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dissertation entitled

BIFURCATION SUBSYSTEM METHOD AND ITS APPLICATION
TO DIAGNOSIS OF POWER SYSTEM BIFURCATIONS
PRODUCED BY DISCONTINUITIES

presented by

Khadija Ben-Kilani

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Ph.D degree in Electrical Eng

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mane-m

BIR'RC
DIAG

BIFURCATION SUBSYSTEM METHOD AND ITS APPLICATION TO
DIAGNOSIS OF POWER SYSTEM BIFURCATIONS PRODUCED
BY DISCONTINUITIES

by

Khadija Ben-Kilani

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1 997

 

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ABSTRACT

BIFURCATION SUBSYSTEM METHOD AND ITS APPLICATION TO DIAGNOSIS OF
POWER SYSTEM BIFURCATIONS PRODUCED BY DISCONTINUITIES

by

Khadija Ben-Kilani

Recurrent problems in diagnosing the cause and location of a stability problem in a
power system are that of (a) the size and complexity of the model, (b) the various kinds
and classes of bifurcations, and (c) the effects of discontinuous hard limit or equipment
outage-induced transitions. Problem diagnosis in power systems is in need of stability
assessment methods that (i) take into account both the existence of discontinuities and
their effect, and (ii) identiﬁes which physical elements or subsystems are associated with a
speciﬁc bifurcation observed on the full model.

First, diagnosis is initiated by performing a classiﬁcation of the modeling
complexity and of the various bifurcations, based on their kinds and classes. The dynamic
behavior of a simulated stability problem depends on the modeling complexity used.

Second, an Epoch-Based Trajectory Stability Assessment Methodology that is aimed
at assessing asymptotic stability within time intervals free of hard limits discontinuities is
presented. In this method, the stability assessment requires testing of quasi equilibria,
limit cycles and trajectories within each epoch, rather than attempting to study stability of
the entire trajectory.

The third major step is to identify the smallest subsystem, a subset of the equations
of the full system model that experiences the same bifurcation in the full system model.

The bifurcation subsystem must satisfy both the linear and the transversality conditions

“1,.
Jitter“
.,,.,

1... :1:

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W ~13 tn
J ‘ .

h..-‘ 5.1

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.e. ..Au

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, .l.‘..
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Th:
~31" tier:

Li

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n, 4...;
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l.~‘J\J‘V L

for the bifurcation that is being experienced in the full system model. The portion of the
full system external to this bifurcation subsystem has no effect on the linear conditions for
bifurcation to be satisﬁed in the full system model, and yet the equilibrium point is
dependent on the variations in the full system equations which results in bifurcation in
both the full system and the bifurcation subsystem. The bifurcation subsystem method
utilizes the geometry associated with the various submatn'ces of the differential-algebraic
Jacobian J and with the eigenvectors associated with the bifurcating eigenvalue to
establish conditions for existence of such subsystems. Two examples for validating and
using the bifurcation subsystem method to identify bifurcation subsystems in a model are
presented, and the results are compared with right eigenvector and participation
information. The bifurcation subsystem method can be in disagreement with both
eigenvector and participation information.

The bifurcation subsystem and the model dependency of test matrices T, KS and KD
can identify the subsystem that experiences the bifurcation, the subsystem that produces
the bifurcation, as well as possibly the location, cause and cure for the bifurcation,

including loading changes or hard limit discontinuities.

Copyright by
Khadija Ben Kilani

1997

To my mother and father Fatma and Azzouz Ben Kilani

5....

ACKNOWLEGMENTS

I thank Allah the most powerful for giving me the power and the patience to
accomplish this dissertation. I am in debt to my parents who although far away, were with
me all steps of the way in every way possible.

I would like to express my sincere gratitude to my dissertation advisor Dr. Robert
Schlueter for his continued academic advising, moral support, and constructive guidance.
1’ am also grateful to my committee members, Dr. Gerald Park, Dr. Hassan Khalil, and Dr.
Jerry Schuur for giving me of their encouragements and their professional opinions.

The ﬁnancial support from the Detroit Edison company and the MSU Computing and

Technology, as well the Graduate School made this work possible.

vi

TABLE OF CONTENT

LIST OF TABLES ........................................................................................ xii
LIST OF FIGURES................ ....................................................................... xiii
LIST OF SYMBOLS ..................................................................................... xvii
CHAPTER l INTRODUTION ......................................................................... l

1.0 Introduction to and Justiﬁcation for

Bifurcation Subsystems .......................................................... 1
1.1 Description of the Static Bifurcation Phenomena ..................... 9
1.2 Dynamic Bifurcation on Power Systems:

A Brief Review ..................................................................... 10
1.3 Objective ............... 13

CHAPTER 2 POWER SYSTEM MODELING

EFFECTS ON STABILITY PROBLEMS .................................. 19
2.1 Description of a Power System ............................................. 20
2.1.1 Synchronous Generator ............................................... 20
2.1.2 Electric Load ............................................................ 22
2.2 General Power System Model ................................................ 24
2.2.1 Model Equations ......................................................... 25
2.3 Modeling Complexity ........................................................... 26
2.3.1 Type 1 Model ........................................................... 27
2.3.2 Type 2 Model ........................................................... 27
2.3.3 Type 3 Model ........................................................... 28
2.3.4 Type 4 Model ........................................................... 29

2.4 Classes and Kinds of Bifurcations

in a Type 2 Model ............................................................. 31
2.4.1 Kinds of Bifurcation .................................................. 34
2.4.2 Classes of Bifurcation ............................................. 38
2.4.2.1 Classes of Loss of Causality .......................... 39
2.4.2.2 Classes of Static Bifurcation .......................... 40
2.4.2.3 Classes of Hopf Bifurcation ........................... 41

vii

CHAPTER 3 POWER SYSTEM STABILITY ASSESSMENT

A TAXONOMICAL APPROACH .............................................. 42
3.1 introduction ........................................................................... 42
3.2 A Taxonomy of Stability Assessment in

Power Systems ....................................................................... 45

3.3 Development of a Taxonomy for Power System
Models with Hard Limits ....................................................... 48

3.4 Stability Security Assessment for Models

with Hard Limits ................................................................... 51
3.4.1 Epoch—Based Trajectory Stability Assessment............... 55.
3.4.2 Bounded Stable Trajectory Feasibility Method .............. 62
CHAPTER 4 BIFURCATION SUBSYSTEMS IN A POWER SYSTEM
DIFFERENTIAL-ALGEBRAIC MODEL .................................. 65
4.0 introduction ........................................................................... 65
4.0.1 Bifurcation Subsystems for Epochs .............................. 69
4.0.2 Existence of Bifurcation Subsystems ............................ 70
4.0.3 Introduction to Bifurcation Subsystem Theory .............. 73
4.0.4 Chapter Objective and Outline ...... y .............................. 75
4.1 Differential-algebraic Power System Model ............................. 78

4.2 Conditions for Bifurcation Subsystems

Experiencing SN and LF Bifurcations ..................................... 82
4.2.1 Equivalent System Models ........................................... 82
4.2.1.1 Equivalent Algebraic System Model ................ 84
4.2.1.2 Equivalent Dynamic System Model ................. 85
4.2.2 Deﬁnition of Bifurcation Subsystems ........................... 87

4.2.3 Decoupling Conditions for Algebraic
Bifurcation Subsystem and Differential

Bifurcation Subsystems ................................................ 94
4.2.4 Geometric Conditions .................................................. 97
4.2.5 Algebraic Bifurcation Subsystem

Experiencing LF Bifurcation ........................................ 105

4.3 Bifurcation Subsystems Experiencing

SN and LF Bifurcation .......................................................... 106
4.3.1 Differential Bifurcation Subsystems

for SN Bifurcation ...................................................... 107
4.3.2 Algebraic Bifurcation Sub-subsystem

for LP Bifurcation ...................................................... 109
4.3.3 Differential-Algebraic Sub-subsystems .......................... 110

4.4 Differential Bifurcation Sub-systems
for Hopf Bifurcation .............................................................. 112

viii

CHAPTER 5

5.3

5.4

CHAPTER 6

6.0

4.4.1 Differential Bifurcation Sub-subsystems
for Hopf Bifurcation ................................................... 114

Application of Algebraic Bifurcation Subsystem
Experiencing LF Bifurcation in Power

System Stability Assessment .................................................. 115
4.5.1 Description of a Load Flow Model .............................. 117
4.5.2 Structural Conditions for Utilizing J), ......................... 119

4.5.3 Comparison of Algebraic Equivalent Model,
Algebraic Bifurcation Subsystem,

and Load Flow Model ................................................. 120
4.5.4 Bifurcation in LP or in Generator Dynamics ................ 126
4.5.5 Conclusion ................................................................. 127

Effect of Hard Limit Discontinuities on Stability

Analysis .............................................................................. 129
Introduction ......................................................................... 129
Linearized Muiti-machine Power System Model ...................... 132
Single Machine Inﬁnite Bus Model with Local Load .............. 134
5.2.1 Model Representation .................................................. 134
5.2.2 Model Implications ...................................................... 136
5.2.3 Model Equations .......................................................... 139
5.2.4 Simpliﬁed Linear SMIB Model ................................... 140
5.2.4.1 The E’q Equation ........................................... 140
5.2.4.2 The electrical torque Equation ........................ 142
5.2.4.3 Terminal Voltage Equation .............................. 143
Application Examples of the Bifurcation Subsystem
Method on a SMIB model ...................................................... 146
5.3.1 Differential Bifurcation Subsystem for SN
(Unregulated Model) .................................................... 146
5.3.2 Differential Bifurcation Subsystem for Hopf
(Regulated Model) ....................................................... 154
Effect of Hard Limit Discontinuities on Stability
Analysis .............................................................................. 158
5.4.1 Synchronizing and damping torques ............................. 159
5.4.1.1 Unregulated System (Ge(s)=0) ......................... 160
5.4.1.2 Thyristor-Type Excitation System .................... 161
5.4.1.3 IEEE dc Type 1 Excitation System ................. 164
5.4.2 Bifurcation Tests for the SMIB ......... ~ ......................... l 67
5.4.3 Conclusions ................................................................. 170

'A Diagnosis for SN and Hopf Bifurcations in a

SMIB Model .......................................................................... 173

Introduction ......................................................................... 173

HENDi)
uPENDix

that

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6.1

6.2

6.3

CHAPTER 7
7.1
7.2

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

APPENDIX El

APPENDIX E2

Diagnostic Models Formulation .............................................. 174
6.1.1 Power Transfer Model (Rsh= X“: inﬁnity) .................................................... 174
6.1.2 Local Load Model ....................................................... 175
6.1.3 Combination Model ..................................................... 176
Diagnostic Study .................................................................... 176
6.2.1 Power Transfer Model ................................................. 179
6.2.1.1 Effect of PM and OM .................................... 179
6.2.1.2 Effect of Transmission Network Inductance ..... 181
6.2.2 Power Local Loading Model ...................................... 183
6.2.2.1 Effect of real-inductive local loading ...................................... 183
6.2.2.2 Effect of real-capacitive local loading ..................................... 184
6.2.2.3 Effect of the transmission network .......................................... 184
6.2.3 Conclusion of the Power Transfer and Local
Load Models ..................... . .......................................... l 85
6.2.4 Combination Local Load-Power Transfer Model ............ 187
6.2.4.1 Effect of Power Transfer ................................. 187
6.2.4.2 Effect of Local Loading ................................. 188
6.2.4.3 Effect of the Transmission Network ................ 189
6.2.5 Discussion of the Combination Model Results .............. 192
Diagnostic Summary .......................................................... . ..... 194
6.3.1 Classes of Hopf Bifurcation in a SMIB Model ............. 194
6.3.2 Classes of Static Bifurcation in a SMIB Model ........... 195
SUMMARY AND FUTURE RESEARCH ................................ 197
Thesis Summary ..................................................................... 197
Future Research ..................................................................... 202
POWER SYSTEM DYNAMIC MODEL .................................... 205

COMMON DEFINITIONS IN POWER
SYSTEM STABILITY ............................................................. 217

DEFINITION OF HOPF AND SADDLE NODE
BIFURCATIONS WITH TRANSVERSALITY
CONDITIONS ........................................................................ 221

INPUT DATA FILES FOR THE BIFURCATION
SUBSYSTEM EXAMPLES ..................................................... 223

FIGURES FOR DIAGNOSTIC ANALYSIS
POWER TRANSFER MODEL ................................................. 227

FIGURES FOR DIAGNOSTIC ANALYSIS
LOCAL LOADING MODEL .................................................... 231

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APPENDIX E3 FIGURES FOR DIAGNOSTIC ANALYSIS
COMBINATION MODEL ........................................................ 235

REFERENCES ............................................................................................... 249

xi

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 6.1 (a)
Table 6.1(b)
Table 6.2(a)
Table 6.2(b)
Table 6.3 (a)
Table 6.3 (b)

Table 6.3 (c)

LIST OF TABLES

Computational results for identifying the bifurcation subsystem for SN

 

 

 

 

 

bifurcation in the SMIB model - - ............................... 149
Participation factor results for the SN bifurcation in the SMIB model ....................... 153
Computational results for identifying the bifurcation subsystem for

Hopf bifurcation in the SMIB model: Inertial mode -- 157
Participation factor results for the inertial Hopf bifurcation in the

SMIB modeL--- - - -- ......................... 158
Results on the power transfer model bifurcation diagnosis ........................................ 181
Power transfer model non-bifurcation results .............................. 182
Results on the local loading model bifurcation diagnosis .......................................... 185
Non-Bifurcation Results on the local loading model Diagnosis ............................. 185
Results on the combination model bifurcation diagnosis ................. 190
Combination model: Effect of Local loading ............................................................. 190
Combination model: Non-bifurcation cases ............. - 191

 

xii

LIST OF FIGURES

Figure 1.1 Identiﬁcation of the Critical Area in 230 kV buses

under area System Stress ......................................................... 5
Figure 2.1 Functional Block Diagram of a synchronous

generator excitation control system ........................................... 23
Figure 2.2 IEEE Type 1 Excitation System ................................................ 23

Figure 3.1 2-D Example portraying the effects of the i’th discontinuous

control action. (xbyj) is the i’th quasi equilibria ......................... 45
Figure 3.2 Response of a system after a severe disturbance:

EI-IV Line fault & line trip which resulted in a

large generator off-line .............................................. _. ............. 61
Figure 3.3 Active power ﬂow between central and south-east regions

of the Swedish test system after the tripping of a

line and generator in the North ................................................... 61
Figure 5.1 Block Diagram of a Power System Model .............................................. 134

Figure 5.2a General Conﬁguration of a single machine connected .
to a large system transmission line ............................................. 137

Figure 5.2b Equivalent system where the transmission network

is reduced to its Thevenin’s equivalent ....................................... 137
Figure 5.2c Circuit Model of (b) where R5 is neglected ................................. 137
Figure 5.3 Simpliﬁed single-machine inﬁnite bus mode. (a) Equivalent

Thevenin model where rm xe and Vi are given by
equations (5.1), (5.2) and (5.3) respectively. (b) Phasor diagram for the ‘
synchronous machine connected to an inﬁnite bus shown in (a) 138

Figure 5.4 Block diagram of a simpliﬁed linear model of a
synchronous machine connected to an inﬁnite

fa,

 

 

Figure 5.5

Figure A1
Figure A2
Figure A3

Figure Bla

Figure Blb

Figure E1.1

Figure El.2

Figure E1.3

Figure 151.4

Figure E2.l

Figure E2.2

Figure E23

synchronous machine connected to an inﬁnite

bus through an external impedance ............................................. 145

Block Diagram of a linearized machine model showing

Bifurcation Subsystems and the test matrix T ............................. 153
Power system stabilizer model ................................................... 214
Speed-governing-turbine system model ....................................... 215
Speed-governing-turbine system model ....................................... 216

Response of a four-machine System during a Transient-,

stable System ................................................................ . ........... 219

Response of a four-machine System during a Transient-

unstable System ........................................................................ 219

Power Transfer Model: Effect of real and reactive power transfer

to the inﬁnite bus on the parameter K5 . (a) Real and reactive power
ﬂow in the same direction, (b) Real and reactive power ﬂow

in opposite directions .......

Power Transfer Model: Effect of real and reactive power transfer
to the inﬁnite bus on the parameter [(4. (a) Real and reactive power
flow in the same direction, (b) Real and reactive power flow

in opposite directions

Power Transfer Model: Effect of real and reactive power uansfer
to the inﬁnite bus on the parameter K1 -K2K5/K6. (a) Real and
reactive power ﬂow in the same direction, (b) Real and reactive
power ﬂow in opposite directions

Power Transfer Model: Effect of real and reactive power transfer

to the inﬁnite bus on the parameter K, -K2K3K4. (a) Real and reactive
power ﬂow in the same direction, (b) Real and reactive power

ﬂow in opposite directions

Local Loading Model: Effect of the shunt local load on the

............ 227

228

229

_ 230

parameters K4, (a) Inductive loading, (b) Capacitive loading ...................................... 231

Local Loading Model: Effect of the shunt local load on the
parameter K5, (a) Inductive loading, (b) Capacitive loading .............

Local Loading Model: Effect of the local load on the parameters

xiv

232

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5

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p .

J.

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p w r

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¥

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h

F: ,

. 3. nﬂ —s —..

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P.- . PM.”

Figure 52.4

Figure 53.1

Figure E3.2

Figure E3.3

Figure 53.4

Figure E3.5

Figtn’e E3.6

Figure E3.7

Figure E3.8

Figure E3.9

Figure E3.10

K1 -K2K3K4, (a) Inductive loading, (b) Capacitive loading ...........................

Local Loading Model: Effect of the local load on the parameters
K. -K21(5/K6, (a) Inductive loading, (b) Capacitive loading“ -

Combination Model: Effect of power transfer on the parameter K5
when the local load is real-inductive. (a) Power ﬂow in the same direction,
(b) Real and reactive power ﬂow in opposite directions

Combination Model: Effect of power transfer on the parameter K5
when the local load IS real-capacitive. (a) Power flow in the same direction
(b) Real and reactive power ﬂow in opposite directions -

Combination Model: Effect of power transfer on the parameter K4
when the local load is real-inductive. (a) Power ﬂow in the same direction,
(b) Real and reactive power flow in opposite directions---

Combination Model: Effect of power transfer on the parameter K4

when the local load is real-capacitive. (a) Power flow in the sane direction.

(b) Real and reactive power ﬂow in opposite directions .......

Combination Model: Effect of power transfers to the inﬁnite bus on
the synchronizing term Kl «21(ng (regulated) when the local load is
real-inductive. (a) Real and reactive power ﬂow in the same direction,

(b) Real and reactive power ﬂow in opposite directions - .......

Combination Model: Effect of power transfers to the inﬁnite bus on
the synchronizing term Kl {21(ng (regulated) when the local load is
real-capacitive. (a) Real and reactive power flow in the same direction,

(b) Real and reactive power ﬂow in opposite directions. ...............................

Combination Model: Effect of power transfers to the inﬁnite bus on
the synchronizing term K r 491ng (unregulated) when the local load
is real-inductive. (a) Real and reactive power ﬂow in the same direction,

(b) Real and reactive power flow in opposite directions --

Combination Model: Effect of power transfers to the inﬁnite bus on
the synchronizing term Kl -K2K3K4 (unregulated) when the local load is
real-capacitive. (a) Real and reactive power ﬂow in the same direction,

(b) Real and reactive power ﬂow in opposite directions - - ................

Combination Model: Effect of local loading on the parameter
K5 (regulated) when the uansfer to the inﬁnite bus is at PmFOS. Qinf=05
(a) real-inductive loading; (b) real-capacitive loading ------ - - -

Combination Model: Effect of local loading on the parameter

XV

............. 233

234

............ 235

236

237

.. 238

............ 239

............ 240

-_. ....... 241

............ 242

............ 243

 

Figs: E311

: F1 1
‘v‘ij-\
r-.J~L—.-l-

.‘all: E}. II

-11

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1:53. ‘

Figure 53.11

Figure E3. 12

Figure £3.13

Figure £3.14

K4 (unregulated) when the transfer to the inﬁnite bus is at PmFOB. Q-mf=0.5.
(a) real-inductive loading; (6) real-capacitive loading. .........................................

Combination Model: Effect of local loading on the parameter
K, -l(2l(3l(4 (unregulated) when the transfer to the inﬁnite bus is at

..... 244

Pint-=08. Olaf-0.5. (a) real-inductive loading; (b) real-capacitive loading .................. 245

Combination Model: Effect of local loading on the parameter
K , «21(ng (regulated) when the transfer to the inﬁnite bus is at

P-mFOB. Qinf=05 (a) real-inductive loading; (b) real-capacitive loading ................. 246

Combination Model: Effect inductive-real local load on the system
synchronizing and damping torque coefﬁcients when PmFOB. Obj-=05

(a) synchronizing torque coefﬁcient KS. (b) damping torque coefﬁcient KD .........

Combination Model: Damping torque coefﬁcient Kd and

 

parameter K5 for different levels of real local loading .............

xvi

247

248

LIST OF SYMBOLS

Synchronous machine

E’q

D

internal bus voltage proportional to d-axis ﬂux linkage of ﬁeld current
damping constant due to damper winding effects (pu / (rad/sec) )
angular speed of the machine (rad/sec)

nominal angular speed (rad/sec)

terminal bus voltage

mechanical power input to the rotor (pu)

direct-axis (d-axis) current

quadrature-axis (q-axis) current

1,, + j Iq

angle difference between internal bus (rotor) and terminal bus
q-axis synchronous reactance

d-axis synchronous reactance

d-axis transient reactance

d-axis open-circuit transient time constant (sec)

resistance of the stator

Excitation system

Vc

output of load (or line drop) compensator

xvii

output of power system stabilizer (PSS)

reference voltage

output of automatic voltage regulator (AVR)

ampliﬁer output voltage

measured voltage

stabilizer output voltage

dc gain of AVR

Time constant of AVR (sec)

dc gain of exciter

time constant of exciter

washout gain of stabilizing feedback of excitation system
washout time constant of stabilizing feedback of excitation system
output of the exciter (ﬁeld voltage of synchronous machine)

compensator adjustable inductive reactance

xviii

(I;

we

I

Introduction

1.0 Introduction to and Justiﬁcation for Bifurcation Subsystems

An important aspect of stability analysis of a power system is to determine which
physical elements or subsystems are associated with a speciﬁc instability phenomena
observed on the full system. The recurrent problem in diagnosing a speciﬁc power system
stability problem is that of size and complexity of the model since a power system may be
composed of 10,000 interconnected buses and 1000 generators. However, although the
models required to study the dynamic effects of interest in a power system seem to
initially involve a large set of variables associated with the many different components of
the system, only a subset of these state variables may actually be critical to problem
diagnosis for a speciﬁc stability problem associated with a speciﬁc eigenvalue or mode.
The challenge in then not to obtain a reduced-order model that incorporates all of the
components in the system, but to obtain a subsystem of the full model which experiences
and causes the same bifurcation (a discontinuous change in system response as a
parameter changes slowly) as the full model, i.e. the smallest subsystem that preserves the
qualitative information of the system instability involving the speciﬁc bifurcating

eigenvalue.

I‘d

System modal reduction methods on a power system have mainly been based on
modal analysis (or eigenanalysis) of the linearized power system model, a technique that
has been found to be promising and of fruitful applications [1,2]. Sparsity, singular
perturbation theory, coherency reduction as well as Kalman minimal realization theory [3]
have also been applied in an attempt to simplify the large and complex dynamical system
models that arise in various power system problems. These traditional methods for
identifying subsystems are generally based on apriori knowledge of (a) the system or its
control design; (b) time scale information on what system state variables are fast and
which are slow variables; (0) eigenvectors and participation factor information from
eigenvectors; or (d) decoupling within the state space model. While such methods are
often useful, they are not always effective in identifying the smallest subsystem or
subsystems that experience and cause the bifurcation in the full system because the
criterion for deciding their effectiveness is their usefulness in model stabilization,
designing controls, and developing operation schedules for a system. Obtaining a
bifurcation subsystem model that experiences and causes the bifurcation in the full model
requires a quite different set of methods.

The singular perturbation approach for problem diagnosis in power system [4] can
not generally be applied for problem diagnosis of a particular stability problem that occurs
after a power system disturbance since the states most involved in an unstable mode in a
power system cannot generally be classiﬁed into either the slow or the fast dynamics
groups. When a power system is subject to undamped inter-area oscillations or local
generator oscillations, this technique seems to be inadequate to establish the bifurcation

subsystem for a single mode since the frequency of all inter-area oscillation modes lie in a

i\

E

narrow band and the frequency of all local generator oscillation modes also lie in a narrow
band. A singular perturbation based subsystem model contains all the modes in the narrow
band and not just the bifurcating mode.

In modal analysis theory, the location and involvement of speciﬁc components of the
system in the system instability may be obtained from the right and left eigenvectors
associated with the bifurcating eigenvalue. While the magnitude of the eigenvalue of the
normalized modal-transformed system is only a relative measure to proximity to
bifurcation (collapse) [2], the components of the left eigenvector can be interpreted as
indicating a direction normal to the operational boundary of the system [5-6], and right
eigenvectors indicate the degree to which given variables are involved in a given mode.
The dimensionless participation matrix, obtained from right and left eigenvector
information associated with the bifurcating eigenvalue, has widely been used as an
indicator to the relative involvement of system components in the unstable system mode or
as a tool for identifying the collapse regions in the system. Although Eigenvalue/modal
analysis has provided promising results, it seems to be insufﬁcient to locate the voltage
collapse regions or system components causing the collapse as can be demonstrated by the

following example.

Example

Modal analysis has been used in [7] on the 1993 BC. Hydro system model to ﬁnd
the suitable locations where reinforcement and remedial actions ought to be taken. The
relevant portion of the system in question as well as the three critical regions are shown in

Figure 1.1. The system participation factor matrix for the critical mode of the normalized

5(1'1“ V
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5

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ilt‘sit’fl

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mimic:

[Anifx a

system has been computed when the entire BC. Hydro system was stressed to determine
the location where reactive power injection would beneﬁt the system the most. Mth a
uniform MVA load increase, modal analysis near the point of collapse has resulted in the
participation factor plot shown in Figure 1.2. The modal information in the plot indicates
that the British Columbia Lower Mainland was the critical area, with North Vancouver
having the next larger participation factor elements. The Vancouver Island region in
Western British Columbia had relatively small participation and thus was rated relatively
ineffective for reactive compensation [7].

A study on the same BC. Hydro system has been conducted by Michigan State
University where the reactive compensation device was a synchronous var compensator
(SVC). The study based on MSU Voltage Stability Security Assessment and Planning
Methodology Programs [8] concluded that the collapse region was caused by reactive
imbalance in the Vancouver Island region. It has been found [9] that in this system, loss of
excitation voltage control on generators in one generating station on Vancouver Island
causes a voltage collapse in Vancouver Island which uncontrollably spreads and brings
about voltage collapse in the entire BC. Hydro system. This result was later conﬁrmed
because ABB [10] installed one SVC and it was located in the Vancouver Island region at

the site suggested by [9].

5;

Participation Factor ('0.001)

 

‘ N

d

 

 

 

0.5

Figure 1.1

Buses Ordered in the Sequence of P Factor

Identiﬁcation of the critical areas in 230 kV buses under
area system-stress

 

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The above example demonstrates that the participation matrix can be misleading or
not sufﬁcient for indicating the portion of the system that is causing the instability in the
full system, since (i) the voltage collapse can occur due to discontinuities in regions where
states have quite small participation for the mode closest to bifurcation or (ii) out of many
states with “high” participation, only few may be crucial for remedial actions. This result
may not be surprising since for most realistic power systems (a) the Jacobian matrix is
very sparse where a node in a 10,000 bus network may be connected to only three to ﬁve
other buses in the system; (b) weak coupling exists between the real power-angle
submodel and the reactive power-voltage submodel in the algebraic network model; (c)
weak coupling of coherent bus groups exists in both the real power-angle and in the
reactive power-voltage network submodels; (d) decoupling in the different dynamics of
the power system model exists and (e) most bifurcations in a power system occur due to a
sequence of discontinuities such as generators reaching ﬁeld current limits, under load tap
changers reaching tap position limits, and switchable shunt capacitors reaching
susceptance limits. All of these factors suggest that loss of control in some coherent bus
groups results in decoupling between some generator dynamics and system algebraic
constraints, and thus will produce a bifurcation for the algebraic, differential, or
differential-algebraic model associated with that coherent group. This has been
demonstrated in [11] for a classical load ﬂow model and is expected to be true in a
differential-algebraic model.

Selective Modal Analysis is a model reduction approach that recognized that the
participation matrix is not sufﬁcient for obtaining the smallest Subsystem that preserves

the qualitative information of the system near collapse. In the modal analysis algorithm

0
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[12], state variables with high participation are divided into “relevant” and “less relevant”
states where the relevant states are preserved in the reduced order model, while the effect
of the less relevant states is incorporated in the ﬁnal reduced model [12]. The relevant
states are obtained either by intuition or from participation factor information. This
approach demonstrates that only a few out of the many states with high participation
factors in a model may be of crucial importance in a particular unstable mode.

Center manifold dynamics may be a subsystem of the actual system that produces
and causes the bifurcation experienced in the full model. The center manifold dynamics
are obtained through a nonlinear u'ansformation of the original state space model. The
center manifold dynamics are not considered a bifurcation subsystem despite the fact that
it manifests the bifurcation similar to a bifurcation subsystem, because
(a) a bifurcation subsystem is a subsystem of the state space model written in terms of

physically meaningful variables and not a nonlinear transformation of these

dynamics;

(b) the nonlinear transformation on very large power system models generally makes it
impossible to analyze how operating condition changes or model parameter changes
will cause the bifurcation;

(c) the needed nonlinear transformation is not quickly and easily computed, which is not
the case for ﬁnding a bifurcation subsystem.

(d) a bifurcation subsystem is not a reduced model that can necessarily be simulated or
analyzed apart from the full system model.

A bifurcation subsystem is a subset of the equations of the full system model that

experiences, produces and causes the full system bifurcation by satisfying the same

L71,

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conditions for bifurcation as satisﬁed by the full system model. The subsystem that
experiences bifurcation may or may not produce the bifurcation in the full system since
the structure or parameters external to the subsystem may help produce the bifurcation in
both the full system and the subsystem. This is certainly true when geometric coupling of
the bifurcation subsystem and external system approaches zero as bifurcation occurs
rather than being zero. If the geometric coupling between the subsystem experiencing
bifurcation and the external system is zero as bifurcation occurs, which is called a
structural geometric decoupling, then the bifurcation subsystem both experiences and
produces the full system bifurcation. Thus the subsystem that produces the bifurcation
experienced by the bifurcation subsystem and the full system may be of larger dimension
than the subsystem experiencing bifurcation. The subsystem causing bifurcation may be
still larger than the subsystem experiencing or producing the bifurcation in the full system
and the bifurcation subsystem. The system that causes bifurcation must include all model
parameters, controls, and discontinuous changes that cause the bifurcation to develop. The
discontinuous changes include under load tap changer tap position changes, switchable
shunt capacitor switching changes, equipment outages, over-excitation limit relays, etc.
that cause instability to occur. The controls include maximum excitation limiter controls,
under load tap changer controls, switchable shunt capacitor controls, and excitation

system controls.

 

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1.1 Description of the Static Bifurcation Phenomena

In a power system, the parameter space breaks up into open connected regions, each
characterized by a particular behavior of the system in the state space. As parameters
move across the boundary from one open region to another, the topology of the state space
changes. This corresponds to the bifurcation phenomena. Although the bifurcation can
take various forms, a great concentration of the study of bifurcation in power systems has
been devoted to the study of saddle node bifurcation. It is only recently that the power
system researchers have recognized the signiﬁcance of other forms of bifurcations in
power system stability analysis above and beyond the static bifurcations. The creation of
limit cycles in the state space through Hopf bifurcation, although not fully investigated,
has nevertheless been a familiar although undesirable phenomena in power systems.

Saddle node bifurcation is a kind of static bifurcation that occurs at the steady state
operating point of a dynamic system. Signiﬁcant research on static voltage stability [App.
B] has resulted in (a) establishing the modeling requirements for static voltage stability,
(b) classiﬁcation of the various kinds of voltage instability such as algebraic bifurcation,
loss of causality, and saddle node bifurcation. Each of these different kinds of bifurcations
can affect different algebraic and/or differential states of a model and each is called a class
of that particular kind of bifurcation. For example, a saddle node bifurcation can occur in
generator ﬂux decay dynamics or generator inertial dynamics. Each of these is a different
class of saddle node bifurcation. A loss of causality is associated with singularity of the
Jacobian of the algebraic equations at an equilibrium. The Jacobian can be singular due to
row dependence of the real power balance equations and is referred to as angle instability,

or due to row dependence of the reactive power balance equations and is termed a voltage

 

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10

instability; both are classes of loss of causality.

Although there is a theory for establishing which kind of these bifurcations occurs,
there is no general method for deﬁning subsystems associated with classes of bifurcation
for each of these kinds of bifurcation. There is no actual test that identiﬁes whether a
subsystem actually causes the bifurcation. A method will be developed that (i) identiﬁes
subsystems that experience and cause saddle node, load ﬂow, and Hopf bifurcations and
the classes of each; (ii) provides tests for whether that speciﬁc subsystem causes the static
bifurcation observed in the system, which in turn may identify the kind and class of static
bifurcation which occurs; and (iii) establishes a diagnostic procedure for determining the
causes in terms of the speciﬁc parameters in parameter space and the model structure
discontinuities that produce this subsystem bifurcations. The bifurcation subsystem
method will also be developed for Hopf bifurcations in power systems where less

literature exists. A brief review of the literature on dynamic bifurcation in differential

power system models is now given.

1.2 Dynamic Bifurcation on Power Systems: A Brief Review

As a continuing effort to the understanding of voltage stability [App B] mechanism
in a power system, recent attention has been focused on dynamic bifurcation where
periodic solutions suddenly appear. It is now evident that poorly damped or unstable low
frequency oscillations are a major threat to the security. of power systems for many

utilities. The modes of oscillations observed are becoming more numerous and complex in
mOdern interconnected systems. Three speciﬁc classes of Hopf bifurcation have been

identiﬁed [25]: The local plant mode is associated with the mechanical dynamics of units

11

at a generating station swinging with respect to the mechanical generator dynamics of the
rest of the system and the frequency of such modes is typically in the range of 0.8 to 2.0
Hz. The inter-area mode is associated with the swing of the mechanical dynamics of one
group of machines in one part of the system against the mechanical dynamics of groups of
machines in other parts of the system. The frequency of the inter-area mode is typically in
the range 0.2 to 0.8 Hz [13]. The voltage mode appears in any machine’s ﬂux decay
dynamics and exciter dynamics. Its frequency is higher than the local plant and inter-area
modes.

In 1984, Abed and Varayia investigated generating units dynamics including both
generator electro-mechanical and ﬂux dynamics [14]. Their investigation revealed for the
ﬁrst time that generating unit dynamics may experience Hopf bifurcation. In 1989, Sauer
and Pai [15] investigated the stability of a two-axis generator model connected to an
inﬁnite bus as load level at the generator terminals increased. The study showed that near
the critical or maximum possible load level, a pair of complex eigenvalues associated with
the excitation system experienced Hopf bifurcation, and that the onset periodic solutions
were locally unstable. From 1990-1992, similar investigations have been carried on by
others [16-19], where system stability was based on eigenvalue analysis at different
loading conditions: The oscillatory instability in all of this literature was due to an
apparent subcritical Hopf bifurcation in the generator ﬂux decay and exciter dynamics.
During 1992-1993, Ajjarapu, Abed and Varayia [20-23] conducted a thorough testing on a
similar single machine—inﬁnite bus voltage collapse model developed by Dobson et al [24]
in 1990. The results revealed that Hopf bifurcation and saddle node bifurcation occur on

this model at different system loading levels. The oscillatory instability was characterized

in

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12

by the interaction of the generator angle dynamics and the induction motor dynamics.

A recent study on bifurcations on system models of increasing complexity has been
completed by Ajjarapu [49]. Two models were of interest: one model included a detailed
machine model with IEEE type DC-l exciter, and network with a constant power load
model, and a more complex model where a generic dynamic load model has replaced the
constant power load model. Eigenvalue analysis on both models revealed that before the
system collapses due to saddle node bifurcation (which occurs at the tip of the PV curve),
the system experienced Hopf bifurcation, node focus bifurcation and singularity induced
bifurcation. The sequence of Hopf, node focus, singularity induced and ﬁnally saddle node
bifurcations occurred on both a 3-generator 9-bus system and a IO-generator 39—bus New
England system model.

Issues that need additional research include (1) identiﬁcation of generic sequences of
bifurcations in a power system; (2) identiﬁcation of a bifurcation subsystem for each
possible class of saddle node, Hopf, node focus, or load ﬂow bifurcation in the system; (3)
development of tests for establishing whether a speciﬁc subsystem causes a speciﬁc
bifurcation observed in the system; and (4) a diagnostic for establishing the speciﬁc cause
of that subsystem bifurcation. Such a subsystem whose bifurcation causes bifurcation in
the full system is called a bifurcation subsystem, and may be composed of a subset of
solely differential, algebraic, or a combination of differential and algebraic equations. The
method presented in this thesis establishes conditions for which a speciﬁc bifurcation
subsystem exists. This subsystem method is ﬁrst developed in Chapter 4 for (1) a general
differential-algebraic model for identifying both differential and algebraic bifurcation

subsystems; then (2) for an algebraic model for identifying algebraic bifurcation sub-

(1)

l3

subsystems; (3) to a differential model for identifying differential bifurcation sub-

subsystems; and (4) to a differential-algebraic model for identifying differential-algebraic

bifurcation subsystems.

1.3 Objectives

1.

Initiate a diagnostic classiﬁcation of power system stability problems that is based on
bifurcation kinds and classes for the different model types that have been used for
bifurcation studies. The necessity for simplifying assumptions used in power system
stability analysis has resulted in four types of power system models of increasing
complexity [25], and which are commonly used to study the different forms of power
system instabilities. Kinds of bifurcation refer to saddle node, Hopf, LF bifurcation,
etc., thus refers to the bifurcation behavior observed, while a class of bifurcation refers
to the speciﬁc dynamical or algebraic states experiencing the bifurcation. It has been
shown in [25] that the type of model used for assessing stability has an effect on the
kind and class of bifurcation observed. This classiﬁcation based on model types as
well as kinds and classes of power system stability problems seem to be necessary to
diagnose the problem and to obtain information on how system stability can be
improved most effectively. This diagnostic classiﬁcation together with a description of
a power system differential-algebraic model is given in Chapter 2.

Brieﬂy review in Chapter 3 the recent development on power system stability which
extends the classical theory of power systems as dynamical smooth nonlinear systems.
to the ongoing development of a method that includes the effects of hard limits and

equipment outages which discontinuously modify the model order and system

14

behavior. The theoretical framework for study of stability of a power system model,
called a “taxonomy”, has been developed for smooth systems by Zaborsky [26]. The
taxonomy addressed three regions that lie in state and parameter space: (i) stability
region composed of the region of attraction deﬁned for every parameter vector in
parameter space; (ii) a feasibility region in state and parameter space composed of all
connected stable equilibria and the parameter vectors that produce them; and (iii) a
viability region composed of a subset of the equilibria and associated parameter
vectors belonging to the feasibility region that lie within the acceptable range for each
current, voltage, real power and reactive power variables.

Since hard limits associated with equipment outages, actions of operator control,
protective devices such as relays and ﬁeld current limit controllers, actions of under
load tap changers and actions of switchable shunt capacitors cause most of the
bifurcations experienced in a power system, the taxonomy is being extended to include
such models. The taxonomy that includes the effects of hard limits [27] deﬁnes and
describes similar stability, feasibility and viability regions as the taxonomy for smooth
systems. However, the taxonomic assessment is only valid if one can predict and
compute the sequence of quasi equilibria or limit cycles as well as the sequence of
hard limits encountered along every system trajectory without simulation. This
requirement and the assumptions made in [27] are not true in a power system with
hard limits, where there is different time delay between the time that a speciﬁed hard
limit surface is encountered and when the action associated with the hard limit is

implemented.

A different approach to stability assessment methodology for systems with hard

15

limit discontinuities is then proposed in Chapter 3. This approach considers an
episodal trajectory stability assessment rather than attempting to study the stability of
all trajectories and stability of all quasi equilibria and limit cycles encountered in some
subregion of state and parameter space. This new approach is aimed at assessing
asymptotic stability of the transient during (5.1;) and the stability of quasi equilibria

or limit cycles in each epoch, where an epoch is a period of time (if, if”

) where no
hard limit transitions occur. This procedure can be applied to a broader class of models
with hard limits that need not obey the assumptions made in [27]. The Epoch Based
Trajectory Stability Assessment Method is based on a stability simulation method
implemented by Van Cutsem in 1992. [28] which does not require time simulation in
(135“) , but only computes the quasi equilibria (El-,9.) for i = 1,...,N in a manner
similar to how a classical load ﬂow simulates the effects of hard limits. Speciﬁc rules
and guidelines decide the sequence of discontinuities that occur using a procedure that
avoids full time simulation that in turn determines the sequence of quasi equilibria
(2.3.) for i = 1..... N which can then be used to test for stability, feasibility and viability
via trajectory segments in epochs (if, rg+ l) , i = 1... N.

A Bounded Trajectory Feasibility Region is then deﬁned in Chaptm 3, which
classiﬁes trajectories produced by continuous parameter changes or by discontinuous
parameter or model structure changes that are induced by continuous operating
condition or parameter changes. Tire classiﬁcation of trajectories is based on (i)
whether each quasi equilibria (21.3.) or encircling orbit is stable and (ii) whether each

trajectory segment in (if, 1.; 1) belongs to the Region of Stability of the associated

quasi stable equilibrium point or limit cycle. The subset of trajectories, where (a) all

l6

equilibria or encircling orbits are stable and (b) all trajectory segments (or only the last
trajectory segment) lie in the region of stability of a stable orbit or equilibrium,
belongs to a Bounded Trajectory Feasibility Region.

Since the episodal stability assessment method is applied after each hard limit is
encountered, the power system model in each interval may be different and yet
analytic so that existing dynamical system theory can be applied. A natural desire is to
ﬁnd a reduced—order model or the smallest subsystem within each epoch that
experiences the same bifurcation as the full model, which motivates the bifurcation
subsystem method investigated in the rest of the thesis. If the bifurcation subsystem is
identical in each epoch as discontinuities bring on the bifurcation, it is a bifurcation
subsystem for the bifurcation that ultimately occurs and changes the system dynamical
behavior.

A power system differential-algebraic model is investigated for the purpose of
identifying the possible bifurcation subsystems for each kind of bifurcation (SN, Hopf,
LF bifurcation) that may occur on a Type 2 power system model where load dynamics
and controls are ignored. A Type 2 model with a constant power load model, tap
changers, switchable shunt capacitors in the transmission, subtransmission and higher
voltage distribution system will be assumed in this thesis because it generally contains
the dynamics that experience bifurcation and cause voltage collapse in a large region
[25]. The bifurcation subsystem method allows one to ﬁnd a speciﬁc subsystem that
experiences, produces and causes the same bifurcation that occurs in the full model.
The bifurcation subsystem always experiences the bifurcation that occurs in the full

system model. The geometric coupling of the bifurcation subsystem may not be null

l7

and may produce the bifurcation in the full system and the bifurcation subsystem as it
becomes zero. The system that causes the bifurcation in the bifurcation subsystem
contains all the discontinuities, operating changes, equipment outages and model
parameter changes that cause the bifurcation to develop in both the full system and the
bifurcation subsystem. Thus, this subsystem may be larger than the bifurcation
subsystem that experiences the bifurcation of the full system model. or the subsystem
that includes the effects of coupling to the external system when this coupling is not
null but approaches zero as bifurcation occurs.

The bifurcation subsystem method is an extension of the diagnostics employed
in producing the taxonomy for feasibility, stability and viability for an equilibrium or
closed orbit developed in Chapter 3, the classiﬁcation of kinds and classes of
bifurcation given in Chapter 2, and the episodal epoch by epoch stability assessment
given in Chapter 3, because it ﬁnds the smallest subsystem that experienced, produced
and caused the loss of stability, feasibility and viability for that equilibrium for
bifurcations produced by encountering hard limit induced discontinuities. The
bifurcation subsystem method allows one to better diagnose the causes of a subsystem
bifurcation and separate causes and effects of this bifurcation for bifurcations brought
on by hard limit induced discontinuities. Since the sequence of discontinuities are the
major cause for many bifurcations, a bifurcation subsystem method based on epoch by
epoch diagnosis appears to be the only way to understand the causes and the
bifurcation subsystem that produced the bifurcation. This is needed in a power system
model when there are hundreds of generators, tens of thousands of dynamic loads, and

a network that couples the load and generator dynamics. The bifurcation subsystem

l8

method should identify bifurcations that occur in

(a) algebraic subsystems composed of generator bus groups, groups of load buses,

and groups of generators and load buses;

(b) differential subsystem dynamics in one or more groups of generators;

(c) differential-algebraic subsystems in one or more groups of generator and in

one or more groups of generator and load buses.

