Dynamic S w i t c h i n g Technique For Load C o n t r o l i n a PV System

Eisham A . E l Khashab Abdel-Kamead Maamoun ?.:. S a i d Abd El-Yotaleb

Abscrac t : Vol tage l o a d c o n t r o l of a s tand-a lone phor ;ovol ta ic (SAPV) system can be an i n d i c a t i o n f o r t h e optimum u t i l i z a t i o n o f t h e sys tem. can be r e a r r a n g e d u s i n g an e l e c t r o n i c s w i t c h i n g c i r c u i t t o o b t a i n t h e d e s i r e d l e v e l o f t h e l o a d v o l t a g e . Nore- o v e r , s u c h a c i r c u i t may be a l s o used t o c o n t r o l t h e c h a r g i n g and d i s c h a r g i n g o f t h e b a t t e r y s t o r a g e sub- sys tem i n t h e SAPV sys tems. The d e t a i l s o f t h i s c i rc- u i t a r e p r e s e n t e d i n t h i s paper . The i n s t a n t i n u o u s optimum numbersof t h e a r r a y series and p a r a l l e l c e l l s a r e deduced f o r a g i v e n e q u a l i t y c o n s t r a i n t . From an economical p o i n t o f view, a c o s t f u n c t i o n i s i n t r o d u c e d f o r t h e proposed sys tem f o r c o s t min imiza t ion r e q u i r e - ments.

The PV a r r a y

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

S tand-a lone PV sys tems f o r remote power p r o d u c t i o n g e n e r a l l y c o n s i s t of a PV a r r a y and b a t t e r y s t o r a g e sub-system. A l a r g e p e n e t r a t i o n o f PV i n t h e marke t w i l l o n l y o c c u r i f s u f f i c i e n t l y low energy c o s t can be achieved . S e v e r a l s t u d i e s f o r t h e a u t h o r s l - 5 had been p u b l i s h e d i n t h e f i e l d o f renewable energy , w h i l e t h i s paper is s p e c i a l l y o r i e n t e d t o t h e c o n t r o l o f t h e PV sys tem.

T h i s paper p r e s e n t s a low c o s t on- l ine c o n t r o l by a s i m p l e e l e c t r o n i c c i r c u i t u s i n g a s w i t c h i n g t e c h n i - que f o r bo th t h e PV a r r a y , and t h e s t o r a g e b a t t e r i e s . T h i s c i r c u i t a l l o w s t o reduce t h e s i z e n o t o n l y o f t h e W a r r a y but a l s o of t h e b a t t e r i e s , whi le m a i n t a i n i n g t h e r e l i a b i l i t y o f t h e power supply .

F i g . 1 . E q u i v a l e n t c i r c u i t of t h e PV c e l l

The ce l l is a n o n l i n e a r d e v i c e and can be r e p r e s e n t e d a s a c u r r e n t s o u r c e model6 a s shown i n F i g . 1 . c u r r e n t s o u r c e c i r c u i t s have t h e d i s a d v a n t a g e of v o l - t a g e f l u c t u a t i o n s w i t h v a r y i n g l o a d . As a r e s u l t , t h e main i d e a o f t h i s paper is t o s t u d y an e l e c t r o n i c s w i t - c h i n g t e c h n i q u e t o c o n t r o l t h e l o a d v o l t a g e t h r o u h a c o n t i n u o u s v a r i a t i o n o f t h e PV a r r a y arrangement3r7r8.

The

The s o l a r c e l l i s an e l e c t r i c a l c e l l o f low- leve l v o l t a g e and power, t h e r e f o r e , t h e ce l l s a r e connec ted i n series and p a r r a l l e l combina t ions i n o r d e r t o form

a a r r a y o f t h e d e s i r e d v o l t a g e and power l e v e l s g . The ?V g e n e r a t o r c o n s i s t s o f s t r i n g s i n p a r a l l e l connection. Each s t r i n g is made up of modules i n series c o n n e c t i o n and each module c o n s i s t s o f series connected ce l l s .

With l o a d v a r i a t i o n s , t h e a r r a y of t h e PV gene- r a t o r can be r e a r r a n g e d by u s i n g a n e l e c t r o n i c swit- c h i n g c o n t r o l c i r c u i t t o o b t a i n t h e d e s i r e d l e v e l o f t h e l o a d v o l t a g e . The l o a d v o l t a g e i s compared w i t h respect t o a r e f e r e n c e v o l t a g e . With t h i s s w i t c h i n g e l e c t r o n i c c i r cu i t , a l l c e l l s o f t h e PV g e n e r a t o r can be f u l l y used w i t h t h e s w i t c h i n g o p e r a t i o n , and t h e s t o r a g e c a p a c i t y can be i n c r e a s e d .

