PREDICTING GENERATOR COUPLING USING POWER SYSTEM IMPEDANCE 
MATRICES 
 
 
 
 
 
Except where reference is made to the work of others, the work described in this thesis is 
my own or was done in collaboration with my advisory committee.  This thesis does not 
include proprietary or classified information. 
 
 
 
 
__________________________________________ 
Kent Alexander Sayler 
 
 
 
 
 
 
 
Certificate of Approval: 
 
 
 
 
________________________________            __________________________________ 
Charles A. Gross                                                 S. Mark Halpin, Chair 
Professor                                                             Alabama Power Distinguished Professor 
Electrical and Computer Engineering                Electrical and Computer Engineering 
 
 
 
________________________________            ________________________________ 
R. Mark Nelms                                                   Stephen L. McFarland 
Professor                                                             Dean 
Electrical and Computer Engineering                Graduate School 
 
 
 
 
 
PREDICTING GENERATOR COUPLING USING POWER SYSTEM IMPEDANCE 
MATRICES 
 
 
 
 
 
 
 
 
Kent Alexander Sayler 
 
 
 
 
 
A Thesis 
Submitted to 
the Graduate Faculty of 
Auburn University 
in Partial Fulfillment of the 
Requirements for the 
Degree of 
Master of Science 
 
 
 
Auburn, Alabama 
May 11, 2006 
 iii
 
 
 
 
 
PREDICTING GENERATOR COUPLING USING POWER SYSTEM IMPEDANCE 
MATRICES 
 
 
 
 
Kent Alexander Sayler 
 
 
 
 
Permission is granted to Auburn University to make copies of this thesis at its discretion, 
upon the request of individuals or institutions and at their expense. The author reserves 
all publication rights. 
 
 
 
                                                                                          ___________________________ 
                                                                                          Signature of Author 
 
 
 
                                                                                          ___________________________ 
                                                                                          Date of Graduation 
 
 
 
 iv
 
 
 
VITA 
 
 Kent A. Sayler, son of Greg and Dianne Sayler, was born June 26, 1980 in York, 
Alabama.  He graduated from Flomaton High School in 1998.  For the next two years he 
attended Jefferson Davis Community College and he received his Associate of Science in 
May of 2000.  He transferred to Auburn University in the Fall of 2000 where he enrolled 
in the College of Engineering.  In December of 2003 he received his Bachelor of 
Electrical Engineering from Auburn University.  He continued his education by 
becoming a graduate student in the Department of Electrical and Computer Engineering 
at Auburn University, where his research effort has been power system stability. 
 
 
 
 v
 
 
 
 
 
THESIS ABSTRACT 
 
PREDICTING GENERATOR COUPLING USING POWER SYSTEM IMPEDANCE 
MATRICES 
 
Kent Alexander Sayler 
Master of Science, May 11, 2006 
(B.E.E., Auburn University, 2003) 
 
69 Typed Pages  
 
Directed by S. Mark Halpin 
 
 The combination of power flows on multiple lines constitutes a flowgate.  
Transmission flowgates are one tool being used to deal with wide-area stability issues 
because experience with dynamic simulation suggests that proximity to a stability limit 
may be approximated using a transmission flowgate.  An inconvenience associated with 
flowgates, however, is the absence of a defined procedure by which to identify where 
they need to be implemented in the power system.  Knowing which machines swing 
together as a result of a disturbance somewhere in the system is helpful when considering 
possible flowgate locations.  The concept of how the system impedance matrix can be 
used to determine which generators in the power system move together for a given 
contingency is discussed in this thesis.     
 vi
 Once a mathematical model of a multi-machine power system is formed, the 
system impedance matrix can be extracted from this model and used to suggest groups of 
machines which are likely to be electrically coupled together when the system is 
perturbed.  The term ?influence? will be used to refer to the degree to which impedances 
affect the total dynamic coupling because several factors affect the coupling between 
machines under dynamic conditions.  This electrical coupling is validated by the in-phase 
behavior of generator rotor angles in the system.  The generator rotor angle plots are 
obtained through time domain simulations and compared via visual inspection to the 
groups of machines as identified by the system impedance matrix values.   
 
 vii
 
 
 
ACKNOWLEDGMENTS 
 
 I would like to express my sincere gratitude to all those people who provided a 
constant source of help and motivation throughout my work.  First of all, I would like to 
thank my major professor and advisor Dr. S. Mark Halpin.  I will always revere his 
patience, expert guidance and ability to solve intricate problems.  He made my pursuit of 
higher education a truly enjoyable and unforgettable experience. I would also like to 
thank my committee members, Dr. Charles A. Gross and Dr. R. Mark Nelms for their 
help and valuable time they spent in reviewing this work.  Last but certainly not least, I 
want to thank my wife, Marisa.  She continually inspires me and is the sole reason I 
pursued a graduate degree.   
 
   
 viii
 
 
 
 
 
Style manual of journal used Graduate School: Guide to preparation and 
submission of theses and dissertations_________________________________________ 
 
Computer software used Microsoft Office 2003, MATLAB 6.5, Microsoft Visual 
Studio .NET 2003, PSS/E 28.0_____ 
 
 ix
 
 
 
 
 
TABLE OF CONTENTS 
 
    LIST OF TABLES????????????...?????????????....x 
    LIST OF FIGURES...????????????????????????.....xi 
1 INTRODUCTION ....................................................................................................... 1 
2 MULTI-MACHINE STABILITY ............................................................................... 7 
2.1 Plant and Machine Models.................................................................................... 7 
2.2 Power System Model ............................................................................................ 9 
2.3 The Complete Mathematical Model ................................................................... 10 
2.4 Reducing the System Model ............................................................................... 11 
2.5 The Multi-Machine Stability Problem................................................................ 12 
3 GENERATOR COUPLING AND THE POWER SYSTEM MODEL..................... 15 
3.1 System Impedance and Electrical Power Output................................................ 15 
3.2 5-Bus System Example....................................................................................... 18 
4 179-BUS SYSTEM EXAMPLE................................................................................ 24 
4.1 WSCC 179-Bus Test System.............................................................................. 24 
4.2 Graphical Depiction of Machine Coupling......................................................... 24 
4.3 Contingencies...................................................................................................... 30 
4.3.1 TL
24,25
........................................................................................................... 30 
4.3.2 TL
92,93
........................................................................................................... 35 
5 CONCLUSION.......................................................................................................... 41 
6 REFERENCES .......................................................................................................... 43 
7 APPENDIX................................................................................................................ 44 
7.1 Fast Fourier Transform ....................................................................................... 44 
7.2 Prony Analysis.................................................................................................... 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x
 
 
 
 
 
 
 
 
LIST OF TABLES 
  
Table 3.1  5-Bus System Bus and Machine Data.............................................................. 19 
Table 3.2  5-Bus System Branch Data.............................................................................. 19 
Table 3.3  5-Bus Impedance Matrix Magnitudes, [Zij] .................................................... 19 
Table 3.4  5-Bus Impedance Matrix with Bus 4 Breaker Failure Contingency................ 21 
Table 4.1  Generator Clusters in the 179-bus system ....................................................... 32 
Table 7.1  Prony Analysis Results for Various Generators .............................................. 56 
 
 xi
 
 
 
LIST OF FIGURES 
 
Figure 1.1  SMIB Positive Sequence Model....................................................................... 1 
Figure 1.2  Single Machine with Multiple Lines ................................................................ 4 
Figure 2.1  Positive Sequence Generator Model:  Thevenin Form..................................... 8 
Figure 2.2  Positive Sequence Generator Model:  Norton Form ........................................ 8 
Figure 2.3  General Power System Model........................................................................ 10 
Figure 3.1  5-Bus Test System.......................................................................................... 18 
Figure 3.2  Rotor Angles for Breaker Failure Contingency at Bus 4 ............................... 23 
Figure 4.1  WSCC 179-Bus Test System.......................................................................... 25 
Figure 4.2  Top Section of WSCC 179-bus System ......................................................... 26 
Figure 4.3  Middle Section of WSCC 179-bus System .................................................... 26 
Figure 4.4  Right Section of WSCC 179-bus System....................................................... 27 
Figure 4.5  Bottom Section of WSCC 179-bus System.................................................... 27 
Figure 4.6  Generator Coupling with All Branches in Service......................................... 29 
Figure 4.7  Generator Coupling with TL
24,25 
Out of Service ............................................ 31 
Figure 4.8  Generator Rotor Angles for TL
24,25
 Out of Service........................................ 33 
Figure 4.9  Generator Coupling with TL
92,93 
Out of Service ............................................ 35 
Figure 4.10  Generator Rotor Angles for TL
92,93 
Out of Service ...................................... 36 
Figure 7.1  Exponential Frequency Spectrum................................................................... 45 
Figure 7.2  Time Domain Plot of a Sinusoid with Exponential Decay............................. 46 
Figure 7.3  Frequency Spectrum of a Sinusoid with Exponential Decay ......................... 46 
Figure 7.4  Frequency Spectrum of Sinusoid with Multiple Exponential Decay Terms.. 47 
Figure 7.5  Frequency Spectra with Muliple Exponential Decay Terms.......................... 48 
Figure 7.6  Time Domain Simulation of Generator 6....................................................... 49 
Figure 7.7  FFT Results for Generator 6........................................................................... 50 
Figure 7.8  Prony Analysis Results for Generators 6 and 11 ........................................ 56 
Figure 7.9  Prony Analysis Results for Generators 4 and 18............................................ 56 
   
 
 
 
 
 
 1
 
 
 
 
 
  CHAPTER 1 
1 INTRODUCTION 
The continual increase of generation in localized areas is resulting in congested 
transmission lines within the power system.  One kind of situation resulting from this 
congestion is the potential to threaten a system stability limit.  One method used to help 
curtail congestion on transmission lines is the implementation of transmission flowgates.  
The idea of a transmission flowgate was originally constructed to deal with loop flows in 
interconnected systems and to manage overall power transfers from one control area to 
another.  Experience with dynamic simulations suggests that proximity to a stability limit 
may be approximately predicted using a transmission flowgate [1]. 
The single-machine infinite bus (SMIB) system can be used to illustrate the 
usefulness of a flowgate, and it is shown in Figure 1.1. 
  
