Solar cell array parameters

~d Sel~k ELSEVIER

Solar Energy Materials and Solar Cells 45 (1997) 377-384

Solar cell array parameters Meena Aggarwal *, A. Kapoor, K.N. Tripathi Department of Electronic Science, Universi~ of Delhi, South Campus Benito Juarez Road, New Delhi 110021, India

Received 21 March 1996; revised 31 August 1996

Abstract A generalized method of combining non identical parameters like series and shunt resistance dependent photocurrents under open circuit conditions (lphoa) and the loaded conditions (Ipha) have been deduced. It is found that in the case of series array, only Ipha is dependent upon shunt resistance, while in the case of parallel array both are dependent upon shunt resistance. Keywords: Solar cell; Array parameters; Shunt resistance

I. Introduction The solar cells in an array are never identical, which complicates the operation of an array under different conditions. So efforts have been made to combine the cell parameters of the array into a single aggregate model. In the previous work, either the array parameters were not derived by completely analytical method [ 1 - 3 ] or only the effect of variation of the parameters of the solar cell array was obtained by simulation [4]. Other works deal only with the series resistance a n d / o r array photocurrents [5-8], but no effort was made to calculate the shunt resistance (which is a parasitic power consuming parameter and can not be ignored, particularly for the polycrystalline PN junction solar cell) and its effect on other parameters.

* Corresponding author. 0927-0248/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PH S0927-0248(96)00085-2

M. Aggarwal et al. / Solar Energy MateriaLs and So&r Cells 45 (1997) 377 384

378

2. Array parameters A real solar cell 1-V characteristics in a single exponential model is described by the following equation. / V

(lp,,

[31-V/R~h+I. I,

log

-IR,+

i

"

(I)

where /3 = 1 + ( R J R ~ h ) . 1 and V are the cell terminal current and voltage respectively, 1o is the reverse saturation current, lph is the light generated photocurrent and R and R~h are the series and shunt resistances respectively. A = q / ( m k T ) , where k T / q is the thermal voltage, m is the diode quality factor included to model the effects of absorption and recombination of electron-hole pairs in the space charge region of the solar cell. We can write the above equation for the nth cell as l,, = lph,, - l.,,(exp(A,,( V,, + I,,R,,) -- 1))

v,, +L&,,

(2)

R ~h n where the subscript "'n'" denotes the nth cell. We can assume that lph >> I,,. When V = V,,c,,, I = 0, so the open circuit voltage is given by

v ..... = Z,, l ° g l

(3)

lph,,R~,,,, + Z,, l°g E ,

and Eq. (2) becomes

't

V,, = - I , , R ,, + ~ , l o g

+ V~..... _ 1 log I a,, ~

1 Iph,,

Iph,, R~h,,

. lph,, R~h,, (4)

Assuming that the I - V equation for the array is of the same form as that for single cell, we can write

,(

V, = - 1.,R~ + - ~ log 1 •

'

lph., ,

/pha R~h,,

i

+ V°~" -- -7--log 1 Aa

/pha g s h a

(5)

where the array open circuit voltage is given by

~:,~,= ~ l o g [ l •

Aa

\

]pht,a R~h,,

+

log

~6)

The array contains two photocurrents, lp~.... and It,h,,. Former is the array photocurrent at open circuit voltage and is current independent. Latter is the array photocurrent at load and is current dependent.

M. Aggarwalet al./ SolarEnergyMaterials and SolarCells45 (1997)377-384

379

3. Series array For the cells connected in series N

V~(Ia) = Y'~ Vn(Ia),

Ia = In = 1.

(7)

n=]

Let 1

N

1

For the identical cells, Eq. (8) becomes Aa/N = A J N . Denoting 6~ (8) now becomes V

=

An/NA a, Eq.

l

6--~ = N.

