Power System Steady State Stability and the Lf Jacobian

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1374IEEE Transactions on Power Systems,Vol. 5, No. 4, November 1990

POWER SYSTEM STEADY-STATE STABILITY AN D THE LOAD-FLOW JACOBIAN

P. W Sauer (Senior Member) M. A . Pai (Fellow)

Department of Electrical and Computer Engineering

Univers i ty of I l l i no is a t Urbana-Champaign1406 WUrbana,

Abstract

This paper presen ts the re lat ion shi p between a

de ta ile d power system dynamic model and a st and ard load-

flow model. The line ar iz ed dynamic model is examined

t o show how the load-flow Ja cobia n appears i n th e system

dynamic s t a t e J acobian for evaluat ing s t eady- s t a t e S t a-

b i l i t y . Two spec ia l cases ar e g iven for t he s i t ua t i on

when s in gu la r i t y of the load-f low Jacobian implies

s in gul ar i ty of t he system dynamic s ta te Jacobian.

Keywords

Steady-s tate s t ab i l i t y, load-f low Jacobian, power

system dynamics.

1 . In t roduct ion

The subj ect s of vol tage col lapse and vol tage ins ta b-

i l i t y have cr ea t ed a r enewed i n t e r e s t i n l oad-f low

J ac o bi a n s i n g u l a r i t i e s a nd t h e i r r e l a t i o n s h i p t o s t ea dy -

s t a t e s t a b i l i t y [1-10]. While load-flow has been theprimary method used to compute ste ady -sta te con diti ons ,

i t s r o l e i n e v a l u a ti n g s t a b i l i t y h as n o t bee n f u l l y

c l a r i f i e d . I n 1975, V . A . Venikov e t a1 publ ished a

paper which proposed tha t under cer tai n condi t io ns ,

t h e r e is a d i r ec t r e l a t ionsh ip be tween the s ingu lar i t y

of the s ta ndard load-f low Jacobian and the s ing ula r i ty

of the sys tem dynamic s t a te Jacobian C 1 1 1 . This paper

has been ci ted as t h e p r im ar y j u s t i f i c a t i o n f o r s t u d yi n g

tne load-flow Jaco bian matrix t o d e t e rm i ne c r i t i c a l

l o a d l e v e l s . I n t h i s p ap e r , we c l a r i f y t h i s r e s u l t i n

the conte xt of a f a i rl y general dynamic model and show

tna t t h e r esu l t should be cons idered opt imi s t i c for any

type of s t eady- s t a t e s t a bi l i t y analys i s . The paper

includes a tu to r i al on the ro le of load-f low in dynamic

analys i s .

2 . A Detailed Dynamic Model Without Stator/Network

Trans i ent s

Thi s s ec t io n present s a bas i c nonl inear mul t i -machine dynamic model which includes the fundamental

fea tur es of vol tage and f requency con trol , but assumes

that a l l s t a tor /ne twork t r an s i en t s have been e l iminated .

I t has been shown th at fo r pure R-L networks and loads,

these t r ans i en t s a r e very f a s t compared to t he s lower

mechanical or voltage type dynamics, and as such can be

formally e l iminated us ing s in gular per turbat ion and the

concept of in te gr al manifolds [12-141. The eli min ati on

of t he s t a tor /ne twork t r an s i en t s l eads t o a lgeb ra i c

equ atio ns which accompany the multim achine dynamic model

as fol lows:

Green St.

I L 61801

dSi-t = 1 si= l , . . . ,m (1)

dui

Mi dt = TMi-[E' q l- X A I di 11 i -CEhitX~ilqi ' ldi

- D i ( w i - w s ) i = 1 , . . m (2 )

dE' .

TAoi 2 -E' qi - ( X di -X ' . ) Il d i + ~ f d ii=l , . . . ,m ( 3 )

i = 1 , . . m ( 4 )

dEfd iTEi 7 -(KEi+SEi(Efdi))Efdi+VRi i= l , . . . ,m ( 5 )

TAi dtv R i -V +K R KAiKFiR i A i f i - E f d i t K A i ( V r e f i - V i )

i = 1 , . . m ( 6 )

dRf KF

TF i dt = - R f i + E f d ii l l , . . m ( 7 )

% P i T R H iTRHi 2 -T + ( I - KHpiTRHi )PCHi+-

CH i TCHi 'SVii

i - 1 ,.. m (8)

i = i , m (9)

i= l , . . . ,m (10)

- [E ; i+(Xh , -Xh i ) I q i+ jE ; i ] e j ( b i - "2 ) i = l , . m (1 1 )

0 = -Pi- j Qi +Vi e' 'i ( Idi - j I q i )e-' (bi"'2)

+PLi ( V i ) + j Q L iV i 1 i = l , . . m (12)

0 = -P i j Q L i iP ( V )+jQLi(Vi) i=m+l,. . n (13)

89 SM 682-6 PWRSby the IEEE Power System Engi neer ing Committee of th e

IEEE Power Engineering Soc iety fo r pres entation a t theIEEE/PES 1989 Summer Mee tin g, Long Beach, C al if or ni a,

July 9 - 1 4 , 1989. Manuscript submitted January 2 3 , 1989;

made available for printing Ma y 23, 1989.

A paper recommended and approved where the notat ion is s tand ard fo r a machine with onedamper winding plu s fi e ld (two-axis model), IEEE type I

ex ci t at i on sys tem, and s imp li f i ed turbine/governor model

C151.

