations, the Newton-Raphson method is applied for solving the
nonlinear equation system. The necessary conditions for the feasibility of solving the
estimation problem are derived and discussed in this chapter.
Although the specific context of the present research is that of estimating synchronous
generator operating states, the estimation procedure developed is of general validity and
applicable to any dynamical system where unmeasured internal state variables are
required for monitoring and control purpose. Given this general nature, the development
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159
of the estimation procedure in its dynamic form will commence with a general nonlinear
dynamical system described by a set of differential and algebraic equations.
Synchronous generators are interpreted, for the purpose of estimating their internal
states, as a particular case of nonlinear dynamical system.
Within the context of this research, the estimation procedure developed is verified for its
effectiveness and accuracy by simulation applied to a synchronous generator in a
representative multimachine power system subject to a fault disturbance. The simulation
study confirms that the estimation procedure has low computing time requirement, and
is suitable for real-time application.
9.3 DEVELOPMENT OF ESTIMATION PROCEDURE
9.3.1 Continuous-time nonlinear dynamical system and estimation requirement
In a general form, a continuous-time nonlinear dynamical system is modelled by the
following set of differential and algebraic equations as derived in the Chapter 2 and
represented as:
))(),(),(()( tttt uyxfx = (9.1)
0))(),(),(( =ttt uyxg (9.2)
In (9.1) and (9.2):
t : independent continuous-time variable
x(t) : vector of state variables
y(t) : vector of non-state variables
u (t) : vector of input variables
f, g : nonlinear vector functions of x(t), y(t), and u(t)
The total number of equations in (9.1) and (9.2) is denoted by N. The total number of
individual variables in x(t) and y(t) is equal to L.
It is taken that the system described by (9.1) and (9.2) operates, in general, in transient
condition, and it is, therefore, not valid to assume that 0)( =tx .
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160
The dynamical system described in (9.1) and (9.2) can be a subsystem operating within
a system, and some of the variables in x and/or y in (9.1) and (9.2) are the interface
variables between the system and its subsystem. An example is that of a synchronous
generator operating in a multimachine power system. The interface variables in this case
are the voltage and current at the generator terminal to which the power network is
connected.
There are direct measurements in the discrete-time domain of some of the variables in
vectors x and/or y. It is required that the remaining variables in x and/or y which cannot
be directly measured be estimated, using the set of available measurements at individual
sampling time instants.
The sampling rate of the measurement is denoted by fs corresponding to the sampling
time interval of ∆t = 1/fs.
It is taken that the sampling rate of the measurement system has been selected, taking
into account the transient phenomena of interest arising from the responses of the
dynamical system following disturbances.
9.3.2 Discrete-time domain system model
With measurements available in the form of time series obtained via data acquisition
systems and PMUs, it is required to transform (9.1) and (9.2) into discrete-time
equations for deriving the estimation algorithm. The time step length used in
transforming into the discrete-time domain is chosen to be the same as the sampling
time interval, ∆t, of the measurement system. By applying the trapezoidal rule of
numerical integration, the differential equation in (9.1) is transformed into:
[ ]))1-(),1-(),1-(())(),(),((2
)1()( nnnnnntnn uyxfuyxfxx +∆
+−= (9.3)
In (9.3), integer n is the discrete time variable from which the actual time is given by
n.∆t.
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161
The algebraic equation in (9.2) transforms directly to, in the discrete-time domain:
0))(),(),(( =nnn uyxg (9.4)
The set of algebraic equations in (9.3) and (9.4) is combined to:
0))1-(),1-(),1-(),(),(),(( =nnnnnn uyxuyxh (9.5)
Vector function h in (9.5) is defined in:
⋅
−−⋅−
=))(),(),((
))1-(),1-(),1-(()1-())(),(),(()(
))1-(),1-(),1-(),(),(),((nnn
nnnnnnnn
nnnnnnuyxg
uyxfxuyxfx
uyxuyxhα
α
(9.6)
where 2t∆
=α (9.7)
The total number of individual discrete-time equations in (9.5) is N.
9.3.3 Estimation problem formulation
With the input vector u being specified, and a subset of the variables in vectors x and/or
y being measured at individual time steps, it is required to estimate all of the
unmeasured variables in vectors x and/or y. With time instant n=0 nominated as the
time reference, the estimation of the unmeasured variables is to be carried out for
successive time instants, starting from n=0. If the total number of measured variables is
denoted by M, then it is required to estimate (L-M) variables at each time instant n ≥ 0.
The problem of estimation does not arise if M=L.
If the dimension of vector g is NG, then there are NG equations obtained from (9.4) at
time n=0:
0))0(),0(),0(( =uyxg (9.8)
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162
With successive time instants j = 1, 2, . . ., K applied to (9.5), the following set of
equations are obtained:
0))1-(),1-(),1-(),(),(),(( =jjjjjj uyxuyxh j=1, 2,..., K (9.9)
As the dimension of h is N, there are K.N individual equations in (9.9).
With (K+1) time instants, there are (K+1).(L-M) unmeasured elements to be estimated.
From (9.8) and (9.9), there are (K.N+NG) equations. A necessary condition for the
estimation to be possible is:
(K.N + NG) ≥ (K+1) (L – M) (9.10) or , if (N – L + M ) > 0 :
)()(
MLNNGMLK
+−−−
≥ (9.11)
It is not feasible to solve the estimation problem when (N – L + M ) ≤ 0.
If there does not exist a non-negative integer K that satisfies (11), then it is not possible
to solve the estimation problem. If the inequality in (11) is satisfied for some non-
negative integer K, then the problem is that of solving the set of algebraic equations in
(8) and (9) to obtain the values for unmeasured variables in x and/or y at time instants 0,
1, 2, . . . , K. In the present work, the smallest non-negative integer K which satisfies
(11) is chosen. Once the values of unmeasured variables at sampling time instants 0, 1,
2, . . . , K have been determined, the estimation process for subsequent time instants is
carried out as described in Section 9.3.5.
9.3.4 Solution method
Equations (9.8) and (9.9) are assembled in:
H (XK) = 0 (9.12)
In (9.12):
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163
XK = unmeasured elements in vectors x(i) and/or y(i) for i = 0,1,2, . . ., K.
and
=−−−
=
Kjjjjjjj
K
,1,2,3,for ))1()1()1()()(),((
))0()0(),0(()(
u,y,x,u,yxh
uyxgXH (9.13)
Based on (9.11), if there is a non-negative integer K such that K = (L – M – NG) / (N – L
+ M), then the number of equations in (9.12) is equal to the number of unknown
variables in vector XK.
The Newton-Raphson method is applied to solve the set of nonlinear equations in (9.12)
for XK. The iterative solution sequence is given in:
⋅
−=
−1-
11-1- p
KpK
pK
pK XHXJXX (9.14)
In (9.14):
p = the Newton-Raphson iteration step counter 1-
,pK
pK XX : values of the elements in vector XK at iteration steps (p-1) and p
respectively
J = Jacobian matrix of vector function H:
1-1- )(
pK
K
KpK
XXXH
XJ
∂
∂=
(9.15)
A necessary condition for achieving the solution using the iterative sequence in (9.14) is
that the Jacobian matrix formed in (9.15) is nonsingular for each iteration. The iterative
sequence in (9.14) is terminated when the following convergence criterion is achieved:
ε≤
pi KH X i = 1,2, . . . , (K.N + NG) (9.16)
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164
Hi in (9.16) is the ith element of vector function H, and ε is a pre-set tolerance for
convergence checking.
On the other hand, from (9.11), if, K is greater than (L – M – NG) / (N – L + M), then
there are more individual equations in (9.12) than unknown variables in vector XK. In
this case, the solution for XK is determined by minimising the following objective
function with respect to XK:
( ) ( )KKtE XHXH ⋅= (9.17)
On linearising H(XK) around the tentative solution at the previous iteration step (p-1)
where p is the current step:
( )
−
+
≅
−−− 111 pKK
pK
pKK XXXJXHXH (9.18)
In (9.18), J is the Jacobian matrix of vector function H as given in (9.15). However, J in
(9.18) is not a square matrix as there are now more equations than unknown variables.
Combining (9.17) and (9.18) gives:
−
+
⋅
−
+
=
−−−
−−−
111
111
pKK
pK
pK
tpKK
pK
pKE
XXXJXH
XXXJXH (9.19)
Minimising the objective function in (9.19) as a quadratic form in XK by Newton’s
method leads to the following optimal solution for XK, at iteration step p:
⋅
⋅
−=
−−−−− 11111 pK
pK
tppK
pK XHXJAXX (9.20)
In (9.20):
⋅
=
−−− 111 pK
pK
tpXJXJA (9.21)
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165
A necessary condition for the feasibility of solving the estimation problem is that
⋅
−− 11 pK
pK
t XJXJ is nonsingular for each iteration step. Convergence is achieved
when, based on (9.17):
ε≤
⋅
pK
pK
t XHXH (9.22)
In (9.22), ε is a pre-set tolerance for convergence checking.
9.3.5 Estimation process for subsequent time instants
The formulation and solution method developed in Sections 9.3.3 and 9.3.4 are for the
set of time instants {0, 1, 2, . . . , K}. The procedure is that which starts the estimation
process. It leads to the set of values {x(i), y(i) for i = 0,1,2,. . . , K} which allows the
estimation for the subsequent time instants greater than K to be carried out.
