A COMPREHENSIVE STUDY OF THE APPLICATION OF POWER
SYSTEM STABILIZERS USING EIGENVALUE ANALYSIS AND
TIME DOMAIN SIMULATION
by
Shawn E. Patterson
B.S.E.E., University of Colorado, 1985
\
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
1995
This thesis for the Master of Science
degree by
Shawn E. Patterson
has been approved
by
William R. Roemish
M. Radenkovic
Date
Patterson, Shawn E. (M.S., Electrical Engineering)
A Comprehensive Study of the Application of Power System Stabilizers using
Eigenvalue Analysis and Time Domain Simulation
Thesis directed by Professor Pankaj K. Sen
ABSTRACT
Maintenance of power system stability is constant challenge to power system
engineers. Since the consumption of electrical power is likely to outpace the growth of
power system networks, stability margins will need to be increased through available
means of control. One such controller which can improve dynamic stability of a power
system is the power system stabilizer (PSS).
In the past, the application of PSS has been an imprecise endeavor, since complete
information about the system has not been available. For years, PSS has been applied
to generators by engineers using whatever analysis tools available such as frequency
response analysis or time domain simulation. The use of eigenvalue of analysis has
been restricted to very small and inaccurate system models.
Recently, due to the developments in power system modeling, computational
power, and software techniques, eigenvalue analysis has become an additional tool for
the engineer to study dynamic stability of a power system. These Electric Power
Research Institute (EPRI) developed computer programs, called MASS, PEALS, and
DYNRED, allow the engineer more insight into the problems and solutions to power
system stability.
This thesis describes a pioneering effort in the use of these software tools in the
study of power system stabilizer application. In this study, the programs were used to
m
calculate eigenvalues which were then used to evaluate PSS application at the Grand
Coulee Power Complex. Time domain simulation studies were also performed to
corroborate eigenvalue analysis. All studies were performed using a full
representational model of the WSCC power system.
Background information on system stability, the development and application of
PSS, and eigenvalue analysis are reviewed in detail. Topics investigated were speed
input versus accelerating power input PSS, the use of transient gain reduction in
voltage regulator circuits, and the replacement of rotating exciters with static excitation
equipment for stability enhancement. Compromises between dynamic and transient
stability are evaluated. Insight is gained into the effectiveness of PSS and the dynamic
behavior of the power system.
This abstract accurately represents the content of the candidate's thesis. I recommend its
publication.
Pankaj K. Sen
iv
ACKNOWLEDGMENTS
I wish to thank Mr. Jeff Hoffman and Mr. J.C. Agee for their valuable input and
support throughout the life of this research project. A special thanks goes to Mr.
Hoffman who provided many of the additional computer programs used to process data
and results.
I would also like to thank Boyd Leuenberger and Vernon Nokes for allowing me
the opportunity to take on such a project.
v .
TABLE OF CONTENTS
CHAPTER
1. Introduction........................................................ 1
2. Background Information .............................................. 7
2.1 PSS Concepts.................................................... 7
2.2 Practical Considerations of PSS ............................... 12
2.3 Application of PSS ........................................... 14
2.4 Eigenvalue Analysis ........................................... 19
3. Study Details ...................................................... 24
3.1 Purpose of Study .............................................. 24
3.2 The Grand Coulee Power Complex ............................... 25
3.3 System Data ................................................... 26
3.4 Analytical Techniques ........................................ 27
3.5 Study Methodology ............................................. 29
4. Study Results .................................................... 33
4.1 General Observations........................................... 33
4.2 Excitation Modifications at Grand Coulee Third Powerplant .... 34
4.2.1 Evaluation of Existing System ............................... 34
4.2.2 Accelerating Power Input Stabilizers ........................ 36
4.2.3 Transient Gain Reduction..................................... 43
vi
4.2.4 Summary .................................................. 49
4.3 Replacement of Rotating Exciters on Generators 118 ...... 49
4.3.1 Evaluation of Existing System .......................... 49
4.3.2 Static Excitation ........................................ 51
4.3.3 Damping of Power System Oscillations ...................... 51
4.3.4 Transient Stability....................................... 53
4.4 Combination of Modifications.............................. 54
4.5 Topics for Further Study ................................... 55
5. CONCLUSION .................................................... 57
APPENDIX
A Generator data for Grand Coulee Power Complex...................... 60
B Time domain simulation results ...................................... 65
C Disturbances used in time domain simulation ....................... 106
D Eigenvalue plots for post disturbance cases........................ 109
REFERENCES ............................................................ 115
FIGURES
Figure
1 Block diagram of single machine, infinite bus model....................... 8
2a Torque due to armature reaction.......................................... 9
2b Torque due to exciter.................................................... 9
3  Excitation system models with transient gain reduction................. 11
4  Power System Stabilizer block diagram .................................. 12
5  Block diagram for accelerating power PSS circuit........................ 14
6a Frequency response plots of generator and exciter without PSS, and the
PSS response tuned for local mode compensation....................... 16
6b Impulse response of generator with and without PSS .................... 17
7a Overcompensated local mode ............................................ 19
7b Undercompensated local mode ........................................... 19
8  Single line diagram of the Grand Coulee complex ....................... 26
viii
9  Eigenvalues existing PSS vs. no PSS (PEALS)
35
10  Eigenvalues existing PSS vs. no PSS (MASS) ........................ 37
11  Eigenvalues existing PSS vs. increased gain (MASS).................. 38
12  Pulse on G19 existing PSS vs. increased gain ...................... 39
13  Pulse on G22 existing PSS vs. increased gain ....................... 39
14  Eigenvalues existing PSS vs. increased gain ....................... 40
15  Eigenvalues exciter with TGR vs. exciter without TGR ............... 45
16  Eigenvalues Generators 118, with PSS vs. no PSS ................... 50
17 Eigenvalues existing excitation vs. static excitation, G1G18 ...... 52
18 Eigenvalues limited gain vs. optimal gain generators 118 ........ 53
19  Eigenvalues existing case vs. combination of modifications ........ 54
IX
1. Introduction
Electric power systems are very dynamic, both in structure and in operation. They
are in a state of constant evolution as more elements are added to the system due to the
increasing demand for electric power. These systems are never truly in a steady state
condition, as the state of the system changes with every flip of a switch. Loads are
continuously changing and generation must increase or decrease correspondingly. Yet
throughout all this change, the electric power system is expected to continuously supply
power at constant voltage and constant frequency, even in the event of large
disturbances to the system.
Historically, the growth of an electric power system has kept in step with the
increasing consumption of electric power. As the demand for power has increased over
the years, the power utilities have responded by increasing the amount of generation in
their systems, and have built new transmission lines to deliver it. The result is a large
interconnected network of powerplants, transmission lines, and loads.
As for any dynamic system, there are boundaries that separate stable operation from
instability. Of major importance to power utilities are the boundaries that define the
maximum amount of power that can be transferred in a system. Historically, these
stability limits were increased by a strengthening of the system or building more
transfer paths.
1
So the evolution of electric power systems can be attributed to the need to increase
their maximum power transfer limits in order to serve more customers. This evolution
continues today, however, the traditional solution of building new transmission lines is
becoming increasingly difficult due to economic, environmental, and political reasons.
It has therefore become the burden of power system engineers to find ways of
increasing the stability, and therefore the maximum power transfers of a system,
without constructing new transmission lines. It is for this reason that stability analysis
is becoming more important an issue for the power industry.
A power system consists of a very large number of electrical, electronic, electro
mechanical, and mechanical devices, which together make a very complex, nonlinear,
timevarying system. These facts alone make stability analysis and control design a
very difficult problem. This problem is further complicated by the physical size of the
system. The whole power system is made up of smaller power systems that are
individually owned and geographically dispersed, which makes centralized control of
the whole system an impossibility. Therefore, stability enhancements to the system
must be made on a decentralized, or local level.
There are three types of stability that are usually referred to regarding power
systems, although they all refer to the same overall system stability. The three terms
arise depending on the time scale of interest. Steadystate stability refers to the ability of
the system to maintain synchronism and operate at a point of constant power flow and
voltage for an indefinite period of time, provided the system is not disturbed. Steady
state stability refers to operation at a stable equilibrium point Transient stability refers
to the ability of the system to recover from a significant disturbance. During the very
short time after a disturbance, about one second or less, the system is operating in a
nonlinear mode, in which the configuration of the system may be changing
2
dramatically. If a portion of a system, such as a powerplant, is able to withstand the
disturbance and remain in synchronism, the plant can be said to be transiently stable.
Between the time scales of transient and steadystate stability exists a period of time
which is referred to by the term dynamic stability. It is often called smallsignal
stability, which means that the system stability is tested by a small perturbation from its
equilibrium point. Since the perturbation is small, the operation of the system remains
linear around the initial operating point. Dynamic stability refers to the ability of the
system to return to its original operating point. In a power system, in the period after a
perturbation in the power flow, the response of generators is characterized by low
frequency oscillations in the range of 0.01 to a few hertz. Above this frequency range,
the generators in the system have essentially no response, due to the large amount of
inherent damping of the generators and system loads.
There are several ways in which stability of the system can be enhanced on the
powerplant level. Transient stability can be improved generally by increasing the speed
of equipment response. Decreasing fault clearing times, fastvalving of generator
turbines, or increasing excitation voltage response are a few things that can be done on
a plant level which can improve transient stability. Dynamic stability can be affected by
the generator controllers, namely, the governor and the voltage regulator. The governor
is used to control the speed of the machine through regulation of the water or steam
through the turbine. The frequency response of the governorturbine control loop is
limited to very low frequencies, usually less than 0.2 hertz. Due to the mechanical
nature of the prime mover of a generator, the power system has a lot of inherent
damping at these frequencies and therefore stability enhancement is not necessary in
this frequency range.
3
The voltage regulator is the controller in an excitation system of a generator which
controls the voltage of the machine. The frequency response of the generatorexciter
control loop has a much higher bandwidth which extends typically up to around 2
hertz. In this higher range of frequency there is very little inherent damping, due only to
the very small amount of resistance in generator windings. Therefore, oscillations in the
0.2 to 2 hertz range of frequency can present a dynamic stability problem. Therefore,
the generatorexciter system of a plant is one place where the enhancement of stability is
often required.
One way to accomplish this is to apply power system stabilizers (PSS) to
powerplant generators. The purpose of the PSS is to increase the ability of a generator
to damp low frequency power oscillations. This is accomplished, very basically in
frequency response analysis terms, by introducing phase compensation of the inherent
phase lag in the system. Although this may seem like a simple solution, in practice it is
neither simple nor straightforward. This is due primarily to the complexity of the
system, whose frequency response characteristics, if completely obtainable, are far too
great in number to be addressed by the simple, available means of control. There are
also other issues that need to be addressed, such as controllability and observability,
which this method of control design fails to. Furthermore, since the system
characteristics are prone to change, a controller designed for one system configuration
may not be satisfactory for other configurations.