The bifurcation subsystem method, developed in Chapter 4 would allow a very
complex behavior to be seen as being very simple so that one can potentially explain
virtually everything about feasibility, viability and stability for each and every subsystem
that can produce loss of feasibility, loss of viability or loss of stability. The bifurcation
subsystem model is produced to study the causes of a speciﬁc bifurcation and may have no
value except for studying that bifurcation. It may not be useful for control design or even
simulating the contingencies which produce that bifurcation, because the subsystem that
produces the bifurcation may be quite small compared to the subsystem needed to
simulate a speciﬁc contingency or design a control system to prevent that bifurcation.

In chapter 5, two examples for validating and using the bifurcation subsystem
method in a model are presented, and the results are compared with the participation factor
information. In Chapter 6 a diagnostic study for the occurrence of SN and Hopf
bifurcations in a single-machine-inﬁnite bus model is performed based on test matrices
derived in Chapter 5. Chapter 7 summarizes the dissertation and suggests topics of future

investigation.

II

Power System Modeling
Effects on Stability Problems

In this chapter, a diagnostic classiﬁcation of power system stability problems that is
based on bifurcation kinds and classes is developed, and a description of the model types
that have been used for bifurcation study is initiated. A speciﬁc model type (Type 2) is
selected for further study in this thesis. A further classiﬁcation of kinds and classes of
bifurcations for this Type 2 model is then given. Kinds of bifurcations include saddle
node, H0pf, load ﬁow bifurcation, thus refer to the kind of qualitative change in dynamical
or algebraic network behavior encountered at the point of bifurcation. Classes of
bifurcations are the speciﬁc subsystem dynamics or algebraic network variables that
experience a particular kind of bifurcation for a given model type. For the model Type to
be used (Type 2 model), there are two classes of saddle node bifurcation, three classes of
Hopf bifurcation, and three classes of loss of causality bifurcation.

The effort to classify model types as well as kinds and classes of bifurcation is
necessary since later, a bifurcation subsystem method will be developed and applied to
saddle node, load ﬂow and Hopf bifurcations and the classes for each. This bifurcation
subsystem method will attempt to diagnose the geometric subsystem structure that

bifurcates. what locations in a system are vulnerable to this particular kind and class of

19

 

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20

bifurcation, what causes it in terms of model parameter changes, equipment outages, or
Operating condition changes. This systematic diagnosis needs a classiﬁcation of kinds and
classes of bifurcation in a given model type. Details of the classiﬁcation to be given in this
Chapter comes from a paper published by the author [25] in 1994.

As a natural start to this diagnostic classiﬁcation, a general power system model
description including models for generators, transmission network and loads are given in
Sections 2.1 and 2.2. Section 2.3 presents the four model types suggested by the author in

[25], and ﬁnally kinds and classes of bifurcations in a Type 2 model are presented in

Sections 2.4.

2.1 Description of a Power System

Electric power systems are comprised of generation, transmission and distribution/
load. Electric power is produced at generating stations and transmitted to consumers
through a complex network of individual components including transmission lines,
transformers, and switching devices. Power systems models, although they vary in size
and complexity, all have three basic structural components: synchronous generators, a

transmission network, and electric load.

2.1.1 Synchronous generator

Prime movers convert the primary sources of energy (fossil, nuclear and hydraulic)
to mechanical energy that is in turn converted to electrical energy by synchronous
generators. The synchronous generator output voltage is usually regulated to maintain a
constant value using an excitation system. A functional diagram of a synchronous

generator with an excitation control system is shown in Figure 2.1 [29]. The subsystems

21

inside the dashed-line box in Figure 2.1 constitute the excitation system, of which a

typical model widely used in literature is the IEEE Type DCl excitation system model

shown in Figure 2.2. For consistency and task simpliﬁcation, this speciﬁc model will be

used in this work. First, a brief description of the indicated functional blocks is given.

(a).

(b)

(C)

(d)

(e)

Exciter: provides dc voltage to the synchronous machine ﬁeld winding based on
input from the regulator control;

Regulator. Processes and ampliﬁes input control signals (as shown in Figure 2.1) to a
level and form appropriate for the control of the exciter. This includes both
regulating and stabilizing functions (rate feedback or lead-lag compensation);
Terminal voltage transducer and load compensator: senses generator terminal
voltage, rectiﬁes and ﬁlters it to do quantity, and compares it to a reference which
represents the desired terminal voltage. In addition, load compensation may be
provided if it is desired to hold constant voltage at some point electrically close to or
distant from the generator terminal bus;

Power system stabilizer: provides an additional input signal to the regulator to damp
oscillations in generator mechanical dynamics. Some common input signals are
rotor speed deviation, accelerating power, and frequency deviation;

Limiters and protective circuits. They include a wide array of control and protective
functions whose role is to insure that the capability limits of the exciter and
synchronous generator are not exceeded. A recent textbook [29] provides a detailed
description of the various limiters and protective devices commonly used on power

systems.

I»)
1‘»)

2.1.2 Electric load

The diversity of electric loads present on a power system, makes it virtually
impossible to represent every individual component in a power system model. The
common practice is to make considerable simpliﬁcations of the composite load
characteristics as seen from bulk power delivery points. Load models have been broadly
classiﬁed into two categories, static loads and dynamic loads. The static load model
expresses the load as algebraic equations, classically in the exponential form

- b
P P0(V)

Q = 9007)”
where P and Q are active and reactive components of the load and the voltage magnitude
is V. The subscript “0” refers to the values at the initial operating condition. If the
parameters a and b are 0, 1 and 2, the models are respectively constant power, constant
current and constant impedance load. The dynamic attributes to motors are usually the
most signiﬁcant aspects of dynamic characteristics of system loads. Typically motors
consume 60-70% of the total energy produced by a power system. Other devices that
require dynamic consideration are load shedding, operation of protective relays,

thermostatic control of loads, response of under load tap changers (ULTC). etc...

7?” ~_—.
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t ”I ‘1

 

 

 

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Protective circuits

 

 

 

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—i Excrter Generator -u—> sysigm

—— v

 

 

 

 

     

 

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C

 

 

 

 

 

 

Figure 2.1 Functional block diagram of a synchronous
generator excitation control system

 

 

 

 

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V
REF Ampliﬁer
KA/(1+sTA)-4| 7L . i—E-“l 0(5) l—q
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Figure 2.2 IEEE Type DC 1 Excitation system model

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24

2.2 General Power System Model

The general power system model consists of a set of ﬁrst order differential equations,

subject to algebraic constraints. The differential equations represent the mechanical

dynamics, ﬁux decay dynamics, and excitation system dynamics of a generator whereas

the algebraic equations represent the real and reactive power balance equations for each

bus in the network. A multi-machine system consisting of m network buses and n

generators with voltage regulation can therefore be written in the parameter dependent

differential-algebraic model

where

x(t)

W)

110)

XxY

I R

I R

i = f(x(t).y(t).u(t)) Mt) e P 2.1(a)
0 = 800). YO). 9(1)) 210’)
n + m + p -> R" , an n-vector of real smooth nonlinear functions.
'1 + m + p —; Rm, an m-vector of real smooth nonlinear functions.
e X cR" , a state vector of dynamic state variables. typically time dependent

values of generator voltages, rotor phases and induction motor variables.

: 6 Y c Rm , a state vector of instantaneous variables, typically time dependent values

of voltage and angle of terminal buses, high side transformer buses and load buses.

: e P c R” , a vector of slow varying operating parameters. It can represent structural

parameters like system components, reactances, under-load tap changers and
switchable shunt capacitors, or operational parameters like reactive power load or
generation, voltage set points on voltage control devices, as well as generation
dispatch. As it varies over the parameter space F, it is assumed that there must be at

least one solution to the above differential-algebraic system of equations.

: state space

P : parameter space.

2.2.1 Model Equations

The differential-algebraic model consists of

a) Differential equations

 

d5i .
W = (”i—(Os [=1 ..... n
dcol.
Mi—d-t— = Pmi—Di((i)i-O)S)—Pei r=l,...,n
d5".

Tdoi dt = “Eqi—(Xdi-Xdi)ld;+Efdi i=1.....n

dE .
J '_ _ -_

TE. m _ (KEi+sEi(Efdi))Efdi+vRi r-l.....n
Tﬂ- —V +V +'(lx) i-l n
Rid: " 11' In JTicil — .....
T dV3i _ v KFIVRi KFiEfdi(SEi(Efdi)+KEi) _1
Pi dt - _ 3i..- T - T l- ,. n

51' El

T {-l—V-B-i v +K v v v '-1

M d: ‘ Ri Ai( REFi’ ii’ 31') l- ~~~~~ n

for the generator and exciter dynamics where the variables and constants are deﬁned in the
list of symbols.
b) Stator algebraic equations
Vicosei + Rsi(ld‘.sin8i + Iqicosﬁl.) — X'diuqisinbi ~1dicosﬁi)

_[E'disinﬁi+(X'qi-X'di)lqisin8i+E' icosbi] = 0

q
—[E'qisin6i—(X'qi-X'di)lqicosbi+E'dicosbi] = 0 t: I, ..... n

that deﬁne the lqi and 14,- at the generator terminal buses.

lll‘ll Cl:

_.
1
(I:

.5”
3

1.11

crab:

Where

SUSCCI

2.3

dimer,
em in:

t. '
WP:

('73

26

c) Network (Power Balance) Equations

Terminal bus equations

,- = Villdisinwi - 9i) + lqicos(5i — 91.)] + PLi(Vi) i: 1..... n
Q.- = Viildicosw. — 0,) + Iqisinrsi - 991+ QLi(V,-) i = 1..... n

that deﬁne voltage and angle at generator terminal buses where PLi(Vi) and Q ”(Vi) are
the loads at the terminal buses.

Load and high side transformer bus equations

The net real and reactive powers at each of the network load and transformer buses

can be expressed in terms of the voltage magnitudes, conductances and susceptance as,

n
pl. = —PU(V'.) = kg, lViv/((Gikcostiik+Biksin0ik) 1=n+l,...,n+m
n
Qi = -QLl-(Vi) = 2 VivkmikSineik—Bikcoseik) i=n+l,...,n+m
k = l

where PLi(Vi) and QLi(Vi) are the loads at bus i and G ik and Bik are the conductance and

susceprance of the ik’th element of the Y-bus.

2.3 Modeling Complexity

In dynamical system theory, the power system is generally a very high-order, high
dimensional multi-variable process, that is operating in a constantly changing
environment. Simplifying assumptions are therefore a virtual necessity for developing and
using power system analytical methods. A classiﬁcation of four types of power system
models of increasing complexity, that are used in the existing literature to study or
simulate. the various forms of power system instability is presented. A model type has

been deﬁned in [25] by the author as the simplest model that captures observability,

€0th

de. el

all}. [if

2.3)

‘-

27

controllability, feasibility region, region of attraction and modal behavior associated with
at least one class of stability problem. This paper [25] discusses the known classes of

stability problems for each model type.

2.3.1 Type 1 model

Type 1 model reduces a complex differential-algebraic model that includes dynamic
models of both generator and loads, to an algebraic model of the generator, network and
load, because the dynamic states are assumed at equilibrium. However this model takes
into account the time sequence of tap changing transformer, switchable shunt capacitor
and low voltage relaying actions as well as secondary voltage control, AGC and over-
excitation limiter (OEL) control actions. A classical load ﬂow model approximates a Type
1 model at steady state ( t=inﬁnity) and often incorrectly estimates the sequence of control
actions of under load tap changers, switchable shunt capacitors, and ﬁeld current limit
controllers after a contingency or due to slow operating condition changes. The recently
developed simulation tool of Van Cutsem [28] is an example of a Type 1 model if relaying

actions are represented.

2.3.2 Type 2 model

Type 2 model is perhaps the most common power system dynamic model used to
simulate and study instability. It currently includes generator inertial, flux decay, exciter,
power system stabilizer, and boiler turbine governor dynamics. However this model
excludes all load dynamics, for they are assumed to be decoupled from the generator

dynamics, and thus do not contribute to the stability problems observed on this model. The

28

general form of the Type 2 model is:

x = f(x.ld.lq.V.O) 2.2(a)
O = ld—gd(x,V.O) 2.2(b)
O = lq—gq(x,V,O) 2.2(e)
0 = PL—gP(x,V,O) 2.2(d)
0 = QL-gQ(x,V,O) 2.2(e)

where
x : generator dynamics
P L, Q L : real and reactive power load

v, 6 : voltage and angle of generator terminal, high side transformers and load buses.

2.3.3 Type 3 model

In this model, the generator dynamics are modeled by constant power sources, while
the load modeling includes all of induction motor and under load tap changers (ULTCs)
controller dynamics, as well as dynamic models of switchable shunt capacitors, and
thermostatically controlled loads. An extensive use of this model is for studying radial
voltage instability, since by deﬁnition, this instability problem occurs at points electrically
decoupled from the generator dynamics [30]. The vector x representing the generator
dynamical states is either a constant vector or not included in the state space model
equations. Therefore the complexity of this model depends on the complexity of the
dynamic model used to represent the induction motor dynamics. A Type 3 power system
model has the following form (2.3a-e), where the induction motor model used is a 3rd
order model recently derived by Sauer and Pai [31] as an implicit load dynamic model,

although other dynamic load models may alternatively be used.

 

it here
SID-d)’ s

trough

23.4

decay,
the 103

M} an.

29

o = PL—gp(x, v, e) 2.3(a)
o = QL—gq(x, v,e) _ 2.3(b)
o = h,(PL, QL, v, m,‘%, %,%) 2.3(c)
o = he(m, 559,-) 2.3(d)
o = hm(PL, QL, v, 03.9%, ‘2—‘f,‘%, ‘13-?) 2.3(e)

where V, q and to are N dimensional state variables. A type 3 model ought to be used to
study saddle node bifurcation in load dynamics and saddle node and Hopf bifurcations

brought on by under load tap changers and switchable shunt capacitors.

2.3.4 Type 4 model

This model combines the complexity in generator modeling of a type 2 model and in
load modeling of a type 3 model. The generator model includes generator inertial, ﬂux
decay, exciter, power system stabilizer, and boiler-turbine-generator dynamics, whereas
the load model may be a separate dynamic model for real and reactive power, or a
composite model of real and reactive power dynamics. A Type 4 model has the form 2.2(
a-c) and 2.3(a—e) if a nonlinear implicit load model is used, although the load model
equations 2.3(c-e) may differ.

Type 1-3 models will later be viewed as speciﬁc subsystem models for the Type 4
power system model. This will be made clear when differential subsystems are discussed
in Chapter 6. Thus it will be clear that a Type 4 model is quite general and that Type 1-3
models are only used when speciﬁc types and classes of bifurcations are to be studied.

A Type 2 model may be able to capture the generator dynamics that ultimately

experience saddle node bifurcation, Hopf, or node focus bifurcaﬁon, and which may be

{a

51 s
n
exp
U011]
01 t’.’

iriild

tier:
dynun
couch

litlie

3()

affected by load ﬂow bifurcation. algebraic bifurcation and singularity induced
bifurcation. A Type 2 model with a constant power load model. tap changers, switchable
shunt capacitors in the transmission, subtransmission and higher voltage distribution
system will be assumed in this thesis because it generally contains the dynamics that
experience bifurcation and cause voltage collapse in a large region, a utility, or even a
county. However, a Type 2 model can not accurately capture the load buildup and
overshoot but just approximates the peak load produced by action of distribution system
load control dynamics and the voltage control dynamics.

A Type 4 model is the only model that can hopefully simulate actual voltage collapse
events. For Hopf bifurcation studies, a Type 4 model must be used because both the load
dynamics and generator dynamics must be considered Load dynamics may strongly
couple to generator dynamics when subcritical (unstable), or low frequency supercritical
(stable) Hopf bifurcations occur. In fact, when a pair of stable conjugate eigenvalues
approaches the jun-axis, the real part of the eigenvalues also decreases, which causes the
orbit size of the associated limit cycles to increase. This causes more control actions of
under load tap changers, switchable shunt capacitors and thermostatically controlled loads
to take place to prevent the system equipment from operating beyond the maximum
tolerable range. The impact of the load dynamics is therefore present and a full Type 4
dynamic load model is necessary to represent them. Although Type 4 models are needed,
they have generally been developed only for small regions within a utility or are only
being developed at the present time for larger regions.

Since a Type 2 model with voltage dependent algebraic load model, with under load

tap changers, switchable shunt capacitors, control in the transmission, subtransmission,

and f

\Olic

Mum
A'CL‘enr
PCP-iii!

T3330;

"m

p435

End

31

and higher voltage rating distribution system is the best model available at present to study

voltage collapse, they will be used in this thesis with the following assumptions

1. The load dynamics are electrically far from the generator such that they are
decoupled from the generator dynamics;

2. There is reasonably good voltage control at the load buses in the system so that the
load variation with voltage is small;

3. If the magnitude of stable orbits associated with generator dynamics remains large
enough and if the frequency of the associated oscillations remains small enough,
then signiﬁcant control actions of tap changers, switchable shunt capacitors and
thermostatically controlled loads should take place in response to these oscillations.
These assumptions indicate a constant power load model would be satisfactory to

determine the equilibrium point set by the actions of tap changers, switchable shunt

capacitor and thermostatically controlled loads, and that a perturbation load component
that is voltage dependent and responsive to orbital dynamics need not exist using this
assumption. Indeed, a perturbation load component model has been validated and used in
recently developed simpliﬁed dynamic load models, that include both steady state and
perturbation components. The perturbation load component model can be viewed as the

Taylor series approximation of the system load model around equilibrium [31].

2.4 Classes and Kinds of Bifurcations in a Type 2 Model

A classiﬁcation of stability problems that are observed and/or may occur on a Type 2
power system model is established in terms of kind and class:

Kind - Kind (SN, Hopf, Load Flow bifurcation...) of local bifurcations that may

32

occur on the speciﬁc model type.
Class - The subset of generators, controls, and/or load dynamics which exhibits the
speciﬁc kind of bifurcation in this model type.

The bifurcation phenomena in the nonlinear power system model (2.1) refers to
characterizing the qualitative change of the ﬁxed points of the differential-algebraic model
in (2.2) for a smooth continuous change in parameter value it over a speciﬁed range.
Although all system bifurcations usually result in a qualitative change in system behavior,
they however have different effects on the dynamics of the system. Kinds of bifurcation
simply refers to saddle node, Hopf etc., while a class is a further division of each
bifurcation kind into the speciﬁc system dynamics experiencing the bifurcation. Kinds of
bifurcations observed in a Type 2 power system model have been reported in [25] and
include saddle-node, Hopf, load ﬂow bifurcation and period doubling bifurcations. Loss of
causality bifurcation may be split into a voltage, angle, or both voltage and angle
bifurcation classes. Similar classes may be found for the other kinds of bifurcations. A
class of bifurcation is necessary to (a) identify the subsystem dynamics and speciﬁc
parameter u or parameters a. of a power system model, that cause a speciﬁc kind and
class of bifurcation to occur in the full dynamic model; (b) to develop methods that can
identify all potential bifurcations that exist in a particular model; and (c) to ﬁnd the
underlying continuous or discontinuous parametric, operaﬁng schedule, or structural
model control changes that produce such bifurcations.

In order to present the kinds and classes observed on a Type 2 power system model,

rewrite the differential-algebraic model (2.1) as

x = f(x.0T. VT. u) 2.4(a)

3 3

t) = (}P(x,9T,VT,1.t) - 2.4(b)
GQ(X, 67" VT’ ll) 2.4(C)

C
II

where

x = [6, 5",. y] : generator dynamic state variables: 6 is the angle at synchronous machine
internal bus excluding swing bus, E", is the voltage behind transient reactance at
generator internal bus and y presents the states of exciter dynamics of synchronous
generator.

VT , GT : voltage and angle at generator terminal, generator high side transformer and load
buses, not including synchronous machine internal buses and the swing bus.

u : parameter of interest that has an impact on the stability property of the equilibrium
points of (2.4), such as real and reactive loads and generator real generation.

Local stability of a steady state equilibrium point of interest for some u e R

Y. = (ram). 91001). V1000)
may be investigated using a linearized version of (2.4) evaluated at the equilibrium,

obtained by solving the system of equations

0 = f(x.GT. V1.11) 2.5(a)
0 = GP(X, 9T, VT, ‘1) 2.5(b)
O = 00(x, 9T, VT’ u) 2.5(C)

The Jacobian matrix of (2.4) at the equilibrium point Y0 = (xo(u), 91 (u), VT (11)) may be
obtained by linearizing the differential and algebraic equations and then combining all the
linearized components as shown below

ax Ax

0 = 100' A91 2.6(a)
o AVT

34

where
"er. 2:. 2:2
ax 39, av,

DFO—(O) 30,, BGP 30,,
’(“)-_ax - ‘37 Te; 5v;
aGQ Bag 809
L 8x BOT 3V1}?

A1 31 C1
A3 33 C3

 

 

 

X = (1(9).9T(i~l). V7~(il))
F(X) = [f(X).Gp(X).GQ(X)]

An equilibrium point X ‘0 = (x3, 9%,, Va.) of (2.4) is called a bifurcation point and u‘ a
bifurcation value if the complete Jacobian matrix J(u") is singular, in which case
existence or uniqueness of solutions is lost. Since the power system model as given in
(2.4) is composed of both algebraic and differential equations, it is clear that an equivalent
differential or equivalent algebraic equation model ought to be obtained to derive the kinds

of bifurcation that may occur in the model, as will be shown in the following section.

2.4.1 Kinds of Bifurcation

From the implicit function theorem [33], a unique solution to 2.4(b) and 2.4(c) exists

locally if and only if the matrix

J (u) ___ 32 c2 5 BGP/BBT aGP/BVT
‘ B3 C. aGQ/BBT aGQ/BVT go

is nonsingular. If Jc(|.l.) is nonsingular for all x, 8T, VT and u in a neighborhood of )‘to
then 2.4(b-c) may be solved uniquely for OT(x(|.1)),V7-(x(u)), which results in the

equivalent differential equation model

Ax = f(x. 91001)). V1061)»

35

which linearization gives the linear equivalent differential model

x = rxru) . Ax 2.7(a)

-1
Jx(ll)=Ar-[Bl c1][:2:2] [2] 2.70»)
3 3

3

Similarly, a unique solution to 2.5(a) exists locally if and only if the matrix 41005 SE l?
x

0

is nonsingular. If A101) is nonsingular for all x, 9T, VT and u in a neighborhood of if"
then 2.5(a) may be solved uniquely for x(6T(u), VT(u)) . which results in the equivalent
algebraic equation model

0 = g(x(97~(u). V100). err»). VT(u)) 2.8(a)

or upon linearization gives the linear equivalent algebraic model

0 = Jy(u) - Ay 2.8(b)
_ 32 C2 A1 -1
Jy(|.l) .. [33 c3] - [Ali/4.1 [3, cl] 2.8(c)

The power system deﬁned by (2.4) with equivalent differential equation model 2.7(a) is
said to be asymptotically stable iff all the eigenvalues of the equivalent differential system
Jacobian matrix J,(l1) have negative real parts. Hence a critical state is a state when one
or a pair of eigenvalues become non-hyperbolic and have zero real part. Kinds of
bifurcation on a Type 2 model are summarized in the following:
(0 WW
Occurs at u= u“ when Jc(u‘) is nonsingular and the system (2.7) has one real
eigenvalue that crosses the imaginary axis of the complex plane and where certain
transversality conditions hold [App C]. In this case the system trajectories are

attracted by the center manifold, and the system behavior becomes monotonic and

(ii)

36

may experience monotonic instability. i.e. collapse type of instability.

II [E .E . Ill:

Occurs at u= u‘ when 1001‘) is nonsingular and the equivalent differential Jacobian
matrix 1.01.) has one pair of imaginary eigenvalues with zero real parts. Additional
transversality conditions must also be satisﬁed [App C]. In the neighborhood of this
point, periodic solutions always exist and the stability of the system is determined by
the stability of the periodic solutions (limit cycles). A Hopf bifurcation can be stable
(supercritical) or unstable (subcritical). Both supercritical and subcritical Hopf

bifurcations have been observed on a single-machine-inﬁnite-bus model [25].

(iii) WW)

Occurs in the system equivalent algebraic model 2.8(a), at u: u‘ when Al(u") is
nonsingular and the equivalent algebraic Jacobian Jy(u‘) in 28(c) has one simple
zero eigenvalue. This bifurcation is called Load Flow (LF) bifurcation because it
indicates a system at equilibrium is unable to satisfy the system power ﬂow
requirements imposed by the given generation and demand requirements.The system
in 2.8(a) is called the Exact Load Flow Model since these equations represent the
power balance equations at all buses in the system, while the ‘classical’ load ﬂow

model will be shown in Chapter 4 to be an approximation of this model.

(M LasaQLCausallnr

Occurs when Jc(u) becomes singular as it varies. Simulation cannot be performed

on the system (2.4) since 2.4(b-c) have no solution or have multiple solutions

37

91",(X(ll))
Vri(x(l1))

i= 1,2...

in the neighborhood of an equilibrium 550 = (x0, 9w VTo) . If there is no solution, the
system trajectory terminates abruptly. If multiple solutions exist, random
discontinuous trajectories can occur as the solution (en-(x00), VTi(x(u))) changes
rapidly from time point to time point. Two types of loss of causality may occur.

(a) Algebraicﬁiﬁrtcaﬁcnﬂil

Occurs when the submatrix of 1(a) .

A a c 1'
1m”: 2 2 2 = f"
A3 33 C3 JQL

becomes row dependent at u: u‘. Hence it requires that the differential-
algebraic bifurcation occurs in a subset of the equations. Note that algebraic
bifurcation may suggest that there are no solutions of the form (x7, 9T, VT) or
multiple solutions (113971,» VTr)’ i=1,2,..
(b) 5' l . I I lE°ﬁ . :51:
Occurs when the full system Jacobian matrix 1.01) is nonsingular with
eigenvalues crossing from the right half plane to the left half plane or vice
versa by going through inﬁnity, rather than across the imaginary axis (jar-axis).
At this point, one real eigenvalue of the system is at inﬁnity and the rest of
them are bounded which makes simulations using the differential-algebraic
model not possible.
(V) W

Occurs in the case of multiple eigenvalues when a pair of complex conjugate

38

eigenvalues become two equal real eigenvalues. A node focus bifurcation (NF) may
be stable or unstable depending on whether the real part of the eigenvalue pair is
negative or positive, respectively.
(vi) Wee: Occurs in the case of multiple eigenvalues when two pairs of
complex conjugate eigenvalues become two equal complex eigenvalues.
At (SN), (H) and (SI) bifurcation points, local system steady state stability around
the equilibrium point changes. At 1:1 resonance point, the system may experience non-
linear oscillatory behavior due to order 1 resonance. In this case, participation factors,

eigenvectors and residues can be misleading because eigenvectors are not unique and they

lose their physical meaning.

2.4.2 Classes of Bifurcation
For each kind of bifurcation, the bifurcation subsystem method will attempt to

identify subsystems of smaller and smaller dimension to determine the smallest
bifurcation subsystem that produces and causes the bifurcation. The bifurcation subsystem
implies the class of bifurcation of the particular kind which will be investigated in
Chapters 4-6. Classes of bifurcation in a Type 2 power system model may be

(a) the power system differential equations;

(b) the power balance equations at load buses;

(c) a generator’s inertial, ﬂux and exciter dynamics;

(d) a generator’s electrical dynamics; .

(e) the control systems dynamics ( PSS, Govemor..) on a generator;

(f) a combination of the above.

39

An investigation to ﬁnd the smallest bifurcation subsystem that experiences and
causes the bifurcation in the full system is initiated theoretically in Chapter 4 and

continued to the end of the thesis.

2.4.2.1 Classes of Loss of Causality in a Type 2 Model

Although the two kinds of loss of causality system bifurcations (algebraic bifurcation,
and SI bifurcation) all produce truncation of trajectory, they have quite different effects on
generator dynamics. Loss of causality may occur in real power-angle variables, in reactive
power-voltage variables, or in both angle and voltage variables.
(i) mislelrcsscmausaliut
Occurs when [32 C2] = [Bap/89, aoP/avT] is row dependent, thus since [32] is
singular, this loss of causality is due to real power-angle instability.
(ii) W
Occurs when [83 C3] = [609/89, aoQ/aVT] is row dependent, thus since the square
matrix [C3] it singular, the loss of causality is due to reactive power-voltage
instability.
(iii) WWW
Occurs when Jc(u) is row dependent but neither [82 C2] nor [B3 C3] is row

dependent resulting in a voltage and angle loss of causality.

40

2.4.2.2 Classes of Static Bifurcation in a Type 2 Model

Static bifurcation in a Type 2 model occurs when the matrix 1.01) is nonsingular

and

-l
B C A
W = [3‘ €11le ll
3

is singular as m changes. Saddle node bifurcation is a special but generic case of static

instability which occurs when a single eigenvalue of 1,.(11) is zero and certain

transversality conditions hold [App C]. Two classes of static bifurcation may occur on a

Type 2 power system model:

(i) S'El .. °'l| .
Static bifurcation in generator inertial dynamics may be associated with the loss of
transient [App. B] stability [36] such as the voltage instability that occurred in the
Czechoslovakian system after tripping lines in a major interface [30]. This class of
static bifurcation is likely to occur when the disturbances are very large and the
transmission interfaces or boundaries are weak, so that loss of stability occurs soon
after the contingency before the ﬁeld current level/duration limits are reached.

(ii) W
This bifurcation has been called classic voltage instability and is due to insufﬁcient
reactive supply in the EHV system, an example of which is the blackout that
occurred in the French system [37]. This class of stability has been analyzed with the
use of a sensitivity matrix between reactive generation and voltage at generator
internal buses 5965' The matrix SQGE is diagonally dominant for inductive networks

(networks that absorb reactive power), and not diagonally dominant for capacitive

41

network (network with long high voltage transmission lines). Classic voltage
instability has been described as occurring when generators reach their ﬁeld current
limits and the ﬁeld current limit controllers disable the exciters. An important
implication to this result is that studying this saddle node bifurcation must include

the ﬁeld current limit controllers action in the model.

2.4.2.3 Classes of Dynamic Bifurcation in a Type 2 Model

Three classes of dynamic bifurcation may exist and have been observed in a Type 2

power system model

(i)

(ii)

IIE' "lfﬂlf°l ':
Hopf bifurcation in generator inertial, ﬂux decay and exciter dynamics and is
associated with low frequency inter-area oscillations in a power system. This mode
of oscillation is usually referred to as a swing mode and appears as stable limit
cycles (supercritical) [15].

Wanna:

Hopf bifurcation in the ﬂux decay/exciter control dynamics. This mode of oscillation
is usually referred to as voltage swing modes and appear as unstable limit cycles

(subcritical) in literature [15].

(iii) Ii E' . °lfﬂ l l . :

Hopf bifurcation in generator inertial/ﬂux decay dynamics and occurs only when the
exciter in completely disabled. This mode of oscillation is called the voltage collapse

mode and has been shown to occur in the single machine inﬁnite bus model [15].

III

POWER SYSTEM STABILITY ASSESSMENT
A TAXONOMICAL APPROACH

3.1 Introduction

Power system stability can be broadly deﬁned as that property of a power system
which enables it to remain in a state of operating equilibrium under normal operating
conditions. and to regain, in a ﬁnite time, an acceptable state of equilibrium after being
subject to a disturbance [29]. Stability analysis of post contingency equilibrium point or
limit cycle using dynamical system theory for smooth nonlinear systems, although
necessary, is not generally sufﬁcient for assessing stability of equilibria or limit cycles of
the power system dynamical model. Stability analysis for smooth nonlinear systems must
also be performed for points in parameter space where parameter change is continuous.
Assuring power system security based on describing local stability of an equilibrium,
determining local connected regions in state and parameter space where local stability is
retained (feasibility region) and region of stability for trajectories when parameters are
ﬁxed, has been investigated in [26]. This work is being extended [27] to include effects of
hard limits and equipment outages which discontinuously modify the model order, the

structure, equilibria and dynamic behaviors

42

43

A realistic power system model is characterized by nonsmooth elements, the most
important of which are the hard limits encountered during system operation. In power
system terminology, “saturation” is usually reserved for denoting smooth effects
associated with saturation phenomena such as ﬁeld saturation; and “hard limits” are
usually associated with nonsmooth effects such as actuation limits, tap changer limits,
relay effects, etc. From a conceptual point of view, there exists three types of hard limits:
(a) windup limits or actuation limits [27]; (b) nonwindup limits or state limits [27]; and (c)
relay type limits or switching limits [27]. Depending on how a system with saturation
limits comes back to the limit, windup and non-windup limits are distinguished. Whereas
windup limits can come off the constraint at any time, this is not possible for nonwindup
limits before the windup variable has been released completely. Hard limits are generally
associated with actions of operator control; actions of protective devices such as relays or
ﬁeld current limit controllers; actions of under load tap changers and actions of switchable
shunt capacitors.

An example that qualitatively portrays the effect of consecutive discontinuities on a
state trajectory is shown in Figure 3.1 as well as Figure 3.2 [40] and Figure 3.3 [41] for
simulations on actual detailed power system models. The standard bifurcation theory for
smooth systems [26], being based on the functions f and g in Equation (2.1) being smooth,
is not applicable to assess the stability of the system experiencing such discontinuous
transitions. Since voltage collapse time scenarios are characterized by all of the
discontinuities described above, it is necessary to extend the stability analysis for models
with discontinuities (hard limits and contingencies).

The stability analysis of a power system is complicated by the lack of smoothness

44

needed to characterize not only the bifurcations existing on smooth systems, but also
because new bifurcations that are directly connected with the hard limits must be
described.'Such bifurcations include hard limit induced static bifurcations and hard limit
induced dynamic bifurcations [27,42-43]. Although the purpose of this thesis is not to
develop an extension of stability theory for hard limits, a brief description of the current
development on this subject is necessary for consistency and a better understanding of the
theoretical framework under which the bifurcation subsystem method is being developed.
In this chapter, a rather brief description of the recent development of the taxonomy of
the large differential algebraic power system model is given. Zaborsky’s taxonomical
work of 1991 [26] which summarizes a method for applying dynamical system theory to a
smooth power system model is reviewed in Section 3.2 and a discussion of the status of
the theoretical development on the extensions of this taxonomy to include models with
hard limits discontinuities [27] is brieﬂy reviewed in Section 3.3. A parametric episodal
approach to security assessment is then proposed that
a. analyzes the stability region, feasibility region and viability region for the
continuous analytical model in intervals (if, rg+ 1) between instants t,- and ti“ where
discontinuities brought on by equipment outages or hard limit induced model
changes occur; and uses this analysis to decide stability, feasibility and viability of
the entire trajectory;
b. enumerates all equipment outages and operating changes for which the trajectories
and attracting equilibria and limit cycles of the system are expected to be stable and

viable and then establishes for which contingencies and operating changes the

trajectories are indeed unstable or nonviable. Viability for equilibria, limit cycles and

45

transients is deﬁned to be those equilibria, limit cycles and transients which (i) do
not cause equipment damage; (ii) do not cause relaying actions for protecting
equipment of the system; and (iii) have voltage, current, frequency and power

variations that are within acceptable design and operating ranges.

)’(1i+l')

 

r I
‘i ti+1 (time)

 

Figure 3.1 2-D example portraying the effects of the ith
discontinuous control action. (yi) is the ith quasi equilibria

3.2 A Taxonomy of Stability Assessment in Power System

A taxonomy theory refers to a general and structural theoretical analysis of the
dynamic behavior of very large nonlinear systems. Assessing local and transient stability
[App B] of a power system as well as maintaining its security for parameter changes in an
operating range is certainly a taxonomical task. Security has been deﬁned in Zaborszky
[26] in parameter and state space as operating conditions which belong to the feasibility

region, stability region (region of attraction) and viability region. The state space includes

46

both the dynamic state vector x and algebraic state vector y, whereas the parameter space

includes structural parameters such as transmission network capacitances and topology,

and operational parameters that could include

a. the level and variation of real and reactive load as a function of voltage and
frequency at each bus;

b. the real generation at every generator and its change with frequency, load level, and
area control error as well as its upper and lower limits;

c. the voltage setpoints on generators exciters, or voltage setpoint upper and lower
voltage limits on switchable shunt capacitors and the upper limit on their
susceptance, and voltage limits and tap setting limits on under load tap changing
transformers;

d. the transfer or wheeling actions taken.

There are numerous parameters above and beyond those itemized above but these
are certainly important parameters in deciding when voltage stability is retained or lost
[11]. Assessing stability of the large power system for a speciﬁc equilibria or limit cycle
requires the assessment of four regions: region of stability, feasibility region, viability
region and security region, which deﬁnitions are based on Zaborszky [26], the
Zaborszky’s discussion [47] of a paper by Schlueter and Benkilani [25], and their reply
[48]. The deﬁnitions are not new but generalize the deﬁnitions found in [26] as suggested
by the discussion and reply.

3.2.1

 

The region in state space where every trajectory originating inside, will ﬁnally con-

verge to the speciﬁc attractor of a particular type (equilibrium point or closed orbit) for a

47

speciﬁc set of operating parameters p = p0. The region of attraction in state space is dif-
ferent for each speciﬁc point pa in parameter space that belongs to a feasibility region of

that speciﬁc attractor of a particular type.

322 E .1]. E . [5 'E E .H. 1..: I

The region in parameter and state space, within which a speciﬁc attractor of a partic-
ular type (operating equilibrium point or stable closed orbit) can be shifted smoothly from
any point in the region to any other point by continuous parameter change while retaining
local stability. Structuralstability holds [34,50] within the feasibility region for each spe-
ciﬁc attractor of a particular type.

323 W

The region in parameter and state space belonging to a feasibility region of a speciﬁc
attractor of a particular type where the values of the state variables, i.e. voltages, current,
frequency, active or reactive power for the speciﬁc attractor of a particular type remain
within prescribed limits. The bounds can be based on physical device limitations on
parameters or states; safe operating limits on parameters or states; hard limits based on
device limitations or relay actions for protection of the device or system.

32.4 W

A trajectory that is stable and belongs to a region of attraction of a speciﬁc simple
attractor of a particular type that belongs to the feasibility region for that speciﬁc attractor
of a particular type is viable if the current, voltage, frequency or other variable trajectory
does not cause equipment damage, does not exceed design limits, and does not trigger

relay actions that protect equipment or system operation.

48

3.3 Development of a Taxonomy for Power System

Models with Hard Limits

The taxonomy developed in [26,27] is certainly the ﬁrst rigorous effort to describe

stability in a differential-algebraic power system model, and yet the taxonomy is still

under development. One major area of development is the inclusion of hard limit-induced

discontinuities, which Occur during operation when

1.

the generator ﬁeld current reaches its magnitude and duration limits so the ﬁeld cur-
rent limit controller either causes a relay to disable the generator exciter or reduces
the exciter voltage setpoint and thus ﬁeld current down to a sustainable level that will
not produce thermal damage to the generator rotor;

the controlled bus voltage remains outside the' prescribed upper or lower voltage lim-
its, and under load tap changers discontinuously and abruptly change the turns ratio
on a transformer to bring the voltage within limits. The action is possible as long as
the tap settings remain within tap position limits. If under load tap changers reach
upper or lower tap position limits, a second hard limit induced discontinuity is
encountered;

the voltage at the controlled bus exceeds lower or upper voltage limits, a switchable
shunt capacitor switches one or more banks of capacitors in or out to bring the con-

trolled voltage within limits.

Voltage collapse in the distribution system can be prevented by actions of under load

tap changers and switchable shunt capacitors as long as the under load tap changers

remain within the tap setting limits and switchable shunt capacitors remain within the

capacitive susceptance limits. However when a sufﬁcient number of generators reach ﬁeld

49

current limits, tap changers reach tap position limits. and switchable shunt capacitors

reach capacitive susceptance limits, voltage collapse will occur due to loss of voltage con-

trollability [11].

The discontinuities associated with generator ﬁeld current limit controllers, under
load tap changer controllers, and switchable shunt capacitor controllers can be modeled
using actuation (windup); state (nonwindup); and relay hard limits. Due to the discontinu-
ous nature of the actuation, state and relay hard limit models [27], the taxonomy described
for smooth dynamical systems can not be used to describe the feasibility region and stabil-
ity region for voltage collapse which develop when generator ﬁeld current limit control-
lers, under load tap changer controllers and switchable shunt capacitor controllers hit hard
limits in response to equipment outages or continuous operating changes.

Zaborszky [27] has started the long process of extending the taxonomy so that it can
include actuation, state and relay limits. The status of taxonomy development and its
structure is now described.

(1) The model used [27] is a differential equation model rather than the differential-
algebraic model used in the taxonomy without hard limits [26]. When hard limits are
ignored in the model, the boundary of the feasibility region has segments due to
Hopf bifurcation, saddle node bifurcation and singularity induced bifurcation,
whereas singularity induced bifurcation can not be part of a feasibility region bound-
ary in the taxonomy being developed for a differential equation model with hard lim-
its. In the taxonomy without hard limits [27], the boundary of the region of stability
allows

(i) stable manifolds of order 1 equilibria;

(2)

(3)

50

(ii) stable manifold of periodic orbits;

(iii) stable manifolds of pseudo saddles, semi saddles and bad anchors;

(iv) singular boundary pieces;

among others where the stable manifolds of pseudo saddles, semi saddles, bad sad-
dles and singular pieces could not belong to the stability boundary for a taxonomy
being developed for a differential equation model with hard limits [27];

the feasibility region for a differential equation model with actuation and state hard
limits have been derived but not for relay limits. The feasibility boundary for actua-
tion (state) limits is composed of segments due to saddle node bifurcation, Hopf
bifurcation, actuation (state) induced dynamic bifurcation, and actuation (state)
induced static bifurcation. The actuation induced static bifurcation and state induced
static bifurcation behaviors near the feasibility boundary are similar to saddle node
bifurcation except that the transversality and bifurcation conditions are somewhat
different. The actuation induced dynamic bifurcation behavior is similar to Hopf
bifurcation except that the transversality conditions at the bifurcation point are dif-
ferent;

When windup limits are modeled in the analysis, equilibria and their stability prop-
erties change and the system transient trajectories are characterized by nonsmooth-
ness (“kinks”). The unstable equilibria which anchor the stability boundary may
correspond to points where hard limits are encountered, hence results from these
equilibria differ from those of the smooth systems. Although the boundary of the
stability region is composed of stable manifolds of unstable equilibrium and periodic

orbits, these manifolds have comers or kinks (actuation) or have different dimension

if state or relay limit are reached. The analysis of the region of stability and charac-

terization of its boundary is only complete for actuation limits. This task is far more

challenging for state and relay hard limits since the state space for state and relay
limits consists of several smooth systems of varying dimensions.

The most important conclusion [27] is that there is indeed a stability region bound-
ary when hard limits are included in the model and where one or more hard limits are
reached either on the trajectories on the stability boundaries or on trajectories that are ini-
tiated at some point within the region of stability and converge to some equilibrium.
Another conclusion is that within the epoch (2;, (5+ 1 ) , the trajectory is smooth and the fea-
sibility and stability regions satisfy all the properties of a model where hard limits are
ignored. A ﬁnal conclusion is that the nonsmoothness of the stable manifolds means that
the construction of Lyapunov functions or energy functions is more difﬁcult and requires

different techniques than the typical u.e.p based methods used in models without hard lim-

its.

3.4 Stability Security Assessment for Model with Hard Limits

The desire in developing a taxonomy for studying stability in a power system model
is to be able to assess retention or loss of stability without time simulation of the system,
or quite possibly with almost no simulation of the contingency or operating change. The
stability region, feasibility region, and viability region and their boundaries are known if
such a taxonomy is complete. Even if the taxonomy were known, its success of avoiding
time simulations appears questionable, since one is unable to predict apriori the post con-

tingency or operating change steady state equilibrium point or limit cycle without time

52

simulation. This is because (1) the set of hard limits and the sequence through time at
which they are encountered is impossible to predict apriori and (2) knowing the set of hard
limits and their sequence over time is necessary to determines the ﬁnal post contingency
equilibrium point or limit cycle. The fact that the set and time sequence of hard limits
encountered not only determine the equilibria or limit cycle trajectories converge to but
whether they are stable or unstable, is documented in [41]. The reasons that it is impossi-
ble to predict the equilibria or limit cycles and their stability are (i) that the hard limit
encountered in a power system model are never enabled instantaneously when the limit is
encountered, (ii) the delay for each hard limit is different and can depend on the magni-
tude of the violation and duration of the hard limit, and (iii) the delay may be large com-
pared to the transient response of the dynamics, (iv) the sequence of future hard limit
control actions are dependent on previous ones, (v) the system is highly nonlinear so dif-
ferent sequence of hard limit control actions give different ultimate results.

