FP5 - m

ESTIMATING THE DOMAIN OF ATTRACTION FOR NONLINEAR FEEDBACK SYSTPIS*

Kenneth Loparo Systems Engineering Department Case Western Reserve University

Cleveland, Ohio 44106

Abstract

An iterative procedure is developed for estimating the domain of attraction of a class of nonlinear systems. The procedure uses the Volterra series representation for the solution and is suitable for automatic computations, The algorithm is applied to the transient sta- bility problem in electrical power systems and a favorable comparison with previous criteria is obtained.

1. Introduction

This paper addresses the problem of esti- mating the domain of attraction of nonlinear systems which have a Volterra.series representa- tion. The paper is divided into three sections. In Section 2 we list assumptions which guarantee the existence of a Volterra series and we con- struct the Volterra kernels. In Section 3 we develop an algorithmic procedure for estimating the domain of attraction for nonlinear systems based on the Volterra series representation. In Section 4 the results of Section 2 and Section 3 are used to estimate the domain of attraction of the classical damped harmonic oscillator, This system may be regarded as a model of a two- machine power system, i.e., a round-rotor gener- ator transmitting power to an infinite buss, in the transient regime [3]. The results obtained are compared with those presented in [3] and [SI and a significant improvement is observed.

Gilmer Blankenship Systems Engineering Department Case Western Reserve University

Cleveland, Ohio 44106

Al: Let hi, i=1, 2 , 3 , . . . , n be the eigen- values of A, then Re(Xi) 2 0, i=l, 2, 3 , . . . , n.

A2: f ( a ) :Rm * Rm is a uniformly bounded ana- lytic map on Rm with fQ) : 0.

Given system (2.1) and assumptions (Al) and (A2) we consider the existence of a Volterra series representation for the solutions of (2.1). To facilitate the discussion, we introduce the p-forms of an n-vector.

Definition 1: Let x E Rn, then x['' is the lexio-

graphic listing of the N(n;p) = ( p ) linearly independent terms of the form

n+p- 1

(2.2)

where xl, ..., x are the components of x and the

terms Cnp(p1,p2, ...,p ,) are normalizing constants

chosen so that I 1x1 I p = I I { ,where 1 1 9 I { is the

Euclidean norm, p > 0, and f: pj = p . In fact,

n

j - j =1

2. General Theory

Consider the nonlinear system

dxO = Ax(t) + Bf (u(t)) dt ; X(O) = xoE~n u(t) = Cx(t) (2.1)

where A, By C are matrices such that A:Rn +. Rn, B:Rm * Rn and C:Rn * Rm. The function f (*):Rm * R".

The following assumptions are required,

Definition 2: Let A:Rn + Rm be a linear map, if y = Ax, then ALP' is defined to satisfy ylP1 = AIPI,[PIe

Definition 3: Let A:Rn * Rn be a linear map, if = Ax(t), x(0) = x then A is defined to

[PI satisfy %[PI = A x[p], x[p1(0) = x. [PI .

[P 1

*This work was supported by National Science Foundation Engineering Grant ENG 75-08613.

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Remark: A is a linear func t ion of the e lements

a i j of the matr ix A. For more information on

t h e s e q u a n t i t i e s i n c l u d i n g t h e i r r e l a t i o n s h i p t o Kronecker products, see [1 ,2] .

D e f i n i t i o n 4: I f g (x) is an R"-valued a n a l y t i c

func t ion on Rn, then we write

[PI

m

Using these quant i t ies , w e can represent the so lu t ions o f the sys tem (2 .1) by a Vol te r r a series converging on any compact i n t e r v a l and provide formulas for computing the Volterra kernels . These r e s u l t s a r e summarized i n t h e Theorem below whose p roof fo l lows t he l i ne o f argument used i n [2 ] fo r " l i nea r ana ly t i c " sys t ems . S ince t he estimates developed i n t h e p r o o f are used i n t h e s t a b i l i t y a l g o r i t h m , we p re sen t t he p roo f i n de - t a i l .

