vournas sauer pai 1997

7/23/2019 Vournas Sauer Pai 1997

1/8

LS VI R

Electrical Pow er Energy Systems, Vol. 18 No. 8 pp. 493-500 1996

Copyright 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved

PII

S0 14 2-0 61 5(9 6)0 00 09 -9 0142-0615/96/ 15.00+ 0.00

e la tionsh ips b e tw ee n v o ltag e and

ang le s tab il it y o f p o w er sys tem s

C D V o u r n a s

Electr ica l Energy Systems Labora to ry Nat iona l

Tech n ica l Un ivers i ty PO Box 261 37

A th e n s G R - 1 0 0 2 2 G r e ece

P W S a u e r a n d M A P a i

Depar tment o f E lect r i ca l and Computer Eng ineer ing

Un ive rs i ty o f I l li no is 1 406 W. Green St Urbana

IL 61 801 US A

This pap er discusses modelling a nd theoretical issues asso-

ciated with vol tage an d angle stabil i ty o f pow er systems. A

time-scale decomposition is per form ed to illustrate how the

critical modes can be identified with reduced-order models

and the bi furcation p henom ena can be explained with these

low order models. E xam ples are given fo r single and muhi-

machine systems. Copyright 1996 Elsevier Science Lt d

Keyw ords. voltage stability, angle stability, bifurcation,

reduced order modelling

I I n t r o d u c t i o n

P ow e r sy s t e m dyna m ic s a r e c lo sely re l a t ed t o t he m e c ha n -

ica l and e lec t rica l dyn am ic s ta te va r iables o f a l l the

sync h r onous m a c h ine s i n t e r c onne c t e d t h r ough the ne t -

w or k . H i s to r i ca l l y , pow e r sy s t e m s t a b i l it y ha s be e n a s so -

c i a t e d w i th t he ge ne r a to r r o to r a ng l e dyna m ic s . I n t h i s

pap er w e re -examine the i ssue of the c lass ic ' s teady s ta te '

(non-osc i l la tory) angle s tabi l i ty in the l ight of r ecent

resea rch re la t ing to vol tage s tabi l i ty problems. We

in t e nd to de m ons t r a t e t ha t t he S a dd le N ode B i f u r c a t i on

( S N B ) a s soc i a t e d w i th t he m a x im um pow e r de li ve r e d by a

sync h r onous m a c h ine unde r c o ns t a n t e xc i t a t ion i s no t a n

'angle ' s tab i l i ty problem , but o n the cont ra ry i t in i t ia te s a

s low de m a gne t i z a t i on p r oc e s s s e nsed in t he ne tw o r k a s a

vol tage decay wi th negl ig ib le f requency e r ror . I t i s only

du r ing t h i s p r oc e s s t ha t t he a c tua l l o s s o f sync h r on i sm

point i s me t .

Vo l tage s tabi l i ty i s pred om inan t ly load s tabi l i ty 1 '2 .

H ow e ve r , i t c a nno t be c om ple t e ly s e pa r a t e d f r om the

dyna m ic s o f t he sync h r onous ge ne r a to r s w h ic h p r ov ide

bo th t he pow e r a nd the vo l t age t o t he l oa d buse s . I n a t r a -

dit ional analysis, voltage stabil i ty was init ia l ly considered

Received 8 Jun e 995; revised 23 No vem ber 995; accep ted

8 February 996

f rom a load f low perspec t ive , in which the g enera tor s w ere

s im p ly r e ga r de d a s ' P V b use s ' . The c on t r a d i c t i on o f u s ing

the t e rm ' s t a b il i t y ' t o r e fe r t o a p r ob l e m w i th no dyna m ic s

was recognized eventua l ly and i t i s now genera l ly

a c c e p te d t ha t ' vo l t a ge c o l l a pse i s u l t im a te ly a dyna m ic

phe n om e non 1 '2 . H ow e ve r , t he dyna m ic s i nvo lve d in

vol tage s tabi l i ty have in many s tudies been res t r ic ted to

loa d bu se s on ly , i nvo lv ing f o r i n s t a nc e l oa d t a p c ha ng ing

(LTC ) t rans formers , r e s tora t ive loads 3 '4 , e tc ., wh ose t ime

f r a m e is o f t he o r de r o f one o r m o r e m inu te s . M a ny

vol tage s tabi l i ty inc idents evolve in th is t ime f rame , for

w h ic h t he ge ne r a to r dyna m ic s c a n be l e g i t im a te ly sub -

s t i t u t e d b y a pp r o p r i a t e e qu i l i b r iu m c ond i t i ons 5,6.

In many cases , however , a vol tage ins tabi l i ty scenar io

or ig ina t ing in the midte rm t ime-sca le gradua l ly man ifes ts

i t se l f a s tr ans ient ins tabi l i ty , and the equi l ibr ium of

gene ra tor dynam ics i s eventua l ly los t 7 leading to a loss

of synchronism. Also , the re i s a t leas t one case repo r ted in

the l i te ra ture s a s a vol tag e col lapse inc ident , for w hich

sync h r ono us ge ne r a to r s s e e m to be t he ke y f a c to r r espon -

sible for the instabil i ty.

