EEAC Wyoming

Power System Transient
Stability Analysis Software
Tool for an Undergraduate
Curriculum
S. MUKNAHALLIPATNA, STANISLAW LEGOWSKI, SADRUL ULA, JASON KOPAS
Department of Electrical Engineering, University of Wyoming, Laramie, Wyoming 82071-3295
Received 12 January 2000; accepted 2 January 2001
ABSTRACT: This paper describes, a technique of the transient stability assessment,
known as the Extended Equal Area Criterion (EEAC) as applied to a multimachine power
system. The EEAC is implemented using the MATLAB program to provide a software tool with
GUI. The software tool of the EEAC is developed to incorporate the transient stability analysis,
into an undergraduate curriculum in power engineering. Another criterion for developing this
software is to reduce the effort normally associated with the instruction of the complex
theory of the transient stability and the use of the software in an undergraduate classroom
environment. The results obtained from the MATLAB software on two multimachine power
systems are compared with the traditional time domain analysis technique for veri®cation
purposes. ß2001 John Wiley & Sons, Inc. Comput Appl Eng Educ 9: 37±48, 2001
Keywords: Transient; stability; machines; education; software
INTRODUCTION
In 1986, the IEEE Spectrum published an article
examining the power engineering education in the
United States [1]. The article reported serious pro-
blems af¯icting power engineering education at the
Universities due to a number of factors. The major
factors cited in this article were shrinking funds for
education and basic research, a slowdown in elec-
tricity demand, and decreasing student interest. In the
Department of Electrical Engineering at University of
Wyoming, one of the major research areas is power
engineering with substantial research funds and
publications. In order to continue the research in
this area, the department had to address the issue of
waning interest of students in the area of power
engineering. A proposal was submitted to the National
Science Foundation to restructure the power engi-
neering curriculum that was funded. One of the
areas addressed in this proposal was the teaching of
transient stability assessment (TSA) to the under-
graduate students and development of software tools.
Power system stability can be classi®ed into two
major types namely: steady-state stability and trans-
ient stability based on the nature of the disturbance,
i.e., large or small.
Correspondence to S. Muknahallipatna.
ß2001 John Wiley & Sons, Inc.
37
Steady-State Stability
For a particular operating condition, the system is said
to be steady-state stable if, following a small dis-
turbance, it reaches a steady-state operating condition
identical or close to the predisturbance condition. A
power system has an upper limit on the load level,
which is termed as the steady-state stability limit. The
steady-state stability limit is a steady state operating
condition for which the system is stable, but a small
change in the operating condition in an unfavorable
direction will cause the power system to lose stability.
Transient Stability
Power system transient stability is a term applied to
alternating current electric power systems, denoting a
condition in which the various synchronous machines
of the system remain in synchronism, or in step with
each other [2]. Conversely, instability denotes a
condition involving loss of synchronism, or falling
out of step. Under normal operating conditions, a
power system will have all the synchronous machines
running at synchronous speed. In the event of a large
disturbance or fault, the machines start swinging
with respect to each other. The period between
the initiation of a large disturbance and restoration
of the synchronous operation or the damping out of
the swings is called the transient period. The motion
or the swings of the machines are governed by
nonlinear differential equations. For the power system
to be transiently stable after a large disturbance,
the disturbance has to be removed or isolated in a
short period of time. The time duration between the
instant of disturbance initiation and the instant of
disturbance removal is termed as the fault clearing
time. This fault clearing time primarily constitutes
the time taken by the relays and the circuit breakers
to operate after a disturbance has been detected.
The upper bound on the fault clearing time is termed
as the critical clearing time (CCT). The CCT is
the maximum time between the disturbance initiation
and its fault clearing time such that the power system
is transiently stable. The process of determining
whether the power system will reach a stable state
following a disturbance is known as the transient
stability assessment. At present, in power engineering
curriculums across the country, the TSA is discussed
using two techniques namely: time domain simulation
and Equal area criterion (EAC). Both these techniques
have various advantages and disadvantages with
respect to use in a classroom. In the next section,
some of the advantages and disadvantages are
discussed.