There i s a n optimum v a l u e f o r t h e d e s i r e d v o l t a g e o b t a i n e d from t h e PV a r r a y which i s a f u n c t i o n o f t h e l o a d c u r r e n t , and t h e number of b o t h series and p a r a l - l e l c o n n e c t e L c e J l s , may be deduced f o r a g i v e n equa- l i t y c o n s t r a i n t . That optimum v a l u e o f t h e g e n e r a t e d v o l t a g e may be de te rmined a t each i n s t a n t a c c o r d i n g t o t h e c o r r e s p o n d i n g v a l u e s o f series and p a r a l l e l c e l l s .

As a r e s u l t , t h e sys tem i n t h i s c a s e i s more con- t r o l l a b l e due t o t h e c a p a b i l i t y o f i n s t a n t i n u o u s l o a d v o l t a g e c o n t r o l d u r i n g t h e dynamic o p e r a t i o n .

From o t h e r i n t e r e s t i n g p o i n t o f view, an o b j e c t i v e c o s t f u n c t i o n i s des igned f o r t h e s u g g e s t e d sys tem un- d e r i n v e s t i g a t i o n i n o r d e r t o minimize t h e t o t a l c o s t o f t h e d i f f e r e n t items o f t h e sys tem. A modern optimum c o n t r o l t e c h n i q u e i s used t o o p t i m i z e t h a t c o s t func- t i o n f o r r e d u c i n g t h e sys tem t o t a l v a r i a b l e c o s t of t h e o v e r a l l system.

System Concepts

A PV system i n which t h e load is d i r e c t l y coupled t o t h e s o l a r ce l l s ( w i t h o u t a Plaximum Power P o i n t Tra- cker,lvlPPT) i s a r e l a t i v e l y s i m p l e one and i s u s u a l l y r e l i a b l e . As t h e l o a d may e x i s t o v e r a l o n g e r p e r i o d of time compared w i t h t h e s o l a r r a d i a t i o n , t h e r e f o r e an energy-accumulating sys tem is r e q u i r e d . The i n t e r - a c t i o n between s o l a r c e l l a r r a y and a c c u m u l a t o r s may be c h a r a c t e r i s e d by overcharge and u n d e r d i s c h a r g e phe- nomena of t h e accumula tors . T h i s f a c t h a s a s e v e r e impact on t h e l i f e - t i n e o f t h e accumula tors .

The system under c o n s i d e r a t i o n i n t h i s s t u d y con-

1 . PV g e n e r a t o r 2 . S t o r a g e b a t t e r i e s 3 . R e s i s t i v e l o a d 4 . The s w i t c h i n g e l e c t r o n i c c o n t r o l c i r c u i t

The s t o r a g e b a t t e r i e s , b e s i d e s b e i n g e x p e n s i v e , a r e fundamental t o g u a r a n t e e t h e c o n t i n u o u s o p e r a t i o n of t h e PV system. It i s n e c e s s a r y t o have a s i m p l e e le- c t r o n i c c o n t r o l f o r c h a r g i n g and d i s c h a r g i n g o p e r a t i o n s of t h e b a t t e r i e s t o m a i n t a i n t h e b a t t e r i e s and t o work always a t a s u i t a b l e l e v e l o f l o a d v o l t a g e .

sists mainly from:

As a r e s u l t , t h e s w i t c h i n g e l e c t r o n i c c o n t r o l

CH2976-9/91/0000-0674 $1.00 0 1991 IEEE 674 I ECON ’91

c i r c u i t under c o r s i d e r a t i o n c o n t r o l s , l o a d v o l t a g e through s e r i e s - p l r a l l e l PV a r r a y - c e l l s c o n n e c t i o n s , c h a r g i n g and d i s c b a r g i n g s t o r a g e b a t t e r i e s c u r r e n t s .

::?e d e c a i l of che g y m n i c s;rie:hing e1ec :xnic c i r c u i c o f a PV sys tem is given i n F ig .? . As a gene- ra l way t o e x p l a i n t h e syscec! o p e r a t i o n , t h e f o i l o w i n g manner i s p r e s e n t e d .

