transient stability analysis of large-scale power systems with speed governor via vector lyapunov...

Transient stability analysis of large-scalepower systems with speed governor via

vector Lyapunov functionsH. Shaaban, M.Sc., Ph.D., and Lj. Grujic, M.Sc, D.Sci.

Indexing terms: Stability, Power systems and plant, Control systems

Abstract: In the paper, transient stability analysis of an iV-machine power system is carried out using thedecomposition-aggregation method. In the study, transfer conductances, mechanical damping, electromagneticdamping and speed governor action are taken into consideration. The system decomposition is performed sothat every subsystem includes three machines, instead of only two machines as for the pair-wise decomposition.

The system mathematical model is derived and decomposed into (N — l)/2 sixth-order and one second-orderinterconnected subsystems. A vector Lyapunov function, whose elements are Lyapunov functions of the free(disconnected) subsystems, is constructed and used for the system aggregation.

A seven-machine power system is used as an illustrative example, and an asymptotic stability domain esti-mate is determined for the system. It is found that the proposed decomposition scheme can lead to a consider-able reduction in the conservativeness of the decomposition-aggregation method. It is found also that thepair-wise decomposition-aggregation approaches developed so far are not suitable for stability analysis of large-scale power systems.

Notation

Pmi = mechanical power delivered to ith machinePei = electrical power delivered by ith machinePgi = variation of mechanical power of ith machine<5, = absolute rotor angleM, = inertia coefficientD, = mechanical dampingM = M^Dj = mechanical damping coefficientMj = Mr 1Dij = electromagnetic damping coefficient£, = modulus of internal voltageYij = transfer admittance between ith and jth

machinesJt ={ii, i / + 1} = set introduced to denote the

machines (except the comparison machine) ofthe Ith subsystem

Au = EtEj Yu; A, = EhENYilN

Ai = Eil + lEN Yi[ + x N; A{ = Liltil + 1 Y(l il +!

°i/ , ii + 1 = &iiN ~ aii + 1, N = °ii, i/ + 1 ~ t̂"/, i; + 1Wi/N = ^ i / ~ (^N 5 Wi, +1, N = CO{, +1 — caNTI = "N ~ Mi '•> Xl = ^-N ~_Mi + 1A/ = /,-,, i, + l — ^N, i ; + l ! ^ / = ^i, + I, i/ — ^Ni,

*i — Mi +1 + AN, i, +1 + 2^ Ait + l j

Q, = rotor speed with respect to synchronous speedH'1 = time constant of first-order proportional speed

governora / i " l = gain of first-order proportional speed governor

a, = a,, - aN; a,_= aI/ + 1 - aw

Z 2 , Z 3 = two functions, defined as follows:Z2(a, (p) = min {^/z max ( | a | , | < ^ | ) ; ( | a | + | ^ | ) }Z3(a, q>, Q = min {2 max ( | a | , \q>\, \C\)',

( | a | + |<jo| + |CD; (Z2(a, (p)+ |CD; (Z2(a, C) + l<p|);(Z 2 (<p ,0+ | a | ) }

Paper 3640D (C8, C9, P9), received 21st June 1984

Dr. Shaaban is with the Faculty of Engineering and Technology, Shebin El-Kom,University of Menoufia, Egypt, and Dr. Grujic is with the Faculty of MechanicalEngineering, University of Belgrade, 11001 Belgrade, Yugoslavia

1 Introduction

Most of the present day stability studies are done by simu-lation on an analogue or a digital computer. In this way,the nonlinear differential equations of the system, for agiven initial operating condition and a specified dis-turbance, are integrated either by electrical analogy ornumerically. A time solution for the generators' rotorangles is obtained, and by examining these angles atvarious instants, stability (or instability) of the system canbe detected. However, to find the boundary of the stabilityregion for a power system, the simulation method is slowand expensive [1].

The scalar Lyapunov function method appeared one ofthe most powerful methods for stability studies of powersystems [2]. However, this method did not seem suitable,owing to the continuous increase in size and complexity ofpower systems, and in particular when the problem of thestability domain estimate of the system is attacked [3].

