Algebraic Models of Power Systems

Cop\l"i~hl COllgress.

© I F.\C 1 11 h Trit"lllli ;tl Il"orld ' Ldlilll\, F.'lolli ;1. l 'SSR . I ~ I~ltl

STABILITY THEORY OF DIFFERENTIAL/ ALGEBRAIC MODELS OF POWER SYSTEMS D.

J.

Hill, I. A. Hiskens and I. M. Y. Mareels

/)''/1''1"1111 1' 111 ut l :"/I'/"IIim/ l': lIgilll'l'I"illg & CUIII/lllln .)cit/IU'. ( ' lIh ',' nilv ut .\" ''''1'11.\1/",

AlISlmli"

Abstract : Lyapunov stahility results an:: given for ditferential/algebraic models of power systems which include the elkct of generaTor damping and nonlinear load s. The glohal dynamical structure of such a system is studied in Tenns of multi valued energy functions defined on so-ca lled 'voltage causal regions' where voltage behaviour is predicted from angle behaviour. These regions are separated hy ' impasse surfaces' related to singularity in the load tlow equaTions. Keywords : Power Tr:\Ilsmi ss ion, stahiliTy, Lyapunov methods, nonlinear SYSTems.

INTRODUCTION

Consider now a network consisting of no buses connected by transmission lines. At m of these buses there are generators. The buses which have load but no generation are labelled i= I , ... , no-m . The m:twork is augmented with m fictitious buses representing the generator internal buses, in accordance with th,. classical machine model. They are labelled i+m where i is the "US number of the corresponding generator bus. The TOtal number of buses in the augmenteu system is therefore no+m:=n,

Direct methous of transient stability assessment for power sysTems rely on simpl itied nonl inear equation model s. The usual moud assumes impeuam:e loaus anu uses a network reuuction to derive a model as a SeT of coupled (swing) differemial equaTions (Pili, 1981). Bergen anu Hill (1981) suggesteu using mouds where The loads and neTwork STruCTure are preserved . This approach !eaus naturally to mouds which are of uifferential/alge hraic type . More recemly such model s have heen used as a ha sis for voltage stability analysis (Kwatry, Pasrija anu Bahar, 1986), Thus the theoretical hasis for direct methods using structure pre serving mouds is uepenuant on stability theory for uiffen:ntial/ algehraic equations. This paper provides funher results in that direction .

The network is assumed lossless, so all lines (including those wrresponding to the machine transient reactances) are modelled as se ries reactances. The bus admittance matrix Y is therefore purely imaginary, with elements Y,j =jBij . Let the complex voltage at the ith bus be the (time varying) phasor Vi = IVil LOi wh.:re Oi is the bus phas.: angle with respect to a synchronously rotating reference frame, Define

Hill and Mareels (1I)gS) haw given some basic result s on Lyapu nov stability of differential/algebraic systems and used these to justify use Of:Ul ent'rgy function for undamped power systems . Hiskens and Hill (19K9) have explon::d more practical aspects of using this energy function: thi s work identities several theoretical extensions which should be made . Of these , the two considered here are as follow s. Firstl y. the theory is easil y extended to all ow for generator damping. Secondly, an improved decomposition of the state-space is presente.d: so-called voltage ca usal region s are defined as open se ts which are separated by 'impa"e surface s' of algehraic s ingularity and within which ordinary differential eyuation theory can be use d. This helps to fonnulaTe praCTical al gorithms for finding the region of Tran s ient stability. Funher, there are useful too" for analysis of shon-tenn voltage collapse .

IVI = [IVll, ... , IVnol]', where 't ' denotes matrix transpose , The bus frequency deviation is given by Wi =

J ,.

Using machin.: n:krence angles , we take the nth bus as th.: reference. We use the internodal angles ai:= Oi - on ' Define

~ = [UI , ... ,an_tl T and C!!.g = [WnO+I , ... ,wnl'· Let Phi and 01" denOTe the toTal real and reaCTive power leaving the ith bus via transmission line s. Then p\d Q., IYI ) =

I

IVillVjlBijs in(ai - aj)

(la)

j=1 n

Qb,(Q.. IYI ) =-

DIFFERENTIAL/ALGEBRAIC POWER SYSTEM MODEL

I

IVillVJIB,los(ai -aj)

(I b)

j= l

In

Model Developmem

IV,I Un

TIle classical machine moud is useu in the uevelopment of this model. Therefore The synchronous machines are repre semed hy a constant voltage IE,I in se rie s with transient reactance . Th is assumpTion corresponds to ignoring tlu x decay and having an exciter which is too slow to al'! i.1l the u .ulsient period .

these

equations .

