STABILITY OF LARGE SCALE POWKR SYSTEMS BY DECOMPOSITION

6.1 Introduction

The rapidly increasing size ot power systems and the complex modes of instabilities that are inherent in such systems has emphasized the need to approach the stability analysis problem in a hicrarchial manner. Such an approach involves decomposi¬tion of the composite system into subsystems which consist ot lower order, isolated (free) subsystems plus the inter¬connect],ons between the subsystems. The decomposition Ctin be 110 r i o rmod mathematically ot [jhysically. A Lyapunov function is constructed for e«.ch of the free subsystems which becomes much easier. After this there arc two approaches possible to ascertain stability of the overall composite system. In the first approach, a scalar Lyapunov function consisting of the weiyhLed sum of the tree subsystem Lyapunov functions is constructed and its negative definiteness is examined along trajectories of the composite system. The interconnections influence the negative definiteness of the weighted sum scalar Lyapunov functions (WSSLF). In the second approach, a vector Lyapunov function (VLF) whose components are the subsystem Lyapunov functions is postulated. The derivative ot the vector Lyapunov function is examined and applying the comparison principle conditions are obtained for the stability of the composite system. Under certain mathematical assumptions, it can be shown that both the approaches yield equivalent results.

There has been a vast literature on the subject of stability of large scale dynamical systems. The concept of vector Lyapunov functions originally proposed by uellman [1J was first applied to interconnected dynamical systems by

J../U

Bailey [2], Since then, for nearly a decade there has been extensive literature on this topic which has been discussed authoritatively in two books, Michel and Miller 13J and Siljak Ml. In this chapter, our focus will be on the application of both the weighted sum scalar Lyapunov function and the vector Lyapunov function approaches to the power system stability problem. Both of these methods share a common principle, namely, the decomposition of the composite system into sub¬systems. In that context, we shall first study methods of decomposing a power system. These range from purely

mathe¬matical ones to those based on physical insight such as coherency, weak tie lines, etc. Next, we consider application of both the weighted sum scalar Lyapunov function and the comparison principle based vector Lyapunov function.

We have seen in Chapter V that estimating the region of stability is a very important step in computing the critical clearing time. In decomposition based methods, it is attractive to compute the region of stability of the composite system from regions of stability of the individual free sub-systems. Wc use results of Weisscnbcrger [5] in this connection.

6.2 Decomposition Techniques

Application of thc vector Lyapunov function approach to the transient stability problem of a power system was first initiated by Pai and Narayana [61. In their approach, a n-machine system was decomposed into n(n-l)/2 subsystems. Thus, for higher order systems, the most important advantage of the vector Lyapunov function, namely, reducing the dimensionality ot thc stability problem is thus lost. However, as wc shall sec later, their decomposition was essentially an overdescription of the problem. Later on, a number of other decompositions were proposed which greatly reduced the number of systems. References [7,8,9,10] propose a number of variations of decompositions into (n-i) or n subsystems depending on whether the system has uniform or

non-unitorm damping. Reference includes governor action also in the subsystems.

These decomposition procedures can take into account transfer conductances in their models. If we ignore transfer conductances, then other types of decomposition such as those preceded by an cquivalencing procedure to group machines into coherent groups arc possible . Since the primary advantage of the stability analysis via decomposition is the possibility of modelling subsystems more accurately, we restrict our treatment to systems which do not simplify the composite model to start with.

6.2.1 Decomposition with Non-uniform Damping

Consider a n-machine system whose swing equation is given in Eq. (3.13) and is reproduced below

d2<s. as.

M. —J- + D. ,r = P. - P ,

i i dt i ci

whore P. - P - - |E.|2 C..

i ma 1 11 11

n

P . ^ I |E. | |E . | | Y. ^ ; cos (6 . . . .) (6.1)

j = l

n

— I C. -sino.- + D..costi.. j^l A3 JO x3 *J

If the parameters i n Eq. (6.1) correspond to the post-fault state, then the post-fault SEP flJ can be transferred to the origin and the Eq. (6.1) can be rewritten as Eq. (3.35), i.e.

M: ^ + Di inr = - pei<!>< 1=1'2 h

dt

(6.2)

= ^ - A.-tcosC*.,-^) - -(^j^j)] - ^ " hihj

where A.. £ |E.I|E.IIY..I

Aij' ^'ij anCi ^ij arG re*ate<* as follows:

q±j = |EV||E.|:^ - |E.||E.|!Y..|sin,.j - A..Sin*..

(6- 3)

D.. = iBiHSjlG^ = Ie.MK.IIy^Icos*^ = A..cos*..

To facilitate the understanding of later material, we write Kq.- (6.2) corresponding to the ntn machine as

d26 96

M 41 + D -rr = P - P (5)

n dL2 n dt en - en*1'

n-3

V An.(coS(6nj-,nj) - ooS(^r*n.)]

Define

D. 1

A. and Lhe state variables as . , w.

1 in' m

(i-1,2,—,n-l) and The (2n-l) dimensional state space

model can be obtained as

■*: . = 1.1 in n

in 1 in n "1'"n

+ M £ A .f . - M. £ A. ,f. .

n nj nj x ^ id ID

'Lin " "Vn " Mn' £ Anjfnj ^ <6'5>

The most eftectivc decomposition which takes into account Lhe transfer conductances in the model is the pair-wise decomposition 17-10]. There are two possibilities.

(i> The choice of subsystem state vector is x^ -

(g.^-gv^r*"^nf«n), i-1,2,...,n-i. We impose the restriction on

the choice of t.he reference machine that % > A.

n - i

(i-1,2,...,n-1). With this choice of subsystem state vector, we have (n-1) subsystems. The state variable o:^ corresponding to the reference machine appears in all thc subsystem descriptions. In the literature 18] this is often called the overlapping decoinpos i t i on.