An approach is now proposed that studies stability of the equilibrium or limit cycle
as well as transient stability epoch by epoch where an epoch is a period of time I: < t < t; + 1
(see Figure 3.1) where no hard limit transitions occur. This method would utilize the tax-
onomy to study stability of equilibria, limit cycles and trajectories within epochs, rather
than attempting to study stability of the entire trajectory as well as the steady state equilib-
rium or limit cycle without simulation. The justiﬁcation for this approach is that the taxon-
omy being developed with hard limits [27] indicates that the taxonomy developed for
models without hard limits [27] applies to time intervals (12.1}, 1) as well as for subinter-
vals in state space where no hard limit transition surfaces are encountered. An Epoch

Based Trajectory Stability Assessment Method is discussed in Section 3.4.1.

53

A second change in stability assessment methodology to be discussed is to consider
an episodal trajectory stability assessment rather than attempting to use a taxonomy which
might determine stability without time simulations for all trajectories resulting from (a)
disturbance initiated changes in initial state but no change in model; (b) continuous slow
operating changes in parameters of that model and (0) power system models with no delay
hard limit so that the set and sequence of hard limits encountered can be known. Utilities
currently utilize an episodal stability assessment rather than a taxonomy based method in
dynamic security assessment and contingency selection for fault contingencies [52]. The
motivation for this taxonomy is based upon the earlier work on fault contingencies. The
use of an episodal rather than a taxonomic method is due to the fact that most realistic
fault contingencies are followed by a number of discontinuities which could not be pre-
dicted to occur without time simulation. Thus taxonomy based methods and episodal
methods are competing in an application where taxonomy based methods held dominance
for years due to their promise in providing accurate stability assessment without the need
for time simulation. An episodal trajectory stability assessment is thus proposed for the
equipment outages and operating changes that cause voltage instability. Utilities currently
specify the contingencies and the range of operating condition changes they expect to sur-
vive without loss of voltage stability. If one can quickly assess stability of trajectories pro-
duced by each of the set of contingencies for the ranges of operating conditions speciﬁed
epoch by epoch to ﬁnd those that are stable and those that are unstable, then a contingency
selection or stability assessment procedure, in the language of power system engineering,
could to be produced. The set of trajectories that are viable and transient stable [App B] to

attracting sets (equilibria and limit cycles) that are feasible and viable for every epoch are

54

said to belong to a Bounded Trajectory Feasibility Set. A discussion of the contingency
selection and Bounded Stable Trajectory Feasibility Methodology is given in Section

3.4.2.

3.4.1 Epoch Based Trajectory Stability Assessment

The Epoch Based Trajectory Stability Assessment is proposed because each hard
limit induced model change may produce

(a) a model
xi = fi(xiryi»l*i) ‘:<’<‘i+r (3.1)
O = 810pr Hi)

where an equilibrium (mi) may or may not exist;

(b) a stable or unstable equilibrium point;

(c) a stable limit cycle encircling the unstable equilibrium point;

((1) a region of attraction for that stable equilibrium point or limit cycle that is sufﬁ-
ciently large to include (x(z;), y(r;)) .

All four questions (a-d) must and will be assessed for each time epoch (If, t'

i+ r) to
be able to diagnose when and why the trajectory does not converge to a ﬁnal attracting set
as time progresses. The Epoch Based Trajectory Stability Assessment is based on a trajec-
tory simulation method implemented by Van Cutsem in 1992 [28] that does not require
time simulation in (1321;, l) , but only computes the quasi equilibria (ivy!) for i = 1,...,N
since it has been found that

(a) the transition to the quasi equilibria (5:9,) is stable because the region of attraction is

generally large;

(b) that xi(t‘-’+ l) = ii and yi(t‘?+ 1) = 5:1. since the time T= lf+ r —ti+ between hard limit

55

model changes is large enough to cause the trajectory to reach steady state at each

hard limit model change;

(c) The lack of a stable equilibrium without existence of a stable limit cycle may be
observed as a diverging set of quasi equilibria.

If one has an unstable sequence of quasi equilibria (fl-,5") but a stable encircling limit
cycle around each quasi equilibria, this rapid simulation method [28] would not indicate
retention or loss of stability. Furthermore it has been found that the short term dynamics
and long term dynamics interact and thus such a rapid simulation method is not always
adequate [41]. The Epoch Based Stability Assessment Method utilizes the rapid simula-
tion method to evaluate quasi equilibria (ivy!) but takes into account not only stability and
feasibility of quasi equilibrium (in?) but also stability of trajectory during each epoch.

In order to illustrate the argument that the rapid simulation method is needed to sim—
ulate the voltage stability trajectories with a large number of discontinuities but is inade-
quate to assess the interactions of short and long term dynamics that cause instability
without Epoch Based Trajectory Stability Assessment, we bring out two examples from
recent literature. Figure 3.2 shows the time simulation of the active power load of a large
utility system after a severe fault where the over-excitation limiter controller (OEL) dis-
abled the exciter on one machine but all other machine protection devices are enabled
[41]. The disturbance was an EHV line fault and line trip which resulted in tripping a large
generator unit off-line. The model includes excitation systems, turbine/govemor systems,
PSSs, generator protection functions, AGC...etc. The transition dynamics which occur due
to the action of voltage protective devices (LTCs, switchable shunt caps.) clearly manifest

a nonsmooth behavior. A number of discontinuities occurred before the system was driven

56

to voltage collapse and thus a rapid time simulation method [28] is needed. This example
would not need to simulate the trajectories in epochs but only needs to determine quasi
equilibria provided by the Epoch Based Trajectory Stability Assessment because the quasi
equilibria capture the response of the system and no short term dynamic behavior is
observed. Feasibility and viability of equilibria could be utilized to assess why the system
experienced voltage collapse.

Figure 3.3 is intended to demonstrate interaction of short term and long term
dynamic behavior via simulation of the active power ﬂow between two regions following
tripping of a line and a generator in a Swedish test system [41]. We notice that due to the
severe fault, the dynamic behavior was initiated by large oscillations, which after the acti-
vation of the OEL and the tap changers reduced to smaller limit cycles. As the voltage
keeps dropping, the system was ﬁnally driven to voltage collapse. Three crucial phenom-
ena are demonstrated by this ﬁgure: (1) Viability of limit cycles is necessary in each
epoch; (2) quasi stable/unstable limit cycles rather than quasi equilibria may exist; and (3)
both long term and short term dynamics must be considered to assess stability and security
of the entire trajectory. Viability, feasibility and region of stability of trajectory segments
and the feasibility and viability of equilibria and limit cycles would be needed to assess
why voltage collapse occurred in this system.

In order to address the stability of the full trajectory (x(t), y(t)) as well as the interac-
tion of short term and long term dynamics, time simulation or taxonomy based stability
assessment within epochs I: < z < :;+ r is necessary. The Epoch Based Trajectory Stability
Assessment that avoids complete time simulation within 031;”) that utilizes a taxon-

omy stability assessment in each epoch to computes

57

(1) X(t;) = J‘c,._I and y(:;) = 5L1 by solving for (ii-l’yi—l) from

fi-1(xi-l'yi—1)= 0 32(8)
85-1(xi-1,)’1_1) = 0 3.2(b)

(2) y(t;‘) by assuming x(t;') = x(t;) = 31.1 using
g;(i.-_1.y(t.-*)) = 0 (3.3)
This assumption is based on the fact that whereas the algebraic state vector y(t) may
change instantly, this is not possible for the dynamic state vector x(t). Inﬁnite power is
needed to cause an instantaneous change in x( t). The Epoch Based Trajectory Stability
Assessment then checks
(i) if solutions (ii—1,534) and (xi_1,y(ti+)) exist;
(ii) if (2,; 1, 5L1) is stable and viable;
(iii) if (x(r,.+_,), y(z;'_ 1)) lies within the region of attraction of (ii-1’ 5L1) if step (ii) is
true;
(iv) establishes if there are attracting sets (closed orbits) surrounding the unstable equi-
librium and whether they are stable if step (ii) is not true;
(v) establishes whether these attracting sets are stable and viable if step (iv) is true;
(vi) establishes whether (x(t;‘_l), y((‘-+_1)) lies in the region of attraction of the attracting
set if steps (iv) and (v) are true.

Stability and viability are assumed to be retained if the answer to step (i) and either
steps (ii) and (iii) or steps (vi), (v), and (vi) are true. Such a procedure would eliminate the
need to perform time simulation and would only require solving y(t;') using equation
(3.3) and checking (i-vi)

A slight expansion of the above (i-v) rules for testing trajectories via trajectory seg-

58

ments in epochs ($117+ 1) for the diagnostics of knowing feasibility, viability and stabil-

ity is given below

(a)

(b)

(C)

(d)

(8)

Every post contingency steady state operating point or stable limit cycle, and every
quasi equilibrium point (ii, 5".) or limit cycle which occurs at the PM discontinuous
control action, must exist, be stable and viable and thus reside in the (non-empty)
intersection of its viability region and its feasibility region. (feasibility, viability)
Assuming (x(t‘-+), y(t;)) accurately reﬂects the unmodeled transition from
(x(t;), y(t;)) to (my), y(t;‘)) for each i, the stability for the transition from
(x(t;), y(t;)) to the stable steady state or quasi equilibrium point (it-,1.) or its stable
encircling limit cycle can be decided by checking whether (x(ti+), y(r;)) lies within
its region of attraction for each i = I , 2 N.

Shrinking unstable limit cycles around the stable steady state or quasi equilibrium
point constrain the size of its region of attraction. A lower limit on the size of the
unstable limit cycles should be prescribed. (viability of unstable limit cycles)

Every trajectory initiating in the region of attraction of each steady state or quasi sta-
ble equilibria does not stall or trip motors, does not cause equipment damage, relay
action to protect equipment , and operator emergency action to trip generators off the
system. (viability of transient trajectory)

Stable limit cycles around unstable steady state or quasi equilibrium points that are
viable must also satisfy the additional condition to be small enough to prevent dam-
age of equipment. An upper limit on the size of the limit cycles is prescribed so that
(e) equipment damage could not occur or (b) so that the region of attraction could

not disappear (as the limit cycles reach the boundary of the region of attraction).

59

(viability of stable limit cycles)

Performing steps (ii - vi) or (a-e) is computationally difﬁcult if not impossible at
present due to the very large dimension of models. Research is underway to make possible
rapid evaluation of steps (ii - vi). This thesis on development of a bifurcation subsystem
method would allow one to quickly assess whether the equilibrium is stable or unstable
and whether the attracting set (.5 one ext-as) is stable or not with very little computation.
The bifurcation subsystem method identiﬁes the small dimension subsystem that experi-
ences the same bifurcation as the entire system and for the same value of the bifurcation
parameter. The natural sparse structure, weak coupling and geometric structure of the
model cause bifurcation subsystems to exist. Furthermore, the bifurcation subsystem
method would indicate not only which parameters of a linearized model cause a speciﬁc
bifurcation to occur, but also what operating changes cause that bifurcation to occur.

The bifurcation subsystem method may also identify the subsystem dynamics that
determine the stable manifold of a particular unstable equilibrium that belongs to the
boundary of the region of attraction. Although investigation of how the dynamics of the
bifurcation subsystem deﬁnes a stable manifold of an unstable equilibrium, is beyond the
scope of this thesis, it is an important subject for the future.

Bifurcation subsystems appear to be omnipresent in a power system model. The
bifurcation subsystem can not generally be identiﬁed using eigenvectors or participation
factors because the eigenvectors indicate the states affected by the bifurcation eigenvalue
and not the subsystem that speciﬁcally experiences the same bifurcation as that experi-
enced in the full system. The bifurcation subsystem of a differential-algebraic model may

be differential, algebraic, or differential-algebraic depending on the kind and class of

6(1)

bifurcation experienced. Each bifurcation subsystem needs to be checked for structural
stability and retention of stability at each quasi equilibrium (it-.53) to assume stability dur-
ing each epoch. If such assessments are rapid, the epoch based trajectory stability assess-
ment is a natural procedure for performing a contingency selection and stability

assessment that does not require time simulation in each epoch.

60000
$0000.
cocoa
soooo

20000

Figure 3.2

61

time (seconds)

System active power load (MW) of a system [40] after a severe

disturbance: EHV line Fault & line trip which resulted in a large generator oﬁ-line

200..

100..

 

 

20

I I l
40. 60 80 lm

time (seconds)

Figure 3.3 Active power ﬂow between Central and Southwest regions of the
Swedish test system [41] after the tripping ofa line & generator in the North

62

3.4.2 Bounded Stable Trajectory Feasibility Method

A Bounded Stable Trajectory Feasibility Method is proposed because the
trajectories initiated by contingencies or operating changes which experience voltage
collapse or other types of instability are marked by a number of discontinuities such as the
contingency itself or a hard limit-induced discontinuity such as described in Section 3.3.
Since the number of such discontinuous control actions is typically large and quite
different, an episodal classiﬁcation based on trajectory segments and thus trajectories and
their attracting sets being stable appeared to be logical and consistent with electric utility
practice. Contingency selection methods have been widely used to
a. determine contingencies where the currents on every branch (element) of the
Hansnussion network lie within thermal limits and those contingencies where the
currents on one or more branches exceed thermal limits.

b. determine contingencies where voltages at every bus (node) lie within a secure range
( 0.95puSViS 1.05pu) and those trajectories where one or more bus voltages lie

outside the range;

.6

determine contingencies where voltage collapse does not occur in any subregion and
determine the contingencies that cause voltage collapse in some identiﬁed subregion.
Each of these subregions can be proven to be a bifurcation subsystem where the
saddle node bifurcation makes it impossible to obtain solutions or may cause
multiple solutions to exist.

All of these contingency selection and security assessment procedures are based on
load ﬂow solution which attempts to ﬁnd ilgn~(ii, 5".) where N is the number of

discontinuities experienced along the trajectory. A Bounded Trajectory Feasibility Method

63

appears far more reasonable because it uses the Epoch Based Trajectory Stability
Assessment that actually simulates

(i) the actual sequence of quasi equilibrium points (3,, 53-) that would occur after

each discontinuity;

(ii) the transitions from ii = x(t;); )7, = y(t;) to x(t;') = x(r;); y(t;') = y(tg);
also evaluates feasibility and viability of each equilibrium and limit cycle, and assess
stability of trajectories (x(t), y(t)) over epochs (131‘; l) . The Bounded Trajectory
Feasibility Method would utilize an Epoch Based Trajectory Stability Assessment rather
that a load ﬂow because the load ﬂow often does not calculate a steady state equilibrium
ileN(ri, 5".) that actually occurs since it does not determine the correct set or sequence of
discontinuities that occur, and thus the correct equilibrium as explained earlier. Therefore
the conclusions concerning its stability, viability and whether the initial conditions after a
contingency lie within its region of stability are irrelevant Even if load ﬂow calculated the
correct steady state equilibrium [lgnNup ii) the hard limit based taxonomy would be
difﬁcult to apply to assess stability and viability of trajectories within each epoch.

The set of equipment outage and operating condition combinations evaluated by the
Bounded Trajectory Feasibility Method is clearly speciﬁed by a utility’s planning
criterion. For each such combination the Bounded Trajectory Feasibility Method would

determine stability and viability based on

(a) existence and uniqueness of a solution or equilibrium point (39.5!) for each trajectory

1

segment;
(b) stability of the equilibrium point or stable closed orbit for each trajectory segment;

(c) asymptotic stability of the trajectory segment towards a stable closed orbit or

equilibrium for the epoch (tf, I; + r ) ;

(d) viability of the equilibrium, stable closed orbit, and trajectory segment transient for
each trajectory segment;

(e) classiﬁcation of the trajectory as being or not being stable, and bounded trajectory
feasible based on (a-d) for each segment.

The Bounded Trajectory Feasibility Method would then attempts to assess why the
bifurcation subsystem was affected by the equipment outage and operating change
combination, why it was unstable or nonviable, and what could be done to prevent the
problem, based on information that can be determined once the bifurcation. subsystem is

known. This diagnostic step of the Bounded Trajectory Feasibility Method is similar to the

diagnostic that can be performed for the voltage stability security assessment procedure in

[11].

IV

Bifurcations Subsystems in a Power System
Differential-Algebraic Model

4.0 Introduction

The focus of this chapter is to initiate investigation on the bifurcation phenomena in
a differential-algebraic power system model for the purpose of identifying the various
bifurcation subsystems for each kind of bifurcation (saddle node, Hopf and load ﬂow
bifurcation) observed on a Type 2 power system model. The bifurcation subsystem
method will attempt to identify subsystems of very small dimension, called bifurcation
subsystems, that are associated with each kind and class of bifurcation.

Diagnostic classiﬁcation of power system instability problems that is based on
bifurcation kinds and classes, as well as model types used for bifurcation studies has been
presented in Chapter 2. A Type 2 power system model was selected for further study in
this thesis. The kinds of bifurcations that have been observed on this model include saddle
node, Hopf, load ﬂow, and each exhibits a different qualitative change in dynamical or
algebraic network behavior encountered at the point of bifurcation. Classes of bifurcations
are associated with the speciﬁc subsystem dynamics that experience a particular kind of

bifurcation for a given model type. Possible bifurcation subsystems of a Type 2 power

65

66

system model are

(a) the power system differential equations,

(b) the power balance equations at load buses,

(c) a generator’s inertial, ﬂux and exciter dynamics,

(d) a generator’s electrical dynamics,

(e) the control systems dynamics ( PSS, Govemor..) on a generator,
(f) a combination of the above

One known class of Hopf bifurcation occurs in generator ﬂux decay and exciter

dynamics and another occurs in generator ﬂux decay and inertial dynamics. Two other

classes of saddle node bifurcation are also known to exist; one in generator flux decay

dynamics and the other in generator inertial dynamics. Similiar classes may be found for

other kinds of bifurcations. Since a power system trajectory after a disturbance is generally

characterized by a sequence of hard limit induced discontinuities, during each epoch

(23.15”) , the bifurcation subsystem method will be applied to a differential-algebraic

Type 2 power system model to

(i)

(ii)

determine the subsystems that experience the same bifurcation as that observed in
the full power system model during each time epoch (93:5,, 1) . These bifurcations
capture the effects of the previous discontinuous or continuous parameter changes
that had occurred in the past time epochs (13‘, (5+ 1) for k = 1.....i-1. The subsystem
that produces the bifurcation in the subsystem and in the full system may at times be
larger than the subsystem experiencing the bifurcation of the full system.

ﬁnd the cause and the remedial action for preventing the bifurcation without actually

ﬁnding the center manifold dynamics. The subsystem containing all the causes

67

including model parameters, controls, and model discontinuities may extend outside
the subsystem that experiences or produces the bifurcation. Even though the research
ultimate objective is to ﬁnd the subsystem experiencing, producing, and
experiencing bifurcation, the present work is solely aimed at ﬁnding the subsystem
experiencing bifurcation as the full system experiences bifurcation. The diagnostics
on the bifurcation subsystem may be more easily performed because retaining how
speciﬁc continuous model parameter changes, how continuous operating, or
scheduling changes affect the center manifold dynamics is not easily retained on
very large power system models. The ability to analyze how the center manifold
change along the whole trajectory when discontinuities are encountered has been
investigated by Zaborszky [27] and is not an easy task.

The traditional methods for identifying subsystems are generally based on apriori
knowledge of (a) the system or its control design, (b) time scale‘information on real
eigenvalues or imaginary components of complex eigenvalues, (c) eigenvectors and
participation factor information from eigenvectors, or (d) decoupling within the state
space model. While such methods are often useful in model reduction that preserves time
or modal behavior at one operating condition or developing and redesigning particular
excitation systems, system stabilizers or FACTS controls, they are not always effective in
identifying the smallest subsystem or subsystems that manifests the same bifurcation in
the full system. These traditional methods (a-d) for determining subsystems are not
effective because obtaining a bifurcation subsystem model which experiences the
bifurcation in the full model has a different objective than preserving modal or coherent

behavior, designing controls or developing operation schedules for a system. The objective

68

of a bifurcation subsystem is to capture the nonlinear behavior that produces and causes a
bifurcation in a speciﬁc eigenvalue or a particular model or operating condition changes.
Time scales are often not effective in deﬁning a bifurcation subsystems for a speciﬁc
bifurcating eigenvalue in a power system model, because the frequencies of all inter-area
oscillations lie in a narrow band and because the frequencies of all local generator
oscillations also lie in a narrow band. A singular perturbation method would preserve all
the modes in a band and not just the bifurcating modes for a particular bifurcation
parameter. Subsystems used for control design must be satisfactory nonlinear model
representations of the full system and the associated linearized model must be a
satisfactory representation for several eigenvalues of the speciﬁc device or process being
controlled and for a range of operating conditions. Despite the fact that eigenvector
methods indicate the system state variables most severely affected by each Hopf
bifurcation or the controls that are controlling that mode, this information does not always
describe the location or cause of the bifurcation as noted in the power system example
presented in Chapter 1. In that case, the modal information (participation factor) showed a
rather large region near the city of Vancouver was the most severely impacted, but did not
capture the fact that the disabling of voltage control on generators on Vancouver Island
could actually cause the voltage collapse. Decoupling of subsystem dynamics by
inspection of the structure of the system Jacobian matrix could also be used for model

reduction and for identiﬁcation of system states with high participation in a particular

bifurcation.

69

4.0.1 Bifurcation Subsystems for Epochs

The great majority of the literature on bifurcation in power systems assumes that the
differential-algebraic power system model is continuous and differentiable. The
discussion in Chapter 3 shows that virtually all bifurcations occur after (1) generators
reach ﬁeld current limits, (2) switchable shunt capacitors act, (3) tap changers reach upper
or lower tap position limits, and (4) switchable shunt capacitors reach upper susceptance
limits. All of these actions are discontinuous and violate the smoothness assumptions
needed to apply bifurcation theory. Thus an extended bifurcation theory is being
developed in [27] to properly describe the bifurcation that occurs in power system models
that experience actuation, state, and relay hard limits [27]. Since inclusion of these
discontinuities is absolutely essential for system stability analysis of power systems, the
framework for the bifurcation subsystem method should accommodate and include the
effects of such discontinuities. If the effects of discontinuities were neglected, then the
bifurcation subsystem method would have limited and possibly little value, since there are
relatively few if any bifurcations that occur in an actual power system without one and
likely several discontinuities.

Therefore, in this chapter we adopt the Epoch Based Trajectory Stability Assessment
method, where system stability analysis is performed epoch by epoch where an epoch is a
period of time (1:45”) free of hard limit discontinuous transitions. In this method
presented in Chapter 3, stability analysis for smooth systems may be performed within
epochs (1;, 117+ l) for i =1,... N, where 6+ 1 — z: is very long compared to the time constants
of the linearized dynamics, rather than attempting to study stability of the entire trajectory.

In each time epoch 031;”), bifurcation subsystems need to be determined and checked

70

for retention and structural stability at each quasi equilibrium (Sq-.5"). Since the progress of
discontinuities during system operation is accompanied by loss of control and loss of
subsystem coupling, bifurcations may be of different kinds, and bifurcation subsystems
may be different for each epoch. Consequently, the bifurcation subsystem method makes
the Epoch Based Trajectory Stability Assessment Method a rapid and natural procedure

for stability assessment and contingency selection in a power system.

4.0.2 Existence of Bifurcation Subsystems

Stability analysis and testing for bifurcations in a power system had always put
special focus on its subsystem structure and on ﬁnding the critical subsystem dynamics or
subsystems which exhibit the bifurcation. For example, it is quite clear that the generator
ﬂux decay dynamics and the generator excitation system are a subsystem based on the fact
that the exciter controls both the ﬂux and the voltage produced by a generator, and that a
subcritical Hopf bifurcation is known to occur in these dynamics [25]. It is also known
that a supercritical Hopf bifurcation can be observed in the generator mechanical, ﬂux
decay, and excitation system dynamics [25]. Since two Hopf bifurcations affect the exciter
and ﬂux decay dynamics, it might be difﬁcult to deﬁne the proper subsystem for the
subcritical and the supercritical Hopf bifurcations. These classes of bifurcation motivate
the investigation of whether a subsystem composed of those dynamics produces and
experiences the same bifurcation as the full system, what produces bifurcation in both and
causes of the speciﬁc bifurcation, thus the name bifurcation subsystem.

A nonlinear transformation can reduce a very complex high dimensional nonlinear

system into a simple subsystem that describes the system behavior along the center

71

manifold. If one is able to retain how the parameters and equilibrium state change with the
changes in operating conditions in the system, one could fully diagnose causes of the
bifurcation and how it might be effectively prevented through direct control actions or
through better adjustment or scheduling of the operating conditions. Although the
subsystem that describes behavior at or near the center manifold is typically of very low
dimension, the dimension of the participation factor subvector (obtained from normalized
right and left eigenvectors of a speciﬁc mode) with large (> 0.1) elements can be quite
large. From the example shown in Chapter 1, the right and left eigenvectors do not
necessarily indicate the cause or the geographical location of the bifurcation, nor the states
in the model where the bifurcation has an effect. In this system, loss of excitation voltage
control on generators in one generating station on Vancouver Island causes a voltage
collapse on Vancouver Island that uncontrollably spreads and brings about voltage
collapse in the entire BC Hydro system. The Vancouver Island subsystem experiencing
bifurcation may not be even retained based on the magnitude of the participation factor
elements of the bifurcating eigenvalue before this loss of excitation control occurs which
is shown in Figure 1.1. Once the loss of excitation occurs, there may be no load flow
solution that can be used to calculate participation factor information, or eigenvectors.

This result indicates that when discontinuities in the model exist and cause the
occurrence of bifurcation [11] in a power system, the magnitude of the eigenvectors and
participation factors may be insufﬁcient or may give misleading information on the
location, cause, or remedial action needed to prevent bifurcation.

The result in [11] indicate that there are small coherent bus groups, called voltage

control areas, with independent voltage collapse problems. Each of these coherent bus

72

groups in the electric transmission network is protected from experiencing a voltage

collapse bifurcation by retaining reactive supply and thus voltage control on one or more

of the generators in the unique subset of generators that protect that voltage control area

from voltage collapse. This result indicates there are indeed small subsystems of a power

system model, called bifurcation subsystems, that produce and cause the bifurcation that is

experienced in the entire power system.

(1)

(2)

(3)

The reasons that bifurcation subsystems exist on a power system model are
Bifurcations in a power system algebraic network model are typically brought on by
a series of hard limit or equipment outage induced discontinuities. While hard limit
actions are result of disablement of system (voltage) protective devices and can
result in loss of voltage control, equipment outages such as the trip of a tie line result
in decoupling or separation of two regions in the system. Therefore in each time
epoch (If, 5+ 1) , the effect of the previous discontinuities which occurred at times (k
for k=1,..,i-1 may fully be captured through the disappearance of certain dynamics,
isolation of some subset of dynamics, increase in matrix sparsity when dynamic
equations are substituted by algebraic equations,.. etc. The resulting subsystems
within each epoch (If, 1,; I) may experience different kinds of bifurcations.
Bifurcation subsystems with different kinds of bifurcations seem to be a natural and
essential way for tracking the subsystem experiencing, producing, and causing
bifurcation, as well as the actual cause for the ﬁnal system instability;

the extreme sparsity of the transmission network where every bus or node in a

10,000 bus network may only be connected to three to ﬁve other buses;

the weak coupling between real power and voltage magnitude as well as the weak

S!

260
in it
PM
deCO

lIdtnt

73

coupling of reactive power—voltage angle. This decoupling implies the existence of a
real power-voltage angle subsystem and a reactive power-voltage magnitude
subsystem. Both subsystems have the extreme sparsity discussed above;

(4) effective decoupling of coherent bus groups within the real power-voltage angle
subsystem and within the reactive power-voltage magnitude subsystem.

The decoupling of the two subsystems, the decoupling of coherent groups within
each subsystem, and the extreme sparsity make a power system vulnerable to loss of
control induced bifurcations;

The bifurcation subsystem model is just a truncated portion of the actual power
system model where the impact of parameter and operating changes simulated on a full

system model can be easily observed and analyzed on the subsystem model.

4.0.3 Introduction to the Bifurcation Subsystem Method

The bifurcation subsystem method utilizes the geometry associated with the various
submatrices of the differential-algebraic Jacobian matrix I and with the eigenvectors
associated with the bifurcating eigenvalue to establish conditions for existence of a
bifurcation subsystem. Conditions for existence of a bifurcation subsystem require
geometric decoupling that encompasses time scales, matrix decoupling and eigenanalysis
in identifying the smallest subsystem that not only experiences the bifurcation but also
produces and causes the same bifurcation observed in the full model. Geometric
decoupling will be shown to be the most effective as well as a generalized method for
identifying bifurcation subsystems.

A bifurcation subsystem can therefore be loosely deﬁned as a truncated portion of

the .11

\E \lf

bli‘dl

ll)

1"
.1

t...
:3”

SW:
[lime

)Iiﬂ/
ﬁler

at”
Osﬁc

74

the actual power system model that experiences and causes the same bifurcation in the full

system model. Special characterictics of a bifurcation subsystem come from the fact that a

bifurcation subsystem

(1) is a reduced model developed for a speciﬁc kind of bifurcation since the geometric
decoupling conditions for a bifurcation subsystem are only on the linearized model
and only on the subspace associated with the left and right eigenvectors of the
bifurcating eigenvalue as the bifurcation parameter p. approaches the bifurcation
value 14*. The bifurcation subsystem should not only experience the bifurcation of
the full system model, but may also be useful in capturing what produces and causes
the bifurcation in any epochs (1;,1;+1) at the quasi equilibrium (39,511.), but also in
tracking the model transitions which uncover the persisting dynamics that ﬁnally
produced the collapse.

(2) is not necessarily an equivalent model to the full differential-algebraic model in the
sense that it preserves all the characteristics of the full model, since it can not be
used to observe and investigate all other bifurcations of the full model.

(3) geometric decoupling can break down at the point of bifurcation so that the entire
system produces the bifurcation rather than just the bifurcation subsystem.

The fact that the geometric decoupling at a quasi equilibrium for an epoch (1331;+ l)
breaks down before the actual bifurcation occurs and does not always solely produce the
system bifurcation may discredit application of the bifurcation subsystem method for
these cases if one views the bifurcation subsystem in the strict sense of its deﬁnition.
However as an engineering tool, the bifurcation subsystem method still provides

diagnostic information on how much the geometric decoupling and how much the full

shit"

from

future
bli’dlt‘
him
since J
Also. i
bifurca

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pattern

4.0.4 1

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an equit
CXlSlCnCe
bifurtan‘g
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75

system behavior after the breakdown contribute to the bifurcation in the full system model.

It should be noted that geometric decoupling does not ever prevent the entire system
from experiencing the bifurcation experienced in the subsystem, but hopefully can in the
future help identify the subsystem that produces and causes the bifurcation. The fact that
bifurcation subsystems experience and could produce most bifurcations even though the
bifurcation affects all dynamic and algebraic states of the system should not be a surprise
since almost all system bifurcations are understood in terms of affecting some subsystem.
Also, it should be noted that one should not and can not assume all bifurcations develop in
bifurcation subsystems, because the bifurcation could be produced by the full system, in
no particular identiﬁable subsystem or subsystems, and in very strange unexplainable

patterns of state variables.

4.0.4 Chapter Objective and Outline

The three main objectives of this investigation are therefore to (a) develop tests for
identifying subsystem dynamics that produce and cause a speciﬁc bifurcation to occur in
an equivalent dynamic or equivalent algebraic system model; (b) utilize the test for
existence of bifurcation subsystems to develop methods that can help identify all potential
bifurcations that may exist in a particular model; and (c) introduce several application
examples of bifurcation subsystems in stability assessment of power systems. One such
example in this thesis is the justiﬁcation of the classical load ﬂow model as an algebraic
bifurcation subsystem for load ﬂow bifurcation of the differential-algebraic model.

These objectives are quite ambitious Since the study of bifurcations usually assumes

this knowledge while it is often unknown and is rather essential for excitation system

red:

km

\01

mt

die

dc

di

CL

ﬁt

Ct

76

redesign and FACTS control that may prevent the bifurcation from occurring. This
knowledge may also be useful in developing unit commitment, generation dispatch and
voltage setpoint schedules that could prevent each speciﬁc kind and class of bifurcation.

The development of this bifurcation subsystem method will be ﬁrst developed for a
general differential-algebraic model and then applied to a Type 2 power system model. A
Type 2 power system model is chosen rather than any other power system model type
because (1) the bulk of bifurcation studies in power systems has been based on this model
and (2) because the kinds and classes of bifurcations are best understood for this model.
The differential equations in this Type 2 power system model represent generator
mechanical and flux decay dynamics, generator excitation system dynamics, and governor
turbine system dynamics. The algebraic equations represent the transmission and
distribution networks as well as the load. The Type 2 power system model is described in
details in Chapter 2 and structurally as a differential-algebraic model in Section 4.1.

The ﬁrst step in this investigation is to assume that it is possible to obtain a
differential (dynamic) system model and/or an algebraic system ’ models, that are
equivalent to the full differential-algebraic power system model. Such assumption is
generally made because most bifurcations do not occur solely in the differential equations
or solely in the algebraic equations of the differential-algebraic model. Formulation of
equivalent differential and algebraic system models will be based on Schuur’s theorem as
will be shown in Section 4.3.1. The conditions for existence of an algebraic bifurcation
subsystem and a differential bifurcation subsystem which experience bifurcation in the
equivalent algebraic system model and the equivalent differential system model,

respectively, are then derived in Section 4.3.2.

77

The conditions for a differential bifurcation subsystem to exist are further reﬁned to
provide conditions for a subsystem of the differential equations to produce bifurcation in
the equivalent differential equation model. When the bifurcation is experienced solely in a
subsystem of the differential equations, then a bifurcation sub-subsystem is said to exist
Conditions for a differential sub-subsystem are given in Section 4.4.1.

The conditions for an algebraic bifurcation subsystem to exist are further reﬁned to
provide conditions for a subsystem of the algebraic equations to experience bifurcation in
the equivalent algebraic system model. When the bifurcation is experienced solely in a
subset of the algebraic equations, then an algebraic bifurcation sub-subsystem is said to
exist. Conditions for algebraic sub-subsystems to exist are given in Section 4.4.2.

The bifurcation produced in the full set of differential and algebraic equations can be
experienced solely in a subsystem composed of a subset of the differential equations and a
subset of the differential equations. When this occurs, a differential-algebraic sub-
subsystem is said to exist. Conditions for a differential-algebraic subsystem to exist are
given in section 4.4.3

Under normal conditions, a power system should never experience bifurcation solely
in the differential equations, if the generators, governor and exciation system are properly
designed. However, during voltage collapse, generator exciters cause the generator to
reach ﬁeld current limits in an effort to produce the reactive power needed by the network,
and theexciter is completely disabled on excitation systems built prior to 1970 [51]. This
can produce bifurcations that are solely in the generator dynamics. A condition for an
algebraic bifurcation subsystem is also derived in section 4.3 where bifurcation in the

equivalent algebraic system model is captured totally in the algebraic equations. Voltage

78

collapse is often studied using an algebraic model and thus its use is theoretically justiﬁed
based on the conditions derived for an algebraic bifurcation subsystem to exist. This
understanding motivates the development of Sections 4.3, 4.4 and 4.5.

Finally, an application example of bifurcation subsystems in stability assessment of
power systems, is given in Section 4.6. A classical load flow model is justiﬁed for the ﬁrst
time in terms of being both a bifurcation subsystem for saddle node bifurcation in the
differential-algebraic model or equivalent differential model and a bifurcation subsystem
for algebraic bifurcation in the algebraic equations of the network far from generator
terminal and internal buses. In Chapters 6 and 7, the concept of bifurcation subsystems
will be used to study several different kinds and classes of bifurcations in a Type 2 power

system model.

4.1 Differential-algebraic Power System Model

From power system stability assessment practice it is currently assumed that the
underlying cause of voltage collapse is typically a bifurcation associated with the
nonlinear power system dynamics, and can be invoked by a small change in system
parameters. First, recall the nonlinear power system differential-algebraic model presented
in Chapter 2, where the differential and algebraic nonlinear equations describe the
generator and control dynamics, the network and load, respectively. The form of the
nonlinear model in each time epoch (1‘1“,1;+ I) after the i’th discontinuity occurs in the
system, 1 gig N — 1 is given by:

Diﬁerential equations: r,- = f1(x1(’))s)’1(’))sﬂr(‘)) te (1;:1;+ l) 4.1(a)

Algebraic equations: 0 = gi(x,-(1), 311(1), 111(1)) 115 (1311:”) 4.1(b)

Mn

1}

IQ

l

where
)3-
81'

xi(t) :

Yr“)

11,-(1)

XxY

2 R

I R

N

n+m+ n - - '
p —> R . an n-vector of real analytic nonlinear functrons

+ . . .
" + m p —> Rm , an m-vector of real analytic nonlrnear functrons
e x cR" , a state vector of dynamic state variables, typically time dependent

values of generator and excitation system voltages, rotor phases and induction

motor variables for (E (1;',1;+ l) .

: e 1’ cRm, a state vector of instantaneous variables, typically time dependent

values of voltage and angle at generator terminal buses, high side transformer
buses and load buses for 1e (1f,1;+l) .

e P cRp , a vector of slow varying operating parameters for the time epoch
1e (1f,1;+1) . It can represent a scaling of any of the parameters discussed in
Chapter 3 as belonging to parameter space. The parameter “1“) is likely to scale
parameters like reactive power load or generation, generator exciter voltage set
points, or active generation or load. As Ill-(t) varies over the parameter space P,

there must be at least one solution to the above differential-algebraic system of

equations.

: state space

P : parameter space

Note (Notation):

The subscript ‘i’ used in the vectors xi(1) , yi(t), f i(x,-(1), yi(1), ui(1)),.. indicates that the

model in valid after the i’th discontinuity, during the epoch (131:+ l) . For example,

I

xi(t) = “‘1’ x12, ""x‘n) e R" is the n-dimentional state vector during (1;,1;+1) .

><|
'tt

euurt

A ﬁn:
ht 0b
series
The J
Uncut

tompt

there

and

[nthe

tjrj1
0CL‘Ur \
and thﬁ

SIand“

80

For an arbitrary ﬁxed parameter u‘. = H.» a steady state quasi equilibrium point

X .= (time), 53115)), is a point of quasi equilibrium deﬁned by locally solving the system of

l

equaﬁons

0 = firm.» £111.). 11.) (49-3)
0 = 342411.). Time). it.) (421’)

A linearized model of (4.1) about an isolated equilibrium point of interest (2., £11,) may
be obtained using Taylor’s theorem where every function is approximated by its Taylor
series expansion of the ﬁrst degree, in the neighborhood of the isolated equilibrium point.
The Jacobian matrix of (4.1) at an equilibrium point (Ivy—1,11,) may be obtained by

linearizing the differential and algebraic equations and then combining all the linearized

[A151] = Ji- [AL] 4.3(a)
O Ayi

J _ afi/axi afi/ayi = f‘x“ fly“
‘ egg/3% agi/ayi (:1" )7? He) gixi 3in

components, as shown below.

where

4.3(b)

and
Axi = rim-ft,-

Ayi = )2“) " )7,
In the rest of the thesis, sufﬁcient smoothness of the functions f i(xi(1)), yi(t)), ui(t)) and
gi(x,-(1)), yi(1)), lit-(0) is assumed to be preserved since no discontinuous system changes
occur within the time epoch (1;',1;+ l) . Therefore, for simplicity, we omit the subscript i

and the argument ‘t’ in the analysis within the epoch. That is, unless otherwise noted, g

stands for gi(xi(t)),y,-(1)), ui(1)), fx stands for fi(xi(t)), yi(1)), [rt-(1)) ,.. etc.

A linearized differential-algebraic power system model has been derived in [52] for

\lller

81

a general Type 2 power system model, where each synchronous machine and its control
systems are described. The control systems include (a) Excitation control with Load
(Line-Drop) Compensator, (b) Power System Stabilizer (P88) [52], and (c) Speed-

Govemor-Turbine System. The full linearized system model [52] is given by:

TXXAXX Axx AXE Axe 0 Axe Axv FAX;
TEEAXE AEX A55 0 Ass A59 AEV AXE
TGGAXG Aax 0 A06 0 0 0 AXG

 

 

 

 

 

 

= +BOAU0 (4.4)
TSSAX'S Asx 0 0 Ass 0 0 AXS
-APC APX 0 0 0 Ape APV A9

Where
AXX : states of mechanical and flux decay dynamics;
AEF : states of ﬂux decay dynamics
AXE : states of excitation system
AX G : states of speed-governing-turbine systems;
AX S : states of power system stabilizer;
A9 : angle variables at network buses;
AV : voltage variables at network buses;
APC : coefﬁcients of non-voltage-dependent active power load demand model;
AQC : coefﬁcients of non-voltage-dependent reactive power load demand model;
T x x : diagonal matrix composed of inertia constants of synchronous machines;
identity matrix, and time constants of flux decay dynamics;
TEE : time constants of the excitation systems dynamics;

Too : time constants of speed-governing-turbine systems;

82

T55 : time constants of power system stabilizers;

4.2 Conditions for Bifurcation Subsystems Experiencing

SN and LF Bifurcations

4.2.1 Equivalent System Models

Although testing for singularity of the full Jacobian matrix J in (4.3b) is a sufﬁcient
indication of a system bifurcation, this test is not sufﬁcient to tell (a) whether either saddle
node bifurcation (when gy is nonsingular), load ﬂow bifurcation (when fx is nonsingular),
or algebraic bifurcation ([gx gy ] is row dependent) occurred; (b) whether the bifurcation is
associated with strictly an equivalent dynamic system model or an equivalent algebraic
system model; or (c) what system dynamics were associated with the bifurcation. Schur’s
theorem [53] is used to establish conditions under which equivalent systems to (4.1)
comprised solely of differential equations or solely of algebraic equations can be
constructed with different eigenvalues than the differential equation model which better
reﬂect the effects of the eigenvalue qualitative changes of the full system model.

Existence of equivalent differential or algebraic system models to (4.1) depends on
the row dependence of the rows associated with the differential equations and/or the row
dependence of the rows associated with the algebraic equations in the linearized full
system (4.3). This will be clear from Schur’s theorem which reduces the computation of a
determinant of a matrix of order n to the computation of a determinant of a matrix of order

r<n [53].

83

Schur’s Theorem
If a square matrix A of order n is represented in partitioned form as
= All A 12
A21 A22 m,
where A, , is a square nonsingular matrix of order r<n, then

(a) IfAu is nonsingular, then Det(A) = De1(A22)-De1(A”-A12Ai!zA2|)

(b) IfA,, is nonsingular, then Det(A) = De1(A”) - De1(A22—A21Al‘llA12)

The reduced order matrices
—r
Arl = A[1‘1412'422'421
-r
ArZ = Azz‘AerrrArz

are commonly called system reduced matrices, system modiﬁed matrices or system
sensitivity matrices [38,51], since they preserve the singularity of A, a property that is very
desirable for system reduction techniques and absolutely essential in bifurcation analysis.
Indeed, intensive power system research has been performed using these low-order
matrices to investigate system sensitivities and proximity to voltage instability bifurcation
where A and .4,1 , A and 11,2, or A, .4,1 and 14,2 approach singularity together. Eigenvalue
analysis, singular value decomposition technique, and system sensitivity analysis to
various operating parameter variations are examples of power system stability analytical
methods that are based on formulating system submatrices in the form of A,l and 14,2 in a
classical load ﬂow model [38]. Letting A22 = AQV, A21 = Age, A12 = Apv , and A“ = Ape
submatrices of the load ﬂow Jacobian A, test results in [38] show that the minimum
singular value of A, A,2 , and A22, are identical down to nearly the point of bifurcation.

The changes in all the minimum singular values of all matrices occur at discontinuities

84

when generator buses become load buses and control of voltage is lost. This result
establishes that A22 is a bifurcation subsystem of the load ﬂow model with Jacobian A and
motivates development of a general theory for describing when a bifurcation subsystem
exists in the differential-algebraic model.

Since we are interested in reduced order equivalent system models that have the
same dynamic and algebraic properties as the full model, we apply Schur’s theorem to the
matrix I in (4.3b), where we can see that two possible equivalent matrices may be used to

obtain equivalent reduced order models, namely

-1
1,. = f,,-fygy s.

—r

Jy = gy—gxfx fy
Jx and Jy are indeed two equivalent system Jacobian matrices associated with an
equivalent dynamic model and an equivalent static/algebraic model respectively, as can be

seen below.