We

= IEi_nol , i = Il{) + I , ... , n

assume

the

substitution

has been made. Also we take

:= 0 ,

Now consider the modelling of loads. Denote the real and reacTive power demand at the ith bus by Pdi and Qdi respectively, In

I ~)

PJi = p"J, QJi = QJ,(IV,I)

(2a)

i = 1. .... 110

(2b)

There are unresolved difficulties in allowing voltage dependant rt!alluads. The last component of the model to he cons idered is the generator dynamit:s. given by the swing t!quatiuns

where M J an: the inertia t:ullstants. DJ the gellerator damping cun-

(7c)

lb,(<'!.IYI) + MYI ) = Q

general these powers are functions of voltage and frequency. For tht! tht!ory tu b.: ,kvdupt:d. the luads must b.: rt!stricteJ tu satisfy

where Mg. Qg are diagonal matrices of inertia. damping constants. Note that use of ~M requires a reference shift for ~g so that (43) remains valid. Partition <'! as <'!' =

[Q] iliz1so the loads

can be identified. Also define fg(<'!g.QJ.IYI) := fg(<'!g.<'!l.IYI) - fM(IYI )

(8a)

EI(~.gl.IYI) + fJ(IYI)

(Rb)

t{gg.GJ . IYI) :=

stants. and P~tJ the mechanical input powers . We assume that

~(gg.QJ.IYI) := llylj-I(2b(<'!g. QJ.IYI) + Qd(IYI ))<8C)

D, ;!L 0 fur at kast une i=1. .. .. m. The usual assumptions of constant met:hanit:al puwer. and the nt!twurk b.: ing in sinusoidal steady state are made.

where lllj denotes ldiag r a, 1 j for vector 1I . Then (7) can be written

Combining the pOWt!T balant:e and swing equatiuns gives the total system representation

fig = I~g

Mi-n"Wi + Di_,~,wi + Pbi(Q..IYI) = P'r:".i-n" ;i = no+ I • .... n

(9b)

(43) (4b) Qb,( ~. IYI)

= - Ow(lV,I) ; i = I • ... nu

Define£;, = [Pbl . .. .. Pb.n-Il = vectors

refening

f(~ = [-

E:'

to

(4c)

lE: f~l where Pd'g are no. (m- I)

loads.

generators

respectively;

E~I where

f~ = [P~I"

P~Jll-tl' ; ~ = IOhI . ... . Oh",I'; and

Equations (9). (10) describe the model which all further results are ba.-;ed on. We note it consists of a set of differential - algebraic (DA) equations. The system variables are clearly ~ E IRm. gg E 1R m-I. <'!l ERn" and IYI E IR~" It is easy to check that the bus power transformation (and associated frequency reference shift) implies the equilibrium points are given by ~ = () and

Q = I-Q"I . .... -0,,""1'. Also set P II := P~m ' From (la)

P~:= - PT

I(Phi - pl') = - I i=l

(5)

fg(<'!g.<'!I.IYI) = Q

(lla)

Q

(lib)

~(~g.~ . IYI) = Q

(llc)

ft(gg.QJ.IYI) =

i=l

It is cunvenient fur this excess bus puwer tu b.: distributed across generator powers in proportion

10

damping. Define Local ODE Representation (6)

Here it is shown that the model is locally equivalent to a set of ordinary differential equations for almost all operating states. The load bus variables QJ.IYI are rdated to the generator angles <,!g by the algebraic equations (lO).ln fact. (10) defines an (m-I)

m

where

Dr := I D i(

;!L

0 by assumption ).

i=l

Define the modified real power vector

E' = [- ElE~l . Then from

_[:! a~~I]

(4b). (5). (6)

I

- manifold on which <'!g can flow. Define the lacobian (12)

hl(Phi - Pi)=O

a~1 aQJ

So wbere.L._I is the (n-I ) vector with uniry entries. Define I g=ll,- I: - In-tl where In-I is the (n-I) identiry matrix. Then (4) can b.: rewritten

(7a)

(7b)

~ alYI

Then. by the implicit function theurem (Fleming. 1977). if det 111 ;!L O. locally the load bus variabks can b.: written explicitly in terms of the generator angles as (13)