(ii) Thе overlapping decomposition can be avoided as indicated in Ref. [10J if wc define n subsystems such that thc state vector for the first (n-1) subsystems is x_ = ^in ^in'^n' and the state vector for thc nfc^ subsystem is the scalar u^.

We first consider decomposition of thc type (i).

In (6.5), the term corresponding to ^n^Ani£ni an(^ Mj^Ain^in ^C expressed as

M'^A . f , = M~ 1 IE. | I K MY. I

n mm n 1 i 11 n" in1

[oos(6ni-*ni' " cos(6ni-?ni,]

" -Mnacin<8in6in-«in*in»

+ M_1D. (cos6. -cos6"? ) (6-6)

n in in j n

Mi\,.fJn " MIlcin(sini'in-Ein^n»

t h ^*

we now have the i subsystem description with =

(*irr4n',-'in",:n) as

x. — -1

fi(o±) i h.(x)

(6.8)

Mnlcin

a. = 11 0 0\x.

i-1,2,...,n-l

where.

fi«*i> - sin(o±^ln) - sin^n

(6.9)

h.(x) =

-1 -1 n~l -1 -1

(M -M .ID. *. + E (M A .J" . -M. A. . f . . )

n I ' in*i n nj nj i i| i3'

r M A .£ -

-1 n_i -1

-M D. $. - v M

*i " (coB(c.+^n»-cos^n)

(0.iO) (6.11)

in the description (6.8), the last equation is common to all

tlie subaystemfi. This can be avoided by defining the subsystem

T S

state vectors as x- = (fi. -6. ,u>. ) i-1,2, .. . ,n-l and x„ = W .

-r. in in in n n

Kow we write the i subsystem description (i-1,2,n-l) as

-A.

-i

.-1

n-1

x = ->. x - M * ): A . t

n nn n j_^nlnl

i-1,2 n-1

(6.12)

Thе difference between this description and (6.8) is easily seen by inspection.

Wе have only second order subsystems to deal with as opposed to third order subsystems. A more general form of this decomposi¬tion with asynchronous damping and governor representation is contained in Rets. [9,101.

6.2.2 Decomposition with Uniform Damping [7]

In this case, A... - A for all i. The state vector for the

in

composite system is xT = (6^-5^ Vl,n~6n-l ,n'i

u. . ,. . . ,ic . ). The state space model is given by

n-l,n'

<5, - ic. in in

n j=1 "3 ^3

-1 n_1

= -w. i M >: A .f..

in in

.-1

- M." £ A. -f . . i j=1 13 *3

n-1

(6.13)

The subsystem vector is X* = C^^n—*S^nr^l and the subsystem state space model becomes

0

1 0

x - +

0 -A -i -1

_ — —

(6.14)

0V = 11 0)xi

f.^) = <V+Mi >Cinlsin(V6In> " sin6inl

(6.15)

0

-1 -1 n~} -1 -1

(M -M. ) D . d> . + I (M A .f .-M. f,,A..)

n i irri n nj rjj i ij i}

ri

(6.16)

where flU is defined as in (6.11). EquaLion (6.14) is the simplest decomposition where the free subsystems do not contain any transfer conductance terms. Two improvements over this decomposition are possible.

(i) Absorbing transfer conductances into the free subsystem [13]

We rewrite (6.14) as

0 1 0

X . - x. +

—1 0 -A. -l -1

0.;| = fl 01 Xj

(6.17)

where

fi<'V - ^"I'^in'^^i^L* " sin6In»

+ (MTI-M"1)D. lcos(o.+:SS ) - C0S6? ]

1 i n in j in' in

(6.18)

Denoting <Mn1+Mi1)Cin = wli and (M^-M^JD^ = u2j , (6.18) can be rewritten as

W * ^li + 1,2i ^^^|f*in> - sin<S!n+*in)}

(6.19)

where tan*. - (MT1-M~1)D. / (M~1+MT1)C .

in in in n 1 in

If |«^n + (^in| % it/2, then f"i(cti) is in the first and third quadrant in an interval around the origin. if is chosen as the machine with the largest inertia, the above condition is likely to be satisfied.

(ii) Modification of (6.18)

In Lhis subsystem description, we delete thc terms corre-

sponding to in in (6.18) and include it in the

interconnection term hj (x). Thus with (6.17) remaining the

same, f. (o: j-9 in (6.18) becomes

W = "i^in^'VO " sin*inl

+ "in^os^+S^) - cos«?nl ) (6.20)

••- MTXA- {sin(a.+5? ) - sin(6^ +y. ) \

I in I in rinin Tin

, , -1 Din ?r

where u>. — tan —— = - d>.

1 in n

<f^n being the angle of the transfer conductance between internal buses i and n. Under these conditions h^ (x) becomes

h.(x) = l^n^^'^i^ln^J^^^A^

-1 -1

^ J: (M XA .f .-M. A..f..)J (6.21)

n nj nj i 13 13'

Wе thus see that there arе quite a few variations in constructing thе free subsystem structure in pair-wise decomposition techniques.

^04

POWKK KYSTKM H TATSLLL 11

6.2.3 Other Decomposition Techniques

(i) In a pair-wise decomposition structure, we can in addition take into account asynchronous damping and first order proportional governors [10].

(ii) One of the restrictions in the non-uniform damping case is that A > A.. This is necessary in order that the linear part of the free subsystem in (6.8) be stable. This restric¬tion can be removed if instead of u*n we introduce wQ which is the velocity corresponding to the center of angle. A new set of decomposition and its variations is possible [14].

