Illlllllllllllllllllllllllllllllllllllllllllll

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This is to certify that the

dissertation entitled

BIFURCATION DYNAMICS AS A CAUSE OF THE
RECENT VOLTAGE COLLAPSE PROBLEMS ON THE
WSCC SYSTEM

presented by

Shuzhen Liu

has been accepted towards fulfillment
of the requirements for

Ph. D. Electrical Engineering
degree in

 

 

 

Date May 12, 1999

 

 

 

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University

 

 

 

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1!” WM“

BIFURCATION DYNAMICS AS A CAUSE OF THE RECENT VOLTAGE
COLLAPSE PROBLEMS ON THE WSCC SYSTEM

By

Shuzhen Liu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

1 999

ABSTRACT

BIFURCATION DYNAMICS AS A CAUSE OF THE RECENT VOLTAGE
COLLAPSE PROBLEMS ON THE WSCC SYSTEM

By

Shuzhen Liu

The voltage collapse problem is considered as the principal threat to power system
stability, security and reliability in many utilities around the world. Recently, three severe
blackouts occurred on the WSCC power system, which have caused loss of power for
millions of customers. Although reasonably accurate models of that complex system
have been assembled and can reasonably accurately capture the sequence of events that
occurred, there is no clear diagnostic theoretical explanation of what occurred and why.
This dissertation is the first attempt to understand the dynamics through sensitivity

analysis.

This work consists of three parts of theoretical explanations of the cause of the recent

blackouts on the WSCC system.

First, a theoretical analysis and simulation study of the excitation control system,
especially the maximum excitation limiter, is presented. This part explains why
maximum excitation limiters fail to reduce field current limit violations when large inter-
area oscillations are present, and why this can lead to switching from an AC regulator to

a DC regulator excitation control, and finally to tripping of the generator off the system.

Secondly, voltage PV controllability, load PV controllability, voltage PQ
controllability and load PQ controllability are defined. The conditions on sensitivity

matrices SQGE’ SQGQL , SQL y and 5,4; that indicate when each type of controllability is

retained and when it is lost, are derived. Furthermore, it is proven and confirmed through
simulation that the occurrence of loss of different controllabilities has certain patterns and

that loss of controllability is affected by the load characteristics.

Finally, a theoretical justification and simulation study for the dynamic response to
bifurcation sequences associated with voltage collapse are presented. This part
establishes that a saddle node bifurcation followed by an inevitable singularity induced
bifurcation produces the characteristic negative voltage spike observed on the WSCC

system blackouts.

In summary, the purpose of the sensitivity analysis is to identify the subsystem
which initially experiences instability, to identify why this instability occurs and how this
instability could cascade to other subsystems, and to identify what can be done to cure the
instability problems and when and where the cure should be applied. This dissertation
provides both diagnostic theoretical justifications and comprehensive simulation results
on the dynamic phenomena similar to those observed during the recent WSCC system

blackouts.

To my mother, father and daughter

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Professor Robert A.
Schlueter, for his valuable academic guidance, continuous spiritual encouragement and
financial support throughout the time taken to complete this work. Without his great help,

I would not have reached this point today.

Special thanks also go to Professor Hassan Khalil, Professor Steven Shaw and

Professor Chichia Chiu for their intellectual contributions to this dissertation.

I am indebted to my husband Zhibo for his continuous assistance.

I also wish to thank Dr. Khadija Ben Kilani for her helping hand and friendship.

TABLE OF CONTENTS

 

 

DEDICATION ............................................................................................................... iv
ACKNOWLEDGMENTS ............................................................................................... v
LIST OF TABLES ......................................................................................................... ix
LIST OF FIGURES ......................................................................................................... x
CHAPTER 1 INTRODUCTION - - -- - - -1
1.1 Voltage Stability ............................................................................. l
1.2 WSCC System Blackouts ................................................................ 2
1.3 Excitation Control System ............................................................... 7
1.4 Bifurcations in Power System ........................................................ 11
1.5 Hypotheses .................................................................................... 18
1.6 Relevant Literatures ...................................................................... 22
CHAPTER 2 POWER SYSTEM MODELING - .............. 23
2.1 A General Power System Model .................................................... 23
2.2 Excitation Control System Model .................................................. 25
2.3 Maximum Excitation Limiter Model ............................................. 26
2.4 Power System Load Model ............................................................ 27
CHAPTER 3 PROBLEM ANALYSIS ........................................................................ 31
3.1 Disablement of Current MXL due to Oscillation ............................ 32
3.1.1 Theoretical Analysis .......................................................... 32

3.1.2 Analysis of Boundary A’ Where MXL Operates
Correctly and Malfunctions ................................................ 36
3.1.3 Simulation Results ............................................................. 39

vi

CHAPTER 4

CHAPTER 5

3.2 Excitation Control System Tripping and Unit Tripping .................. 45

 

3.3 Uncontrollable Spreading Voltage Collapse ................................... 46
VOLTAGE AND LOAD CONTROLLABILITY. ..... - - -- 47
4. 1 Introduction ................................................................................... 47
4.2 Sensitivity Matrices ....................................................................... 48
4.3 Voltage and Load PV Controllability ............................................. 52
4.3.1 Voltage PV Controllability ................................................. 52
4.3.2 Load PV Controllability ..................................................... 53
4.4 Voltage and Load PQ Controllability ............................................. 53
4.4.1 Voltage PQ Controllability ................................................. 53
4.4.2 Load PQ Controllability ..................................................... 54
4.5 Effects of Loss of Load PQ Controllability .................................... 55
4.6 Effects of Load Characteristics ...................................................... 60
4.7 Validation of Theory ..................................................................... 63

THEORETICAL AND SIMULATION JUSTIFICATION FOR THE
DYNAMIC RESPONSE TO BIFURCATION SEQUENCE

ASSOCIATED WITH VOLTAGE COLLAPSE .................................... 76
5. 1 Introduction ................................................................................... 77
5.2 Test Matrices for Static Bifurcation in Flux Decay Dynamics ........ 79

5.3 Effects of Loss of Voltage PQ Controllability on Stability
of Generators with Exciters ........................................................... 82

5.4 Effects of Loss of Voltage PV Controllability and Disablement
of the Excitation Control System on Stability of Generator
Dynamics ...................................................................................... 83

5.5 Justification of the Bifurcation Sequence that Occurs due to Loss
of Voltage PV Controllability and Disablement of the Excitation
Control System on a Specific Generator ........................................ 84

5.6 Justification of the Large Negative Voltage Spike that Develops
due to Loss of Voltage PQ Controllability on Generator Response

vii

with or without Excitation Control ................................................. 86

 

5.7 Simulation Study ........................................................................... 89
CHAPTER 6 CONCLUSIONS AND FUTURE WORK .............................................. 95
APPENDIX AzTHEOREM PROOFS ......................................................................... 99
APPENDIX BzPUBLICATIONS ........ - . ............. . - . 107
REFERENCES . -- ...... ................... 108

 

viii

Table 4.1

Table 4.2

Table 4.3

Table 4.4

LIST OF TABLES

Bus data ....................................................................................... 66
Line data ....................................................................................... 66
Generator data ....................................................................................... 67
Exciter data ....................................................................................... 67

ix

Figure 1.1

Figure 1.2

Figure 1.3

Figure. 1 .4

Figure 1.5

Figure 2.1
Figure 2.2

Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5

Figure 3.6

Figure 3.7

LIST OF FIGURES

July 2, 1996 WSCC system blackout recordings: (a) Voltage recording (b)
Power recording ....................................................................................... 5

August 10, 1996 WSCC system blackout recordings (a) Voltage recording

(b) Power recording ................................................................................. 6
December 14, 1994 WSCC blackout voltage recording ............................ 6
Block diagram of excitation control system .............................................. 8

A typical bifurcation sequence (a): P-V Curve; (b): Critical modes

movement ....................................................................................... 14
Excitation control system model ......... 25
Model of maximum excitation limiter .................................................... 26
Ifl, with oscillation component ............................................................... 33
Plot for determining A' ......................................................................... 37
Relationship between 10 and Il .............................................................. 38
The MXL action when I0 < R ................................................................ 40
The MXL action when I0 = R ................................................................ 41
The MXL action when I0 > R but Al < A2 ............................................. 42
The MXL action when 10 > R but Al > A2 (The effect of

increasing I0 for certain Il ) .................................................................. 43

Figure 3.8

Figure 4.1

Figure 4.2

Figure 4.2

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8
Figure 5.1
Figure 5.2

Figure 5.3

The effect of increasing 11 for certain I0 ................................................ 44

A three-machine 9-bus test system ......................................................... 68
I _
Eq, SQLV I, SQGEaSVE and SQGQL for case 1 ..................................... 69
I —1
Eq, SQLV , SQGEaSVE and SQGQL for case 2 ..................................... 70
Say—1 for case 1 .................................................................................. 71
SQL fl for case 2 .................................................................................. 72
SQLf‘ , SQGEaSVE and SQGQL for case 1 .............................................. 73
SQLy—l , SQGE,SV£ and SQGQL for case 2 .............................................. 74
Effects of load characteristics ................................................................. 75
Simulation result for theorems 5.1 and 5.8 .............................................. 92
Simulation result for theorems 5.2 and 5.3 .............................................. 93
Simulation result for theorems 5.4, 5.5, 5.6, 5.7 and 5.9 ......................... 94

xi

CHAPTER 1

INTRODUCTION

1 . 1 Voltage Stability

Voltage stability problem is a subset of overall power system stability problem.
Voltage instability results in progressive voltage increase or decrease that can spread
uncontrollably throughout a power system. A power system at a given operating state and
subject to a given disturbance undergoes voltage collapse if post-disturbance equilibrium
point is unstable or does not exist. Voltage collapse may be total (blackout) or partial [2].
Voltage collapse problems are considered the principal threat to power system stability,

security and reliability in many utilities around the world.

The classic voltage collapse problem in a load flow model is always considered a
reactive supply demand problem. Two types of voltage collapse problems have been
identified to exist on a load flow model: loss of control voltage collapse and clogging
voltage collapse. A loss of control voltage collapse is caused by exhaustion of reactive
reserve with resultant loss of voltage control on a particular set of generators, synchronous
condensers, or SVC’s. The loss of control voltage collapse not only cuts off the reactive
supply to a subregion requiring reactive power, but also increases reactive network losses
that prevent sufficient reactive supply fiom reaching that region. A clogging voltage
collapse occurs due to 12X series reactive losses, tap changers reaching tap limits,

switchable shunt capacitors reaching susceptance limits, and shunt capacitive reactive

supply withdrawal due to decreasing voltage. These network reactive losses can completely
choke off the reactive flow to a subregion needing reactive supply without any exhaustion
of reactive reserves and loss of voltage control on generators, synchronous condensers or

SVC’s [11].

The loss of control voltage collapse should have occurred in the algebraic model of a
differential algebraic power system model since the load flow model is a bifurcation
subsystem model if excitation systems are sufficiently high gain, the maximum excitation
limiters act correctly, and the excitation control systems are not disabled [32]. Although
there is yet much to discover how the loss of control voltage stability problem in a load
flow model is evidenced in a differential algebraic power system model, this thesis is on a
totally different voltage stability problem. This voltage stability problem was observed in
the Western System Coordination Council (W SCC) system on December 14, 1994 [20],
July 2, 1996 [18] and August 10, 1996 [21] and was neither a clogging nor a loss of
control voltage instability problem because it was initiated in the differential submodel

rather than in the algebraic submodel of the differential algebraic power system model.

1.2 WSCC System Blackouts

WSCC system is the entire power system network west of Denver including Canada.
In each of the December 14, 1994, July 2, 1996 and August 10, 1996 blackouts [18, 20,
21], there were large interarea oscillations observed before the blackouts, and the blackouts
occurred after some fault or contingency. Based on the existing operating and design
criteria, the contingency should not have resulted in blackouts and certainly not have
produced such severe blackouts that each caused loss of power for millions of customers.
Although reasonably accurate models of that complex system have been assembled and can

reasonably accurately capture the sequence of events that occurred, there is no clear

diagnostic theoretical explanation of what occurred and why. On the August 10, 1996
blackout, it was clear that the excitation control system failed on the McNary station
generators that led to the tripping of generators at that station [21]. Since (1) these
blackouts have led to significant reduction in power transfer limits in the WSCC system,
(2) there were significant deregulated power flows from one subregion to another on
August 10, 1996 that raise the possibility that this type of problem could affect other
regions of the US and other countries experiencing deregulation of their utilities, and (3)
the causes and cures for these stability problems are so poorly understood, this particular
voltage stability problem was singled out for study in this thesis. Moreover, the study will
be the first attempt to understand the dynamics after each bifurcation in the sequence of

bifurcations that possibly produced these blackouts.

From the records of the July 2, 1996 [18] and August 10, 1996 [21] power blackouts
in the Western United States, it is observed that although the cascading tripping of units
and lines has been described as voltage collapse, neither blackout was due to the classical

or any known kind of voltage instability problem.

These three loss of voltage stability incidents couldn’t be loss of control voltage
instability because the time fiarne was too short for tap changers and capacitors to act that
are associated with the recovery of voltage and load in the distribution system [11, 33].
Recovery of voltage and load causes the generators to exhaust reactive reserves and causes
the Maximum Excitation Limiter (MXL) control to act that allows generator excitation
system voltage setpoints to decline as reactive load or reactive network losses increase.
This decline of generator exciter setpoints due to increased network reactive load and losses
begets more reactive load and losses and more voltage decline that causes the cascading
voltage collapse associated with loss of control voltage instability. The recent two blackouts
couldn’t be clogging voltage instability because the system was not heavily loaded when

the outages occurred [1 l].

Furthermore, the stability controls were effective and responded correctly to the early
contingencies. The response of these stabilizing controls should have ensured stability and
prevented further outages if these stability problems were loss of control or clogging
voltage stability problems. In each of the July 2, 1996 and August 10, 1996 blackouts,
sharp spike reductions in both voltage and power were observed as shown in Figure 1.1

and Figure 1.2. A sharp spike reduction in voltage was also observed in December 14,

1994 blackout as shown in Figure 1.3.

 

  

600

 

Garrsion

500

Voltage [KV]
A
G
c

300

  

 

 

l L

0 10 20 30 40 50
Time [second]

   

200

 

(a) Voltage recording

5000

Power [MW]
&
U!
c
c

A
Q
G
G

3500

 

 

California-Oregon Intertie Power

 

 

 

 

10

20

30
Time [second]

(b) Power recording

40

50

Figure 1.1 July 2, 1996 WSCC system blackout recordings

 

 

 

   

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

515
John Day
_ — I PI 1
505‘ I t
E “'1! ll l
g 510 ‘ ‘ i
g Big Eddy 1
505 .
500 L '
495 .12,
15:47:20 15:47:35 15:47:50 15:48:05 15:48:20 15:48:35 15:48:50

Time

(a) Voltage recording

Voltage [KV]

 

 

 

   

 

 

 

 

 

 

250
200 -
E 150 -
'2'
‘3'
a 100 -
m I
50 P
0 ._ ~ I I I I I I h I
0 10 20 30 40 50 60 70 80 90 100
Time [second]
(b) Power recording
Figure 1.2 August 10, 1996 WSCC system blackout recordings
605 l
Kinp-Midp. 345 KV line trips
Measured Wheelon-American falls 138 KV line trips l
555 1- L
‘HELJAA—rij—JA--.\tti
505 ’ ‘I
455 Simulation Garrison 500 KV reactors trips
405 I I I I
0.00 10.80 21.60 32.40 43.20 54.00

Time [seconds]

Figure 1.3 December 14, 1994 WSCC blackout voltage recording

Since malfunction of the excitation control system is hypothesized to be at the center of
these blackouts, the description on how it has been modified and improved over time to

handle voltage instability problems is discussed in the next section.

1.3 Excitation Control System

The functional block diagram of a synchronous generator excitation control system [1]

is shown in Fig.1.4.