Attempts to overcome the drawbacks of the scalar Lya-punov approach have led to the decomposition-aggregation method, which is based on Bellman's conceptof vector Lyapunov functions [4]. Application of themethod to a power system is carried out by decomposingthe system into a number of subsystems. Stability proper-ties of these subsystems are established by scalar Lyapu-nov functions. These functions are used as components inconstructing a vector Lyapunov function by which thesystem is aggregated.

Recently, the decomposition-aggregation method hasbeen used for stability analysis of power systems in anumber of papers [5-9]. In these papers, an iV-machinepower system was considered, and it was decomposed(except in the first paper) into N — 1 interconnected sub-systems using the pair-wise decomposition. In Reference10, the authors introduced a decomposition scheme bywhich a power system can be decomposed into subsystemswhich may have more than two machines.

In the present work, an /V-machine power system isconsidered and the triple-wise decomposition [11], bywhich a power system is decomposed into subsystems,each consisting of two machines in addition to the com-parison machine, is used for decomposing the system into(N — l)/2 interconnected subsystems. Considering transferconductances, mechanical damping, electromagneticdamping and speed governor action, the mathematical

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 2, MARCH 1985 45

model of the system is derived, and it is decomposed into(N — l)/2 sixth-order and one second-order interconnectedsubsystems. Each of these systems is decomposed into afree (disconnected) subsystem and interconnections. Eachof the sixth-order free subsystems is assumed to include thelargest number of nonlinearities, i.e. six nonlinearities. Forthis subsystem we adopt a scalar Lyapunov function in theform 'quadratic form + sum of the integrals of the six non-linearities'. A vector Lyapunov function, whose com-ponents are Lyapunov functions of the free subsystems, isconstructed, and used for the system aggregation. A squareaggregation matrix of the order (N + l)/2 is obtained, andstability of this matrix implies asymptotic stability of thesystem equilibrium.

2 Power system model

Consider an TV-machine power system (the transfer con-ductances are included) with mechanical and electromag-netic dampings in addition to the first-order proportionalspeed governor. The motion of the ith machine takenseparately is expressed by (£, and M, are assumedconstant)

iN = Wi -coN N

ijicot - coj) + Mr1 p.

Pi = -ftPi - oiiCOi for i = 1, 2, . . . , N (7)

where

fij{Oij) = cos (au + dfj - Oij) - cos {dfj - 9^ (8)

These nonlinear functions satisfy the following conditions:

foTi±j,i,j=l,2,...,N (9)

on the 0^ compact intervals, which are defined as

- 2(TT - 0u + dfj) ^ Oij ^ 2(du - dfj) (10)

In eqn. 9, £,v are positive numbers, and may be determinedas

Obviously, the state vector of the whole system is

= P9i + Pmi ~ Z EtEj Yu cos (du - By) x = [<7lN, (olt Px, a2N, co2, P2, . . . , aN.u N, CDN,

Pgi= -mPgi-*i&i f o r i= 1, 2 , ...,N (1) (12)

Choosing the Nth machine as a comparison machine, andintroducing the (3N — 1) state variables Power system decomposition

(2)

the mathematical model of the whole system is derived as

$.N = Q. - QN for i ± N

M," 1

i + Pmi- I Aij cos (5U - 6U)j=i

=fi(SiN,SjN,6i,6j,Pgi)

gi = -lii Pgi ~ « A for i = 1, 2, . . . , N (3)

Now, to make the origin of the state space coincide withan equilibrium position of the system, we adopt the newstate variables

tN = 8tN-8?N, cot = ni-Q° and Pt = Pgl, - P°gi

(4)

where

In this paper, an iV-machine (N is odd, without losinggenerality) power system is considered, and it is decom-posed into (N — l)/2 interconnected subsystems, each con-sisting of two machines and the comparison machine,using the triple-wise decomposition. It is to be noted thatnone of the system machines (except the comparisonmachine) can be included in more than one subsystem.

