79-83!87!103 Analysis of Wind Driven Grid Connected Induction

8/2/2019 79-83!87!103 Analysis of Wind Driven Grid Connected Induction

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IEEE Transactions on Energy Conversion, Vol. 9, No. 2, June 1994 217

ANALYSIS OF WIND DRIVEN GRID CONNECTED INDUCTIONUNDER UNBALANCED GRID CONDITIONS

A.H. Ghorashi, 8.8. Murthy*, B. P. Singh", BhimDepartment of Electrical Engineering

Indian Institute of TechnologyNew Delhi-110016 India

* Senior Member IEEE

Abstract - Wind ariven induction generator feedingpower to the grid has been analyzed under theabnormal condition of unbalanced grid voltages.Using the symmetrical component and doublerevolving field theory, appropriate equivalentcircui ts and model equations have been derived forthe generating mode with suitable realisticmodifications. It is emphasized that the activeand react ive power components and their directionsfor both positive and negative sequence systemsneed to be properly identified in order to obtainthe cumulative response of the generator underdifferent wind power conditions. In view of thefact that the reactive power is drawn from thegrid while the active power is fed into the grid,the extent of variations in power fed to the gridand th e reactive VAR due t o unbalanced gridvoltages for different wind power conditions needto be estimated to provide guidelines in the

design and operation of wind energy conversionsystem. Both experimental and theoretical resultsfor a 3.7 kW laboratory model have also beenpresented, to validate the theoreticalformulations, extendable to large units. Extensivedata have been presented and discussed for a 5 5 kWunit installed in site.

Ke y words : Wind energy, Induction generator,Unbalanced voltages

NOMEWCLATURE

R,, Rr and X,, Xr, Xm, Rm :

V1, V2, Vo and 11, 12, Io :

Per phase parameters of equivalent circuitdepicted in Fig. 2 (referred to stator)

Per phase positive, negative and zero sequencevoltages and currents

Irlt 'r2 :Positive and negative sequence rotor currentszl, z2 :Per phase positive and negative sequenceimpedances

Line voltages and currents of gridv,br vbcr Vca and Ila# Ilc :

'ab' Ibc'Ica :Phase currents in stator windings

'q.max' Ir.max :Highest of the three phase stator and rotorcurren ts under unbalance.

'in11 'in2Power fed to positive and negative systems

9 3 SM 545-4 ECby the IEEE Energy Development and Power GenerationCommittee of the IEEE Power Engineering Society forpresentation at the IEEE/PES 1993 Summer Meeting,Vancouver, B.C., Canada, July 18-22, 1993. Manu-script submitted August 2 8 , 1992; made availablefor printing May 12, 1993.

PRINTED IN USA

A paper recommended and approved

GENERATORS

Singh

Pin : Power fed to the shaftPout : Power fed t o the gridQ : VAR drain from the gridVAR : Voltamperes reactive

V2/V1 : Degree of unbalancePloss : Power lossEff : Efficiency(Phas or values ar e indicated in bold letters.)

S : Slip

INTRODUCTION

Due to the energy crisis, search and exploitationof alternative renewable energy resources haveassumed increased importance leading to relevanttechnological efforts. Wind energy has beenidentified as a promising resource for such anexploitat on.

Induction Generator (IG) has been found to be veryappropriate for wind energy applications due toits low unit cost , reduced maintenance, rug gedand brushless rotor(squirre1 cage type),etc. In atypical Wind Energy conversion System (WECS) theIG is driven by the wind turbine through a gearbox. There may be several such units in a windfarm whose output is fed to the local 11 kV gridthrough step-up transformer. Presently individualunits of rating from 100 to 1000 kW have beenstandardized.

WECS ar e now installed in large numbers in severalcountries in Europe and North America in highwind density areas. India has also embarked on amajor wind energy program. A wind survey has beenrecently completed for the whole country based o nwhich prospective locations have been identified.Wind farms are in operation in the coastal areaof Gujarat, Tamilnadu and Orissa. Based on theexperience of the authors in working with t heagencies operating the wind farms, it was realizedthat efficient and reliable operation of the unitshave been considerably hampered by abnormal gridconditions. This is also due to the fact that theconcerned utilities to whose grid the wind poweris fed have not been able t o maintain the purityof supply invariably due to distant locations ofthe wind farms from the main power generatingcenters normally therma1,hydro or nuclear.

-7

--I L

-,~ 7 I

Fig. 1. Schematic of wind driven GCIG system.

0885-8969/94/$04.00 0 1993 IEEE


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A s a n i n i t i a l p h a s e o f t h e s t u d y t h e p e r f o r m a n c eo f G r i d C o n n e c t e d I n d u c t i o n G e n e r a t o r ( G C I G ) u n d e rv a r y i n g v o l t a g e a n d f r e q u e n c y c o nd i t i o n s w a su n d e r t a k e n a n d i t s o u t c om e w a s r e p o r t e d [ l ] .F u r t h e r , t h e t h e o r e t i c a l m o d e l w a s r e f o r m e d b yi n v o l v i n g t h e v o l t a g e d e p e n d e n t p a r a m e te r s o f I G .T h e p r e s e n t p a p e r i s t h e e x t e n s i o n o f t h e a b o v ei n v e s t i g a t i o n w h i c h c o n c e r n s w i t h t h e e f f e c t o fu n b a l a n c e d g r i d v o l t a g e s , a r e a l i s t i c c o n d i t i o nf o u n d t o b e c o m m o n l y p r e v a l e n t i n t h e f i e l d. T h i ss t u d y w a s i d e n t i f i e d a s v e r y c r i t i c a l s i n c e t h e

e n e rg y a g e n c i e s f e lt t h a t g r i d u n b a l a n c e i s am a j o r c a u s e o f po o r p e r f o r m a n c e o f c u r r e n t l yi n s t a l l e d w i n d s y s t e m s , d u e t o a d v e r s e e f f e c t o nI G. T h e y w e r e i n t e r e s t e d t o k n o w t h e e x t e n t o f