At time instant (K+1), there are N equations derived from (9.5) with n = K+1:
0))(),(),(),1(),1(),1(( =+++ KKKKKK uyxuyxh (9.23)
There are (L – M) unknown (i.e unmeasured) variables in vectors x(K+1) and y(K+1).
The values of all of the elements in vectors x(K) and y(K) are completely known, using
the estimation results at time instant K, combined with the measured data at that time
instant.
As (N – L + M) > 0, there are more equations in (9.23) than unknown variables in
x(K+1) and y(K+1). Therefore, the unconstrained minimisation algorithm developed
and described in (9.17) to (9.22) is applied for solving the estimation problem for time
instant (K+1). The vector function H in (9.17) to (9.22) is now replaced by h in (9.23),
and XK in (9.17) to (9.22) by the vector of unknown variables in x(K+1) and y(K+1).
The above estimation process is then repeated successively for individual time instants
(K + l )’s for l ≥ 2.
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166
9.4 APPLICATION TO SYNCHRONOUS GENERATOR
9.4.1 Generator dynamic model
The fifth-order dynamic model of a 3-phase synchronous generator [163] formed in the
rotor d-q axes is adopted in the present work. On discounting the stator flux transients,
the following stator algebraic equations have been derived [163]:
0))((cos)())((sin)(
))((sin)())((cos)())((
)()()(
)())((cos)())((sin)(
))((sin)())((cos)(
=
⋅+⋅−
⋅+⋅⋅+⋅
+
⋅⋅−
⋅+⋅−
⋅+⋅
ttIttI
ttIttIt
ttt
tttVttV
ttVttV
QD
QD
R
kq
kd
fd
R
QD
QD
δδ
δδω
ψψψ
ωδδ
δδ
ss
s
RL
P
(9.24)
In (9.24):
VD(t), VQ(t): D- and Q- components of stator terminal voltage in the network D-Q axes
ID(t), IQ(t): D- and Q- components of stator current in the network D-Q axes
δ(t): angular separation between the rotor d-axis and the network D-axis (rotor angle)
ωR(t): rotor angular speed
ψfd(t): field winding flux linkage
ψkd(t): d-axis damper winding flux linkage
ψkq(t): q-axis damper winding flux linkage
Matrices Ps, Ls and Rs are defined in the Appendix in terms of generator parameters.
The rotor flux linkage transients are described by the following set of differential
equations [163]:
⋅+⋅−
⋅+⋅⋅
+
+
=
))((cos)())((sin)(
))((sin)())((cos)(
00
)(
)()()(
)()()(
ttIttI
ttIttI
tV
ttt
ttt
QD
QD
fd
kq
kd
fd
kq
kd
fd
δδ
δδ
ψψψ
ψψψ
m
m
F
A
(9.25)
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167
In (9.25), Vfd is the field winding voltage. The coefficient matrices Am and Fm are
formed from generator parameters as shown in the Appendix.
The relationship between the rotor angle and angular speed is expressed in the following
differential equation:
0RR ωωδ −= (9.26)
In (9.26), ωR0 is the nominal rotor angular speed.
9.4.2 Procedure for generator internal state estimation
The generator dynamic model given in (9.24), (9.25) and (9.26) is that of a continuous-
time nonlinear dynamical system of the form described in (9.1) and (9.2), with the
following definition for the state variable vector, x, and non-state variable vector, y:
[ ]t)(),(),(),()( ttttt kqkdfd δψψψ=x (9.27)
[ ]t)(),(),(),(),(),()( tVttItItVtVt fdRQDQD ω=y (9.28)
The total number of variables in vector x is four, and that in vector y is six. The
superscript t in (9.27) and (9.28) denotes vector transpose. The input vector u(t) in this
case is zero. The generator state equation set is that given in (9.25) and (9.26) are
assembled to (9.29), using the definitions in (9.27) and (9.28):
))()()( ttt y,f(xx = (9.29)
where f is the vector function described in (9.25) and (9.26). The total number of
individual differential equations in (9.29) is four. The complete generator model is
described by the state equation in (9.29) and the stator algebraic equation set in (9.24)
which is expressed in the following compact form:
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0))()( =tt y,g(x (9.30)
where g is the vector function defined in (9.24). The number of individual algebraic
equations in (9.30) is NG = 2. The total number of individual equations in (9.29) and
(9.30) is N = 6, and the total number of individual variables in x(t) and y(t) is L = 10.
Equations (9.29) and (9.30) are of the general form in (9.1) and (9.2) with u(t) = 0. The
estimation procedure developed in a general form in Section 9.3 applies directly for
estimating the generator internal operating states, using external measurements.
9.4.3 Measurement requirements
As derived from Section 9.3.3, a necessary condition for estimation feasibility is (N – L
+ M) > 0, which means in this case the number of measured variables, M, is to be 5 or
greater. It is not practically feasible to measure the state variables in vector x of (9.27)
which comprises rotor flux linkages and rotor angle. However, the measurements of the
non-state variables in vector y of (9.28) present no difficulty. These six variables are the
generator terminal voltage and current phasors, rotor speed and field winding voltage.
Key aspects relevant to the measurement are discussed in the following:
9.4.3.1 Voltage and current phasor measurements
There are synchronised PMUs at terminals of individual generators operating in the
power system. Phasors representing stator terminal voltage and current at the supply
frequency for each generator are formed in the positive-phase sequence.
The power system waveforms from which the phasors are formed are expressed as
discrete time series with the time origin nominated to be at the time reference n = 0 at
which the estimation process starts. All of the phase angles of the generator terminal
voltage phasors formed by the synchronised PMUs at any sampling time instants are
relative to the time origin as nominated. For the purpose of defining the network D-axis,
the voltage phasor at time n = 0 of the terminal of a generator is selected. The angular
separation between the voltage phasor and the D-axis is the phase angle of the voltage
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phasor. In this way, the D-axis is defined, and the Q-axis is that separated from the D-
axis by 900. With the time origin common to all of the power system waveforms, all of
the network voltage and current phasors at any sampling time instants formed from the
waveforms are relative to the network D-Q axes as established.
VD and VQ in Section 9.4.1 are the components of the stator voltage phasor VD + j.VQ in
the positive-phase sequence formed in the network D-Q axes. Similarly, ID and IQ are
the components of stator current phasor ID + j.IQ in the positive-phase sequence formed
in the network D-Q axes.
The measured values for VD, VQ, ID and IQ at successive sampling time instants are used
in the estimation process described in Sections 9.4.1 and 9.4.2.
9.4.3.2 Availability of rotor speed measurements
Rotor speed measurement is required if there is a power system stabiliser (PSS) with a
rotor speed input signal. In general, rotor speed measurement is also required as an
input to the governor system. Therefore, advantage can be taken of the available rotor
speed measurement which is used for PSS and/or governor input in forming the set of
measured variables needed in the estimation procedure of Section 9.4.2.
With six measured variables, the necessary condition identified in Section 9.3.3 for
estimation feasibility is satisfied. On this basis together with the practicality of
measuring the variables in vector y of (9.28), the present work applies the procedure for
estimating generator internal states with measurements of stator voltage and current
phasors, rotor speed and field winding voltage.
9.4.4 Discussion
9.4.4.1 Representing magnetic saturation
When it is required to represent magnetic saturation, the d- and q-axis magnetising
inductances (Lmd and Lmq respectively) in the generator dynamic model of Section 9.4.1
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are modified, based on the generator magnetisation characteristic and air-gap flux
linkage [6]. The calculation of air-gap flux linkage, taking into account the contribution
of the negative-phase-sequence component of the stator current in unbalanced
operation, is described in [6]. The estimation procedure in Section 9.4.2 would be
iterated, using modified magnetising inductances at each iteration.
9.4.4.2 Initial values
The Newton-Raphson iterative solution method presented in Section 9.3.4 requires
starting or initial values of the variables. The following scheme for forming the starting
values is proposed, in the context of a synchronous generator. With rotor speed set to its
nominal value or measured value, and discounting the flux linkages in d-axis and q-axis
damper windings, equation (9.24) is solved for the rotor angle for each sampling time
instant.
With rotor angle and speed being known, equations (9.24) and (9.25) are linear in terms
of rotor flux linkages. These linear continuous time-domain equations transform into a
linear equation set in the discrete time domain, using the trapezoidal rule of integration.
For (K+1) sampling time instants, individual linear equation sets are formed, and then
solved simultaneously to give approximate solution values for rotor winding flux
linkages.
With respect to the estimation process based on a minimisation method for subsequent
time instants described in Section 9.3.5, the initial values for the unknown variables at
time instant (K + l) (for l ≥ 1) are set to be equal to the values obtained in the previous
time instant (K + l - 1). The above procedure for forming starting values for the iterative
solution methods based on the Newton-Raphson algorithm and/or minimisation
algorithm does not require any specification or knowledge of the steady-state condition
of the generator and power network.
Extensive case studies have been carried out to verify the effectiveness of the above
procedure for forming the starting values of the variables required in the estimation
procedure of Section 9.4.2. They lead to the convergence of the iterative sequence
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derived from the estimation procedure. The results of a representative study are
presented in Section 9.5.
9.5 REPRESENTATIVE CASE STUDY
The estimation procedure developed in Sections 9.3 and 9.4 is applied to generator 9 in
the 10-generator, 39-node New England power system [158] augmented with two
TCSCs (thyristor-controlled series compensators) as shown in Fig.F1 of Appendix F.
