observable island identification for state estimation using incidence matrix

8/9/2019 Observable Island Identification for State Estimation Using Incidence Matrix

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Automatica. Vol. 24. No. I. pp. 71-75, 1988

Printed in Great Britain.

0005-1098/88 $3.00 + 0.00

Pergamon Journals Ltd.

© 1988 International Federationof AutomaticControl

Brief Paper

Obs e rvab le I s lan d I de n t i f i c a t io n f o r S t a t e

E s t i m a t i o n U s i n g I n c i d e n c e M a t r i x *

YOUNG H. MOON,t YOUNG M. PARK~: and KYUNG J. LEE§

K e y W ords - -S t a t e e s t im a t ion ; obse rvab i l i t y ; t opo log i ca l ana lys i s ; i n t ege r G auss e l im ina t i on ; m e te r

p l acem en t ; i r r e l evan t m easurem en t ; obse rvab l e i s l and i den t i f i ca t i on .

A b s t r ac t - -T h i s pap e r dea l s w i th t he t opo log i ca l obse rvab i l i t y

ana lys i s and t he deve lopm en t o f an obse rvab l e i s l and

iden t i f i ca t i on a lgo r i t hm in l a rge pow er sys t em s by us ing t he

inc idence m a t r i x .

T he t opo log i ca l obse rvab i l i t y i s ana lyzed by us ing t he

pa r t i t i oned i nc idence m a t r i x w h ich has a l l t he i n fo rm a t ion o n

t he sys t em topo logy and m easur ing po in t s a s w e l l . T he

topo log i ca l obse rvab i l i t y t e s t i s pe r fo rm ed by app ly ing t he

in t ege r G auss e l im ina t i on m e thod t o t he t opo log i ca l

m e a s u r e m e n t e q u a t i o n .

F or t he i den t if i ca t ion o f obse rvab l e i s l ands , a bus g roup ing

a lgor i t hm i s deve loped on t he bas i s o f t he t opo log i ca l

obse rvab i l i t y ana lys i s unde r t he a s sum pt ion o f a l l m easu re -

m en t s be ing ze ro . T h i s a lgo r i t hm rem arkab ly r educes t he

com puta t i ona l bu rden com pared w i th t he conven t iona l

a lgo r i t hm s . A m e te r p l acem e n t a lgo r i t hm i s p r e sen t ed t o

r ecove r t he obse rvab i l i t y o f an unobse rvab l e m easurem en t

system.

1. Introduction

THE OPERATION of m odern pow er sys t em s i s based on s t a t e

e s t i m a t i o n p e r f o r m e d b y p r o c e s s i n g p r o p e r m e a s u r e m e n t

da t a . T he r e l i ab i l i t y o f s t a t e e s t im a t ion e s sen t i a l l y depends

o n t he obse rvab i l i t y o f t he sys t em s t a t e . T h e ob se rvab i l i ty o f

t he sys t em s ta t e i s r e l a t ed t o t he nu m be r o f m easurem en t s

and t he i r geograph i c d i s t r i bu t i on . E ven i f t he sys t em i s

des igned t o be obse rvab l e , t em pora ry unobse rvab i l i t y m ay

be encoun te red due t o t opo log i ca l ne tw ork changes o r

f a i l u r e s on m e te r s , R T U s (R em ote T e rm ina l U n i t s ) e t c . I n

cases w here t he sys t em i s unob se rvab l e , i t i s des i r ed t ha t t he

fo l l ow ing p rocedures a r e t aken .

( i ) I den t i fy obse rvab l e i s l ands o f t he sys t em and pe r fo rm

the s t a t e e s t im a t ion fo r each obse rvab l e i s land un t i l t he

obse rvab i l i t y fo r t he w ho le sys t em i s r ecove red .

( i i) R eco ve r t he obse rvab i l it y o f the w ho le sys tem as soon

as poss ib l e by p l ac ing a m in im a l s e t o f add i t i ona l

m easurem en t s .

Im po r t an t con t r i bu t i ons t o t he o bse rvab i l i t y ana lys is have

b e e n m a d e b y C l e m e n t s et al. (1982a, b , 198 3) . The se

au thor s e s t ab l i shed t heo re t i ca l f ounda t i ons fo r t he obse rv -

abi l i ty analys is wi th the use of the graph theory.

* R ece iv ed 4 F ebrua ry 1986 ; r ev i sed 6 J anua ry 1987 ;

r ev i sed 23 June 1987 . T he o r ig ina l ve r s ion o f t h is pape r w as

p r e s e n t e d a t t h e I F A C S y m p o s i u m o n P o w e r S y s t e m a n d

P ow er P l an t C on t ro l w h ich w as he ld i n B e i j i ng , P . R . o f

C h ina du r ing A ugu s t 1986 . T he P ub l i shed P roceed ings o f t hi s

I F A C M e e t i n g m a y b e o r d e r e d f r o m : P e r g a m o n B o o k s

L i m i t e d , H e a d i n g t o n H i l l H a ll , O x f o r d O X 3 O B W , E n g -

l and . T h i s pape r w as r ecom m ended fo r pub l i ca t i on i n r ev i sed

fo rm by A ssoc i a t e E d i to r R . V . P a t e l unde r t he d i r ec t i on o f

E d i t o r H . K w a k e r n a a k .

1 D epa r tm en t o f E l ec t r ica l E ng inee r ing , Y onse i U n ive r -

s i t y , 134 S h incho n-D on g , S eodae m oo n-K u , S eou l , K orea .