DISCUSSION OF EXISTING TECHNIQUES
The two techniques mentioned in the previous section
are commonly found in textbooks dealing with power
systems. The implementation details of both the tech-
niques are not presented in this paper, since it is
discussed in great detail, in a number of textbooks and
in journal articles [2±5]. The advantages and draw-
backs of the two methods are presented.
Time Domain Simulation Technique
In this method, for a multimachine power system,
nonlinear differential equations are developed for
each generator present in the power system. During
a disturbance, these nonlinear differential equations
are solved using a suitable numerical integration
technique to generate swing curves. The swing
curves represent the variation of the rotor angles with
respect to time for each generator. These swing
curves of each generator is analyzed to determine
the stability of a power system. If, the difference
between the rotor angles of any two generators during
the transient period does not exceed 180
·
, the power
system is said to be stable else unstable. The most
popular software tool available to perform the TSA
using the time domain simulation technique is the
Extended Transient Mid-term Stability Program
(ETMSP) [6].
Advantages. The time domain simulation technique
has a number of advantages, which are listed below:
1. It can incorporate any level of system modeling.
2. It is the most accurate method.
3. It is the industry standard technique.
Drawbacks. There are a number of drawbacks of the
time domain simulation technique especially for
classroom use. The drawbacks are:
1. For a simple power system model, the com-
putations involved are unwieldy and time
consuming.
2. The computation of the CCT is an iterative
process, rather than a closed form solution.
3. If used in the undergraduate curriculum, the
focus tends to be on understanding the dyna-
mics of the generators rather than analyzing the
transient behavior of the power system.
4. Due to lack of mathematical background,
the students invariably cannot solve the non-
linear differential equations to obtain the swing
curves.
38 MUKNAHALLIPATNA ET AL.
5. Most of the software tools available are not user
friendly. Majority of the software tools avail-
able are on UNIX platform only.
Equal Area Criterion
In this method power angle curves representing the
change in electric power output with respect to the
rotor angles of a single generator connected to an
in®nite bus, known as one machine in®nite bus system
(OMIB) is constructed graphically [7]. A typical
OMIB system and the power angle curves for a fault at
the center of the transmission lines are shown in
Figures 1 and 2. In Figure 2, the horizontal line
represents the input mechanical power P
m
that
remains constant for the entire period of the fault,
due to the high inertia of the prime movers. The three
curves represent the prefault, during-fault, and post-
fault electric output power of the generator. Basic idea
of the EAC technique is to compare the two areas
shown in Figure 2.
*
Area of acceleration (A
acc
): The electric output
power is less than the input power during a fault
causing the generator to accelerate. The difference
between the input mechanical power and the elec-
trical output power during a fault within a certain
rotor angle interval is represented as the A
acc
.
*
Area of deceleration (A
dec
): The electric output
power is greater than the input power immedi-
ately following a fault removal. The difference
between the input mechanical power and the
electrical output power after a fault is removed
within a certain rotor angle interval is repre-
sented as the A
dec
.
Using these two areas, a stability margin and the
critical clearing angle is described as follows:
*
The stability margin indicating, how far the
system is from becoming unstable can be
expressed as:
= A
dec
÷A
acc
where,
Z > 0 indicates the system is stable:
Z < 0 indicates the system is unstable:
Figure 1 One machine in®nite bus system.
Figure 2 Power angle curves for the OMIB system.
TRANSIENT STABILITY ASSESSMENT 39
*
The critical clearing angle, d
cc
can be expressed
as:
Z = 0 <=> A
dec
= A
acc
Advantages. The EAC technique has a number of
advantages, which are listed below:
1. The EAC provides a metric representing the
power system transient stability. The metric can
be either the critical clearing time or the energy
margin.
2. The computations are simple and an iterative
process is not required to compute the CCT.
3. The sensitivity of the power system transient
stability to changes in the system can be
measured using the energy margin.
Drawbacks. There are a number of drawbacks of the
EAC especially for classroom use. The drawbacks are:
1. The EAC method can work only with a OMIB
system.
2. In classroom use, this technique cannot be used
to demonstrate the effect of different generators
on the power system transient stability.
3. There are no commercially software tools
available for a classroom use.