A t low s o l a r i n s o l a t i o n , when t h e a r r a y c u r r e n t i s lower t h a n a s p e c i f i e d v a l u e Idn , t h e c u r r e n t t o RL is s u p p l i e d , by e i t h e r t h e PV g e n e r a t o r o r t h e b a t - teries, o r bo th . To p r e v e n t t h e b a t t e r i e s from over- c h a r g i n g a t V B ~ ~ ~ , o r under -d ischarg ing a t V E ~ ~ ~ , t h e b a t t e r i e s a r e d i s c o n n e c t e d from t h e PV g e n e r a t o r and from t h e l o a d ( R L ) , r e s p e c t i v e l y . The l o a d r e s i s t a n c e RL remains connec ted t o t h e PV g e n e r a t o r .

The system o p e r a t i o n f o r a f u l l c y c l e o f opera- t i o n w i l l now be d e s c r i b e d , a s i l l u s t r a t e d i n F i g . 3 . It is assumed t h a t t h e o p e r a t i o n is f o r d a i l y c y c l e , when t h e b a t t e r i e s are f u l l y charged a t day time t o t h e v o l t a g e o f V B ~ ~ ~ , and d i s c h a r g e d t o a v o l t a g e o f V B ~ ~ ~ ~ . During t h e day , t h e power t o R L is s u p p l i e d by e i t h e r , t h e PV g e n e r a t o r o r t h e b a t t e r i e s , o r bo th . A t n i g h t time, t h e l o a d RL is s u p p l i e d by t h e b a t t e r i e s o n l y - l q i a t J), t h e r e f o r e , t h e v o l t a g e o f t h e b a t t e r i e s d e c r e a s e s t o Vp 7 V B ~ ~ ~ ~ ( p o i n t L), where t h e modules o f t h e PV a r r a y are a r r a n g e d t o r e g u l a t e t h e o u t p u t v o l t a g e ( v b u s ) t o h i g h e r l e v e l ( f rom t h e n e x t morning) . By t h e n e x t morning, t h e v o l t a g e h a s d e c r e a s e d t o r e a c h to p o i n t .A . From t h a t time (as t h e sun rises) t h e l o a d i s s u p p l i e d by both t h e PV g e n e r a t o r and t h e b a t - teries. The b a t t e r i e s c o n t i n u e t o d i s c h a r g e b u t a t a s l o w e r rate. With t h e i n c r e a s e i n i n s o l a t i o n , t h e

block I n s d l o d a

acr anoemen' o f s e r l e s L p a r a l l e l

module8

s a n e r a t o r 7

1, c w r e n

I-, min

' b i n "Bmean "Bmax F i g . 3 . I - V c h a r a c t e r i s t i c o f a PV g e n e r a t o r ,

l o a d r e s i s t a n c e ( R L ) and s t o r a g e b a t t e r i e s i n F i g . 2 .

o p e r a t i n g p o i n t s move a l o n g t h e AB l i n e u n t i l p o i n t B r e a c h e s V B ~ ~ ~ . The b a t t e r i e s are d i s c o n n e c t e d from t h e load t o avoid- t h e d i s c h a r g i n g a t V B ~ ~ ~ . Then, t h e l o a d is s u p p l i e d from t h e PV g e n e r a t o r o n l y ( t h e t r a j e c t o r y B C D ) . A t p o i n t D, t h e PV g e n e r a t o r s u p p l i e s t h e l o a d R L and s i m u l t a n e o u s l y c h a r g e s t h e b a t t e r i e s . While a t p o i n t E ( V B s V ~ ~ ~ ~ ~ ) , t h e modules o f t h e PV a r r a y are r e a r r a n g e d t o r e g u l a t e t h e o u t p u t v o l t a g e ( v b u s ) to lower l e v e l . ' d i t h t h e i n c r e a s e o f i n s o l a t i o n , p o i n t F is reached . The b a t t e r i e s are d i s c o n n e c t e d from t h e PV g e n e r a t o r , when VB 5 V B ~ ~ ~ as shown i n F i g . 2 . S i n c e