, u n d e s i r a b l e e f f e c t s o n t h e I G a n d a s s o c i at e ds y s t e m s d u e t o g r i d u n b a l a n c e so t h a t s u i t a b l er e m e d i a l m e a s u r e s c a n b e t a k en i n d e s i g n a n do p e r a t i o n s .T h e s y s t e m h a s b e e n m o d e l e d u s i n g t h e w e l l k n o w nt h e o r y o f s y m m e t r i c a l c o m p o n e n t s ( 8 1 a n d t h er e l e v a n t e q u a t i o n s a r e s i m u l a t e d o n a c o m p u t e r.B a s e d o n s i m i l a r c o n c e p t s u n b al a n c e d a n da s y m m e t r i c a l o p e r a t i o n s o f I n d u c t i o n M o t o r s h a v eb e e n w e l l d o c u m e n t e d i n l i t e r a t u r e [ 4 - 7 1 , w h i l et h e p e r f o r m a n c e a n a l y s i s o f I n d u c t i o n G e n e r a t o rd e t a i l i n g d i f f e r e n t a d v e r s e c o n s e q u e n c e s k e e p i n gi n v i e w t h e r e a l i s t i c p r i m e m o v e r c o n d i t i o n s a n dn e w e m p h a s i s o n w i nd a n d m i n i h yd r o s y s t e m s h a sn o t so f a r b e e n r e p o r t e d . T h e r e a r e s u b t l e

d i f f e r e n c e s b e t w e e n m o t o r i n g a n d g e n e r a t in gc o n d i t i o n s . A i r g a p v o l t a g e o f g e n e r a to r i se l i g h t l y h i g h e r a s c o m p a r e d t o m o t or, w h i c h m a yc a u s e i n c r e a s e d s a t u r a t i o n . E q u i v a l e n t c i r c u i tp a r a m e t e r s f o r p o s i t i v e a n d n e g a t i v e c i r c u i t s m a yd i f f e r d u e t o d i f f e r i n g o p e r a t i n g v o lt a g e s.

T H E O RY

U n d e r u n b a l a n c e d g r i d v o l t a g e s :The basic scheme under consideration is shown inFig. 1. Capacitors and transformer are omitted fors i m p l i c i t y , t h o u g h t h e y c a n b e e a s i l yi n c o r p or a t e d . D e l t a c o n n e c t e d I O i s c o n s i d e r e d a sp e r f i e l d c o n d i t i o n s .U s i n g s y m m e t r i c a l c o m p o n e n t t r a n s f o r m a t i o n s [ 2 , 8 ]t h e p o s i t i v e , n e g a t i v e a n d z e r o s e q u e n c e v o lt a g e s ,V1,V2, Vo can be obtained by

S i m i l a r t r a n s f o r m a t i o n y i e l d s s e q u en c e l i n ecu rr en ts Il l , 112, Il0 from Ila, I lb, I lc andsequence phase currents 11, 12, Io fr om I&, Ib c,IFa. The zero sequence quantities would be absentsince sum of th e line voltages is zero.Knowing th e magnitudes of Vab, Vbc, Vca it ispossible to obtain their relative phasor positionsthrough a closed triangle. If a and /3 are thephase angles of Vbc and Vca with respect to Vab,Va b = Va b k , bc = V b c k n d Vca = Vca@ .E q u i v a l e n t c i r c u i t s o f I G f o r p o s it i v e a n dn e g a t i v e s e q u e n c e s y s t e m s a r e g i v e n i n F i g . 2 ,w h e r e s l i p S i s n e g a t i ve . F i g u r e 2 ( a ) d e t e r m i n e sthe pos i t ive s equen ce phase cur ren t I1 for g iven

S and circuit parameters, similarly Fig. 2(b)z k d s I2 for t he givenAlthough th e equivalent Vc?i;cuit parameters forp o s i t i v e a n d n e g a t i v e s e q u e n c e s y s t e m s a r en o r m a l l y a s s u m e d t o b e s a m e , a c o r r e c t a n a l y s i sm u s t t a k e i n t o a c c o u n t t h e d i f f e r e n c e s w h e r e v e rt h e i r i m p a c t i s s i g n i f i c a n t . F o r e x a m p l e , t h er o t o r r e s i s t a n c e a n d l e a k a g e r e a c t a n c e s d i f f e rw i d e l y i n c u r r e n t d i s p l a c e m e n t r o t o r s [ 4 ] , w h i c ha f f e c t s r o t o r c u r r e n t c a l c u l a t i o ns . S i m i l a r l ym a g n e t i z i n g r e a c t a n c e a n d c o r e loss v a r yc o n s i d e r a b l y d e p e n d i n g o n t h e o p e r a t i n g a i rg a pv o l t a g e w h i c h d e c i d e s t h e c o r e f lu x . D e p e n d i n g o nthe degr ee of unba lance V1 and V2 vary and thecorrect value of X & Rm for these voltages mustbe used. Further, for low values of slip t & Xare very large compared to (Rr/2-S) and hence themagnetizing branch can be omitted in the negativesequen ce circui t of Fig.2(b).I n t h e m o d e l in g s i m u l a t i o n p r es e n te d h e r e t h ea b o v e f a c t o r s h a v e b e e n i n c o r p o r a t e d t o o b t a i nr e a l i s t i c p e r f o r m a n c e p r e d i c t i o n s.Wi t h v a r y i n g w i n d s p e e d t h e i n p u t p o w e r w o u l dc h a n g e a s c u b e o f t h e s p e e d w h i c h w o u l d a l t e r t h er o t a t i o n a l s p e e d o f t h e s h a f t c a u s i n g a d j u s t m e n to f t h e n e g a t i v e s l i p s u i t a b l y. F r o m t h e c i r c u i t so f Fig. 2 ro t or cur ren ts I r l , I r2 a re de terminedfor the given S. Power fed to positive sequencesystem is given by ;

slip and parameters.

Vo l t a g e d e p e n d e n t p a r a m e t e r s m u s t b e c a r e f u l l yc h o s e n a f t e r i d e n t i f y i n g c o r r e c t s a t u r a t e dc o n d i t i o n s .A p r a c t i c a l g e n e r a t i n g s c h e m e o f 55 KW ra t ing hasb e e n t a k e n f o r t h e s t u d y , a n d t h e r e s u l t s h a v eb e e n p r e s e n t e d u s i n g i t s d a t a. T h e s y s t e mp e r f o r m a n c e b o t h u n d e r b a l a nc e d a n d u n b a l a n c e dc o n d i t i o n s a r e c o m p a r e d a n d d i s c u s s e d . T h er e l e v a n t p e r f o r m a n c e e q u a t i o n s h a v e b e e n w r i t t e nf o r p o s i t i v e a n d n e g a t i v e s e q u e n c e s y s t e m s t oa r r i v e a t t h e c u m u l a t i v e r e s p o ns e . A p p r o p r i a t ec o m p u t e r a l g o r i t h m h a s b e e n d e ve l o p e d t o p r e d i c t

t h e p e r f o r m a n c e f o r d if f e r e n t w in d p o w e rc o n d i t io n s . T h e i n v e s t i g a t i o n s w e re a l s o c a r r i e do u t o n a 3 .7 k W l a b o r a t o r y m o d e l o n w h i c h d e t a i l e de x p e r i m e n t s w e r e c a r r i e d o u t u n d e r s i m u l a t e du n b a l a n c e d c o n d i t i o n s . B o t h t h e o r e t i c a l a n de x p e r i m e n t a l r e s u l t s h a v e b e e n c o m p u t e d a n dc o m p a r ed . T h e s t u d i e s a r e a l s o e x t e n d e d f o r t h ee x t r e m e c a s e o f u n b a l a n c e c a u s e d by d i s c o n n e c t i o no f o n e o f t h e l i n e s r e s u l t i n g i n " s i n g le - p h a s in g " .