Based on the disadvantages of the two methods, it
was decided to explore other possible TSA techni-
ques. Two techniques belonging to the class of direct
methods for TSA were explored. The techniques are
transient energy function (TEF) method and the
extended equal area criterion (EEAC). These two
techniques were used to perform TSA on two large
power systems. The results obtained by both the
techniques were compared with the results obtained
by the time domain simulation technique [8]. Based
on the comparison, the EEAC method was found to be
most promising.
EXTENDED EQUAL AREA CRITERION
Y. Xue, Th. Van Cutsem, and M. Ribbens-Pavella
introduced the EEAC method of TSA in 1988 [9]. It is
of®cially classi®ed as an offshoot of the EAC method,
as it extends the EAC method to the case of a mul-
timachine power system. Various researchers [10,11]
have developed similar versions of the EEAC method,
but almost every additional version is based on the
same idea. The original version introduced by Xue,
Van Cutsem, and Pavella is considered here.
Between 1988 and 1993, Y. Xue, Th. Van Cutsem,
and M. Ribbens-Pavella [9, 12±15] have developed
their version of the EEAC method. In this method, the
generators of the multimachine power system are
considered to be a member of one of the two clusters
of machines, namely the critical and the noncritical
group. An equivalent generator replaces each of these
groups of machines. Lastly, the two-machine equiva-
lent power system is further reduced to an OMIB
power system. A slightly modi®ed version of the EAC
is then applied to the equivalent OMIB system to
generate the CCT and energy margin for the given
critical cluster of machines. In 1993, they developed
a method to assist in identifying the critical cluster
of generators, called the critical machine ranking
(CMR) method. In this method, the generators of
the power system are ranked based on a metric of
their criticality with respect to the fault on the
power system. The CMR method has been demon-
strated, to be successful on various power systems
[15]. The power systems chosen as examples in their
identi®cation of the actual critical cluster of machines
happen to display an obvious separation between
critical and noncritical machines. The line separating
critical and noncritical generators, even with a sorted
list of generators obtained by the CMR method, is
still an exercise in heuristics for a practical power
system.
The hit-and-miss nature of detecting the critical
cluster of generators makes this method a good
candidate for the use in a classroom. The under-
graduate students can experiment with the power
system transient stability behavior by selecting dif-
ferent generators for the critical cluster. The resulting
analysis can be computed quickly and the student
can view the data regarding the TSA almost im-
mediately. Thus, this is the new method selected for
teaching TSA in the undergraduate power engineering
curriculums.
EEAC Implementation Details
A multimachine system modeled in the classical mode
is considered for developing the OMIB system. The
motion of the ith machine of an n-machine system,
reduced to the internal nodes of the generators is
described by:
M
i
d
2
d
i
d t
2

= P
m
i
÷P
e
i
\ i = 1; 2; . . . ; n (1)
d d
i
d t
= !
i
\ i = 1; 2; . . . ; n; (2)
40 MUKNAHALLIPATNA ET AL.
where
P
e
i
= E
i
X
K
j=1
E
j
[G
ij
×cos(d
i
÷d
j
)÷B
ij
×sin(d
i
÷d
j
)[;
P
m
i
= Mechanical input power of the ith machine;
M
i
= Moment of inertia of the ith machine;
E
i
= Internal EMF of the ith machine:
Even after the system has undergone a fault, the
parameters P
m
i
; E
i
, E
i
, and M
i
are assumed to stay
constant for the entire transient period.
Equivalent Two-Machine System Formulation. The
notations used in representing the two-machine
system are:
*
S, the set of machines composing the critical
cluster.
*
s, its equivalent, aggregated machine.
*
A, the set of all remaining machines.
*
a, its equivalent, aggregated machine.