Given these restrictions on the design of the PSS, it is easy to understand that in the
past the application of PSS has been a somewhat imprecise endeavor, at times more of
an art than science. The application of PSS has proceeded, nonetheless, primarily
through the use of very simple system models, usually consisting of one detailed
machine model and an infinite bus. These models provide fairly accurate information
4
regarding the dynamics of a single machine, but obviously no insight into the system
dynamics can be gained. Analysis using these simple models have resulted in PSSs
designed to increase damping of the host machines natural oscillations while at the
same time slightly increasing the damping torque of the machine at other frequencies of
oscillation and, hopefully, improving system stability. Since the rest of the system is
treated as a black box, a likely result of this design method is either too conservative a
controller or one that does more damage to overall.stability than good.
For most of its history, this has essentially been the design method for PSSs. The
stabilizer is applied to improve damping of one or two modes of oscillation, and the
solution is verified by comparing the oscillation before and after the PSS is added. If
the oscillation is more damped with the PSS, the design is a success. As the use of
computers evolved, and consequently the development of power system simulation
programs, the ability to perform this verification prior to implementation became
possible. Using a simulation program, several design alternatives can be compared
using several different system configurations and several system perturbations. This
tool therefore allows much more information to be used in the design problem,
resulting in a more effective and robust controller.
Still, even if the power system can be modeled in great accuracy, there are many
more possible modes of oscillation than can be studied using time domain simulation.
Furthermore, the factors that affect these modes remain a mystery in this form of
analysis. What is needed is another analysis tool that can provide this additional
information. Eigenvalue analysis provides this tool.
In a linear system, a wealth of information regarding the oscillatory behavior of the
system can be obtained from the systems eigenvalues. All that is required are some
5
mathematical calculations. The ability to perform these calculations using data from a
detailed system representation has only recently become possible, since the
computational power required is very large, requiring both a powerful computer
processor and a large amount of memory. Also required are some software tools
designed for the specific application of power system analysis. Recently, some tools
have become available through the research efforts of the Electric Power Research
Institute (EPRI), and have begun to be used by some power system engineers.
Since these software tools have been available for only a short period of time, then
use has remained fairly limited. Most of the published works involving these programs
has been involving further development or understanding of the tools themselves, and
these studies are usually performed on small, hypothetical systems consisting of only a
few machines. To date, only a very small number of engineers are using these tools to
analyze real problems on real systems.
The purpose of this thesis is to perform a comprehensive stability analysis of the
PSSs installed at the Grand Coulee Power Complex using eigenvalue analysis tools in
conjunction with traditional analysis techniques, such as time domain simulation. A
general discussion of PSS application will first be presented, followed by some
background material on the analytical methods to be used. A presentation of the system
to be analyzed will then be given. Finally, a detailed analysis of the test system will
then be performed and results and conclusions will be presented.
6
2. Background Information
2.1 PSS Concepts
For about 30 years electric power utilities have been applying power system
stabilizers to generating units to improve dynamic stability of the system. These
stabilizers are intended to solve a specific problem. The problem began to occur with
the advent of automatic voltage regulators, which when incorporated into an excitation
system of a generator, tends to decrease the little amount of natural damping torque
present in the generator [1], This problem became even more evident when remotely
located generators were connected to the power grid via long transmission lines, or
when power systems in different geographical areas began to be tied together [24].
This problem manifested itself in the form of poorly damped, low frequency power
oscillations in some generators in the system.
The erosion of the natural damping of generators is further exacerbated through the
use of high gain, high initial response exciters. The effects of these excitation systems
can be best understood by a brief examination of figure 1. This figure shows the
simple, single machine, infinite bus block diagram presented in [1], which is one of the
most referenced diagrams in the literature. The complete definitions of the constants on
the figure will not be discussed here, but are detailed in [1], The feedback term
7
representing miscellaneous damping contributions such as amortisseur effects, load
damping, friction, windage, etc., commonly denoted as D, has been neglected.
Figure 1 Block diagram of single machine, infinite bus model.
From the diagram we note that there are two feedback torques. The first involves
the constant Ki, which is the synchronizing torque coefficient for constant field flux
linkages. Ki is the slope of the torque angle curve usually associated with steadystate
and transient stability concerns. The other torque involves the armature reaction effects
and the effects of the excitation system. In this diagram, the exciter is shown consisting
of a gain and a time constant, which would be proper for a static, rectifiertype exciter.
A rotating exciter model would contain a second time constant
In the first loop, the torque component is simply KiA5, which is in phase with the
rotor angle and, therefore, defined as a synchronizing torque. A damping torque is
defined as a torque in phase with rotor speed variations (which lead rotor angle changes
8
by 90 degrees). Neglecting the exciter for a moment (constant Efd), we can see that the
torque component from the other loop will, for a given oscillation frequency, lag the
rotor angle changes by an angle between zero and 90 degrees, but with negative sign.
Therefore, the effective phase angle of this torque component relative to the rotor angle
will be between 90 and 180 degrees leading. As can be seen in Figure 2a, this torque
adds a positive component of damping torque, but a negative component of
synchronizing torque.
Figure 2b Torque due to exciter
When the effects of the exciter are considered, the effects of the K4 branch in the
diagram can be neglected at low frequencies due to the high exciter gain. If this is done,
then the equation for the torque component in the lower loop is
K2KeK5
AT =
Td0'Tes2 + (TJ K3 + Td0')s + (1IK3 + K6KJ
A5
(21)
The result is a torque component which lags rotor angle changes by an angle between
9
zero and 180 degrees, depending on the frequency of oscillation. Figure 2b shows
typical torque phasors for low frequency oscillations for both a static and a rotating
exciter, which has additional lag.
If equation (21) is broken into its synchronizing and damping torque components
at a particular oscillation frequency, the following torque expressions result:
K2KeK5
ATs=j AS (22)
VK3 +K6Ke co Td0'Te
(neglecting small components due to high values of gain and low frequencies) and
K2K,Ks(.T,IK, + TM'Xa (23)
AT0=5:tA5
(,llK3+K6Ka Td;T,f + (TJK2 + Td0') (a
Since the equations for the two torques are different in sign, there is a tradeoff
between adding synchronizing torque and damping torque. Since K5 is negative in most
cases where stability is a concern, it is the damping torque component that is usually
negative. In this case, as the voltage regulator gain is increased, synchronizing torque is
increased and damping torque is decreased, which can leave the generator prone to
oscillatory behavior.
A high gain is a desirable feature of a regulator because of the low steadystate
error. The positive effect on synchronizing torque is also valuable since the transient
stability of the machine is also improved. However, too high a gain can result in
unstable performance of the regulator. The common remedy of this problem has been to
add compensation into the voltage regulator loop using classical ratefeedback or lag
lead compensation techniques, in order to limit the transient gain. The compensator
parameters are then used to tune the regulator response for a reasonably fast, but well
10
damped response [5]. The tuning is carried out by applying a voltage step into the
voltage regulator and monitoring terminal voltage response. Block diagrams of voltage
regulators incorporating transient gain reduction (TGR) are shown in figure 3.
Excitation System Generator
Transient Gain Reduction using leadlag compensation
Generator
Transient Gain Reduction using ratefeedback compensation
Figure 3 Excitation system models with transient gain reduction
The generator is shown represented in its most simplest form for convenience. The
inclusion of TGR in the voltage regulator control loop helps to maintain stability in the
higher frequency ranges (by decreasing high frequency gain), but does nothing to
improve stability at lower oscillation frequencies. In fact, the phase lag at lower
frequencies is actually slightly worsened by TGR. From figure 1, what is needed to
improve damping of low frequency oscillations is either another control loop supplying
more damping torque component, or a controller which cancels the negative damping
introduced in the system. Since the only portion of the system which is accessible to
adding a controller is the excitation system, the latter choice is the obvious one. A
11
controller can be added as an input to the excitation system which compensates for the
negative damping torques due to the excitation system. This controller is the power
system stabilizer.
2.2 Practical Considerations of PSS
A block diagram of a typical PSS is shown in figure 4. The basic objective of the
PSS is to add lead compensation (zeros) to cancel the inherent lags (poles) in the lower
feedback loop in figure 1, which will result in torque components which are in phase
with speed and rotor angle changes. Ideally, this controller would consist entirely of
phase lead elements. Since in reality each phase lead must be accompanied by a phase
lag, the PSS consists of two or three leadlag pairs. Also, since it is not desired to have
the PSS respond to normal system speed or load changes, a reset, or washout stage is
Limiter
Figure 4 Power System Stabilizer block diagram
added. The output signal of the stabilizer is limited to prevent serious impact on
transient stability, since the stabilizer may prevent the exciter from holding field voltage
to ceiling. The limiter also serves to prevent large voltage fluctuations in terminal
voltage due to a malfunction in the PSS circuitry. The PSS gain is shown in the same
12
block as the time constant due to the input signal transducer.
The input signal for the PSS circuit has been the subject of much discussion. There
is a variety of measurable quantities that are possible candidates for the input signal,
since system oscillations are observable in most machine and system variables. Since
the goal of the PSS is to provide torque components in phase with rotor speed changes,
a speed signal input is an obvious choice and was indeed used in early
implementations, as were frequency deviation signals [69]. Speed input stabilizers
remain the most widespread type in use today, however, power input type stabilizers
have received a great deal of attention, particularly those using accelerating power as an
input signal.
Using speed as a PSS input signal has some inherent drawbacks. The signal can
contain several unwanted components of noise due to the power system, shaft runout
and other mechanical vibrations, and the frequency transducer. The noise amplitude is
worsened by the phase lead compensation in the PSS circuit. Also present are shaft
torsional modes of oscillation, which are of particular concern for thermal units. The
result is a very noisy input signal into the voltage regulator. In order to overcome some
of these problems of the speed input stabilizer, a considerable effort has been devoted
to the development of accelerating power input stabilizers [1013]. Since changes in
accelerating power precede speed changes by 90 degrees, less phase lead compensation
will be necessary in the PSS, and therefore, noise problems will be reduced.
Since accelerating power is the difference between mechanical power and electrical
power, measurement of both components are necessary. Measurement of mechanical
power is a difficult endeavor, however, a mechanical power signal can be synthesized
as shown in figure 5 [11]. A speed signal is differentiated to produce the rotor
13
acceleration, then filtered and added to a filtered electrical power signal to produce a
filtered mechanical power signal. This low bandwidth signal will be an accurate
representation of mechanical power if the governor response time is not very fast. An
unfiltered electrical power signal is then subtracted from this signal resulting in an input
less noisy than speed. Also, since both mechanical power and electrical power are
being measured, a washout function is not necessary to filter out slow load or
frequency changes.
Limiter
Speed
Figure 5 Block diagram for accelerating power PSS circuit.
2.3 Application of PSS
Since their inception, power system stabilizers have been used, in most cases, to
provide some improvement of system stability. This is due to a simple understanding of
the problem and the controller, as reviewed so far. A truly optimal controller using
lead/lag control techniques is not realizable due to the system complexity,
nonlinearities, and time variance of the system parameters. It is in order to overcome
some of these difficulties that recently so much effort has been spent in the development
of an adaptive PSS. However, to date, these devices remain in the research stage.
