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STRUCTURAL BIFURCATION TESTS FOR
VOLTAGE COLLAPSE AND LOW FREQUENCY OSCILLATION
IN MULTIMACHINE POWER SYSTEMS

By

TZONG-YIH GUO

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1992

ABSTRACT

STRUCTURAL BIFURCATION TESTS FOR
VOLTAGE COLLAPSE AND LOW FREQUENCY OSCILLATION
IN MULTIMACI-IINE POWER SYSTEMS
By
TZONG-YIH GUO

A differential-a1 gebraic power system model is developed that is more complete
than previous models utilized to study bifurcations of power system dynamics. The
added modeling complexity and accuracy is essential for studying both voltage col-

lapse and low frequency oscillation in a power system.

This dissertation systematically classifies the bifurcation and stability problems,
establishes necessary conditions for each bifurcation and stability problem, and
identifies the generic types of bifurcation associated with steady state angle stability,
voltage instability, and low frequency oscillation. The three generic types of bifurca-
tion in the differential-algebraic power system model are shown to be: (1)
Static/Algebraic Bifurcation due to the singularity of the equivalent static/algebraic
Jacobian matrix; (2) Static Bifurcation and (3) Dynamic Bifurcation due to the
equivalent system Jacobian matrix having eigenvalues with zero real parts. Voltage-
angle bifurcation in the load flow model is shown to be an excellent approximation to
the static/algebraic bifm'cation. The prerequisite for the static/algebraic bifurcation test
is the single machine stability; the one for the static and dynamic bifurcation tests is
the causality condition. These two special stability problems are argued to be improb-

able in a practical sense, even though they are not argued to be non-generic.

The static/algebraic and static bifurcations are shown to be equivalent under the
assumption that single machine stability and causality condition hold. Three types of
static bifurcation are identified to occur and necessary conditions for each are derived;
one in mechanical dynamics, one in flux decay dynamics, and the third one in both
mechanical and flux decay dynamics. The load flow bifurcation is shown to be nearly
identical to the static bifm'cation in mechanical dynamics; but not the other two types.
This dissertation suggests utilizing the static/algebraic Jacobian matrix rather than the
test matrix for static bifurcation as the test for identifying static bifurcation. The
static/algebraic bifurcation test avoids inverting causality matrix, and can be performed
by slightly modifying the load flow program Both analytical and simulation results
for the load flow bifurcation test and the static bifurcation test illustrate the important
role of maintaining the reactive generation reserves at generators in preventing voltage
collapse and steady state stability. These results also point out that both loss of

causality and loss of single machine stability are not likely to occur.

Three types of dynamic bifurcation are identified and necessary conditions for
each are established; in mechanical dynamics, in flux decay dynamics, and in both
mechanical and flux decay dynamics. Experimental results indicate that tests for vari-
ous types of dynamic bifurcation actually classify the type of bifurcation in terms of
the dynamics (mechanical, flux decay, or both) in which it occurs. Such classification
is shown to be more precise than eigenvalue analysis. It is expected that the structur-
ally represented test conditions for various types of static/dynamic bifurcation will be
useful in establishing the factors of these bifurcations.

The proposed approach provides a basis for static/dynamic point of
collapse/oscillation method, that is similar to a V—P or Q -V point of collapse method
in the load flow model, but detects the first mode to become unstable as stress is added
to the network. This unstable mode can be associated with any of the stability prob-
lems classified and identified in this dissertation.

Copyright by
TZONG-YIH GUO
1992

To my parents

ACKNOWLEDGEMENTS

I am indebted to my parents, wife, brother, and sisters for their support and
encouragement over the years. Special remembrance goes to my mother, who passed
away during the pursuit of my degree. I would like to express the sincerest thanks to

my wife for her understanding, support, patience, encouragement. and love.

I would like to thank my dissertation advisor, Dr. Robert A. Schlueter, for his
valuable and constructive guidance. I am also grateful to my committee members, Dr.
Gerald Park, Dr. Hassan Khalil, and Dr. Wei-Eihn Kuan, for their professional opin-
ions and advice.

The financial support of Taiwan Power Company (Taiwan, ROC) is gratefully
acknowledged for making the completion of my Ph.D. dissertation possible. The
research support of Electric Power Research Institute (EPRI), and Detroit Edison Com-

pany (DECO) is also appreciated.

TABLE OF CONTENTS

LIST OF TABLES ..................................................................................................... x
LIST OF FIGURES ........................................................................ _ ........................... xi
1. Introduction ........................................................................................................... 1
1.1 Literature Review about Dynamic Stability .......................................... 3

1.2 Literature Review about Voltage Collapse ............................................ 7

1.3 Organization and Main Contributions of This Dissertation .................. 9

2. Power System Dynamic Model ........................................................................... 17
2.1 Load Flow Model ................................................................................... 17

2.2 Transformer and Phase Shifter Model ................................................... 21

2.3 Load Demand Model .............................................................................. 25

2.4 Synchronous Machine Model ................................................................. 26

2.4.1 Power Equation at Intemal Bus .............................................. 29

2.4.2 Power Equation at Terminal Bus ............................................ 31

2.4.3 Flux Decay Equation ............................................................... 32

2.4.4 Excitation System Model ........................................................ 33

2.4.5 Load Compensator Model ....................................................... 36

2.4.6 Power System Stabilizer Model .............................................. 38

2.4.7 Swing Equation ........................................................................ 40

2.4.8 Speed-Governing-Turbine System Model .............................. 41

2.5 Equilibrium point of Power System Dynamic Model ........................... 44

2.6 Linearized Power System Model ........................................................... 48

3. Bifurcation in a Load Flow Model ...................................................................... 55
3.1 Linearized Load Flow Model ................................................................. 56

3.2 Load Flow Bifurcation ............................................................. ‘ .............. 64

3.2.1 LF-T Bifurcation ? ............................................................... 66

3.2.2 LF-I-I Bifurcation ? .............................................................. 68

3.2.3 LF-L Bifurcation ? ............................................................... 69

3.3 Generic Load Flow Bifurcation ............................................................. 70

3.4 Simulation Results .................................................................................. 72

4. Stability Problems in a Differential-Algebraic Model ........................................ 80
4.1 Linearized Power System Dynamic Model ........................................... 81

4.2 Bifurcations in Power System Dynamic Model .................................... 89

4.3 Prerequisites of Bifurcation Study ......................................................... 93

4.3.1 Causality Condition — gy ...................................................... 95

4.3.2 Single Machine Condition — f x ............................................ 97

4.4 Structure of Static/Algebraic Jacobian Matrix — J, ............................ 103

4.5 Generic Static/Algebraic Bifurcation Test ............................................. 113

4.6 Simulation Results .................................................................................. 114

5. Static/Dynamic Biftucation in Dynamic Model .................................................. 123
5.1 Static/Dynamic Biftu'cation .................................................................... 125

5.2 Mechanical and Flux Decay Dynamics ................................................. 130

5.3 Mechanical Dynamics ............................................................................ 135

5.4 Flux Decay Dynamics ............................................................................ 139

5.5 Simulation Results .................................................................................. 142

viii

6. Conclusion and Discussion .................................................................................. 153

6.1 Main Contributions of This Dissertation ............................................... 155
6.2 Relationship to Literature ....................................................................... 171
6.3 Future Work .............................................................................. A .............. 173
APPENDIX A — Schur’s Formula and Inverse Matrix .......................................... 176
APPENDIX B —- A53 ............................................................................................... 173
APPENDIX C — K33 and Km ............................................................................... 182
APPENDIX D — C3,: and cat; ............................................................................... 186
LIST OF REFERENCES ........................................................................................... 189

LIST OF TABLES

4-1 Static Data of Synchronous Machines .............................................................. 116
5-1 Dynamic Data of Synchronous Machines ......................................................... 145
5-2 Eigenvalue Analysis for 9-bus System .............................................................. 146

LIST OF FIGURES

A General Branch Between Two Buses ............................................................ 18
Power Flow Diagram ......................................................................................... 18
Equivalent u-circuit of a General Branch Between Two Buses ...................... 22
Synchronous Machine Model ............................................................................ 27
IEEE-DCl Excitation System Model ................................................................ 34
Load Compensator Model .................................................................................. 37
Power System Stabilizer Model ........................................................................ 39
Speed-Governing—Turbine System Model ......................................................... 42
9-bus 3-machine Power System Diagram ......................................................... 73
Simulation Results of Load Flow Bifurcation .................................................. 75
Block Diagram of Power System Jacobian Matrices ....................................... 83
Steady State (S = 0) Block Diagram of f I and Afar ....................................... 99
Steady State (5 = 0) Block Diagram of f;1 f, .............................................. 105
Simulation Results of Static/Algebraic Bifurcation .......................................... 117
Equivalent Power System Dynamic Model ...................................................... 129
Equivalent Model of Mechanical and Flux Decay Dynamics .......................... 134
9-bus 3-machine Power System Diagram ......................................................... 143
Simulation Results of Bifurcations of Mechanical Dynamics .......................... 147
Simulation Results of Bifurcations of Flux Decay Dynamics ......................... 150
Flow Chart of Mechanical and Flux Decay Bifurcation Tests ........................ 169

Chapter 1
Introduction

Low frequency oscillations have been one of the most common and frequent sta-
bility problems in utilities around the world. Voltage collapse is a stability problem
that is less common and frequent than low frequency oscillations, but has captured
widespread attention because its causes and cures were less understood. The bifurca-
tions of a power system mid-term transient stability model, that are observed and
labeled as low frequency oscillations and voltage collapse, are the subjects of this
dissertation. The determination of different types of bifurcation and stability problems
that could happen in multimachine power system load flow and dynamic models, and
the development of necessary condition tests for existence of the identified types of
bifurcation are the main purposes of this dissertation. An improved power system
mid-term transient stability model is developed and structurally organized in this
dissertation, which has the modeling accuracy needed to perform the classification and
identification of bifurcations leading to low frequency oscillations and voltage collapse.
Therefore, each developed bifurcation test structurally shows the effects of power sys-

tem components.

The power system dynamic model is characterized by a set of differential-
algebraic equations (or differential-algebraic model). The set of algebraic equations
describes the active and reactive power balance equations of the transmission network,
i.e., the power balance equations at the internal and terminal buses of machines, at the
transformer high-side buses of machines (simply called as high-side buses), and at the
load buses. The set of differential equations can include (1) synchronous machine

mechanical and flux decay model, and (2) control systems of machines: excitation

systems with load compensators, power system stabilizers, and speed-governing—turbine
systems. The coupling between the differential equations and the algebraic equations
is through the terminal bus of each machine. The load flow model describes the
power system steady state operation 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 excluding the reactive power balance equations at generator
terminal PV-buses, since the excitation systems of these generators are assumed to
have infinite gain. Note that the load demand model adopted in this dissertation can
be voltage dependent.

Based on the representation of the power systems, this dissertation will study the
bifurcations that lead to low frequency oscillations and voltage collapse by systemati-
cally developing simple tests for each class of bifurcation identified. These tests not
only avoid the intensive computation of eigenvalue analysis, but also analytically and
structurally show the effects of transmission network as well as each control system.
Based on the load flow model, the load flow bifurcation is defined, and the generic
form of the load flow bifurcation test matrix is derived and is proven to be consistent
with the generic voltage-angle bifurcation in the load flow model. In power system
dynamic model, static and dynamic bifurcations will lead to voltage and angle instabil-
ity, and will occur due to the coupling of the network, generator, and control systems.
The singularity induced static bifurcation and dynamic bifurcation will be systemati-
cally analyzed in this dissertation. The static/algebraic bifurcation will be defined and
shown to be identical to the generic singularity induced static bifurcation. The static
and dynamic biftu'cations will be defined and shown principally associated with the
generator mechanical and flux decay dynamics, and test conditions for these bifurca-
tions will be established. The proposed approach to assessing the studies of voltage
and angle dynamic stability in multimachine power systems provides a static/dynamic

point of collapse/oscillation method. The proposed approach clearly identifies the

modes mainly associated with mechanical and flux decay dynamics and structurally
shows the effects of the control systems and transmission network, yet avoids the com-

putationally intensive eigenvalue analysis.

1.1 Literature Review about Dynamic Stability

Time-domain simulation has being popularly used to assess the dynamic stability
problems of a power system differential-algebraic model (e.g., [1,2]). However,
recently most of the literature dealing with the problem of dynamic stability is filled
with eigenvalue analysis orientation [3-22]. For eigenvalue analysis, the linearized
power system dynamic model is used, and the elimination of network variables in the
algebraic equations has to be feasible such that the equivalent power system dynamic
model can be obtained and is well defined. Major part of these eigenvalue analysis
studies concentrated on the solution of eigenvalues of the whole equivalent power sys-
tem dynamic model; others on the computation of some particular modes of oscilla-
tion, especially the electromechanical modes which in general are of the low fre-
quency oscillatory modes [3,4,21,22], since the electromechanical modes of low fre-
quency oscillation is more common if oscillations would occur in power systems.
Some special eigenvalue analysis programs have been developed for the above pur-
poses, such as the AESOPS (Analysis of Essentially Spontaneous Oscillations in
Power Systems) [3] and the PEALS (Program for Eigenvalue Analysis of Large Sys-
tems) in the Small Signal Stability Package (SSSP) [4] for particular modes of eigen-
value analysis, and the MASS (Multi-Arca Small Signal Stability Program) in SSSP
[4] for the eigenvalue analysis of the whole equivalent power system dynamic model.

However, eigenvalue analysis of a linearized power system dynamic model has

some drawbacks, such as:

(l) The dimension of the system matrix for a large power system model

becomes extremely large since each synchronous generator has at least

eleven states (see Section 2.4 of Chapter 2: Power System Dynamic
Model).

(2) The computation required to compute all the eigenvalues of such a large
system matrix is beyond the capability of most computers. Numerical error
can make the results invalid when the dimension of the system matrix is so

large.

(3) Eigenvalue analysis provides little information about the factors causing the
existence of zero or positive real part of any eigenvalue. Computation of
left and right eigenvectors and participation factor matrix for each eigen-
value for iterative variation of parameters can provide some information but

at extremely high computational cost.

As mentioned above, this dissertation will develop bifurcation tests for the power sys-
tem dynamic model, especially on the mechanical and flux decay dynamics, that avoid
the eigenvalue analysis and provide insight into the effects of power system com-

ponents on each identified bifurcation.

Another group of the literature about dynamic stability emphasized the computa-
tion on the damping and synchronizing power coefficients of a power system [23-30].
The concept of damping and synchronizing power coefficients is very important to the
study on dynamic stability. This concept has been utilized extensively since chello
and Concordia extended it to study the stability of a synchronous machine affected by
excitation control [23]. However, most of the studies applying this concept were lim-
ited to a single machine case with or without excitation system dynamics [23-25]. In
order to compute the damping and synchronizing power coefficients of multimachine
power systems, some studies even needed to use some time domain signals or to com-
pute the equivalent loading condition of each machine [27-29]. Eliasson and Hill com-
puted damping power coefficient matrix for a power system where each generator was

represented by the classical model [30]. The application of Lyapunov or energy

function method is another approach to study the stability problems of a power system.
Lyapunov methods have been applied for the most part when the electrical dynamics
and the control systems dynamics were ignored [31-35]. In this dissertation, damping
and synchronizing power coefficient matrices associated with the mechanical dynamics
for multimachine power systems will be derived by taking the effects of network and
control systems into account. Lyapunov stability dreary will be applied to obtain con-
ditions on these test matrices for prevention of bifurcations of the mechanical dynam-
ics.

The theory of bifurcation [36-38] has been applied to the study of steady-state
stability and low frequency oscillation based on the differential-a1 gebraic power system
model [39-45]. Similar to the eigenvalue analysis, the studies of static and dynamic
bifurcations need the equivalent linearized power system dynamic model, provided that
the elimination of network variables in the algebraic equations is feasible. This
requirement is called as the causality condition of the differential-algebraic model
[39,40]. The equilibrium point is a bifurcation point when the system satisfies one of

the following conditions:

(a) The complete system Jacobian matrix of the differential-algebraic model is
singular,

(b) The equivalent static/algebraic equation Jacobian matrix is singular, which
is obtained by aggregating the differential equations into algebraic equations
provided that this reduction is feasible;

(c) The equivalent system Jacobian matrix has eigenvalues with zero real parts,

provided that the causality condition holds.

Note that in terms of the static bifurcation due to the singularity of the equivalent sys-
tem Jacobian mauix, these three conditions are equivalent to each other. The static
bifurcation could be saddle node bifurcation if the equivalent system Jacobian matrix

has a simple zero eigenvalue, and the transversality condition of this matrix at the

bifurcation point holds. The dynamic bifurcation could be Hopf bifurcation if the
equivalent system Jacobian matrix has a pair of pure imaginary eigenvalues, and the

nondegeneracy condition of this matrix at the bifurcation point holds [36-38].

The static bifurcation on the study of steady-state stability of the power system
was proposed to analyze the singularity of the Jacobian matrix associated with the
differential-algebraic model or with the equivalent power system dynamic model
[40.41.45]. The static bifurcation leading to loss of steady-state stability (and/or vol-
tage stability) was shown to occur in a power system model where only mechanical
dynamics was involved [40,41]. Hopf bifurcation theory has also been applied to
study the dynamic stability of a power system by taking the effects of nonlinearity into
account, assuming that the system is at the operating condition with a pair of pure
imaginary eigenvalues associated with the equivalent system Jacobian matrix [41-44].
The focus was on determining the condition for stability of Hopf bifurcation of a sim-
ple power system model for changes in the bifurcation parameter. These Hopf bifurca-
tions were shown to be pertaining to the mechanical and/or flux decay dynamics due to
the effects of control systems. This dissertation will focus on detecting occurrence of
instabilities due to low frequency oscillations, particularly associated with the mechani-
cal and flux decay dynamics, rather than on determining the behavior of the power

system when Hopf bifurcation occurs.

The role of load demand modeling on power system stability studies has been
discussed [46,47]. However, even though it is obvious that load demand modeling
plays an immrtant role on the stability studies, a proper load demand model for each
time frame of stability study has not yet come to a fairly clear point. The load

demand model adopted in this dissertation will be voltage-dependent.

1.2 Literature Review about Voltage Collapse

Basically, voltage collapse problem has been considered as either a static
phenomenon by analyzing the load flow model or a dynamic voltage behavior by tak-
ing the differential equations of synchronous machines and/or of load into account as

well.

Voltage collapse is a phenomenon where the voltage at some bus(es) of a power
system is slowly declining over a one to several minute interval and then is observed
to drop rapidly. Voltage collapse has been claimed to be a power supply-demand
problem, and has been associated with the power transfer over long and/or weak
transmission systems because of the trend of siting generators far from load centers
(mostly due to the location of power resources and environmental concerns) and
because of insufficient enhancement in the load center or weak transmission lines [48-
52].

Most of the studies on voltage collapse were based on point of collapse methods,
such as the V-P , V-Q , and Q-V cm've approaches, sensitivity analysis approaches,
and optimization approaches, that attempt to establish a load flow bifurcation point by
increasing load or transfer [48-61]. These load flow voltage collapse methods are
assumed to be associated with a lack of convergence to a load flow solution. These
methods determine the additional load or transfer at some bus(es) or on some
branch(es) before voltage collapse occurs. The additional load or power transfer that
can be added or shifted before voltage collapse occurs is the voltage stability margin.

Testing a sign change of the determinant of the load flow Jacobian matrix was
also proposed to indicate voltage instability of a power system [62]. The existence of
a pair of closely related multiple load flow solutions was shown to be an indication of
the singularity of load flow Jacobian matrix and be a proximity to voltage instability
[58]. Modal analysis is another approach by applying the SSSP algorithms [4] to com-

pute the eigenvalues, eigenvectors, and participation factors of an equivalent Q -V

Jacobian matrix. The magnitude of the small eigenvalues approaching zero is used as
a proximity measure to voltage instability for a large power system [63].

Dynamic voltage instability methods take into account the dynamics of synchro-
nous machines and their control systems, and the dynamics of load modeling. How-
ever, most of the research on dynamic voltage stability did not include the complete
control systems dynamics. Some only included the mechanical dynamics of the gen-
erator in the dynamic model [39-41,64-68]; while others neglected the mechanical
dynamics, and only kept the flux decay dynamics and/or excitation system dynamics
[69-72]. Still others included both flux decay dynamics and excitation system dynam-
ics [41,48,525333]. The effects of load (line drop) compensator and saturation on the
problem of voltage stability was investigated in [52,53]. The effects of load demand
model on voltage stability was included in [46,64,65,71,74-79].

Time domain simulation was used to show that the proximity to voltage collapse
in a single machine case could occur because of the dynamics of induction machine
load or tap changer [76,77]. Eigenvalue analysis was proposed to capture the proxim-
ity to voltage collapse by tracking the eigenvalue of a simple dynamic system [71-75].
Singular perturbation theory was also applied to investigate voltage stability problem in
[78], where difl'erent types of voltage instability patterns were shown to be related to
the singularity of load flow Jacobian matrix, to eigenvalue analysis, or to direct non-
linear analysis. The relationship between the singularity of the power system dynamic
Jacobian matrix and that of the load flow model Jacobian matrix was presented in
[45,48,525333].

The theory of bifurcation [36-38] has also been applied based on power system
differential-algebraic model, to investigate dynamic stability and voltage collapse prob-
lems. Most of the research analyzed the existence (and/or uniqueness) of the equili-
brium point of the system by testing the singularity of the system Jacobian matrix

directly or indirectly. The research was generally confined to a simple power system

where excitation systems were simplified or neglected, and speed-governing-turbine
systems and power system stabilizers were omitted [239-4152516440].

One of the necessary conditions for the static bifurcation in a power system lead-
ing to voltage collapse and/or loss of steady-state stability is the singularity of the
equivalent system Jacobian matrix [52,53,39-41,45,64-67,73,78,79]. Schlueter
developed test matrices for static voltage collapse that included the flux decay dynam-
ics and excitation systems dynamics of generators, and the effects of network [51-53].
One of the necessary conditions for the dynamic bifurcation in a power system leading
to dynamic voltage-angle instability is that the equivalent system Jacobian matrix has
eigenvalues with pure imaginary parts. Schlueter also mentioned that Hopf bifurcation
can likely occur in the mechanical dynamics [53]. Hopf bifurcation was shown to
exhibit dynamic voltage instability in a small system [41,42,67-72,79]. The bifurcation
theory was also intensively applied to the study of what occurs just prior to or just

after bifurcation of a dynamic model [41].

In this dissertation, the tests for load flow bifurcation and static/algebraic bifurca-
tion will provide a simply generic approach to assessing voltage collapse and steady
state stability in multimachine power systems. The tests for dynamic bifurcations will
precisely identify that the modes of oscillation are mainly associated with mechanical
and flux decay dynamics. Moreover, the effects of the control systems and the

transmission network will be structurally shown in the test matrices.

1.3 Organization and Main Contributions of This Dissertation

(A) Power System Dynamic Model

In Chapter 2, the representation of the components of power systems is discussed.
First, the general form of the active and reactive power flow between any two buses

connected by any kind of branch is shown. The general branch can be any

10

combination of the following network apparatus: (a) a regular transmission line with or
without series capacitor, (b) a tap changing transformer, and (c) a phase shifter. The
models utilized in previous studies of voltage collapse and low frequency oscillation
omitted the series capacitor, tap changing transformer, or phase shifter. Since these
components can affect voltage stability and low frequency oscillation, their models
should be included. The equivalent n-circuit of a general branch is derived, and then
the active and reactive power balance equations at any bus can be formulated. The
voltage-dependent load demand model at any bus is adopted in this dissertation, which
could be any combination of (1) constant power load, (2) constant impedance load, (3)

constant current load, and (4) any other voltage dependent load.

The differential equations of the power system will include synchronous
machines, and the control systems of machines. These control systems are adopted
from general IEEE models, and will be (a) excitation systems with load compensators,
(b) power system stabilizers, and (c) speed-governing-turbine systems. The model of
synchronous generators takes the armature and salient-pole effects into account. The
active and reactive power balance equations at the synchronous machine terminal
buses, and the representation of the load compensator which describes the feedback
from the terminal bus to the excitation system will also be derived in order to model
the coupling between the differential equations and the algebraic equations. These
models of power system dynamics are much more detailed than those generally util-
ized to study voltage stability and low frequency oscillation [23-35,39-45,48-79].
Again, such modeling detail is necessary to study these stability problems.

The equilibrium point representing the steady state operation condition is dis-
cussed by organizing the algebraic power balance equations of the transmission net-
work and the algebraic equations obtained by setting all the derivatives of the
differential equations to be zero. This chapter is ended with the linearized power sys-

tem model to set up the foundation of Jacobian matrices that are necessary for the

ll

discussion of bifurcations in the power system model.

(B) Voltage/Angle Bifurcation in Load Flow Model

The simplified model of the power system is the load flow model, where the
differential equations are omitted and generators are only modeled by their terminal
buses. The load flow model and its linearized representation in matrix form is dis-
cussed in the first section of Chapter 3. The linearized load flow model is structurally
described based on the three bus types: terminal bus of generator, generator-

transformer high-side bus (simply as high-side bus), and load bus.

The load flow bifurcation point is defined as the operating condition where the
load flow Jacobian matrix is singular. The voltage, angle, and voltage/angle bifurca-
tions will be shown to be the three types of bifurcations that could occur in a power
system load flow model. The voltage bifurcation and angle bifurcation, which occur
when the rows of the load flow Jacobian matrix associated with reactive power balance
equations and active power balance equations are respectively dependent, are shown to
be non- generic biftucations which can only occur when a decoupled load flow model
is used. Moreover, the load flow bifurcations due to the row dependence of some par-
ticular rows of the load flow Jacobian matrix associated with single buses are shown to
never occur. Hence, analogies between bifurcations in a two-bus model have limited
usefulness in understanding how voltage instability develops in the large power system
load flow model. Thus, the generic form of load flow bifurcation test matrix is proven
theoretically and experimentally to be the voltage-angle bifurcation which is due to
row dependence of all the rows associated with both active and reactive power balance
equations. This result indicates the important effects of the coupling between active
power and voltage, and the coupling between reactive power and angle on the

voltage-angle bifurcation.

12

Finally, the simulation for the load flow bifurcation will be performed on a 9-bus
3-machine power system. The simulation results for the load flow bifurcation illustrate
the important role of maintaining reactive power generation reserves at PV-buses in

preventing voltage collapse.

(C) Stability Problem in a Differential-Algebraic Model

Chapter 4 is devoted to the classification of stability problems in a power system
differential-algebraic model, and to the discussion of the conditions of these identified
stability problems. The linearized differential-algebraic power system dynamic model
is first developed and the properties of its Jacobian matrix are discussed in Section 4.1.

Next, based on the complete power system Jacobian matrix, basic bifurcation
theory is reviewed and six possible types of bifurcation in the power system
differential-algebraic model are identified, and two related non-bifurcation stability
problems are also identified. Necessary condition tests for each of the bifurcation and
stability problems are given. However, only three of these bifurcations are then
identified to be generic:

(a) The static/algebraic biftucation is defined when the equivalent
static/algebraic Jacobian matrix is singular, which is obtained by aggregat-
ing the linearized equilibrium equations associated with the differential
equations into the algebraic equations provided that this reduction is feasi-
ble.

(b) The static bifurcation occurs when the equivalent system Jacobian matrix is
singular, which is produced by the elimination of the network variables in
the linearized algebraic equations provided that this reduction is feasible.

(c) The dynamic bifurcation occurs when the equivalent system Jacobian

matrix has pure imaginary eigenvalues.

13

Note that the reduction in (a) of the linearized equilibrium equations associated with
the differential equations requires the stability of the single machine dynamics; the
reduction in (b) and (c) is the so-called causality condition [39] of the differential-
algebraic model. These two special stability problems, loss of single machine stability
and loss of causality, will be argued to be improbable in a practical power system
model.

Both the static/algebraic bifurcation and the static bifurcation are the type of
singularity induced static bifurcation, and will be shown to be equivalent to each other.
In this chapter, testing singularity of the static/algebraic Jacobian matrix for the singu-
larity induced static bifurcation is suggested rather than testing the singularity of the
equivalent system Jacobian matrix. This static/algebraic bifurcation test avoids the
inverse of the transmission network Jacobian submatrix (causality matrix) which is
non-sparse and has high dimension, and yet preserves the structure of the causality
matrix which is very similar to the load flow J acobian matrix. The static/algebraic
Jacobian matrix corresponds to a set of algebraic equations that are similar to those of
a load flow model, and yet is the Jacobian matrix for computing the (dynamic) equili-
brium point of the differential-algebraic model. Moreover, this approach shows that
each synchronous machine can be represented by the single-axis model, and that only
the DC. gain of the excitation system of each machine is required to model the control

systems.

The similarity and the difference between the static/algebraic Jacobian matrix, the
causality matrix, and the load flow Jacobian matrix will be comprehensively discussed.
It is concluded that if the DC. gains of the excitation systems are very large which is
generally true, the singularity induced static bifurcation test will have similar results as
the load flow bifurcation test. Furthermore, if the reactive power generation limits are
included in the model, the static/algebraic Jacobian matrix will more likely be singular

than the causality matrix. The analysis is conducted for the case where generator flux

14

decay and exciter dynamics is replaced by an algebraic equation that specifies the reac-
tive power generation and models the reactive power balance equation at the generator
terminal bus when the reactive power generation limit is reached. An alternative is to
model the field cun'ent limiter that adjusts the voltage reference on the excitation sys-
tem to hold the field current constant. When this alternative is used, static bifiucation
may occur in flux decay dynamics and in both flux decay and mechanical dynamics for
the generators which reach field current limits. This possibility is eliminated if the
generator flux decay and exciter dynamics is replaced by the algebraic reactive power
balance equation. The static bifurcation is solely associated with the mechanical
dynamics of the generators that reach field current limits when their flux decay and
exciter dynamics are replaced by the algebraic reactive power balance equations. No
discussion of the comparison of the static/a1 gebraic Jacobian matrix and the load flow
J acobian matrix is conducted in this dissertation when the field current limiter is used
to adjust voltage reference on the excitation system to control the field current to be

constant.

The simulation results utilizing the static/algebraic bifurcation test on the same
power system as used for the load flow bifurcation test show that static bifurcation and
load flow bifurcation occur at approximately the same operating point and for the same
reason. The results for the load flow model and the differential-algebraic model both
indicate that exhaustion of reactive power generation reserves causes the bifurcation to
occur if the generator electrical dynamics are replaced by the PQ-bus bus load flow

model when the reactive power generation limits are reached.

(D) Static/Dynamic Bifurcation

The static bifurcation and dynamic bifurcation are generic types of bifurcation in
the power system difl'erential-algebraic model. The static or dynamic bifurcation can

occur when the equivalent system Jacobian matrix has eigenvalues with zero real parts,

15

provided that the causality condition holds. Based on (a) the proof that the control
system themselves should have asymptotically stable eigenvalues, and (b) the experi-
mental results of the eigenvalue analysis [4] that most oscillations occur in the
mechanical and/or flux decay dynamics, Chapter 5 mainly focuses on the derivation of
simple necessary condition tests for static and dynamic bifurcations in multimachine
power systems associated with mechanical dynamics and with flux decay dynamics.
First, the portion of the equivalent system Jacobian matrix associated with the
control systems themselves is shown to have asymptotically stable eigenvalues. A
reduced Laplace transformed (S = jQ) model of the equivalent system Jacobian matrix
pertaining to solely mechanical and flux decay dynamics is produced by the elimina-
tion of the control system dynamics since they are proven to be nonsingular for any
(I 2 0. This resultant matrix provides the test of static bifurcation if it is singular at
Q = 0, and the test of dynamic bifurcation if it is singular at some I) > 0. Note that

this test matrix also structurally shows the effects of network and control systems.

Damping and synchronizing power coefficient matrices associated with the
mechanical dynamics will be further derived by further elimination of flux decay
dynamics. Test conditions on these matrices for identification and prevention of static
(Q = 0) and dynamic ((2 > 0) bifurcations of the mechanical dynamics will be esta-
blished. Test matrices for identification and prevention of flux decay bifurcation will

be similarly generated from the reduced model.

A static/dynamic point of collapse/oscillation method is proposed that would test
for the closest bifurcation as the system is continually stressed, as is the case when a
Q—V or P—V curve is produced in a load flow model. The method would increase a
transfer or loading level for some active power loading pattern or reactive power load
at some bus(es), determine when the first bifurcation occurs and its type. This metho-
dology would assess proximity to both steady state voltage-angle stability and low fre-

quency oscillation.

tion
tage

Oll‘

16

The simulation results on a power system [4] in the proximity to experiencing
dynamic bifurcation leading to low frequency oscillation illustrate that the proposed
approach clearly identifies the modes associated with mechanical dynamics and flux
decay dynamics without performing eigenvalue analysis. The proposed approach is
shown to identify whether the bifurcation is in the mechanical or flux decay dynamics
when eigenvalue analysis can only shows that the mode affects both mechanical and

electrical dynamics.

(E) Conclusion and Discussion '

This last chapter ends this dissertation with the brief summary of the contribu-
tions of the proposed approaches to assessing the generic bifurcations leading to vol-
tage collapse and low frequency oscillation in the multimachine power system based
on two different models: (1) the voltage/angle bifurcation in the load flow model, and
(2) the static and dynamic bifurcations in the differential-algebraic power system
model. A flow chart for the identification and prevention the generic types of bifurca-
tion is provided. SOme potential future research about this topic is also proposed.

