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U T T E R W O R T HI N E M A N NElectrical Power & Energy Systems, Vol. 17, No. 5, pp. 335-346, 1995Copyright 1995 Elsevier Science LtdPrinted in Great Britain. All rights reserved0142-0615(95)00050-X 0142-0615/95/$10.00+0.00

S im ple and e ffic ien t m etho d fo rload f lo w so lu tio n o f rad ia ld is t r ibut ion networksD D a sDepartment of Electrical and Electronic Engineering,Birla Institute of Technology and Science, Pilani,Rajasthan-330 031, IndiaD P K o t h a r iCentre for Energy Studies, In dian Inst i ti tue ofTechnology, Ha uz Khas. Ne w D elhi-11 0 01 6, IndiaA K a l a mDepartment of Electrical and Electronic Engineering,Victoria University of Techn ology, Footscray,Melbourne. Australia

The paper presents a simple and ef ficient m ethod or solvingradial distribution networks. The pro pos ed meth od involvesonly the evaluation o f a simple algebraic expression o fvoltage magnitudes and no trigonometric functions asopposed to the stan&~ rd load f low ca se . Thus, co m-putat ionally the proposed method is very ef f ic&nt and i trequires less computer m emo ry. The proposed m etho d caneasily handle different typ es of load characteristics. Seve ralIndian rural distribution networks have been successfullysolved by using the propos ed method.Keywords: distribution load low, mathematical techniques,radial netwo rks

I . N o m e n c l a t u r eN BL N 1P L ( i )Q L ( i )I v ( i ) lR ( j )x(j)I ( j )P(m2)Q(m2)

total number of nodestotal number of branches ( L N 1 = N B - 1 )real power load of ith nodereactive power load of ith nodevoltage magnitude of ith noderesistance of jt h branchreactance ofjth branchcurrent flowing through b ran chjtotal real power loadTed through node m2total reactive power load fed through node m2

R e c e i v e d 1 7 M a y 1 9 9 3 ; r e v i s e d 1 7 N o v e m b e r 1 9 9 3 ; a c c e p t e d 4A u g u s t 1 9 9 4

fi(m2) voltage angle of node m2L P ( j ) real power loss ofb ra nch jL Q ( j ) reactive power loss of br an ch jI S ( j ) sending-end node of branch jI R ( j ) receiving-end node of b ranch jP L O S S total real power lossQ L O S S total reactive power loss

I1. I n t r o d u c t i o nLoad flow analysis of distribution systems has notreceived much attention unlike load flow analysis oftransmission systems. However, some work has beencarried out on load flow analysis of a distribution net-work but the choice of a solution method for a practicalsystem is often difficult. Generally distribution networksare radial and the R / X ratio is very high.Because of this, distribution networks are ill-conditioned,and conventional Newton-Raphson (NR) and fastdecoupled load flow (FDLF) methods 1-4 are inefficientat solving such networks.Man y researchers 5-7 have suggested modified versionsof the conventional load flow methods for solvingill-conditioned power networks.Recently researchers have paid much attention toobtaining the solution of distribution networks.Kersting and Mendives and Kersting9 have presented aload flow technique based on the ladder network theoryand it appears to work very well. Shirmohammadi et al.lOhave presented a compensation-based power flow

3 3 5

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3 3 6 M e t h o d f o r s o l u t i o n o f ra d i a l d i s t r ib u t i o n n e t w o r k s . D . D a s et al.m e t h o d f o r w e a k l y m e s h e d d i s t r i b u t io n a n d t r a n s m i s s io ns y s t e m s . B a ra n a n d W u 11 a n d C h i a n g u h a v e o b t a i n e d t h el o a d f l o w s o l u t i o n i n a d i s t r i b u t i o n s y s t e m b y t h e i t e r a t i v es o l u t io n o f t h re e f u n d a m e n t a l e q u a t i o n s r e p r e s e nt i n g r e a lp o w e r , r e a c t i v e p o w e r a n d v o l t a g e m a g n i t u d e . Th e s et h r e e e q u a t i o n s a r e v e ry u s e fu l a s t h e y a p p e a r t o b eu s e fu l i n r e a l p h y s i c a l s y s t e m s . R e n a t o 13 h a s p ro p o s e do n e m e t h o d f o r o b t a i n i n g l o a d f l o w s o l ut i o n s o f r a d ia ld i s t r i b u t i o n n e t w o rk s . H i s t e c h n i q u e s e e m s t o b e q u i t ep ro m i s i n g b e c a u s e i t g i v e s a s o l u t i o n fo r b u s v o l t a g em a g n i t u d e o n l y . G o s w a m i a n d B a s u 14 h a v e p r e s e n t e d ad i r e c t s o l u t i o n m e t h o d fo r s o l v i n g r a d i a l a n d m e s h e dd i s t r ib u t i o n n e t w o r k s . H o w e v e r , t h e m a i n l i m i t a t io n o ft h e i r m e t h o d i s t h a t n o n o d e i n t h e n e t w o rk i s t h e j u n c t i o no f m o re t h a n t h r e e b r a n c h e s , i .e . o n e i n c o m i n g a n dt w o o u t g o i n g b r a n c h e s . J a s m o n a n d Le e 15'16 h a v ep r o p o s e d a n e w l o a d f l o w m e t h o d f o r o b t a i n i n g t h es o l u t i o n o f r a d i a l d i s t r i b u t i o n n e t w o rk s . T h e y h a v ed e r i v e d t h e f u n d a m e n t a l e q u a t i o n s f o r s o l v i n g a l o a df l o w p ro b l e m o f a d i s t r i b u t i o n n e t w o rk u s i n g a s i n g le - l in ee q u i v a l e n t .I n In d i a , a l l t h e 11 k V ru ra l d i s t r i b u t i o n f e e d e r s a r er a d i a l a n d t o o l o ng . T h e v o l t a g e s a t t h e f a r e n d o f m a n ys u c h f e e d e r s a r e v e ry l o w w i t h v e ry h i g h v o l t a g ere g u l a t i o n .I n t h is p a p e r t h e m a i n a i m o f t h e a u t h o r s h a s b e e n t od e v e l o p a n e w l o a d f l o w te c h n i q u e fo r r a d i a l d i s t r i b u t i o nn e t w o r k s . T h e p r o p o s e d m e t h o d i n v o l v e s o n l y th ee v a l u a t i o n o f a s i m p l e a l g e b ra i c e x p re s s i o n o f v o l t a g em a g n i t u d e a n d n o t r i g o n o m e t r i c te r m s , a s o p p o s e d t o t h es t a n d a r d l o a d f l ow c as e. C o m p u t a t i o n a l l y t h e p r o p o s e dm e t h o d i s v e ry ef f ic i en t . A n o t h e r a d v a n t a g e o f th ep ro p o s e d m e t h o d i s t h a t i t r e q u i r e s l e s s c o m p u t e rm e m o r y . C o n v e r g e n c e i s a l w a y s g u a r a n t e e d f o r a n yt y p e o f p r a c t i c a l r a d i a l d i s t r i b u t i o n n e t w o rk w i t h areal is t ic R / X r a t i o w h i l e u s i n g t h e p ro p o s e d m e t h o d .Lo a d s , i n t h e p r e s e n t f o rm u l a t i o n , h a v e b e e n r e p re s e n t e da s c o n s t a n t p o w e r . H o w e v e r , t h e p r o p o s e d m e t h o d c a ne a s i ly i n c l u d e c o m p o s i t e l o a d m o d e l l i n g i f t h e b r e a k u p o ft h e l o a d s is k n o w n . Th e p r o p o s e d l o a d f l o w te c h n i q u e h a sb e e n i m p l e m e n t e d o n a n I B M P C - A T . S e v e ra l pr a c t ic a lru r a l r a d i a l d i s t r i b u t i o n f e e d e r s i n In d i a h a v e b e e ns u c c e s s ful l y s o l v e d u s i n g t h e p ro p o s e d m e t h o d . R e l a t i v es p e ed a n d m e m o r y r e q u i r e m e n t s o f th e p r o p o s e d m e t h o dh a v e a ls o b e e n c o m p a r e d w i t h th e m e t h o d s p r o p o s e d b yB a r a n a n d W u ~1.