(9)

n=l

From Eqs. (3)-(6), (8) and (9), we get following parameters for the series array: N Rsa =

ERsn,

(10)

n=l N

E Rsh.,

Rsha =

(ll)

n=l

loa =

=

( I o . ) 1/~"

Iphoa ----

(/phn)

(12)

,

"

,

(131

n=l

va + IaRsa Rsha lpha =

(( 1 1-FI rt=l

I/N"

(14)

la V" + IaRsn ) 1/~°) iphn iphnRshn

If we assume the shunt resistance to be infinite, Eq. (14) reduces to

Ia Ipha =

[a 1/8, I/N"

(15)

Eqs. (9)-(12) and (15) are in conformity with earlier result of Shechter et al. [6]. Thus if the shunt resistance is taken into account, the array photocurrent becomes quite complicated and a function of both the array current and the array voltage.

380

M. Aggarwal et al. / Solar Energy Materials and Solar Cells 45 (1997) 377-384

4. P a r a l l e l

array

For the cells connected in parallel N

v,=v,,=v, /,(v)= ~l,,(v),

(16)

tl-- [

1

I

I~= ~

(17)

( ( 1

~-log

t v)

lpha -- 3~ 1, - V/R~h ~ ~ , -

(18)

Using Eqs. (16)-(18), we get 1

R~.

(19)

E,u tl/R~' ,,

(20) I

- -

Aa R ~.,

log Ipho~

~ N

1 --log

loa = E

-- A,, R,,, log

Iph,, -- /3,1,, -- - -

v ),

(21)

R~h,,

1

A,, R,,-----~, log 1,,,,.

(22)

Denote gs,

(23)

Substituting Eq. (23) into Eq. (19), we get - - = N.

(24)

n = I /:n

Substituting Eq. (23) into Eq. (22), we get I

A~

,v

I

log Io~ = ,~,__ A,, p,,-~-Nlog/.,,.

(25)

For a parallel array of identical cells, A~ = A,,. Preserving this condition for nonidentical cells, we write ] N 1 --

A~

= ~

--N.

,,---'lA,,u,,

(26)

Denote /z,, =

A,, ~ Aa

(27)

M. Aggarwalet al. / SolarEnergyMaterialsand Solar Cells45 (1997)377-384

381

Substituting Eq. (27) into Eq. (26), we get N 1 ~--=N.

(28)

n = l ~'Ln

Substituting Eq. (27) into Eq. (22), we get 1o; -

=

(29)

(/on) '/""

and Eq. (27) into Eq. (21) gives us ~hoa 1 - - ¢ a l a

(30)

n=' l p h n - - f n l n - - - Rshn

~hoa

Under open circuited conditions, I, = 0, V = Voca,

I/N

a + ~VOC

(31)

Rshn ' where the asterisk denotes an intermediate value. For identical cells, we obtain Io~ = Ion,

(32)

Ip~o~= Iph,,

(33)

g~h n = Rshn. (34) However, Eqs. (32) and (33) are N times smaller than the expected values, obtained by identical cells, while Eq. (34) is N times greater than the expected value. Multiplying the first two equations by N and dividing the third by N does not effect the open circuit voltage Eq. (6). Therefore, we can write Iphoa= Nlphoa, Ioa = Nloa,

Rsha=

Rsh~

(35)

N

Array current is obtained by substituting Eq. (31) into Eq. (28) Iph. = ( t~a/a(v) -- V/R~ha) -l

((n~=l(lphn--~nln(Voca) -- Woca//eshn)1/~n)

V°ca

Rsha (36)

When

Rsh tends to infinity I.(V) lpha

I/N ' 1-

fl

Iphn ~nln(Voca)

(37)

M. Aggarwal et al. / Solar Energy Materials and Solar Cells 45 (1997) 377-384

382

which is the same as derived earlier by Shechter et al. [6]. So, /pha f o r parallel array depends upon the array voltage (or current). Since 1,, ( V ) and 1,, (V,,~) must be determined for each cell in the a r r a y /phoa a n d /pha becomes difficult to calculate.