:~ D ep a r tm e n t o f E l ec t r i ca l E ng inee r ing , S eou l N a t iona l

U nive r s i t y , K orea .

§ R e s e a r c h a n d D e v e l o p m e n t C e n t e r , K o r e a E l e c t r i c

P o w e r C o r p o r a t i o n , K o r e a .

71

M ont i ce l l i and W u (1985a , b ) p r e sen t ed a com ple t e t heo ry

of ne tw ork obse rvab i l i t y w i th theo rem s , and p roposed a

s im ple a lgo r i t hm fo r t he obse rvab l e i s l and i den t i f i ca t i on and

the m easurem en t p l acem en t , based on t he ca l cu l a t i on o f

phase angles .

B a rg i e l a et al. (1985) p ropose d an i n t e r e s t i ng obse rvab i l it y

a lgo r i t hm us ing a ne tw ork f l ow t echn ique , an d H or i sbe rge r

(1985) p re sen t ed an a lgo r i t hm o f m ax im a l obse rvab l e i s l and

iden t i fi ca t ion based on sea rch ing fo r an ob se rvab i l i t y dec is i on

t ree wi th the complete theoret ical analys is .

In this s tudy, the topological observabi l i ty i s analyzed by

us ing t he pa r t i t i oned i nc idence m a t r i x w h ich has a l l t he

in fo rm a t ion o f m easur ing po in t s. A n ana logy o f t he D C

pow er f l ow m e thod t o t he D C c i r cu i t ana lys i s i s i n t roduced

s ince t he l a t t e r can be w e l l - fo rm ula t ed by us ing t he i nc idence

m a t r i x . W i th t h i s ana logy , a l l r e l a ti ons be tw een p ow e r f low s

and phase ang l e s a r e r ep l aced by t he co r r e spond ing

cur r en t -vo l t a ge r e l a t ions . A s a r e su lt , a se t o f T M E s

(T opo log i ca l M easurem en t E qua t i ons ) i s de r ived fo r t he

topo log i ca l ana lys is . T he f ea tu re o f t he T M E se t i s t ha t eve ry

e l em en t o f m a t r i x i n t he equa t i on i s an i n t ege r .

T he obse rvab i l i t y t e st i s ca r r i ed ou t by exam in ing t he r ank

of t he m easurem en t m a t r i x i n t he T M E se t . T he i n t ege r

G auss e l im ina t i on m e thod i s i n t roduced fo r t he de t e rm ina -

t i on o f m a t r i x r ank .

A p rec i se a lgo r i t hm fo r t he obse rvab l e i s l and i nden t i f i ca -

t i on i s deve loped w i th an app l i ca t i on o f a bus g roup ing

m ethod . T he bus g roup ing m e thod i s based on t he

ca l cu l a t i on o f bus vo l t ages unde r t he a s sum pt ion o f a l l

m easurem en t s be ing ze ro .

T he op t im a l m e te r p l acem en t a lgo r i t hm i s p r e sen t ed t o

r ecove r t he obse rvab i l i t y o f unobse rvab l e m easurem en t

sys t em s . T he a lgo r i t hm i s e s t ab l i shed by com bin ing t he bus

g roup ing m e thod and H or i sbe rge r ' s obse rvab i l i t y dec i s i on

t r ee a lgo r i t hm s . T he a lgo r i t hm p roduces t he m ax im a l

obse rvab l e i sl ands even fo r t opo log i ca l l y sym m et r i c sys t em s ,

and sea rches fo r t he m in im a l s e t o f add i t i ona l m easurem en t s

w hich w i l l r ecove r t he obse rvab i l i t y o f t he w ho le sys t em .

2. Topological obseroab ility analysis by incidence matrix

This sect ion deals wi th the analysis of topological

obse rvab i l i t y . W i th t he use o f i nc idence m a t r i x , t opo log i ca l

analysis i s carr ied out on the basis of measurement sys tem

m ode l ings by t he bus vo l t age r ep re sen t a t i on .

2.1. Topological observability analysis. T he obse rvab i l i t y o f

a m easurem en t sys t em depends on t he sys t em conf igu ra t i on

and t he m easurem en t p l acem en t on ly . T he p r inc ip l e o f new

topological analys is i s presented, which i s based on the

analogy of the DC load f low analysis to the c i rcui t analys is .

T he pow er sys t em pa ram e te r s have t he fo l l ow ing

prope r t i e s :

( i ) a l l l i ne pa ram e te r s a r e know n,

( i i ) l oad im pedances and gene ra to r im pedances a r e

unknow n s ince t hey change w i th s t a t e s o f l oads and

gene ra to r s .

Consider a s ix-bus system wi th e ight l ines given in F ig. 1 .

R ea l and r eac t i ve i n j ec t ion pow er s a r e m easu red a t a l l buses .

T he f l ag s i gn r ep re sen t s pow er m e te r p l ac ing . T he sys t em i s


2/5

7 2 B r i e f P a p e r

. . . - . X F ~

Z S

P,

F i G . 1. S ix-bus system.

P~

®

®

P6

obv ious ly obse rvab l e , s i nce a ll bus i n j ec t ions a r e know n. T he

system observabi l i ty means that a l l l ine f lows and bus

in j ec t ion pow er s can be ca l cu l a t ed f rom the m eas ured da t a .