I I I . A s s u m p t i o n sW e a s s u m e t h a t t h e t h r e e -p h a s e r a d i a l d i s t r i b u t i o n n e t -w o rk s a r e b a l a n c e d a n d c a n b e r e p re s e n t e d b y t h e i re q u i v a l e n t s i n g l e l i n e d i a g ra m s . Th i s a s s u m p t i o n i sq u i t e v a l i d fo r 11 k V ru ra l d i s t r i b u t i o n f e e d e r s in In d i aa n d e l s e w h e re . L i n e s h u n t c a p a c i t a n c e (d i f f e r e n t f r o ms h u n t c a p a c i t o r b a n k s t h a t a r e c o n s i d e re d a s l o a d s ) i sn e g l ig i b le a t t h e d i s t r i b u t i o n v o l t a g e l e v e ls a s i s f o u n d i nm o s t p r a c t i c a l c a s e s.

I V . S o l u t i o n m e t h o d o l o g yF i g u re 1 s h o w s a s i n gl e - li n e d i a g ra m o f a n e x i s t i n gru ra l d i s t r i b u t i o n f e e d e r . B ra n c h n u m b e r , s e n d i n g -e n da n d r e c e i v i n g -e n d n o d e s o f t h i s f e e d e r a r e g i v e n i nTa b l e 1 . F i g u re 2 s h o w s t h e e l e c t r i c a l e q u i v a l e n t o fF i g u re 1 .

S u b s t a t i o n

I '7 12

1 3

- ~ L I 4

)- - 1 5 5

F i g u r e 1 . S i n g l e l in e d i a g r a m o f a n e x i s t i n g d i s t ri b u t i o nf e e d e rT able 1 . B ranch num ber , s end i ng - end and rece i v i ng - endn o d e s i n F ig u r e 1B ra n c h S e n d i n g - R e c e i v i n g -n u m b e r e n d n o d e e n d n o d e( j ) IS( j ) IR( j )

1 1 22 2 33 3 44 4 55 2 96 9 l07 2 68 6 79 6 810 3 l lll 11 1212 12 1313 4 1414 4 15

, v2 ,~ 2I V l l ~ l I ( I ) _ . . . . . .R ( I ) + j x ( I ) - ~ P ( 2 ) , Q ( 2 )

F i g u r e 2 . E l e c t r i c a l e q u i v a l e n t o f F i g u r e 1F r o m F i g u re 2 , w e h a v e t h e fo l l o w i n g e q u a t i o n s :I( l ) = ]V(1)]16(1) - ] V(Z)]L6(2)R(1) + iX (l) (1)P ( 2 ) - j Q ( 2 ) = v * ( 2 )I ( 1) (2)

Fr om equat ion s (1) and (2) we have (de ta il s in Append ix 1 ):[ V(2 )I = {[ (P(2 )R (1) + Q(Z )X (1) - 0 .5 ] V(1 )]2 ) 2

- (R 2 (1 ) + x z (1 ) ) ( p 2 ( 2 ) + Q 2 (Z) )]U 2- (P (Z)R (1 ) + Q (Z)X (1 ) - 0.5]v(1)[2)} /2

(3)w h e re P (2 ) a n d Q (2 ) a r e t o t a l r e a l a n d r e a c t i v e p o w e rl o a d s f ed t h r o u g h n o d e 2 .

(P 2 ) = s u m o f t h e re a l p o w e r l o a d s o f al l t h e n o d e sb e y o n d n o d e 2 p l u s t h e r e a l p o w e r l o a d o fn o d e 2 i ts e l f p l u s t h e s u m o f t h e r e a l p o w e rl o ss e s o f a l l t h e b r a n c h e s b e y o n d n o d e 2 .