Theorem: Let E > 0 and T = be given and l e t x ( t ) be the so lu t ion to (2 .1) under assumpt ions (Al ) and (A2). Then t h e r e e x i s t s an in teger p=p(c ,T) , l i n e a r maps A and C and a vec to r yo such tha t

P P '

sup I l x ( t ) - x a ( t ) 1 I < E w h e r e x a ( t ) s a t i s f i e s O < t T

& N(n;k) we have A :B -+ R, % Nx P

(2.4)

Proof of Theorem 1: L e t E > 0 and T > 0 be given.

By assumption (A2) and d e f i n i t i o n 4 we have:

f ( o ) i~ 2. $(k)x[kl where $(k) I F(k)C[kl

k= l (TI )

Define the funct ion i ( o ) to be the t runca t ion of f @), t h a t is:

where p=p(~,T) is an

D e f i n e z ( t ) by

z ( t ) = e-Atx(t)

Then z(t) s a t i s f i e s

in teger to be chosen la ter .

@3)

y - g ( t , z ( t ) ) , z ( 0 ) = zo = x.

where g ( t , z ( t ) ) = e-AtBf(CeAtz(t)),

Let Z(t ) represent the "approximate" solut ion t o (T4) i n t h e s e n s e t h a t i ( t ) s a t i s f i e s

-I d p i ( t , i ( t ) ) , i ( 0 ) - Zo = x. (T5)

where w e d e f i n e i ( t , i ( t ) ) = e-A'Bt(CeAti(t)) with f ( * ) given in (T2).

Consider the fol lowing sequences of i terates:

z$) = zo

z l ( t ) = zo + g(x , zo ( s ) )ds * l 2 z n W - zo + g(s,zn-l(s))ds

and for the "approximate" system:

io(t) - z

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where J(T) E K(s)ds and M(T) is given by (T6) a and K( t ) by (T7). Furthermore, i f we d e f i n e

z ( t ) = zo + 2 [ z k ( t ) - z k - l ( t ) ] , t h i s series has

nth p a r t i a l sum zn(t) and we have tha t k = l

I I z ( t ) - z n ( t ) 1 1 2 M(T)T[exp(J(T))-ll rn n!

Thus, on any compact i n t e r v a l t h e series converges uniformly.

(T9)

Our o b j e c t i v e now is t o estimate t h e e r r o r between z n ( t ) and i n ( t ) g i v e n by (SI) and (S2) r e spec t ive ly .

From (Sl) and (S2) we have:

/ [ z l ( t ) - a l C t ) 1 f 5 g(S,Zo(S>)-~(sazo(S)) 1 Ids x i 1 (TI01

Since (T2) r ep resen t s t he t runca t ion o f t he series rep resen ta t ion of f up t o o r d e r p , g i v e n q > 0, t o be spec i f ied la ter , t h e r e e x i s t s a n i n t e g e r p (n) such that :

I l f ( o ) - I(o) I I < q f o r a l l u E R~ (TU)

Remark: } l g ( t , z ) - i ( t , z ) I t i I I

T[ l K ( ~ ) d s ] ~ - ’ T[ p ( s ) d s ] n - l +

(n-2) ! (n-1) ! 1 (TI51

Let z ( t ) b e d e f i n e d as t h e s o l u t i o n of (T4) and i ( t ) t h e s o l u t i o n of (‘€51, then by our previous estimates, (T9) and (T15),

I I z ( t ) -Z( t ) l I= l i m l I z n ( t ) - i n ( t ) 1 1 5 YTlexpJ(T)-ll n+m

0 1 6 1 If w e select q > 0 such tha t ,

n < EbA(T)T[expJ( t ) - l l I IBI I (TI71

and p(e ,T) to sa t i s fy (TU) then:

I l z ( t ) - z ( t ) l I < E f o r a l l t E [O,T] (T18)

Define

F(T) = noA(T)TI I B I I [expJ(T)-lI (T19)

t h e n f o r any E > 0 an& T given w e must s e l e c t p(c ,T) such that F(T) 2 E to guarantee uniform convergence.