The disturbance condition is that of a 3-phase-to-earth fault on transmission line L6 and
close to node N37, with a fault clearing time of 160ms. The estimation starts at 20ms
subsequent to the fault clearance. For the purpose of illustration, the results and
computing time of the following case study by simulation are presented in the following
sections.
9.5.1 Results
In the study, it is taken that the generator terminal voltage and current phasors, rotor
speed and field winding voltage are measured and available at successive sampling time
instants. The set of measurements satisfies the necessary condition as discussed in
Section 9.4.3. In the simulation study, time-domain transient stability analysis with the
time step length of 10ms is carried out to give the values of the generator terminal
voltage and current phasors, rotor speed, and field winding voltage at successive
sampling time instants. The field, d-axis damper and q-axis damper winding flux
linkages and rotor angle are to be estimated.
With time instants denoted by 0 and 1 used for starting the process, there are eight
individual equations in (9.8) and (9.9) (when applied to generator model) with eight
unknowns in rotor flux linkages and rotor angle at time instants 0 and 1. The Newton-
Raphson method described in Section 9.3.4 is applied to solve the estimation problem.
For estimation of each subsequent time instant greater than 1, there are four unknowns
(i.e. rotor flux linkages and rotor angle at each time instant) and six equations. The
unconstrained minimisation method developed in Section 9.3.5 is applied to estimate
the values for the unknown variables.
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In Figs. 9.1 to 9.4 are shown the comparisons between the estimated rotor flux linkages
and rotor angle and their actual values obtained from time-domain transient stability
simulation. The comparison indicates that very high accuracy has been achieved with
the estimation procedure. In addition, the procedure has high numerical stability. There
is no evidence of error accumulation throughout the estimation process performed in
many successive sampling time instants. For each estimation, the typical number of
iterations required for convergence is three, with the tolerance of 10-6 pu. The Jacobian
matrix in (9.15) or the A matrix formed in (9.21) is nonsingular in each iteration.
Fig.9.1 Comparison between actual and estimated rotor angle
Fig. 9.2 Comparison between actual and estimated main field flux
(pu on generator rating)
0.4 0.5 0.6 0.7 0.8 0.9 1 1.160
70
80
90
100
110
Time (s)
Rot
or A
ngle
(Deg
)
ActualEstimated
0.4 0.5 0.6 0.7 0.8 0.9 1 1.13.8
3.9
x 10-3
Time (s)
d-a
xis M
ain
Fiel
d Fl
ux (p
u)
ActualEstimated
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Fig.9.3 Comparison between actual and estimated d-axis damper winding flux
(pu on generator rating)
Fig.9.4 Comparison between actual and estimated q-axis damper winding flux
(pu on generator rating)
9.5.2 Computing time
With Intel (R) 2-core CPU having E6550 processor, the computing time required for the
estimation of sampling time instants 0 and 1 is about 5ms, and that for each subsequent
time instant is about 3.6ms. This computing time requirement confirms that real-time
0.4 0.5 0.6 0.7 0.8 0.9 1 1.13
3.1
3.2
3.3
3.4
3.5
3.6x 10-3
Time (s)
d-a
xis D
ampe
r Win
ding
Flu
x (p
u)
ActualEstimated
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-10
-8
-6
-4
-2x 10-4
Time (s)
q-a
xis D
ampe
r Win
ding
Flu
x (p
u)
ActualEstimated
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estimation is feasible, even with a relatively high sampling rate of 100Hz (i.e. the
sampling time interval of 10ms) in the measurement system.
9.6 CONCLUSIONS
The chapter has derived in a general form a systematic procedure based on dynamic
modelling and numerical techniques for solving nonlinear equations for estimating the
internal states of a nonlinear dynamical system, using available measurements of
variables external to the system. The procedure is applied to estimate the synchronous
generator rotor flux linkages and rotor angle, using available measurements of generator
terminal voltage and current phasors, rotor speed and field winding voltage. The key
advantages of the procedure developed include:
(i) Initialising the estimation process. The estimation process can be started at any
time instant, without the need of specifying or knowing the steady-state operating
condition.
(ii) Generator dynamic responses. They are taken into account in the procedure where
transients in rotor flux linkages of the field and damper windings, rotor angle and
speed, and field winding voltage are fully represented.
(iii) Accuracy and numerical stability. The high accuracy and numerical stability of
the procedure are verified by many simulation studies, the representative results of
which are presented in the chapter.
(iv) Computing time requirement. The low computing time requirement of the
procedure would allow real-time estimation, even when the measurement system
has a high sampling rate of 100Hz. Drawing on this key advantage, the potential
applications of the estimation procedure include real-time transient-stability
control and monitoring.
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Chapter 10
Conclusions and Future Work
10.1 CONCLUSIONS
This chapter brings together and summarises the original contributions and advances
that have been made in the research and presented in the body of the thesis.
A new MPC based TCSC controller is developed for a single-machine-infinite-bus
system. The control scheme is developed using a detailed dynamic model of a
synchronous generator and is coordinated with exciter, prime-mover and governor
controllers. The MPC scheme developed generates a required control output (i.e. TCSC
input reference for series compensation requirement) based on present and future
operating scenarios. The control scheme has shown its effectiveness in restoring rotor
angle stability even when the fault is cleared after critical clearing time. The research
and results presented in the thesis have confirmed that it is possible to form a control
law which is adaptive to power system operating conditions, and effective in improving
or maintaining its transient stability. This is achieved by directly deriving the
relationship between the relative rotor angle and the control variable through
linearisation in individual time horizons, which leads to the objective function to be
minimised for forming successive optimal values for the control variable.
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As a further extension of transient stability improvement using FACTS devices, the
single machine problem is extended to a multimachine system. Considering the wide-
area network requirements, an online control coordination scheme is successfully
developed for a multimachine system which is adaptive to the changing power system
topology.
The developed method draws on transient stability time-domain simulations and
constrained optimisation for deriving the control algorithm. Optimal input references for
FACTS device controllers are determined to maximise the transient stability margin,
taking into account communication channel time delays in obtaining the power system
operating state, and computing time required in executing the control algorithm.
An optimisation problem is formulated using available data, predicted performance, and
future system states, including the future possible control actions and solved for
obtaining the optimised value of necessary TCSC reactance. The time required for the
entire process, of getting the data, predicting, formulating the optimisation problem and
solving it, is considered as computation delay. Even after solving the optimisation
problem and getting the output of necessary compensation, there can be delay in
communication in passing on this signal to TCSC and actual insertion of this TCSC
reactance in the system. The computation delay for the controller to check with the
future predicted system state, formulate the optimisation problem with given constraints
and give output as a required value of TCSC reactance compensation to make the
system stable, is considered in terms of the computing time and the form of the
computing constraints. The dynamic performance of the control method together with
its feasibility in terms of computing time requirements is investigated in detail. The
analysis outcome is a set of feasible constraints which are to be satisfied by computer
systems used for online control coordination of FACTS devices with the aim of
enhancing or maintaining power system transient stability following disturbances.
With reference to a 10 generator New England system, having TCSCs, the study
presented in this thesis has indicates that it is feasible, within the current technology, to
implement the control coordination algorithm in real time, using a cluster of processors
operating in parallel. Offline simulation, using realistic timing parameters for the
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control scheme, confirms its effectiveness in maintaining transient stability following a
fault disturbance.
The online controller coordination scheme is developed with two scenarios based on the
system model updating strategy. With option 2 the system model used in control law
formation, is updated more frequently to reflect changed system topology considering
the fast dynamics after severe disturbance. With many experimental simulation studies
it is found that the length of the control window can be increased to generate a feasible
control law if the communication and computation delays are greater.
The last part of the thesis gives a procedure for estimation of internal states of
synchronous generator using the available measurements. As in practical scenario, it is
not possible to measure the internal state variables of the synchronous generator and the
only measurements available are voltage, current phasors and speed. Considering all
voltage, current measurements availability, along with rotor speed and field
measurements as complete measurements, a case study is carried out for estimation of
internal variables of rotor fluxes and rotor angle. The simulation results are validated
with a representative machine from 10-generator power system network of the New
England system. The plots of estimation and actual values show the close tracking and
correctness of the method developed. The key features of the developed method is that,
it does not need to start with a steady-state assumption and that it can be started at any
instant, including before or after fault in post-fault scenarios.
10.2 FUTURE WORK
With the foundation of work provided by the new concepts and developments presented
in the thesis, further research is envisaged and outlined in the following sections.
10.2.1 MPC based controller for transient stability using multiple FACTS devices
This proposal is that the MPC algorithm will be further developed for accommodating
various shunt and series types of FACTS devices coordinated together for transient
stability improvement. The proposed method can be converted to general form to handle
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multimachine problems of any size. The method can be extended to accommodate
various time delays to make it adaptive to wide-area network requirements for a real-
time controller.
10.2.2 Online control coordination scheme for multiple FACTS devices
This proposal is that of re-hosting the software systems which have been developed in
the thesis on a cluster of high-performance processors, and then carrying out extensive
testing in the real-time environment to provide effective compensations using shunt as
well as series compensations with various FACTS devices. In the testing, real-time
dynamic simulation of power systems will be performed, and provide the interactions
between the FACTS devices and the power systems together with their controllers.