Pinl=-3 Ir12Rr (1-S) S

Fig. 2 . S e q u e n c e e q u i v a l e n t c i r c u i t r.

P o w e r f e d t o t h e n e g a t i v e s e q u e n c e s y s t e m i so b t a i n e d b y r e p l a c i n g I r l b y I r 2 a n d S by (2-S) ineqn. (2).

P i n 2 = 3 I rZ2 Rr( l -S) / (Z-S) ( 3 )

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

'in = 'in1 + 'in2 ( 4 )

T h u s , t h e s h a f t p o w e r P i n f e d f r o m w i n d g e t sd i v i d e d i n t o t w o c o m p o n e n t s P i n l a n d P in 2. s i n c e si s n e g a t i v e P i n i s p o si t i ve . U n d e r b a l a n c e dc o n d i t i o n s P i n 2 i s z e r o a s I r 2 = 0.


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S u b s t i t u t i n g f o r 1, i n e q n . ( 1 2 ) a p o l y n o m i a lequa t ion of o rder 4 i s ob ta ined as ;

AS 4 + BS 3 + C S 2 + D S + E = 0 ( 1 3 )

T h e c o e f f i c i e n t s A , B , C , D a nd E a r e g i v e n i n t h eAppendix I .Eqn . ( l3 ) is s o l v e d f o r S u s i n g p o l y n o m i a l r o o tf i n d i n g t e c h n i q u e w h i c h y i e l d s 4 v a l u e s a m o n gw h i c h a r e a l i s t i c o n e i s a c c e pt a b l e. K n o w i n g t h er o t o r s p ee d an d t h e se q u e n c e q u a n t i ti e s t h ep e r f o r m a n c e o f t h e g e n e r a t o r u n d e r s i n g l e p h a s i n gc o n d i t i o n i s s i m i l a r l y o b t a i n e d w i t h d i f f e r e n tinput power, as g iven in preceding sec t ion .

We m u s t n o t e t h a t s l i p h a s t o a d j u s t t o a v a l u es u c h t h a t t h e s h a f t i n pu t p o w e r P i n i s e q u a l t ot h e s u m o f P i n l a n d P i n 2 . T h e pr o b le m o fp e r f o r m a n c e p r e d i c t i o n c e n t e r s a r o u n d d e t e r m i n i n gS f o r t h e g i v e n P i n a n d t h e u n b a l a n c e d g r i dvol tages . Thro ugh a computer p rogram s imula t ingt h e e q u a t i o n s o b t a i n e d f r o m F i g . 2 , i t will bep o s s i b l e t o f i n d S for the g iven input powert h r o u g h a n i t e r a t i v e p r o c e s s. S i m u l t a n e o u s l y

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

a c t u a l p h a s o r a n d l i n e c u r r e n t s a r e o b t a i n e d a s ac u m u l a t i v e r e s p o n s e . K n o w i n g t h e c u r r e n t s a n d t h es l ip , bo th ac t ive and reac t ive powers , e ff ic iency,e t c. o f t h e s y s t e m u n d e r u n b a l a n c e d c o n d i t i o n s a td i ffe ren t wind power can be de te rmined us ing thefo l lowing equa t ions .P o w e r f e d t o t h e g r i d ;

Reactive power drain from the grid ;

U n d e r "single-phasing" c o n d i t i o n :The single-phasing is caused by the accidentaldisconnection of one of the lines. Assuming theline C in Fig. 1 is open i.e. ILc=O. Ap plyingK i r c h o ff ' s l a w s t h e t e r m i n a l c o n s t r a i n t s a r eo b t a i n e d f r o m Fig. 1 [ 3 ] . A c c o r d i n g l y ;

Using

'ab - Ica='La'bc - Iab='Lb ( 8 )

' 'bc

the symmetrical component eqn.(8) yields;

I l = I2

la b = 211 (9)

Ib c = -I1

Knowing th e line voltage Vab we get

Va b = v1 + v 2

Since V1 = Ilzl and V2 = 12Z2, hence,

I l = VAB/(2 + x (11)1 2 )

Equations (2-4) are valid after finding V1,V2 and11,12. Equation (4) can be solved for S afteromitting the magnetizing branch in the negatives e q u e n c e c i r c u i t a n d s h i f t i n g t h e m a g n e t i z i n gb r a n c h o f t h e p o s i t i v e s e q u e n c e c i r c ui t t o t h eo u t p u t t e r m i n a l s . C o m b i n e d w i t h t h e e q n s . ( 9 - 1 0 )a n d t h e a b o v e a p p r o x i m a t i o n s , a s i m p l i f i e dequiva le n t c i r cu i t o f F ig .3 i s ob ta ined . Hence

eqn. (4) i s modi f ied as ;

P i n = - 3 1 r 2 K R r ( 1 2 )

w h e r e K=2 (l-s)'/s ( 2 - 6 )

= v ab 2 / ( ( R ~ + K R , ) ~ + X ~ ~ )Rt = 2 ( R s + R r ) and Xt = 2( Xs + X r )

Fig . 3 . Equiva len t c i rcu i t under s ing le -phas ingc o n d i t i o n a f t e r c o m b i n i n g s e q u e n c e n e t wo r k s .

MACHINE DETAILS

Two machines have been considered for the study, a

3.7 kW laboratory model and a 55 kW field model,d e s i g n a t e d a s m a c h i n e 1 a n d m a c h i n e 2r e s p e c t i v e l y. T h e d e t a i l s o f t h e s e m a c h i n e s a r eg i v e n i n t a b l e 1. T h e f i x e d p a r a m e t e r s o f m a c h i n e1 w e r e ob t a i n e d b y s t a n d a r d t e s t s, w h i l e t h ev o l t a g e d e p e n d e n t p a r a m e t e r s s u c h as m a g n e t i z i n gr e a c t a n c e a n d c o r e - l o s s r e s i s t a n c e w e r e o b t a i n e db y a p p r o p r i a t e v a r i a b l e v o l t ag e no load tes t s , andis shown in F ig .4 . The paramet ers of machine 2w e r e o b t a i n e d f r o m t h e m a n u fa c t u re r s .