(iii) If we do an apriori coherency analysis according to some criteria and group the machines, then the number of subsystems could be reduced. Lyapunov function can be applied to the reduced model. This technique discussed in Kefs. [11J and 112] however does not strictly fall into the scope of our discus¬sion .

(iv) All the px~evious techniques rely on a mathematical decomposition of Lhe composite system. Since the description is based on Lhe bus admittance matrix at the internal nodes after eliminating all physical nodes, any type of physical decomposition is therefore ruled out. However, if a structure preserving model is used as in Ref. H5], then a physical decomposition is possible 116].

Although a great number of decomposition techniques have been discussed in the literature, there is no clear cut evidence to indicate that any one method is preferable. Since our main focus is on methods which have met with some success in practical application, detailed discussion of decomposition techniques has been limited to such cases only.

oii S Weighted Sum Scalar Lyapunov Function Approach

We first state some general results before applying it for the power sysLem problem. The treatment in Lhis section follows largely that in Michel and Miller- [3].

Consider the composite system

x = f(x) (6.22)

where x and f are N-vfifit.ors. Let the system (6.22) be decomposed into k subsystems with x^ of dimension n^ as the state vector of the i*"*1 subsystem. Hence,

I n. - N 1=1 1

mm T T

and x - (x1,x2, — ,xk) .

Fret the decomposition have the form

x. = f^lxy + g.(x) i=l,2,...,k (6.23)

where g^(x) represents the interconnection term in the i*"*1 subsystem. A particular form of g^(x) is

k

Z q-.tx)

m

ki = fi(xi) (6.24)

represent.s the i*"*1 free subsystem.

The following theorem due to Michel [17] is fundamental to the weighted sum scalar Lyapunov function approach.

Theorem 6.1

The equilibrium x = 0 of the composite system (6.23) is a.s.i.l. if the following conditions arc satisfied.

(i) Each free subsystem (6.24) is asymptotically stable in the large with a Lyapunov function v^'x^^ - u an^

v(x.) < —cr. |i x - 11 2 (o.>0) for all x. (6.25)

_1 (6.24) 1 ~y 1 _1

ni

where || * || denotes any one of thc equivalent norms in R

(ii) Given v^(x^), there exists constants a^. such that the followinq inequality is satisfied with respect to the inter¬connection terms

VV<x ) g (x) < ||x || I a . -||x

1 -1 1 j=l 11 1

k

(6.26)

(m) Given o. of hypothesis (i) there exists a k-vector T

a = (aa ) > 0 such that the test matrix S = ls..J

~ 1 K - 13

specified by

ai(-°i+aii' w

s±1 = ^ (6.27)

(aiaij+ajaj.i.)/2 is negative definite.

Proof: For the composite system, choose the Lyapunov function

k

V(x) - % ct.v. (x.) (6.28)

i=l 1 1 ~1

Clearly V(x) > 0 in Lhe entire region by virtue of hypoLhesis (i) and the fact Lhat > 0 (i=l,...,k). Along the solutions of (6.23) we have

k T

V(x) = I a-lVV. (x.r[f. (x.) + g.(x)1)

k ,2 k M

< & QLA-0, X, I + x. || I a..||x.||l (6.29)

i-1 1 1 1 1 j = l 13 3

T

Now let w = ( ,| x, |! , .. . , |! x. ||) and R - [r..] be the k*k maLrix

-1'" ;-k'" - ij

specified by

& . {-<>. +a . .) i=j

1 1 11 J

rij = *

ai^.j m

We now have

T

T T 8+£ r 1, ll2

V(x) < w R w = w ( 2 ) w = w S w <. &„(£) |lw|| (6.31

where S is the tost matrix of hypothesis (iii) and A^ denotes the maxinium eigenvalue. Hence V(x) is negative definite if S is negative definite. We also note that since S is symmetric,

all its eigenval ties are real. Since S is negative definite, *„<§> < 0.

The condition for negative definiteness of S is given by the following inequalities

Sll ' ' ■ * Sli

(-1 >

0

1-1,2,...,fe

(6.32)

y!il S«U

0.4 Region of Attraction

Suppose that the hypothesis (i) of Theorem 6-1 is not valid in the entire state space. In other words, the origin of the free subsystem is asymptotically stable in a region around the state space and the region of attraction is defined as

v.(x.)

vQ^. Then, as shown by Weissenberger 15], the region

of attraction of thc composite system is given by

V(x)

i or

Y whore y - Min (u.v .)

Ki<k

(6.33)

This rcqion of attraction is in general conservative and one way ot improving it is to optimize the "^'s, a topic to be

discussed later.

6.

Results Involving M-Matrices

It may be observed that in the application of Theorem 6.1, wc need the existence of suitable cu's > 0 (i=l,2,...,k). In the special case when thc off diagonal elements of the test matrix S are > 0 (non-negative) we can utilize properties ot the so-called M-matrices [3] to establish results which are easier to apply because (i) they do not involve usage of weight age vector ii and (ii) under certain conditions the results are equivalent to those obtained by the comparison principle and vector Lyapunov function. Wc first review certain properties of M-matrices.

M-Matrices [3]

Dof in ition

A teal k«k matrix D - ] is sa.id t.o be an M-matrix if

(i) d. . < 0 i^j (i.e. all off diagonal elements of D are non-positive) and (ii) if all principal minors are positive./ i.e. it Dg denotes the determinant

dn .... a

£=1,2,...,k (6.34)

h:l dtl

The following equivalent statements bring out some of the properties of M-matrices.

a) D is an M-matrix.

b) There is a k-vector u > 0 such that Du - 0.

—T

c) There is a k-vector v > 0 such that D v > 0.

d) The real parts of all eigenvalues of D are positive.

e) D is nonsinqular and all elements of D 1 are non-negative (in fact, all diagonal elements of p are positive).