"bus

Fig.2. The dynamic switching electronic circuit o f a PU s y s t e m

I ECON '91 675

R L resla1r.s connec ted t o t h e PV g e n e r a t o r , t h e o p e r a t i n g p o i n t i s now a; p o i n t G on t h e RL l i n e . As t h e i n s o l a - t i o n v a r i e s - i c c r e a s i n g o r d e c r e a s i n g - t h e o p e r a t i n g p o i n t s move a l o n g t h e i i ~ l i n e ( p o i n t s G , E a n a I ) . A t 2 o i n t I , Vz 7 'damax, t t e b a t t e r i e s are reconnec ted t o t h e a r r a y , w h i l e t h e l o a d is s u p p l i e d by bo th t h e ?V ger!er=tor a ~ c be b a t t e r i e s a g a i n . The b a t t e r ' i e s s t a r : :s d i s c t r - s e sizk ELY i z c r e a s i n i ; raTr2, w:il zcir.: is r o ~ c c e 2 , x z r e :te C S ~ K :i=e s:~r:s. -2s :rzjec::ry

c y c l e o p e r a t i o n o f t h e system. The stares of t5e sys- L e 3 a t d i f f e r e n t o p e r a t i n g p o i z t s i n F ig .3 d L r i r 5 d a i l y c y c l e is g iven i n T a b l e 1 .

.SC>;TGFz'-I,L, ; x s : d e s c r i k e e , is a c e 2sss i c i l i : j - 35 ::e

! J a t h e r a t i c a l F o r n u l a t i o n o f t h e System

The I - V e q u a t i o n o f a s i n g l e c e l l i s g iven i n most o f p u b l i c a t i o n s byio:

V = -ZRs+l/> . I n ( 1 + - I p h - ' ~ ( 1 ) I O

where,

Vc : c e l l v o l t a g e

Iph : t h e p h o t o c u r r e n t ( a m p s ) , p r o p o r t i o n a l t o t h e

Io Rs : c e l l series r e s i s t a n c e

1 : e q u a l s q/AKT q : e l e c t r o n c h a r g e A : comple t ion f a c t o r K : Boltzman c o n s t a n t T : a b s o l u t e t empera tu re

i n s o l a t i o n : t h e r e v e r s e s a t u r a t i o n c u r r e n t

The I - V e q u a t i o n o f a PV g e n e r a t o r which c o n s i s t s

A < U P -

/ ' % .," I Off I Off I

I I I

on o f f

G&H / > ' e - I On I O f f

L -- O f f o f f ' 0 .I"

of N, c e l l s i n serres and N p c e l l s i n p a r a l l e l i s g i v e n by :

wrern -,

?ower and Cos t Op t imiza t ion A n a l y s i s

K i t h i n c r e a s i n g c o s t s o f c o n v e n t i o n a l f u e l s and e v e r i n c r e a s i n g g e o p o l i t i c a l c o n t r o l o f t h e ene rgy r e s o u r c e s , i t i s f a s t becoming obv ious t h a t one c a n n o t c o n t i n u e t o n e g l e c t t h e development o f t h e abundant and non-dep le t ing ene rgy of t h e sun i n i ts v a r i o u s manife- s t a t i o n s . From a n economical p o i n t o f view t h e t o t a l power g e n e r a t e d from t h e p h o t o v o l t a i c a r r a y a t any o p e r a t i n g p o i n t and s p e c i f i c l o a d demand shou ld be maximum t o e n s u r i n g minimum o v e r a l l cos t o f t h e system. Using t h e v o l t a g e r e l a t i o n g iven i n e q u a t i o n ( 2 1 , t h e power d e l i v e r e d from t h e PV s o u r c e o f ene rgy may be deno ted by:

N - I - - I P Ph L )

Np I o (3)

To maximize P a n o b j e c t i v e f u n c t i o n must be i n t r o d u c e d i n t h e form:

relay notes mode of the switching mode for batteries (3) the PU modules

t o i n c r e a s e o u t p u t w o l t a a e l o a d Is s u p p l i e d from 1 b o t h

d i s c h a r g i n g

I

~ uO,, o f f t o ~ n c r e a s e o u t p u t v o l t a p e PU g e n e r a t o r ( L b a t t o r i o s

l e v e l

e n d o f

d i s c h a r g i n g

o f OPeTbtlOn < off t o ~ n c r e a s e o u t p u t w o l t a o e PU g e n e r a t o r is t h e o n l y

l 0 " D l SUPPlY

s t a r t o f c h a r g ~ n a i l :::I 1 i-f 1 t o I n c r e a s e o u t p u t v o l t a g e 1 load a battOri.s a r ~

e n d of c h a r g i n g > Ug,_

l e v e l s u p p l i e d f r o m t h e

PU g e n e r a t o r .