Using the above notations the following inertia
equivalents are de®ned:
M
s
=
X
k÷S
M
k
; M
s
is the moment of inertia of the
equivalent aggregated machine of the critical
cluster; S: (4)
M
a
=
X
j÷A
M
j
; M
a
is the moment of inertia of the
equivalent aggregated machine of the
remaining machines; A: (5)
M
T
=
X
i=1
M
i
; M
T
is the sum of the moment of
inertias of the machines (6)
M =
M
a
M
s
M
T
; M is the moment of inertia of the
OMIB: (7)
The generator rotor angles that are commonly
referred with respect to the synchronous frame is
transferred to the partial center of angle (PCOA) as
follows:
d
s
= M
÷1
s
X
k÷S
M
k
d
k
d
s
is the PCOA of the critical
group: (8)
d
a
= M
÷1
a
X
j÷A
M
a
d
j
; d
a
is the PCOA of the
remaining machines: (9)
In this new PCOA, the deviations of the individual
rotor angles from their respective PCOA is computed
as follows:
x
k
÷ d
k
÷d
s
; k ÷ S: (10)
&
j
÷ d
j
÷d
a
; j ÷ A: (11)
Using the expressions of the PCOA for the two groups
in (8) and (9), the motion of the two PCOA can be
described as follows:
M
s
d
2
d
s
d t
2
=
X
k÷S
(P
mk
÷P
ek
) (12)
M
a
d
2
d
a
d t
2
=
X
j÷A
P
mj
÷P
ej
À Á
(13)
where, P
mk
is the mechanical input power of the
generators \k ÷ S.
P
mj
is the mechanical input power of the generators
\j ÷ A.
P
ek
= E
2
k
Y
kk
cos
k
÷E
k
X
l÷S;l,=k
E
l
Y
kl
cos
×(x
k
÷&
l
÷
kl
) ÷E
k
X
j÷A
E
j
Y
kj
×cos(d
s
÷d
a
÷x
k
÷&
j
÷
kj
) \k ÷ S: (14)
A similar expression can be obtained for the electrical
power output of the remaining generators. The
equivalent machines of the critical cluster and the
remaining group are represented by (12) and (13),
respectively.
OMIB Equivalent. Using the two equivalent
machine expressions computed in the previous section
an OMIB equivalent is formed. Some of the important
parameters pertaining to the OMIB equivalent are
described as follows:
d ÷ (d
s
÷d
a
); d is defined as the rotor
angle of the OMIB: (15)
P
m
= M
÷1
T
M
a
X
k÷S
P
mk
÷M
s
X
j÷A
P
mj
!
; (16)
P
m
is de®ned as the mechanical input power of the
OMIB.
TRANSIENT STABILITY ASSESSMENT 41
P
e
= M
÷1
T
M
a
X
k÷S
P
ek
÷M
s
X
j÷A
P
ej
!
(17)
= P
e
÷P
max
sin(d ÷n); is the electrical output
power of the OMIB:
Using the above important parameters the expres-
sion for the OMIB can be written as:
M

d = P
m
÷[P
c
÷P
max
sin(d ÷n)[ (18)
where
P
c
=
M
a
M
T
X
k÷S
X
l÷S
g
kl
cos(x
k
÷x
l
)
÷
M
s
M
T
X
j÷A
X
l÷A
g
jl
cos(&
j
÷&
l
) (19)
P
max
=

(C
2
÷D
2
)
q
(20)
n = ÷tan
÷1
C
D

(21)
C =
X
k÷S
X
j÷A
b
kj
sin x
k
÷&
j
À Á
÷
(M
a
÷M
s
)
M
T
X
k÷S
X
j÷A
g
kj
cos x
k
÷&
j
À Á
(22)
D =
X
k÷S
X
j÷A
b
kj
cos x
k
÷&
j
À Á
÷
(M
a
÷M
s
)
M
T
×
X
k÷S
X
j÷A
g
kj
sin x
k
÷&
j
À Á
(23)
g
ij
= E
i
E
j
G
ij
; b
ij
= E
i
E
j
B
ij
All these above parameters of the OMIB vary with
the reduced bus admittance matrix. The reduced bus
admittance matrix will change according to the pre-
fault, during-fault, and post-fault con®gurations of the
system.
Application of the EAC to the OMIB. The EAC
described in Equal Area Criterion is applied to
describe the dynamics of the OMIB expressed by
(18). Figure 3, shows the (P-d) curves in the prefault
(P
eO
), during-fault (P
eD
), and postfault (P
eP
) con®g-
urations.