4.2.1.1 Equivalent Algebraic System Model (Exact Load Flow model)
If fx is nonsingular at steady state (me = 0 ), the differential equations in (4.2a) model may
be aggregated into the algebraic equations (4.2b) by setting x = 0 and solving for x( y)
from [(11, y, u) = 0 by the lrnplicit Function Theorem [33], resulting in a model where no
saddle node bifurcation in the differential equations occurs, i.e. the full system bifurcation
may be determined from the equivalent reduced order algebraic equivalent model given in
4.5(a), or locally from the linearized model about )7: (31, y, it) given in 4.5(b):

g(x(.v). MA) = 0 4.5(a)

-l
0 = Jy-Ay = [gy-gxf, fyl - Ay 4.5(b)

This result suggests using an algebraic model g(x( y), y, u) = 0, called an Exact Load Flow

85

model is obtained. This Exact Load Flow model g(x(y), y, u) = 0 is implemented as a load
ﬂow where generator and exciter algebraic equations from f(x, y, u) = 0 are included in
the model. The usual procedure of incorporating generator PV and PO buses is an
approximation of g(x(y), y, u) = O and is referred to as a classical load ﬂow model. If the
Exact Load Flow model g(x(y), y, u) = 0 is valid because there exists a unique solution
x( y) then test for bifurcation on the full model using (4. 1) is equivalent (if and only if) to

the test for bifurcation on the Exact Load Flow model g(x( y), y, u) = 0 using Jy

4.2.1.2 Equivalent Dynamic System Model

From 4.3(a) and 4.3(b), we can see that if gy is nonsingular, the algebraic equations may be
aggregated into the set of differential equations by obtaining y(x), resulting in an
equivalent dynamic model x = f(x, y(x), p). This reduced order model consists only of
differential equations, making this system suitable for assessing static and dynamic
bifurcations associated with the full system using 4.6(a), or locally from the linearized

model about X: (it, 5», u) given in 4.5(b): ~

2 = f (x, y(X). 11) 415(3)
Ax = Jx. Ax = [fx—fyg;lgx] - Ax 4.6(b)

Based on these two equivalent models, we can see that a qualitative change in the full
linearized system behavior, may be determined from the algebraic or the differential
equivalent models depending on the singularity of the matrices fx and gy . An equilibrium
point ft“: (x*, y*, 11*) of 4.3(a) can experience bifurcation at a bifurcation value of u , 11*
when J is singular and either :

(i) g), is nonsingular, and Jx = fx— f ygfgx is singular. This bifurcation in the equivalent

dynamic system model is called static bifurcation, which is generically a saddle node

86

bifurcation.

(ii) fx is nonsingular, and J y = g). - gxfI‘f, is singular. This bifurcation in the equivalent
algebraic system model called a load ﬂow bifurcation since it occurs in the Exact
Load Flow model g(.r( y), y, u) = 0.

(iii) Both fx and g), are singular. There is no known description of this case in dynamical
system literature on systems composed of differential and algebraic equations, such
as a power system.

There is no theory to establish that the dynamics experience, produce, or cause the
bifurcation in (i) or that the load ﬂow would experience, produce, or cause the bifurcation
in (ii), but convension has assumed such is the case. This result suggests that the ﬁrst step
in the investigation of bifurcation subsystems is generally to determine (if possible)
whether the bifurcation occurs in the equivalent differential or algebraic system model. It
may be possible that fx and gy are both nonsingular when I is singular and one may study

the bifurcation in both the equivalent differential mode]

x = f (x. y(X). ll)
and the equivalent algebraic model (Exact Load Flow model)

g(x(y).y.u) = 0
Our subsequent results will show that some of the kinds and classes of bifurcations being
investigated in this research can be studied in both the equivalent dynamic and algebraic
system models. Identiﬁcation of such differential-algebraic bifurcation subsystems will be
investigated in this thesis and will be shown to be quite useful in determining voltage
instability/collapse regions in a power system.
Now, a conceptual and formal deﬁnitions of bifurcation subsystems are given, using

the notation and deﬁnitions of systems (4.1) and (4.3).

87

4.2.2 Deﬁnitions of Bifurcation Subsystems

Deﬁnitions of Hopf and saddle node bifurcations with transversality and nondegeneracy
conditions are given in Appendix C. For saddle node bifurcation, Hypotheses (8N2) and
(8N3) given in [App C] are the transversality conditions which control the nondegeneracy
of the behavior with respect to the parameter u and the dominant effect of the quadratic
nonlinear term. For Hopf bifurcation, hypothesis (H2) is the transversality condition.
Although testing for SN and Hopf requires testing all hypothesis (SNl-3) and (HI-2)
respectively, in practical investigations, only (SNl) and (H1) are tested since transversality
conditions are generally satisﬁed [59]. Therefore, in the rest of the thesis and without loss
of generality, we assume that as the system eigenvalues cross the imaginary axis, the
transversality conditions are satisﬁed so that a Hopf or a saddle node bifurcation occurs.
An Exact Load Flow model is a model that describes an equilibrium situation
g(x(y), y, u) = 0 that exists when x = 0, and the dynamic model f(x, y, u) = O at
equilibrium has a unique solution x( y ) for each y in some set The bifurcations in the Exact

Load Flow model are called load ﬂow bifurcations and occur when Jy is singular.

(a) Conceptual Deﬁnition of a Bifurcation Subsystem

A bifurcation subsystem is any subsystem of the original power system model equations
that experiences the same saddle node (Hopf, load ﬂow bifurcation) that occurs in the full
model. Therefore, the dynamic behavior in the full model is associated with the dynamic
behavior in the subsystem and thus investigating the speciﬁc bifurcation occurring in the

full model may be greately simpliﬁed by investigating the lower order bifurcation

subsystem.

88

Formal deﬁnitions of bifurcation subsystems for saddle node, Hopf, load ﬂow
bifurcation, differential sub-subsystems, algebraic sub-subsystems, and differential-
algebraic bifurcation subsystems are now given. These deﬁnitions are based on the
dynamic relation of the physically based subsystem of the full model, as the system

approaches bifurcation. These deﬁnitions may be extended to other kinds of bifurcations.

(b) Formal Deﬁnition of a Bifurcation Subsystem

Consider the systems in (4.1) and (4.3) where we partition Ax, Ay, f, g, and the full

system Jacobian matrix I as follows:

 

 

 

 

 

 

 

 

 

 

x-l Pf1(x9 Yr l1;
i2 = f2(x, y, ’1) 1(a)
0 81(x.y.u)
_o‘ _g2(x, y, u)_
' 1
_ 1 at: II: .—
Ax} fxnﬂl) fx12(u) Ax:
. x ( x ( a: at:
W = f 2‘ u) f 22 u) _ Ax2 Kb)
0 * * 8,,1101) 8,1204) Art
- 0 - * * 11,2101) smut) is”;
f (u) = fxnw) f‘rzw) g (u) = 3,1101) 3,1201)
1“ Y
122101) fxntu) yum) 3,2201)

with equivalent differential model 4.6(a) when gy(tt) is nonsingular and with equivalent
algebraic model 4.5(a) when fx(|,t) is nonsingular, and assume that the full system
experiences a bifurcation at a parameter value at e ([11, 11.2) . Also assume that

n n
x=[x[.x{lT, xleR ‘, xzeRZ, y=ly[.yngT

"‘1 ”'2
yleR , yzeR , nl+n2=n, m1+m2=m,

89

frill) = af/axj
1!
and note that the “*” in the matrices indicate unspeciﬁed elements or submatrices. Then

(A) the subset of differential equations of I(a)

x = f(x(t). yuan) “(21)

A .
[ 11,] = f.(u)- [M‘] II(b)
sz sz

where the algebraic state vector y is not a function of t, but only function of the

with linearization

parameter p. , is called a bifurcation subsystem of 4.6(a) if the dynamic and/or static

behavior of 4.6(a) is related to the dynamic and/or static behavior of [I(a) in one of

the following ways:

(Al) If fxﬂl) in II(b) has an eigenvalue Mu) approaching Mu") = 0 with
eigenvector u(tt) e R" , then the equivalent differential Jacobian matrix
1,,(11) has an eigenvalue Mu) approaching 1101*) = 0 with eigenvector
u(u) e R". At 2*: 0101*), y(u*), 11*) , transversality conditions for SN must
hold [App C].

(A2) If fx(u) in II(b) has eigenvalues 11,201) approaching xl’zuﬂ) = iju) for
some m¢0,to e R with associated conjugate eigenvectors u(u)e C",
u*(u)e C", then the equivalent differential Jacobian matrix Jx(u) has
eigenvalues 711.2(11) approaching i1,2(u*) = :1:th) and with conjugate
eigenvectors 110.1) 6 C", u*(u) e C" and 13:0, (be R. At
2*: 0101*), y(u*), 11*) , transversality conditions for Hopf must hold [App
C]

(B) the subset of algebraic equations of I(a)

(C)

90

mm, y(u). u) = 0 [II(a)

0 =’ . A)” [Nb
[0] gym) [4Y2] ()

is called a bifurcation subsystem of 4.5(a) if the equilibrium behavior of (4.5) is

with linearization

related to the equilibrium behavior of (IIIa) in the following way:

If gym) in [II(b) has an eigenvalue Mu). approaching Mir“) = O with eigenvector
u(u) e R’" , then Jy(u) has an eigenvalue 1(a) approaching Mu") = O, with
eigenvector u(u) e R’"

the subset of differential equations of I(a)

x'. = f(x1(t).x2(u). y(u). u) IV(a)

with linearization

Ail = fxuur) . Axl IV(b)

where the algebraic state vector y and the dynamic state subvector x2 are not
function of t but only function of the parameter p, is called a bifurcation sub-
subsystem of 4.6(a) if (i) x = f(x(t), y(u),u) is a bifurcation subsystem of (4.5)
when gym) is nonsingular, (ii) 322‘“) is nonsingular, and (iii) the dynamic and/or
static behavior of 4.6(a) is related to the dynamic and/or static behavior of IV(a) in
one of the following ways: -
(Cl) If fxuur) in IV(b) has an eigenvalue Mu) approaching 7t(u*) = 0 with
eigenvector u(tt)e Rnl , then the equivalent differential Jacobian matrix
Jxxﬂl) given by
Juan = fxnut)-f,.12(u)f;‘22(u)fx21(u)

has an eigenvalue 1(a) approaching 1101*) = 0 with eigenvector

91

u(u)e R"l . At X": (.t(u*). Ni“). 11*), transversality conditions for SN
must hold [App C]

(C2) If fxnﬂl) in IV(b) has eigenvalues 11.20;) approaching lam") = ijw
for some to $0.11) 6 R with associated conjugate eigenvectors u(u) 5 CR1,
u*(u)e C"l , then the equivalent differential Jacobian matrix Ju(u) has
eigenvalues £1,201) approaching X1,2(u*) = :th and with conjugate
eigenvectors u(u) e C"1 , u*(u) e C"1 and (3:0, the R . At
ft": (x(u*), y(u*), 11*) , transversality conditions for Hopf must hold [App

C]

(D) the subset of algebraic equations of I(a)

810601). y(u). 11) = 0 V(a)
with linearization

0 = smut) - 4y] V(b)
is called a bifurcation sub-subsystem of (4.5a) if (i) g(x(tt), y(u), u) = O is a
bifurcation subsystem of (4.5) when fx(p.) is nonsingular, (ii) 8mm) is nonsingular,
and (iii) the equilibrium behavior of (4.5) is related to the static behavior of V(a) in
the following way:
If 3mm) in V(b) has an eigenvalue Mu) approaching Mu“) = 0 with eigenvector
u(u)e le , then Jyy(u) given by

5,0» = gynw)-g,12(u)g;‘22(u)gy21(u)

has an eigenvalue Mu) approaching 101*) = 0 , with eigenvector 1101) e RmI

(E) the subset of differential-algebraic equations of I(a)

x', g f 161(1). x200. y(u). u) Wu)
0 g1(X(u).y(u). u)

92

. fx ([1) "'
Axl = l _ AX] Vl(b)
0 * gylw) Ayl

where the algebraic state vector y and dynamic state vector x2(u) are not function of

with linearization

I but only function of the parameter u , is called a bifurcation subsystem of I(a) if the
static behavior of (la) is related to the static behavior of Vl(a) in the following way:

If the subsystem in Vl(b) has an eigenvalue Mu) approaching 101*) = 0 with

n +ml

eigenvector u(u)e Rl , then 101) has an eigenvalue 71m) approaching

n+m

5411*) = 0 with eigenvector v(u) e R

From the above deﬁnitions, possible bifurcation subsystems are:

(a)

(b)

(C)

[if 'IE'E . S! E . . 51!
If (Al) holds, then [1(a) is a differential bifurcation subsystem experiencing saddle
node bifurcation that causes saddle node bifurcation in the equivalent differential
system model (4.6)

DEE 'IET . 5! E . . H E

If (A2) holds, then [1(a) is a differential bifurcation subsystem experiencing Hopf
Bifurcation which causes Hopf bifurcation in the equivalent differential system
model (4.6)

If (B) holds, then IIl(a) is an algebraic bifurcation subsystem experiencing load ﬂow

bifurcation which causes bifurcation in the Exact Load Flow model (4.5).

(d)

(e)

(f)

(g)

93

[ii 'lEi . Sl-l , E . . $11
If (Cl) holds, then [II(a) is a differential bifurcation sub-subsystem experiencing
saddle node bifurcation that causes saddle node bifurcation in the equivalent
differential system model (4.6)

E'Eﬁ 'lE'E . 51-! E . . H E

If (C2) holds, then [II(a) is a differential bifurcation sub-subsystem experiencing
Hopf Bifurcation which causes Hopf bifurcation in the equivalent differential system
model (4.6)

El I '51 . Sl-l EIEE'E .

If (D) holds, then IV(a) is an algebraic bifurcation sub-subsystem for load ﬂow (LF)
bifurcation which causes load ﬂow bifurcation in the Exact Load Flow model (4.5).
[if °l-l!°E'E .5!

If (E) holds then Vl(a) is a differential-algebraic bifurcation subsystem which causes
bifurcation in the full differential-algebraic model I(a).

Finally, it is important to note that in the case of differential-algebraic bifurcation

subsystems, the kind of bifurcation could not be determined, unless an equivalent

differential or algebraic model exists.

Note that as u approaches u*, x(u) and mu approach mm and y(u*) where

bifurcation occurs. The algebraic bifurcation subsystem depends on both x(u) and y(u)

determined from the full model even though the bifurcation subsystem equations may only

deﬁne y(u) or some subset. The remainder of the equations not in the bifurcation

subsystem help deﬁne the approach to bifurcation but do not have effect when load ﬂow

bifurcation occurs in the bifurcation subsystem. Similarly, the saddle node or Hopf

94

bifurcations that occur in the bifurcation subsystem dynamics and simultaneously in the
full system model due to satisfaction of hypotheses SNl or H1 in the bifurcation
subsystem model depends solely on the bifurcation subsystem differential and possibly
algebraic equations, even though the differential and algebraic equations not in the
bifurcation subsystem are used to determine x(u) and y(u) but not whether hypotheses
SNl and H1 are satisﬁed in the bifurcation subsystem model. The time behavior of the
components of x(1,u) and y(t, u) not in the bifurcation subsystem model is ignored in

testing for SNl or H1 in the bifurcation subsystem model.

4.2.3 Decoupling Conditions for Algebraic Bifurcation
Subsystems and Differential Bifurcation Subsystems

An equivalent differential subsystem model x = f(x, y(x), u) or an equivalent algebraic
subsystem model g(x(y), y, u) = 0 exist if gy or fx are nonsingular respectively, and may be
used to study static or dynamic bifurcations in the differential-algebraic model from
Schur’s Theorem. In fact, to understand the bifurcation physically, the use of equivalent
models is imperative. The usefulness of the equivalent models is universally known in
power systems and certainly not the subject of this thesis.

The bifurcation subsystem method to be developed states conditions where the
subspace in which the bifurcation subsystem experiences bifurcation is totally
independent and unaffected by coupling to the rest of the model. Thus geometric
decoupling conditions which formally state this principle, is the core of the bifurcation
subsystem method to be developed. This result does not solely attempt to show that f; = J;

for differential bifurcation subsystems or that g y a J y for algebraic bifurcation subsystems,

95

but rather the far more feasible condition that near the bifurcation value p = p.“ and in
some subspace of x, Jx - u = fx - 11 because [fygjgx] - u = 0 for some u e Null(fx) and that
in some subspace of y, 1y. v = gyo v since [gxfjfy] -v = 0 for some ve Null(gy). lfthese
geometric decoupling conditions for a bifurcation subsystem hold, then in Null(.lx) and
Numfx) the equivalent linearized dynamic model becomes

Ax = Jpr = fx-Ax
and in Nulluy) and Null(gy) the equivalent linearized algebraic model becomes

0 = Jy-Ay = gy-Ay
for the speciﬁc differential bifurcation subsystem or algebraic bifurcation subsystem
respectively.

Consequently, it is clear that a trivial decoupling condition is the case when the
generator and control dynamics are fully decoupled from the network equations, i.e. when
the off-diagonal matrix fy in 4.3(b) is null near the bifurcation value u = 11*. Similarly,
when the off-diagonal matrix gx in 4.3(b) is null near u* , the network equations are fully
decoupled from the generator and controls dynamics equations. These two conditions can

formally be stated as two corollaries to Schur’s Theorem, Corollary 4.1 and Corollary 4.2.

Corollary 4.1

Given 1, fx, fy, gx, and gy as in 4.3(b) where gy is nonsingular and near 11*.fy or gx are
null, then x = f(x(t), y(u), u) is a drﬁerential bifurcation subsystem experiencing SN
bifurcation for the full model in (4.1).

Proof:

It is clear that if fx has an eigenvalue Mu) approaching Mgr") = 0 with eigenvector

96

u(u*)e R" then Jx-uﬂl) = lfx-fyg;'g,l - um) = f,,- um) = Mutual) aPPmaChe-Y
Mart) - u([.t*) = O, which says that J,‘ has an eigenvalue int) approaching int“) = 0 with
eigenvector u(u*) e R". Hence assuming that transversality conditions (for SN
bifurcation) hold near 101*) = 0, deﬁnition A] imply that x = f(x(1), y(u),u) is a
bifurcation subsystem for (4.1 ) experiencing SN bifurcation

The decoupling condition given by Corollary 4.1 implies that the system dynamic
behavior is totally dictated by the machine and control dynamics. This type of modeling
which appears to be inappropriate to real situations, is encountered in system modeling
techniques where study of stability of a speciﬁc synchronous machine in the system is
based on a single machine inﬁnite bus model. The load bus and generator terminal bus are
aggregated back to the generator terminal bus and the swing bus, so that the network
seems to be virtually decoupled from the generator dynamics. If the shunt admittance at
the load bus is zero, the single machine inﬁnite bus model is a differential bifurcation

subsystem since fx = 1x .

Corollary 4.2
Given J, fx, fy, gx, and gy as in 4 3(b) where fx is nonsingular and near 11* , fy or gJr are null,
then g(x(|.t), y(u), u) = O is an algebraic bifurcation subsystem experiencing LF

bifurcation for the full system in (4 .1 ).

Corollaries 4.1 and 4.2 imply the equivalent differential and algebraic system models
could be decoupled so that an algebraic bifurcation subsystem and a differential

bifurcation subsystem exist. Total decoupling of the algebraic and differential subsystems

97

is of course never true for the differential-algebraic power system model and thus this
thesis needs not pursue non-geometric decoupling. The more general geometric

decoupling is now investigated

4.2.4 Geometric Conditions

Since the power system model is nonlinear, it is clear that a bifurcation subsystem of a real
system should also be nonlinear, so that the effects of the nonlinearities would be
captured. A bifurcation subsystem could but may not have to contain the exact center
manifold dynamics as the full system model, but this is difﬁcult to verify since computing
the center manifold of the speciﬁc critical operating point, has been argued (in Section
4.0.2) to be impractical if not impossible to compute for the 10,000 state variable power
system model. Whether center manifolds of the bifurcation subsystem and the full system
are identical and whether the effects of the nonlinear terms of the normalized power
system model on the system dynamics are identical is impossible to know without further
research. Although the description of the center manifold is crucial for stability analysis
near the nonhyperbolic equilibrium point, their study is beyond the scope of this work and
are not of high concern to us for establishing a rather fast and practical method for
identifying bifurcation subsystems of the full model.

The transversality conditions (H2) for Hopf bifurcation and (8N2) and (SN3) for
saddle node are generally assumed to hold without testing so that only (SN 1) is generally
tested for saddle node bifurcation and (H1) is tested for Hopf bifurcation. Since the

theoretical results that follow only relate to (SNl), (H1) tests on a linearized model and do

98

not attempt to prove that the transversality conditions that hold on the full system model
are sufﬁcient to assure that they hold in the bifurcation subsystem (which can be assumed
due to genericity of these transversality conditions), the existence of a bifurcation
subsystem of the system model are limited to a ﬁrst order (linear) satisfaction and
simultaneity of SNl and H1 tests in the full model and the bifurcation subsystem model. If
the transversality conditions (8N2) and (SN3) for saddle node and (H2) for Hopf
bifurcation were checked for the bifurcation subsystem (second order conditions), then the
bifurcation subsystem would be the nonlinear model found in the bifurcation subsystem
deﬁnitions, rather than a bifurcation subsystem of ﬁrst order (the linearized bifurcation
subsystem model) found in the propositions which follow.

Proposltlon 4.1

Let fx(l.l*) in 4 .3(b) be singular for some u* e (111,112) where transversality conditions for
SN hold at 2*: (x*, y*,u*), gym.) nonsingular near u*, and let N1 = Null(fx(l.t*)) and
N2 = Null(fI(u*)). Then Ax = fx(|.l.) - Ax is a diﬁerential bifurcation subsystem of ﬁrst

order for (4.1) experiencing SN bifurcation if and only if near 11*, condition (i) or (ii) is

satisﬁed
. -r
(I) [fygy gxl-u, = 0 forsome 111 15 Nl
(ii) vtr - [fyg;lg,,l = 0 for some v1 6 N2
Proof:
Necessity:

Given that f,(14*) is singular, we need to show that if (i) or (ii) is satisﬁed, then the
equivalent differential Jacobian matrix Jx(u*) is also singular with eigenvector ul .
(i) Let [fyg;lgx] - u1 = 0 for some ul 5 N1. Since foI“) - u1 = 0 by deﬁnition of ul

and the assumption that f,(u*) is singular, we have

(ii)

99

“(rm-u! = [f,,(u*)-fyg;'g,] -u, = 0
by condition (i). Therefore Jx(u*) is singular with eigenvector 111 as required by
deﬁnition Al. Therefore, Ax‘ = fx(u) - Ax is a bifurcation subsystem of (4.3) and a
bifurcation subsystem of ﬁrst order for (4.1) experiencing SN bifurcation.
Let v? - [1,g;‘g,1 = 0 for some v16 N2. Since vI-fxut'“) = 0 by deﬁnition of V]
and the assumption that fxﬂl‘“) is singular, we have

$14,111:“) = th - tf,(u*) -f,g;‘g,1 = o
by condition (ii) . Therefore Jx(p.*) is singular with eigenvector 111 as required by
deﬁnition Al. Therefore, Ax = f,(ll) - Ax is a bifurcation subsystem of (4.3) and a

bifurcation subsystem of ﬁrst order for (4.1) exoeriencing SN bifurcation.

Sufﬁciency:

(0

Given that Ax = fx(p.)-Ax is a bifurcation subsystem, we need to show if

singularity of f,,(tl*) is associated with singularity of J,(u*) as in deﬁnition A1,
then we must have condition (i) or (ii) satisﬁed. Suppose Jx(u*) is singular, and let
N and NT be the Null spaces of 1,,(11‘) and Jx(u*)T respectively.

Suppose there exists 111 6 N1 such that
fx(l4*)u1 = 0 4.7(a)
and
Irma, = [f,(u*)-f,g;1gxlru1 = o 4.70»
Which implies that the singularity of f, is associated with the singularity of Jx,
because the vector 11l satisfying 4.7(a) also satisﬁes 4.7(b). From 4.7(a) and 4.7(b)
we have

lf,(u*)-f,x;'gxl-ul = f,(ll*)-ul-[fyg;lgxl'u1

-l

100

Therefore u, e N, , and [f'yg;lgx] -u, = 0, as required.

(ii) Suppose there exists v, 6 N2 such that

v:- .fx(u*) = 0 4.8(3)

and T
v, -J,,(u*) = v[- [fx(u*)-fyg;lgx] = 0 4.8(b)
Which implies that the singularity of fx(u*) is associated with the singularity of

Jx(u*) as in deﬁnition A1, because the vector v, satisfying 4.7(a) also satisﬁes
4.7(b). From 4.8(a) and 4.8(b) we have
vf- tf,<u*)-f,g;‘g,1 = v[1,(u*)-v[~tf,g;‘g,1 = -v[ - rf,g;‘g,1 = 0

Therefore v, 6 N2 and v{- [fyg;lgx] = O, as required.

Implication of Proposition 4.1
Proposition 4.1 implies that singularity of the matrix fx(u*) is associated with the
singularity of the equivalent differential system Jacobian matrix Jx(u*) if and only if the
system submatrices fy , gx, and g, satisfy at least one of the conditions (i) and (ii) as
u —> 11* . Under these conditions, singularity of the reduced order matrix fx becomes crucial
for analyzing and studying the occurrence of saddle node bifurcation in the equivalent
dynamic model of (4.3). The reduced order system is called a bifurcation subsystem of
ﬁrst order for (4.1), since its change in dynamic behavior experiences the change in
dynamic behavior in the full system. Recall conditions (i) and (ii) of Proposition 4.1

v,’ - tf,g;‘g,1 = o tf,g;‘g,1 - u, = 0 as u-w" (4.9)
These equalities are seldom perfectly satisﬁed in a real power system Type 2 model.
However taking into account the extreme sparsity, decoupling of reactive power and

voltage from real power and angle, and the decoupling of coherent bus groups in both the

(It

In.

mal

101

real power and angle as well as reactive power and voltage model, (i) and (ii) are likely to
be satisﬁed to an 0(8) as discontinuities occur that cause J to be singular when fx is nearly
singular. Hence, the subsystem Ax = f, - Ax , may effectively be used to assess the causal
factors of the bifurcation in the full model. Equations in (4.9) are approximately satisﬁed

for the following ﬁve conditions:

(a

v

1ng n is 0(a) and llg;l||"fy|| - "u,“ is 0(1) near 11*

(b) Ilfy n is 0(a) and ||g;‘||"g," - "u,“ is 0(1) near 11*

(c) up u is 0(a) and “hung," . "a," is 0(1) near to

(d) None of the above but near it“ , the vector v, satisﬁes vf- [fyg;‘gx] z 0
(e) None of the above but near 11* , a vector u, e N, satisﬁes [fyg;lgx] - u, = 0

where H . II is any suitable matrix norm.

Proposition 4.2
Let gy(tt*) in 4.3(b) be singular for some 11* e ([1,, u?) , fx nonsingular near 11* and let
N, = Null(gy(lt*)) and N2 = Null(gy(p.*)T). Then, = gyul) - Ay is an algebraic
bifurcation subsystem of ﬁrst order for (4 .l ) experiencing LF bifurcation if and only if
near 111* , condition (i) or (ii) is satisﬁed

(1') [gxf;lfy] - u2 = 0 for some 11215 N,

(ii) v§.[gxf;1fy] = 0 forsome v26 N2
Implication of Proposition 4.2
In contrast to (4.9), conditions (i) and (ii) of Proposition 4.2 may be satisﬁed when the
matrix fx associated with the machine dynamics, exciter and governor controls is

diagonally dominant, a condition which is generally satisﬁed in a real power system Type

102

2 model. It can be shown that conditions (i) and (ii) of Proposition 4.2 can be satisﬁed to
an O(e) only in Section 4.6.5 if the bifurcation occurs near or in the distribution system
and electrically far from generator terminal and internal buses. The resulting bifurcating
model 0 = gym) - Ay may then effectively be used to assess the causal factors of the
bifurcation in the full model. Structural and operating conditions and constraints for the

validity of using the algebraic bifurcation subsystem will be discussed in Section (4.3.5).

Discussion of Propositions 4.1 and 4.2

The geometric conditions that near 11* , [gxfjfy] - u2 = 0, v; - [gxfjfy] = 0,
v? - [fyg;lgx] = 0, and [fyg;lgx] ~ u, = O generalize the case of full decoupling of the
differential equations and the algebraic equations stated in Corollaries 4.1 and 4.2, while
Corollaries 4.1 and 4.2 give conditions for the existence of equivalent reduced-order
systems to the full differential-algebraic model and would never occur. The more general
geometric conditions given in Propositions 4.1 and 4.2 are for the existence of bifurcation

subsystems and can indeed occur.

(a) Sparsely connected subsystems
Although these conditions seem to be a strong requirement, they are indeed a quite
easily satisﬁed requirement since the submatrices of J and especially g, have few
nonzero elements, so that there is a natural column or row dependence in the
submatrices of J, because so many elements of each column and row are identical or
are zero. In a 10,000 node network, each node may be connected to at most ﬁve other

nodes making gy virtually null except for ﬁve nonzero elements in the block

(b)

(C)

103

submatrices corresponding to the Jacobians dF/de , dP/uv , dQ/de and dQ/dv . The

submatrices of fx in (4.4) are diagonal which makes f, very sparse as well.

Weakly connected subsystem

Weekly connected may be expressed by elements in off-diagonal matrices being
multiples of a small scalar e > 0. If some of the few nonzero elements in different
rows and columns are 0(8) and can be approximated by zero, then the combination
of sparsity and weak coupling can produce row or column dependence. The natural
decoupling of real power-angle submodel and reactive power-voltage submodel as
well as the decoupling within bus groups of both these models satisﬁes the weak

coupling assumptions.

Discontinuous system changes

Time epochs (1,.+,1,r+ ,) are deﬁned in Chapter 3 as the time interval after the i’th
discontinuity and before the i+l hard limit or equipment outage discontinuities.
These nonsmooth system transitions described in Chaptre 3 usually result in loss of
voltage control of certain buses in the system and/or decoupling and islanding of
certain areas or group of buses of the system. Loss of control implies that the
generator dynamics and controls or other voltage protective devices of the system
are disabled which results in a decoupling of the generator dynamics and the system
algebraic operational constraints. Loss of control induced discontinuities occur when

generators reach ﬁeld current limits that cause the exciter to effectively lose control

of voltage, tap position limits to cause tap changer controls to lose control of voltage

01

If

\"C

de

104

and switchable shunt capacitor susceptance limits that also cause loss of control of
voltage discontinuously affect the structure and in some cases change the dimension
of f, and g, . On the other hand an equipment outage such as the trip of a tie line, i.e
a line that connects two major areas in a‘power system results in decoupling between
the two regions or even an isolation or islanding of a major part of a system.
Therefore in each time epoch (1;,1;,,) , the effect of the previous discontinuities
which occurred at times 1,, for k=l,..,i-l may fully be captured through the
disappearance of certain dynamics, decoupling or isolation of some subset of
dynamics, increase in matrix sparsity when dynamic equations are substituted by
algebraic equations,.. etc. For example, the effects of the discontinuities in epoch
(1 ,t, 1;, , ) , in addition to the extreme sparsity in the network create the weak coupling
of the reactive power and voltage and real power and angles as submodels in g,,, as
well as the weak coupling of coherent bus groups in both models. Therefore, the
resulting model within each epoch (1;,1;, ,) may experience separate bifurcations in
the real power inertial (angle) dynamics, ﬂux decay, exciter, voltage and reactive
power dynamics within coherent groups. The evolution of the bifurcation
subsystems with different kinds of bifurcations for each discontinuity is a natural
and essential way for tracking the actual cause for the ﬁnal system instability. The
relative isolation of the real power and angle coherent groups and reactive power and
voltage coherent bus groups in submodels of g), allows Proposition 4.1(i) and (ii) to
be satisﬁed for eigenvalues associated with the inertial dynamics and for the ﬂux

decay exciter dynamics in coherent groups of buses.

U34

105

(d) Geometric Structural Decoupling

The decoupling may only be geometric and thus occurring in the direction of the
bifurcating eigenvalue. The geometric decoupling may occur as u —+ (1* , or may be such
that it holds for all u as u—>u*. This geometric decoupling, asymptotic as u—iut' or
structural for all u, is quite easy to achieve given factors (a-c).

It is often impossible to determine all of the bifurcation subsystems without
comprehensive stress tests at every bus in the network and possibly every subsystem of the
generator model since the coordination of controls in different bifurcation subsystems
mask the existence of such subsystems. Comprehensive stress test cause the
discontinuities that disable the controls to allow one to observe the subsystem or sub-
subsystem that experience bifurcation, which result when conditions of Proposition 4.1

and 4.2 are satisﬁed.

4.2.5 Algebraic Bifurcation Subsystem Experiencing LF Bifurcation

Until the last decade, the classical method for analyzing local power system stability has
been limited to the use of a load ﬂow model. This model commonly called classical or
standard load ﬂow model, will be shown in Section 4.5 to be an approximation of a
linearization of the Exact Load Flow model
0 = g(x(y).y(u).u)

Under the Conditions of Proposition 4.2, the linearized Exact Load Flow model is shown
to be a bifurcation subsystem of ﬁrst order for the differential-algebraic model. The P-V or
Q-V curves computed using the classical load ﬂow model have been the principal methods

used to study voltage stability [App B] and proximity to voltage collapse. Studies [54]

106

have shown that voltage collapse occurs at a bifurcation in the classical load ﬂow model
and results either in no solution or multiple solutions in the model equations. The results
in [l 1] show that each coherent bus group where the Q-V curve have nearly identical
minima have unique voltage collapse problems. These voltage collapse problems occur
whenever the generators that exhaust reactive reserves and lose control of voltage in
computing the Q-V curve minima for each coherent group exhaust reserves for equipment
outages or operating changes. The fact the classical load ﬂow is an approximation of an
algebraic bifurcation subsystem of the differential-algebraic model supports the discussion
of how bifurcations occur due to extreme sparsity, decoupling of real power-angle and
reactive power-voltage submatrices, weak coupling of coherent groups in each submodel
and loss of voltage control discontinuities actually cause bifurcation in the differential-

algebraic model as claimed in Section 4.2.4.

4.3 Bifurcation Sub-subsystems Experiencing
SN and LF Bifurcation

The results presented in 4.6.5 indicate the classical load ﬂow model is a bifurcation
subsystem when generator exciters have high gain and are not disabled and results
presented in Chapter 5 indicate that the reduced differential equation model Ax = J,- Ax
can be a bifurcation subsystem of ﬁrst order. when generator exciters are disabled. The fact
that the classical load ﬂow is a bifurcation subsystem that produces and cause bifurcation
in the differential-algebraic model does not end the application or investigation of
bifurcation subsystems. The bifurcation subsystem method will attempt to identify

algebraic sub-subsystems of smaller and smaller dimension to determine the smallest

l()7

bifurcation sub-subsystem (a coherent bus group) that produces and causes the bifurcation
in the complete system as will be shown in Chapter 5. Similarly, differential bifurcation
sub-subsystems containing (a) generator mechanical dynamics, (b) electrical dynamics,
(c) generator ﬂux decay dynamics, ((1) the power balance equations at terminal buses, (e)
the control systems dynamics or (f) a combination of the above would be investigated to
ﬁnd the smallest bifurcation subsystem once it is conﬁrmed that a differential subsystem
bifurcation has occurred. If a differential or algebraic bifurcation subsystem does not exist
when conditions of Propositions 4.1 and 4.2 do not hold, a differential-algebraic
bifurcation subsystem containing both differential and algebraic equations may exist For
the following results on ﬁnding the smallest algebraic subsystem, the smallest differential
bifurcation subsystem and the smallest differential—algebraic subsystems, the Jacobian

matrix J must be partitioned as follows

x11 ler12 fyrr fy12
f f
J fxzr x22 yzr fyzz

g
g,” gle Yu gym

 

 

ngI g‘22 gyzr gyzz

4.3.1 Differential Bifurcation Sub-subsystem for
Saddle-Node Bifurcation

Not all system dynamical states are generally involved in a differential subsystem
bifurcation. The differential subsytem Jacobian matrix fx includes the machine electrical
and ﬂux decay dynamics, excitation system dynamics, governor control dynamics and

power system stabilizer dynamics. The matrices associated with each set of dynamics are

108

shown in the partitioned fx matrix. The matrix fx“(p.) in

1 (WI (u)
1]=1,.<u>.sx= ‘11 ’12 [Ail] (4.12)

2 13,2101) fxnm) Ax,

[Ax
Ax’ = ,
can represent any subsystem of the the differential equation in (4. 12), and the conditions
for differential bifurcation sub-subsystems of ﬁrst order are given in the following two

corollaries.

Corollary 4.3.1(a)

Given that (4.1) has a differential bifurcation subsystem for SN bifurcation because
conditions of Proposition 4.] hold. Let fx”(u*) in (4.12) be singular for some
11* 5 (111.112) where transversality conditions for SN hold at if" , fx,, nonsingular near u*
and let N, = Null(fx“(,u*)) and N, = Null(fx”(tt*)T). Then Ax, = fx“(u)-Ax, is a
diﬁerential bifurcation sub-subsystem of ﬁrst order for (4 .1 ) experiencing SN bifurcation if
and only if near 11*, condition (i) or (ii) is satisﬁed

(1') [3,12an fle] - u, = Oforsome 11,6 N,

.. T -1 .

Corollary 4.3.1(b)

Given that (4.3) has a drﬁferential bifurcation subsystem for SN bifurcation because
conditions of Proposition 4.1 hold. Let fx22(u*) in (4.12) be singular for some
11* e (u,, u,) where transversality conditions for SN hold at X“ , fx” nonsingular near 11*

and let N, = Null(f([.t*)) and N, = Null(f(l.t*)T). Then Ax, = f

Jr220.1) - Ax, is a diﬂ'erential

bifurcation sub-subsystem of ﬁrst order for (4.1 ) experiencing SN bifurcation if and only if

[(19

near 11* , condition (i) or (ii) is satisﬁed

- —1
(z) [“21fo [X12] - 11, Oforsome u, e N,

.. -1
(ll) V{'[fx2,fx,, [X12] = Ofor some v, 6 N2

4.3.2 Algebraic Bifurcation Sub-subsystem for Load-Flow Bifurcation

Let the algebraic bifurcation subsystem 4.3(b) be partitioned as

0 = gym = [g’118’12J.[Ay1] (4.13)

g’zr gyzz Ay, -
where gm , gym , g),21 , g,22 are arbitrary submatrices of 3),. Depending on the singularity
of g,“ and gy22 , algebraic bifurcation sub-subsystems experiencing LF bifurcation may be
obtained by applying Propositions 4.1 and 4.2 to the system in (4.13) as given in the two

corollaries below.

Corollary 4.3.2(a)

Given that (4 .3) has an algebraic bifurcation subsystem for LF bifurcation because
conditions of Proposition 4.2 hold. Let gy”(|.t*) in (4.13) be singular for some
we (u,,u,), 11,2201) in (4.13) nonsingular and let N, = Null(gy“(tt*)) and
N2 = Null(gy”(|.t‘)T). Then 0 = gynut) . Ay, is an algebraic bifurcation sub-subsystem of
ﬁrst order for (4.1) experiencing LF bifurcation if and only if condition (i) or (ii) is

satisﬁed

. —1
(l) [gylzgyzzgyu] - u1 = Ofor some ul 6 N,

-- T -1
(u) v [ ]=0 orsomeveN
1 smgyzzgyu f 1 2

110

Corollary 4.3.2(b)

Given that (4 .3 ) has an algebraic bifurcation subsystem for LP bifurcation because
conditions of Proposition 42 hold. Let gy22(|.t*) in (4.13) be singular for some
at e (u,,11,), gin nonsingular near 111* and let N, = Null(gy22(tt*)) and
N, = Null(gy22(tt*)T). Then 0 = gynm) - Ay,2 is an algebraic bifurcation sub-subsystem of
ﬁrst order for (4 .1 ) experiencing LF bifurcation if and only if, near 11* , conditions (i) or
(ii) is satisﬁed

0 forsome u, e N,

- —1
(I) [gyzrgyugylz] “1

(ii) vT-[g 3“,; ]

er yrl'ylz

Ofor some v, 6 N2

The thesis will show that the disablement of generator excitation control produces voltage
collapse in a load ﬂow model, which is an algebraic bifurcation subsystem and will also
cause bifurcations in the differential subsystem composed of the generator inertial and
ﬂux decay dynamics. These bifurcations in this differential bifurcation subsystem can

explain the dynamical behavior observed as a system experiences voltage collapse.

4.3.3 Differential-algebraic Bifurcation Sub-subsystem

Differential-a1 gebraic subsystems may be the most common type of bifurcation subsystem
that produces and causes bifurcation in the full differential algebriac model. It has been
reported that the transfer of real power across long transmission network is the reason
behind the low-frequency oscillations, which may affect one or many generators in the
system, or may affect more than one coherent generator groups. Differential-algebraic
subsystems may therefore exist to represent not only the generator and control dynamics

involved in the bifurcation, but also the contribution of the power balance constraints in

lll

causing the failure. These subsystems may be derived by ﬁrst permuting the order of the
state variables as for instance,

1x ylr= [x115 Y1 by —>[x1y1x2 ill

Then the full system (4.3) now has the form

 

 

 

 

 

 

 

 

r -
~ f f f f r . r -
,7 x x y
Ax, g 11 gym 8 12 g 12 Ax, .. . Ax,
0 = .x11 )’11 x12 in , Ay, = fx fy , Ay, (414)
. t t
sz fx fy fx fy M2 8x g), M2
0 21 21 22 22 Ay A)’
“ ‘ g g g s L 2‘ L 2‘
_ ‘21 hr ‘22 222

Differential-algebraic sub-subsystems may be obtained by applying Propositions 4.1 and

4.2 on the Jacobian matrix as partitioned in Equation (4.14), where f;, f; , g; and g; are

 

 

given by:
pr1 f)"; 1 fX" fr”
gx = -1 ..l ’ 8y «2 ~~ 4.15(a)
£3112, g)’31_ ng22 gy22
- f - f
e x y " x y
fx = 11 11 ’ fy 12 12 4.15(b)
‘11 8)”, x,2 gy12
Corollary 4.4.3(a)

Let f;(u*) in (4.14) be singular for some 11* 6 (111,112). g;(u) in (4.15) nonsingular near
11* and let N, = Nuu(f;(11*)) and N, = Nu11(f;(u*)7) . Then
Ax, e AX,
= f (11) ~
[ 0 i x [Mr]
is a differential-algebraic biﬁtrcation subsystem of ﬁrst order for (4 .1 ) if and only if near
11* , condition (i) or (ii) is satisﬁed

‘ ‘ ﬁ-l . -
(I) lfyg, gxl-u, = Oforsome u, e N,

112

(ii) 1" 1 - [ﬁgf‘sfél = 0 for some V, 6 N2
Corollary 4.4.3(b)
Let g;(u*) in (4.14) be singular for some 11* e (11,, 11,). f;(u) in (4.15 ) nonsingular near
11* and let N, = Null(g;(tt*)) and N, = Nu11(g;(11*)T). Then
. . A
0 Ah
is a diﬂerential-algebraic bifurcation subsystem of ﬁrst order for (4 .1 ) if and only if near

11* , condition (i) or (ii) is satisﬁed

(i) 1g;1;"1;1 . u, 0 for some it, e N,

—1e

-- T e ‘
(II) VI ~111fo fyl Oforsome v, e N,

4.4 Differential Bifurcation Subsystems for Hopf Bifurcation

The analysis and results carried out so far in this chapter have been based on Propositions
4.1 and 4.2 which focused on the system’s change in behavior when a simple real
eigenvalue of the differential-algebraic Jacobian matrix crosses the jw—axis. It has been
shown that under certain system operating conditions, two bifurcation subsystems of
4.3(a) are of interest:
(a) An algebraic bifurcation subsystem for studying load ﬂow bifurcation (also called
loss of causality) in the equivalent algebraic system model (4.5) is given by
O=Jy-Ay = [gy-gxf;lf,l-Ay = gy-Ay
(b) A differential bifurcation subsystem for studying saddle node (SN) bifurcation in the
equivalent differential system model (4.6) is given by
Ax’=Jx-Ax=[fx-fy8;18x1'Ax = f, Ax

Such results are desired and may easily be extended to the case where the system behavior

113

is characterized by the onset of unstable (subcritical) or stable (supercritical) oscillations.
With no further analysis Proposition 4.1, Corollaries 4.4.1(b) and 4.4.1(b) may be
generalized to cover the case when the equivalent dynamic system Jacobian matrix J, has
a pair of pure imaginary eigenvalues is, , = iiwo. If (00:0, the equivalent dynamic system
model experiences SN bifurcation, otherwise the system experiences undamped
oscillatory behavior whose frequency depends on too. Also it is evident that if the system
Ax = 1,01!) - Ax has a pair of pure imaginary eigenvalues "1.2 = :in with associated
conjugate (complex) eigenvectors 11,, , e C" then we have

l,-u, = two-u, = Jx(u*)-u,

[Jx(|-‘*)"“°o’nxn] ' “r = 0
which says that when the matrix I, has a pair of pure imaginary eigenvalues 71,, = :in ,
the matrix [1,(u* )riwolm] has a zero eigenvalue, i.e. singular. Therefore, with the

following deﬁnitions,

ffﬂu) = f,(u)-iw,l,,,,,, 4.16(a)
Jf‘out) a 1410-1111051”, 4.16(b)
_| .
= f,(u)-f,g, grifﬁn/"rm,

= ﬂaw-1,59,

similar proposition and corollaries to Proposition 4.1 and Corollaries 4.4.1(a-b) may be
stated for the existence of differential bifurcation subsystems of 4.3(a) for Hopf
bifurcation.