Therefore, it is important to maximize the benefits of existing stabilizers and stabilizer
14
technology, since they are likely to remain in use for quite some time.
Much has been written over the years about how to tune a PSS to provide the most
stable and robust control. One early set of guidelines was set by the Western System
Coordinating Council (WSCC) [14]. These recommendations, based upon online
frequency response measurements of the voltage regulating system performed at full
load, suggest that the two lead time constants of the PSS be set at the frequency where
there is 90 degrees of phase lag in the plot of the response. Lag time constants should
then be set as small as possible as noise restrictions permit. The washout time constant
should be set at a slow value of 30 seconds. It is then recommended that the gain value
be chosen by increasing the gain until undamped oscillations of around 2 to 4 hertz are
observed, then decreasing the gain to one third of this value.
The standard method of setting the parameters of a PSS used by the Bureau of
Reclamation [15] is a refinement of the guidelines developed by the WSCC. Since the
WSCC procedure often results in undercompensation of the local mode, or natural
frequency oscillation of the generator rotor, a modified procedure is used. In this
procedure, instead of simply setting the lead time constants for compensation of the 90
degree phase lag point, the lead and lag time constants are set to provide near optimal
compensation of the local mode by aligning the peak of the PSS phase lead at the local
mode frequency (see figure 6a). This is essentially a trial and error method, but some
experience can yield a fairly quick procedure. The gain setting is then increased in small
steps while executing impulse response tests and noting the damping ratio and
frequency of the response. The gain with the best damping ratio is then selected as
final. The washout time constant is set to the WSCC recommended value of 30
seconds. The output limits may vary with different equipment applications, but are
usually set near 8 percent.
15
Phase (degrees) Gain (dB)
20
O
20
40
GO
80
Generator and F.xoiter Rs^nnsp.
lif
f
101
lOO
50
50
lOO
150
100 Frequency (rad/sedp1
Generator anii F.rriter Response.
102
:i
l ;.!
: v
 ; v
^
101
100 Frequency (rad/sec}01
102
80
60
CA
U 40
fcb
t3 20
CA
Â£
o
20
40
PSS Response
c.<
y
* N
. \
V
.. e.
101
102
io Frequency (rad/sec/01
Figure 6a Frequency response plots of generator and exciter without PSS, and the PSS
response tuned for local mode compensation.
16
An impulse response of a generator with a PSS tuned using this method is shown
in figure 6b comparing the response without PSS. As can be seen, the PSS is very
Figure 6b Impulse response of generator with and without PSS
effective in damping the local mode oscillation.
Frequency response methods of tuning stabilizers, as discussed here, were used
early on and are still widely in use today. This is because a frequency response
measurement is still, in spite of recent developments using system identification
techniques, the only real way of actually measuring the oscillatory nature of a
generator. In this method of measurement, a sine wave input is swept through two or
three decades in frequency while measuring the terminal voltage. The ratio of the
output to the input signal defines the frequency response. However, since this is done
at a small signal level, the dynamics, and therefore the characteristics of the system are
not measured. With some accurate representation of the generator and excitation
system, a single machine, infinite bus computer model can adequately duplicate the
results of these measurements. Therefore, PSS tuning methods using frequency
17
response measurements and single machine, infinite bus models are essentially
equivalent in the amount of information they supply. Both tools are adequate in
compensating the local mode behavior of a generator, but that is about all they are good
for.
As the application of PSS began to be studied and written about more, additional
theoretical tools such as root locus and eigenvalue analysis were used more often to
better describe the dynamics of the system [16,17]. The use of eigenvalue analysis
ushered in the study of several generators or groups of generators oscillating against
each other, usually at frequencies on the range of 0.2 to 0.7 hertz [18]. So began the
study of these low frequency, interarea modes of oscillation, only on a very small
scale and usually on a more theoretical level.
These tools were also used to shed more light on the local mode behavior of
generators using a single machine, infinite bus representation. In addition to the local
mode of oscillation, which is due to the rotor dynamics of the machine, there is also a
mode which arises due to the interaction of the generator and the excitation system.
This so called exciter mode of oscillation is shown in the root locus plot in figure 7a
along with the local mode for a typical generator with a PSS. The curves are loci of the
eigenvalues of the local and exciter modes as stabilizer gain is varied. From the local
mode curve, it is evident that the local mode is slightly overcompensated, since the
angle of the eigenvalue becomes more negative (and the locus turns downward) as the
gain is increased. If this mode were undercompensated, the loci would curve upward as
shown in figure 7b. From both figures, we note that there is a value of gain which
causes one of the modes to cross into the righthand plane and become unstable at a
frequency of a few hertz. This is the high frequency oscillation as described in the
WSCC tuning procedure.
18
Overcompensation of local mode Undercompensation of local mode
As a result of parameter studies such as these that use root locus plots, several general
observations regarding the tuning of PSS have been made [12]. One is that, for both
speed and power input types of PSS, as the system strength increases, so does the
sensitivity to the PSS gain. Therefore, the PSS gain should be set under strong system
conditions to ensure that the gain will not be set too high. A similar situation exists for
the loading of the machine, where PSS gain settings would be lower if tuned under full
load conditions. Therefore, the tuning conditions for these types of stabilizers would
be with the generator at full load into a strong system. Setting of the PSS gain under
these conditions will ensure that the PSS will not cause instability at other machine
loadings or system conditions.
2.4 Eigenvalue Analysis
Having stated that eigenvalue analysis can be a valuable tool in the study of PSS
application, a brief background of the subject will be presented to introduce some of the
concepts and terms used in this study.
19
The oscillatory behavior in a system is dependent upon the roots of the
characteristic equation that describes the system. From modem control theory, we
know that we can describe a linear system in state space format as follows:
x =Ax + Bu
y=Cx+Du
(2.4)
A power system is not a linear one, however. Therefore in order to describe the
system in state space, it must first be linearized about a steady state operating point. For
small system disturbances the linearization approximation is a valid one.
We also know that the eigenvalues of the matrix A are the roots of the characteristic
equation. There is also much more useful information contained in this representation
which can be extracted using simple modem control theory, such as controllability and
observability information.
The state matrix A has dimension nxn. An nxl column vector vj is called the right
eigenvector of the matrix A associated with the eigenvalue X if [19]:
Avj = X.V; (2.5)
If for a lxn row vector w;, the following is true:
w,A = (2.6)
then the vector wi is the left eigenvector of the matrix A associated with the eigenvalue
^i. The right and left eigenvectors are orthogonal, and since they are not unique they
can be normalized so that
vfw,= 1
(2.7)
20
The vector Vj is the i111 column vector of the matrix V, such that
V = [v, v2 v3 vj (2.8)
Similarly, the vector w\ is the ith row vector of the matrix W:
W = [w, w2 w3 wn]T (29)
Using the matrices V and W, we can make the following change in coordinates:
(X = Wx (210)
x = Va (2.11)
The state space representation using the new state variable a then becomes the set of
first order decoupled equations:
a = Aa+Tu (2.12)
y = Qa (2.13)
where
A= WAV (2.14)
t=wb (2.15)
a=cv (2.16)
The matrix T contains controllability information. The ith mode of the system is
controllable from the input if the vector WjiB is nonzero. Similarly, the i* mode of
21
the system is observable in the output if the vector Cvi is nonzero. Therefore, the
matrix Q contains the observability indices.
The transformation from the state variable x to the state variable a results in a new
state matrix A which is diagonal. This uncoupling of the state matrix means that each
state variable a is associated with only one mode. The matrix V in Equation 2.11
therefore gives information on how each mode is distributed over the state variables.
This information is known as the mode shape.
If Equation 2.5 is differentiated with respect to the kth row and jth column, a^j, the
result is
3 A dVj
V: +A'
B a
kj
5akj
dX. ByJ
Vi + X,
d\i
' dakj
(2.17)
(2.18)
Since w^ = 1 and w;(AXiI)=0, this equation simplifies to:
B A 3X.
w, v, =L
d akj B akj
Since all elements of dA/day are zero except for the element in the k* row and the jth
column, which is 1, this results in
6X.
B at
= wikvji
(219)
This sensitivity of eigenvalue to the element akj of the state matrix is known as a
participation factor. It is equal to the product of the left eigenvector element w^ and the
right eigenvector element vj, and is a measure of the relative participation of the k^ state
22
variable in the i111 mode and vice versa. These terms and concepts will be used liberally
in the analysis of the study results.
23
3. Study Details
3.1 Purpose of Study
One of the most challenging problems facing power system engineers today is the
pursuit to increase power system stability limits. This study is a further attempt to do so
through simple, inexpensive means. Improvements to the excitation systems of
generators is one simple way to increase the stability limits of a power system.
Additional damping can be gained by the addition of PSS or the improvement in its
application. Synchronizing torque in a generator can be improved by replacing slow
exciters or regulators with faster ones.
These modifications to excitation systems, when made on the scale of one typical
plant alone, are not likely to provide much noticeable improvement in stability.
However, when they are made on a much larger scale, some significant benefit might
be derived. The Bureau of Reclamation's Grand Coulee Power Complex, with an
output capacity of about 4 percent of the total WSCC generator capacity (the largest
single site in the WSCC system) is well known to play an important role in the
dynamics of the WSCC system. Therefore, modifications made to improve stability of
the Grand Coulee generators should have a significant impact upon the stability of the
western power system.
24
The modifications to the Grand Coulee excitation systems that were studied are:
1. Replacement of the existing speed input type PSSs with accelerating power
input type PSSs for the six Third Powerplant generators.
2. Removal of the transient gain reduction (TGR) circuits in the voltage regulators
of the six Third Powerplant generators.
3. Replacement of the rotating exciters with static excitation systems for the Right
and Left Powerplant generators.
3.2 The Grand Coulee Power Complex
A general singleline diagram of the Grand Coulee power complex is shown in
figure 8. Grand Coulee Dam is located on the Columbia river in the central part of
Washington state. The Right and Left Powerplants consist of a total of 18 125 MVA
hydrogenerators. The Third Powerplant houses three 718 MVA units and three 615
MVA units. The total output capacity of the three plants is over 6,500 MVA, with the
Third Powerplant making up over twothirds of the total. There are also six
pump/generator units which are rarely used for generation. The Third Powerplant
generators are equipped with static excitation systems with PSS. The other generators
use rotating excitation systems, but are also equipped with PSS. Generator constants,
exciter, PSS, and governor parameters for all Grand Coulee generators are listed in
Appendix A.
The switchyard contains several ties to area substations at 500 kV, 230 kV, 115
kV, and lower voltages. The plant serves as a major power source to both northwestern
and southwestern load centers. The switchyard also provides a major transfer path from
the western to eastern parts of the system.