10.2.3 Control coordination for power system stability improvements in case of
loss of communication signal information/data
The online control coordination considered in the thesis is for dynamic mode i.e.
transient state of system operation with an assumption that all information is available
through WAM without any loss of signal. However, in reality, because of the
widespread nature of power system networks, where huge data information is travelling
over many channels through different modes of communication, there are possibilities
of loss of communication signal. In such circumstances the controller will have to
generate a correct control law even with this incomplete input data information. With
high-speed computing facilities and FACTS controllers, it is proposed to investigate the
feasibility of deriving the control law and its implementation for FACTS devices
coordination in a shorter time frame related to system stability even with loss of
communication signal, with the objective of preserving system stability following a
large disturbance.
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10.2.4 Estimation of internal state variables for synchronous generator with
incomplete/inaccurate measurements
The case study represented in Chapter 9 shows that using rotor speed, field voltage
measurements and terminal voltage and current phasors, the internal state variables of a
synchronous generator can be estimated accurately. However, it is not possible to have
all the measurements available all the time or the available measurements might have
some measurement errors. To consider such a scenario, the present estimation algorithm
is proposed to be developed for two cases: firstly, with incomplete measurements and
secondly with errors in measurements. In the case of errors in measurement, based on
the standard accuracy and acceptable errors in speed, voltage and current measurements
the robustness of the controller will be verified. In the case of incomplete
measurements, as the number of known inputs is less than the unknown variables to be
calculated from the same set of equations, it is necessary to consider a sufficient number
of time instants to form a complete, feasible set of equations which can be solved.
Depending on the availability of speed or field voltage, the algorithm is divided in two
cases- Case 1 deal with calculation of unknown field voltage along with rotor flux and
rotor angle while case 2 considers a scenario where the field voltage is known and rotor
speed is unknown.
The problem is formed using dynamic equations of rotor flux and rotor angle equations
combined with algebraic equations of system voltages. The procedure for forming this
set of four main equations is the same as explained in Chapter 9 with a case of eight
equations and eight unknowns. However, in this particular case the number of
unknowns is more, due to the equations for the next successive instants of two and three
being added to previous model, making it twenty equations and twenty variables.
Although the main set of equations to be solved remains the same, the problem
formulation procedure for two cases will differ based on unknown variables.
10.2.5 Estimation of internal state variables for exciter, turbine and governor
system
The algorithm developed in the Chapter 9 is very general and can be applied to any
dynamical system at any instant for estimation of internal variables. Considering the
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effect and importance of exciter, turbine and governor systems in dynamic studies,
when a complete dynamic model is formed including these state variables, the
developed algorithm can be used for estimation of internal state variables of exciter and
turbine system.
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______________________________________________________________________A.1
Appendix A
Generator Rotor Dynamics
A.1 GENERATOR STATOR VOLTAGE EQUATIONS
Assuming a synchronous machine with one main field winding, one direct-axis and one
quadrature-axis damper winding, the total voltage and current vector of synchronous
machine v and current vector i can be given as follows:
=
=r
s
kq
kd
fd
q
d
vv
vvvvv
v ;
=
=r
s
kq
kd
fd
q
d
ii
iiiii
i
(A1)
The relation between this voltage vector and current vector can be given by
RiiGdtdiLv r ++= ω (A2)
where, the coefficient matrices L, G, and R are as follows in voltage equation v in
terms of current i
______________________________________________________________________A.2
=
−−−
−
=rrrs
srss
kqkdfdqd
kqmq
kdmdmd
mdfdmd
mqq
mdmdd
s
s
LLLL
LLLLLLLL
LLLLL
kqkdfdqd
L
ss
0000000
00000
[ ]srss
kkfqd
mqmdd
mqq
s
s GGLLL
LLqd
G
qddss
=
−
−=
00000
(A3)
=
−
−
=rr
ss
kkfqd
kq
kd
fd
a
a
q
d
d
s
s
RR
RR
RR
R
kkfqd
R
qddss
00
00000000000000000000
Equation (A1) can be expressed explicitly in complete matrix form as
×
−
−
+
−
−
×
−−−
−
=
kq
kd
fd
q
d
kq
kd
fd
a
a
rmqmdd
mqq
kq
kd
fd
q
d
kqmq
kdmdmd
mdfdmd
mqq
mdmdd
fd
q
d
iiiii
RR
RR
R
LLLLL
pipipipipi
LLLLLLLL
LLLLL
vvv
ω00
000
0000000
00000
00
Partitioning (A4) in terms of stator and rotor circuit equations
(A4)
[ ]
+
+
=
r
s
r
s
r
ssrssr
r
s
rrrs
srss
r
sii
RR
ii
GGpipi
LLLL
vv
00
ω
(A5)
ssrsrrsssrrsrssss iRiGiGpiLpiLv ++++= ωω
(A6)
rrrrrsrsr iRpiLpiLv ++=
(A7)
Using flux linkages expression
______________________________________________________________________A.3
=
r
s
rrrs
srss
r
sii
LLLL
ψψ
(A8)
rsrssss iLiL +=ψ (A9)
and
rrrsrsr iLiL +=ψ (A10)
Substituting (A9) in (A6) for rsrssss piLpiLp +=ψ and (A10) into (A7)
[ ] ssrsrrsrssrrrrrsrss iRLLGGLGpv ×+−++= −− 11 ωψωψ (A11)
At this stage, advantage can often be taken, in practical stability studies based on the
synchronously rotating reference frame, of reducing the computing time expanded in
analysis. This is achieved by eliminating the terms of stator-voltage transients
corresponding to the rate of change of stator flux linkages with respect to time. These
terms contribute very little dynamic response analyses that are dominated by the inertial
characteristics of rotating machines and they may be readily eliminated. Hence,
eliminating stator transients and simplifying it using:
1−= rrsrrm LGp ω ; ( )[ ]srsrrsrssrm RLLGGz +−−= −1ω (A12)
smrms izpv −= ψ (A13)
Using notations of (A12) and expanding (A13) to separate d-axis and q-axis voltage
equations as:
qmdmkqmkdmfdmd izizpppv 1211131211 −−++= ψψψ (A14)
qmdmkqmkdmfdmq izizpppv 2221232221 −−++= ψψψ (A15)
However, examining the structure of matrices pm and zm as given in (A17) - (A18),
taking advantage of sparsity, vd and vq equations can be further simplified as (A19-
A20).
______________________________________________________________________A.4
−
−+
−−−
−−
−
−
=
−
−
q
d
a
a
mq
md
md
kq
kdmd
mdfd
mdmd
mq
d
qr
kq
kd
fd
kq
kdmd
mdfd
mdd
qr
q
d
ii
RR
LLL
LLLLL
LLL
LL
LLLLL
LLL
vv
00
000
0000
000
00
0000
000
1
1
ω
ψψψ
ω (A16)
where,
1
0000
000
−
−
=
kq
kdmd
mdfd
mdd
qrm
LLLLL
LLL
p ω (A17)
−
−+
−−−
−−
=
−
a
a
mq
md
md
kq
kdmd
mdfd
mdmd
mq
d
qrm R
R
LLL
LLLLL
LLL
LL
z0
0
000
0000
000
00
1
ω
(A18) as pm11 = 0; pm12 = 0; pm23 =0, eliminating these three terms from (A14) and (A15), final
expression for stator voltages in terms of d-axis and q-axis can be obtained as:
qmdmkqmd izizpv 121113 −−= ψ (A19)
qmdmkdmfdmq izizppv 22212221 −−+= ψψ (A20)
A.2 GENERATOR ROTOR FLUX EQUATIONS
Using (A11) rrrsrsr piLpiLp +=ψ and ( )srsrrrr iLLi −= − ψ1 (A21)
simplifying (A7) with (A21) will lead to
[ ]sssrrrrrr iLLRpv −+= − ψψ 1 (A22) Rearranging with 1−−= rrrm LRA and rsrrrm LLRF 1−−= rotor flux equations for one
main field winding and two damper windings, one on d- axis and one on q-axis can be
given as
rsmrmr viFAp ++= ψψ (A23)
______________________________________________________________________A.5
which can be explicitly written for expressing rotor dynamics as follows:
fdmqmdmkqmkdmfdmfd EKiFiFAAA 111211131211 +++++=•
ψψψψ (A24)
qmdmkqmkdmfdmkd iFiFAAA 2221232221 ++++=•
ψψψψ (A25)
qmdmkqmkdmfdmkq iFiFAAA 3231333231 ++++=•
ψψψψ (A26)
A.3 ELECTROMAGNETIC TORQUE EXPRESSION
[ ]
=
r
ssrsstrse I
IGGIIT
00refω (A27)
[ ] [ ]
=
r
ssrss
ts I
IGGIrefω (A28)
rsrtssss
ts IGIIGI refref ωω += (A29)
Substituting for ( )srsrrrr iLLi −= − ψ1 from (A15)
( )( )srsrrrsrtssss
ts iLLGIIGI −+= − ψωω 1
refref (A30)
[ ] rsrtssrsrrsrss
ts GIILLGGI ψωω ref
1ref +−= − (A31)
Let, rsrrsrss LLGGAA 1−−= and rsrGBB ψ=
BBIIAAIT tss
tse refref ωω += (A32)
______________________________________________________________________A.6
Appendix B
Exciter and Prime-Mover
Modelling
B.1 EXCITER AND AUTOMATIC VOLTAGE REGULATOR MODELLING
Assuming a synchronous machine with one main field winding, one direct-axis and one
quadrature-axis damper winding, the total voltage and current vector of synchronous
machine v and current vector i can be given as follows:
The complete exciter system consists of five major components:
(i) Main exciter: being a power stage of the excitation system, exciter provides dc power
to the synchronous machine field winding;
(ii) Regulator: processes and amplifies input control signals to a level and form
appropriate for control of the exciter;
(iii) Terminal voltage transducer: This senses generated terminal voltage, rectifies and
filters it to dc quantity and compares it with a desired terminal voltage set as
reference;
(iv) Power system stabilizer: PSS provides additional input signal to the regulator for
damping power system oscillations. The input given to PSS can be rotor speed
deviation, accelerating power, and frequency deviation;
______________________________________________________________________A.7
(v) Limiters and protective circuits: these include a wide array of control and protective
functions which ensures that the capability limits of exciter and synchronous
generator are not exceeded;
Generator Exciter Excitation Controller Regulator
Terminal voltage transducer and
load compensator
Power system stabilizer
Vref
VUEL
VOEL
VR
Vs
Ifd
Efd
|Vt|It
Fig.B1: Functional block diagram for synchronous machine exciter control system
Fig.B1 summarises the main blocks for synchronous machine exciter controller
system. VUEL and VOEL are the upper and lower exciter limits while Vt and It is the
output voltage and current which is feedback to control and adjust the regulator and
exciter output.