Chapter 2
Power System Dynamic Model

Figure 2-1 represents a general model of any element of the network that can
connect any two buses i and j. The general model for a network element or branch
can represent a transmission line, transformer, phase shifter or any combination. A
specific example is a step-up transformer between a generator terminal bus and its
high-side bus, which allows a general discussion of the modeling of both power injec-
tion (the mismatch between power generation and load demand) at a bus and power

flow on a general branch.

This general branch can be represented by an equivalent n-circuit, which will be
discussed in Section 2.2. The power injection at each bus t“, P,- + 1 Q“ is the alge-
braic sum of power supply, PC,- + j Q6“ and power demand, PD,- + j QDt- The
power demand model, in general, can be any combination of constant power model,
constant current model, constant impedance model, and other types of voltage depen-
dent model, which will be discussed in Section 2.3. Section 2.4 shows the generator
model. The load flow model or the power balance equations that integrate the expres-

sion of a power system is now discussed in the next section.

2.1 Load Flow Model

Figure 2-2 shows the power flow diagram of any two buses, t’ and j. The con-
nection between them is represented by an equivalent general rt-circuit. Note that the
equivalent charging admittance at bus 1', 170,-, could be different from that at bus 1',
17C},- , and that the equivalent transfer admittances, 17,-,- and 17,-,- might not be the same.

These effects are due to the branch types of transformer and phase shifter (see Section

17

18

 

 

 

 

 

 

 

 

 

 

 

 

V1.20}.
Step-up General
Branch
Terminal High-Side
Bus .
P l+jQi Pj TJQJ'
P1=PGFPDi Qi=QGi‘QDi
Fig. 2-1 A General Branch Between TWO Buses
e o 0‘ > ‘7 ’0 o o
1 G]. i+iji= inzyji *
P i + le. . [’14-ij
I 7Cij| = GCij +jBCij GCji+jBCji = l YCjil
_ 2
V101} "Vi VJ 't'jT

_ 2
9;... -ViBij- V.VjUij

Fig. 2-2 Power Flow Diagram

19

2.2). The generality of this branch model is introduced to permit modeling of

‘1’

transformer and phase shifter [80].

Assume that there are n buses in a power system. Then the power flow between
bus 1' and bus j, and the power balance equation at each bus 1' can be expressed as
follows [81]:

Pi} = V32 GU - Vi Vj [GU 008(0; - 01-) +30- sin(9,- - 61)]

= V52 03} - Vi Vj Y5]: 008(95 - 91' - 731')

= Viz Gij — Vi Vj Tr” (2'13)
Q,j = - V} 19,-,- — v,- v,- [0,-j sin(e,- - 9,.) - 3,]. cos(9,- — 9p]

= - Viz Bij - Vi V]- Yij SID(0" " 9" "‘ 731')

= — Viz Bi] - V‘- V] U”: (z'lb)

n
Pi=PGi-PDI=.1(V1'20Cij+Pij)
I:

u n
= V32 2 (CCU + Gij) '- 2 Vi Vj Ti] ; j #1. (2'23)
j=1 j=1

Q1 = 961 - Q0: = f; (‘ V12 BCij 4' Qt'j)

j=1
= — 11,2 1:1 (so-,- + By) 2:1 V,- V,- U,- ; j at (2-2b)
where
751' = m ; ‘Yt'j = tan‘1 (Bij/Gt'j) (2'3)
TU = 05,- cos(6,- - 0,) + Bii sin(0,- - 01-) (2-4a)

= Yij °°S(9t° ' 9; - 71;)

U5]: = Gij sin(9,- '- 9") - Bij (308(65 - 6") (24b)

20

= Y1} Sin(9t' " 91' " 7:1)

Note that the above branch power flow is expressed without taking the equivalent
charging admittance into account, since in the ”dumped" model of a general branch the
equivalent charging admittances is considered to be connected at the buses.

During solving the power flow equations, the following variables are specified:

(1) V,- (1' = l, 2, - - - , m), i.e., voltages at the swing bus (bus number 1 with
01 = 0) and the m-l PV-buses.

(2) P,- (i = 2, . - - , m) which are the model of net active power injection at
the PV-buses.

(3) P,- and Q,- (t' =m+l, m+2, , n) which are the model of net active
and reactive power injection respectively at the load buses (PO-buses). .

Notethat
Pi=PGi—PDi , i=l,2, "’ ,n; (2'5)
Pai=0,i=m+1,m+2, :--,n;

QI=QGI-QDI' 9 i=1929 ... an;
Q65=0 , i=m+l,m+2, ,n;

and that PD; and Q0,- are the load demand which may be functions of V,- . Then the
Zn -m —lunknowns(V,-.t° =m+l, ,n,and0,-.i =2, ,n)canbesolved
from 211 '- m - 1 equations (n - 1 active power equations (2-2a) and n — m reactive
power equations (2-2b)), if they are independent. After that, the voltage and angle
(with respect to swing bus) at each bus are known, and the proper reactive power gen-
eration at each PV-bus and active and reactive power generation at swing bus can be
obtained from (2-2). Note that when the reactive power generation of any PV-bus hits
its limits, it becomes a PQ-bus with fixed reactive power generation equal to the limit

and with varying voltage. Thus, the voltage and reactive power generation at this bus

21

become an additional dependent and independent variable, respectively; and the load
flow equations have an additional reactive power balance equation at this bus. Then,
the load flow is re-solved until every variable converges within its limits; otherwise the

load flow diverges.

2.2 Transformer and Phase Shifter Model

Figure 2-3 shows the physical one-line diagram of a general branch between bus
1' and bus j, where bus 1' is called tap-side bus, bus j is called impedance-side bus,
and the bus between them is a dummy bus, bus k [80,82]. Between the tap-side bus
and the dummy bus, an ideal transformer with phase shifter is represented by the com-
plex value 57,-,- = a3,- exp (i «ii ), where 0,7 is the tap ratio and (1,-1- is the phase shift.
The n-circuit between the dummy bus and the impedance-side bus represents the leak-
age irnpedance and magnetizing admittance of a transformer, or the line impedance

and line charging admittance of a transmission line, and hence

7kj=7jk=l7=Yexp(jy)=G+jB (2-6a)
for} = for = Ger ‘1'} BC] (2-6b)
I7ij = i702 = Gcz 4'} Bcz (2-60)
Note that usually

(1) 17 represents the leakage impedance of a transformer, or the line impedance

of a regular transmission line;
(2) for a transformer, 17“ = 0 and Ya represents the magnetizing admittance;
(3) for a transmission line, 17cl = in = 17C /2, where it} is the total line charg-
ing admittance.

Thus, Fig. 2-3 can represent a general branch between any two buses.

P.+'

"CI

“<l

ji

h;

22

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

YA‘Y = G+jB ‘-
. P. ' .
Pi+JQi J+JQJ
l YCl I l YC2 l
V; .
Vizei J 9]
I Fiji’Qij G Mi: 1 Yzj‘ln' 1017+}jS

- -< e t >-

Pi+jQi Pj-i-jQJ-

 

 

lYCijl = GCij+jBCij GCji+jBCji = IYCjil

— ”Y Y - 1 - ..
i} ail 1]

— Y_ Y - .. 1 _

11 01.]. aij 7 ji ji 11 2 aij

Fig. 2-3 Equivalent n-circuit of a General Branch Between TWO Buses

23

The current-voltage relationship between bus 1' and bus j can be derived by con-
siderin g the relationship between bus It and bus j first:
Ff,- - —
C1 + Y — Y [5"]:

V.-
‘ -17 1762+? t7, (2'7”

5".

J
(701 + ?)/a‘% - 7/5,. V.
= — _ - — —‘ (2-7b)
' -' Y/aij Ycz + Y Vj

From (2-7b), we can represent a general branch as an equivalent rt-circuit, Fig. 2-2,

I

H-N' 07"

 

with the following parameters:

- Y Y .
.Yij =T=ZT¢XPIJ (7+at'j)]
5}; u

= Yij CXp (I Yij) = Gij + Bij (2'83)

- 17 Y
Y.. =—=—ex . _a..
,. 5t- 00. p110! .,)l

= ’31 “P U 71.“) = Gji + 3}: (2'3b)
_ 1701 + l7

Yi=YCij+Yij= 2

0,-1-

_ 7c1+(1-b'tj)17

 

YCij = 2 (2-8C)
0,-1-
7} = _Cjt' +717 =f02+f
_ .. 1 _
YCjt' = Ycz + (1 - -_—) Y (2-8d)

a”
From the above, the following important phenomena is observed:

( l) for a phase shifter, the equivalent u-circuit is completely asymmetric

because the transfer admittances, 17,7 and Yfi, are different, and the

equivalent shunt admittances, 170-,- and 17c],- , are also different.

(2)

(3)

(4)

(5)

(6)

24

for a transformer (ay- = 0), the equivalent rt-circuit is partially asymmetric

in that the uansfer admittances, Yij and 17-

,5 . are identical. but the equivalent

shunt admittances, 170,-] and 176,-,- , are different.

in IEEE Common Format [82], the bus at which its voltage is controlled at
specific value by adjusting the tap ratio if the tap ratio is still within its lim-
its, is called controlled bus. If the controlled bus is the tap-side bus, to
raise its voltage, the tap ratio is set to be greater than 1.0; to lower its vol-
tage, the tap ratio is set to be less than 1.0. If the controlled bus is the
impedance-bus, the setting of tap ratio is reversed.

for a transformer (11,-,- = 0), the magnetizing admittance 17a is approxi-
mately zero, so (- B) > B“. Thus, from (2-8), it is observed that a
transformer has opposite effects on the equivalent shunt admittances, 170-,-
and 170,-. Regardless whether the tap-side bus and the impedance-side bus
are step-up and step-down, respectively; or reverse, the step-up side always
has capacitive equivalent shunt, and the step-down side always has induc-
tive equivalent shunt. These transformer effects are interesting and impor-
tant in the problems of voltage collapse and low frequency oscillation

[52,53,77].

if 0,-1- = l and (13,- = 0, then the dummy bus is no more needed and the ori-
ginal it-circuit (Fig. 2-3) becomes the same representation as the equivalent
1t-circuit (Fig. 2-2).

after computing all the parameters of the equivalent n-circuit of any general

branch, i.e., Y5}, 1’17, 175,-, and 170,-, we can formulate the load flow equa-

tions by substituting them into (2-1) - (2-2).

25

2.3 Load Demand Model

Since the models derived in this dissertation are to be used in the studies of vol-
tage collapse and inter-area oscillation, it can be assumed that frequency deviation is
small, and thus the effects of frequency deviation on the load model can be ignored.
Then a fairly general load demand model at each bus can be put into a voltage depen-

dent form as follows:

1D,, = PL + 1,. V + Gs V2 + kp V"' (2-9a)
QD = QL +19 V -Bs V2+ k9 v’tz (2-9b)
where
PL + j QL represents constant power load model;
(1p + j IQ) V represents constant current load model;

(Gs — j BS) V2 represents constant impedance load model

(BS < 0, if inductive; 8,; > 0, if capacitive);
kp V"' + j [:9 V"9 represents other load model different from the above.

Substituting (2-9) into (2-2), we have the following load flow model.

For each bus 1' ,

II n
PGi — (Pu +Ipi Vi +kp‘: Va”) = V52 [ 631' + 2 (GCij + Gij) ] — 2 Vi Vj Tij
i=1 i=1

n
= V} 0,,- - 21 V,- Vj Ti,- ; j at: (2-10a)
1:

Q0.- - (Qu +19.- V1 ‘1' km V”) =" Viz [3st' ‘1' i (Bct'j +3.3) ] - i Vi Vj Ut'j
i=1 i=1

5
I:

where

26

iii = Gir' +1. Bit
with

n
G“: =05;- ‘l" 2 (GCt'j +60) ; j #1.
i=1
n o o
Bii =85; + 2 (ECU +3“) ; j #1
i=1
is the so called self admittance or the diagonal element of Yb,” matrix under the
assumption that only the combination of constant-current-and-constant-impedance load
model is used in the network. Then - (Gy- + j BU) is the off-diagonal element of

Yb“, matrix. For a constant impedance load, Gs is nonnegative; but BS is negative if

it is inductive, and positive if capacitive.
Note, from (2-10), that load demand model plays an important role in solving the

load flow equations, since it is voltage dependent [46,47].

2.4 Synchronous Machine Model

The synchronous machine model adopted in this dissertation is the single-axis
model which is very common for the representation of synchronous machines
[26,83,84]. Figure 2-4(a) and Figure 2-4(b) represent its one-line diagram and phasor
diagram respectively. The notation is as follows:

17, : terminal bus voltage

1., = 1,, + j [q : current output of the machine at terminal bus
1,, : direct-axis (d-axis) current

[4 : quadrature-axis (q-axis) current

8 - 0, : angle difference between intemal bus (rotor) and terminal bus

P, + j Q, : power output from the machine at terminal bus

27

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eq’zs PDt+jQDt
M Gen P E H QE ‘ PM +1. QWT P"! ”Q”!
D 4 we}- - 4..
T, f q , 5 iin=nmt
‘10 Efd | Load Compensator j
Excitation - 1 VC *
P M System | Voltage Detector ]
Speed-
Governing- I
Turbine
System
A Power
> System
P Stabilizer
o

 

 

Fig. 2-4 Synchronous Machine Model

28

PD, + j Q0, : load demand at terminal bus

P”, + j Q”, : power flow from terminal bus to its transformer high-side bus
P; + j Q5 : power output from the machine at internal bus

R, : armature resistance

X 4 : d-axis synchronous reactance

X q : q-axis synchronous reactance

X4 : d-axis transient reactance

Eq : internal bus voltage proportional to d-axis flux linkage of field circuit
7“,; : d-axis open-circuit transient time constant (sec)

E ,4 : field voltage of the machine

PM : mechanical power input to the machine

to : angular speed of the machine (rad lsec)

H : moment of inertia constant (MW -sec IMVA)

M = 2 H loin : moment of inertia (sec/(rad/sec))

too : nominal angular speed (at too)

D : damping constant due to damper winding effects (pu l(rad/sec)).

Note that all the above parameters of each synchronous machine are in per unit (pu)

based on machine base, except those with specified units.

The dynamics of the induced stator voltage (at internal bus), 54', is characterized
by the flux decay equation through the open-circuit time constant, Td'o, and is con-
trolled by the excitation system whose output is the field voltage of the machine, E ,4 .
The angular speed of the unit, to, is governed by its speed-governor-and-turbine (or
simply governor-turbine) system whose output is the mechanical power, PM . The

detailed dynamic models of the synchronous machine and its control systems will be

discusso
nous m2
termini

parame‘

2.4.1

I
from I
2.4.7)

POWC‘.

WhCl

29

discussed later in this section to complete the whole dynamic model of the synchro-

nous machine. Before them, the power balance equations at the internal bus and at the

terminal bus are expressed in terms of bus variables (voltage and angle), and branch

parameters in the following subsections.

2.4.1 Power Equation at Internal Bus

The power balance equation at internal bus describes the electrical power output

from the stator, and its active power output will be used in the swing equation (Section

2.4.7) which expresses the rotor dynamics due to the unbalance between mechanical

power input and electrical active power output.

From the phasor diagram shown in Fig. 2-4(b), we can obtain:
13,;— V, cos(6 - 9,) = R, 1, +x,’ 1,

V, sin(5- 9,)+R, I, =x, I,

 

 

1 X, [5,; -V, cos(8—6,)]—R, V, sin(5-9,)
‘ " R,2+x,' X,

R, [12,;— V, cos(8-6,) ] +X,’ V, sin(5- 9,)
" ' R,2+X,; x,

= 3; Ga "’ Vt Tdt
where
T4, = G, cos(8 - 0,) +8; sin(8 - 9,)

0,, =6, sin(5-9,)—B, cos(5— 9,)

 

 

 

Ra I -Xd' -Xq
a = 2 I ; Bd = 2 I ; Bq = 2 I
R, +x, x, R, +x, x, R, +x, x,

(2-11a)

(2-1 1b)

(2-12a)

(2-12b)

(2-13)

30

The electrical power output at internal bus (for the swing equation (see Section 2.4.7»

is

P;- +j Q5 = '1, 72=1 5,01 +11,)‘ =5, 1, +1 E. I.

E, =E,+(X, —X,,')I,

PE = 5,1, =5, 1, + (X, -X,)I, 1,

Q5 =13, 1, =15, I, +(X, —X,; )1,22
Substituting (2-11) into (2-14), we have

P, = 5,2 GE - E,’ V, 1)., + V,2 0;,

Q, =-E,'ZBEE -E,’ V, HQ,- -V,2BE,
where

GEE =1;1 G, ; k, = 1 —(X, —X,')B, >1

Tm =1, Ta +0. or, -X.;> U,

=0”,- cos(5-6,)+Bp5 sin(6-9,)

G”,- =sz, ;k2=1-2(X, -X,;)B, >1

3,, = k1 B,’ + 6,2 (X, -X,;)

GE, = (X, —X,;) 1,, U,,

3,, =le,

UQ, =1:2 U,, = 09, sin(5- 90439, cos(6- 9,)

Ga, =k2 0, =6”;

HQ, =k23,

BE, =-(X, -X,;) U;

 

(2-14)

(2-14a)

(2-l4b)

(2-15a)

(2-15b)

31

2.4.2 Power Equation at Terminal Bus

It is observed. from Fig. 2-4(a), that the power balance equation at the terminal
bus have three components: (1) net power injection which is the negative sign of the
local load demand, PD, + j Q0,; (2) power flow from the terminal bus to its high-side
bus through a step-up transformer, P,,, + j Q,,,, whose modelhas been shown in Sec-
tion 2.2; (3) power flow from the terminal bus to the internal bus, P,, + j Q,,, which

is the negative sign of the power sent from the internal bus and received at the termi-

nal bus, - (PC, + j QG,).
Thus, from Fig. 2-4 and (2-11) - (2-13), we have
PG: +1 QC. =V, 17“
P0, = V, sin(5 — 0,) I, + V, cos(8 — 0,) 1,
P,, = - Pa, = V,2 0,, - V, E,’ 7,, (2-16a)
Q, = V, cos(8 - 0,) I, - V, sin(8 - 0,) I,
Q,, = - Q, = - V3 3,, - V, E,’ U,, (2-16b)
where
T,, = G, cos(8 — 0,) - B, sin(5 - 0,)
= G, cos(0, - 5) + B, sin(0, — 5) (2-l7a)
U,, = - G, sin(8 - 0,) - B, cos(5 - 0,)
= G, sin(0, - 6) - B, cos(0, - 8) (2—l7b)
G,, = 0,, sin(5 - 0,) + 1,, cos(5 - 0,)
= G, — (3,; — 3,) sin(0, - 5) cos(0, - 5) (2-l8a)
3,, = T, sin(5 - 9,) - U, cos(5 - 9,)

= B,’ sin2(0, - 5) + B, cos2(0, - 5) (2-139)

32

Note that since B, (Xi) and B, (X,) are different, both G,, and B,, are functions of

the angle difference between the terminal bus and the internal bus, 0, - 5.

2.4.3 Flux Decay Equation

The internal bus voltage, E, , is the induced stator voltage through the flux link-
age from the field winding with field voltage 5,, at the rotor of a synchronous
machine. For the single-axis machine model, the dynamics of E, is governed by the
following flux decay equation through the d-axis open-circuit transient time constant,
T,’,.

7,, 5,’ = 5,, — 5, = 5,, — ( 5,’ + (X, - X,') I, ] (2-l9a)

Substituting (2-11) into (2-19a), we can express the flux decay equation in terms of

bus variables (voltage and angle):
T,’, 5,’ = 5,, — 5,3 5,’ + (X, —X,’ ) U,, V, (2-l9b)
where
K53: 1 -(X, -X,)B, >1

We can also describe the flux decay dynamics as a function of reactive power genera-

tion at the internal bus, Q5, by substituting (2-l4b):

 

-5,+~15,'2+4(X, -X,’)Q,

 

 

 

 

Id: ,
2(X,-X,)

T'5'—5 1 xd-X‘; 5'

do «7" f4‘[ ‘20, -X,')] q

4
Xd-Xd' v
- . NIE'Z+4(X -xd)Qg (2496)
2(X,-X,) " "

This expression has been used in [52,53] to see the role of Q,- in the problem of

voltag

that l

2.4

USE

dtr

33

voltage collapse due to the instability of the flux decay dynamics under the assumption

that X, = X,':

Talc ér=Efd’E¢;’(Xd ‘Xb?

95

4

2.4.4 Excitation System Model

For adjusting the generator field voltage, E f, , in order to control the terminal vol-

tage properly, there are many control schemes to do so. Basically, the terminal vol-

tage and/or current feedback has been applied. A typical excitation control model is
IEEE Type DCl Excitation System Model [85], as shown in Pig. 2-5(a) that will be

used in this dissertation. Most of other types of excitation control model can be

derived from this IEEE-DCl excitation system model.

The notation is as follows:

VC:
VS:

V

ref:

output of load (or line-drop) compensator (see Section 2.4.5)
output of power system stabilizer (PSS) (see Section 2.4.6)
reference (set point) voltage

output of voltage detector

DC (Direct Current) gain of voltage detector

time constant of voltage detector

transient gain reduction

lagging time constant of TGR

leading time constant of TGR

 

Mile-t‘;

 

\-

34

382 82% 8:564”. 6a - EH 3 .5

 

ce
, a -
glam: _ > 3

#3.. _

 

 

 

 

 

 

IF
’3
a;
+
U
a:
V
+
a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

amt
.5 * be

 

 

 

_ $152.5 +5 _

 

 

 

ME

 

 

 

5+—
9%:

 

 

 

 

 

 

can

It]

 

35

VA: output of automatic voltage regulator (AVR)

K A: DC gain of AVR

TA: time constant of AVR

E“: output of exciter or excitation system (field voltage of synchronous
machine)

l/K $51 DC gain of exciter (Note: K 55 is function of the saturation effect of
exciter)

TE IKSE :time constant of exciter

K 53 washout gain of stabilizing feedback of excitation system

Tp: washout time constant of stabilizing feedback of excitation system

In order to transfer the above excitation system model into state space form, we

can express both lead—lag and wash-out transfer function blocks as a gain block paral-

lel to a lagging block, as shown in Fig. 2—5(b). Then the state space form of the

IEEE-DCI excitation system model is as follows:

 

 

 

 

 

 

TD YD -1 o o o o
r, V, o -1 o 0 -Kp/Tp
TA VA = "KA Tc’TB "KA Tc/Ta ‘1 KA 'KA (Tc”5)(KF”5)
T, V, -(l-Tc/Ta) -(1-Tc/T5) 0 -1 -(1-TC/TB)KFITF
TE Efd . 0 0 1 0 _KSE J

"V, ' .

K0 Vc

VF 0

VA 4' KA (Tc ”5) Wu] + Vs) (2-20)

VB (1" Tcn'B) (Va, 4' Vs)

0 .
.514

 

 

36

2.4.5 Load Compensator Model

This section and next one will respectively describe the models of the common
accessory functions of excitation systems, load compensator and power system stabil-
izer [85]. Load compensator usually uses the terminal voltage (and current) as feed-
back measurement of the excitation system model. The block diagram of a load com-

pensator is shown in Fig. 2.6(a). Fig. 2-6(b) shows its phasor diagram.
From the phasor diagram, we have
VG =‘7: +(Rc +17%”:- =VCd +f VCq
= V, sin(5 - 0,) +j V, cos(5 — 0,)
+(RC +j XC) (I, +j 1,)
Substituting (2-11) for I, and 1,, we obtain
VC, = V, sin(5 — 9,) +RC (- 5,’ 5, - V, U,,) -XC (5, G, — V, 7,)
= 5,, 5,’ + 5,7 V, (2-21a)
VC, = V, cos(5 - 9,) +RC (5, G, - V, 1,) +Xc (- 5,’ B, - V, U,,)
= 5,, 5,’ + 5,7 V, (2-219)
V, = W (2-21c)
where,
K52 =- (Xc Ga +Rc 3,)
5,, =Rc G, -Xc B,
5,7 = (1 +Xc 5,’ -RC 6,) sin(5 - 9,) - 5,, cos(5 - 9,)

Kq-z =<1- 5,) cos(5 - e.) - (Xc G. we Bi) sin(a - 9,)

37

<l

 

t V
_ n, _ _ C
V,+(RC+jXC)I,| ___,
i

 

 

 

 

var = Kdqu'+Kd7Vz

_ 2 2
Vc - JVCd‘“ VCq

Kdz = -(XCGa+RCBq) qu = RCGa—XCBq

Kd7 = (1+XCBd’._RCGa) sin (8-0,) + (XCGa+RCBq) cos (5—0t)

VCq = quEq +Kq7V,

K47 = (_1+XCBq-RCGa) cos (8-9!) — (XCGa+RCBd') sin (8-0,)

Fig. 2-6 Load Compensator Model

wh

2-f

bi]

na

ch

38

Note that when load compensator is not employed (RC = 0, X C = 0), then
VC = V,. In other words, the magnitude of terminal voltage is sensed and is
transferred to be a DC (direct current) quantity, then is fed into the voltage detector
whose filtering can usually be modeled as a very small time constant (TD = 0 in Fig.

2-5(a)).

On the other hand, when load compensator is desired, the appropriate values of
RC and XC are entered. Both RC and XC can be positive or negative, depending on
how the synchronous machine is connected to other units and to the power system
[85], and how the synchronous machine is set to control the voltage of a certain point
in the system. In most case, the RC component is negligible and only a value for XC
is required. For these cases, the coefficients, K,” and K ,7, are functions of the angle

difference, 8 - 0, , so is the load compensator voltage, Vc.

2.4.6 Power System Stabilizer Model

The function of power system stabilizer (PSS) is mainly to improve dynamic sta-
bility of a synchronous machine and/or power system, using other regulator input sig-
nals in addition to terminal voltage through its excitation system. These signals are
chosen to provide positive damping to oscillations. Figure 2-7(a) shows the general
model of such a power system stabilizer. Some common stabilizer input signals are:

accelerating power, speed, frequency, and terminal voltage [85].

Transferring the lead-lag blocks and washout block into first order lagging blocks,

as shown in Fig. 2-7(b), we have the state space form for PSS model.

 

 

 

 

f . . _ -
782 V32 = 1 '- Tsr’Tsz - 1 0 Vs,
LTS4 V54. (1 ‘ 7'83”“) (TM/TS?) 1 - T337134 - l V84

ex»

_ hank. + _

_ -

~ M..,...,h+ _

K

twilph {k

 

39

 

 

 

 

 

 

 

 

 

 

 

 

 

o .— > - .
_ ea: 859% as” mag.— » a ma a>+a>fi+a>mwmm+a>amww u as
\> an «a I, fix [J fix vm um va 1/ xx ,/
«MIALJV 5,. H - 7...; IALLV 3.5
as as as as as
a. a.
a .5 lb 2.. L 1 a. «a
> A JV + > c H - A i - > a
as as
gt .5» c c H .. 35¢
/ y / \ / \ / \
3V 5,. a. a a.
m h+~ > “5+—
AJ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

may + _
was + _

 

 

 

 

 

 

was + _
3Q

 

 

aux»

 

 

+ (1 - 751/737) Ks V51
(1 ' Tsan'sa) (731/132) Ks
753 751 733 731 7'53
=——K +—-—V +—V +V (2-22)
5 s s: so 32 S4
T84 Tsz Ts4 Tsz TS4

2.4.7 Swing Equation

The mechanical dynamics of a synchronous generator is at the rotor of this unit.
The motion of the r0tor is due to the acceleration torque/power, mainly which is the
difference between the mechanical power input to the rotor and the electrical power
induced in the stator. The acceleration power will be used:

(1) to change the kinetic energy, or angular speed, of the unit;

(2) to overcome the damping power that develops mainly in the damper wind-

ings and in the form of mechanical frictions.

Equation (2-23) is the so called swing equation which describes the above

 

phenomena:
4 5
+ T, = TM - TE (2-23a)
d t
2 H 2 5 d 5
— — — = P — P -
(.00 dtz +D dt M E (223b)
j—f = a) - (00 . (2-23c)
where,

T = (90 P: T represents torque, and P represents power;
(90: nominal angular speed (rad lsec);

PM: mechanical power input to the rotor (pu );

41

P 5: equivalent electrical power output from the rotor (pu ), which includes the
power loss due to armature effect and the power received at the terminal

bus (PC, + j Q6“ see Section 2.4.2);

P,: damping power (Ru);

WK = (Wm/S 3,) (99/913): total kinetic energy of the generator and turbine
(Pu -SOC);

Wm: nominal WK (at (90);

S3,: 3-phase MVA base;

H = Who/S3,: per unit inertia constant (sec);

to: angular speed (rad/sec);

8: the angular rotor position, measured in electrical radians of the rotor rela-
tive to a synchronously rotating reference (rad);

1: time (sec);

D: approximate damping coefficient (pu /(rad lsec)).

M = 2 H/coo: moment of inertia (sec/(rad/sec»

2.4.8 Speed-Governing-Turbine System Model

The mechanical power PM is controlled by the speed- governing system through
turbine system, using rotor speed as an input signal. Even though there are
mechanical-hydraulic control (MI-1C) and electro-hydraulic control (EHC) types of
speed- governing system, basically, their models are very similar. Figure 2-8(a) and
Figure 2-8(b) are the general models of speed-governing system for steam-turbine and
hydro-turbine, respectively [86].

Pay is the power at gate or valve and is the input of turbine system. The general

models of steam-turbine and hydro-turbine systems are shown in Fig. 2-8(c) and Fig.

42

f’ UP P MAX

 

 

 

KG(1+T,S)
1+T,S

 

Cal—-

 

 

 

 

 

w _|

 

 

 

 

 

 

1" DOWN Pym

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\\\\\\\\\\\\\\\\\\\\\\\\\\

 

 

 

 

 

 

 

 

 

 

 

 

1
1+ TCOS
P MAX
5,(1+ 7,5) , Pav
(1+T1S) (1+T3S)
(b) P MIN
an " (013021 ' “11023) TWS p
1+ auTWS ‘ I M
((1)

Pi g 2-8 Speed-Goveming-Turbine System Model

2-8I

gox

ICU

43

2-8(d), respectively [86].

After transferring the lead-lag blocks into first-order lagging blocks in the speed-

governing and tm'bine systems, the state space model of speed-governing-nu'bine sys-

tem can be obtained by combining their models together.

(a) Steam Governor-Turbine

 

 

 

 

 

 

ms, . up, ‘
.TIS -l 0 0 0- 0 0
3 G," -1—1 0 o o o PGV
TCHPVHR 0 1 -l 0 0 0 PVHR
TRHIPHP - 0 0 1 '1 0, 0 PHP
' 0 O O l -l 0 p
T P H’
"”2.” _o o o o 1 -1, P
LTcoPu’ . LP .

KG (1 -T2/T1) ((1)-(1)0)
‘KG(T2,T1)(m-(°O)+PO

COCO

 

 

L .1

PM =FVHP PVHP +FHR P55 “”715 PIP +FLP PLP

(b) Hydraulic Governor-Turbine

 

 

 

 

F . 1 - '1
TIPI -1 o 0 P1
T3PGV = -l -l 0 Pay

_ TW )5“, . 0 0W6 " “WW PW

KG (1 “Ta/T1)(¢°-wo)
+ "KG (T2/T1) (OJ-(Do)+Po
0

Fur-Fay PGV+PW

013 021
011

 

- 2 . .. . ..
awc; -013 021/011 . aww -1/011 . FGV -023-

(2-24a)

(2-24b)

44

2.5 Equilibrium point of Power System Dynamic Model

This section will organize the models derived in the previous sections to clearly
describe the dynamic model of a power system, and to discuss obtaining its equili-
brium point(s).

Without loss of generality, it is assumed that the connection between the terminal
bus of a synchronous machine and its high-side bus is equivalently one-to—one, as
shown in Fig. 2-1, and that the power system has n buses, including m machines with
m high-side buses, and n - m other (load) buses. From previous derivation, the
dynamics of the power system can be described by the differential equations associated
with rotor dynamics and control system dynamics, and the algebraic equations associ-
ated with active power and reactive power balance (load flow) equations, as follows.
(3) Swing Equation:

For each synchronous machine, from (2-23),

3 = a, _ mo (2-25a)

M (i)=-D (to-(oo)-PE +PM (2-25b)

where PM is the mechanical power output from the speed-governing-turbine system of
the unit (see (2-24)).
(h) Flux Decay Equation:

For each synchronous machine, from (2-19),
7,, 5,’ = 5,, - 5,3 5,’ + (X, —X,' ) U,, V, - (2-26)
5,,=1-(X, -X, )3, >1
U,, = G, sin(8 - 0,) - B, cos(5 — 0,)

where, 12,, is the output of the excitation system of the unit (see (2-20) - (2-22)).