To compute the approximate so lu t ion we ob- s e r v e t h a t ,

x a ( t ) = e H(t) A t (T20)

s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n g i v e n i n (T21), P (E ,TI

a d t x a ( t )= Axa( t ) + P(k)xa [k l ( t ) a Xa(O) = x. k-1

Ef: (T21)

I f w e def ine y( t )ER by

x:’ ( t ) I T (T22)

where the matr ices B(k), k - 2,3 , . . . , s a t i s f y t h e r e l a t i o n s h i p s ,

- d x[k1 = B‘k)y(t) d t 0 2 4 )

and the ma t r ix B( l ) i s given by B( l ) P [F(’), p ( 2 ) , s .. , F(P) ] (T25)

and cp = [Inxn, 01

The proof of the theorem is complete. E From the theorem we observe that the approx-

imate t r a j e c t o r y x a ( t ) c a n b e w r i t t e n i n t e r m s of A t

t h e t r a n s i t i o n m a t r i x e of Equation (T23) as i s g iven in (2 .5) .

A t P x a ( t ) = C e

Using the representat ion of x a ( t ) g i v e n i n (2.5) we want to inves t iga t e t he a sympto t i c s t ab i l - i ty p roper t ies o f sys tem (2 .1) by examining the t r a j e c t o r i e s of x a ( t ) . To accomplish our object- i v e , i n t h e n e x t s e c t i o n we d e v e l o p a n i t e r a t i v e procedure which allows us to estimate the domain of a t t r a e t i o n of t he equ i l ib r ium so lu t ion x - 0 of (2.1).

P YO (2 * 5)

3. An I te ra t ive Procedure for Es t imat ing the

S tep 1: Select any z= 0 so t h a t t h e set Domain of Attraction of System (2.1)

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D = { x cRn:I 1x1 I 5 c 0 ) (3.1)

is a domain of a t t rac t ion of (2 .1) .

S tep 2: With c0 > 0 given by Step 1 s e l e c t a n i n - t e g e r p 2 1 which i s the degree of the approxima- t i o n ?(a) given by Equation (T2) i n t h e p r o o f o f t h e Theorem. Given and p determine rl from (T11) so t h a t

l l f b ) - n(u) l l < l l (E0 ) (3.2)

I l a l I 5 E0/2 (3.3)

f o r a l l a such tha t

Using Equation (T19) and observing that F(T) given by (T19) is a s t r i c t l y monotone increasing func- t i on o f T , determine the maximum value of T = T k O , p) such that ,

F(T) - rloA(T) 1 I B I IT[expJ(T)-l] 2 E ~ / Z (3.4)

Denote t h i s v a l u e o f T by To. Then, as a conse-

have t h a t quence of (3.4) with E ~ , p, r l ( ~ ~ ) and To given, we

I I x ( t ) - x a ( t ) 11' c 0 / 2 , t E [O,To] (3.5)

Step 3: Note t h a t i f I Ixa(T,) 1 1 < c 0 / 2 , then by t h e time invariance of Equat ion 72.1) , from time To on the solut ions of (2 .1) are asympto t i c t o zero

By Equation (2.5) w e have tha t I I Xa(To) I ILEo/2 i f and only i f

Thus, a s u f f i c i e n t c o n d i t i o n f o r I Ixa(To) 1 I 5 E ~ / Z i s A T

I l Y o l I 2 E0/2(1 le O1 I F (3.7)

From t h e d e f i n i t i o n o f y ( t ) g i v e n by (T22) we have tha t

Hence we s e e k t o f i n d a 6 o > 0 such tha t I lxol 156

A T impl ies I I yo/ 12 Eo/ (2 I I e '1 I ) . It s u f f i c e s t o

take 6 o = 6 , t h e maximum solu t ion of A T

6(@-1)/(6-1) = E0/(211e ' 1 1 ) (3 .9 ) Remark: If 6, s a t i s f i e s ( 3 . 9 ) , t h e n I l y o l l s a t i s - f i e s (3 .7 ) and t he set D = {xO€ Rn:I JxoII 5 6,) is an e s t ima te o f t he domain of a t t r a c t i o n of (2.1).

Step 4: L e t E~ = 60, t hen r ep lace i n S t ep 1 by

cl and repeat the procedure.

Remark: The algori thm produces a sequence

= N(ck) , E given, k - 0,1 ,2 , ... with each value Ek being an approximation from below of t h e r a d i u s of t h e domain of a t t r a c t i o n .

(3.10)

The example given i n t h e n e x t s e c t i o n i l l u s - t r a t e s t he a lgo r i thm and t he computa t ions i t e n t a i l s .