Now, by introducing the set J\ = {i,, it + 1} and defin-ing the state vectors x ; and xN as follows:

XI = [ f f i ,JV' " " i j + l , IV' W i / N ) W i ; + 1 , N> M/IV) * i ; + l , J v J

= [x7l, xl2, xl3, x/4, xIs, xl6]

and

xN = [a)NPNY = [xNlxN2Y (13)

we can decompose the system mathematical model (eqn. 7)into S = (N — l)/2 sixth-order interconnected subsystemsand the second-order interconnected subsystem, which hasthe general form

xN = &NxN + hN(x) (14)

where0C;

Q = QO = _HL for i = 1, 2, . . . , N (5) and

and where SfN, dJN are solutions of the equations

ffi°ul,d%,no,I»d = 0 fori=\,2,...,N (6)

Hence the overall system motion is governed by the stateequations

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 2, MARCH 1985

hN =

N-l

Z (*NJ u>jN ~ MN 1 ANj fNj{aN)}

0

Each of the sixth-order subsystems may be written in thegeneral form

x7 = 0>,x, + B, Fjiaj) + /J7(X) for / = 1, 2, . . . , S (15)

46

and it can be decomposed into the free (disconnected) sub-system given by

wherefor / = 1, 2, ..., (16)

and the interconnections hj(x).In eqn. 16, the matrices ^ 7 , Bj and Cj are constant

matrices, and ^(07) is a nonlinear vector function. Referr-ing to eqn. 7, we can define the matrix ^ 7 as

(17)

Assuming that the free subsystem of eqn. 16 contains thesix nonlinearities (see eqn. 8) given by

5?N -

-o0000

-0

00

0000

10

- r ,A,

- « . 70

01

A;

-r,0

-<*i /+i

00Mr*

0

-nu0

00

0

MF,ii0

-^•7+1

= cos (ailN

- cos (5flN -

ilN)

= COS

c o s (ffi

- c o s

= C 0 S (ffJ

- cos

= cos (CT

. if +1 )

, , , - , + 1

I/ + 1

(18)

we can define the following matrices (see notation):

00

-MF,1AIB,=

h,(x) =

101

- 1

- 10

r,xNl

01

- 110

- 1 0

00

0 00 00 00 00 00 0

/v-i

00

-MrK000

000000 J

- Mh

— dj xNl —Nl

It is obvious that the state vector of the whole system is

given now by

x = i>r,xi,...-,xj,x£r (23)Now, to establish stability of the original sytem (eqn. 7), wecan apply the methods of vector Lyapunov functions tothe expansion of eqns. 14 and 15, which is given as aninterconnection of low-order subsystems [10].

4 Power system aggregation

Following the aggregation procedure in Reference 12, anaggregation matrix A = [a7J] is constructed. The elementsof the matrix (constant matrix) obey the inequality

s + ij) for / = 1, 2, . . . , S + 1 (24)

where K/(x7) is the total time derivative of the functionV^Xj), which is a Lyapunov function for the 7th free sub-system, along the motion of the 7th interconnected sub-system. Uj and Uj are positive definite (comparison)functions, which are chosen, in this paper, to be of the form[9]

Uk(xk) = || xk || = (xlxk)112 for k = 1, 2, 3, . . . , S + 1 (25)

Now, we can rewrite the left-hand side of eqn. 24 as

* ft MYh^x) (26)where V^x^j is the total time derivative of Vl along themotion of the 7th free subsystem.

4.1 Construction of Lyapunov functions for the freesubsystems

For each of the sixth-order free subsystems, we adopt aLyapunov function of the form [10, 13]

6

V,(Xl) = xjH,Xl

for 7 = 1, 2, ...,S (27)

1 = 1

00 J

(19)

(20)

(21)

MN xANjfNj{oNjj)

fai, +1, j) + MJV l ANj fNj{^Nj)}

— a,

(22)


where Ul is a sixth-order symmetric positive-definitematrix, \\ih are arbitrary positive numbers, and the nonlin-earities//((<7//) are given by eqn. 18.

For the last (S + l)th second-order free subsystem, weconstruct the Lyapunov function

V(x)-xT\C °lxV\XN) — XN Q 1 N

(28)

where C is an arbitrary positive number.

4.2 Stability criterionAccording to theorem 1 of Reference 12, stability of theaggregation matrix A = [a/y] or, equivalently, if it satisfiesthe Hick's conditions

(-1) '21

, • • • , *2k > 0 V/c = 1, 2, ...,S+ 1 (29)

implies asymptotic stability of the system equilibrium.