(c) Algebraic Load Flow Equation:

45

[1] At internal bus, for each synchronous machine, from (2-15),
3, = 5,2 G,, — 5,’ V, 1,, + V,2 G,, (2-27a)
Q, = - 5,2 3,, - 5,’ V, U,, - V,2 3,, (2-279)
The expressions of parameters are listed in (2-15).
[2]Atterrninalbus,t =n +1, ,n +m;h = 1, ,m,from(2-16),

= V,2 G,, - V, 5,’ 1,, + V,2 (0,, + 0,) — V, V, T,,, (2-28a)
- Q»: = Q, + QM
= - V,2 3,, — V, 5,’ U,, - V3 (3,, + 3,) - V, V,. U,,, (2—28b)

The'expressions of T,,, U,,, G

,,, and 3,, can be found in (2-17) and (2-18). The

expressions of T,,, and U”, are similar to those of T,, and U,,, as follows:

Ta, = G,,, cos(0, - 0,) + B”, sin(0, - 0,) (2-29a)
Uth = G»: sin(9: ' 0,) - But cos(et - 9h) (2‘29b)
[3] At high-side bus, h = l, . - - , m; t = n + h, from (2-1) and (2-16),

" PD}: = V1.2 (Gem 4’ Gm) " V1: V1 Th:
2 " " .
+ V,. 2 (Gen. 4' Ghj)- 2 VII Vj Thj , 1 ¢ ’1 (2-308)
jsl i=1
- QDh = - V1.2 (3cm +3») - V). V: Um

u a
- V1.2 2 (Ben; + 311;) - 2 VI: Vj Uhj . J' 9* ’1 (2-30b)
i=1 i=1

[4]Atloadbus,i =m +1, ,n,from(2-1),

'8
"P01 = V12 2 (GCij 4' 0110' 2 V.- Vj 7}, . j 6” (2-3la)
i=1 '

J=1

dy.
igr

wl'.

F0

45

n u
- Q01 =- V12 21 (Boy +3.7) - 21 Vi Vj Uij . j 5“ (2-31b)
l= 1=

Note that in the load flow equations, the load demand at each bus PD + j QD

may be function of its bus voltage, see (2-9), and that we have used the following gen-
eral formulas.

Tij = GU COS(9,° - 6") +80 sin(0,- - 61') , j #1. (2'323)
Uij = 63'} sin(0,- - 9") -3” COS(9,° - 6") , j #1. (2'32b)

The above power system dynamic model is nonlinear, except the control system
dynamics of govemor-turbine systems and excitation systems, if their limits are
ignored. The equilibrium point(s) can be solved by the set of algebraic equations
which include the load flow algebraic equations and those obtained by setting all the
time derivative terms in dynamic equations to be zero. This is equivalent to solve for
the steady state operating condition of the power system dynamic model. At steady
state, each synchronous machine can be equivalently represented by single-axis model,
and its control systems are represented by their D.C. gains. Note that the reactive
power output equation at internal bus is only a "side-output" output the synchronous
machine, and has to be replaced by the algebraic equation governored by the DC. gain
of excitation control K, IKSE and voltage set point V,,, . Since at steady state a) = (90,
the electrical active power output P, at the internal bus is the same as the mechanical
output PM or the set point P0 of the speed-governing—turbine system which does not
have any effects on the equilibrium point, neither does the power system stabilizer due
to its washout characteristics. Thus, the set of algebraic equation for the equilibrium
point(s) can be described as follows.

For m machines:
a) = (00 (2-33a)

P0=PM =P5

47

= 5,2 G, — 5,’ V, T, + V,2 G,

5 —K‘ (V 5 V V
= — +
= 5,3 5,’ - (X, -X,) 0,, V,
Vs = O

 

V, = «5,, 5,’ + 5,, V,)2 + (5,, 5,’ + 5,, V,)2
- PD: = V12 0,, - Vt E, T,, + V12 (GCrh + Grit) '" Vt VI: Tut

— Q». = - v3 3,, — v, 5,; U, — v,2 (30:11 + 3111) - V. v). 3,,

For n high-side and load buses: i = l, - - - , rt,
2 Ill-HI Ill-HI . .
—PDi = V,‘ 2 (CCU '1' GU)- Vi Vj Tij , j #1
i=1 j=l
2 M44! DIM . .
’Qot ="V1 21(3074’311)" 21 V1 Vj Us} . J *1
j: J:

For n+m loads:
3, =3, +1, V +Gs V2+kp v"'

Q, =Q,+1, V-3, V2”, W

(2-33b)

(2-34a)
(2-34b)
(2-340)
(2-35a)

(2-35b)

(2-36a)

(2-36b)

(2-37a)

(2-37b)

Let the swing bus be the terminal bus at machine 1, i.e., bus 11 + l, with specified ter-

minal voltage V“, (0,”, = 0). Then there are 4 (m - l) + 2 1: equations to be solved

for §,=E,exp(ifi) and V,=V,exp(j0,) atbusn-t-Z, ,n+m, and for

l7,=V,exp(j0,-),i=l, ,n,inwhichr'=l, .111 male»: high-side

buses associated with m terminal buses under the assumption that the connection

between each terminal bus and its high-side bus is equivalently one-to-one.

However, the above algebraic equations are also nonlinear. The Newton method

is a general approach by applying the Jacobian matrix of the system, generated by

C0

C11

48

taking partial derivative for each equation with respect to each variable. Doing so, the
linearized model of the system is obtained, and the linear system theories can be
applied to analyze the characteristics of the system in a neighborhood of the isolated
equilibrium point of interest. In next section, we will develop the linearized dynamic
model of a power system based on the apparatus dynamic models derived before.
Then, the following studies on the static and dynamic bifurcation leading to voltage
collapse and oscillation, especially associated with fiux decay dynamics and mechani-

cal dynamics, will be investigated on the base of the linearized dynamic model.

2.6 Linearized Power System Model

In analyzing equivalent solutions of a nonlinear system, we generally start by
linearizing the system in a neighborhood of the isolated equilibrium point of interest.
Of course, every function describing the above power system dynamic model has a
Taylor series expansion of the first degree plus higher order rest terms in the neighbor-
hood of of the equilibrium point; linearizing means that we leave out the higher order

terms.

Due to the linear representation of the control apparatus of every synchronous
machine, linearization work for speed-governing-turbine system model, power system
stabilizer model, and excitation system model is not needed, but it is needed for load
compensator model. The linearization work will be done in the following sequence:
(a) load demand model; (b) power balance equation; (c) swing equation; (d) flux decay
equation; (e) load compensator. Note that to simplify the notation, a variable with A
l in front means the deviation of this variable from the equilibrium point of interest, and
its coefficient (in front of A) consists of variables that are evaluated at their equili-

brium values.

(a) Linearized Load Demand Model:

49

For each load, from (2-9),
APD = AP, + FPV AV (2-38a)
AQD = AQC + FQV AV (2-38b)
where
va =1, +20, V +11, 5,. V""’1
- : coefficient of active power load functional of bus voltage;
5,, =1, —23, V+nQ 1:, WV1
: coefficient of reactive power load functional of bus voltage;
AP, =AP, +V A1,. +V2AGS +V"' Ak, +5, v" (1nV)An,
: coefficient of active power load not functional of bus voltage;
AQ, =AQ, +V A1, -V2ABS 4w"!2 A5, +5, V” (an)AnQ
: coefficient of reactive power load not functional of bus voltage.
(b) Linearized Power Balance Equation:
[1] At internal bus, for each synchronous machine, substituting the following

property

ij - ij - U . 0*. 239
_30,---39,- 11.11 (-8)
0H,. _ av, _T . . . 2399
99,-"59."i"¢’ (‘)

1

into (2-1 1) and (2—15), we can derive

A1, = - 3, A5, - U,, AV, — T, V, (A5 - A9,) (2-40a)
AI, = G, A5,’ - T, AV, + U, V, (A5 - A9,) (2409)
T,, = G, cos(5 -— 0,) + B, sin(5 - 0,) (2-40c)

U, = G, sin(5 - 9,) - 3,; cos(5 - 9,) (2-40d)

50

AP, = Kfl'l (A5 - A9,) + K35 AE,’ + K}?7 AV, (2-41a)
AQ,E = K3, (A5 — A9,) + Kg, AE,’ + K3, AV, (2.415)
where,
[(33 = E,’ V, UP, + V,2 (x, -x,;) (T, T, - U, U,,)
K33 = 2 E,’ 0,, - V, T”,
Kfi', = 2 V, GE, -1§,,’T,.,E
UPE = 1‘1 U4: "' Ga (X4 “'Xd') th
T”, = k, T, + G, (x, -x,') U,
Kg, = - E,’ V, TQ, + 2 V,2 (x, -x,') U, T,
Kg, =-2E,’B,, -V, 0,,-
K37 =—2V, B, —E,' U,,-
T9, = k2 T,
UQE = k2 th
[2] At terminal bus, for each synchronous machine, from (2-28) and (2-29),
AP, + APD, + AP, = 0
AQ,, + AQD, + AQ,,, = 0
— AFC, = AP, = K91, (A9, — A5) + K55 AE,’ + K1347 AV, (2-42a)
- AQG, = AQ,, = Kg, (A9, - A5) + Kg, AE,’ + Kg, AV, (2425)
- AFC, = -K,‘¥1 A5 +sz AE,’ + (K9!l +A,,,) A9, +31, A9,,
+ (Kifi, + C1,) AV, +Dl,,, AV, (2-43a)

—AQC, =-K5l A8+K§2 AE,’ + (K5?l +A3,,) A9, +33, A9,,

51

+ (K37 + C3,) AV, + D 3,, AV, (2435)

where, the same properties of (2-39) and (2-40) are used such that

K3, = V, 3,’ U,, - V,2 (3,; - 3,) cos[2(9, - 5)]

Kg, = — V, T,,

Kill, =2V, 0,, -3,' T,,
Ala = V: VI. U».
31:}. =" V: V» Um ="Am
Cm = FPV, + 2 V: (Gem + Ga.) - V); T»:
D 1m = ' V: T».
Kg, = — V, 3,’ T,, - V,2 (3,; - 3,) sin[2(9, — 5)]
K32 = - V, U,,

I

U

Kg, =-2V, 3,, —3 ,,

q
43:: =" V: VI. Ta.
33:). '-" V: V]: T». =‘A3a

C31: = FQV, ' 2 V: (3cm +33.) - VI. Um
D3,,. =- V, U",

[3] At high-side bus,h =1, -~ ,m,j ¢h,from (2-30),

3
-APCII =A2~ A0, +821”, A9,. + 2 32’”- A0]-
i=1

n
+ C2” AV: +02”. AV'. 4' E Dz,”- AVJ' (2443)
”=1

l
-AQCII =Am A9, +84”, A6,. + 2 8410' A9}
i=1

52
n
+ C4,, AV, +04”, AV,l + 2 D4,,- AV,-
i=1
where, we use the same property of (2-39) and have

A2» ='Vh V: Um

n
Bm=vh V, UM+2VII Vi Ullj
i=1

82h] =- VII Vj Uh}

C2” =—Vh Th!

fl
DZhh=Fth+2VII [GCM+GM+2(GChj+Ghj)]-VI Th:—

i=1

D7},-

1 ='Vh Th1

A4» =Vh V: Th:

n
34». ="Vh V: Tin-Z VI. Vj Thj
-1

Bug = VII Vj Thj

C4» =-Vh Uh:

a
D4». =FQV, "ZVI. [BCM +33: + 2 (BChj +th)]-Vx Uh: "

i=1

DMj=-Vh Uhj
[4]Atloadbus,i=m+1, ,n,j¢i,from(2—31),

I! II
- Ape; =82fi A6,- + 2 82,,- A9,- +0217 AV; 4' 2 D2,? AV,-
j=l i=1

n n
”AQc: =3“: A9: + 2 34:} A0j +044? AVI + 2 D40 AV}
i=1 i=1

where, we use the same property of (2-39) and have

(2-44b)

n
2 V; Thj
i=1

n
Z V} Uhj
i=1

(2-45a)

(2—45b)

53

32,-- =- V, V,- U,-

n n
029' =FPV, + 2 Vi 2 (GCij + Gij) " Z Vj Tij
i=1

i=1
Dzij =‘ V: Tij
n
34.; =" Z V: V} Tij
i=1

’3 '3
D4,,- =FQVa' — 2 V,- 2 (ECU +3”) - 2 Vi Uij
i=1 j=1

D4,,-

= - V,- U,-
(c) Linearized Swing Equation:
For each synchronous machine, from (2-25),
A5 = A99 (246a)
MA(b=—DA(o-APE+APM (246b)

where, APM can be obtained from (2-24); APE can be substituted by (2—41a).
' (d) Linearized Flux Decay Equation:

For each synchronous machine, from (2- 19) and the properties of (2-39) and (2-
40).

Td’o AB; = AEfd - K53 AEq' -' K54 (A8 — A6,) + K57 AV, (247)
where,
K,3 =1- (x, -x,;) 3,

K,4 = - (X, -x,;) T, V,

K37 = (X4 " Xi) U4:

Note that each K 33 is a constant coefficient only functional of the machine parameters.
AB], can be obtained from (2-20) - (2-22), and from the linearized load compensator
model (see (e)).

(e) Linearized Load Compensator Model:

For each excitation system, fiom (2-21),
2 Vc AVC = 2 VC, AVC, + 2 Vc, AVC,
AVC = K,1 (A5 — A9,) + KC2 A3, + K,7 AV, (2-48)
where,

KC1=(Kdl Vca + K41 ch)/Vc

Kcz = (K42 Vca + K42 VCq)/Vc

KCT = (K47 VCd + K47 ch)/Vc

K,l = [ (1 + x, 3,; —Rc 0,) cos(5 - 9,) + K,2 sin(5 - 9,) 1 V,

K,1 =- [ (1 -K,,) sin(5- 9,,+ (X, G, +3, 3,) cos(5— 9,)] V,

Having the complete linearized power system dynamic model, the following
chapters are going to study the characteristics of the power system based on three
different approaches:

(1) voltage-angle bifurcation in the load flow model,
(2) static/algebraic bifurcation in the dynamic model, and
(3) static/dynamic bifurcation in the dynamic model.
Matrix forms of linearized load flow model and linearized power system dynamic

model will also be formulated Different kinds of test matrix for the above bifurcation
tests will then be established.

Chapter 3
Bifurcation in a Load Flow Model

The bifurcation in a load flow model (simply called as Load Flow Bifurcation)
will occur when the Jacobian matrix of the load flow model is singular. The load flow
Jacobian matrix describes the linearized load flow model, and is a square matrix,
which will be structurally discussed in the next section. In this chapter, a necessary
and sufficient condition test matrix for the load flow bifmcation is derived, that
includes effects of PV-bus to PQ-bus changes. It will be shown that the load flow
bifurcation can not occur due to the row dependence of the load flow Jacobian matrix
associated with the rows of the active and reactive power balance equations at a single
bus or at a subset of buses; solely associated with all the rows of active power balance
equations; or solely associated with all the rows of reactive power balance equations.
The result that load flow bifurcation of a large power system model can not occur at
single buses shows that the two-bus analogies used to understand large power system
voltage stability problems have very severe limitations. Thus, the load flow J acobian
matrix is the generic test matrix for the load flow bifurcation. The generic load flow
bifurcation is also the so called voltage/angle bifurcation, since it occurs only when the
load flow Jacobian matrix is singular due to the row dependence associated with active
and reactive power balance equations at all the buses. Special cases of the load flow
bifurcation, voltage bifurcation and angle bifurcation, will also be discussed to provide
better understanding of the causes of the load flow bifurcation in term of active and
reactive power transfer and limitations. Finally, the simulation results of a 9-bus 3-
machine power system demonstrate the voltage collapse due to the load flow bifurca-

tion, and show the important role of the reactive power generation reserves.

55

56

3.1 Linearized Load Flow Model

The load flow model is a simplified model of power system dynamic model under

the following assumptions: (1) the coupling between internal bus and terminal bus (or

between rotor and stator) of each generator is neglected, (2) the excitation gain of each

generator is infinite, and (3) all time constants in the differential equations are zero.

Hence, for a generator, its terminal bus is the only bus required to represent it.

All the buses in the load flow model can be classified as the following types of

bus.

(0) swing bus:

(1) PV-bus:

(2) PQ-bus:

It is one of the generator terminal bus with infinite active and reactive
power capacity (reserve), so that its voltage can be specified and its
angle is always set to be zero as the angle reference of the system.
Its active and reactive power generations are dependent variables, that
match the power mismatch in the rest of the system.

This type of bus has its active power generation or injecfion (genera-
tion - load demand) and voltage specified as independent variables,
and its reactive power generation and angle to be dependent. This
bus type is specified when the bus has adjustable reactive power gen-
eration reserves, which implies that its reactive power generation lim-
its have not yet been reached. Generator buses and buses with
switchable shunt capacitors are PV-buses.

This type of bus has its active and reactive power injection specified
as independent variables, and its voltage and angle are dependent
variables. High-side buses, load buses and PV-buses at their reactive
power generation limits are all PQ-buses. Note that the active and

reactive power generations at high-side and load buses are zero.

57

Thus, the load flow model consists of active power balance equations at both PV-buses
and PQ-buses, and reactive power balance equation at PQ-buses. The reactive power
generation at each PV-bus, and the active and reactive power generations at swing bus
are the output equations of the load flow model. The load flow model can be

described as follows.

0=81()’1;)‘2,)’3’Hr) (3-18)
QPV = 82(Y1;)’2 , Y3 . N2) —’ PV-Bus 0WD!“ Equation (3-1b)
P5 = g3, (yl , y3 ) -> Swing Bus Output Equation (3-lc)
Q3 =g3Q (y1 ,y3) —9 Swing Bus Output Equation (3-ld)

where,

g 1: active power balance equations at PV-buses and PQ-buses, and reactive

power balance equations at PQ-buses;
g2: reactive power balance equations at PV-buses;
g3p: active power balance equation at the swing bus;
g 39: reactive power balance equation at the swing bus;
y 1: angles at PV-buses and PQ-buses, and voltages at PQ-buses;
y2: specified voltage at PV-buses;
y3: voltage at the swing bus;

u]: active power demand at PV-buses and PQ-buses, and reactive power

demand at PQ-buses;
“23 reactive power load demands at PV-buses;
va: reactive power generations at PV-buses;

PS: active power injection at swing bus;

58

Q5: reactive power injection at swing bus.

Note that since the swing bus is assumed to have infinite active and reactive power
generation capacity, the characteristics of the swing bus will not be further discussed
in the load flow model, and the voltage at swing bus is assumed to be always constant,

i.e., y 3 is a specified constant.

Consider an n bus power system with one of the generator terminal buses
selected as the swing bus (bus# 1), m - 1 generator terminal PV-buses and n - m
PQ-buses. It is assumed that no generator PV-bus has become a PQ-bus, referring to
the power balance equations derived in Chapter 2, the linearized load flow model and
the load flow Jacobian matrix I“: can be structurally represented as follows:

 

 

 

 

 

 

 

 

381 381
- =—A +——A (3-2a
Alli ah Y1 ah Y2 )
382 882 .
AQpV -Au2 = — Ay1+ —— Ayz -9 Output Equation (3-2b)
3Y1 3Y2
. ' "A 3 o D o ' t - - -
APT 1 I” 1” A91 C1
-1513,” A23 32"” 32m. 02m! 02111. A6,, c2” ‘
-APCL =' 0 8w! Bu 02L” Du AOL + O AVT (3'28)
_ C
_:gc” A411 B433 B431. D411” D431. 2‘6” 6"
. CL. _0 3m, 34,, D4“, 0a.. L. ‘
A9,
A9,,
AQT =[A3 B3” 0 03H 0] AOL +C3 AVT —) Output Equation
AV”
AVLJ

 

 

APQ =‘AF1=[APT' 'N’CH' ‘APCL' "AQCH' 'AQCL' 1‘
Ay1=[AOT‘ A9,,' A9,,‘ AVH‘ AV,‘ 1‘ ; Ay2=AVT

AQPV=AQGT ;-Ach="Al>12

where,

59

APT = APO? - AFC! : active power injections at terminal buses;

AQT = AQGI- - AQCT : reactive power injections at terminal buses;

APQ: P and Q injection vector of the left hand side of (3-2a).

subscript T: represents the generator terminal PV-buses (including switchable

subscript H:

subscript L:

subscript G:

subscript C:

shunt capacitor buses), i.e., P0,, PC, , 0,, Q6“ QC“ V,,
t = 2, ..., m .

represents the generator-transformer high-side PQ-buses. If the con-
nection between each generator and its transformer high-side bus is
equivalently one-to-one, then PC," 0,, QC," V,,, h = m + 1, ...,
m + m .

represents the rest of the PQ-buses. If the above one-to-one con-

nection is ”SUM then PCI’ 61, QCI’ V1, I = 2»: +1, ..., n.
active or reactive power generations;

not-voltage-dependent (or constant) active or reactive power load

demand or injection;

The above matrices can be partitioned and defined as follows:

[

 

 

 

. C,-
J J 3g, [139 JPV] J 3g, C8"
“P "-ayl- Joe 19v ’ ”-3y2_ c,”
0 .
382 882
J =—-=[A B OD 0];J =—=C
21 an 3 an 311 22 3,2 3
, . .
A, B, = A211 32m: 32m. 3 13v: D, - 02311 Dan.
. 0 3,”, 3m, 3,“, 3,1,.

 

 

 

 

JQ0=[A4 B4]=[

60

A4” Bum 34m.

 

Dam D4113]

];JQV=D4= D411! D411.

9 341,34”,

where, J”; is referred as to the load flow Jacobian matrix. The output equation then

becomes

AQT = SQV AVT + SQP Ma (32!»

SQV =122-12113't1 112 3 Sm =JZIJl-Fl

where, SQV is the equivalent Q-V sensitivity matrix of PV-buses when the load flow

Jacobian matrix J”: of (3-2) is nonsingular.

Based on the assumed one-to-one connection between each generator bus and its

transformer high-side bus (simply called the high-side bus), the following properties

are observed.

(1)
(2)

(3)

(4)

A1, A3, C1, and C3 are all (m - l) x (m - l) diagonal matrices;

A2", A4”, C2”, and C4” are all m x (m - 1) matrices. Each has the first
row being zero and the rest being an diagonal (m - l) x (m — 1) subma-
trix.

B1”, B3”, DI”, and D3” are all (m — 1):: m matrices. Each has the
first column being zero and the rest being an diagonal (m - 1) x (m - l)

submatrix.

Since the power balance equation of each bus is function of the angle
differences between this bus and other buses that connect to it, each diago-
nal element of the power-angle Jacobian matrices (Jpo and 199) of the
above load flow Jacobian matrix 1,}, shown in the matrices of A and B , is
the negative sum of its corresponding off-diagonal elements; except the first
diagonal elements of the matrices B 21!!! and B um , which are associated

with the swing bus. Thus, if the angle of the swing bus is not set to be

(5)

(6) '

(7)

61

zero as the reference of the system, the load flow Jacobian matrix J u will
be singular [45].

With the angle reference of the swing bus, the diagonal matrix A, is equal
to the diagonal submatrix of - B m; and the diagonal matrix A3 is equal to
the diagonal submatrix of -- B 3,, , where the submatrix means the matrix

without the row and/or the column corresponding to the swing bus.

APC and AQC are respectively the coefficients of active and reactive power
load demand, PD and Q0 , which are not function of bus voltage. Hence,
the effect of the voltage dependence of load demand model, mentioned in
Chapter 2, is included in each diagonal element of the power-voltage Jaco—
bian matrices (pr and JQV) of the load flow Jacobian matrix Jur, shown

in the matrices ofC andD.

Recalling (2-1) and (2-2)

Pij = Viz Gij — Vi V] [GU cos(G, - 6") +3” sin(0,- - 0,)]

= V52 01'} "' Vi Vi Ti} (3'33)

Qij = " Viz Bij " Vi Vj [ Gij Sin(9t - 9;) "' Bij 008(95 " 9;) ]

= — Viz Bij - V,- Vj Ugj (3'3b)

we have the following properties shown in (2-48):

fl—-fl-_U” - igej (3-4a)
30, 39,- " ’

3U”- 3U-- '
__=_.__‘L-_—, .. - ° ’

39‘. 80,- T,, , rat 1 (34b)
31’” ‘1

3'0..— —- - 39': — Vi V] ”U (3'53)
3 .. ..

_Q_"J = - .321. = ._ v, v, T, (3-59)

62

93,. 93..

__ =2 V- 0.. _. v. 1.. ; ._‘L = — V- T.. (3-5c)

av,- ' '1 I *1 av, ' '

3‘7?— =- 2 Vi Bi}. - Vj Uij ; a—V‘j = - Vi Uij (3'5d)
t 1

Note that in general, T,- ¢ :1: Tii and U,,- at i U 1,. However, if the network
does not have phase shifter branch so that 6,,- = G,- and B,-,- = ,,, and if
the series resistance of each network branch is small enough and is

neglected (Gij = 0), then Tij = - Ti; and Uij = Uji' Thus,

93,-- 93.-

379;— = 30;; = - Vi Vi "ii = ’ Vi Vi Uji (3'63)
3Q.“ 3Q'i

33’— = — Bi- = V,- V,- T, = - V,- V,- T, (3-6b)

so that the Jacobian matrix In is symmetric, B4 is screw-symmetric, and
A4,, = - B 3,, where superscript t means matrix transpose. This presents
the sparse property of the load flow Jacobian matrix.

(8) If the voltage variable is chosen as per unit deviation, i.e., AV/V , then,

without the assumption 0,,- = O, we have

apt] aQij
-—a j W - Vi Vj Uij (3'73)
aQij apij _

so that A2” = C4", B1” =03”, and both 82 and D, have the same off-
diagonal elements, and that A4” =- C2”, B3,, =—Dw, and each off-
diagonal element of B4 is the negative sign of that of D 2. This further
shows the another sparse property of the load flow Jacobian matrix.

63

When any generator PV-bus becomes a PQ-bus due to exhausting its reactive
power reserve or hitting its reactive power generation high or low limit, the defining
equations for the load flow model change by adding a reactive power balance equation
at the generator terminal bus, because the voltage magnitude at the generator terminal
bus is no longer specified. At this generator PQ-bus, its reactive power generation is
set to be fixed at the limit. Hence, in the case that any generator PV-bus becomes a
PQ—bus, using the following matrix partitions associated with terminal buses:

APP A9,”
APT = APFQ ; A61 = AGTPQ

AQP’ AVP’
Af" Am 0 3 3f};
Al-A‘fQ- OAffla’m'Bffi
€va 0 0P1};

=WPQ= - =
C1[C1 C1] O (3ng :01}! 0‘1"?

Azu=tA§uV A581 ; cw=tc§¥ 6581

A Alf" Ag, 9 3 35};
3" A59 " o 3599 ’ 3”' 358
05% 0 or;

= W m = ‘ =
C3 [C3 C3 1 O nga .0311 pm

A,” =[Az‘,’ Am] ; c4, =[c£,," cm]
then the linearized load flow model and its corresponding Jacobian matrix Ju- would

become as follows.

64

 

 

 

 

 

 

 

 

 

 

_ A, 3,, 0 cf9 D,” 0 _ q . .
AP A9 PV
APT A23 32311 32111. C53 0211” 02m AOT C},
_ C” o 3 3 o D D ” C2”
AAQQTEQ A59 Egg 0 C399 D33 0 (2‘39 C2,,
- 4H
—AQ::I A43 B4HH 34111. C53 D4311 D4111. AV]: 0
" 0 841.” Ba 0 D411; D4LL ‘ " b .
"A9,?
A9,,
A9
AQfV=IA§V 8‘3”]? 0 0 05X 0] ,,,; wilt/AV?”
AVH
Av,“
= SQV AVP’ + SQ, APQ 4 Output Equation
139 JPV
11.17: 199 JQV (3'8)
P I .- P -
A, 31 A1 311! 0 CIQ D,” 0
139: A2 32 = A2” 3233 3231. ; JPV= C23 D2113 02m.
0 3,1,, 32,, _ 0 32,, 32,,“

 

 

 

 

Age 353 0 C599 04.58 0

JQG .. A43 B4HH 34m. 3 JQV= C53 D4HH D4HL

 

 

 

 

019me _OD,,,D4,,‘

3.2 Load Flow Bifurcation

Load flow bifurcation will occur when the load flow Jacobian matrix I”: is
singular. This section will comprehensively discuss the subclasses of load flow bifur-
cation based on the the structure and the properties of the load flow Jacobian matrix.
These subclasses are due to row dependence associated with different types of buses

leading to the singularity of load flow Jacobian matrix, when some of generator PV-

65

buses become PQ-buses (see (3-8)). which is more generic than when all generators
have reactive power reserves and are PV-buses (see (3-2)).
Based on the matrix partitions associated with terminal buses shown in (3-8), the

load flow Jacobian matrix Ju: (3-8) can be rewritten as the following form:

 

 

 

 

.an A1 3111 0 Gig 013 0
1552 A59 353 o c§3Q 353 o
J ’33 A211 32m: 3231. C53 DZHH 02m. (3 9)
LP- ’01! — A43 Bum 34111. C59 Dam Dam.
1n. o 32,, 32,, o 32,, on,
I}? Af" 3”,}; o o 3”,}; 0
’PT' 132 ‘ Afoafgocfp, pmo
ormoreprecisely:

J5¥=[Af}'v 9 3f}; 9 o of}; 9]

133=[o 11% 3f3 0 C53, D',’3 or

155?:[0 3% 353 o (:53, 353 9]

JP]! =lA§Hv A53 321m 32m. 0’2? Dana 0231.]

JQH =[Afnv Am Bum 34m. €53 D4311 0431.]

JPL=[0 0 32111 3212 0 021.3 0211.]

JQ,=[9 9 34m 34,, 03,, Du]

Note that this arrangement of J”.- according to the type of bus shows that 1;?

and 15?, JP” and IQ”, and JP, and la, respectively have the same structure. Hence,

basically from the structure and the properties of the load flow J acobian matrix Ju-
(3-9), there are three possible single-bus subclasses of load flow bifurcation leading to

66

the singularity of load flow Jacobian matrix. These single-bus bifurcations, if they
exist, would establish the correspondence between the load flow bifurcation that occurs
due to the singularity of J” and the bifurcation that is evident in a transfer versus vol-
tage nose curve for a two bus model. These three subclasses are due to row depen-
dence associated with the power balance equations at PQ-buses: terminal (T), high-
side (H), and load (L).
LF -T: This type of bifurcation is solely due to row dependence associated with
the active and reactive power balance equations at terminal PQ-buses
(J5? and 153).
LF —H: This type of bifurcation is solely due to row dependence associated with
the active and reactive power balance equations at high-side buses (J p”
and JO}! ).
LF -L: This type of bifurcation is solely due to row dependence associated with
the active and reactive power balance equations at load buses (J FL and
JQL ).
Note that since the diagonal properties of A and C matrices shown in Section 3.1
(property (1) and property (2)), other types of bifurcation due to the combination from
any two of above three are not possible at all. However, the following subsections
will prove that none of the above subclasses of load flow bifurcation can happen, and
will derive a reduced form of J” based on the generic form of 1“.» with the structure

of (3-8) or (3—9) [87].

3.2.1 LF—T Bifurcation ?

Since under the assumption that the connection between each generator terminal

bus and its transformer high-side bus is one to one, all the submatrices of If? and

153, i.e., M3,, 313, 013,, 3(3, 14%, 353, C53,, and 353, are diagonal matrices.

67

The only condition for row dependence leading to the LF -T type of bifurcation is that
the ratios of each corresponding pair of elements of these diagonal matrices must be

thesame:
kT=—=-—=——=——— (3.10)

where, each small character represents the element corresponding to the matrix with
the same subscript, and the superscript PQ is omitted. Using the following facts and
definitions from Chapter 2 and Section 3.1:

Tij = GU COS(9,' "' 01') +8” sin(0,- - 01-)
= Yij 005(6i " 6" -75).) ; i #1. (3°lla)

Uij = Gij sin(0,- "' 6") -35] 008(9i -9,-)

= Yij Sin(9i - 61' " yij) ; i #1. (3'llb)
-1 Bij . .
Y,,- = 516,324-802 ; 7;,- = tan —G— ; t at} (3-11c)
u

3T,- 87},-

— = - —- = — U... ; . . -11
39,- 39} ,1 r at] (3 d)
Uij anj . .

36,- -- 80,- -T,-,- . rat} (3-lle)

we can obtain each element as follows:
a, = V, V, Y,,, sin(O, - 0,, - 7”,) (3-12a)
b,” = - V, V,. Y», sin(e, - 6,, - 7”,) (3-12b)
c, = 2 V, (G, + Ge”, + G,,,) — V, Y, cos(0, - 0,, - 7,) + 1p, (3-12c)
d1” = ' Vt Ytlr 005w: - 91. " 7th) (342(1)

03 =- Vt VI. Yul 008(9t " 91. '73.) (3-13a)

68

by, = V‘ V,. Y“ 008(9‘ - 0,, - Ya.) (3'13b)
C3 = "' 2 V, (35‘ +ch '1' Ba.) " V,. Y”. sin(e, " 6h " Y!!!) + [Q (3'13C)
d3H = " Vt Yr): 5mm: " 0,. "’ Ya.) (343(1)

Note that the load demand model adopted here is the combination of constant power,

constant impedance, and constant current models. It is clear that

k”: fl=£yi_= Y». Sin(9x ‘9). '7m) (3-14)
T 03 b3" "' Ya. cos(O, " 9h ’71:.)

 

d Y oosG -0 -
kj4= II! = a. . (r h 711.) (3-15)
‘1315! Y». 8111(9: - 9;. " Ya.)

 

~ which makes the satisfaction of (3-10) impossible unless Y”, = 0. Computing the ratio
of the element in C1 to that in C 3 is not necessary. Thus, it is concluded that the
LF —T type of load flow bifurcation due to the row dependence associated with the
active and reactive power balance equations at a single generator terminal PQ-bus can

never happen. This proof holds for every generator terminal bus.