Voltage stability is affected greatly by the excitation control system. The basic function
of an excitation control system is to provide direct current to the synchronous machine field
winding and thus to control the internal induced voltage of the generator. In addition, the
excitation control system performs corrective and protective functions essential to the
satisfactory performance of the power system by controlling the field winding voltage and

thereby the field current.
The corrective functions include:

(a) Prevention of transient instability for faults, line outage, or generator outage
contingencies using a high ceiling voltage, short rise time exciter, and sufficient damping to

prevent multiple swing instability;

(b) Effective control of generator or network voltage through use of a line drop
compensator. Effective voltage control requires selecting a control bus or point in the
network where voltage is to be controlled and using a high gain exciter to maintain control

bus voltage at the desired setpoint; [1, 3]

(c) Effective coordination of generator exciter setpoints to assure effective primary

control of voltage at every network bus within limits of 0.95 5 IV] S 1.05 pu [33]. This

 

Woltage sensing &
I load compensation

 

Power system

 

 

stabilizer

 

 

 

‘ adjusit
———'

r?

AC voltage
adjust

    

DC

regulator
DC voltage

AC
regulator

I—

 

Ik

 

 

Voltage
sensing

 

 

 

 

Exc . system 7 ‘
Stab. circuits] ' I

 

 

I. _ _ Overexcitation I
limiter

Underexcitation l‘

_ _ - i limiter

 

 

_ _ _ V / Hz limiter
& protection

Fig. 1 .4 Block diagram of excitation control system

Generator l—l

 

. um

function is supplemented by under load tap changers and switchable shunt capacitors
especially at subtransmission and distribution system buses that are electrically distant from
generators. This coordination of generator exciter setpoints is also required to prevent a
generator fi'om exceeding its field current limit and thus from exceeding generator reactive

capability in the process of holding control bus voltage to setpoint values.

The protective functions [1, 2] associated with an excitation control system are:

(a) A dc regulator is a backup system for the ac regulator in case that the field current is
not reduced below the continuous rating limit after a suitable duration. This dc regulator
was first installed in the mid 70's as a means of preventing the kind of voltage instability
that caused the blackout on the Electricite de France in 1978. It replaces a manual system
which requires the operator to adjust field voltage and current on this backup manual
controller in response to stress on the system that reduces generator internal voltage via
armature reaction. The dc regulator provides a backup control that automatically adjusts
field current and voltage to continuously track the changes produced by the ac regulator
without exceeding field current limitation. If the ac regulator is disabled by an
overexcitation limiter relay, the dc regulator ensures that reactive supply and generator
voltage are maintained at what the ac regulator would require. In manual control, operators
might not continually adjust field voltage to values being produced by the ac regulator due
to inattention or other pressing matters. Reduction of reactive supply due to ineffectiveness
of manual control when the ac regulator is tripped by a field current limiter relay, can cause
a cascading trip of excitation control systems and reduction of voltage setpoints and thus
reactive outputs on other nearby generators that produce voltage collapse. Thus, the dc
regulator is a significant advance by preventing the reduction of generator reactive supply
on one or more generators when the excitation control systems are tripped by overexcitation

limiter relays.

(b) A Maximum Excitation Limiter (MXL) reduces field current on a generator below
continuous rating limits by (i) overriding but not disabling the ac regulator (ii) effectively
reducing the exciter's voltage setpoint through the summer as is shown in the excitation
control system block diagram (Figure 1.4). This MXL action prevents disabling the
stabilization of an excitation control system. The MXL allows the excitation control system
to retain the stabilizing effects of the power system stabilizer on oscillations and the effects
of the excitation control system on steady state stability. Without MXL, which was first
introduced more recently than the dc regulator, the ac regulator was disabled by a field
current overexcitation limiter relay whenever continuous field current limit violations
occurred for a sufficiently long duration or whenever the ac regulator was not acting
properly. Loss of the excitation systems' stabilization control could trigger voltage
instability. The MXL accomplishes reduction of field cmrent without losing the
stabilization effects of the excitation system or by providing the stabilizing signals if the

excitation control is overridden.
There are two relay based protective functions of the excitation control system [1, 2]:

(l) The field current limiter relay that disables the ac regulator with or without MXL
when the field current remains above continuous limit or when the excitation system/MXL
controller is faulty. The field current limiter relay should operate infrequently when a MXL
is present compared to when a MXL is not present. The field current limiter can be a two
step device that reduces field current to continuous rating in 10 seconds if field current
exceeds 208% of rating or in 30 seconds if field current exceeds 145% rating. An inverse

time overcurrent relay can also be used that keeps the field current below the specified level
[1].

A second inverse time field current relay can trip the generator if the dc [regulator does

not maintain field current below the actual field current limit curve that would produce

10

equipment damage if exceeded. This curve allows larger field current or duration than the

inverse time curve that trips the ac regulator with or without MXL [1].

(2) An armature current limiter relay that acts on armature currents in a similar manner
to field current relay. Both relays act to prevent overheating and equipment damage on the

generator. The armature current capability of the machine [1] is shown below:

Time(seconds) 10 3O 60 120
Armature current (percent) 226 154 130 l 16

1.4 Bifurcations in Power System

The bifurcation phenomena in the nonlinear power system refers to characterizing the
qualitative change in the response of the system for a smooth continuous change in

parameter 11 over a specified range. For the following model:

i = f(x.9.V..u)

O = GP(x’9’ Vmu)

O = GQ(x,6, V,,u)
where x is the generator dynamic state variables, 6 and V, are the angle and voltage at
generator terminal, high side transformer and load buses, u is the parameter of interest.
The fimction f represents the generator and excitation control systems on all generators in
the system, GP is the active power balance equation that describes the network power

distribution from generators to loads. GQ is the reactive power balance equation that

describes how reactive power is supplied and how voltage is controlled in the network.

The full Jacobian matrix of the above model is:

11

A1 Bl C1
J ([1) = A2 32 C2
A3 33 C3

32 C2

It has been proven [32] that if matrix Jc(,u)=|:
B3 C3

] is nonsingular, then an

equivalent differential equation model at = f (x, 0(x(11)), V,(x(/.t))) can be uniquely obtained

with equivalent differential Jacobian matrix

B, C, “ A,
W... a], c111]

Similarly, if matrix A101) is nonsingular, then an equivalent algebraic equation model
0 = g(x(0(p), V,(}1)),6(}1), V,(,u)) can be uniquely obtained with equivalent algebraic

J acobian matrix

32 C2 A2 -1
Jy(#)=[B C]-|:A:|[A1][Bl C1]

Saddle node bifurcation occurs at 11 = 11' when (i) Jem‘) is nonsingular, (ii) Jx(,u')
has one real eigenvalue that crosses the imaginary axis of the complex plane, and (iii)

certain transversality conditions hold.

Hopf bifurcation occurs at p = 11" when (i) Jc(,u') is nonsingular, (ii) .403) has
one pair of imaginary eigenvalues with zero real parts, and (iii) certain transverselity
conditions hold. There are two different Hopf bifurcations: (a) supercritical Hopf

bifurcation which occurs when a stable limit cycle exists around an unstable equilibrium

point and the stable limit cycle grows as 11 moves away from ,u‘; (b) subcritical Hopf

bifurcation occurs when an unstable limit cycle exists around a stable equilibrium point and

the orbit size shrinks to zero as ,u approaches 11'.

12

Singularity induced bifurcation occurs when the equivalent differential J acobian matrix
Jx(,u) is nonsingular with eigenvalues crossing from the right half plane to the left half

plane or vice versa by going through infinity, rather than across the imaginary axis. At this

point, one real eigenvalue of the system is at infinity and the rest of them are bounded [45,

4].

A sequence of bifurcations generally causes voltage collapse. A commonly observed

sequence of bifurcations [35] is shown in Figure 1.5.

 

 

 

 

 

 

1.00 A
B

,_ 0.75 1.
=1
3:
0
30 0.50 '
'e"
>
"3;: 0.25 -
A

0.00 I I I I

0 1 2 3 4 5
Load Voltage [p.u.]
(a) P-V Curve

l3

 

 

 

 

 

 

 

3
i:
a
a.
t‘
a
a
a:
E.
-2 - ............... ._ ......
_3 1 1 J 1 1
-1 1 2 3 4 5
Real Part

(b) Critical Modes Movement

Figure 1.5 A Typical bifurcation sequence

Hopf bifurcation is followed by node focus bifurcation, singularity induced bifurcation
and saddle node bifurcation. Such a bifurcation sequence has been observed in several
example systems [9, 35]. The exact point where blackout is inevitable has not been
established and may be at the point of node focus bifirrcation, and may be at the point of
singularity induced bifurcation. The point of singularity induced bifurcation is certain to

result in a blackout. Since [6] has shown that

(l) the algebraic model can be broken into coherent bus groups called voltage control

areas where each has a unique voltage collapse problem.

(2) There is a unique eigenvalue 1,, of the Jacobian J associated with each coherent

bus group.

(3) The voltage collapse for any coherent bus group occurs where certain subsets of

the generator excitation controller are overridden by maximum excitation limiters, and thus

14

lose control of voltage, and unable to provide additional reactive supply. The subset of
generators associated with the voltage collapse in a voltage control area is called the

associated reactive reserve basin.

(4). The reactive reserve basins are nested subsets of generators. It has been proven
[l 1] that exhaustion of reactive reserve via MXL action on each of the generators in the
reactive reserve basin causes the eigenvalue associated with the voltage control area to

approach zero.

(5). It has been shown via simulation [1 1] that exhaustion of some small reactive
reserve basin in a nested set causes a cascading exhaustion of reserve and MXL action that
loses voltage control on all of the generators in larger and larger reactive reserve basins.
This causes several eigenvalues of voltage control areas associated with larger and larger
reactive reserve basins to approach zero. This implies a sequence of singularity induced

bifurcation can occur.

The classical voltage instability problem, that caused first known voltage collapse in
Electricite de France in 1978, occurred long before the development of MXL and DC
regulators. This classical voltage instability problem occurred when overexcitation relays
disabled that excitation control system because there was no MXL to override the excitation
control system without disabling its stabilizing effects that include (i) damping oscillations
(ii) overcoming armature reaction effects and (iii) manual control effects that can reduce that
reactive power output of the generator after overexcitation limiter relay disable the excitation
control system. The bifurcation subsystem in a generator without an excitation control

system is for the saddle node bifurcation that produces instability has been shown to be the
flux decay state M3 in [32]. The instability in AE; is assumed to describe the voltage

collapse produced in Electricite de France. It will be shown that instability in AE; describes

a voltage instability that in itself is not voltage collapse but inevitably produces the
singularity induced bifurcation that does produce voltage collapse. The response of the AE;

15

state as this voltage collapse occurs due to the bifurcation sequence of saddle node
bifurcation followed by singularity induced bifurcation will be described in this thesis. The
analysis of how the algebraic equations can experience voltage instability is needed in

Chapter 4 before the resultant dynamic response can be described in Chapter 5.

The development of the dc regulator and the maximum excitation limiter was felt to
eliminate the possibility that this classical voltage instability could occur. The analysis and
simulation results in Chapter 3 shows that sufficiently large interarea oscillation produced
by Hopf bifurcation will disable the MXL and cause the excitation control system to be
disabled. The classical voltage collapse produced by saddle node bifurcation inevitably
followed by singularity induced bifurcation thus not only describe the Electricite de France
voltage collapse but also the three voltage collapse problems on the WSCC system where
large interarea oscillations existed before the equipment outage that produced the three

cascading voltage collapse problems on this system.

The results in [5] suggest that the loss of control and clogging voltage instability have
a counterpart in the differential or differential algebraic model. Clogging voltage instability
or algebraic bifurcation which occurs in the distribution system and remote to the location
of any generator has little or no impact on generator reactive generation as would
singularity induced bifurcation. Clogging voltage instability or algebraic bifurcation is
understood solely from load flow based results in [5] and is not studied in this thesis.
Singularity induced and saddle node bifurcation are two distinctly different differential
algebraic model manifestation of loss of control voltage instability in a load flow model.
Singularity induced and saddle node bifurcation have very different dynamic behavior
when they can occur [45]. The differences in dynamic behaviors of saddle node bifurcation
and how saddle node bifurcation may inevitably cause singularity induced bifurcation are
investigated in this thesis. This sequence of bifurcation may provide a recognizable

signature, a negative spike in bus voltage and real power transfer, that has been observed

16

on the WSCC system for each of the three major blackouts on December 14, 1994, July 2 ,
1996 and August 10, 1996.

A recent paper [11] established that the saddle node bifirrcation which occurs in the
flux decay dynamics is the description of the classic voltage collapse problem. It concluded
that: when the excitation control system is completely disabled via an field current limiter
relay without loss of PO or PV controllability, a reduction in generator internal voltage is
proven to occur due to increase of reactive load where an increase in internal voltage would
occur if the excitation control system (ac regulator) were present. If loss of PV
controllability occurs when the field current limiter relay disables the ac regulator, the
internal voltage Ext) dynamics are proven to be unstable. A slow decline in E;(t) is
evident in a simulation [1 1]. This voltage decline may be due to saddle node bifurcation.
Conditions for loss of PV controllability are not well understood and will be investigated
further in this thesis. Conditions for the excitation control system to be disabled by an

exciter limiter relay is also not well understood and will be investigated in this thesis.

The loss of PV controllability may cause the slow decline in E;(t) dynamics; undoing

the effects of the dc regulator to maintain voltage and reactive output. Although results in

[8] suggest such behavior, there are no theory or simulation results to confirm such

behavior. This thesis will attempt to prove that E;(t) is reduced if PV controllability is lost

and the ac regulator is disabled by the field current limiter relay.

It has not been proven that voltage collapse will necessarily occur when a single
generator exciter is disabled by field current limiter relay and a loss of PV controllability
occurs. The instability in E; (t) reduces voltage dynamically and may reduce Qg(t) out of
the generator that can ultimately cause loss of PO confiollability. Loss of PO controllability

is proven to cause instability in generator flux decay dynamics and cause a dynamic

increase in E50) [11]. A simulation result in [8], however, appears to suggest a very rapid

drop in E; (t) before PQ controllability is lost followed by a rapid rise in both after PQ

l7

controllability is lost. This behavior and the reason for it will be investigated theoretically
and experimentally. This rapid change in voltage appears as a spike and could trigger

armature current limiter relays or undervoltage relays that trip generators off line.

1.5 Hypotheses

According to the literature study concerning the WSCC system blackouts above, we

have the following hypotheses:

(1) There may exist a new kind of voltage collapse problem which is totally different
fiom the voltage stability problems which are currently understood and for which the dc

regulator and Maximmn Excitation Limiter have been designed to overcome;

(2) The current MXL is unable to limit the field current level due to the existence of
interarea oscillations, as will be proven theoretically and confirmed through simulation in

Chapter 3;

(3) The overexcitation protection relays trip off the MXL and ac regulator to use. the dc
regulator or manual control due to the malfunction of MXL in reducing field current. These

relays prevent the generators from damage due to rotor overheating;

(4) When the excitation control system is disabled without loss of PQ controllability or
PV controllability, a reduction in voltage in the transmission network can be initiated due to

increase in reactive load. Since the dynamics are stable, the dc regulator can increase VF,

I fd and Q6 if needed;

(5) If a slow loss of PV controllability [15] occurs when the excitation control system

is disabled, it will cause immediate saddle node bifurcation in flux decay dynamics and thus
loss of stability. The instability drives the internal voltage E;(t) toward zero. It is

18

anticipated that generator armature current relays may but not necessarily trip the unit off
line. It is believed that the generator armature current relays may be one explanation on

what tripped the McNary station generators off the system in the August 10, 1996 blackout;

(6) If PQ controllability [15] is slowly lost as E;(t) decline, E; (t) will decline more

rapidly as loss of PO controllability approaches;

(7) If a loss of PO controllability occurs, singularity induced bifurcation can occur.
The internal voltage response E;(t) becomes stable and rapidly increasing for generators

with active excitation control systems as well as for those generators where excitation
control systems are disabled. The negative side of the voltage spike in voltage E;(t) thus
occurs as loss of PQ controllability approaches and finally occurs, the positive side of the
spike occurs after loss of PQ controllability occurs. This spike is very likely the spike
observed on the WSCC system during the July 2, 1996 and August 10, 1996 blackouts;

(8) Afier the unstable generators are tripped off line, additional generators will try to
pick up the reactive power from the unstable generators as well as the decreased shunt
capacitive supply and increased network losses, the field current of these generators will
reach their limits and these additional generators will experience the same sequence of
bifurcations on those generators that initially get tripped. As a result, a spreading

uncontrollable voltage collapse will occur.