4.3 Aggregation matrixLet us consider first the (N — l)/2 sixth-order subsystems.Along the motion of the free subsystem of eqn. 16, wecompute

*/(*/)/ = xi(- G7)x7 + 2Fj {o,)B\ Hj x7

6+ V (K f, ( o A o , V / = 1 2 <? HOI

where G7 is a symmetric matrix, defined as

<J7 — ^r i n i T nj^i \JL)

Substituting the matrix ^ 7 from eqn. 17 in eqn. 31, thematrix G7 is computed, and it is found that this matrixcannot be positive definite (the first two diagonal elementsare zeros).

Now, under the condition A7 = A7 = 0, and by choos-ing the matrix Hj in the form

H,=

where

rMio

oM5o

0

0

M4o

M6

"13

0

M3ooo

0hi24

0

^44

00

r ' l 5

000

oMe000

(32)

' 1 3

we obtain the matrix G7 in the form

G,=

"00000

.0

000000

00

2Kjh[3

000

000

2K7fc24

00

0000

2/*,-,M50

00000

2^i, + i«66-

(33)

It is to be noted that positive definiteness of the matrix Ht

of eqn. 32 can be guaranteed (Kj is an arbitrary positivenumber) only under the two conditions hl3 > 0 andh2A > 0.

Now, substituting from eqns. 19-21, 32 and 33 into eqn.30, and selecting the positive numbers

(34)

we obtain

- 2MrlAIfl3(ol3)(h[3xIl

[4xl2 h'33xl3)

Let us now introduce the positive constants e7| £ (0, £7

/ = 1, 2, 5, 6 (£7( are determined by eqn. 11), for which

o f (a \> E o2 for / = 1 2 5 6oiiJii\uii) ^ fc//u/i lKJl l — l' *••> -*•> u

is satisfied on a compact interval Uu of ah, i.e.

l 2 4

(35)

), for

(36)

U,, = [V_u, CjJ (37)

where U_It and Vh are the negative and positive solutions,respectively, of

/ / , K ) = £/,*/, f o r / = 1 , 2 , 5,6 (38)

It is important to note that if the value of e7 is takensmaller, the interval U/ given by eqn. 37 becomes larger,and so we obtain a larger estimate for the system stabilitydomain.

Now, by using the so-called S-process [13], that is byadding and subtracting from the right-hand side of eqn. 35the non-negative expression (see expr. 9)

/J] (39)

where £/3 and £u are determined, using eqn. 11. as

r,' 2 4

£«.

' 2 4

and

£/, = sin(0lf,lf + 1 + a ? l i l f + 1) (40)

we can 'majorise' the right-hand side of eqn. 35, aftertrivial calculations, as

*i(x/)/< ~ t f ll*i II2 V/ = 1 ,2 , . . . ,S (41)

48 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 2, MARCH 1985

where Xf is the minimal (positive) eigenvalue of the sym-metric matrix Gf, which is given as

[grad Vfr

Gf =

Oilg[2

0

0 1 4

000

g\ fi

flf'l2

g!22

0 2 3

000

gii0

0

0^3

0000

g™

gU00

gi*00

0 4 7

0

0000

0 5 5

000

00000

0 6 6

00

0

0

g\i

00

077

0

0 1 8

0

0 3 8

0000

0 8 8

(42)

where

gl22 =

g[4 = -

038 = -MrlJjhi^

glni = IM^A^J^

and where (see eqn. 11)

£Nil = sin (0ilN + dflN)

and

=

g'8B

V.b + l n ( 9 i l + ltN + dfI+UN) (43)

It is important to note that the numbers Kj, h[3, h'24, e7l

and sl2 (sl5 = el6 = 0.001, and are chosen to be slightlylarger than zero) should be chosen so that we can guar-antee positive definiteness of the matrix Gf.

Now, we proceed to majorise the second term (the firstterm is majorised by eqn. 41) in the right-hand side of eqn.26 as follows:

[grad KXx,)

rL

rj,xN2

/i4 4x/ 4) Tj

2Z/w||x/||||xw|| + X jxJ 2

+ 2£zIJC||xJ||xlc||

V/ = 1,2 , . . . ,S (46)

where (see notation)

ZEN = Z2(ZlNa, ZlNb) (47)

ZINO = Z2 [Z 3 {(T/ h[ 3 - a, h[ 5), T, h3 3 , a, h'5 5 } ;

Zsiiii h'24, - a7 h'26), xj / J 4 4 , a, /i66}]

ziNb = Z2[Z3{(riih[3-Hih[s), ^1/133^/^55};