0885-8950/90/1100-1374$01.00 1990IEEE



1375

The notation f o r an m-machine, I-blls

Viejei = vol tage a t bus i i =

v i e j e i = (vdi+jv . )ej(6i-n '2) i=q l

system i s ,

,..., (15)

,..., (16)

Yikejaik = stan dard load-flow bus admittance matrix

e n t r yi , k = l , ..., (18)

P i + j Q . = net inj ect ed power i nt o bus i from a path

not included i n the bus admittance matrix

i = 1 , . . n (19)

The alge brai c variab les Pi and Qi are introduced s o t h a t

th e standard load-flow equ ations (14) are preserved for

any dynamic lo ad model (1 2) and (1 3). As w r i t t e n , t h i s

full dynamic model contains 10m dynamic s t a t e s ( 6 , w , E ; ,

E&,Efd,VR,Rf,Tn,PCH,Psv) and 2m + 4n real a l g e b r a i c

s t a t e s ( I , I ,P ,Q,e ,V) , and 2m inputs (Vref,PC). Equa-d q

t i o n ( 1 1 ) i s t h e s t a t o r a l g e b r a i c e q u a t i on w hi ch isnormally expressed as a phasor d iagram i n t he l i t e r a -t u r e . W prefe r the c i r cu i t fo rm shown in F igure l .

Figure 1 - Synchronous machine dynamic ci r cu i t

Note t ha t PLi(Vi) and QLi(Vi) ar e assumed t o be

given func t ions which descr ibe th e loads a t a l l n busesas a f u n c t i o n of t h e i r r e s p e c t i v e v o l t ag e s . T h i s

rep res ent at ion al lows most common load models inclu ding

cons tant impedance. There are m+2n complex algebraic

equa t ions which should i n pr in c ip le be so lved for the

2m+4n rea l a l g e b r a i c s ta tes as func tion s of t he 10m

dynamic s t a t e s . The machine currents I d , I c a n e a s i l y

be e l imina ted by so lv ing (11) and subs t i tu t i ng in to

(2)- (12) . The P and Q a l g e b r a i c s t a t e s can eas i ly be

e l imina ted by subs t i tu t i ng (12) and (13) in to (14) ,

leav ing only n complex a lgebra ic equa t ions (14) t o be

s o l ve d f o r t h e 2 n r ea l a l g e b r a i c s t a t e s 8 an d V. These

remaining a lge bra ic equations cannot normally be solved

expl i c i t ly . In the spec i a l case of cons tan t impedance

l o a d s , i t i s customary t o use an inte rna l generat or bus

model and inc lude a l l lo ad s and the machine impedance

R s + j X ; i n th e bus admi t tance mat r ix (en la rged to m+n

buses). With th e add it io nal assumption of X ' = Xh, a l l

a l g e b r a i c s t a t e s can be expl ic i t ly e l imina ted wi th areduced (mxm) admittanc e matr ix. For nonl inea r load

models, th e alge bra ic equation s must be reta ined . This

paper does not introduce internal machine buses.

q

q

3. Standard Load-Flow

Standard load-flow has been the t rad it i ona l mecha-

n i sm fo r computing a proposed stea dy-s tate ope rati ng

point . For th i s paper , we define standard load-flow asthe following algorithm C16-181:

(a ) Spec ify bus volta ge magnitudes numbered 1 t o m.

(b) Specify bus voltage angle number 1 ( s lack bus) .

:c )

:d )

Spec i fy ne t in j ec ted r ea l power Pi a t buses num-

bered 2 t o m.

Specify load powers PLi and QLi a t a l l buses num-

bered 1 t o n.

Solve t he fo l lowing equa t ions ( ( 1 3 ) and (14) rewr i t t en)

f o r e 2 , ..., n , v m + l* . v n t

n

o = - P . + 1 V.V Y . cos(8 - 8 -a i = 2 ,..., ( 2 0 )k= l lk ik) (PV buse s)

no = -P + z v v Y ik c o s ( e . - e -CL ik ) i=m+l , . . . ,n (21)

k=1 (PQ buses)i

n

0 = -QLi+ Z V . V Y . sin(ei-ek-aik) i=m+l,..., (22 )

where P. (i =2 ...,m), V. ( i = l ,..., ), P . ( i=m+l,..., ) ,

Q . i=m+l , . . n) , and are sp ec if ie d numbers. The

standard load-flow Jacobian matrix i s t h e l i n e a r i z a t i o n

of ( 2 0 ) - ( 2 2 ) w i t h r e s p e c t t o e 2 , ..., n , V m + l , . . . , V n .

4 f t e r t h i s s o l u t i o n , c om pu te

k=l (PQ buse s)

L1

L1

n

k=l

Q. = 2 V V Y ik s i n ( 8 . - 8 -a k ) i =2 , ..., (24)

This standard load-flow has many variat ions including

the addi t i on of o ther dev ices such as tap changing

under lo ad (TCUL) tran sfor mers , sw itch ing var sour ces

and HVDC conver te rs . I t can a l so inc lude inequa l i ty

cons t ra in t s on quan t i t i e s such as Q i , and be rev ised t o

di st r i bu te the sl ack power between a l l genera tors .

W would l ik e t o make one important poi nt aboutload-flow. Load-flow i s normally used t o eva lu a te

s p e r a t i o n a t a s p e c i f i c l o a d l e v e l ( s p e c i f i e d by a

given s e t of powers). For a spe cif ied load and genera-

t i o n s c h e d u le , t h e s o l u t i o n i s independent of the

act ual load model. That i s , i t is c e r t a i n l y p o s s i b l e

t o e v a l u a t e t h e v o l t a g e a t a constant impedance load

f o r a s p e c i f i c case where t ha t impedance loa d consumes

a sp ec if ic amount of power. Thus th e use of 'Iconstant

power" i n load-flow anal ysis does not requi re or even

imply tha t th e load i s t r ul y a con stant power device.

I t mere ly g ives the vo l tage a t th e buses when th e load s

(any type) consume a specific amount of power. The

l o a d c h a r a c t e r i s t i c i s important when th e anal ys t wants

t o s tudy th e system in response t o a change such ascontingency analysis or dynamic ana lysi s . For thes e

purposes, s tandar d load-flow usu ally provides th e

" i n i t i a l c o nd i ti o ns . I1

4 . In it ia l Condit ions for Dynamic Analysis

For any dynamic ana lysi s using (1) -(14 ), i t i s

necessary t o compute the in i t i a l va lues of a l l dynamic

s ta tes and t o spec i fy the f i xed inputs (Vref and pc).