These hypotheses are developed based on (a) observations of the time responses of the
three recent blackouts on the WSCC system, (b) partially on the theory developed in [11]

that describes (1) what causes instability in the generator flux decay dynamic when
excitation control systems are disabled and the effects on AE;(t), (ii) what causes

instability in generator flux decay dynamics when the excitation control systems are active,

and (0) partially on the extensions of that theory given in this thesis that more clearly

l9

describes what appears to have caused these three blackouts. The extensions fall into three

separate contributions.

(A) Theoretical analysis and simulation results, that show the maximum excitation
limiters are disabled by sufficiently large oscillations, are given in Chapter 3 of this
proposal. The theory establishes why and when (the size of the oscillation compared to the
size of the violation of the continuous rating field current limit) the disablement occurs.
Large interarea oscillations existed on the WSCC system before the December 1994, July
1996 and August 1996 blackouts as shown in subsection 1.4 suggests that the maximum
excitation limiters may have been disabled and the overexcitation limiter relays may have
acted to disable certain excitation control systems and ultimately caused the tripping of
generators. The tripping of the McNary station generator unit prior to the voltage spike are

very likely to have occurred due to the phenomenon discussed in chapter 3.

(B) Extension of results in [15] that separate PV controllability of AQG into voltage
PV controllability and load PV controllability based on control of AQG using AE; and AQL
respectively. Similarly, PQ controllability of load bus voltages AV is broken into voltage
PQ controllability and load PQ controllability based on control of AV using AE; and AQL
respectively. It is proven that if load PQ controllability is lost and other conditions are
satisfied, then voltage PQ controllability, voltage PV controllability and load PV
controllability are lost. It is also proven that loss of load PQ controllability can occur if
shunt capacitive susceptance is added to every bus in a voltage control area, and that the
margin to the point of loss of load PQ controllability increases for the addition of shunt

inductive susceptance. It is shown that loss of voltage PV controllability can occur before
loss of load PQ controllability occurs. Finally, the test matrix elements for SQL 12.1 , Sn;

and SQGQL associated with a voltage control area, that slowly progresses toward loss of

load PQ controllability, can approach infinity, discontinuously change to negative infinity,

and then increase toward zero. These properties of the algebraic submodel are all needed to

20

prove the reasons why stability is lost and why a sharp narrow spike in voltage occurs that
was observed in the recorded and simulated responses of the WSCC system for the three

recent blackouts. These results are given in Chapter 4 of this thesis.

(C) Prove where and why stability in M5 dynamics is lost when the MXL is disabled

by oscillations and the overexcitation limiter relay disables the excitation control system.
Loss of voltage PV controllability in a voltage control area, that can happen long before
loss of load PQ controllability occurs in that voltage control area, causes instability in AB;
dynamics The system is shown to be stable if voltage PV controllability holds in the voltage

control area of the generator internal bus where the excitation control system is disabled.

. I
The unstable response rs proven to cause voltage AEq on the generator to approach zero.

(D) Prove that this instability in E; (t) is like computing a Q-V curve at the generator

M
internal bus if {SQGE h < 0 and 2", [S965 ], < 0 and load PV controllability still holds.
,.

This dynamically administered Q-V eln've stress test is proven to lead inevitably to (i) loss
of load PQ controllability and bifirrcation of the algebraic submodel, and (ii) singularity
induced bifurcation of the differential algebraic model.

(B) Prove that loss of load and voltage PQ controllability in a voltage control area

causes instability in generators with active excitation control systems in that voltage control

area and a rising voltage AE; .

(F) Prove that slow continuous loss of load PQ controllability in a voltage control area

causes a sharp negative spike in M3 on a generator with its excitation control system
disabled. The slow development of the loss of load PQ controllability causes voltage M;
to drop sharply, and a sharp rise in AE; occurs after load PQ controllability is lost.

21

(G) Show that a saddle node bifurcation followed inevitably by a singularity induced
bifurcation occurs when the excitation control system is disabled and a loss of voltage PV

controllability occurs.

The results in (C-G) are contained in Chapter 5 of this thesis and are proven

theoretically and confirmed via simulation results.

1.6 Relevant Literatures

Sensitivity analysis has been used to assess proximity to bifurcation at an equilibrium
point. Sensitivity becomes large at the point of bifurcation [56, 57]. Trajectory sensitivity

has been used to assess proximity to the boundary of region of attraction [56, 57 ].

Costi [15] used specific sensitivity matrices to assess proximity to bifurcation in the

algebraic model of a power system.

Hu [8] related the network sensitivity matrices to assessing stability of the trajectory on
a linearized transient stability model. The network sensitivity matrices were related to
controllability of the voltage at load buses and the generation at generator buses. The
criteria for controllabilities were based on experience of the requirements for stable

behavior in the network.

The sensitivity matrices for controllability are further investigated in chapter 4 of this
thesis in order to describe their behavior which affects the stability of the linearized
transient stability model. The effects of the behavior of the controllability matrices are

investigated in chapter 5.

22

CHAPTER 2

POWER SYSTEM MODELING

2.1 A General Power System Model

A general power system model, which includes mechanical dynamics, flux decay
dynamics, excitation control system dynamics, and real and reactive power balance
equations, can be used to test for power system voltage instability. This general power
system model consists of two different kinds of nonlinear equations: one is a set of
nonlinear differential equations which represents the dynamics of the generator, the other is
a set of nonlinear algebraic equations which represents the real and reactive power balance

equations for each bus in the network. This model has the following form:

M05+D08+FG(6,6,E;,VL,y,)—PG =0 (2.1)
F,(5,6,E;, VL,y2) — P, = 0 (2.2)
E; + 60(6,9,E;,V,,y,) = 0 (2.3a)
01(5. 9. 15;. Vuyz) - Q. = 0 (2.31:)
y, + H(6, 0,E;, V,,y,) = 0 (2.4)

An output equation indicates the reactive generation at the generator internal buses

satisfies:

09(6, 9,15; V,, y,) — Q, = 0 (2.3c)

23

where 6 is the generator angle referenced to a synchronous rotating reference flame and

E 5 is a voltage proportional to generator flux linkage; 6 and VL are the angle and voltage
at generator terminal, generator transformer high side, and load buses; y2 is the state of the
excitation control system and power system stabilizer dynamics. Equation (2.1) represents
generator internal dynamics, equation (2.2) represents real power balance at all other buses
in the system, equation (2.3a) is the flux decay differential equations, equation (2.3b) is
reactive power balance at all other buses in the system, and equation (2.4) is the generator

excitation control system and power system stabilizer dynamics.

The real and reactive power balance equation (2.2) and (2.3b) are:

P, = V1.20... V, cos 9,, + 3me sin 9:...)
m=l

Q, = V,2(G,U"Vm sin 9,,"I —B,m Vm c059,“)
m=l

At any parameter pattern and level 11 = [1,, the solution (Xe) is a solution to:

 

 

 

 

F(Xe9ue) = 0
where:

_ 5 _ "PG-

0 P:
X: E; and F (5. 9.19;, Vuyz) = Go (2.5)

V1. Gr

-yz . _ H _

The Jacobian of (2.5) is:

24

3F ,
J = 5(69 GaEqs VLayz)

fA, B1 C1 D1 E1
A2 32 C2 D2 E2 (2 .6)
= A3 33 C3 D3 E3
A4 B4 C4 D4 E 4
_A5 35 C5 D5 E5 _

 

 

2.2 Excitation Control System Model

The block diagram of a excitation control system model is shown in Figure 2.1.
Ger (5) is represents the transfer function of the AVR (Automatic Voltage Regulator) and

is the

 

 

any type of exciter, K3 is the transfer function of the field circuit,
1 + 5T3 1+ ST R

transfer function of the voltage transducer, and GP” (5) is the transfer function of the power

system stabilizer.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1 Excitation control system model

25

It is similar to Figure 1.1 except that the dc regulator and the overexcitation limiter,
underexcitation limiter, and volt/hertz relays are omitted. The model of the generator inertial
dynamics (2.1), flux decay dynamics (2.3a) and generator excitation control system, power

system stabilizer and voltage sensor dynamics (2.4) are shown in Figure 2.1.

The maximum excitation limiter is now described that works through the summer of
the excitation control system as shown in Figure 2.2. The role of the maximum excitation

limiter is described in section 2.4.

2.3 Maximum Excitation Limiter Model

A typical maximum excitation limiter model is shown below

 

Generator

 

Voltage .
egulator

 

 

 

 

 

 

 

 

 

 

 

 

  

Overexcitation limiter

 

 

 

Figure 2.2 Model of maximum excitation limiter

26

During normal conditions with field current less than the setpoint (typically 105% of
rated field cmrent), path 1 and 2 both force the integrator to its lower limit(-M). The input
to the voltage regulator summing junction (X5) is zero. If field current is above the
setpoint, path 2 drives the integrator towards a positive value. Once the integrator output
becomes positive, the voltage regulator will start to reduce field current. For a step increase

in field current above the setpoint, the time to the start of current limiting will be:

M
K2 ' K3 ' (Ifd " l’OSIfdrated)

 

The parameters can be adjusted so that this time approximates the desired time to
obtain field current limiting. Once field current is reduced to the setpoint, path 1 rapidly

resets the integrator output X 4 to —M.

2.4 Power System Load Model

Load dynamic response is an important mechanism of power system voltage stability,
load modeling is a difficult problem because power system loads are aggregates of many
different devices such as fluorescent and incandescent lamps, refrigerators, heaters,
compressors, motors, furnaces, and so on. The exact composition of load is not only
difficult to estimate, but also changes depending on many factors including time, weather

conditions and state of the economy.

The load models are traditionally classified into two broad categories: static load

models and dynamic load models.

27

The static characteristics of load are usually classified into three categories: constant
power, constant current and constant impedance load. A widely used load characteristic is

the well-known exponential load, which has the general form:

P 2 211(1)"
V0
B
V
“Eli/z]

where z is a dimensionless demand variable, V0 is the reference voltage, and the

exponents a and fl depend on the type of load(motor, heating, lighting, etc.). Note that
2P0 and zQ0 are the active and reactive powers consumed under a voltage V equal to the
reference V0 and relate to the amount of connected equipment. These have been called

nominal load powers, in contrast to the consumed powers P and Q.

2
P = zP0 {-5—}
0

When a=fi=2,

it is called constant impedance load.

When a=B=l,

it is called constant current load.

Whena=13=0,

28

P=zP0
Q=ZQ0

it is called constant power load.

An alternative load representation is based on summing up load components which
have the same exponents. A special case is made up of three components: constant

impedance, constant current and constant power, it is given by the following quadratic

P = zP0[:aP(—I:] + bp[£] + CF]
V0 V0
Q = 2Q0[0Q(-:;—) + [94%] + cg]

where aP+bP+cP=aQ+bQ+cQ=L while zP0 and on are the load real and reactive

expressions:

powers consumed at the reference voltage V0.

Even though the use of static load models is often satisfactory, there are, however,
many cases where it is necessary to account for the dynamics of load components. For
example, study of systems with large concentrations of motors requires representation of
load dynamics. Researchers developed various kinds of dynamic load models [30, 38, 39,
40, 41] to represent the dynamic characteristics of the system loads. Since this subject is

beyond the scope of this thesis, it is not going to be discussed in detail here.

It is worth noting that load can transition between dynamic load model and static load
model. For example, the static models mentioned above can be accomplished using the

generic non-linear dynamic model which satisfies:

29

. V a, V a,
TPZP=i7i 447)
0 0

13. B.
V0 V0

This change in load models is needed later to discuss why specific kinds of

biflucations occur and do not occur.

30

CHAPTER 3

PROBLEM ANALYSIS

A large negative voltage spike was observed on the WSCC system shortly after the
McNary station generators were tripped. This large negative voltage spike could have been
resulted when the AC regulator portion of the automatic voltage regulator is disabled and
the network violates rather weak controllability conditions at generator buses [11, 57, 58,
59]. The theory in [11] indicates the unstable flux decay dynamics on the McNary
substation generators, that resulted from disablement of the AC regulator and loss of
network controllability on these generators, caused a self destructive stress test to be
administered. This stress produces the negative voltage spike and ultimately causes the
cascading instability of additional subsystems that may have produced the August 10, 1996
blackout on the WSCC system. The network controllability conditions are often violated in
many power systems but have no negative consequences because the AC regulator
overcomes this loss of controllability. Failure of the MXL to reduce field cmrent followed
by disablement of the AC regulator via an Overexcitation Protection Relay leaves the
system vulnerable to a cascading instability of bifurcation subsystems that lead to blackout.
The understanding of how the MXL fails to reduce field current is studied in this chapter to
understand how and why this sequence occurs. This chapter also provides suggestions on
what may be done to avoid such failures in the Maximum Excitation Limiter that lead to
OXP relay tripping of the AC regulator, instability in the generator dynamics that
administer a self destructive stress test and finally cascading instability of bifurcation

subsystems and associated network and generator protective relaying actions.

31

3.1 Disablement of Current MXL due to Oscillation

The traditional overexcitation limiting and protection mechanism is as follows: The
MXL detects the high field current condition and acts through the ac voltage regulator to
return the level of excitation to a preset value after a time delay during which overexcitation
is permitted. The MXL is supplemented by the Overexcitation Protection (OXP) device. If
the MXL finlction is unsuccessful, the OXP trips the ac regulator after a preset time
interval, and then transfers control to a dc regulator and repositions the setpoint to a value
corresponding to the rated value. If the tripping of the ac regulator and the repositioning the
setpoint has not reduced the field current below I fd med, the OXP will also initiate a unit
trip after an additional time delay. When oscillation components exist in field current,
sometimes the MXL cannot work correctly. This results in the trip of the excitation control
system, sometimes, followed by the generator unit trip, and even voltage collapse of the

system if the conditions exist as derived in Chapter 5.
3.1 1 r ' An sis

The analysis here is based on the maximum excitation limiter system which is shown

in Chapter 2 (Figure 2.2).

Suppose that the field current I fl, is composed of a constant component 10 and an

oscillating component Il cos271fi, that is,

Ifd(t) = 10 + II cos2717’t

(3- 1)
= I0 + II cos 5(t)

where

6(t) = 277?

32

and f is the frequency of the oscillation observed in voltage, angle and field current.

 

 

 

 

Figure 3.1 1,4 with oscillation component

As is shown in Figure 3.1, A is determined by
10+],cosA=R (3.2)

where R is the continuous field current rating where field current limiting fi'om the MXL
occurs. Usually R = 1.051fmwd.

The expressions for the area a, (above R) and area (12 (below R) are now derived,
these expressions are then used to analyze the effects on X 4 and X5 in Figure 2.2. Note
that the positive change in X, in Figure 2.2 for If, > R over (-A, A) is A, = K,a, and the
negative change in X4 for Ifd <R over (A, 27t-A) is A2 =(K, +K2)a2. If A, >A2 for
each cycle, then X 4 increases, and X 5 will eventually become positive and reduce the

excitation setpoint. If A, < A2 for each cycle, then X 4 decreases toward the lower limit

- M, and X5 will stay at or decrease to zero.

Area 0, satisfies

a, = flu, + 1, coer - R)d6

(3.3)
= 2(10A - RA + II sin A)

33

Noting that R = 10 + 1, 003A, (1, becomes

a, = 2(1, sinA — AI, cos A)
= 21,(sinA - AcosA)

The change in X, over interval (—A, A) is

A1 = K201
= 21,K2 (sinA — Acos A)
= 21,8,
where
B, = K,(sinA — Acos A)
Area a2 satisfies

a2 = I:”_A(R— I0 — I, cosd)d6
= (R- 1,,)(2n:— 2A) +21, sinA

Again noting that R = 10 + 1, 008A, a2 becomes

a2 = 2(71:I,cosA — AI, cosA + I, sinA)
= 21,(sinA - AcosA + 7rcosA)

The changein X, over interval (A, 27r-A) is

A2 = (Kl + K2 )0,
= 211 (K, + K, )(sinA — AcosA + JrcosA)
= 21,13,
where

B2 = (K, +K,)(sinA — AcosA + IrcosA)

34

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

According to the value of the constant component I0 and the preset value R (usually

R = 1.051 f, med), it can be classified into three cases:

Case 1: 10 <R

When 10 < R, A. must be less than g. Since A, < A, always holds, the oscillation

component does not affect X 5 in Figure 2.2 and it remains zero, The voltage regulator set
point is not affected and the excitation system works fine. (It may make X 4 go to the lower

limit value faster).
Carrel: 10 > R
When 10 > R, A. must be greater than ~25. For this case, we have A, > A,, A, = A,

and A, < A, three subcases.