ZM, hi* ~ V, hie), h KA, Jit h'66)}l

and A/ is the maximal eigenvalue of the fourth-order sym-metric matrix Qj, whose elements are given as

q[ 1 = 022 = 033 = 044 = q[2 = 014 = 023 = 034 = 0

0713 = AV*33 Z i^inA.iK + ̂ ,..ir+l4i*+l)

S

0 2 4 = ^ i / + l / J 4 4 Z J ( ' ^ i / + l , « K < s i / + l . i j C

In eqn. 46, ZIK is defined as

ZIK = Z2\Z2(ZlK, ZIK); Z2(ZIK,

where

(49)

(MN ANiKQNiK + MI/ + 1 • Ail + l iKi;il + 1 i

^ / K = Z2[[MN ANiK+lc>NiK + l-\-Mil

xZ2{h[3,h33);

{MN ANiK + lQNiK + l + M1-/ + 1/4,- / + 1 ) i j c +

MN 1 ANjfNj{aNj)}

rj,xN2

N 1

+ Z {^/J^jw - Mr,l 1 ̂ i , +1, j fi, +1, Mi, +1, j) + MN 1ANj fNj{

+ 2(^/15x/l +h'55xl5)(-(xIxNl -

oNj)}J

-Ji,xN2) (44)

N- 1 S

Noting that £ is equivalent to £ £ , and introducingj * . / / K*I jeJk

the following majorisations (see eqn. 11):

I ; j = Sin (0,.,; - (5?

2,K = Z2l\XlllK-XMlK\Z2{h[3,h33);

7 — 7 r i y — ) \7 (hl h1 v

•WK — ^ 2 L I fHi, IK+ 1 AN, iK+l I ' ^ 2 ^ 1 3 ' "33^'

I 'Hi + l , i K + l ~ ^N, iK -tI - • (a _ s? ) (45) Combining eqns. 41 and 46 and comparing the result withd, + i,j — s i n ( il + i j i,+ i,j) I ) e q n 24, We can define the first (N - l)/2 rows of the aggre-

we can majorise the right-hand side of eqn. 44 in the form gation matrix A = [a / x ] as follows:


K =

2Z IN(50)

N - 1for / = 1, 2, ...,S =

To define the last {(N + l)/2th) row of the matrix A, thetotal time derivative of V(xN), which is given by eqn. 28,along the motion of the subsystem of eqn. 14, is computedas

= -2CXNx2Nl - 2fiNx

N- 1

-2XNI X (*Nj ^Nj

- cc

(51)

Choosing the constant C = MNaN, we can eliminate theterm in xNlxN2, and then eqn. 51 is rewritten as

Kx) = -2(MNocNXNx2Nl

s

K=l

(52)

The left-hand side of eqn. 52 may be directly majorised as

Kx) ^ - 2 min (MNuNAN, 2

s

This implies

aS + 1, K ~

- 2 min (MNaNXN,

2Z\M Z{AQ , AN

(53)

= / f o r / = 1,2, . . . , 5 =

The system aggregation matrix is completely defined nowby eqns. 50 and 53.

It is important to note that we can determine a newaggregation matrix, in which only the first (N — l)/2diagonal elements are different from those in eqn. 50, forthe case in which the first four nonlinearities (eqn. 18) areincluded in every free subsystem.

5 Numerical example

In this example, the seven-machine system shown in Fig. 1is considered as a simple power system for an applicationof the developed approach. Choosing machine 7 as thecomparison machine, the system is decomposed into threeinterconnected subsystems, as indicated in Fig. 2.

The following parameters are selected:

A,-= 8.0, i= 1, 2, . . . , 6 ; A7 = 8.1;

Ay = 0.10, i ^=y, i,y = 1, 2 7;

H. = 20; a, = 25, i = 1, 2, . . : , 7;

h{3 = h^ = 1-0,7 = 1, 2, 3; Ky = 2.0; K2 = 2.6; K3 = 2.1;

Sji = Ej2 = 0.70, j = 1, 2; e3l = e32 = 0.65

50

Then the matrices G* and Q of eqns. 42 and 48, respec-tively, are constructed for each of the three subsystems,

E=U5/0M=1O.O

0.50/-80

0.50/-76"\

E=1.40/U."M=0.316P =0.75

Fig. 1 Seven-machine system

All values in p.u.