T' -T



1376

In power sys tem dynamic an aly s is , the f ix ed in puts and

in i t ia l condi t ions are normally found f rom a base case

load-f low solut ion. That i s , t he va lues of Vref a r e

computed such that the m genera tor vol t ages ar e as spe-

c i f i e d i n t h e l oa d- fl ow . The values of Pc are computea

such tha t t he m generator re al power outputs ar e as

SDecif ied and computed in th e load-f low fo r ra ted speed

,us). To se e how th i s is done, we assume th at a load-

f low solu t ion ( as def ined i n s ec t ion 3) has been found.

T he f i r s t s t e p i n c om pu ti ng i n i t i a l c o n d it i on s i s nor-

mally the c a l cu la t io n of genera tor cur r ent s from (12)and (17) as ,

IGie jy i = ( (pi-pLi (vi) ) - j (Q,-Q,, (v i ) ) / ( viejei

i = l ,. m (25)

and machine re la t i ve r oto r angles f rom manipulat ion of

( 1 1 ) and the algebraic equat ion f rom ( 4 1 ,

6 i = angle of (vie Jei + ( R . + j x . ) I e j Y i )s i q i G i

i = l , m (26)

With the se qua nti tie s, th e remaining dynamic and alge-

bra ic s ta te s can be found by,

i = l , . . m (27)

i = l , .

.m (28)

followed by E f d from (31, ( 4 ) , ( 1 1 ) and (16)

di d i + V q i + R . I i = 1 , . . m (29)

With th i s f i e l d vol t age , R f i , VRi and V r e f i can be found

from ( 5 ) - ( 7 ) a s ,

s i q ifd i = X I

KF

TFi Efdif i =- i = l , . . m (30)

VRi = (KEi+SEi(Efdi))Efdi i= l ,. m (31)

V re f i = Vi+(VRi/KAi) i = l , m (32)

'The in i t ia l values of E' . and Eh i are then found f rom

( 3 ) and ( 4 ) ,

E ' . = - (Xd i -Xh i ) Id i+Efd i i = l , . . m (33)

E& = (Xqi-Xii)Iqi i = 1 , . . m ( 3 4 )

Note tha t i f t he machine s a tura t ion is i n c l u d e d , t h i s

c a l c u l a t io n f o r Eh i and Eh i may be i ter at iv e.

mechanical s t a t es and Pci ar e found f rom ( l ) , ( 2) and

(8) - (10) as ,

P i

91

The

w. = w i = l , . . m (35)

( 3 6 ?

'CHi = T M i i = l , . . m (3 '1

i = l , . . m (38)'SVi ' C H i

Pc i = PSVi+(l/Ri) i - 1 , . . m (39)

1 s

TMi = (E;i-X;iIdi)Iqi+(E;i+X;iIqi)Idi i = l . . . . , m

This completes the computation of a l l dynamic sta te

i n i t i a l c o n d i t i o n s a nd f i x e d i n p u t s .

For a given dis turba nce, th e inpu ts remain f ix ed

throughout t he s imula t ion . I f t he d i s turbance occur s

in the a lgebra i c equat ions , t h e a lgebra i c s t a t es mustc ha ng e i n s t a nt a n e o u s l y t o s a t i s f y t h e i n i t i a l c o n di t io n

of t he dynamic s ta te s and the new algebrai c equat ions .

Thus i t may be neces sary to r e- solve the a lgeb ra i c

equat ions wi th the dynamic s t a t e s speci f i ed a t t he i r

i n i t i a l c o n d i t i o n s t o d e te r mi n e t h e new i n i t i a l v a l ue s

of t he a lgebra i c s t a t e s .

From the above descr ipt ion i t is clear t ha t once a

s t andard load- f low solu t ion i s found, t he remaining

dynamic s t at es and inputs can be found in a s t r aig ht-

forward way. The machine rel at iv e angles 6 i can always

be found provided,

v i e j e i + ( R ~ ~ + ~ x1 ) ~ ~ ~ e J ' io i = l , . .. m ( 4 C )

I f cont ro l l imits a r e e n f o rc e d , a s o l u t i o n s a t i s f y i n n

these l imits may not exi s t . I n t h i s case , t he s t a t e

which i s l im ite d would need t o be f ixe d at i t s l i m i t i n g

value and a corresponding new s tead y-s tat e sol ut i on

would have t o be found. This would req ui re a new load -

f low speci fy ing e i th er d i f f er ent va lues of genera tor

vol t ages , d i f f er e nt genera tor r ea l power s, or possibly

speci fy ing genera tor r ea c t ive power in j ec t io ns , t hus

a l lowing genera tor vol t age t o be a p a r t of the load-

f low solu t ion . In f a c t , t he use of r eac t ive power

limits i n l oad-f low can usual ly b e t race d back t o an

at t empt t o cons ider exc i t a t i o n sys tem l imits or genera-

t o r c a p a b i l i ty limits.

5. Angle Reference

In any ro t a t ion al sys tem, the r ef er enc e for angles

i s ar bi t rar y. Examinat ion of ( 1 ) - ( 1 4 ) clear ly shows

th at the orde r of t hi s dynamic system can be reduced

from 10m t o 10m-1 by int rod uci ng t he new re l a t iv e ang les

( a r b i t r a r i l y s e l e c t i n g 6 1 a s r e f e r e nc e )

61 = 6 i - 6 1 i = l ,. m ( 4 1 )

e; = ei-61 i = l , . . n ( 4 2 )

The fu l l sys tem remains exac t ly th e same as ( 1 ) - ( 1 4 )with each 6 i replaced by 61, each €Ji replaced by e; and

w Dur ing a t r ans i e nt , t he angle

6 s t i l l changes from i t s i n i t i a l c o n d i t i o n (as found

i n t h e l a s t s e c t i o n ) as w1 changes, s o t ha t each or ig i -

na l 6 . and Bi can be ea s i l y recovered i f needed. The

angle 6 ; r em ai ns a t z e r o f o r a l l t i m e.

amic s imula t ion , t he d i f f er e nt i a l equat ion fo r 6

normal ly replaced by the algeb raic equat ion which s imply

s t a t e s 6' = 0. Not ice tha t e i s n or ma ll y a r b i t r a r i l y

se l ec t ed as zero for t he load flow analys i s . This means

t h a t t h e i n i t i a l v al ue o f 61 i s normally not zero.