R, l l m l t r t h e c h a r o r n o

t o d e c r e a s e o u t p u t v o l t a g e

l a v a 1 c h a r a lng

c u r r e n t to d o c r c r r c o u t p u t v o l t a g e

l O W S l

t o d e c r e a s e o u t p u t v o l t a g e P U g e n e r a t o r 1s t h e o n l y

l e v e l I SUPPlY

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

t o i n c r e a s e o u t p u t v o l t a g e b a t t e r l o s a r e the o n l y

d i s c h a r p r n g SUPPlS

Table 1. Daily cycle of system states at different operating points in Fig.3

676 I ECON '91

t h i s o b j e c t i v e f u n c t i o n i s desiqLed such k a t aL ezcn p o s s i b l e s o l u t i o r . an e q u a l i t y c s n s t r a i n t ir. t h e f o l - lowing form is i x u r e d :

Ns . N = N t ( 5 1 P

where

N t : t o t a l number o f series and p a r a l l e l c e l l s

A : L a g r a n g e ' s m u l t i p l i e r

A t a n extreme c o n d i t i o n a t which t h e power g e n e r a t e d w i l l be maximumtwe have t h e f o l l o w i n g t h r e e e q u a l i t y r e q u i r e m e n t s f o r o p t i m i z a t i o n purpose:

2 N I -1L Rs I L - + - l n ( 1 + ph - I L ) + p N p I o ( 6 )

N p .'A Np I o "

N I I h - N I h I o + I I o + ) . t N = o ( 7 ) P o p

N i 1:

Ns )Ip - N t 0 ( 8 )

* * * t h e r e q u i r e d c o o r d i n a t e s ( N p , N s , & ) c o r r e s p o n d i n g t o t h e d e s i r e d optimum o p e r a t i n g p o i n t may be deduced from t h e above set of e q u a t i o n s (6) - ( 8 ) by u s i n g G a u s s e l i m i n a t i o n method. T h i s method depends on t h e e l i p i n a t i o n o f N p , NS,& i n s u c c e s s i v e two c o n t i n u o u s procGssesunt i loneequat ion w i t h one c o o r d i n a t e i s o b t a i n e d j t h a t can be s o l v e d r e a d i l y f o r t h a t c o o r d i - n a t e . The remain ing c o o r d i n a t e s can be o b t a i n e d by back s u b s t i t u t i o n i n t h e i n t e r m e d i a t e r e l a t i o n s . Sone- times t h i s method is n o t a p p l i c a b l e s i n c e one o r more c o o r d i n a t e s cannot a lways be e l i m i n a t e d from t h e non- l i n e a r s e t o f e q u a t i o n s . I n t h i s case an i t e r a t i v e method i s i n i t i a t e d by s e l e c t i n g a n approximate s o l u - t i o n and t h e p r o c e s s i s cont inued u n t i l a l l changes i n t h e c o o r d i n a t e s N p , NS,& i n s u c c e e d i n g i t e r a t i o n s are w i t h i n a s p e c i f i c t o l e r e n c e . I f more a c c u r a t e s o l u - t i o n is r e q u i r e d e s p e c i a l l y when t h e c o o r d i n a t e s r e p - r e s e n t p r a c t i c a l items a s i n o u r c a s e , t h e Newton- Raphson method must be used . That l a s t ne thod depends on t h e c o n s t r u c t i o n of both t h e J a c o b i a n m a t r i x and t h e change v e c t o r . The p r i n c i p a l p r o c e s s i s r e p e a t e d u n t i l two s u c c e s s i v e v a l u e s f o r each c o o r d i n a t e d i f f e r o n l y by a s p e c i f i c t o l e r e n c e .

From o t h e r i n t e r e s t i n g p o i n t o f view a n a l g e b r a i c r e l a t i o n f o r N p r Ns,& may be o b t a i n e d i n t h i r d o r d e r form a s :

T h i s r e l a t i o n may be s b l v e d f o r N p a t d e f i n i t e v a l u e s f o r t h e s y s t e a p a r a m e t e r s by u s i n g t h e well known me- thod F e r r a r i s 'o r Cardan ne thod f a r n o n l i n e a r e q u a t i o n s of h i g h e r o r d e r w i t h t h e a i d of D i c a r d ' s r u l e of signs.