25
Olympia
Columbia Potholes Bell#4 Midway Bell#5 Chief Joseph
BeU #2A A A A A A A A 1&2A
230 kV
Bell #1
XX
11X1'
18 125 MV A
500 kV
115 kV (^) Chief Raver Han
P/G 712 isePh Third Powerplant
Generators 1924
#3
3615 MVA 3718 MV A
Figure 8 Single line diagram of the Grand Coulee complex
3.3 System Data
The power system data used for the study was compiled by the WSCC and its
members. It is one of about a dozen cases that are compiled each year for use by
WSCC members. This data base is a representation of spring 1994 with heavy system
loading. The case is modeled with high generation in the northwest with maximum
transfers to California. The data case consists of over 4400 buses and 8000
transmission lines, including three DC interties. There are over 700 generators, of
which about 300 of these generator models are of the classical type, which consists of
the rotor swing equation only. The rest of these generators are detailed, third order dq
26
models, with detailed excitation and governor models, such as those shown in
Appendix A.
The Grand Coulee Third Powerplant is modeled as running at 115 percent of rated
capacity, which is typical. The combination of Right and Left Powerplants are modeled
running at 85 percent capacity. This loading of the plants reflects the maximum power
output, of the complex. For simplicity, the generators of the Right and Left Powerplants
are modeled as one large equivalent generator, since they are all identical. All six of the
Third Powerplant generators are represented individually.
3.4 Analytical Techniques
Several software tools were used in the study of the Grand Coulee excitation
modifications. The primary tool to be used is that of eigenvalue analysis. This was
implemented through the use of the two computer programs which comprise the EPRI
Small Signal Stability Package (SSSP) [20]. The first program in the package is the
MultiArea Small Signal Stability Program (MASS). This program, using system
powerflow data and dynamic device data, forms the complete system state matrix. It
then can calculate all system eigenvalues, right and left eigenvectors, and participation
factors for each mode. The program is limited to a maximum of 800 states. Since a
typical WSCC data base case representation would consist on the order of 10,000
states, a reduced system model is usually required for use in the MASS program. This
reduced order model can be obtained from the full scale representation using the
program DYNRED [21].
The EPRI program DYNRED is capable of reducing the size of both the
transmission network and the number of generators. The number of generators is
reduced by aggregating coherent groups of generators into a single equivalent
27
generator. A coherent group of generators is one in which all the relative differences in
rotor angles of the generators remain constant following a disturbance. The program
offers a few different methods for determining this coherency. One method is to simply
simulate a disturbance and compare rotor angle swings. Another method identifies
weak connections in the system state matrix and aggregates those machines which are
strongly coupled, which implies a strong synchronizing torque between them.
The second program in the SSSP package, the Program for Eigenvalue Analysis of
Large Systems (PEALS), does not suffer the same restriction on size of system
representation as the MASS program. However, since the size of the full system
representation is entirely impractical for calculation and analysis, this program does not
form the full system matrix A. Instead, reduction techniques are used to limit the
calculations to only the portions of the matrix which are of interest. This can be done
using two different methods. One is the Analysis of Essentially Spontaneous
Oscillations in Power Systems (AESOPS), which computes eigenvalues associated
with rotor angle modes only. This method has limited use, since a significant amount of
prior knowledge is required about the mode of interest, such as approximate frequency,
damping and the general mode shape. The other method, the Modified Amoldi Method,
allows the computation of a few eigenvalues around a specific point in the complex
plane, or shift point. The program also will also use this method repeatedly to find all
eigenvalues in a specific frequency range.
Time domain simulation was also used in the study for verification of the
eigenvalue analysis results. This was done using the power system simulator program,
PSS/E from Power Technologies, Inc. This program uses the Modified Euler method
of numerical solution. The Extended Transient Midterm Stability Program (ETMSP)
[22], also an EPRI program, was also used to a limited extent. This program was only
28
used in order to provide an additional verification of damping changes through the use
of Prony analysis [23], which is a feature of the output analysis portion of this
program. Prony analysis is similar to Fast Fourier Transform (FFT) analysis, except
that the output quantities of the mode fitting process are exactly what we are interested
in, namely, frequency and damping ratio.
3.5 Study Methodology
Since the purpose of this study is to determine whether or not the proposed
modifications improve stability of the system, comparisons of the cases before and after
modifications will be used for evaluation. The criterion to be used in the evaluation of
the modifications using eigenvalue analysis is the relative position of the eigenvalues in
the complex plane. The damping ratio is defined as:
C = 7=r (3D
y G + (D
This value is numerically equal to the negative of the sine of the angle, from the
imaginary axis, of the line that passes through the origin and the eigenvalue in the
complex plane. If the modification shifts the eigenvalue to a point which is at a greater
angle from the imaginary axis, then the damping ratio of the mode has been increased.
If the eigenvalue is shifted to a point which is at a smaller angle from the imaginary
axis, the damping of the mode has decreased. A damping ratio of less than 0.05 is
usually considered to be poorly damped. Comparisons of the cases using time domain
simulation was done both visually, by overplotting before and after plots of responses
to'disturbances, and through Prony analysis of the responses. Evaluation of transient
29
stability was performed through time domain simulation.
Eigenvalue analysis was performed using the MASS and PEALS programs. The
MASS program facilitates the stabilizer tuning process by allowing detailed examination
of state space information on the plants of interest and a reduced network model.
PEALS can then be used to refine the stabilizer tuning and examine all eigenvalues in
desired ranges of values. Therefore, eigenvalue sensitivity to parameter modifications
can be obtained and modes of interest can be investigated in detail. Time domain
simulations were then used to provide some verification of the eigenvalue study results.
Because modifications at Coulee impact all locations in the power system in different
amounts and in different ways, selecting monitoring points to observe the significant
changes in responses is a difficult proposition. The locations that were chosen as
monitoring points were limited primarily to the Northwest area. Voltages at selected
locations in the Northwest were monitored, as was power flow in most of the major
transmission paths between the Northwest and other areas. The output results included
in Appendix B are limited to just a few locations per disturbance, and are to be
considered as representative of most of the other locations observed. The disturbances
used for the simulations are listed in Appendix C, which were chosen as test cases by
Bonneville Power Administration.
A complete state space representation of this data case requires more than 6000
states. Only up to 800 states are allowed in the MASS program, so a dynamic
equivalent of the full system was obtained using DYNRED. A reduction was performed
using the weak link method, retaining the Grand Coulee details and all transmission
lines over 300 kV. This reduction produced a system with 722 states, which was the
base system model used for all MASS studies. PEALS studies were performed on the
entire unreduced system.
30
MASS was used initially to find system eigenvalues in which the Grand Coulee
units had a significant participation. These values were then used as initial search
points, using PEALS to verify the eigenvalue location and Coulee unit participation.
Although in general, the values found with MASS tended to be slightly higher in
frequency and higher in damping, a reasonably good correspondence existed between
the outputs of the full and reduced models. Except for figures 10 and 12, all eigenvalue
plots included in this report are outputs from PEALS using the full system
representation.
Although the criteria used for analysis of the results of the study are fairly
straightforward, the actual process involved was quite difficult and time consuming.
First of all, the EPRI software used in this study (MASS, PEALS, DYNRED,
ETMSP) is still in its infancy and is not of commercial grade. It therefore required
hours of modifying and tuning before it could be used. In addition, the data format
required for the programs is not very compatible with the WSCC data base, both in
structure and in modeling philosophy. Therefore, numerous hours were spent
converting and modifying the data so that the system was modeled correctly for the
software.
A second major obstacle occurs when attempting to analyze the results of the
program outputs. The outputs of MASS and PEALS consist only of the calculated
eigenvalues and participation information for all machines or states. When the system is
modified in some way, the eigenvalues change as does the participation information.
This makes it very difficult to compare two different cases, since the eigenvalues of
interest are different between cases and the modeshapes may have changed drastically.
This problem is further compounded by the very large amount of data which results
from a large number of calculations necessary to find all the eigenvalues of interest.
31
Much preliminary reduction of the results was accomplished by several computer
programs that were developed over the course of the study. These programs aided in
limiting the amount of data that needed to be analyzed by hand. A program was then
written that distilled the eigenvalues from the data so they could be plotted for visual
comparison. Therefore, the eigenvalue results are not directly obtained from the
computer programs, but also are the result of a painstaking human evaluation process.
32
4. Study Results
4.1 General Observations
From eigenvalue analysis on the different post fault cases, about two dozen modes
exist in which each Grand Coulee plant has any noticeable participation. Of these
modes, only a few exhibit Coulee participation of more than 1 percent. The eigenvalues
vary somewhat between different post fault configurations, but basically the results
show participation in modes near 0.45 hertz, 0.67 hertz, 0.76 hertz, 0.9 hertz, 1.0
hertz, 1.04 hertz, 1.1 hertz, 1.17 hertz, 1.24 hertz, and 1.35 hertz. Prony analysis of
the Grand Coulee power output in the time domain simulations of these disturbances
shows that some of these modes become excited to varying degrees for each
disturbance case, with a few that tend to predominate. These predominate modes are at
1.0 hertz, 1.04 hertz, and 1.17 hertz, which MASS and PEALS show to have damping
ratios between 0.05 and 0.07, although Prony analysis of the time domain responses
shows the damping ratio of the 1.0 hertz mode to be greater and the other modes to be
slightly less. The local modes from both eigenvalue analysis and Prony analysis are
1.36 hertz, 1.21 hertz, and 1.27 hertz for generators 1 through 18, generators 19
through 21, and generators 22 through 24, respectively.
Only a few modes of oscillation below 1.0 hertz occur in which the Grand Coulee
plants appear to participate in the base case or in the post disturbance cases. PEALS
33
results show these modes to be 0.88 hertz, 0.87 hertz, 0.83 hertz, 0.66 hertz, and 0.45
hertz. Only the latter two modes appear significantly in the time domain simulations.
The time domain simulations display a small oscillation of about 0.7 hertz which
appear throughout the system after around 10 seconds. This is not an unusual
phenomena in simulations of the WSCC system. The exact cause of this oscillation was
not found, but does not negatively affect the study results. The modifications studied
have no apparent effect on this oscillation.
4.2 Excitation Modifications at Grand Coulee Third Powerplant
4.2.1 Evaluation of Existing System
Before studying the effects of modifications to equipment at Grand Coulee, the
behavior of the existing system must first be assessed. Initially, the location of all the
eigenvalues of interest were found using the base case model with the existing
equipment at Coulee. Using PEALS, a frequency scan of all eigenvalues between 0.2
hertz and 1.5 hertz with damping ratios of less than 0.15 was performed. The results
showed a large number of eigenvalues above 1 hertz, but just a few dozen lower
frequency modes, almost all of which have damping ratios greater than 0.05, with a
few notable exceptions. Of these less damped modes, only one shows any noticeable
participation of the Third Powerplant units (generators 19 through 24), namely, a 0.45
hertz mode with a damping ratio of 0.027, in which the units have about a 2 percent
participation factor. This mode is a well known Canadian oscillation and Grand Coulee
is one of the largest nonCanadian participants. The higher frequency modes in which
the Third Powerplant units participate significantly have damping ratios of 0.05 or
greater.