Proposition 4.3

Let ffﬂu) in (4.16) be singular for some 11* e (11,, 11,) where transversality conditions for
Hopf hold at 2*, gym) nonsingular near 11* and let N, = Null[f:)°(|.t*)]ch and

N, = Null[ffO(u*)T] c C". Then, Ax = fx(u) - Ax is a diﬁ’erential bifurcation subsystem of

114

ﬁrst order for (4.1) experiencing Hopf if and only if near 11*, condition (1‘) or (ii) is
satisﬁed
(1') {1,518,} ' u, = 0 for some u, e N,

.. T . —1
(U) v r-[fygy 12,] = Oforsome v, 6 N2

4.4.1 Differential Bifurcation Sub-subsystems for Hopf Bifurcation

The Hopf bifurcation in a differential bifurcation model for Hopf, may exist in only a
subset of the generator inertial dynamics, ﬂux decay dynamics, exciter system control
dynamics, governor control dynamics...etc. Condition for bifurcation of Ax = fxur) . Ax or
bifurcation sub-subsystems of (4.1) for Hopf are given below. Let the matrix ffom) be
partitioned as

(D
f,"(11) f
f?"(u)-Ax= ” “2 [1111,] (4.18)
. m0 Ax,

1,2, 52,111)

then conditions for differential bifurcation sub-subsystems for Hopf bifurcation are given

in the following two corollaries.

Corollary 4.4.1(a)

Given that (4 .3) has a dijferential bifurcation subsystemof ﬁrst order for Hopf bifurcation

exists because conditions of Proposition 4.3 hold. let 1'2", (11*) in (4.18) be singular for

some 11* e (11,, 11,) where transversality conditions for Hopf hold at 2*, figm)

nonsingular near 11*, and let N, = Null|:f:)"(u*)] and N, = Null[(f:)0(u*))1]. Then
11 11

Ax, = fxuut) - Ax, is a differential bifurcation sub-subsystem of ﬁrst order for (4.1) for

Hopf if and only if near 11* , condition (i) or (ii) is satisﬁed

115

(i) folzfégﬂl) 1,2,] - u, = 0 for some u, e N,

.. 1' to -1
(ll) v, -[fxlzf,2"2(u) [X21] = Ofor some v, e N,

Corollary 4.4.1(b)

Given that (4 3 ) has a dtﬁerential bifurcation subsystem for Hopf biﬁtrcation because
conditions of Proposition 4.3 hold. Let f2; (11*) in (4.18) be singular for some
11* e (u,, 11,) where transversality conditions for Hopf hold at 71*, fzolm) nonsingular
near 11* and let N, = Null[f:;02 (11*)] and N, = Null[(f:)2°2(|.t‘))r]. Then
Ax, = fx22(t1)-Ax,, is a dijferential bifurcation sub-subsystem of ﬁrst order for (4.1) for
Hopf bifurcation if and only if near 11* , condition (i) or (ii) is satisﬁed

(i) 1%,,fo (1104fo - u, = Ofor some 11, e N,

l

.. T 020 —l _
(II) V, -[fx2,fx”(u) fx12]- Oforsome v, e N,

4.5 Application of Algebraic Bifurcation Subsystem Experiencing
LF Bifurcation in Power System Stability Assessment

When the linearized system model (4.4) satisﬁes conditions of Proposition 4.2, the
algebraic bifurcation subsystem of ﬁrst order 0 = g, . Ay or consisting only of the network
equations 0 = g(x, y, p.) is a bifurcation subsystem of the Exact Load Flow model

0 = g(x(y), y, u) that is used to study load ﬂow bifurcation occurring in the full

[if] 1:1 -

Considering the similarity between g, and the classically used approximate load ﬂow

differential-algebraic model

model with Jacobian matrix 1,; [52], such a result would justify why experience has

shown that a classical load ﬂow model is such a useful tool in assessing stability that

116

occurs in a differential-algebraic model. The classical load ﬂow model was the ﬁrst and for
some time the only tool for studying voltage collapse or voltage stability [App B]. The P-
V or Q-V curves computed using a classical load ﬂow model have been the principal
methods used to study voltage instability and proximity to voltage collapse. Studies [54]
have shown that voltage collapse occurs as a bifurcation in the classical load ﬂow model
and results either in no solution or multiple solutions in the model equations. Over the past
ten years far more accurate Type 2-4 differential-algebraic models have been developed.
However the classical load ﬂow model is still the model and computer program being used
in power system voltage stability assessment programs despite the existence of far more
accurate models and simulation tools. The objective is therefore to use the present
knowledge of bifurcation subsystems, to assess when and why a classical load ﬂow model
is a bifurcation subsystem for the full differential-algebraic model. The discussion will
start with a brief description of a classical load ﬂow model, then take hierarchical steps by
ﬁrst looking into the preliminary system condition (f, nonsingular) for utilizing J, as a test
matrix that is equivalent to J for testing when a static or algebraic bifurcation occurs in the
differential-algebraic model. The second step is to compare the Exact Load Flow model
0 = J, - Ay ( 0 = g(x(y), y(tt), u) ) with the algebraic bifurcation subsystem 0 = g), - Ay
(0 = g(x(11), y(u), 11) ), and the classical load ﬂow model 0 = JLF ~ Ay (0 = g’(y, 11)).
Finally, based on the bifurcation subsystem conditions, some operational and geometric
grounds are established for the use of the classical load ﬂow power system model when
algebraic bifurcation subsystem experiences the same algebraic bifurcation as the Exact

Load Flow model.

117

4.5.1 Description of 3 Classical Load Flow Model

The classical load ﬂow model describes the power system steady state condition of the
transmission network and is a set of algebraic equations composed of the active and
reactive power balance equations of the transmission network at load buses and only the
active power balance equations at generator terminal PV buses. The reactive power
balance equations at generator buses are neglected since the excitation systems of these
generators are assumed to have inﬁnite gain and thus hold the generator terminal voltage
to set point value. The classical load ﬂow model can also be viewed as a simpliﬁed model
of a power system differential-algebraic model under the following assumptions: (a) the
generator armattu‘e 1esistaz..'e is zero; (b) fx is invertible (c) the excitation gain of each
generator is inﬁnite; and (d) all time constants in the differential equations are set to zero
since the time interval over which the model is to be valid is inﬁnite.

Buses in a classical load ﬂow model are conventionally divided into three types: (1)
a swing bus is a generator bus with inﬁnite active and reactive power reserves so that its
voltage and angle are ﬁxed while its active and reactive power generations are dependent
variables that provide the power mismatch between generation and load plus losses in the
rest of the system; (2) a generator PV—bus represents a generator bus operating under over-
excitation limits (OEL), and thus its reactive power generation and angle are dependent
variables while its voltage and real power generation are independent variables. (3) load
PQ-bus is any bus with no reactive or real power generation, and thus instead active and
reactive power injections are speciﬁed at this bus and are independent variables in the
model. The voltage and angle at PQ buses are dependent variables that must be solved for

using the model.

118

The classical load ﬂow model consists of active power balance equations at all PV-
buses and PQ-buses, and reactive power balance equations at all PQ-buses. The reactive
power generation at each PV-bus and the active and reactive power generations at the
swing bus are output variables (dependent variables) of the load ﬂow model. In addition, if
a generator PV-bus exciter is disabled by ﬁeld current limit controllers, the generator PV-
bus becomes a load PQ bus due to action of ﬁeld current limit controller that disables the
exciter and controls the reactive power to be the continuous rating limit. Thus, one more
reactive power balance equation at that generator terminal bus needs to be added, since the
voltage magnitude at its terminals is no longer held ﬁxed. The general structure of the

classical load ﬂow Jacobian has been derived in [52] and presented in Appendix A as:

 

 

 

 

’1”. Ar 31H 0 C19 011-! 0
3P 8P 1”” A21! BZHH ‘3an C519 021m 02111.
JLF = E W ___ J“ = 0 BZLH 321.1. 0 DZLH DZLL (4.18)
ﬁg 9!: 1’"? A59 3179 0 C13? 0139 0
393V 19” A B B CPﬁD D
JQL 4H 4HH 41.11 4 41m 4111.
L - 0 B B 0 D 0
Where _ 4LH 4LL 41.14 41.1.,
I
9 = 191‘ 9”! 9L1] : angle variables at network buses;
t
v = [vat V”! VL‘] : voltage variables at network buses;
t
P = [P1, PH, [31“] : voltage-dependent active power load demand;
I
Q = [QfQ' Q”: M2131 voltage-dependent reactive power load demand;

The real and reactive power balance equations in ( 4.18) have been arranged according to
the types of bus variables in the transmission network, i.e generator terminal (T) buses,
high side (H) transformer buses and load (L) buses. The matrices in the fourth row block

and column blocks are solely associated with the generator PV buses which became PQ

119

buses, thus the superscript notation PQ.

4.5.2 Structural Conditions for Utilizing Jy

The structural condition (f, nonsingular) imposed by Proposition 4.2 for formulating the
algebraic bifurcation subsystem 0 = g, . Ay may be understood by observing the matrix f,
as presented in (4.4). The derivation of the model used is given in [52] and summarized in
Appendix A. The matrices fx and ACC are given by

r r
Axx Axe Axo 0

~ “ A15x ABE 0 ”‘53
OX 66

Asx O 0 Ass

 

 

ABS 0 Ass
0 0 ASS

Matrix fx is composed of diagonal block submatrices where each diagonal element is
associated with one machine. The elements of each submatrix of f, is deﬁned based on the
single machine inﬁnite bus matrix, since it represents the operating condition when the
terminal bus of each machine is assumed to be an inﬁnite (swing) bus with constant
terminal voltage. Note that each diagonal block submatrix of fx can be viewed as an
element of a single machine model, since the dynamics of each machine are uncoupled in
fx. The control systems form an upper triangular block matrix Ace and its eigenvalues are
those of each control system: AEE , AGO , and Ass- Based on IEEE general models of
speed-goveming-turbine system and power system stabilizers [52], both Ace and ASS are

lower triangular matrices with real and negative eigenvalues. Regarding the excitation

120

system matrix AEE, when the excitation system is “properly tuned” and is able to be
manually operated, all the eigenvalues of Aer—2 are nonzero negative real values. Moreover,
under the assumption that each synchronous machine is operated below its steady state
stability limit, the matrix f, can be shown to be nonsingular, with stable eigenvalues. It
will be shown in Chapter 6 that fx can be singular if the mechanical dynamics or ﬂux decay
dynamics experience saddle node bifurcation. Operationally neither saddle bifurcations
should occur if the machine is operated within its real power generation limits and the
exciter is properly designed. Therefore, under the above assumptions, the equivalent

algebraic Jacobian matrix Jy = gy — gxfjfy can be analytically deﬁned.

4.5.3 Comparison of Exact Load Flow model, Algebraic
Bifurcation Subsystem, and Classical Load Flow Model

The Exact Load Flow model 0 = g(x(y), y(u), 11) , algebraic bifurcation subsystem
0 = g(x(u), y(u), 11), and the classical load ﬂow model 0 = g’(y,u) are all quite similar
because they all have real and reactive power balance equations at generator high side
transformer buses. The associated linearized models with J acobians J), g, and J”: are also
quite similar. The diﬂerences in J, and g), is investigated ﬁrst to establish (i) when and if
load ﬂow bifurcation occurs due to loss of causality so that 0 = g(x(u),y(1t), u) is an
algebraic bifurcation subsystem of 0 = g(x( y), y(u), u), and (ii) when the classical load
ﬂow model is a bifurcation subsystem model of the Exact Load Flow model.

The geometric conditions of existence of an algebraic bifurcation subsystem of ﬁrst

order for LF bifurcation.

—1
[exfx f,l-u,=0 when gy-u,=Jy-u,=0

121

1"2-[gxf;'fy] = 0 when v, -gy = v, J, = 0

where u, and v, are the right and the left eigenvectors associated with the zero eigenvalue.
may be approximately satisﬁed in different ways such as

(Cl) 1ng II is 0(8) and "ff““fy” - “a," is O(1) near 11*

(C2) llfy II is 0(8) and ||f;‘||||g,|| - “a," is O(1) near 11*

(C3) riff" is 0(a) and "f,““gxu - "u,“ isO(1) near 11*

(G1) None of the above but near 11* , the vector v, is such that vT2- [,g,,f;l fy] = 0.

(G2) None of the above but near 11* , the vector u, is such that [gxfffy] - u, z 0.
where 11 . II is an appropriate matrix norm. Conditions C1 and C2 pertain to the case when
the coupling matrices g, and f, are negligible so that the interaction between generator
internal and terminal buses is insigniﬁcant to the bifurcation study. Likewise, condition
C3 implies that the single machine matrix f, must be diagonally dominant. The
eigenvalues of j; are in fact proportional to the eigenvalues of the control systems AGC ,
A S S and A 125 , and the system K -matrices (will be derived in Chapter 6). These eigenvalues
may be high when the system is operating below its steady state stability limits and with
high excitation gain. Despite the implication of these conditions, it is clear that the
geometric conditions G1 and G2 not only encompass conditions C1-C3, but also are more
likely to occur due to the sparsity and weak coupling property of the Jacobians as noted
earlier. G1 and G2 say that near 11* , [gxf;lfy] need not be zero, but only that gy-Jy be
nearly singular and that its null space (approximately) overlap the null space of gym") .

Conditions G1 and G2 require examination of the structure and characteristics of the
matrix [gy—Jy ] = gxfjfy near 11* and its geometric interaction with the right and left

eigenvectors u, and v, of the null space of gy(|.l*)

122

In order to demonstrate and characterize the validity of the use of the algebraic
bifurcation subsystem and subsequently the classical load ﬂow model, analytical
expressions of g, and [11,, —Jy ] = gxfjfy derived and presented in [52] are

ng FAIK BIH 0 CIK Dru 0 8
£5” A211 32111! 321.11 CZH 021m Dzm.
_ 145’” 3 0 321.11 321.1. 0 DZLH 021.1. (420)
35” A3K 33H 0 C3K 03H 0

QJQJQJ

9.5110 CD11:
on we)

Q)
<|w <14

3,9” A4” 34111-1 341.}! C411 D411” D4HL

 

 

 

 

£321. ,0 341.11 341.1. 0 041.11 D4LL_

Kgooxgod
oooooo

[g,f;'f,l=g,-J,= O 00 0 00 (4.21)

K3100C3x00
o 00 0 00
[0 00 o 0Q

 

 

Where
t
6 = [A6,' A9”! ABL'] : angle variables at network buses;
l
v = AVrt AVH‘ AVL‘] : voltage variables at network buses;

l
P = [Mort APH' APL‘] : voltage-dependent active power load demand:

Q [AQr‘ AQn‘ A941! :voltage-dependent reactive power load demand;

Note that the submatrices in the fourth row and fourth column blocks of g, model the
reactive power Jacobians associated with the generator terminal buses, while in the
classical load ﬂow model (4.18) the fourth row and fourth column blocks are associated
with unregulated generator PV-buses which became PQ-buses. Therefore, the difference
between the two matrices resides mainly in the fourth row and fourth columns, since the

row dimension of submatrices in the fourth row of JLF associated with Q’T’Q and the

column dimension of the submatrices in the fourth column of J U, associated with Vf‘l
equal to the number of generators with PV to PO bus changes, but those in gy are
proportional to the number of generators in the model. These are also differences in the
terms Am, Am, C”, and C3,, of g), and A, , AfQ, CW and C5? of JLF .The relations
between these terms as well as their relation to the equivalent algebraic Jacobian matrix
J y is now discussed when the dimension of uPQ and the number of rows associated with
V§Q and Q’T’Q in (4.18) are zero.
Note that the submatrices A”; and A3,, of gy are given by
AIK = Kipqr+41 ; A311 =K31+A3
where A, and A3 do not include the effects of the K131: aPT/ae, and K3,: aQT/ae,
coupling between generator terminal and internal buses. Thus A, and A3 are identical to
the matrices A, , A§Q of a classical load ﬂow model Jacobian (4.18) since this model
includes terminal buses but not internal buses. Also note that
ClK = Kits-1C, ? C311 = K317+C3

where C, and C3 do not include the effects of the K #17: ap,/av, and K547: aQT/av,
coupling between generator terminal and internal buses. Therefore C, and C3 are
identical except in dimension to the matrices CfQ and C5]? of a classical load ﬂow model
Jacobian (4.18) since this model includes terminal buses but not internal buses.
Expressions for K51, , K51, , K fat, and K3, are easily derived [52] and can be found in
Appendix A. When adding [1, -g,, ] of (4.21) and g, of (4.20) to obtain Jy , it turns out
that A, K and A3,, of g), are replaced [52] by A, and Ag’Q of the classical load ﬂow
Jacobian J L F which implies that the exciter and ﬂux decay dynamics have no effect on

aPT/ae, and aQT/ae, terms of J y . Furthermore, in J, the submatrices C , K and C3,; of

124

g, are replaced by

Cir = Cir—C111

Cir = CsK‘Csx C3+K37 ‘Crx

Matrices C 1x and C3,, are derived in [52] and are dependent on the DC gain of the
excitation system KEE which makes diagonal matrices CfK and c), have very large
negative elements when all exciters are active. However if the armature resistance of the
generator R0 = O , then [52]

C”, = Kg, thus ch = C,.

This result implies that when the armature resistance of the generator is neglected, the
control systems do not have any effect on the active power-angle Jacobian matrix
CfK = C,. Since R, is generally small so that CL, = C, and 1:1,, is assumed to be large
when exciters are operating at high gain at every generator, the conditions for static

bifurcation of J y

0 = 1y. )4, 4.23(a)
where
n,” = (air. 11;”. uh. 115,. 115),. 115,) 4.23(b)
Equation (4.20), (4.21) and (4.23) imply that

A3uPT+B3uPH+C§KuQT+D3HuQH z C§KuQT z 0 4.24(a)

since or, is very large due to its being linearly dependent to KEE. From equation 4.24(a)
and the fact that cat, is diagonally dominant when all generators have excitation control,
4.24(a) suggests that

em = 0 4.24(b)
Since 4.24(b) is associated with the fourth row of J y not in J L r when all generators have

excitation control and uQT = 0 eliminates the fourth column of J, , the remaining

125

equations of 4.23(a), (which do not include 4.24(a), require that
J T T T T T T :3 0
LF' “pr “PH “PL “Po “QH “or.
where uPQ has dimension equal to zero.
Thus Propositions 4.2 implies that the classical load ﬂow model is a bifurcation subsystem

of

and
u = , u
0 Ay
for LP bifurcation for the case where all exciters are active and operating with high gain,
Ra on every generator is assumed to be zero, and own is invertible.

An option has been added to some load ﬂow algorithms to model the generator ﬂux
and exciter loop. In this case, the exciter gain on every generator does not have to be large
for the load ﬂow to be a bifurcation subsystem for the differential-algebraic power system
model, since in this case J L r = J y regardless of whether the exciter is present or is
disabled by ﬁeld current limit controllers. The only difference in J u, = J y when exciters
are present or disabled is in the terms C, x and C3,, that depend on whether excitation
system gain K or = 115,511,,“ is zero or nonzero, where ng is the inverse diagonal matrix
of the inverse DC gain of the exciter, K A is a diagonal matrix of the ampliﬁer gain of each

excitation system, and K o is the diagonal matrix of the sensor gain for each generator

exciter.

126

4.5.4 Bifurcation in LF Model or in Generator Dynamics

Recall that the conditions for g, to be a bifurcation subsystem of J, and J require that

[1y —gy] - u2 = O (4.25)

when

J,- u, = o (4.26)
From (4.21), condition (4.25) can be written as
0 4.27(a)

K1141 '“PT+C1X'“QT

O 4.27(b)

K31 '“PT + Csx ' “QT
where u, is the eigenvector associated with the zero eigenvalue,
T .. T T T T T
“2 " (“PT’ “PH’ “er “QT’ “on: “91.)

The J y condition in (4.26) for terminal buses can be written

A,uPT+B,uPH+(C,+K;¥7-C,X)uQT+D3HuQH = 0 4.28(3)

A3“PT + B3uPH + (C3 + [(8, - CBX)“QT + D3HuQH = 0 4.28(b)
There is very little chance that J y and J will be singular simultaneously with g, since the
C3,, term in equations 4.27(b) and 4.28(b), which is dominant, occurs with different signs.
On the other hand, if 11,, , up” , um and 110,, were zero for a speciﬁc bifurcating
eigenvalue, then from (4.26) and (4.27) we can see that the bifurcation occurs strictly in
the load buses far from the generators in the system, which indicates that J and I, can
reach bifurcation when gy reaches bifurcation [52]. If 11", “PH , um. and 110,, are nonzero
then g, is never singular when J and J), are singular, and singularity in J always occurs
due to singularity in 1,. This bifurcation in generator dynamics that occurs whenever
uPT, up” , um. and 11,,” are nonzero, will occur when load ﬂow bifurcation occurs if all
generator exciters have inﬁnite gain, all generators have zero armature resistance, and f, is

nonsingular, or if the load ﬂow modeling option has been selected that makes J, = J ,F.

4.5.5 Conclusion

(1)

(2)

(3)

(4)

The bifurcation subsystem method suggests that
the classical load ﬂow model is a bifurcation subsystem for the differential-algebraic
model for all static bifurcations when all exciters are active and high gain at the point
of bifurcation; R, is zero on every generator, and f, is invertible.
the classical load ﬂow model with complete DC modeling of generator exciter and
ﬂux decay effects is a bifurcation subsystem for LP bifurcations for both

J -Ay = o
and

11 = ,_ u

0 Ay
the set of algebraic equations for a differential-algebraic model is not likely to be a
bifurcation subsystem of all the differential-algebraic model whether exciters are
active or disabled, unless an , u H, , uQT and uQH are zero. This result implies that
bifurcation of these algebraic equations can only occur if it occurs solely in the
algebraic equations of load buses in the system, which produce an algebraic
bifurcation subsystem.
as a result of (3), static bifurcations of the differential-algebraic model are either
static bifurcations of the reduced differential model with Jacobian JJr when an ,
up” , uQT and 11,, H are non zero, or an algebraic bifurcation in the real and reactive
power balance equations at load buses when 11,-, , up” , “QT and 11,,” are null. The
algebraic bifurcation is referred to as clogging voltage instability in [l l] and is due
to inability to ship reactive power to load buses. The clogging occurs due to reactive
losses that consume the reactive supply. The static bifurcation of the reduced

differential model is a saddle node bifurcation because the generic static bifurcation

128

is a saddle node bifurcation. The static bifurcation is most often a loss of control
voltage instability that occurs due to exhaustion of reactive reserves and voltage
control at generator buses since the generator terminal bus is affected by the
bifurcation. There can be clogging voltage collapse problems [11] that are due to
static bifurcation in the reduced differential equation model since clogging voltage
collapse can exhaust reactive reserves and cause loss of voltage control, but this is by
deﬁnition not the cause of the instability. The exhaustion of reactive reserves of these
generators could only occur if the generator terminal buses were affected and

uQTrtO.

V

Effect of Hard Limit Discontinuities on
Stability Analysis

5.0 Introduction
The Bifurcation Subsystem Method developed in Chapter 4 and the Epoch Based Trajec-
tory Stability Assessment Method developed in Chapter 3 provide a framework for initiat-
ing a structural stability analysis on a single—machine—inﬁnite bus (SMIB) model that
encompasses local shunt loads, with emphasis on the effects of one type of hard limit dis-
continuous system transition, the disablement of excitation control. This discontinuous
control action is shown to have an effect on the kind, class and the tests for bifurcation
occurring on the SMIB model. The goals of this chapter are to:

(a) Demonstrate two examples of using the bifurcation subsystem method to identify
bifurcation subsystems in a model. The generator angle-speed dynamics is the bifur-
cation subsystem for Hopf bifurcation in the regulated SMIB model, and the angle-
speed-ﬂux decay dynamics is the bifurcation subsystems for SN bifurcation, in the
unregulated SMIB model, respectively.

(b) Analyze the effects of hard limit discontinuities by assessing the effect of excitation
control or lack of excitation control on the (i) types, (ii) classes of bifurcations pro-

duced, and (iii) the stability test needed for detecting the occurrence of bifurcation.

129

130

(c) Derive expressions for the synchronizing and damping torques in a linearized power
system model with no exciter, a thyristor exciter, and type 1 exciter models in terms
of the model parameter K.-K,. The damping torques are derived for f<<l and for
f>> l.

(d) Derive conditions on linearized model parameters [(4 and K, for the occurrence of
Hopf and on parameter K,, K, K3, K... K, and K, for Saddle Node bifurcations, and
preservation of stability of the equilibrium. Such conditions are derived for the unreg-
ulated model and for the regulated model (thyristor and type 1 exciter) SMIB model.

The conditions on model parameters K, and [(5 set the stage for establishing diagnostic

tests for speciﬁc kind and class of Hopf bifurcation in terms of active load, reactive load,

shunt capacitive susceptance, and real power transfer levels in Chapter 6.

The stability analysis in this chapter is based on the principle of synchronizing and damp-

ing torques or a bifurcation sub-subsystem

MAS + 0116‘ = Arm — AT e
AT, = [(5135 + KDAS 5 ATS + ATD
where
AT m : incremental change in mechanical power input to the generator
AT e : incremental change in the electrical power output of the generator
AT s : incremental change in synchronizing torque
AT D : incremental change in damping torques of the machine

The use of damping and synchronizing torque model for the model shown in Figure 5.3
was developed in [55] where the generator inertial and exciter dynamics are assumed to
produce equivalent damping and synchronizing torques on the inertial dynamics. This

type of analysis performed in [55] was later extended in [56], in both classical papers on

131

power system dynamics. Matrices KS and K,, were derived in [52] for multi-machine
power systems given that the subsystem external to the mechanical dynamics is invertible.
The Ks and KD tests for saddle node and Hopf bifurcations respectively, do not necessarily
imply that the bifurcation occurs in the generator mechanical dynamics, nor that the
inertial dynamics are a bifurcation subsystem, and neither the test matrices nor the
bifurcation subsystem indicates anything about the center manifold for the bifurcation in a
non-classical power system model. The system is stable if and only if both the
synchronizing and damping torques are positive. If a classical generator model is used,
where no ﬂux decay or exciter dynamics are modeled, then the damping and
synchronizing torques are namely ATS = K, , the machine real power coefﬁcient, and
ATD = D, the natural damping of the machine. In the case of more complex generator
models, ATS and ATD depend on the single machine inﬁnite bus model constants Kl'KG.
The form of ATS and ATD in terms of K,-K6 have been derived for nonclassical generator
model with different exciter models and for the unregulated model [56]. The system Kl-K4
constants (unregulated) and Kl-K6 (regulated) are derived as part of this thesis and are
expressed as functions of the system operating point, the network parameters, and the
inﬁnite bus voltage. Although the parameters or matrices ATS and ATD do not imply that
the inertial dynamics contain the center manifold or even a bifurcation subsystem, they are
tests for saddle node and Hopf bifurcations in the full system model utilized as effective

measures of stability and security, and are for that reason used in this dissertation.

132

5.1 Linearized Multi-machine Power System Model

The power system model used in this chapter is based on linearizing the differential and
algebraic equations of a multi-machine power system model. The modeled system consists
of n synchronous generators represented by a two-axis models, m network buses, and an
inﬁnite bus for each area in the model. The model takes the multivariable block diagram
form shown in Figure 5.1, where the state variables are the electrical angle and rotor speed
deviations of the n generators A5 and A11) , and the q-axis transient EMF deviation AE', .
In this model, T M is a diagonal matrix of inertial time constants TM; = 2H ,, T’d is a
diagonal matrix of ﬁeld open circuit time constants T'dai, and D a diagonal matrix of
damping coefﬁcients. The inputs to the multi-machine model are the mechanical power
deviations AP", and the excitation voltage deviations AE, of the n generators.

The nxn linearization coefﬁcients matrices K,, K2, K3, K4, K5 and K6 depend upon the
network parameters, the system quiescent operating point, the inﬁnite bus voltage. They
change as the system conditions change, and therefore can decouple angle and voltage
dynamics or can strongly couple voltage and angle dynamics for other operating condi-
tions. Speciﬁc system operating characteristics, such as heavy local loading conditions,
large transfer of active power, or a capacitive network can dramatically effect K1'K6 and
the kind and class of the bifurcation produced. As the signiﬁcance of these constants will
be revealed throughout this chapter, it is important to explain their deﬁnitions. In the next
section a linearized SMIB model will be derived, where these constants Kl-K6 are scalar.
(a) K, = BPG/ (8801p : Change in electric power for a small change in rotor

411 = 15,411"

angle with constant ﬂux linkage in the direct axis °. The component of

 

E’q,| =

 

Eq,

 

torque (or power) K,A51‘ is in phase with A5 and hence represents a synchronizing

(b)

(C)

(d)

(e)

(f)

133

power component. Indeed K, is the synchronizing power coefﬁcient in a classical
generator model where no exciter or ﬁeld circuit is present.

K, = arc/(a

 

Eq,

 

)| , _ : Change in electric power for a small change in the d-

5: = 51°

axis ﬂux linkage at constant rotor angle 81' = 81°.
K 3 = [1 +((26,,,-—x',,;)(xq;+11,,,.))/(A1')]"l :K3 is an impedance factor that takes into
account the effect of the external impedance that relates the generator transient

closed circuit time constant T’ (,0 to the transient open circuit time constant T’ d,.

1

K4 = E313

 

 

E'qi)/(88i)1efd : K, represents the natural coupling between

i: constant

AlE’,,-| and A81. It is related to the demagnetizing effect of a change in the rotor

angle.

K5 = (alv,,| )/(a81')llE' , 0: K5 is the change in terminal voltage V,, for a small
qt =

15'

 

qi
change in rotor angle with constant ﬂux linkage in the direct axis

E' °

qi=E'

 

 

 

 

qi

K, = (alv,,|)/(a : Change in terminal voltage V,,- for a small change in

 

Eq,

 

’1

i=5i°

the d-axis ﬂux linkage at constant rotor angle 81’ = 81°.

134

 

 

 

 

 

 

 

 

 

 

 

 

 

AlV,l +
<—-——4 K,
. Ks
A8
+ + _, AIE'ql _>
(Ta) 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.1 Block Diagram of a Power System Model

5.2 Single Machine Inﬁnite Bus Model With Local Load

5.2.1 Model Representation

Considering how large and complex a general power system is, understanding the basic
effects and concepts of power system stability phenomena using analytical techniques
requires a simple conﬁguration and a reasonably low-order model. A single machine con-
nected to a large system through a transmission network and supplying a local constant
impedance load (Z,,,) is shown in Figure 5.2 (a). This system may be simpliﬁed by using a
Thevenin’s equivalent of the transmission network external to the machine and its trans-
former as shown in Figure 5.2(b). Furthermore, because of the relative megawatt (MW)
generator capacity and inertia of the machine that represents a generating station compared
to the generator MW capacity and inertia of the rest of the large interconnected system,
dynamics associated with the single machine will cause virtually no change in the voltage
and frequency of a Thevenin equivalent source (V0,) used to represent the large intercon-

nected system. Such a source of invariable frequency and voltage has been called an inﬁnite

135

bus [29]. The described single machine inﬁnite bus model has widely been used in the lit-
erature as a simple yet powerful tool for understanding of power system stability of far
more complex systems under small disturbances [55]. Accordingly, in order to develop an
appreciation for the dynamical behavior of the system as it undergoes bifurcations, their
causes and effects, it is virtually necessary to investigate such a low-order system before
dealing with very large complex systems. The Thevenin electric circuit of a complex trans-
mission network model is shown in Figure 5.2(a) with simpliﬁcation shown in Figure
5.2(c). This single machine-infinite bus model includes

(1) A generator and its transformer represented by a voltage source

 

5,128 behind a
transient reactance x',, + x, where x7 is the transformer reactance. This generator
and transformer are radially connected to a large interconnected system, via a long
transmission corridor with equivalent reactance X E that is large. The large
interconnected system represented by the inﬁnite bus (i) may be a major bus or
group of buses that are stifﬂy interconnected so that they act as a single bus and (ii)
contains very large generation capacity and inertia compared to the single machine
connected to this bus group via the transmission corridor reactance X E;

(2) A load center at the generator terminal is represented by a constant shunt resistance
R,,, in parallel with a shunt capacitance or inductance X 3,, that can include line
charging of the transmission line, local shunt capacitive compensation, or inductive
load.

(3) X E is the equivalent transmission network inductance shown in Figure 5.2(b);

(4) The equivalent transmission network resistance R E shown in Figure 5.2(b) that is

neglected in Figure 5.2(c);

136

(5) The inﬁnite bus is modeled as a voltage source with voltage 1V°°1 40°.

5.2.2 Model Implications

Despite the simplicity of its conﬁguration, the single-machine inﬁnite bus model can be

used to represent

(a) A generator connected to a large system, via a long equivalent transmission corridor
with equivalent reactance x); that is large. The large system may be a major bus or
group of buses of a power system of very large capacity compared to the rating of the
machine under consideration, or a large load center. Real power is shipped from the
generator to the large load center, whereas reactive power may be shipped in either
direction across the transmission network;

(b) A generator serving a heavy local constant impedance load that can include line
charging of the transmission line, local shunt capacitive compensation, or inductance
load. With x5 large, the dynamics of the generator serving a local load center can be
investigated. Effects of transfer to and from the large remote load center can also be

included.

137

 

Loca—l load __ T

 

 

 

 

 

 

 

 

(a)
Vt le|40°
/G\ l e
V =RE+1XE E
Zsh
Locjload 1
(b)
V, Ia 1V°°1 40°
Xd'+XT ———’ ‘ XE I?
(11111 ' 1 011111 E
[154’ 28 a
Rsh g: xsh
E— —'L“— —="

 

Local load
(0

Figure 5.2 Single machine with local load connected to an inﬁnite bus through
a transmission network: (a) General conﬁguration of a single machine connected
to a large system through transmission lines; (b) Equivalent system where the
transmission network is reduced to its Thevenin’s equivalent; (c) Circuit model of
(b) where the RE is neglected

138

V, I, v;
liﬂ—M—tlxil___i

 

 

 

lEq’28
(a)
q-axis
E
d-axis E ' . (Xd-Xqﬂd
qa ‘
Eq, ‘. ,
‘. 1(xq+xe)lq
I.
8 Ref
, ‘ , (X ’d‘ Xe) Id /////
Id a rela
(b)

Figure 5.3. Simpliﬁed single-machine inﬁnite bus model. (a) Equivalent
Thevenin model where r, x, and v; are given by equations (5.1). (5.2) and
(5.3) respectively, (b) Phasor diagram for the synchronous machine connected
to an inﬁnite bus shown in (a)

139

5.2.3 Model Equations

The circuit model in Figure 5.3 (a) can obtained from Figure 5.2 (c) by ﬁrst transforming
the Thevenin’s equivalent XE and 1V°°1 40° to a Norton’s equivalent circuit, aggregating XE,
X,h and R5,. and then transforming the resulting Norton’s equivalent to a Thevenin’s equiv-
alent. This produces the circuit model shown in Figure 5.3 (a) where re and xe and V; are
the equivalent thevenin’s resistance, inductance and inﬁnite bus voltage. The parameter re,
x8 and V; are expressed in terms of the original line inductance XE, and the shunt loads R,.,
and X,,, in equations (5.1), (5.2) and (5.3) respectively.

Rsh

rc = 2 (5.1)
1+ii,,,(1/x,,,+1/XE)2

Z
R3,,(1/X3h-1- 1/x,.)

 

 

 

 

,, = 2 (5.2)
1+R (1/x +1/x )2
sh :11 E
e = 1le a Ive|z_B (5 3)
°° l+XE/Xs,,+jXE/RS,, °° '
The differential equations for the single synchronous machine are
M8+08 = Pm-PG(8, IE'ql) (5.4)
T‘d0(d|E'q|/dt)+|Eq| = of, (5.5)
Where PG has the form
5' v; vg,2
P615454) = 1 , '1 lsin8+| I ( 1 — , 1 )sin2(8—B) (5.6)
xd+x, 2 14"": xd-t-x,

when r, = 0. An equation of PG is derived when r, at 0. The algebraic equations represent-
ing the network shown in Figure 5.3 are

5,, = V;+rela+j(xd+x,)ld+j(xq+xe)lq (5.7)

E'q = v; +rcla+j(x'd+x,,)ld+j(xq+x,)lq ld<0 (5.8)

l 40

5.2.4 Simpliﬁed Linear SMIB Model
A simpliﬁed linear model for the synchronous machine connected to an inﬁnite bus
through a transmission line with reactance x, shown in Figure 5.3(a) is derived, based on
the phasor diagram in Figure 5.3(b). In the linearized model, the following assumptions
have been made:

(i) The amortisseur effects are neglected;

(ii) Stator winding resistance is neglected;

(iii) Balanced conditions are assumed and saturation effects are neglected;

(iv) The synchronous machine can be represented (electrically) by a constant volt-

age source behind its direct-axis transient reactance.

Under these assumptions, the linearization of the E ', equation (5.2), the electric torque/
power equation and the terminal voltage equation will be performed in the next subsection.
It should be noted that the equations describing the system are expressed in p.u.
5.2.4.1 The E’q Equation

From the system algebraic equations (5.7) and (5.8), E q and E ’q are related by

E“, = Eq+ j(xd"x.d)ld (5.9)
Using the phasor diagram representation of (5.7)-(5.9), we express 1,, (<0) and Iq as func-
tions of 15.41 and lv‘ml. Projecting E 'q and V; on the q-axis and the d-axis and keeping in

mind that Id is negative yields

-(x’d+x,)ld+rclq = [E'ql-lvglcosw-B) (5.10)
(xq+x,)lq+reld = lVf,|sin(5-B) (5.11)
|qu = [E'ql-(xd-x'dﬂd (5.12)

Assume steady state operating condition at an operating point such that

s = 6" 1, =1; 15,1 =15'.1° 1v1=1v1°

141

B = B ’d = ’4° 514 = 514° Pa = (”13° = Po(8°’ IE'41°)
A180 assume a small perturbation around the operating point, such that

5=8+A6 lq

B=BO+AB Id

lq°+Alq |V,| = |V,|°+A|V,|

lo° + No IE'ql = IE'ql° + A1511

Substituting these into equations (5.10) and (5.11), and using Taylor’s expansion method

around the nominal solution as the steady operating point yields the following linear equa-

tions (5.10’)-(5.12’)

—(x',, + x,)AI,, + rem, = AIE’ql + |v;| sin(8° — 15°) - A8 (5.10’)
reAld+(xq+x,)Alq = lV,|cos(8°—B°)~A8 (5.11’)

Solving for A1,, and A1,, and writing the result in matrix form as

_,, -(xq+xe) relVeoolcos(5°-B°)—(xq+xe)|Veoolsin(5°-B°) AlEiq, (5.13)

re ’91Ve°°151"(6°‘13°) + (x'd+xe)|Vem|cos(5°—ﬁ°)

A“ = r} + (x, + x,)(x',, + x,)

Replacing Alq from equation (5.13) into equation (5. 12’), an expression of the incremental
quadrature voltageAIE (11 as solely function of AIE’ 41 and A5 is obtained

Alqu = [1+(xd—x'd)(xq+x,)A-'] -A|E'

«l
+ [(114 - .11',,)A’l IVfol ((xq + x,)sin(6° - 13°) - r,cos(6° — B°))] - A8

which can be written as
l

A15111 = [(3

~A|E'q| +K4-A6 (5.14)
Now, substituting into an incremental version of equation (5. 12) expressed in the s-domain,

the following transfer function is found

K3K,

. — —3_. _____.
A1541 - 1+K3T’dos AIEfdl 1+K3T’dos

A5 (5.15)

where

142

K, = [1+ A-1(x,,_x',,)(x,+x,)]" (5.16)

K4 = A—l(xd-X'd)

 

V"

 

'[(xq+x,,)sin(5°-15°)—recos(8°-B°)] (5.17)

5.2.4.2 Electrical Torque Equation
The electric torque T e in numerically equal to the three-phase generated electric power PG
when expressed in per unit, thus
T, = PG = Re(v,-1,,*) = led+ v41, p.u (5.18)
where V, and V, are the projections of the generator terminal voltage V, on the q-axis and
the d-axis, respectively, i.e
v, = Vq-i-de (5.19)
Vq and Vd can be expressed in terms of I,I and 1,, by projecting IV.) and I, on the q-axis and
the d-axis as shown in the phasor diagram in Figure 5.3, we obtain
V, = v; + (r, +jx,)l,,
... |vg| cos(8° - 13°) - j|v;| sin(8° -13°)+ (r, + jx,)(1,, + jld)
= |vg|cos(8° — 11°) + 51,, —x,1,,—j[|v:,|sin(8°— 13°) -x,1,,-r,1,]
Which may be expressed using 5.10 and 5.11 simply as
v, = lay + {41,- jqu,, (5.20)
Setting real and imaginary components in these two representations of V, in equation 5.19

and 5.20 equal, the following is obtained

Vq = 15.41 +x'dld (5.21)
Thus,

= [lqu "' (xq‘x'd),d]lq (5.22)

Linearizing PG as expressed in equation (5.22), we get

I43

All)G = A115.4114o " (x, " x'dlA’d’qo + Al4115.111o ‘ (x, ‘ x'dlAlq’do

= A|E',,|I,,° + |Eqa|°Alq — (x, -—x',,)AI,,I,,°

where we used the q-axis voltage 13,, deﬁned in Figure 5.3 (b) as

with initial conditionvalues
lqul° = 1qu° + (“"d’i‘qﬂdo

Substituting for 15410 from Equation 5.12, one obtains

15 o =1Eq’

= 15410 " (x, ‘ x-d),do

 

qal °—(xd—x'd)ld°+(xd—xq)ld°

Substituting expressions from equation (5.13) into APO to ﬁnally compute

APG = [1,,°+A-IIE,,,|°r,+n-l(x,-x',,)(x,,+x,)(l,,°)] . AIE'ql
+ A-1|v;| [|E,,|°(r,sin(6° — 3°) + (x'd + x,)cos(8° — 3°))
_(xq-x',,)lq°(r,cos(8° - 13°) - (x, + x,)sin(8° - B°))] . A8

a K, - A8 + K, - A|E,,’1 (5.23)

where
K1 = 4"lVilllqul°(resin(5° - 6°) + (x'd+ x.)cos(5° - B°))]

+ A-1|v;|1,,°[(x, + x,)(x,, - x'd)sin(8° — 3°) — r,(x,, - x',,)cos(8° — 3°)1 (5.24)
K, = rc|Eqa|°A‘1+lq°[l+ A"(x,, - x'd)(x,, + xe)] (5.25)

5.2.4.3 Terminal Voltage Equation

The square of the norm of V, as expressed in equation (5.19) is given by the equation

2 2 2
V, = Vq+vd (5.26)
Linearizing equation (5.25) yields
V v"
AV =—1;-AV +é-Av (5.27)
, l " IV) 1 MI ..
AV d and AV ,, can be obtained from an incremental version of equation (5.21) as
AV =

q AIE'ql + x'dAlq

144

Finally, substituting AV d , AV 4 , and then Ala, and A1,, from (5.13) with re = 1) into equation

 

 

(5.27) we obtain
V,° o v °
A|V,| = “61°.qu +Xe|Vfolcos(8 —3°-) A8+,—9—V V,° -'A|E ,|
v ° ' v ° '
" x" -A|E' |-—"—-—,x—--|v; |sin(8°— 3°) A8

lV,|°.x',,+x,, IV ,|° x'd+x,,

 

 

1V;1.( x" Vd°cos(8°-B°)- .xd V,°Siﬂ(5°-B°1)'A5

 

 

IV,|° x,,+x,, xd+xe

V° x,

+_‘l_. __

|V,l° x'd +x A1541

= K5 - A8+K6 - AIE' 41 (5.28)

where
K = _|Vf°|.-(—x‘1——V °cos(8°—B°)- x'd V°sin(8°-B°)) (529)
5 IV,|° xq+xe d x'd-t-xe q "
O

K, _ 1’3. x. 15.30)

+|V,|°'x'd+xe

Equations (5.30), (5.31) and (5.32) below constitute the basic equations for the simpliﬁed lin-
ear model, which is implemented in the block diagram shown in Figure 5.4 where the

dynamic characteristics of the system are expressed in terms of the K constants K.-K6

MA8+DA8 = ATm-AT (5.30)

. K 3K4
APG =K,-A8+K,-A|E,,’| (5.32)
AIV,| = K5-A8+K6-A|E',,| (5.33)

145

 

 

41

 

 

 

 

Ks
A1V,1
{ 3F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

K
- , , ’ + + AP“,
Avref 5x31321211 + K3 , NE q K2 _ . l
'1' _ 1+K3T do 5 + AI, M82 + DS
ux Decay Dynamics APO Rotor angle Dynamics

 

 

K4

 

 

 

 

 

Figure 5.4 Block diagram of the simpliﬁed linear model of a synchronous machine
connected to an inﬁnite bus. through an external impedance.