From (b) and (c) it also follows, respectively that

(i) There exist positive constants \\ (j—l,...,k) such that

k

£ A .a. ■ > 0 i=l.,;2, .. . ,k

(ii) There exist positive constants Mj (j-1,2,...,k) such that k

>: n-d.. ■• 0 i-1,2,...,k

j=l J W

Another useful property of M-matrices which we require is the following.

Corollary. Let D - Id^J be a k*k matrix with non-positive off diagonal elements, i.e. d^.. < 0 i#j. Then D is an M-matrix if and only if there exists a diagonal matrix A = diag(o^,...,a^), ct^ % 0 i=l,2,...,k such that thc matrix

B - DTA + A D

(6.35)

is positive definite.

This concludes the discussion on M-matrices and its propcrtic! Wc now reformulate the stability Theorem 6.1 in terms of M-matriccs following Michel and Miller \3].

Theorem 6.2 [3]

Suppose that conditions (i) and (ii) of Theorem 6.1 are true with the additional assumption that a^ > 0 tor all i/j. Then the equilibrium x - 0 of (6.23) is a.s.i.l. if any one of the following conditions hold

(i) The successive principal minors of the k*k Lest matrix D = l^^^l sixe all positive, where

°i " aii 1=3

d. . =

lj

(6.36)

(ii) The real parts of Lhe eigenvalues of D are all positive, (iii) There exist positive constants (i-l,...,k) such that

k X

(o,-a..) - T. a.. > 0 i=l,2,...,k (6.37)

1 11 j=l Xi 1J ri

(iv) There exist positive constants (i=l,2,...,k) such that

k r,

(a,-a,,) - 7. -r1 a.. > 0i=l,2,...,k (6.38)

1 11 j=l ni J1 3*1

Proof. Since by assumption a^.. > 0 tor i#j and since the successive principal minors of D are positive, it follows that D is an M-matrix. Hence, from the property of M-matriccs, there exists a diagonal matrix A - diag («,,..., a.) s 0 such

— "1 K

that.

-2S - DTA + A D (6.39)

is positive definite.

Hence -2S -

2 Is. . ] where

s

i

i=3

(6.40)

is positive definite. If -2S is positive definite, § is negative definite. This tantarnounts to satisfying condition (iii) of Theorem 6.1. Hence, the equilibrium is a.s.i.l. Conditions (ii) - (iv) are true since D is an M-matrix. liquations (6.37) and (0.38) are referred to as row dominance and column dominance conditions, respectively.

It may be noted that condition (i) is independent of c^'s. Also, (6.37) and (6.38) can be used to optimize the region of attraction as shown below.

6.6 Optimizing the Region of Attraction

The region of attraction for the composite system has been defined to be

V (x) < y where

in (a.v„.) . „, 1 oi

it follows from (6.37) that - a^j > 0 if the origin is asymptotically stable. As suggested by Chen and Schinzinyer [131 the row dominance property (6.37) (called also as "negative diagonal guasidominance" [4]) can be used to produce thc following two criteria for optimizing the region of attraction.

Criterion 1

Minimize the trace of the test matrix S

Min z =

a

subject to (6.37) namely

k

(6.42)

I a. = 1 i=l 1

Criterion 2

Maximize the weighted sum of the boundaries of the stability region of the subsystems

Max ZV- E aivoi (6.43)

o i-I

subject to (6.42).

Let ___ be the optimized value of ____ by either criterion 1 or 2. The region of stability is defined to be

I *

7. a.v. (x.) < V° (6.44)

i«l 1 1 1

and V6 = Min (n*v .)

l<i<k 1 01

In view of the numerous equivalent properties of M-matrices,

there arc quite a few other possibilities for optimizing the

n .'s. l

6.7 Application to Power Systems [13]

Since the best results to date have been obtained by the weighted sun scalar Lyapunov function method using decomposi¬tion, we present a numerical example in detail. We consider a uniformly damped system and first state the general formula¬tion for an n-machine system before presenting the example. The presentation follows closely that of Chen and Schiny.iriger [13], The subsystem modelling is the same as in (6.17) which involves absorbing the transfer conductances in the free subsystems.

^i = -i-i + & fi W * -i(x)

ip (6.45)

where -

xii

X2i

in

in

- 6?

u> -in

A. -1

0 1 0 -A

T

b - 10

-U

i-1,2, ,n-l

so that

fi(ci) and hi<x) are defined in (6.18) and (6.19). The machine having the largest inertia is designated as the n machine and chosen as the reference machine.

in (0.19) is almost equal to - t). where 6.

Tin ' ~"J~ * 2 in

the transfer admittance angle between machine i and n.

KT1

1

is

6.7.1 Subsystem Analysis

The i1**1 free subsystem is described by

*± - -i-i + 6 fiK»

i-1,2,...n-1

(6.40)

fJi = ^i*i

i1^ lies in the first and third quadrants in a region around the origin defined by -v - 2^*n4'r'^n* < °j < 'n "

2(6? . )-v.(x.) is constructed followinq the procedure in in i -l

= M

in Sec. 4.6 since Eq. (6.46) is analogous to Eq. (4.37) with the followinq equivalences

in in

eg

1

-D /M = A ; M -

J,Jli + P2i

(6.47)

The most general form of the Lyapimof function is used, namely Eg. (4.44). Choosing n=2 and dividing by M, we obtain from Eq. (4.44)the Lyapunov function for the system (6.44) as

Vi(x-i> = 2

). 2 - 2 .