N ow w e w i l l exam ine how a l l r ea l l i ne f l ow s can be

ca l cu l a ted f rom the m easured da t a . F o r t he s im pl i c i ty o f

ca l cu l a ti on , i t is a s sum ed tha t a l l li ne im pedan ces a r e p u re ly

im ag ina ry . S ince t he re i s no l o s s o f. r ea l pow er , t he re a r e

e igh t unknow n r ea l l i ne f l ow s f rom F , t h rough Fs . By

app ly ing t he pow er ba l ance cond i t i on t o each bus , w e have

s ix equa t i ons . H ow e ver , on ly fi ve equa t i ons am o ng them a re

l i nea r ly i ndependen t . F rom the phase ang l e r e l a t i ons , l oop

equa t i ons can be se t up us ing t he D C f low m e tho d . L oo p 1

gives an equ at ion of phase angles , 0 ,z + 023 + 03~ = 0 or

7-qFx ZzFz-

7-~F = 0 wi th l ine imp eda nce Z 1 through Z s .

S im i l a r l y , tw o m ore l oop equa t i ons can be se t up fo r l oop 2

and l oop 3 . A l l o f t hese l oop equa t i ons a r e i ndependen t .

Independen t equa t i ons t o t a l e i gh t .

C onsequen t ly t he e igh t unknow n l i ne f l ow s a r e de f in i t e ly

c a l c u l a t e d f r o m t h e a b o v e - m e n t i o n e d e i g h t i n d e p e n d e n t

equa t i ons . T h i s m eans t he sys t em i s obse rvab l e .

F or t he sys t em a t i ca l t e s t o f t opo log i ca l obse rvab i l i t y , t h i s

s t udy adop t s t he fo l l ow ing p rocedures .

( i ) A ssum e a com plex cu r r en t i s m easu red fo r eve ry po in t

w here r ea l and r eac t i ve pow er s a r e m easured .

( ii ) S e t up a ll i ndepen den t l oop and noda l equa t i on s by

us ing t he i nc idence m a t r i x .

( i i i ) C heck w he the r o r no t a l l l i ne cu r r en t s can be

ca l cu l a t ed f rom the m easured cu r r en t s . I f pos s ib l e,

t hen t he sys t em i s t opo log i ca l l y . obse rvab l e . O the rw i se ,

t he sys t em i s unobse rvab l e .

2.2.

Topolog ical observability analysis algorithms u sing the

inc idence matr ix . The P-O

obse rvab i l i t y a lgo r i t hm can be

appl ied to the analysis of Q - V obse rvab i l i t y a s d i scussed

ea r l i e r by C lem en t (1982b) and H or i sb e rge r (1985). I n t h i s

con t ex t , on ly t he P - O obse rvab i l i t y w i l l be ana lyzed . T he

topo log i ca l obse rvab i l it y a lgo r i t hm i s deve lope d by r ep l ac ing

rea l pow er m easurem en t s by cu r r en t m easurem en t s .

Bus vol tage formulat ion. B us i n j ec t i on cu r r en t s can be

class if ied into m easur ed inject io n curre nts I m and un-

m easured i n j ec t i on cu rr en t s Inz . W e have t he fo l l ow ing bus

vo l t age -cu r r en t r e l a t i on by appropr i a t e o rde r ing o f bus

num ber s :

w h e r e y n u s : b u s a d m i t t a n c e m a t r i x

V B : bus vo l t age vec to r .

S imi lar ly the l ine currents can be c lass i f ied into

unm e asured l i ne cu r r en t s I , , and m easu red l i ne cu r r en t s 1 , 2,

a n d w e h a v e t h e f o l ow i n g p ri m i t iv e e d g e c u r r e n t - v o l t a g e

rela t ion:

i, 1= v.=#ATVB (2)

Ie2J

w here ) ,: p r im i t i ve adm i t t ance m a t r i x

V , : edge vo l t age vec to r

A : i nc idence m a t r i x .

W i th t he use o f t he pa r t i t i oned i nc idence m a t r i x , t he bus

in j ec t i on cu r r en t s can be expres sed a s fo ll ow s :

= 3 )

l ~ J L A 2 1 I A = J L I . 2 J

w i t h

u n m e a s u r e d m e a s u r e d

l ines l ines

r A 1 , i A , 2 ] } m easu red buses

A = L . . . ~ i ; ; ' ~ ' - ' - ~ J ) u n m e a s u r e d b u s e s.

T he p r im i t i ve m a t r i xy i s a l so pa r t i t i oned a s fo l l ow s :

u n m e a s u r e d m e a s u r e d

[ y n ~ y ,2 ] ) u n m e a s u r e d

Y = L y 2 ~ ' , . . . . ~ ' - J } m e a s u r e d . ( 4)

Subst i tut ion o f par t i t io ned A an d y into (2) gives:

y,4r4 L4 l,,

l . z J - L ~ 7 Y ~ ; ; J La T ~ , A ~ . I - w ( 5 )

B y us ing t he above equ a t i on , t he un m easu red l i ne cu r r en t 1 , ,

in (3) can be e l iminated as fol lows:

Im AH T T , T

F rom (5) and (6 ) , w e have t he fo l l ow ing equa t i ons w i th

re spec t t o V s :

_ T

I m [ A , , , ~ , : A , 1 A , , , ~ , 2 A T I A ~ , ,~ t tA 2 T t

+ X l , y , , A ~ ] V s + A I ~ I,2 7 )

In t he above equa t i ons , bus vo l t ages V s a r e t he on ly

unknow n va r i ab l e s .