3.2.2 LF—H Bifurcation ?

Under the same one-to-one connection assumption, among the J acobian subma-
trices opr,, and IQ”, both A2,, = [A53 A28] and A4,, = [14% A28 ] are diago-
nal matrices, and both CW and C 23 have the same structure with diagonal subma-
trices. Hence, to determine whether the U” -H bifurcation can occur or not, one can
first check if the corresponding elements of the above four matrices have the same
ratio or not. Similarly, letting the notation of small character with the same subscript

represent the element of that matrix, and using the facts stated in (3-11), we have
02H = — VI: V: Yb: Sin(oh “ 9: ’7») (3-163)

Cm = — V), Y” 008(6), " 6, " 7“) (3461))

69

 

 

04H = Vlr V: Y): 008(9). ‘ 9: ‘7») (3-178)
c4” = - V,. Y,“ sin(e, - e, --y,,,) (3-l7b)
kfi=-::”;-='f::::§2::3:::::
k}; = Ci = Y» 008(9). - 0. '7») (349)

C41! Yin 8M9}. ' 9: '7»)

Since kfi at 1:}; , the LF -I-l type of load flow bifurcation due to the row dependence
associated with the active and reactive power balance equations at a single high-side
bus can never occur either. This proof holds for every generator-transformer hi gh-side

bus.

3.2.3 LF—L Bifurcation ?
To check whether the U“ —L bifurcation is possible to occur or not, one can first
compare the corresponding elements of off-diagonal Jacobian submatrices of J FL and

JQL, i.e., 82”,, Du”, B4”, and Du”. Similarly, using the facts of (3-11), we have

 

 

bzm = " V1 V1: Y0. Sin(91 ' 9h " Tu.) (3-203)

d2”, = - V, V,. Y”, cos(e, - 6,, - 7”,) (3-20b)

bu.” = V! VI. YU. 008(91 " 9h ' 70.) (3-213)

44111 = ' V1 VI. 1’0. Sinai: - 9h '70.) (3-21b)

kf = bzur = ' Ya. SW91 " 91. " Ya.) (3_22)
baur Ya. 008(91 ' 9h - 71:.)

kg = dun YD: cos(G, - 9n — Ya.) ' (3_23)

 

 

441.” = Ya. sin(O, " 9h ‘70.)

Since kf at 1:5, the LF -L type of load flow bifurcation due to the row dependence

associated with the active and reactive power balance equations at a single load bus

70

can never occur. This proof holds for every load bus.

3.3 Generic Load Flow Bifurcation

From the above, it can be concluded that there are no load flow bifurcation subc-
lasses due to the row dependence of some particular rows of the load flow Jacobian
matrix associated with single buses. Thus, analogies between bifurcation in a two-bus
' model have limited usefulness in understanding how voltage instability develops in the
large power system load flow model. The only necessary and sufficient test condition
for the load flow bifurcation is the test for the singularity of the load flow Jacobian
matrix J L,- (3-9) due to joint row dependence at all buses. This generic load flow
bifurcation is called the voltage/angle bifurcation because it occurs due to not only the
active power-angle coupling (J p a) and the reactive power-voltage coupling (JQV), but
also due to the active power-voltage coupling (J pv) and the reactive power-angle cou-
pling (199) at all the buses [53,87].

If the active power-voltage coupling and the reactive power-angle coupling can be
ignored, or a decoupled load flow model is assumed, then the voltage bifurcation
and/or the angle bifurcation can approximately occur. The voltage bifurcation [53],
due to the row dependence of the reactive power balance equations of IL]: or the
singularity of Jay, is a bifmcation that solely leads to voltage instability. The angle
bifurcation, due to the row dependence of the active power balance equations of J”-
or the singularity of Ira, is a bifurcation that solely leads to steady state angle instabil-
ity. The voltage bifurcation [53] when a bus or a group of buses can not be loaded
with an additional increment of reactive power load, because the reactive power supply
from the generator PV-buses with reactive power reserves that would attempt to supply
this reactive power load is totally consumed in the network reactive power losses. The
angle instability would occur when a transmission interface or boundary has reached

its active power transfer limit. Note that since there always is some active power-

71

voltage coupling and reactive power-angle coupling, the load flow Jacobian matrix Ju:
can be singular, even though the matrix Jpo is nonsingular or the matrix JQV is non-
singular. Thus, the generic voltage/angle load flow bifurcation can be understood to be
caused by the active and reactive power transfer limitation, and can occur before the
active power transfer limitation due to the angle bifurcation or the reactive power
transfer limitation due to the voltage bifurcation is reached in a decoupled load flow

model.

It should be pointed out that the above proof not only indicates that the singular-
ity of the load flow Jacobian matrix J” leading to the load flow bifurcation can not
occur due to the row dependence associated with the active power and reactive power
balance equations at a single bus, but also implies that the load flow bifurcation can
not occur due to the row dependence associated with any subset of mws of JLF. The
singularity of J”- requires that v' I“: = 0, where v is the left eigenvector of J”
associated with the zero eigenvalue of J”- . The requirement that only a subset of the
rows of J”.- be row dependent amounts to finding a point of singularity where
v‘ J”- = 0 such that elements of v are specified to be zero and not free. Thus, load
flow bifurcations due to the row dependence associated with any subset of rows of I”:
can be considered as very rare types of the load flow bifurcation since the constrains
associated with forcing elements of v to be zero for the operating conditions where
v‘ Ju- =Omakeit so.

It has been proven [53] that a Q —V curve test is a test for voltage/angle bifurca-
tion that occurs at the Q —V curve minimum. The Q-V curve has a minimum when
the load flow Jacobian matrix J".- is singular due to the row dependence of both active
and reactive power balance equations at all the buses [53,87]. The minimal of Q-V
curve test for voltage collapse is the test for the singularity of the equivalent Q —V
sensitivity matrix SQV (38), where the test bus (pattern) is treated as a (set of) PV-

bus. It has also been shown that the V—P (or V—Q) curve test is a test for

72

voltage/angle bifurcation which occurs when the nose of the V-P (or V-Q) curve is
reached, where the load flow Jacobian matrix J”- is singular also due to the row
dependence of both active and reactive power balance equations at all the buses. The
nose of V-P (or V—Q) curve test for voltage collapse is the test for the singularity of
the load flow Jacobian matrix I“: ((3-8) or (3-9)) solely due to the row dependence
between the rows associated with the active (or reactive) power balance equation at the

test bus pattern and the rest rows of load flow Jacobian matrix.

It should be noted that the singularity of the load flow Jacobian matrix J”.- is
approached continuously with added active or reactive power transfer. across an inter-
face or boundary as long as generator PV-buses do not experience exhausting reactive
power reserves. When a generator PV-bus exhausts its reacrive power reserve and
becomes a PQbus, the addition of another reactive power balance equation and
another voltage variable causes a discontinuity in the measure of the singularity of
JLF. This discontinuity often occurs at the point of voltage collapse because the load
flow Jacobian matrix for the case where this bus is still a PV-bus are row independent
but yet no voltage instability would occur because the load flow Jacobian matrix is
nonsingular. If this PV-bus became a PQ-bus so that the Jacobian of the reactive
power balance equation at this bus was added to the load flow Jacobian that existed
prior to this bus becoming a PQ-bus, the expanded Jacobian matrix would be singular
and the loss of voltage stability would begin immediately. This phenomenon will be
observed in the simulation results of a 9-bus 3-machine power system shown in the

following section.

3.4 Simulation Results

The 9-bus 3-machine power system of Fig. 3-1 [26] is adopted to confirm the
load flow bifurcation test. Figure 3-1 shows the transmission network configuration,

and gives the base case data, where the resistance, reactance, and line charging

73

100.00

 

 

18.4 xv 230 KV (9‘65) 230 KV 138 KV
'0. 250 '0. 172
Gen“ 18 0.0255+j0.2160 0.0357+j0.3024 J S $2
1900 < B/2=j0.0373 B/2=j0.0523 85 00
(23.41) , (5.32)
Bus#3 Bu ;#7 Bus# 8 B11 s#9 Bu s#2
1.0250 1.o0161 1.0200 1.0235 1.0250
(13.86) (2. 510°... C (-5.48) g ,,, (0 .35) (5.79)
:2 m m 05
. V3 . 00
a. g 92. g
s a é a
8 m 5 an
O O
Bus#5 Bus#6
125-00 <——— 0.9720 0.9543 ——> -00
(50.00) (-6.16) (-909) (30.00)
O C
a a g a
Q 8 a 8
it as
8, on C2 an
O O
230 KV Bum
04997836
VV j0.1152 ( ' )
16.5 KV Bus#l
1.0400

 

 

 

 

 

 

 

 

 

 

 

74.38
(40.400 (0.00)
Gen#1

Fig. 3-1 9-bus 3-machine Power System Diagram

74

susceptance of the transmission network is in pa based on 100.00 MVA and the rated
bus voltage as shown in Fig. 3—1; the bus voltage is shown in the format: V ( 0 ) with
V in pa and 9 in degree; and the active and reactive power injection at each bus is
givenintheformatzl’ (Q )withP inMW andQ inMVAR. Thesethreegenerators
are operated as follows: the generator terminal bus#l is the swing (infinity) bus; the
generator terminal bus#2 is a PV-bus with reactive power generation limit
:1: 40.00 MVAR; and the generator terminal bus#3 is a PV-bus with reactive power
generation limit :t 50.00 MVAR . Note that the resistance, reactance, and line charging
susceptance of this power system are adjusted and are different fi'om the original data
in [26].

The Q—V curve test of the point of voltage collapse method [53] is used to show
the load flow bifurcation as shown in Fig. 3-2(b). The bus#8 is treated as a fictitious
PV-bus — a dummy generator test bus — whose voltage is assigned and changed
from the base case voltage 1.02 pa with reactive power "generation" (its negative sign
is the actual reactive power load) 0.0965 pu . Figure 3-2(a) shows two matrix deter-
minants: one with the noration Der (LFO ) is for the load flow Jacobian matrix I“:
((3-2) or (3-8)) excluding the fictitious PV-bus; the other Der (LFl ) is for the load
flow Jacobian matrix including the fictitious PV-bus. Figure 3-2(b) shows the Q-V
curve at the fictitious PV-bus, and its slope Der (QV ) (a scalar) which is the
corresponding diagonal element of the sensitivity matrix SQV ((3-2) or (3-8)) of this
Q -V curve. Figure 3-2(c) gives the Q -V curve profile of the generator buses with
reactive power generation high limits. and Figure 3-2(d) represents the voltage profile
of each bus. The voltage angle (in degree) of each bus, and the active power genera-
tion of each generator bus are respectively shown in Fig. 3-2(e) and Fig. 3-2(f), which
should help understand the phenomena of the power system when the voltage and the

reactive power injection at the test bus (fictitious PV-bus) are changed.

Det(LFO) & Det(LFl)

QstB & Dct(QV)

x10”

75

Fig. 3-2(a) - oextm)» ; 15mm):

 

2.5

1.5 P..........

0.5..-

 

r l 1

7‘lxxxl‘

 

 

0.8 0.85 0.9 0.95
V(FPV-bus: 8)

Fig 3-2(b) -- QgtS:x; Det(QV):"

1.05

 

f r T
t . .
:5 .. .......................... .fl ..... .....4p ..... ;. .4
. U
I
. .
C

 

txxaaaaxigalitaxlli‘7“.

O . . .
O ‘ '
. ... . . ..... ,.... .... . . ..... ....... ................ .... _

' ' ...... t ............................. . I
0 .u-nu...-o..u--......~...o...................u.......-.....-.-oo.....g ....... ‘. . ..... .0: uuuuuuu . . . . .‘...‘-..‘...

 

 

0.75

0.8 0.85 0.9 0.95
V(FPV-bus: 8)

l

l .05

Qsl (Quin)

VW

76

 

l 4 Fig. 3-2(c) - Qgt (Qmax):[bus: 8x : 3+(-.) ; 2‘(~) ; 10]

o . §
..................... 9 . .... . .. ,. . r ..
1.2 . ' o :
o . :
0
1,. .............. Qua.-. ................... _‘
o o .
o E
0.8.. ........... . . . .....o. 9 .. .. _. . . . _ _.

, - O '
*“-‘.'-. *'¢. *OOWD- *. ‘.".*'4.-*0"0"*-*--§‘¢. 020-00-.o~o--°O o-o O C o o o c
. . - ‘ .

. O 0 g
0.4 h ------ .-.---."CO'OO-OOOO--.---.O¢C~OOOC'O: .............. .00.... OOOOOOOOO 9-0 q
o 9
. _ ' . 7 9
' ~ 0 . - O .
0.2 p .................................... ....... ...................................... .... ....................... 2. ...... ............... . ...... . d
. . .
’ . C
_ . . . ‘
‘ O
0.... ................. ‘ . fl
l
I
X
.03.. ......... .‘... 2
I
‘ X
g X
.04.. .. I g ‘ X ‘ ‘ ..... , X I I .1
‘ i ‘ x a a a a x ‘

 

 

 

0.7 0.75 0.8 0.85 0.9 0.95 l l .05
V(FPV-bus: 8)

 

Fig. 3-2(d «- Vpu(bus# 4-,5-,6:,7-.,9. ; 3+,2“,lo)

 

1.05 17 T U T '
0 O 0 0 0 O O 0 O O O O O 0 0 O O 0 9 O 0 O O 0 0 O O O 0 O 0
o e e o g o o c o o o o o o o
O - . .
l . a
0.9 a
0.85 -
o ,8.-
g .‘ I
0.8 -.. .:.,' ....... '. .............................................................................. an:
0.75 ‘

 

 

 

0.7 0.75 0.8 0.85 0.9 0.95 l 1.05
V(FPV-bus: 8)

Angle (Degree)

PSI

Fig. 3-2(e) «- Angle(bus# 4-.5-.6:.7-..8it.9. : 3+.Z‘.lo)

77

 

15 _......

 

 

f ...... .....
§
O
O
0
T +
. 0
. a. ‘ .
¢ 0 O O 0 §
0
O . . . . .
' 0 e o o o 0 0 o o o 0
0000000000000 0006-0 0
x t ‘ ‘ .‘O-!-5-.&.§-.‘O05.1.0‘.--.-.-5-.x..§-.‘..‘..--.E.--’

 

 

1.7

Fig. 3.20) Pgt(bus

0.85 0.9
V(FPV-bus: 8)

0. 95

# 3+, 2", lo)

 

1.05

 

0.00900000000009000000000000

1.6 _

1.4»

1.3--

1.1.-..“

 

0009000ecooeooeooeéoeeogoooo

0.85....

.006 ....... ~ ,
.000 - . : .
: 090000000000990060000

fi

.......

f

i

1.

l

 

 

0.7 -

0.85 0.9
V(FPV-bus: 8)

p.-

78

It is observed from Fig. 3-2 that the load flow bifurcation occurs at the minimum
of the test bus#8 Q-V curve when the reactive power load about 0.467 pu is added at
the bus#8 whose voltage is assigned at 0.87 pa. Note from Fig. 3-2 (a) - Fig. 3-2 (c)
that the generator bus#3 hits its reactive power generation high limit 0.50 pu before
the generator bus#2 does, and that whenever one generator hits its reactive power gen-
eration (high) limit and becomes a PQ-bus, the dimension of the load flow J acobian
matrix Ju- changes by increasing one; whereas the dimension of the Q—V sensitivity
matrix SQV reduces by one. This causes the jump phenomena of the determinant of
the load flow Jacobian matrix and that of the slope of the test bus Q-V curve. Note
also that, from Fig. 3-2(c), the load flow bifurcation occurs at about the point where
the generator bus#3 becomes a PQ-bus (the generator bus#2 has already become a
PQ-bus), even though the swing bus#l has infinite active and reactive power reserves.
This result shows that the voltage collapse is a problem of (reactive) power demand
and supply. Each time a PV-bus changes to a PQ-bus, the supply rate from the PV-
bus is lost and the rate of increase in reactive power losses increase with further vol-
tage decline. The effective reactive power supply rate to the bus with voltage drop
decreases until the reactive power load demand at the bus#8 is 0.467 pu , where the
transmission network does not have the ability to transfer enough reactive power to
meet the requirement at bus#8 and the losses in the transmission network, even though

this system still has reactive power reserve at bus#l.

It should be pointed out that from Fig. 3-2(e), the angle differences between any
two contiguous buses are never more than about 40.00" . In addition, after bus#3
changes from a PV-bus to PQ-bus, its voltage angle starts dramatically increasing
when the system is more stressed. The voltage angle at bus#2 also begins to increase
rapidly after it becomes a PQ-bus. Whenever a generator PV-bus becomes a PQ-bus,
its voltage is no longer constant and begins to decrease when the system gets more

stressed, but it still keeps constant active power output (see Fig. 3-2(f)). The generator

79

PQ-bus then needs to increase its angle to compensate the voltage drop in order to
keep its active power output to be constant. When both voltage and angle change as

stress is added, a voltage/angle bifurcation is being approached.

Chapter 4
Stability Problems in a Differential-Algebraic Model

The theory of bifurcation [36-38] has been applied to investigate dynamic stabil-
ity and voltage collapse problems, under the assumption that the causality condition
[39,40] holds in the differential-algebraic power system dynamic model. The
equivalent system Jacobian matrix can then be obtained after inverting the causality
matrix and aggregating the transmission network back to the generator internal buses.
One of the necessary conditions for static/saddle-node or dynamic/Hopf bifurcations is
that the equivalent system Jacobian matrix has eigenvalues with zero real parts (others
are transversality and nondegeneracy conditions) [36-38]. The above bifurcations have
been shown to lead to loss of dynamic stability and/or voltage collapse, and been
shown to be mainly associated with both mechanical and electrical system dynamics of
generators [3945.52.53,64-73,78,79,87,88]. However, most of the studies have used
eigenvalue analysis on the equivalent system Jacobian matrix to establish if these
bifurcations can occur. Eigenvalue analysis is not only computationally memory inten-
sive, but also computationally time intensive for a large power system dynamic model.
Furthermore, eigenvalue analysis has not generally been applied to the very large data
bases to study voltage stability problems. Moreover, the main assumption that the
causality condition holds may not be true, such that the equivalent dynamic system
J acobian matrix can not be simply defined. The stability problems due to loss of
causality are not described and are not associated with the eigenvalues of the

equivalent system Jacobian matrix.

This chapter will classify the types of bifurcation and stability problems in the
differential-algebraic model. Necessary conditions are determined for each of these

80

81

types of bifurcation and stability problems. Several of these bifurcations are shown to
be non-generic. The static and dynamic bifurcations associated with eigenvalues of the
equivalent system Jacobian matrix having zero real parts are shown to be generic
bifurcations. The stability problems due to loss of causality and due to loss of stability
of the generator dynamics are also identified as possible, if not improbable, stability
problems. The static/algebraic bifurcation will be shown to be equivalent to the static
bifurcation. The static/algebraic Jacobian matrix has similar structure to the load flow
Jacobian matrix. This static/algebraic Jacobian matrix is associated with a set of
linearized algebraic equations that produces the equilibrium point of the differential-
algebraic model, which is very similar to the load flow model. In addition, this
static/algebraic bifurcation test matrix avoids the inverse of causality matrix. Finally,
the static bifurcation test due to the singularity of the static/algebraic Jacobian matrix
will be shown to occur based on the simulation results of a 9-bus 3-machine power

system.

4.1 Linearized Power System Dynamic Model

From Chapter 2, the dynamics of a power system is characterized by two types of
equations: (1) the differential equations of rotor and flux decay dynamics of machines,
and those of their control systems (that include (a) Excitation System with Load
(Line-Drop) Compensator [85], (b) Power System Stabilizer (PSS) [85], and (c)
Speed-Governing—Turbine System [86]), and (2) the active and reactive power balance
(algebraic) equations at the internal and terminal buses of machines, at the transformer
high-side buses of machines (simply called high-side buses), and at the load buses.
Hence, the specific power system dynamic model can be represented as follows:

i=f(x.y;ll) (4-1a)

0=g(x.y;u) (4-1b)

82

where,
x : the M ~vector of dynamic states of generators and control systems;

y : the N -vector of network dependent variables, i.e., the voltage and angle at
each terminal bus (VT, 07), high-side bus (V3, 0,, ), and load bus (VL, 0L );

u : the vector of independent parameters, such as reference voltage of excita-
tion system (V,.,f ), mechanical power setting of speed-governing-turbine
system (Po), and load demand parameters (PD , QD );

f : the function describing the relationship among the dynamic states, network
dependent variables, and control parameters, through the coupling branch

between internal bus and terminal bus of each generator;

g : the function describing the power balance equation at generator terminal

buses, at high-side buses, and at load buses.

All the solutions of (4~1) must belong to the M-dimensional surface Sg which is
defined as all points x and y where (4-lb) is satisfied. The Jacobian matrix of (4-1) at
any equilibrium point (1:, y:; u: ) can be obtained from the linearized power system
dynamic model, that has the following matrix form, by combining the linearized equa-
tion of each component in the power system shown in Section 2.6. The relationship

between the Jacobian submatrices is shown in Fig. 4-1.

 

 

'E’L 2L.

J— J! _[fx fy]_ 81: By

‘ J. ' s. s. ' 2:. 22 (4’2”
_3x 3y‘(x.°.y.':u:)

- Arc: Axe Axe 0 Axe Axv Axx

' TXx 102,. 1
TE AX‘E Ag ABE 0 A3 A59 AEV AXE Bo
' A 0 A 0 0 0 AX 5°
Too AXG GX ‘36 G 360

 

 

 

 

 

 

rss arts as,‘ 0 0 Ass 0 0 AXS + (0) AUG (4‘2”)
- AP C Apx 0 0 0 Ape Apv A9 0
" - AQC ‘ AQX 0 0 0 A09 AQV . AV a b -

 

 

83

30322 5582 883m .838 ..e 8835 0.02m 70 .wE

 

 

 

A.\. l —E¥ .vmk RU¥ aruk “ 1'

 

f + -2082 0.82on

- _
h 11.— fikafiglvradsiw _ e

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OU< samhm g i mu<
.l' ..w 0585. , Baa—35m
55050 3.0.2 sank :mk 2&5 um um cm 8293
a 608m . .539—
g e _ _ 9.4.

peter mm... a
_ a 8on m _
w nonsmoxm + x3.

   

 

 

 

 

84
where,

Axx AXE Axe 0
f x = (diagl Tn TEE Toe 735 l)" Aax 0 A60 0

ASXO 0.133

 

Axe Axv

. _‘ Are My
fy=(dlangrcr TEE T66 735]) 0 0

_0 0
Apxooo
82‘ AmOOO

Are APV
g, = Age AQV

 

 

AXX = [ 1301‘ A5‘ A5; 1‘

: states of mechanical and flux decay dynamics;
AXE = [ AVD' AV,‘ AVA' AVB‘ AEfd‘ 1‘

: states of excitation systems (see Fig. 2-5);
AXG =[AP1' APGV' APVHP' APHP' Apr!" “’0' 1'

: states of speed-governing-mrbine systems (see Fig. 2-8);
M8 = [ AVso' AVsz‘ AVS4' ]‘

: states of power system stabilizers (see Fig. 2-7);
A0 = [ 2101' A0,,‘ AOL‘ 1‘

: angle variables at network buses (terminal: T, high-side: H , load: L);
AV = [ AVT‘ AVH‘ AVL‘ ]‘

: voltage variables at network buses;

85

APC = [ Apart APCH' APCL' 1‘
: coefficients of non-voltage-dependent active power load demand model;

AQc = I Ach' Ach' Ach.' 1'
: coefficients of non-voltageodependent reactive power load demand model;

M 0 0
rxx =diag[M 1 1],]: 0 I 0
0 0 1;,
M =diag[M1,M2,...,Mm]
: inertia constants of m synchronous machines;
I : m x m identity matrix;
T40 : m x m diagonal time constant matrix of flux decay dynamics;
TEE = ding [To TF 7.4 Ta 75]
: time constants of excitau'on systems (see Fig. 2-5);
Tea = 41°08 [T1 Ta Terr TRHl Tarn Tco ]
: time constants of speed-governing-turbine systems (see Fig. 2—8);
Tss = 41°08 [Ts Tsz Tsa]
: time constants of power system stabilizers (see Fig. 2-7);
Note that TEE , TOG , and T33 submatrices above are defined using a similar notation to
that used in defining Txx, and the submatrices T 4'0, - ° - , TM are m dimensional diag-
onal matrices composed of the appropriate time constants of the generator and control

systems models. The A -matrices of each Jacobian submatrix are defined as follows.

‘0 ‘Kfl'r “”5
A“ = I O 0
0 'K54 "Krzs

[A503 AECI Aecz ] =

P

—I
0

 

=71?ch 3’36 =1 "TBC

= T's—41753 3 721 = 7521731

0
-I

00
00

 

= KA Tac T.137'21Ks ; ABm = lac T4372119;

0
- TF- 1K1:

“KATBC ’KATBC '1 KA ”1973chle

" Inc
0

0
0

KA Tac 743721 K4 Tnc T43 KA Tac

lac T43721
1 0

 

[A60 00]:

-IBC

0

0 I 0

O
0

lac T43
0

' 71-17219;
0

 

0
0
0

-l

r
(1 -Tf1Tz)KG 0 0

' lac TF- 1K1?

' Kse

0
0

13c
0

 

00
00

 

GOO
GOO

 

OOOOO

 

 

001-1]

”KS 0 0
ASX =[ASm 0 0]: Iles 0 0
I43T21KS 0 0

143:1-743 3 ’21=1-Tzr
-I 0 0
Ass= ’21 ’1 0

143721 143 ‘1

AUO = [ Aan‘ APO‘ ]'

 

 

 

 

 

 

 

 

0 0H
0 0
BEO=IBEE 0]: KATEITC 0 :Boo=IOBaa]=
143-11¢ 0
0 0_
. , .
[K133 0 0 -Kfl'-, O 0
Axe= 0 00 :Axv= 0 00
LKuOOJ _KE, 00J
l-Kchr 0 0‘ r111,103, 0 0'
0 00 o 00
A50: 0 00 9AEV= 0 00
0 00 0 00
0 00. _ 0 00,
0 -K,‘¥1Kfi‘72 O —K31K62
Apx= 0 0 0 mg: 0 0 0
0 0 0 0 0 0

 

 

 

 

COCOOO
COCONO

 

 

88

Au: 3m 0 C11: Dru 0

AP0= A211 32m! 32m. 3 Apv= C2” DZHH 02m

0 32”, am 0 D2“, Du

A1K=KF1 +A1;C1K=K;rq7 +C1

ASK 33H 0 Cart 03H 0

AQ0= A411 Bum 34m 3 AQV= C411 Dam D4111.

A3K =K31+A3 ; C3K =K37 +C3

The following properties of the submatrices of the linearized power system model

help provide understanding of the model.

(1)

(2)

(3)

(4)

All the K matrices are m x m diagonal matrices. Among them, the
matrices, K33, K53, K54, K57, [ff-‘5, K32, and Km, are associated with the
flux decay dynamic states 8,; of the single-axis modeled synchronous

machines.

The matrices th and K“? of the power balance equations at internal buses
and terminal buses of synchronous machines represent the coupling between
both buses. In general, K '9 a: K 4‘ , because of the armature and salient-pole
effects of synchronous machines, that cause X4, Xq , and x; are different.
If the armature resistances (Rd) and salient-pole effects are neglected, then
PE =PG, = -P,., such that [(33 =Kfi41,Kfl'2 =-—K;92, and K33 = -K}."-,.
The load compensator is characterized by the diagonal matrices: KC 1, KC 2,
and Kc 7. If a load compensator is not employed, then VC = VT, i.e.,
Km: 0, Kc; =0, and Km =1.

Due to the effects of transformers, phase-shifters, and network conduc-

tances, each of these network submatrices, Ape, AW, A99, and AQV,

89

composing of the network Jacobian matrix is not symmetric.

(5) The coefficients of voltage dependent load demand model are embedded in
the diagonal elements of C and D submatrices.

(6) Under the assumption that the connection between each generator terminal
bus and its transformer high-side bus is one-to-one, the submatrices A,- , C,- ,
i =1, 2, 3, 4, and Bi ,Dj”,j =1, 3 are diagonal.

(7) . Other properties of the network Jacobian matrix can be found in Section

3.1, that are associated with the submatrices of the load flow Jacobian

matrix in the load flow model.

4.2 Bifurcations in Power System Dynamic Model

At any parameter pattern 11 = u, of the power system dynamic model (4-1), an

equilibrium point (xe, y,; 11,) on the surface 88 defined by (4-lb) is a solution to
0=f(x.y;u) “30
0=s(X.y;u) (4-3b)

Note that the precise formula of (4—3) for solving for the equilibrium point(s) of the
power system dynamic model has been discussed in Section 2.5. The bifurcation prob-
lem is to characterize the solution set of (4-3) in a neighborhood of an equilibrium
point (x,, y,; 11¢ ). The equilibrium point (x; y:; 11:) is a bifurcation point and it: is
a bifurcation value of the parameter if the system satisfies one of the following condi-

tions.

(a) The complete system Jacobian matrix

2L.fll

J ft fy 8x By
]- f - - 4.4
- Jr - 3133 - 28.28. ( )

8): 3y (xi-yiwi)

is singular.
(b) The equivalent static/algebraic J acobian matrix

Jy = 8y - 81 :1 fy (4'5)
is singular, provided that f, is nonsingular.

(c) The equivalent system Jacobian matrix

Jx = fx "fy gy-l gx (4'6)
has eigenvalues with zero real parts, provided that gy is nonsingular.

Note that Jacobian submatrices, f ,, f y , g,, and g, , can be found from the linearized
power system dynamic model (4—2).

From the above, six subclasses of bifurcation in power system dynamic model are
identified.

(1) A static/algebraic bifurcation occurs when the equivalent static/algebraic Jaco-
bian matrix J, is singular, but J, is row independent and g, and f, are non-
singular [52.53.87.88].

(2) A static bifurcation occurs when the equivalent system Jacobian matrix J, is
singular and does not have pure imaginary eigenvalues, but J I is row
independent and g, and f, are nonsingular. The static bifurcation could be
the saddle node bifurcation if the matrix J, has a simple zero eigenvalue, and

the transversality and nondegeneracy condition of J, at the bifurcation point
holds [36- 38].

(3) A dynamic bifurcation occurs when the equivalent system Jacobian matrix J,
is nonsingular and has pure imaginary eigenvalues, but 3, and f, are non-
singular. The dynamic bifurcation could be Hopf bifurcation if the matrix J,
has a pair of pure imaginary eigenvalues, and the nondegeneracy condition of

J, at the bifurcation point holds [36-38].

91

(4) A differential bifurcation occurs when the equivalent system Jacobian matrix

J, is singular due to the row dependence of J, (but 3, is nonsingular) [53].

(5) An algebraic bifurcation occurs when the equivalent static/algebraic Jacobian
matrix J, is singular due to the row dependence of J, (but f, is nonsingular)
[53].

(6) A differential-algebraic bifurcation occurs when the complete system Jacobian
matrix J is singular due to b0th the row dependence of J I and the row depen-

dence of J, .
Note that two specially stability problems occur along with the above bifurcations:

(a) A single-machine instability occurs when the machine Jacobian matrix f, is
singular, or when f, has complex eigenvalues with zero real parts, but J ,- is

row independent.

(b) Loss of causality occurs when the transmission network J acobian matrix g, is

singular, but J, is row independent.

The above stated conditions are necessary conditions for these specific types of bifur-
cation. Transversality condition for saddle-node bifurcation and nondegeneracy condi-
tion for Hopf bifurcation [36-38] which control the nondegeneracy of the behavior
with respect to the p. parameter change and the dominant effects of the nonlinear terms

of power system dynamic model will not be discussed in this dissertation.

Note that type 1, type 2, and type 3 bifurcations require that both f, and g, be
nonsingular and both If and J, be row independent, and that type 4, type 5, and type
6 bifurcations are due to the characteristics of row dependence either in J I , or J, , or
both. The first three types of bifurcation require both f, and g, be nonsingular mak-
ing both J, and J, well defined. The type 4 differential bifurcation requires J, be row
independent and g, be nonsingular so that J, is well defined. This differential bifurca-
tion also requires f, be singular and If be row dependent leading to the singularity of

92

both J and J,. The type 5 algebraic bifurcation requires that f, be nonsingular so that
J, is well defined, and that g, be singular and J, be row dependent resulting in the
singularity of both J and J, . The type 6 differential-algebraic bifurcation is restricted
to the case where both f, and g, are singular and both J, and J, are row dependent
causing J to be singular and both J, and J, to be undefined. The above restrictions
on the above bifurcation tests make sure that the classifications are distinct. The rela-

tionship and difference between the six classes of bifurcation are now discussed.