During a t rans ient , 8' and e change as a l l angles1 1

except 6 ; change.

t o i n f i n i t y , w1 and 6

replaced by w1 i n ( 1 ) .

1

Thus fo r dyn-

1 i s

1 1

I f t he i ne r t i a of machine 1 i s s e t

r emain cons t ant fo r a l l t ime.1

6 . Steady-Sta t e S t abi l i t y

The s tead y-s ta te s ta b i l i t y of mult imachine sys tems

i s usual ly evaluated by computing the eige nvalues of

the sys tem dynamic s tate Jacobian matr ix which is th e

l ine ar i ze d ver s ion of (1) - (14) wi th a l l a lgebra i c equa-

tio ns elimi nated . This dynamic model has one zeroeigenvalue cor responding t o th e angle r ef e r ence d i s cus s -

ed above. Elimin ation of 6 1 through the use of (41)

and (42) would el iminate th is zer o eigenvalue. The



system i s l i n e a r i z e d a r o u n d a s t e a d y - s t a t e o p e r a t i n g

point found us ing s tandard load-flow. The Jacobian

mat r ix for t h i s s t andard load- flow appears as a sub-

matr ix

below:

d t

in the lower r igh t b lock and i s denoted as JLF

r

( 4 3 )

where v conta ins t he load f low var i ables 0*, e 3 , . . . , e n ,

V m + l, . . .r V n . I n o rd e r t o e v a l u a te t h e s t a b i l i t y of t h e

dynamic sys tem, the alge brai c eq uat ions must be el imi-

nat ed . Thi s r equi r es t he nons ingular i t y of t he a lge bra i c

equat ion J acobian ( J A E ) ,

The s t ab i l i t y of t h e s t eady- s t a t e equi l i br ium point is

then determined by the sys tem dynamic s ta te Jaco bian

( J s y s ) '

( 4 5 )

Specia l cases where these thr ee J acobians J sys , JA E nd

JL F can be more ex pl i c i t l y r e l a t e d ar e g iven in t he

fol lowing s ec t ion .

7 . Special Cases

-1J = A-B JA E

SYS

There ar e two spe cia l cases where the s tandard loa d

f low Jacobian can be d i r e c t ly r e l a t ed to t h e sys tem

dynamic s t a te Jaeobian. We do not claim th at the se ar e

n e c e s s a r i l y r e a l i s t i c c a s e s , o nl y t h a t t h e y l e a d t o

cases where the three Jacobians can be related.

( a ) The f i r s t sp eci a l case makes the fo l lowing assump-

t i o n s :

( a l ) S t a tor r e s i s t a nce of every machine i s n e g l i -g i b l e ( R s i = 0. i = 1 , ..., ).

n e g l i g i b l e ( X A i = 0 , X'

(a3) Fi eld and damper winding t ime c onst ants f or

e v er y ma ch in e a r e i n f i n i t e l y l a r g e (E;i= con-

s t a n t , Eh i = c o n s t a n t , i = l , ..., ).

each generator (TMi = c o n s t a n t , i = l , ..., ) .

( a5) Genera tor number one has i nf in i t e i n er t i a .

Thi s t ogether wi th ( a l ) - ( a ) ) makes V 1 = con-

s t a n t , e l = c o n s ta n t ( i n f i n i t e b u s ) .

( a2) Trans i ent r eac t anc es of every machine are

= 0 , i = 1 ,..., ) .q i ,

( a ll ) Cons t ant mechanica l t orque to t he sha f t of

( a 6 ) A l l loads are constant power ( P (V. ) = con-L i i

s t a n t , Q L i ( V i ) = c o n s t a n t , i = l , ..., ) .

With the se assumptions , each 6i i s equal t o i t s cor r e-

sponding Bi plus a consta nt , and each V i is c o n s t a n t .

1377

Choosing w, a s ws and 8 , as ze ro. th e dvnamic model fo r

t h i s s p e c i a l c a s e ( a f t e r e l i m i n a t i n g Pi and Qi) i s ,

dei- = i = 2 , . . m (4 6

1 s

dw n

dt w . - w

dtMi+PLi- ,;, v . v Y . cos (e . - e -a )-Di(wi-ws)

i k i k 1 k i kM > = T

i = 2 , . . m

n

o = - P . + z V . Vk i k cos (e . - ek i ka 1L1 k= l

i=m+l , . . . ,n

n

o = - Q ~ ~ +V . V Y . s in(Bi-ek-aik)i k i kk= 1

i=m+l, n

w i t h

TMi = cons t ant i = l , . . m

V . = cons t ant

PLi = cons t ant

QLi = cons t ant

e l = 0

i = l ,.. m

i = l , . n

i = l , . . n

The l ine ar iz ed form of t h is model i s ,

dA0

d t

dAw

M - d t

0

-1

K 3 0 1 K4 1[ij

A e ~ V ~

( 4 7

( 4 8

where A e L V L i s t he vector L A O L AV , l t of incremental

load angles and vol t ages . For t h i s case ( a ) , t he

a lgebra i c equat ion Jacobian (JE) i s K4 . Fo r nonsin-

g u l a r K 4 , t he syst em dynamic s t a t e J acobian for t h i s

c a se ( a ) i s ,

( 56)

The determinant of is (see appendix A ) ,SYS

The s tandard load f low Jacobian as previously def ined

can be wri t t en in terms of t hese subma tr ices as ,

(58)



1378

Again for nonsingular K,,] the determinant OS t h e l oad

det JLF = d et K 4 det(K1-K2Ki1K3) -1 )m-l

0 = - : V P e"'ic8kLalk)+P . + j Q L if low Jacobian is (see appendix A ) : k = l ik L1

i=m+l, n (65:

(59) with

TMi = cons t ant

Vi = constant

For t h i s c ase , a c l ear r e l a t ion ship be tween the de t er -

minant of t he s tan dard load-f low Jacobian and the de t -

erminant of t he sys tem dynamic s ta te Jacobian e xi s ts as!

de t JL F

d e t = (60)sys de t K 4 d e t M

T h s means tha t under th ese assumptions , moni tor ing the

load-f low Jacobian determinant can detect a poss ible

dynamic ins ta bi l i t y. This is d i sc u s se d i n s e c t i o n 8 .