The t he o r e t i c al que s t i on ske t c he d a bove c a n b e s t a t e d

as fo l lows: S ince vol tage s tabi l i ty i s d i rec t ly or indi rec t ly

l inked to the ' s teady s ta te ' angle s tabi l i ty problem is

the r e a f r a m e w or k f o r c o r r e c t l y de c oup l ing a ng le a nd

vo l t a ge s t a b i li t y p r ob l e m s? A n a nsw e r t o a s im i l a r que s -

t ion i s a t tem pted in Refe rence 9 for osc i l la tory ins tabi l i ty

us ing t he c onc e p t o f a n ' i nc r e m e n ta l r e a c ti ve c u r r e n t f l ow

ne tw or k ' . I n ou r a na ly si s , t he a nsw e r is sough t b y a pp ly -

ing s ingu la r pe r tu r ba t ion a na ly s i s t o a n un r e gu la t e d

m ul t im a c h ine pow e r sy s t e m in o r de r t o de c om pose i t

f o r m a l ly i n to a s l ow a nd f a s t subsys t e m . The phys i c a l

c onc e p t o f t im e -sc a le de c om p os i t i on w a s i n t r oduc e d in

R e f e r e nc e 10 , w he r e t he pow e r sy s t e m w a s de c om pose d

in to a midte rm and a t r ans ient t ime-sca le . In our

a pp r o a c h w e p r oc e e d f u r the r t o de c o m po se the ge ne r a to r

dyn amics ( in the t r ans ien t t ime-sca le ) in to v ol tage ( f lux)

4 9 3


2/8

4 9 4

Relat ionships between voltage and angle stabili ty of po we r systems. C. D. V oum as e t a l .

and s ha f t ( e l ec t rom echan i ca l ) dynam i cs . U s i ng t h i s

app roach i t i s c l ea r l y dem ons t r a t ed t ha t t he re i s on l y

one m ech an i s m fo r gene ra t o r S NB , wh i ch i nvo lves bo t h

vo l t age and ang l e , bu t no t f r equency o r power .

A fo rm al decom pos i t i on o f t he dynam i cs o f a m u l t i -

m ach i ne , un reg u l a t ed s y s t em i n t o f a s t e l ec t rom echan i ca l

osci l l a t ions and s low f lux-vol tage response modes i s

und er t ak en i n S ec t i on II . S ec t i on I I I s uggest s an app rox -

i m a t e m e t hod o f ana l y si s fo r a pa r t l y r egu l a t ed s y s tem ,

and Sect ion IV presents three i l lus t ra t ive examples .

I I D e c o m p o s i t i o n o f m a c h i n e d y n a m i c s

I 1 . 1 Slow ma ni fo ld in mul timachine systems

A p owe r s y s tem cons is t ing o f m s ync h ronou s m ach i nes

can be d escr ibed in the t rans ien t t ime-scale by the fo l low-

i ng s e t o f equa t i ons , w i t h t he m ach i nes r ep res en t ed u s i ng

t he one -ax i s m ode l U:

6 = , , . , 1 )

~ o l TM t~

= P m

- P (6 , E q ) - ~ o i D w

( 2 )

T d l ~ = E f - E ( 6 , E~q) ( 3 )

wh ere 6 , to , E~ , and E r are m x 1 vectors represent ing the

ro tor angles , speed deviat ions , in ternal vol tages ( f i e ld

f luxes) , and exci ta t ion vol tages of the m ma chines , respec-

t ively , TM, D, and Td are m x m diagonal mat r ices con-

t a i n i ng t he m echan i ca l s t a r t i ng t i m es T M i = 2Hi) , the

dam pi ng t e rm s , and t he f i el d open c i r cu i t t i m e cons t an t s ,

respect ively , and f inal ly mo i s the syn chro nou s speed . The

m ach i nes cons i de red i n t h i s s ec t i on a re w i t hou t au t o -

m at i c vo l t age r egu l a to r s and t he m echa n i ca l pow er i npu t

P m i s a s s um ed t o be cons t an t . The ope ra t i on wi t hou t

AVR s i s pos s i b l e when t he m ach i nes a re under m anua l

vo l t age con t ro l , o r when t hey have r eached t he i r exc it a -

t ion limit s. Th e funct ion s P an d E are m-valu ed funct ion s

of the s ta te var iab les 6 and Eq, depending on the in ter -

connec t i on be tween m ach i nes and l oads .

T h e m u l t im a c h i n e m o d e l ( 1 ) -( 3 ) c a n b e d e c o m p o s e d

in to two subsys tems , a s low one cons i s t ing of f lux-decay

mo des , and a fas t one descr ib ing e lect rome chan ical osci l-

l a t ions . To achieve th i s , the fo l lowing parameters are

i n t roduced :

c

V 4 )

1 m

I I = m I I

5 )

w = cw (6)

H = d i a g [ H i / H o ] (7)

Us i ng t he above n o t a t i on , t he s y s t em (1 ) -(3 ) t akes t he

fo l l owi ng s t andard fo rm fo r s i ngu l a r pe r t u rba t i on ana -

lysis12:

e 6 = , , / 8 )

e J

= H - 1 [ P m - P ( ~ , E ; ) - ~ H o D J ] (9)

E ~ = T d 1 [ E f - E(6, Eq)] (10)

In (8)- (10 ) 6 , w are the fas t var iab les an d E q are the s low

variables.

C ons i de r t he m -d i m ens i ona l m an i fo l d in t he s t a t e s pace

o f t he s y s t em (8 ) - (10 ) de f i ned by t he 2m equa t i ons :

6 = h l (Eq) =

h l o

+ eh n+ O( c 2 ) (11 )

Of = h 2 ( E q ) = h 2 o + eh : l+O (e 2) (12)

The mani fo ld def ined by h i , h2 wi l l be an in tegral or

i nva r i an t m an i fo l d fo r t he f a s t s ha f t dynam i cs i f t he

fo l l owi ng cond i t i ons ho l d , wh i ch gua ran t ee t ha t a

t r a j ec t o ry s t a r ti ng o n t he m an i fo l d wi ll r em a i n on i t fo r

al l t ime:

Ohl 1

~ q q T d [ E f - E ( h b E q )] = h 2 ( 13 )

0h2

~ -1 r ~

C ~q q I d L l L f

E ( h l E q ) ]

= H - 1 [ P m - P ( h l , E q ) - ~ H o D h 2 ] ( 1 4 )

No t e t ha t h i , h 2 a re func t i ons o f E~ .