(i) If A, > A,, the MXL is able to detect the high field current condition and reduces

the voltage setpoint on the AC regulator as expected( it may speed up the upper limiting);

(ii) If A, < A,, the MXL will mistaken the high field current (1,, > R) as low field

current and not reduce the voltage setpoint on the AC regulator, so it fails. This is where
the oscillation component will disable and confuse the MXL, and cause the overexcitation

protection relay to trip the excitation control system and the maximum excitation limiter;

(iii) If A, = A,, the oscillation effects are cancelled out, the MXL does not increase or
decrease X4 thus X5 even though 10 > R. If X, = 0, then the field current limit is going to

be violated, and the MXL does not reduce 1,. As a result, the excitation control system

will be tripped by the overexcitation protection relay. If X5 > 0 and this value is sufficiently

large, then X, will reduce If, to below R. to prevent unit tripping If X5 > 0 and this value

is not sufficiently large, then the excitation control system must be tripped by the

35

overexcitation protection relay to prevent thermal damage to the generator field winding or

the rotor.

Case 3: I0 = R
When 10 = R, A must be less than {25' Since A, < A, always holds, the MXL always

goes to lower limiting as it should. In this case, X 4 can be affected, but X5 is not affected.

0
3.1.2. ALL ° 03 Cat A‘W ‘ ‘ 11);.” Drug-m :9 01‘ - :10 M 1-1 Hr.

Suppose that A’ is the boundary point where A, = A,, or equivalently, B, = B,. The

value of A' can be determined by the following equations:

10 + I, cosA‘ = R
(3.11)
B1 = Bz
These equations are equivalent to
10 +1, cosA” = R
- a a s - a e a a (3’12)
K,(srnA —A cosA )=(K, +K,)(s1nA —A cosA + 7rcosA)

A‘ depends on K, and K, based on (3.12). For example, if we set K, = 100 and

K, = 1, we obtain

B, = sin A" - A" cos A‘
B, =101(sinA‘ — A‘ cosA’ + ncosA“)

36

 

 

 

2 2.2 2.4 2.6 A" 2.8 3 A

Figure 3.2 Plot for determining A‘

The plots of B, and B, are shown in Figure 3.2. From the plot, we get A’ = 2.698. If
we set 11mm! = 1.0, R: 1.05, then, from equation (3.12), we obtain the relationship

between 10 and I, when A' = 2.698
10 = 1.05 — cos(2.698)1, (3.13)

Equation (3.13) that provides a boundary in 10 1, space for when MXL operates

correctly and when it malfunctions is shown graphically in Figure 3.3.

37

I” A 2),.

(lo - R)

--------------------------------- A=rt/ 2

 

0.8 _

 

 

0.6 1 ' ' 1 up
0 0.2 0.4 0.6 0.8 11

Figure 3.3 Relationship between 10 and I,

Since the Maximum Excitation Limiter works incorrectly over 71'/ 2 < A < A' , and the
relationship between 10 and I, is given in Figure 3.3, we can conclude that the area

indicated by the arrows in Figure 3.3 is where the MXL works incorrectly. When 10 > R, a

complete description of when the MXL works correctly and incorrectly is given below in

terms of A’:

(a) With oscillation component, MXL works correctly when A 5 7r/ 2 (or equivalently
A, < A, with 10 S R) and when A _>. A' (or equivalently 10 > R with A, 2 A,). The second

case A Z A' suggests that the MXL will act to alleviate the field current violation;

(b) MXL does not work correctly when rr/ 2 < A < A'. (or equivalently 10 > R with
A, < A,);

38

(c) When A 2 A', so that the MXL is acting correctly, increase in size of oscillation 1,

given a specific value of 10 will cause A < A' and thus the MXL to malflmction as shown

in Figure 3.3;

(d) When It/Z < A < A' , so that the MXL is working incorrectly, increase in average

field current value 10 given a specific value of 1, will cause A .>_ A' and thus the MXL to

operate properly.

.1 ' ' Rslts

The problem of MXL disablement can be shown clearly by the following simulations.
Case 1; (10 < R, A <15)

For this case, the simulation parameters of the MXL shown in Figure 2.2 and the field
current expression 3.1 are: M=1.5, K,=100, K,=l, K,=l, R=1.05, 10:1.0, I,=0.2.
Here, I0 < R, A <1; the MXL should not act, that is, X, should remain zero. Figure 3.4
shows that X 4 reaches its lower limit after a time delay because of the integration, and X5

remains zero for all the time regardless of the existence of the oscillation component. The

MXL works correctly.

39

 

2.5 r 1 1 l 1 1 r r

 

2 - .4
1.5 _ "
R Ifd (t)
1 _\//—\\//—\\/<
0.5 - "
X5

 

-0.5

Magnitude
o
J
I

 

 

 

 

 

X:
-1.5 '-
-2 — h
_2.5 I I L M l I l I
0 2 4 6 8 10 12 14 16 18

Time
Figure 3.4 The MXL action when 10 < R

7:
MI. (10 = R: A = 3)
For this case, the simulation parameters are: M=l.5, K,=100, K,=l, K,=1,
R=1.05, 10:1.05, I,=0.2. Here, 10 =11, 13:; the MXL should not act, that is, X,
should remain zero. Figure 3.5 shows that X , and X5 go above zero a little for a short time

at the very beginning because X, = 0 at t= 0, then X, decreases and reaches its lower
limit — M, and finally oscillates around that lower limit. X5 remains zero after a short
interval where it is positive. Since the positive period for X, and X5 is very short, and the

magnitude is negligible, the MXL works correctly as intended.

40

 

2.5 l T I I I I f T

 

1.5 »- 1,,(t) «
\ R A /\ /

1 — V V V
0.5 - -

Imagnnude
O
3::

 

 

 

 

 

(15 - s
1 .- -1
X4 /\
-15 - “‘
.2 .. -
-25 l 1 1 4 l l 1 l
0 2 4 6 8 10 12 14 16 18
Tune

Figure 3.5 The MXL action when 1,, = R

mapa D125)
For this case, the simulation parameters are: M=l.5, K,=100, K,=1, K,=l,

R=1.05, I,=1.055, I,=0.2. Here, 10>R, 13>; with A, <A,, the MXL should act

since 10 > R, that is, X, and X, should be positive and go to the upper limit +M after

some time delay if it works correctly. Note that 10:1.055 is just slightly above R=l .05,
thus A is just slightly above 375 and less than A'. A is within the range where the

oscillation disables the MXL. This analysis suggests that the oscillation component keeps
A, < A,, this makes it impossible for X, to keep positive and reach its upper limit.

Therefore, X5 remains zero most of the time and the MXL works incorrectly. Figure 3.6

displays the result that confirms this point.

41

 

2.5 I l I I I I I I

1.5— 1,110) 7

0.5 - A

Magnitude
O
l

l

-0.5 -

 

 

 

_2.5 1 L l 1 l 1 r l
0 4

Time
Figure 3.6 The MXL action when 10 > R but A, < A,

Ce§e4;(10>R,A>A*)
For this case, the simulation parameters are: M=1.5, K,=100, K,=1, K,=1,
R=l.05, Io=1.07, I,=0.2, except that 10 is increased from 1.05 to 1.07, all the other

parametersarethe sameasthoseincase3. Here, 10>R, A>§ with A, >A,, theMXL

should act correctly since 10 is sufficiently greater than R for the particular I, value. X,
and X 5 should be positive and go to the upper limit after some time delay. Since the
oscillation component remains small for the 10 - R value, A, > A, , X, remains positive

and near its upper limit. Therefore, X5 reaches and remains at its upper limit after a short

42

time delay and the simulation confirms that the MXL works correctly. Figure 3.7 confirms

this analysis.

 

2.5 I I I I I I T I

X41XS

 

1.5 -
1,,(1)

 

 

Magnitude

0.5 ~

-0.5 -

 

 

 

J
,—
)—
r—
,_
.—
)—
i-
.—

Figure 3.7 The MXL action when 10 > R but A, > A,

(The effect of increasing 10 for certain I, )

Qa§_e_5_;_(IO>R, A < A * due to increase in 1,)

For this case, the simulation parameters are: M=1.5, K,=100, K,=1, K3= l ,

R=1.05, 10:1.07, I,=0.4, except that I, is increased from 0.2 to 0.4, all the other

18

parameters are the same as those in case 4. Here, 1,, > R, A >325 with A, > A,. This case is

the same as case 3. From Figure 3.8, it is clear that the MXL does not work correctly.

43

 

2.5 r , , T ,

"N . i /\1 /
\/

-0.5 -

 

 

 

 

 

 

 

Magnitude
“ o
I I l
,><
L X l I

 

 

 

-1.5
.2 .. A
-2-5 1 l l 1 1 l l
0 4 6 8 10 12 14 16 18
Time

Figure 3.8 The effect of increasing 1, for certain 10

These results show that when 10 is slightly above R with certain amount of oscillation
component, the MXL is disabled, but the MXL will again be enabled if 10 increases
sufficiently. Another result is that for any 10 value, if the magnitude of the oscillation
component increases sufficiently, the MXL will be disabled. The large magnitude of the
oscillation component (1,) in the WSCC system for a light stress (10 - R small) would
suggest why the McNaly station unit excitation control systems to be disabled and the use

of overexcitation protection relays to disable the excitation control system and its MXL.

3.2 Excitation Control System Tripping and Unit Tripping

If MXL is unable to limit the field current level due to the existence of oscillations that
disable it, the OXP (Overexcitation Protection Relay) will trip the ac regulator to do
regulator or manual control and disable the excitation control system. When the excitation
control system is disabled without loss of PO controllability or PV controllability, the
armature reaction on the generator is no longer masked by the excitation control system due
to increased load on this generator. Generator internal voltage is reduced due to increased
reactive generation output. The reduction in E; (t) can cause a significant reduction in
voltage in the transmission network, and increased network reactive losses and reduced
shunt capacitive supply. This, in turn, is followed by further reduction of internal voltage
E g (1) due to armature reaction in order to override possible generator response for decrease
in network shunt capacitive reactive supply and increase in network reactive losses due to
the drop of network voltage. Manual control requires the operator to adjust the field voltage
manually so that the generator terminal voltage E g (t) and reactive power are about the same
as what are produced out of the generator through excitation control system action. A dc
regulator attempts to maintain field current at rating level but is not a feedback controller on
generator terminal voltage as the excitation control system is. The use of a DC regulator or

manual control does not overcome armature reaction that reduces internal voltage.

The complete disablement of the excitation control system and MXL via an
overexcitation protection relay for a generator together with a slow loss of PV
controllability causes immediate saddle node bifurcation in flux decay dynamics and thus
loss of stability [22]. The instability drives the internal voltage to decrease slowly toward

zero. As E,’ (t) is reduced, loss of PQ controllability may result in E;(t) to decrease more

rapidly. Undervoltage relays or armature cmrent relays may trip the writ off line.

45

If a loss of PQ controllability occurs, singularity induced or saddle node bifurcation
can occur. The internal voltage response E;(t) is stable and rapidly increasing. The MXL

is not generally fast enough to overcome the increase in E g (t) by driving the generator

excitation control system setpoint to zero and E g (t) continues to increase rapidly. As a

result, the OXP relay would trip the generator. This is likely what caused the tripping of the
McNary station generators during the August 1996 blackout on the WSCC system.

3.3 Uncontrollable Spreading Voltage Collapse

After the unstable generators were tripped off line, additional generators will try to
pick up the reactive power from the unstable generators as well as the decreased shunt
capacitive supply and increased network losses, the field eturent of these generators will
reach their limits and these additional generators may experience the same sequence as
those generators that initially get tripped. As a result, a spreading uncontrollable voltage

collapse will occur.

46

CHAPTER 4

VOLTAGE AND LOAD CONTROLLABILITY

4.1 Introduction

Sensitivity indices have been used to help assess proximity to instability in a load
flow model and in linearized mid-term power system stability models from the inception
of the investigation of voltage instability at an equilibrium. The use of sensitivity
matrices other than sensitivity indices was first adopted in [15] and was applied to both
load flow and mid-term stability models. More recently, sensitivity matrices have been
used to assess proximity to instability along a transient trajectory [12, 56, 57]. Sensitivity
matrix elements have been shown to increase rapidly as the algebraic model, differential
algebraic, or differential model equilibrium approaches local bifurcation[12, 56, 57, 58]
or when the trajectory of the differential model approaches the boundary of the region of

attraction[12, 56, 57].

This chapter utilizes the sensitivity matrices upon which PQ controllability of load
buses and PV controllability of generator buses are defined and are related to the
sensitivity indices that were developed at the inception of voltage instability research [33,
62]. The condition for PQ controllability of load buses and PV controllability of
generator buses is based on common engineering judgement of the conditions that reflect

proper control capability and response of a transmission or distribution network.

47

In this chapter, firstly, the results on PO and PV controllability in [15] are extended.
PV controllability of AQG is separated into voltage PV controllability and load PV
controllability based on AE; and AQ, changes respectively. Similarly, PQ controllability
of load bus voltages AV is broken into voltage PQ controllability and load PQ
controllability based on AE; and AQ, changes respectively. Secondly, it is shown that

-l . .
the test matrix elements for SQLV , Sn; and SQGQL assocrated wrth a voltage control

area slowly progress toward loss of load PQ controllability, (a) approach infinity, (b)
discontinuously change to negative infinity, and (c) then increase toward zero. Thirdly, It
is proven that if load PQ controllability is lost, then voltage PQ controllability, voltage
PV controllability and load PV controllability are lost. It is also proven that loss of
voltage PV controllability can occur before loss of load PQ controllability occurs.
Finally, it is proven that loss of load PQ controllability can occur if shunt capacitive
susceptance is added to every bus in a voltage control area, and that the margin to the
point of loss of load PQ controllability increases for the addition of shunt inductive
susceptance. Each of these results are related to instability in generator flux decay
dynamics through simulation results where these behaviors are observed. The simulation
results also validate the theoretical results. These theoretical and simulation results are
derived in order to theoretically establish (in chapter 5) the time behaviors that lead to
production of the sharp negative voltage spike observed in December 1994, July 1996
and August 1996 blackouts on the WSCC system.

4.2 Sensitivity Matrices

For a non-decoupled differential algebraic model

48

 

 

 

 

 

'AP,‘ "A, B, C, D," A6 1
AP, A, B, C, D, A0
= . . 4.1
AQ, A, B, C, D, AEq/E, ( )
_AQ,_ _A, B, C, D,__AV/V_
APO

A6
[A6] can be determined by first setting [ ]= 0 in the first two equations in (4.1)

AP,

and solving to obtain:

A6 A, B, "‘ C, D, A5,;
=- (4.2)
A9 A, B, C, D, AV
. . A5 . . .
Substituting [A6] in the last two equations 1n (4.1) produces a reduced model
AQG = AE; 43
AQ, AV (')
where
J=[Jll J12-
J21 J22,
,6. 1),, FA. all!!! B.]"'[Ct 0.] (4'4)
C4 D4 _A4 B4 A2 32 C2 D2

 

Here, the sensitivity model can now be derived from (4.4) and has the form

AV/V = SQLV“AQ, + SV, (AE; /E,;) (4.5)
AQG = 59.5 (Mi/ED " 59th AQ, (4-6)

The sensitivity matrices are defined as

49

SQLV = 22

SVE = —SQLV—IJ21
SQGQL = -J128 LV—l (4'7)
SQGE = Jll " JIZSQLV—IJZl

The J acobian can be based on a constant power model as in the Newton load flow or
as in a decoupled load flow where C,, C,, D,, D,, A,, A,, B3 and B, are null. In a
decoupled load flow, one could assume that angle differences are small and voltage
differences are small so that the remaining J acobian elements are proportional to the
adrnittances connecting buses in the network. A further step is to assume that the matrices
include shunt admittance or constant current load [54]. A combination of constant

current, constant impedance, and constant power model is ofien used.

In this thesis, the non-decoupled constant power load model is used when saddle
node bifurcation is studied. Saddle node bifurcation occurs when an eigenvalue of the
equivalent differential model approaches zero which implies that the response is
infinitely slow at the bifurcation, all tap changers and switchable shunt capacitors in the
distribution system have time to complete their action to cause voltage and load recovery.

Thus, a constant power load model is appropriate for study of saddle node bifurcation.