In the present research, a standard IEEE Type I exciter is used as shown in Fig B2.
Practically, Efd is the output of the exciter which is input to the generator as shown
in Fig B1. However, it should be noted that independent selection of per unit system
is necessary for modelling the excitation system and for proper interfacing between
the low voltage exciter system and high voltage generator circuit. The exciter output
Efd and generator input Vfd are related to each other by a factor of Km11 (= Rfd/Lmd).
RV
FV
F
F
sTsK+1
( )fdE EfS =
fdE+
+ +
−
−maxRV
minRV
refV
tV −
+
PSSV
∑ ∑ ∑EE sTK +
1
A
A
sTK+1
Fig.B2: IEEE Type I Excitation system model
______________________________________________________________________A.8
The dynamic equations describing the excitation control system can be derived from the
above block diagram.
( )( ) RfdfdEEE
fd VEESKT
E ++−=• 1
(B1)
( )
−+−+−=
•trefAfd
F
FAfAR
AR vVKE
TKK
RKVT
V 1
maxminRRR VVV ≤≤
(B2)
+−=
•
fdF
Ff
Ff E
TKR
TR 1
(B3)
where, FVETKR fd
F
Ff −= and
F
Ffd T
sKEV+
=1F
and, Efd is known as exciter field voltage, VR as automatic voltage regulator output and
Rf as rate feedback. The meaning and explanation of all other symbols is as explained in
the list of symbols.
Simplifying the block diagram of Fig.B2 and with the meaning of other symbols
explained in the list of symbols, the set of above dynamic equations can be rewritten in
compact notation as:
ExcExcExcExcExc VBXAX +=•
(B4)
where,
=
f
f
ExcRVE
X R
d;
−
−−
+−
=
FFF
AAFA
EEE
TTTK
TK
TTTKK
TTSK
A
10
1
01
F
AF
EE
Exc
(B5)
=
0
0A
ExcAT
KB ; [ ]tVVV −= refExc
(B6)
______________________________________________________________________A.9
B.2 PRIME-MOVER AND GOVERNOR COMBINED SYSTEM
In the present research, a steam turbine with nonreheat type is chosen as the prime
mover. Figure B3 shows the block diagram representation of turbine and governor
combined model.
dR1
CHsT+11
+
-PSV
Psv(max)
Psv(min)
PC
ω(pu) TM∑ SVsT+1
1
Fig.B3: Turbine and governor model
The steam chest dynamics and the effect of the steam valve position (PSV) on the
synchronous machine torque (TM) are of major concern from transient stability
modeling point of view. Equations (B7)-(B9) gives a complete model of turbine and
governor system which can be represented in compact notations as shown in (B10)
( )SVM
CHM PT
TT +−=• 1
(B7)
(B8)
( )( )puSVCSV PPP ω−−=•
where, ( )ref
refωωω
ω−
=pu (B 9)
CGovGovGovGovGovGov PDCBXAX +++=•
ωω ref (B10)
where,
=
SV
MGov P
TX and AGov, BGov, CGov, and DGov are matrices dependent on
system design for a given gain and time constants of controller.
______________________________________________________________________A.10
B.2 POWER SYSTEM STABILIZER (PSS)
The PSS is one the most commonly used controllers for damping rotor oscillations.
Using rotor speed as input denoted by X in the PSS block diagram given in Fig. B4, the
equations can be derived from three major blocks as:
From block 1:
XKXT
X PSSPSSPSS
PSS += 111 (B11)
From block 2:
PSSPSS
PSSPSS
PSSPSS
PSSPSS X
TTX
TX
TX 1
2
12
21
22
11 +−= (B12)
From block 3:
PSSPSS
PSSPSS
PSSPSS
PSSPSS X
TTV
TX
TV 2
4
3
42
4
11 +−= (B13)
which can be simplified and represented in final form as:
KPSS
Vpss(min)
X2pss
pss
psssTsT
2
111+
+
pss
pssT
sT+1 pss
psssTsT
4
311+
+X1pss
Vpss(max)
Vpss⋅
X
Fig.B4: PSS schematic block diagram
X
TTTTTK
TTK
K
VXX
TTTTT
TTTT
TT
TTTTT
T
VXX
PSSPSS
PSSPSSPSSPSSPSS
PSSPSSPSS
PSS
PSS
PSS
PSSPSSPSS
PSSPSS
PSSPSS
PSSPSS
PSS
PSS
PSSPSSPSS
PSSPSSPSS
PSS
PSS
PSS
⋅
⋅⋅⋅⋅
⋅+
−⋅−
⋅−
⋅
−⋅−
−
=
42
4312
12
1
42
32
2
1
4
3
22
12
1
1
01
001
(B14)
The complete PSS model can be represented in final compact form as:
XBXAX PSSPSSPSSPSS += (B15)
where, XPSS is the vector of state variables of PSS and, APSS, BPSS are matrices elements
depending on the gain and time constant of the PSS controller while X is the speed
variation.
______________________________________________________________________A.11
Appendix C
Dynamic modelling of FACTS
devices
C.1 SVC DYNAMIC MODEL
With reference to the block diagram presented in Fig. 2.5
From block 1:
( ) s
s
SDCtref sTK
XVVX
+=
−− 11
(C1)
( )( )SDCtrefs
sXVVKX
TX −−+−= 11
1 (C2)
From block 2:
( ) s
s
SDCtref sTK
XVVX
+=
−− 11
(C3)
( )( )SDCtrefs
sXVVKX
TX −−+−= 11
1 (C4)
2
1
1 11
sTsT
XB
++
= (C5)
______________________________________________________________________A.12
( )111
2
1 XTXBT
B ⋅++−= (C6)
With proper substitution of (C2) in (C4) and simplifying, the final SVC dynamic model
can be represented in compact matrix form as:
ref
s
ss
s
t
s
ss
s
SDC
s
ss
s
s
ss V
TTTK
TK
V
TTTK
TK
X
TTTK
TK
BX
TTTTT
TBX
⋅
−
−+⋅
−
−+⋅
−
−+
⋅
−−
−=
2
1
2
1
2
11
22
11
1
01
(C7)
The complete model can be expressed in compact notation as:
refSVCtSVCSDCSVCSVCSVCSVC VDVCXBXAX ⋅+⋅+⋅+⋅= (C8)
C.2 DYNAMIC MODELLING OF TCSC
Iline
Inductive limits
Iline
Xmin
Iline Capacitive limits
Iline
Xmax
∑ csc
csc1 t
tsT
K+
Xreftcsc
-
+
XSDC
Xtcsc
Xtcsc(max)
Xtcsc(min)
Xtcsc(min)
Fig.C1: TCSC dynamic model
( ) tcscSDCreftcsc 1
1 XsT
XXc=
+− (C9)
etttt PBXAX csc1csc1csc11csc1 +=
(C10)
______________________________________________________________________A.13
ettttt PtBXAXAX csc2csc2csc22csc1csc21csc2 ++=
(C11)
etttttttt PBXAXAXAX csc3cscSDCcsc33csc2csc32csc1csc31cscSDC +++= (C12)
( )cscSDCreftcsctcsc
1t
cXXX
TX −+=⋅
(C13)
Dynamic reactance limits
With reference to the Fig.C1 for transient reactance limits, the TCSC model permits
operation anywhere within the enclosed region. These boundaries are due to a number
of constraints on both the capacitive as well as the inductive side as explained in
Chapter 2.
In the capacitive region, the constraints are due to
(a) limit on the firing angle, expressed as a constant reactance limit (Xmax0);
(b) limit on a voltage across the TCSC;
(c) limit on the line current (ILtran) at which point the TCSC will go into a protective
bypass mode.
XTCSC
Inductive-2
12
Xmin0
Xmax0Capacitive
Xmin VL
Xmax VC
Iline=1.0puXmax ILine
ILine
Thyristor Bypass
3
XTCSC=1.0pu
Xmin ILT
Xbypass
Fig.C6: Dynamic reactance Limits
Once the TCSC is bypassed on this overcurrent constraint, it is subject to a time delay
on reinsertion after line current falls back below ILtran. In a multi-module TCSC, it is
possible that only some of the modules will bypass, since one module has to stay in
______________________________________________________________________A.14
capacitive mode. For simplicity in typical stability studies, it is suggested that this
nuance be neglected. The final capacitive reactance limit is the minimum of these
individual constraints.