The above cond i t i ons (13 ) and (14 ) a r e ob t a i ned by

subs t i tu t ing (1 l ) an d (12) in to (8) and (9) and m akin g use

of(10) . I t i s no t poss ib le in gen eral to so lve equa t ions (13)

and (14) in ord er to obta in analy t ical ly the in tegral

ma ni fo ld h i , h2 (a lso cal led the s low ma ni fo ld) o f the

s ys t em . An app rox i m a t e s l ow m an i fo l d can be found ,

how eve r, by su bst i tut ing hi , h 2 as a~aower series in c, as in

(1 l ) and (12) and eq uat ing the c and eI t erms . This

process g ives the fo l lowing set of equat ions :

O h l o T _ I rr,

C~-E-7-q d [~f - E( hl o, E q) - eM 4h ll ] ~ h20 -4- eh21

0 h 2 o T - 1

,~,

e ~ -E -~ q d L r - f - E ( h , o , E~) - c M 4 h t , ]

= H - 1 [ P m - P (h l0 , E q ) - e M , h , , - ~ H o D h2o ]

where m a t r i ces M1 and M 4 a re de f i ned as t he J acob i ans o f

func t i ons P an d E w i t h r e s pec t t o 6 ca l cu l a t ed a l ong t he

m a n i f o l d hi0:

0 P 6 = h o O ~ 6 =h to

1 = ~ - ~ M 4

Equ at ing the coeff ic ien t s of e :

0 = h 2 o ( 1 5 )

0 =

H - I [ p m - P ( h l 0 , E q ) ] ( 1 6 )

Equ at ing the coeff ic ien t s of e l:

0 h i 0 T - 1

h 2 1 ~ -- ~ q d [ E f - E ( h l 0 , E / q )]

(17)

0 = - H - I M l h l l ( 1 8)

f rom which i t i s c lear that

h u = 0

Note that (15) was used to obta in (18) . Di f ferent ia t ing

I .

(16) w ith respe ct to Eq.

0 h t o

n t - - f f~ , + n 2 = 0

( 1 9 )

U l ~ q


3/8

Relationships between voltage and angle stabi l i ty of powe r systems. C. D. Vournas e t a l . 4 9 5

wh ere M2 i s def ined as :

OP 6=hi0

2 = ~ q q

F ro m (19 ) we can wr i t e t he J acob i an o f the s l ow m an i fo l d

h i0

fo r t he gen e ra t o r ang l es a s a func t i on o f t he m a t r i ces

M1, M 2 :

0 h I 0 = - M I I M 2

(20)

F i na l ly , t he O(e 2) app rox i m at i on o f t he i n t eg ra l m an i -

fo ld for the shaf t var iab les i s g iven by:

6 s =

h lo

(21)

J s = - e M i l M 2 T i I [E t E (h l0 , Eq )] (22 )

w h e r e h lo i s the impl ic i t funct ion of E~ def ined by the

solu t ion of :

P h l o ,

Eq) --

a m 2 3 )

Variables 6s , wts are the s low components of the fas t

va r i ab l es co r res pond i ng t o t he s ha f t dynam i cs and a re

funct ion s of E~ .

No t e t h a t a cco rd i ng t o (22) t he f r equency dev ia t i ons

du r i ng s l ow t r ans i en t s a re o f o rde r e , a l t hough t he ang l es

m a y va ry con s i de rab l y to ach i eve t he pow er ba l ance (23 ).

This i s character i s t i c of the c lass ical descr ip t ion of vol tage

s t ab i l i t y p rob l em s , where l a rge vo l t age va r i a t i ons a re

accom pan i ed by neg l i g i b l e f r equency e r ro r s . As t he

va l ue o f t he f r equency e r ro r i s s m a ll i t is r eas onab l e t o

as s um e t ha t t he f r equenc y con t ro l l oop i s no t exc i ted

dur in g s low f lux-vol tage t rans ients .

1 1 . 2 Slow f lux an d vol tage dynamics: SNB

An app rox i m at e decom pos ed ve r s i on o f t he s l ow f l ux

dynam i cs is de r i ved f rom (10 ) by r ep l ac i ng 6 by

h l 0 ,

w h i c h

is obta ine d by so lv ing (23). Thus :

1 ~1 = T d I [ E l - E h l 0 , E t q )] 2 4 )

The l i nea r i zed ve r s i on o f (24 ) a round an equ i l i b r i um

poin t E qo g ives the fo l lowing s ta te ma t r ix , obta in ed

us ing the def in i t ions of the M ma t r ices and (20) :

As = T i I [M3o + Ma oMllM 2o] (25)

where:

0 E

eh lo

3 - - 0 E ~

and the subscr ip t 0 deno tes evalu at ion a t the equil i-

br iu m po in t E~o.

The m at r ices M 1 - M4 def ined in th is and the previous

sect ion , are d i rect general izat ions of the l inear izat ion

coeff ic ien ts K 1 - K 4 used in the He ffron -Phi l l ip s mo del

13,14

of a s ynch ro nous m ac h i ne . There ex is ts an exac t one

t o one co r res po ndence , w i t h t he excep t i on o f M 3 wh i ch

co r res ponds t o - 1 /K3 o f the s i ng le m ach i ne m ode l .

No t e t ha t t he ap p rox i m at e s t a t e m a t r i x o f t he sl ow

ma chin e dyna mic s (251) i s the sam e ma t r ix tha t w as

in t rod uce d in Refer ence 15 as a vol tage s tab i li ty

ma t r ix . Fol low ing the abov e analys i s , th i s mat r ix

app rox i m at es wi t h an c 2 e r ro r t he f l ux dynam i cs o f a

m u l t i m ac h i ne sy s t em wi t hou t AVR s . M oreover , a ze ro

e i genva lue o f t h i s m a t r i x c o r res ponds exac t l y t o a s add l e

node b i fu rca t i on o f t he o r i g ina l s y s t em , as can be eas i ly

ver i f ied by l inear iz ing (1)- (3) as in Reference 15 . The

SNB con di t ion o f th i s sys tem i s g iven by:

det[M3o + M4oM lolMzo] = 0 (26)

A s add l e node b i fu rca t i on in a m u l t i m ac h i ne s y s t em wi ll

resu l t in a s low f lux decay fe l t by a l l genera tors . This w i ll

have a s i m i la r d r i f ti ng e f f ec t on t he ge nera t o r t e rm i na l

and l oad bus vo l t ages l ead i ng t he who l e s y s t em e i t he r to a

vo l t age co ll apse , o r t o t he l o ss o f s yn ch ron i s m be t wee n

t he genera t o r s .