34
An identical scan of the case with the stabilizers for units 19 through 24 removed
was then performed. The resulting eigenvalues are shown graphically in figure 9, along
with the eigenvalues for the case with the stabilizers in service. The plot shows that for
the eigenvalues that change between cases, that is, the modes in which the Coulee units
have a significant participation (controllability), the general effect of adding the
stabilizers is to shift the eigenvalues further into the left hand plane, therefore increasing
the damping of the modes. Eigenvalues which change with the removal of the PSS are
circled on the plot. Only two eigenvalues are noticeably shifted to the right. One is a 1.2
hertz oscillation with a damping ratio of 0.078 which decreases to 0.071. This mode
35
involves machines in the Northwest, Idaho, and Montana. This mode is interesting
because Coulee only participates in it when the stabilizers are in service. The
participation factor of Coulee with the stabilizers in service is 10 percent. Furthermore,
the addition of stabilizers at Coulee increases the number of significant participants in
the mode. The other mode that becomes less damped is a 1.17 hertz oscillation with a
damping ratio of 0.0216 which decreases to 0.0198. This mode consists of a small
number of Northwest plants and Grand Coulee has a participation factor of less than 1
percent.
The eigenvalues shown in boxes on the plot are the local modes for the units
without stabilizers. The furthest left are the local modes for generators 22 through 24,
and the rightmost are the local modes for generators 19 through 21. The middle one is
a plant mode, where the two groups of units oscillate against each other. The local
modes for the case with stabilizers are shifted left beyond the range of the plot in figure
9, but can be seen in figure 10. Also in this plot can be seen the exciter modes, which
move from the real axis with the addition of the stabilizers. The addition of the
stabilizers increases the local mode damping ratios from less than 0.05 to more than
0.26. All of the above results give some verification that the existing speed input
stabilizers at Grand Coulee Third Powerplant are adequately tuned and add more
positive damping than negative damping and are beneficial to the stability of the Grand
Coulee generators and the power system.
4.2.2 Accelerating Power Input Stabilizers
Since the local and exciter mode eigenvalues are not aligned along the same angle in
figure 10, the values of stabilizer gain are not at their optimal values. The optimal
values of gain will be higher than the existing settings. However, the existing settings
36
represent the maximum stabilizer gain that could be reached before the signal became
too noisy. Therefore, the effectiveness of the existing speed input stabilizers is being
reduced by the susceptibility to noise.
16
14
12
10
jco
(r/s)
8
6
4
2
0
14 12 10 8 6 4 2 0
'Exciter Modes 1921
Exciter Modes 2224
+ Without PSS
o With PSS
/
Local Modes 1924
+
it
ee
a (seci)
Figure 10 Eigenvalues existing PSS vs. no PSS at Third Powerplant (MASS)
An accelerating input stabilizer can be tuned to provide a phase response which is
equivalent to that of a speed input stabilizer [24], If this is done, the only essential
difference in performance between the two stabilizer types would be their susceptibility
to noise in the input signal. The output of the accelerating power input stabilizer would
be much more noise free and would therefore be capable of higher gain settings than the
speed input stabilizer.
37
With the existing stabilizers, the local and exciter modes have respective damping
ratios of 0.26 and 0.68. If the gains of the stabilizers are increased, the damping ratios
of the local and exciter modes become equal at gain settings which are about twice those
of their existing values. Figure 11 shows that the increase in gain has placed the
eigenvalues of the two modes at about the same angle from the imaginary axis. The
Figure 11 Eigenvalues existing PSS gain vs. increased PSS gain at Third
Powerplant (MASS)
damping ratios of both modes become about 0.46 at this point. By increasing the gain,
the damping of the local mode is increased at the expense of the damping of the exciter
mode. This effect can be illustrated in the time domain by pulsing the generator
excitation input, which excites both of these modes. Figures 12 and 13 show that the
38
Figure 12 Pulse on G19 existing PSS Figure 13 Pulse on G22 existing PSS gain
gain vs. increased PSS gain vs. increased PSS gain
higher frequency exciter mode becomes slightly visible in the response (about 3.2 hertz
for units 19 through 21 and 2.2 hertz for units 22 through 24) and the local mode
oscillation becomes more damped.
Figure 14 shows a comparison of eigenvalues of the existing no disturbance case
and the case with increased stabilizer gain. As can be seen, most of the modes below
the local mode frequencies in which the Coulee units have any controllability have been
shifted to the left with the increase in gain. As seen before, the only exceptions are
some modes which are near the local mode frequencies. Because very little phase lag is
present in the generator excitation control loop at low frequencies (i.e., interarea
frequencies), a pure increase in the stabilizer gain can be expected to increase the
damping contribution from these generators at these frequencies. This effect is shown
in figure 14: low frequency eigenvalues have experienced an almost perfect shift in
damping with no change in frequency. Of particular note in the plot is the most poorly
damped 0.46 hertz mode, which increases in damping ratio from 0.028 to 0.33 with
the increase in gain.
39
Shown in Appendix D are the eigenvalue comparisons for a post fault
configuration for the five primary disturbances studied. As can be seen in the plots, the
overall effect of the gain increase is similar to that of the base case system.
Figure 14 Eigenvalues existing PSS gain vs. increased PSS gain
The higher frequency eigenvalues that have become less damped are worthy of some
investigation. A number of the modes in this frequency range are local modes of
machines with rotating exciters. Several classically modeled generators also participate
in these modes, which may cause these modes to appear less damped than in reality. Of
the modes which become less damped as Coulee stabilizer gain is increased, one is the
local mode for the model of the 18 generators of the Grand Coulee Right and Left
40
Powerplants. This mode is a 1.35 hertz oscillation with a damping ratio that decreases
from 0.059 to 0.054. As shown in the next section (4.3), this mode becomes very well
damped when the rotating exciters are replaced with static excitation with appropriately
tuned stabilizers. Another, a 1.24 hertz mode, is due primarily to a Bonneville
Powerplant unit, which is modeled with a noncontinuously acting voltage regulator.
The mode which shows the most serious degradation in damping is a 1.14 hertz mode
for which the damping ratio reduces from 0.071 to 0.052. Although this mode is due
mostly to a few machines, a large number of plants in many areas have a small but
significant amount of participation. The participation of the Third Powerplant units are
6 percent with the existing gain and 3 percent with the increased gain. The modes that
become more undamped as the Grand Coulee stabilizer gains are increased generally
consist of one very dominant machine with a few other minor participants. As the
stabilizer gain is increased, the participation of Grand Coulee units 19 through 24
decreases from what is already a small amount. This decrease results in a shift of the
participation factors of some of the other participants to those of higher values. The net
result is that the number of significant participants in the mode decreases, and so does
the number of machines with controllability of the mode. Alternatively stated, the
number of machines capable of providing damping decreases. Therefore, although the
magnitude of these oscillations decreases as seen at Grand Coulee, the damping of the
mode decreases. The increase in stabilization at Coulee causes the modes to become
less observable and less controllable from the Coulee generators. In contrast, the modes
that become more damped as the stabilizer gain is increased have a much higher amount
of Grand Coulee unit participation, which remains the same or increases with the
increase in gain. These effects can be seen in the time domain responses of the
disturbances that were examined (see figures A through L in Appendix B).
41
Figure A shows the power output for generator 19 in response to the Coulee
Hanford outage. With the existing stabilizer settings, this response is dominated by a
1.0 hertz oscillation with a damping ratio of 0.10 which lasts for about 4.5 seconds.
After this time, a different oscillation near 1.0 hertz dominates but has a smaller
damping ratio of 0.05. This dominance continues until this oscillation is exceeded in
magnitude by the 0.7 hertz oscillation that appears. Measurement of the lower
frequency modes present in the responses was difficult with the Prony analysis tool
used, but these components appear to be about 0.35 hertz, and 0.45 hertz. Increasing
the stabilizer gains improves the initial 1 hertz oscillation to a damping ratio of 0.15 and
slightly improves the second 1 hertz oscillation that appears. This effect can be seen in
figures B through D. The initial 1 hertz oscillation can be seen throughout a very large
percentage of the system, and the modifications of the Coulee stabilizers improves the
damping of this oscillation at these locations. Slight improvement of the lower
frequency modes also occurs.
The responses to the other disturbances are very similar to this case. The modes
that seem to be excited the most are a few in the 1.0 hertz to 1.15 hertz range. Using
figure 14 as a reference, the initial 1 hertz oscillation corresponds to the eigenvalue
which has the largest leftward shift when the stabilizer gain is increased. Eigenvalue
analysis shows this mode to have a large number of participating machines in several
areas. The secondary oscillations observed in Coulee generator output in this frequency
range seem to be a combination of the other 1.0 hertz to 1.15 hertz modes in which
Coulee participates.
Observation of different points in the system for all the different disturbances show
either one of two results. Either the response is noticeably improved, as in figures B
through D, or there is no net effect. The latter results from the combination of several
42
effects. At some locations, one mode may become more damped, and another may
become less damped. The mode shape and participation factors may be changed so that
even though the damping is increased, the magnitude of the oscillation at different
locations is increased. Therefore, the peaks may be higher in some points in the
response while they are lower in others. After extensive examination of responses in
the system, these effects tend to be the rule rather than the exception.
One of the drawbacks to time domain simulations is that the disturbances (faults
and line outages) used to perturb the system excite only some modes of oscillation, in
different combinations and to different degrees. Therefore, the effects of a system
modification cannot be fully examined in the time domain. As the cases above have
shown, the disturbances used so far have excited some of the modes of interest, but not
all. Because the effects on lower frequency modes are not very apparent from the five
disturbances used above, two additional disturbances were applied. One was a loss of
the Intermountain Adelanto dc tie and the other was a tripping of one of the Palo
Verde generators. The plots in figures M through P show power output of one of the
Third Powerplant generators and the bus voltage at Malin substation for both of these
disturbances. As can be seen, some low frequency modes of oscillation are excited in
these cases at these locations. Prony analysis shows the dominant modes to be near
0.23 hertz and 0.45 hertz for both disturbance cases at both locations. It also shows the
damping ratios are increased for both modes by about 20 percent with the increase in
stabilizer gain.
4.2.3 Transient Gain Reduction
The use of transient gain reduction in the exciter control loop when PSS is being
used has been well discussed [25]. It is common practice to use some method of TGR
43
to add damping to the transient response and local mode of a generator. Reclamation
policy currently includes the application of TGR to provide local mode damping in the
absence of PSS. Some of the benefits of not using TGR are pointed out in [25], one of
which is that with PSS, the same amount of local mode damping can be obtained with
much less stabilizer gain required than when using TGR. This point is relevant to the
objectives of this study because stabilizer gain limitations are of primary interest.