An equivalent differential equation form can therefore he obtained lucally by substituting (13) into (9a). Setting

£;(~) := £g( Q:g,re(Q:g). Jt(Q:g))

It follows from the implicit function theorem and Assumption A2

gives the mood

~ ~ =- M~ IQiQg - M~ I!~( E;(~) - E~)

ag = IgW.g

that

(x, y) E Q G

given

there

is

some

neighbourhood

U C Rn of X and a unique twice differentiable function

(14a)

u : Rn

--+

R"', u E C 2 (U)

such that

(14b) 0= g(x. u(x»)'Vx E U and (U x u (U»G C Q G

Equations (14) define ordinary differential equations (ODEs) which are locally equivalent to the DA system .

with lacobian

This idea of local solvability will he extended later to solvability uver disjoint regions . Let A:= U x u(U)

STABILITY THEORY OF DIFFERENTIAL/ALGEBRAIC SYSTEMS

Lemma 1. In the neighourhood AG(X) E QG , the system (13) reduces to

This section develops a useful result on the stahility of equilihria in general DA systems . The general topi<.: of Lyapunov stability for such systems has heen studied in Hill and Mareels (19118). The result reyuin:d hen:: is a LaSalk lnvarian<.:e version of an asymptOTic stahility criterion. This is easily developed using ideas given in Hill and Mareds (1988).

x = f(x, u(x»

We assume that the system (15) has a unique (isolated) equilibrium in Q , which we regard to be the origin, without loss of generality.

We consider DA in the general fonn

x = f(x,y)

(15a)

A3. In Q, f(x, y) = 0 and g(x, y) = 0 iff(x, y) = 0

0= g(x . y)

(l5b)

Remark

with sume <.:ompalible initial conditions, (xo, yo) , i.e. 0= g(XO,yo), where f: R" x Rill

--+

R".g : R" x Rill

--+

In order to satisfy (A3) it may be necessary to shrink the set Q of (AI) and (A2) to a smaller subset.

Rm .

When discussing stability in the DA system context it should be clear that we only consider stability with respect to penurbations which satisfy the algebraic constraints. When using RS representation, this feature has been accounted for.

We assume tluoughout : A I. f and g are twice continuously differentiable in some open COIUlected set Q C R" X Rm. i.e. Cg E C 2 (Q)

We now present the fonnal definitions of stability of the trivial solution (x(t 0,0), y(t,O,O» '" (0,0) of the DA system (15).

A2. The Jacohian of g with respect to y has constant full rank on Q , i.e. rank (D2g(X. y»

=m

Definition 1. The trivial solution of (15) is called stable if given €> 0 , there exists a 6 > 0 such that for all (xu, Yu) E Q G n Bd

'V(x . y) E Q

then (x(t, xo.yo), y(t, xo.Yu» E Q G n B f ,'Vt E R+ .

We use the following notations: x(t, xu, Yo), y(t , xu, Yu) are solutions of (13) and initial conditions Br { (x , Y) E Rn

G

= /(x,y ) E Rn

IT = dosure of K = { a : R,

~

X

R m : II (x,Y ) II
X Rm

:

g(x.y ) =

Q in Rn

X

a.~

L>elinition 2. The trivial solution of (15) is called asymptotically stable if it is stable and there exists TJ > 0 such that for all

a fun<.:tion of time

(xu.Yu) E Q G n B~ then

I lim

0)

t --+ ao

R"'

R t continuous . strictly increasing. 3(0)=0

(11 )

II (x(t, xu, Yo) , y(t, Xo, YU» 11= 0

It is straightforward to derive versions of the basic Lyapunov stability arguments for DA systems. Some basic results are given in Hill and Mareds (1988). In the later power system analysis we need a LaSalle Invariance type result.

I

Theorem V :Q

V

(16)

= derivative of the function V with respect to time along

--+

1.

Suppose

there

exists

semi-definite derivative on Q G ,i.e.

the solUTion of the system with equations (n). V(X , y) ~

We now <.:onsider stability propenies of equilibria of the general DA system (15). Firstly note that 3 local ODE description - exemplified by (13), (14) - can be given.