"2 Xli 4 ?'xlix2i 1 x2i>

+ 'o

vi'-i^ ;1 lenq (6.46) is evaluated as

vi STABILITY BY DECOMPOSITION 213

• , . A fi(xli) 2 X 2

= - 2 xu Xli ' 2 X2i

(6.49)

Since the rionlinearity f. (x..) is in a region around

x

fi(xii>

origin, we get an upper bound on —- as 3 where

li

Lhe

ii

>fl(*li>

ax

n = (l/M )cos(6? )

x^^=0 ' eq in vin'

Hence

vi*xiJ - ~-i--i' where Q = Diag(>. \S/2, A/2)

II II 2 < - o . x. l"-i "

(6.50)

where ci is min (A li/2 , A/2) .

The region ot stability of the free subsystem is given using Eq. (4.47) and the equivalences (6.47) as

in in

ir-2(6T )

M 0

eq

f.(o.)da.

(6.51)

i.e.

vi(xi) i voi

6.7.2 Stability of Composite System

The weighted sum scalar Lyapunov function is

n-1

V(x) = I a.v.(x.) 1*1 1 1 1

(6.52)

n-1

V(x) = X {a [v. (x.)

i-i 11-1 1(6.46)

Applying the obvious inequalities

+ [grad v.(x.))lh±(x)]}

xli * X

lk1 - ,Xli

+ x

lk

and

x

li

Xi

and *X2i' - H-i

and another inequality suggested in Ref. [71

co8<Xli-xki+^k-9.k) - cos(6?k-^k)

< Isinf^-e^JMx^ - xlk| (6.54)

We majorize the second term in the right-hand side ot (6.52).

0

h. (x) =

(6.55)

h.2(x)

n-1

i2

nj nj

where h.-<x) = M I A .lcos(fi .)

j-1 nJ

.-1

- oos(6nj-$nj)]

n-1

M. I A. . [cos (6 . .-d>. .)

i ID ID yiD

(6.56)

T A

Now [grad V/^'fec^J'J h^(x) - x]_j_",'x2i^'1i2

£ (1 4 fjlM"1 V Aij|sin(6^j-*ij)||xli-xlj; (6.57)

¥i

-1 n"x s

+ M I A . sin(6 .) Ix, . }• x.

n n]1 nD rnD 1 ' ID 1 1 '"l"

n-1

2_.

13 -ij 1 "-1

(1 + ^{M,1 Z A. .1 sin <<*?.-$. .)

ij~Tij" »*i»»*j

1 j=l ^ n-1

- n-1

+ M. I A. . I sin(6? .-<p, .) I * llx.

i

^ M" % A . I $ in (6S ) I ■ 'Ix.

ri '_j n j 1 n j 1 nj 1 " -l

Hence from (6.52) and using (0.58)

n-1 - n-1 n-1

V(x) < -7 a.o.||x.|| + I Z a.a. . Ilx. I; ' |!x. I! (6.59)

1=1 1=1 ]=1 J J

mm au = M^V Aij|sin(««.-,pij)|

«ii - (1 +1' {m;1 Rnji8in(S»r*»i) (6-601

* M_1 A. . I sin . .) I) i^j

By the majorization process, we naturally have a'^g > 0 for i^j. Also, a^ > 0. We can now apply any of the conditions (i), (ii) or (iii) of Theorem 6.2 to ensure asymptotic stability of the composite system, the (n-1)x(n-1) test matrix being f> = [tl^-j] where

r o. - a.. i=i

\ I ii J

d. . = { (6.61)

13

In particular, we utilize the row dominance property, i.e., there must exist > 0 (i=l,2,...,n-l) such that

n-1

w. (a.-a. . ) - 7. a.a.. > 0 i=l, 2,. . . ,n-l (6.62)

l I 11 j _j j i-j

This implies c«|.fc^a^j > 0 and since > 0, we have > a^, i.e.

> (1+A/2> V A. . I sin (fiS .-<[>. .) ] (6.63)

1 Mi j=l W 13 13

Equation (6.63) indicates that ti. which is a function of damping as well as the pre-fault state of the individual sub¬systems should be such as to overcome the factors arising through the network interconnections Y^_.. This equation also reveals that the weaker the value of Y„ (i^j) the easier it

is for the c. - a.- to be > 0 which is only a sufficient l ii

condition. It also means that the decomposition should he performed in such a mariner that the resultant subsystems are weakly coupled 13). In the decomposition based on the internal node description, v^j's are fixed and hence there is very little flexibility. Decomposition techniques based on struc¬ture preserving models [15] might be helpful.

6.7.3 Numeri cal Example [1.3]

7

The four machine example of Fig. 6.1 is taken from El-Abiad and Nagappan [IB] and modified to provide uniform damping characteristics. Tables 6.1-6.6 give the pertinent data. Table 6.7 shows that four different fault locations are examined. For example, in Test No. 1 we assume that a 3-phase fault occurs near bus 8 and that line 8-9 is tripped after fault is cleared. We assume A = .001 and fJ = 1.0. The numerical results are adapted from Chen and Schinzinger [13].

From Bus To Bus P.U. Impedance

5 6 0.2 + jO.4

5 7 0.2 + J0.5

5 6 10 0.1 + 30.3 0.1 + 10.15

7 8 0.1 + J0.5

7 10 0.05 I J0.2

8 9 0.2 4 J0.8

Table 6.1 Line Constants

Generator Number MVA Capacity in P.U • M in P.U. D in P.U.

1 15 1.0 1,130.0 1.13

2 40 0.5 2,260.0 2.26

3 30 0.4 1,508.0 1.508

4 100 0.004 75,350.0 75. 35

NOTES: 1. MA - 4 nfHi P.U. - rad /rad

2. D^ = 2 rfdi P.U. - rad/rad

3. h\ ■ inertia constant in P.U./sec

4. d. = damping coefficient in P.U.-sec

Table 6.2 Machine Constants

Generator Number

E. 6 . x in

1.057/5.30" 1.155/11.08° 1.095/5.48° 1.000/0.08°

Table 6.5 Internal Voltage of Pre-Fault System

Generator Number

E. 6. l in

1.057/5.69° 1.152/14.39 1.095/2.27° 1.000/0.08"

Table 6.6 Internal Voltage for Post-Fault System (line 8-9 tripped)

With machine 4 as reference and letting = 1, i-1,2,3, the following p matrix is obtained using 16.60) with ■ .0005.