-1.50130.68410.68800.4839

0.6213-2.0747

0.62780.4480

0.56880.5727

-1.51260.4051

9.509412.996111.9691

-40.0

Fig. 2 Graph of the decomposed system

and the eigenvalues A* and A for these matrices are deter-mined. Finally, the aggregation matrix of eqns. 50 and 53is computed as

A =

This matrix satisfies the conditions of eqn. 29, and thus itis a stable matrix. This implies asymptotic stability of thesystem equilibrium.

We can now go further, to determine a stability domainestimate for the whole system. Therefore the matrixATB + BTA, with the matrix B = BT, is in the form

B = diag [1 1 1 20]

and is computed and found to be negative definite.According to theorem 4 of Reference 9, we conclude that

where

V2(x2) + K3(x3) + 20K4(x4)

and

= m i n ( K ° , V°2, ^


is an estimate of the system asymptotic stability domain.Using the following equation (see Appendix of Refer-

ence 11):

° = min minm = 1, 2 6 xim 6 {xim, xim) and

6

Z </ = 1 JO

(55)

we get

F ° = 10.37, V°2 = 21.12 and K? = 13.82

Using these values (V% = oo), the constant yt = 10.37 isdetermined.

In terms of the original physical variables d, co and P ofthe system (eqn. 7), the estimate Sl is written as

S, = {(S, co, P): (V,(5, co, P) + V2(3, co, P) + V3(5, co, P)

+ 20V4(co1,P7))^ 10.37}

where

K,(<5, co, P)

= 11.7(<517 + 0.09)2 + 11.8(<527 - 0.19)2 + 0.35

x {(a>1 - co7)2 + (co2 - co7)

2} + 0.04

x {(Pi ~ Pi)2 + (P2 ~ Pi)2} + 2 0 ! 7 + 0.09)

x {(co, - co7) + 0.12(Pl - P7)} + 2(S21 - 0.19)

x {(co2 - co7) + 0.13(P2 - P7)} + 2.0

x sin (<517 - 1.74) + 2.03 sin (S21 - 1.8) + 2.25

x sin {(<§17 - S21) - 1.71} + 2.36

x sin {(<527 - <517) - 1.71} - 0.08

x {sin (<517 + 1.74) + sin (<S27 + 1.8)}

+ l.O8<517 - 1.205<527 + 8.85

V2{5, co, P)

= 12.8(<537 - 0.16)2 + 12.7(<547 - 0.24)2 + 0.41

x {(co3 - co-j)2 + (co4 - co-,)2} + 0.06

x {(^3 - Pi)2 + {P* ~ Pi)2} + 2(^37 - 0.16)

x {(co3 - co,) + 0.16(P3 - F7)} + 2((547 - 0.24)

x {(o>4 - co7) + 0.16(P4 - P7)} + 3.31

x sin (331 - 1.83) + 3.19 sin (<547 - 1.75) + 3.44

x sin {(<537 - (547) - 1.73} + 3.34

x sin {(c547 - <537) - 1.73} - 0 . 1 0

x {sin (<537 + 1.83) + sin (<547 + 1.75)}

+ 0.904<537 - 0.863<547 + 13.395

V3(S, co, P)

= 12.3(<557 - 0.17)2 + 12.6(^67 + 0.11)2 + 0.36

x {{co5 - Co-,)2 + (co6 - co-,)2} + 0.05

x {(P5 - P 7 ) 2 + (P 6 - P7)2} + 2(S51 - 0.17)

x {(co5 - co7) + 0.14(P5 - P7)} + 2(S61 + 0.11)

x {(co6 - co7) + 0.16(P6 - P7)} + 2.07

x sin (<557 - 1.82) + 2.26 sin (<567 - 1.75) + 2.2

x sin {(<557 - <567) - 1.71} + 2 . 4


x sin {((567 - <557) - 1.71} - 0.072

x {sin (<557 + 1.82) + sin (361 + 1.75)} - 1.167<557

+ 1.90<567 + 9.144

K4(co7, P7) = 250o>7 + P7

It is important to note that for the case in which the freesubsystem includes only four nonlinearities, we can ensurestability of the system aggregation matrix for the par-ameters £,-, = ej2 = 0.75, j = 1, 2, 3. For this case, we canobtain

S, = {(3, co, P): (9,(3, co, P) + 92{S, co, P)

+ V3(3, co, P) + 20V4(co1, P7)) ^ 8.03}

as an estimate for the system asymptotic stability domain.