It can be shown, using Schur’s formula (Appendix A), that the static/algebraic
bifurcation due to the singularity of J, (4—5) can be proven to be equivalent to the
static bifurcation due to the singularity of J, (4-6), provided that both f, and g, are
nonsingular which is assumed in the definition of the static and static/algebraic bifurca-
tions. Thus, one could consider making the static and static/algebraic bifurcations as a
single bifurcation class. The two reasons this wasn’t done was (a) a proof is required
to establish they are completely equivalent, and (b) at an initial review one would con-
sider singularity of J, and singularity of J, as different stability problems, which
indeed they would be if all of the possible stability problems due to singularity of J,
or J, were considered as one class of stability problem. The classification of the
bifurcations and the stability problems indicates that there are different stability prob-
lems associated with singularity of J, (static; differential) and different stability prob-
lems associated with singularity of J, (static/algebraic; algebraic); but when f, and
g, are nonsingular, then there is one stability problem associated with J, and J, being
singular. If one chose to make the static and static/algebraic one class of bifurcation
problem based on the fact that f, and g, are assumed nonsingular in the definition of
these stability problems, it would be called a static bifurcation, static/algebraic bifurca-
tion, or singularity induced static bifurcation, since both J, and J, are singular for this
bifurcation to occur. A singularity induced static bifurcation Will be shown to be

~ caused by the transmission network Stress (see Section 4.6), in a manner almost

93

identical to that causing the load flow bifurcation as shown by the simulation results in

Section 3.4.

Note that the type 4 differential bifurcation is not equivalent to the type 2 static
bifurcation even though both occur when J, is singular. The difference between them
is that the differential bifurcation is solely due to the instability of the single machine
dynamics (J f is row dependent and f, is singular), whereas the static bifurcation is
due to the instability of the transmission network coupled with the dynamic states
represented by J, . Likewise, the type 5 algebraic bifurcation is not equivalent to the
type 1 static/algebraic bifmcation even though both occur when J, is singular. The
difference between them is that the algebraic bifurcation is solely due to the lack of
solutions or multiple solutions of the transmission network (J, is row dependent and
g, is singular), whereas the static/algebraic bifurcation is due to the instability of the
transmission network coupled with the dynamic states represented by J, . The similar-
ity of J, and the load flow Jacobian matrix I” is discussed in Section 4.4. It sh0uld
also be pointed out that the row dependence of J, and loss of causality (g, singular)
indicate that the set of algebraic equations (4-3b) may have either no solutions or mul-
tiple solutions; but the static/algebraic bifurcation is a bifurcation in both machine
dynamics and transmission network. The two special stability problems due to the
singularity of f, and that of 3, will be discussed in Section 4.3, even though they are
not bifurcations because they indicate loss of stability and because the tests for the

static/algebraic, static, and dynamic bifurcations require that they be nonsingular.

4.3 Prerequisites of Bifurcation Study

Having classified the simplest types of bifurcation for a differential-algebraic
power system model, it will now be shown that the type 4 differential bifurcation due
to J I being row dependent, the type 5 algebraic bifurcation due to J, being row

94

dependent, and the type 6 differential-algebraic bifurcation due to both J I and J,
being row dependent are not generic types of bifurcation. The singularity of J
requires that v‘ J = 0, where v is the left eigenvector of J associated with the zero
eigenvalue of J . The requirement that an only subset of the rows of J be row depen-
dent amounts to finding a point of singularity where v' J = 0 such that the elements of
v are specified to be zero and not free. Thus, the differential bifurcation, the algebraic
bifurcation, and the differential-algebraic bifurcation can be considered as very rare
types of bifurcation since the constraints associated with forcing elements of v to be

zero for the operating conditions where v' J“.- = 0 make it so.

Therefore, although we started with the possibility of six different types of bifur-
cation in the power system dynamic model of (4-1) and (4-2), only three types, type 1,
type 2, and type 3, of bifm'cation appear to remain as likely, and will be discussed in
the rest of this dissertation. Both type 1 and type 2 are singularity induced static
bifurcations and equivalent to each other when f, and g, are always nonsingular,
which are required to be based on their definitions. The type 3 of bifurcation is the
dynamic bifurcation. It should be pointed out that both f, and g, might still be singu-
lar [87] and produce non-bifurcation stability problems. The behavior of the system
due to loss of causality will now be discussed further in Section 4.3.1. The singularity
of f, will be further addressed in Section 4.3.2, and be argued to be improbable, but
the proof is not strong enough to prove that the singularity of f, is non-generic. Thus,
singularity induced bifurcation, dynamic bifurcation, and loss of causality will remain
as generic stability problems. Although loss of causality must be considered as a gen-
eric stability problem, discussion concerning the structure of 3, compared to the
static/algebraic Jacobian matrix J, addressed in Section 4.4 and the simulation results
of the static/algebraic bifurcation test for a power system with the reactive power gen-
eration reserve constraints addressed in Section 4.6 suggest that loss of causality may

not be a generic stability problem.

95

Chapter 5 will concentrate on systematically developing test conditions for the
static and dynamic bifurcations (type 2 and type 3), based on the equivalent dynamic
Jacobian matrix J, , under the assumptions that causality condition holds (g, is non-
singular) and that transversality/nondegeneracy condition holds. Test conditions for
the static and dynamic bifurcations pertaining to mechanical and flux decay dynamics
will also be established [88], based on the property that each of control systems itself
does not have eigenvalues with zero real parts which will also be proven to be true in

Section 4.3.2.

4.3.1 Causality Condition — g,

In Section 4.1, the M —dimensional surface S, is defined, which requires that all
the solutions of the power system dynamic model (4-1) x and y must satisfy (4-lb)
0 = g (x , y ; u ) at any n parameter. The causality condition of the power system,
g, = 8g [By being nonsingular, means that every solution x and y must belong to the
regular part of S, ,

5,, = {0=g (x ,y :11) ; J, is row independent.)
which is an M -dimensional submanifold in S, [89].

When J, is row dependent, the solutions 1: and y on the surface S, are no longer
regular and do not belong to 3,. Thus, there can not only be more than one y solu-
tion for each x but several x solutions with more than one y solution for each 1:.
These 1: and y , when J, is row dependent, lie on the geometric boundary separating
the regular part of S, fi'om the remainder of S, . Note that the row dependence of J,
implies the singularity of 3, leading to loss of causality. Since loss of causality can
still happen even when J, is row independent, this boundary is characterized by the
particularly complicated type of stability problems observed as being associated with

loss of causality.

96

Loss of causality (g, is singular) occurs on the so-called fi'ontier

F, = [0=g (x ,y ;|.t) ; g, issingular.)
which is an M -1-dimensional surface [89]. On the frontier, y in (4-1b) can no longer
be solved as a unique function of x . Loss of causality results in uncertain dynamic
behavior of two different types: singular points and impasse points can occur on the
F, surface. At an impasse point, a solution may not be continued for all time after an
impasse point is encountered on the frontier. This is not the case for a singular point.
If the initial condition is a singular point, then solutions exist but are not unique. A
solution can switch from component to component of the regular part of the solution
set via singular points. Jump phenomena and complicated periodic solutions can occur

at these impasse and singular points.

From the linearized power system dynamic model (4-2), the causality matrix g,
can be structurally represented as follows:

Ape APV

g, = 3‘90 AQV

All: 3111 0 Cur Dru 0
A211 32101 32m. Czu DZHH 02m.
= 43x 33H 0 Cart 0311 0 (4’7)
A4” 34m! 34111. C411 Dam Dun.

 

 

Note that the structure and properties of the causality matrix g, and the Jacobian
matrix J, = [ g, g, ] are very similar to those of the generic load flow Jacobian
matrix Ju- shown in (3-8). The only differences are that there are reactive power bal-
ance equations at all terminal buses and that the inequality of the diagonal matrices
Am #31”, CIK #01”, A“ #83”, and C3; ¢D3H holds in g, but not in Jur,

because Am, Cm, A“, and C3,; have the effects of the coupling between the

97

internal buses and terminal buses in g, . This results in a significant diagonal domi-
nance in 3, that is not present in Ju.~ if all the generator terminal buses were PQ-
buses. Note that since the reactive power balance equations at all the generator termi-
nal buses are included in g, , the eigenvalues of g, will not change discontinuously
when reactive power generation limits are reached and PV-bus to PQ-bus changes
occur. The test for the static/algebraic bifurcation on J, will be shown to be similar to
J”- , and will experience discontinuous change when any generator exhausts it reactive
power generation reserve and changes to a PQ—bus. The simulation results will also
show that the singularity of g, is nOt likely to happen when the reactive power genera-

tion limits are included in the model for the simple power system considered.

Loss of causality (g, singular) will occur due to the row dependence of J, . The
discussion of the various possible subclasses of the load flow bifurcation due to the
singularity of J“.- can be viewed as possible subclasses of loss of causality. The dis-
cussion in Section 3.2, that illustrated that (a) the bifurcation at a single bus due to the
row dependence of the rows of J“.- associated with the active and reactive power bal-
ance equations at this bus, (b) the voltage bifurcation due to the row dependence of
rows of .1” associated with the reactive power balance equations, (c) the angle bifur-
cation due to the row dependence of rows of IL]: associated with the active power bal-

ance equations can not happen, can be carried over to J, and loss of causality.

4.3.2 Single Machine Condition — f,

From the linearized power system dynamic model and the Jacobian submatrices
of (4-2), the matrix f, is defined as

f. =(diagl Txx TEE Too Tss l)“l Fx (4-8a)

98

Axx Axe Axe 0
45x 431: 0 Ass [Axx Axe]

 

 

FX = A6)! 0 466 0 Acx Ace (4.813)
As, 0 0 Ass
where,
A5, A55 0 A53
AXC = [AXE AXG O] ; ACX = AGX ; ACC = 0 A66 0 (4-8C)
Asx 0 0 Ass

It. is observed that the matrix f, represents the operation condition where each syn-
chronous machine is assumed to have a swing (infinite) bus at its terminal bus, since
f, is formed by diagonal block submatrices composed of the K -coefficients of each
machine, as shown in the steady state block diagram Fig. 4-2. The nonsingularity and
lack of complex eigenvalues with zero real parts of f, or the singularity of (f, - jfl}
for some fi'equency Q 2 0 characterizes the "proper" or "nonproper" basic design of
each machine when its terminal bus is treated as the swing bus of the power system.
The matrix f, can now be argued to be nonsingular and has no complex eigenvalues
with zero real parts because the system is designed so that loss of stability will not
occur on the machine dynamics only (f, ), but rather on the complete system dynamics
(J, ). It should be noted that the excitation systems and power system stabilizers are
designed to prevent stability problems and thus the stability problems in f, are not
likely to occur. However, the stability problems may occur in J, , especially when the
system is heavily stressed or when some machines are operated in underexcited condi-
tion (leading power factor at the terminal bus) [27]. A simple test matrix for testing
the (non)singularity of f, will also be derived for the case where the machine is not
sure to meet the basic operating and design practice for a single machine operated

independent of the transmission network.

4%. Ba ..x e E235 seem a u 3 3% €an 3 a...

 

2.50:3 +4.3

 

~l

 

 

 

 

 

ck 833m
059.5.
-wEEchO
68am

 

 

 

 

 

 

 

 

 

 

 

.535 + Quench + 80:
MI

+

~U¥mm¥+ vflk —

»

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a
c
a < .3. 1 mm
c. c: - a It 3333
8 823m
housefl—
< a .w _ ax»
wa8u5+nm¥ fink.»
_ 80% m
nonsmoxm n+ VS
a — ~U¥Q¥ —NU¥Q¥

 

 

 

 

r.

a ..m

 

100

Note that each block submatrix of f, is diagonal and can be viewed as an ele-
ment, since the dynamics of each machine is uncoupled in f, . From (4-8), it is obvi-
ous that the control systems form an upper triangular block matrix ACC , hence the
eigenvalues of Ace are those of each control system, A 53 , AGO , and A $3 . Moreover,
- based on the IEEE general model of speed-governing-turbine systems (Fig. 2-8) [86],
for both steam and hydro turbines, and the model of power system stabilizer (Fig. 2-7)
[85], both A66 and Ass are lower triangular matrices with real and negative eigen-
values. Regarding the excitation system matrix A 55 , if the excitation system is "prop-
erly tuned" and is able to be manually operated, A55 should not have any eigenvalue
with zero real part. Hence, the control system matrix Acc is virtually certain to be
nonsingular. Therefore, the control systems matrix Acc . can be aggregated into the
mechanical and flux decay matrix A“, such that the singularity of f, is equivalent to

the singularity of the following matrix (Appendix B: A56):

4%. = An - Axe 456 Acx

" (D + KG) " Kfl'r ‘ Kfl'z
= I 0 0 (4'9)
0s " (K54 + K51) - (K53 + K152)

where, the following matrices are defined
KG = A006 453; A600
: effects of speed-governing-turbine system;
05 = Arr; 4511”an '43s ASS] As... ] = 0
: effects of power system stabilizer;
K121 =AFE AEEIAEC1=KEE Kcr
: effects of excitation system on A8;
K52 = An Air} Aecz = K35 Kcz

: effects of excitation system on A5, ;

101

K55 = [($721 KA KB-
The block diagram of Age is also shown in Fig. 4-2.

Note that KG is the D.C. gain of speed-governing-ttu'bine system (Fig. 2-8), and
K 55 is the D.C. gain of excitation system (Fig. 2-5). The power system stabilizer does
not show any effects on 4,5, due to its D.C. washout transfer function (Fig. 2-7).
Hence, the singularity of f, is fully dependent upon the D.C. gains of excitation sys-
tems K 55 and speed-governing-turbine systems KG , and upon network effects through
the six diagonal coefficient matrices: Kfl‘l , K ,‘1‘2, K 33, K54, K013 and Kcz- The nota-
tion of these diagonal K -matrices are adopted so that they are consistent with those
K -matrices [23,26] of the single-machine-to-infinite-bus power system where synchro-
nous machine is represented as the single-axis model and does not have control sys-
tems (K55 = 0):Kp1= K1, Kn = K2, K53 = K3, K54 = K4, and K5 = 0, K6 = 0 since
in this case the generator terminal bus is the infinite bus.

Note also that based on the theory of synchronous machine, when all the deriva-
tives of dynamic states are set to be zero (at steady state), all the induced currents at
damper windings are also zero. Hence, the synchronous machine can be represented
by single-axis model, such that only the dynamic state 5, of flux decay dynamics
needs to be preserved. Therefore, Afx represents the Jacobian matrix of (single-axis
modeled) synchronous machines at steady state condition with the D.C. gains of con-
trol systems [87]. It will now be shown that f, and .45, would not be singular and
would have asymptotically stable eigenvalues, provided that each generator is operated
at the usually overexcited condition (lagging power factor) and at the power levels
where steady state angle stability limit, and its excitation D.C. gain is not "badly"
tuned [2326].

Each synchronous machine is assumed to be operated below its steady state sta-
bility limit so that each element of the diagonal matrix K ‘3'] which is the power-angle

102

coefficient at internal bus is not zero (Appendix C). The matrix

AM — (4-10)

 

- (D +KG) 'Kfl'r]
l 0

is thus nonsingular, assuming that K A is nonsingular. This is also true even though

the damper constant D and the D.C. gain of speed-governing-turbine system K6 are

neglected, because from (4-8) setting D + KG to be zero does not affect the singular-

ity or the determinant of A5,. Hence, the singularity of f, can be shown to be

equivalent to that of the following matrix:

K 1
Km =(KE3+K52)+[0 (KE4+K51)]A,;1[ 52]

= (K53 + K52) " (K54 + K151) Kfl'r-l K33 .(4-11)

assuming that Kfl'l is nonsingular. Thus, the singularity of f, is solely a function of
the D.C. gain matrix of excitation systems K 55, and the six diagonal coefficient
matrices: Kfl‘l , K33, K53, K54, Km, and Kcz- In Other words, the singularity of KFD
is a necessary and sufficient condition for the singularity of f ,, provided that K ’7‘] is
nonsingular. Checking the singularity of f, can thus be performed neglecting the
effects of damping constant D , speed-governing—turbine systems, and power system

stabilizers, which tremendously reduces the computational burden.

Note that the characteristics of f, and 4,5, describes the steady state operation
condition of each synchronous machine when its terminal bus is assumed to be the
infinite (swing) bus. The matrix K FD provides an equivalent singularity test of f, and
Afi associated with the flux decay dynamics when K 3', is nonsingular. At the steady
state condition, f, , Afx, and K [:0 (Appendix C) should be nonsingular and have
(asymptotical) stable eigenvalues, except when the synchronous machine is operated at
the steady state angle stability limit and/or the D.C. gain of the excitation system is

"badly tuned". Thus, it is likely that f, is nonsingular.

103

It should be pointed out that the test matrix K FD will be shown to be the same as
the test matrix TF0, (Q = 0), derived in Chapter 5, for the static bifurcation associated
with the flux decay dynamics except K FD neglects the network effects. The test con-
ditions of singularity of [ f, - jQ} for all 0 2 0 were not established in a manner that
test conditions of singularity of (J, - jQ} are obtained in Chapter 5. It is important
to make certain that (f, -j£2} has neither singularity for (2:0, nor for all n > 0
since stability problems could well result in the full model that includes the effects of

the transmission network.

4.4 Structure of Static/Algebraic Jacobian Matrix — J,

In this section, the structure of the static/algebraic Jacobian matrix J, will be
derived based on the assumption that f, is nonsingular as indicated in previous sec-
tion. The structure of the matrix J, will also be shown to be similar to that of the
causality matrix g, (4-7).

The relationship between the power system Jacobian submatrices, f ,, f, , g, , and
g,, has been represented by the block diagram as shown in Fig. 4-1. The

static/algebraic Jacobian matrix is defined as
Jy = 8y " 8x :1 fy (4‘12)

provided that both g, and f, are nonsingular. Based on the properties of f, shown in
previous section, the matrix f, only directly affects the internal bus and terminal bus
variables of each generator, and can be represented by its reduced form Afx as shown
in Fig. 4-2. Likewise, the Jacobian matrices f, and g, composed of diagonal K -
submatrices show the coupling between the internal bus and terminal bus of each gen-
erator. Thus, the second part of the matrix J, , { g, f,"1 f, I, should only show the
effects on the terminal bus variables, and the static/algebraic Jacobian matrix J, would

in turn keep the same structure as that of the causality matrix g,, where only the

104

diagonal block submatrices associated with the active and reactive power balance equa-

tions at the terminal buses are expected to be modified.

From Fig. 4-1 and Fig. 4-2, the relationship between f, and f, can be illustrated
by the steady state block diagram of each machine shown in Fig. 43, where the incre-
mental symbol A in front of each variable is omitted. At steady state condition,
A00 = 0 and only the D.C. gain of the excitation system shows the effects of the com-
plete control systems. Based on Fig. 4-3, the mechanical and flux decay states A8 and

AE,’ can be represented in terms of the terminal bus variables A01 and AV,~ as fol-

 

 

 

 

lows:
APO Kfltl K33 A5
K5751 KA AV", — L [ K54 + K515 Kcr ] [K33 + K55 Kczl AE,’
- Kfl'r KIT? A67 4
+ - [ K34 +1955 Ker] - [K57 "KEE K67] AVT (J33)
' —1
A5 _ Kfltl K312 . APO
A5,; - th“ 4' K55 Kcr 1 [K53 4' K55 Kc2] K37} K, Aan
" K131 K197 A07 4, 3
-L-[KE4+KEEKC1] ’lKar-KEEKC'I] AVT ) ( 1b)
Note that f ,‘1 f, is embedded in the above formulation:
-1
_1 K31 K33
0" f’)’ _ [KE4+KEE Ker] [K53+KEE Kc2]
' K33 K39-
- K K K K K K (4'14)
[154+ 55 C1]-[ 57" 55 c7] .

where, (f;1 f, ), represents that f;1 f, has only effects on the internal bus variables

105

3.5%. no Eflwfin 0.85 8 u 2 85m ,8an m

.4 am

 

 

2.

 

  

 

 

 

 

 

 
  

 

: SMNMVNIEVN—

-..,\

VMQu

 

 

 

 

106

A8 and AE,’, since the rest of block matrices of f ,‘1 f, are zero. Note that each of the
matrices in each of the sub-blocks is diagonal. Based on the assumption that f, and
Kfl‘l are nonsingular, and on the two equalities K31 =KEE KC] and K52 = K35 KC2
defined in Afx (49) and in Km (4-11), and applying the inverse matrix formula pro-
vided in Appendix A, we have the further formulation for (f ,‘1 f, )1 as follows. Let
-1
K31 K32

[K£4+K51][K53+K£2] = 021022

 

011 012]

then

622=([K£3 +K521-[Kr4‘l'K511Kfl'1-1K3'2 )'1
_ -1
- FD

021 = " Kiri [K54 + Ker 1 Kiwi-1

012:- Kfl'r-l K33 Kiri

Grr=K13'1-1[1 —Kfi’2 021]
= 1091-1 + Kiri-1 K172 Kiri [K54 + K5111033-l

Hence, (f;1 f, ), can be written as

-1 611 612] ’Kfltl K37
x fy)1- _021022 -[KE4+K51] -[K57-Krz£ Kcv]
”—1 H12
= _ 0' H22 (4-14a)

 

where,
”21 = Kfl'1'1Kfl‘7 + Kfl'r-l Kfl'z Kiri [K34 4' K51 ] K1914 K193
+ XXI-1K3; Kiri [K157 - K515 Kev]

= meg-1K3; + Kg'1‘1K3', K53 K50 (4-14b)

107

”22 = ' Kit-1311954 4‘ Ker ]Kfl'1’1Kfi'-, " Kiri [K37 ‘ Ker Ker]
= — K53 KEC (4-14C)
Ker: = (K154 + K515 Kcr)K3'r-1KF'7 + (K57 - K55 K07)
Note that K51 = K55 Kcr-

Likewise, the active and reactive power balance equations show the coupling

between the internal bus and terminal bus of each machine as follows (from (4-2)):

A8 [A 1K C 1K J .39,
= o 4'
AE, ABK C3K AVT

F
'APcr] -K}.41 K32

 

 

 

— AQC'I' - K51 K52
_ 3311 Dan AVH

 

where, the main submatrices of g, associated with the internal bus variables A5 and
A5,, Kfifll , Kfa‘lz, K31, and K32, are included. Note that each submatrix in the above
representation is diagonal. Then substituting (4-13b) into (4-15), we have

[arm]: [Am C1K]- m c... )[Aer]

“‘3ch A3K C3K K31 C3x AVr
3m Din A0 AP
11 0

+ +6 4-16

[33H 03”] AVH . 7 K37} KA AV",- ( )

 

 

since the computation of g, f ,‘1 f, shows the first part of (4-16) and its sub-block
matrix associated with the terminal bus variables A01» and AVT, (g, ,‘l f ,)1 has the
form:

-1 _ “K191 K1542 -1
(8x 1 fy)T"' -K31K32 '(fx fy)!

Ki’qr Cu: (417)
K51 Csx

108

where,
Clx =-—K[.41 Klg'r-l KIT-I - (K1541 Kfl'l-l K192 +1092) Kiri KEC
Csx = ‘Kal Kfi‘l-l K35 - (K31 Kfl'l-l Kfl'z 4' K32) Kiri Kec
Note that the rest of block matrices of g, ,‘1 f, are zero. The expression for GT is

omitted without loss of continuity.

It is observed from (4-16) and (4-17) that g, ;1 f, only affects the diagonal

submatrices of g, associated with the terminal bus variables. Thus, substituting (from
(4-2))

AIK =K§1+Al 3C1K =19"? +Cl

A3, =K31+A3 ; C3,, =K§7 + C3
and (4-17) into (4-12) by combining the Jacobian submatrices of g, associated with

the power balance equations at high-side buses and load buses, we have the final form

of the static/algebraic Jacobian matrix J, as follows:

 

 

 

 

PA] 31,, 0 Cf, D”, 0 . ’5”
A2}! 32m 32m. C2" 02111! Dznz Jim
0 32“, BM 0 02”, Du 1;”-

Jy: A3 B3,, 0 C5“ D3,, 0 = 1,97 (4’18)
A411 Bum 34111. C411 D4111! D4111. 1,9”
0 3.”, Ba 0 D4“, Du. 1’01.

where,
C¥K=Kitrql +C1-C1X
C§K = K37 + C3 - C31:

Note that both Cfx and C4,} are still diagonal matrices, and that A1, C1, A3, and C3
are the appropriate diagonal sub-blocks of the load flow Jacobian matrix I” defined in

109

(3-9).

The above shows that J, can be obtained from the causality matrix g, by slightly
modifying the submatrices associated with the (active and reactive) power-angle and
power-voltage diagonal Jacobian matrices at terminal buses. Note that the active and
reactive power-angle Jacobian matrix of g, are A“ = Kffll + A1 and A3,, =
Kg, +A3; but those of J, become A1 and A3 that are the same as those in the load
flow model of the specific case where all the generator terminal buses are represented
as PQ-buses. This is because in each generator, except the swing bus, the power-angle
Jacobian matrices associated with the internal bus angle 5 and the terminal bus angle
0, are opposite in sign. This also implies that the control systems at steady state do
not have direct effects on the terminal bus angle through the coupling coefficient
matrices K H and Kg 1. Moreover, in the active power-voltage Jacobian matrices Cfx
and C m of J, and g, respectively, the difference between them is due to the fact that
K 9‘ at K ‘4 which represents the active power losses in the armature resistance R, and
the reactive power losses in the reactances of synchronous machines. Note that if R,
is neglected, then P5 = P0, = - ,, such that Kfl'l = K1191, Kfi‘z = — Kl‘flz, and K39, =
-Kp‘l,. Hence, C1}! = -Kfl‘-, = K1577, and C’fx = C1 whether the generator has
reached reactive power generation limits and disabled the excitation control or not.
This result implies that if R, is neglected, the control systems at steady state do not
have any direct effects on the active power-voltage coefficient matrix C’fx = C 1.
However, C {5: of J, is not equal to C 3 of J u- , which indicates the excitation control
has efl‘ect on the reactive power generation of the synchronous machine by controlling

the generator terminal voltage.
Thus, the structure of the static/algebraic Jacobian matrix J, is like the load flow
Jacobian matrix Ju- except for C§K, since 0;], = K37 + C3 - C3x. Note that since

- C 3x is a diagonal matrix with positive elements (see Appendix D) that increase

110

almost linearly with K 55 when it is assumed that the load compensators are not used
in the model, the matrix C’3‘K is then also a diagonal matrix with positive elements that
increase almost linearly with K 55 because the D.C. gain of the excitation system K 55
is in general very large compared to the rest of terms in ng. Hence, the load flow
bifurcation is an excellent predictor of the singularity induced static bifurcation
because

(a) there are rows associated with the reactive power balance equations at all
generator terminal buses in J, , but only in J“.- when the reactive power

generation reserves are exhausted;

(b) the reactive power-voltage Jacobian matrix in J, is extremely diagonally
dominant when the reactive power generation limits are not reached and
exciters are not disabled (C§K is a diagonal matrix with very large positive
elements), which is much like adding equations VT = V10 or AVT = 0 for

the generator terminal buses in the load flow model;

(c) the rows of J, and Jut- associated with the reactive power balance equa-
tions at the generator PQ-buses, when the reactive power generation
reserves are exhausted, are identical.

This similarity between the static/algebraic bifurcation test and the load flow bifurca-
tion test will be shown in Section 4.6.

The difference between the static/algebraic Jacobian matrix J, (4-18) and the
causality matrix g, (4-7) is that the four diagonal submatrices associated with the
power balance equations at the terminal buses: A1, C1, A3, and C§K =K§7 +
C3- C3,, of J,; and Am =A1+K,'¥1, Cm = C1+Kfi7, A3; =A3+K§1, and
C3,, = C3 + K37 of g, . When generators do not reach their reactive power genera-
tion limits, J is more diagonally dominant than g, in the reactive power-voltage Jaco-

bian matrix at terminal buses (8QT/3VT) since C35, in J, is a diagonal matrix with

111

very large positive elements proportional to K 35 . When generators exhaust their reac-
tive power generation reserves and become PQ-buses, C§K in J, then becomes C3
which is less dominant than C 3K in g, . Furthermore, the active power-angle
(EFT/807) matrix in J, is A 1, but the matrix in g, is A “g which is more diagonally
dominant. Hence, J, is less diagonally dominant than g, in both EFT/807 and
BQT IBVT matrices when the reactive power generation limits are reached. Thus, when
the reactive power generation limits of all the generators in a region are reached, J,
should generally become singular before g, . If the reactive power generation limits
are ignored, loss of causality (g, singular) may occur before the static/algebraic bifur-
cation (J, singular), since BQT 13V,» is more diagonally dominant in J, than in g, , but
BPT IBGT is more diagonally dominant in g, than in J, . Moreover, the discontinuous
changes will occur in J, causing the static bifurcation, but do not occur in g, which
becomes singular as a continuous function of the imposed stress on the system. This
will be shown in the simulation results of the static/algebraic bifurcation test in Section
4.6.

Testing the singularity of the static/algebraic Jacobian matrix J, for identifying
the singularity induced static bifurcation rather than testing the singularity of the
equivalent system Jacobian matrix J, is recommended since both tests are valid and
should theoretically always become singular at the same point as long as f, and g, are
nonsingular. This proposed static/algebraic bifurcation test matrix is proven to have
similar structure to the causality matrix g, , and is virtually identical to the load flow
Jacobian matrix J”. A set of nonlinear equations is thus proposed with J, that com-
putes the equilibrium point of the power system differential-algebraic model. The
equivalent load flow model would represent generators which have not reached reac-
tive power generation limits by the steady state model of generators and excitation sys-
tems, and by the reactive power balance equations when reactive power generation

limits are reached This equivalent load flow model for the static/algebraic Jacobian

112

matrix J, would precisely determine when a static bifurcation occurs in the
differential-algebraic power system model. Testing for a static bifurcation using J,
does not require inverting g, which is required when J, is used.

The load flow model for the load flow bifurcation test on J”.- and the equivalent
load flow model for the static/algebraic bifurcation test on J, are shown to be virtually
identical for all rows of J“.- and J, if the D.C. gains of the excitation systems are very
high so that the generator terminal voltage can be kept constant, and if the flux decay
equations are replaced by the reactive power balance equations at terminal buses for
those generators exhausting reactive power generation reserves and being modeled as
PQ-buses like in the load flow model. However, they would not be identical if the
equivalent load flow model includes the effects of the generator flux decay dynamics,
excitation system dynamics, and field current limit controller dynamics at steady state
condition when the reactive power generation limits or field current limits are reached.
The reason is that a static bifurcation in flux decay dynamics or in both flux decay
dynamics and mechanical dynamics would be captured by the equivalent load flow
model if steady state representation of flux decay dynamics, excitation system dynam-
ics, and field current limit controller dynamics is used when field current limits are
reached; but would not be captured in the load flow model or in the equivalent load
flow model if the generator and excitation system dynamics is replaced by a reactive

power balance equation (simply called the conventionally equivalent load flow model).

This result is quite important because it indicates that the conventionally
equivalent load flow model can only capture one of the three generic static bifurcations
(in mechanical dynamics, in flux decay dynamics, and in both) that occurs in mechani-
cal dynamics [39,40,57,62]. The static bifurcations that occur in flux decay dynamics
[52,53], and mechanical and flux decay dynamics are most likely to occur when field
current limits are reached and the field current limit controller works through the exci-

tation system to fix field current to the continuous rating value. A modified load flow

113'

model that properly reflects the steady state behavior of the generator flux decay
dynamics, excitation system dynamics, and field current limit controller dynamics
would accurately determine the equilibrium point(s) of the differential-algebraic model,
and experience all of the static bifurcations of the differential-algebraic model.

The comparison of the static/algebraic Jacobian matrix J, , the causality matrix
g,, and the load fiow Jacobian matrix J”, carried out in this dissertation, is for the
case where- the flux decay dynamics and excitation system dynamics are replaced by
the reactive power balance equations when the reactive power generation limits are
reached The use of the reactive power balance equations is justified based on the
approximation of operation action to maintain the units close to their capability curves

when the reactive power generation limits or field current limits are reached.

4.5 Generic Static/Algebraic Bifurcation Test

This section will discuss the subclasses of static/algebraic bifurcation due to the
singularity of equivalent static/algebraic Jacobian matrix J, , which is structurally simi-
lar to that of the load flow Jacobian matrix 111:. Based on the structure of J, shown
in (4-15), basically it seems that there will be three possibilities leading to the singu-

larity of J, that can causes static bifurcation [87].

SA -T: This type of bifurcation is solely due to row dependence associated with
the active and reactive power balance equations at terminal buses (J57
and 1,97).

SA -H: This type of bifurcation is solely due to row dependence associated with
the active and reactive power balance equations at high-side buses 0;”
and 1,9").

SA -L: This type of. bifurcation is solely due to row dependence associated with

the active and reactive power balance equations at load buses (JfL and

114
1,“).

Note that since A and C submatrices of J, (4-15) are diagonal, other types of bifurca-

tion due to the combination from any two of above three are not possible at all.

The SA -H and SA -L types of static/algebraic bifurcation are identical to the
LF-H and [F -L types of load flow bifurcation, respectively. The LF-H and LF —L
bifurcations have been shown to never happen in Section 3.2.2 and Section 3.2.3, and
the SA -H and SA -L bifurcations will never occur based on the same argument. The
SA —T type of bifurcation is similar to U“ -T type of bifurcation, and will also never
occur due to the fact that qub at 1:74 shown in Section 3.2.1, even though the C matrices
associated with SA -T and LF-T types of bifurcation are different. Hence, none of
the above subclasses of static/algebraic bifurcation will happen. The analogies
between the bifurcations in a two—bus model and those of the model considered in this
dissertation are shown to have limited validity. It was shown that voltage bifurcation
due to the row dependence of the Jacobian rows associated with the reactive power
balance equations will not occur, and that angle bifurcation due to the row dependence
of the Jacobian rows associated with the active power balance equations can never
happen. Thus, the generic type of static/algebraic bifurcation is shown to be the one
that has row dependence in all the rows of the static/algebraic Jacobian matrix J, [87].