1 1 . 3 Fast dynamics. electrome chanical oscil lations

O n c e a n a p p r o x i m a t e s lo w m a n i f o l d h as b e e n f o u n d f r o m

(21 ) and (22 ) , t he f a s t dynam i cs o f t he m u l t i m ach i ne

s ys tem can be r econs t ruc t ed u s i ng t he o f f -m an i fo l d va r i-

ab les def ined below:

6r = 6 - 6s (27)

(28)

f ---~0.2 -- W

The o f f -m an i fo l d dynam i cs a re des c r i bed wi t h an O(e 2 )

app rox i m at i on by t he fo l l owi ng d i f f e ren t i a l equa t i ons ,

which are der ived by d i f fere nt ia t ing (27) an d (28) as in

Reference 12 and omi t t ing the 2 and h i ghe r o rde r t e rm s .

No te that the s low c om pone nts 6s, W s are funct ions of E q

and t he re fo re t he ir t i m e de r i va ti ves have t o be eva l ua t ed

using the chain rule, i .e. (21) and (22.) are different iated with

respec t to E q an d mu l t ip l ied b y E~ g iven by (10) .

err = eh21 + w~ + ~Mi1M 2Td 1 [Ef - E(h lo + 6 I Eq)]

(29)

= n - I [ Pm - P h l o + 6 t , E q ) -

~--~ Dwt] (30)

In l inear iz ing equa t ions (29) and (30) arou nd a po in t ly ing

on t he s l ow m an i fo l d , fo r wh i ch ~ t = 0 , t he dependence o f

h i0 , h21 on E q has to be tak en in to acc oun t by app ly ing the

chain ru le o nce mo re, i .e .

+ e M l l n 2 T d - I [ ( M 3 - M 4 ~ ) A E q - M 4 A r f J

Subs t i tu t ing h21 f rom (17) and us ing (20) it becom es c lear

t ha t t he sl ow va r i ab les Eq a re e l i m i na ted f rom the above

equat ion . Th e sam e is t rue wh en l inear iz ing (30) .

Now, i n o rde r t o r e t u rn t o t he o r i g i na l va r i ab l es we

define:

Wr= (1/e)w~ (31)

and the fo l lowing l inear ized s ta te equat ion for the off -

m an i fo l d , e l ec t rom echan i ca l o s c il l at i on dynam i cs i s

ob t a i ned :

Ad. fJ = [ -T~l lM 1 -T~I1D LA~f]

where I , , is the m m i den t i t y m a t r i x . The M m at r i ces fo r

t he f a s t dynam i cs s t a te equa t i ons (32 ) a re com p u t ed a t an

e q u i l ib r i u m p o i n t o f t h e o f f - m a n i f o l d d y n a m i c s 6 t = 0 ,

i . e . a poin t ly ing on the s low mani fo ld . This i s not

neces s a r il y an equ i l ib r i um po i n t o f t he s y s tem , becaus e

t he s l ow dynam i cs m ay no t be a t equ i l i b r ium a t t h i s po i n t.

One in teres t ing aspect of (32) i s that i t demonst ra tes

m at hem at i ca l l y t ha t t he f i e l d wi nd i ng i s i n t roduc i ng


4/8

4 9 6 R e l a t i o n s h i p s b e t w e e n v o l ta g e a n d a n g l e s t a b i li t y o f p o w e r s y s t e m s : C . D . V o u r n a s et al.

p o s i t i v e d a mp in g t o r q u e t o t h e f a s t e l e c t r o me c h a n ic a l

o sc i ll a t io n s t h r o u g h th e ma t r ix b lo c k - M i - I M z T ~ I M 4 .

T h e r e f o r e , a n u n r e g u la t e d sy s t e m i s n o t e x p e c t e d t o

dem ons t ra te o sc i l la tory ins tab i l i ty , a s d iscussed in grea te r

deta il in Reference 16.

11.4 R e v i e w o f t h e a s s u m p t i o n s

L e t u s c o n s id e r n o w th e a s su mp t io n s imp l i c i t l y ma d e

dur ing th is br ie f p resenta t ion .

( 1) F o r n o n - imp e d a n c e l o a d s t h e f u n c t i o n s P a n d E a r e

n o t u n iq u e , b e c a u se t h e y d e p e n d o n t h e so lu t i o n o f

th e n e tw o r k . O u r a s su m p t io n i s t h a t , s t a r ti n g f r o m a

n o r ma l o p e r a t in g p o in t , t h e sy s t e m r e ma in s w i th in o n e

causa l r eg ion 17. As the a lgebra ic cons t ra in ts ha ve no

s ingula r po in ts ins ide the causa l r eg ion , the func t ions

P a nd E remain unique an d they a re the ones cor re -

sp o n d in g t o t h e n o r ma l o p e r a t i o n o f th e sy s t em.

( 2 ) T h e d e c o mp o s i t i o n p r e se n t e d h e r e i s p o s s ib l e o n ly

whe n the fa s t dy nam ics o f the sys tem ( i. e. the e lec t ro-

mechanica l osc i l la t ions) a re s tab le . I t i s a l so c lea r

f rom Sec t ion I I . 1 tha t a cond i t ion for the ex is tence of

a s low manifo ld i s tha t equa t ion (23) has a so lu t ion .