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M eth od fo r so l u t i on o f rad i a l d i s t r ibu t i on ne two rks : D . Das et a l . 337(Q 2 ) = s u m o f t h e r e a c t iv e p o w e r l o a d s o f al l t h en o d e s b e y o n d n o d e 2 p lu s t h e r e a c ti v e p o w e rl o a d o f n o d e 2 i t se l f p l u s t h e s u m o f t h er e a c t i v e p o w e r l o s se s o f a l l t h e b r a n c h e sb e y o n d n o d e 2 .Eq u a t i o n (3 ) c a n b e w r i t t e n i n g e n e ra l i z e d fo rm :I V ( m 2 ) l = [B ( j ) - A(j)] 1/2

w h e r e(4 )

A ( j ) = P(m2)*R( j ) + Q(m2)*X( j ) - 0 .5"1V(m l) l 2

B( j ) = { A 2 ( j ) - [ R 2 ( j ) + X 2 ( j ) ] *( 5 )

[PZ(m2) + QZ(m2)]}l/2 (6 )j i s t h e b r a n c h n u m b e r , r n 1 a n d m 2 a r e s e n d i n g -e n d a n dre c e i v i n g -e n d n o d e s r e s p e c t i v e l y (m l = I S ( j ) a n dm2 = IR( j ) ) .R e a l a n d r e a c t i v e p o w e r l o s s es in b r a n c h 1 c a n b e g i v e nby :

LP(1) = R(1 )* [p2(2 ) + Q2(2 )]IV(2)] 2

LQ (1 ) = X (1 )* [P 2 (2 ) + Q 2 (2 ) ]Iv (2 ) l 2

Eq u a t i o n (7) c a n a l s o b e w r i t t e n i n g e n e ra l iz e d fo rm a s :

(7 )

L P ( j ) = R ( J )* [ p 2 ( m 2 ) + Q 2 ( m 2 ) ]] V ( r n 2 ) I 2L Q ( j ) = X ( J )* [ P Z ( m 2 ) + Q 2 ( m 2 ) ]

I V ( r n Z ) l 2( 8 )

In i t i a l l y , i f L P ( j ) a n d L Q ( j ) a re s e t t o z e ro fo r a l l j , t h e ni n i t ia l e s t i m a t e s o f P (m 2 ) a n d Q (m 2 ) w i ll b e t h e s u m o ft h e l o a d s o f a ll t h e n o d e s b e y o n d n o d e m 2 p l u s t h e l o a d o ft h e n o d e m 2 i t se l f .V. Ex p l a n a t io n o f t h e p r o p o s e d t e c h n i q u eB e fo re g i v i n g t h e d e t a i l e d a l g o r i t h m , w e w i l l d i s c u s s t h em e t h o d o l o g y o f i d e n ti f y i n g t h e n o d e s a n d b r a n c h e sb e y o n d a p a r t i c u l a r n o d e w h i c h w i l l h e l p i n f i n d i n g t h ee x a c t l o a d f e e d i n g t h r o u g h t h a t p a r t i c u l a r n o d e .F i r s t w e w i l l d e f i n e t h e v a r i a b l e s :J

I K ( ~ )

j = 1 , 2 . . . , L N 1 ( j i n d i c a t e s b r a n c h o fF igure 1 ; see a l so Tab le 1 )n o d e c o u n t ( id e n ti f ie s n u m b e r o f n o d e sb e y o n d a p a r t ic u l a r n o d e )n o d e i d e n t i f ie r ( h e lp s t o i d e n t i fy t h e s e n d -i n g -e n d a n d r e c e i v i n g -e n d n o d e s w h i c h a r eg i v e n i n t h e i t h b r a n c h o f Ta b l e 1 ( i > j )LL(ip) s t o re s s e n d i n g - e n d n o d e o f i t h r o w o fT a b l e 1 ( i > j )KK(ip) s t o r e s r e c e i v i n g - e n d n o d e o f i t h r o w o fT a b l e 1 ( i > j )N ( j ) t o t a l n u m b e r o f n o d e s b e y o n d n o d e I R ( j )p l u s 1 (n o d e I R ( j ) i t s e l f )IB(j , ip + 1 ) s e n d i n g -e n d n o d eIE ( j , ip + 1 ) r e c e i v i n g -e n d n o d e

W e w i l l n o w e x p l a i n IB( j, ip + 1) and IE (j , ip + 1) .Co ns id er the f i r s t b ra nch in F ig ure 1 , i . e . j = 1 , ther e c e i v i n g -e n d n o d e o f b r a n c h 1 i s 2 ( se e a l s o Ta b l e 1 ),t h e r e fo re , IB(1 , ip+ 1) and I E ( 1 , i p + l ) wil l he lp toi d e n t i fy a l l t h e b r a n c h e s a n d n o d e s b e y o n d n o d e 2 a n dnode 2 i t se l f .Th i s w i l l h e l p t o f i n d t h e e x a c t l o a d f e e d i n g t h ro u g hnod e 2 . S imi la r ly , cons id er b ran ch 2 , i .e . j = 2 , ther e c e i v i n g -e n d n o d e o f b r a n c h 2 is 3 . Th e re fo re ,IB(2, ip + 1 ) a n d IE (2 , ip + 1 ) w i l l i d e n t i fy n o d e s a n db ra n c h e s b e y o n d n o d e 3 a n d n o d e 3 it s el f . Th i s w i l l h e l pc o m p u t e t h e e x a c t l o a d f e e d i n g t h r o u g h n o d e 3 . F o r e a c hn o d e a n d b r a n c h i d e n t i f i c a t io n ip w i l l b e i n c r e m e n t e d b y 1 .N o t e h e r e t h a t b e fo re i d e n t i f i c a ti o n o f n o d e s a n d b r a n c h e sb e y o n d a p a r t i c u l a r n o d e , ip h a s t o b e r e s e t t o z e ro .F o r j = 1 , ( f ir s t b r a n c h i n F i g u re 1 , Ta b l e 1 ),I R ( j ) = 2 , c h e c k w h e t h e r I R ( j ) = I S ( i ) o r n o t f o ri = 2 , 3 , . . . ,LNI-1 . I t i s seen tha t I R ( j ) = IS(2 ) = 2 ;I R ( j ) = IS(5 ) = 2 ; I R ( j ) = IS (7 ) - - 2 ; c o r r e s p o n d i n gr e ce i vi n g- e nd n o d e s a re I R ( 2 ) = 3; I R ( 5 ) - - 9 a n dIR (7 ) = 6 .T h e r e f o r e I B ( j , 1 ) = I, I E ( j , 1 ) = 2 ; I B ( j , 2 ) = 2 ,I E ( j , 2 ) = 3 ; I B (j , 3 ) = 2 ; I E ( j , 3 ) = 9 ; I B ( j , 4 ) = 2 ,I E ( j , 4) = 6 .W e w i l l m i s s o u t IB( j , 1 ), b e c a u s e w e w a n t t o i d e n t i fyt h e n o d e s a n d b r a n c h e s w h i c h a r e b e y o n d n o d e I R ( j ) a n dw e s t o r e t h e r e c e i v i n g -e n d n o d e i n t h e n a m e o f a v a r i a b l e ,sa y KK(ip), i . e . K K (1 ) = 2 , K K (2 ) - - 3 , K K (3 ) = 9 a n dK K (4 ) = 6 .N o t e t h a t t h e r e s h o u l d b e n o r e p e t i ti o n o f a n y b r a n c ho r n o d e w h i le i d e n t if y i n g n o d e s a n d b r a n c h e s a n d t h isl o g i c h a s b e e n i n c o r p o r a t e d i n t h e p r o p o s e d a l g o r i t h m(A l g o r i t h m 1 ). Th i s i s f u r t h e r e x p l a i n e d i n t h e f l o w c h a r t