4.6 Simulation Results

The 9-bus 3-machine power system with the base case data as the one shown in
Fig. 3-1 [26] is adopted to confirm that the following three tests for the static bifurca-
tion in the power system dynamic model are equivalent to each other:

(1) testing the singularity of the static/algebraic Jacobian matrix J, for the

singularity induced static bifurcation;

115

(2) testing the singularity of the complete system Jacobian matrix J, that allows
the detection of all the types of bifurcation except the dynamic bifurcation
(differential algebraic, differential-algebraic, static/algebraic, and static
bifurcation);

(3) testing the singularity of the equivalent dynamic system Jacobian matrix J,
for the static bifurcation due to the static/algebraic or differential bifurca-
tion;

(4) testing the singularity of f, for single machine instability;

(5) testing the singularity of g, for loss of causality.

The purpose of these results is to show that the static/algebraic bifurcation due to the
singularity of J, will be the generic singularity induced static bifurcation that is
equivalent to the static bifurcation due the singularity of J,. and that the differential,
algebraic, differential-algebraic bifurcations and loss of causality do not occur in a
practical sense. Note that if the static/algebraic bifurcation is the generic static bifur-
cation, testing for the singularity of J, is far easier to implement than other tests, since
it is easy to compute from the load flow Jacobian matrix and because it does not

require the inversion of g, as does J,.

It should be noted that these simulation results are only searching for the static
bifurcation even though the theory suggests that both static bifurcation and dynamic
bifurcation be generic. The static and dynamic bifurcations are studied and classified
in the Chapter 5 of this dissertation. Simulation results in Chapter 5 compute both the

various types of static bifurcation and dynamic bifurcation.

Along with the base case shown in Fig. 3-1, the dynamic system data required for
the static/algebraic bifurcation test is the single-axis parameter data of each synchro-
nous machine as listed in Table 4-1 [4].

116

Table 4-1 Static Data of Synchronous Machines

 

 

R, X, x; x, K55
Gen#l: 0.0000 1.7900 0.3000 1.7000 19.000
Gen#2: 0.0000 1.9400 0.2900 1.9100 44.000
Gen#3: 0.0000 1.7200 0.2400 1.6600 270.00

 

 

 

 

 

 

 

 

* Data is on machine base.

* Gen#1: 150.0 MVA; Gen#2: 150.0 MVA; Gen#3: 250.0 MVA .

Note that the D.C. gain K E,- = K 3751 K A K D of the excitation system of each machine
can be obtained from the complete data machine parameter data set (see Table 5-1,

where the dynamic data of the power system is adopted from [4]).

Similar to the load flow bifurcation test used in Section 3.4, the static/algebraic
point of voltage collapse method [53] is used to find the static/algebraic bifurcation by
making bus#8 a fictitious PV-bus. Figure 4-4(a) shows the determinant of f, with the
n0tation Der (F, ), that of g, with the notation Def ( G, ), and that of J, with the
notation Der (J, ). Figure 4-4(b) shows the Q—V curve at the fictitious PV-bus, and
its slope De: ( QV ) (a scalar) which is the corresponding diagonal element of the sen-
sitivity matrix similar to SQV of (3-8) of this Q-V curve. Figure 4-4(c) gives the
Q -V curve profile of the generator buses with reactive power generation high limits,
and Figure 4-4(d) represents the voltage profile of each bus. The voltage angle (in
degree) of each bus, and the active power generation of each generator bus are respec—
tively shown in Fig. 4-4(e) and Fig. 4-4(f), which should help understand the
phenomena of the transmission network when the voltage and the reactive power injec-

tion at the test bus (fictitious PV-bus) are changed. Figure 4-4(g) shows the profile of

MR) & M61!) & Delay)

QKISCLMQV)

117

 

Fig. 440) ~- Det(Fx)/E+02:x; DettGy)/B+16:o ; Det(Jy)/E+16:"

 

' x I .

x x ‘ .

‘ ‘ o o
. I
o O :

T0 °

 

..‘H...£....‘....k...i...i..3...‘....‘.......uu...§........o..............f..n e one... u... . . . a. . ..

3 i o o a :
-.........'..'..... 0003090.3‘3“&..".."‘.. ..
’ ‘ o : .

.

. .
. .
. . . .
. . .
O . _
. . . .
O . .
. .
O _ .
O , .

p .............................................................. ‘ ........................................................................................

. .
o . a
. .
. . .
. .
- r
. .
.
.
' I

 

 

i i i L
.7 0.75 0.8 0.85 0.9 0.95 l

V(FPV-bus: 8)

1.05

 

-2

"8.

Fig. 44(b) -- Qgt8:x ; Det(QV):" ‘

:xxxxsxxxxxxxxxixxxi

booli - no. a ...-o o I o-u ...-...: ... a a on... n u

 

 

.— ...... r'rxmi ....................... . ............................................... ,‘grti 1...! .....

 

 

7 0.75 0.8 0.85 0.9 0.95
V(FPV-bus: 8)

fl

1 .05

0:! (Own)

Vpu

118

 

1.4

1.2 ......u .. .

0.8 r-'
0.6 r'
0.4 ...

0.2 --

 

Fig. 4.4(c) -- Qst (omnibus: 8X: 3%.) : 2’(-) ; lol

' i . 0 .
h-¢-~¢--¢ ctr-b. 0-0 oo- «o- tutmfi--t--¢-¢--¢--«h-fo-O-ofi-‘grevgwamo- 01-0-0 . ...
. r r o
. 0 O
......c.-’-.o--.o.-..-.o-o.---.-.O’-.-.'.-.-.: oooooooooooooo .-‘ ....... .- -0-0
. . . . , .
. . ¢
0
g ‘ O
O 9
...................... ...... . ........
U
, O . .
. I
x O
o v a .a . ‘
X
‘ X
8 x
I .. , . ‘ . ..
X , . I
X , x
" x
x , , . , x
......... ‘ux. ‘ ‘ ‘ t ..

 

 

1.05

0.75 0.8 0.85 0.9 0.95 l
V(FPV-bus: 8)

1.05

 

0.95

0.9

Fig. 4-4(d) ..- Vpu(bus# 4-,5--,6:,7-.,9. ; 3+,2*,lo)

‘00

. ,t oo°é°°°
6922 , ,tttttt

O
o
+

0.8 ,,,. .

0.75-- -

 

 

. a}
u ”
.'.
O A
I
"r
9 g'
, o
[I
'6‘
0

L

 

 

0.7

0.75 0.8 0.85 0.9 0.95 1
V(FPV-bus: 8)

1.05

Angle (Degree)

98‘

119

 

30

 

0 ‘ r V
3 5
0 E
, .4
* :
e E
. I
. I
......... f. .. ,, .41.. —r
+ , ,
9' , r
4 , .
’ 0
. .. 1- , . .1
e . . .
O . . . .
. . ' ' o . '
.‘eeoeeooooeeeoooe
00000000000000000-0-6600066606000 -
‘ -4
‘ x X ‘ x x x k ‘ .5..‘O-‘.-.x.-‘.-£-.‘.’5'}.-‘--§--
‘ X ‘ ‘ x050-‘o ooooooooooooooooooooo
x E}. ----------
........ _,
L 1

Fig. 4-4(e) «- Angle(bus# 4-.5-.6:,7-..8x.9. ; 3+.2‘,lo)

 

 

 

1.7

0.75 0.8 0.85 0.9 0.95 l l .05
V(FPV-bus: 8)

Fig. 4-4(f) --- Pgt(bus# 3+, 2*, lo)

 

1.4-.

1.2

0.8 -

 

*‘.’..‘f.H‘..“**+.;..**;.j+f.‘.*
_.
..
..
........ _‘
...... ...
cocoon.neeoeeeeeeeéeooaeoeeeeoe
°o h ....... '
oéoooooooooeooooeoooogo0,0,0, ‘

 

 

0.7

0.75 0.8 0.85 0.9 0.95 l 1.05
V(FPV-bus: 8)

Vref

120

 

Fig. 44(3) .— Eqp(bus# 3+. 2*, lo)
103 1' V I T I

1.2 r°o,. , ,. . . . .1

O O , .....
O o o
o 0 ° ° ° ° 0 0
1p ..................... .... ... ... ..........E ......... .. ............................................................... d
. o
a ' ° 0
. C ' C
, 0
O . . I Q 0 . . t
e +
0.9 ........... _ e 7 . ,_ 0 . t : ..., , _
. 0 . , 0 o . . ,
e . 9 + o
O f
. ¢ . .' .
03}- ...: ................................. - ......... . .......... . ......... . 1
0 - +
. .
‘ 0
O
0

 

 

 

0. m l l 1 L l
3.7 0.75 0.8 0.85 0.9 0.95 1 1.05
V(FPV-bus: 8)

Fig. 44(h) -- Vref(bus# 3+. 2". lo)
1.15 . . . , fl ,

oooooooooooooOooooooooooooooooo

 

1.1“ ....... . . . . . . ., ...,

00.00.000.000...

1.05 ._ . ,,. ,. , g

, : ++++¢0+0+0
to

I

U
D
f
l

0.95 . -

C . I
C 9 .
’ 2
O :
'. §
0.85 in. ..... .t , ............. -‘
4*
g I
0.8 ......... ,. _,

 

 

 

.7 l l 1 r L
0 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

V(FPV-bus: 8)

121

the internal bus voltage 8,, and Figure 4-4(h) gives the voltage set points V",- of the
excitation systems which control the voltage at terminal buses, that are not provided in
the load flow bifurcation test. Note that the reactive power generation limits (Gen#2:
i 40.00 MVAR; Gen#3: 1:50.00 MVAR) are still used as the constraints whose viola-
tion causes the excitation systems on these generators to be disabled. If the generator
exhausts its reactive power reserves, it is operated as a constant active and reactive

power generation unit.

It is observed from Fig. 4-4(a) that this power system experiences a
static/algebraic bifurcation when 43.11 MVAR reactive power load is added at bus#8
which occurs when the voltage at this bus is about 0.86 pa . At the bifurcation point,
the static/algebraic Jacobian matrix J, becomes singular; while both the matrix f, and
the causality matrix g, are nonsingular. Note that the matrix f, is always nonsingular,
whose determinant starts with 0.0057 at the base case, and graduately increases as the
system is more stressed and before any generator operates in PQ-bus mode. When
Gen#3 becomes a PQ-bus when bus#8 is at 0.92 pa voltage and 35.26 MVAR load,
the determinant of f, jumps to 2.1407, and then increases slowly until Gen#2 becomes
a PQ-bus. When Gen#2 exhausts its reactive power reserve, the determinant of f,
jumps more dramatically to 97.0080 where bus#8 is at 0.86 pa voltage and
43.11 MVAR load. The static/algebraic bifurcation Jacobian matrix J, becomes singu-
lar at this point. The causality matrix g, is nonsingular until the voltage at bus#8 is
down to about 0.73 pa with 14.14 MVAR load far after the static/algebraic bifurcation
point. This confirms that in general the matrix f, is not singular and that g, is more
diagonally dominant than J, if reactive power generation limits are reached. More-
over, this proposed approach for assessing the test for the static bifurcation by testing
the singularity of the static/algebraic J acobian matrix J, rather than the dynamic Jaco-
bian matrix J, provides the advantages that no test for causality condition is necessary.

In addition, this test can be performed by slightly modifying the load flow program,

122

since the structure of J, is similar to that of the load flow Jacobian matrix Jun.

It should be pointed out that comparing Fig. 4-4(a) - Fig. 4-4(f) with Fig. 3-2(a) -
Fig. 3-2(f), the results of the load flow bifurcation test and the static/algebraic bifurca-
tion test are very similar. This is because of the high D.C. gain of the excitation sys-
tem of each machine kept the generator terminal voltage almost constant until the gen-
erator exhausts its reactive power reserve and becomes a PQ-bus. Note that when a
generator operates in a manner of a PQ-bus and the system is getting more stressed, its
internal voltage E, should be controlled by the voltage set point V", of the excitation
system to produce constant power output at its terminal bus, as shown in Fig. 4-4(g)
and Fig. 4-4(h). This type of control reflects the operator action that attempts to keep

the generator near its capability curve.

Chapter 5
Static/Dynamic Bifurcation in Dynamic Model

In Chapter 4, the basic necessary conditions for a power system dynamic model

to experience static and/or dynamic bifurcation were categorized:

f f
(a) The system Jacobian matrix J = [8‘ g, is singular.
x r

(b) The equivalent static/algebraic Jacobian matrix J, = g, - g, f;1 f, is singu-
lar, provided that f, is nonsingular and J, is row independent. The singular-
ity of J, is the generic test for the static/algebraic bifurcation or what has

been called the singularity induced static bifurcation.

(c) The equivalent system Jacobian matrix J, = f, - f, g,‘1 g, has eigenvalues
with zero real parts, provided that g, is nonsingular and J] is row indepen-
dent. When J, has zero eigenvalues, a static bifurcation is said to occur.
This is the generic type of static bifurcation due to the singularity of J, , since
J f has been shown to be row independent and f, has been shown to be non-
singular. This type of static bifurcation is equivalent to the static/algebraic
bifurcation, and is also referred to as a singularity induced static bifurcation.
A dynamic bifurcation occurs when J, has complex eigenvalues with zero

real parts.

The differential, algebraic, and differential-algebraic bifurcations have been
identified that are respectively associated with the row dependence of J ,- , of J, , and of
both J, and J, . These three bifurcations have been shown to be nongeneric and thus
extremely rare. Moreover, it has been argued that both f, and g, should be

123

124

nonsingular in practical power systems, and that f, should have no complex eigen-
values with zero real parts if the control systems of each generator are designed prop-
erly and each generator is operated within design limits. Thus, the singularity induced
bifurcation and dynamic bifurcation are the most probably generic bifurcations of the

power system dynamics.

This chapter will thus assume that the system is causal, and will be devoted to the
discussion of static/dynamic bifurcation [88] that occurs when J, is singular or has
pure imaginary eigenvalues. The static bifurcation could be saddle node bifurcation if
J, has a simple zero eigenvalue, and if the transversality condition of J, at the bifur-
cation point holds. The dynamic bifurcation could be Hopf bifurcation if J, has a pair
of pure imaginary eigenvalues, and if the nondegeneracy condition of J, at the bifurca-
tion point holds. Note that the auxiliary necessary conditions — transversality condi-
tion for saddle-node bifurcation and nondegeneracy condition for Hopf bifurcation —

will not be discussed in this dissertation.

In general, the dynamic angle and voltage stability problems are associated with
the mechanical and flux decay dynamics of the power system equivalent dynamic
model, and are particularly in the modes of low frequency oscillation. This chapter
will study low frequency oscillation and voltage collapse bifurcations by identifying
static and dynamic bifurcations associated with mechanical and flux decay dynamics,
and by developing tests for these bifurcations, based on the fact that the control sys-
tems themselves have stable eigenvalues as proven in Chapter 4. A reduced Laplace
transformed (S = jfl) model is produced that is composed of solely mechanical and
flux decay dynamics, after the elimination of the control system dynamics since they
are not singular for any 0 2 0. A test matrix for bifurcations of mechanical and flux
decay dynamics is then developed. Damping and synchronizing power coefficient
matrices associated with the mechanical dynamics will be further derived, and test con-

ditions on these matrices for identification and prevention of bifurcations of the

125

mechanical dynamics will be established. Test matrices for identification and preven-
tion of flux decay bifurcation will be similarly generated from the reduced model.
These tests not only avoid the intensive computation of eigenvalue analysis, but also
structurally show the effects of transmission network and control systems. Finally, the

simulation on a 9-bus 3-machine power system model confirms the results [88]. -

5.1 Static/Dynamic Bifurcation
The equivalent system Jacobian matrix is defined as
Jx =f. -f, g," 8.
provided that the causality matrix g, is nonsingular. If the equivalent system Jacobian
matrix J, is obtained by directly substituting the corresponding Jacobian submatrices,
f ,, f ,, g,, and g,, from the linearized power system dynamic model (4-2), significant
matrix manipulations would be required. In this section, an equivalent system Jaco-

bian matrix J, will be derived in more physically understandable manner based on the

relationship between these Jacobian submatrices shown in Fig. 4.1.

Under the assumption that the causality condition holds (3, is nonsingular), from
the linearized power system dynamic model (4-2) and Fig. 4-1, one can represent the
variables at terminal buses, A01 and AVT, in terms of mechanical and flux decay

states, A5 and A135,, as follows:

A0 _ AP AX
[AV]=gy1'([AQ:]-gx AX: )

Ara Arv-
3y‘_A,, A,“ '31:: A9, 000

”A A "1
A0 _ '9 ”V . APC Apx
[AV]-LAQ9 AQV. ([AQCJ-[AQXJAXX) (5'13)

 

 

126

 

 

491
A0,,
A9, 110 A9 100000 A9,
[AVT]=[0 If] AV]=[0 0 0 I 0 0] AVT
AV,
Av,
L .
s7 5, A, so, 5,, AP,
= $5 56 A5, + SVP SVQ AQC (S-lb)

Substituting (S-lb) into the active power output equations at internal bus (for the swing
equations), flux decay equations of machines, and load compensators of excitation sys-

tems, we have:
PE =Kfl‘1(A5-A01)+Kfl‘2AE,'+Kfl‘7 Av,
= 5,, A5 + SP2 AE,’ + s". APC + 5,, AQC (52)
r, A13, =AEFD -K,~4(A8-AOT )-KE3 AE,’ +KE7AVT
=AE,.-D — s4 A8-s3 AE,’ + 5,5,. APC +s,, AQC (53)
TD av, = KD AVC (5-4a)
AVC = KC1 (A8 - A01- )KC2 AE,’ + KC7 Av,
= 5,, A6 + s,2 AE,’ + SC, APC + 5,, AQC (5-4b)
where,
Sp1=Kfl'1-Kfl‘IS-7 “”97 55
SP2 =K17'2 " K33 ss +1007 56
S4=K£4-K£4 57”er 55
53=K£3"K£4 Ss'Km S6
Sc1= Kcr - K0157 + Km 55

Scz =Kc2— K0158 4‘ K67 56

127

Note that these S -matrices might not be symmetric if the network is not symmetric
(see Section 2.6). The notation of these S ~matrices are adopted so that they are con-
sistent with those K -matrices [23,26] of single-machine-to-infinite-bus power system
where the synchronous machine is also represented by single-axis model: SP1 = K 1,
Sp, = K2, S3 = K3, S4 = K... S, = K5, and S, = Kg. Recalling that if the load com-
pensator is not employed, then Km = 0, KC; = 0, and Km = I , such that
5c1=55 =K5, and SC2=S6=K5.

It should be noted that the above derivation for the active power equations at the
internal buses, the flux decay equations, and the voltage dewctor and load compensator
equations by aggregating the terminal bus variables (A01 and AVT) through the
transmission network is equivalent to performing the computation of f, g,‘1 g, , the
second part of the equivalent dynamic J acobian matrix J, . Thus, after the elimination
of the network variables, the equivalent dynamic model and the equivalent system

Jacobian matrix J, can be structurally written as follows:

 

 

 

 

 

PTxx AXx - 43h Axe Axe 0 T FAqu
TEEAKE A3, A5,- 0 A53 Ax,
Tea ,,,6 = AG, 0 Ads 0 AXG +3 AU (5-5)
LTSSAJES . far 0 0 Ass. _AXs_

 

Jx =fx -fy gy-l 8x
= X-1 17.; (5'53)

TX = drag [ Txx TEE TOG TSS ] (S'Sb)

S _

 

 

where,

128

 

 

 

 

 

 

. 4 0 Kb Scr K0 Scz
'0 “SP1 “SP2 0 O 0
A§x= 1 0 0 ;A,§~x= AA, 0 0
0 -St ~53 . AB, 0 0
' . 0 0 0 .
’ 0 0 0,, Bx,‘ AV",
3,, 0 3,5,. 3,, APO
B = O BOG 0 0 ; AU= AFC
_ 0 0 0 0 . LAQC .

 

 

The complete expressions of the A submatrices can be found from (4-2). The detailed
form of the B matrix is omitted without loss of continuity. The equivalent power sys-
tem dynamic model can also be shown in Fig. 5-1, which help provide understanding
of the model.

It is observed that the structure of the equivalent dynamic Jacobian matrix J, is
the same as that of the matrix f, and that of the matrix F}, defined in (4-8). The
difference between them is shown by the superscript S and by the sensitivity S -
matrices replacing the coefficient K -matrices. This indicates the physical meaning that
the effects from the transmission network through the terminal buses on the internal
buses are responded by the corresponding dynamics that governs or feeds back the ter-

minal bus variables.

The static and/or dynamic bifurcation may occur when the equivalent system
J acobian matrix J, of (5-5) has eigenvalues with zero real parts. Taking the Laplace
transform, S =jQ where j =‘J—-_1 and Q: 2 rtfo, on (5-5) and on (4-2), it can be
shown that for all 0 2 0, the singularity of the matrix

1,09) =J, - jm ‘ (5-6)

is a test condition for static and dynamic bifurcations, since, by Schur’s formula

(Appendix A), the singularity of J,(i£2) and that of

129

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00¢. 839$ cc
05959. A ' ecu—55m
tw£80>00 em a 883m
13% .BBOA—
m
>
m as + an uu< L
. lr_ll Banzai]
noes—cum + e3

 

 

 

 

 

 

 

 

 

 

 

130

jot 0
JIUQ)=J-[ 0 0] (5-7)

are equivalent if causality holds. In Other words, assuming that causality condition and
transversality/nondegeneracy condition hold, the singularity of J, (j 9) is a necessary
condition for either static/saddle node bifurcation if Q = 0, or dynamic/Hopf bifurca-
tion if (2 > 0. The static and dynamic bifurcations due to the singularity of the matrix
1,09) (5-6) is associated with the mechanical dynamics, the flux decay dynamics, the

control systems dynamics, and their couplings.

5.2 Mechanical and Flux Decay Dynamics

In general, from the experimental results of SSSP [4,21], low frequency oscilla-
tions often link with mechanical and flux decay dynamics, but are rarely associated
with control systems dynamics. This characteristics of the mechanical and flux decay
dynamics can be self-explained by the properties of f, shown in Section 4.3, and by
the structure of the static/dynamic Jacobian matrix J, (5-5). It is observed that J, has
the same structure as f, (4-8), and that the transmission network effects only show on
the mechanical and flux decay dynamics — AjEx, and on the dynamics of voltage
detectors of the excitation systems through the load compensators — A Ex Moreover,
the excitation systems govern the field voltage Em and in turn affect the internal bus
voltage 5,. However, the main diagonal matrix Ace (4-8c) of both f, and J, does
not change, which indicates that the bifurcations or low frequency oscillations would
not be associated with the dynamics of control systems, if the bifurcations or low fre-

quency oscillations are caused by the transmission network effects.

In other words, the static and dynamic bifurcations due to the singularity of
J,(iQ) (5-6) associated with all the dynamic states of synchronous machines could be

further reduced to a Jacobian matrix solely pertaining to the mechanical and flux decay

131

dynamics. In order to obtain the equivalent Jacobian matrix associated with mechani-
cal and flux decay dynamics, the effects of exciter, governor, and stabilizer control
systems are going to be aggregated. As discussed in Section 4.3, the control systems
themselves are usually asymptotically stable, i.e., no eigenvalues with zero real parts.
Thus, the Laplace transformed diagonal block matrix of the control systems (from (4-
89))

 

 

"A5, -jo T5,; 0 Ass l
ACCUQ) = 0 Ace "' 19 Too 0 (5-8)
0 0 Ass - jo T53

should be nonsingular for all 9 2 0, and its inverse is as follows.

 

 

' [A55 -jf2 7551-1 0 A5509) 1
A5509) = 0 [A6, - ja 1061-1 0
L o 0 [Ass - in Tssr‘.

45509) = - [4155 '1"2 7551'] ArslAss - in Tris]-1

Therefore, aggregating control systems from (5-5), one can produce the following
equivalent model solely associated with mechanical and flux decay dynamics where the

input matrix B is omitted without loss of continuity.

Air
1,, in Ax, =(A,~§x -1Ax, Ax, MAC-gun) A,x )AX, (5-9a)
As, .
M j!) Am A00
'1'!) A6 ’ =A§xum A5, (5.91;)
1,, ,1) AE, ,
Afatja) = A§x -AXG [A6, - in To, 1'1 AG, (5-9c)

-AXE [Ab-SUI!) A5, + [A55 -jf2 1551-1 A5,]

= Ar‘hf 4,3900) "Afstm 4,3509)

132

 

 

r 'l
-[D +DGUQ)] ‘SPI "SPZ
= I 0 0
- $500) "' [ S4 ‘1' 55100)] - [ S3 4' 55209)]

where, the following complex matrices are defined

4:560“): Axe [A66 ‘19 Too 1'1 Acx

page) 0 0
= 0 00
0 00

Do (i m= DGr (9) + 1'9 Dar (9)
=Atua [Ass ’19 Too 1'1 A00

: effects of speed-governing-turbine system;

4.550% Axs Arson) Asx

0 00
= 0 00
ssqo)00

5509) = 50(9) + 1'9 5.0“!)
”ANSI ABE ‘19»ng 1'1 [A500 ”4551455 "'J'‘2 733 1'1 ASm]
: effects of power system stabilizer;

AfoQF AXE [ Arr: - in TEE 1'1 Alix

0 0 0
= 0 0 0

0 $5100) 35200)
5510.9)" 851,61) +19 Salim)
=AFE [ABE "J.Q T1513 1'1 A1561
: effects of excitation system. on A5;
$5200)= 552(0) + j!) 5523(9)
= APE [Are '19 Tar; 1’1 Arcz

: effects of excitation system on A5,.

133

Note that AF,- is a submatrix of AXE defined in (4-2), and that A5,, Aecrt and Aacz
are submatrices of A 3, (4-2). Each matrix with (9), that is defined as a function of
Q, is a real matrix. Figure 5-2 shows the equivalent model of mechanical and flux
decay dynamics.

From Schur’s formula (Appendix A), it is observed that the test for the singularity

of the matrix
Jxxom = Afro!» - 1'9 Txx (540)
Txx=diag [M I 1],]

is equivalent to the test for the singularity of J,(jQ) = J, - j 9 1 (5-6), if ACCUQ)
(5-8) is nonsingular for all 0 2 0 of interest. Thus, adding one more condition,
ACCOQ) is nonsingular for all 9 2 0 (which has been proven to be true in Section
4.3), to these two preliminary conditions: causality and transversality/nondegeneracy,
allows one to focus the necessary test condition on India) (5-10) for the static and
dynamic bifurcation associated with the mechanical and flux decay dynamics, rather
than on 1,09) (5-6) or on J,(j£2) (5-7).

In addition, Afio'fz) (5-9) structurally shows the effects of different control sys-
tems and those of transmission network sensitivity matrices. It should also be pointed
out that Ace (j 0) represents the frequency response of the control systems (in general,
the frequency range of interest is from 0.00 Hz to 2.00 Hz for the studies of low fre-
quency oscillations), is only function of the parameters of the control systems that are
usually constant, and is an upper-triangular matrix preserving the structure of each
component of control systems. The inverse of Ace (j 9) (5-8) keeps the structural
sparse property, and needs to be performed only once. This indicates the savings on
the computation and memory if the bifurcation test is performed based on J,“ (i 52)
(5-10). Furthermore, the structure of JxxUQ) can be analyzed either analytically or

computationally to establish the causes of various static bifurcations, dynamic

134

Ag. u .3

ceases 38o as... one 3582 ..o coo: seoazeem 3 mi

 

 

L

 

 

«2.33.43 Iran:
—Il

 

 

 

 

 

 

 

 

985 + A Ab New + 5
—|

 

 

 

 

 

 

135

bifurcations or even complex dynamical phenomena.

5.3 Mechanical Dynamics

In this section and next section, test conditions solely associated with the mechan-
ical dynamics and solely associated with the flux decay dynamics will be established,
respectively. The objective is to more precisely determine to "main" dynamics that
could be vulnerable to experience the static and/or dynamic bifurcations, if the cou-

pling between the mechanical dynamics and flux decay dynamics is "weak".

In this section, damping and synchronizing power coefficient matrices will be
derived. and test condition for the static and dynamic bifurcations strongly associated
with the mechanical dynamics will be developed from these two matrices. This pro-
vides a measure of proximity to static/dynamic bifurcation pertaining to the mechanical
dynamics. To accomplish the above purpose, it is assumed that the main block matrix
of the flux decay dynamics of (5-9)

S309) = 53 + 5520(2) + 19 Td',
=53r(9) +J'953i(9) (5-11)
is nonsingular, or that both real and imaginary components
s3,(o) = 53 + 35,,(12) (51 1a)
835(0) = Tr; + Sta-(n) mm

are nonsingular for all 0 2 0 of interest. These conditions along with the conditions
that Acc (i (2) (5-8) is nonsingular and that causality and transversality/nondegeneracy
hold assure that if the power system (4-2) experiences a bifurcation, it will mainly

occur in the mechanical dynamics.

Thus, from (5-9), aggregating the flux decay states back into mechanical states we

have

136

MjQAO) -[D+DGUQ)] -SP1
j0A5 ]=( 1 0

P

—s
_ 0P2] $5109) [-Ss(jfl) —(S4+Sgl(jfl))]) [13%)]

L

 

'- 100m - 1600) A,
= l . I 0 J A5] (5'12)
K4012) = D + 0,00) + 0,300) (5-12a)
’90 £2) = SP1 + 135500) (5-12b)

where,
0,300) = - 5,, [ 53 + 5,200) + j0 7,, r1 5,00)
= DES, (0) + 10 D55,- (0)
0,500) = - 5,, [ 53 + 55200) + 10 1;, 1'1 [5. + 55100)]
= DEE, (0) + ,0 DEE,- (0)
Using the following properties of the rotor angle and rotor speed
A0300) = )0 A500) (5-13a)
- 02 A500) = ,0 A0100) (5-13b)

we define the following damping and synchronizing power coefficient matrices in a
manner similar to that of Concordia and deMello [23] for a single-machine-to-infinite—

bus system.

[‘2 6‘0“]

 

-Kmm) -K,,,(0)] [Am]

I 0 A5 (5'14)

Km (0) = D + D6, (0) + 05,, (0) + D55,- (0) (5-l4a)

Ksync (Q) = SP1 + DEEr (Q) - 92 1 Dot (9) 4" Dram) ] (5-14b)

137

It is observed that excitation system, power system stabilizer, and speed-governing-
turbine system have effects on both damping and synchronizing power coefficient
matrices K ,,,, (Q) and K ,,,, ((2), which are functions of the six sensitivity matrices:
Sp], SP2, S3, S4, SCI, and S62. If load compensators are not employed, SCI and Scz
will respectively become SS and S5 in (5-1), since Kc1= 0 , Kc, = 0, and Km = 1.
Note that for all (2 2 0, the singularity of (5-14) is equivalent to that of
0 '= - j0 Km(0) - K,,,(0) + 02 M
= — jfl Kdamp (Q) - Kmm) (5-l4c)
Km (0) = Km, (0) - 02 M ’ (5-14d)
It can be concluded that the necessary condition for static and dynamic bifurcations
associated with mechanical dynamics is that
(l) for static bifurcation: Q = 0,
Km“) = 0) is singular;
(2) for dynamic bifurcation: (I > 0,
both Km (0) and K, (0) - ,, (0) - 02 M are singular;

when
(a) causality is assumed,
(b) transversality/nondegeneracy is assumed,
(c) ACCUQ) is assumed nonsingular, and

((1) S300) = S3 + Sn, (0) + j (1 [ Ti, + 852,-(9) ] is assumed nonsingular.

It should be noted, for Q = 0 , that the imaginary component of each matrix with

subscript i is equal to zero, such that
1),, (0 = 0) = Aw A55 A6, = K, (5-15a)

$3, (0 = 0) = A"; A5,} [A5, - A5, A531 A5,, 1 = 0 (5-15b)

138

5121'“2 = 0) = An: Air! AEC1= K515 Scr (5-156)

ng,(fl = 0) = Ans Alt-151 Aecz = K55 Scz (5-15d)
D55, (0 = 0) = — 5,2 is, + 5,2,(0 = 0) 1'1 ss,(0 = 0) = 0 (5-15e)

0,5,(0 = 0) = - s...2 [ .33 + 5,2,(0 = 0) 1'1 [s.+s,,,(0 = 0) 1

=-5102[53+Kr.‘rasczl’l [54+KEE Scr] (5-150
K55 = Ks? KA Kb (545%)
and that
K4,, (0 = 0) = D + KG (5-16a)
Km, (0 = 0) = 5,.1 + D55, (0 = 0) (S-léb)

where, K6 is a diagonal matrix composed of the D.C. gains of the speed-governing-
turbine systems (see Fig. 2-8) [86]; and K 55 is the D.C. gain diagonal matrix of the
excitation systems (see Fig. 2-5) [85]. The power system stabilizers do not show any
effects on damping and synchronizing power coefficient matrices when (2 = 0
(SS, (52 = 0) = 0 is the same as the Os shown in Afx (4-9)), due to their D.C. washout
transfer functions (see Fig. 2—7) [85]. Since D > 0 and K6 > 0 (positive definite) such
that Km ((2 = 0) > 0, the stability of the mechanical dynamics at Q = 0 will fully
depend on K ,,,, (Q = 0) which in turn depends on the D.C. gains of excitation systems
K 55 and the transmission network effects through the six sensitivity matrices: Sp],
S”, S 3, S 4, Set: and Scz- It is obvious that from the Lyapunov stability theory, the
sufficient condition for this system being locally stable is that both K ,,,, (£2 = 0) and
K ,,,, (Q = 0) are positive definite [22-35].