Figure 1.1  SMIB Positive Sequence Model 
??E
+ 
- 
'd
jX
S
jX
+ 
- 
VV ??
VV ??
?
+ 
- 
P
E  
, Q
E 
 2
 
The terminal complex power can be expressed as a function of terminal voltage, machine 
internal voltage, machine reactance and system reactance as shown in (1.1).   
d'
2
V
d'
V
X
V) ?-? ( cos VE
j
X
) ?-VEsin(?
S
?
+=                                   (1.1) 
It is important to recognize the assumptions associated with the SMIB system: 
1. The machine under study is not large enough to influence the frequency of the 
bulk system.  Therefore, the phase angle of the equivalent system is constant and 
chosen to be the reference angle, 0
o
. 
2. The machine under study is not large enough to influence the voltage of the bulk 
system.  Therefore, the voltage magnitude of the equivalent system is constant.  
 
The amount of real and reactive power that will move from the generator to the 
system is represented by the first and second terms of (1.1), respectively.  The real part of 
(1.1) is rewritten in (1.2).  Notice in (1.2) that 
V
?  is no longer present because it is 
chosen as the reference angle. 
) sin(?
X
VE
P
E
=                                                          (1.2) 
Observation of (1.2) reveals that there is a maximum amount of real power that can be 
moved from the machine under study to the rest of the system.  This maximum amount of 
real power is obtained when the rotor angle, ?, reaches 90
o
.  The rotor angle and speed of 
the machine are determined by the classical swing equation that relates the machine shaft 
angle with applied power as shown in (1.3). 
 3
{}
?
dt
d?
PP
H
f?
dt
d?
EM
=
?=
                                                (1.3) 
In (1.3): 
 H is the inertia constant in seconds; 
 f is the nominal frequency in Hertz and is constant (not affected by changes in ?); 
 P
M 
is the mechanical power input for the machine and is assumed constant; 
 P
E 
is the electrical power output for the machine; and 
 ? is the relative angular velocity (with respect to a synchronous reference) of the 
 rotor of the machine. 
 For the SMIB case, the worst-case stability scenario is when the electrical power, 
P
E
, drops to zero during a fault [2].  For this situation, the swing equation in (1.3) reduces 
to that shown in (1.4) by using a double integration with respect to time.  
    
0
2
M
?tP
2H
?f
?(t) +=                                                   (1.4) 
Substituting ?
cc
 for ?(t) and solving for t produces (1.5), where t
cc
 represents the 
maximum time available to clear the worst-case fault, or the critical clearing time. 
?
?
?
?
?
?
?
?
?=
M
0cccc
?fP
2H
)?(?t                                             (1.5) 
 In (1.5), the angles ?
cc
 and ?
0
 represent the clearing time dependence on pre-
contingency and post-contingency electrical system conditions, and P
M
 represents the 
pre-contingency mechanical power input to the generator, which is very nearly equal to 
P
E
.  It is clear from (1.5) that the critical clearing time is inversely proportional to the 
square root of P
M
.  In cases of practical interest, the critical clearing time t
cc
 is fixed by 
 4
circuit breaker and relay operating times.  For a generation owner, the power output (and 
input P
M
) is varied to meet economic goals.  Therefore, it is a simple matter to solve (1.5) 
for the maximum permissible power output (and input, P
M,max
) such that generator 
stability is maintained for a given t
cc
[1].   Because P
M
 is approximately equal to P
E
, it is 
possible to monitor the power flow on the branch from the machine under study to the 
system and accurately predict P
M
.  For the SMIB system shown in Figure 1.1, this single 
measurement of P
M
 constitutes the most basic flowgate because it is limiting only one 
element.  The resulting equation for the SMIB flowgate is shown in (1.6). 
maxM,M
PP ?                                                          (1.6) 
 The case where a single machine is connected to the power system through 
multiple lines is shown in Figure 1.2.  As with the SMIB example, only one machine is 
limited.  This means the flowgate completely surrounds the machine under study, G
1
.  
For the case where G
1
 is connected by multiple branches, (1.6) is slightly modified to 
account for both lines connected to bus 1 and is shown in (1.7). 
maxM,1312M
PPPP ?+=                                                 (1.7) 
 
Figure 1.2  Single Machine with Multiple Lines 
 
 5
 For a multi-machine case, it also is possible to monitor the power flow on 
multiple transmission lines and accurately predict P
M
, because P
M
 is approximately equal 
to the electrical power injected into the network.  The combination of power flows on 
multiple lines constitutes the flowgate; assuming there is a negligible amount of load 
inside the flowgate, this flowgate total is an accurate representation of P
M
 and can 
therefore be compared to P
M,max
 to assess generator stability[1].   
One problem with flowgates, however, is that no real procedure to define and 
place one in the system exists.  One possible way to help identify potential flowgate 
locations would be to determine which generators in the system will ?swing together? 
when the system is perturbed.  The concept of how the system impedance matrix can be 
used to determine which generators in the power system move together for a given 
contingency is discussed in this thesis.        
The first step of building a mathematical model of the power system is discussed 
in Chapter 2.  The mathematical model will be a set of nonlinear coupled differential 
equations and includes models for system generators, loads, and the transmission 
network.  After developing the model, a relationship for the electrical power output for a 
machine in the system in terms of all known quantities except machine rotor angles can 
be formed.    
Once the system is modeled, a relationship between system impedance and 
generator coupling will be formed in Chapter 3 from examining the power system model 
equations.  This generator coupling will then be verified by performing dynamic 
simulations on a 5-bus power system and comparing machine rotor angle plots to the 
coupling relationships specified by the system impedance matrix. 
 6
In Chapter 4, the generator coupling method will be further tested by introducing 
the Western States Coordinating Council (WSCC) 179-bus test system.  The methods 
introduced in Chapter 3 will be applied to the larger 179-bus system for two cases:  when 
a disturbance is close to a particular group of machines and when a disturbance is far 
away from the same group of machines. 
A group of strongly coupled machines represent a stability concern because if any 
machine within the group goes unstable, the strong coupling between it and the other 
machines in the group will most likely result in multiple machines going unstable.  
Therefore, generation within this coupled group most likely needs to be curtailed.  The 
amount by which generation needs to be curtailed can be correlated to some P
M,max
 that is 
equal to the flow on the flowgate which surrounds the group of strongly coupled 
machines.  The system impedance matrix is one factor of the total coupling that exists 
between generators and observation of the equations to be developed will show that it is a 
significant one.       
   
 7
 
 
 
CHAPTER 2 
2     MULTI-MACHINE STABILITY  
 A mathematical model of the power system must be constructed if stability 
studies are to be done.  This involves determining models for system generators, loads, 
and the transmission network.  After developing the model, a relationship for the 
electrical power output for a machine in the system in terms of all known quantities 
except machine rotor angles can be formed. 
2.1 Plant and Machine Models 
 The generator model depicted in Figure 2.1 [3] is a positive sequence ?voltage 
behind a reactance? model of a synchronous machine.  This model is used to study the 
stability of a power system for a period of time of one second or less because during this 
period the system dynamic response is dependent largely on stored kinetic energy in 
rotating masses [2].  Studies can be conducted in a relatively short time because the 
classical model is the simplest model used in studies of power system dynamics and 
requires a minimum amount of data. 
The reactance X
d
? is the direct axis transient reactance and is used when the 
machine has just experienced a transient but has not yet reached steady state.  This value 
is applicable from a time period of about three 60 Hz cycles to 1 second after a 
disturbance has occurred.  The machine internal voltage magnitude, E, is controlled by 
 8
 
Figure 2.1  Positive Sequence Generator Model:  Thevenin Form 
 
the machine rotor speed and the field excitation.  It is considered constant due to the facts 
that speed changes are minimal and exciter response is limited during the first second 
after a disturbance.  The internal rotor angle, ?, depends on the angle between the stator 
and rotor magnetic fields of a synchronous machine [2]. 
 When the power system is extended to a multi-machine case, the model shown in 
Figure 2.1 along with the assumptions discussed previously still applies, recognizing that 
several of these models will be included.  It is convenient to convert (via source  
 
Figure 2.2  Positive Sequence Generator Model:  Norton Form 
 
??E
+ 
- 
'd
jX
+ 
- 
VV ??
iI ??
'd
jX
E ??
+ 
- 
'd
jX
+ 
- 
VV ??
iI ??
 9
transformation) the Thevenin model in Figure 2.1 into the Norton model shown in Figure 
2.2. 
2.2 Power System Model 
The power system transmission network will be modeled as an admittance matrix.  
Consider the n-bus power system with each node specifically identified as shown in 
Figure 2.3. 
 An admittance matrix model of the bulk power transmission system can be 
formed that mathematically relates the nodal current injections and voltages as shown in 
(2.1).  The current and voltage vectors are of dimension (nx1) and the admittance matrix 
is of dimension (nxn). 
[ ]V
~
YI
~
=                                                          (2.1) 
The admittance matrix, [ Y ], can be built as follows: 
1. The diagonal entries, iiY , are the sum of all admittances connected to node i. 
2.  The off-diagonal entries, ijY , are the sum of the negative of admittances between 
node i and node j.             
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
n
j
i
1
nn,jn,in,n,1
nj,jj,ij,j,1
ni,ji,ii,i,1
n1,j1,i1,1,1
n
j
i
1
V
V
V
V
yyyy
yyyy
yyyy
yyyy
I
I
I
I
M
M
LL
MOMMNM
LL
LL
MNLLOM
LL
M
M
                             (2.2) 
 10
 