D =

3. 013 1. 431 0.757

-0.733 4.010 -0.4b0

-0.506 -0.623 3.976

x 10

•4

(6.64)

It may be verified that D is indeed an M-matrix. Hence the equilibrium is asymptotically stable. Betore optimizing the ci^'s, we compute vQ^ (i=l,2,3) the stability regions of the three free subsystems. To illustrate, the free subsystem for machine 1 i s

li

dx

dF

x2i

(6.65)

2i

dx

dt

-.001 x21 - 6.251*10 [sintc^+0.0786) - 0.0786]

xli

0j_ = [1 0]

x2i

computed as 1.1x10

3 are computed as v respectively.

01

Similarly the region for machines 2 and - 5.477*10"4 and vQ3 = 9.281*10~4

6.7.4 Optimizing a^'s and Computing the Region of Attraction

We use the results of section 6.6 to optimize the rcqion of attraction by an Appropriate choice of weight ages 's>. Using Criterion 1 gives

Min z - (3.0130^4.01o2+3.976o3)lG

(6.67)

Subject to

al + a2 + a3 ~ *

3.013^ - 0.733v-*2 - 0.580a3 > .0

(6.68)

-1.431a] + 4.01a2 - 0.623a3 > 0 -0.757^ - 0.456a? + 3.976«3 > 0

The solution to the above linear programming problem is found

to be a, = 0.6139, a.. --0.2415, a, = 0.1446 and

1 * * 5

Min (a. v .) turns out to be 1.32*10 . Hence the . . - - 1 oi 1^1,2,3

stability region is

3 , _4

Z a. V.(x.) < 1.32x10 i-1 1 1 1

For this particular post-fault condition, the solution obtained

by using Criterion 2 yields the same as tt^'s in Criterion 1.

The critical switching time determined using the procedure

discussed in Chapter V is 0.36 sec. The value of by

actual simulation is found to be 0.37 sec. Thus there is a good agreement by both the methods.

Table 6.7 summarizes the list of results at the several fault locations in this four machine system. It appears that Criterion 2 gives the results which are less conservative. However, by both criteria the results agree very well by actual simulation method for tests 1 and 2. The discrepancy is larger for tests 3 and 4. This means that the decomposi¬tion technique gives favorable results only for certain faults and not for other ones. This is not a desirable feature and more research is needed to remove this drawback.

6.8 Vector Lyapunov Function Approach Using the Comparison Principle

Historically the vector Lyapunov function method using com¬parison principle was the first technique to be used for stability analysis of large scale systems via decomposition [l,2j. In one sense, it is a very general approach and has naturally generated a lot of interest in control theory and

mathematical circles. However, for most of thc specific cases considered thus far, the method reduces to the weighted sum scalar Lyapunov function method where usage of the comparison principle is not required. In power systems, comparison principle based vector Lyapunov function (VLF) was used initially together with the requirement that the free subsystem be exponentially stable [0,7,8]. This condition is, however, stringent so far as power systems ate considered and hence thc results were extremely conservative. Therefore, the method did not result in practical applications to realistic systems, in spite of the relaxation on the conditions on the subsystems [9]. In this section, we briefly review the VLF method and indicate the application to the power system problem. The comparison principle is discussed extensively in the litera-ture [3,4]. Our presentation is along the lines of Michel and Miller [3] .

6.8.1 Comprtr i son Principle

Consider a system of k differential equations

i = s<2>. x'V " yQ (6-69)

Usually we assume that (0.09) possesses unique solutions for every yQfR and for all t > 0.

Definition

H(y) is said to be quasi-monotone if for each component H^ 1=1,2,.-.,J( the inequality

tij(y) < tu(z)

is true whenever y^ < for all ij'j and y^ ■ If H(y) -

P y where P is a k*k matrix, then the quasi-monoticity condition reduces to the condition > 0 (ij^j).

Comparison Principle Theorem

If r(t) is a continuous function and satisfies

such that l(t-o) < y(t ), then a necessary and sufficient condition for the inequality

r(t) < y(t) for t > tQ (6.71)

to hold is that H be quasi-monotone.

In the case H(V) - P V, the condition reduces to p.. > 0 (i/j). For the linear case it also follows that if P has eigenvalues whose real part < 0, then y(t) and r(t) -> 0 as t > #i This, together with the property that p^ > 0 (i/j) is equivalent to -P being an M-matrix.

We now state the theorem for a.s.i.l. of the system (6.23) using the comparison principle and VLF.

Tjicqrem 6 ■ 3

Let the free subsystems (6.24) have the Lyapunov function v^(x^) and let

v.(x.)

< -3 v (X ), $

(6.24 iii

Define the vector Lyapunov function V(x) as

T

(v^ (x1),...,v^(xk)) . Now

v. (at. ) = [Vv. (x.) ]T[f - (x-)+g. (x. )1

1 _:L (6.23) 1 * "x "x _1 -1

- -fii vi(xi) + Vv.(xi)Tgi(xi) (6.72)

Let us assume that there exists a k*k matrix P — lp^.], p.. > 0 i^i such that

13 1 -1

k

T. p, . v, (x. ) (6.73)

(6.23) j=l

Note that !?- is absorbed in p... Hence V(x) < P V(x)

If the matrix P has eigenvalues with negative real parts, the equilibrium of the composite system is asymptotically stable.