g i v e n in F i g u re 3 .F ro m t h e a b o v e d i s c u s s i o n i t i s s e e n t h a t n o d e 2 i sc o n n e c t e d t o n o d e s 3 , 9 a n d 6 a n d t h e c o r r e s p o n d i n gb ra n c h e s a r e b r a n c h 2 , (2 ~ 3 ) , b r a n c h 5 (2 ~ 9 ) a n dbra nch 7 (2 - --+ 6 ) . S im i la r ly , the p ro pos ed log ic wi l li d e n t if y t h e n o d e s a n d b r a n c h e s w h i c h a r e c o n n e c t e d t on o d e s 3 , 9 a n d 6 . F i r s t , it w il l c h e c k w h e t h e r n o d e 3a p p e a r s i n t h e le f t - h a n d c o l u m n o f Ta b l e 1 . I t is s e ent h a t n o d e 3 i s c o n n e c t e d t o n o d e s 4 a n d 1 1 (b r a n c h e s 3a n d 1 0 i n F i g u re 1, Ta b l e 1 ). Th e re fo re IB( j , 5 ) = 3 ,I E ( j , 5) = 4, IB( j , 6) = 3, I E ( j , 6) = 11 an d KK( 5) = 4 ,K K (6 ) = 1 1. Th e n i t w i l l c h e c k w h e t h e r n o d e 9 a p p e a r s i nt h e l e f t - h a n d c o l u m n o f Ta b l e 1 . I t i s s e e n t h a t n o d e 9 i sc o n n e c t e d t o n o d e 1 0 (b r a n c h 6 i n F i g u re 1 , Ta b l e 1 ) .T h e r e f o r e , IB(j , 7) = 9, IE (j , 7) = 10 , K K ( 7 ) = 10.S i m i l a r l y , n o d e 6 i s c o n n e c t e d t o n o d e s 7 a n d 8 .T h e r e f o r e , IB( j , 8) = 6, I E ( j , 8) = 7, IB( j , 9) = 6,I E ( j , 9 ) = 8 , K K (8 ) = 7 , a n d K K (9 ) = 8 .F r o m t h e a b o v e d i s c u s s i o n , a g a i n i t i s s e e n t h a t n o d e 3i s c o n n e c t e d t o n o d e s 4 a n d 1 1 , n o d e 9 i s c o n n e c t e d t on o d e 1 0 a n d n o d e 6 is c o n n e c t e d t o n o d e s 7 a n d 8 .S i m i l a r l y , t h e p ro p o s e d l o g i c w i l l c h e c k w h e t h e r n o d e s4 , 1 1 , 10 , 7 a n d 8 a r e c o n n e c t e d t o a n y o t h e r n o d e s . Th i sp ro c e s s w i l l c o n t i n u e u n l e s s a l l t h e n o d e s a n d b r a n c h e sa r e i d e n t i fi e d b e y o n d n o d e 2 .T a b l e 2 g i v es th e n o d e s a n d b r a n c h e s b e y o n d n o d e 2 .C o m p u t e r l o gi c ( A l g o r i t h m 1) w il l a u t o m a t i c a l l y s k ipt h e n o d e a n d b r a n c h b e h i n d n o d e 2 , i. e. n o d e 1 a n db r a n c h 1 .T o t a l l o a d f e d th r o u g h n o d e 2 is th e s u m o f th e l o a d s o fa l l t h e n o d e s b e y o n d n o d e 2 p lu s t h e l o a d o f th e n o d e 2i t s el f ( r i g h t - h a n d c o l u m n o f T a b l e 2 ) p l u s t h e s u m o f t h el o s se s o f a ll t h e b r a n c h e s b e y o n d n o d e 2 ( se e Ta b l e 2 ) .

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3 3 8 M e t h o d f o r s o l u t i o n o f r a d i a l d i s t r ib u t i o n n e t w o r k s . D . D a s et a l .

FromQ

From@From~

R e a dR e c e i v i n g - e n d & s e n d i n g - e ndn odes an d t o ta l n u m b e r o fn o d e s

I , ,_ - o I_1

F r o m

tE (j,,p+l)=1Rq)In clip+ 1)=/s q)

YosY ~ = INo

I , n - - , . " IY e s ~ -

N o

IR(j)=KK(iq)K=lK(iq+ 1

ip=ip+ l/ K (ip)=iLL ( ip)=lS ( i )K K (ip)=/R (i)/ E q , ip+l)=lR (i)IB ( j , ip+l)=/S ( i )N ( j )=ip+ I

Y es

N o

Yesm q , i t , + l ) = m q) ]ta q, ip+~)=m q)lU (j)=ip+l [

Y e s

I Y e s ~ N o[ E ( L N 1 , I ) = IR ( L N 1 )(~ ) I m (t~vl , l )=m (t .m )I N ( L N I ) = - !