Therefore, the effects of not using TGR were compared to the effects of using
accelerating power input stabilizers.
For the comparison, the same stabilizer tuning technique was used for the non
TGR model as was used for cases with TGR. The peak of the open loop stabilizer
response was set at the local mode frequency and then the lead/lag ratio was adjusted to
reduce the peak to provide near zero phase lead. This resulting lead/lag ratio for the non
TGR case is about half that of the case with TGR. The gain was then adjusted to
optimize the local mode and exciter mode damping ratios. The final gain setting was the
same as the existing gain setting for units 19 through 21, but was slightly lower than
the existing gains for units 22 through 24. The resulting damping ratios for the local
modes are about 15 percent lower than for the case with TGR and optimum stabilizer
gain, but are sufficiently close for comparison. The stabilizer parameters used are
shown in Appendix A.
One of the first noticeable differences between the case with TGR and the case
without is that the root loci of the local and exciter modes are much closer to the
imaginary axis and are much more sensitive to variation in gain. This sensitivity
impedes optimization of the stabilizer parameters because small changes result in large
changes in eigenvalues for these modes. The inability to exactly match the damping
ratios between the TGR case and the non TGR case is partly caused by this difficulty.
44
As expected, one of the main differences between the eigenvalues of the case with
TGR and the case without TGR would be due to the magnitude of the stabilizer gain.
Because the optimum gains of the case without TGR are about the same as the existing
case (about half those of the optimized case with TGR), a plot of the eigenvalues for the
case without TGR should resemble the existing case more than the case with TGR and
the optimized gains (accelerating power input stabilizers). As figure 15 indicates, this
expectation is generally true, particularly for eigenvalues of frequency below 1.1 hertz.
At higher frequencies, the comparison results are mixed. A few eigenvalues which
increase in damping ratio with the removal of the TGR are unaffected in the cases with
TGR. These eigenvalues lie in the 1.3 hertz to 1.4 hertz range and the improvement is
slight.
Figure 15 Eigenvalues exciter with TGR vs. exciter without TGR
45
For lower frequency modes, however, the eigenvalue behavior differences between
cases seem to be dictated by stabilizer gain. Therefore, in the case with no TGR, with
the lower values of stabilizer gain, the benefits of the increased stabilizer gain shown in
the case with TGR will not be present. This difference can be observed in the time
domain responses. Figures Q through V show that the 1 hertz oscillations that dominate
the responses to the disturbances are more poorly damped for the case with no TGR. In
fact, because the stabilizer gains of generators 22 through 24 are actually lower in the
non TGR case than in the existing case, the damping of these modes is actually worse
than in the existing case.
Another significant difference between using transient gain reduction in the exciter
and not using it is in the transient responses of the machines. As shown in figure Q, the
machine without TGR is capable of providing more power output during the
disturbance and therefore should have less power angle swing. Figure A shows that the
increase in gain with TGR added actually decreases the first swing power output a
small amount. Therefore, one of the advantages of not using TGR is some
improvement in first swing stability. This can be seen in figure W, which shows a
comparison of rotor angle swings for the CouleeHanford disturbance. Although
increasing the stabilizer gain in the case with TGR results in a slower transient
response, any negative effects on stability were not apparent in any of the time domain
simulation results. The amount of transient stability margin gained by not using TGR
was not studied in detail. However, comparisons between the case with increased
stabilizer gain using TGR and the case without TGR were made by varying the fault
clearing time for a fault on the Coulee 500 kV bus. In simulations involving the
removal the Coulee Hanford line or no line removal, the increase in fault clearing time
gained by removing the TGR was only about 0.1 cycles. Figure X shows the rotor
46
angle responses of one of the Third Powerplant generators for a fault on the Coulee 500
kV bus which was cleared at 6.2 cycles with no line tripping. The generator models
with TGR and increased stabilizer gain went unstable after a few seconds while the
generator models without TGR remained stable. When the fault clearing time was
increased to 6.3 cycles, both cases went unstable.
It has been demonstrated that by simply increasing the gain of the existing
stabilizers at Grand Coulee, the damping of the power system can generally be
improved. However, to try to define a true optimum tuning of the stabilizers, the
damping of all affected modes over all possible operating conditions would have to be
considered. Although this task is beyond the scope of this study, a few alternate
stabilizer tunings were examined to try to improve the performance over those cases
already examined. The settings examined involved only the shifting of the lead/lag pairs
higher and lower in frequency, varying the lead/lag ratios, and varying the gain.
Because the root loci indicate that the local modes are slightly overcompensated for
this case, the lead/lag ratio was adjusted to provide near zero phase lead at the local
mode. As compensation becomes closer to ideal, a greater amount of damping becomes
possible for the local and exciter modes, and the required gain setting for optimum
damping of these modes increases. An increase in gain increases the eigenvalue shift
and, therefore, the damping can be further improved for some modes while worsened
for others. With the compensation adjusted, the optimum damping of the local and
exciter modes occurs with a gain value which is 50 percent higher than the optimum
settings examined, or three times the existing gain values. The damping of the local and
exciter modes improves to a damping ratio of 0.54.
Shifting the lead/lag pairs higher and lower in frequency led to some interesting
47
results. Some of the eigenvalues are affected differently by these stabilizer differences.
When the pairs were shifted higher in frequency, the root locus became less sensitive to
variation in gain. Therefore, the optimum value of gain increased, as did the damping
ratios of the local and exciter modes. When the center frequencies of the lead/lag pairs
were nearly doubled, the optimum values of stabilizer gains were over five times the
existing values. These values are so high that the maximum practical settings must be
questioned. How much gain is possible before noise limitations arise again because of
the remaining lead compensation? In addition, how high in frequency can the stabilizer
gain peak be shifted before higher frequency signals and noise are no longer attenuated?
These questions need to be investigated further if the full range of possible stabilizer
parameters are to be studied in detail.
As a final comparison between the cases with and without TGR, a single machine,
infinite bus model was used to calculate the best possible damping ratios for local and
exciter modes for all possible lead/lag and gain settings. The lead time constants for
generators 1921 can be set in the range of 0.2 to 2.2 seconds. The tuning range of the
lag time constants is 0.02 to 0.22 seconds. While keeping both lead/lag stages identical,
an exhaustive calculation of damping ratios was performed over the entire possible
ranges of lead, lag, and gain settings. The results indicate that similar damping ratios of
local and exciter modes can be obtained with or without using TGR. However, in every
case comparing similar damping ratios, the optimal gain value for the case with TGR is
about twice as high as the case without TGR. Therefore, if both cases are tuned for
similar damping of the local and exciter modes, the case with TGR will result in more
control of interarea modes of oscillation due to the higher value of gain. This is an
interesting result, since past discussions of the use of TGR have neglected this fact.
48
4.2.4 Summary
Although one single optimum tuning of the stabilizers was not determined, two
significant conclusions can be drawn from the efforts described. The first is that an
increase in the damping of both the low frequency and local frequency modes will
require an increase in the stabilizer gain beyond what is obtainable with the existing
stabilizers. Secondly, the resulting modifications to the stabilizer settings do provide a
significant increase in the stability of some power system oscillations. Therefore,
accelerating power input stabilizers by increasing the amount of allowable stabilizer
gain will provide more control of power system oscillations than speed input
stabilizers. However, because the increase in stabilizer gain decreases the amplitude of
the first power swing following a disturbance, the increases in damping are
accomplished at a cost of a slight reduction in the transient stability margin of the
generators. This reduction has had negligible effects on the results of this study. The
removal of TGR in the excitation control circuit can add to the transient stability margin,
but control of interarea modes of oscillation is then sacrificed.
4.3 Replacement of Rotating Exciters on Generators 118 with Static
Excitation Equipment
4.3.1 Evaluation of Existing System
Generators 1 through 18 are 125 megawatt units with rotating exciters which are
equipped with power system stabilizers. For this study, the generators were modeled as
one unit with a 1660 megawatt output. Figure 16 shows a comparison of eigenvalue
scans of the existing system and the case with the stabilizer on this generator removed.
The plot shows that the stabilizer provides the most damping to the local mode, which
is 1.34 hertz. The damping ratio is marginally increased from 0.058 to 0.060 with the
49
addition of the stabilizer. Several other modes are also slightly affected with the
addition of PSS, some positively, some negatively. The modes which these 18
generators have any control over are generally those in which generators 19 through 24
have been shown to participate, so much of the analysis for these units is the same as
that of the Third Powerplant units. One major difference is that the impact of these
generators, being of much smaller rating, have much less impact on the system than
those units in the Third Powerplant. As can be seen in the plot, the addition of the
stabilizers on these generators is relatively ineffectual.
10
9
8
i
j0)7
(r/s)
6
5
4
3
2
1
I11T
ir
* %
e
'
+ No PSS G1G18
o With existing PSS
6
+o
+ e ^
e
e K)
e 0
40
0 j
e 
_ 04
_J________________L_
O'1111
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
a (sec1)
Figure 16 Eigenvalues Generators 118, with PSS vs. no PSS
50
4.3.2 Static Excitation
Because static excitation systems can have a much faster response time than a
rotating exciter, they can provide greater transient stability of a generator and the power
system. The fast acting static exciter with a power system stabilizer can also provide
much more damping of power system oscillations than the rotating exciters. Because
the Grand Coulee generators 1 through 18 represent a substantial amount of energy and
have a significant impact upon the power system, an examination of the potential
benefits of replacing the excitation systems is of interest
The single machine model representing generators 1 through 18 was modified to
include a static exciter identical to that of generators 19 through 21. A new power
system stabilizer model was added and tuned as shown in Appendix A to provide near
zero phase lag for the local mode. The gain of the stabilizer was then adjusted to
optimize the damping of both the local and exciter modes. This point occurred with a
gain of 14. Assuming that the noise limitations for this stabilizer are similar to those of
generators 19 through 24, the highest practical gain setting for a speed input stabilizer is
about one half this value. With the stabilizers adjusted for optimum local mode
damping, the local mode frequency is 1.27 hertz and the damping ratio is 0.39. This
setting should be easily obtainable with an accelerating power input type stabilizer.
With a speed input stabilizer and the gain limited to the lower value, the damping ratio
of local mode would be 0.27.
4.3.3 Damping of Power System Oscillations
The eigenvalue plots in figure 17 show how the replacement of the rotating exciters
with static excitation affects the damping of the modes in which the generators
participate. Note the very large increase in the damping of the local mode. The pattern
51
10
9
8
j0)7
(r/s)
6
5
4
3
2
1
0
"* <$> *
^____+
a
*
V
+ <9
o
+o
3
40
Ah *>
*
+ Existing excitation, G1G18
o Static Excitation, G1G18
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
a (sec1)
Figure 17 Eigenvalues existing excitation vs. static excitation, generators 118
of the other eigenvalues which show a change in location bears a resemblance to that
observed in the study of generators 19 through 24. The modes of frequency less than
about 1 hertz show an improvement in damping while those in the range of 1 hertz to
1.15 hertz are mixed, with some increasing in damping, and others decreasing in
damping. This plot reflects the highest practical stabilizer gain considering noise
restrictions. Figure 18 shows a comparison of these stabilizer gain settings against the
optimal value of gain possible with accelerating power inputs. As shown in this plot,
the increase in stabilizer gain slightly increases the shift in eigenvalues.