Proof: The proof involves using Theorem 6.1 concerning

weighted sum scalar Lyapunov function. The two assumptions,

namely that P has eigenvalues with negative real part and

p.. ^ 0 , imply that -P is an M-matrix. From the

properties of M-matrices in Sec. (6.5), we have thc equivalent computable condition

11 K1JI

(-1)

£-1,2,...,k

(6.74)

P

11 y%l

Alternatively, we can use property (ii) and equate T|.'S with

n.'s. Thus, we can postulate the existence of a vector

3 >J> m

u = (u ,...,a ) > o such that a P < 0. We can define the

scalar Lyapunov function

k

V(x) = Z a. v. (x.)

■

(6.75)

V(x) is positive definite and

V(x) - J! a.v. (x.) ^ a V(x)

ii=i

< a P V(x)

(6.76)

Since <^P< 0, a p V is < 0 and hence V(x) < 0. Hence the origin is asymptotically stable. We have also thus shown that in the case of a linearized version of the comparison principle, the VLF and weighted sum scalar Lyapunov function are really equivalent.

- 9 Power System Application of Vector Lyapunov Function Me thod

It has been pointed out that the VLF method was the first one to be applied to power systems [6] and hence there is a good amount of literature using this method. To construct the.P matrix which satisfies the inequality V(x) < P V(x), it is generally necessary to have vjtx^) which are quadratic functions of x^. Hence, it poses a problem in power system application where v^tx^) consists of a quadratic plus an

integral of thc nonlinearity. Ways of circumventing this difficulty often resulted in very conservative results. We briefly sketch a number of approaches to the problem.

6.9.1 Approach Based 011 Exponential Stability of Subsystems (20.7]

The uniform damping case is discussed. With each subsystem we associate a Lyapunov function v^{x^) k v^^(x^) where v^(??^) is given by (6.48). Each subsystem is assumed to be exponentially stable with the following estimates on v^fx.) and v^(x^).

0±1 II ^1II 1 -vA <xd) < rii2l|x:||, < -ni3l|xi|j (6.77)

{W <x .)]Th (x) || < t F |Jx |;

3 1 1 j=l '3 3

(6.78)

The vector Lyapunov function is now V(x) =

n ft /> T1

{v (x-),v,.{x0),...,v. (xv)) . It can be shown that in the inequality V(x) < P V(x), P is given by

"ni2ni3 + ^iinIl

p.. = < (6.79)

13

In (6.77) and (6.78) n.. Ts are positive numbers and £..'s are

IK IT

non-negative numbers. Hence -P in (6.79) is an M-matrix and if the conditions (6.74) are satisfied, the equilibrium of (6,23) is asymptotically stable.

6.9.2 Region ot Attraction [7]

Let thc region of attraction of the free subsystem be defined by

v. (x.) < vQi (6.80)

Define v? = y£$. > Following Weissenberger [5], the region of attraction is defined by

v(x) < y (6.81)

where v(x) = Max jv. (x.)|/p. 1 < i< k

and f - Min (vV/p.)

Ki<k 1 1

p is computed using the properties ot -P which is an M-matrix (see Sec. 6.5) as follows: Since -P is an M-matrix, -P~ is non-singular and all elements of -P are non-negative (all diagonal elements of -P * are positive). Hence, given any vector r > 0, there exists a p > 0 defined by

P ■ £

The application of this method for a power system is briefly sketched below for the uniform damping case given by (6.45). The subsystem Lyapunov function v(X;L' satisfies the following inequalities which can be deduced from (6.48)

*i Si xi « Vxi> t 2i ^ Si

(6.83)

where

Si

1

1

\

T x

2

and

where 3 is the slope of f^to^) at = 0. The bound on ^(x^) is given by (6.50) as

vi(xi> K- "xi 2i x

(6.84)

Since v. (X^J ■ vV2(x.) , the numbers r)^, ii2, n.^ are chosen

as

"il = "If2^' *12 = >M/2<»i>'

m -l

ri3^/2(«;

(6.05)

where A and X_, are the minimum and maximum eigenvalues of the

m M

indicated matrices. The C^j's are computed using a procedure similar to the computation of <*^j's in (6.60). Details are to bo found in Ref. [71.

This methodology has been extended to Lhe non-uniform damping case in Ref. 18] with the stability region defined as

n-1

7. a. v. (x.) < v

i=i 1 1

cc^'s are chosen to satisfy {6.39) with D — P i.e.

where

T

P A. + A P is positive definite with A = Diag (a.). y is

computed as Min (<*.v°). Therefore, the method essentially l<i<n-l 1 1

reduces to that~of the weighted sum scalar Lyapunov function method. However, the majorizing process is different in the two oases. The VLF majorization seems to be more restrictive than the WSSLF method. This perhaps explains why the results are better in the latter case.

6.9.3 Combined VLF and Weighted Sum Scalar Lyapunov Function Approach 19]~

in the weighted sum scalar Lyapunov function method, we chose

majorization on v.{x.) to be v. (x. ) < -o. ||x. || . Instead of

|'x.|' , we can choose other functions such as positive

definite functions u. (x^) so that v^(x^) < -o ^ u.j (x^) . The

analysis can proceed along the same lines as in Sec. 6.3. In

the weighted sum scalar Lyapunov function method (Theorem 6.1)

we imposed no sign restrictions on a^j- *n the case of power

systems, the very nature of majorization yields a^.. > 0 j.