21

a

CI(Q) function

R + such that V is positive definite and has negative

a(ll (x,Y) III

whe:re:!J is a scalar and lpq de:note:s a pX4 matrix with all its elcment> c4ual to I. ( To sunplify notatiun , the: dum:nsion~ will Ix omitted .) The scalar!J is chosen to ensure £I(,U) > () . Note that

un QG fur ,um, a E K. L<:l

EI(O)

= 4M• . Thi ~ function can Ix evaluated as

( 17) and M Ix th, i;.lfg<:>t invariaJll ><:t within S . Funh,r defin,

Then the trivial solution (0,0) of the DA system is stahle,

n V;;I

x(t) -- M

a~ t -- cc and the dumain uf attractiun cun-

Define the constraint manifold

tains v;;t . Proof: Lenuna I

givt:~

that thert: t:xi~b a nt:ighbuurhuud 1\(0) uftht: uriDifferentiating V on G gives

gin in which DA ,y,tem (15) is equivalent to the ODE (16) Let r E IR+ be ~uch that B rG C 1\ n Q G Within Br. the correspon-

_ 1

V

ding arguments for ODE systems can he used (LaSalle, 1<)76; Ruuche: , Halxb and Laluy, 1977).

(20)

whe:re

o

(21)

Rl:lllark~

A funher re4uirement on!J is to make 'YJ(iJ) :S O. Note that

a) In general, solution of ( 15h) yields multiple values ofy for t:a<.:h x. Fur t:ach branch, systt:m (16) and V(x,u(x» i~ wdldefined. However, on Q , we must regard these as multivalued. For ul~t,ulce. V may be repre sented by multiple surfaces.

Zo(O) :S

14 .

more close ly related tll the damping. This is considered in much more de:tail by Hill and Chong (1989) following the results for impedance load systems hy Willems (1970). In the special case of zero or unifonn dwnping , the: kinetic energy tenn becomes the familiar

Stability Result An ohvious starting point for large disturbance stability assessment is the detennination of a~ymptotically stable equilibrium point s (EPs) The following result establishes the connection betwe:en small disturbance stability and asymptotic stability for EPs. Small disturbance stahility refers to asymptotic stabiliry of the line:arize:d sy~tem.

Energy Function (Lyapunov function candidate) Tht: development of energy functions fur the DA modd has been studied el sewhere (Narasimhamunhi and Mu savi. 1984; Hill and Chong, I <)R9) using first inte:gral and Lur'e: problem analysis methods. Here we s ummari se from Hill and Chong (1989) . A valid energy function is

Theort'nt 2. If an equilibrium poult ~c is "small disturbance stable", then it is asymptotically stable in the sense of Definition 2.

I

z,

and

gives the simple kinetic energy function

negative) values of dwnpulg However, better estimates of stahility regions can he ohtained a with a value of f.i which is

[n this and the following sections, we provide some has ic methodulugy fur de:te:nnining large: di~turbance: stability of the EPs in the DA power system model. Emphasis will he given to ne:w insight~ on the: nature: of e:ne:rgy surface:s in the: pre:se:nce of multiple equilihria :lI1d the statement of stahility re su lts for the difft:n:ntial-a1ge:braic e:4uation mode:1.

~ = (~~'~I,r\{.I),l.!W = (fg
=0

~fQ;MgfQg . TIlis re:mams a valid ene:rgy function for any (non-

STABILITY RESlLT

V(fQg.z) =~fQ~I(,U~g+ J

O.

Clearly f.i

b) Clearly. the: ~tability re~uil folluw~ ea~ily fwm one: fur the re duced system. However, the reduced system is nO! u~ually known . So ~tability I.'onditiuns which wurk dire:ctly on functions f, g in ( 15) are needed .

where

1

-"2 ~ZdJ.t '!l!4

( 18)

~s denO!es

Th.: proof of this theorem is given in Hill , Hiskens and Mareds ( 1989).

a stahle EP. £1 is given hy (19)

l), )

Similar re:sults havo: Ixe:n uo:rive:u Ixforo: lDo:Marro anu Bagen. 19K4). hur have relied on singular perrurhation results. In so uoing Iho:y have placo:d conditions on the sign of III which are not required . (; I.( m\1.

surface. voltage behaviour can no longer Ix predicted from the DA model. The impasse surface I is given by

1>":\.\.\11(:,\1. STRU TUn:

In this so:ction, wo: IllOVO: Ixyonu the local ODEo:yuivalence given hy Lemma I to study the DA system as glohally decomposed into multipk ODE systo:ms on ro:gions bounueu by surfaces of algo:hraic singularity. Then the energy function picture is extended to iIlustrato: tho: significanco: of V Ixing multivaluo:u.