F ig u r e 3 . F lo w c h a r t f o r i d e n t i f i c a t i o n o f n o d e s a n d b r a n c h e s b e y o n d a p a r t i c u l a r n o d e

Simi la r ly , we have to cons ide r the r ece iv ing-end n odeso f b r a n c h 2 , b r a n c h 3 , b r a n c h 4 , . . . , b r a n c h L N I - 1 inF igure 1 , and in a s imi la r way to tha t d iscussed abov e , then o d e s a n d b r a n c h e s h a v e t o b e i d e n t i f i e d b e y o n d t h e s erece iv ing-end nodes Table 3 and 4 a lso g ive the nodesa n d b r a n c h e s b e y o n d n o d e s 3 a n d 4 .N o te t h a t i f t h e r e ce iv in g - e n d n o d e o f a n y b r a n c h i nF igu re 1 ( see a l so Tab le 1) i s an en d no de o f a pa r t icu la rla te ra l , then the to ta l load fed through th is node i s theload o f th is node i t se lf For exam ple , cons ide r nod e 5 inF igure 1 (branch 4 , Tab le 1). This i s an end nod e ,t h e r e f o r e t h e t o t a l l o a d f e d t h r o u g h n o d e 5 i s t h e l o a dof nod e 5 only . S im i la r ly no des 7 , 8 , 10, 13 , 14 and 15 a ree n d n o d e s i n F ig u r e 1 . T h e p r o p o s e d c o m p u te r l o g i c w i llau tomat ica l ly ident i fy a l l the end nodesT a b l e s 2 , 3 a n d 4 g iv e t h e p a t t e r n o f c o mp u te r o u tp u t F o r a n y e n d n o d e , t h e c o mp u te r o u tp u t w i l l s h o w o n lyth i s e n d n o d e .

T h e c o n c e p t o f i d e n ti f y in g th e n o d e s a n d b r a n c h e sb e y o n d a p a r t i c u l a r n o d e w h ic h h e lp s i n c o mp u t in g t h ee x a c t lo a d f e e d in g t h r o u g h t h e p a r ti c u l a r n o d e h a s b e e nrea l ized us ing an a lgor i thm (Algor i thm 1) g iven be low .

V I . A l g o r i t h m 1" I d e n t i f i c a t i o n o f n o d e sa n d b r a n c h e s b e y o n d a p a r t i c u l a r n o d eS te p 1 r e a d s y s t e m d a t aStep 2 j = 1S t e p 3 k = j + lS t e p 4 i p = O ; i q = OStep 5 i = kS t e p 6 n c = 0i f { I R ( j ) = I S ( i ) } g o t o s t e p 7o th e r w i s e g o t o s t e p 1 5S tep 7 i f { i p = 0} go to s tep 13o th e r w i s e g o t o s t e p 8

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M e t h o d f o r s o l u t i o n o f r a d i a l d i s t r ib u t i o n n e t w o r k s : D . D a s et al . 33 9T a b l e 2 . N o d e s a n d b r a n c h e s b e y o n d n o d e 2 in F i g u r e 1

S e n d i n g -e n d n o d e s

St ep 14 ip = ip + 1I K ( i p ) = iR e c e i v i n g - L L ( i p ) = I S ( i )e n d n o d e s K K ( i p ) = I R ( i )I E ( j , ip + 1) = I R ( i )I B ( j , i p + 1) = I S ( i )U ( j ) = ip + 12 S t e p 15 i = i + li f { i < ~ L N 1 } g o t o s t e p 63 o t h e r w i s e g o t o s t e p 1 64 S t e p 1 6 i f { i p = 0 } g o t o s t e p 1 75 o t h e r w i s e g o t o s t e p 1 89 S t ep 17 I E ( j , ip + 1) = I R ( j )10 I B ( j , i p + 1) = I S ( j )6 N ( j ) = ip + 17 g o t o s t e p 2 08 S t ep 18 i q = i q + l11 if {iq > ip} g o t o s t e p 2 01 2 o t h e r w i s e g o t o s t e p 1 913 S t ep 19 I R ( j ) = K K ( i q )

14 k = I K ( i q ) + 115 i f {iq

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3 4 0 M e t h o d f o r s o l u t i o n o f r a d ia l d i s tr i b u t io n n e t w o r k s . D . D a s et al.

Step 8S tep 9S tep 10Step 11S tep 12

Step 13

m 2 = I R ( j )P ( m 2 ) = P L ( L 2 )Q ( m2 ) = Q L ( L 2 )go to s tep 13L 1 : I B ( j , i )L 2 = I E ( j , i)i n = li f {L1 = IS ( i n ) a n d L 2 = I R ( in ) } g o t o s t e p 1 1othe rwise go to s tep 12P ( m2 ) = P ( m2 ) + P L ( L 2 ) + L P ( in )Q ( m2 ) = Q ( m2 ) + Q L ( L 2 ) + L Q ( i n )in = in + li f { in

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M etho d fo r so lu t i on o f rad ia l d i s t r ibu t ion ne tworks : D. Das et al . 341

Step 18Step 19

S L Q = Q L O S Sgo to step 2solution has converged, write voltage magnitudes,power losses, feeder current etc.stop

IX. ExamplesIX.1 Example 1In th is example, we consider Figure 1, which is an existingrural distribution feeder. Line data and load data of thissystem are given in Appendix 2.It took three iterations to converge by the proposedmethod. The proposed method has also been comparedwith that of Baran and Wu II. Their method also tookthree iterations to converge. The memory requirement ofthe proposed method is 63% of Baran and Wu's method.The proposed method is 1.37 times faster than that ofBaran and Wu. It is worth mentioning here that tocompute the Jacobian matrix, Baran and Wu's methodneeds a series of matrix multiplications and the number o fthese multiplications increases with the increase in size ofthe radial distribution networks. In addition, Baran andWu's method needs at least one matrix inversion.However, the proposed method needs no matrixmultiplication and inversions.The solution of this system has been given in Table 5.The total real and reactive power losses of this system are61.79 kW and 57.30 kVAr respectively.The proposed method has also been compared withthe methods proposed by Renato 13 and Kersting8,respectively. The proposed method is 1.1 times fasterthan the method proposed in Reference 13 and 2.3times faster than the method proposed in Reference 8.However, the memory requirement is the same as that ofthe methods given in References 8 and 13.IX.2 Example 2In this example, we consider a rural distribution feeder of85 nodes. The line and load data of this system are givenin Appendix 3.The proposed me thod to ok four iterations to converge.Baran and W U ' S 1 1 ' 1 2 method also took four iterations for

T a b l e 5 . S o l u t io n o f E x a m p l e 1Node no. I V I

1 (substation)23456789101112131415

1.000000.971280.956670.950900.949920.958230.956010.956950.967970.966900.949950.945830.944520.948610.94844

convergence. The mem ory requirement o f the proposedmethod is approximately 65% of Baran and Wu'smethod. The proposed method is also 1.93 times fasterthan that of Baran and Wu.It is worth mentioning here that the proposed methodis much faster than Baran and Wu's ll' 12 method when thesystem is very large.The solution of this system is given in Table 6. The to talreal and reactive power losses of this system are286.52 kW and 180.52 kVAr respectively.Further, it is found that the proposed method is 1.39times faster than the method in Reference 13 and 4.2times faster than the method in Reference 8. Again, thememory requirement is the same as in References 8 and13.