Figure 2.3  General Power System Model 
 
2.3 The Complete Mathematical Model 
 To complete the model of the power system, the effects of machines and loads 
must be considered with the bulk transmission system in a single relationship.  The 
effects of the machine are accounted for as follows: 
1. The machine impedance, X
d
?, is converted to an admittance and included in the 
appropriate diagonal entry of the admittance matrix. 
2.  The machine current is included in the current injection vector, I
~
.  
 Loads are modeled as simple impedances at their respective buses.  This can be 
done due to the fact that complex power and voltage is known for a given load.  As a 
result, the equivalent shunt admittance of a load, LY , can be calculated using (2.3). 
2
L
L
2
L
L
L
V
Q
j
V
P
Y ?=                                                    (2.3) 
 11
2.4 Reducing the System Model 
 Careful evaluation of (2.2) reveals that for an n-bus power system, if i buses are 
machine buses, then n-i buses are non-machine buses.  All the buses except for the 
generator buses can be eliminated, thus reducing the dimension of the admittance matrix.  
This reduction, known as Kron reduction, is performed by matrix operations considering 
the fact that all buses which do not contain a generator have a current injection that is 
equal to zero. 
 The first step is assigning machine buses the numbers 1 through m and all other 
buses (ones without machines) are numbered (1+m) through n.  The revised version of 
(2.2) is shown in (2.4). 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
+++++
+
+
+
n
1m
m
1
nn,1mn,mn,n,1
n1,m1m1,mm1,m1,1m
nm,1mm,mm,m,1
n1,1m1,m1,1,1
n
1m
m
1
V
V
V
V
yyyy
yyyy
yyyy
yyyy
I
I
I
I
M
M
LL
MOMMNM
LL
LL
MNLLOM
LL
M
M
               (2.4) 
Using the new subscripts denoted in (2.4), (2.5) is written and shown below, where the 
subscript ?G? is equal to m and the subscript ?S? is equal to (n-m). 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
S
G
SSSG
GSGG
S
G
V
~
V
~
][Y][Y
][Y][Y
I
~
I
~
                                      (2.5) 
Recognizing that there are no current injections at buses that do not contain machines, the 
current vector,
S
I
~
, is equal to zero.  Therefore, (2.5) can be expanded as shown in (2.6) 
and (2.7). 
SGSGGGG
V
~
][YV
~
][YI
~
+=                                         (2.6) 
 12
SSSGSG
V
~
][YV
~
][Y0
~
+=                                          (2.7) 
Equation (2.7) can be solved for 
S
V
~
 and the result can be substituted into (2.6) to obtain 
(2.8) and (2.9). 
{ }
GSG
1
SSGSGGG
V
~
][Y]][Y[Y][YI
~
?
?=                                (2.8) 
[ ]
GGG
V
~
YI
~
=                                                  (2.9) 
Equation (2.9) is the admittance matrix model that includes the effects of system 
components but is a relationship strictly between generator current injections and 
generator terminal voltages. 
2.5 The Multi-Machine Stability Problem 
 With all machines, power transmission equipment, and system loads included in 
(2.2), the entire model can be solved to determine all nodal voltages.  The swing equation 
presented for the SMIB case in Chapter 1 is shown in (2.10) for a machine at bus i in the 
power system.  All assumptions presented in Chapter 1 are still applicable for the multi-
machine case. 
{ }
i
i
iE,iM,
i
i
?
dt
d?
PP
H
f?
dt
d?
=
?=
                                                (2.10) 
An expression can be developed for P
E,i 
in terms of the power system component 
impedances and other machines using (2.9).  The electrical power output based on Figure 
2.3 for machine i is given in (2.11). 
)?cos(?IVP
ii,iv,iiiE,
?=                                              (2.11) 
 13
Recognizing in (2.9) that each current injection is associated with a specific machine 
internal voltage and impedance allows (2.12) to be calculated for the i
th
 machine. 
id,
ii
i
jX'
?E
I
?
=                                                        (2.12) 
From (2.12), the current magnitude and angle in (2.11) are known for each machine.  The 
terminal voltage, however, is unknown and must be expressed using (2.9).  Assuming 
[ Y ] is non-singular and can therefore be inverted, (2.9) can be rewritten as (2.13), where 
[ Z ] is known as the impedance matrix. 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
m
j
i
1
mm,jm,im,m,1
mj,jj,ij,j,1
mi,ji,ii,i,1
m1,j1,i1,1,1
m
j
i
1
I
I
I
I
zzzz
zzzz
zzzz
zzzz
V
V
V
V
M
M
LL
MOMMNM
LL
LL
MNLLOM
LL
M
M
                             (2.13) 
 Using (2.11)-(2.13), the machine electrical power output can be expressed as a 
summation involving all known quantities except machine rotor angles as shown in 
(2.14). 
()
?
=
?
?
?
?
?
?
?
?
?
?
??=
m
1j
ji,jiij
jd,
j
id,
i
iE,
???cosz
X'
E
X'
E
P                               (2.14) 
Substituting (2.14) into (2.10) yields (2.15), which is a complete set of equations of 
motion for machine i in terms of the power system component impedances and other 
machines. 
 14
{ ()
i
i
m
1j
ji,jiij
jd,
j
id,
i
iM,
i
i
?
dt
d?
???cosz
X'
E
X'
E
P
H
f?
dt
d?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???=
?
=
                 (2.15) 
 A relationship for the electrical power output for a machine has now been derived 
and is used to write the complete set of equations of motion in (2.15).  The next step is to 
use the equations of motion in (2.15) to form a relationship between system impedance 
and generator coupling.  Once developed, this relationship can be used to help identify 
potential flowgate locations within a power system.
 15
 
 
 
CHAPTER 3 
3 GENERATOR COUPLING AND THE POWER SYSTEM MODEL 
 A relationship between system impedance and generator coupling can be formed 
by examining the equation that models the electrical power output for a machine in the 
system in terms of all known quantities except machine rotor angles.  This generator 
coupling can then be verified by performing dynamic simulations on a given power 
system and comparing machine rotor angle plots to the coupling relationships predicted 
by the system impedance matrix. 
3.1 System Impedance and Electrical Power Output 
 The equations of electrical power output, bus voltage (in terms of impedance), 
and the swing equation developed in Chapter 2 are repeated in (3.1) ? (3.3).   
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
m
j
i
1
mm,jm,im,m,1
mj,jj,ij,j,1
mi,ji,ii,i,1
m1,j1,i1,1,1
m
j
i
1
I
I
I
I
zzzz
zzzz
zzzz
zzzz
V
V
V
V
M
M
LL
MOMMNM
LL
LL
MNLLOM
LL
M
M
                           (3.1) 
()
?
=
?
?
?
?
?
?
?
?
?
?
??=
m
1j
ji,jiij
jd,
j
id,
i
iE,
???cosz
X'
E
X'
E
P                            (3.2) 
{ ()
i
i
m
1j
ji,jiij
jd,
j
id,
i
iM,
i
i
?
dt
d?
???cosz
X'
E
X'
E
P
H
f?
dt
d?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???=
?
=
                  (3.3) 
 16
 Direct examination of (3.2) brings insight into how generation at a particular bus 
affects conditions at other buses.  For instance, the influence of generator j on real power 
injected at bus i is quantified by
?
?
?
?
?
?
?
?
?
?
ij
jd,
j
id,
i
z
X'
E
X'
E
.  Also, the voltage at bus i due to the 
current injected at bus j is seen in the
ij
jd,
j
z
X'
E
 term.   
 Further examination of (3.2) shows that there is a relationship between system 
impedance and the electrical power output of each machine in a given system.  Consider 
a disturbance represented by a 1 per-unit current injection at bus i in the system.  From 
(3.1), this injection at bus i (and only at bus i) will result in voltages being produced at all 
other buses j in the system based on the product z
ij
*I
i
.  In (3.2), the current injection 
magnitude at bus i is represented by the 
id,
i
X'
E
term and the bus voltage magnitude at bus i 
is represented by the 
?
=
?
?
?
?
?
?
?
?
?
?
n
1j
ij
jd,
j
z
X'
E
 term.  Therefore, a voltage produced at bus i will have 
the effect of increasing the electrical power output from the machine at bus i. 
 Consider the effect on the terminal (i.e. bus) voltage i due to a current injection at 
bus j that is quantified by
ij
jd,
j
z
X'
E
.  As stated in Chapter 1, there is a maximum amount of 
real power that can be moved from a machine to the rest of the system.  This maximum 
power is defined in (1.2) where V is the terminal voltage magnitude, E is the machine 
internal voltage magnitude, X is machine and system impedance, and ? is the machine 
rotor angle.  The variables E and X are constant, which means anything that affects V 
will impact stability more significantly than something that has little affect on V.  The z
ij
 