There fore , M 1 shou ld be nons ingula r . W e wi ll r e turn

to th is po in t when d iscuss ing the f i r s t i l lus t ra t ive

example .

( 3 ) F o r a n y su d d e n d i s tu r b a n c e , t h e p r e - d i s tu r b a n c e

c o n d i t i o n s mu s t b e lo n g t o t h e r e g io n o f a t tr a c t i o n

o f t h e p o s t - d i s tu r b a n c e s t a b l e e q u i li b r iu m o f t h e o f f -

ma n i f o ld d y n a mic s .

I I I S y s t e m w i t h r e g u l a t e d m a c h i n e s

T h e in t r o d u c t io n o f A V R s in t h e mu l t ima c h in e sy s t e m

(1) - (3) in te r fe res wi th the t ime-sca le decomposi t ion pe r -

f o r me d in t h e p r e v io u s s e c t i o n f o r t h e f o l l o w in g r e a so n s .

(1) I t i s we l l known tha t exc i ta t ion cont ro l le r s cont r i -

b u t e , u n d e r c e r t a in c o n d i t i o n s , n e g a t i v e d a mp in g t o

the e lec t romech anica l osc i l la t ions . This may re su l t in

a v io l a t io n o f th e s e c o n d a s su m p t io n o f S e c t io n I I .4

tha t r equi re s the s tab i l i ty of f a s t sha f t dynam ics .

(2) Hig h exc i ta t ion sys tem ga ins tend to forc e the

machine f lux va r iab les E~ to become fas t . Thus , the

b a s i s f o r t h e d e c o mp o s i t i o n i n to f a s t a n d s lo w

d y n a mic s i s d e s t ro y e d .

A s a c o n se q u e n c e , t h e p r o b l e m o f t ime - sc a le d e c o m-

p o s i t i o n w h e n a l l t h e g e n e r a to r s i n a p o w e r sy s t e m a r e

u n d e r a u to ma t i c v o l t a g e r e g u l a t i o n , h a s n o t b e e n

addressed in th is pape r .

A n o th e r , p e r h a p s mo r e c h a l l e n g in g , p r o b l e m i s t h e

a n a ly si s o f a p o w e r sy s t e m w h e n so m e o f i t s g e n e r a to r s

a re r egula t ing and some a re not , be ing e i the r under

m anua l cont ro l , o r a t the i r exc i ta t ion limi t. Th is problem

was d iscussed fo r ins tance in Refe renc e 18 . In th is s i tua -

t i o n t h e r e a r e so me s lo w s ta t e v a r ia b l e s a s so c i a te d w i th

the unregula ted machine f lux dynamics , so tha t a t ime-

sca le decomposi t ion i s poss ib le , a t leas t in pr inc ip le ,

prov ided tha t the s tab i l i ty of the e lec t romecha nica l osc i l -

la t ions i s ma in ta ined .

A t t h i s s t a g e a f o r ma l d e c o mp o s i t i o n s imi l ar t o t h e o n e

presented in Sec t ion I I i s no t ava i lab le for the pa r t ly

regula ted case . Ins tead we sugges t an a l te rna t ive , le ss

r i g o r o u s f o r mu la t i o n , t h a t g iv e s g o o d a p p r o x ima te

resu l t s under ce r ta in c i rcumstances .

T h i s a p p r o x ima te me th o d a s su me s a n a lg e b r a i c

e q u iv a l e n t o f t h e r e g u l a te d m a c h in e s s imi l a r t o t h e P V

b u s r e p r e se n t a t io n o f t h e c l as s ic a l l o a d f lo w . F o l lo w in g

th is the s low dynamics s ta te equa t ions (24) and (25) for

th e r e d u c e d sy s t e m c o n s i s ti n g o n ly o f t h e u n r e g u la t e d

ma c h in e s a r e f o r mu la t e d . F o r l o w e x c i t a t i o n g a in s t h i s

p r o c e s s is n o t a c c u r a t e b e c a u se o f t h e A V R v o l t a g e

dro op s 5 . Fo r h igh ga in exc i te r s , how ever , a co ns tan t

v o l t a g e e q u iv a l e n t f o r t h e r e g u l at e d ma c h in e s c a n p r o v id e

a f i rs t approx im a t ion , a s wi ll be seen in the next sec t ion .

I V I l l u s t r a t i v e c a s e s t u d i e s

At th is po in t th ree case s tud ies a re in t roduced as i l lus-

t r a t ive exam ples . Th e f i r s t case i s a s imple s ing le -machine

in f in i t e - b u s sy s t e m u se d t o e x p l a in t h e me c h a n i sm o f

v o l t a g e i n s t a b il i t y a n d l o s s o f sy n c h ro n i sm. I n t h e

se c o n d c a se t h e v a l i d i t y o f t h e t ime - sca l e d e c o mp o s i t i o n

in t r o d u c e d i s d e mo n s t r a t e d o n a mu l t ima c h in e C I G R E

test system. The third case study again involves a single

ma chine and i t is se lected to represen t the sa lient features of

a r ea l wor ld inc ident documented in Refe rence 8 a s a

voltag e collapse .

IV.1 S i n g l e m a c h i n e i n f i n i te b u s s y s te m

In th is example we wi l l r e -examine in a new l igh t the

c la ss ica l pow er -angle curve of a s ing le -machine infin ite-

bus sys tem. The machine i s r epresented a s a one -axis

mo d e l ( 1 ) - ( 3 ) , h a v in g b o th a n g l e a n d f l u x d y n a mic s . T o

make th ings a s s imple a s poss ib le we cons ide r a loss le ss

sy s t e m a n d a r o u n d r o to r ma c h in e w i th

X d = Y q .

T h e

equi l ibr ium curve o f th is sys tem is g iven as:

P m E f V sin 6

d

(where V is the con s tan t vo l tage o f the inf in ite bus) and

c o r r e sp o n d s t o t h e f a mi l a r s i n u so id a l c u rv e d r a w n in th e

a n g le - p o w e r p l a n e f o r c o n s t a n t e x c i t a t i o n v o l t a g e ET,

with the SN B exac t ly a t 6 = 90 , a s show n in F igure 1 .