52
10
9
8
ja>7
(r/s)
6
5
4
3
)
0
*
.
#
6
+ limited gain setting
o optimal gain setting
_]_________________1_
d+
6
CH O
4?
o+
o +
Of
_1______________1 1______________l_
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
ct (sec1)
Figure 18 Eigenvalues limited gain vs. optimal gain generators 118
4.3.4 Transient Stability
The eigenvalue plots show that the replacement of the excitation systems on
Generators 1 through 18 can provide some benefit to damping local mode and interarea
oscillations. The most tangible benefit as seen in time domain simulations is due to the
increased transient response of the static excitation systems. The fast acting exciter
increases the electrical power output of the generator during the disturbance, which
decreases the amplitude of the power angle swings of the generators after the transient.
This effect is shown in the power output plot in figure a and the angle plot in figure e.
The decrease in power swing magnitude is reflected throughout the power system, as
53
can be seen in figures b through 1. In an attempt to vaguely quantify the increase in
transient stability gained with the static excitation, the CouleeHanford single line
outage disturbance was simulated with varying fault clearing times. The rotating exciter
case became unstable when the clearing time was increased to just over 6 cycles. With
the static excitation systems modeled, the critical fault clearing time was hearly one
cycle longer.
4.4 Combination of Modifications
Figure 19 shows an eigenvalue scan for the ease with static excitation replacing
rotating exciters on generators 1 through 18 and accelerating power input stabilizers on
Figure 19 Eigenvalues existing case vs. combination of excitation modifications
54
all 24 generators compared to the existing equipment case. The effect of all the
modifications is basically similar to those that have been observed individually so far,
namely, that the majority of the eigenvalues that are shifted significantly are done so in
the direction of increasing damping factors. However, when this figure is compared to
those of figures 14, 17, and 18, the effects of the modifications at each plant are
obviously not exactly additive. Each modification affects the eigenvectors in different
ways and the combination produces yet a different result. This result is very illustrative
of the nature of system dynamics and the value of studies such as this one. Figures m
through p show time domain responses to the various disturbances at various locations
in the system. As can be seen, the combination of the transient stability improvements
due to the static exciters and the increased damping of oscillations due to the higher
stabilizer gain afforded by the accelerating power input stabilizers has resulted in a
noticeable improvement in the system responses to the disturbances applied.
4.5 Topics for Further Study
Many issues remain to be examined which would make a study of Grand Coulee
excitation modifications more complete. One area requiring further examination is that
of the effects on low frequency modes of oscillation. Because the results of this study
show that these excitation modifications would be especially beneficial in controlling
these modes, more well defined problems with these modes should be modeled and
studied.
More transient stability studies should also be performed, particularly to further
compare rotating exciters versus static excitation for generators 1 through 18. In
general, if actual replacement of equipment is to be truly studied, more specific stability
problems will need to be studied and comparisons will need to be made between these
55
options and other solutions, such as remedial action.
Further studies involving application of PSS on excitation with and without TGR
should also be pursued. Of particular interest are any significant tradeoffs between
transient and dynamic stability when considering these two control philosophies.
Like all research oriented projects, this study has raised questions as well as
answered some. Some of the more intriguing ones involve the relationships between
the participation vectors and the suitability of different stabilizers in damping specific
oscillations. Perhaps a further understanding of the relationships between PSS
parameters of certain machines and the effects on system modes of oscillation will lead
to the ability to coordinate excitation tuning between many machines and increase the
effectiveness of stabilizer applications.
This study was Reclamation's initial application of eigenvalue analysis tools. There
are many areas which require further study using these tools, such as sensitivity of PSS
parameters and the application of PSS, in general. Underdamped modes should also
be studied in detail, which may lead to solutions for poorly tuned equipment or
correction of modeling errors.
56
5. Conclusion
This thesis has investigated the application of power system stabilizers using
eigenvalue analysis and time domain simulation. It has presented a discussion on the
development and current use of PSS along with a discussion of tools for analysis. A
detailed study of PSS at Grand Coulee Power Complex and its effects on the stability
of the power system has also been conducted.
Although the eigenvalue analysis tools used were difficult in implementation, many
conclusions have been reached. First of all, it can be said that eigenvalue analysis is
indeed a valuable tool in power system stability assessment and improvement. It has
been proven that these analysis methods can be of great use in investigating
modifications to controllers such as power system stabilizers. It has also been observed
that there is much left to be understood regarding the dynamic behavior of the power
system, and that eigenvalue analysis will play a critical role in gaining this
understanding. However, there is still some improvement to be made in the software
tools used in this thesis before this will be realized.
Much has also been learned about PSS through the efforts presented here. It has
been shown that the present methods of applying PSS are overly simplified, and that
eigenvalue analysis can increase the effectiveness of PSS applications.
57
5
The results of the study of Grand Coulee excitation equipment have shown that
improvements to system stability can indeed be made by the modifications examined.
The additional stabilizer gain that can be provided by replacing the existing speed input
stabilizers at Grand Coulee with accelerating power input stabilizers can effectively
double the damping of the local mode oscillations of the Grand Coulee generating units.
The increase in gain also improves the damping of most lower frequency modes of
system oscillations. Only in a very few locations were any cases observed where the
effects of the decreases in damping were more apparent than those of the increases in
damping. This effect was only observed for modes in which the Coulee units have very
little control, which suggests that improvement in damping of these modes can be better
accomplished by modifications to other machines in the system.
The use of speed input stabilizers without TGR in the excitation control loop can
provide nearly as much damping of the local mode oscillations as the application of
accelerating power input stabilizers, and can provide a slight increase in transient
stability margin. However, the lower stabilizer gains resulting from the elimination of
TGR have less impact on other modes of oscillation and make the stabilizer tuning
problem more difficult because of the increase in sensitivity to the stabilizer gain.
Replacing the existing rotating exciters on Grand Coulee Generators 1 through 18
can provide an increase in transient stability of these generators, the results of which
can be seen in responses throughout the system. In addition, the power system
stabilizers on these units can be much more effective with static excitation equipment,
providing several times more damping of the local mode oscillations and more impact
on other modes of system oscillations. These effects can be further enhanced by using
accelerating power input stabilizers. In general, the addition of accelerating power input
stabilizers on these generators has similar results to those of the Third Powerplant
58
generator study, except that the system is impacted less because of the smaller size of
these generators.
59
APPENDIX A
Generator data for Grand Coulee Power Complex
60
Generator models
Generators 118
T'do = 7.2 sec. T"do = 0.05 sec.
D = 0 Xd = 0.638 Xq = 0.424
Generators 1921
T'do = 6.9 sec. T"do = 0.04 sec.
D = 0 Xd = 0.79 Xq = 0.54
Generators 2224
T'do = 7.6 sec. T"d0 = 0.08 sec.
D = 0 Xd = 0.86 Xq = 0.58
T'q0 = 0.79 sec. H = 4.58 p.u.
Xd = 0.228 X"d = 0.161 Xi = 0.127
Tqo = 0.03 sec. H = 5.14 p.u.
X'd = 0.289 X"d = 0.24 Xi = 0.152
T'q0 = 0.04 sec. H = 4.99 p.u.
Xd = 0.32 X"d = 0.22 X! = 0.21
61
Exciter models
Generators 118
Tr = 0 sec. Ka = 17.8 TA = 0.08 sec. TB = 0 Tc=0
Vrmax = 3.0 Vrmin = 3.0
Ke=1.0 Te = 0.4 Kp = 0.03 TF = 0.127
Generators 1921
Tr = 0 sec. Ka = 200 Ta = 0.033 sec. Tb = 0 Tc=0
VIMAX =0.18 Vimin = 0.18 Vrmax = 12.0 Vrmin = 12J
Kf = 0.009 Tf = 1.2 Kc = 0
Generators 2224
Tr = 0.012 sec. Ka = 200 Ta = 0.012 sec. Tb = 0 Tc=0
Vimax = 175 Vimin = 1.75 Vrmax = 8.0 Vrmin = 6.8
KF = 0.009 TF = 0.75 Kc = 0
62
Power System Stabilizer Models
Generators 118
II o to II H II O on CD O T2 = 0.02 sec T3=31 sec T4=31 sec
T5/7 = 0.6 sec. T6,8 = 0.0175 sec T9,10 = 0 sec
Lsmax = 005 Lsmin = 0.05 Vcu = 0.06 Vcl = 0.095
Generators 1921
Ki=0 K2 = 6.3 Tj=0sec. T2 = 0.02 sec T3=30 sec T4=30 sec
T5J = 0.22 sec. Tgi8 = 0.041 sec T9,10 = 0 sec
Lsmax = 01 Lsmin = *0.1 Vcu = 008 Vcl = 0.08
Generators 2224
Kx = 0 K2 = 5.0 Tj = 0 sec. T2 = 0.02 sec T3=30 sec T4=30 sec
Tsj = 0.30 sec. Tg>8 = 0.04 sec T9,10 = 0 sec
Lsmax = 01 Lsmin = 0.1 Vcu =0.095 VCL = 0.095
Generators 118, retuned for static excitation
K1=0 K2 = 14.0 Ti = 0 sec. T2 = 0.02 sec T3=30 sec T4=30 sec
T5J = 0.237 sec. T6<8 = 0.0475 sec T9,10 = 0 sec
Lsmax = 01 LSmin = 0.1 Vcu =0.08 VCL = 0.08
63
Governor models
Generators 118
K = 23.36 Ti = 15.0 sec. T2 = 3.0 sec
Pmax = 1.168 Pmin = 0 T4 = 0.66
Generators 1921
K = 23.69 Ti = 32.0 sec. T2 = 5.3 sec
Pmax =1.185 Pmin = 0 T4 = 2.0
Generators 2224
K = 23.01 T! = 50.0 sec. T2 = 6.4 sec
Pmax =1.150 Pmin = 0 T4 = 2.4
Po
PMAX
Aca K(1 + sT2) r . 1sT4
SPEED (1 + sT,) (1 + sT3) 1 0.5 sT
PMIN
T3=0.10 sec
T3=0.25 sec
T3=0.35 sec
P
PMECH
64
APPENDIX B
Time domain simulation results
65
7 GRAND COULEE STABILIZER STUOV 1994 HSP BASE CRSE
JV'/'/v COULEE HANFORO SINGLE LIME OUTAGE C19 ELEC POWER OUTPUT
iccurnttio.
c._______
Figure A
66
0.0 3.0000 6.0000 9.0000 12.000 15.000
I.SOOO 1.5000 7.5000 10.500 13.500
TINE(SECONOS)
Figure B
'o
*o
TIME (SECONDS)
GRRND COULEE STABILIZER STUDV 1994 HSP BASE CASE
COULEE HANFORD SINGLE LINE OUTAGE POWER MALINROUND HTN
GRAND COULEE STABILIZER STUDY 1994 HSP BASE CASE
COULEEHANFORD SINGLE LINE QUTAGEPOHERHATNAILONERCRANITE
Unwttttia.
tiC*
Figure C
O.o 3.0000 b.OOOD 9.0000 12.000 15.000
1.5000 0.5000 7.5000 10.500 13.500
TIME (SECONDS)
7 GRRND COULEE STRBILIZEP. STUDY
JIAV JOHN DRY GRIZZLY SINGLE LINE OUTAGE
itCMHOlbCJCS.