 17
term in 
ij
jd,
j
z
X'
E
is an indicator of how a current injection at bus j affects V at bus i and 
therefore the response of the generator at bus i.     
 From (3.3), it can be seen that because the mechanical power input from machine 
j is constant, the equilibrium between the input and output power will be upset if 
electrical power output is increased.  This loss of equilibrium causes the rotor angle of 
machine j to increase or decrease according to the laws of motion of a rotating body [2].  
During a disturbance, two machines are said to be electrically coupled if their rotor 
angles swing in phase with each other.   
 For a current injection from a generator at bus i, one factor that could indicate if 
machines in a system will swing together (i.e. are strongly coupled) is the magnitude of 
the off-diagonal z
ij
 terms in the impedance matrix.  This is due to the fact that a larger z
ij
 
entry in the column corresponding to bus j will result in a larger change in rotor angle 
because of the direct relationship between angular velocity and impedance seen in (3.3).  
If machines at various buses are not strongly influenced by each other, then they may not 
swing together when disturbed.  The extent to which a disturbance at bus i is felt at bus j 
is shown in[]Z .  The same principle applies to indicate the extent to which the behavior 
of a generator at bus i has on bus j.  Because i and j both contain generators, []Z  can be 
considered to show at least part of the influence generator i has on generator j.  Therefore, 
by looking at the magnitude of the z
ij
 terms in (3.1), some indication of how much impact 
a current injection at bus j has on all other buses i=1?m can be obtained. 
 Further observation of (3.2) reveals that X?
d 
can also have a similar influence on 
how the behavior of a generator at bus i can possibly affect bus j.  This is because the 
 18
effects predicted by the magnitude of the z
ij
 terms in (3.1) are altered by X?
d
 in the 
current injection.  Each current injection in the vector in (3.1) contains an X?
d 
that scales 
each row of the system impedance matrix by its corresponding machine impedance.  
Therefore, for power systems where large variations in X?
d 
exist, X?
d 
becomes very 
important when considering how a current injection from a generator at a bus can 
possibly affect other buses in the power system.    
3.2 5-Bus System Example     
 The 5-bus system [1] in Figure 3.1 will be used to illustrate the generator 
influence concept introduced in Section 3.1.  The corresponding system information is 
detailed in Tables 3.1 ? 3.2. 
 The admittance matrix, [ ]Y , can be formed from the information given in Tables 
3.1 and 3.2 as described in Section 2.2.  Because[ ]Y  is non-singular, the system 
impedance matrix,[]Z , can be formed by inverting[ ]Y .  Kron reduction is then performed 
on[]Z , which reduces it to a 4x4 matrix.  Because bus 2 is the only non-generator bus, it  
 
 
Bus 4
Bus 1 
Bus 2 
Bus 3
Bus 5 
 
Figure 3.1  5-Bus Test System       
 19
 
Table 3.1  5-Bus System Bus and Machine Data 
Bus # V ? P
G
 Q
G
 P
L
 Q
L
 
1 0.975 -15.98 110.0 53.2 350.0 87.7 
2 0.978 -17.19 - - 350.0 87.7 
3 0.997 -15.61 160.0 251.8 350.0 87.7 
4 1.0 -12.43 200.0 93.0 - - 
5
*
 0.995 -4.77 596.0 44.4 - - 
*Slack bus 
Table 3.2  5-Bus System Branch Data 
From To R X B Rating 
1 2 0.0014 0.0138 0.0284 502 
1 4 0.0056 0.0552 0.1136 502 
4 5 0.0042 0.0414 0.0858 502 
2 3 0.0014 0.0138 0.0284 502 
3 4 0.0014 0.0138 0.0284 502 
1 5 0.0070 0.0690 0.142 502 
  
Table 3.3  5-Bus Impedance Matrix Magnitudes, [Zij] 
1534
1 0.0159 0.0033 0.0082 0.0072
5 0.0033 0.0094 0.0029 0.0037
3 0.0082 0.0029 0.015 0.0098
4 0.0072 0.0037 0.0098 0.0144
 
 
was simply swapped with bus 5 in the renumbering procedure.  It is shown in Table 3.3, 
however, by its original label of ?5? so it will correspond with the bus labels seen in 
Figure 3.1.  The magnitudes of the resulting impedances are calculated and depicted in 
tabular form in Table 3.3, where z
1,1
=0.0159, z
1,5
=0.0033, and so on.       
 As stated in Section 3.1, looking at the size of the z
ij
 terms in Table 3.3 will give 
an indication of how much impact a current injection from a generator at a particular bus 
will have at the other buses in the system.  Inspection of the off-diagonals in Table 3.3 
doesn?t reveal much disparity in magnitude between the terms.  For example, the 
 20
impedance magnitudes z
3,5 
and z
4,5 
differ by 0.0008 pu.  The difference is even smaller 
when other generator buses are compared.  It appears that the behavior of a generator at a 
particular bus will have similar effects on all generators in the 5-bus system, because all 
relevant entries in the impedance matrix have magnitudes whose values are close in 
proximity to each other.   
 Also, all machines in the 5-bus system have the same value for X?
d
, which means 
all the terms in [ ]Z  are all effectively scaled by the same amount.  The relative size of the 
entries in []Z  to each other is an indicator of how generators can possibly influence one 
another to a greater or lesser degree.  If all entries in [ ]Z  are scaled by the same amount, 
X?
d
 has no overall effect on generator influence for the 5-bus system.   
 The impedance matrix entries will be different, however, if one or more branches 
are taken out of service due to a contingency.  It is this scenario where the size of the z
ij
 
terms will be quite informative.   
 A typical contingency that can create stability problems is a three-phase fault with 
breaker failure [1].  Suppose that bus 4 is a straight bus with two bus sections connected 
with a tiebreaker.  The generator and TL
45
 are on one section, while the lines TL
41
 and 
TL
43
 are on the other section.  The contingency considered here is that a three-phase fault 
occurs on TL
43
 and is just outside the substation of bus 4.  For this breaker failure 
contingency, let the bus 4 line breaker fail to clear.  Assuming pilot relaying is present 
and three cycle breakers are used, the bus 3 end of the faulted line clears in 
approximately 4 cycles (66.67 ms for 60 Hz systems).  Due to the stuck breaker, the 
entire bus section is cleared in approximately 12 cycles (200 ms for 60 Hz systems) by   
   
 21
Table 3.4  5-Bus Impedance Matrix with Bus 4 Breaker Failure Contingency 
1534
1 0.0195 0.0026 0.0115 0.0015
5 0.0026 0.0096 0.0015 0.0054
3 0.0115 0.0015 0.0232 0.0009
4 0.0015 0.0054 0.0009 0.0262
 
opening the tiebreaker and initiating a transfer trip operation to bus 1.  Generator G
4
 
remains in service and connected to the system through TL
45
. 
 The loss of TL
41
 and TL
43
 change the impedance matrix magnitudes as seen in 
Table 3.4.  Inspection of the off-diagonals with the two branches out of service yields a 
much different result than before.  The impedance magnitudes z
3,5 
and z
4,5 
differ by 
0.0081 pu ? a 1000% increase from the pre-fault system conditions.  Observation of the 
other columns of the impedance matrix also reveals larger magnitude disparities than in 
the case with all lines in service.  In the column corresponding to generator 1, the entries 
corresponding to generators 4 and 5 both have a magnitude less than one-fifth the size of 
the entry corresponding to generator 3.  In the column corresponding to generator 3, the 
entries corresponding to generators 4 and 5 both have a magnitude less than one-fifth of 
the size of the entry corresponding to generator 1.  The opposite effects are seen in the 
columns corresponding to generators 4 and 5.  In each case, the entries for generators 1 
and 3 have much smaller magnitudes than the entry for the other generator in question.   
 As stated in Section 3.1, looking at the size of the z
ij
 terms in Table 3.4 can give 
some indication of how much impact the behavior of a generator at a particular bus will 
have on the other machines in the system.  Because of the direct relationship between 
impedance and angular velocity in (3.3), the largest off-diagonal magnitude in column j 
of the system impedance matrix is indicative of the machine most affected by a current 
 22
injection from a machine at bus j.  Observation of the current injection vector reveals that 
if a generator at bus j experiences a change in rotor angle, of the remaining generators in 
the system, the generator that will experience the greatest change in rotor angle 
corresponds to the largest off-diagonal entry in column j of the system impedance matrix.  
The larger that this magnitude is, the more influence that the generator at bus j has on this 
other generator.  If the influence is large enough, the affected generator rotor angle can 
move in phase with the rotor angle of machine j.  In this case of in-phase behavior, it can 
be said that these two machines could be strongly electrically coupled to each other.   In 
the 5-bus system example with two lines out of service, the large magnitudes of bus 4 in 
column 5 and bus 5 in column 4 indicate that the generator at bus 5 has a much stronger 
influence on the generator at bus 4 (and vice-versa) than on the generators at buses 1 and 
3.  Also, the large magnitudes of bus 3 in column 1 and bus 1 in column 3 indicate the 
generator at bus 1 has a much stronger influence on the generator at bus 3 (and vice-
versa) than on the generators at buses 4 and 5. 
 Therefore, it would seem from observation of Table 3.4 that generators 4 and 5 
strongly influence one another, while generators 1 and 3 also strongly influence each 
other.  The presence of these two distinct groups can be verified by observing the 
frequency and phase characteristics of the machine rotor angle plots in Figure 3.2 which 
were obtained via dynamic simulation.   
 It is clear from observation of Figure 3.2 that if the generator at bus 5 experiences 
a large change in rotor angle, the generator at bus 4 will also experience this large change 
and possibly go unstable.  The same can be said for generators 1 and 3.  Reducing the 
output power of a generator increases the amount of time available to clear a worst-case  
 23
-60
-40
-20
0
20
40
60
80
0
0.
0
8
0.
1
7
0.
2
6
0.
4
9
0.
7
7
1.
0
4
1.
3
2
1.
5
9
1.
8
7
2.
1
4
2.
4
2
2.
6
9
2.
9
7
Time (s)
M
a
ch
i
n
e R
o
to
r
 A
n
g
l
e (d
eg
)
Gen 1
Gen 3
Gen 4
Gen 5
 