146

5.3 Application Examples of the Bifurcation Subsystems
Method on a SMIB Model

Assume that under some system stress, no bifurcation in the model algebraic equa-
tions occurs in the SMIB model shown in Figure 5.4. Then, the resulting system bifurca-
tion, if one exist in some subsystem must reside in the model differential equations
representing the generator angle, speed, and ﬂux decay dynamics. Using a power system
toolbox for Matlab [59] will allow us to demonstrate the existence of bifurcation sub-
systems in the model differential equations. Example 1 shows that when the system in
unregulated, the bifurcation subsystem for SN is the machine angle-speed-ﬂux decay
dynamics (A8, A10. AEq’). 1n Example 2, the bifurcation subsystem for Hopf is the generator
angle-speed dynamics (A8, Aw), called the inertial mode [25,31]. These examples are now
presented.

5.3.1 Differential Bifurcation Subsystem for SN (Unregulated Model)

A linearized differential equation model of the unregulated SMIB model is obtained
under heavy local loading system stress, i.e. when the shunt inductance Gs, is increases as
can be seen in Appendix C. The state vector Ax , the A matrix as well as the critical eigen-
value N" and its associated eigenvector are given by

Ax = [A8 Ate AEq' Awkd AEd' Awkq]T

- '1

0 376.9911 0 0 0 0
—0.1980 —0.0000 40.2289 -00000 -00091 0.0000
A = —0.4740 0.0000 —0.5519 0.0000 0.0000 —0.0000
—2.1235 —0.0000 30.6816 —32.2581 0.0000 0.0000
-0.1333 -00000 .00000 —0.0000 -4.0476 0.0000
_-0.0422 0.0000 0.0000 00000155923 46.3934

 

 

147

 

 

__ 0.2767 1 j8.6336_ 60.6281
_4.0431 151000,,23
“SW = .00030 2* = —0.0030 as = 0-
—32 2581 0.5572
-l6.3934 0.0207
1110213-

 

 

The eigenvalues of A indicate the existence of static bifurcation (SN), provided that trans-
versality and degeneracy conditions for this SN bifurcation [34] hold at this point Since at
this operating point the system differential equations experience bifurcation, the Bifurca-
tion Subsystem Method is used to possibly ﬁnd a smaller differential bifurcation sub-sub-
system for SN. In order to test whether we have a bifurcation subsystem, we need to

partition the square matrix A as A = [A“ A,,] where A,, is ixi, A,, is ixj, A,, is jxi, and

A21 A22
A,, is jxj, i+j = 6, i = l, ..,5. The Bifurcation Subsystem Method (Proposition 4.1) says
that when the full matrix A is near singularity and for all possible subsystems of order
j=2,3..,n, A22 is nonsingular, the term [A ,, - A,,l - A,,] - u,* z 0 as 11 -> 11* for u,* e Null(A,,) ,
then the dynamics conﬁned in Ax, = A ,, - Ax are a bifurcation subsystem for SN. In a prac-
tical sense [A , ,] - u,* is never null as u approaches the bifurcation value 11* since we never
ﬁnd the exact value of the bifurcation parameter 11* , and thus
”511141111531
should be a measure of how close the subsystem is experiencing saddle node bifurcation.
The coupling term expressed by
C E 111A12A221A21] ' “* 111

is also never null at 11 close but not equal to 11* , and thus is a good indication of how little
the system external to [A,,] contributes to producing the saddle node bifurcation in [A].

This is indicated by

148

R E 11[A12A21A21] ' “I 111
111A”) ‘ “1‘11
When saddle node bifurcation is close to occur in [A,,] as measured by N, then if R << 1,
the saddle node bifurcation in [A,,] is producing the saddle node bifurcation in [A] with
little or no coupling or help from the system external to [A,,] . Bifurcation subsystems exist
if N is small and approximately equals 1* as u —) 11* , and C is very small and much less
than 71* for all u —~> 11* . This second condition requires R to be much less than 1.0.

The subsystem used for testing are decided by the ranking of the participation factor
or right eigenvector elements except that the dynamics 8 and or are considered together
since they are part of the inertial dynamics. Note that both N and C are equal to or less than
1* = —0.0030 for all subsystems of order greater or equal to third order, which implies that
the bifurcation subsystem is (A8, Aw, AEq‘). Note that N = 71* and R = C/N is less than one
for any subsystem of order equal to or greater than three, indicating that the third order
model is the smallest bifurcation subsystem, but that the fourth and ﬁfth order models are
all bifurcation subsystems. This third order model is thus the bifurcation subsystem since it
is weakly coupled (R <<1) to the dynamics of AE,‘ and A111,“, from the results in rows 4 and

5 in Table 5.1.

149

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

subsystem -1
A,, 141242242111,- [Arr] ' "i N C R
145] 4.8397e + 12 inf
4.8397 6+12 0 0
0 0.1237 0.9944
A5 1 0019 0.1244
[Am] 10-1237 10.12431
' As 0 0.0019 0
-3 . 1.88731: - 04 .0754
A2,“ —0.1887 10 0:0002 000”
1 4,3000 -0.0016
:05] 0 0.0019
—0.1887 -3 0.0002
10 0.003 _
AEq 0,0000 -0.(X)16 1.8873e 04 0.0626
A111,“, 0.0000 .0.0017
- A8 ’ ‘ ' 1
Am $0043 0001: 0.003
' -13 _ _
AE‘q 0.0218 ,0 -0.0016 4.47e 14 1.49e 14
AW)“, —0.2474 .0.0017
, A54 1-0-3719. 1-0-0001.

 

Table 5.1 Computational results for identifying the bifurcation subsystem
for SN bifurcation in the SMIB model.

 

150

This bifurcation subsystem is quite obvious from Figure 5.5 since three integral
blocks have inﬁnite gain when A = s = 0 which implies that the states A8, A0), AE,‘ are con-
nected by inﬁnite gain blocks in a loop when saddle node bifurcation occurs. This explains
why the ﬂux decay and mechanical dynamics have both been argued to be associated with
saddle node bifurcation when in fact both are part of the bifurcation subsystem. It should be
noted that the saddle node bifurcation could be due to inertial dynamics or the ﬂux decay
dynamics experiencing bifurcation and yet the bifurcation subsystem (A8, Am. AEq‘) would
still be from Figure 5.5 because there is a loop with inﬁnite gains relating (A8, Am, AEq’).

In order to see whether the information obtained from the participation matrix indi-
cates similar conclusions, the participation factor have been computed for this case and
summarized in Table 5.2. This information indicates the only state with signiﬁcant partici-
pation in this SN bifurcation is the AE,’, which is one of the states in the bifurcation sub-
system. The eigenvector u* is also a poor prediction of the bifurcation subsystem since (A8,
A0). AEq‘) are predicted to be associated with the dynamics experiencing bifurcation.

It should be noted that the bifurcation subsystem does not necessarily contain the
center manifold dynamics since no formal theory has been presented that suggests so for
this saddle node bifurcation. The fact that R<<1 as u -> 11* so that the coupling of the exter-
nal system is not affecting the conditions for bifurcation
[A,, —A,, - A,, -A,,] - u,* = [A,,] . u,* long before bifurcation occurs in the full model
matrix [A I and in the bifurcation subsystem matrix [A ,,] at the value of 11* , suggests but
does not prove that (i) the center manifold of the full system is contained in the bifurcation
subsystem, or (ii) that the bifurcation subsystem necessarily contains any portion of the

center manifold dynamics. The proof is beyond the scope of this thesis.

151

Several factors suggest the bifurcation subsystem contains the center manifold

beyond the discussion above are:

(i)

(ii)
(iii)

(W)

(V)

(Vi)

a root locus of the unregulated SMIB model shows that the eigenvalue associated
with AEq’ is the one approaching zero;

the participation factor indicates AEq‘ is by ar the largest element;

the test matrix [52] Ks for saddle node bifurcation in the full system to be derived for
the unregulated model is K, - K,K,K4 < 0 and thus depends on the parameters of the
bifurcating subsystem; .

a test matrix T developed by [51] for an equivalent ﬂux decay model AE’, = TAE’, is
T = (—1/T’,,,,)[1/K3 — K,K,/K,] for the SMIB model shown in Figure 5.5, is singular
when the full system is experiencing saddle model bifurcation. This test matrix also
depends on all the parameters of the bifurcating subsystem. This test matrix if it
exists assures that loss of causality has not occurred and that saddle node bifurcation
has not occurred [51] in mechanical dynamics K, at 0;

the test matrices T and KS for saddle node bifurcation in the full system, do not theo-
retically guarantee that bifurcation occurs in ﬂux decay or inertial dynamics respec-
tively as their derivations might suggest, but solely that a saddle node bifurcation in
the full system exists. This conﬁrms the above result that as long as K, at 0 , both T and
Ks and are singular at the same point, i.e. the point of bifurcation. This would suggest
that the bifurcation subsystem (A8, Am. AEq’) containing all the parameters in both T
and Ks and no others, contain the center manifold of the full system model;

the fact that the ﬂux decay eigenvalue in a root locus is the one approaching zero [59]

as the term K,K4 increases and that the participation factor so heavily weighs AEq'

152

might suggest the center manifold is just the AE,‘ dynamics, but the fact that both T
and K3 are both singular at saddle node bifurcation, and both depend on all the
parameters of the bifurcation subsystem and no others, the center manifold would

appear to lie within the bifurcation subsystem.

Without determining the center manifold, it is impossible to say whether it is tangent
to the eigenvector of AE,’ in the (A8, Am, AEq’) bifurcation subsystem subspace, or even
whether it lies outside the bifurcation subsystem. It is clear that the test matrices T and K5
are diagnostic in saying the center manifold lies solely in the inertial or ﬂux decay dynam-
ics since both are zero at the bifurcation. The above results (i-vi) suggest that the bifurca-

tion subsystem may well contain the center manifold but no proof exists that it does.

AIE’ql

153

 

 

 

 

 

 

 

 

A8

 

 

V

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)

Figure 5.5 Block diagram of the a linearized machine model showing the
bifurcation subsystem and the test matrix T.

 

 

 

State A8 A11) 45,, A‘de A54,
Participation 0.(X)00 0.0000 0.2088 0.0000 0.0000
factor

 

 

 

 

 

 

 

Table 5.2 Participation factor results for the SN bifurcation in the SMIB model

 

154

5.3.2 Differential Bifurcation subsystem for Hopf (Regulated Model)

A linearized differential model of a regulated SMIB model is obtained when the
transmission network inductance x, or the power transfer to the inﬁnite bus has been grad-
ually increased. The input parameters for this bifurcations are given in Appendix E. The

state vector Ax , the A matrix as well as its eigenvalues and the critical one 71* , are given by

T
Ax = [A6 A0.) A5," Awkd AEd' Aqu AVA: AVR]

' 0 376.9911 0 0 0 0 0 6

4.1395 0 -0.0868 -0.0781 —0.0072 —0.0318 0 0
4.0542 0.0000 -2.0818 2.0781 —0.0240 -0.1056 0.0000 1.9373
A = -2.9066 0.0000 30.7985 --33.5716 0.0000 0.0000 0.0000 0.0000
4.0934 0.0000 0.0000 0.0000 -8.3982 6.5047 0.0000 0.0000
—0.9893 0.0000 0.0000 0.0000 15.6517 49.6571 0.0000 0.0000
3.2771 0.0000 4.5117 -8.5604 3.0798 1.5531 4.1499 0.0000
47.0239 0.0000 -163.5591 486.1236 56.3153 247.8250 14.9613 —l72.986Q

 

 

471.59

42.99

eig(A) = ‘2559 N" = 0.09: j7.16
0.09 :1: j7.16

-394

__ 1.46 :1; j0.4l_

 

 

The eigenvalues of A indicate the existence of a swing mode of frequency

to = 7.16 rad/s
Since at this operating point the system differential equations are at bifurcation, it is possi-
ble to ﬁnd a diﬁ’erential bifurcation sub-subsystem for Hopf. Proposition 4.3 implies for-

mulating the matrix

155

 

— 171.59 - 17.16 0.3285 + )08108

— 32.99 _ j7.16 — 0.0153 + j0.0064

— 25.59 — )7. 16 0.0424 + 10.0114

6,3,3, = 0.09 ,. = 0,09 ,,., = - 0.0022 - )0.0591
0.09 _ )1432 — 0.0349 + )00082

- 3.94 - j7.16 - 0.0499 - j0.0160

— 1.46 + j6.75 0.3991 + j0.0264

_ — 1.46 - )6.57_ _ 0.3991 + j0.0264_

 

 

 

In order to test whether a bifurcation subsystem exists, the B matrix is partitioned as
3 = [8“ 8‘2] where 3,, is ixi, 3,, is ixj, 3,, iiji, and 3,, iijj, i+j= 8, i = 1, ..,7, and

821 322

form column vectors a, = [u,...u,-], i=1....7. Note that

N = "[B,,]-u,*||, ; Cs||{B,,B,,B,,].u*,|| and R = C/N

are deﬁned as for saddle node bifurcation for all subsystems where 3,, is nonsingular. N
must be less than 2* and C must be << N“ as u —> 11* for a bifurcation to exist. The values
of N and C are less than 21* = 0.09 and R<<l only for the second order subsystem, indicat-
ing that (A8, A10) is the bifurcation subsystem for this mode of oscillation. Since for this sub-
system R = 0.0551 the bifurcation subsystem (A8, A10) produces the bifurcation in the full
subsystem without coupling to the external system.

The participation factor information for this bifurcation is presented in Table 5.4.
These results indicate that the generator inertial dynamics {A8, A10} have higher participa-
tion in the occurrence of the bifurcation then the rest of the states. It is worth noting that
these two states form the bifurcation subsystem obtained using the bifurcation subsystem
method, and thus the participation factor can indeed sometimes agree with bifurcation sub-
system method results. Although participation factors can sometimes indicate the states in

a bifurcation subsystem, there is no theoretical assurance the bifurcation subsystem is cor-

rect in assessing the center manifold dynamics. The large drop off in participation factor

156

elements for states other than (A8, A01) may indeed contain the center manifold. However
the fact that KD will be shown in Section 5.4 to depend on parameters outside (A8, A18) sug-
gests that the bifurcation subsystem may only contain a portion of the center manifold
dynamics. Since may only be true since D = 0 and the bifurcation subsystem must be (A8,

A00). If D ¢ 0 then the bifurcation subsystem may be larger if one exists.

157

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i subsystem -1
Art [812322321] ' “i 311'“,- N C R
, [A5] 5.7393 - 12.5040 5.7393 - j2,5040 6.2636 6,2617 09997
M - o 0045 0 0551
Am [ 0] 0.0313 + 10.0748 0.0812 - .
2 00017- 10 0041 0.0003 — 10.00351
45 0 0.0313 + j0.0748 0 8 740
3 Am -.0 0020— 10.0051 - 0.0034 - 10.0045 '3 07 0-3 09324
135 — 0. 0280- 103729 - 0-0243 -1’03729
1 A8 ,
4 Am 0 0.0313 + )00748
A,, -0. 0018- )0 0004 - 00°32 - 100002 0.5011 0.4967 0.9911
‘1 __ 0 0164-10 4964 — 0.0288 - [0.4937
“"14 — 0.0003 - )00055
1- A5) . P T
A,,, l 0 0.0313 + j0.0748
., A3,, - 0.0016 — 10.0005 - 0-0030 - 100001 0.6046 0.6023 0.9961
‘ A — 0.0153 _ j0.4953 - 0-0280 - 10.4939
“’14 0,0000 _ 0.0003 — 10.0055
_ N54 _ 0.3261 + )0.1039_ . 0.3214 +1'O-10481
1 A8] , , ,
Am ' 0 0.0313 + j0.0748
6 AB 0 - 0.0014 + 0.0006
q _ _ - —0.0227— 10.4922 0.4994 0.4936 0.9884
A,,, 0.0106 )04953
M 00000 -00003 — 10.0055
45,) 0.0000 — 0.0032 + j0.0008
A111,“, _ 0.0000 _- 0.0046 - j0.0014j
- - r .
A8 r 07 0.0313 + j0.0048
A0) 0 - 0.0014 + j0.0006
A,,, 00000 - 0-0003 - 10.0055 0.5008 0.4942 0.9868
"4 0 0000 — 0.0032 + j0.0008
AE - .
d 011m - 0.“)46 — [0.0014
A111,, 0000,, _ 0.0370 + 10.0020
bAVA _,

 

 

 

 

 

 

 

 

Table 5.3 Computational results for identifying the bifurcation subsystem
for Hopf bifurcation in the SMIB model: Inertial mode.

 

158

 

 

 

 

 

 

 

 

 

 

state Participation factor
A8 0.0132-j0.0101
Acu -0.0129+j0.()105
AEq’ 0.0004-j0.0008
AW I1 1! j0.0001
AEd’ j0.0002
Av)“, 10.0002
A V A, -j0.0001

AV R 0

 

 

 

 

Table 5.4 Participation factor results for the
inertial Hopf bifurcation in the SMIB model

5.4 Effect of Hard Limit Discontinuities on Stability Analysis
Diagnosing and assessing stability problems in the SMIB model is dependent on whether
the system is regulated by an active excitation control or whether it is unregulated. The
effect of the hard limit discontinuity on stability analysis can then be viewed by means of
the damping and synchronizing torque coefﬁcients KD and Ks since both KD and Ks are
determined in terms of K,-K(, constants when the system is regulated and unregulated. Con-
ditions for system stability in terms of constraints on the K-constants are derived and sum-
marized in proposition 5.1-3 for before and after the hard limit discontinuity that disables
the generator exciter occurs.

The damping and synchronizing torque coefﬁcients KD and K8 are derived for the
case when (a) the generator terminal voltage is unregulated, (b) when a thyristor type exci-

tation system is used, and (c) when an IEEE dc Type 1 excitation system is used. The

159

effects of measurement or sensor dynamics are included in this analysis. Finally important
conclusions are determined on how the variation of the system K-constants affect the
damping and synchronizing torques to cause a Hopf or SN bifurcation in the SMIB dynam-

ics. First, a general formula for AT D , ATS , K o and K s will be derived in the next section.

5.4.1 Synchronizing and Damping Torques

Consider a change in electrical torque following a disturbance at an arbitrary frequency.
The restoring forces that act to this change are in form of braking torques that develop in
phase with machine rotor angle and in phase with machine rotor speed [55]. Indeed, the
change in electrical torque of a synchronous machine can be resolved into two main com-
ponents

AT, = KSA8 + KDA8 s ATS + ATD (5.34)
where ATS is the component of torque change in phase with rotor angle perturbation A8
and thus referred to as the synchronizing torque, and ATD is the component of torque in
phase with the speed deviation A00 and thus referred to as the damping torque. K D and K s
are called the damping and the synchronizing torque coefﬁcients, respectively. System sta-
bility depends on the existence of both components of torques for each machine. Positive
synchronizing torques assure restoring of the rotor angle after a small incremental change
of this angle, whereas positive damping torque is necessary to damp out any oscillation fol-
lowing a disturbance. A necessary and sufﬁcient condition for the system to be stable is that
both damping and synchronizing torque coefﬁcients K o and K s be positive [51]. A
decrease in either coefﬁcient implies a destabilizing effect and therefore, evaluation of

these coefﬁcients can be used as a proximity measure to oscillatory instability or to an

160

increasing change in the machine rotor angle. First, ATD, ATS , K o and K S are deﬁned as:

 

 

 

AT, _ AT:
ATS=Re[ A8 , J and ATD=Im[-E-5- _ , 1 (5.35)
s = )0) s — jto
. S = __S
KS - —A8 and KD 01A8 (5.36)

A transfer function ATe/A8 that is independent of the type of excitation system used, may
be derived from the block diagram in Figure 5.4, in which G,(s) refers to the transfer func-
tion representation of the AVR and the exciter, and 03(5) represents the transfer function of
the ﬁeld circuit. From the block diagram we see that

E!

A q

 

 

 

5'41}

E!

A «1

 

 

= _ 11,6,(s) . A8 - K,o,(s)G,(s) - AS 4.0.0103“) ' A115311

By grouping terms involving A

 

3",] and rearranging,

— K,G,(s) — KSG,(s)G3(s),
1+ K60,(s)G,(s)

 

A

 

E’ql = A5 (5.37)

Replacing (5.35) in the expression of AT, given in equation (5.31),

K4G3(S) + KSG‘(S)G3(S) .

1 + K6o,(s)o,(s) A5

 

 

Finally the general transfer function ATS/A8 is given by equation (5.38).
AT, K,G3(s)(K4 + K50,(s))

T8 1 ‘ 1 + K6G,(s)G,(s)

 

(5.38)

5.4.1.1. Unregulated system (G,(s) = 0)
A system that is not under an active excitation control is called unregulated. In such condi-
tion, K 5 , K6 and G,(s) in the block diagram in Figure 5.4 can be set to zero. The transfer

function ATS/A8 is then equals to

AT,

[61

AT K2K3K4

__ _. K __.______
A5 1 1+K3T'dos

AT, K2K3K4 ° (1 - ijBT’do)

.—- = K1-

AS . 2 , 2
s=1w 1+0)(K3Tdo)

 

(5.39)

The damping and synchronizing torques ATS and ATD are the real and imaginary parts of

ATe/AS in (5.39) respectively and are equal to

6 K2K3K4 8 0
AT = K-A ———-A 5.4a
s 1 1+(DZT'42 ( )
AT K K ——K3 ° T,“ A8 54%
= . - a) . .
D 2 4 1-1-0)2T’d2 ( )
where T’d = K3T’d0
From Equation 5.34. Ks and K0 are given by
K3'T'd
2 , 2
1+0) T d
K KqK
[(5 = K,-—3—'—4— (5.4113)
2 , 2
1+0) T d

From these two expressions, we can see that the damping and synchronizing coefﬁcients

are positive if they satisfy the conditions:

KD>O {fandonlyif K4K2>0
. . K2K3K4
1+0) T’d

5.4.1.2 Thyristor-Type Excitation system

This is a simpliﬁed thyristor type excitation system that includes only the elements that are
considered necessary for representing a speciﬁc excitation system, namely the terminal
voltage transducer time constant TR with gain K R , the exciter with gain K E and time con-

stant TE. The transfer function is given by

G (S) ___ IAEfdl = "KEKR
t’ A|Vt| (1+sTE)(l +sTR)

 

(5.42)
For typical values for the terminal voltage transducer time constant TR and for the gains and
time constants in fast exciters, acceptable bounds for TR, the exciter gain TE and exciter
time constant TR satisfy
KE»1; TE<<1 and TR<<1
Substituting G,(s) in (5.42) into the general transfer function in equation (5.38), and col-
lecting the powers of s in the numerator and the denominator, we obtain
AT, 00 + ass + azs2

__ = K1 _ (5.43)
A5 b0~1»bls+bzsz44:353

 

Where

a1 = K2K3K4(TE+TR)

02 = K2K3K4TETR

b0 = 1+K3K6KEKR 5K3K6KEKR
b1: T’d+TE+TR

()2 = TETR+T’d[TE+TR]

b3 = T’dTETR

Evaluating ATS/A6 at s = j0), we get

AT 0 -a (oz-1'0 0)
_55 =Kl— "22, 1‘ 2 (5.44)
A ,,]-a, bo—bza) +1(b10)—b30))

 

Therefore ATS and ATD are given by:

[an - azwz] [b0 — b21021 + alwlblw - b3w3]

AT = K ~A5-
S ‘ [ha-b2w2]2+[blm-b3w3]2

A8 (5.45a)

 

[aO—aZwZHbl - 17302] —al[b0— 12202]

AT =
D [ho-b20212 + [blm-b3m312

 

. (ma) (5.451»

163

(a) Frequency Range (0)« 1)

Setting the of terms to zero in the numerator and the denominator of K o and K 5 yields,

ATD aobl -alb0

D: — _
wA503=0 bzo

(K2K3K4 + K EKRK2K3K5)(T’d + T5 + TR) — K2K3K 4(15 + TR)( 1 + K3K6KEK
(1 + K3K6KEKR)2

R)

or approximately
(T’d+ TE+ TR) K (TE+ TR)

 

 

 

KDEK5K2 — 4 2 (5.4621)
. K3K125KEKR K6KEKR
by assummg [(5 >> 1
Similarly, the synchronizing torque coefﬁcient is
K _ K aobo _ K2K3K4 + KEKRK2K3K5
s - 1 — - 1 —
b2" 1+K3K6KEKR
K 2 K 2
or, K s s K 1 K (5.46b)

—K .____ ._
4 K6KEKR 5 [(6

Therefore positive damping and synchronizing torques are obtained for the following con-

ditions:
T£+TR
K >0 ifwehave K >KKK -,———
D 5 4 3 6 Td+TE+TR
K K K
K >0 ifwehave K >K ---3+—“.-—2
5 1 5 K6 KEKR [(6

(b) Frequency Range (0)»1)
In this range, the terms involving the highest power of 0) dominate. Hence, keeping only

these terms from equations S.45(a—b), approximate expressions for K 1) and KS may be

obtained as,

- azb3 1
K0 - W =KZK4‘m (5.473)

Similarly, the synchronizing torque coefﬁcient may obtained by retaining only the highest

powers of 0) in the numerator and the denominator

164

 

s - 1‘ [232036 " 1 13ng
K K K
or, K5 5 Kl——2——3—5§ (5.4713)
“32(7’40)

5.4.1.3 IEEE dc Typel Excitation System

The gain of the voltage transducer block is usually close to unity, it time constant is small,
in the 0.2 to 0.4 range. The ampliﬁer gain KA is typically on the range 25 to 400, its time
constant TA is typically in the range 0.02 to 0.4 sec. The power system stabilizer parameter
values are KP in the range 0.02 to 0.1, and T; in the range 0.35 to 2.2 sec. The block dia-

gram of this system has been shown in Figure 2.2, from which the transfer function is given

by
25995400 0.0251As0.1 0585513
—l.0$KESI.0 0.551‘5310 550.05

The transfer function for this excitation system is given by

-KAKR(1+ sTF)

09(5) = [(1+sTA)(1+sTF)(55+KE+STE)+‘KAKF]'(l+STR)

 

(5.50)

Substituting Ge(s) into the transfer function in equation (5.38) and collecting terms of pow-

 

 

ers of “s” in the numerator and denominator we obtain
AT! r10+ass-1-0252+03.33+a4s4
= - 5.51)
A5 'bbb2b3b4b5 (
0+ l5+2s-1-3.9-1-454-55
Where

a0 = MK2K3K4+K2K3K5KAKR
5K2K3K5KAKR since KAEM

[65

a4 = K2K3K4TATETFTR a T04
b0 = M + K3K6KAKR
s K3K6KAKR since K3K6KAKR » M

5K3K6KAKRTF+Tbl
b = T

4 b4

b5 = T’dTATETFTR = Tb5
M = SE+KE

= K3T’do

'5 KAKF+TE+M(TA+TF+TR)

T
T
T = KAKFTR + (TE-1» MTR)(TA + TF) + MTATF + TETR
T TATETF+TETR(TA+ TF)+MTATFTR

T

- KAKF+K3K6KAKRTF+M[T’d+TA+TF+TR]+TE

@-
I

T13, = TE[T+TR]+M[TATF+T’4TR]+[TE+MT+MTR][TA+TF]
Tb? = [T’dTE+MT’dTR+TETR][TA+TF] +TATF[TE+MT’d+MTR] + T’dTRTE

Now, evaluating AT/AS at s = j0), we get

 

ATc A0 ~1-jAl
= K -..—___... (5.52)
A8 ”1.0) ‘ 30"131
Where
2 4
A0 = ao-azw +040)
Bo = bo- b20)2 + b41104
B1: b10)-b30)3+b50)5
The synchronizing and the damping torques can ﬁnally be deduced.
148 +A£i
ATS = [Kl -°—‘2%]-A5 (5.5323)
BO+B1
A03, - .4130
ATD = —-2——2—— - (0)A8) (5.5313)

BO+Bl

166

(a) Low" Frequency Range ( 0) « 1)

In this range of frequency, we can neglect the terms in ATS and ATD of higher exponents.
“obt — (11 b

b

0

0 (5.54)

Ill

K D
Evaluating the numerator yields to

Replacing in equation (5.54) gives an approximate expression for K D, for 0)<<1.

 

 

 

 

 

K K T T Tb‘ K2 Ta‘ 555
D ‘ 5 K F’ R+KAKRK3K6 ‘ KGKAKR ("20
Similarly,
5‘ 1 132., ‘ 1 K3K6KAKR
K
or, KS :—_-K,—K5 - 1?? (5.5513)
6
Therefore,
Ta
K >0 ifwehave K5>K4K1K6- ‘
D - T +K K K K T —T
131 A R 3 6( F R)
KS>0 ifwehave K1>K5-,—(3
6
(b) Frequency Range (0) »1)
TaTb Ta KKK
_ 4 5 _ 4 _ 2 3 4
K0... 2 2 ..T 2 .. 2T, (5.56a)
7’wa b5“) (1) d

Similarly, the synchronizing coefﬁcient may obtained by retaining only the highest powers

of 0) in the numerator and the denominator

4
(azbz—alb3)m _ K (azbZ-alb3) ~ K2K3K4
1‘ 2 6 " 1' 2 2 = 1' 2 , 2
b30) b30) 0) Ta,

KS=K

 

 

(5.56b)

167

5.4.2 Bifurcation Tests for the SMIB

Given that the system K—constants K2, K3 and K6 are positive, bifurcation tests for SN and
Hopf Bifurcations in both the regulated and unregulated SMIB models have been derived
in the previous section and are stated in the form of two propositions and two corollaries,

followed by their proofs.

Proposition 5.1 (Unregulated system)

(i) A sufﬁcient (and necessary) condition for a Hopf bifurcation to occur in the unregu-
lated SMIB model is that [(4 < 0.

(ii) A sufﬁcient (and necessary condition) for a saddle node bifurcation to occur prior to
Hopf bifurcation in the unregulated SMIB model is that K 1 - K2K3K4 < 0 when [(4 > 0

Proof:

(i) Since all the terms in the expression of the damping coeﬁicient for the unregulated
SMIB given in equation (5 .4 I a) are positive except K4, then its sign depends on the
sign of the parameter K.,, i .e. KD<0 when K, < 0.

(ii) Similarly, the expression for the synchronizing torque coefﬁcient K s in equation
(5 .4 I b) indicates that it is negative if and only if K 1 - K2K3K4 < 0 whenmw=0 as is the
case for saddle node bifurcation. The condition K4 > 0 assures that Hopf bifurcation

has not occurred.

Corollary 5.1 (Unregulated System)
The unregulated system is stable if (and only if) the constant [(4 lies in the range

0<K < K1 (1+0)2T’2)
4 K2K3 d

 

168

Proof:

Equations 5 .41 (a) and (b) indicate that suﬁicient synchronizing forces exist in the system

 

KKK
ijf Ks: K1-—2-—23—‘1- > 0,whichcanberewrittenas
1+0) T’d K
1 2 2
K < (1+0)T’)
4 K2K3 d

Since stability is assured only when both synchronizing and damping torque coefﬁcients
are positive, this condition combined with K4>0 prove the claim of the corollary.
Proposition 5.2 (Regulated system, Inter-area mode)

When f<<1 and K4 >0 under voltage regulation, (i) A necessary condition foraHopfbifur-
cation to occur in the SMIB model is that K5 < 0 and (ii) A suﬁicient condition for a saddle

node bifurcation to occur in the SMIB model is that K ,, K 2, K 5 and KO satisfy
K 5" 2

l - K6

K 5" 2 K 4" 2
1-.-KT — K 3" 6K R

 

< 0 for Type I exciter

 

< 0 for Thyristor exciter

Proof:

(i) Equation 5 .46(a) and 5 .55 (a) indicated that sufﬁcient damping forces exist in the

 

SMIB systemif
(T’d+TE+TR) (TE+TR) _
KDEKSKZ 2 - 4 2m— 0 for Thynstor type
K3K6KEKR 6 E R
K K K2[T T Tb‘ ] K2 Ta‘ > 0 f T I 't
s -— - +—— — -—— or ype excz er
0 5 K6 F R KAKRK3K6 4 KéKAKR

which can be rewritten as

-1
K , and K > K - 1 T -T +——-‘——
5 (Td+TE+TR) 5 4 KAKR( F R KAKRK3K6]

 

 

for a thyristor type exciter and type 1 exciter, respectively. By observing the terms on
the right hand side of both inequalities of K 5, we can conclude that they are generally
small and either positive or negative depending on the sign of K4. Thus, K 5 is my??-

ciently positive, no Hopf bifurcation occur; and when K 5 is suﬁiciently negative Hopf

169

bifurcation occurs.

(ii) Similarly. equation 5 .46(b) and 5 .55 (b) indicated that sufﬁcient synchronizing forces

are positive in the SMIB system if

: _ . ___— - - — > 5 _. . .—
KS_Kl K4 K6KEKR K5 K6 0 and KS K1 K5 K6 > 0
for a thyristor type exciter and type I exciter, respectively. If K 5 is suﬁ'iciently positive,

large and increasing, saddle node bifurcation occurs.

Corollary 5.2 (Regulated System)

The SMIB model is stable for f< <I if the constant K 5 lies in the range
T

- a' < K < [(1—K6 Type I exciter
6 Tb1+KAKRK3K6(TF-TR) 5 K2

 

K 4K3K

TE+TR K1K6 K4

K4K3K6'ﬁ75—TT]; <"s< x, -KEKR

 

Thyristor exciter

Proof:

(i) For thyristor exciter, equations 5 .46( a ) and (b) indicate that suﬁicient damping and

synchronizing forces exist in the system if
(T’d+ TE+ TR) (TE+TR)
_ 4 2—
K3K§KEKR K 6K 5" R

 

KD=~=K5K2

K K K K2 K K2 >0
8— 1 4 KGKEKR 5 K6

These two conditions may be rewritten respectively as

TE+TR

K1K6 K4
6'T",4~T,:.+TR

K 2 _ K 5" R
(ii) For a Type I exciter, equations 5 .55 (a) and (b) indicate that suﬁ‘icient damping and

K5 > K4K3K

 

and K 5

synchronizing forces exist in the system if
K :: K ...?[7' ..7‘ +._n,l__]-g.153i >0
0 5 K6 F R KAKRK3K6 4 K6KAKR

K,

KsK—K-—" >0
S 15K6

170

These conditions may be rewritten respectively as

a <
+KAKRK3K6(TF- TR) 5 K2

 

 

K5 > K4K3K6 - Tb]
Since stability is assured only when both synchronizing and damping torque coeﬁi-
cients are positive, conditions for positive damping are combined with conditions
for positive synchronizing for the thyristor type exciter and for the Type I exciter,
which leads to the desired result of the corollary. If K 5 is below the lower limit or K 5

exceeds the upper limit Hopf bifurcation occurs.

5.4.3 Conclusions

It is our desire to hopefully ﬁnd a subsystem of the full model which experiences, produces,
and causes the same bifurcation in the full system model. Determining the center manifold
dynamics requires ﬁnding a nonlinear transformation that is often given in terms of a Tay-
' lor series and may be inﬁnitesimally linked to all the states of the model even if it virtually
lies in a subsystem of the full model which experiences, produces, and causes the bifurca-
tion. Determining such a subsystem if one exists is essential to diagnosing the location,
class, cause, and cure for a bifurcation, and that is important from an engineering sense.
Determining the center manifold is thus not what is desired, but a subsystem of the full
model which experiences, produces, and causes the bifurcation, and which is normally of
larger dimension than that of the center manifold. One may be able to diagnose the loca-
tion, class, cause and: cure without resorting to ﬁnding the center manifold transformation,
or its approximation.

The subsystem which experiences bifurcation may or may not produce bifurcation in

the full system since the structure or parameters external to the subsystem may help pro-

171

duce the bifurcation both in the full system and subsystem. This is especially true if the
geometric coupling of the bifurcating subsystem and the external system in the direction of
the bifurcating eigenvector is not zero but approaches zero. Thus the subsystem that pro-
duces the bifurcation experienced by the subsystem and the full system may be of larger
dimension than the subsystem experiencing bifurcation. The subsystem causing bifurca-
tion may be still larger than the subsystem experiencing or producing the bifurcation in the
full system model and the subsystem. The subsystem that causes bifurcation must include
all model parameters, controls, and discontinuous changes that cause the bifurcation to
develop. The discontinuous changes include under-load tap changes, switchable shunt
capacitors switching changes, equipment outages, over-excitation limit relays,...etc. The
controls include maximum excitation limiter controls, under-load tap changer controls,
switchable shunt capacitor controls, and excitation system controls.

The results of this chapter along with chapter 4 imply the bifurcation subsystem
method may provide such a tool for identifying the subsystem that experiences, produces,
and causes a speciﬁc kind and class of bifurcation, since
(a) the theoretical conditions for existence of bifurcation subsystem was applied to show

that they exist and can be determined. The bifurcation subsystem showed some prom-

ising results in identifying the subsystem that experiences the bifurcation in the full
model;

(b) the sensitivity matrices K5, KD and T were derived in two recent Ph.D dissertations
with the purpose of identifying the subsystem that causes and produces the bifurca-
tions, and which may also contain the center manifold. It has been shown that the test

matrices Ks, and T are not necessarily associated with the subsystem that experi-

(C)

(d)

172

ences, produces, or causes the bifurcation, but the parameters of the system they
depend on can help identifying the bifurcation subsystem;

The results on Hopf bifurcation indicate that the bifurcation subsystem obtained
(Example 2) which experiences the bifurcation of the full system model may be too
small to track what produces and even less capable of tracking what causes and cures
that bifurcation, since the test matrices suggest thath depends on K, and K5 which
lie outside the bifurcation subsystem (A5, A03). This result may be true since the
(A8, A00) dynamics are experiencing bifurcation for all u , and this may imply that the
bifurcation subsystem producing and causing the bifurcation should be identiﬁed as
u —> m , and not at the point of bifurcation, so that the subsystem captures the causes
and cure for that bifurcation;

the determination of what [(5, KD and T depend on for a generator with exciter, a thy-
ristor based exciter, and a dc] model exciter, gives implicit information for determin-

ing the subsystem which experiences the bifurcation in each case.

VI

A Diagnosis for SN and Hopf Bifurcations
in a Discontinuous SMIB Model

6.0 Introduction

The goal of this chapter is to perform a diagnostic study to assess the structural causes,
operational stresses, and stabilizing requirements for the occurrence of SN and inter-area
modes of oscillations that occur in the generator inertial and flux decay dynamics of a sin-
gle-machine-inﬁnite-bus model (S MIB). The tool for this study is based on the bifurcation
tests derived and concluded in propositions 5.1-2 and corollaries 5.1-2 of Chapter 5.

The diagnostic study is performed by formulating three study models: (1) a power
transfer model where the effects of both the network inductance xe and real and reactive
power transfer are investigated: (2) a local loading model where the eﬂect of local loading
on system stability may be investigated; (3) a power transfer-local loading model where
the effect of local load model and transfer can be investigated. The SMIB local load may
be capacitive, inductive or some combination. The generator exciter may be enabled (regu-
lated case) or disabled (unregulated case).

The objective of the analysis in this chapter is to establish that (i) increased local
load causes quite different bifurcation phenomena in the regulated and unregulated load

center models and (ii) that increased power transfer causes quite different bifurcations in

173

174

the regulated and unregulated power transfer model: When the exciter is enabled, the

regulated model is vulnerable to Hopf bifurcation for increased power transfer, whereas
when the exciter is disabled, the system is vulnerable to saddle node bifurcation in the ﬂux
decay dynamics as power transfers increase. The analysis also shows that when the local
loading in light-real, heavy-capacitive, the regulated local load center model is vulnerable
to Hopf bifurcation, and the unregulated local load center model is vulnerable to saddle
node bifurcation. The combination local load center/transfer model is then used to
validate, extend, or possibly disagree with the diagnostic conclusions obtained from the

local load center or the transfer models.

6.1 Diagnostic Models Formulation

Based on operational and structural parameter variations of the SMIB, three operationally
and structurally different models may be obtained. This kind of diagnostic modeling is per-
missible by the SMIB model since this model represents a generator connected to a large
system, via a long equivalent transmission corridor with equivalent reactance x, The large
system may be (a) a major bus or group of buses of a power system of very large capacity
compared to the rating of the machine under consideration, or (b) a large load center. Such
a structure allows formulating three study models as follows.

6.1.] Power Transfer Model (Ra: X“: inﬁnity)

In a power transfer model, no local load or equivalent line shunts are modeled, so that the
generated real and reactive power are solely used to satisfy the power transfer requirements
to the ‘large’ system at the side of the inﬁnite bus. Real power is shipped from the generator

to the inﬁnite bus, whereas reactive power may be shipped in either direction across the

175

transmission network. The effects of the power transfer as well as the network inductance
may then be investigated. In the power transfer model shown in Figure 6.1, the system

steady state conditions are computed for given values of P” , Q“, and V,° = 1.010 p.u.

 

V,, = 1.040
9 Vt
XT+Xd I Xe
mo 1 Tom—ﬂ
15'qu _~__,
Q...

Figure 6.1 Power transfer model

6.1.2 Local Load Model

A local loading model is obtained when a shunt constant load is connected to the
generator terminals while the power transfer requirements to the inﬁnite bus are negligible.
The local constant impedance load can include line charging of the transmission line, local
shunt capacitive compensation, or inductance load. “ﬁth xe large, the dynamics of the
generator serving a local load center can be investigated. This results in a model where
stability problems are solely associated with the local load center. Note that since the power
absorbed by the local load in proportional to VZ/Zsh, the local load is capacitive when
X,h<0 and the reactive absorbed is proportional to l/Xm, it is inductive when X,h>0 and the

load is proportional to l/Xsh. Similarly, the real shunt load is proportional to 1/R,,,. The local

load is shown in Figure 6.2

176

 

 

 

V“, = LOZO
V
XT+X‘d t
15000 ‘ 015615 E
E IqZS Rsh xsh Xe large

LoEai-‘Ead

Figure 6.2 Local loading model

6.1.3 Combination Local Load/Power Transfer Model

Under normal operation, power system models serve both local loading and power
transfer requirement patterns, and therefore a diagnostic study of this model is necessary.
The more realistic SMIB model consists of real local loading in combination with power

transfer model.

6.2 Diagnostic Study

From Propositions 5.1 and 5.2, it is clear that since the system constants K2, K3 and K6 are
positive for a generator mode of operation of the machine, the occurrence of Hopf bifurca-
tion in the unregulated and regulated SMIB model depends on the parameters K, and K5
which approximate the sign change behavior of the test matrix KD, respectively. The oper-
ating condit: ‘218 where K5<0 cause undamped oscillations in the generator inertial dynam-
ics when the exciter is active. The operating conditions where K4<O cause negative
damping in the SMIB model with a local load when the exciter is disabled. Similarly, the
terms K 1 —K2K5/K6 and K1 — K2K3K4 which approximate the sign change behavior of the
test matrix K8 are indication to the occurrence of SN bifurcation for the regulated and the

unregulated systems, respectively. These conclusions are based on results in Chapter 5.

177

The scope of this section is to investigate the change in the parameters K4, K5 as well
as Kl - K 2K3K 4 and K 1 — KZKS/K6 , for the purpose of establishing a diagnosis for the occur-
rence of Hopf and saddle node bifurcations in systems subject to relay-type discontinuous
hard limit action. It should be noted that these I(l-K,S parameters are associated with the
bifurcation subsystems that experience saddle node and Hopf bifurcations in a single-
machine-inﬁnite-bus model. The bifurcation subsystem would approximately indicate the
generator and dynamics (bifurcation class) experiencing bifurcation. Investigating how the
key indices for sign changes in test matrices (KS,KD) for occurrence of Hopf (K4, K5) and
saddle node (K 1 - K2K3K4 , K 1 — K2K5/K6) change with load parameters provides important
diagnosis of why the bifurcation occurs and what can be done to prevent it. The parameter
and stress tests that will be investigated are:

(a) Power transfer to or from the inﬁnite bus (P0,, > O and Q” 2 O or Q” s 0)

(b) Local loading: inductive or capacitive shunt X3). > 0 or Xsh < 0 in parallel with a

real load (R,.,)

(c) Network inductance (x,).