T he t opo log i ca l obse rvab i l i t y can now be cons ide red by

se t t i ng t he p r im i t i ve adm i t t ance m a t r i x u t o an i den t i t y

m a t r i x . T ha t i s y l l = l , y i z = O, y2 , = O, y22 = L Subst i tut ion

of t hese m a t r i ce s i n to (7 ) l eads t o

[ [ 'a l 'a 3 ~ ' : d ' ~ ' 4 : d

s i - A ,21e21 T ' T

= / A v ,

AT j - - s = M s V s

( 8 )

l e 2

a L 12 ' 22

w i th

T , T

r A , , A 1 , , A n A 2 , ]

M s = l--~---: --~,r---J.

L 12 ' 22

T he m a t r i x M s w i l l be ca l l ed t he t opo log i ca l bus

m e a s u r e m e n t m a t r ix .

C onsequen t ly , t he t opo log i ca l obse rvab i l i t y can be

d e t e r m i n e d b y c h e c k i n g t h e r a n k o f m a t r ix M s . I f t he r ank o f

M s is (N - 1 ), t hen t he m easu rem en t sys tem is obse rvab l e .

O the rw i se , t he sys t em i s unobse rvab l e .

I t i s n o t e d t h a t t h e t o p o l o g i c a l m e a s u r e m e n t m a t r i x M s

can be d i r ec t l y com posed by us ing t he pa r t i t i oned i nc idence

m a t r i x g i v e n i n ( 8 ) . M o r e o v e r , e v e r y e l e m e n t o f M s i s a n

i n te g e r . T h e r a n k o f a n i n t e g e r m a t r i x ca n b e d e t e r m i n e d b y

reduc ing t he m a t r i x t o a row eche lon fo rm by e l em en ta ry

row ope ra t i ons i nvo lv ing i n t ege r s , i . e . by p rem ui t i p l i ca t i on

by an i n t ege r m a t r i x w h ich i s un im odu la r ove r t he r i ng o f

in t ege r s . F o r t he r ank t e s t o f m a t r i x M s , t h i s s t udy has

adop ted t he i n t ege r G auss e l im ina t i on m e thod , w h ich

enab l e s us t o avo id t he nea r ly -ze ro -p ivo t p rob l em .

3. Ob serv ab le island identification

W hen a m easurem en t sys t em i s unobse rvab l e , i t i s des i r ed

to f i nd m ax im a l obse rvab l e i s l ands and pe r fo rm the s t a t e

e s t im a t ion fo r each i s l and . T he obse rvab l e i s l and i s an

o b s e r v a b le s u b n e t w o r k o f a p o w e r s y s te m . A n u n o b s e r v a b l e

pow er sys t em can be d iv ided i n to obse rvab l e i s l ands .

S uppose a p ow er sys t em cons i s t s o f t h r ee o bse rvab l e i s l ands.

T h e n , t h e m e a s u r e m e n t e q u a t i o n s c a n b e d i v i d e d i n to t h r e e

d e c o u p l e d g r o u p s o f m e a s u r e m e n t e q u a t i o n s a n d a g r o u p o f


3/5

Brief Paper 73

i r r e le v a n t m e a s u r e m e n t e q u a t i o n s :

l r o l i z

. . . . . . . . . . . . . o f

0 . 1 / o z . 9 )

I 0 : 0 : n / l / : O =

P h a s e a n g l e s i n e a c h i s l a n d a r e i n d e p e n d e n t o f t h e

m e a s u r e m e n t s i n o t h e r g r o u p s w h e n t h e i r r e l e v a n t

m e a s u r e m e n t s a r e r e m o v e d . E a c h o b s e r v a b l e i s l a n d h a s a

s l ack bus . Al l phase ang le s a s we l l a s a l l l i ne f lows in the

o b s e r v a b l e i s l a n d a r e u n i q u e l y d e t e r m i n e d f r o m t h e

c o r r e s p o n d i n g g r o u p o f m e a s u r e m e n t s i f th e p h a s e a n g l e o f

t h e s l a c k b u s is a s s i g n e d . C o n s e q u e n t l y , i f we h a v e z e r o

r e a d i n g s f o r a l l m e a s u r e m e n t s , t h e n a l l l i n e f l o ws m u s t b e

z e r o a n d a l l b u s v o l t a g e s i n t h e i s l a n d m u s t b e t h e s a m e .

U n d e r t h e s e o b s e r v a t i o n s , s e v e r a l t h e o r e m s w i l l b e

p r e s e n t e d .

A s s u m e t h a t a l l m e a s u r e m e n t s a r e z e r o . T h e n , w e h a v e

t h e f o l l o wi n g f r o m ( 8 ) :

M BVB = 0 o r M ~ M a V a = 0. (10)

I r r e l e v a n t m e a s u r e m e n t s m a k e u s e n c o u n t e r g r e a t

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

d e c o u p l e d g r o u p s . F o r t h e o b s e r v a b l e i s l a n d i d e n t i f i c a t i o n , i t

i s n e c e s s ar y t o r e m o v e a l l i r r e l e v a n t m e a s u r e m e n t s f r o m t h e

m e a s u r e m e n t e q u a t i o n .

F o r t h e d e v e l o p m e n t o f t h e o b s e r v a b l e i s la n d i d e n ti f ic a t io n

a l g o r i t h m , t h e f o l l o wi n g t h e o r e m s a r e e s t a b l i s h e d .