Tab le 1

Details I Machine 1 Machine 2

Rating : 3.7 KW 55 KWLine voltage : 415 V 415 VLine current : 1.6 A 93 AHZ. : 50 50Pole No. : 4 6Base power : 3.7 KW 55 KWStatorconnection : Delta DeltaR s (P.U.) : 0.057 0.019Rr I , : 0.053 0.0164x s I , : 0.093 0.069x r # , : 0.093 0.087

x m ,, : 1.937 3.0*R m n r : 25.98?* 47.85*

~~

( * At nominal voltage)

RESULTS &NJ DISCUSSIONS

Results of the investigations are presented inFigs. 5-8. While both theoret ical and experim entalresults are presented for machine 1, only thesimulated results are presented for machine 2.Experimental results are shown by points. All theresu lts ar e presen ted in p.u. values.


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05 0 6 0 7 0 8 09 1 11 12v, (Pa.)

Fig. 4. Va r i a t i o n o f R m a n d X m w i t h V1 .

E f f e c t o f g r i d v o l t a g e u n b a l a n c e :Considering the realistic grid conditions in thef i e l d i n w h i c h t h e d e g r e e o f u n b a l a n c e c o u l de x t e n d u p t o 20% , t h e e f f e c t o f s u c h a r a n g e o fu n b a l a n c e f o r b o t h t h e m a c h i n e s w e r e s t u di e d. B o t ht h e e x p e r i m e n t a l a n d s i m u l a t e d r e s u l t s o b t a i n e df o r m a c h i n e 1 are presen t ed in F ig . 5 (a -b). A veryc l o s e a g r e e m e n t b e t w e e n t h e o r e t i c al a n de x p e r i m e n t a l r e s u l t s i s o b s e r v e d w h i c h v a l i d a t e st h e t h e o r e t i c a l m o d e l a n d t h e c o m p u t e rs i m u l a t i o n s .F i g u r e 5 ( a ) s h o w s t h e v a r i a t i o n o f p o w e r o u t p u ta n d m a x i m u m s t a t o r c u r r e n t w i t h p o w er i n p u t a t 0%and 15 .6% unba lance . Compa r ing curves a an d b , areduction in power output is noticed for thegiven power input decided by the wind speed. T hisis due to the increased losses in the systemcaused by the negative sequence components. At 1.0p.u. input power the reduction in output power is0.09 p.u. The major effect of unbalance is on thewinding currents. Curves c and d indicate themaximum current, Is -max. A considerable increasein th e current can be noticed due to unbalancew h i c h i s d e f i n i t e l y a c a u s e f o r c o n c e r n . F o rins tan ce a t 1 .0 p .u. input power I s em ax increa sesf r o m 0.84 p.u. to 1.47 p.u. and the oth er tw ow i n d i n g c u r r e n t s a r e 0 .9 7 p.u. a n d 0 .6 5 p .u .r e s p e c t i v e l y.F i g u r e 5 ( b ) s h o w s t h e v a r i a t i o n of p o w e r l o s s a n dVA R d r a i n f r o m t h e g ri d w i t h p ow e r i n p u t f o rd i f f e r e n t d e g r e e of u n b a l an c e . I t i s i n t e r e s t i n gt o n o t e t h a t VA R d r a i n d e c r e a s e s i f t h e u n b a l a n c ei s d u e t o u n e q u a l a n d r e d u c e d v o l t ag e s , w h i l e i ti n c r e a s e s i f t h e u n b a l a n c e i s d u e t o u n e q u a l a n di n c r e a s e d v o l t a ge s . T h i s i s d u e t o t h e f a c t t h a ta t r e d u c e d v o l t a g e s t h e m a c h i n e o p e r a t e s a t " u n d e rf l u x " c o n d i t i o n a n d a t r e d u c e d s a t u r at i o n , w h e r e a sa t o v e r - v o l t a g e s t h e m a c h i n e i s " o v e r f l u x e d " a n ds a t u r a t e d c a u s i n g i n c r e a s e i n t h e m a g n e t i z i n gc u r r e n t s . C u r v e s a an d c show the difference inthe amount of VAR drain for the same degree ofu n b a l a n c e o f 8 % a t o v e r v o l t a g e s a n d u n d e rv o l t a g e s r e s p e c t i v e l y. A u s e f u l c o m p a r i s o n c a n b em a d e w i t h c u r v e b which corresponds to the VARdrain at balanced rated voltages. Curves d and eindicate the trend of variation in power loss withPin from 0% to 15.6% unbalance. At the rated powerinput the losse s are increa sed from 0.148 p.u. to

0.238 p.u. Since the losse s contribute to heati ng,th e over heating of the machine under unbala ncehas been a matter of concern [5-71. These resultsc o n s o l i d a t e t h i s w e l l k n o w n c o n ce p t o f o v e rh e a t i n g u n d e r g e n e r a t i n g m o d e a n d c a n b e u s ed f o rd e r a t i n g a n d o t h e r o p e r a t i o n a l c o n s t ra i n t s .F i g u r e 6 ( a - c ) d e p i c t t h e a n al y t i c a l r e s u l t s

1 2

08

1 5

O B

1

0 4

050 2

n n-0 02 0 4 7"0 PU l08 1

Fig. 5 ( a) V ar ia ti on o f P o u t and I w i t h P i nfor m/c 1.a : Pout at 0% unbalance ; - test pointsb : ,, t r 15.6% r r i A A t r I ,

I ,

,,

0 02 O B 12

Fig. 5(b)Variation of Q & Ploss with Pin for m/c 1.a :Q at 8% unbalance(over-voltage); * * test pointsb : ,, 0% ,, (rated voltage); U 0 ,, ,,c : ,, 8% ,, (under-vooltage); + + ,, ,,d , e : Ploss at 15.6% & 0% unbalance; A, ,, ,,

0 0 02 00 4 008 008 0 1 0 12

V J V ,

Fig. 6(a ) Variation of power capacity and VAR withunbalance at rated current for m/c 2.a, b , f : Pin, Pout,

c t dt e : r r ,, ,, I , I , (under-voltage)Q with unbalance(over-voltage)