In this diagnostic study, the real power transfer P” is varied between 0.1 and 1.0 p.u,
(Pa = 0.1, 0.2, 1 p.u.) , and the reactive power transfer to the inﬁnite bus Qm is either pos-
itive (Q,, = 0.0.1, 0.2, 1 p.u.) indicating reactive power absorption by the inﬁnite bus, or
negative (Q,o = —1.0, —0.9, —0.8, 0 p.u.) , indicating reactive power supply by the inﬁnite

bus. Power transfer levels of $1.0 indicate maximum system rating since for [le = 1 and

Poo 1.0 x s1n(zV‘ AV”) — 2.S|Vt|srn(zV‘ AV”)
—|Vi|2+|Vm||VAcos(zV,-ZV”)

xe

1.0 =

(Q
8
ll

 

= 2,5(_|v°°|2 + IVNHVJ cos(zv, — eve,»

178

so that the system is operating at
[Vt—AV” = 15.1°
lVll = 1.45 p.u.

This level of terminal voltage indicates heavy reactive supply to the inﬁnite bus with large
reactive losses sufﬁcient to bring on instability in the network equations let alone the SMIB
dynamics. Similarly, when P0:, = 1.0 and Qm = —1.0 the system operates at:

4v,_zv,, = 33.5°
[th = 0.72 p.u.

and this level of angle difference indicates heavy real and reactive power loads at the inﬁ-
nite bus and very large 12X losses sufﬁcient to bring on instability in the SMIB dynamics.
Thus P” = 1.0 and Q” = :10 indicate the system is heavily stressed and it is operating at
marginal values of voltages and angle differences.

Similarly, The chosen ranges of the local load R5,, Xsh, and x, are such that
1 s l/Rsh s 6 , and 1 s l/Xsh s 6 result in acceptable system operating conditions and viable
system voltages and angles required for normal operation: When X,.,= 0.17, the resulting
voltage at the terminal bus shown in Figure 6.1 is approximately V t = 0.45 , since
x’d+xT = 0.2 p.u.. Operating: below this voltage level, is considered outside the system
capabilities [59]. However parameter studies in literature [55,56] usually consider the
entire range of RM, X3}: = 0.1.1.0 p.u. i.e, ls l/Rshs 10, and Is l/Xshs 10, as it is the
approach in this study. For this kind of study, when solvability was not possible, the cases
were skipped as will be observed in the tabulated results and the associated ﬁgures.

The results of this diagnostic study are summarized in tables for operating conditions
resulting in bifurcations, and tables for operating conditions where no bifurcations

occurred. The tables indicate the test parameter, the kind of bifurcation tested, the network

179

and the power transfer operating conditions used. The arrows in the tables point to the

range where the test parameter was more sensitive, i.e was more negative.

6.2.1 Power 'ITansfer Model
6.2.1.1 Effect of P,,,, and Q,,.,

The effect of P” and Q,° on the parameters [(4, K, as well as K, -K2K3K4 and
K, —K2K5/K6 is simulated in Figures El.1-E1.4 in Appendix El, while the bifurcation
results and the associated ﬁgures are summarized in Table 5.1. In these ﬁgures, the param-
eters K4, K5 as well as K, — Ksz/K,5 and K, -K2K3K4 are plotted versus sequential 0.1 p.u
step increases in real power transfer P, , and for increases in levels of reactive power trans-
fer Q“, = O, 0.1, 0.2, ..., 1 p.u. or Q” = —l, —O.9, —0.8, 0 p.u.. The simulation results on
bifurcations of this study are summarized in Table 6.1(a) for cases resulting in bifurcations,
and in Table 6.1(b) for non-bifurcation cases. A careful review of these tables indicates
ranges of P” , Q” 2 0 , Q” s 0 when oscillatory and non-oscillatory instability occurs, the
points (pointed by arrows) where the measure of instability of a particular kind (K4, K5,
K, -K2K5/K6, K, —K2K3K4) is most negative over the range of P,° and Q”, the kind of
bifurcation (SN or Hopf) that occurs, the values of (P9,, , Q”) where the bifurcation occurs
(in general at points opposite to where the arrows point to) and whether the instability
occurs in the regulated or unregulated model. The tables are so complete so that the discus-
sion that follows may be superﬂuous. The results indicate that Hopf bifurcation occurs in
the regulated model but not in the unregulated model, and saddle node occurs in the unreg-

ulated model but not in the regulated model.

Namely, when the model is regulated, Figure El.1(a) shows that (i) as the real power

180

transfer P“ increases and the reactive power is such that 0 s Qm<0.3 , K5 is positive and
decreasing; (ii) for 3 3 Q0, < 0.7 , K 5 = a|V,|/E)8 decreases and becomes negative as P“,
increases; and (iii) for higher levels of reactive power transfer, i.e when Q” > 0.7 pu , K5<0.
This implies that the regulated SMIB model is subject to undamped oscillations in the iner-
tial dynamics when the generator terminal voltage begins to decrease for increases in 8 and
occurs for the range of values (Po, , Q”) shown in Table 6.1. The fact that Hopf bifurcation
occurs when K 5 = Blvd/35 is negative for a regulated system due to large P” and Q” indi-
cates that Hopf bifurcation in the inertial dynamics is due to voltage if not voltage instabil-
ity. .

Diagnosis for the causes of undamped oscillations in the system is also performed
when the reactive power is withdrawn from the inﬁnite bus, i.e. real and reactive power
transfer are in opposite directions. From Figure El.1(b), we see that (i) when
Q” = —0.3, 0 p.u. K5>0 and (ii) when Qm = —1.0, ...,—0.5 p.u. K5<0. An oscillatory mode
in the regulated SMIB model may then be brought on by negative reactive power transfer,
i.e generator reactive absorption.

Power transfer also affected the synchronizing term K, — K2K3K4 for the unregulated
machine, where it became negative for heavy-real and heavy-reactive power transfer to the
inﬁnite bus, as indicated by Table 6.1(a).

Note that the non-bifurcation cases are summarized in Table 6.1(b). Real and reactive
power transfer to the inﬁnite bus has no effect on the parameter K4, since K, is positive and
increasing for the whole range of (Po, ,Q” ). If no local load is present at the generator side,

K4-associated oscillatory stability problems in the unregulated model do not seem to

appear. The synchronizing torques for the regulated model are not affected by the transfer

181

of real/reactive power, whereas when the system is heavily stressed by real/reactive power
supply to the inﬁnite bus, the synchronizing coefﬁcient in the unregulated model is driven
to zero, indicating the occurrence of saddle bifurcation. The bifurcation subsystem of (A5.
A0). AEq‘) explains why heavy real and reactive power transfer cause saddle node bifurcation
because K, decreases with real power transfer and K2 and K4 coupling between inertial
(real power) and ﬂux decay (reactive power) increases with real and reactive power transfer
causing K, - K2K3K4 to approach zero.
The diagnosis study will now focus on the effect of the transmission network inductance.
6.2.1.2 Effect of transmission network inductance x,,

The effect of the transmission network inductance is simulated when P” = 0.8 p.u.
and Q” = 0.5 p.u. in Figure 6.3. As the network inductance increases, the constant K,
decreases, and ﬁnally changes sign at xe=0.4 p.u. The system damping coefﬁcient also
decreases and changes sign, indicating that the system damping forces are insufﬁcient to
damp the ongoing oscillatory modes. This implies that long transmission corridors, cause
undamped oscillation in the regulated SMIB model. The inter-area oscillations often occur
in a multi-machine model between two groups of machines connected by a long line carry-

ing signiﬁcant transfer.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Excitation Bifurcation Network Power Transfer Figure

System

Reg/Un Kind Test x. 11R... 0x... P... G... t

R H K5<0 0.4 0 0 .—l‘—. ‘ El.1(a)
0.1 1 0.3 1

R H K,<0 0.4 0 0 ,.L‘___. ,1— El.1(b)
(r1 1 .1 .05

U SN K. -K,K,K,<o 0.4 o 0 \ ,__\,~ El.4(a)
0.65 l 0.8 l

 

Table 6.1 (a) Results on the power transfer model bifurcation diagnosis

 

182

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-

Excitation Bifurcation Network Power 'ltanster Figure
System
Reg/Un Kind Test x. HR... 1/X,,, PM 0",, #
U H x,>0 0.4 0 0 57—1 .'——, o 1 131.2
U SN K| “K2K3K4>0 0.4 0 O (ST—‘1 .1 0 E1.4
R SN K. -K2K,/K,>0 0.4 0 0 6.1—___) fro—1 51.3
Table 6.1(b) Power transfer model non-bifurcation results
04 1 . T r 1 1 I !
. : : : : Pint-0.8 f
z 3 =. 2 z taint-0.5 2 3
005 ........ .. . ... .........

 
 

 

—0.05

-O.‘l

 

—O.15

 

 

 

.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
xe (DU)

Figure 6.3 Power Transfer Model: Effect of transmission inductance
on the parameter K5 and the damping torque Kd.

 

183

6.2.2 Local Load Model

The effect of the local constant impedance load parameters X,,, and RS, on the occur-
rence of system bifurcations is simulated in Figures E2. l—E2.4. The bifurcation results, the
associated operating conditions and ﬁgures are summarized in Tables 6.2(a-b). In the local
load model, Rsh is a positive parameter between 0.1 3 R3,, 51 and Xsh may be negative or
positive, depending on whether it is capacitive or inductive, respectively. Note that x3 = 0.9
which effectively isolates the local load bus from the inﬁnite bus. In this case, no transfer of
power to or from the inﬁnite bus is permitted. Therefore, three cases are considered: the
effect of (i) real-inductive load and (ii) real-capacitive load. The simulation results on
bifurcations of this study are summarized in Table 6.2(a) for cases resulting in bifurcations,
and in Table 6.2(b) for non-bifurcation cases. A careful review of these tables indicated
ranges of (l/RM , l/Xsh ). when oscillatory and non-oscillatory instability occurs, the
points (pointed by arrows) where the measure of instability of a particular kind (K4, K,,
K, —K2K5/K6 , K, -K2K3K4) is most negative over the range of ( “Ru. , l/Xsh ), the kind of
bifurcation (SN or Hopf) that occur, the values of (l/Rsh , UK“) where the bifurcation
occurs (in general at points opposite to where the arrows point to) and whether the instabil—
ity occurs in the regulated or unregulated model. The tables are so complete so that the dis-
cussion that follows may be superﬂuous.
6.2.2.1 Effect of real-inductive local loading

When the shunt local load was real-inductive, the parameter K, was negative for the
entire R,,,-X,,, range. Furthermore, K4 was most negative at l/Rsh = 2.5 and decreased as 1/
X5,, was decreasing. This implies that the unregulated SMIB model was subject to

undamped oscillations in the inertial dynamics, as summarized in Table 6.2(a).

184

6.2.2.2 Effect of real-capacitive local loading

When the shunt local load is real-capacitive load, Table 6.2(b) shows that the param-
eter K, is negative and decreasing for the entire R,,,-X,,I range from Figures E2.1(a-b)
results. Furthermore, K4 is decreasing when the inductance l/Xsh is negative and ll/X,,,l
increasing. This implies that the unregulated SMIB model is subject to undamped oscilla-
tions in the inertial dynamics. Table 6.2(a) also indicates that the synchronizing term
K, —- K 2K 5/K 6 became negative for light-real, heavy capacitive local loading, implying that
the regulated model is vulnerable to SN bifurcation. The capacitive shunt is larger than
x’d+.tT = 0.2 p.u. when Ixshl = 1 and becomes the same size as x'd-l-xT as [X34 becomes
smaller, causing heavier capacitive loading. This appears to cause the network to induce the
saddle node bifurcation.
6.2.2.3 Effect of the transmission network

We notice that K,, is negative if no power transfer to the inﬁnite bus occurs, regardless
of the local real power load (R,,,), or the shunt capacitance/reactance (X,,,) at the generator
side, which indicates that undamped oscillations in the unregulated SMIB model are asso-
ciated with (i) heavily loaded load centers; (ii) local shunt capacitive system compensation;
(iii) transmission lines with high capacitive/inductive charging; and (iv) when power trans-
fer from or to the inﬁnite bus is not possible.
On the other hand, under the real-inductive and real-capacitive local loading operating con-
ditions, Table 6.2(b) summarizes the non-bifurcations cases. Both K5 and the synchroniz-
ing term K, - K2K3K4 are positive for the local load variations. This diagnostic study
implies that in a local model, it is unlikely to experience (i) HOpf in the regulated model; or

(ii) SN in the unregulated model.

185

 

 

 

 

 

 

 

 

 

 

 

 

Excitation Bifurcation Network Poster Transfer Home

System

Reg/Un Kind Test x. 1m... 11x... PM 0..., s
u H K4<0 0.9 1‘- 10 [4.1—m 0 o EZ.l(a)
u H K4<0 0.9 7,—“10 145—.1 0 0 152103)
R SN K. -K2K,/K.<0 0.9 [_A_’___w .ﬁ.’__,.6 0 0 E2.4(b)

 

 

 

 

 

Table 6.2(a) Results on the local loading model bifurcation diagnosis

 

 

 

 

 

 

 

 

Excitation Bifurcation Network Power Tiansfer Figure

System

Reg/Un Kind Test x. 1/R,.. 1/X... P... 0.... #
R H K,> o 0.9 1 ,0 l——'10 o o 152.2(a)
R H K5> 0 0.9 {___—'10 ’17—: 1 0 0 52.2(b)
U SN K. -K2K3K.>0 0.9 5——-,,, {—10 o o 132.3(a)
U SN K. -K,K,K,>0 0.9 0'__'“10 INT—'1, 0 0 152.303)
R SN K, -K,1<,/1<,>o 0.9 (3—“10 ,,,,'——_, o o 132.4(a)

 

 

 

 

 

 

 

 

 

 

 

Table 6.2(b) Non-bifurcation results on the local loading model diagnosis

6.2.3 Conclusion of the Power Transfer and Local Load Models

From the diagnostic stress tests results of the power transfer and local loading models, we
deduce the following three conclusions: (i) undamped oscillations in the regulated SMIB
model are highly associated with the transfer of active and reactive power (supply or
absorption) along transmission corridors: (ii) undamped oscillations in the unregulated
SMIB model are associated with heavy capacitive local loading; and (iii) power transfer

causes different kinds of bifurcations in the SMIB depending on whether the generator is

186

regulated or unregulated: the effect of relay-type control action.

The regulated SMIB model is vulnerable to Hopf bifurcation when considerable reac-
tive power transfer is absorbed by the generator —1 s Q” « 0. The unregulated model is vul-
nerable to Hopf bifurcation in the generator inertial dynamics under local capacitive and
local real loading operating conditions. Therefore, shunt capacitive load (generator absorp-
tion) causes Hopf in the unregulated model, and transfer of reactive power to the generator
(generator absorption) causes Hopf in the regulated model. The same conclusions hold in
the regulated model for (Q,° > 0) and in the unregulated model for (l/Xsh > O).

Hopf bifurcation occurs both in the regulated and unregulated models for heavy
absorption of reactive power. This is not surprising due to armature reaction that reduces
the rotor ﬁeld ﬂux for reactive absorption and is shown to cause instability due to insufﬁ-
cient synchronizing torque between rotating rotor stator ﬂux waves.

The unregulated SMIB model is vulnerable to SN bifurcation when considerable real
and reactive power is supplied by the generator 0.4 SPgs 1, 0 « gas 1. The regulated
model is vulnerable to SN bifurcation in the generator inertial dynamics under heavy-local
capacitive, and heavy-real local loading. Therefore, transfer of reactive power to the inﬁ-
nite bus causes SN in the regulated model, and shunt capacitive load (generator absorption)
causes SN in the unregulated model.

Finally, no saddle node bifurcation occurs in the regulated power transfer model, and
no saddle node in the unregulated local loading model. In correlation, no Hopf bifurcation

occurs in the unregulated power transfer model, and no Hopf in the regulated local loading

model (l/R,,,, l/Xsh).

187

6.2.4 Combination Local Load-Power Transfer Model

The diagnostic study now focuses on the more realistic model consisting of a local
load in combination with power transfer requirement patterns. The variation of the machine
damping and synchronizing torques under the effect of the various local loading patterns,
and power transfer patterns is investigated for both the case where the generator is
regulated (K5,K, — K 2K S/K,,) and the case where the generator is unregulated
(K4,K, -K,K3K4). Various transfer and loading patterns are considered: (i) the effect of
power transfer for a local loading pattern, (ii) the effect of local loading for a power transfer
pattern, and (iii) the effect of the transmission network. Simulation for the combination
model bifurcation diagnosis are shown in the ﬁgures in Appendix E3, and the bifurcation
results are summarized in Tables 6.3(a-c).
6.2.4.1 Effect of Power Transfer

First, the local shunt load and the transmission network inductance are ﬁxed
respectively at R5,, = 0.65 p.u. , X,,, = $0.17 p.u. and x,2 = 0.4 p.u. , while the power
transfer to the inﬁnite bus (Pm , Q”) is varied. Note that x, = 0.4 p.u. rather than 0.9 p.u. as
in the local load center model. The bifurcation results for the effect of P” and Q0 on the
parameters K4 and K5 as well as K, -K,K5/K6 and K, --K2K3K4 are summarized in Table
6.3(a). The occurrence of bifurcation as shown in the table indicates that (i) when the load
Xsh is inductive, K5<0 for both the case when Q” 2 0 and the case when Q” s 0; (ii) when
X,,, is capacitive, K5<0 for both when Q” 2 0 and Q“ s 0; and (iii) the parameter K, >0 for
the power transfer stress tests except when the local load is capacitive and reactive power is
being supplied by the inﬁnite bus (X3). = —0.17 ,Qm s 0 ). Note that while (i) and (ii) conﬁrm

the power transfer model diagnosis results, (iii) occurred only in the combination model,

188

and thus extends the diagnostic conclusions. This result (iii) may be due to the effect of the
large shunt capacitive load which caused Hopf bifurcation in the unregulated model for
shunt capacitive load and no power transfer.

Power transfer stress on the system seem to also ultimately cause (i) SN bifurcation in
the regulated model when the local load is real—capacitive X3, = -0.17 and P” increasing,
Q0, decreasing; and (ii) SN in the unregulated model when the local load is inductive
X3, = 0.17 and Po, and 0,20 are increasing. These results in (11) above extend the SN
bifurcation results obtained from the power transfer model.
6.2.4.2 Effect of Local Loading
The power transfer to the inﬁnite bus, the local shunt load and the transmission network
inductance are ﬁxed respectively at P,° = 0.8 p.u. , Qco = 0.5 p.u. and xe = 0.4 p.u. , while the
local shunt load at Rs]: and X,,, are varied. The bifurcation results for the effect of R3,, and
X“, on the parameters K, and K, as well as K, -K2K3K4 and K, — Ksz/K,5 , are summarized
in Tables 6.3(b-c). The occurrence of bifurcation as shown in the table indicate that when
the model is regulated, (i) the parameter K, changed sign for the light- real, heavy-inductive
local loading conditions (approximately l/Rshs 1.67, 1/X3h22.0), and is decreasing for
lighter l/Rsh loading conditions as can be seen from Figure E3.9(a); and (ii) the parameter
K, is positive with no change in sign when the local load is real-capacitive from Figure
3.9(b) results.

Given the system is unregulated, (i) when the shunt load is real-capacitive, the
parameter K, changes sign under the heavy-capacitive loading conditions (l/Xsh s -1.67 ,
generator reactive absorption); and (ii) when the shunt load is real-inductive, the parameter

K, is positive which implies that shunt real-inductive real loading patterns are unlikely to

189

cause undamped oscillatory behavior in the unregulated machine dynamics in the SMIB.
Figures E3.ll and E3.12 in Appendix E3, show the variation of. the machine
synchronizing torques under the effect of the machines local loading patterns, for both
when the system is regulated (K ,-K2K5/K6) and unregulated (K,—K2K3K4). For the
regulated system, the summarized ﬁgures in Table 6.3(b) indicate that (i) the synchronizing
torque coefﬁcient is positive for real-inductive loading conditions; and (ii) negative for
light-real, heavy-capacitive loading (l/Rsh small, |1/X,h| 2 1.67 ). When the generator
exciter is disabled, the synchronizing torque is positive for the entire R,,,-X,ll range. This
, implies that the system is unlikely to experience static kind of instability when the
generator is unregulated.
6.2.4.3 Effect of transmission network
The effect of the system’s transmission network inductance x, on the system synchronizing
and damping forces is simulated in Figure 6.4. The system operating conditions were at
P, = 0.8 p.u., Q, = 0.5 p.u., X3, = 0.4 p.u., and two real loading levels were tested,
R5,, = 0.17 and R3,, = 0.87. For this case, the damping torques are plotted versus x, We see
that (i) both K, and K(, both decrease as x° increases, and (ii) K0 is very much more negative
for heavy loading conditions, but K5 is slightly more negative for light loading conditions.

This behavior conﬁrms the results obtained from the power transfer model.

190

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Excitation Bifurcation Network Power Transfer Figure
System
Reg/Uh Kind Test x, R... x... P... G... #
R H K5<0 0.4 0.65 +0.17 6?)? 65—1 53101)
R H K5<0 0.4 0.65 +0.17 (ST—:51 1’ M 53103)
R K5<0 0.4 0.65 0.17 (3?)? 0 ‘1 53.20.)
R H K,<o 0.4 0.65 .017 6‘7“)? .l—‘L—M 53.20»
R SN K, -1<,K,/K,<0 0.4 0.65 .017 (S's—:1 (Sf—“o 5 E3.6(a)
R SN K, -K,1<,/K,<0 0.4 0.65 .017 68—“, .1 ‘0 4 53.60»
0 H K,<0 0.4 0.65 -017 137—)? T—A .o.1 53.4(13)
‘ i
U SN K, ~K2K3K, <0 04 0.65 +0.17 ﬁr 5;——-, 53.7(a)
U SN K, —1<,1<,1<,<0 0.4 0.65 41.17 675—153 0.3 E3.8(a)
Table 6.3 (a) Results on the combination model bifurcation diagnosis
Excitation Bifurcation Network Power Transfer Figure
System
Flag/Uri Kind TOSt X‘ 1 IR.“ 1 IX“, Pint Q.“ #
U H K,<0 0.4 1 1. 6., 3%: 05 53.1003)
R H K5<0 0.4 1‘1—1 67 2,—2le 05 53.903)
R SN K, -K,1<,/K,<0 0.4 L. ,t_ 05 53.1203)
l 3.33 -2.5 -i.6"l

 

 

 

 

 

 

 

 

 

Table 6.3 (b) Combination model: Effect of Local loading

 

 

191

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Excitation Bifurcation Network Power Transfer Figure

System

Reg/Uh Kind Test x. R... X... P... Q... ff
U H K,>0 0.4 0.65 +0.17 0, , 6—‘ E3.3(a)
U H K,>() 0.4 0.65 +0.17 5.1—1 :5— E3 .3(b)
U H K,>0 0.4 0.65 .017 (3.1—1 6?"— E3.4(a)
U SN K. -K,K,1<,>o 0.4 0.65 +0. 17 5,—1 .173“ 53.7(13)
0 SN K. -1<,K,x,>o 0.4 0.65 -017 57—, .T_"‘- 53803)
R SN K. -K.K,/K,>0 0.4 0.65 +0.17 (ST—1 F E3.5(a)
R SN K. -K2K,/K.,>0 0.4 0.65 +0. 17 67““, (fl—“*7 E3.5(b)
R H K5>0 0.4 {—10 127—11 0.8 0.5 E3.9(b)
R H K,>0 0.4 1 ,0 f ,0 0.8 05 5310(3)
R SN K.-1<,1<,/1<,>0 0.4 1 m i 10 0.8 0.5 53.1203)
U SN K.-K2K,K,>0 0.4 1 '10 1 ,0 0.8 0.5 E3.1l(a)
0 SN K.-K2K,K,>0 0.4 r 810 2,, T. 0.8 0.5 53.1103)

Table 6.3 (c) Combination model: Non-bifurcation cases

0.1 1

 

 

 

 

 

 

    

 

 

_04l- ............................................. _.
---— K5. lash-0.17
_0-5-H,"n"Kd,'.RSh*o.l1?HU-'j .......................... .1
'-"K5._RSh-0.87 .
_OIG- -.—.——-.Kd..RshsO.87.. . -f ..................... ....
1' i i
0 0.2 0.4 0.6 0.8
X903“)

Figure 6.4 Combination Model: Damping torque coefficient Kd
and parameter K5 for different levels of real local loading

 

192

6.2.5 Discussion of the Combination Model Results

How do the power transfer and local loading models predict, disagree, or possibly

extend the results obtained from the combination model?. The combination model

diagnostic study shows that combining both models may extend or conﬁrm the results from

each individual model, depending on Operating conditions or stress is dominating. The

following conclusions are drawn:

a)

b)

Although local capacitive or inductive local loading (XS.,<0 or Xsh>0,P°° = Q” = 0)
did not cause Hopf bifurcation in the regulated model in the local load diagnostic
Study for reactive supply (Q2 0) to the inﬁnite bus, local inductive or capacitive
loading in combination with reactive power supply to the inﬁnite bus (X31. = $0.17 ,
Q0, 2 0) appears to drive the system to oscillatory instability (Figures E3.1(a) and
E3.2(a)). This extends the local loading model results and allows the diagnostic
conclusion that the regulated model Hopf bifurcation can occur when the reactive
power is supplied to the inﬁnite bus. The problem is aggravated when real power
supply P” increases and real loading (UR...) decreases. This result may be due to the
transfer of power to the inﬁnite bus over a weak but extremely weak line.

Although local capacitive loading (X,.,<0 or X,.,>0,P,, = Q... = 0) did not cause Hopf
in the regulated model, for reactive power supply from the inﬁnite bus (Q, s 0) local
capacitive or inductive loading (Figure E3.1(b), E3.2(b)) in combination with
reactive power supply from the inﬁnite bus (X3, = $0.17 , Q“, s 0) seem to drive the
system to oscillatory instability. This extends the local loading model results and
allows the diagnostic conclusion that the regulated model Hopf can occur when

reactive power is supplied from the inﬁnite bus. The problem is aggravated when real

C)

(d)

193

power supply P” increases and real loading (l/R,.,) decreases. This result may be due

to the large active power transfer to the inﬁnite bus, and reactive power transfer from

the inﬁnite bus.

For saddle node bifurcation, although power transfer stress seems to affect the

unregulated model synchronizing torque, and not the regulated model, in the

combination model the following bifurcation occurred:

0)

(ii)

SN bifurcation in the regulated model occurs when the local load is real-
capacitive(xs,, = -0.17 ), active power transfer to the inﬁnite bus is increased
and reactive power from the inﬁnite bus is increased (Figures E3.6(a-b)). This
bifurcation was not observed in the power transfer model diagnosis, but appears
to be caused by the presence of the heavy capacitive local load ll/Xshl == 6 , as
diagnosed by the local loading model.

SN in the unregulated model for power transfer (P002 0,Q” 2 0) to the inﬁnite
bus and either X,..>0, or X,.,<0). These results (Figures E3.7(a), E3.8(a)) are
consistent with the SN bifurcation results obtained from the power transfer
model.

The effect of local loading stress seems to be less severe in the presence of real
and reactive power transfer to the inﬁnite bus. The local loading model
diagnosis Showed that K 4 s 0 for the entire range of R,.,-X,.,. In the combination
model, Figure 133.10 (a) shows that the parameter K4>0 for a Signiﬁcant

portion of the operating range- of Rsh-Xs...

194

6.3 Diagnostic Summary

The diagnostic study demonstrated that the cause and effeCt of a speciﬁc kind of bifurcation
in the SMIB model are dependent on (1) the effect of the relay-type control action, i.e
whether the system is regulated or unregulated; (2) local loading-power transfer operating
requirement patterns; and (3) transmission network inductance. The comprehensive power
transfer and local loading pattern variations in the three study models, has been the mean
for drawing (if possible) conclusions for diagnosing the causes of SN and Hopf
bifurcations in the SMIB. Furthermore, the results establish the classes of saddle node and

Hopf bifurcations that were observed in the SMIB model.

6.3.1 Classes of Hopf Bifurcation in a SMIB Model

Ill; Power Swing Mode (Regulated)
This oscillation mode occurs when K5 is negative in the single machine inﬁnite bus
model, when the voltage regulator is still operating within its ﬁelds current limits.
This oscillation is observed in the generator angle/speed/ﬁux decay dynamics, and is
associated with low frequency inter-area oscillations. It is a supercritical (stable)
Hopf bifurcation since these oscillations are stable inter-area oscillations. It is often
referred to as a power swing mode, Since it is brought on by considerable power
transfer between two areas. The power swing mode can be associated with a single
generator, two generators or with groups of generators. There are two subclasses:
l. Reactive Power Supply Swing Mode (regulated)

2. Reactive Power Absorption Swing Model (regulated)

195

ﬁg; Voltage Collapse Mode (Unregulated)
Occurs in the inertial and ﬁns decay dynamics of the generator in a Single machine
inﬁnite bus model, when the exciter is completely disabled, and the single machine
parameter K, is negative. These classes of I-Iopf bifurcations are likely to be the limit
cycles that develop as a precursor to voltage collapse and thus are called voltage
collapse modes. It is not clear whether voltage collapse oscillatory modes are
supercritical or subcritical Hopf bifurcations. The diagnostic study showed that these
undamped oscillations in the unregulated SMIB model are associated-with (i) heavy
loaded local load centers; (ii) local shunt capacitive system compensation; (iii)
transmission lines with high capacitive line charging; and (iv) when power transfer
from or to the inﬁnite bus is not possible. There are two subclasses:
l- Reactive Power Supply Voltage Collapse Mode (unregulated)
2- Reactive absorption Voltage Collapse Mode (unregulated)
It should be noted that there is a Voltage Mode that is subcritical Hopf bifurcation that

depends solely on the exciter and ﬂux decay dynamics.

6.3.2 Classes of Static Bifurcation in a SMIB Model (SN)

Similarly, two classes of Static bifurcation may occur on a Type 2 power system model:
SNJ Static Bifurcation in generator inertial dynamics (Regulated)
Static bifurcation in generator inertial dynamics may be associated with the loss of
transient Stability [36] such as the voltage instability that occurred in the
Czechoslovakian system after tripping lines in a major interface [30]. This class of

static bifurcation is likely to occur when the disturbances are very large and the

l 96

transmission interfaces or boundaries are weak, so that loss of stability occurs soon
after the contingency before the ﬁeld current level/duration limits are reached. It
would appear from the results in this chapter that the Static bifurcation in generator
inertial dynamics occurs the local network is capacitive near a local generator, real
power is Shipped away from that generator, and reactive power is shipped to that
generator.
5N1 Static Bifurcation in generator inertial dynamics(Unregulated)

This bifurcation has been called classic voltage instability and is due to insufﬁcient
reactive supply in the EHV system, an example of which is the blackout that occurred
in the French system [37]. This class of Stability has been analyzed with the use of a
sensitivity matrix T between reactive generation and voltage at generator internal
buses [51] that is diagonally dominant for inductive networks (networks that absorb
reactive power), and not diagonally dominant for capacitive network (network with
long high voltage transmission lines). Classic voltage instability has been described
as occurring when generators reach their ﬁeld current limits and the ﬁeld current limit
controllers disable the exciters. An important implication to this result is that study-
in g this saddle node bifurcation must include the ﬁeld current limit controllers action

in the model.

VII

Thesis Summary and Future Research

7.1 Dissertation summary

This thesis addressed three major difﬁculties in diagnosing the cause of a speciﬁc
Stability problem in a power system. These difﬁculties are (a) the size and complexity of
the model, (b) the various kinds and classes of bifurcation that can occur in the model, and
(c) the lack of smoothness in the dynamic behavior after a disturbance or during operation,
(d) the inability to determine a subsystem of the full model that experiences the same
bifurcation as the full system for each kind and class of bifurcation. These different
problems in a power system have been and Still are in need of better methods for Stability
analysis, developing application software, and problem diagnosis in a large power system.
This thesis will address all four problems and hopefully provide the basis for methods that
can be utilized to establish where, why, and what can be done to prevent loss of Stability
for three kinds (kinds) of bifurcation: Hopf, saddle node, and Load Flow bifurcations
observed on a Speciﬁc power system model as well as all of the speciﬁc subsystems of the
model affected by each particular kind of bifurcation (classes). The contributions of this

work can be summarized in the following four sections:

197

198

A) Diagnostic Classiﬁcation

First, the diagnostic study started by classifying (a) the model types used in power system
studies, (b) the kind and (c) class of bifurcations. This classiﬁcation is performed in
Chapter 2 of the thesis. The kind and class of a bifurcation observed depends on the model
used. Four model types are classiﬁed: Typel model is the simplest model where both
generator and loads are modeled by algebraic equations and under load tap changers
(ULTCS) and switchable shunt capacitors controls are modeled. Type 4 model represents
both generators and loads by differential equations and yet all the ULTCS controls
dynamics and switching capacitor inductors, etc. are included. Type2 and Type 3 models
are simpliﬁed Type 4 models that eliminate the load dynamics and generator dynamics
respectively. The complete Type 4 dynamic model has been developed only for small test

systems.

B) Epoch Bases Trajectory Stability Assessment Method

Virtually every local bifurcation is brought on by a sequence of equipment outages and
hard-limit discontinuities enabling or disabling various controls. Although dynamic
system theory is fairly well developed for smooth systems, it is currently being developed
for nonsmooth systems. These hard limit discontinuities do not cause immediate
enablement or disablement of a particular control. The dynamical system theory for
nonsmooth systems [27] is not sufﬁciently well developed to be able to determine the
sequence of hard limit discontinuities when each discontinuity has a different length time
delay. Current dynamical system theory for smooth system can be applied to intervals

(1315”) called epochs where discontinuities occur only at [1,] , the discontinuous change

199

at [1,] can be computed; and the time T, = 1,:+1 4; is long compared to the settling time
of the dynamics. Therefore, a method aimed at assessing asymptotic stability of the
transient during 0,7,1?) and the Stability of quasi equilibria or limit cycles in each epoch is
proposed. This method called the Epoch Based Trajectory Stability Assessment, is based
on a trajectory simulation method implemented by [28] that does not require time
simulation in (131;, ,) , but only computes the quasi equilibria (22,53) for each epoch for
1' = 1.... N. This method allows the assessment of the stability of individual trajectories for
a power system by assessing stability for every epoch through (a) assessing stability of the

sequence of quasi equilibria or the limit cycles, and (b) the asymptotic stability to each

quasi equilibrium or limit cycle.

C) Bifurcation Subsystem Method:

A third a major contribution in this diagnostic framework is the establishment of the
Bifurcation Subsystem Method, This method attempts to identify the smallest subsystem,
a subset of the equations of the full system model that experiences and causes the same
bifurcation that occurs in the full system model. A bifurcation subsystem can loosely be
deﬁned as a truncated portion of the actual power system model that experiences the same
bifurcation in the full system model. The Bifurcation Subsystem Method utilizes the
geometry associated with the various submatrices of the full system differential-algebraic
Jacobian J = [f‘ f5] since

x 8 y
Det(J) = Det(gy) - Det(fx—fyg;lgx) when g), is nonsingular;

De1(1) = Det(fx)-Det(gy—gxf;lfy) when f, is nonsingular;

and with the eigenvectors associated with the bifurcating eigenvalue to establish

2110

conditions for existence of such systems. The fact that the Jacobian of the truncated
portion of the system (gy) must be Singular somewhat restricts the set of possible
subsystems that one is using to ﬁnd the bifurcation subsystem. The bifurcation subsystem
must satisfy both the linear and the transversality test conditions for bifurcation that is
used to test for bifurcation in the full system model. The null space of the effects of the
truncated system (fyg;lgx) must be common with the null space of the bifurcation
subsystem (15,) in which the right or left eigenvector of the bifurcating eigenvalue resides.
The portion of the full system external to this bifurcation subsystem has no effect on the
linear conditions for bifurcation to be satisﬁed in the full system model only at bifurcation
(asymptotic), or as bifurcation develops for 11¢ 11" (structural), and yet the equilibrium
point is dependent on the variations in the full system equations which results in
bifurcation in both the full system and the bifurcation subsystem. Propositions for the
existence of bifurcation subsystems for a particular kind of bifurcation were derived and
proved. Possible bifurcation subsystems in a power system Type 2 model are:

(a) Differential Bifurcation Subsystem Experiencing SN;

(b) Differential Bifurcation Subsystem Experiencing Hopf;

(c) Algebraic Bifurcation Subsystem for LF;

(0) Differential Bifurcation Sub-subsystem Experiencing SN;

(e) Differential Bifurcation Sub-subsystem Experiencing Hopf;

(g) Differential-algebraic Bifurcation Subsystem.
It should be noted that the bifurcation subsystem is not the center manifold of a speciﬁc
bifurcation since the same bifurcation subsystem could exist for saddle node bifurcation in

the ﬂux decay dynamics, and for saddle node bifurcation in the inertial dynamics. The

201

sensitivity matrices Ks, KD and T also do not necessarily identify the center manifold
dynamics, and thus do not necessarily identify the exact class and location of a speciﬁc
bifurcation. The research has shown that what is desired is not to determine the center
manifold dynamics, but rather a subsystem of the original model that experiences the same
bifurcation as the full system. The bifurcation subsystem may in most cases be an
excellent approximation of the possibly larger subsystem that causes and produces the
bifurcation in the full model, as it is the desired result. The root locus of the bifurcating
eigenvalue and the participation factor information provides additional diagnostic
information on such a subsystem.

Determining sensitivity matrices Ks, KD or T as a function of the model parameters
K.-K., and then studying how K3, KB or T vary with changes in operating conditions and
hard limit discontinuities also can assist in indicating the subsystem that produces, causes

and cures a speciﬁc bifurcation, in special cases.

D) Stability Conditions and Diagnosis:

Lastly, the Epoch Based Trajectory Stability Assessment Method and the Bifurcation
Subsystem Method are the framework for initiating a diagnostic study for Hopf and saddle
node bifurcations on a single-machine-inﬁnite bus (SMIB) model to (a) computationally
demonstrate and validate the use of the bifurcation subsystem method to identify
bifurcation subsystems in the SMIB; (b) identify the classes of saddle node and Hopf
bifurcations that may occur in the SMIB model: the generator angle/Speed dynamics and
angle/speed/ﬂux decay dynamics are bifurcation subsystems for Hopf and SN bifurcations

in both the regulated and unregulated SMIB models, respectively; (0) gain better

202

understanding on the effects of hard limits by assessing the effect of excitation control on
the types and classes of bifurcations produced; and (d) establish an understanding to the
causes, operational and structural stresses, and stabilizing requirements for SN, local and
inter-area modes of oscillations that occur in generator inertial, ﬂux decay and exciter
dynamics of the SMIB model. This study was performed by formulating three study
model: (1) a power transfer model; (2) a local loading model; and (3) a combination
power-transfer/local loading model. The diagnostic study demonstrated that the cause and
effect of a speciﬁc kind of bifurcation in the SMIB model are dependent on (1) the effect
of the relay-type control action, i.e whether the system is regulated or unregulated; (2)
Local loading or power transfer operating patterns; and (3) transmission network

inductance across interfaces.

7.2 Future Research

The following topics are subjects for further investigation:

1. Develop a load flow program based on the Bifurcation Subsystem Method to obtain
load ﬂow bifurcation subsystems that experience the same bifurcation as the full sys-
tem, but also include what produces and causes the bifurcation. Evaluate how load
ﬂow subsystems change with (a) decoupling of P-V and Q-O Jacobian submatrices;
(b) generator reaching ﬁeld current limits; (0) under-load tap changers reaching tap
limits; ((1) switchable shunt capacitors reaching shunt susceptance limits; and (e) the
effects of multiple contingencies. This knowledge allows the prediction of the most
critical bifurcation subsystem in the system for a load ﬂow model, given any equip-

ment outage or operating condition combination using proximity measures and diag-

203

nostic tools. A heuristic method has been developed for identifying bifurcation
subsystems that experience the bifurcation of the full system, given certain assump-
tions are true. The work would establish the bifurcation subsystems when the

assumptions are not made and when they are true.

Develop a dynamic stability program based on the Bifurcation Subsystem Method to
obtain dynamic bifurcation subsystems that not only experience the bifurcation of the
full model, but also produce and cause the bifurcation. Evaluate these subsystems
change with (a) decoupling the generator/load dynamics of the system; (b) hard limit
discontinuities in generator and network controls; and with (c) the effects of multiple
contingencies. This knowledge allows the prediction of the most critical bifurcation
subsystem in the system for a differential-algebraic model for any equipment outage

or operating condition combination using proximity measures and diagnostic tools.

Develop proximity measures for each class of bifurcation subsystem found in a load
ﬂow or a differential algebraic model. These proximity measures can be (a) reactive
reserves on a particular group of generators; (b) measures of properties of sensitivity
matrices like K5, KB or T; (c) measures of properties of the system matrices K.-K.,; or
((1) measures of properties of the J acobian of the bifurcation subsystem that produced

the bifurcation.

Develop diagnostic procedures for identifying the kind, class, location, cause, and

remedial action needed for each bifurcation subsystem or class of bifurcation identi-

204

ﬁed in a load ﬂow or differential algebraic model. The diagnosis Should be based on
identifying how the system matrix A; sensitivity matrices like K5, KB or T; sensitivity
matrices [(1-K6; or Jacobian matrix J are affected by load, transfer, reactive reserve

exhaustion, etc. that produce each bifurcation class.

APPENDICES

APPENDIX A
POWER SYSTEM DYNAMIC MODEL

Appendix A

Power System Dynamic Model

A.1 Nonlinear Model

The dynamic model of the full power system [52] is described in this section. Without loss

of generality, we assume that the connection between each generator and its high-side bus

is equivalently one-to-one, and that the power system has 11 buses total buses including m

machines with m high—side buses and (n-m) other (load) buses. The subscripts used in the

various variables used in the model equations, are described here:

Subscript t: represents the generator terminal PV—buses (including switchable shunt
capacitor buses. Eg: P0,, P0, V. etc, t = 2, m.

Subscript H: represents the generator-transformer high-side PQ-buses. Eg: P0,, PC,

Subscript L: represents the rest of the P-Q buses. Eg: PC,

Subscript G: represents the active or reactive power generations

Subscript C: represents the non-voltage (i.e constant) active or reactive power load or

injection

A.1.1 Swing Equation

For each synchronous machine,

8 = (1)—0),,

M0) = —D(0)—0)(,)+Pm—Pe
Where

5 : angular rotor position, measures in electrical radians of the relative to a syn-
chronous rotating frame

0) : angular speed

205

(no : nominal angular speed

D : approximate damping constant due to damper winding effects( pu/(rad/s) )
M = 214/(1)0 : moment of inertia (sec/(rad/sec))

l—l : moment of inertia constant (MW - sec/MVA)

Pm : mechanical input power to the machine

P6 : electric power output of each generator
A.1.2 Flux Decay Equatlon

For each synchronous machine,

K53 = l-(xd-x'd)-Bq > 1
U4, = Gasin(5—9,)-chos(5—9,)
R -x
Ga = 7—“—— ; B = _T—L—
Ra-l-x’dxq Ra+x’dxq

Where
T’do : d-axis open-circuit transient time constant
Efd : ﬁeld winding voltage

Eq’ : induced stator voltage due to the ﬂux linkage of the ﬁled winding

A.1.3 Excitation system

A typical excitation system in shown in Figure Al, is responsible for controlling the
terminal voltage. Using the following notation:

VC: output of load ( or line drop) compensator

V3: output of power system stabilizer (PSS)

Vref: reference (set point) voltage
VD: voltage detector output

KD: dc (direct current) gain of voltage detector

Efdi
l/KSE:

207

time constant of voltage detector

transient gain reduction

lagging time constant of TGR

leading time constant of TGR

output of automatic voltage regulator (AVR)
dc gain of AVR

time constant of AVR

output of exciter (ﬁeld voltage of synchronous machine)

dc gain of exciter. Note that KSE is function of the saturation effect of the

exciter.

TE/KSE:time constant of exciter

KF:
TF:

washout gain of stabilizing feedback of excitation system

washout time constant of stabilizing feedback of excitation system

The state space model of the [BEE-DC] excitation system is given by

.%DV;
roF
rAv,
rev,

 

 

P

-KA

 

L O

:2
TB

7'.
_(1__E
TB

H

A.1.4 Power System Stabilizer

A power system stabilizer improves the dynamic stability of a synchronous machine and/
or a power system, using other regulator input signals in addition to terminal voltage
through its excitation system. Some common stabilizer input signals are accelerating
power, machine speed, frequency and terminal voltage. A state space model for the PSS

model shown in Figure A2 may be written using V50, V52 and V54 shown in the block

diagram.

 

 

 

'- '1

KDVC
0

TC

TC
(1 - ﬁ)(an + VS)

 

 

L 0

d

208

 

 

 

 

P ..1 o 0) ‘ —KS
TSVSU l—Z‘ﬂ —1 0 v30. (1 TS')K
Tszvsz = Tsz Vsz + 752 5 VS!
T. V54 T33 TSl T53 _1 V54 T33 T51
54 “FT—1’?— 1__ —KS
L S4 52 54 L T54 T57
T53751 753751 753

s = EEKS 51+‘fsj1'T—SZ so 754
A.1.5 Speed-Governing-Turblne System Model

The speed—goveming system controls the mechanical input power, using a turbine system
and rotor speed as an input signal. Figure A3 present the general model of speed
governing system, for steam-turbine and hydro-turbine. In the ﬁgure, PGV is the power at
the gate or valve and represents the input of the turbine system. The state space model of

the speed-goveming-turbine system may be obtained by combining both models.