The bas i c s t ruc t ure of t h i s case ( a ) i s used f r equent ly]but the assumptions (al ) - (a 3) ar e s l i g h t l y d i f f e r e n t .

The same s t r uc tur e of (46)-(49) can be obtained b y

assuming a cons t ant vol t age behind t r an s i ent r eac t a nce

model with the terminal buses el iminated. T h i s leadst o a non s t anda rd load-f low Jacobian matr ix which

inclu des machine parameters i n th e bus admit tance

matrix.

( b ) A second case where such a relat ionship can be

f i rmly es t abl i shed was proposed i n pr in c ip l e by

Venikov i n [ill. This spe cial case makes the

following assumptions:

S t a t o r r e s i s t a n c e is n e g l i g i b l e ( R S i = 0 ,

i = l , . . m )

No damper windings or speed damping ( T I

D = 0 , i = l , ..., )

High gain and f a s t exci t a t i o n sys t ems so t h a t

a l l generator t e rminal vol t ages ar e cons tant

( V i = cons t ant , i = 1 , . . . , m )

Constant mechanical torque t o the sh af t of

each generator (Tni = c o n s t a n t , i = l , m )

Generator number one has in f i ni te i ne r t ia and

negl ig ib l e r eac t ances . Thi s t ogether wi th

( b l ) - ( b 3 ) mak e s V1 = cons t ant ] = constant

( i n f i n i t e b us )

A l l loads a re constant power (PLi(V i) = con-

s t a n t , Q ( V ) = c o n s t a n t , i = l , ..., )

qoi=

i

L i i

Nith the se assumptions, th e spe ci al ca se dynamic model

( a f t e r e l i m i n a t i n g P and Qi ) is,i

d6 i

d t = w i - w s

dw

-i

Mi dt = TMi-[Eii+(X -X ' ) I 3 1q i d i di q i

i=2, . . . ,m (61)

1=2, . . m (62)

i = 2 , . . m (63)

i = 2 , . . m (64)

PLi = constant

QLi = constant

i = 1 , . . m (66)

i = l , m (67)

i = 1 , . . n (68)

i = 1 , . . n (69)

e l = 0 (70)

The m+n-2 complex al ge br ai c e qua tio ns must be used tc

el iminate the 2m+2n-4 real a lgebra i c var i ables E'

I j i , I q i ( i = 2,..., ), oi ( i = 2,..., ) , V ( i=m+l , .., ) ,

W begin by f i r s t not ing tha t f rom (63) and (641,

q i '

i

n

k=C E ~ i + ( X q i - X ~ i ) I d i I I q i = -PLi+ E V V Y ik cos(8i-8k-aik)

i = 2 , . . m (71)

This can be sub st i t ute d in to (62) . Secondly, we note

th at (63) and (64) can be rewri t te n (us i ng cos(A+B) =

CosAcosB-sinAsinB and sin(A+ B) = sinAcosB+cosAsinB),

X I = -V i s i r 1 ( 8 ~ - 6 ~ ) i = 2 ] .. m (72)

i=2, . . m (73)

q i q i

Ehi-XhiIdi = Vi cos(8i i6

n

k= 1

V i I d i = E V i V k Y i k s i n ( 6 -8 -a )+PLi s i n ( 8 -6 )i ik i k

- QLi cos(8 -6 ) i = 2 , . . m (74)i i

n

k=

V i I q i = E V i V k Y i k cos(6 -8 -a )-PLi cos(8 -6 )i k i k i i

- QLi s i n ( 8 i6 ) i = 2 , . . m (75;

El iminat ing E i i 1 I d i , Iq i (s imply equat ing Iq i i n ( 7 2 )

and (75 )) give s

- XqiPLi cos(8 -6 )-XqiQLi s in(e i -g i )i i

i = 2 , . . m (76)

Using (71 ) , (76) and (65) th is special case dynamic

model with E i i , Id i , Iq i ( i = 2 , ..., ) el iminated i s ,

152,. . m (77)

n

L i - kt l ViVkYikcoS(8i-8k-aik)

i = 2 , . . m (78)

= ViSin(8i-6i)-XqiQLisin(8i-6i)-X P . cos (8 -6 )q i L 1 i i

n

+X v V Y . cos(6i-8k-aik)

i = 2 , . . m (79)i k=l i k l k



a

0 = -P + E V i V k Y i k cos(e -e -aLi k=l i k i k )

i=m+l, n

n

k= 1

0 = - QLi + E Vi VkYi k sin(ei-ek-aik)

i = m + l , n

t i i h

TMi = constant

Vi = constant

PLi = constant

QLi = constant

i = l , . .