Instead of considering c. and a.. separately, we can assume

l i j

what is called a "one-shot" aggregation by assuming the existence p.. such that

vi(x.) * tgrad v. (Xj^) ]Tlfi(xi) +hi<xi)]

k

< I p. . vi. (x. )u. (x.) (6.86

We can now restate the stability theorem, 6.1, as follows [9]. Theorem 6.4

Given v.(x.) and u.{x.) > 0 and v.(x.) satisfying (6.86), if

X — 1 J. — J.XI rp

there exists a diagonal matrix B - (a^,...,d^) > 0 such that

T

P B + B P is negative definite, then the origin of (6.23) is asymptotically stable.

in the above theorem, p^ may be of any sign. A closer look at this theorem reveals that it is equivalent to Theorem 6.1 with

V^i* = H%ll

+ a i=j

T

Furthermore, p B + B P = 2S. Hence, the two theorems are entirely equivalent. 1f in addition P is an M-ttiatrix, then properties of the M-matrix can be used to get optimum value of a.

The application of thc preceding theorem to power systems has been done in Ref. [10J whose main features are

(i) In the power system model, asynchronous electro¬mechanical damping in addition to mechanical damping has been taken into account.

(ii) Governor action is incorporated for each machine.

(iii) The decomposition is done along the lines discussed in 6,2.1(ii), i.e. there is no overlapping decomposition.

(iv) Thc region of attraction is obtained as the union of Lhe two reqions, one obtained by weighted sum scalar Lyapunov function method and the other as in VLF method.

However, the computation results reported are quite conserva¬tive. The main reason is the type of ma jori ?.at ion used which removes information about the fine physical structure of the power system and the absence of any optimal choice of «j_'s-

Conclusions

In this chapter, we have presented a systematic methodoloqy of stability analysis by large scale power systems by decomposi¬tion. As shown, the weighted sum scalar Lyapunov function approach is equivalent, to thc vector Lyapunov function

approach in most applications considered so far. While applying the VLK approach, the requirement of exponential stability gives rise to overly conservative results.

Further research in large scale power system stability should be directed towards improved methods of decomposition retaining the physical integrity of the system and optimizing the region of attraction.

References

1. Bellman, R. , "Vector Lyapunov Functions", SIAM Journal of Control, Vol. 1, 1962, pp. 32-34.

2. Bailey, F. N., "The Application of Lyapunov's Second Method to Interconnected Systems", SIAM Journal of Control, Vol. 3, 1966, pp. 443-462.

3. Michel, A. N. and Miller, R. K., "Qualitative Analysis of Large Scale Dynamical Systems", (Book) Academic Press, New York, 1977.

4. Siljak, D. D., "Large Scale Dynamic Systems - Stability and Structure", North Holland, New York, 1978.

5. weissenberger, 5., "Stability Regions of Large Scale Systems", Automatics, Vol. 9, 3979, pp. 653-663.

6. Pai, M. A. and Narayana, C. L., "Stability of Large Scale Power Systems", Proceedings of 6th IFAC world Contress, Boston, 1975, Paper 31.6, pp. 1-10.

I. Jocic, Lj. B., Ribbens-Pavella, M, and Siljak, D, D.,

"Multimachine Power Systems; Stability, Decomposition

and Aggregation", IEEE Trans., vol. AC-23, 1978,

pp. 325-332.

0. Jocic, Lj. B. and Siljak, D. D., "Decomposition and

Stability of Multimachine Power Systems", Proceedings Of 7th IFAC Congress, Helsinki, 1978, pp. 21-26.

9. Gruijc, Lj. Ti and Ribbens-Pavella, M., "Relaxed Large Scale Systems Stability Analysis Applied to Power Systems", Proc. 7th IFAC World Congress, Helsinki, 1978.

10. Ribbens-Pavella, M., Gruijc, Lj. T. and sabatel, J. , "Direct Methods for Stability Analysis of Large Scale Power Systems", Proc. of IFAC Symposium on 'Computer Applications on Large Scale Power Systems', New Delhi, India, Aug. 16-18, 1979.

II. Gruijc, Lj. T. , Darwish, M. and Fantin, J., "Coherence,

Vector Lyapunov Functions and Large Scale Power Systems",

International Journal of Control, Vol. 10, 1979,

pp. 351-362.

12. Araki, M., Mohscn-Mctwally, M. and Siljak, D. D., "Generalized Dccomposions for Transient Stability Analysis of MulLi-machinc Power Systems", Proceedings of »7ACC, San Francisco, Paper No. TA3-B, 1900.

13. Chen, Y. K. and Schinzingcr, R., "Lyapunov Stability of Multimachine Power Systems Using Decornposition-Agyregation Method", Paper A-80-036-4, IEEE Winter PES Meeting, New York, Feb. 1980.

14. Vittal, V., "Lyapunov Stability Analysis of Large Scale Power Systems", M. Tech. Thesis, I.I.T., Kanpur, July 1979.

lb. Bergen, A. R. and Hill, I). J., "A Structure Prcscrvinq

Model for Power Svstem Stability Analysis", IEEE TranG., Vol, PAS-100, Jan. 1981, pp. 25-35.

16. Pai, M. A., "Stability of Large Scale Interconnected Power Systems", Proc. IEEE International Conference on Circuits and Computers, Oct. 1-3, 1980, Port Chester, New York, pp. 413-416.

17. Michel, A. N., "Stability Analysis of Interconnected Systems", SIAM J. Control, Vol. 12, No. 3, Aug. 1974, pp. 554-579.

18. Michel, A. N. and Porter, D. W., "Stability of Composite Systems", Fourth Asilomar Conference on Circuits and Systems, Monterey, California, 1970.

19. El-Abiad, A. ll. and Nagappan, K. , "Transient Stability Region for Multi-machine Power Systems", IEEE Trans., Vol. PAS-85, No. 2, Feb. 1966, pp. 169-178.

20. Siljak, D. D., "Stability of Large Scale Systems", Proceedings of the 5th IFAC World Congress, Part IV, June 1972, Paris, France.