T a b l e 6 . L o a d f l o w so lu t io n o f E x a m p l e 3Node Voltage Node Voltageno. magnitude no. magnitude

1 (substation) 1.00000 44 0.883162 0.99587 45 0. 00 21 33 0.98977 46 0. 88 15 34 0.98197 47 0. 88 14 35 0.97816 48 0.880506 0.96443 49 0.880247 0.95599 50 0. 87 97 98 0.91950 51 0. 87 94 59 0.91778 52 0. 87 81 810 0.91394 53 0. 87 77 511 0.91122 54 0. 87 74 912 0.09897 55 0. 87 77 713 0.90799 56 0.8801414 0.90769 57 0.9156215 0.90751 58 0. 9097 516 0.99557 59 0.9096217 0.98916 60 0.9062418 0.97580 61 0. 90 51 719 0.97439 62 0.9044420 0.97370 63 0. 90 53 321 0.97300 64 0.9018422 0.97234 65 0.9016723 0.97427 66 0.9015424 0.95559 67 0. 90 00 825 0.91410 68 0.8972226 0.90996 69 0.8951027 0.90453 70 0. 89 45 528 0.90202 71 0.8943029 0.89741 72 0.8999430 0.89319 73 0.8958031 0.89122 74 0.8956032 0.88998 75 0. 89 53 333 0.88900 76 0. 89 41 534 0.88476 77 0.9016635 0.88248 78 0.9131936 0.88239 79 0.9129437 0.90969 80 0.9074438 0.90380 81 0.9069439 0.89700 82 0. 90 68 840 0.88934 83 0. 90 62 541 0.88841 84 0.9060642 0.88828 85 0. 90 76 143 0.88819

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3 4 2 M e t h o d f o r s o l u t i o n o f r a d i a l d i s t ri b u t i o n n e t w o r k s . D . D a s et al.X . A d d i t io n a l a p p l i c a t io n o f t h e p r o p o s e dm e t h o dX . 1 L o a d m o d e l l i n gA l l lo a d s , i n c l u d i n g s h u n t c a p a c i t a t o r s f o r r e a c t iv e p o w e rc o m p e n s a t i o n a r e r e p r e s e n t e d b y th e i r a ct i v e (P 0 ) a n dr e a c t i v e ( Q 0) c o m p o n e n t s a t 1 .0 p .u . T h e e f f e c t o f v o l t a g ev a r i a t i o n i s r e p r e s e n t e d a s f o l lo w s :

P = Pol vI ~ (10)Q = Q01Vl k (11 )

w h e r eI v l i s t h e v o l t a g e m a g n i t u d ek = 0 f o r c o n s t a n t p o w e r l o a d sk = 1 f o r c o n s t a n t c u r r e n t lo a d sk = 2 f o r c o n s t a n t i m p e d a n c e l o a d sT h e v a l u e o f k d if f e r a c c o r d i n g t o t h e l o a d

c h a r a c t e ri s t ic s . T h e l o a d f l o w s o l u t io n d e p e n d s o n t h et y p e o f r e al a n d r e a c t iv e l o a d s . I t i s e x t r e m e l y e a s y t oi n c l u d e a re a l a n d r e a c t i v e p o w e r l o a d s r e p r e s e n t a t i o n i nt h e p r o p o s e d a l g o r it h m . F o r c o n s t a n t c u r r e n t a n dc o n s t a n t i m p e d a n c e l o a d s , r e a l a n d r e a c t i v e p o w e rl o a d s h a v e t o b e c o m p u t e d a f t e r e v e r y i t e r a ti o n .

X I . C o n c l u s i o n sA s i m p l e a n d e f f i ci e nt lo a d f l o w t e c h n i q u e h a s b e e np r o p o s e d f o r so l v i n g r a d i a l d i s tr i b u t i o n n e t w o r k s . I tc o m p l e t e l y e x p l o i ts t h e r a d i a l f e a t u r e o f t h e d i s t r i b u t i o nn e t w o r k . T h e p r o p o s e d m e t h o d a l w a y s g u a r a n t e e sc o n v e r g e n c e o f a n y t y p e o f p r a c t i c a l r a d i a l d i s t r ib u t i o nn e t w o r k w i t h a r e a l i s t i c R / X r a t i o . C o m p u t a t i o n a l l y ,t h e p r o p o s e d m e t h o d i s e x t r e m e l y e ff ic i en t c o m p a r e dw i t h B a r a n a n d W u ' s m e t h o d a s i t s o l v e s a s im p l ea l g e b r a i c e x p r e s s io n o f v o l t a g e m a g n i t u d e o n l y .A n o t h e r a d v a n t a g e o f t h e p r o p o s e d m e t h o d i s t h a t i tr e q u ir e s le ss c o m p u t e r m e m o r y . T h e p r o p o s e d m e t h o dc a n e a s i ly h a n d l e t h e c o m p o s i t e l o a d s i f t h e b r e a k u p o ft h e lo a d s is k n o w n . T h e p r o p o s e d m e t h o d h a s b e e ni m p l e m e n t e d o n a n I B M P C - A T . S e v e r a l I n d i a n r u r a ld i s t r i b u t i o n n e t w o r k s h a v e b e e n s u c c e s s f u l l y s o l v e d b yu s i n g t h e p r o p o s e d l o a d f l o w t e c h n i q u e .