Moreover, testing the singularity of K ,,,, (Q = 0) leading to static bifurcation
associated with mechanical dynamics is equivalent to testing the singularity of the

static/algebraic Jacobian matrix J, discussed in Chapter 4. However, for K ,,,, (Q = 0),

139

inverse of causality matrix g, is needed; for J, , there is no further information about
participation dynamics or states provided. Note also that the dependence of
K ,,,, (Q = 0) on the the D.C. gains of excitation systems and on network sensitivity

matrices is similar to the dependence of J, on them shown in Section 4.4.

These tests for static and dynamic bifurcations associated with the mechanical
dynamics are the first time such tests have been derived for a multimachine power sys-
tem model, although such tests for static bifurcations have existed for single-machine-
to-infinite-bus models [23]. These tests clearly show the roles of various control sys-
tems and the effects of transmission network on the development of static and dynamic
mechanical stability problems. These results open up the possibility of analytically
and computationally investigating the generic causes of such stability problems in a
manner heretofore impossible. In addition, these test matrices allow sensitivities to be
analytically derived or to be computationally calculated to help understand the specific
cause of the stability problem of the mechanical dynamics and the most appropriate
remedial actions. These results may be extremely useful and further research is neces-

sary to investigate their usefulness.

5.4 Flux Decay Dynamics

In this section, a necessary condition is developed for the static and dynamic
bifurcations associated with flux decay dynamics. This test condition provides an
identification and an prevention measure of proximity to static/dynamic bifurcation
strongly pertaining to the mechanical dynamics, likewise the one for the mechanical
dynamics.

From (5-9), if for all (2 2 0 of interest, the main block matrix of the mechanical

dynamics

-[D +0009” '591
A5109): I 0

 

j0M 0
0 )0]

140

—[D +DGUQ)]-jQM —s,.,

is nonsingular, which is equivalent to require the matrix
SMUQ) = Spl +19 10 + 0009) +111 M l
= 5143(9) + 10 Slur-(Q) (5-17b)
be nonsingular, or both real and imaginary components
5mm) = SP1 - 02 t M + Bra-(fl) 1 (5-l7c)
SM.- (9) = D + Dorm) (5-l7d)

be nonsingular, we can aggregate the mechanical states back into the flux decay states

in order to investigate the equivalent flux decay dynamics:

7;, ,0 AE,’ = — 1,000) AE,’ (5123)

-S
TFDUQ)=S3+ng(jQ)-[Ss(jfl) S4+SEIUQ)]AMUa)-l[ 0P2]

= 53 + 3,200) - [ ,0 5,00) + 54 + 5,100) 15,5100)»,
= TFDr (a) + f 9 TI-‘Dt' (9)
For all 9 2 0, the singularity of (5-18) is equivalent to that of
0 = 10 1.}, + 1,000)
= TF0.- (Q) + i Q Tm: (9) (5-138)
Tm, (0) = 1,, + rm,- (0) (5-18b)

Thus, the necessary condition for static and dynamic bifurcations associated with flux
decay dynamics is that
(1) for static bifurcation: fl = 0,

Tm, (Q = 0) is singular;

141

(2) for dynamic bifurcation: 9 > 0,
both T”), (0) and T”), (0) = 7,, + T”),- (0) are singular,

when
(a) causality is assumed,
(b) transversality/nondegeneracy is assumed,
(c) ACCUQ) is assumed nonsingular, and
(e) 5,00) = 5,1- 02 [M + D,,(0)1+ 1011) + 1),,(0) 1 is nonsingular.
Note that for Q = 0, using the facts shown in (5-15) and SM, (9 = 0) = Sp], we have
7,0,.(0 = 0) '= [ 53 + 5,2,(0 = 0) 1 - 1 54 + 5,1,(0 = 0)1s,;,1(0 = 0) 5,.2

= [ 53 +1925 5C2 1 " [ 54+Krf£ Scr 151711 SP2 (5'19)

This test matrix TF0, (Q = 0) for static bifurcation in flux decay dynamics shows
the dependence on the the D.C. gains of excitation systems K E,- and on the transmis-
sion network sensitivity matrices: Sp], SP2, S3, S4, SCI, and Sc23 similarly to the syn-
chronizing coefficient matrix K ,,,, (Q = 0) (5-16), and to the equivalent static/algebraic
Jacobian matrix J, (Chapter 4). In addition, TFD, (Q = 0) has another similar charac-
teristic to Kayne (Q = 0) about the inverse of the causality matrix g, . Testing the
singularity of Tm, (Q = 0) leading to the static bifurcation associated with flux decay
dynamics is also equivalent to testing the singularity of the static/algebraic Jacobian
matrix J, . However, for T”), (Q = 0), inverse of causality matrix g, is needed; for

J, , there is no further information about participation dynamics or states provided.

It should be pointed out that the test matrix Tm, (Q = 0) is consistent with the
one previously derived in [52,53] for the flux decay dynamics of a power system
without both power system stabilizers and speed-goveming-turbine systems, but using
a very different procedure. The static bifurcation in flux decay dynamics due to the

singularity of T“), (Q = 0) has also been experimentally shown to cause voltage

142

collapse, especially when field current limits are reached and then the normal excita-
tion controls are disabled [52,53].

The Hopf bifurcation of flux decay dynamics has been shown to occur when
mechanical dynamics, speed-goveming~turbine dynamics, and power system stabilizer
dynamics are ignored [42,68]. Whereas, the test condition on TF0, (Q) and TFD;(Q)
for (2 > 0 can test for this Hopf bifurcation, including all the effects of mechanical and

control systems dynamics.

The tests for the static and dynamic bifurcations of the generator flux decay
dynamics are the first tests of their kind ever developed for multimachine power sys-
tems, although such problems have been studied in small power system models with
simple excitation controls {39-42,45,52,53,64-73,78,79]. These tests clearly show the
effects of various control systems and transmission network on the development of
static and dynamic stability problems associated with flux decay dynamics. These
results probably make the analytically and computationally investigating the generic
causes of such stability problems possible. Moreover, these test matrices provide the
significance of the sensitivity matrices of the control systems and transmission net-
work, which allows further analytical development or computationally calculation on
these sensitivity matrices to help understand the cause of flux decay dynamic stability

problem and the most appropriateremedial measures.

5.5 Simulation Results

The 9-bus 3-machine power system shown in Fig. 5-3 [4] is adopted to confirm
the above bifurcation tests associated with mechanical and flux decay dynamics. Fig-
ure 5-3 also shows the operating condition of this power system. This power system
is very similar to the ones used in the previous two chapters. The difference between

these two systems is that the system of Fig. 5-3 has less transmission line impedance

143

100.00

 

 

 

 

 

 

 

 

 

 

 

 

18.4 KV 230 KV (33.00) 230 KV 13.8 KV
'0. 25 jO. 586
Gen” ’26 0.0085+j0.0720 0.0ll9+j0.1008 % 683“”
1 .00 B/2=j0.0745 B/2=jo.1045 85.00
(6.65) __ {-10.86)
Bus#3 Bu .#7 Bus#8 Bu s#9 Bu s#2
0.9998 1.0035c 0.9947 1.0142 1.0067
(11.85) (4 .§08) c (0.94) § 3 (2.21) (5.01)
"" CIR "1 b
2 "1 $2. a
.9, .

é a i a
o an o. an

O O
Bus#S Bus#6
12500 <———— 0.9824 1.0011———> 90.00
(50 00) (4 .01) (-370) (30.00)
C O
Q 53 g 8
. OO . 8
.9. °- 9. o.
8 E é "g
5: an 5. an
O O
230 KV 1,3381%
(-2 24)
vv j0.0576
16.5 KV Bus#l
1.0400
71.88
(39.82)0 (000)
Gen#1

Fig. 5-3 9-bus 3-machine Power System Diagram

144

and has more line charging susceptance, so that the generators Gen#2 and Gen#3
operate at lower terminal voltage and provide less reactive power generation. The
dynamic data of the synchronous machines is listed in Table 5-1. Table 5-2 shows the
eigenvalue analysis [4] for this system data which is provided for reference. These
results will be compared with the bifurcation test results from the proposed approach.
It should be nOted that the causality condition of this system operation condition holds.
It is observed, from Table 5-2, that the system has two pairs of complex eigenvalues
with very small real parts — one pair associated with mechanical dynamics (Mode#1),
and the other associated with both mechanical dynamics and flux decay dynamics
(Mode#3). Nore that the mechanical and flux decay dynamics associated with these
two pairs of complex eigenvalue are also coupled with other states (control systems
dynamics).

Figure 5-4(a) and Figure 5-4(b) respectively show the determinant of Km ((2)
and that of Kmm) = Kmm) - 92 M where o = 21tfo for f0 from 0.0 Hz to
2.0 Hz, which is in general the range of low frequency oscillation. It is observed that
these two determinants are simultaneously approaching zero in these two frequency
ranges: 0.0 —> 0.03 HZ and 1.1 -) 1.5 Hz, which are consistent with the oscillation
frequencies associated with mechanical dynamics as obtained from the eigenvalue
analysis shown in Table 5-2. Figure 5-4(c) and Figure 5-4(d) more precisely show
both determinants in the above two ranges, respectively. It is observed from Fig. 5-
4(c) that both the determinants of Km (52) and Km (9) are zero at about 0.01 Hz.
These results indicate that the system is close to experiencing low frequency (interarea)
oscillation in the mode of 0.01 Hz associated with mohanical dynamics. Compared
with the eigenvalue analysis, this proposed approach clearly points out that there is
only one low fiequency oscillation mode at about 0.01 Hz in the mechanical dynam-
ics. Some other (asymptotically) stable modes in the same frequency range are also

associated with the mechanical dynamics, Mode#9 (0.05 Hz) and Mode#12 (0.007 Hz)

Table 5-1 Dynamic Data of Synchronous Machines

145

 

 

 

 

 

 

 

 

 

 

 

Gen: H D x, qu x; 1,,
...... 'l
#1 4.00 1.00 1.79 1.70 0.30 5.70
#2 3.80 1.00 1.94 1.91 0.29 7.30
#3 3.00 0.00 1.72 1.66 024 7.10
—r 4 —
Exciter: K, 7', [(35 r; K,- r,
#1 19.0 0.12 1.00 0.65 0.05 2.00
#2 44.0 0.10 1.00 0.30 0.075 2.00
#3 270.0 1.20 0.00 0.00 0.00 0.00
Exciter: 1, TC K0 r0 RC xc
Wm
#1 0.00 0.00 1.00 0.06 0.00 0.00
#2 0.00 0.00 1.00 0.00 0.00 0.00
#3 2.00 1.00 1.00 0.05 0.00 0.00

 

 

 

 

* Data is on machine base, and R, = 0.

"' Gen“: 150.0 MVA; Gen#2: 150 MVA; Gen#3: 250 MVA .

Table 5-2 Eigenvalue Analysis for 9—bus System

146

 

 

 

 

 

 

fie-l Eigenvalue Frequency Major States“
......
1 - 0.0219 : j 0.0483 0.0077 51 , 5, , 5,
3 - 0.0038 i j 3.9144 0.6230 to, , to, , to, , 5;, , VA, , P55,
5 - 0.1714 1: j 7.6557 1.2184 to, , to, , 0), , P55,
7 -0.263lij 9.1871 1.4622 (02,82,003 ,5, ,P55,
9 - 0.3946 :1: j 0.3210 0.0511 51 , 5, , 5, .54, , V”
11 - 0.5206 54’, .11,-1
12 - 0.9983 :1 0.0448 0.0071 54', , v,l , (o3 , 5, , v,, , P55,
14 - 1.2839 :1: j 1.2844 0.2044 54', , Em, , (1)3 , P55,
16 - 6.3489 :1: j 6.1678 0.9816 VA, , (5,,,
18 - 8.4695 ' 5;, , VD, .14,1
19 - 16.429 v,,
20 - 20.539 VD,
21 - 44.405 P55,
22 - 55.192 P55,

 

* due to high participating factor and high magnitude of eigenvector

 

147

 

Fig. S-4(a) : Kdamp ‘

 

 

 

 

 

 

_ _ . . q 1 2
-3
.l
H 6
t 1 ................... .. .
V 1
- g. .......... .4.
. l
......................... -2.
.1
. .8
r 0
, H ‘0
fl t H .x x x. 40
‘ IA x .IAIA .
Xxlxx
llxx
A”). . 1 2.
m , 0
All!” m . .
. 1 x rm: H ..
. h . I... .
n n n . ll“
. .p .r _ . ..- 0
m“ MW m... m w m w 0 mm.
3 1.- 2 2 1 I .

Frequency --- Hz

x105

 

Fig. 5-4(b) : Ksm

 

 

1.4 1.6 1.8

1.2

0.4 0.6 0.8

0.2

 

 

 

-10?”
-12

53—

Frequency -- Hz

Kdarnp&Ksm

Kdamp& Ksm

148

 

,zm .. ..

 

Fig. 54(0) - "x":'l(damp ; ”o”: Ksm

Y Y T '. r

 

 

-1200
0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Frequency - Hz

 

0.15

0., ;

0.05

-005 1.

-0.11 -

-0. 15

0.2

-0.25 -

Fig. 5-4(d) m "x”: Kdamp ; "o": Ksm/E4-05

r 1' fl

0 xxixxxxaxxasll
i 0 -1

 

 

 

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Frequency «- Hz

149

from the eigenvalue analysis results in Table 5.2. However, the determinants of
Kdamp (Q) and K ,,, ((2) do not approach zero at these frequencies, since these modes
are not close to or on the fa axis and thus not close to causing bifurcations. The
eigenvalue analysis shows two other modes (Mode#5 and Mode#7) are associated with
mechanical dynamics. From Fig. 54(5) (16mm) is rescaled by 10*), these two
modes at 1.22 Hz and 1.46 ”2 have effects on the determinants of Km (9) and
K ,,,, (0). Since these two modes are asymptotically stable modes, the determinants of
Km (9) and K ,, (9) do not approach zero simultaneously. When one of the deter-
minants is zero, the other is above or below zero, and the determinant of K,” ((1) is on
the order of 105. Note that from the eigenvalue analysis, Mode#l4 (0.20 Hz) and
Mode#16 (0.98 Hz) are also asymptotically stable modes associated with flux decay
dynamics. Note also that at these two frequencies, the determinants of Km (0) and
Km (0) do not approach zero or even change significantly since they are not coupled

to the mechanical dynamics.

It should be pointed out that from Fig. 5-4(a) and Fig. 5-4(b) both the deter-
minants of Kdamp (Q) and K ,,,, (0) have sudden changes and have sign changes in the
frequency range of 0.55 —> 0.62 Hz. This indicates that there exists one eigenvalue
with small real part and with imaginary part whose frequency is in the the frequency
range or close to these two boundary frequencies associated with mechanical dynamics
and other dynamics or solely associated with other dynamics. From the eigenvalue
analysis, Mode#3 (0.62 Hz) is associated with mechanical, flux decay, and other (con-
trol systems) dynamics. This mode will be shown to be a mode close to
dynamic/Hopf bifurcation (low frequency oscillation) mostly pertaining to flux decay
dynamics in Fig. 5-5.

Figure 5-5(a) and Figure 5-5(b) respectively show the determinant of T”), (Q)

and that of TH), (Q) = T" + firm-(Q). Both determinants are close to zero in the

der

150

 

25m 1 f Y

 

~ . , . o .
- > - » . . e .
r . ‘ . , o o .
- . . _ . o .
n . . . - u .
. - - . _ , o .
. . . . . _ . o‘ .
. . . _ . n .
.p..o..occ.--t..o-..... c...-....o...............-o-.4.... ... ......cu....-...-...co'..o.......................-on.-un....-oc-....--.oo--c-v.....u.....c-..........~..1
. . . . .

 

 

 

L l 1 L

-500 L i
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

 

Frequency - Hz

Fig. 560)) : der

 

160 T, Y ,
140 ..., ......... .. ................ x ................ .' ............... ; ................ ... .............. J

120 ,. ................ . ................ ................ 1 ................ 1 ................ g ................ , .............. ..

A
W

 

 

 

! .
i

 

 

 

 

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

151

 

fig. 5-5(c) --. "x”: Trait/em ; "6": derIE+02

g l l l l T.

 

1.5

dex & der

 

c o O
J; 4 41

 

 

 

-0.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Frequency - Hz

 

10

Fig. 5-5(d) - "x": dex; "o": der

v ' f 1 l

 

 

dex St deI

 

 

 

 

 

 

 

 

152

frequency range 0.05 -) 2.00 Hz. Figure 5-5(c) and Figure 5-5(d) more precisely
show how both determinants are approaching zero simultaneously. It is observed that
there is a mode about 0.62 Hz of a pair of complex eigenvalue with very small real
part associated with the flux decay dynamics. This proposed approach precisely points
out that the system experiences the 0.62 ”2 mode of low frequency oscillation, or is
close to experiencing dynamic/Hopf bifurcation of flux decay dynamics at this fre-
quency. The eigenvalue analysis indicates that this mode affects both mechanical and
electrical states of the generator. But these proposed tests for the dynamic bifurcation
of flux decay dynamics in Fig. 5-5(c) and Fig. 5-5(d), and the dynamic bifurcation of
mechanical dynamics in Fig. 5-4(a) and Fig. 5-4(b) clearly identify this mode as being
associated with the dynamic bifurcation of flux decay dynamics. These tests are thus

more precise in establishing the cause of bifurcation mode than eigenvalue analysis.

It can be observed from the eigenvalue analysis that Mode#9 (0.05 Hz) and
Mode#12 (0.007 H2) are asymptotically stable modes associated with mechanical and
flux decay dynamics. The tests for the flux decay dynamic bifurcation in Fig. 5-5(c)
and the mechanical dynamic bifurcation in Fig. 5-4(c) show that these two modes are
unrelated to the dynamic bifurcation in either mechanical dynamics or flux decay
dynamics. Note also that two sudden changes shown in Fig. 5-5(d) at 1.20 Hz and
1.40 Hz reciprocally illustrate that there are two modes around these two frequency
associated with other dynamics, which has been shown from Fig. 5-4 to be the asymp-

totically stable modes in terms of mechanical dynamics.

Chapter 6
Conclusion and Discussion

The principal contribution of this dissertation is the systematic categorization of
the stability problems and the establishment of necessary test conditions for different
types bifurcations leading to voltage collapse and low frequency oscillation in mul-
timachine power systems, based on a differential-algebraic power system model.
Bifurcations in a load flow model are also classified. since the load flow model is an
approximation to the steady state behavior in the differential-algebraic model. Both
the linearized load flow model and the linearized differential-algebraic power system
model, and their Jacobian matrices are systematically and structurally formulated in
this dissertation. This dissertation concludes that the generic types of bifurcation in
the power system model can be

VAB:Voltage/Anglc Bifurcation in a Load Flow Model,

SAB: Static/Algebraic Bifurcation in a Differential-Algebraic Model,

SDB: Static/Dynamic Bifurcation in a Differential-Algebraic Model.

This dissertation establishes a test condition for each kind of bifurcation identified
above. Singularity of the full load flow Jacobian matrix J“: is the generic form for
the voltage/angle (load flow) bifurcation test. Singularity of the equivalent
static/algebraic Jacobian matrix J’ is the generic form for the static/algebraic bifurca-
tion test. The load flow Jacobian matrix Jut- is shown to be similar to the
static/algebraic Jacobian matrix J, , and to have similar bifurcation properties if the
generator and excitation system model is replaced by the reactive power balance equa-

tion when the reactive power generation limit or the field current limit is reached.

153

154

The replacement of the generator and excitation system dynamics by the reactive
power balance equation is an approximation to the insertion of the field current limit
controller. If the field current limit controller is inserted into the model, then the
static/algebraic Jacobian matrix Jy can experience static bifurcation in flux decay
dynamics, in mechanical dynamics, and in flux decay and mechanical dynamics as can
be observed from the results of Chapter 5. However, if the generator and excitation
system dynamics is replaced by the reactive power balance equation to reflect the
operator actions to hold the generator near its capability curve, then the static/algebraic
bifurcation can only occur in mechanical dynamics. It is possible to conclude that
bifurcations in the equivalent load flow model only approximate the static bifurcations
in mechanical dynamics but not in flux decay dynamics or not in mechanical and flux
decay dynamics. This equivalent load flow model certainly can not indicate the

occurrence of the dynamic bifurcations.

Singularity of the reduced equivalent dynamic system Jacobian matrix Afxo'fl)
solely associated with mechanical and flux decay dynamics at any Q 2 0 is the generic
form for the static/dynamic bifurcation test at that Q. The static/algebraic bifurcation
and the static bifurcation are equivalent to each other, and are generic types of singu-
larity induced static bifurcation. Although loss of causality (singularity of gy) and loss
of stability of single machine dynamics (singularity of f J,) have not been proven to be

non-generic stability problems, arguments have been made that suggest they be highly

improbable.

A second contribution of this dissertation is the identification of three different
static bifurcations and three different dynamic bifurcations. A static bifurcation can
occur in mechanical dynamics, can occur in flux decay dynamics, and can occur in
both mechanical and flux decay dynamics if they are "strongly" coupled. Specific test
matrices for identification when these specific bifurcations have occurred are derived

which may indicate why, where, and how each of these three static bifurcations occurs.

155

An equivalent test, - singularity of the static/algebraic Jacobian matrix, for the static
bifurcation is also established, that has similar structure to the load flow Jacobian
matrix so that this test can be performed by slightly modifying the load flow program.
A dynamic bifurcation can occur in mechanical dynamics, can occur in flux decay
dynamics, and can occur in both mechanical and flux decay dynamics if they are
"strongly" coupled. Specific test matrices for identification when these specific bifur-
cations have occurred are developed which may indicate why, where, and how each of
these three dynamic bifurcations occurs. A static and a dynamic bifurcation can occur
in b0th mechanical and flux decay dynamics. An algorithm for testing the
static/dynamic bifurcation in mechanical and flux decay dynamics is proposed that
allows one to classify bifurcations into dynamic or static and whether they affect

mechanical, flux decay, or both mechanical and flux decay dynamics.

6.1 Main Contributions of This Dissertation

(A) Power System Model

Both the load flow model and the power system dynamic model are used and are
systematically formulated in this dissertation. The load flow model describes the
power system steady state operation condition of the transmission network. The load
flow model is a set of algebraic equations composed of the active and reactive power
balance equations of the transmission network excluding the reactive power balance
equations at generator terminal PV-buses. The power system dynamic model is
characterized by a set of differential-algebraic equations (or differential-algebraic
model). The set of algebraic equations describes the active and reactive power balance
equations of the transmission network including all generator terminal buses. The set

of differential equations can include

156

(l) synchronous machine mechanical and flux decay dynamics, and
(2) control systems dynamics of machines:
(a) excitation systems with load compensators,
(b) power system stabilizers, and
(c) speed-governing-turbine systems.
Power system stabilizers and specd-governing-turbine systems have not been used in
any of the previous studies of bifurcations in a power system model. Furthermore, the

excitation system model is often quite simple and the load compensator is omitted in

most of the previous studies of bifurcations in a power system model.

Note that the load demand model adopted in this dissertation can be voltage

dependent, including any combination of '
(1) constant power load,
(2) constant impedance load,
(3) constant current load, and
(4) any other voltage dependent load.

This detailed modeling of load demand is necessary to study voltage collapse and low
frequency oscillation.

In addition, the model of a general branch between two buses is also derived,
which can be any combination of

(1) regular transmission line,

(2) transformer, and

(3) phase shifter.
The general branch model is used since tap changing transformers, phase shifters, and

switchable shunt capacitors affect voltage collapse and low frequency oscillation.

These devices have often been ignored in the studies of voltage collapse, low

 

157

frequency oscillation, and bifurcations in power systems.

Both the linearized load flow model and the linearized power system model, and
their Jacobian matrices are systematically and structurally formulated in this disserta-
tion. The matrix form of the linearized power system model shows the structure of the
J acobian matrix and the Jacobian submatrices show the coupling between components
of the power system. The properties of these Jacobian matrices are also clearly dis-
cussed and organized This structural representation of power system Jacobian matrix
provides valuable evidence for classifying the various bifurcation and stability prob-

lems, and establishing those generic bifurcations in the power system model:
(1) Voltage/Angle Bifurcation in a Load Flow Model,
(2) Static/Algebraic Bifurcation in a Differential-Algebraic Model,
(3) Static/Dynamic Biftucation in a Differential-Algebraic Model.
It has also been shown that
(1) single-machine stability generally holds,
(2) control systems stability holds, and
(3) causality condition is generally maintained.

Having provided a brief summary of contributions above, a more complete set of

contributions follows.

(B) Load Flow Bifurcation

The Load Flow Bifurcation occurs when the load flow Jacobian matrix Jut- is
singular. The voltage, angle, and voltage/angle bifurcations have been shown to be the
three types of bifurcations that could occur in a power system model. The voltage
bifurcation and angle bifurcation are shown to be non-generic bifurcations which can

only occur when a decoupled load flow model is used. The voltage bifurcation occurs

158

when only the load flow Jacobian rows associated with the reactive power balance
equations are row dependent and the load flow model is decoupled. The angle bifurca-
tion occurs when only the load flow Jacobian rows associated with the active power
balance equations are row dependent and the load flow model is decoupled. The angle
bifurcation occurs due to the active power transfer limits across some boundary or
interface. The voltage bifurcation occurs due to the reactive power transfer limits
across some boundary or interface. Thus, the voltage/angle bifurcation, which occurs
when the load flow Jacobian rows associated with the active and reactive power bal-
ance equations become row dependent, is due to both the active and reactive power
transfer limits across some boundary or interface. This theoretical result corresponds
with observation and with design and operating practice that constrains power transfer
and loading, and constrains the lower reactive margin from a Q—V curve that assures
reactive power transfer capability. It is also shown that bifurcations can not occur due
to the row dependence of the load flow Jacobian rows associated with the active and
reactive power balance equations at a single bus. Hence, the analogies between the
bifurcations in a two bus model to the bifurcations in a large power system load flow
model have limited validity. Finally, it is shown that the bifurcations due to the row
dependence of subsets of the load flow Jacobian rows are nongeneric bifurcations.
This result indicates that bifurcations generally affect a very large region, if not the
entire system, when under voltage relaying, load shedding, and control actions do not

prevent the bifurcations from occurring or from spreading.

The simulation for the load flow bifurcation test is performed on a 9-bus 3-
machine power system by applying the Q -V curve point of voltage collapse method.
The simulation results for the load flow bifurcation illustrate the important role of
maintaining the reactive power generation reserves at PV—buses in preventing voltage
collapse. Whenever a generator PV-bus exhausts its reactive power generation reserve

and becomes a PQ-bus, the determinant of load flow Jacobian matrix decreases

159

discontinuously toward zero and then load flow bifurcation occurs resulting in voltage
collapse or loss of steady state stability. The more generators at which reactive power
generation reserves are exhausted. the closer the system comes to experiencing the

load flow bifurcation.

(C) Classification of Stability Problems in a Differential-Algebraic Model

Eight distinct types of stability problems are identified and necessary conditions
for each of these stability problems are given based on the Jacobian matrix of the

differential-algebraic power system model:

If - fx fy]
It]‘[8x st “’1’

The differential, algebraic, and differential-algebraic bifurcations, that could occur due

 

to the row dependence of J,. of ’5' and of both J, and 18 respectively, are shown to
be non-generic bifurcations just as the bifurcations which occur due to the row depen-
dence of subset rows of the load flow Jacobian matrix are non-generic. Loss of
causality due to g, being singular and loss of stability in the single machine dynamics
due to f 3 having eigenvalues with zero real parts are two special stability problems.
Although these two stability problems and are not proven to be nongeneric, they are
proven to be improbable when the generator controls are properly designed, the gen-
erators are properly operated within design limits, and the generator reactive power
generation limits are included in the model. The remaining three generic bifurcations

assuming that f x and g, are nonsingular are
(a) the static bifurcation which may occur when the equivalent system Jacobian
matrix
I: =fx ’fy gy-l 8x (6’13)

is singular;

160

(b) the dynamic bifurcation which may occur when J, has complex eigenvalues
with zero real parts;
(c) the static/algebraic bifurcation which occurs when the equivalent

static/algebraic Jacobian matrix
1) =8) .81 1.1 fy (64b)
is singular.

It is proven that a static bifurcation is identical to a static/algebraic bifurcation
which indicates that J,. J,. and J are singular for this bifmcation where J, and J, are
row independent, and f x and gy are nonsingular. This static or static/algebraic bifur-
cation is also called a singularity induced bifurcation, and results in voltage collapse
and/or loss of steady state stability.

The prerequisite for the static/algebraic bifurcation is that the single-machine sta-
bility Jacobian matrix f x is nonsingular in order to obtain the equivalent
static/algebraic Jacobian matrix. The matrix f x is shown to equivalently (1) model
each synchronous machine as the single-axis model, (2) represent excitation systems as
D.C. gains K55, and (3) neglect all the effects of machine damping constant D,
spwd-goveming-turbine systems, and power system stabilizers. This prerequisite for
f 3 being nonsingular is argued to be true, since each synchronous machine should be
designed to be operated in a stable manner below its steady state stability limit when
its terminal bus is operated as the swing bus. The prerequisite for the static and
dynamic bifurcations is that the transmission network Jacobian matrix g, is nonsingu-
lar in order to obtain the equivalent system Jacobian matrix J,. This prerequisite is
. also referred to as the causality condition of the differential-algebraic model, and is
discussed in this dissertation. The transversality and nondegeneracy conditions for
static/saddle-node and dynamic/Hopf bifurcations, controlling the nondegeneracy of the

behavior with respect to the parameter change and the dominant effects of the

161

nonlinear terms of power system dynamic model are not discussed in this dissertation.
The system behavior when the system loses the causality condition is briefly discussed
in this dissertation.

Testing the singularity of the static/algebraic Jacobian matrix J, for identifying
the singularity induced static bifurcation rather than testing the singularity of the
equivalent system Jacobian matrix J, is recommended since both tests are valid and
should theoretically always become singular at the same point as long as f x and g, are
nonsingular. This proposed static/algebraic bifurcation test matrix is proven to have
similar structure to the causality matrix g,, and is virtually identical to the load flow
Jacobian matrix J”. A set of nonlinear equations is thus proposed with J, that com-
putes the equilibrium point of the power system differential-algebraic model. 9 The
equivalent load flow model would represent generators which have not reached reac-
tive power generation limits by the steady state model of generators and excitation sys-
tems, and by the reactive power balance equations when reactive power generation
limits are reached. This equivalent load flow model for the static/algebraic Jacobian
matrix J, would precisely determine when a static bifurcation occurs in the
differential-algebraic power system model. Testing for a static bifurcation using J,
does not require inverting g, which is required when J, is used.

The load flow model for the load flow bifurcation test on J”: and the equivalent
load flow model for the static/algebraic bifurcation test on J, are shown to be virtually
identical for all rows of Jut- and J, if the D.C. gains of the excitation systems are very
high so that the generator terminal voltage can be kept constant, and if the flux decay
equations are replaced by the reactive power balance equations at terminal buses for
those generators exhausting reactive power generation reserves and being modeled as
PQ-buses like in the load flow model. However, they would not be identical if the
equivalent load flow model includes the effects of the generator flux decay dynamics,

excitation system dynamics, and field current limit controller dynamics at steady state

 

162

condition when the reactive power generation limits or field current limits are reached.
The reason is that a static bifurcation in flux decay dynamics or in both flux decay
dynamics and mechanical dynamics would be captured by the equivalent load flow
model if steady state representation of flux decay dynamics, excitation system dynam-
ics, and field current limit controller dynamics is used when field current limits are
reached; but would not be captured in the load flow model or in the equivalent load
flow model if the generator and excitation system dynamics is replaced by a reactive

power balance equation (simply called the conventionally equivalent load flow model).

This result is quite important because it indicates that the conventionally
equivalent load flow model can only capture one of the three generic static bifurcations
(in mechanical dynamics, in flux decay dynamics, and in both) that occurs in mechani-
cal dynamics [39,40,57,62]. The static bifurcations that occur in flux decay dynamics
[52,53], and mechanical and flux decay dynamics are most likely to occur when field
current limits are reached and the field current limit controller works through the exci-
tation system to fix field current to the continuous rating value. A modified load flow
model that properly reflects the steady state behavior of the generator flux decay
dynamics, excitation system dynamics, and field current limit controller dynamics
would accurately determine the equilibrium point(s) of the differential-algebraic model,

and experience all of the static bifurcations of the differential-algebraic model.

The comparison of the static/algebraic Jacobian matrix J, , the causality matrix
g,, and the load flow Jacobian matrix I”, carried out in this dissertation, is for the
case where the flux decay dynamics and excitation system dynamics are replaced by
the reactive power balance equations when the reactive power generation limits are
reached. The use of the reactive power balance equations is justified based on the
approximation of operation action to maintain the units close to their capability curves
when the reactive power generation limits or field current limits are reached. The

comparison carried of J , g, , and J“.- is now reviewed.