On the inductive side similar constraints apply:
(a) limit on the firing angle, expressed as a constant reactance limit (Xmin0)
(b) limit on the harmonics, approximated as a constant voltage across the TCSC;
(c) limit on the thyristor current: As an approximation, the fundamental component of
thyristor current is limited to that at which the TCSC can operate in thyristor bypass
for duration of the transient.
C.3 DYNAMIC MODELLING OF STATCOM
The first state equation of STATCOM is started from the capacitor voltage on dc side and given by:
dc
dcdc I
CV 1
=
(C14) The current Idc can be determined using STATCOM active power flow equation as:
dcdcdcC IVPP ⋅== (C15)
Where, ( )*stastaC IVReP ⋅= and (C16)
φstadc VAV = (C17)
SDCstaCqstaTstaTrefsta XEIDVCVBV +++= (C18)
φstaSDCstaTstaTrefstadcstastaC VLXKVJVHVGVFX +++++= (C19)
φNXMφ staCsta += (C20)
______________________________________________________________________A.15
C.4 SUPPLEMENTARY DAMPING CONTROLLER (SDC)
S D C
S D C1 sT
sT+ SD C2
SD C111
sTsT
+
+
SD C4
SD C311
sTsT
+
+PeSDC1K XSDCX1SDC X2SDC
XSDC(max)
XSDC(min)
Fig.B5: Supplementary damping control block diagram
ePKXT
X SDC1SDC1SDC
SDC11
+−= (C21)
ePKTT
XTTTT
XT
X SDC1SDC2
SDC1SDC1
SDCSDC2
SDC1SDCSDC2
SDC2SDC2
1+
−+−= (C22)
( )e
SDC
PKTTTT
XTTTTTT
XTT
TTX
TX
SDC1SDC4SDC2
SDC3SDC1SDC1
SDCSDC4SDC2
SDC1SDCSDC3
SDC2SDC4SDC2
SDC3SDC2
SDC4SDC
1
+−
+−
+−= (C23)
Substituting,
SDCSDC11
1T
A −= ; SDCSDC2
SDC1SDCSDC21 TT
TTA
−= ;
SDC2SDC22
1T
A −=
SDC1SDC1 KB = ; SDC1SDC2
SDC1SDC2 K
TT
B = ; SDC1SDC4SDC2
SDC3SDC1 KTTTT
;
SDC1SDC4SDC2
SDC3SDC1SDC3 K
TTTT
B =
Using above symbolic notations
ePBXAX 11111 += (C24)
ePBXAXAX 22221212 ++= (C25)
ePBXAXAXAX 3333232131SDC +++=
(C26)
______________________________________________________________________A.16
Appendix D
Optimisation problem
formulation for MPC
Considering a linear, discrete-time, state-space model of the system
( ) ( ) ( )kBukAxkx +=+1 (D1) ( ) ( )kxCky y= (D2)
( ) ( )kxCkz z= (D3)
where,
x is nx dimensional state vector
u is nu dimensional input vector
y is ny dimensional vector of measured outputs
z is nz dimensional vector of output which are to be controlled
The MPC controller will produce ( ) ( ) ( )1−−=∆ kukuku which will be passed on to the
system as input. One of the ways to include this ‘integration’ in a state space model is
by augmenting the state vector.
For example defining the state vector by ( ) ( )( )
−
=1ku
kxkξ
______________________________________________________________________A.17
Modifying the model of (D1) and (D2)
( )( )
( )( ) ( )
110
x k x kA B Bu k
u k u kI I+
= + ∆ − (D5)
( ) [ ] ( )( )
01
x ky k C
u k
= − (D6)
When whole state vector is measured such that
( ) ( ) ( )kykxkkx ==ˆ so C = I (D7)
The components of y and z may overlap and may be the same.
With this assumption y = z, and Cy= Cz
Defining the cost function as
( )( ) ( ) ( )( )
( )( )
21
0
2
1,J
j
CN
jj
N
jk kjkkjkkjkk
RQref uyyux ∑∑
−
==+∆++−+= (D8)
Starting prediction with assumption that whole state vector is measure, so that in the
circumstances of no information about any disturbances or measurement noise; the only
option is to predict by iterating the model (D1) and (D2).
Predicting by iterating (D1) and (D2)
( ) ( ) ( )kBukAxkkx +=+1 (D9)
( ) ( ) ( )kkBukkAxkkx 112 +++=+
( ) ( ) ( )kkBukkABukxA 12 +++= (D10)
( ) ( ) ( )kNkBukNkAxkNkx 11 −++−+=+
( ) ( ) ( )kNkBukkBuAkkxA NN 11 −++++= −
Summarising it, ( ) ( ) [ ]( )
( )
−++=+ −−
kjku
kkuBIAAkxAkjkx jjj
1
21
(D11)
At the time of computing the prediction ( )ku is unknown so using ( )kku instead of
( )ku
______________________________________________________________________A.18
Assuming that input may only change at times 1,,1, −++ CNkkk and will remain
constant after that. This will lead to ( ) ( )1−+=+ CNkukjku for 1−<< NjNC
As the ( )1−ku is already known at time k, the prediction can be expressed in terms of
( )kjku +∆ rather than ( )kjku +
( ) ( ) ( )kjkukjkukjku 1−+−+=+∆ (D12)
With this
( ) ( ) ( )1−+∆= kukkukku (D13)
( ) ( ) ( ) ( )111 −+++∆=+ kukkukkukku
( ) ( ) ( ) ( )111 −+++−+∆=−+ kukkukNkukNku CC
Substituting value of (D14) in (D9) for ( )ku in terms of ( )kku
( ) ( ) ( ) ( )[ ]11 −+∆+=+ kukkuBkAxkkx (D14)
( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]1112 2 −+∆++∆+−+∆+=+ kukkukkuBkukkuABkxAkkx
( ) ( ) ( ) ( ) ( ) ( )112 −+++∆+∆++= kBuIAkkuBkkuBIAkxA
At the end of the control horizon
( ) ( ) ( ) ( )( ) ( ) ( )11 1
1
−++++−++
+∆++++=+−
−
kBuIAAkNkBu
kkuBIAAkxAkNkx
C
CC
NC
NNC
(D15)
In general, for (k+j) instant predicted at kth instant
( ) ( )( )
( )( )1
1
1
0
1
0−+
−+
+=+ ∑∑
−
=
−
=kBuA
kjku
kkuBBAkxAkjkx
j
i
ij
i
ij
(D16)
For CNj ≤
( ) ( ) ( ) ( )( ) ( ) ( ) ( )11
1 1
−++++−+∆++
∆++++=++ +
kBuIAAkNkuBIA
kkuBIAAkxAkNkx
C
CC
NC
NNC
(D17)
( ) ( ) ( ) ( )( ) ( )( ) ( )1
11
1
−++++
−+∆++++
+∆++++=+
−
−
−
kBuIAA
kNkuBIAA
kkuBIAAkxAkNkx
NC
NN
NN
C
______________________________________________________________________A.19
Summarising in short for NjNC ≤< ,
( ) ( )( )
( )( )1
1
1
00
1
0−+
−+
+=+ ∑∑∑
−
=
−
=
−
=
kBuAkNku
kkuBABAkxAkjkx
j
i
i
C
Nj
i
ij
i
ijC
Combining all state predictions in one expression to give a matrix-vector form as
( )
( )( )
( )( )
( ) ( )
( )
( )( )
Future
kU
C
G
NN
i
iN
i
i
N
i
i
N
i
i
Past
N
i
i
N
i
i
N
i
i
N
N
N
ky
C
C
kNku
kku
BABA
BABBA
BBA
BABB
ku
BA
BA
BA
B
kx
A
AA
A
kNkx
kNkxkNkx
kkx
y
C
C
C
C
C
C
C
∆
−
=
−
=
=
−
=
Γ
−
=
=
−
=
Φ
+
−+∆
∆
+
+
+
−
+
=
+
+++
+
∑∑
∑
∑
∑
∑
∑
1
00
11
1
0
1
0
0
1
0
1
0
0
1
0
1
(D18)
The prediction of y is now obtained for j = 1… N as
( ) ( )kjkCxkjky +=+ (D19)
Now rewriting the objective function of (D8) as
( ) ( ) ( ) 22RQrefk kUkYkYJ ∆+−= (D20)
where,
( )( )
( )( )
( )
( )( )
( )
( )
−+∆
∆=∆
+
+=
+
+=
kNku
kkukU
kNky
kkykY
kNky
kkykY
Cref
ref
ref1
;11
(D21)
and the weighting matrices Q and R are given by
______________________________________________________________________A.20
( )( )
( )
( )( )
( )
−
=
=
100
010000
;
00
020001
CNR
RR
R
NQ
Q
(D22)
Y (k) has the form of
( ) ( ) ( ) ( )kUGkukxkY y∆+−Γ+Φ= 1 (D23)
For suitable matrices ΓΦ, and yG error vector E(k) can be written as
( ) ( ) ( ) ( )1−Γ−Φ−= kukxkYkE ref (D24)
E(k) can be thought of as ‘tracking error’ which represents the difference between the
future target trajectory and the ‘free response’ of the system. The free response is
nothing but the response that would occur over the prediction horizon if no input
changes were made that is – if ( ) 0=∆ kU . Now modifying the objective function of
(D16) in terms of tracking error:
( ) ( ) ( ) 22RQyk kUkEkUGJ ∆+−∆= (D25)
( ) ( )[ ] ( ) ( )[ ] ( ) ( )KURkUkEkUGQkEGkU Ty
TTy
T ∆∆+−∆−∆= (D26)
( )[ ] ( ) ( ) ( ) ( ) ( )kQEkEkUQGkEkURQGGkU Ty
Ty
Ty
T +∆−∆+∆= 2 (D27)
This has the form
( ) ( ) ( ) constkUfkUHkUJ TTk +∆+∆∆=
21 (D28)
where,
( ) ( )kQEGfRQGGH Tyy
Ty 22 −=+= (D29)
and neither H nor f depends on - ( )kU∆
Finally writing the simple relation between input increment u∆ and control input u as:
( )( )
( )
( )( )
( )( )1
1
1
1
1−+
−+∆
+∆∆
=
−+
+kfu
kNku
kkukku
M
kNku
kkukku
CC
(D30)
______________________________________________________________________A.21
=
=
I
II
f
III
III
M
000
(D31)
Therefore from equation (D21) the following optimisation problem is to be solved
( )( ) ( ) ( )kUfkUHkU TT
kU∆+∆∆
∆ 21min (D32)
It is to be remembered that only the first step of the above solution will be used as per
the receding horizon strategy.