1994 HSP BASE CASE
VOLTAGE RAVER
Figure D
69
3.0000 6.0000 9.0000 12.000 IS.OOO
.5000 9.5000 7.5000 10.500 13.500
TIME(SECONDS)
fyvw
ItCKitOtOClCS.
]'C
GRANO COULEE STABILIZER STUDY
JOHN DAY GRIZZLY SINGLE LINE OUTAGE
1994 HSP BASE CASE
VOLTAGE MCNAP.Y 230
Figure E
70
3.0040 6.0000 9.0000 12.000 15.000
SOOO 9.5000 7.5000 10.500 13.500
TINE (SECONDS)
7 GRRND COULEE EXCITRTION STUDY' 1994 HSP BRSE CRSE
JV'/'/v JOHN DRY GRIZZLY SINGLE LINE OUTAGE POWER COULEE 118
vrueuciQ.
Figure F
71
fyV/W
POdt*
TEEMMBCIES.
iu:A________
GRAND COULEE EXCITATION STUDY
COULEE ROVER SINGLE LINE OUTAGE
1991 HSP BRSE CRSE
POWER BOUNDRRYWONETR
Figure G
72
3,0000 6.000D 9.0ODO 12.000 15.00(1
.5000 4.5000 7.5000 10.500 13.500
TIME (SECONDS)
GRAND COULEE EXCITATION STUDY
COULEE ROVER SINGLE LINE OUTAGE
1994 HSP BASE. CASE
VOLTAGE GRIZZLY
CHNL 73: CVGRIZZLY]
1.1500 FILEi couraver_opt3rd.plt CHNL 73s CVGRIZZLY3 ^ 0.95000
1.1500 FILE: couraver.pLt B O 0.95000.
Figure H
73
3,0060 b.OOOD 9.0000 12.000 15.000
.5000 9.5000 7.5000 10.500 13.500
TIME (SECONDS)
fyvw
KU(*
TOCKKOLfiCTC.
WC*_________
GRAND COULEE EXCITATION STUDY L994 HSP BRSE CASE
HRNFOROCOULEE,HANFORDVANTAGE DBL LINE OUT PCOULEERAVER
Figure I
74
3.0000 b.OOOD 9.0000 12.000 15.000
.5000 9.5000 7.5000 10.500 13.500
TIME (SECONDS)
GRRND COULEE EXCITRTION STUDY L991 HSP BASE CRSE
HRNFOROCOULEE,HRNFOROVRNTRGE DBL LINE OUT VOLTRGE MRLIN
TOKHOlOtJC.
Ir c?
Figure J
75
3.0000 b.OOOD 9.0000 12.000 15.000
5000 9.SO00 7.5000 10.500 13.500
TIME (SECONDS)
n GRRND COULEE EXCITRTION STUDY 1991 HSP BASE CRSE
Vw HRLINROUND MTN DBL LINE OUT POWER HELLSCRNYONENTERPRISE
tcmmwcjo,
l.C*______
__________________________________CHNL. 91: CHELENTO__________________;_______________
110.00 FILE: nlnrndatn_opt3rd.plt * 10.000
___________________________ CHNL 99; rHELENTH________________________________________
110.00 FILE: nlnrndntn.plt s 10.000
110.00 FILE: nlnrndatn.plt s 10.000
Figure K
3.0000 6.0000 9.0000 13.000 15.000
5000 9.5000 7.5000 10.500 13.500
TIME (SECONDS)
fywv
teckmumo.
nc^______
GRAND COULEE EXCITATION STUDY
HALINROUND MTN DOUBLE LINE OUTAGE
1994 HSP BASE CASE
VOLTAGE JOHN DAY
Figure L
77
0.0 3.0000 6.0000 9.0000 12.000 15.000
l.SOOO <1.5000 7.5000 10.500 13.500
TIME (SECONDS)
1994 HSP BASE CASE
POWER COULEE G19
GRAND COULEE EXCITATION STUDY
LOSS IF IPP DC LINE
f.
Figure M
78
2.0000 4.0000 6.0000 8.0000 10.001
1.0000 3.0000 5.0000 7.00d0 9.0000
TIME (SECONDS)
Figure N
TIME (SECONDS)
K>
O
o
o
et
T
l
l
i
o
CD
o
o
o
o
*n<'
GRAND COULEE EXCITATION STUDY 1994 HSP BASE CASE
LOSS OF ipp DC LINE VOLTAGE MALIN
Figure O
El
K
0
GRAND COULEE EXCITATION STUDY 19 g 4 HSP BASE CASE
LOSS OF 1 PALOVERDE GENERATOR POWER COULEE G19
^/VVv
GRAND COULEE EXCITATION STUDY
LOSS OF 1 PALOVERDE GENERATOR
1994 HSP BASE CASE
VOLTAGE MALIN
f.
Figure P
81
3.0000 6.0000 9.0000 13.000 15.000
S000 4.5000 7.5000 10.500 13.500
TIME(SECONDS)
w*:
GRAND COULEE STABILIZER STUDY
COULEE HflNFORO SINGLE LINE OUTAGE
1994 HSP BHSE CASE
G19 ELECT POWER OUTPUT
ii r.*
Figure Q
82
3,0000 b.GOOD 9.0000 12,000 15.000
.5000 4.5000 7.5000 10.500 13.500
TIME(SECONDS)
GRAND COULEE STABILIZER STUDY 1994 HSP BASE CASE
COULEE HANFORD SINGLE LINE OUTAGE POKER HAL IN P.OUNO MTN
TorHvsLCOcz.
IC.________
Figure R
83
3. QOC'O b.OOOO 9. i'ODO 12.000 15.000
.5000 M.5000 7.5000 10.500 IS.500
TIRE(SECONDS]
7 GFiflMD COULEE STABILIZER STUDY 1994 HSF BRSE CRSE
kw COULEEHANFORO SINGLE LUTE OUTAGE POHERHATITAI LOITERGRAMI TE
_________________________________CHNL 93; CHRTL0HG3___________________________________
1100.0 FILE: couKnrrd_notgr3rd.plt *< 500.00
_________________________________CHUL 93; CHRTL0HG3___________________________________
1100.0 FILE: couhnfrd_opt3rd.plt ba 500.00
Figure S
84
3.0000 5.0000 9.0000 12.000 J5.000
.5000 il.SOOO 7.5000 IG.SOO IS. 500
T [ME (SECONDS)
1994 H5P BRSE CRSE
VOLTAGE RfiYER
T GRAND COULEE STABILIZER STUOY
1^ JOHN DAY GRIZZLY SINGLE LINE OUTAGE
Kf
icrHMeibocs.
if._______
___________________________________CHML 71: CVftftVERD ._________________________________
1.1500 FILE.* Jnd9rz_n0t9r3rd.plt *** 1.0000
___________________________________CHNL 71; CVRRVERD_____________________________________
1.1500 FILE: jndgrz_opt3rd.plt e 1.0000
Figure T
85
3.0000 6.0*0& 9.0ODO 12.000 15.000
.5000 4.5000 7.5000 10.500 13.500
GRRNO COULEE STABILIZER STUOY
JOHN DAY GRIZZLY SINGLE LIME OUTAGE
1994 HSP BRSE CASE
VOLTAGE MCNARY
V'a
Figure U
86
3.00*0 6.0000 3.0000 12.000 15.000
.SOOO 4.5000 ?.5000 10.500 13.500
TIME(SECONDS)
GFiRUO COULEE STRB1LIZEB STUDY
JOHN DAY GRIZZLY SINGLE LINE OUTAGE
iggy hsp brse case
POWER COULEE 13S
_________________CHH1 99: CP 1 COULEE 20__________________________________
20.000 FILE: jndgrz_notgr3rd.plt * " 8.0000
__________.______________________CHHL. 90; CP 1 COULEE 23_______________;___________________
20.000 FILE: jndgrz_opt3rd_jnore.plt s 0 8.0000
Figure V
87
o 0 3.0000 6.0000 9.0000 12.000 15.000
I.SOOO 9.9000 7.SOOO 10.500 13.SOO
TIME(SECONDS)
n GRAND COULEE EXCITRTION STUDY 1994 HSP BRSE CASE
J\/Wv COULEE HANFORO SINGLE LINE OUTAGE ANGLE COULEE G1Q
TETHMICCIES.
iir.fi______
CHNL lr CRNC1COUL 19D
90.000 FILE: couhnfrd_notgr3rd.plt < p 40.000
CHNL 1: rflNCICOUL 191
90.000 FILE: couhnfrd_opt3rd.plt ^ 40.000
CHNL Is CflNGlCOUL I9D
90.000 FILE: couhnfrd.plt a 40.000
Figure W
88
1.QO0O 2,0406 3.40D0 4.0000 5.0000
.SOOOO l.SOOO 2.5000 3.5000 4.5000
TIME (SECONDS)
Figure X
TIME (SECONDS)
73
o
3
gvo
VO
O
r
w
I ^3
O D
o >
q w
r m
w
w o
^ >
o w
)tn
vo
GRAND COULEE EXCITATION STUDY
COULEEHANFORD SINGLE LINE OUTAGE
1994 HSP BHSE CASE
POWER OUTPUT COULEE 118
CHNL 99: CP 1 COULEE 21 1
23.000 FILE: couhnfrd_18statIcopt.plt q 5.0000 
CHML 99: TP 1 COULEE 21 
23.000 FILE: couhnFrd.plt 5.0000 
Figure a
90
1.6000 3.2000 4.8000 b.4000 6.0000
. 80000 2.4000 8.0000 S. Â£>000 7.2000
TIME (SECONDS)
TOTTtO;)C2.
uc.________
GRAND COULEE EXCITATION STUDY
COULEEHANFORD SINGLE LINE OUTAGE
1994 HSP BRSE CASE
POWER HAL INROUND MTN
_________________________________CHHL 97: CHLHRHDHT3_________________________________
2000.0 FILE: couhnrrd_18statlcopt.pi t >> 1000.0
_________________________________CHML 97: CHLNRHPHT]_____________________________
2000.0 FILE: couhnfrd.pl t a  a lOOO. 0
Figure b
91
3.0000 5.000D 9. 00 0 0 12.000 15.000
.5000 9.5000 7.5000 10.S09 13.500
TINE(SECONDS)