Singularity induced bifurcation occurs in the network equations. This bifurcation can
develop as tap changers, switchable shunt capacitors, and generator electrical and inertial
dynamics respond to result in an equilibrium in an infinitesimal time interval. A constant
power load model reflects the effects of distribution level under load tap changers and
switchable shunt capacitors over an infinite time interval, but does not reflect the
transition from saddle node bifurcation to singularity induced bifurcation if the excitation

control system is disabled by an overexcitation protection relay and loss of voltage PV

50

controllability has occurred on the generator. The conventional combination of constant
current, constant impedance, and constant power load model is not adequate to represent
the aggregated distribution network and load over this transition from an infinitely slow
instability to an infinitely fast instability. This is because the nature of the load model
must change dynamically over time due to tap changer and switchable shunt capacitor
controls aggregated in the load that change constant impedance and constant current
loads to behave as constant power loads. The induction motor load dynamics and
thermostatic load dynamics are also ignored if the conventional combination of constant

power, constant impedance and constant current load models are used.

Recent changes of power engineering load modeling [49] practice has witnessed use
of a short time transient and a long term steady state load model to handle induction
motor load and thermostatic load dynamics. as well as the action of tap changer controls
and switchable shunt capacitor controls in the distribution system. This short term and
long term load modeling allows a transition from one to the other over time as discussed
in Chapter 2. The long term model is very heavily constant power reflecting the effects of
thermostatic load recovery over time and the load recovery due to underload tap changers
and capacitors actions on voltage. The short term load model is often constant current and
constant impedance combinations. The long term model is used in computing a stable
equilibrium and the eigenvalues near saddle node bifurcation. The short term model is
used in computing the eigenvalues and eigenvectors of the differential equation model

near singularity induced bifurcation.

The difficulty in studying a bifurcation sequence is that the equilibrium is unstable
after the first bifirrcation. The use of a constant power model to compute the eigenvalue
and the long term changes in the equilibrium response before and shortly after bifurcation

for a saddle node bifurcation with an infinite steady state response time at bifurcation is

51

appropriate. Once saddle node bifurcation occurs and is transitioning to singularity
induced bifurcation, one is no longer in equilibrium, a short term load model is
appropriate in attempting to analyze the response. This is especially true if the unstable
response afier saddle node bifurcation is becoming ever faster producing the negative

voltage spike.

4.3 Voltage and Load PV Controllability

4.3.1 Veltage PV Centlollebility

Voltage PV Controllability relates control of reactive generation( Q) to generator

internal voltage( EJ) at generator(PV) buses. For a system to be voltage PV controllable,

the reactive power injection at the PV buses should increase when the load at PQ buses is

fixed and the voltage at some PV buses is increased

Definition 4.1

A system is voltage PV controllable at any time t along the transient trajectory or at

an equilibrium point if: when AQ, = 0, any nonzero nonnegative control AE causes the

M
PV state AQG to become such that XAQG, > 0 and if AE, > 0 with AE, = 0 for all

i=1

j¢i then AQG, >0 and AQQ, S Oforalljati.

Theorem 4.1
A system is voltage PV controllable if and only if the matrix SQ G E is such that

M
2189,51,, >0, j=1,...,M
i=1

52

>O,i=j
{SQGEh-{s 0, lat j

This proof is omitted but can be obtained from results in [15].

4.3.2 L ad PV ntrolla ili

Load PV Controllability relates control of reactive generation( Q) at generator(PV)
buses to load( Q,) at load(PQ) buses. For a system to be load PV controllable, the reactive

power injection at the PV buses should increase when the voltage at PV buses is fixed

and the load at PO buses is increased.

Definition 4.2

A system is load PV controllable at any time t along the transient trajectory or at an
equilibrium point if: when AB = 0, any nonzero nonpositive disturbance AQ, cause the

PV state AQG to become nonzero nonnegative.

Theorem 4.2
A system is load PV controllable if and only if the matrix SQ G Q L satisfiing

Q; =- SQer AQ, is nonnegative and no zero columns.

This proof is omitted but can be obtained from results in [15].

4.4 Voltage and Load PQ Controllability

4.4.1 V P ontr lla ili

53

Voltage PQ Controllability relates control of voltage( V) at load(PQ) buses to

generator internal voltage( E ,I) at PV buses. For a system to be voltage PQ controllable,

the voltage at PQ buses should increase when the load is fixed and the voltage at some

PV buses is increased.

Definition 4.3
A system is voltage PQ controllable at any time t along the transient trajectory or at

an equilibrium point if:

(a) When AQ, = 0, any nonzero nonnegative control AE causes the PQ state AV to

become nonnegative and

(b) When AQ, = 0, for each j there is a nonzero nonnegative control AE that

causes AV, to become positive.

Theorem 4.3

A suflicient condition for voltage PQ controllability is that SVE satisfying

A V = SVEAE is nonnegative with no zero rows.

This proof is omitted but can be obtained from results in [15].

44,2 Leed PQ Centrellability

Voltage PQ Controllability relates control of voltage( V) to load( Q,) at load(PQ)

buses. For a system to be load PQ controllable, the voltage at PO buses should increase

when the voltage at PV buses is fixed and the load at some PQ buses is increased

Definition 4.4

54

A system is load PQ controllable at any time t along the transient trajectory or at an

equilibrium point if:

(a) when AE = 0, any nonzero nonnegative disturbance AQ, causes the PQ state

AV to become nonnegative and

(b) when AE = 0, for each j there is a nonzero nonnegative disturbance AQL that

causes AV}. to become positive.

Theorem 4.4
A suflicient condition for load PQ controllability is that SQ, V is a M—matrix.

This proof is omitted but can be obtained from results in [15].

4.5 Effects of Loss of Load PQ Controllability

Theorem 4.5
As load PQ controllability is lost and SQLV loses its M matrix property by a single

-1
eigenvalue 11m,“ (11) of SQLV approaching zero and becoming negative, matrix SQLV
elements in the voltage control area associated with 11m," (11) approach infinity, become

negative infinity, and then increase toward zero if the constituent matrix is continuous,

does not change its magnitude as rapidly as T, and does not change sign in the

min

interval [11,, - 8,11,, + 8] when 8 is sufliciently small and [A —) 110.

Proof:

The M matrix SQLV has eigenvalues 11,, 1': 1,2...N, and the matrix SQLy—l has all

nonnegative elements by the proven properties of an M matrix. SQ, V can be written as

55

N
SQLV = 2’1er (4.8)
i=1

where A, is the eigenvalue of S Ly and Z, is the constituent matrix of SQtV° Matrix

—1
SQL y can be expressed as

N
_ 1
Soy ‘= 2 i'rTZ" (4.9)

Since the constituent matrices satisfy

ZiZi =Zi
Z,Z,.=O, i¢j (4.10)

As one eigenvalue 1, approaches zero,

1

—1

59,1» z T2]: (4.11)
k

in the voltage control area [12] associated with that eigenvalue 21,.

. . . . . -l . .
The constltuent matrlx Z, must be nonnegative matrlx srnce S Ly 1s nonnegatrve
as long as load PQ controllability holds and SQLV is an M matrix. When A, approaches

-1
zero, elements of SQ, y must approach infinity if 11—1- grows faster than the elements of
k

matrix Z, could ever approach zero. If Z, does not have all zero elements when 11,, = 0
(because they must be nonnegative since SQLV is an M matrix until l, = 0), there are
some elements of SQL y—1 in the voltage control area associated with the bifurcating
eigenvalue 1,, that may approach infinity [45, 12]. If there is a small operating change

that causes 1,, to move from zero continuously to negative infinitesimal value where Z,

56

elements have no perceptive change (because they are also continuous), then elements of

-l . . . . . .
SQLy can be near negatrve 1nfin1ty 1n the voltage control area assocrated wrth

.] .
eigenvalue 1,. The elements of SQL y may mcrease and approach zero as 11, becomes

more negative.

Theorem 4.6
Loss of voltage PV controllability, load PV controllability and voltage PQ

controllability occur in the voltage control area associated with 11m if (i) the

nonpositive continuous elements in J,,, J,, and 21",,“ changes imperceptibly in
[110 - 8,11,, + 8] when 8 is sufliciently small and (ii) the elements of SQL y—I approach
infinity, become negative infinity and increase in [[10 - 8,110 + 8] when 8 is sufficiently
small. Loss of voltage PV controllability may occur long before loss of load PQ

controllability occurs, but loss of load PV controllability and loss of voltage PQ

controllability generally occur simultaneously.

proof:

The sensitivity matrices are defined as

SQLV = J22
SVE = _SQLV-]J21
Scoot = "JrzSQLV—l

__ -1
SQGE ‘ Jll -J12SQLV J21
-1 . . . . .
where SQLV 1s the test matrix for load PQ controllabrllty, SVE rs the test matrix for

voltage PQ controllability, SQ

G

Q, is the test matrix for load PV controllability, and

SQoE is the test matrix for voltage PV controllability.

57

When SQLV is singular, SW" is not technically defined, its elements may be very

large positive numbers before the singularity point and very large negative numbers in a
particular voltage control area after the singularity point as shown in Theorem 4.5. Since
it is assumed that J,, and J,, elements are nonpositive and change continuously and

._1 .
imperceptibly in [110 -8,fl0 +8] and elements of S Ly 1n the voltage control area

associated with A approach infinity, go to negative infinity and then approach zero as

,u —) 11,, in [110 —8,fl0 +8], elements of SW; and SQGQL in that voltage control area
. . -1 .
assocrated With Jim,“ behave the same as elements of S L y in that voltage control area.

Loss of load PV controllability and voltage PQ controllability occur in [11,, — 8,11,, + 8] in

the voltage control area associated with Am," as ,u —-) 110.

The conditions for voltage PV controllability are

M
259.511; > 0.1' =1..--,M (4.12)
i=1

S >O,i=j
{ 905i:- so, i¢ j (4°13)

SQGE = J11 —J12SQLV_IJ21

where

The diagonal elements of SQGE will approach zero and become negative because

the second term should dominate at PV buses in the voltage control area associated with

11

approaches zero since

min

(i) Z)“, must have all nonnegative elements by assumption that change

imperceptibly in [[10 — 8,11,, + 8] if load PQ controllability holds. This is true because

SQ, y" must have all nonnegative elements and Z ,1 min is continuous;

58

(ii) J12. and J,, have all nonpositive elements because they are off diagonal

submatrices of matrix J and change imperceptibly in [p10 - 8,11,, + 8] by assumption and

continuity;

(iii) The il th elements of _']12SQ, 11—1le in the voltage control area associated with

eigenvalue A,,,i are negative and grow toward infinite value as A,,, approaches zero

n 11

since every element of matrix
1
E Z —{J12}ij A. . {ZAm,,, }jk{J21}k1
J min

is negative because the elements in the sum associated with the voltage control area that

is nonzero is negative and approaches negative infinity as Amin approaches zero due to

assumed properties (i) and (ii);

1
(iv) SQGE is approximated by J,, when Amin is large and by -J,2 72,,“me at all

ma

buses in the voltage control area as A,,, is reduced. The diagonal elements in J,, are

['1

positive but the elements in -J,2 A——Z’1"““J21 in the voltage control area become more
min

negative as Am,n decreases toward zero;

1
(v) Since all of the elements in -J _ZA..:.J2| are negative and the fact that

12 Amin
condition (4.12) is violated before condition (4.13) is violated indicates that loss of

voltage PV controllability occurs substantially before loss of voltage PQ controllability

for buses in the voltage control area associated with A

I
min ’

(vi) The elements of SQGE in the voltage control area associated with Amin should

approach negative infinity, become positive infinity and then decrease as ,u —-) [.10 in

59

[110 — 8,11,, + 8] if J,,,, J,, and 21",," elements in the voltage control area associated

Amin are continuous and thus do not change perceptibly as assumed.

In summary, when load PQ controllability is lost, voltage PV controllability, load PV
controllability and voltage PQ controllability can be lost. But loss of voltage PV

controllability can occur before load PQ controllability is lost.

4.6 Effects of Load Characteristics

Theorem 4.7
Assume that load PQ controllability and voltage PV controllability hold at some
operating point close to loss of load PQ controllability when the excitation control

system is disabled

(i) [f inductive load susceptances increase at all buses, then elements of SQL y—1 will
become smaller .
(ii) If shunt capacitive susceptances increase at all buses, then SQ, y-1 will become
larger.

If the SQLV off-diagonal elements and all elements of Zion. change imperceptibly in

[#0 " 8.110 + 8]

Proof:

From [15], for a decoupled load flow model

AQG _ A5; _ Jn le AE;
[A91].- AVl_iJ21 JzziiAVl

v

60

where J has positive diagonal elements, nonpositive off-diagonal elements. And when

E; is specified, AB; = 0, and
AQL = J,,AV = SQLVAV (4.14)

When V. z 1 and 5,, z 0, the Jacobians [54] can be expressed as

1

AQ, = B”AV (4.15)

where

B” = B’ -— 2diag{Bsmm ,BMM W3 } (4.16)

where B’ has positive diagonal elements and nonpositive off-diagonal elements. The fact

that off-diagonal elements of SQ], are assumed to change imperceptibly justifies use of

the decoupled model. The diagonal elements of B’ are the negative of the sum of its off

diagonal elements in a row plus the negative sum of the elements connecting that bus to

PV buses. Element Bsii's are negative when the susceptance is inductive, B,,,'s are

positive when the susceptance is capacitive.
From (4.14) and (4.15), we obtain

5,, = B” = B’ — 2diag{B

sm¢Lm+l, sm+Lm+2,.

..,B,M} (4.17)

(i) If inductive load susceptances increase at all buses, all the B,’s are negative and

becomes more negative. This will make the minimum row sum of SQLV increase since

the diagonal elements increase from (4.17), so that the lower bound on the minimum

eigenvalue Amin of SQLV’ which is the minimum row sum of S Ly, will increase. Since it

was assumed that the operating point is close to loss of load PQ controllability, the

minimum eigenvalue term in the modal expansion dominates in the voltage control area

61

l

I O I —1 U

associated with that mrmmum eigenvalue. Therefore, S ,y z “—1 Z 11...... wrll become
min

smaller in the voltage control area associated with eigenvalue Amin for increase in shunt

inductive susceptance at all buses as long as changes in the discontinuous function of ,u

 

A are greater than changes in Z1"... which are a continuous at 11,, and Amin = 0.

min

(ii) If shunt capacitive susceptances increase at all buses, all the B,'s are positive and
increase. This will make the minimum row sum of SQLV decrease since diagonal
elements of SQLV are decreasing from (4.17). The minimum eigenvalue of S Ly should
also decrease since it is assumed that the operating point is close to loss of load PQ

controllability and Amin is small, the minimum eigenvalue term in the modal expansion of

-l . . . . . . .
SQ, y dominates 1n the voltage control area assocrated wrth that mrmmum ergenvalue.

-1 . .
This suggests that the elements of SQLV in the voltage control area assocrated wrth

Amin are increasing toward infinite value as shunt capacitive susceptances are increasing.

Based on Theorem 4.6 and Theorem 4.7, the conclusion is that a network that has
sufficiently large line charging and shunt capacitive susceptance will experience loss of
voltage PV controllability long before it has experienced loss of load PV controllability,

loss of voltage PQ controllability and loss of load PQ controllability.

It should be noted that the ith diagonal element of SQL y-l should describe the slope
of the Q-V curve which is positive and decreases toward zero as Q,, = [1 -) [10. Since a

Q-V curve is really a reactive load stress test and the system should be stable as

Q,, = p —) [J,,, the elements of vector AQG = -SQGQLAQL,e, should be positive since

AQL, < 0 for increase in load at bus i, e, is a unit vector with l in the i th element and
zeros elsewhere, the ith column of SQer has all non-negative elements when load PV

controllability holds. This last condition on SQGQL is important since the increase in

reactive load should cause reactive generation increase and exhaustion of reactive

62

reserves on certain generators in the reactive reserve basin at generator i. When the

system is stable, if a Q-V curve is run at generator i, the i th diagonal element of SQoE

must be positive and decrease toward zero as 11 —) ,uo since it is the slope of the Q-V

curve.

To establish that the Q-V curve stress test is administered under stable operating

conditions and that reactive load would produce the same result as reducing voltage AE,
to reduce AQG, to more negative values, all elements of SQGQL should be positive
indicating reactive load increase at any bus j would cause increased generation at every

generator.