Note that G

=(

~' C;) UI .

1=<)

Define augmented algebraic constraint function

ODE Decomposition Our first ro:sult will o:stabJish tho: ODE do:composition. Define open sets C I as

Fal·t. Suppose rank Di = 2no+ I at a point p in I. Then in a neigh bourhood of p, I is a differentiable (m-2) - manifold.

TIle se sets may not be connected. Partition each Cl into its conk

nc:rtc:u componc:nb C II .

CIk

.

i.c: . Cl

= U Ch

and

From this fact we can build a picture of I as composed of intersecting differentiahle (m-2)- manifolds. On each of these manifolus, hi has exactly one zero eigenvalue. They intersect at lower dimensional manifolds where 2 or more eigenvalues are zero and rank D1 < 2no+ I . It fI!mains to determine whether Di has full rank at all non- intersection points, i.e. does some (m-2) - manifold segment have boundary ?

i= l

Now projc:u the sc:ts Ch onto the generator angle components. Define (23)

Thus the glohal structure of DA is established: the constraint set G consists of disjoint open sets Ch which are separated by the impasse surface and within which the dynamics is given by a local ODE description.

Theorem 3. On each Cli . the set G is represented by unique continuous functions 'fh: Ati ..... IR f!1 =

tP';(f!~).IYI

=

1p"(f!~ )

n

,

tE,h: Ah ..... IR

nO

such that

. The DA system (9). (10) is equivalent

to the local ODE representation (13). (14). In Theorem 3. local solvability of (ID) was extended to solvability over voltage causal regions. The same concept can be used to extend the region of validity of the local representation of V. An estimate for the region of attraction for a stable EP, b , of the DA model can then be detennined . Let the number of negative eigen-

Proof l Outl ine): For any point (~ . gj , Iyl*) E Cli, the uefUlition of Cli implies the result in a neighbourhood. i.e . there exists functions which relate

~.1YI

to

~~

tli,tE,ii

values of

in the nc:ighbourhood.

!JJ

Ib

be I . By Theorem 3 there exist unique con~I

tinuous functions

= 'f' ,(l!g), Iyl = tE" i(l!g) such that over a

voltage causal region Ch, V at (18) can be written

Since Cli is open. there is such a neighbourhood for each point contained in C" .

=+~PI(,u~g +

The result follows from uniqueness of 'f~ ' tE,~ on each neighbourhood and a continuation argumc:nt.

o

J


(~»),~ >

(24)

~

Define the sets

(a) It is easy to generate examples of systems where the sets C, are not connected (Hiskens and Hill , 1989b). (b) The sizes of sets Cli dc:pend greatly on the load model parameters . Conditions can be given for ensuring C, I "" 0 are empty (Hiskens and Hill. I Y~Ya) .

Note that the elements of

sh are simply points in

(Qg,~, IYI)-

space which correspond to elements of Rh (i.e., points in l!Qg , ~) -

(c) This result sharpens one given earlier by Hiskens and Hill.1989a)

space) .

An estimate of the stabil ity region is obtained via the following theorem.

The boundaries of the sets C" are referred to as "impa.sse surface s" - a term borrowed from circuit theory (Hasler and Neiryck . 19116).

Tht!ort!m 4. Let

~s

= (Q, ~e) be an asymptotically stable EP of

the DA model. Then , for all k>O such that Rh is bounded and

sb

TIle regions Cli are referred to as voltage causal regions . Within any C" ,the load hus voltages and angles are continuously dependent on the: gene:rator angles . If trajectories meet an impasse

CJ

Cli any trajectory s (t, xu) with initial conditions ~

E Rri has the! following properties:

23

The proof of this theorem is given in Hill. Hiskens and Mareels (1989).

While not hard to illustrate by example (Hiskens and Hill, 1989a), a complete theoretical discussion of these issues remains to be studied. Nevenheless it is already clear that DA models change the traditional view oflarge disturbance stability substantially. For instance, the phenomenon of shon-term voltage collapse can be explained in tenns of "jumps" between different energy levels (Hiskens and Hill, 1989a; DeMarco and Overbye, 1989).