Figure 3.2  Rotor Angles for Breaker Failure Contingency at Bus 4 
 
fault, according to the inverse relationship between power output and critical clearing 
time in (1.5).  Therefore, the 5-bus system would be more likely to remain stable for the 
above contingency if the combined output power of generators 4 and 5 or 1 and 3 is held 
below some limiting value.  One way this limiting value could be enforced is by defining 
a maximum amount of power allowed to flow on the lines that connect a group of 
generators to the rest of the power system.  This means constraining the power flowing 
on lines TL
51
, TL
41
, and TL
43
, and this constraint can be represented as a flowgate.  The 
system impedance matrix concepts introduced in this chapter helped in identifying the 
two groups of coupled generators in the 5-bus system, as well as a possible flowgate 
location. 
 The relationship of generator coupling and system impedance appears to be valid 
at least for this particular 5-bus system.  The next step is to consider a much larger power 
system to further verify the validity of this method.
 24
 
 
 
 
CHAPTER 4 
4 179-BUS SYSTEM EXAMPLE 
 In this chapter, the effect of system impedance on generator coupling is further 
discussed by introducing the WSCC 179-bus test system.  The methods introduced in 
Chapter 3 will be applied to the larger 179-bus system for two cases:  when a contingency 
is close to a particular group of machines and when a contingency is far away from the 
same group of machines.  
4.1 WSCC 179-Bus Test System 
 The approach presented using the 5-bus system in Chapter 3 was applied to the 
larger WSCC 179-bus system.  The single line diagram for the 179-bus system is shown 
in Figure 4.1.  Closer looks at the four sub-sections of Figure 4.1 are shown in Figures 
4.2 ? 4.5.  Figure 4.1 is necessary to give overall feel of the topography of the system 
while the smaller maps are necessary for clarity of discussion of individual buses, lines, 
and generators. The methods introduced in Chapter 3 will be used to identify possible 
flowgate locations in the WSCC 179-bus system.   
4.2 Graphical Depiction of Machine Coupling 
 For the 5-bus system introduced in Chapter 3, it was a trivial task to observe the 
impedance matrix and determine how a current injection at one bus affected the other 
buses in the system.  This is due to the fact that the Kron-reduced [ ]Z  for the 5-bus 
system is only a 4x4 matrix.  The WSCC 179-bus system presents a problem, however,  
 25
 
 
Figure 4.1  WSCC 179-Bus Test System 
 26
 
Figure 4.2  Top Section of WSCC 179-bus System 
 
 
Figure 4.3  Middle Section of WSCC 179-bus System 
 
 27
 
Figure 4.4  Right Section of WSCC 179-bus System 
 
 
Figure 4.5  Bottom Section of WSCC 179-bus System 
 
 28
due to its larger size.  When the system is reduced via Kron reduction, effects of 
generator current injections still need to be examined at twenty-nine buses.  Due to the 
overwhelming nature of this problem, a graphical tool was developed that - for a 
particular impedance matrix [ ]Z  - displays the buses that have large impedance 
magnitudes (relative to the other magnitudes in the column) in every column of [ ]Z.  
These magnitudes correspond to the buses that are most affected by a generator current 
injection at a particular bus.  As described in Section 3.2, this effect can be correlated to 
the amount of influence a particular machine has on another, which can result in two or 
more machines being electrically coupled together.       
 The graphical tool template is a picture approximating the locations of the twenty-
nine machines that are depicted in Figure 4.1.  The effects of a current injection from a 
generator at a particular bus were represented as a line drawn between that machine and 
the machines affected by the current injection.  In other words, for a generator at bus i, 
lines were drawn between that bus and the buses corresponding to the largest magnitudes 
in column i of []Z .  Because the effect that a current injection from a generator at a 
particular bus has on another bus is related to how much influence the machine at the 
injected bus has on the machine at the other bus, these lines represent which machines 
strongly influence other machines in the power system.  Some way to visualize this 
influence was needed, so it was determined that the strongest influence would be 
represented by a red line, influence of some lesser degree by a yellow line, and even 
weaker influence by a green line.  Experience with the impedance characteristics of the 
179-bus system led to threshold values of 0.00025 per unit, 0.00008 per unit, and 0.00001 
per unit to be chosen for the red, yellow, and green lines, respectively.  Therefore, the red  
 29
 
Figure 4.6  Generator Coupling with All Branches in Service 
line drawn between generators 36 and 162 in Figure 4.6 means that the magnitude of 
z
36,162
 in the 179-bus system impedance matrix is greater than 0.00025 per unit and it  
represents a strong influence between generators 36 and 162.  These lines are drawn for 
all twenty-nine generator buses in Figure 4.6 and are extremely helpful in identifying 
groups of machines that influence one another by visually representing the strength of 
this influence with red, yellow, and green lines.  Accordingly, there appears to be four 
areas where generators could be strongly coupled together, as indicated by the groups of 
red lines in Figure 4.6. 
 30
 The situation in Figure 4.6 is one with all branches in service.  Notice that 
changes to the system impedance matrix could result in changes in the colors and 
directions of the lines in Figure 4.6.  For losses of various branches, the Kron-reduced 
[]Y  is formed as outlined in the procedure in Chapter 2.  The impedance matrix is then 
formed by inverting []Y  and used to get a graphical output similar to Figure 4.6.  As with 
the 5-bus system in Chapter 3, dynamic simulation is needed to verify the results 
predicted by the graphical tool. 
4.3 Contingencies 
 Several contingencies were placed on the system for purposes of further verifying 
the generator coupling concepts discussed in Chapter 3.  The area of interest in the 179-  
bus system is the pocket of five generators (4, 6, 9, 11, and 18) that appear to be strongly 
coupled to one another, according to Figure 4.6.  The first contingency to be considered is 
one close in proximity to this generation pocket, while the second is one further away 
from the same group of machines. 
4.3.1 TL
24,25
  
 
 The first contingency used to evaluate the generator coupling concepts is placing 
a three-phase fault on the transmission line, TL
24,25, 
very close to bus 24.  Assuming pilot 
relaying is present and three cycle breakers are used, the bus 24 and bus 25 ends of the 
faulted line clear in approximately 3 and 4 cycles (50 ms and 66.67 ms for 60 Hz 
systems), respectively.  The graphical depiction of the system machine couplings if 
TL
24,25 
is taken out of service is shown in Figure 4.7.  Notice that generators 4 and 15 
were represented as being weakly coupled according to the green line drawn between  
 31
 
Figure 4.7  Generator Coupling with TL
24,25 
Out of Service 
them in Figure 4.6, but upon removal of a piece of the transmission system that links 
them (TL
24,25
), they are no longer weakly coupled as indicated by the absence of a 
connection between them in Figure 4.7.  This appears to be the only major change 
between Figures 4.6 and 4.7.  Visual inspection is used to separate the machines into six 
clusters by only crossing lines that are not red (remembering that a red line indicates the 
possibility of two generators being strongly influenced by each other), then crossing 
green lines before yellow lines as necessary.  These clusters are denoted by the blue lines 
that surround the various groups in Figure 4.7 and are listed in tabular form in Table 4.1. 
F 
B 
D 
E 
A 
C 
 32
  
Table 4.1  Generator Clusters in the 179-bus system 
Groups A B C D E F
Generators 4 36 13 103 40 65
6 45 15 112 43 70
9 159 138 116 47 77
11 162 140 118 140 79
18 144 147 30
148 35
 
Machine rotor angle plots from the dynamic simulations are shown in Figure 4.8.  The 
fault was placed on TL
24,25 
at t=0 seconds.  The plots show the rotor angles from t=0-5 
seconds, and they are grouped to correspond to the six clusters seen in Figure 4.7 that 
were produced by encircling areas while not crossing a red line. 
 Observation of the rotor angle plots verifies much of what inspection of the 
graphical depiction of system machine couplings initially revealed.  Currently, visual 
inspection is the method used to compare the rotor angle plots with the results obtained 
from the impedance matrix.  Other means for grouping the rotor angle plots in Figure 4.8 
have been attempted but will not work for the general cases presented in this chapter.  
Descriptions of these methods and reasons for their ineffectiveness are presented in the 
Appendix.   
  Cluster A (4, 6, 9, 11, and 18) has rotor angles that move in phase with each 
other, while also exhibiting similar frequencies.  The rotor angles of the machines in 
cluster B (36, 45, 159, and 162) also move very closely together.  The rotor angles of the 
machines cluster C (13, 15, 138, 144, and 148) move well together except for generator 
15.  The rotor angles of the machines cluster D (103, 112, 116, and 118) also appears to 
move well together except for generator 103.  In each of the cases mentioned above, there
 33
 
 
Figure 4.8  Generator Rotor Angles for TL
24,25
 Out of Service 
were red lines linking several generators in each group, meaning there were indicators of 
possible strong coupling between the generators in each pocket according to the system 
impedance data. 
 Furthermore, the observation that two generators are moving differently than the 
rest of their respective groups (15 and 103) could mean they aren?t as strongly coupled to 
the machines in their group.  This observation is in line with the information obtained 
 34
from the system impedance matrix shown in Figure 4.7.  In both cases, the generators in 
question (15 and 103) are linked to the others in their groups by yellow lines.  These 
yellow lines represent a weaker coupling than red lines that link the remaining generators 
in each group.  However, another factor that could be causing generator 15 to separate 
from the rest of the group is the fact that the fault in question is very close in proximity to 
that specific generator.  This, along with the possibility that generator 15 isn?t as strongly 
influenced as the other machines in its group could be cause of its separation as shown in 
the rotor angle plots. 
 For the TL
24,25 
contingency, the impedance matrix concepts introduced in Chapter 
3 hold up fairly well ? especially when considering the generators that are predicted to be 
strongly coupled to one another as shown in Figure 4.7.  As with the 5-bus system, the 
machine impedances X?
d
 are not a factor in the generator coupling of the 179-bus system.  
The WSCC 179-bus system is an equivalent of a much larger power system.  Because it 
is an equivalent system, the 29 generators in the system all have a very large machine 
base relative to the system power base of 100 MVA.  The original X?
d
 values are in per 
unit on machine MVA base and they are converted to 100 MVA base for system 
modeling in []Z .  Because the machine base is much larger than the system base, the 
machine impedances are reduced dramatically in the conversion from machine base to 
system base.  Therefore, X?
d
 is not considered when examining generator influence in the 
179-bus system.  The next step, however, is to see what happens when the machines on 
the right side are caused to swing by a fault that is further away in the system. 
 35
4.3.2 TL
92,93
     
 The second contingency used to evaluate the generator coupling concepts is 
placing a three-phase fault on the transmission line, TL
92,93
.  The fault description and 
subsequent clearing follow the same procedure as the TL
24,25
 fault, with the offending 
line being taken out of service in approximately four 60 Hz cycles.  Again, visual 
inspection is used to separate the machines into six clusters by the method described in 
Section 4.3.1.  The graphical depiction of the system machine couplings if TL
92,93 
is taken 
out of service is shown in Figure 4.9.    
 