4.7 Validation of Theory

The objective of this section is to (a) validate that SQ, fl approaches positive
infinity, jumps to negative infinity, and goes back to zero when a power system
experiences a negative voltage as observed in the WSCC system blackouts (b) validate
that infinity phenomena can occur both when all the exciters present and when one
exciter is disabled by an over-excitation protection relay (c) validate that SVE, SQGE and
SQer all show that same infinity phenomena, and (d) establish the effects of load
characteristics on S Liz-l when an exciter is disabled by the over-excitation protection

relay.

The example system studied here is taken from F ouad [55]. It is a 3-machine 9-bus
system, and the network is shown in Figure 4.1. The bus data, line data, generation data

and exciter data are summarized in Tables 4.1-4.4 respectively.

63

Two different cases are investigated here. The first one is the network shown in
Figure 4.1 where all the excitation control systems are active on all three generators. The
second case is the same network but with the excitation control system on generator 3

disabled by an over-excitation protection relay.

For the first case, a disturbance, which is a sudden reactive load increase (1.56 pu) on
generator bus 3, is applied at t= 1.1(s). Then the internal bus voltage E; at generator 3
exhibits the behavior as shown in Figure 4.2(a). The sensitivity matrices corresponding
to each point of E g are computed, and the results are shown respectively in Figure 4.2(b),
(C). (d) and (e).

In the second case where the excitation control system on generator 3 is disabled by
an over-excitation protection relay, a similar disturbance as in the first case (i.e. a sudden
reactive load increase of 1.56 pu) is applied at t=1.1(s) on generator bus 3. Now the
internal bus voltage E; at generator 3 exhibits the behavior as shown in Figure 4.3(a).
The sensitivity matrices corresponding to each point of E; are computed, and the results
are shown respectively in Figure 4.3(b), (c), (d) and (e). The fact that the infinity
discontinuity occurs respectively in Figure 4.1 and Figure 4.2 indicates that the trajectory

is on a singular surface or continues to cross the singular surface.

In order to perform the validation clearly, different points on Figure 4.2 and Figure
4.3 are zoomed. Figure 4.4 and Figure 4.5 show the behavior of SQL V"l around point B,
respectively for case 1 and case 2. It goes to positive infinity, jump to negative infinity
and then goes back to zero. Theorem 4.5 that states such behaviors in S LV_1 is

confirmed.

Figure 4.6 and Figure 4.7 show the occurrence pattern of loss of different

controllabilites for case 1 and case 2 respectively. At point A, the diagonal elements of

SQGE on generator 3 changes sign from positive to negative. Voltage PV controllability is
lost first, which apparently triggers the instability in E; at generator 3. This instability
then causes the sharp rise in SQL fl , SQG E and SQGQL as shown in Figure 4.6. Figure
4.7 shows the similar phenomena for case 2. At point B, S ,y-1 approaches positive
infinity, becomes negative infinity and increase toward zero, load PQ controllability is
lost. At the same time, SyE and SQGQL go through the same infinity transition process
and become negative afterwards, voltage PQ controllability and load PV controllability
are lost simultaneously. Theorem 4.6 which states all of the above behaviors in SQGE ,

Sn; and SQGQL is confirmed.

In Figure 4.8, there are three curves, the curve with circles is SQ, f] with the same
original settings as in case 2, the curve with dots is SQ, 11-] with 0.05 pu B increase on
every bus. The curve with stars is S Ly-l with 0.05 pu G shunt increase on every bus,
and all the other parameters are the same as in case 1. From Fig.4.8, it is shown clearly
that SQ, y-l becomes smaller when G shunt is increased, and SQL y—l becomes larger

when B shunt is increased. Theorem 4.7 is thus confirmed.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

bus voltage( angle p _gen q_gen p_load q_load G B bus
No. pu) (deg) (pu) (pu) (pu) (pu) shunt shunt type
1 1.04 0.00 0.716 0.27 0.00 0.03 0.00 0.00 l
2 1.025 9.30 1.63 0.07 0.00 -0.02 0.00 0.00 2
3 1.025 4.70 0.85 —0.1 l 0.00 -l .56 0.00 0.00 2
4 1.026 -2.20 0.00 0.00 0.00 0.00 0.00 0.00 3
5 0.996 -4.00 0.00 0.00 1.25 0.50 0.00 0.00 3
6 1.013 -3.70 0.00 0.00 0.90 0.30 0.00 0.00 3
7 1.026 3.70 0.00 0.00 0.00 0.00 0.00 0.00 3
8 1.016 0.70 0.00 0.00 1.00 0.35 0.00 0.00 3
9 1.032 2.00 0.00 0.00 0.00 1.60 0.00 0.00 3
Table 4.1 Bus data
From To resistance reactance line tap
bus bus (pu) (pu) charging ratio

1 4 0.0000 0.0576 0.0000 1 .0

4 5 0.0100 0.0850 0.0880 1.0

5 7 0.0320 0.1610 0.1530 1.0

7 2 0.0000 0.0625 0.0000 1.0

7 8 0.0085 0.0072 0.0745 1.0

8 9 0.0119 0.1008 0.1045 1.0

9 3 0.0000 0.1586 0.0000 1.0

9 6 0.0390 0.1700 0.1790 1.0

9 4 0.0170 0.0920 0.0800 1.0

 

Table 4.2 Line data

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Gen Base x_l r__a x d x' d T'_d x__q x'_q d_o d_l
No. mva
1 247.5 0.0336 0.01 0.145 0.061 8.96 0.097 0.097 13.54 9.6
2 191.9 0.0521 0.01 0.896 0.119 6.00 0.865 0.197 5.40 2.5
3 127.9 0.0742 0.01 1.313 0.865 5.89 1.258 0.250 3.01 1.0
Table 4.3 Generator data
Gen exc. T_R K_A T_A V_R V_R K_E E_l S_E E_2 S_E
No. type max min 1 2
l 1 0.06 25.0 0.06 5.0 -5.9 -0.02 0.75 0.12 1.0 0.30
2 0 0.06 25.0 0.00 5.0 -1.0 -0.05 0.75 0.07 1.0 0.30
3 2 0.06 1.00 0.01 5.0 -5.0 -0.06 0.75 0.09 1.0 0.36
Table 4.4 Exciter data

67

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1 A three-machine 9-bus test system

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ _ 8 8 8 a 8 8
A. w _ _
A” L" AW .#
1 r7r A. 171 L" 17T um 471 A_ 17
A ”w. A _
_ . A _
.. 1.11 A1 A.
1 r61 161 161 16.. 16
1.. 1L
1 15fi fl 151 151 15
O
i 14. 14- 14.. 44m
T
r 131 r31 131 13
1 121 121 121 12
r A r F
J W J J
1 111 1.111 111 11
_ _ 0 0 0 0 0
1 5 0 0 0 o 0 0 0 o 0 o 0 o 0
. 0 00 00 00 0

0
oEnaluw E
mwmm>z_ 38 3V meow E m>m e. .608 E

. ' -l
F1g.4.2 E , S Ly , SQGE,SVE and SQGQL for casel
69

 

 

 

 

L _

 

0
oEtalaw E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m
T1
..|
n1
q
r1]
1|
1 1
o o 00 o 0
0 00 0
1 11 d_l

.
mmmm>z. Bow 3 meow E

10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100

0

m>m A3

-100

10

 

 

 

 

 

 

 

1 L _ 11111JJ1H1L11111
T

 

 

 

 

 

 

 

m

0

-100

.508 3

10

Time

, SQGE, Sn; and SQGQL for case 2

l

I
F1g.4.3 E , SQLV

70

SQLV INVERSE

100

80

60

20

-80

-100

 

T

 

I

 

 

 

 

 

1.1

1.2

1.3 1.4 1.5 1.6
Turns

—1
Fig. 4.4 SQLV for casel

71

1.7

1.8

1.9

 

SQLV INVERSE

100

80

60

20

-60

-80

-100
1

 

I

1

 

l

I

I

 

 

 

 

1.5 2 2.5 3 3.5

Fig. 4.5 SQL V4 for case 2

72

4.5

 

SVE SQGE SQLV INVERSE

SQGQL

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. —l
Flg.4.6 SQLV ,

QUE,

 

73

 

 

 

 

 

 

 

 

SVE and SQGQL for case 1

 

 

 

 

 

SQLV INVERSE

.5

 

 

 

 

 

 

 

 

 

 

 

 

 

‘v-uu1v.vr.11.wuv11r-.r-r.uvvvrw—v .........................................................................

SQGQL
o

 

 

 

 

.5

. —1 ‘
F1g.4.7 SQLV , $965,514; and SQGQL for caseZ

74

 

SQLV INVERSE

2.5 1 1 1 1

 

 

 

 

 

 

——9— Original /
-—*—-— Increased G
Increased B

2 ~ 4

 

 

 

 

TIME

Fig.4.8 Effects of load characteristics

75

CHAPTER 5

THEORETICAL AND SIMULATION
JUSTIFICATION FOR THE DYNAMIC
RESPONSES TO BIFURCATION SEQUENCES
ASSOCIATED WITH VOLTAGE COLLAPSE

This chapter establishes that a saddle node bifurcation followed by an inevitable
singularity induced bifurcation produces the characteristic negative voltage spike observed
on the December 14, 1994, July 2, 1996 and August 10, 1996 WSCC system voltage

collapse problems. The theory uses previously defined matrices T and N [8] to show that:

(a) Disablement of the excitation control system is not enough to cause instability, but
disablement of the excitation control system and slow loss of voltage PV controllability
cause saddle node bifurcation and thus loss of stability. This loss of stability causes
AE‘; (t) to decline. This can be observed in Figure 1.2(a) starting from 15:47:50 ignoring
the oscillations observed in that figure. The loss of stability can be observed in an

eigenvalue of T going to zero and slowly assuming positive value;

(b) The decline in voltage A5; (I) should cause a generation decrease given that
{5965} <0. A dynamic Q—V curve is then produced as long as SQGQL has all

nonnegative elements and load PV controllability holds as 13E; (I) is decreasing;

76

(c) The dynamic Q-V curve at a generator bus in a voltage control area should cause

-1
elements of SQLV associated with that voltage control area to approach infinity and
{5120511, to continually decrease and thus further violate voltage PV controllability

conditions;

(d) A more and more rapid decline in AB; (1‘) should occur as the positive unstable

eigenvalue of T associated with the loss of voltage PV controllability and loss of the
excitation control system on generator 1' becomes more positive because the magnitude of
the positive eigenvalue depends on the degree of violation of PV controllability at bus i.
-1
The acceleration of SQLV elements to infinity causes acceleration in the magnitude of
violation in voltage PV controllability and increase in positive value for the unstable
-1

eigenvalue of T. This acceleration of SQLV elements to infinity causes the rate of
A5210) decline to increase resulting in a spike and the eigenvalue approaching infinity.
The evolution of the slow decline in AE; (t) at saddle node bifurcation to the more and
more rapid decline associated with singularity induced bifurcation can be observed in
Figure 12(3);

—1
(e) The chan e in S V values from +00 to — oo in the volta e control area causes
3 QL g

SQGE ‘0 change from -°° t0 +°°. and T from +00 to -<>o [15, 48, 40]. This change in T
causes a stable but infinitely fast rise in AE; (t). This can be observed in Figure 1.2(a).

5.1 Introduction

Sensentivity methods were very thoroughly investigated as a means of assessing
proximity to voltage instability in a loadflow model [15, 33, 62] and for assessment of the
stability of an equilibrium in a differential algebraic modol [15, 33, 62]. More recently
trajectory sensitivity methods have been applied [12, 56, 62] to assess proximity to the

77

boundary of the region of attraction and to assess how discontinuities due to contingencies,
field current limiters, switchable shunt capacitors, and tap changers affect proximity to
instability of a trajectory. In all cases, the sensitivity measures become large as proximity
to instability become small. Network sensitivity matrices were first defined in chapter 4.
PQ contollability of voltage (V) at load buses and PV controllability of reactive genration
(Q6) at generator or voltage control buses was defined based on the understanding of the
proper response and contol of tese variables that not only reflected stable operation but
acceptable operaton. The concept of PQ controllability and PV controllability was extended
in chapter 4 to differentiate the response of the variables (V, Q) to reactive load QL and
generator internal voltage (E). The properties of test matrices SQLV-l and SVE for load
and voltage PQ contolability respectively and test matrices SQGQL and SQc; E for load and
PV controllability were defined in chapter 4, are the foundation for the theory presented in
this chapter on how and why large negative voltage spikes can occur as a system
experiences a blackout. These large negative voltage spikes are believed to have
contributed to or possibly caused the three recent blackouts on the WSCC system because
(1) they represent a series of cascading bifurcations that make the dynamics unstable and
ultimately destroy the network's ability to transfer power and (2) the large negative voltage
spike triggers equipment protection limiters and relays that outage equipment that ultimately

cascades the blackout throughout the system.

A sharp negative voltage spike has been observed on the WSCC system prior to the
December 14, 1994, July 2, 1996 and August 10, 1996 blackouts. The voltage spike is
proven to occur due to Maximum Excitation Limiter (MXL) action that causes an unstable
trajectory crossing a singular surface due to loss of load and voltage PQ controllability.
This is quite possibly the voltage spikes that occurred on the first two WSCC blackouts. A
voltage spike is also proven to occur when a MXL is disabled by large interarea oscillations

and an Over Excitation Protection Relay (OXP) trips the AC regulator over to the DC

78

regulator of the automatic voltage regulator. The voltage spike that is observed during the
August 10, 1996 blackout quite possibly occurred due to a sequence of saddle node
bifurcation that is inevitably followed by the unstable trajectory crossing a singular surface.
The saddle node bifurcation occurs due to (a): disablement of the Maximum Excitarion
Limiter due to large interarea oscillations as proven in [57,60], (b) OXP relay tripping of
the AC regulalor of the AVR [57, 60] and loss of voltage PV controllability [60, 63]. Loss
of voltage PV controllability is proven to occur on networks where the network provides
the reactive supply via line charging and/or heavy shunt capacitive compensation. The
instability following saddle node bifurcation is proven to administer a dynamic Q—V stress
test that results in sigular surface via loss of load and voltage PV controllability [57, 62].
The voltage decline portion of the negative voltage spike is proven to occur due to the
saddle node bifurcation and the voltage increase is proven to occur due to the unstable .
trajectory crossing the singular surface. This bifurcation sequence is quite possibly what
produced the voltage spike that was observed on the August 10, 1996 blackout when the
large interarea oscillations apparently disabled Maximum Excitation Limiters on the

McNary station generators that ultimately caused tripping of these generators.

5.2 Test Matrices for Static Bifurcation in Flux Decay

Dynamics

A test matrix T for static bifurcation in flux decay dynamics is defined in [8]:

 

 

 

A1 Bl Di E11 C1
A2 B D E C
T=- C,—[A, B3 D3 E3]A4 B: D: E: C: (5.1)
5 _A5 BS D5 E5_ _C5__

 

 

 

A second expression for T is

 

 

X —X’
T=diag -1— diag —E;_ + QG°‘( d: d‘)
TdOi 0' 401'
r (5.2)
—K K V X —X
+diag A" R‘. 0" WSVE—diag ———d" , a" Q E
K51 + $51 + SEiEfdm E401 G

where KA', is the excitation control system gain, K A.- is the voltage sensor gain, K5, and

S 5’. are the excitation control system parameters, and N is defined as:

 

N =d1ag —- drag ‘ ’. ’ WS QLV +d1ag —‘—-—' SQ Q
T4,, K51 +SE‘, +S£,Efdo, E G L

401'

(5.3)

The derivation in [8] shows that the reduced differential equation model for a system

experiencing static bifurcation in flux decay dynamics is

 

I
40

AE’
O , — q
AEq — T + NAQL (5.4)
The following Lemma taken from [8] qualifies derivation of the linearized model (5.4)

when T can be written as either (5.1) or (5.2) and N is defined by (5.3).

Lemma 5.1

Given that Jc is nonsingular so that A is defined and J1 is nonsingular so that T is

defined, then matrix T is singular if and only if both A and J are singular.