5/7

22

05

obta ine d for the f ie ld model i.e. machine 2 . Thev a r i a t i o n o f p o w e r i n p u t , p o w e r o u t p u t a n d t h e VA Rd r a i n fr o m t h e gr i d w i t h di f f e r en t d e g r e e o fu n b a l a n c e at r a t e d m a x i m u m c u r r e n t a r e s h o w n i nFig. 6 (a ) . T he per formance a t ra ted cur ren t i so b t a i n e d u s i n g U n i v a r i a t e M e t h o d o f O p t i m i z a t i o n[ 9 ]. T h e u n b a l a n c e i s s i m u l a t e d by c o n s i d e r i n g t w ol i n e v o l t a g e s a t r a t e d v a l u e a n d t h e t h i r d v o l t a g ev a r i e d b o t h a b o v e a n d b e l o w t h e r a t e d v al ue . S a m ed e g r e e o f u n b a l a n c e c a n b e o b t a i n e d at a s e t o fincreas ed and reduced th i rd vo l tages . In o therw o r d s t h e r e i s a n o v e r - v o l t a g e a nd u n d e r v o l t a g es t a t e y i e l d i n g t h e s a m e v o l t a g e u nb a l a n ce . T h e s er e s u l t s a r e v e r y u s e f u l a s t h e y p r o v i d ei n f o r m a t i o n r e g a r d i n g t h e c a p a b i l i t y o f t h eg e n e r a t o r t o h a n d l e p o w e r a t d i f f er e n t u n b a l a n c e dc o n d i t i o n s w i t h o u t o v e r l o a d i n g t h e w in d i n g. F o re x a m p l e a t u n d e r v o l t a g e u n b a l a n c e t h e p ow e r i n p u tcapa city dr ops from 1.147 p.u. at 0% u n b a l a n c e t o0.334 p. U. a t 10% unbalance . S imi la r ly i f theu n b a l a n c e i s c a u s e d d u e t o o v e r v o l t a g e t h e i n p u tpowe r l im ita tion s are from 1.147 p.u. to 0.24 p.u.f o r t h e s i m i l a r c on d i t i o n s o f u n b a l a n c e ( c u r v e s a& c ) . For th e power output limitations at ratedmaximum current the same trend is obtained asevident from the curves b and d. This imp lies t hatat a particular unbalance there is a maximum windspeed beyond which t he generator should not bea l l o w e d t o o p e r a t e . S i n c e t h e p o w e r i n p u t i sp r o p o r t i o n a l t o t h e c u b e o f w i n d s p e e d t h e m a x i m u m

w i n d s p e e d a t 1 0 % o f u n b a l a n c e d s h o ul d b e a r o u n d6 6 % o f t h e w i n d s p e e d a t 0% unbalanced . Thi s i s au s e f u l i n f o r m a t i o n i n s e t t i n g t h e p r o t e c t i n ge q u i p m e n t s . A f t e r m e a s u r i n g t h e d e g r e e o fu n b a l a n c e t h e s y s t e m m u st b e s e t n o t t o o p e r a t ea b o v e t h e t h r e s h o l d w i nd s p ee d . C u r v e s e a nd f i nFig. 6(a) indicate that the trend of variation inVA R d r a i n f r o m t h e g r i d i n c r e a s e s w i t h o v e rv o l t a g e u n b a l a n c e a n d d e c r e a s e s w i t h u n d e r v ol t a g eunbalance . Thi s can a le r t the u t i l i t i es , eo t h a tt h e y m a y k n o w t h e e x t e n t of VA R d r a i n d u e t ou n b a l a n c e . T h i s c a n a l s o b e a c c o u nt e d i n c h o o s i n gc a p a c i t o r s .F i g u r e 6 ( b ) s h o w s t h e v a r i a t i o n o f P o u t , Ia n d I r e m a X w i t h t h e d e g r e e o f u n b a l a n c e a t f i x e di n p u t p o w e r o f 1. 0 p. u. a n d 0 .5 p . u. I t i sobserve d th a t a t the input power power of 1 .0 p .u ,I s am ax increas es f rom 0.9 p ,u. a t 0% t o 1.5 p.u at10% unbalance . S imi la r incre ase i s found in ro torc u r r en t . T h e s a m e o b s e r v a t i o n s c a n be m a d e a t t h er e d u c e d w i n d s p e e d c a u s i n g 5 0 % p o w e r i n p u t ( s e ec u r v e s a, b , c , d).At rated power input the power output decreasesfro m 0.95 t o .92 p.u. whi le at 0.5 p.u. powerinput th e power output drops from 0.5 p.u. to 0.44

I

0 002 00 4 008 008 0 1 012v dv,

Fig. 6(b) Variati on of Is.max, & Pout withunbalance at constant Pin for m/c 2.at b , 0 : I s e m a x , I ~ . ~ ~ ~ ~out at pin = 1.0 p.u.C, d, f : t, I ,, Pi n = 0 . 5 P.U.

p.u. Ther e would be fur the r d rop in power fed tot h e g r i d i f t h e l o s s e s i n t h e t r a n s f o r m e r s a n dt r a n s m i s s i o n l i n e s a r e co n s i de r e d . A n i m p o r t a n to b s e r v a t i o n i s t h a t b o t h t h e s t a t o r a nd r o t o rw i n d i n g c u r r e n t s a r e v e r y s e n s i t i v e t o g r i du n b a l a n c e . A 10% v o l t a g e u n b a l a n ce c a u s e s a ni n c r e a s e o f a r o u n d 6 0 % i n t h e s t a t o r m a x i m u mc u r r e n t a n d 6 4 % i n t h e r o t o r m a x i m u m c u r r e n t a t1.0 p.u . powe r input. F igure 6(c) show s thev a r i a t i o n o f c u r r e n t u n b a l a n c e ( i. e. r a t i o o fn e g a t i v e t o p o s i t i v e s e q u e n c e c u r r e n t s ) i n s t a t o ra n d r o t o r w i t h in p u t p o w e r a t 4% and 12.5%u n b a l a n c e . I t i s o b s e r v e d t h a t t h e c u r r e n tu n b a l a n c e i s h i g h e r a t l o w i n pu t p o w e r w h i c hi n c r e a s e s w i t h v o l t a g e u n b a l a n c e a n d d e c r e a s e s a sthe input power increases . For example a t 12 .5%unbalanc e , a s P in increas es f rom 0 .51 t o 1 .03 p- U.s ta tor cur ren t unba lance drop s f rom 1.4 to 0 .78a n d r o t o r c u r r e n t u n b a l a n c e d r o p s f r o m 1 .6 3 t o0 .81. I t i s impor tan t to no te tha t the ro to rc u r r e n t u n b a l a n c e i s v e r y h i g h e s p ec i a l l y a t l o ww i n d p o w e r w h i c h w o u l d h a v e d e t r i m e n t a l e f f e c t s o nt h e r o t o r v i b r a t i on a n d losses c a u s e d b yo s c i l l a t o r y t o r q u e s a n d n e g a t i ve s e q u e n c ec u r r e nt s . B a s e d o n t h e s e d a t a t h e s y s t e m n e e d t ob e de s i g ne d t o t a k e c a r e of t h e s e e f f e c t s i n t h ep r o t e c t i o n s c h e m e s.