Theorem

1 . E v e r y b o u n d a r y b u s i n je c t i o n m e a s u r e m e n t is a n

i r r e l e v a n t m e a s u r e m e n t .

Proof.

F o r t h e s t a t e e s t i m a t i o n o f e a c h i s l a n d i n F i g . 2 , t h e

f l ow t h r o u g h l i n e ,~ i s c o n s i d e r e d a s a b u s i n j e c t i o n . T h e l i n e

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

s i n c e b u s e s i a n d j b e l o n g t o d i f f e r e n t o b s e r v a b l e i s l a n d s .

Th e r e f o r e , t h e t o t a l b u s i n j e c t i o n a t b u s i o r j is u n k n o w n .

T h i s m e a n s a n i n j e c t i o n m e a s u r e m e n t a t a n y b o u n d a r y b u s i s

a n i r r e l e v a n t m e a s u r e m e n t . I t i s n o t e d t h a t a n y l i ne

m e a s u r e m e n t c a n n o t b e a n i r r e l e v a n t m e a s u r e m e n t .

Theorem

2 . A s s u m e a l l m e a s u r e m e n t s a r e z e r o . I f d i f f e r e n t

p h a s e a n g l e s c a n b e a s s i g n e d t o t w o b u s e s u n d e r t h e

a s s u m p t i o n , t h e n t h e t w o b u s e s b e l o n g t o d i f f e r e n t

o b s e r v a b l e i s l a n d s .

Theorem

3 . A s s u m e a l l m e a s u r e m e n t s a r e z e r o . I f a

s u b n e t w o r k i n w h i c h a l l b u s v o l ta g e s c a n n o t b e d i f f e r e n t h a s

n o i r r e le v a n t m e a s u r e m e n t s , t h e n t h e s u b n e t w o r k i s a n

o b s e r v a b l e i s l a n d .

T h e t w o t h e o r e m s a b o v e c a n b e p r o v e d f r o m t h e d e f in i t io n

o f o b s e r v a b l e i sl a n d a n d ( 1 0 ) w i t h t h e a n a l o g y o f t h e D C

p o we r f l o w a n a l y s i s t o t h e c i r c u i t a n a l y s i s a s m e n t i o n e d

ea r l i e r .

Theorem

4 . I f a b u s m e a s u r e m e n t I B r is r e m o v e d , t h e

m o d i f i e d m e a s u r e m e n t e q u a t i o n i s g i v e n b y d e l e t i n g t h e

c o r r e s p o n d i n g ro w . U n d e r t h e a s s u m p t i o n o f z e r o r e ad i n g s

o f a l l m e t e r s , t h i s i s e q u i v a l e n t t o a m e a s u r e m e n t e q u a t i o n

o b t a i n e d b y r e p l a c i n g t h e r o w c o r r e s p o n d i n g t o t h e

m e a s u r e m e n t b y a z e r o r o w v e c t o r .

By a p p l y i n g t h e a b o v e t h e o r e m s t o ( 1 0 ) , t h e f o l l o wi n g

a l g o r i t h m i s d e v e l o p e d f o r t h e o b s e r v a b l e i s l a n d

iden t i f i ca t ion .

( i ) As s i g n b u s n u m b e r s t o m e a s u r e d b u s e s f i rs t a n d t h e

o t h e r b u s e s n e x t . A s s i g n l i n e n u m b e r s t o u n m e a s u r e d

l ines f i r s t and the o the r l i nes nex t . In i t i a l i ze a

m e a s u r e m e n t s e t c o n s i s t i n g o f a l l a v a i l a b l e m e a s u r e -

m e n t s . S e t u p a b u s g r o u p c o n s i s t in g o f a ll b u s e s o f t h e

s y s te m a n d e s t a b l i s h t h e m e a s u r e m e n t m a t r i x M a . T h e

wh o l e s y s t e m i s c o n s i d e r e d a s a c a n d i d a t e f o r a n

o b s e r v a b l e i s l a n d .

( i i) Ta k e a n o b s e r v a b l e is l a n d c a n d i d a t e . S e l e c t a s l a ck b u s

f o r t h e i s l a n d a n d s e t t h e s l a c k b u s v o l t a g e t o 1 .

C o m p o s e r e d u c e d m e a s u r e m e n t m a t r i x M s R b y

s e l e c t i n g c o l u m n v e c t o r s o f M 8 c o r r e s p o n d i n g t o b u s e s

o f t h e c a n d i d a t e i s l a n d . I f t h e c a n d i d a t e i s l a n d h a s o n l y

o n e b u s , g o t o s t e p ( v ) .

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

c a n d i d a t e b y a p p l y i n g t h e i n t e g e r Ga u s s e l i m i n a t i o n t o

( 1 0 ) . W h e n e v e r a z e r o p i v o t i s e n c o u n t e r e d , s e t t h e b u s

v o l t a g e t o ( t h e l a r g e s t b u s v o l t a g e + 1 ) . I f o n l y o n e

z e r o p i v o t i s e n c o u n t e r e d , g o t o s t e p ( v i ) .

Co m p o s e b u s g r o u p s b y c l a s s if y i n g b u s e s w i t h s a m e b u s

v o l t a g e s i n t o o n e g r o u p ( c f . Th e o r e m 2 ) . Ea c h b u s

g r o u p i s r e g a r d e d a s a n e w o b s e r v a b l e i s la n d

c a n d i d a t e .