6 Conclusions

The developed triple-wise decomposition-aggregationapproach is applied in the present work to a seven-machine power system, and the following conclusions areobtained:

(a) Decomposing a power system, such that every sub-system includes three machines, instead of only twomachines as usual, we can reduce the conservativeness ofthe decomposition-aggregation method. The reason is thatthe triple-wise decomposition allows strong intercon-nections among machines (thick lines in Fig. 2) to beincluded in the subsystems, instead of exposing them asinterconnections among subsystems for the pair-wisedecomposition

(b) The pair-wise decomposition-aggregationapproaches of References 9 and 11 are not suitable for usein stability studies of power systems with large numbers ofmachines (say more than three machines). Note that theapplication of any of the two approaches did not lead to astable aggregation matrix in the present example

(c) The developed triple-wise decomposition-aggregation method can be used for stability studies ofmultimachine power systems (number of machines may bemore than seven) which are composed of tightly coupledgroups of machines with weak interconnections, acommon situation in practice

(d) We can obtain larger stability domain estimates for apower system by increasing the number of nonlinearities inevery free subsystem.

7 References

1 WILLEMS, J.L.: 'Direct methods for transient stability studies inpower system analysis', IEEE Trans., 1971, AC-16, pp. 332-341

2 RIBBENS-PAVELLA, M., and LEMAL, B.: 'Fast determination ofstability regions for online transient power-system studies', Proc. IEE,1976, 123, (7), pp. 689-696

3 GRUJIC, LJ.T., DARWISH, M., and FANTIN, J.: 'Coherence, vectorLyapunov functions and large-scale power systems', Int. J. Syst. Sci.,1979, 10, pp. 351-362

4 BELLMAN, R.: 'Vector Liapunov functions', SI AM J. Control &Optimiz., 1962, 1, pp. 32-34

5 PAI, M.A, and NARAYANA, C.L.: 'Stability of large-scale powersystems'. Proceedings of 6th IFAC World Congress, Boston, 1975, pp.1-10

6 GRUJIC, LJ.T., and RIBBENS-PAVELLA, M.: 'Large-scale powersystems: decomposition, aggregation and stability'. Internal report,University of Liege, Belgium, 1977

7 JOCIC, LJ.B., RIBBENS-PAVELLA, M., and SlLJAK, D.D.: 'Multi-machine power systems: stability, decomposition and aggregation',IEEE Trans., 1978, AC-23, pp. 325-332

51

8 JOCIC, LJ.B., and SILJAK, D.D.: 'Decomposition and stability ofmultimachine power systems'. Proceedings of 7th IFAC WorldCongress, Finland, 1978, pp. 21-26

9 GRUJIC, LJ., RIBBENS-PAVELLA, M, and BOUFFIOUX, A.:'Asymptotic stability of large-scale systems with application to powersystems. Part 2: transient analysis', Int. J. Electr. Power & EnergySyst., 1979, 1, pp. 158-165

10 ARAKI, M., METWALLY, M.M., and SlLJAK, D.D.: 'Generalizeddecompositions for transient stability analysis of multimachine powersystems'. Proceedings of joint automatic control conference, Califor-nia, August 1980, pp. 1-7

11 SHAABAN, H.: 'Transient stability analysis of electric power systemsunder structural perturbations via vector Lyapunov functions'. Ph.D.thesis, University of Belgrade, 1983

12 GRUJIC, LJ.T., and RIBBENS-PAVELLA, M.: 'Asymptotic stabilityof large-scale systems with application to power systems. Part 1:domain estimation', Int. J. Electr. Power & Energy Syst., 1979, 1, pp.151-157

13 TOKUMARU, H., and SAITO, N.: 'On the absolute stability of anautomatic control system with many nonlinear characteristics'. KyotoUniversity, Japan, 1965, pp. 347-379

52 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 2, MARCH 1985

transient stability analysis of large-scale power systems with speed governor via vector lyapunov...

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