80

The matrix JC is a test matrix for causality[27] and causality is assured if

J 82 D2
c — B4 D4 (5.5)

is nonsingular. A is defined only when the system is causal, it acts as the text matrix for

static bifurcation, a static bifurcation occurs when both J and

A1 C1 E1 Bl D1
A: A, C, 53—33 D3[
A5 C5 E5 BS DS

32 Dz]-I[A2 C2 E2] (5 6)

34 04 A4 C4 54

are singular. Jl is the system jacobian matrix when generators are modeled by a classic
machine model with constant voltage behind transient reactance and the network is not

aggregated back to generator internal buses. Matrix

-A' ..
Jr = A2
A4

 

 

Bl DI
32 D, (5 .7)
B4

is a test matrix for static bifurcation in generator inertial dynamics.

This Lemma shows that when T is defined, if loss of causality (Jc nonsingular) and
angle instability (Jl nonsingular) can not occur and if both J and A are
singular(nonsingular), then T is singular(nonsingular). The analysis in [30] shows that
when the exciter is disabled and the generator flux decay mode experiences saddle node
bifurcation, and the equivalent model (5.4) should capture this bifurcation. The
computational result in [32] on bifurcation subsystem theory suggests that on at least one

example, the bifurcation subsystem for a saddle node bifurcation when the excitation

81

control system is disabled is just the generator flux decay dynamics AE; (t). All these

results suggest that model (5.4) contains the subsystem experiencing bifurcation and T

actually tests for this bifurcation that occurs in the subsystem model.

5.3 Effects of Loss of Voltage PQ Controllability on Stability

of Generators with Exciters

Lemma 5.2, Theorem 5.1 and Theorem 5.2 are taken from [11] where the proofs of

each result are given. Simulation results are also provided that help substantiate the theory.

Lemma 5.2:

If matrix W that specifies the voltage controlled buses is selected so that WSW; is

positive definite when voltage PQ controllability holds, then T is negative definite when

the excitation control system gain K A, is assumed large.

Theorem 5.1

Loss of voltage PQ controllability for one or more generator control buses causes
WSVE and T to be singular or have an eigenvalue change sign when K141 is assumed
large. This implies that loss of voltage PQ controllability causes instability in the flux decay

dynamics.

The results of Lemma 5.2 and Theorem 5.1- indicate that when the excitation control
system are active, preservation of voltage PQ controllability is sufficient to preserve
stability in generator flux decay dynamics, but loss of voltage PQ controllability is
sufficient to cause loss of stability in the generator flux decay dynamics. Simulation results
in [11] suggest that the loss of voltage PQ controllability bring on Hopf bifurcation
followed by node focus, and finally singularity induced bifurcation when loss of voltage

82

PQ controllability occurs. Such a sequence of bifurcation is shown in Figure 1.5, A saddle
node bifurcation in generator flux decay dynamics follows singularity induced bifurcation
since the unstable real eigenvalue crosses the jet axis, it is solely associated with the flux
decay dynamics. Theorem 5.1 theoretically confirms these experimental results. The result
also indicates slow loss of voltage PQ controllability can produce saddle node bifurcation
followed by the singularity induced bifurcation.

The remainder of this chapter will describe and justify the negative voltage spike that

characterizes voltage collapse incidents on the WSCC system.

5.4 Effects of Loss of Voltage PV Controllability and
Disablement of the Excitation Control System on Stability

of Generator Dynamics

Theorem 5.2:
Given that -T is irreducible, diagonally dominant, and positive definite when voltage

PVand voltage PQ controllability hold, T becomes singular or indefinite when (i) voltage
1 2

E
PV controllability is lost such that SQGEii — Q01 + -)—(—qi—X'— < 0 (ii) field current
d,- " d,-

limits are reached on the same generator and (iii) the excitation control system on that

generator is completely disabled by an over excitation relay and placed under manual

control.

Theorem 5.2, taken from the result in [l 1], establishes that disablement of a generator
excitation system and loss of voltage PV controllability on that same generator produces
loss of stability in generator flux decay dynamics. If the loss of voltage PV controllability

occurs very slowly due to addition of switchable shunt capacitors that causes loss of

83

voltage PV controllability (Theorem 4.7), then a saddle node bifurcation appears to

develop.

Theorem 5.3:

Matrix T is negative definite if an excitation control system is disabled but voltage P V

controllability and voltage PQ controllability hold.

Theorem 5.3 establishes that disablement of the excitation control system alone is not a
sufficient condition for loss of stability in generator flux decay dynamics. The proof of

Theorem 5.3 is given in Appendix A.

5.5 Justification of the Bifurcation Sequence That Occurs due
to Loss of Voltage PV Controllability and Disablement of

the Excitation Control System on a Specific Generator

Theorem 5.4

m
Given that loss of voltage PV controllability occurs ( z SQa 51,- < 0,
i=1
1 2

(It . .

— —— on a rator contr l b h r

SQGE“ Q01, + Xd —X;1 < 0) gene 0 us w e e the excrtatlon
i i

control system is disabled by an overexcitation limiter relay, load P V controllability, load
PQ controllability and voltage PQ controllability hold at all buses. then AE; (t) is

unstable and decreasing toward zero for AEJ' = 0, j ii and a load increase
AQL', < O.

The proof of Theorem 5.4 is given in Appendix A. Theorem 5.4 states that the loss of
stability caused by loss of voltage PV controllability and disablement of the excitation

control system on the same generator causes AE; (t) to be unstable and approach zero.

84

Theorem 5.5 that follows proves that the loss of stability in AE; (I) does not cause

instantaneous voltage collapse, but inevitably leads to a cascading voltage collapse by
causing a singularity induced bifurcation that occurs in the algebraic submodel but affects
dynamics in the differential model. The slow decline in A1220) has exactly the same

effect as computing a Q-V curve at the internal bus i.

Theorem 5.5
Given that {SQG E }ii > 0 and load PV controllability holds until load PQ

controllability occurs. When voltage PV controllability is lost ( XSQG 13,, < 0.
j=l
1 2

E .
q‘ . < O) on generators where the excitation control system is

S 5.. ‘— Q0. '1‘
Q6 " ' Xdi ' th
disabled and AEJ- = O for j it i, then a dynamic Q-V curve stress test occurs on the

generator internal bus that inevitably produce loss of load PQ controllability and singularity
induced bifurcation.

The proof of Theorem 5.5 is given in Appendix A.

Theorem 5.6

. . . -l . .
The dynamic Q-V curve, that 15‘ observed vra elements m SQLV assocrated wrth the

voltage control area approaching +00, causes continually increasing violation of voltage PV
controllability, causes an eigenvalue of T associated with the unstable generator in flat
voltage control area where the excitation control system is disabled to approach +00, causes
the more and more rapid decline in AE; (t) to producing the negative side of the voltage
spike.

The proof of Theorem 5.6 is given in Appendix A.

85

5.6 Justification of the Large Negative Voltage Spike That
Develops due to Loss of Voltage PQ Controllability on

Generator Response with or without Excitation Control

Theorem 5.7

Given that (i) loss of voltage PV controllability (XSQGE< Eu 0,
i=1
E' 2
_ +__q_'_
SQGEii QG,+ Xd -Xd < 0) and load PV controllability occur on a generator

control bus i in a voltage control area where the excitation control system is disabled by an

overexcitation limiter relay, and (ii) load and voltage PQ controllability hold at all other
buses, then the system is unstable and the change in E; is positive when AQL, < O

andqu =0, ] at LAB,” >0.

Theorem 5.8
Given that
(i). loss of load PQ controllability has occurred at the generator control bus i;

—1
(ii). {WSQLV }ij approaches positive infinity, jumps to negative infinity but

increases toward zero after load PQ controllability is lost at bus j where

(iii). {WSVE} ii for generator i approaches positive infinity, jumps to negative
infinity , increase to zero and positive value after voltage PQ controllability is

lost at bus i,

(iv) the exciters on all machines are active where Aqu - _,0 j ¢ 1

(v) the system is stable;
then the steady state change AE; for increased load ( AQL < O ) is positive.

86

Theorem 5.7 and Theorem 5.8, which are proven in Appendix A, deal with the effects
after the singularity induced bifurcation.

If a generator excitation control system is disabled via an overexcitation limiter relay
and loss of voltage PV controllability slowly develops at that same generator, then
singularity induced bifurcation inevitably follows which implies that load PV controllability
may ultimately be lost, as noted from results of Theorem 5.4. Theorem 5.7 states that if
voltage PV controllability and load PV controllability are lost on the generator where the
excitation control system is disabled, then AB; increases. This is initially surprising
because the voltage decline that occurred before singularity induced bifurcation is reversed

to produce a voltage rise.

Observation of SQGQ 1. as singularity induced bifurcation (singularity of SQLV)
develops may indicate that SQG QL elements in the voltage control area(elements of both Q6
and Q) will increase toward infinity, become negative infinity, and then increase to zero.
SQGE elements approach negative infinity, become positive infinity, and then decrease.
From the proof of Theorem 5 .7, large positive SQGQL elements can have a dominant effect
on AE‘; < 0, thus AE; will decrease more and more rapidly as loss of load PV

controllability develops due to elements of S00 QL approaching infinity, AE; then will

increase extremely rapidly after loss of load PV controllability occurs because elements of
S
Q

0

Q1 are negative infinite. This could produce the large sharp negative voltage spike that

has been characteristic of the WSCC system blackout observed in December 14, 1994, July
2, 1996 and August 10, 1996.

The singularity induced bifurcation effectively causes the real eigenvalue of T
associated with the voltage control area to become positive, approach positive infinity,

become negative infinity and then decrease when this eigenvalue depends on the elements

of SQGE that approach zero and proceed to negative infinite, become positive infinite and

87

then decrease toward zero. This suggests a bifurcation sequence of saddle node bifurcation,
that occurs if shunt susceptance is slowly added at bus i where the excitation system is

disabled, and singularity induced bifurcation that inevitably follows.

Theorem 5.8 indicates that if loss of load PQ controllability occurs for generators with
active excitation control, and the singularity induced bifurcation causes eigenvalues to

transverse through the right half plane toward infinity, enter the left half plane at infinity,

the stable response will cause AE; to increase for reactive load changes. Thus the voltages

on both generators with active excitation control may enhance the voltage spike observed

on generators where excitation has been disabled.

Theorem 5.9
As [.1 increases in the interval [#0 — 8, #0 +8] and loss of load PQ controllability

occurs at no,

(i) The elements of SQLV associated with the voltage control area having the

dynamic Q-V curve go to +00, discontinuously change to —00, and then increase toward

zero.

(ii) The elements of SQGE associated with the voltage control area having the dynamic

Q-V curve approach —00, discontinuously change to +00, and then decrease toward zero as

long as J”, 1,2 and J21 are continuous in [110 - 8, #0 + 8].

(iii) The eigenvalue of T associated with one unstable generator where the excitation

control system is disabled, approaches +00 causes rapid decrease in AE; producing the

positive part of the spike and making the eigenvalue move through +00 to the left half

plane. The switching of SQGE elements from +00 to -00 causes the eigenvalue of T to

o o o I O C ’
swrtch discontinuously from +00 to -00 and causes a rapid nse tn AEqi.

88

The proof of (i) and (ii) is from Theorem 4.6. The proof of (iii) is based on the proof

of Theorem 5.6 and the results in Theorem 5 .7 and Theorem 5.8.

5.7 Simulation Study

Based on the simulation study performed on the 3-machine 9-bus power system in

chapter 4, test matrix T for static bifurcation in flux decay dynamics is computed here.

For casel, where all the exciters are active, the simulation results for SVE, SQLV-l , T
and E; are shown in Figure 5.1. The diagonal element of T plotted is for generator 3, the
diagonal element of SQLV-l is for load bus 9, the element of SVE represents the coupling
of internal generator bus 3 and load bus 9. At point X, the element of SVE approaches
infinity, jumps to negative infinity and remains negative until the next spike point at
t = 5.4s as shown in Figure 5.1(a). Loss of voltage PQ controllability thus occurs at point
X. At the same point of time, the element of T goes to zero and become positive as shown
in Figure 5.1(c). Instability in flux decay dynamics associated with generator 3 is brought
about at point X that is observed in E;. This confirms Theorem 5.1. In the same figure, it
is also shown in (d) that E; reaches the lowest point and increases after point Y. Theorem
5.8 is confirmed since all the conditions (i—v) are met just prior to the increase in E; on

generator 3.

For case 2, where the excitation control system on generator 3 is disabled, the
simulation results for the appropriate elements of SQGE , SVE and T are shown in Figure
5.2 at the same buses as in case 1. At point A, the diagonal element of SQoE associated
with generator 3 changes from positive to negative as shown in Figure 5.2(a), so that
voltage PV controllability is lost. Around the same time, the diagonal element of T changes

from negative to positive. This confirms Theorem 5.2. From Figure 5.2, it is also shown

89

 

that before point A, 590 E and SVE are positive, voltage PV controllability and voltage PQ
controllability hold, and the element of T is negative. This confirms the theory in Theorem

5.3.

In order to confirm the theories in Theorem 5.4, Theorem 5.5, Theorem 5.6, Theorem
5.7 and Theorem 5.9 clearly, the sensitivity matrices, the test matrix and the internal
voltage of generator 3 are plotted in Figure 5.3. On this graph, it is shown that between
point A and point B, the element of SQaE associated with generator 3 is negative, but the
appropriate elements of SQGE’ SVE and SQLV—l are positive, E; at generator 3 is unstable
and decrease toward zero. This confirms Theorem 5.4. At point B, SQLV—l approaches
positive infinity, jumps to negative infinity and then go back toward zero, load PQ
controllability is lost, and singularity induced bifurcation occurs. This confirms Theorem
5.5. Between point B and point C, the element of SQLV-l and T go through the infinity
process, E; at generator 3 declines more and more rapidly and produces the negative
spike. Theorem 5.6 is thus confirmed. A dynamic Q-V curve is generated that produces

singularity induced bifurcation and the negative spike.

At point B, load PQ controllability at bus 9 occurs since the element of SQLV--l goes to
positive infinity, discontinuously changes to negative infinity, and then increase toward
zero. At the same time, SQGE at generator 3 goes to negative infinity, discontinuously
changes to positive infinity, and then decrease toward zero. And the diagonal element of
matrix T associated with generator 3 switches from positive infinity to negative infinity and

causes a rise in E; as is shown at point C. This confirms Theorem 5.9.

At point C, the diagonal element of SQoE associated with generator 3 goes to negative
infinity, and then positive infinity, the appropriate element of SQer that relates generator
bus 3 to load bus 9 goes to positive infinity to assure loss of voltage and load PV

controllability as required by Theorem 5.7. It is not shown that load and voltage PV

90

controllabilities are maintained because all elements of SVE and SQGQL are not shown but
they are because there are no spikes on any of the other sensitivity matrix elements. It is
then clear that the change in E; is positive at point C when these confirming conditions and

results of Theorem 5.7.

The results show that a sharp negative voltage spike is a result of singularity induced
bifurcation brought by a dynamic Q-V curve. This occurs when the exciter is disabled on a

generator.

A sharp negative voltage spike also occurs when the exciter is active and a maximum

excitation limiter is present as noted in the discussion of case 1 results.

91

 

 

 

..............

 

 

 

 

 

 

m>m E

 

 

 

 

 

 

 

 

 

p o
o 0 o
1 1

 

.
mmmm>z_ Bow 3

 

 

 

.........................

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5.1 Simulation result for theorems 5.1 and 5.8.

92

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 1 1.5 2 2.5 3 3.5

Fig. 5.2 Simulation result for theorems 5.2 and 5.3.

93

 

 

 

 

 

 

 

 

20

o

moow

0
4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

31 ............... 1 31 ................ L 31 .............. l 31 ............... 1 31 ...... H ......... 1

51 ............. -51 -5- ....... w ..... 151 ............... 151 ..... w .......... 1

2 EA .......... 1 2 1A... ........ 1 2 1AA. .1 2 12A ........ 1 2 1 ..... WA ...... 1

51 ............... 151 ....... m ........ 151 ............... 15- ............... 15- .............. 1

1 1 m 1 4| 1

1 1 1 1 1 m .
0 o 0 0 0 o 0 0 o o o 0 1 5 0

_
JOOOw wmmm>z_ >1.

Ow w>m

._.

 

3

2.5

1.5

Trrrie

Fig.5.3 Simulation result for theorems 5.4, 5.5, 5.6, 5.7 and 5.9.