( v ) F i n d i r r e l e v a n t m e a s u r e m e n t s i n t h e i s l a n d u n d e r

c o n s i d e r a ti o n b y T h e o r e m 1 , an d r e m o v e a l l t h e

i r r e l e v a n t m e a s u r e m e n t s f r o m t h e m e a s u r e m e n t s e t .

M o d i f y th e m e a s u r e m e n t m a t r ix b y T h e o r e m 4 . G o t o

s t ep ( i i ) .

Th e o b s e r v a b l e i s l a n d c a n d i d a t e i s a n o b s e r v a b l e

i s l a n d . De l e t e t h e b u s g r o u p f r o m t h e s e t o f c a n d i d a t e

i s l a n d s . I f a n y o b s e r v a b l e i s l a n d c a n d i d a t e r e m a i n s , g o

t o s t e p ( i i ) . O t h e r w i s e , t e r m i n a t e t h e a l g o r i t h m .

( i i i )

i v )

(v i )

Special Case.

Co n s i d e r t h e f o l l o wi n g f o u r - b u s s y s t e m ( F i g . 3 )

w i t h t w o b u s m e a s u r e m e n t s a n d o n e l i n e m e a s u r e m e n t . T h e

m e a s u r e m e n t s y s t e m i s o b s e r v a b l e s i n c e a l l o f t h e t h r e e

m e a s u r e m e n t s a r e i n d e p e n d e n t .

Ho r i s b e r g e r ' s t o p o l o g i c a l a l g o r i t h m y i e l d s t h e c o r r e c t

a n s w e r f o r o b s e r v a b i li t y . H o w e v e r , c o n d u c t i n g t h e p r o p o s e d

a l g o r i t h m y i e l d s t h e f o l l o wi n g r e s u l t : t h e m e a s u r e m e n t

s y s t e m i s u n o b s e r v a b l e , a n d t h r e e o b s e r v a b l e i s l an d s { 1 , 2 } ,

{ 3 } , ( 4 } a r e f o u n d . Th i s r e s u l t i s o b v i o u s l y wr o n g . M o n t i c e ll i

a n d W u ' s a l g o r i t h m ( 1 9 8 5 b ) a l s o y i e l d s t h e s a m e wr o n g

a n s we r . T h e w r o n g r e s u l t f o r t h i s s p e c i a l c a s e c o m e s f r o m t h e

t o p o l o g i c a l s y m m e t r y o f t h e g i v e n s y s t e m . S i n c e e v e r y l i n e

a d m i t t a n c e i s a s s u m e d t o b e 1 f o r t h e t o p o l o g i c a l a n a l y s i s ,

t h e s y s t e m i s s y m m e t r i c a l . Th i s s y m m e t r i c a l s t r u c t u r e m a k e s

t h e m e a s u r e m e n t a t l i n e 3 h a v e n o i n f o r m a t i o n o n b u s 3 o r

b u s 4 . I f we g e t r i d o f t h e t o p o l o g i c a l s y m m e t r y b y a s s i g n i n g

l i n e a d m i t t a n c e s w i t h 2 t = 1 , y - - 2 , y 3 = 1 , y 4 = 2 a n d y s = 2 ,

t h e n t h e p r o p o s e d a l g o r i t h m y i e l d s t h e c o r r e c t a n s we r .

H o w e v e r , n o e f f ic i en t m e t h o d s h a v e b e e n f o u n d y e t t o s e a r c h

f o r t h e t o p o l o g i c a l s y m m e t r y o f s y s t e m s a n d t o g e t r i d o f t h e

s y m m e t r y s y s t e m a t i c a l l y .

Th e n e x t s e c t i o n w i l l g i v e a s i m p l e a l g o r i t h m o f m a x i m a l

o b s e r v a b l e i s l a n d i d e n t i f i c a t i o n f o r t h e s e s p e c i a l c a se s .

s L a n d

s t nd 2

F IG . 2. Ob s e r v a b l e i s la n d s .

®

I 2

FxG. 3 . Spec ia l ca se o f fou r -bu s sys t em.


4/5

74 Brief Paper

4. Meter placement

W h e n a m e a s u r e m e n t s y s t e m i s u n o b s e r v a b l e , i t i s

n e c e s s a r y t o p l a c e a d d i t i o n a l m e t e r s i n o r d e r t o m a k e t h e

s y s t e m o b s e r v a b l e . I n t h e e c o n o m i c a l s e n s e , i t i s d e s i r e d t o

f i n d a m e t e r p l a c e m e n t p o l i c y w i t h a m i n i m a l m e a s u r e m e n t

s e t wh i c h e n s u r e s t h e s y s t e m o b s e r v a b i l i t y .

Th e s y s t e m o b s e r v a b i l it y c a n b e a t t a i n e d b y c o m b i n i n g a l l

o b s e r v a b l e i s la n d s w i t h a d d i t i o n a l m e a s u r e m e n t s , w h i c h

s h o u l d b e c r i t i c a l b o u n d a r y b u s i n j e c t i o r , o r l i n e m e a s u r e -

m e n t s . I n o r d e r t o r e d u c e t h e n u m b e r o f c r i t i c a l

m e a s u r e m e n t s , i t i s m o r e e f f i c i e n t t o s e l e c t b o u n d a r y b u s

i n j e c t i o n s wh i c h a r e c o n n e c t e d t o a s m a n y o b s e r v a b l e i s l a n d s

as poss ib le .

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

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

Ho r i s b e r g e r ' s a l g o r i t h m .