Figure 4.9  Generator Coupling with TL
92,93 
Out of Service 
A 
F 
B 
D 
E C 
 36
 Machine rotor angle plots from the dynamic simulations are shown in Figure 4.10.  
The fault was placed on TL
92,93 
at t=0 seconds.  The plots show the rotor angles from t=0- 
5 seconds, and they are grouped to correspond to the six clusters seen in Figure 4.9 that 
were produced by encircling areas as described in Section 4.3.1. 
 Again, observation of the rotor angle plots verifies much of what inspection of the 
graphical depiction of system machine couplings initially revealed.  Cluster A (4, 6, 9,   
 
Figure 4.10  Generator Rotor Angles for TL
92,93 
Out of Service 
 
 37
11, and 18) still has rotor angles that move in phase with each other, while also exhibiting 
similar frequencies.  The rotor angles of the machines in cluster B (36, 45, 159, and 162) 
also move very closely together.  However, generator 36 appears to have a much more 
dramatic initial swing than the others in the group.  As with the generators in the first 
contingency, this could be due to the fact that generator 36 is close in proximity to 
TL
92,93. 
 Also, observation of Figure 4.9 reveals some intermediate influence (as 
represented by the yellow lines) between generator 36 and three generators that are quite 
close to the TL
92,93 
contingency.  This influence could also be part of the reason that 
generator 36 separates from the group designated by visual inspection of Figure 4.9. 
 The rotor angles of the machines in cluster C (13, 15, 138, 144, and 148) move 
together better for this contingency than for the contingency at TL
24,25
.  This could be due 
to the fact that generator 15 is no longer strongly influenced by the close proximity of the 
fault at TL
24,25
.  Notice also how generator 15 is strongly coupled to the rest of the group 
as denoted by the red line between it and generator 13.  This was not the case in the 
TL
24,25
 contingency.  Observation of the graphical depiction of the impedance data leads 
to the conclusion that moving the fault away from generator 15 results in a stronger 
coupling between it and generator 13 and subsequently the rest of the group.   
 The rotor angles of the machines in cluster D (103, 112, 116, and 118) also 
appears to move well together except for generator 103.  This was the same result as 
when TL
24,25
 was taken out of service, which means intermediate influence indicated by 
the yellow lines extending from generator 103 to the rest of the group could be 
insufficient to influence the generator to  move with the others in the group.  
 38
 More interesting results are obtained when observing cluster F (30, 35, 65, 70, 77, 
and 79) of the system.  For both contingencies, the green lines are indicative of weak 
coupling or influence between generators in this group.  As demonstrated by observation 
of the results of the contingency at TL
24,25
 and how the contingency affected machines 
(13 and 15) that were intermediately influenced by each other, proximity of a fault will 
have an effect on how well machines move together.  When the fault was close to 
generators 13 and 15, they didn?t move very well together in the dynamics.  However, 
when distance was put between the contingency of interest and these two machines, they 
grouped together quite nicely as seen in Figure 4.10.   
 A similar situation has taken place for the group of machines in cluster F.  For the 
contingency at TL
24,25, 
they moved together very closely as seen in Figure 4.8.  When the 
fault was moved to TL
92,93
, however, they separated rather dramatically as shown in 
Figure 4.10.  TL
92,93
 is relatively close to this group, but isn?t as severe a contingency due 
to the parallel transmission line adjacent to the line which TL
92,93 
helps comprise.  
Nonetheless, the group still breaks up as seen in Figure 4.10.  This could be due to the 
weak coupling displayed between generators of the group in Figure 4.9.  The fact that 
they aren?t very strongly coupled as a group to begin with could lead to a minor 
contingency causing them to not move closely with each other. 
 For the TL
92,93 
 contingency, the impedance matrix concepts introduced in 
Chapter 3 do appear to conservatively help reveal which machines move together, while 
at the same time showing that proximity of a group of generators to a fault can have an 
appreciable effect on the group coupling.  In terms of the lines drawn in Figures 4.6, 4.7, 
and 4.9, the red lines represent influence that appears to be immune from fault location, 
 39
the green lines represent influence that could be affected by fault location, and the yellow 
lines represent influence that may or may not be affected by fault location.  Overall, 
observation of results from the 179-bus system reveals that there appears to be reasonable 
correlation between system impedance and generator coupling. 
 As previously discussed in Chapter 3, this relationship between system impedance 
and generator coupling can be used to help determine possible flowgate locations in a 
power system.  Implementing this concept on the larger 179-bus system reveals more 
interesting results.  Observation of Figures 4.6, 4.7, and 4.9 reveals that the system 
impedance matrix doesn?t change very much for the two contingencies considered in this 
chapter.  This means that generator influence doesn?t change very much with respect to 
contingency location, according to the correlation between system impedance and 
generator influence developed in Chapters 3 and 4.  This is especially true for the cases of 
?strong? influence, as indicated by the red lines in the graphical depictions of the 
impedance matrix.  In particular, groups A, B, and D exhibit the same ?strong? influence 
for both contingencies considered in this chapter.   
 Having the ability to identify these ?strong? influences is quite helpful when 
considering possible flowgate locations because these groups of coupled machines 
represent wide-area stability concerns where generation may need to be curtailed.  For 
example, the red lines connecting the machines in group A (4, 6, 9, 11, 18) are indicative 
of the strong influence exhibited by the machines in group A.  Observation of Figure 4.4 
reveals the three branches connecting this group of generators to the rest of the system 
are TL
5,160
, TL
8,163
, and TL
7,28
.  The wide-area stability concern involving group A could 
possibly be lessened if the total flow on these three lines is constrained using a 
 40
transmission flowgate.  Dynamic simulation could then be used to determine if a flowgate 
is actually needed, and if so, what the limiting value of the flowgate should be.        
 
 41
 
 
 
  CHAPTER 5 
5 CONCLUSION 
 The continual increase of generation in localized areas has the potential to 
threaten a system stability limit due to congested transmission lines.  Transmission 
flowgates are one tool being proposed to help manage this congestion.  One problem with 
flowgates, however, is the absence of a defined procedure used to place one in the 
system.  Developed in this thesis is a method that conservatively identifies potential 
flowgate locations by determining which generators will swing together due to a 
contingency in the system.  This act of swinging together is denoted as electrical coupling 
and it is affected by several things, one of which is the system impedance matrix.  
Detailed in this thesis is the explanation of how system impedances influence the total 
dynamic coupling.  The ability to identify this influence is quite helpful when considering 
possible flowgate locations because generators that are strongly coupled together will be 
so without regard to contingency location. 
 The first step was to build a mathematical model of the power system.  The 
mathematical model obtained was a set of nonlinear coupled differential equations.  Once 
built, the system impedance matrix was extracted from the mathematical model and used 
to help determine generator coupling by the theory that a 1 per-unit current injection at 
bus i in the system will result in voltages being produced at all other buses j in the system 
based on the product z
ij
*I
i
.  The results produced by this method agreed with the dynamic 
 42
results as tested on the 5-bus system.  The WSCC 179-bus system was then used to 
further examine this idea of generator coupling by placing a contingency in various 
places in the power system relative to a group of electrically coupled generators.   
 Future work on this method includes abandoning the current method of visual 
inspection of the dynamic responses of the generators.   Some method should be used to 
quantify the rotor angle plots so they can be properly compared to the clusters produced 
by the impedance matrix data.  Currently, Prony and fast Fourier transform analyses are 
being explored as ways to classify the rotor angle plots by phase and frequency.  The 
results of the progress made with these methods are detailed in the appendix.  Other 
future work consists of learning more about how the impedance matrix values can be 
used as a part of some influence coefficient that can be derived and calculated for a given 
power system under study.  Once derived and calculated, much needs to be learned about 
how these coefficients can be better utilized to locate flowgates. 
 43
 
 
 
6 REFERENCES 
[1] Valenzuela, Jorge, Halpin, S. Mark, and Park, Chan S., ?Generation Expansion 
 Planning in Stability Limited Power Systems,? National Science Foundation 
 EPNES Workshop, Puerto Rico, July, 2004. 
[2] Kundur, P., ?Power System Stability and Control,? McGraw-Hill, New York, 
 1994. 
[3] Wildi, Theodore, ?Electric Machines, Drives, and Power Systems,? Prentice Hall, 
 New Jersey, 2002. 
[4] Kamen, Edward, and Heck, Bonnie, ?Fundamentals of Signals and Systems using 
 MATLAB,? Prentice Hall, New Jersey, 1997. 
[5] Hauer, J.F., ?Initial Results in Prony Analysis of Power System Response 
 Signals? IEEE Transactions on Power Systems, Vol. 5, No. 1, Feb., 1994. 
 