 

 

 

 

a. Speed Governing-Turbine
.- ' W P — T1 q
TP~ —1 o 0 0 0 0 P ' T1
3 f” -1—1 0 0 o 0 6V T,
TCHPVHP = 0 1 -l 0 O 0 PVHP + _KG(1'F‘)(°”(°0)+P0
3 3 3" ° 3 °
1 -1
T P PIP 0
RH?” _o o o o 1-1, I, 0
_TCOPLP_ - LP 0

 

 

 

 

PM = FVHPPVHP+FHPPHP+FIPPIP+FLPPLP

209

b. Hydraulic Governing-Turbine

 

 

 

 

T
. K (iv—Warm)
TIP! -1 0 0 P: 0 T1 "
TPGV= “1‘1 0 Pav“
3 _ 0 a ~-a —KG(l——3)(w-wo)+l’o
TWPW W0 WW PVHP l
_ 0 _
. “13021 . “13021
PM=FGVPGV+PW 9 “WC: 0211 ’ “WV/'51— , FGv-aza‘ a”

A.1.7 Load Compensator Model

The load compensator uses the terminal voltage and current as feedback measurement of
the excitation system model. A state space representation of this model is given by:

‘ K4254 + Kd7V:

<
C
a.

l

Where,
K42 = -(XcGa+Rch)
K42 = Rcaa-XCBq
K47 = (l +XCB’d-RCGa)sin(8—9,)—Kd2cos(6-9,)

Kq-, = (1 -Kq2)cos(8-9,) -(XCGa +RCB’d)sin(8-9,)
A.1.6 Algebraic Load Flow equation

(1') At Internal Bus: for each synchronous machine

e

’2 I 2
P = E 40515-5 thTPE+VtGEt

,2 , 2
Qc E 4355+‘5quUQE—VzBEx

A.2

210

(ii) At terminal Bus: for t = n+1. n+m; h = 1.,m

2 , I 2 ,,
‘PDt Vt (th ‘ VtE thq + V! “’Cth + Gm) ' Viv/1TH:

2 , 2
"Q0: _Vtth+-VtE qth—Vt(BCth+Bth)_ vrvhUth

(iii) At high-side bus:for h = l, m; t = n+h, we have

2 w 2 n w I!
J: I:

_QDh
1:] 1:

(iv) At load bus: for 1': m+1, ..., n, we have

2 n ‘ n
‘PDi = Vi .21((’Cij+ Gij)‘ .ZlVngng
/= 1=

’3 ' n
2
‘Qoi = _Vi .ZI(BCij+Bij)" _leiVjUij
j: I:

“‘1
ll

i]- Gijcos(9,- — 91-) + Bijsin(9i- 9}.)

Q
ll

Q
ll

Q
ll

Linearized Power System Model

2 2 n n
"Vh(ch+Bm) —VthUht'Vh 2- (HChj+th)' 2 VthUtj
. . 1

1:}

i¢j

i¢j

i¢j

i¢j

i¢j

Taylor series expansion is used to obtain a linear model about a nominal isolated

equilibrium point. Linearization is necessary for the load demand model, power balance

equations, swing equation, ﬂux decay equation and load compensator. The jacobian matrix

full linear model is expressed in terms of submatrices that are explicitly given below.

F . 7 — '1
TXXAXX AXX AXE Axr; 0 Are Axv AXX

TEEAXE AEX AEE 0 A55 A59 AEV AXE

 

 

 

 

 

 

= +BOAU0
TSSAXS ASX 0 0 ASX 0 0 AXS
—APC APX 0 0 0 Are APV A9
L-AQc_ LAQX 0 0 0 AQa AQV‘LAV‘
Where,

r n-1c w '- —l-

Txx o 0 0 AXXAXEAXG 0 Txx 0 0 oi AMA”)
f.= 0 TEE 0 0 AEXAEE 0 A55 :0: o TEE 0 o AseAev
' 0 O TGG 0 AGX 0 AUG 0 O 0 TGG O 0 0

_0 o 0 rs, LASX o o ASX‘ L0 o o TSS‘ Lo 0_

 

 

 

 

 

 

 

 

t I
AXX = [Amt A5‘ AE'F] : states of mechanical and ﬂux decay dynarrucs;

L.

r

I
AEF = LAB"; AE’d‘ Atyld' Atqu’] :states of ﬂux decay dynamics

F t
- t t t t t . ‘ '
AXE _ LAVD AVF AVA AVB ABM] . states Of CXCltathl'l system

 

I
AX“ [AFC “’th APVHPt APHP‘ APIP‘ APLP’]

:states of speed-governing-turbine systems;

I
= I t l : . . ;
AXS [Avso Avsz (”54] states of power system stabilizer
t
A9 = [AGT' A9”! 491.1] : angle variables at network buses;

l
AV = [AVT' AVH' AVL‘] :voltage variables at network buses;

I
Me = [APCT‘ APCH‘ APCL‘]

: coefficients of non-voltage-dependent active power load demand model;

1
I
AQC [AQCT AQCHI AQCLI]

: coefﬁcients of non-voltage-dependent reactive power load demand model;

diug[M I T’do] : diagonal matrix composed of inertia constants of

synchronous machines, identity matrix, and time constants of ﬂux decay

dynamics;

TEE = diag [TD Tr T A TB TE] : time constants of the excitation systems dynamics;
Tar; “mg [Tr Ts TCH TRHl TRHi Tea]

: time constants of speed-governing-turbine systems;

T55 = diug[T5 T52 T54] : time constants of power system stabilizers;

M = diag[Ml M2, Mm] :inertia constants for m synchronous machines

1 .° m x m identity matrix

Tdo : m x m diagonal matrix of time constants of ﬂux decay dynamics

—D _Kj’.‘l «if, 0 0 0 0 0 0 0 Fvnp Fur Fm FLP
=1 0 o Ax15=00000 Axa=oo 0 o o 0
0454-19,, 00001 00 o 0 o 0
“KS 00 -l 0 0 [($100
‘ 1211‘s 00 Ass-'- ’21 ‘I O Axe: o ()0
O—Kﬁ'l—Kg’z 04(31ng Am BIH 0
0 0 0 AQX" 0' o o APB: Azu 3211:1321”.
0 0 0 O 0 0 0 32w BZLL

Cur 01H 0 Ant 33H 0 CM 03H 0
C2}! 02m: 02m. A06 = A411 34m: 34m. AQV = Cm D4HH 04m.

0 02LH Out. 0 34w B4LL 0 04”! 041.1.

EX

GX

Axv

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ q -I 0 0 0 0
0 0 0 —1
-l 0 _
0 0 0 0 0 TFKF
-1
KATBCT43T21 KATBCT43 KATBC AEE = —KATBC -KATBC 4 KA —KATBCTF KF
I .T T l .T l .
3c 43 21 BL 43 3t -1
"BC "BC 0 " “’BCTF KP
L 0 0 0 .
L 0 0 I 0 4‘55 .
P 0 K K- K K.“ ~ 1 - ,
D c1 DLLZ —KDKC100 -KDKC7OO
0 0 0 0 00 0 00
AAw 0 0 Axe“ 0 00 AEV‘ 0 00
A 0 0 0 00 0 00
Bw L
L0 0 0 . L 0 00 0 00
I T—IT K 00 ' r - - r -
[‘12]G 400000 0 0 00
..1 -I-I 0 0 O 0 0 o 01
‘T1T2KG 00 A = 01—1000 3 = KATBCO B = 00
0 00 00 001—100 50 IT 0 GO 00
0 00 0 0 0 I .1 0 ’ BC 00
0 00 _0 0 0 0 ’-. L 0 0_ _00_
. 0 00.
qt
-KP100
0 00
K5700

tq . _ tq . __ tq . _ '4
AlK=KPl+Al . AIK-KP7+C1 . A3K‘KQ1+A3 . C3K-KQ7+C3

. . ‘1 . ‘1 . ‘1
’43 =1+T43 .12,= ”721 ~ T43 = T S4Tss . T21= T SZTSI ’ TBC = TB Tc

’Bc = "THC ; AM = KATBCT43T21KS ? BBw = [BCT43T21KS

K54 =

UPE

_(xd _ x’d) ' thvl

= klUdt-Gauq-x’dﬂq,

= kZUdt

, 2 ,
E qlePE + V! (xq -x d)(Tdth(- Udtht)

, 2 ,
-E qv,TQE+ 2v,(xq—x (1)0414,

-(xd -x’d) - U4,

led, — Ga(xq —x’d)Uq,

kZTdt

4‘ I
Km 2‘5 qGEE‘ VITPE

45’qu — mugs

—2V,BE,—E’qUQE

 

KD/( l+s$ )

 

 

 

 

 

 

 

 

 

 

KD/( l+s'§ )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PSS
VAmax
+ T ' Ef
+ 1+5 L KA/(1+STA) -——~q 1/(Ksr~:+S"E) d >
l+sTB
TOR V_/ Exciter
Amin
SKF/(l-l'STF)
(a)
PSS
'1‘
i‘zc VAmax
+ V5 + VA Efd
- KA/(l-l-STA) —q l/(KSgI-S'E)
HSTC + Exciter
1+sT3 VB
VAnu'n 5F
Tr
VF sKFJ(1+sTF]' ‘
(b)

Figure Al. [EBB-DC] Excitation System Model

V

215

 

Vsr ——u

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SKSTS l + STSI 1 + STS3 VS
1 + 31‘s 1 + 5T3; l + sTS4 '
T51 1'51 .
T52 T54
+
l'TSi/r ,2 . 1°Tsﬂ54
l + sTsz V82 1 + 8T4 v84

 

 

 

 

Figure A2. Power system stabilizer model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(00 + 1)0 PU a PllSMAX
P
0) +‘ KG(1 + s13) 1__ 1 av
’ l + 8T1 - T3 3
pDOWN PMIN
+ P
PGV Fvap +
A 1 Pvnp
l + STCH m
l _ . 1
1+ Slum Pup 1+ srm , Prp 1+ s'rCO Pu.
ﬁe?) ...........................................................................................
Hydrp
mo P° PMAX
m_.$‘__ KG(1 + 5T2) (g r ’ Pov
(1+sT1) (1 + 5T3) J

 

 

 

 

 

P a23 '3 (313321 ' “1323)T pM
GV 1 + sauTw

Figure A3. Speed-Governing-Turbine System Model

APPENDIX B
COMMON DEFINITIONS FOR POWER SYSTEM STABILITY

APPENDIX B
COMMON DEFINITIONS FOR POWER SYSTEM STABILITY

In order to facilitate analysis of stability problems, identify the essential associated factors
and to formulate methods of assessing and improving stable operation, stability problems
have been classiﬁed into appropriate categories. A classiﬁcation recently presented in
[29], summarizes the most widely used stability deﬁnitions, the associated concepts, and
the related terms. The classiﬁcation is based on the following considerations: (a) the
physical nature of the resulting instability, (b) The size of the disturbance considered, (c)

the devices, processes and time span required to determine stability.

8.1 Rotor angle stability

Any unbalance between the generation and load results in a net accelerating (or
decelerating) torque exerted on the rotors. From the swing equation, we see that this
transient causes the rotors of the synchronous machines to “swing”. If the net torque is
sufﬁciently large to cause the rotor to swing far enough so that one or more machines “slip
a pole”, synchronism is lost. However an appreciable increase in the machine’s rotor
angular velocities does not necessarily mean that synchronism will be lostThe important
factor is the angle difference between machines, where the rotor angle is measured with
respect to a synchronously rotating reference. Rotor angle stability is the ability of

interconnected synchronous machines of a power system to remain in synchronism after

217

218

being subject to a transient disturbance.

3.2 Small signal ( or small-disturbance) stability

It is the ability of the power system to maintain synchronism when subject to small
disturbances. The disturbances are considered small enough to cause a small change from
the operating equilibrium point, so that linearization of system equations is permissible.
The powerful tool of linear system theory may therefore be explored for purpose of
analysis.

8.3 Transient stability

Transient stability is the ability of a system to maintain synchronism when subjected to a
severe disturbance. The system response involves large excursion of the generator rotor
angle. Stability depends on both the initial state and the severity of the disturbance.Usually
the system is altered so that the post-disturbance steady state operating point differs from

that prior to the disturbance.

219

f,

 

Rotor angle, rad

 

 

 

 

time (s) >
(a)
A
1i A
2
DO
5 B
5
o
CZ.
C
D
time (s) ’
(b)

Figure B]. Response of a four-machine system during a transient [57]
(a) stable system; (b) unstable system

8.4 Voltage stability and voltage collapse
A power system at a given operating state is voltage stable, if an injection of reactive
power causes the voltage magnitude at this node increases and all other voltages also

increase or at least do not decrease. A system enters a state of voltage instability if a

220

disturbance. increase in load demand, or a change in system condition causes a

progressive and uncontrollable drop in voltage.

APPENDIX C

DEFINITION OF HOPF AND SADDLE NODE
BIFURCATIONS WITH TRANSVERSALITY CONDITIONS

Appendix C

Deﬁnition of Hopf and Saddle Node,
Bifurcations with 'Ii'ansversality Conditions

Hopf and Saddle node bifurcations have been recognized as the two generic bifurcations

in variety of power system models, and are generally detected by monitoring the

eigenvalues at a certain operating point, as a particular parameter of interest is varied

slowly. Saddle node involves the coalescence and disappearance of two neighboring

equilibrium points, while Hopf involves the emergence of a small amplitude periodic orbit

from an equilibrium as the parameter is varied

(a1) Saddle Node Bifurcation

Let x = f(x,u) be a system of diﬂerential equations in R" depending on the single

parameter p. At 11 = u‘ , assume that there is an equilibrium point X: (2(tt"), u‘) for

which the following hypotheses are satisﬁed

(SNI) fx = af/axam‘), if) has one simple eigenvalue Mir“) = 0 with right
eigenvector v and left eigenvector w. fit all other (n-l) eigenvalues with negative
real part

(5N2) w- [Bf/3u(i(u‘). u')] #0

(SN3) w- [(af/allb’cw'). ll'))(v. V)] #0

Then the system undergoes a saddle node bifurcation from Y: (2(u"), u“)

221

37')

(a2) Hopi-Andronov Bifurcation
Let x = f (x, u) be a system of dzﬁerential equations in R" depending on the single
parameter it. At ii = u‘, assume that there is an equilibrium point Y= (201‘), u‘) for
which the following hypotheses are satisﬁed
(HI) f; = af/ax(i(p.‘),u‘) has a simple zero eigenvalue Mu") = O and no other
eigenvalues with zero real part.

(H2) £1(Re7t(u))|u=p,¢0 (Transversality condition)

Then a small amplitude non-constant periodic orbit of x = f(x,l,t) emerges from

7= (301‘). 11‘) at 11 = 11‘

APPENDIX D

INPUT DATA FILES FOR THE BIFURCATION
SUBSYSTEM EXAMPLES

APPENDIX D

INPUT DATA FOR BIFURCATION SUBSYSTEM EXAMPLES

EXAMPLE 1 (SN)

09

The single machine infinite bus system for power system

% stabilizer design
% datasmib.m
% bus data format
% bus: number, voltage(pu), angle(degree), p_gen(pu),
% q_gen(pu), p_load(pu), q_load(pu), gshunt, bshunt,
% bus_type
% bus_type - 1, swing bus
% - 2, generator bus (PV bus)
% - 3, load bus (PQ bus)
% the voltage on bus 2 is adjusted for zero Q at machine 1
bus = [ l 1.05 0.0 0.7 0.4 0.0 0.0 0.00 0.300 2;
2 1.08103 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1;
3 1.0 O 0 0 0 0 0.0 0.0 3];
% line data format
% line: from bus, to bus, resistance(pu), reactance(pu),
% line charging(pu), tap ratio, phase shifter angle
line = [l 3 0.0 0.61 0. l. 0.;
2 3 0.0 0.1000 0. 1. 0.;
2 3 0.0 0.1000 0. l. 0.];

% Machine data format

leakage reactance x_l(pu)

r_a(pU)

sychronous reactance x_d(pu)
transient reactance x'_d(pu)
subtransient reactance x'_d(pu)
open-circuit time constant T'_do(sec)
open-circuit subtransient time

% machine: 1. machine number
% 2. bus number

% 3. base mva

% 4.

% 5. resistance

% 6. d—axis

% 7. d-axis

% 8. d-axis

% 9. d-axis

% 10. d-axis

% constant T‘_do(sec)
% 11.

q-axis sychronous reactance x_q(pu)

223

224

% 12. q-axis transient reactance X'_q(pU)
% 13. q-axis subtransient reactance x"_q(pU)
% l4. q-axis open-circuit time constant T'_qo(sec)
% lS. q-axis open circuit subtransient time
% constant T'_qo(sec)
% 16. inertia constant H(sec)
% 17. damping coefficient d_o(pu)
% l8. dampling coefficient d_1(pu)
% 19. bus number
% 20. S(1.0) - saturation factor
% 21. S(l.2) - saturation factor
% note: machine 1 uses mac_sub model, machine 2 uses mac_em
% model
mac_con = [
l 1 991 0.15 0 2.0 0.2 0.2 5.0 0.031 1.91 0.2 0.2
0.66 0.061 2.8756 0.0 0 1 0 0,

2 100000 0.00 0 0. 0.01 0 0 0 0 O 0

0 3.0 2.0 0 2 0 O],

Exciter data format
exciter: 1. exciter type - 3 for ST3
2. machine number
input filter time constant T_R
voltage regulator gain K_A
voltage regulator time constant T_A
voltage regulator time constant T_B
voltage regulator time constant T_C
. maximum voltage regulator output V_Rmax
9. minimum voltage regulator output V_Rmin
10. maximum internal signal V;Imax
11. minimum internal signal V_Imin
12. first stage regulator gain K_J
13. potential circuit gain coefficient K_p
14. potential circuit phase angle theta_p
15. current circuit gain coefficient K_I
16. potential source reactance X_L
17. rectifier loading factor K_C
18. maximum field voltage E_fdmax
l9. inner loop feedback constant K_G
20. maximum inner loop voltage feedback V_Gmax
exc_con = [
3 1 O 7.04 0.4 0.00 1.0 7.57 0 0.2 -0.2 200 4.365 20
4.83 0.091 1.096 6.53 1 6.53];

(DQOWU'Iobw

deddedePo‘Po‘PoPdeOdePdePdePdePo‘PoPo‘PdOdPON

EXAMPLE 2 (Hep!)

% The singl machine infinite bus system for power system
stabilizer design
datasmib.m

W09

bus data format
bus: number, voltage(pu), angleidegree), p_gen(pu),

q_gen(pu), p_load(pu), q_load(pu), gshunt, bshunt,
bus_type

090909019

% bus_type - 1, swing bus
% - 2, generator bus (PV bus)
% - 3, load bus (PQ bus)
% the voltage on bus 2 is90djusted for zero Q at machine 1
% # V ang Pg Qg Pl Ql Gsh Bsh type
bus = [ l 1.05 0.0 0.90 0.80 0.0 0.0 0.0 0.0 2;
2 1.08103 0.0 0.0 0.0 0.0 0.0 0.0 0.0 l;
3 1.0 0 0 0 0.7 0.4 0.0 0.0 3],

% line data format
% line: from bus, to bus, resistance(pu), reactance(pu),

% line charging(pu), tap ratio, phase shifter angle
% fr to r x chg tap pshft
line = [1 3 0.0 0.6 0 1 0.;

2 3 0.0 0 01 0 1 0.;

2 3 0.0 0 01 0 1 0.];

% Machine data format

% machine: 1. machine number
% bus number
% base mva

leakage reactance x_l(pu)

resistance r_a(pu)

d-axis sychronous reactance x_d(pu)

. d-axis transient reactance x’_d(pu)

d-axis subtransient reactance x'_d(pu)

d-axis open-circuit time constant T'_do(sec)

d-axis open—circuit subtransient time
constant T'_do(sec)

11. q-axis sychronous reactance x_q(pu)

12. q-axis transient reactance x'_q(pu)

13. q-axis subtransient reactance x'_q(pu)

l4. q-axis open—circuit time constant T'_qo(sec)

15. q-axis open circuit subtransient time

constant T‘_qo(sec)

l6. inertia constant H(sec)

17. damping coefficient d_o(pu)

18. dampling coefficient d_l(pu)

19. bus number

H
OKDmQO‘lUlewN

oPo‘PdOoWdPoWoPdﬂdPonPdPoPdePdePdPoPo‘OWdP

20. S(1.0) - saturation factor
21. S(1.2) - saturation factor
note: machine 1 uses mac_sub model, machine 2 uses mac_em
model
mac_con = [
l l 991 0.15 0 2.0 0.245 0.2 5.0 0.031 1.91 0.42 0.2
0.66 0.061 2.8756 0.0 0 l 0 0;
2 2 100000 0.00 0 0. 0.01 0 0 0 0 0 0
O 0 3.0 2.0 0 2 0 0];
% Exciter data format
% exciter: l. exciter type - 3 for ST3
% 2. machine number
% 3. input filter time constant T_R

o‘PoOdPanf’o'PopoPoOo'PoPoP

o‘Po‘PdePo‘P

exc_con = [

004mm»

9.
10.
ll.
12.
13.
14.
15.
l6.
17.
18.
19.
20.

226

voltage regulator gain K_A

voltage regulator time constant T_A
voltage regulator time constant T_B
voltage regulator time constant T_C
maximum voltage regulator output V_Rmax
minimum voltage regulator output VﬂRmin
maximum internal signal V_Imax

minimum internal signal V_Imin

first stage regulator gain K_J
potential circuit gain coefficient K_p
potential circuit phase angle theta_p
current circuit gain coefficient K_I
potential source reactance x_L
rectifier loading factor K_C

maximum field voltage E_fdmax

inner loop feedback constant K_G
maximum inner loop voltage feedback V_Gmax

3 1 0.00 1.9 0.19 3.7 0.08 7.57 0 0.2 -0.2 400 4.365 23
4.83 1.0 1.096 6.53 2.949 6.53];

APPENDIX El

FIGURES FOR DIAGNOSTIC ANALYSIS
POWER TRANSFER MODEL

APPENDIX E1

FIGURES FOR DIAGNOSTIC ANALYSIS
POWER TRANSFER MODEL

 

T V T V I

Fowor Transfor Model

glocnl load-O
arc-0.4

 
   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

It) .
x I
gains-0.0
o .—
§ hint—1.0
"0'10 0.2 o 4 0.6 0.3 1
Pinf (pu)
(a)
0-4 '. r I .6
:Powor Transfdr Model _
0.3 ,. .............. ............ ........... no“. '0..d.-o .. . . . . .
0.2 - »
“x7 o _ .
_0'1 _ .............. ....................................................................
_0‘2 .. ........... ................................................. ...............
-O.3 .. ............. I ..........................................................
—o.4 l 1 z 1 a
O 0.2 0.4 0.6 0.8 1
Pint (pu)

(b)
Figure 81.1 Power Transfer Model: Effect of real and reactive power transfer

to the infinite bus on the parameter Ks . (a) Real and reactive power flow in
the same direction, (b) Real and reactive power flow in opposite directions.

227

K4

K4

228

 

 

 

 

r r I l I
2.5._..Qinfa°_.7 ................ ;""W"T ......... l A.” ............... ...

  
    
   

15_..........;, ........3. ............. ................ ; ............. _.
.

flooalload-Oé
gxesOA
QSL .............. ............... ................ ............... .

Qinf=0.0 ;

 

 

 

 

2'5 T. l E ! ﬁ

15.. .. . .. .. ............ ant-'51JO'H"

 
 

Power Transfer Model

 

 

 

: Elocal load-O i

0.5 i— .................... . .......... 3.x..e'.’.o...4. ............................ . ......... -
o 1 l l J A
O 0.2 0.4 0.8 1

0.6
Pint (pu)

(b)

Figure E1.2 Power Transfer Model: Effect of real and reactive power transfer
to the infinite bus on the parameter K4 . (a) Real and reactive power flow in
the same direction, (b) Real and reactive power flow in opposite directions.

 

 

  

 

 

 

 

 

 

 

3 f I I T '
pint-0.7
2 5 _ _ Power Transfer Model .-
local load-O 3
Eire-0.4 '
2 ~- ............................ _
<0
‘2‘.
g 1.5L .......... .
’f
SE 3
QIDI80.0
‘ ._ ........................................... . ............... q
0.5 - . . ...... ..
0 l l 4 l l
O 0.2 O 4 0.6 0.8 1
Pint (nu)
(a)
l :6 l i r l I
1 4 Power Transfer Model Glut-0.0
' local load-O ' ‘
ire-0.4
1 .................................................... / -.. .............. ..
g S
, Oink—1 .0
$0.8 ............ ............... ...
| .
SE ;
0.6 . . . — , . . . ............................................... ..
0‘4 . ............. § ................................................ ..
0.2 .......................... . . .............. ............................................... q
o L 1 1 l l
0 O 2 0.4 0.6 O 8 1
Pint (w)

(b)

Figure E13 Power Transfer Model: Effect of real and reactive power transfer
to the infinite bus on the parameter K1 -K2K5/K5. (a) Real and reactive power flow
in the same direction. (b) Real and reactive power flow in opposite directions.

2M)

 

I I I T I

V . ............... Fdwé}. Tr'a'ns‘fe'r'M'o’d-el ...... . . . . ......... '1
~ : f rocaI load-0 f '
\\ ; Exec-0.4 '

: : : §oinr.0.o
0.5.... .m g ............... .L .............. _.

 

K1-K2K3K4

 

 

 

 

. .
. .
._ .............. _. ................ .-..
. . . a
; . . . .
’ ‘ Oil
' -10
, n .

 

 

 

_ .5 i i i i i
O 0 0.2 0.4 0.6 0.8 1
Pint (pu)

(a)

 

‘ r f T I I

0,9f- ....... . ...... Poww.Tr.Mr.M°dd ....... ., ................ g.nt-..-.1:.o.....(
Flocat load-O ; i i

e

 

 

 

 

 

 

O 0.2 0.4 0.6 0.8
Pint (pu)

(b)

Figure E1.4 Power Transfer Model: Effect of real and reactive power transfer
to the infinite bus on the parameter K1 ~K2K3K4. (a) Real and reactive power
flow in the same direction, (b) Real and reactive power flow in opposite directions.

APPENDIX E2

FIGURES FOR DIAGNOSTIC ANALYSIS
LOCAL LOADING MODEL

APPENDIX E2

FIGURES FOR DIAGNOSTIC ANALYSIS
LOCAL LOADING MODEL

 

 

 

 

 

K4

f r : 3
_O.1’- ............ ... ..... u ..:... .... .:. ....1... 2 .. .. ...d

—O.15‘- ............... .............. ;. ..... _3

..... LocalLoadModel
: Pinte-Qint-O ‘
are—OB

 

 

1 1 1 4 1
“0'20 0.2 0.4 0.6 0.8 1
Fish (pu)

(a)

 

 

-o.5)-- .. .. . .. ..

1 5 ... .............................. . ........... . .............................................. ....
- . . . . .

K4

ﬁPinr-Qinr-of : . I
_235- ............... "-09 ............... ............... ........... 4

_2_ ............... Leca.|.l.-9e.d.Mec19! ....... .............. ...........

-3p--~ -- .3.
XSh--—O.6

_3_5_.. , ...... _ . .. . ..2. .L

 

 

_4 1 1 1 1 1
O 0.2 0.4 0.6 0.8 1
Fish (pu)

(b)

 

Figure E2.1 Local Loading Model: Effect of the shunt local load on the
parameters K... (a) Inductive loading, (b) Capacitive loading

231

K5

232

 

I I I I l

: ; LocalLoadModel 3
oo25......... .............. ...

. ore-0.9 .
002... ..... ......... ........... .............. ............. .4
0.015? ............. : . ......... : ....... .....E..... ........é .......... ............... —1

0.01,. .. . . ..... ..

0.005

    

 

 

stem-0.1

 

 

 

—o.oos 1 1 1 1 1
o 0.2 0.4 0.6 3 0.8 1

 

0-5 T. r r. 1 !
0.46~-~~ .............. ........... §.x.h._o,a.....

0.4» .......... ,...,......Locglteadqual ..... ......

i Pint-Qiht-O : § ;
0_35 .. .. ............. E ........ x..0_9§- ................ ............... .3 ............... ............... _
0.3L. .............. -............. .. ............. . ....o: ................ ............... 1
gozs- ......... .............. ............... _

o_2L. ............... ............... ...... ........... ............... —1

Q1. .............. 5 ............... 3. . . ................ Xeh-—10_

 

 

 

Figure E2.2 Local Loading Model: Effect of the shunt local load on the
parameter K5, (a) Inductive loading, (b) Capacitive loading

233

 

0.12 l I I

iLooalLoadMiodel 3 .
0'1? ............... P'nmn'ﬂ. ......... .......... ......1
19-0.9 I

 

 

 

 

 

 

0.04,. ...................... . ...............................................................
0.02? ....................................... 3................................: ...............
___...——— i EXsh-o1
0... ................................................ ................................. C' .............. .
0 02 0.4 . 03 1

 

o_5_....... ................ ..... ...........; ....... .1 ............... .1
LocalLoadModel ; g / ;
0.4l—. Wxa-D.9M'. ............... ............... r' ............. ...............

DAL. ........... ............. ................ ............... ...

 

 

 

 

 

(b)

Figure E23 Local Loading Model: Effect of the local load on the
parameters K1 -K2K3K4 , (a) Inductive loading, (b) Capacitive loading

Kl—K2K5/K6

Kl-KZWKG

0.1

L- . ................ -
PinIf-Qinf-O
XO-O.9
0.05 " ..
./ 3 z 2 2
1 1 1 1 1
O 0.2 0.4 0.6 0.8 1
95" (W)
(a)
" XShI—I .0

—4

—5

l»)
b.)
A

 

I I

I

.Lesell-eedMede'f ........... ....... ..........

I I

 

 

 

 

 

 

 

 

 

................................

 

 

.............................................. 3. . . . . . ..
Xsh--0.1 3
l i

(b)

Figure E2.4 Local Loading Model: Effect of real-inductive loading on the
parameters K1-K2K5/K5 (a) Inductive loading, (b) Capacitive loading

 

APPENDIX E3

FIGURES FOR DIAGNOSTIC ANALYSIS
COMBINATION MODEL

APPENDIX E3

FIGURES FOR DIAGNOSTIC ANALYSISLOCAL
LOADING MODEL

 

r ' T l 7
:Combination Model _

" \, 1 § Xsh-O.1 7 3 ’
. ‘ . 5"“0-4 . :

_0.04L .............. ............... . . .. - ...-......é ................ 2: ............... -

K5

E 1 1 "II-1.0
_o.061.... ............ ........ . . .. . Q .

..........................

—0.08

I

.................................................

. .
I h I I I
0.12... ............. : .......... ................ ; ........................... _ ............... _.

_0.14L..... ....... . ........ ........... ...;.QIDIQ0.0...._
1 '1 i i 1
O 0.2 0.4 0.6 0.8 1

Pinf(pu)

(a)

 

 

 

 

r I I I I

.f ........... ....sqwoereww .......... ..............

  
   
 

-

, §x:n-o.17 _
_o_05...... ........ . “-0.4, ................ .......... _,

..............

.. ;
x QIHI-0.0
- d

 

 

_o.25._ ............... .. .......... . ..... 2................3................S ............... _(

 

 

 

0.6
Pint (pu)
(b)
Figure E3.1 Combination Model: Effect of power transfer on the parameter K5

when the local load is real-inductive. (a) Power flow in the same direction.
(b) Real and reactive power flow in opposite directions.

235

 

2 1 ! I 1 I

Combination Model
..Rshe-Q..65. . i ............. _
‘ Xshs—OJ 7

         
 

 

 

 

 

 

(a)
2 r 1 r r I
1 . Combination Model:
1- ..;.. --~~-Fish=-0-.65 ........... .............. ..
' ' Xsh-—0.1 7
0 _ XO-O.4 . . _
_1 l' <

 

 

 

 

(b)

Figure E32 Combination Model: Effect of power transfer on the parameter K5
when the local load is real-capacitive. (a) Powerflow in the same direction.
(b) Real and reactive power ﬂow in opposite directions.

K4

1

237

.5 r

 

    
  

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

f F .' '
:Cornblnatlon Model
ﬁsh-0.65 '
§Xsh-o.1 7
30-0.4
1 ’- .1
0.5-... .....E...,t. ....§.-.. : .: ............... d
ant-t .0
o ; ; g 1 ;
0 0.2 0.4 0.6 0.8 1
Pint (pu)
(a)
1 -5 I r 1 l f
. 1 g E OWN-0.0
1 .. ..... HH'HEHQAih'f-H—‘OIBI q
:5 . .
Combination Model
chh-O.65 . 3
0.5....”H. XSh-O.17. ................ .............. 4
gxo-OA . O
O l I l 1 1
O 0.2 0.4 0.6 0.8 1
Pint (pu)
(b)

Figure 63.3 Combination Model: Effect of power transfer on the parameter K4
when the local load is real-inductive. (a) Power ﬂow in the same direction,
(b) Real and reactive power flow in opposite directions.

t0
'9.)
OC

 

 

  
 

 

 

 

 

 

      

 

 

 

30. . . . . ., f. . . . .. ,,§.Qinr-o.a....
.................. 1
............ f-Qinf-tt'.0""‘
ECombinatlon Model
S :Rsh-0,65....g ............... ..
szh--o.t7 '
4 _ p E Site-0.4
o l L i l l
O 0.2 0.4 0.6 0.8 1
Plnf (pu)
(a)
0 1. T ' 3 !
Combination Model
Rah-0.65
Xsh.-_0.17 . ,
_1o_............. . ................ ........... ...-
- anfI—LO
x
_15.. ......... q
_20. ..... q
"flu-0.1
_25 ; t L 1 i
0.2 0.4 0.6 0.8 1
Pinf(pu)

Flgure E3.4 Combination Model: Effect of power transfer on the parameter K4

(b)

when the local load is real-capacitive. (a) Power flow in the same direction,
(b) Real and reactive power ﬂow in opposite directions.

Kl-K2K5/K6

Kl-KZKSIKB

239

 

 

 

 

 

   
     

 

 

 

‘ form-0.0
9h- --------------- I .............. l ................ t.................:. .............. j ............... q
8"""" ....... Co.mbin.ationModel ......... ....... ....... ........ ........ ,......—
7_ .......... ....fFish.-Q-.6§....§ ................ 3. ............ ........... _
:XSh-0J7 ‘ '
6,. . fire-0.4 a
Sr... ................ _l
4i. ........................................................ Oi "181......0...
3... ....................................... _://./. ............................... ﬂ
2.. ................................. _,/ . .............................. ..
‘LPW.."_______:—:—::"”r" ..E ....... . ........ i................é ................ i ............... Cl
0 i i L J j
0 0.2 0.4 0.6 0.8 1
Pinf(pu)
(a)
30 r r r r r
‘ EON--03
Combination Model _ . §
EX.h-O.17 '
Ezra-0.4
20,. .:.. . ...... ..
15'- -4
10w » «
5'- r
o L L
O 0.2 0.4 0.6 0.8 1

Pint (pu)

(b)

Figure 53.5 Combination Model: Effect of power transfers to the infinite bus on
the synchronizing term K, -K2K5/K6 (regulated) when the local load is real-inductive.
(a) Real and reactive power flow in the same direction. (b) Real and reactive power
flow in opposite directions.

Kl-K2K5/K6

240

 

I I l 1' I

35 iComblnatlon Model .
:Rsh-O-GS ‘ ._ . a . l .
3_ Xsh-—017 , -
PXB-OA’ .
2_5 ,. : .,. . “4

 
     
  

0mm .0

 

 

 

 

  
   
  

g
Q
Q
J.
x
ngnt-O.3
..1 i .
0 0.2 0.4 0.6 0.8 1
Plnf (pu)
(a)
1 F .T I T !
:Combinatlon Model
gRsh:O.65 _ Z
05? ....... Xsh-v-Otz ............... ............... _

xe-O.4 _
‘ faint-4 .o

 

 

_15 L ; ; ;

 

t
0 0.2 0.4 0.6 0.8 1
Pint (pu)

(b)

Flgure E3.6 Combination Model: Effect of power transfers to the infinite bus on

the synchronizing term K, -K2K5/K6 (regulated) when the local load is real~capacitive.
(a) Real and reactive power ﬂow in the same direction, (b) Real and reactive power
flow in opposite directions.

241

 

 

 

 

 

 

1.5 r i I I ‘
1_ _ _ j .. x 'g .:.Qinf-0.0 a

 

 

 

 

 

 

 

 

 

Kl -K2K3K4

 

 

 

    

 

Combination Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_o_5-._...,RshuO‘;65 .......... _ ,. ... _. f, ;
XSh-O.17 OWN-1.0
XB-O.4. ‘ ‘ ‘
_1 a 1 z i i
0 0.2 0.4 0.6 0.8 1
Pinf (pu)
(a)
T 3 f r ‘.
geomblnation Model
1 RSh-O.65
.Xsh-O.17
,xe-OA , . r 3
o’er ............... g .............. ............ ........ .......... ......§.Q'.'.‘_".'.‘.9-§.._
0.6 L- : i ........ -
0.4... .................................... - ‘4 .4
f . : : ;
0.2... ..... W ................ ................ .” .............. _,
/” j ‘ ‘ Qtnf=0 0
OF. ................ .................................................................. a
O 0.2 0.4 0.6 0.8 1
Pint (pu)

(b)

Figure E3.7 Combination Model: Effect of power transfers to the infinite bus on

the synchronizing term K, -K2K3K4 (unregulated) when the local load is real-inductive.
(a) Real and reactive power flow in the same direction, (b) Real and reactive power
flow in opposite directions.

K1-K2K3K4

242

 

‘ ' V T l
1 '8 - ............ - ... .......... . e . . . i ........... - . s . E ........ 'o - v ... o . e ’M. ............... Eu - . u , .......... d
; 3 ;Combination odel ;

 

 

 

 

 

 

 

° 0 0.2 0.4 0.6 0.3 1
Pinf (pu)

(a)

 

F Y t ', F
1.2i— ............... ;,...............é................E................é ................ é. .............. _1
Contain-tion Model : 5 Q,ﬂ,__,_o
{Fish-0.65 i ' 1
1n- .............. :xghro.’.17. ............................................................ .-

   
  

  

 

 

 

:2 :
g g
'1 0.6» s ..... ..
a: ? :
pink—0.1
o_4.. ...... . A ...,
i ”‘/
/' é ; s 2
o 4 ; t i i
O 0.2 0.4 0.6 0.8 1

Pinf (pu)

(b)

Figure E33 Corrbination Model: Effect of power transfers to the infinite bus on

the synchronizing term K, -K2K3K4 (unregulated) when the local load is real-capacitive.
(a) Real and reactive power flow in the same direction, (b) Real and reactive power
flow in opposite directions.

K5

0.15

0.1

0.05

-0.05

-O.1

0.8

243

 

I V I l I

.

ﬁPlnt-oa Quit-0.5

 

 

_ ............... , ................ . ................ f-XSh-O.~1-----
.

1 l l _L

 

 

 

O 0.2 0.4 0.8 1

0.6
Rsh (pu)

(a)

 

 

 

 

7 7 Y f Y
: 3 Combination Model _
0.7- ............... .............. iPinf-O.8..Oirr:f-0.5 .......... g ............... _
- 5 bro-0.4 ' 3
QGPH. . ............. .......... . ..... ..... . ......... _
; ; ; : XSh'-°-‘
0.5.. ............... .............. ................ ................ ............... _
0'4. ........... _
0.3)....”W .............,:.... ..; ........... ......... ............... .4
0.2,. ....................................... ................................ E ............... _,
ﬁXSh-—1.0
0.1 l l I l l.
O 0.2 0.4 0.6 0.8 1
Rsh(pu)

(b)

Figure E3.9 Combination Model: Effect of local loading on the parameter

K5
(a)

(regulated) when the transfer to the infinite bus is at P,,,,=O.8, Q...=0.5.
real-inductive loading; (b) real-capacitive loading.

 

i ' ! F !
1 .4 _. ............... ........ . ..... : ........ . ....... Combination Mbdﬁl .......... ............... .4
' gPinr-oe cameos

  
 

{XS-0.4
1.2” : " ' ‘

SXSh-ﬂ .0

...............

 

 

 

 

 

 

 

 

0'20 o 2 o 4 0.6 o e 1
Fish (pu)

 

 

 

 

 

 

 

 

 

3 i I f r l

‘x’ 0.. .................................................................................. ._(

_1L.... ..... CombinauonModei ............... _
Pintﬁ-Oﬂ aim-0.5 ' ’ '
xe-0.4 S : t i

' ' i 3 X8h--O.4
-2- ............. ..... . ....... .............. 3 ...............

_3 1 1 l l l
O 02 0.4 06 08 1

Figure £3.10 Combination Model: Effect of local loading on the parameter
K4 (unregulated) when the transfer to the infinite bus is at Pim=0.8, Q...=0.5.
(a) real-inductive loading; (b) real-capacitive loading.

245

 

I
, . :XSh-LO
0.95 ._ ............ ....... V ...... ; ............... $ ,,,,,,,,,,,,,, ~ ............ _(

 

0.85” ............. ...... _ .. _ ................ ............... .4
3 / '/ i i :Xsh-Q1

.0
O
l
i

K1—K2K3K4
.0
\l
on

/ goombination Model

/ ' " ' ﬁpinreo‘e amigos“ Q " ""
2 ; EKG-0.4 2 2

0.65). ............. ................ ..... . .......... x ........... r. .; ............... —:

.0
V
1

°~55F ............. ................ g ................ ................ ............... _

 

 

05 i i i L 1'

 

0.6
Fish (pu)

(a)

 

4 1 V I U I

3.5 ~ Ram—0:4 . . . A

combination Model 2
3 b .............. germane. . QQfHO.-5. .......... ............... g .............. g ............... 4
xe-O.4 I f
2.5 p ........... _ 1 . ................ g ................ .............. j ................ § .......... , . . d

K1-K2K3K4

Xsh-—1.0
1 .. ........................................................................ I ............... .1

0.5). ........... . ................ ................ Z ............... -1

 

 

 

(b)

Figure E3.1 1 Combination Model: Effect of local loading on the parameter
K1 -K2K3K4 (unregulated) when the transfer to the infinite bus is at PiM=0.8. Q...=O.5.
(a) real-inductive loading; (b) real-capacitive loading.

K1-K2K5/K6

10

246

 

I T l I I

,........ ..,...b°mb|naﬂgnMode| ..... .. ................ ............. ...... . ........ .4
Pint-0.8 emf-0.5 ' '

q

E :Xsh-OJ

 

..........................................................................................

 

 

 

 

 

O 0.2 0.4 0.6 0.8 1
Fish (pu)

(a)

 

Kl-K2K5/K6
/
/

_2_ ............. Piaf-0.8..Qini.o.s. .......... .............. g ............... a
Eire-0.4 ' ' '

 

 

' 3 : : Xsh-—04
_3,_ ............... ........ . ...... . ............... ............... -4
_4 i ; i 4' ;

 

Figure E3.12 Combination Model: Effect of local loading on the parameter

K1 -K2K5/K5 (regulated) when the transfer to the infinite bus is at Pw=0.8, Q...=O.5.

(a) real-inductive loading; (b) real-capacitive loading.

 

 

247

 

 

 

 

 

Ks

 

 

 

 

 

1.2P—"" .... ..E .. ..... . , ..: ....... . ....... ,.....W ... .4 ............... x4 Sh...1.'Q-I
1i— ............................................. _
0.8- . ct
0.6? ................. ........................................................... ...
0'4 ' gramme ‘
KIM-0.5
i : ate-0.4 .
02..., ................. . ....... ,... ....... _(
4 i L L '
O 02 0.4 0.6 08 1
RSh(PU)
(a)
0.1 f . r . i
0.08-. ...... .. . ..é... . ....... Pin-f;o-I‘8 ........ I ................. .......
ﬂint-0.5
0.06
0.04

 

 

 

—0.02

 

 

 

 

-0.04._. - .' ............. . .............. Xs‘ﬁiO”.1'-l

 

 

 

_0-06 1 1 l 1 1
O 0.2 0.4 0.6 0.8 1

Fish (pu)

(D)

Figure £3.13 Corrbination Model: Effect inductive-real local load on the
system synchronizing and damping torque coefficients when Pm=0.8, 05“,:05
(a) synchronizingforque coefficient KS, (b) damping torque coefficient KD

248

 

 

 

 

 

 

Figure £3.14 Combination Model: Damping torque coefficienth and
parameter K5 for different levels of real local loading

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