( i ) F i n d a l l o b s e r v a b l e i s l an d s b y t h e o b s e r v a b l e i s l a n d

i d e n t i f i c a t i o n a l g o r i t h m . Th e o b s e r v a b l e i s l a n d s m a y

n o t b e m a x i m a l o b s e r v a b l e i s l a n d s i n c a s e s wh e r e t h e

s y s t e m h a s t o p o l o g i c a l s y m m e t r y .

( i i) Re l e a s e a ll i r r e l e v a n t m e a s u r e m e n t s wh i c h h a v e b e e n

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

p r o c e d u r e .

( ii i) Co n d u c t Ho r i s b e r g e r ' s a l g o r i t h m wi t h th e f i n al r e s u l t

o f s t e p ( i ) . Th i s s t ep y i e ld s m a x i m a l o b s e r v a b l e i s l an d s .

( i v ) F i n d a n u n m e a s u r e d b o u n d a r y b u s wh i c h is c o n n e c t e d

t o a s m a n y o b s e r v a b l e i s l a n d s a s p o s s ib l e .

( v ) Ad d a b u s i n j e c t io n m e t e r t o t h e b o u n d a r y b u s .

( v i ) Co n d u c t t h e Ho r i s b e r g e r ' s a l g o r i t h m b y m o d i f y i n g th e

o b s e r v a b i l i t y d e c i s io n t r e e f o r t h e a d d i t i o n a l m e a s u r e -

m e n t s . I f t h e n u m b e r o f m a x i m a l o b s e r v a b l e i s l a n d s i s

m o r e t h a n 1 , g o t o s t e p ( i v ) . O t h e r w i s e , p r i n t a l l t h e

a d d i t i o n a l m e a s u r e m e n t s a n d s t o p .

5. Numerical example

The 14 -bus sys t em conf igu ra t ion i s g iven in F ig . 4 . Al l

m e a s u r e m e n t s a r e i n d i c a te d b y f la gs .

T h e o b s e r v ab i l it y ha s b e e n t e s t e d b y t h e b u s m e a s u r e m e n t

m a t r i x a l g o r i t h m . Th e a l g o r i t h m y i e l d s t h e s o l u t i o n t h a t t h e

s y s t e m i s o b s e r v a b l e . F o r t h e t e s t o f t h e o b s e r v a b l e i s l a n d

i d e n t i fi c a t io n a l g o r i th m , i t is a s s u m e d t h a t t h e m e t e r a t b u s 5

f a i l s t o wo r k . By c o n d u c t i n g t h e o b s e r v a b l e i s l a n d

i d e n t i f ic a t i o n a l g o r i t h m , t h e m a x i m a l o b s e r v a b l e i s l a n d s h a v e

b e e n f o u n d w i t h t h e f o l l o wi n g g r o u p s o f b u s e s :

G~ = (1, 2, 3, 4, 5, 7, 8, 9}

( ind ica ted wi th a do t t ed l ine in F ig . 4 ) .

Ind iv idua l bus i s l ands : (6} , (10} , (11} , { 12} , {13} , {14} .

Th e m i n i m a l m e a s u r e m e n t s e t s wh i c h m a k e t h e f a i l e d

m e a s u r e m e n t s y s t e m o b s e r v a b l e a r e g i v e n b y :

Z l = ( b u s m e a s u r e m e n t / 1 4 } o r

Z 2 = { b u s m e a s u r e m e n t / t o } .

Co mm ent 1 . Co mputat ional e f fi c iency. A s m e n t i o n e d

e a r l i e r , Ho r i s b e r g e r ' s a l g o r i t h m r e q u i r e s e x c e s s i v e c o m p u t a -

t i o n t i m e d u e t o i t s c o m b i n a t o r i a l n a t u r e . I t i s i n f e r i o r t o

M o n t i c e l l i a n d W u ' s a l g o r i t h m i n c o m p u t a t i o n s p e e d .

Th e M - W a l g o r i t h m h a s a r o u t i n e t o a s si g n b u s a n g l e s

u n d e r t h e a s s u m p t i o n o f a l l m e a s u r e m e n t s b e i n g z e r o

£ d d

8? :~ i ~$ @ ?

C [ n c h c a t e s b u s n ~ r n b e r Ill [ s o n r e j e c t i o n m e a s u r e m e n t

l £ n d lc a t e s L in e n u m b e r - ¢ - I s a L i n e - f l o w

F I G. 4 . I E E E 1 4 - b u s te s t s y s t e m w i t h m e a s u r e m e n t s e t .

w h e n e v e r a n o r s o m e i r r e l e v a n t m e a s u r e m e n t ( s ) a r e f o u n d

a n d r e m o v e d . I n t h a t r o u t i n e , a l l b u s v o l t a g e s o f t h e wh o l e

s y s t e m s h o u l d b e r e a s s i g n e d b y a p p l y i n g t h e Ga u s s

e l i m i n a ti o n to t h e f o l lo w i n g e q u a t i o n: [ G ] 0 = 0 w i t h a n

( n x n ) i n f o r m a t i o n m a t r i x [ G] .

