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.

REFERENCES

[1] K. Pyko and G. Zorpette, Can power engineering

education be reenergized? IEEE Spectrum, 26±31,

Dec. 1986.

[2] E. W. Kimbark, Power System Stability, Vol. I. New

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[3] P. M. Anderson and A. A. Fouad, Power System

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Systems 1(20) (2000), 7±19.

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

[9] Y. Xue, Th. Van Cutsem, and M. Ribbens-Pavella,

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[17] EPRI, DYNRED Users Manual, 1993.

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.