.m

i = l ,.. m

i = l , . . . n

i = l , . . n

The li ne ar iz ed form of th is dynamic model is ,

For th i s case ( b ) , t h e a l g e b r a i c

(J::)) i s ,

%fining B t and C ' a s ,

equation Jacobian

(87)

B' = [K 1 K23C' =

[5 I]

f o r nonsingular J l i ' , the system dynamic st at e Jacobian

l o r t h i s c as e ( b ) is,

and (s ee appendix A )

det(-B'J::)-lC' )

de t M(- 1 Y-le t J ( b ) =

SYS

(88)

(89 1

(90)

Note that the eigenvalues of Jk;: w i l l a l l be e i t h e r

pure imaginary or w i l l include one or more which i s

pos i t ive rea l . We w i l l consider the dynamic systemcase ( b ) t o be s ta b le as long as no e igenva lues a re

po si ti ve re al . By rea rra ngi ng rows and columns of the

mat r ix i n (86) (see appendix A ) .

det Jii) det(-B'J:i)-lCt) = det K5 det JLF

1379

(91)

where JL F i s as i n ( 5 8 ) . T h i s gives the following

rel at i ons hip between the determinants of t he standard

load-flow Jacobian, t he algeb raic equation Jacobian,

and th e system dynamic st at e Jacobian,

de t K det JLFde t J ( b ) = 5

sys d e t M det JL:

(92)

This means th at under the se assumptions, monitoring t heload-flow Jac obian determinant can det ect a poss ib le

aynamic in s ta b i l i t y . This is discussed i n the fo l low-

ing sec t ion .

8. In st ab il it y and Maximum Loadab ility

When studying a proposed load or in te rchange leve l ,

a load-flow solution i s requi red before s teady-s ta te

a t a b i l i t y c a n be analyzed. If a load-flow sol uti on

cannot be found, then i t is normally assumed that the

proposed l oa di ng exceeded th e "maximum power tr an sf er "

ca pa bi li ty of th e system. This maximum power tra ns fe r

poin t i s normally assumed to co incide wit h a zero

aeterminant fo r the standard load-flow Jacobian. Using

th is as a c r i t e r i a , any load leve l which produces azero determinant for th e standard load-flow Jacobian i s

an upper bound and hence an op ti mi st ic value of t he

maximum loadability. This upper bound has been analyzed

i n t h e p a s t , a nd i s r e g a i n i n g i n t e r e s t a s v o l t a g e

col lapse is associated with this point [ l-11,19-211.I t is al so important t o note t hat non-convergence of

load-flows is a lso a mat te r o f so lu t i on technique .

Cases have been ci te d where Gauss-Seidel rou tin es con-

verge when Newton-Raphson routines do not.

If a standard load-flow so lut ion and associat ed

dynamic system equil ibrium p oint ar e found (a s described

i n s e c t i on s 3 and 4 ) the s t ab i l i t y of the po in t mus t be

determined. In order to do th is , t h e a lgebra ic equa t ion

Jacobi an must be nonsingular. This matrix i s given by

1 4 4 ) in genera l , by K 4 fo r case ( a) and by (87) fo r

' as e ( b ) . Assuming these alg ebr aic equation Jacobians

are nons ingula r fo r a g iven case , s teady-s ta te s ta b i l i ty

must be evaluat ed from the eigenv alues of t he system

dynamic s ta t e Jacobian. T h i s matrix i s given by (45)

i n genera l , by (56) fo r case (a ) , and by (89) fo r case

( b ) . A system is a t a c r i t i c a l p o i n t when t h e r e a l p a r t

of one of i t s eigenvalues i s zero . I f a rea l e igenva lue

i s zero then the determinant is zero . In the general

case of (45) . t he ze ro e igenva lue due to the angle

re fe rence can eas i ly be removed by using a dynamic model

reduced i n order by 1 ( see sec t ion 5) . Clear ly many

cases can be found where an equilibrium point can be

c r i t i c a l l y u n st a bl e ( a t l eas t one e igenva lue ha s a zero

r e a l p a r t ) and th e load-flow Jaco bian i s nonsingular.

In cases (a ) and ( b ) , a l l d eta i le d model dynamic

st at es have been el iminated by making rat her dr ast ic

aSSUmptiOnS. In sp eci al case (a ) , as long as de t M and

de t K4 a re non ze ro and bounded, a dynamic equilibrium

point exi sts and has a system dynamic st at e Jacobian

which i s s i n g u l a r if and only if the load-flow Jacobian

is s i n g u l a r . I n s p e c i a l c a se ( b ) , we need t o look a t

the mat r ix K5 .

diagonal with the ith diagonal e ntry equal to:5 is

xamination of (79 ) shows th at K

Kg i = -V f cos(8i-6i)+XqiQLi cos(bi-bi)

n

E V i V k Y i k s i n ( 6 . - 8 -aqi k=l 1 k i k )

xqiPLi sin(Bi-8 )-X

i = 2 , . . m (93)

T



Kg i = - xqiviIdi-v; C 0 S ( € p i )

and from (73)

K5.i = - Vi(E' i + ( X q i - X h i ) I d i )

I n s t e a d y - s t a te , (4 ) and (11) give (w ith RS

o = vie je i + jx i ( Id i+ j Iq i ) e j (6 i -* '2)

i = 2 ,

i = 2 ,

. m

. m

=0)

This means that Kg i can only be zero ( f or nonzero Vi)

if ( s e e ( 1 7 ) )

viejei +jXqiIGieJYi * o i d , . m (97)

Thi s a l so shows th a t Kgi is propor t ional t o t he magni -

tude of th e vol tage behind X i n s t e a d y - s t a te . T h i s

was d i s cus sed i n s ec t ion 4 a s a c o n d i t i o n f o r t h e

exi s t ence of a dynamic equilibrium from a load-flow

solut io n . Thus, i f a dynamic equi l i br ium point exi s t s

( equat ion ( 4 0 ) i s s a t i s f i e d ) , t h en K cannot be singu-

l a r . Thus i f det M and de t J a r e non z er o and

bounded, then th e sys tem dynamic s t a te Jacobian of case

( b ) is s ing ular i f and only i f t he load- f low Jacobian

is s i n g u l a r .