94

CHAPTER 6

CONCLUSIONS AND FUTURE WORK

The recent WCSS power system blackouts have some unique characteristics compared

with the classical voltage instability problems, which include:

0 Interarea oscillations existed before the blackout

0 Loss of control voltage stability and clogging voltage instability do not occur
because the dynamic submodel experiences instability before the loss of control in
algebraic submodel occurs

0 The stability controls were effective and responded to the contingency and became
unstable

0 A sharp voltage spike was observed

This dissertation has addressed three major aspects of the cause of the recent WCSS

power system blackouts.

1. It explained why maximum excitation limiters fail to reduce field current limit
violations when large inter-area oscillations are present and why this can lead to
switching from an AC regulator to a DC regulator excitation control and finally to
tripping of the generator off the system through theoretical analysis and simulation

studies. It proved that if the generator field current limit is exceeded and if the field

95

current magnitude of the interarea oscillations slightly exceed the magnitude of the
generator field cunent limit violation, then the maximum excitation limiter will not
recognize the field current limit violation and would allow thermal damage to the
generator. The failure of the maximum excitation limiter will cause the over
excitation protection relay to trip out the AC regulator in an attempt to avoid the
thermal damage to the generator rotor. The tripping of the AC regulator with PSS
and stabilization compensator to a DC regulator, which has neither PSS nor
stabilization compensator, could lead to unstable generator dynamics that inevitably

cause a blackout to occur.

. It defined voltage PV controllability, load PV controllability, voltage PQ
controllability and load PQ controllability, and derived the conditions on sensitivity
matrices SQLV-l , SVE, SQaE and SQGQL that indicates when each type of the
controllability is retained and when it is lost. It has been proven that element of
SQLV—l approaches positive infinity, jump to negative infinity, then increase
toward zero when load PQ controllability is lost. It is further proved that loss of '
voltage PV controllability may occur long before loss of load PQ controllability
occurs, but loss of load PV controllability and loss of voltage PQ controllability
generally occur simultaneously. It is also proved that loss of load PQ controllability
is affected by the load characteristics of the network. Elements of SQ L114 becomes

larger when shunt capacitive load increases. All the theories are proved both

theoretically and through simulation.

. It has been established that a saddle node bifurcation followed by an inevitable
singularity induced bifurcation produces the characteristic negative voltage spike
observed on the WSCC power system blackouts. The thorough theoretical

justification and simulation studies present a clear understanding of the dynamic

96

responses to bifurcation sequences associated with voltage collapse. It has also
been proven that disablement of the excitation control system by over excitation
protection relay is not enough to cause instability, but disablement of the excitation
control system and slow loss of voltage PV controllability cause saddle node
bifurcation that results in slow unstable decline in E} This unstable decline in E;
causes a reactive generation decrease, a dynamic produced Q—V curve occurs as E q’
is decreasing. The dynamic Q-V curve at a generator bus in a voltage control area
causes elements of SQG 5 associated with that voltage control area to further violate
voltage PV controllability conditions as E; decreases and voltage PV controllability
is finally lost. This dynamic Q-V curve also causes the elements of SQLV-l
approach infinity, become negative infinity and then increase toward zero. A more
and more rapid decline in E; occurs as load PQ controllability is lost. The
accelerated rate of E ; decline causes a sharp negative voltage spike in E; . The
change in SQLV-l values from +00 to -00 in the voltage control area experiencing
instability causes SQG g to change from +00 to -00, and the unstable eigenvalue of T

to change from +00 to -00. Such a change in T causes a stable but fast rise in E; .

The purpose of the sensitivity analysis is to identify the subsystem which initially
experience instability, to identify why this instability occurs and how this instability could
cascade to other subsystems, including the algebraic model, and to identify what can be
done to cure the instability problem and when and where the cure should be applied. This
trajectory sensitivity approach is quite different because it focuses on the network
Sensitivity and controllability behaviors and how loss of one or more controllability

properties produce the instability in dynamical subsystems.

Based on the achievements of this dissertation, future research is defined as follows:

97

To apply the sensitivity approach to the generator inertial dynamics, or the inertial
dynamics and flux decay dynamics of generators together because some bifurcation

actually produce instability in these subsystems of a power system model.

To apply the sensitivity approach to the inertial dynamics, or the inertial dynamics

and flux decay dynamics of inductions motors together.

To apply the sensitivity approach to underload tap changer and switchable shunt

capacitor control dynamics.

To link the trajectory sensitivity method to bifurcation subsystems that experience
the instability produced by bifurcation.

98

APPENDIX A

THEOREM PROOFS

Proof of Theorem 5.3

When voltage PV controllability holds, SQc E is a diagonally dominant, and
I 2

diag ——qi—F ’ Q60. Al
Xdi _ Xd, , ( )

is a diagonal matrix with positive elements since

 

I 2 I 2
Q0 < 41 < 41 , (A2)
0' Xd, Xd, - Xe,
Therefore, the matrix
El 2
diag ——5’-"-—— — QC + S
, ,. Q E (A3)
Xd — Xdi 0 G

i

should also be diagonally dominant. Since

diag .1— 111. 3L—
Trio, and 8 E4, (A4)

have all positive elements, matrix

99

 

 

d' d {X X3 d E, 2 Qe +5
rag lag -—' lag ——:—,-— 0. G A5
do, Eq‘. Xd‘. Xd , Q E ( )

is diagonally dominant and has positive eigenvalues, and

 

1
—dia
8 T’

do.

I

d {X X‘; d { En Qc}+s
rag‘—,d" tag —— 0 A6
E Xdi— x; . QE ‘ )

has all negative eigenvalues. From [8,11], matrix (A6) is matrix T when KAi = 0, for all

i, matrix (A5) is matrix -T when K141 =0 forall i.

If field current limits are reached on a generator and the excitation control system is
disabled, the row of —T associated with that generator is replaced by a row of (A5), and
this row has the smallest row sum since the elements of this row are the elements of —T
with KA'. = 0. This row sum is positive since (A5) is diagonally dominant. Thus the
minimum row sum is a lower bound on all eigenvalues of —T , all eigenvalues of —T are
positive. If the smallest row sum of —T is positive with -T having all positive

eigenvalues, the largest row sum of T is negative and T has all negative eigenvalues.

In summary, if an excitation system is disabled, matrix T is diagonally dominant
and has all negative eigenvalues when voltage PV controllability and voltage PQ

controllability hold, that is, matrix T is negative definite.

Proof of Theorem 5.4

From[11], we know that

I

M?
, + NAQL (A7)

40

 

AE;=T

100

 

 

 

 

X - X '
T : diag J— diag —E;0i + QGot( 4; d1)
T401 «101
I (A8)
. "KA.KR V0 . X11. _ Xd
+diag ' ’. ‘ WSVE — drag '—,—'— SQG
K51 + 581' + SEtEfdor E401 E
—K K V X _ X’
N = diag J— diag A‘ R‘. 0‘ WS'lgLv +diag —d‘ 4“ SQ Q
Trio, K5,. +S51 +SEiEfdot- Em» C L

(A9)

When the excitation system on the ith generator is disabled by over field current

 

limiter relay,K - '0 ,the ith row of T rs replaced by the ith row of (A6),
x, X’ E’ 2
—diag diag —i-:—d+' diag —— — Q, + S
I o, QGE (A10)
Tdoi E; Xd‘ X
the i th row of N is replaced by the i th row of
. 1 . X11. " X11,
d‘ag '71" mag 90121 (Al 1)
dm 401'

When loss of voltage PV controllability occurs, SQGE is not diagonally dominant

. 2
and SQGE~ — QG-+ +——3‘———,— < 0, so matrix (A6) will not be diagonally dominant
" ' Xd "Xd

AE’ Ale"

4
and the ith row sum of TEE—"L E, will be negative if

2 t'
SQGE —QG- +___'_<O for AE; <0 and AEj :0, J i l Note that
u 1 Xd, _ Xd,

101

 

AB} = 0 j ¢ 1 is an approximation of the small changes in internal generator voltage
for generators with exciters compared to the unstable generator j= i. The increased

generation causes (a) generators to reach field current limits, (b) action of field current

limiters, and (c) ultimately singularity induced bifurcation.

When the system is load PV controllable, SQ

GQL rs nonnegative and no zero

columns, so the elements in ith row of matrix (A7) are nonnegative, the i th element of

NAQL {NAQL },. < O for AQL < 0.

From Theorem 2, we know that the generator that loses its excitation control and
those voltage control buses which experience loss of voltage PV controllability becomes

unstable.

 

., AE;
Therefore, M4, = T E' +{NAQL},- <0 for AE; <0, and AQL<0.

‘10,

' I
and thus Eq‘. is unstable and decreases toward zero.

Proof of Theorem 5.5:

The computing of a Q-V curve implies adding reactive load by decreasing voltage at

a bifurcation generator bus. Letting A5120) decrease at bus i with AEJ. = 0 for j¢i and
r 2

assuming SQGEii — Q61. + X (1" X. < O with loss of voltage PV controllability but
d, '" d,-

 

without loss of load PV controllability implies instability in flux decay dynamics causing

AE; (t) to continue to decrease producing a dynamic Q-V curve at bus i that causes
increasing reactive generation at bus j AQGJ- = {SQGE}jiAEi > 0 when AQL S 0 for
j at i. This analysis confirms that a Q—V curve stress test is administered through the

unstable generator dynamics resulting in inevitable bifurcation in the algebraic submodel of

102

the differential algebraic model and singularity induced bifurcation in the differential

algebraic model because the effects are so dramatic on generator flux decay dynamics. This
-1
dynamic Q—V curve causes SQLV , SQGQL and SVE to approach infinity, become

negative infinity, and then increase, and SQoE to approach zero, become negative infinity,

then become infinity and then decrease in the voltage control area associated with Jim, as

noted in Theorem (4.6).

Proof of Theorem 5.6

The slope of the Q—V curve at generator i where the dynamic Q—V curve occurs is

{SQLvh , it approaches zero from Theorem 5.5 and the discussion of Q—V curves at the
end of Chapter 4. SQGE = 11 — J ”SQLV-l] 21 elements associated with the voltage
control area containing generator i where the dynamic Q-V curve occurs become negative
infinite as SQLV-l elements approach +00 because J”, .1,2 and J,, are continuous

—1
functions of the bifurcation parameter AQ“ and change little compared to SQLV elements

approaching +00. The eigenvalue of T associated with generator i, where loss of PV
controllability occurs and the excitation control system is disabled, approaches +00, since
(i) the lower bound eigenvalue estimate computed via a minimum column sum of SQGE is
associated with the generator bus i experiencing the dynamic Q-V curve in the voltage
-1
control area where SQLV elements approaches infinite, and (ii) the minimum eigenvalue
of SQGE is related to this large eigenvalue of T by a negative constant diagonal. The
I
eigenvalue of T approaching + 00 explains the rapid decline in AEqi as the dynamic Q-V
curve ultimately causes loss of load PQ controllability from Theorem 5.4. Note that the
minimum column sum of SQoE is also a measure of violation for voltage PV

controllability condition, and the condition {SQGE},',‘ > O is never violated before
N
condition Z{SQGE}U > 0 is violated.

J=|

103

 

Proof of Theorem 5.7

When the excitation system on the ith generator is disabled by over field current

limiter relay, KAi = 0 , the i th row of T is replaced by the i th row of (21),

 

. 1 X —x;, E; 2
—drag T; diag T diag 77 — 9011.- + Soar ( A12)
the ith row of N is replaced by the i th row of (22)
d ' 1 d ——X X, S
rag _ rag 12691 A13
Tdot' E401 ( )

When loss of voltage PV controllability occurs in a voltage control area such that

 

2
SQGE" — Q01" YJ——_ < O for a bus in that voltage control area, matrix (A12) rs
ll 1
d d,-
’ I
. . . T AE‘I T_AE‘I . . .
not diagonally dorrnnant and the 1th row sum of EI E' rs posrtrve for
40 40 1'

41

AE’. >Oand AE'qj z0,j¢l

When loss of load PV controllability occurs at generator bus i, elements of SQo QL

in row i associated with the voltage control area containing the dynamic Q-V curve are
negative, elements in row i of matrix (A13) are negative, the i th element of NAQL

{NAQLL is generally then positive for AQL < 0 when i belongs to that voltage control area

experiencing the dynamic Q-V curve once load PV controllability is lost..

104

 

From Theorem 5.4, we know that the generator that loses its excitation control and
those voltage control buses which experience loss of voltage PV controllability becomes

unstable.

.’ A_E€1. I ' . .
Therefore, AE,,= T E, +{NAQ1},>0 for A5,, >o,111_~:qj so, 1:1,

40

and AQL < 0, the system is unstable and the change in E; is positive once load PV

controllability is lost in the voltage control area containing bus i.

Proof of Theorem 5.8

If the system is stable after loss of load PQ and load and voltage PV controllability,

at steady state,
TAE; = —NAQL (A14)
for the i th row of the equation above
{TL-AB; = -{N}.-AQ1 (A15)

Since the excitation control systems on all machines are active, T and N can be

represented as

 

 

T ~ dia J— dia _K""K”1V°i WS
g Tdm g KB.- + 551 +SE.Efd11 VE (A16)
—K K V
N z diag ——1— diag A‘ R‘- 0‘ WS—IQLV (A17)
Tdo, K5.- + SE: + SEiEdei

105

 

From theorem 4.6, when loss of load PQ controllability occurs, loss of voltage PQ

controllability may also occur, therefore, from given (i), (ii), (iii) and the above equations

for T and N , we know that {T},.,. will dominate the i th row of T as long as generator i is

the only generator in the voltage control area and W measures voltage in that voltage

control area that experiences loss of load PQ controllability and loss of load and voltage PV
controllability, {N }‘.j will dominate the i th row of N as long as generator bus i and load

bus j lie in the voltage control area where loss of load PQ controllability and loss of load

and voltage PV controllability occur, and (a) can be represented as

{T} ii Mt},- “ keg/CA "{N}ik AQLk (A18)

since AE'qj = 0, j¢ i, where {T},,. and {N}ik will be negative, and Nij z 0, AQLk

is negative. As a result, AE; is positive.

106

 

[1]

[2]
[3]

[4]

[5]

[6]

[7]

[8]

[9]

APPENDIX B

PUBLICATIONS

R. A. Schlueter and S. Liu, "Intelligent control for a power system in a deregulated
environment", Proc. of the North American Power Symposium, MIT, November,
1996, pp. 81-88.

S. Liu and R. A. Schlueter, "Intelligent control for a power system ", Proc. of
American Control Conference, Dearbom, Sept. 1996, pp. 456-460.

R. A. Schlueter, S. Liu, K. B. Kilani and I. P. Hu, "Static voltage instability in
flux decay dynamics of a midterm stability model ", Accepted for publication in the
J. of Electrical Power Systems Research.

R. A. Schlueter, S. Liu, K. B. Kilani and I. P. Hu, "Static voltage instability in
generator flux decay dynamics as cause of voltage collapse", Submitted to IEEE
Trans. on Power Systems.

R. A. Schlueter and S. Liu, "A structure-based hierarchy for intelligent voltage
stability control in operation planning, scheduling, and dispatching power
systems", Submitted to IEEE Trans. on Power Systems.

R. A. Schlueter, K. B. Kilani, S. Liu, "Justification of the voltage stability security
assessment and diagnostic procedure using bifurcation subsystem method",
accepted by the 1998 large Engineering Conference on Power Engineering, Nova
Scotia, Canada, June 7-9, 1998.

R. A. Schlueter, and S. Liu, "A structure based intelligent voltage stability control
in scheduling and distaching power systems", accepted by the 1998 large
Engineering Conference on Power Engineering, Nova Scotia, Canada, June 7-9,
1998.

R. A. Schlueter, S. Liu, "Justification of the voltage stability security assessment as
an improved modal analysis procedure", accepted by the 1998 Large Engineering
Conference on Power Engineering, Nova Scotia, Canada, June 7-9, 1998.

R. A. Schlueter, S. Liu, N. Alemadi, "Intelligent voltage stability assessment,
diagnosis and control of power system using a modal structure", accepted by Bulk
Power System Voltage Phenomena: Dynamics and Control IV, Santorini, Greece,
Aug. 1998.

107

 

[1]

[2]
[3]

[4]

[5]

[6]

[7]

[8]

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T. V. Cutsem and C. Voumas, "Voltage Stability of Electric Power Systems",
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R. A. Schlueter, A voltage stability security assessment method", IEEE 1997
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R. A. Schlueter, K. B. Kilani and U. Ahn, "Impact of modeling accuracy on type,
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