This is a standard optimisation problem known as the Quadratic programming (QP)
problem and standard algorithms are available for its solution.
______________________________________________________________________A.22
Appendix E
Data sheet for SMIB system used
in Chapter 5
E.1 BASE QUNATITIES
• Base MVA = 37.5 MVA
• Base stator voltage = 11.8 kV
• Base rotor voltage = 154 kV
E.2 GENERATOR PARAMETERS
• Armature resistances: Ra = 0.0020 pu
• Field resistance: Rfd = 0.00107 pu
• Direct-axis damper resistance: Rkd = 0.00318 pu
• Quadrature-axis damper resistance: Rkq = 0.00318 pu
• Direct-axis magnetising reactance: Xmd (Xad) = 1.859 pu
• Quadrature-axis magnetising reactance: Xmq (Xaq) = 1.560 pu
• Armature leakage reactance: Xl = 0.140 pu
• Field leakage reactance: Xfd = 0.140 pu
______________________________________________________________________A.23
• Direct-axis damper leakage reactance: Xkd = 0.140 pu
• Quadrature-axis damper leakage reactance: Xkq = 0.140 pu
• Inertia constant: H=5.3 sec.kW/KVA
E.3 TRANSFORMER PARAMETERS
• Resistance: rT = 0.0056 pu
• Reactance: xT = 0.1328 pu
E.4 TRANSMISSION LINE PARAMETERS
• Positive-sequence resistance: RL1 = 0.0075 pu
• Positive-sequence reactance: XL1= 0.5076 pu
• Zero-sequence resistance: RL0 = 0.0225 pu
• Zero-sequence reactance: XL0 = 0.1458 pu
______________________________________________________________________A.24
Appendix F
Data sheet of 10 generator 39
node New England system used
in Chapter 8
F.1 SYSTEM DETAILS USED FOR SIMULATION CASE-STUDY OF
CHAPTER 8
The case study of online controller coordination for multi-machine system is carried out
on 10 generators 39 nodes power system. The IEEE standard New England system is
adapted and the power system network is as shown in Fig. F1. The transmission line
details are given in Table F1, while the generator and load details are given in Table F2.
As shown in Fig.F1, the first TCSC is inserted in branch 11 which is a transmission line
from node 11 to node 15, and the second TCSC is inserted at node 12 and node 16. It is
assumed that this insertion of TCSC in long transmission lines will give rise to
additional intermediate nodes which are numbered in sequence for sake of simplicity as
node 13 and node 14 respectively. Following the numbering sequence as first generator
nodes, followed by FACTS devices node (TCSC in this case) and finally all load nodes
will make the system look like 41 nodes instead of 39 nodes.
______________________________________________________________________A.25
11
10
8
2
3
4 5
7
6
9
W
N12
N10
N11
N2
N8
N25
N26
N28
N29
N9N24N27N38
N37
N13
N15
N19
N18
N17
N1N16
N14
N31
N3
N20
N32
N33
N34
N35
N36
N21
N39
N30
N4 N5
N7
N23
N22
N6
TCSCL10
L1L2
L3
L4
L5
L6
L7
L8
L9
L11
L12
L13
L14
L15
L16L17
L18L19
L20
L21
L22
L23L24
L25
L26
L27
L28
L29
L30
L31
L32
L34
L33
W W
W
W W
W
W
W
W
W
W
TCSC
N40
N41
Fig.F1: 10 Generators 39 nodes New England System
______________________________________________________________________A.26
Table F1: Transmission line data
Sending end Node1
Receiving end Node2
Resistance (pu)
Reactance (pu)
Suspetance (pu)
37 27 0.0013 0.0173 0.3216 37 38 0.0007 0.0082 0.1319 36 24 0.0003 0.0059 0.068 36 21 0.0008 0.027 0.2548 36 39 0.0016 0.04 0.304 36 37 0.0007 0.018 0.1342 16 36 0.0009 0.0094 0.171 14 16 0.009 0.1085 1.830 33 12 0.0009 0.0101 0.1723 28 15 0.0014 0.0151 0.249 13 15 0.0057 0.0625 1.029 11 28 0.0043 0.0474 0.7802 11 27 0.0014 0.0147 0.2396 25 11 0.0032 0.0323 0.513 23 24 0.0022 0.035 0.361 22 23 0.0006 0.0096 0.1846 21 22 0.0008 0.014 0.2565 20 33 0.0004 0.0043 0.0729 20 31 0.0004 0.0043 0.0729 19 2 0.001 0.025 1.2 18 19 0.0023 0.0363 0.3804 17 18 0.0004 0.0046 0.078 35 31 0.0007 0.0082 0.1389 35 17 0.0006 0.0092 0.113 41 18 0.0008 0.0112 0.1476 41 35 0.0002 0.0026 0.0434 34 12 0.0008 0.0129 0.1382 34 41 0.0008 0.0128 0.1342 29 38 0.0011 0.0133 0.2138 29 34 0.0013 0.0213 0.2214 40 25 0.007 0.0086 0.146 40 29 0.0013 0.0151 0.2572 26 40 0.0035 0.0411 0.6987 26 2 0.001 0.025 0.75
______________________________________________________________________A.27
Table F2: Load data
Node No
Pload
(MW) Qload
(MVAr) 1 9.2 4.6 2 1104 250 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 10 0 0 11 139 17 12 0 0 13 0 0 14 0 0 15 283.5 26.90 16 320 153 17 233.8 84 18 522 176.6 19 0 0 20 0 0
Node No
Pload
(MW) Qload
(MVAr) 21 274 115 22 0 0 23 274.5 84.60 24 308.6 92.20 25 224 47.20 26 0 0 27 281 75.50 28 206 27.60 29 322 2.4 30 680 103 31 0 0 32 8.5 88 33 0 0 34 500 184 35 0 0 36 329.4 32.30 37 0 0 38 158 30 39 0 0 40 0 0 41 0 0
______________________________________________________________________A.28
Table F3: Transformer data
Sending end Node1
Receiving end Node2
Resistance (pu)
Reactance (pu)
Suspetance (pu)
39 30 0.0007 0.0138 0 39 5 0.0007 0.0142 0 32 33 0.0016 0.0435 0 32 31 0.0016 0.0435 0 30 4 0.0009 0.0180 0 15 9 0.0008 0.0156 0 25 8 0.0006 0.0232 0 23 7 0.0005 0.0272 0 22 6 0 0.0143 0 20 3 0 0.0200 0 35 1 0 0.0250 0 40 10 0 0.0181 0
Table F4: Generator data
Node Pgen Qgen Qgen(min) Qgen(max) 1 0 0.98 0 0 2 1000 1.03 -500 500 3 650 0.98 -325 325 4 508 1.01 -254 254 5 632 1 -316 316 6 650 1.05 -325 325 7 560 1.06 -280 280 8 540 1.03 -270 270 9 830 1.03 -415 415 10 250 1.05 -125 125
______________________________________________________________________A.29
Table F5: Generator parameters
Gen No.
Xd (pu)
Xq (pu)
Xmd (pu)
Xmq (pu)
Xfd (pu)
Xkd (pu)
Xkq (pu)
1 0.1305 0.0474 0.1125 0.0294 0.1254 0.1315 0.03740 2 0.1305 0.0474 0.1125 0.0294 0.1254 0.1315 0.03740 3 0.1864 0.0677 0.1607 0.042 0.1792 0.1878 0.05350 4 0.2175 0.0790 0.1875 0.049 0.2091 0.2191 0.06240 5 0.1864 0.0677 0.1607 0.042 0.1792 0.1878 0.05350 6 0.1864 0.0677 0.1607 0.042 0.1792 0.1878 0.05350 7 0.2175 0.0790 0.1875 0.049 0.2091 0.2191 0.06240 8 0.2175 0.0790 0.1875 0.049 0.2091 0.2191 0.06240 9 0.1450 0.0527 0.1250 0.0327 0.1394 0.1461 0.0416 10 0.4350 0.1580 0.3750 0.0980 0.4181 0.4383 0.1247
Gen No.