The steady state or prefault condition of the power
system is characterized by the rotor angle, d
O
located
at the intersection of the P
m
represented by the
horizontal line and the P
eO
curve. The post-fault
stable (d
p
) and the unstable equilibrium (d
u
) points are
determined at the intersection of P
m
and the P
eP
curve.
In Figure 3, the area between the P
m
line and the P
eD
curve for the interval (d
O
, d
e
) represents the accel-
erating area (A
acc
). Similarly, the area between P
m
and
the P
eP
curve for the interval (d
e
, d
u
) represent the
decelerating area (A
dec
). The angle d
e
represents the
rotor angle of the OMIB at any fault clearing time T
e
.
The algebraic expressions for the two areas obtained
in a closed form is given below:
A
acc
d
c
( ) =
Z
d
e
d
O
P
m
÷ P
cD
÷P
maxD
sin (d ÷n
D
) ( ) [ [dd
= (P
m
÷P
cD
(d
e
÷d
O
) ÷P
maxD
[cos(d
e
÷n
D
)
÷cos(d
O
÷n
D
)[ (24)
A
dec
d
e
( ) =
Z
d
u
d
e
P
cP
÷P
maxP
sin (d ÷n
P
) ( ) ÷P
m
[ [dd
= (P
cP
÷P
m
)(p ÷d
e
÷d
P
÷2n
P
) ÷P
maxP
[cos(d
e
÷n
P
) ÷cos(d
P
÷n
P
)[ (25)
where, P
cD
, P
cP
are the modi®cations of (19) to
represent the during-fault and the post-fault condi-
tions of the power system respectively.
P
maxD
, P
maxP
are the modi®cations of (20) to re-
present the during-fault and the post-fault conditions
of the power system respectively.
n
D
, n
P
are the modi®cations of (21) to represent the
during-fault and the post-fault conditions of the power
system respectively.
d
u
= p ÷d
P
÷2n
P
, is the unstable equilibrium
point of the power system for the particular fault.
Computation of the Critical Clearing Angle and
Critical Clearing Time. The CCT of the multi-
machine power system is the normal metric for tran-
sient stability. If the CCTexists for ®rst swing stability
consideration, it is the absolute time for which the
disturbance on the power system can remain. If the
disturbance remains any longer, the power system
cannot recover and will become unstable.
Figure 3 Equal area criterion applied to the OMIB of the
multimachine power system.
42 MUKNAHALLIPATNA ET AL.
A Taylor series expansion for the rotor angle
equation of the OMIB equivalent power system is
shown in the equation below.
d t ( ) = d
0
÷
_
dt ÷

d
t
2
2
÷
_
d
t
3
6
÷

d
t
4
24
÷. (26)
The Taylor series is then truncated after the fourth
order term, and a correction factor, a, is introduced
to the resultant truncated series [13] representing the
OMIB equivalent rotor angle. The CCT is then com-
puted using the critical clearing angle. The areas of
acceleration and deceleration are plotted in a scale
of 0±180
·
, to represent ®rst swing stability. A
numerical method is used to determine the angle at
which A
acc
and A
dec
intersect. The angle of intersec-
tion represents the critical clearing angle. If A
acc
and
A
dec
do not intersect, the stability problem is not of the
®rst swing type.
SOFTWARE SIMULATION IN MATLAB
A Graphical User Interface (GUI) MATLAB v5.3
program was constructed to be a graphical, user-
friendly method of incorporating transient stability
analysis of practical, multimachine power systems
into an undergraduate electrical engineering curricu-
lum in power systems. The GUI MATLAB v5.3
simulation of the EEAC method of transient stability
analysis will be subsequently referred to as the
MATLAB simulation. It should be mentioned that
the MATLAB simulation at this time can simulate
only three-phase to ground faults at only busses on the
multimachine power system. There are two reasons
for this limitation. They are:
*
Reduction of the admittance bus matrix to the
generator nodes is complicated, if the fault is on a
transmission line.
*
The students are exposed only to the three phase
to ground fault and this fault is the most severe of
all other types of faults.