163

It should be noted that all elements that are not associated with the generator ter-
minal bus variables of the static/algebraic J acobian matrix J, and those of the causality
matrix g, are identical. The only difference between J, and g, are those diagonal
matrices of 0PT/001, EFT/8V1, 50,011,, and EQT/BVT. In g,, the elements of these
four diagonal matrices are composed of the elements in load flow model and the ele-
ments of the coupling between generator terminal buses and internal buses. In J, , the
elements of EFT/801, EFT/0V7, BQT/BGT, are identical to those in load flow model.
However, if the generators do not reach their reactive power generation limits, the ele-
ments of BQT IBVT include not only the elements of g, , but also the elements propor-
tional to the D.C. gains of the excitation systems; if the generators reach the limits,
these elements become the same as those in load flow model. Hence, the diagonal ele-
ments of BQT /VT of J, where generators have not reached their reactive power genera-
tion limits are much larger than the same elements of g, which do not change whether
generators reach the limits or not. When generators reach the limits, the diagonal ele-
ments of J,. which are much larger than the same elements of g, before the limits are
reached, then become the same elements as those in load flow model which are
smaller than those of g, . Thus, in the region where reactive power generation limits
are reached and the reactive power transfer limits are violated, the static/algebraic
Jacobian matrix J, is closer to singularity than the causality matrix g,. This result
indicates that loss of causality is not a likely cause of loss of stability compared to a

static bifurcation which is tested by the singularity of J,.

The values of these diagonal elements of BQT/VT of the static/algebraic Jacobian
matrix J, are very large when reactive power generation limits are not reached, that
are the only different elements compared to the load flow J acobian matrix Juc. These
diagonal elements (a) are very diagonally dominant compared to the rest elements of
the rows associated with reactive power balance equations at generator terminal buses,

causing the generator terminal voltage to remain constant when reactive power

 

164

generation limits are not reached, and (b) are identical to the elements of J“: when
reactive power generation limits are reached. Thus, J u:- and J, are virtually identical

whether reactive power generation limits are reached or not reached at any bus.

The results of the classification of stability problems in the differential-algebraic

power system model are important contributions because
(1) A static bifurcation was not known to be identical to a static/algebraic bifurcation. '

(2) The occurrence of a static bifurcation can be determined using an equivalent load
flow model rather than a transient mid-term stability model (differential-algebraic

model).

 

(3) The bifurcation properties of the load flow model approximate the static bifurcation
properties of the differential-algebraic model if generator terminal buses are
modeled as PQ-buses when their reactive power generation limits are reached. The
static bifurcation properties of solely mechanical dynamics would be captured using
such a differential-algebraic model (or equivalent load flow model). The static
bifurcation of flux decay dynamics could also be detected if the equivalent load
flow model includes the effects of flux decay dynamics after the generator reactive

power generation limits are reached.

(4) Loss of causality has been considered as an important type of loss of stability
based on the results in [70]. The results derived in this dissertation also shows that
loss of causality can occur before the static/algebraic bifurcation occurs if the gen-
erator reactive power generation limits are ignored in the model. Questions arose
whether the network algebraic model is valid when loss of causality occurs and
whether the algebraic model should be replaced by a partial differential equation
model. The validity of the differential-algebraic model on the frontier where loss
of causality occurs is uncertain, because the reality of the lumped model that

includes the concept of reactive power may be invalid on the frontier. Significant

165

research effort is expected on replacing the differential-algebraic model by a
differential-partial differential model either to confirm the validity of the
differential-algebraic model on the frontier or to show that a differential-partial
differential model is required if loss of causality is closely approached or occurs.
However, the result that loss of causality is not probable if the reactive power gen-
eration limits are included, suggests that this effort at focusing on a differential-
partial differential model may be of minimal value practically. If loss of causality
can be shown to be non-generic, the investigation of bifurcations in a differential-

partial differential model may be of limited value theoretically and practically.

(5) The differential bifurcation has been shown to be a non-generic class of power sys-
tem stability problems, despite the fact that much of the study has been on a single
machine to infinite bus model where the network equations are nonexistent. Such
simple models can only have differential bifurcations which are the same as the
single machine stability problems due to the singularity of f x , being proven

improbable in this dissertation.

(6) The algebraic bifurcation has been previously identified as a separate type of stabil-

ity problem [9,10]. This dissertation has shown it to be nongeneric.

(7) The dynamic bifurcation is shown to be a second generic class of bifurcation prob-
lems, and be principally associated with the mechanical and flux decay dynamics,

since the bifurcations in control systems are argued to be nongeneric.

The simulation for the static/algebraic bifmcation test on the same power system
used to test for the load flow bifurcation shows similar results. This is expected since
the D.C. gains of the excitation systems are enough high to keep the generator voltage
almost constant as they are PV-buses in the load flow model until they exhausted reac-
tive power generation reserves. In addition, these simulation results also point out that

both 1055 of causality (g, singular) and loss of single machine stability (f , singular)

166

are not likely to occur in a practical power system where the reactive power generation

limits are taken into account.

(D) Classification of Static and Dynamic Bifurcations

The static/dynamic bifurcation is identified when the equivalent system Jacobian
matrix J, has eigenvalues with zero real parts. It could be the static/saddle-node bifur-
cation if J,. has a simple zero eigenvalue, and could be the dynamic/Hopf bifurcation
if J, has a pair of pure imaginary eigenvalues, provided that

(a) causality condition holds, and j

(b) transversality/nondegeneracy condition holds.

The research has identified that there can be static or dynamic bifurcation associ-
ated with

(1) solely mechanical dynamics;

(2) solely flux decay dynamics;

(3) both mechanical and flux decay dynamics.

Test matrices are developed for identifying and preventing each of these bifurcations.
The structure of these test matrices can indicate the causes of the bifurcations and the
sensitivity of a particular type of bifurcation to various operating changes or control
system design changes. These test matrices are derived using the procedure described
below.

The Jacobian matrix of the control systems themselves ACC is shown to have

stable eigenvalues, meaning that the assumption:
(c) ACCUQ) is nonsingular,

holds. Then the reduced Laplace transformed (S = jQ) model of J,. Afxqfl),

167

M 19 Am A0)
m 55 =10?qu A5 (62)
TPP 1'9 AEF AEP
' 1
’10 +DGUQ)] "SP1 "SPZ
4,3052): I ' 0 0
"SsUQ) ’1544'5510‘2” —[S3+ng(ifl)]

 

 

is produced that is composed of solely mechanical and flux decay dynamics, after the
elimination of the control system dynamics since they are not singular for any 0 2 0.
The matrix A1500) provides the test of static/dynamic bifurcation if it is singular at
some (2 2 0: static if Q = 0, and dynamic if 9 > 0. Note that this test matrix structur-

ally shows the effects of network and control systems:
network: SP1, SP2, S 3, and S 4;
control: DG, SS, 551, and 552;
DC: effects of speed-governing system
SS: effects of power system stabilizer
S51: effects of excitation system on A8
852: effects of excitation system on A5,.-

where, 551 and 852 include the effects of load compensator, SC, and 5C2 respectively.
These six sensitivity matrices: SP1, SP2, S 3, S 4, Scr and 8C2, are consistent with those
K constants of the single-machine power system without load compensator: Sp, = [(1,
SP2 = K2. 53 = K3: 54 = K4: Sc1= K5» 5C2 = K6-

Damping and synchronizing power coefficient matrices, Kdanp (Q) and K ,,,,c ((2),
associated with the mechanical dynamics are further derived, under the assumption:

(d) 5,09) = S, + Sn, ((2) + j 9 [ T pp + 5525(9) ] is nonsingular.

Then test condition on these matrices for identification and prevention of bifurcations

of the mechanical dynamics are established. Similarly, flux decay matrices, T pp, ((2)

 

168

and Tm,- (Q), are derived, and test matrices for identification and prevention of flux

decay bifurcation are also generated from the reduced model, under the assumption:

(e) 5,012) = 5,, - £22 [ M +DG,(Q) 1 +152 [D + 06,02) 1 is nonsingular.

Figure 6-1 shows the flow chart of static and dynamics bifurcation tests on
mechanical and flux decay dynamics. When the above assumptions (a)-(c) hold, the
reduced model pertaining to mechanical and flux decay states, A§x0m, is obtained by
eliminating the network variables and control systems states. Then damping K damp (Q)
and synchronizing K ,,,,c (9) power coefficient matrices of mechanical dynamics, and
flux decay matrices T“), (Q) and T”),- (Q) are derived when further assumption ((1) and
(e) hold respectively. Based on these matrices, necessary test conditions for
static/dynamic bifurcations associated with mechanical dynamics and flux decay
dynamics are initiated as shown in Fig. 6-1.

Option 0: mechanical and flux decay dynamics

(a) for static bifurcation: Q = 0,
4%,“) = 0) is singular,
(b) for dynamic bifurcation: 9 > 0,
4,3012) is singular. ‘
Option 1: mechanical dynamics
(a) for static bifurcation: Q = 0,
K3,“. ((2 = 0) is singular;
(b) for dynamic bifurcation: (2 > 0,
both Km (0) and Kmm) = ch(fl) — 122 M are singular.
Option 2: flux decay dynamics
(a) for static bifurcation: Q = 0,

TFDr (Q = 0) is singular;

 

169

assumption (a) w No Loss of Causality

Yes

 

[for all (2 2 0]

Not “Properly Tuned
Control Systems

 

 

   

|Kdmp(fl)| = 0
|Km(Q)| = O

        

No1 Yes1 Yes" Yes2 No2

0 : Bifurcation of Mechanical and Flux Decay Dynamics
1 : Bifurcation of Mechanical Dynamics

2 : Bifurcation of Flux Decay Dynamics
Yes : assumption (b) for Transversality/Nondegeneracy Condition

Fig. 6-1 Flow Chart of Bifurcation Tests for
Mechanical and Flux Decay Dynamics

170

(b) for dynamic bifurcation: D > 0,
both Tm, (a) and 1mm) .-. Td'o + rm,- (:2) are singular.

It should be noted that the flow chart shown in Fig. 6-1 provides a static/dynamic
point of collapse/oscillation method [53] that would test for the closest bifurcation as
the system operation condition is continually changed in a manner that could cause one
or more of the bifurcation or stability problems to occur. The point of
collapse/oscillation method would be similar to computing a Q -V curve by adding
reactive load at buses until bifurcation occurs, or a P—V curve where active power
transfer or active power load level is increased at some pattern buses until bifurcation
occurs. This approach indicates the stress level in reactive power load, active power
transfer, or active power load at some pattern buses that brings about the closest bifur-
cation in that subregion of the network. This method also indicates that the bifurcation
is static and/or dynamic, and points out that the bifurcation is associated with mechani-
cal dynamics and/or flux decay dynamics. The test condition matrices structurally
show the effects of operation conditions and control systems as well. Analysis of the
components of these test matrices should help understand the factors causing bifurca-
tions and help design controls and change operating limits to prevent bifurcations.
This approach avoids the computational burden of all eigenvalues of the equivalent

system J acobian matrix J, , provided that causality holds.

The simulation for the static/dynamic bifurcation tests associated with mechanical
and flux decay dynamics is performed on the same power sy5tem where load flow
bifurcation was tested but with dynamic data of the synchronous machines and control.
systems provided from [4]. At some operation condition, the system is in the proxim-
ity to experiencing dynamic bifurcation leading to low frequency oscillation in both
mechanical dynamics and flux decay dynamics. The proposed approach clearly
identifies the mode mainly associated with mechanical dynamics, and the mode princi-

pally pertaining to flux decay dynamics without having to perform any eigenvalue

 

171

analysis. The eigenvalue analysis is shown to have difficulty identifying whether a

bifurcation is possibly occurring in mechanical or in flux decay dynamics.

6.2 Relationship to Literature

Regarding the studies of voltage collapse, the point of voltage collapse method is
a very popular approach to establishing a bifurcation point based on the load flow
model by increasing active or reactive load or transfer at a pattern of bus or branch in
order to understand the margin in terms of the additional load or transfer that can be
added at the bus or branch pattern before loss of voltage stability will occur [53]. The
point of voltage collapse method is usually referred to the V—P, V—Q , or Q-V curve
approach, or to the sensitivity matrix approach [48-59], or to the optimization approach
[60,61]. This dissertation derived the generic load flow bifurcation that has not been

pointed out in the literature.

In the power system differential-algebraic model, the voltage collapse problem
has been investigated by applying the theory of bifurcation on the complete system
Jacobian matrix [3945.52.53.64-79]. The research was generally confined to a simple
power system where excitation systems were simplified or neglected, and speed-
governing-turbine systems and power system stabilizers were omitted. Under the
assumption that the causality condition holds, the static bifurcation leading to loss of
steady-state stability and/or voltage collapse has been studied based on testing the
singularity of the equivalent system Jacobian matrix [3941,45,52,53,64.
67,69,70,73,78,79], where transversality/nondegeneracy condition for the static/saddle-
node bifurcation was not discussed. A test matrix for the static bifurcation solely asso-
ciated with flux decay dynamics has been derived and defined as flux decay bifurcation
[52,53] for a power system without both power system stabilizers and speed-
governing-mine systems, provided that the causality condition holds. If this flux

decay bifurcation test matrix is approaching singularity, experimental results show that

 

 

172

the system is at the proximity to voltage collapse, especially when field current limits
are reached and then the normal excitation controls are disabled [52,53].

In this dissertation, static voltage stability is investigated based on the two
different representations of the power system: (a) algebraic load flow model, and (b)
differential-algebraic power system dynamic model. The load flow bifurcation due to
the singularity of the load flow Jacobian matrix is studied from the viewpoint of the
point of voltage collapse method for the load flow model. The load flow Jacobian
matrix is also proven to be the generic test matrix for load flow bifurcation. In the
power system dynamic model, the equivalent static/algebraic Jacobian mauix is
developed, and is also shown to be the generic test matrix for the singularity induced
static bifurcation due to the singularity of this test matrix. The static/algebraic bifurca-
tion test is equivalent to the static bifurcation test due to the singularity of the
equivalent system J acobian mauix. However, testing the singularity of the
static/algebraic Jacobian matrix for the singularity induced static bifurcation, rather
than testing the equivalent system Jacobian matrix, avoids the inverse of the causality
matrix. Moreover, this test matrix preserves the form of the linearized load flow
model so that this static/algebraic bifurcation test can be performed by slightly modify-
ing the load flow program '

This dissertation further develops test matrix for the static bifurcation associated
with mechanical dynamics and flux decay dynamics, after proving that the control sys-
tems themselves have asymptotically stable eigenvalues. The static flux decay bifurca-

tion test matrix is consistent with the one proposed in [52,531.

In the studies of dynamic stability, especially for low frequency oscillation, both
bifurcation theory and eigenvalue analysis have been applied. A dynamic/Hopf bifur-
cation can occur when the power system is at the operating condition such that the
equivalent system Jacobian matrix has a pair of pure imaginary eigenvalues, provided

that causality and transversality/nondegeneracy conditions hold. Hopf bifurcation of

 

 

173

flux decay dynamics has been shown to occur when mechanical dynamics are ignored
[42,68]. Low frequency oscillation has experimentally been shown to be associated
with mechanical and flux decay dynamics [3.4.21], mainly due to lack of damping and
synchronizing power [22-30].

This dissertation establishes more precise test conditions for the static/dynamic
bifurcation pertaining to mechanical and flux decay dynamics by proving that the con-
trol systems themselves have stable eigenvalues. Moreover, damping and synchroniz-
ing power coefficient matrices are derived, and test conditions based on these two
matrices are established for the static and dynamic bifurcation in mechanical dynamics.
Flux decay test matrices are also defined for the static and dynamic bifurcation in flux
decay dynamics. This approach not only precisely identifies the modes principally
associated with mechanical dynamics and flux decay dynamics, but also does not

involved in the computationally intensive eigenvalue analysis.

6.3 Future Work

Although a classification of the types of stability problems has been developed
and generic stability problems have been identified, much additional work is necessary.
The generic static and dynamic bifurcations have been studied solely for identifying
and preventing generic static/saddle-node and dynamic/Hopf bifurcations in power sys-
tem models. Research is needed to more thoroughly describe the center manifold and
the behavior of the system dynamics and the network response near the bifurcation
point. Loss of stability in single machine dynamics and loss of causality have been
argued to be improbable. Research on these two special stability problems is also
needed to show whether they are generic or not. If loss of causality is a generic sta-
bility problem, more effort must be put on the study of the validity of the differential-
algebraic power system model. The validity of the differential algebraic model on the

frontier, where loss of causality occurs, can be assessed by using a differential-partial

 

174
differential model. If the differential-algebraic model is invalid on the frontier, it may
also be invalid as one approaches the frontier. A differential-partial differential model

would be used rather than a differential-algebraic model wherever the differential-
algebraic model proves invalid.

Although some understanding of these stability problems exists for the single
machine to infinite bus model, little understanding exists for the multimachine power
system stability model. Moreover, no methods exists for establishing where, why, and
when certain stability problems occur. The test matrices proposed in this dissertation
for the generic stability problems structurally show the effects of the transmission net-
work and control systems, that may be useful in understanding the causes of the stabil-
ity problems. Methods for preventing and correcting the various stability problems
should be undertaken once the above understanding is comprehensive. The computer
program for the point of collapse/oscillation method also needs to be developed.

Regarding the parameters of the control systems, it should be noted that only the
D.C. gain of excitation system affects the necessary condition of the static bifurcation;
but each control system has the effect on the necessary condition of the dynamic bifur-
cation. It can be concluded that the proper tuning of each parameter of each control
system, especially the excitation system, is also another main concern of power system
planning. Moreover, the particular parameters of control systems that have major
effects on the damping margin before the power system experiences static/dynamic
bifurcation should be identified, and their relationship should also be established.
Some particular parameters of major control systems are expected to be the D.C. gain
of the excitation system, the gain and time constant of the stabilizing feedback of the

excitation system, and the lead-lag time constants of the power system stabilizer.

Based on the simulation results shown in this dissertation and the experimental
results fi'om previous research [48-53], the reactive power generation reserve of the

power system plays a very important role in the studies of voltage collapse and low

 

175

frequency oscillation. Sufficient and properly placed reactive power supply in the sys-
tem should be carefully studied to prevent the generic bifurcations identified in this

dissertation.

 

 

APPENDICES

Appendix A
Schur’s Formula and Inverse Matrix _

Consider an (m + n)-dimensional square matrix A which can be partitioned as

follows:

A11 412
A: 14214422 (A-la)

where, A11 is an m x tit-dimensional square matrix; A12 is an m x n-dimensional
matrix; A2, is an n x m-dimensional matrix; and A22 is an n x n-dimensional square
matrix.

If the matrix A is nonsingular, its inverse, A", has the following form:

A’1 [3,, Biz] B (Alb)
- 321322 - -
satisfying
1 0
AB=BA=[;I] (A-lc)
,-

where, each 8 -submatrix has the same dimension as A -submatrix with the same sub-
script; 1,, and I, are the m - and n-dimensional identity matrices, respectively; and 0 is
the zero matrix with appropriate dimension.
(1) 11A,, is nonsingular, the determinant of the matrix A, (Det[A 1), can be
written as
Der[A 1=Det[A,,1-Dez[A,,-A,,A,-,1 A12] (A-2)

Then the submatrices of B , the inverse of A , can be represented as

176

177

322=[A22'A21A111 4112]"1 (A-Za)
321="322 112141-11 (A-Zb)
312=-Ai'11A12322 (A-ZC)
311 =Aii1 [1m ‘A123211 . (A-Zd)

(2) If A22 is nonsingular, then

 

Det[A'1=Det[A,,1-Det[A,,—A,,A,-,1 A2,] (A-3)
and
311= [All "Alei'rl A21]-1 (A-3a)
312=-311A12Az-21 (A-3b.)
B21="A2-21 A21311 (A-3C)
3,,=A,-,1 [1,, -A,,a,,1 (A-3d)

(3) If both A,, and A2, are nonsingular, then both case (1) and case (2) are

valid, and the B -submatrices with the same subscripts are equal to each other.

 

Appendix B

-1
Ace

Rewrite the matrix Acc (41-80), which is formed by the control systems of the

synchronous machines, as follows:

'11,, 0 A5,-
Acc= 0 A AGO 0 (3'1)
0 0 AS,

 

 

where, A 55 represents the block matrix of the excitation systems; A06 represents the
block matrix of the speed-governing-tlu-bine systems; As,- represents the block rmuix
of the power system stabilizers; and A53 represents the block matrix of the PSS con-
trols through the excitation systems. The complete forms of these control systems
matrices are rewritten from (4-2) with some short notations for the excitation systems

and power system stabilizers:
Tao =78.ch ; [ac =1 " Tue
721 :1.521751 3 ’21=’ "'T21

T43=Ts-41753 3 143:1 “T43

 

 

F - I 0 0 0 0 '
0 — I 0 0 - 7‘;le
A5,,» = - K, TB, - K, 71,,- - I K, - K, 1'fole (B-la)
- Ix — 1,, 0 - I - 1,,c 7,-1K,
0 0 I 0 - K55 ‘

178

 

 

 

"-1 0 0 0 0 0 '
-I —I 0 0 0 0
0 l -I O 0 0
A66: 0 0 l -1 0 0 (3‘1”)
0 0 O I -l 0
L 0 0 0 0 I -1_
-I 0 0
A55= 12, —I 0 (B-lc)
.l43T21 143 ‘1
0 0 0 “
0 0 0
A155 = KATBCT43T21 KATBCT43 KA Tac (B-ld)
IacT4stl IscT43 lac
t 0 0 0 .

 

Note that the submatrices of each control system block matrix are all diagonal.

The matrix Acc (B-l) is an upper-triangular matrix, and has been proven to be

nonsingular based on the nonsingularity of individual control system. Thus, its

inverse, A56, can be represented as follows:

45% 0 - Arr} Ass Ass‘
Aaé= 0 455
1 0 0

 

0

A531

(B-Z)

 

d

Since A66 and A 55 are lower-triangular matrices, we have

 

#-
8;
ll
NNNNN

”-1 0 0 0 0
—10 0 0
-1-10 0
-1-1-10
-1-1-1-10
—1—1—1-1-1,

0
0
0

0

03-3)

 

180

-I 0 0T
4551: ”I21 '1 0
1—143 “143 "1

 

 

(34)

Regarding A57}, one can apply the Schur’s formula shown in Appendix A. Based on

the structure of A55 (B-la), A35 can be partitioned and has its inverse as follows:

Ell £12
‘38: Ba Ba
F11F12
A“:
55 [F21 F22]

where,

-I O 0
E11: 0 "I 0 :E12=
”Kn Tac ‘KA Tac "1

'IBC ”lac 0
521: 0 0 1

 

3522: 0

F11=[E11"El2E2-21 15.21]-1
F12=-F1151252'21
F22:[522”52151-11E12]-l

F21=-F2252151'11

 

' 0

0

'1 ‘18c Til KP
'KSE

0

__--1
F

Kr

_KA "KA Tac Til KF .

(B-Sa)

(B-Sb)

 

Note that the main diagonal block submatrices, E 11 and £22, in the above partition are

all nonsingular, since E 1, is lower-triangular; 522 is upper-triangular; and both have

nonzero diagonal elements. Moreover, the matrix manipulations, [512 E 2'21 E 2,} of

F1, and {E21 Efil En] of F22, respectively keep the same structure as E1, and 522.

Thus, after applying the Schur’s formula and performing the matrix manipulations, one

can obtained the final form of A5,}:

181

- I 0
“TF5 KA -I+TFE KA
Ara-a1 = KA KA
lac lac
, K575l KA " K5721 KA

 

where

Tn; = Til KP K575l

0 0 0 1
TH; TPE KA TPE
- I - K A 0
0 - I 0
"’ K5751 " K5751 KA "’ 5751 _

 

(B-SC)

Having the main diagonal block matrices of A551- (B-2), one can obtain the only

one off-diagonal matrix of A53} by substituting (B-ld), (B-4), and (B-5c) as follows:

ABS A551 =

4511 Ass A551 =

 

’ 0 0 0
O 0 0
" KA T13c - KA Tac " KA Tao
" lac " Iac " [BC
0 O 0 .
0 0 0 '
" TF5 KA " Tn: KA " TF5 KA
K, K, K,
lac Iat: Iac
_ K575l KA K5731 KA K575l KA _

 

 

(B-6a)

(B-Gb)

Appendix C
Kr, and Km

This Appendix will show that the diagonal matrix Kfi‘, , representing the
coemcient of the active power generation at the internal bus with respect to the rotor
angle, is in general positive definite if each synchronous machine is operated in the
condition of lagging power factor. Then the diagonal matrix K FD , representing the
steady state operation condition of each synchronous machine, should also be positive
definite, when it is assumed that the load compensators are not used in the power sys-
tem model.

Rewrite the matrix [(33 from (2-41):
Kfl‘, = a; V, U”, + 11,2 (x, -x,;) (r, r, - U, U,,) (or)
where,

UPE =kt Uta -G. (X, ~10?) 7,. ; kt=1— (X, Jim,
T, = G, cos(5 - 0,) + a; sin(5 :- 0,)
U, = G, sin(5 - 0,) 43; cos(5 - 0,)
T4, = G, cos(5 - 0,) + B, sin(8 - 0,)
0,, = G, sin(8 - 0,) - B, cos(8 - 0,)

R, , -x,; -x,
= 2 ' - BU: 2 , ; 84 = 2 ’
Ra +Xd Xq Ra +Xd Xq Ra +Xd Xq

 

 

 

Assuming that the armature resistance R, of synchronous machines are small and

neglected, then

182

183

G,=0 ; B,’=—1/Xq<0 ; Bq=-1/x;<0
1,: 1 — Of, -X;)Bq =Bq/B; >1

Up; =—Bq cos(8— 0,) > 0

1,, =8; sin(5-0,)<0

U4, =—B,' cos(8-0,)>0

T,” = B, sin(8 - 0,) < 0

U4, = — B, cos(8 — 0,) > 0

rd, 7,, — U, U, = - B,’ B, cos[2 (5 — 0,)]

Hence,
Kfi', = - E,’ V, a, cos(5- 0,) + V} (X, -X,E) (—B,' a, cos[2 (5 - 0,»)

= —B, V, [5,, cos(8— 0,)

+ v. (x, -x,') -;—‘ ( cos2<6— 6.1- sin2<8- 6.) )1
(I

I

, X
=-B, V, [(5,, —V, (1- ii) cos(5-91))c08(5-91)
q

I

X41
+ V, (1 - —) sin2(5 — 0,) 1 > 0 (02)
X4

Note that the following factors causing K 3', > 0:

I

. x
(a) x, 221,, such that0< [1- 31k 1.
‘I

(b) If the synchronous machine is operated in lagging power factor, then
8 - 0, < 90° implies cos(8 — 0,) > 0, and from the phasor diagram shown in
Fig. 4-4(b), [154' - V, cos(5 - 0,)) > 0, hence,

I

. X4
[8, — V, (1 — Z) cos(8 - 0,)} > 0.

184

Rewrite the matrix Km from (4-11):
Km = (K53 ‘1' K52) - (K54 4' K51) K1"’ 1.1 K132 (03)
where,
K51 = Kara Kcr 3 K52 =Ka£ Kcz

Note that the diagonal matrix K55 represents the D.C. gains of excitation systems.
Under the assumption that the load compensators are not employed, which implies that

KC1= O, 811d Kcz = 0, WC have
KPD = K53 " K154 K1314 K172 (04)

Each K -matrix (K 3‘, is discussed above) can be found from Section 2.6 as follows:

K33 = 2 Eq' GEE - V, TPE (C-Sa)
K54 = — (x, with T, V. (C-5b>
K53 = 1 - (X, - 21,) B, (C-5c)

where,
0,, =k, G, ; k,=1—(Xq —xj)Bq
7p, =k, T, + G, (x, -x,;) U,

With the assumption R, = 0, and the properties of the coefficients shown in the dis-

cussion of K 3‘, above, we have
GEE =0 ; TPE =84 sin(8-0,)< 0
K5, = - V, a, sin(5 - 0,) > 0 (C-6a)

. K54 = — V, (x, - 21,38, sin(5 - 0,) (C-6b)

X
=V, (—“,--1)sin(5-0,)>0
Xd

 

185
K 1 (X X ) X4 0 (C 6C)
= + - — = — > _
53 d 4 X1, X,
Note that X, > Xi. If Kfl‘, > 0 (from (02)). and since each K-submatrix is diagonal,
then the (positive) definiteness of Km is equivalent to that of Kfl‘, Km . Thus,
Kfl‘r Km = K33 K53 - K54 Kfl'z

=—Bq V, [5,; cos(8-G,)

X
- V, (l - 31-) (cos2(8 - 9,) - sin2(8 - 8,) )] —i,-
X, 4

X
—v, (x—{-1)sin<5-e,)[-V. 8, sin(5-901
d

,,,X
_-.—Bq V,q'[E c—-os(8- 9,)
X4

X4 2 X4 2
-V, -—-;-cos (8- 0,)+V,— sin (8— 9,)
X4 X4

V-X—d 28-9 x, 28-6
+-X—cos( ”-7-: ( .)

X
- V, 7(2- sin2(8 - 9,) + v, sin2(8 - 6,)]
d .

=—Bq V,[(Eq' V, cos(8 9,)):{ cos(8- 9,)
d

x,
+V, x cos[2(8- e )]+V, sin2(8- 9,)] (c-7)

Conservatively, if 8 — 9, s 45" and from Fig. 4-4(b) which assure K 3', > 0, then the

first two terms of K 3’, K FD are nonnegative, and hence K FD > O.

Appendix D
C3,, and car,

This Appendix will show that - C3,, is a diagonal matrix with positive elements
that increase almost linearly with the D.C. gains of the excitation system K 55 that are
in general very large compared to the rest of terms in 03x, when it is assumed that the
load compensators are not used in the power system model. Then the matrix C’3‘K is

also a diagonal matrix with positive elements that increase almost linearly with K55.

Rewrite the matrix C§K from (4-18):

65‘, = Kg, + C3 — C3,, (1)-1)
Cax = 'Kar XVI-1K3!) 419% K1334 K192 4' 1‘32)de K150 (D-la)
Kac = «,,, + Ku- Kc,)Kfl‘,'1Kfi‘-, + (K31 - Km Km) (ID-lb)
Km = (K53 +Kw-<K54+K51>K3'1"Krz <D-1c)

where,
K£1=KEE Kcr ; KEZ=KEE Kcz

Under the assumption that the load compensators are not employed, which implies that

KC, = 0, Kc; = O, and Km = I, and if the D.C. gains of the excitation systems K5,;

are very large, we have

Km = K53 - K54 K9 1 -1 KIT: (D-ZC)
Ksc = K154 K171 -1 K35 + K57 - KEE

z .. K55 (1)-2b)
Cax =<K31Kr1“K3'2 +K32) K56 K515 (D-2a)

186

187

As discussed in Appendix C, [(33 and Km should be positive definite, and since K 33
is positive definite with very large elements, then the definiteness of C 3x is equivalent
to that of KPQ a {K31 [(33 + K32 Kfl‘l }, where [(33 and [(35 are shown in Appen-
dix C, and

((3, = - V, 3,’ 1,, - V,2 (3,; - 3,) sin[2(e, - 8)] (D-3a)
K3, = - v, U,, (D-3b)
7}, = G, cos(O, — 8) + B, sin(G, — 8)
U,, = G, sin(0, - 8) - B, cos(O, - 8)
Assuming R, = 0, then
T,, =-B, sin(8-0,) >0

U,, =-B, cos(8-6,)>O

Kg, = V, 3,’ 3, sin(8 - 9,) + V,2 (3,; — 3,) sin[2 (8 - e, )] (D-4a)
K32 = V, B, cos(8 - 0,) < O (D-4b)

Hence, substituting (from Appendix C)

K33 = - 3,’ V, 3, cos(8 - 0,) - v,2 (3,; - 3,) cos[2 (8 - e, )1 (D-Sa)

K35 = - V, 3, sin(8 - 9,) > o (D-Sb)
we have

KPQ =K5'1 Kfl'z 4' K32 Kfl'r (1)-6)

= —3, V,2 [ E,’ 3, sin2(8- 9,) + V, (B; —B,) sin[2(5 - 9,)] sin(5 - 9:)
+ 3,’ 3, cosz(8 - 6,) + v, (3,; - 3,) eos[2(8 - 6.)] cos(8 - 9,)]
=43, v,2 [ 5,3, +v, (3,; -3,)(2 sin2(5-e.)cos<6- 9.)

+ cosz(8 - 6,) cos(8 — 9,) - sin2(8 - 9,) cos(8 - 0,) ) 1

188

=—3, V,2[E,' 3, +V, (Bi—8,)c08(5-9.)l
=-Bq V,2[B, [E,'—V, cos(8-9,)]+V,B,; oos(8—9,)]<0

Note that KPQ < 0 implies {K5, K3, '1 K” + K5,} < 0. Hence, - C3,, is a diago-
nal and positive definite matrix with positive elements that increase almost linearly
with K55. Note also that egg, - Kg, + c3 — C3,, and that both K3, and C3 are the
coefficients of reactive power-voltage Jacobian matrix at the terminal bus that are finite
and are neglectable compared with - C 3): when K 33 is very large. Thus, the matrix
C§K is approximately to — C 3x and is also a diagonal matrix with the positive ele-

ments that increase almost linearly with K 55.

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