4. Appl ica t ion of t h e Theory

The problem examined i n t h i s s e c t i o n is t h a t of e s t i m a t i n g t h e domain of a t t r a c t i o n of t he damped harmonic o s c i l l a t o r which in the contex t o f e l e c t r i c a l power systems represents a round r o t o r g e n e r a t o r t r a n s m i t t i n g p a r e r t o a n i n f i n i t e b u s unde r t r ans i en t s t ab i l i t y cond i t ions 141. Spec- i f i c a l l y , a two-machine system may be modelled by

2 M- 6 ( t ) + a u + s in6 ( t ) -Pm( t ) I 0 (4.1) d t d t

where M i s t h e i n e r t i a o f t h e g e n e r a t o r , 6 ( t ) is the ang le ( i n e l ec t r i ca l deg rees ) be tween t he rotor of the generator and a sha f t runn ing a t synchronous speed, a is t h e damping c o e f f i c i e n t r e s u l t i n g from mechanical effects, and Pm is t h e mechanical power input which can be taken as con- s t a n t f o r t h e d u r a t i o n o f t he t r ans i en t [3 ] and [4]. For the sake of comparison with [3], select

M = 1 .0 , a = 0.5, 6' = 7113 (4 * 2)

where 6* is t h e p o s t - f a u l t e q u i l i b r i u m s t a t e . A t equi l ibr ium in (4 .1) we must have

P = sin8'. m (4.3)

Using ( 4 . 2 ) and (4.3), we can write Equation (4.1) i n s ta te space representa t ion by de f in ing

then we have

which has equi l ibr ium solut ion xe= ( O , O ) T . For s i m p l i c i t y , we s e l e c t t h e o r d e r o f t h e approxima- t i o n t o b e p = 2 .

. .

Fol lowing the resu l t s g iven by t h e Theorem, we compute the ma t r i ces A2 and C2 induced by Equation (4.5). Equations (4.6) and (4.7) re- spec t ive ly g ive t he f i r s t o rde r and s econd o rde r approximations to (4 .5) .

(4.7) where x[11 and x['] are given by Def in i t i on 1, Sect ion 2. By d i r ec t computa t ion , t he ma t r i ces A2 and C2 a s g iven i n t he Theorem a r e

A2 = 1 1 -1.0 -2:osd. 1 (4.8)

-.5 -cos6O 0 0 112sin6'

0

0 0 JT - . 5 -ficos6"

L O O 0 2IJT O J

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c- = P O o o O l (4.9) - z Lo 1 0 0 o j

.~

The a p p r o x i m a t e s o l u t i o n x a ( t ) s a t i s f i e s t h e d i f - ferent ia l equat ion (4 .10) where A2 and C2 are given by (4.8) and (4.9), respectively, and x. i s the i n i t i a l cond i t ion fo r sys t em (4 .5 ) .

Theorem.

4.1 Estimates Based on the Eucl idean Norm

Refe r r ing t o t he a lgo r i thm g iven i n s ec t ion 3 , we observe that the computat ion of the domain of a t t r a c t i o n depends upon the vec to r norm, 1 1 - 1 1 , which is used . In t h i s s ec t ion I I 1 l 2 is t h e Euclidean norm on Rn. I n t h e n e x t s e c t i o n we con- s i d e r t h e u s e o f a weighted norm i n t h e a l g o r i t h m

i . e . , 11x1 I Q = < x , Q x > ~ / ' where Q = Q > O and

C. , e > represents the s tandard inner p roduct on Rn. The a p r i o r i s e l e c t i o n of the norm to be used is no t an en t i r e ly a rb i t r a ry ma t t e r . I n t he ca se o f t h e damped harmonic o s c i l l a t o r t h e t r a j e c t o r i e s i n t he phase p l ane are e s s e n t i a l l y e l l i p t i c a l and

1 f * I I w i l l y i e l d good r e s u l t s . I n g e n e r a l , t h e algorgthm w i l l produce the "best" estimates f o r

t h e domain of a t t r a c t i o n when t h e norm 1 1 - I 1 con-

forms with the contours of t h e t r a j e c t o r i e s i n t h e phase space.

S tep 1: We f i n d t h a t i f I I x o I l 2 5 0.4 then

I I x ( t ) l l 2 + 0 as t -+ a. Thus we t ake = 0.4.