Effect of Single Phasing ISingle phasing is the undesirable condition which

i s n o r m a l l y a v o i d e d . F i g u r e s 7 a n d 8 d e p i c td i f f e r e n t p e r f o r m a n c e c h a r a c t e r i s t i cs o f t h es y s t e m u n d e r s i n g l e p h a s i n g c o n d i t i o n, i nc o m p a r i s o n t o 3 - p h a s e b a l a n c e d c o n d i ti o n .T h e v a r i a t i o n of power ou tput , VAR, losses andc u r r e n t s w i t h p o w e r in p u t f o r m a c h i n e 1 i s s h o w nin F ig . 7 (a -b) . At ra ted wind speed caus ing ra tedp o w e r i n p u t t h e p o w e r o u t p u t s h o w s a d r o p o f 0 .11p.u. w h i c h c o r r e s p o n d s t o t h e l o s s e s i n t h em a c h i n e . C u r v e s c and d show the variation of VARdrain from the grid under 1-phase and balancedc o n d i t i on s . S i m i l a r t o t h e s i t u a t i o n u n d e rVo l t a g e u n b a l a n c e t h e c u r r e n t s a r e t h e d e ci d i n gf a c t o r s f o r t h e l i m i t s o f i n p u t p o w e r f or s a f eo p e r a t i o n o f t h e w i nd s y s t em u n d e r s i n g l e p h a s i n gc o n d i t i o n . F o r e x a m p l e a t 5 0 % i n p u t po w e r t h e l i n eand the maximum ph ase cur ren ts a re 1 .0 p.u. and1.16 p.u. resp ect ivel y ( the p.u. valu es of l inea n d p h a s e a r e c a l c u l a t e d b y t a k i n g b a s e v a l u e s a sr a t e d l i n e a n d p h a s e c u r r e n t s ) . T h e u n b a l a n c er o t o r c u r r e n t r a t i o s h o w n i n c u r v e j of the FIG.7(b) is very high at low input power and decreaseswith input power. For instance at 10% input powerth e Ir2/Irl is 6.0 and at 50% input power t hisratio falls t o 1.5 .

0 2 0 4 O B 12 1 4Qn G.1

Fig. 6(c) Variation of current unbalance with Pinfor m/c 2.a, c :(Ir2/Irl)l(12/Il) at V2/V1 = 0.04bt d : t t , ,, ,, ,, =0.125


6/7

222

F i g u r e 8 d e p i c t s t h e p e r f o r m a n c e o f m a c h i n e 2u n d e r s i n g l e p h a s i n g c o n d i t i o n w h i c h s h o w s s i m i l a rt rend of var ia t ion in the resu l ts . I t i s observe dt h a t t h e f i e l d m o d e l c a n h a n d l e a h i g h e r c a p a c i t yof power input a t th i s ex t reme case of unba lance .F o r i n s t a n c e at 5 0 % i n p u t p o w e r t h e l i n e c u r r e n ta n d I a r e 1.06 p.u a n d 0 . 9 17 p.u.respec t rve ly, P out i s 0.451 p.u and the unba lan cedi n t h e r o t o r c u r r e n t i s 1 .1 9 . T h e VA R d r a i n f r o mt h e g r i d a t r a t e d m a x i m u m c u r r e n t i s 0.44 p.u.

CONCLUSIONS

E x t e n s i v e p e r f o r m a n c e c h a r a c t e r i s t i c s o f t w om a c h i n e s p r e s e n t e d i n t h i s p a pe r h a v e d e m o n s t r a t e dt h e f e a s i b i l i t y a n d l i m i t a t i o n s o f o p e r a t i n g w in dt u r b i n e d r i v e n i n d u c t i o n g e n e ra t o r s u n d e rr e a l i s t i c u n b a l a n c e c o n d i t i o n s o f t e n o b s e r v e d i nthe f ie ld .W i n d d r i v e n I G f e e d i n g p o w er i n t o t h e g r i d ,o p e r a t e s w i t h v a r y i n g p o we r i n pu t , d e p e n d i n g o nwind speed .Deta i led per formanc e chara c te r i s t ics o f 3.7 kW and5 5 k W ( f i e l d m o d e l ) a r e p r e s e n t e d u n d e r u n b a l a n c e dg r i d v o l t a g e s w i t h v a r y i n g w i n d s p ee d .A c l o s e a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l a n ds i m u l a t e d r e s u l t s h a s v a l i d a t e d t h e m o d e l i ngm e t h o d a n d s i m u l a t i o n s .T h e c o n c e p t o f s y m m e t r i c a l c o m po n e n t s, a s s o c i a t e dw i t h s e q u e n c e e q u i v a l e n t c i r c u i t s h a s b e e n s h o w n