N e w ton l i ke me thod ' IEEE Trans. gol PAS-101 (1982)3648 36578 Kersting, W H a n d M e n d i v e , D L 'A n a pp l i c a tion o f l a dde rne two rk theory to the solu t ion of three phase radia l load f lowprob le m" IEEE PES W in te r Me e t ing , N e w Y ork , Ja nua ry1976, pap er no A 76 044 -89 Kersting, W H 'A me tho d to t e a c h the de sign a nd ope ra t iono f a d i s t r ibu t ion sy s t e m ' IEEE Trans. Vol PAS-103 (1984)1945 1952I0 S h i r m o h a m m a d i , D , H o n g , H W , S e m l y e n , A a nd L n o , G X ' Ac ompe n sa t ion -ba se d pow e r f low me thod fo r w e a k ly me she dd i s t r i bu t ion a nd t r a nsmiss ion ne tw orks ' IEEE Trans. V olPW R S-3 (1988) 7 53 -7 6211 Baran, M E a n d W u , F F 'O pt im al s iz ing of capac i tors p lacedon a radia l d is t r ibut ion system' IEEE Trans. V o l P W R D - 2(1989) 735-74 312 Chiang, H D 'A decoupled load f low method fo r d ist r ibut ionpow e r ne tw ork : a lgo r i t hms , a na ly s is a nd c onve rge nce s tudy 'Int. J. Electr. Power Energy Syst. Vol 13 No 3 (1991) 130-

13813 R e n a t o , C G 'N e w me thod fo r t he a na ly s i s o f d i s t r i bu t ionn e t w o r k s ' IEEE Trans. PWRD Vol 5 No 1 (1990) 391-39614 Goswami, S K a n d B a s u , S K 'D i re c t so lu t ion o f d i s t r i bu tionsys t e ms ' IEEProc. C. V ol 188 N o 1 (1991 )7 8 -8815 J a s m o n , G B and Lee , L H C C 'Stabi l i ty of loadflowtechniques for d is t r ibut ion system vol tage stabi li ty ana lysis 'lEE Proc. C. Vol 138 No 6 (1991)16 J a s m o n , G B and Lee, L H C C 'D istribu tion netwo rk reductionfor voltage stabili ty analysis a nd load flow calculations' Int. J.Electr. Power Energy Syst. Vol 13 No 1 (1991) 9-1 3

A p p e n d i x 1F r o m F i g u r e 2 , w e h a v e t h e f o l l o w i n g e q u a t i o n s :

I ( 1 ) = I V ( 1 ) l l r ( 1 ) - I v ( 2 ) 1 l ~ ( 2 ) ( 1 2 )R ( 1 ) + j X ( 1 )

a n dI ( 1 ) - P ( 2 ) - j Q ( 2 ) ( 1 3 )v * ( 2 )

X l I . R e f e r e n c e s1 Tinney, W F a n d H a r t , C E 'Po w e r f low so lu tion by N e w ton ' sm e t h o d ' IEEE Trans Vol PAS-86 (1967) 1449-14562 Stot t , B and Alsae , O 'Fa s t de c oup le d loa d f low ' IEEE Trans.Vol PAS-93 (1974) 859-8693 S t a g g , G W an d EI-Abiad, A H Computer methods in powersystem analysis M c G ra w H i l l (1968 )4 Nag ra th , I J and Kothar i , D P Modern power system analysis2nd e dn , Ta t a M c G ra w H i l l , N e w D e lh i (1989 )5 Rajicie, D a n d B o s e , A A modif ica t ion to the fast decoupledpow e r f l ow fo r ne tw orks w i th h igh R / X ra t ios IEEE Trans.Vol PWRS-3 (1988) 743-7466 Iw a moto , S a n d T a m u r a , Y 'A loa d f low c a l c u l a tion me th odfo r i ll c ond i t ione d pow e r sy s t e ms ' IEEE Trans. Vol. PAS-100(1981) 1736 17137 T r i pa th y , S C , D u r g a p a r a s a d , G , M a l i k , O P a n d H o p e , G S'Lo a d f low so lu t ions fo r i ll c ond i t ione d pow e r sy s t e ms by a

F r o m e q u a t i o n s ( 1 2) a n d ( 1 3) w e o b t a i n :I V ( 1 ) I L r (1 ) - I V ( 2 ) I L r (2 ) _ P ( 2 ) - j Q ( 2 )

R ( 1 ) + j X ( 1 ) V * (2 )t h e r e f o r e

IV (1 ) I IV(Z)ILr(1) - t5 (2 ) - I V ( 2 ) I 2= [ P ( 2 ) - j Q ( 2 ) ] [ R ( 1 ) + j X ( 1 ) ]

t h e r e f o r eI V (1 ) I IV (2 ) I c os [6 (1 ) - 6 (2 ) ] - I V (2 ) I 2

+ J l V (1 ) I I V (2 ) I s in [6 (1 ) - 6 (2 ) ]= [ P ( 2 ) R ( 1 ) + Q ( 2 ) X ( 1 ) ]+ j [ P ( Z ) X ( 1 ) - Q ( Z ) R ( 1 ) ] ( 1 4)

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M eth od fo r so l u t i on o f rad i a l d i s t r ibu t i on ne tworks : D . Das et al .T a b l e A 2 . 1 L i n e d a t a o f E x a m p l e 1

3 4 3

B r a n c h S e n d i n g - R e c e i v in g - R Xn u m b e r e n d n o d e e n d n o d e ( o h m ) ( o h m )1 1 2 1.35309 1.323492 2 3 1.17024 1.144643 3 4 0.84111 0.822714 4 5 1.52348 1.027605 2 9 2.0131 7 1.357906 9 10 1.68671 1.1377 07 2 6 2.5572 7 1.724908 6 7 1.08820 0.7340 09 6 8 1.25143 0.8441 010 3 11 1.79553 1.211 1011 11 12 2.44 845 1.651 5012 12 13 2.0131 7 1.3579013 4 14 2.23081 1.5047014 4 15 1.19702 0.8074 0

S e p a r a t i n g r e a l a n d i m a g i n a r y p a r t s o f e q u a t i o n ( 1 4 ) w eo b t a i nI v ( 1 ) l I v ( 2 ) l c o s [ 6 ( 1 ) - 6 ( 2 ) ] - I v ( 2 ) l 2

= P ( 2 ) R ( 1 ) + Q ( 2 ) X ( 1 )t h e r e f o r e

I v ( 1 ) l I v ( 2 ) l c o s [ 6 ( 1 ) - 6 ( 2 ) ]= ] v ( 2 ) ] 2 + P ( Z ) R ( 1 ) + Q ( z ) x ( 1 ) ( 1 5)

a n dI V(1) ] IV(Z)] s in (6(1) - 6 (2) ) = P (2) X (1) - Q(2 )R(1 )

(16)S q u a r i n g a n d a d d i n g e q u a t i o n s ( 1 5 ) a n d ( 1 6) w eo b t a i n :