 44
 
 
 
7 APPENDIX 
 Currently, visual inspection is the only method used to compare the known 
behavior of the machines (the machine rotor angle plots) with the estimated behavior of 
the machines (the machine influence derived from the system impedance matrix).  As 
noted in Chapter 5, some preliminary effort has been made to try to numerically quantify 
and group the machine rotor angle plots using two methods:  fast Fourier transform and 
Prony analysis.  Neither of these methods has produced reliable results to date.  The fast 
Fourier transform is unable to produce reliable frequency and phase characteristics due to 
exponential decay in the machine rotor angle plots.  Prony analysis is unable to produce 
reliable results for several reasons, including input data precision and the nonlinear nature 
of the power system dynamics.  However, the progress made with both methods is 
documented for future reference. 
7.1  Fast Fourier Transform 
 The fast Fourier transform, or FFT, takes a discrete signal in the time domain and 
transforms that signal into its discrete frequency domain representation [4].  Because all 
data points for the machine rotor angle plots are readily available, it is a simple task to 
use a signal analysis program (such as MATLAB) to calculate the FFT of each rotor 
angle plot.  Once calculated, the dominate frequency of the rotor angles could then be 
observed from the FFT spectral plots.  The machines in the 179-bus system could then be 
 45
grouped by their dominate frequency using cluster analysis and compared to the groups 
observed using the graphical tool in Chapter 4. 
 An arbitrary exponential function is shown in (7.1) and its corresponding 
frequency spectrum is shown in Figure 7.1.  The frequency spectrum seen in Figure (7.1) 
is calculated using the FFT. 
( )taexp)t(y ??=                                                       (7.1) 
 
Figure 7.1  Exponential Frequency Spectrum 
 
When the exponential in (7.1) is multiplied by a sine function as in (7.2), the result is an 
exponentially decaying sine wave of the form in Figure 7.2.  The frequency spectrum of 
this sine wave with exponential decay is shown in Figure 7.3, where
1
?  in (7.2) is given 
the arbitrary frequency value of 
0
?  .  Observation of Figure 7.3 shows that the sine 
function shifts the frequency spectrum by the amount of 
0
?? , which is the dominant 
frequency in (7.2).  An additional exponential decay term is added to (7.2) and the result 
is shown in (7.3).   
)tsin(
t
exp)t(y
1
??
?
?
?
?
?
?
?
?
=                                                 (7.2) 
0
Frequency
Magnitude
 46
 
)tsin(
t
exp)tsin(
t
exp)t(y
21
??
?
?
?
?
?
?
?
?
+??
?
?
?
?
?
?
?
?
=                               (7.3) 
 
Figure 7.2  Time Domain Plot of a Sinusoid with Exponential Decay 
 
Figure 7.3  Frequency Spectrum of a Sinusoid with Exponential Decay 
  
 The frequency spectrum shown in Figure 7.4 is plotted from the FFT of (7.3).  
The frequency of the additional term 
2
?  is twice the value (
2
? =2
0
? ) of the original sine 
function frequency in (7.2).    Notice in Figure 7.4 that in addition to the original ?peaks?  
-Wo 0 Wo
0
Frequency
Magnitude
 
Time
Magnitude
 
 47
 
Figure 7.4  Frequency Spectrum of Sinusoid with Multiple Exponential Decay Terms 
  
at 
0
??  (which represent a high amount of frequency content at 
0
?? ), there are now 
peaks at 
0
2??  which are indicative of the second exponential decay term being 
introduced in (7.3).  The FFT is a very effective tool for distinguishing between the two 
dominant frequencies in (7.3). 
 Complications arise, however, as the frequencies of the exponentially decaying 
terms in (7.3) approach the same value.  This scenario is depicted in Figure 7.5, where the 
frequency of the original sinusoidal term (
1
? ) is kept at
0
? , and the second term?s 
sinusoidal frequency (
2
? ) begins at 
0
2?  and is systematically reduced so it approaches 
0
? .  Observation of Figure 7.5 reveals that as the frequencies of the two terms get closer 
and closer together (i.e. as
2
?  approaches 
0
? ), the spectral plots produced from the FFT 
become more and more cluttered.  There appears to be one dominant frequency at 
0
? in 
the final plot, but in actuality the second decay term is at 1.1
0
? .  This is very hard to 
distinguish in the plot, and is a significant problem when attempting to classify the  
-2Wo
 
-Wo
 
0 Wo
 
2Wo
Frequency
Magnitude
 48
 
 
Figure 7.5  Frequency Spectra with Muliple Exponential Decay Terms 
 
machine rotor angles of the 179-bus system by frequency and phase using the FFT.  
Therefore, when multiple exponential decays are introduced in a signal, the potential is 
present for the continuous nature of the exponentials to greatly distort the frequency 
spectrum as calculated by FFT to the point where the actual dominant frequency or 
frequencies are masked.  This is particularly the case when the frequencies of the 
multiple exponentially decaying terms are very close in value. 
 Complications can also arise if the exponentials change in (7.3), because this 
affects the rate of decay in frequency.  This discussion will remained focused on the 
-2w
o
-w
o 0
w
o
2w
o
 
 
 
 
 
 
 
Frequency
Magnitu
de
-2w
o
- w
o 0
w
o
2w
o
 
 
 
 
 
 
 
Frequency
Magnitu
de
-2w
o
-w
o 0
w
o
2w
o
 
 
 
 
 
 
 
Frequency
Magnitu
de
-2w
o
-w
o 0
w
o
2w
o
 
 
 
 
 
 
 
Frequency
Magnitu
de
 49
frequencies of the sinusoids, but it should be noted that the multiple exponentials in (7.3) 
have the potential to present the same problems as the multiple sinusoids. 
 The time domain response of the rotor angle of generator 6 in the 179-bus test 
system is plotted in Figure 7.6.  The FFT of the same generator rotor angle is shown in 
Figure 7.7.  Notice that, unlike the previous spectrum plots, the continuous nature of the 
exponential frequency spectrum is not depicted in Figure 7.6.  This is the result of 
window function that is applied to the data before the FFT is calculated.  The window 
function used in the analysis is the rectangular window, which has the frequency 
spectrum of the well-known ?sinc? function.  The spectral content of the ?sinc? function 
is what is disrupting the continuous nature of the exponential frequency spectrum.  Zero-
padding was also implemented when taking the FFT of the machine rotor angle plots.  
Zero-padding places an equal number of zeros between all terms calculated by the FFT, 
thus increasing the resolution of the FFT. 
 