Equa t ion (23) de f in ing the s low manifo ld co inc ides in

th is case wi th the t r ans ien t pow er -angle curve g iven by:

V 2 ( 1 , 1 )

E q V

sin 6 sin 26

em

The t r ans ien t power -angle curves for th ree d i f fe ren t

v a lu e s o f , q ranging f rom 0 .85 to 1 .0 , a re a lso show n in

F igu re 1 wi th d ot ted l ines.

1

0 . 8

n E 0.6

0 . 4

0 . 2

0

/ E ' q = 1 . 0

S N B . / / ~ \ \ \

A ;

\ %',

/ / /

/

/ i f

/ ..;>

I

E'q---0.85 x

? ? U

2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0

r o to a n g l e ( d e g )

F igu r e 1 . S t eady state and t r a n s ie n t p o w e r a n g l e c u r v e s


5/8

Relationships between voltage and angle stabil ity of powe r systems: C D Vournas

e t a l: 4 9 7

O

M I

8 O k V

ISOkV

_m l

q o

F i g u r e 2 C I G R E t e s t s y s t e m

Starting from no load and slowly increasing the

machine loading the trajectory of the system will follow

the equilibrium curve, along the solid line, up to the SNB

point. At this point the machine will not lose synchro-

nism, because the synchronizing coefficient, which is the

slope of the transient power-angle curve defined above, is

still positive. However, because the excitation level is

inadequate to maintain the required power transfer at

steady state, the trajectory will depart from the equili-

brium curve. The field flux will begin to decrease, and the

machine angle will start increasing very slowly, with

negligible frequency error according to 22), along the

constant power line. If this voltage degradation does not

bring about a serious disruption of the system operation,

the machine will eventually lose synchronism at the point,

where the transient power-angle curve becomes tangent

to the constant power line. At this point the synchroniz-

ing coefficient K1 will become zero, which in the single

machine case is equivalent to the singularity of the

synchronizing matrix MI.

We return now to the question that was left open before

closing Section II, i.e. the possibility of a singular M1. For

a single machine connected to an infinite bus through a

lossless line it can be shown that Ml becomes singular

always at a loading level that is higher than that producing

a SNB t9. Although a similar strict proo f is not available

for the general case, it is reasonable to expect that due to

the highly reactive nature of the power networks) the

SNB of an unregulated mult imachine system will precede

the loss of synchronism through a singularity of the

synchronizing matrix, as shown in Figure 1.

IV.2 ultimachine test system

As mentioned above, the system studied here is taken

from a CIGRI~ task force 2. The same system has been

studied extensively in Reference 7 for various loading

scenarios. A single line diagram is shown in Figure 2. The

operat ing point studied here is one for which machine M2

has tripped. Therefore, this is a five-machine system with

two infinite buses, shown in the lower part of Figure 2.

Two cases are analysed for this system. Suppose first

tha t all five machines are wit hout AVR. The eigenvalues

of the full model of the system are shown in Table 1

together with those of the reduced fast and slow sub-

systems described by equations 32) and 25), respec-

tively. The e2 accuracy obtained by the time-scale

T a b l e 1 . E i g e n v a lu e s w i t h o u t A V R

Full model Fast subsystem Slow subsystem

-0.240 +j6.81

-0.280 +j 6.59

-0.250 ij6.16

-0.189 +j3.98

-0.204 j4.94

+0.283

-0.302

-0.226

-0.276

0 . 1 9 9

-0.241 +j6.82

-0.279 +j 6.59

-0.250 +j6.16

-0.190 j3.98

-0.204 j4.9 4

+0.285

-0.302

-0.226

-0.276

-0.199

T a b l e 2 . E i g e n v a l u e s w it h A V R s o n m a c h i n e s M 5 a n d M 6

Full model Three mach. appr. Full model

low gains slow subsystem high gains

-0.014 -0.056 -0.052

-0.221 -0.222 -0.221

-0.276 -0.276 -0.276

-0.325 +j3. 90 -0.187 +j3. 75

-0.226 j4 .93 -0.220 +j4 .89

-0.243 +j6 .16 -0.286 +j6. 13

-0.279 + j 6.59 -0.264 + j 6.60

-0.242 j 6.82 -0.251 + j 6.75

-1.80, - 2.13 -4.94 + j 10.04

-4.66 i j 1.39 -5.10 +j4 .84

decomposition applied is self-evident. Note how the

time-scale decomposition is possible even after the SNB

of the unregulated system. In fact, only the stability of the

electromechanical dynamics is required.

Let us look now at the same system when some of the

machines are regulating: In the simulated scenario,

machines M 1, M3 and M4 have reached their overexcita-

tion limits and therefore they have lost voltage control.

This leaves three unregulated machines connected to the

rest of the system. In a first attempt to model this

situation, the dynamics of the regulated machines are

ignored, as was discussed in Section III, and the slow

dynamics subsystem 25) of the remaining three machines

is formed. The eigenvalues of the full system and those o f

the three-machine slow subsystem are shown in the first

two columns of Table 2.

As one real eigenvalue is very small, the system is close

to a SNB, which is due to the field limitation of the three

machines. This situation is reflected in the simplified

three-machine formulation, although the actual value of

the critical eigenvalue is not predicted accurately. As

discussed in Section III, this is due to the small value of

the excitation gains of the two remaining controlled

generators: Machine M5 has an excitation gain of

15p.u. and a time constant of 0.3 s, and machine M6 a

gain of 35 p.u. and a time cons tant of 0.1 s. By increasing

the gains to 95 p.u. and 135 p.u., respectively, and redu-

cing the time constant of M5 to 0.1 s, the eigenvalues of

the third column of Table 2 are obtained. Clearly in this

case the three machine simplified equivalent which is still

the one shown in the second column of Table 2 gives quite

accurate results.