t o b e e f f e c t i v e i n m o d e l i n g t h e s y s t e m.T h e u n b a l a n c e i n g r i d v o l t a g e h a s b e e n f o u n d t om a k e s i g n i f i c a n t i m p a c t o n t h e g e ne r a t o r a n ds y s t e m p e r f o r m a n c e .To a r r i v e a t p r o p e r r e a l i s t i c r e su l t s , c a r e h a s t ob e t a k e n t o i n c l u d e v o l t a g e d e p e n d en t m a g n e t i z i n gi m p e d a n c e c o r r e s p o n d i n g t o o p e r a t i n g f o r w a r d f l u x .T h e s e p a r a m e t e r s a r e i n s i g n i f ic a n t f o r b a c k w ar df ie lds .T h e r e i s a d r o p i n I G p o w e r o u t p u t f o r g i v e n p o w e ri n p u t d u e t o u n b a l a n c e , c a u s e d b y n e g a t i v es e q u e n c e c o m p o n e n ts . A f u r t h e r d e c r e a s e i n p o w e rf e d t o t h e g r i d i s e x p e c t e d i f t h e t r a n s f o rm e r a n dl i n e i m p e d a n c e s a r e i n c l u d e d .T h e w i n d i n g c u r r e n t s o f t h e i n d u c t i on g e n e r a t o ra r e f o u n d t o b e v e r y s e n s i t i v e t o u n b a l a n ce d g r i dvol tage s a t a l l wind speeds . At ra ted power inputt h e h i g h e s t o f t h e t h r e e w i n d i n g c u r r e n t s w a sf o u n d t o b e n e a r l y 1.5 p e r u n i t a t 1 5 . 6 %u n b a l a n c e .Wi t h o u t o v e r l o a d i n g t h e w i n d i n g s t h e m a x i m u m p o w e rt h a t c a n b e h a n d l e d d r o p s c o n s i d e r a b l y w i t hunbalance . Ther e i s a maximum wind speed a t eachu n b a l a n c e w h i c h c a u s e s r a t e d m ax i m um c u r r e n tb e y o n d w h i c h t h e s y s t e m s h o u l d n ot o p e r a t e .R e l e v a n t d a t a a r e p r e s e n t e d i n t h e p a p er a t e a c hu n b a l a n c e .C o n s i d e r a b l e i n c r e a s e i n c u r r en t u n b a l a n c e h a sbeen observe d , espec ia l ly a t low wind speeds . Th ism a y c a u s e r o t o r v i b r a t i o n s a n d o s c i l l a t o r yt o r q u e s .T h e VA R d r a i n f r o m t h e g r i d i s m o d e ra t e l y e f f e c t e db y u n b a l a n c e a n d d e p e n d s m o r e o n f o r w a r d a n db a c k w a r d f l u x l e v e l s t h a n o n u n ba l a n ce . T h e VA Rd r a i n c a n b e c o m p e n s a t e d b y u s i n g a p p r o p r i a t es t a t i c VA R c o m p e n s a t o r s .S i m i l a r m o d e l i n g s i m u l a t i o n s a n d e x p e r i m e n t a li n v e s t i g a t i o n s w e r e c a r r i e d o u t f o r t h e e x t r e m ec a s e o f u n b a l a n c e u n d e r s i n g l e ph a s in g . T h e

t y p i c a l r e s u l t s o b t a i n e d r e v e a l t h e p e r f o r m a n c ec h a r a c t e r i s t i c s o f i n d u c t i o n g e n er a t o r u n d e rs i n g l e p h a s i n g o p e r a t i o n. A s c a n b e e x p e c t e d t h ew i n d i n g g e t s c o n s i d e r a b l y o v e r l o a d e d a n d p o w erh a n d l i n g c a p a c i t y i s c o n s i d e r a b l y r e d uc e d .T h e o v e r a l l r a t i n g o f t h e s y s t em i n h a n d l i n g p o w e rinput a t ra ted maximum cur ren t under 10% v o l t a g eu n b a l a n c e a n d s i n g l e - p h a s i n g w o u ld b e 3 0 % a n d 40%o f t h e b a l a n c e d n o r m a l r a t i n g r e s p e c ti v e l y.

I t i s h o p e d t h a t t h e d a t a p r e s e nt e d i n t h i s p a p e rresu l t i ng f rom an in-depth s tudy would be foundu s e f u l i n t h e d e s i g n o f wi n d s y s t e m s s u b j e c t e d t ounbalanced gr id condi t ions .

, Pout -6- * Q 1P.dC

,

I I

0 0 2 0 4 05 08 1 12 145" (P.U)

Fig . 7. Var i a t io n of sys tem quant i t i es wi th P inu n d e r s i n g l e - p h a s i n g c o n d i t i o n f o r m / c 1.a : Pout at balanced condition ; g g test pointsb : ,, ,, 1-phase I , i 0 0 ,, ,,C : Q ,, , I I , i + + ,, , Id : ,, ,, balanced ,, i 0 0 ,, I ,e : Ploss at 1-phase ,, ; * *f : ,, ,, balanced ,, i 0 0 ,, ,,

h : IL a 8 , e t ,, i 0 0 ,, # #i : Ibc ,, ,, i 0 0 I , I ,j : Ir2/Irl t I ,, i I , I 1

t , ,,

g : Iab at 1-phase condition i 0 0 ,, ,,

O 2t -, , , I O 50 0

0 0 2 0 4 05 0 8 1 12 145" ( P d

Fig. 8 Variation of system quatities with Pinunder 1-phasing for m/c 2.


7/7

223

Prof. S.S. Murthv was born inKarna t aka in 1946 . He rece iv edhis B.E. degree f rom Bangalo reUniver s i ty, M.Tech f rom I ITBombay and Ph.D. deg ree fro mIIT- Delh i in 1967 , 1969 and1974, respec t ive ly.H e w a s a t B I T S P i l a n i d u r i n g1969-70 and has been a t I ITD e l h i s i n c e 1 9 7 0 w h e r e he i s a

p r o f e s s o r w i t h t h e D e p ar t m e n t o f E l e c t r i c a l

e n g i n e e r i n g . h e w a s a v i s i t i n g s t a f f a t t h eU n i v e r s i t y o f n e w C a s t l e U p o n Ty n e d u r i n g 1 9 7 5 -7 6and a t the Univer s i ty of Calgar y dur ing 1980-82 .H e h a s h e l d t h e p o s t o f D i r e c t o r a t E l e c t r i c a lR e s e a r c h a n d D e v e l o pm e n t A s s o c i a t i o n ( E R D A ) ,B a r o d a ( I n d i a ) d u r i n g 1 9 90 - 92 .H e h a s p u b l i s h e d a l a rg e n u mb e r o f p a p e r s a n d h a se d i t e d t w o l a b o r a t o r y m a n u a l s .He i s a Senior Memb er of IEEE, Fe l low of t h eI n s t i t u t e o f E n g i n e e r s ( I n d i a ) a n d a l i f e M e m b e ro f t h e I n d i a n s o c i e t y f o r Te c h n i c a l E d u ca t i o n .H i s a r e a s o f i n t e r e s t i n c l u d e e l e c t r i c a l m a c h i n e sa n d d r i v e s , e f f i c i e n t e l e c t r i c a l u t i l i z a t i o n a n de n g i n e e r i n g e d u c a ti o n .