The MATLAB simulation requires the student to
provide proper input data for simulation; namely a
text ®le containing the solved load ¯ow data in the
common data format [16] and another text ®le, unique
to this MATLAB simulation, containing the machine
data. A typical machine data ®le is provided in the
Appendix. The text ®le containing the solved load
¯ow data is expected to have a ®le extension DAT, and
this ®le will be hereby referred to as the load ¯ow data
®le. The text ®le containing the generator data is
expected to have a ®le extension MCH, and this ®le
will be hereby referred to as the machine data ®le.
Both data ®les are expected to have identical ®le-
names, for example fouad.DAT and fouad.MCH
would describe the 9-bus power system.
As the MATLAB simulation is initially executed,
the dialog box shown in Figure 4 will appear. The
student is then expected to enter the ®lename shared
by the load ¯ow data ®le and the machine data ®le into
the textbox provided, fouad in the above-mentioned
example. When a valid ®lename is entered, the bus at
which the fault will occur can then be entered into the
remaining textbox. If a valid ®lename and fault bus are
entered, the MATLAB simulation will parse the load
¯ow data and machine data and reveal a checkbox
along the left hand side of the dialog box for each
generator of the power system, as shown in Figure 5,
numbering the generators according to their respec-
tive bus numbers within the load ¯ow data ®le. The
two buttons will also be enabled as necessary. The
student may change the ®lename or Faulted Bus at any
point in operation of the MATLAB simulation, and
validity checks on the user-entered data will be per-
formed upon each change in the textboxes.
The student now has several options in using the
MATLAB simulation, excluding simply changing the
®lename or Faulted Bus. The student may perform:
*
Manual selection of the critical cluster of
generators using the checkboxes shown to the
left hand side of Figure 5;
*
Exhaustive check of every possible combination
of generators to ®nd the actual critical cluster of
generators using the button provided; or
*
Load the data saved from a previous exhaustive
check using the button provided.
Assuming the student elects to manually select the
critical machines and data for a previous exhaustive
check that has not been loaded, the MATLAB
simulation will compute stability of the power system
using the EEAC method for each valid change in the
critical cluster. If the student manually selects the
critical group and data for a previous exhaustive check
that has been loaded, the MATLAB simulation uses
the data from the previous exhaustive check to display
the data, rather than computing the results again.
Assuming the student wishes to perform an
exhaustive check of the possible critical groups of
generators to ®nd the actual critical cluster, the
MATLAB simulation will compute the stability of
the power system using the EEAC method for every
TRANSIENT STABILITY ASSESSMENT 43
Figure 4 The MATLAB Simulation before user input.
Figure 5 The MATLAB Simulation after the ®lename and faulted Bus Inputs.
44 MUKNAHALLIPATNA ET AL.
possible critical cluster of generators to ®nd the actual
critical cluster with the smallest CCT. The MATLAB
simulation will then bold the text beside the check-
boxes representing the actual critical cluster. Selecting
those critical generators using the checkboxes will
display the results of the stability analysis to the
student. The data for the exhaustive check is saved
into a text ®le and loaded into memory after the
exhaustive check completes, so that if the student
changes the critical group, the data from the exhaus-
tive check is used, rather than recomputing the results.
Assuming the data from a previous exhaustive check
for the critical group exists and the student wishes to
load that data, the MATLAB simulation loads the data
from the previous exhaustive check, bolds the text
beside the checkboxes representing critical cluster
with the smallest CCT. Selecting those checkboxes
will display the results of the previous stability
analysis to the student. The dialog box displayed in
Figure 6 depicts the MATLAB simulation after the
data generated by a previous exhaustive check,
FOUAD2.OUT, has been loaded. The EEAC stability
analysis computed by the MATLAB simulation dis-
plays a variety of information in the dialog box. The
numerical value of the CCT, which is the metric of the
EEAC stability analysis of this MATLAB simulation,
is displayed ®rst and foremost. The numerical values
of the critical clearing angle, the area of acceleration,
A
acc
, the area of deceleration, A
dec
, the prefault and
postfault OMIB rotor angles, d
0
and d
P
, the UEP, d
U
,
and the constant mechanical input power, P
m
, are all
displayed and plotted on the right-hand side of the
dialog box.