Since J( b ) must have a l l pure imaginary eigen-

v a l u es t o b e s t a b l e , d e t J ( b ) mu st b e p o s i t i v e t o beSYs

s t able . Reference [ l l ] or i g in a l l y proposed the moni-

tor i ng of t he load- flow Jacobian det erminant during

l oa d- fl ow i t e r a t i o n s t o s e e i f i t changed sign between

the in i t i a l gues s and the converged solu t i on . The

impl i ca t io n was th a t i f i t did , t hen the c ase (b) dyna-

mic model would be unst able a t th at so lut ion , and i f i tdid not t hen th e ca se (b ) dynamic model would be st ab le

( a l l pure imaginary e igenvalues ) . Our in t erp re t a t io n

indica t es t h a t t hey d id not account for pos s ib l e va lues

of d e t K and det J A change i n s ign of e i t h er of

tnese would af f ec t s t ab i l i t y i s sues . We have shown tha t

d e t K would probably never change sign, but whether the

d e t J i i ) c ha ng es s i g n or not remains an open quest ion.

q i

5

SYS

5

9. Conclusions

Standard load-flow is used t o f in d sys tem vol t ages

f o r a s p e c i f i e d l e v e l o f l o a di n g or i n t er change ( r egard-

le ss of the dynamic load model) . I t is a l s o th e s t a r t -

i n g p o i n t f o r d e t e r mi n in g t h e i n i t i a l c o n d i ti o n s f o r

dynamic ana lys i s . The s tan dard load-flow Jacobian can

provide informat ion about the exis te nce of a s teady-

s t a t e e q u i li b r iu m p o i n t f o r a s p e c i f i e d l e v e l of l o a d i n g

or interchan ge. There ar e two very speci al cases when

the determinant of the s ta nda rd load-flow Jacobian

impl i es something about t he s t eady- s t a t e s t ab i l i t y of adynamic model . Both of the se cases involve very dra s t ic

assumptions about th e synchronous machines and th e ir

con trol sys tems. The load lev el which produces a s i n -

gula r load-f low Jacobian should be considere d an opt i -

mi st ic upper bound on maximum lo ad ab il it y. The ac tu al

upper bound would be ei th er th e same or lower s ince i tr e q u i r e s b o t h t h e e x i s t e n c e of a s o l u t i o n an d s t a b l e

dynamics.

For vol t age COlMp3e an5 v l t a ge in s t a bi l i t y ana-

ly s i s , any conclus ions based on th e s ingula r i t y of t he

s tandard load-f low Jacobian would apply only t o the

phenomena of v ol ta ge be hav ior ne ar maximum power tr an s-

Per . Such an aly s is would not dete ct any vol tage in s ta -

b i l i t i e s as socia t ed wi th synchronous machine charac t e r -

i s t i c s or t h e i r c o n t ro l s .

10 . Acknowledgements

This work was supporte d in pa r t by th e Nat ional

Sc ie nc e Foun dat ion G rant NSF ECS 87-19055.

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APPENDIX A

The following facts of determinants can be found

in most l ine ar a lgebra tex ts .

1 . If a square matrix J i s block upper tr iangular

then the determinant of J is equal to the product

of determi nants of th e diagonal blocks.

Example

J = [ r] ( A and D square) (A.1)

de t J = det A det.D (A.2)

2 . If a square matrix J is p a r t i t i o n e d w i t h A and D

s q u a r e a s ,

(A.3)

3. I f A and B are sqaure of the same dimension then

(A.5)e t AB = det A det B

If any two rows (or two columns) of a square matrixare interchanged, then the determinant of the

resu l t ing mat r ix i s the negative of the determinantof the or ig ina l mat r ix .

The determinant of a square matrix is equa l t o the

product of i t s eigenvalues.

4.

5.

6 . If each element of a row (o r column) of a squ ar e

matrix i s multipled by - 1 , the determinant of t he

matrix changes sign.

P e t e r W Sauer (S'73, M'77, s " 8 2 )

was born in Winona, Minnesota on

September 20, 1946. He received

th e B.S. degree in e l e c t r i c a l

engine ering from the Univer si ty ofMissouri at Rolla, and the M.S.

and Ph.D. degrees i n el ec tr ic al

engine ering from Purdue Unive rsi ty

in 1969, 1974 and 1977, respec-

t i v e l y .

From 1969 t o 1973 he was t he

e l e c t r i c a l e n g i n e e r on a design

g s s i s t a n c e team f o r t h e T a c t i c a l Air Command a t Lan gle yAPB Virginia, working on the design and construction of

a i r f i e l d l i g h t i n g and e l e c t r i c a l d i s t r i b u t i o n s y st em s.

He has done con sul tin g work for the U.S. Army Corps ofEngineers Con struct ion Engineering Research Laboratory,

developing an in te rac t ive e l ec t r i c power d i s t r i bu t i on

pl an ni ng program. He is cur ren tly a Grainger Associate

and a Professor of Ele ct r ic al Engineering and the

Coordinated Science Laboratory at t he Univer si ty ofI l l i n o i s a t Urbana-Champaign.

He is a member of Eta Kappa Nu, Sigma X i , Ph i Kappa

Phi , IEEE and is a reg is te r ed Profess iona l Engineer inth e Commonwealth of Virg in ia and the S ta te of I l l i no is .

U. A . Pa i was born i n Uanga lore,

India on October 5 , 1931. Afterr e c e i v in g h i s 0.B. i n ElectricalEngineer ing a t Madras Uni vers i ty

i n 1953, he worked in Bombay

Ele c t r ic Supply and Transpor t

u n d e r t a k i n g t i l l 1 9 5 7 . He o b t a i n -

ed hi s M.S. and Ph.D. degr ees from

t h e U ni v e r s i ty o f C a l i f o r n i a ,

Berkeley in 1958 and 1961 resp ec t-

i v e l y . A f t e r t e a c h i n g for a year

year each a t t h e U n i v e r s i t y o f

Cal i forn ia Berke ley and U.C.L.A.,

he joined 1.I.T. Kanpur i n 1963 where he was on t h e

f a c u l t y o f Electric81 Engineering t i l l 1981. He was

v i s i t i n g P r o f es s or a t Iowa Sta te Univers i ty f rom 1979-

1981 and since 1981 he is P r o f e ss o r o f E l e c t r i c a l an dComputer Engineering a t t h e U ni v er s it y of I l l i n o i s a tUrbana-Champaign.