S t e p 2: With c0 given by Step 1 and p = 2 w e com- pu te t he maximum value of T, i . e . , To, such tha t

1 (x(To) - xa(T0)( I 2 ~ ~ / 2 . We f i n d t h a t T

s a t i s f i e s

T

Te'5t[exp{2(e'5t-1)~ -11 - < 2 4 / ~ ~ (4.11)

Maximizing y i e l d s T = 2.

Step 3: To compute t h e f i r s t estimate f o r t h e 0

domain o f a t t r a c t i o n we obse rve t ha t I / y o / I =

l l x o / 1 2 + I I x ~ ~ ) ~ +...+ Ilxo112p , t h e n t o com-

pute 6 we so lve ,

2 2

2 4 2

6' + g 4 = c 0 / ( 2 / / e P "1 1 ) A T

(4.12)

which y i e l d s 6, = .49 upon s u b s t i t u t i n g To = 2 i n (4 .11) . T h e r e f o r e , o u r i n i t i a l estimate f o r t h e domain of a t t r a c t i o n i s

Step 4: Return to Step 1 wi th eo rep laced by ~ ~ = 6 ~ and continue the process. Continuing the process f o r p = 2 y i e l d s a l i m i t i n g domain a f t e r s i x i t e r - a t i o n s

(4.14) D = {x ER : I Ixo l12 5 ,7351

The c i r c l e i n t h e s ta te space descr ibed by che set D is t h e l a r g e s t c i r c l e f o r o r d e r of ap- proximation 2 which l i e s i n t e r i o r t o t h e a c t u a l

2

domain of a t t r a c t i o n f o r ( 4 . 1 ) . By examining Figure 4.1, it is apparent tha t a c i r c l e y i e l d s c o n s e r v a t i v e r e s u l t s , s i n c e t h e r e s u l t s of [3] and [5] given by c u r v e ( i i i ) e n c l o s e a l a r g e r area. I n t h e n e x t s e c t i o n we i n v e s t i g a t e t h e a p p l i c a t i o n of the a lgor i thm us ing a weighted norm.

4.2 Estimates Based on Weighted Norms

The a lgo r i thm p re sen ted i n Sec t ion 3 re- q u i r e s a n i n i t i a l e s t i m a t e f o r t h e domain of a t t r ac t ion . In gene ra l , t he p rob lem o f ob ta in ing t h i s domain depends upon the contours of the tra- j e c t o r i e s i n t h e s ta te space and consequently the norm which one uses. For the damped harmonic o s c i l l a t o r t h e c o n t o u r s are i n h e r e n t l y e l l i p t i c a l and the cho ice of a weighted norm is suggested. An i n i t i a l e s t i m a t e of t h e domain of a t t r a c t i o n is t h e e l l i p t i c a l domain

where Qo and 6, are chosen so t h a t De l i es in- t e r i o r t o r e g i o n ( i i i ) i n F i g u r e 4 . 1 . F o r sim- p4iciFy we de f ine a "primed" coordinate system xl, x2 by ro t a t ing t he xl, x2 system clockwise by

an angle of 0 , 0 - arcsin ( . 75 ) . An e l l i p t i c a l domain descr ibed by the equat ion,

(4.16)

g ives rise t o a n i n i t i a l estimate

De = Cx'CR : I ( x ' ( l Q 0 2 a 1 2 1 (4.17)

where Qo =I d i a g [ l , 71 and x - [x1 , x2] . The 1 9 I T

i t e r a t i v e p r o c e d u r e f o r e s t i m a t i n g t h e domain of a t t r a c t i o n g i v e n i n S e c t i o n 3 w i l l genera te a se- quence of pos i t ive def in i te mat r ices Qk, k = 0 , 1,

2, 3 , . . . with Qo as a n i n i t i a l estimate and con-

s t a n t s 6k, k = 0, 1, 2, 3 , ... where 6o i s f ixed .

To s i m p l i f y t h e i t e r a t i o n s , w e r e q u i r e a t t h e k s t e p t h a t

t h

and that q l l (k) /q22(k) i s a cons tan t . This re- q u i r e m e n t i n s u r e s t h a t a l l t h e e l l i p t i c a l domains generated by the a lgori thm have the same eccen t r i -

c i t y . With 11x1 I Q as a vec to r norm, the induced

mat r ix norm i s given by 1 /AI I Q = (maximum eigen-

va lue L-lAQAL) 'I2 where Q=LTLEL2.