Ra (pu)
Rfd (pu)
Rkd (pu)
Rkq (pu)
H
1 0.0003 0.0001 0.0016 0.0012 500 2 0.0003 0.0001 0.0016 0.0012 30.30 3 0.0004 0.0001 0.0022 0.0017 35.80 4 0.0005 0.0001 0.0026 0.002 26 5 0.0004 0.0001 0.0022 0.0017 28.60 6 0.0004 0.0001 0.0022 0.0017 34.80 7 0.0005 0.0001 0.0026 0.0020 26.40 8 0.0005 0.0001 0.0026 0.0020 24.30 9 0.0003 0.0001 0.0017 0.0013 34.5 10 0.001 0.0003 0.0052 0.0040 42.0
Armature resistances: Ra
Field resistance: Rfd
Direct-axis damper resistance: Rkd
Quadrature-axis damper resistance: Rkq
Direct-axis magnetising reactance: Xmd (Xad)
Quadrature-axis magnetising reactance: Xmq (Xaq)
Direct-axis armature reactance: Xd
Quadrature-axis armature reactance: Xq
Field leakage reactance: Xfd
Direct-axis damper leakage reactance: Xkd
Quadrature-axis damper leakage reactance: Xkq
Inertia constant: H (sec.kW/KVA )
______________________________________________________________________A.30
Table F6 Excitation controller parameters
KA TA KE TE KF TF SE1 SE2 VRmin VRmax
25 0.06 -0.0445 0.5 0.16 1 0.0011 0.3043 -10 10
Table F7 Governor and prime mover parameters
TCH TSV RD PSV(max) dPmin dpmax
4 2 0.05 10 -1.0 1.0
Table F8 TCSC dynamic modeling data
TC K1(SDC) TW(SDC) T1(SDC) T2(SDC) T3(SDC) T4(SDC)
0.01 0.1 0.2 0.2 0.1 0.05 0.2
Table F9 TCSC details
Node1(tcsc) Node2(tcsc) Line no Xref(tcsc) Xmin(tcsc) Xmax(tcsc)
11 13 11 1e-3 -0.05 0.0075
12 14 8 1e-3 -0.075 0.0113
Note: All resistance, reactance and susceptance data is in pu on 100 MVA.
______________________________________________________________________A.31
Appendix G
Problem Formulation for Internal
State Estimation of Synchronous
generator
G.1 SYNCHRONOUS GENERATOR INTERNAL STATE VARIABLE
ESTIMATION
Starting with the dynamic rotor flux linkages equations from Chapter 2 and using the
detailed derivation as given in Appendix A:
fdmqmdmkqmkdmfdmfd EKiFiFAAA 111211131211 +++++=⋅
ψψψψ (G1)
qmdmkqmkdmfdmkd iFiFAAA 2221232221 ++++=⋅
ψψψψ (G2)
qmdmkqmkdmfdmkq iFiFAAA 3231333231 ++++=⋅
ψψψψ (G3)
refr ωωδ −=⋅
(G4)
______________________________________________________________________A.32
Using trapezoidal rule of numerical integration, the set of (G1) to (G4) can be
transformed in to as following:
++
+++
+
+
++++
∆+=
−−−
−−−−
)1()1(12
)1(11
)1(13
)1(12
)1(11
)()(12
)(11
)(13
)(12
)(11
)1()(2
nfdm
nqm
ndm
nkqm
nkdm
nfdm
nfdm
nqm
ndm
nkqm
nkdm
nfdm
nfd
nfd
EKiFiF
AAA
EKiF
iFAAA
t
ψψψ
ψψψ
ψψ
(G5)
+
+++
+
+
+++
∆+=
−−
−−−−
)1(22
)1(21
)1(23
)1(22
)1(21
)(22
)(21
)(23
)(22
)(21
)1()(2
nqm
ndm
nkqm
nkdm
nfdm
nqm
ndm
nkqm
nkdm
nfd
nkd
nkd
iFiF
AAA
iFiF
AAAm
t
ψψψ
ψψψ
ψψ (G6)
+
+++
+
+
+++
∆+=
−−
−−−−
)1(32
)1(31
)1(33
)1(32
)1(31
)(32
)(31
)(33
)(32
)(31
)1()(2
nqm
ndm
nkqm
nkdm
nfdm
nqm
ndm
nkqm
nkdm
nfdm
nkq
nkq
iFiF
AAA
iFiF
AAA
t
ψψψ
ψψψ
ψψ (G7)
( ) ( )[ ]refn
refnnn t ωωωωδδ −+−
∆+= −− )1()()1()(
2 (G8)
As it can be seen that (n) and (n-1) states of the system are both unknown, in the above
set of equations, the total number of unknowns are eight while the number of equations
available are just four. To solve this problem, another set of algebraic equations can be
organised using the available bus voltages and currents for (n) and (n-1) instant and
transforming it from system D-Q axis to rotor electrical d-q axis as given in Chapter 2.
)(
12)(
11)(
13)(
12)(
11)( n
qmn
dmn
kqmn
kdmn
fdmn
d iZiZPPPv −−++= ψψψ (G9)
)(22
)(21
)(23
)(22
)(21
)( nqm
ndm
nkqm
nkdm
nfdm
nq iZiZPPPv −−++= ψψψ (G10)
)1(12
)1(11
)1(13
)1(12
)1(11
)1( −−−−−− −−++= nqm
ndm
nkqm
nkdm
nfdm
nd iZiZPPPv ψψψ (G11)
)1(22
)1(21
)1(23
)1(22
)1(21
)1( −−−−−− −−++= nqm
ndm
nkqm
nkdm
nfdm
nq iZiZPPPv ψψψ (G12)
______________________________________________________________________A.33
Combining (G5) to (G8) and (G9) to (G12) to form a complete set of equations
considering n = 1 and (n-1) = 0 instants to represent the first two successive states:
++
+++
+
++
+++
∆++−=
0012011
013012011
1112111
113112111
01 21
fdmqmdm
kqmkdmfdm
fdmqmdm
kqmkdmfdm
fdfd
EKiFiF
AAA
EKiFiF
AAA
tfψψψ
ψψψ
ψψ
(G13)
+
+++
+
+
+++
∆++−=
022021
023022021
122121
123122121
01 22
qmdm
kqmkdmfdm
qmdm
kqmkdmfdm
kdkd
iFiF
AAA
iFiF
AAA
tfψψψ
ψψψ
ψψ
(G14)
+
+++
+
+
+++
∆++−=
032031
033032031
132131
133132131
01 23
qmdm
kqmkdmfdm
qmdm
kqmkdmfdm
kqkq
iFiF
AAA
iFiF
AAA
tfψψψ
ψψψ
ψψ
(G15)
( ) ( )[ ]refreftf ωωωωδδ −+−
∆++−= 0101 2
4
(G16)
01201101301201105 qmdmkqmkdmfdmd iZiZPPPvf −−+++−= ψψψ (G17)
02202102302202106 qmdmkqmkdmfdmq iZiZPPPvf −−+++−= ψψψ (G18)
11211111311211117 qmdmkqmkdmfdmd iZiZPPPvf −−+++−= ψψψ (G19)
12212112312212118 qmdmkqmkdmfdmq iZiZPPPvf −−+++−= ψψψ (G20)
where,
))((sin)())((cos)( ttVttVv QDd δδ ⋅+⋅=
))((cos)())((sin)( ttVttVv QDq δδ ⋅+⋅−=
))((sin)())((cos)( ttIttIi QDd δδ ⋅+⋅=
))((cos)())((sin)( ttIttIi QDq δδ ⋅+⋅−=
and, pm and zm are speed dependent as derived in Appendix (A12)
______________________________________________________________________A.34
1−= rrsrrm LGp ω ; ( )[ ]srsrrsrssrm RLLGGz +−−= −1ω (G21)
The convergence and accuracy of the optimisation problem depend on the initial values
given for optimisation.
G.2 INITIAL GUESS VALUES FOR STARTING ESTIMATION ALGORITHM
To find the initial values to start the estimation process, the relationship between the
generator terminal voltage and the current expressed in the d-q axis of the rotor at 0
instants can be used.
daqqd iRiXv ⋅−⋅= (G22)
qaddfdmdq iRiXIXv ⋅−⋅−⋅= (G23)
In (G22) and (G23), id and iq are expressed in terms of rotor angle delta for 0 instant.
Using the rotor angle relation with speed measurements, the next successive initial
values for rotor angle at instant 1 is calculated using (G16). Once the rotor angle delta at
0 and 1 instant are calculated, it can be used to solve a set of six algebraic equations
formed by (G13) to (G15) and (G17) to (G20) can be solved to get the initial estimate of
rotor fluxes as explained in Chapter 9.
______________________________________________________________________A.35
Appendix H
Publications
1. T.T. Nguyen, S. R. Wagh, “Model Predictive Control of FACTS Devices for
Power System Transient Stability”, the Proceedings of the IEEE Transmission
and Distribution Asia Conference, Seoul, Korea, October, 2009.
2. T.T. Nguyen, S. R. Wagh, “Predictive Control-Based FACTS Devices for Power
System Transient Stability Improvement”, the Proceedings of the 8th IET
International Conference on Advances in Power System Control, Operation and
Management, APSCOM 2009, Hong Kong, November, 2009.
3. T.T. Nguyen, S. R. Wagh, “Application of Dynamic Modelling for Estimating
Internal States of a Synchronous Generator in Transient Operating Mode from
External Measurements”, submitted to IEEE Trans. Power Systems, 2011 (under
review).
4. T.T. Nguyen, S. R. Wagh, “Online Control Coordination of TCSCs for Power
System Transient Stability”, submitted to IET Generation, Transmission and
Distribution, Dec 2011 (under review).