The OMIB equivalent computations are also shown
both numerically and graphically using (18), in the top
plot of Figure 6. Finally, A
acc
and A
dec
are plotted, in
the bottom plot of Figure 6, for a range of possible
clearing angles representing ®rst swing stability, 0±
180
·
, graphically showing the critical clearing angle
at the point of intersection.
EVALUATION
The developed software was used to perform TSA on
two different power systems. The ®rst power system is
presented in [3]. The system is referred as Fouad
power system. The machine data, as required by the
MATLAB simulation, is provided in the Appendix.
Using the MATLAB simulation, faults were created at
Figure 6 The MATLAB Simulation loads data from FOUAD2.OUT.
TRANSIENT STABILITY ASSESSMENT 45
all nine busses, and an exhaustive search for the
critical cluster of generators was done for each case.
Table 1 shows the data reported by the MATLAB
simulation.
In Table 1, it can be seen that generator-1 is always
a noncritical generator. The generator-1 being a non-
critical member always is due to the H constant. The
generator-1 has the highest per unit inertia constant
(H) compared to the other two generators. This high
value prevents the generator from responding imme-
diately to the faults. Next, the ``Dynamic Reduction
Program (DYNRED)'' from EPRI [17] that deter-
mines the islanding of a power system due to
disturbances was applied on the Fouad power system.
Faults were created at all the nine buses and the
islands were obtained. The DYNRED program for all
the faults indicated two islands containing generators
2 and 3 in one island and the generator-1 in the other
island. This breakup characteristics matches exactly
with that critical and noncritical group generated by
the MATLAB simulation for each fault. This demon-
strates the capability of the software tool to identify
critical and noncritical generators with suf®cient
accuracy.
To evaluate the accuracy of the CCT, the MATLAB
simulation was performed on the New England Test
System. The one-line diagram, the load ¯ow data, and
the machine data are provided in [8]. The test results
are tabulated in Table 2. In Table 2, it can be seen
that the CCT obtained from the MATLAB simulation
is close to that obtained from the time domain
simulation.
The EEAC method cannot achieve the accuracy of
the time domain simulation in regard to the CCT. The
accuracy of the CCT obtained from the EEAC is
suf®cient for demonstrating the TSA of a power
system to the students. In Table 2, the generator-39 is
always a non-critical generator. The generator-39 has
the largest per unit inertia constant (H). The critical
generator composition was again compared with
the results obtained from the DYNRED program.
The islands indicated from the DYNRED program
matched most of the critical groups generated from
the MATLAB simulation.
CONCLUSIONS
The transient stability analysis of a power system
is rigorous and complicated. In undergraduate
classes the students lack the necessary background
in mathematics and, to a certain extent, in system
stability analysis. Due to this, it is not possible to
use the time domain analysis of transient stability in
an undergraduate classroom. At the same time
the EAC method, for all the simplicity of the
method, cannot be used for a multimachine power
system.
In this paper, a new approach to introduce transient
stability analysis to undergraduate classes is provided.
The new approach involves using the EEAC method
along with a MATLAB package for simulation. The
EEAC method incorporates the simplicity of the EAC
method while solving the transient stability problem
of a multimachine power system. In addition to this,
the students have the opportunity to learn dynamic
reduction of the power system while using the
MATLAB simulation package. Dynamic reduction
of a power system coupled with the EEAC method of
power system stability involves eliminating nodes and
generators that are not going to be affected by a
disturbance. This reduces the complexity of a multi-
machine power system by transforming it to an
equivalent power system more suited to the under-
graduate electrical engineering curriculum. The
simulation package also demonstrates to the student
which generators are most affected by a fault on
the multimachine power system, and how to empha-
size those machines in the stability assessment of
the system. This will demonstrate how a power system
breaks into a number of islands, and which sec-
tions of the overall multimachine power system are
important.
The MATLAB simulation makes it relatively
easy to incorporate reliable transient stability studies
into the undergraduate electrical engineering curri-
culum. Studies can include a practical power system,
and the focus can be put upon the transient stability
analysis, rather than understanding and implementing
the theories involved in transient stability studies of a
multimachine power system.