The following conclusions can be drawn from this

experience: The slow dynamics matrix 25) predicts accu-

rately a SNB of a multimachine system with no AVRs. I t


6/8

4 9 8 Relat ionships between voltage and angle stabil i ty of po we r systems C D Vournas et al.

V

Xl

l

~

VI 1 0 1

P I I O P t

F i gure 3 ,

Single m achine tes t sys tem

Table 3 . S ing l e m ac hine t es t sys tem data

Generator Lines

z ~d q XId Zldo X 1 X 2

1.0 1.0 0.3 9 s 0.3 1.0

Initial conditions Loads

Rotor

Ef Pg angle F 1 GA GB

1.4 0.76 55 0.9811 0.63 0.50

can also serve as an approximate model of a part of the

system that is left without automatic voltage regulation

due to field limitation.

IV .3 A realistic SNB

The system in this third case study is a single-machine

system inspired from a B.C. Hydro, North Coast Region

incident of 1979 reported in Reference 8. The parameters

chosen for the test system are not those of the actual

system, so that the similarity is limited only to the

structure of the system (an isolated generating plant

feeding a large load, weakly connected to the rest of the

system), the instability scenario (partial load tripping in

an aluminium plant), and the loading levels. This case

study is meant to demonstrate the following points.

An SNB of a synchronous generator can be felt as a

voltage instability.

A voltage instability is possible even with constant

impedance loads (whereas an algebraic singularity is

impossible in this case).

When a disturbance takes the operating point very

close to an SNB it takes minutes for the instability to

evolve.

The single line diagram is shown in Figure 3. The test

system consists of an equivalent generator under manual

excitation control, a load bus feeding a constant impe-

dance, unity power factor load, and an infinite bus

connected to the load bus through a long tie-line. The

generator is modelled with one-axis rotor flux dynamics,

as in Section II 1] . The parameters of the system and the

initial conditions are shown in Table 3 on a 1000 MVA

basis.

The equilibrium P- V curve of the load bus is drawn in

Figure 4 for constant generator output and excitation

voltage E f because the machine is on manual voltage

control. This curve is not similar to the tradit ional nose

curve . The main difference is that there are two possible

SNB points for a system operating initially at point A,

one for high load (point C) and one for light load (point

SNB). The light load bifurcation point is brought about

by the power transfer limit on the long tie line. Note that

1.1

1.05

0 9

0.~

:3

~0 . 8 ~

O.E

0.7~

7 1 / /

7

o.6s / /

0.6 / /

2OO 3OO 4OO 5O0 600 700 80O

PI MW)

Figure 4. Load bus P V curve

Q

/ B /

/ ^ /

S N B ~

/ / /

I i I

9 00 1 0 0 0 1 1 0 0 1 2 00

this SNB point is in the upper part of the P-V curve,

meaning that the security margin would be larger for a

constant power than for a constant impedance load.

A sudden load admittance reduction at time t = 0,

from GA = 0.63p.u. (corresponding to point A) to

GB = 0.5 p.u. (corresponding to the a lmost tangent load

characteristic close to the SNB point in Figure 4) is

simulated next. The results are shown in Figure 5. The

following points are worth noting.

All responses have a fast transient part corresponding

to the electromechanical oscillation mode, which dies

out in a few seconds after the disturbance.

The effect of the SNB is evident both in the machine

demagnetization and in the rotor angle upward drift.

Synchronism is eventually lost 142 seconds after the

disturbance.

During the two minute interval prior to the loss of

synchronism there are no observable effects either in

frequency, or in the generated power of the synchro-

nous machine.

As soon as synchronism is lost, the generator power

output drops abruptly and the frequency rises rapidly.

Before the loss of synchronism, the apparent power

flow on the tie line has reached 650 MVA, becoming

double the value immediately after the disturbance,

and the load bus voltage has dropped close to 0.7 p.u.

It is interesting to compare the above remarks on the

simulated test case of Figure 5 with the following com-

ments on the actual B.C. Hydro incident taken from

Reference 8.

Although constant impedance load characteristic is

considered soft and least aggravating in voltage stabi-

lity studies, this case shows the fallacy of such general-

iza tio ns. .. The output of generators was fairly steady

within the observed duration, which excludes the pos-

siblity of angle instability... The system separated

when (i) the current at one end exceeded the over-

current relay setting; (ii) the apparent impedance at the

other end was within the out-of-step relay setting.

V . C o n c l u s i o n s

This paper has shown how models containing both angle

dynamics and voltage dynamics can be decomposed into


7/8

Relat ionships between voltage and angle stabi l i ty of pow er systems

C D V o u r n a s e t a l 4 9 9

2

1.5

1

0.5

r o t o r a n g l e r a d )

0 50 100 150

f i e l d f l u x p u )

t .05

1

0.95

0.9

0.85

50 100 150

L o a d b u s V o l ta g e

pu)

0.

0.8

0.7

37~

377.5

377

376.5

37E

L

700

650

0

5OO

40O

300

200

0 50 100 150 0

f r e q u e n c y rad/sec)

g-

50 100 150

G e n e r a t e d p o w e r

MW)

le

50 100 150

T i e l ine active power MW)

L o a d p o w e r MW)

500 7001

450 ~ 6 1

40O ~ 5001

350 4 1

30O 300

25O 2O0

50 100 150 0 5 0 100

t i m e s e e ) t ime sec)

Figure 5

Simulation results for case

N 3

50 100 150

T i e l i n e a p p a r e n t

power MVA)

150

sepa ra te subsys tems. In pa r t icu la r , the process was pe r -

f o r me d o n a mo d e l w h e r e t h e e l ec t ro me c h a n ic al d y n a m ic s

make up the fa s t subsys tem and the f ie ld f lux dynamics

ma k e u p t h e s l o w su b sy s t e m. T h e se r e d u c e d - o r d e r su b -

sys tems prese rve the c r i t ical nonl inea r i tie s and the l inea r

s tab i l i ty in form at ion . Th e linea r ized mu l t imachine vol -

tage subsy s tem has the same s tab i l i ty cond i t ion tha t has

been pro pos ed in the pas t a s a vo l tage s tab i l ity ind ica tor .