He was involved wi th se t t in g up th e undergradru t .

and graduate research programs a t I.I.T. .Kanpur i npower and control. H i s r e s e a rc h i n t e r e s t s ar e power

s y st e m s t a b i l i t y an d c o n t r o l . He is a f e l l o w of I n d i a nNat iona l Sc ience Academy, I ns t i tu t i on of Engineers

( I n d i a ) an d t he IEEE.and if det D I O , then

de t J = det D det(A-BD-lC)

This i s often c a l l e d SChUr's formula.

( A . 4 )



1382

Discussion

Adam Semlyen (University of Toronto): I would like to congratulate the

authors for their interesting and thought provoking paper. Certainly,

many power system engineers have wondered whether the load flow

Jacobian could provide information on the stability of the system, in

addition to its loadability. It appears from the paper that the answer is

almost completely negative since, if there is any relation betweenJ pand Jqs (the dynamic state Jacobianof the system), it is only related to

their singularity, i.e. one real eigenvalue becomingzero. Before thishap-

pens, as the system is more and more heavily loaded, complex eigen-

value pairs ofJq, may cross the imaginary axis into the unstable domain.

Even the singularity of Ju; or of Jv. is not a definitive indicationwhether the system is stable or not sincein that case the stability analysis

can not be based solely on linearized models. As the authors correctly

point out, the exact load flow solution can be obtained even for a tuming

pointA (i.e. extreme loading condition). Even Newton's method (in a

modified form) can be used for that purposeAg.

The fact that system stability is not solely related to the load flow

equations is well known but the general formulation of a dynamic prob-

lem in the form

i =f x )

may be suggesting the opposite, since it leads directly to the equations of

the steady state

f x)=O (2)

There are no distinct Jacobians in this formulation. In reality, what we

use for the load flow problem is not eqn.(2) but is an equation embedded

inf x w . This will be shown in more detail below. For this purpose we

first rewrite (1)and (2) in the more general form

i-N X J ) (3a)

w(x,y)=O (3b)

MX,Y1=0 (4a)

and

W(X,Y)=0 (4b)

where y are auxiliary variables to be eliminated using the second set of

equations.

In the particular case of power systems, the dynamics of the syn-chronous machines and of the induction machines at load buses isdescribed by

( 5 )

when D and pmch are shown to depend, in addition to 6.1, on other statevariables x as well (reflecting thus the effect of controls);pe r depend onthe machine angles6 and the bus voltages v and their angles 0. Thus the

system dynamic equations are

M A D X, a ) ( w ) = P m c h k a)-Pe/(Vt.6)

x = x o

Substitution of (7) into (6a) and (6d) gives

Equations (8) are the load flow equations. They have been obtained

directly from the right hand side of the dynamic equations but part of the

content of the latter has disappeared in the process. Clearly, the load flow

equations are embedded in the dynamic equations as a kernel and are

devoid of the information which is essential to the dynamics of the sys-

tem. Most importantly, the damping term withD x ,a) s missing and

P m c h does not include the effects of the controls. These remarks apply

also to the linearized forms of the paper, in particular as shown in

eqns455) and (86).

[A] R. Seydel, "From Equilibriumto Chaos - Practical Bifurcation andStability Analysis", Elsevier Science PublishingCo., Inc., 1988.

[B] F.L. Alvarado and T.H. Jung, "Direct Detection of Voltage Col-lapse Conditions", Proceedings of "Bulk Power System Voltage

Phenomena: Stability and Security", heldon 18-24 September 1988

at Trout Lodge, Potosi, Missouri; published by EPRI, Report EL-

6183, January 1989.

Manuscript received August 2 , 1 9 8 9 .

K . R. Padiya r (11s Banga lore, India ): The authors of this

paper are t o be compli-mented for providing a clear e xposition of

the relationships between the standard load-flow Jacobian and

system dynamic sta te Jacobian. Ther e is renewed interest in th e

analysis of steady-s tate stability due t o the concerns regarding

voltage instability problems. Although the system dynamics are

usually neglected in t he analysis of voltage instability, thi s is not

always realistic.

The ste ady-stat e instability is associated with th e crossing of a

real eigenvalue of the system from LHP to RHP, and hence t he

condition for singularity of the syste m Jacobian is of interest . As

power flow is increased, th e load flow Jacobian a nd consequently

the system Jacobian tends to be singular. I would appreciate if

the a uthors can respond t o t he following queries:

1. Is it possbile to predict the critical load pattern which

would lead to t he singularity of the Jacobian? For lossless

systems with PV buses it is possible to invoke the concept

of minimal cutset s (see Ref. A).

2. Is it possible to relate the determinant of J f k with that

of J LF , at lea st in the simplified case of systems with P V

buses?

Reference [A] A. Arapostat his, S. Sastry and P. Varaiya, "Anal-

ysis of Power Flow Equat ion", Memorandum No. UCB/ERL

M80/35, 980.

Manuscript received August 30 , 1989 .

P. W. SAUER and M. A. PAI: We thank Professors Semlyen

and Padiyar for their interest in th e paper and added discussion.

Both discussers comment on the critical load level and th e pres-

ence of algebrai c equations. In regard to critica l load level, the

funda mental issue is normally how load is distributed between

buses (both real and reactive). If the distribution is specified in a

meaningful way such as some percentage of total load, a critical

load level can be computed. If the distribution is not specifiedin a meaningful way, the critical load patterns can be physically

unreasonable.



1383

In the relation of Jacobians, one issue which has caused consid-

erable discussion is the notion of “PV” buses. In load flow this

means bus voltage magnitude and real power are fixed while bus

voltage angle and reactive power vary as required. In dynamic

analysis we would assume this to mean th at the bus voltage was

controlled by an automatic voltage regulator with some source

of reactive power such as a synchronous machine. If all buses

are synchronous machines in special case (b) , then JyL is sim-

ply K6 nd JLF is simply -K1 . We have not as yet found any

relationship between th e determinants ofK6 an d K I

Manuscript received August 30, 1989.

Documents

Power System Steady State Stability and the Lf Jacobian