] V(1)[2[ V (2)[ 2 = [[V ( 2 ) [ 2 - ~ - P ( Z ) R ( 1 ) + Q ( Z ) X ( 1 ) ] 2+ [ P ( Z ) X ( 1 ) - Q(Z)R(1) ] z

T a b l e A 2 . 2 L o a d d a t a o f E x a m p l e 1N o d e s K V A N o d e s K V A1 0 . 0 9 100.02 63.0 10 63.03 100.0 11 200.04 200.0 12 100.05 63.0 13 63.06 200.0 14 100.07 200.0 15 200.08 100.0P o w e r f a c t o r o f t h e l o a d i s t a k e n a s c o s q~ = 0 . 7 0 .R e a l p o w e r l o a d = PL = K V A * c o s ~b.R e a c t iv e p o w e r l o a d = QL = K V A * s i n ~b.

o r

] V ( 2 )I 4 + 2 . 0 [ P ( 2 ) R ( 1 ) + Q ( 2 ) X ( 1 )- 0.50] V(1)[2]I V(2)[ 2 + (R 2(1 )+ x z ( 1 ) ) ( p 2 ( 2 ) + Q 2 ( 2 ) ) = 0 ( 1 7)

E q u a t i o n ( 1 7 ) h a s a s t r a i g h t f o r w a r d so l u t i o n a n d d o e sn o t d e p e n d o n t h e p h a se a n g l e , w h i c h s i m p l i f i e s t h ep r o b l e m f o r m u l a t i o n . I n a d i s t r i b u t i o n sy s t e m , t h ev o l t a g e a n g l e is n o t so i m p o r t a n t b e c a u se t h e v a r i a t i o no f v o l t ag e a n g l e f r o m t h e su b s t a t i o n t o t h e t a i l- e n d o f ad i s t r i b u t i o n f e e d e r is o n l y fe w d eg r e e s. N o t e t h a t f r o m t h et w o so l u t i o n s o f I V (2 ) ] 2 o n ly t h e o n e c o n s i d e r i n g t h ep o s i t i v e si g n o f th e sq u a r e r o o t o f t h e so l u t i o n o f t h equa dra t i c equ a t ion g ives a real i s t ic va lue . The s ame i sappl i cab le when so lv ing for IV(2) ] .T h e r e f o r e f r o m e q u a t i o n ( 1 7 ) , t h e so l u t i o n o f I V ( 2 ) Ic a n b e w r i t te n a s :

] V(2)I = {[(P(2)R (1) + Q(2 )X (1) - 0.51V(1)]2) a- (R2(1) + X2(1) ) (p2(2) + Q2(2) ) ]1 /2- (P(Z )R(1 ) + Q( Z)X (1) - 0.5] V(1)[2)} U2

(18)E q u a t i o n ( 1 8) c a n b e w r i t t e n i n g e n e ra l i ze d f o r m :

V ( m 2 ) = [ B ( j ) - A ( j ) ] U 2w h e r e

(19)

A ( j ) = P ( m 2 ) * R ( j ) + Q ( m 2 ) * X ( j ) - 0 . 5 * ] V ( m l ) l 2(20)

B ( j ) = [ A Z ( j ) - ( R Z ( j ) + X 3 ( j ) ) * ( P Z ( m 2 ) + Q Z ( m 2 )) ]l /2(21)

j i s t h e b r a n c h n u m b e r , m l a n d m 2 a r e s e n d i n g - e n d a n dr e c e i v i n g - e n d n o d e s r e sp e c t i v e l y ( m l = I S ( j ) a n dm 2 = I R ( j ) ) .

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~

~

~X

ECq

c C c

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M eth od fo r so lu t i on o f rad ia l d is t r i bu t ion ne tworks : D. D as et a l . 34 5T a b l e A 3 . 1 . C o n t i n u e dBranch Sending- Receiving- R Xno. end end (ohm) (ohm)41 41 42 0.273 0.11342 41 43 0.455 0.18943 34 44 1.002 0.41644 44 45 0.911 0.37845 45 46 0.911 0.37846 46 47 0.546 0.22647 35 48 0.637 0.26448 48 49 0.182 0.07549 49 50 0.364 0.15150 50 51 0.455 0.18951 48 52 1.366 0.56752 52 53 0.455 0.18953 53 54 0.546 0.22654 52 55 0.546 0.22655 49 56 0.546 0.22656 9 57 0.273 0.11357 57 58 0.819 0.34058 58 59 0.182 0.07559 58 60 0.546 0.22660 60 61 0.728 0.30261 61 62 1.002 0.41562 60 63 0.182 0.07563 63 64 0.728 0.30264 64 65 0.182 0.07565 65 66 0.182 0.07566 64 67 0.455 0.18967 67 68 0.910 0.37868 68 69 1.092 0.45369 69 70 0.455 0.18970 70 71 0.546 0.22671 67 72 0.182 0.07572 68 73 1.184 0.49173 73 74 0.273 0.11374 73 75 1.002 0.41675 70 76 0.546 0.22676 65 77 0.091 0.03777 10 78 0.637 0.26478 67 79 0.546 0.22679 12 80 0.728 0.30280 80 81 0.364 0.15181 81 82 0.091 0.03782 81 83 1.092 0.45383 83 84 1.002 0.41684 13 85 0.819 0.340

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3 4 6T a b l e A 3 . 2 . L o a d d a t a o f E x a m p l e 2

M e t h o d f o r s o l u t i o n o f r a d i a l d i s t ri b u t i o n n e t w o r k s : D . D a s e t a

4 56.0 46 35.286 35.28 47 14.08 35.28 50 36.2811 56.0 51 56.014 35.28 53 35.2815 35.28 54 56.016 35.28 55 56.017 112.0 56 14.018 56.0 57 56.019 56.0 59 56.020 35.28 61 56.021 35.28 61 56.022 35.28 62 56.023 56.0 63 14.024 35.28 66 56.025 35.28 69 56.026 56.0 71 35.2828 56.0 72 56.030 35.28 74 56.031 35.28 75 35.2833 14.0 76 56.036 35.28 77 14.037 56.0 78 56.038 56.0 79 35.2839 56.0 80 56.040 35.28 82 56.042 35.28 83 35.2843 35.28 84 14.044 35.28 85 35.2845 35.28Power factor of the load cos = 0.70.Reactive power load = QL = PL* tan 8.Those nodes with no power are not shown.

Node P L Node P Lno. (kW) no. (kW)