Figure 7.6  Time Domain Simulation of Generator 6 
 
 50
 
Figure 7.7  FFT Results for Generator 6 
 
 Observation of the rotor angle plot in Figure 7.6 reveals that the dominant 
frequency of the signal should be about 0.5 Hz.  However, no dominant frequency 
appears to be present in Figure 7.7.  This is the case even though some content is clearly 
present between 0 and 0.5 Hz as evidenced by the distortion of the sinc function.  Similar 
results were obtained when looking at the spectral content of other machines.  In many 
cases, the only significant spectral content was observed in the dc frequency component, 
as seen in Figure 7.7.  As a result, grouping the machine rotor angles via results obtained 
by FFT methods does not appear to be generally useful. 
7.2 Prony Analysis       
 Prony analysis extends Fourier analysis by directly estimating the frequency, 
damping, magnitude, and relative phase of the modal components present in a given 
signal [5].  This estimation is possible due to the fact that any periodic signal can be 
represented as a sum of exponentials.  Prony analysis uses this fact to curve fit an 
observed signal y(t) to the expression in (7.4). 
 51
()( )
?
=
?+??=
n
1i
iiii
tf2costexpA)t(y?                                    (7.4) 
The terms in (7.4) can be rewritten as shown in (7.5), where ?
i
 represents the modes of 
the estimated signal (t)y? .   
()
?
=
?=
n
1i
ii
texpB)t(y?                                               (7.5) 
 The signal y(t) is sampled N times and in this derivation N is chosen to be twice 
the number of modes, or N=2n.  These samples are evenly spaced by an amount t? .  At 
the sample times t
k
 (where k=0, 1, ? , N-1) the exponential in (7.5) is rewritten as shown 
in (7.6), which produces (7.7).  The terms in (7.7) are expanded as shown in (7.8).   
)tkexp(z
i
k
i
??=                                                     (7.6) 
?
=
=
P
1i
k
ii
zB)k(y?                                                      (7.7) 
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
)1N(y
)1(y
)0(y
B
B
B
zzz
zzz
zzz
1n
2
1
1N
n
1N
2
1N
1
1
n
1
2
1
1
0
n
0
2
0
1
MM
L
MOMM
L
L
                          (7.8) 
The vector B
~
is the collection of complex magnitudes that partly comprise (7.5).   The 
rest of (7.5) is comprised of the values in the matrix [Z] and are the complex modal 
frequencies for the signal.  Notice that because )texp(
i
?? is raised to some power, as 
indicated by the term z
i
k
, the exponents add. So, z
i
k
 is equivalent to )tkexp(
i
?? and 
)tk( ? is equivalent to time progressing as k increases.  Also notice that [Z] is not the 
impedance matrix discussed earlier.  It is simply a matrix of coefficients used to represent 
)tkexp(
i
?? .  If the matrix [Z] and vector B
~
 can be solved for then a )k(y? can be found 
 52
that is equal to y(k) (assuming the signal is noise-free).  To solve for [Z] and B
~
, the roots 
of the nth-order polynomial in (7.9) are calculated, where the terms a
i
 are unknown 
coefficients.   
0)zazaza(z
0
n
2n
2
1n
1
n
=+++?
??
L                                  (7.9) 
 The nth-order polynomial in (7.9) can be written because of the assumption that 
y(t)  is the output of some linear nth-order system.  This means that there are n 
eigenvalues that can be found from the roots of the characteristic equation of the system.  
Observation of (7.6) reveals that the z
i
 terms in (7.8) are associated with the system 
eigenvalues due to z
i
 simply being shortcut notation for )texp(
i
?? .  Because of this, each 
z
i
 in (7.8) is a root of the nth-order polynomial in (7.9).  Finding the unknown 
coefficients a
i
 in (7.9) enables the roots of the polynomial and the modes of the response 
signal to be calculated. 
 A (1xN) vector of the unknown a
i
 coefficients in (7.9) is formed in (7.10) and 
multiplied to both sides of (7.8).  The right side of the resulting equation is seen in (7.11) 
and the left side is (7.12), which is expanded to the form shown in (7.13).    
[]001aaaA
~
11nn
LL ???=
?
                              (7.10) 
( ) ( ) ( )[ ]0ya1nyanyY
~
A
~
n1
++??= L                                      (7.11) 
B
~
]Z[A
~
Y
~
A
~
=                                                            (7.12) 
1
BY
~
A
~
= [
n
1
z-( )
0
1n
2n
12
1n
11
zazaza +++
??
L ] 
      + B
2
[
n
2
z -( )
0
2n
2n
22
1n
21
zazaza +++
??
L ]+...= 0   (7.13) 
 53
The summation of terms in (7.13) that is multiplied by B
1
 is equal to zero because z
1 
in 
(7.13) is a root of the characteristic equation in (7.9).  Additionally, summation of terms 
in (7.13) that is multiplied by B
2
 is equal to zero because z
2 
in (7.13) is a root of the 
characteristic equation in (7.9).  The equation in (7.14) can be formed from (7.11) due to 
the fact that 0Y
~
A
~
= as shown in (7.13). 
0)n(y)1n(ya)1(ya)0(ya
11nn
=+?????
?
L                           (7.14) 
 Observation of (7.12) reveals that the entries in A
~
can be arbitrarily placed 
because A
~
 is multiplied by both sides of the equation in (7.12).  Because y(0)?y(n) are 
known, (7.14) is one equation in n unknowns.  Additional equations are necessary to 
determine the coefficients a
i
.  One approach is to shift the non-zero entries in (7.12) right 
by one column (and subsequently inserting a zero into the first column) in A
~
 effectively 
samples the data one increment later.  This shift results in the new (1xN) vector shown in 
(7.15).  The vector in (7.15) is applied to (7.8) to get (7.16).  As before, the right side of 
the resulting equation is seen in (7.16) and the left side is (7.17), which is expanded to the 
form shown in (7.18).  Note that the terms in (7.18) are still equal to zero because all 
terms in (7.18) are simply the terms in (7.13) multiplied by a factor equal to z
i
1
. 
[]001aaa0A
~
11nn
LL ???=
?
                              (7.15) 
( ) ( ) ( )[ ]1yanya1nyY
~
A
~
n1
++?+= L                                      (7.16) 
B
~
]Z[A
~
Y
~
A
~
=                                                            (7.17) 
1
BY
~
A
?
= [
1n
1
z
+
-( )
1
1n
1n
12
n
11
zazaza +++
?
L ] 
+B
2
[
1n
2
z
+
-( )
1
2n
1n
22
n
21
zazaza +++
?
L ]+...= 0        (7.18) 
 54
As seen before in the formulation of (7.14), (7.19) can be formed due to the fact that 
0Y
~
A
~
= . 
0)1n(y)n(ya)2(ya)1(ya
11nn
=++????
?
L                           (7.19) 
The two equations (7.14) and (7.19) have been derived to help determine the n unknowns 
a
i
.  The steps in (7.10)-(7.19) can be applied repeatedly to form all equations that 
comprise (7.20), which can be solved for the coefficients a
i
 that were unknown in (7.9).    
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
????
+
?
??
)1N(y
)2n(y
)1n(y
)n(y
a
a
a
a
)1nN(y)3N(y)2N(y
)2(y)n(y)1n(y
)1(y)1n(y)n(y
)0(y)2n(y)1n(y
n
3
2
1
MM
L
MOMM
L
L
L
                (7.20) 
 The solution of (7.20) enables the coefficients of the nth-order polynomial in (7.9) 
to be specified and rooted to find the z
i
 coefficients in the first row (k=0) of [Z].  The 
additional rows in [Z] can be calculated by simply taking each entry in the first row to the 
kth power (which corresponds to the kth row in [Z]).  Once [Z] is calculated, the vector 
B
~
 can be calculated and solved for in (7.8). 
  Knowing B
~
and [Z], the estimated curve fit )k(y? of the original signal y(t) can be 
obtained.  Once a curve fit is obtained, the frequency and phase of the dominant term of 
the exponential fit (the dominant mode) can be calculated.  As with the FFT method, the 
machines in the 179-bus system could then be grouped by their dominate phase and 
frequency using cluster analysis and compared to the groups observed using the graphical 
tool in Chapter 4. 
 The curve fit and original plot of two machines in the 179-bus system (generators 
6 and 11) are shown in Figure 7.8.  Observation of Figure 7.8 reveals that these two 
 55
generators have very similar phase and frequency characteristics.  Therefore, the 
dominant terms of the curve-fit of the two rotor angle plots should also have similar 
characteristics.  The dominant terms and their corresponding magnitudes, exponential 
decay, frequency, and phase are seen in the top half of Table 7.1.  As an example of the 
form of the resulting estimated equations, the calculated coefficients that correspond to 
the dominant eigenvalue of generator 6 in Table 7.1 are substituted into (7.5) and the 
result is shown in (7.15).  Note that only one term is shown in (7.15).  The actual curve fit 
is comprised of multiple terms of the form of (7.15) that sum to produce the curve fit seen 
in Figure 7.8.  The positive exponential decay displayed in Table 7.1 and seen again in 
(7.15) is a concern and is discussed later in this section. 
( ) ( )??????= 86.109t42374.2cost20661.exp2037.4)t(y?              (7.15) 
 Observation of Table 7.1 reveals that the dominant exponential terms of the 
curve-fit calculated by Prony analysis share similar magnitudes, exponential decay, 
frequency and phase.  This is the expected result due to the fact that visual inspection of 
the rotor angle plots concludes that they are indeed similar waveforms because of their in 
phase behavior.  Therefore, Prony analysis has the capability of accurately curve fitting 
the machine rotor angle plots. 
 Unfortunately, similar results were not obtained when all twenty-nine rotor angle 
plots from the 179-bus system were curve fit with Prony analysis.  The curve fit and 
original plot of two different machines (generators 4 and 18) are shown in Figure 7.9.  As 
in Figure 7.8, these two machines again have very similar phase and frequency 
characteristics.  However, observation reveals that the curve fit is not as accurate as it 
was for the first set of generators - especially after 2.5 seconds into the simulation.  Still,  
 56
 
Table 7.1  Prony Analysis Results for Various Generators 
Dominant Curve Characteristics
Generator Magnitude Exp Decay Frequency (Hz) Phase (Deg)
6 4.2037 0.20661 0.42374 -109.85797
11 4.43115 0.21257 0.42325 -110.65684
4 17.48716 -0.29689 0.19715 106.77783
18 39.34433 -0.6588 0.26692 63.14528
 
  
 
Figure 7.8  Prony Analysis Results for Generators 6 and 11 
 
 
Figure 7.9  Prony Analysis Results for Generators 4 and 18 
 
 57
even with this inaccuracy, the curve fits appear to be quite similar to each other and 
should exhibit similar characteristics.  
 The characteristics of the Prony analysis curve fit of the machine rotor angles are 
seen in the bottom half of Table 7.1.  The results are not as favorable as they were for 
generators 6 and 11.  Even though the curve fits for the two rotor angles appear to be 
quite similar in Figure 7.9, the results seen in the dominant terms calculated using Prony 
analysis do not exhibit similar characteristics.  For example, the two waveforms appear to 
be in phase with one another in Figure 7.9, but the dominant terms of the curve fit are 
more than 40 degrees out of phase with one another.  This large disparity in the phase of 
the dominant terms leads to these two generators being placed into separate groups when 
a k-type clustering algorithm is used to sort the generators into groups according to the 
information gathered from the Prony analysis curve fit.  This should not be the case 
according to their actual time-domain response seen in Figure 7.9.  Also, the positive 
exponential decay calculated and shown for generators 6 and 11 in the top half of Table 
7.1 is another indication that somehow the Prony analysis isn?t working in this specific 
case.   
 The inaccuracies of the Prony analysis can possibly be attributed to several 
different issues.  One issue is the precision of the data used to plot the machine rotor 
angles.  The simulation tool used to generate the plots, PSSPLT, only outputs the data to 
10
-4
 precision.  This lack of precision has the potential to compound when using a large 
number of modal terms to estimate a particular curve.    
 Also, the fact that Prony analysis is associated with eigenvalues means adopting a 
linear assumption set for the system response.  Power system dynamics are not linear for 
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large disturbances [2], so the responses seen in rotor angle plots don?t have any guarantee 
of fitting the linear eigen-based analysis assumptions.  The disturbances used when 
analyzing the WSCC 179-bus system were balanced three-phase faults left on the system 
for a relatively lengthy amount of time, therefore attempting to apply Prony analysis 
when fitting the rotor angle plots for the large disturbances could be another reason for 
the inaccuracies seen in Figure 7.9.   
 While many characteristics of many generators in the 179-bus system are 
accurately estimated using Prony analysis, there are several others that exhibit problems 
similar to the ones seen by generators 4 and 18.  These inaccuracies result in clusters that 
do not accurately depict the groups of generators seen in Figures 7.8 and 7.9.  Until an 
accurate method of numerically describing the plots in Figures 7.8 and 7.9 is obtained, 
visual inspection appears to be the best way to separate the machine rotor angle plots into 
their respective groups.