From o t h e r ecor.ozica1 p o i n t of view when t h e t o t a l c z s f 3f t h e c o n s i d e r e d ?V s y s t e - 3 u s t be minimum as 3ossi:le a s u i t z b l e C n k u m - c o s t a r r a l y s i s !nay be taker. -..I,* 3 ..-. ac3Lr.t. Ir. rRis .=,er --- t h e :re&-even a a l y s i s is 3reser . ted f o r p r e s e n t h g c o s t s ~ ? a p r o f i t s i n a form desi,r.ed t o a i d i n t e r y e t a t i o n a n d a n a l y s i s of p h o t o v c L t a i c sys tem and a l l t h e a u x i l i e r i e s subsystems. I n t h e proposed t e c h n i q u e of c o s t - a n a l y s i s t h e mathemat ica l r e l a t i o n s h i p s t a k e a very s i m p l e form i f t h e f o l l o w i n g a s s m p t i o n s a r e made f o r t h e purpose o f a n a l y s i s " .

1 . The v a r i a b l e c o s t sf t h e i subsystem V i is cons- t a n t , hence N i Vi is l i n e a r l y dependent on power produced ( N I i s t h e t o t a l u n i t s o f t h e i sub- s y s t e m ) .

2. Fixed c o s t s are independent o f power produced.

3. There are no f i n a n c i a l c o s t s .

4 . A l l u n i t s a r e s o l d a t t h e same p r i c e p e r u n i t .

Using t h e above assumpt ions we have t h e f o l l o w i n g c o s t - a n a l y s i s r e l a t i o n s :

t h e t o t a l c o s t J i s g i v e n by: n

i= 1 J = 1 C f i + Ni v i )

and t h e g r o s s p r o f i t Zg i s

n

Zg = Ni Si - J

i= 1

n

and t h e n e t g r o s s p r o f i t Z n e t is

Znet = ( 1 - t ) zg

( 1 1 )

( 1 2 )

where:

C f i : t h e f i x e d c c s t of t h e i

Si t : decimal t a x r a t e

subsys tem

: t h e n e t power produced p e r u n i t

I n t k i s c a s e t h e C o o r d i n a t e s o f t h e break-even p o i n t a r e g i v e n f r o n t h e r e l a t i o n :

n n

2 N i (Si - .Ii) = f i i = l i= 1

( 1 3 )

Determining t h e coordLnates o f t h e break-even p o i n t of t h e p h o t o v o l t a i c system under c o n s i d e r a t i o n is a r e s u l t o f g r e a t i m p o r t a n t . c r i t i c a l index between two considerableboundariesof n e g a t i v e and p o s i t i v e p r o f i t s . The a r e a o f + ve p r o f i t is t t e r e q u i r e d domain of d e s i g n i n which t h e power g e n e r a t e d from t h e p h o t o v o l t a i c a r r a y - b a t t e r y subsys- tems is c o r r e s p o n d i n g t o p o s i t i v e p r o f i t .

Such t h a t t h i s p o i n t d e t e r m i n e s t?.e

I ECON '91 677

C o n d u s i o n s

A low c o s t on l i n e c o n t r o l by a s i m p l e e l e c t r o r . i c c i r c u i t u s i n g a s w i t c h i n g techf i ique f o r b o t t t h e PV a r r a y and t h e s t o r a g e b a t t e r i e s is p r e s e n t e d i n t h i s paper . T h i s c i r c u i t a l l o w s t o c o n t r o l t h e b a d vol- t a g e , t h e c h a r s i n g and C i s c h a r g i n g c u r r e n t s -3 t h e s t o r a g e S a c t e r i e s . The c i r c u i t car. a l s o r e - r a r g e z t e PlJ ~ ~ r a y ce l l s -<aria+'- ..-uns.

The series and p a r a l l e l ce l l s a r e d e t e r n i n e d i n The c o o r d i n a t e s t h i s paper a t each o p e r a t i n g p o i n t .

of t h e p h o t o v o l t a i c a r r a y )Ip and Ns a r e o b t a i n e d s u c h t h a t t h e t o t a l power i s g e n e r a t e d a t extrem,o c o n d i t i o r . . From o t h e r i n t e r e s t i n g poifi t of view, t h e break-ever, p o i n t o f t h e o v e r a l l sys tem is dsduced as an index between t h e two b o u n d a r i e s of power g e n e r a t e d .

R e f e r e n c e s

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[ 4 ] M.S. Abd El-Motaleb

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678 I ECON '91