Table 1 The Fouad Power System Results
Fault Critical Clearing Time Critical
Location (Bus #) (secs.) Generators
1 0.330 2
2 0.214 2,3
3 0.229 2,3
4 0.271 3
5 0.231 2,3
6 0.240 2,3
7 0.204 2,3
8 0.221 2,3
9 0.205 3
46 MUKNAHALLIPATNA ET AL.
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APPENDIX: FOUAD POWER SYSTEM DATA
Machine data:
NAME Bus# Sb H Ra x'd x'q xd xq T'd0 T'q0 xl RPM
GEN1 1 100.0 23.64 0.0000 0.0608 0.0969 0.1460 0.0969 8.960 0.000 0.0336 0180
GEN2 2 100.0 06.40 0.0000 0.1198 0.1969 0.8958 0.8645 6.000 0.535 0.0521 3600
GEN3 3 100.0 03.01 0.0000 0.1813 0.2500 1.3125 1.2578 5.890 0.600 0.0742 3600
Table 2 The New England Power System Results
Fault Location Critical Clearing Critical Generators
(Bus #) Time (s) (s) Time Domain CCT
1 0.4325 32, 33, 34, 35, 36, 0.4199
37, 38
2 0.2874 38 0.2780
3 0.2675 37, 38 0.3359
4 0.2623 31, 32 0.2850
5 0.2532 31, 32 0.2480
6 0.2532 31, 32 0.2390
7 0.2651 31, 32 0.2850
8 0.2639 31, 32 0.2520
9 0.7450 31, 32 0.6379
10 0.2640 31, 32 0.2500
11 0.2615 31, 32 0.2540
12 0.6680 31, 32 0.5349
13 0.2680 31, 32 0.2700
14 0.2664 31, 32 0.2950
15 0.2784 33, 34, 35, 36, 38 0.2866
16 0.2097 33, 34, 35, 36, 38 0.2110
17 0.2737 33, 34, 35, 36, 38 0.2390
18 0.2800 33, 34, 35, 36, 38 0.2940
19 0.1840 34 0.2020
20 0.2058 33, 34 0.1890
21 0.2963 33, 34, 35, 36, 38 0.2700
22 0.2552 33, 34, 35, 36, 38 0.2370
23 0.2622 33, 34, 35, 36, 38 0.2460
24 0.2834 33, 34, 35, 36, 38 0.2840
25 0.2564 38 0.2288
26 0.2165 38 0.1490
27 0.2341 38 0.1960
28 0.2062 38 0.1520
29 0.1996 38 0.1310
TRANSIENT STABILITY ASSESSMENT 47
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BIOGRAPHIES
Suresh Muknahallipatna received his BE degree in electrical
engineering and ME degree from the University of Bangalore,
Bangalore, India, in 1988 and 1991, respectively. He completed his
PhD degree at the University of Wyoming in 1995, with an emphasis
in neural networks. He is currently an assistant professor in the
Department of Electrical Engineering at the University of Wyoming.
Stainslaw F. Legowski received his BSc in electronics engineering
and his MSc and PhD from the Technical University of Gdansk,
Poland, in 1957, 1962, and 1971, respectively. He is currently a
professor in the Department of Electrical Engineering at the
University of Wyoming. His areas of expertise are analog circuits,
power electronics, and digital systems and microprocessors. He has
published more than 50 refereed papers in the area of analog
electronics and power electronics.
Sadrul Ula received his BSc in electrical engineering, University of
Rajshahi, Bangladesh, 1968, and MS in electrical engineering,
University of Engineering and Technology, Bangladesh, 1973. He
completed his PhD degree at the University of Leeds, England,
1978. He is currently a professor in the Department of Electrical
Engineering at the University of Wyoming. His areas of expertise
are electric motors, power electronics, and power systems. He has
published more than 10 refereed papers speci®cally in the ®eld of
control of electric motors.
Jason Kopas received his BSc and MS degree's in electrical
engineering from the University of Wyoming in 1998 and 2000,
respectively. His area of emphasis is software tool development for
power engineering education. He is currently employed at the Intel
Corporation.
48 MUKNAHALLIPATNA ET AL.