I n t h e p r o p o s e d a l g o r i t h m , b u s e s o f t h e n e t wo r k a r e

d i v i d e d t o b u s g r o u p s , e a c h o f wh i c h i s c o n s i d e r e d a s a

c a n d i d a t e o f t h e o b s e r v a b l e i s l a n d . W h e n a n i r r e l e v a n t

m e a s u r e m e n t i s f o u n d , t h e r e m o v a l o f t h e i r r e l e v a n t

m e a s u r e m e n t a f f e c t s o n l y t h e o b s e r v a b i l i t y o f b u s g r o u p , s a y

B u s G r o u p i , w h i ch t h e r e m o v e d m e a s u r e m e n t s b e l o n g e d t o .

Th e b u s v o l t a g e s o f Bu s Gr o u p i a r e r e a s s i g n e d f r o m t h e

f o l lo wi n g e q u a t i o n :

[ M ~ M n ] V s = 0

w h e r e M ~ M R i s an (n × n i ) ma t r ix and n i i s t he no . o f bu ses

i n Bu s Gr o u p i .

I t i s n o t e d t h a t t h e d i m e n s i o n a l i t y o f [M~MR] i s m u c h

s m a l l e r t h a n t h a t o f [ G] . Th e r e f o r e , t h e p r o p o s e d a l g o r i t h m

r e d u c e s c o m p u t a t i o n t i m e r e m a r k a b l y c o m p a r e d w i t h t h e

M - W a l go r it h m .

Co mm ent 2 . Topological symm etry problem.

Ho r i s b e r g e r ' s a l g o r i t h m e s t a b l i s h e s a n o b s e r v a b i l i t y

t r e e w i t h c o n s i d e r a t i o n o f t h e n u m b e r o f i n d e p e n d e n t

b o u n d a r y m e a s u r e m e n t s r e l a t e d t o o n l y t h e o b s e r v a b l e

i s l a n d s u n d e r c o n s i d e r a t i o n . Th e o b s e r v a b l e i s l a n d s c a n b e

p u t i n t o o n e o b s e r v a b l e i s l a n d i f M i s g r e a t e r t h a n o r e q u a l t o

( N - 1 ) . ( M is t h e n u m b e r of t h e i n d e p e n d e n t b o u n d a r y

m e a s u r e m e n t s , a n d N i s t h e n u m b e r o f t h e o b s e r v ab l e

i s l a n d s . ) Th u s , Ho r i s b e r g e r ' s a l g o r i t h m h a s n o p r o b l e m i n i t s

a p p l i c a t i o n t o a n e t wo r k w i t h t o p o l o g i c a l s y m m e t r y .

Ho we v e r , Ho r i s b e r g e r ' s a l g o r i t h m i s s o m e wh a t t i m e -

c o n s u m i n g s i n c e a l l p o s s i b l e c o m b i n a t i o n s o f b o u n d a r y

m e a s u r e m e n t s s h o u l d b e c o n s i d e r e d . I n o r d e r t o r e d u c e t h e

e x c e s s i v e c o m p u t a t i o n t i m e we h a v e d e v e l o p e d t h e b u s

g r o u p i n g a l g o r i t h m , a n d i n o r d e r t o s o l v e t h e p r o b l e m o f

t o p o l o g i c a l s y m m e t r y we a p p l i e d Ho r i s b e r g e r ' s a l g o r i t h m a t

t h e f in a l s t a g e p r o d u c e d b y t h e p r o p o s e d a l g o r i t h m . Th i s

c o m b i n a t i o n o f t h e t wo a l g o r i t h m s r e q u i r e s c o n s i d e r a b l y -

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

p r o b l e m .

6. Conclusion

To p o l o g i c a l o b s e r v a b i l i t y a n a ly s i s h a s b e e n p e r f o r m e d w i t h

t h e u s e o f t h e p a r t i t i o n e d i n c i d e n c e m a t r i x . An e f f i c i e n t

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

d e v e l o p e d o n t h e b a s i s o f a b u s g r o u p i n g m e t h o d . A m e t e r

p l a c e m e n t a l g o r i t h m h a s b e e n p r e s e n t e d t o f i n d a m i n i m a l s e t

o f a d d i t i o n a l m e a s u r e m e n t s . Th e r e s u l t s o f t h i s s t u d y a r e

s u m m a r i z e d a s f o l l o ws .

( i ) A n e w a p p r o a c h t o t o p o lo g i c a l o b s e r v a b i l i t y a n a l y s is

h a s b e e n i n t r o d u c e d w i t h t h e u s e o f i n c i d e n c e m a t r i x .

( i i) Th e p r o p o s e d a l g o r i t h m f o r i d e n t i f ic a t i o n o f o b s e r v a b l e

i s l a n d s c o n s i d e r a b l y r e d u c e s c o m p u t a t i o n t i m e b y

a d o p t i n g a b u s - g r o u p i n g m e t h o d .

( i i i ) Th e i n t e g e r Ga u s s e l i m i n a t i o n m e t h o d h a s b e e n

a d o p t e d f o r t h e o b s e r v a b i l i t y te s t , wh i c h e n a b l e s u s t o

a v o i d t h e n e a r l y - z e r o - p i v o t p r o b l e m .

( iv ) Spec ia l ca ses o f topo log ica l ly sym me t r i ca l s t ruc tu re s in

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

o b s e r v a b l e i s l a n d i d e n t i fi c a t io n .

( v ) A n e w m e t e r p l a c e m e n t a lg o r i th m h a s b e e n p r e s e n t e d

wi t h t h e u s e o f t h e o b s e r v a b i l i t y d e c is i o n t r e e , wh i c h

s a v es a b o u t 2 0 % o f c o m p u t a t io n t im e .

References

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observable island identification for state estimation using incidence matrix

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