The a lgo r i thm be low i l l u s t r a t e s t he computa- t i ons r equ i r ed when weighted norms are used.

S t e p 1: From Figure 4 .1 w e o b t a i n t h e i n i t i a l weight ing matr ix Qo and parameter eo such that

Dl = {xAcR2: ~~x~~~~~ 5 E ~ )

is a domain of a t t r a c t i o n . From our previous ana- l y s i s we have,

Qo = d i a g [ l , . 2 5 ] , eo = .25 (4.18)

S t e p 2: Consis tent with the a lgori thm of Sect ion 3 , we de f ine a t a r g e t se:

Eo = CxAcR2: I lx0l lTo 5 ~ ~ / 2 )

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where Qo = diag[.707, .177] . Next w e compute t h e maximum value of T , i . e . , T o , such t ha t

-

1 Ix(To) - xa(To) I 1 - < c0/2 Qo -

(4.19)

We must f i n d t h e maximum value of T so that (4 .20) holds

~ ~ / [ 8 ( 3 ! ) ] u , ( T ) T [ e x p J ( T ) - l ] '< E0/2 (4.20)

Solv ing (4 .20) y ie lds To = 2.71.

Step 3: With To given we obse rve t ha t by t h e tri- angle inequal i tyn

I / Y o I l 2 5 c I / xoI I 2k

k = l

(4.21)

To compute 6 we f i n d t h e maximum value of 6 which s a t i s f i e s O

(4.22)

Solving (4 .22) yields

6 o = .595 (4.23)

Our estimate f o r t h e domain of a t t r a c t i o n a f t e r o n e i t e r a t i o n is,

where

Q, = diagr1.336, .384]

Step 4: Return to S tep 1 with replaced by ~ ~ = ; 6 ~

and cont inue the process . I f we cont inue the pro- c e s s , we f i n d t h e l i m i t i n g domain of a t t r a c t i o n f o r degree of approximation (p=2) to be

D = c X ~ E R : I lxAl I Q 5 .7351 1 2 (4.25)

where

Q = diag[1.716, ,4291

In Figure 4 .1 we p r e s e n t o u r r e s u l t s f o r

11x1 l 2 = (x x) and 11x1 I Q = ( X ~ Q X ) ~ ' ~ f o r t h e

degree of approximation p = 2. Curve ( i ) is an exact representat ion of the boundary of the domain o f a t t r a c t i o n i f a = 0; a r e su l t ob ta ined u s ing t he equal a rea c r i te r ion deve loped by Kimbark. Curves ( i i ) and ( i i i ) a r e e s t i m a t e s o f t h e domain of at- t rac t ion ob ta ined us ing the second method of Lyapunov [ 3 ] and [5]. Curves (iv) and (v) are our e s t i m a t e s f o r p = 2 using the Euclidean and weighted norms, r e spec t ive ly . Our b e s t e s t i m a t e f o r t h e domain of a t t r ac t ion fo r Equa t ion (4 .1 ) i s t h e union of regions (iv) and (v) , This i s f o r p = 2. Inc reas ing p by a small number does no t necessar i ly r e s u l t i n a l a r g e r e s t i m a t e f o r t h e domain. How- eve r , fo r p s u f f i c i e n t l y l a r g e , i n c r e a s e s i n p do r e s u l t i n b e t t e r e s t i m a t e s [ 7 ] .

T 112

Figure 4.1 Es t ima tes fo r domain of a t t r a c t i o n of system (4.1).

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Willems, J . L . , "A P a r t i a l S t a b i l i t y Approach to the Problem of Trans i en t Power System Sta- b i l i t y , " I n t e r n a t i o n a l J o u r n a l of Control, Vol. 19, NO. 1, 1974, pp. 1-4.

Willems, J . L . , "Direct Methods fo r T rans i en t S t a b i l i t y S t u d i e s i n Power System Analysis," IEEE Transactions on Automatic Control, Vol. AC-16, NO. 4, August 1971, pp. 322-341.

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