Remarks

I{eferences

(i) set, xo) E Rt, for all t ~ 0 (i.e. Rh is invariant with respect to the DA model). (ii) set. xo)

-+ )«

as t

-+ 00

a) As k is increased, a value will be attained where one of the two conditions on Rr, .

st

Bergen,A.R. and DJ. Hill (1981). A structure preserving model for power system stability analysis. IEEE Trans. Power Apparatus and Systems, Vo!. PAS-lOO, No. 1. pp. 25-35.

breaks down. Either

(i) R~ becomes unbounded. i.e. Vii is no longer locally

DeMarco. C.L. and T.J. Overbye (1989). An energy based security measure for assessing vulnerability to voltage collapse. IEEE Paper 89 SM 712-I-PWRS.

positive definite. (ii)

sh et

Cli. i.e. there are points in

st

for which the Fleming, W. (1977). Functions of Several Variahles. SptingerVerlag. New York.

local model is no longer valid. These phenomena are consistent with definitions of power system stability (Hiskens and HilL 1989a). The li..m.it placed

Hasler, M. and J. Neirynck (1986). Nonlinear Circuits. Artech House, NOfWood, MA.

on k by (i) ensures that all points in Rh are attracted to the stabk EP, b ,i.e. , if k was allowed to increase, then for some 1>.0

E Rt. set, 1>.0)

r

)(.s

as t

-+ 00 .

Hill, DJ. and CN. Chong (1989). Lyapunov functions of Lur'ePostnikov fonn for structure preserving models of power systems, to appear Automatica.

111is is angk instabil-

ity. The limit pla<.:ed by (ii) ensures that the local model and energy functions are valid for all point in

st. In this ca~e, if

Hill, DJ., I.A. Hiskens and I.M.Y. Mareels (1989). Stability theory of differential / algebraic models of power systems. University of Newcastle, Technical Report EE8941.

k was to increase, then for some ~u E St,detIJI = 0, i.e., la<.:k uf voltage <.:ausality.

Hill, DJ., I.A. Mareels (1989). Stability Theory for differential / algebraic systems with application to power systems. University of Newcastle. Technical Report EE8941.

b) Let the largest value ofk satisfying Theorem 4 be kent. This value <.:ould be used in the traditional way as the critical value of energy able to be attained by the distturbed system with stability still guaranteed. This of course is likely to be quite conservative. A practical algorithm will employ information on fault location (Pai, 1981).

Hiskens, I.A. and DJ. Hill (l989a). Energy functions, transient stability and voltage behaviour in power systems with nonlinear loads, IEEE paper 89 WM 152-0 PWRS.

c)A result similar (() this. but requiring all eigenvalues of III to be positive has been derived hy DeMarco and B.:rgen( 1984). Singular penurbation results were used in that case.

Hiskens. LA. and DJ. Hill (l989b). Unified stability theory of differential/ algebraic power system models. University of Newcastle. Teclmical Report EE8942. Kwamy, H.G., A.K. Pasrija and L.Y. Ballar (1986). Static bifurcations in electric power networks: loss of steady-state stability and voltage collapse". IEEE Trans. on Circuits and Systems, Vo!. CAS-33 , No.IO, pp. 981-991.

Multipk Em:rgy Fun<.:tion Sheets If the energy function ( Ill) is treated in the usual way as the sum of kinetic and potential energy tenns, then it is only the potential energy tenn which is dependent on the set C li .111e local potential energy functions are functions of £., only. and so can be conceptualized as (111-1 ) - hypersurfaces (or sheets) in

~g -

LaSalle. J.P. (1976). The Stahility of Dynamical Systems. SIAM.

space. (Re-

Narasimhamunhi, N. and M.T. Musavi (1984). A generalized energy function for transient stability analysis of power systems. IEEE Trans. on Circuits and Systems, Vo!. CAS-31, No.7, pp.637-645.

call the potential energy well concept in energy function methods (Pai,1981).) For each region Cl; defined by Theorem 3, a unique local potential energy function exists. each one a sheet in

~g -

Rouehe, N., P. Habets and M. Laloy (1977). Stability Theory by Liapunov's Direct Method . Springer- Verlag. Applied Mathematical Sciences, Vo!. 22.

space. It is not

difficult to imagine therefore how it is possihle to have a number of asymptotically stabk EPs. (Those sheets with a locally positive definite section must have an asymptotically stable EP at the lowest point of that section.) Note that not all sheets need contain an EP however.

Willems. J.L. (1970). OptinlUl11 Lyapunov functions and stability regions for multimachine power systems. Proc. lEE, Vo!. 117, No.3.

All the PE sheets join on the impasse surface. The sheets can be thought of as approaching each other infinitesinlally closely at the impasse surface.

24