A s in g l e ma c h in e e x a mp le sh o w e d h o w a s a d d l e - n o d e

b i f u r c a t i o n o c c u r s d u e t o t h e v o l t a g e su b sy s t e m b e f o r e

a n y a n g l e i n s ta b i li t y . H o w e v e r , a n o n l in e a r s imu la t i o n

sh o w s th a t w h e n a sy s t e m r e a c h e s t h i s s a d d l e - n o d e

bi furca t ion poin t , the sys tem wi l l eventua l ly lose syn-

chron ism as the gene ra to r f ie ld f lux s lowly decays . The

imp o r t a n t p o in t h e r e is th a t t h e s a d d l e - n o d e b i f u r c a t i o n

w a s a s so c i a t e d w i th v o l t a g e d y n a m ic s a n d n o t a n g l e

d y n a mic s . T h u s i t sh o u ld b e c o n s id e r e d a v o l t a g e i n s t ab i l -

i ty . However , a s the s low vol tage ins tab i l i ty progresses ,

t h e a n g l e d y n a m ic s e v e n tu a l l y c o n t r i b u t e t h r o u g h a

s ingula r i ty in the synchroniz ing ma tr ix , caus ing angle

ins tab i l i ty . I t was shoran a lso tha t the s low vol tage decay

p h e n o m e n o n ma y l a s t f o r min u t e s , w h ic h imp l i e s t h a t t h e

in t e r a c ti o n o f s lo w ma c h in e d y n a m ic s w i th o th e r s l o w

a c t in g d e v i c e s su c h a s L T C s ma y b e o f c o n s id e r a b l e

in te res t . This i s a s t imula t ing problem for fur the r

research.

W h e n so me ma c h in e s h a v e a c t i v e v o l t a g e r e g u l a to r s ,

t h e r e d u c e d - o r d e r s l o w v o l t a g e mo d e l co n s i s t in g o f o n ly

the f ie ld f lux dyn am ics of exc i ta t ion- l imi ted m achines c an

se r v e a s a n a p p r o x im a t io n o f th e u n r e g u la t e d su b sy s t e m

s lo w d y n a mic s . T h i s a p p r o x ima t io n g iv e s g o o d r e su l t s

o n ly f o r h ig h e x c i t a t i o n g a in s o f th e r e g u l a t i n g m a c h in e s.

F u r th e r r e se a rc h i s n e c e s sar y in o r d e r t o a c h i e v e a mo r e

a c c u r a t e mo d e l b y u s in g a f o r ma l t ime - sc a l e d e c o m p o s i -

t i o n o f th e p a r t l y r e g u l a t e d sy s t e m.

Vl A c k n o w l e d g e m e n t s

T h i s w o r k w a s su p p o r t e d i n p a r t b y g r a n t s f r o m th e

N a t io n a l S c i e n c e F o u n d a t io n , N S F E C S 9 1 - 1 9 4 2 8 a n d

N S F E C S 9 3 -1 8 69 5 , t h e U n iv e r s i t y o f Il li n o is P o w e r

A f f il ia t es P r o g r a m , a n d t h e G r a in g e r E n d o w me n t s t o

the Un ive rs i ty of I ll ino is .

V I I R e f e r e n c e s

1 Ta y lo r , C W c onve ne r o fC IG R I~ TF 38 .02 .10 ,

M o d e l l i n g o f

v o l t a g e c o ll a p s e in c lu d in g d y n a m i c p h e n o m e n a

Pre p r in t

(March 1993)

2 T a y l o r C W V o l t a g e s t a b i l i ty E P R I / M c G r a w H i ll ( 19 94 )

3 H i l l, D J N on l ine a r dyna mic loa d mode l s w i th r e c ove ry fo r

vol tage s tabi l i ty s tudies I E E E T r a n s . PW R S-8 (1993)

pp 166 -176

4 Xu, W a n d M a n s o u r Y Voltage s tabl i ty ana lys is us ing

ge ne r i c dyna mic loa d mode l s

I E E E T r a n s .

P W R S - 9

(1994) pp 479-493

5 K u n d u r P P o w e r s y s t e m s t a b i l i t y a n d c o n t r o l E P R I /

M c G ra w -H i l l (1994 )

6 V a n C u t s e m T J a e q u e m a r t Y M a r q u e t J - N a n d P r u v o t

P A c ompre he ns ive ana ly s is o f mid - t e rm vo l t a ge st a b i l it y

I E E E T r a n s . PWRS-10 (1995) pp 1173-1182

7 Van Cutsem, T a n d V o u r n a s C D Vol tag e s tabi l i ty ana lys is

in t r a ns i e n t a nd mid - t e rm t ime s c a l es

I E E E T r a n s .

P W R S -

11 (1996)pp 146-154

8 I E E E C o m m i t t e e, Voltag e s tabi l ity of powe r systems: co n-

cepts , ana lyt ica l tools , and indus try experience publ ica t ion

90TH 0358-2 -PW R , IEE E (1990)

9 Pa d iya r , K R a nd R a o , S S , D yn a m ic a na ly si s o f s ma ll

s igna l vol tage ins tabi l i ty decoup led fro m angle ins tabi l i ty

I n t . J . E l e c t r . P o w e r E n e r g y S y s t .

Vol 18 No 7 (1996) 445 -

452

1 0 Y o r i n o N S a s a k i H M a s u d a Y T a m u r a Y K i t ag a w a

M and Oshimo, A An inves t iga t ion of vol tage ins tabi l i ty

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