Prof . B.P. S inah was born in S inghiya , in 1940 .He rece i ved h is B .Sc. (Engg. ) degree in 1963 f romBITS, S ind r i , ME in Elec t r ica l Engg. in 1966 f romCalcut ta Univers i ty and Phd . in 1974 f rom I IT

Delh i .H e w a s a S e n i o r F e l l o w a t B E C o l l e g e , H o w r a h( 1 9 6 3 - 1 9 6 6 ) a n d a f t e r s e r v i n g M I T M u z a f f a r p u r a s af a c u l t y m e m b e r o v e r a d e c a d e ( 1 9 6 6 - 78 ) , h e j o i n e dI I T D e l h i i n 1 9 7 8 , w h e r e h e i s a P r o f e s s o r w i t ht h e D e p t. o f E l e c t r i c a l E n g g. H e w a s a v i s i t i n gP r o f e s s o r a t C a l i f o r n i a S t a t e U n i v e r s i t y, L o n gBeach dur ing 1988-1990.H e i s a S e n i o r M e m b e r o f I E E E , F e l l o w o f t h eI n s t i t u t e o f E n g i n e e r s ( I n d i a ) a n d a l i f e M e m b e ro f t h e I n d i a n s o c i e t y f o r Te c h n i c a l E du c a t i on .H i s r e s e a r c h i n t e r e s t s a r e i n d e s i gn , a n a l y s i s a n dc o n t r o l o f e l e c t r i c a l m a c h i n e s.- -

r. Bhim Sinah , w as born a t Rahama pura in U.P. in1956. He rece iv ed h is B .E . degree f rom Roorke eUniver s i ty, and M.Tech and Ph.D. degree f rom I IT-D e l h i i n 1 9 7 7 , 1 9 7 9 a n d 1 9 8 3 r e s p e ct i v e l y.F r o m 1 9 8 3 t o 1 9 9 0 he w a s w i t h t h e d e p a r t m e n t ofe l e c t r i c a l e n g i n e e r i n g , U n i v e rs i t y o f R o o r k ee . A tp r e s e n t h e i s A s s i s t a n t P r o f e s s o r a t I n d i a nI n s t i t u t e o f Te c h n o l o g y, D e l h i.H e h a s o v e r 6 0 p a p e r s t o h i s c r e d i t s in t h e fi e l do f C A D , P o w e r E l e c t r o n i c s a n d A n a l y s i s a nd C o n t r o lO f E l e c t r i c a l M a c h i n e s .

R E F E R E N C E S

S.S. M u r t h y , C. S . J h a , P. S. N. R a o ," A n a l y s i s o f g r i d c o n n e c t e d i n d u c t i o ng e n e r a t o r d r i v e n by h y dr o l w i n d t u r b i n e su n d e r r e a l i s t i c s y s t e m c o n s t r ai n t s " ,-- -rans . o n enerav conv ers ion , Vol . 5 , No. 1,m a r c h 1 9 9 0 , p p 1 -7 .J.E. Br own and C.S. J h a , '' G e n e r a l i z e dr o t a t i n g f i e l d t h e o r y a n d p o l y p h a s ei n d u c t i o n m o t o r a n d i t s r e l a t i o n s h i p t o

s y m m e t r i c a l c o m p o n e n t t h e o r y " , P r o c. I E E ,vol. 1 09, Par t A, NO. 43, Feb. 1962.J.E. B row n, C.S. Jha, "The starti ng of a 3-p h a s e i n d u c t i o n m o t o r c o n n e c t e d t o a s i n g l e -pha se supply sys tem",Proc . IEE, Vol . 106 , Par tA, No. 26, Apr il 1959 , p. 183.J .E . W i l l i a m s , " O p e r a t i o n o f t h r e e p h a s ei n d u c t i o n m o t o r s o n u n b a l a n c e d v o l t a g e s " ,Tr a n s . A I E E , PAS, Vol. 7 3, No. 11, April195 4, p. 1 25.B.N. G a f f o r d , W.C . D u e s t e r h o e f t , J r. , C .C .M o s h e r, 111 , " H e a t i n g o f i n d u c ti o n m o t o r s o nunbala nced vol tages" ,Trans . AIEE. PAS, Vol.7 8 Pt. 111-A, J un e 1959, p. 2 8 2 .M.M. Bernd t, N.L. Sch mit z, "Derat ing of poly-p h a s e i n d u c t i o n m o t o r s o p e r a t ed w i t hu n b a l a n c e d l i n e v ol t a g e s ", Tr a n s . A I E E PAS,Vol. 81, Feb. 1 963 , p. 680.N . R a m a r a o , P. A. D. J y o t h i r a o , " R e r a t i n gf a c t o r s o f p o l y - ph a s e i n d u c t i o n m o t o r s u n d e ru n b a l a n c e d l i n e v o l t a g e c o n d i t i o n s " Tr a n s.IEE E PAS, Vol. 07 , No. 1, Jan. 1963, p. 240.C .F. W a g n e r, R .D . E v a n s , " S v m m e t r i c a lc o m D o n e n t s a D D l i e d t o t h e a n a l v s i s ofu n b a l a n c e d e l e c t r i c a l c i r c u i t s , ( b o o k ) M cGraw-Hi l l .S.S. R a o , " O D t i m i z a t i o n t h e o r va p p l i c a t i o n s " , ( b o o k ) Wi l l e y E a s t e r n L i m i t e d ,1987 .

A P P E N D I X

A = Rt2+4Rr2-4RtRr+Xt2-6Vab2Rr/Pin

B = -4Rt2-16Rr2+16RtRr-4Xt2+24Vab2R,/Pin

C = 4Rt2+24Rr2-20RtRr+4Xt2-30Vab2Rr/Pin

D = -16Rr2+8RtRr+12Vab2Rr/Pin

E = 4Rr2

B I O G R A P H Y

A. H. Ghoras hi was born in Kasha n,I r a n i n 1 9 57 . H e r e c e i v e d h i sB . Te c h a n d M .E . d e g r e e s i ne l e c t r i c a l e n g i n e e r i n g f r o m I n d i a .P r e s e n t l y h e i s w o r k i n g t o w a r d s h i sPh.D. degree in the depar t ment ofe l e c t r i c a l e n g i n e e r i n g , I n d i a nI n s t i t u t e o f Te c h n o l o g y , N e w

Delhi.H i s a r e a o f i n t e r e st i n c l u de , n o n c o n v e n t i o n a le n e rg y s y s t e m s ( w i n d , w a v e , e t c .) , c o m p u t e r a i d e dm o d e l i n g a nd s i m u l a t i o n of e l e c t r i c a l m a c h i n e s a n dm u l t i - p h a s e t r a n s m i s s i o n o f p ow e r.

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