5. Example The various parameters of a series array., made up of two nonidentical solar cells were calculated considering the same data of Shechter et al. [6]. Parameter

Cell 1

Cell 2

1,,(At )t ( I / v ) R~ (~]) /ph(l~.)(A)

3.3558×10 26.874 0.01 2 1 . 4 × 10 ,t

~)

Array Measured 1 1 . 0 2 3 x 1 0 '~ 6 . 1 7 0 × 1 0 9 25.428 13.110 0.01 0.0193 13.2X 10 " lf~h,,~, = 1 5 . 9 × 10 ~

Calculated 6.263×10-9 13.07 0.02 1 6 . 8 0 7 × 10 ~)

6. Result and discussions The array photocurrent lpha a s a function of array current 1, tor values of array voltage V,, and array shunt resistance R~h. for an nonidentical series solar cell array is shown in Figs. 1-3. It is found that the l~,h,, goes on decreasing with the increase in 1~, and V~. It is found that for R~h~ varying from 103 t o 1 0 4 ~ /pha shows a decline of

16.0 15.5 I

15.0

E

14.5

c,-.. ~ . . . . . . . .

.o

O JE

.....

"0 . . . . . . . . . . . . . .

0..

~-~

.

".,

13,5 \

13.0 12.5 0

I 4

I 8

\

\

\

\

\

12

I a (mAt Fig. I. Variation of array photocurrent /ph~ with array current 1~ for array shunt resistance R~h~ = 10 3 II and (at array voltage Va = 0.2 volts, (b) V~ = 0.5 volts. (c) E~ = 0.8 volts, (d) V~ = 1.0volts.

M. Aggarwal et al. // Solar Energy Materials and Solar Cells 45 (1997) 377-384

383

16.8

16.4

l

16,0

~

=o

-.a____~ ....... .~...~ ~ . " ~ _

o-

E

_

_..~, _ - ------.-- ~ , ~ . ~ . . . . . .

~- ~"

15.6

14.6

0

""'.2;-.

I

I

I

I

I

2

4

6

8

10

"~ 12

14

[cl (BA} - - ~ Fig. 2. Variation of array photocurrent lpha with array current 1a for (a) Rsha =oc, (b) Rsha = 104~'~, 'g~ = 0.2volts, (C) R~na = 104 ~ , Va = 0.5 volts, (d) Rsh~ = 10 + f~, V~= 0.8volts, (e) Rsha, V~= 1.0volts. 3 . 4 4 % ( f o r Va = 0 . 2 v o l t s ) o r 1 3 . 6 0 7 6 % ( f o r Va = 1.0 v o l t s ) to 0 . 9 8 4 % ( f o r Va = 0 . 2 v o l t s ) or 2.2755%

( f o r Va = 1.0 v o l t s ) . A s t h e v a l u e o f Rsh a g o e s o n i n c r e a s i n g , t h e v a l u e o f

Ipha b e c o m e s 0.0615%

less dependent

( f o r Va = 0 . 2 v o l t s )

i d e a l c a s e , t h a t is w h e n

u p o n Va. F o r R~ha = 105 f~, t h e v a l u e o f lph a v a r i e s o n l y to 0 . 4 7 1 5

( f o r Va = 1 . 0 v o l t s )

Rsh a b e c o m e s

infinite.

I 4

I 6

a n d is m u c h

c l o s e r to t h e

16.5

16.G

t ,,¢[ E

15.5

"-6 J::

O.

15.0

14.5

I 2

I 8 lo(mA)

I 10

I 12

,-

Fig. 3. Variation of array photocurrent lpha with array current t~ for Rsha = .Q.. Solid curve: Va = 0.2 volts, dashed-dotted curve: V~= 0.5 volts, dotted: Va = 0.8 volts, dashed: V~= 1.0 volts.

384

M. Aggarwal et al. / Solar Energy Materials and Solar Cells 45 (1997) 377-384

7. Conclusion

The incorporation of array shunt resistance (Rsh a) results in dependence of array photocurrent (Ipha) on voltage (V,,), whereas in ideal situation Ipha is independent of V,a. For values of R~h~ > l0 5 II. it is observed that lr,h. is almost independent of V~,, whereas for all values of R~h,~ < 10 + II, the dependency of Iph, on V~ goes on increasing with deceasing value of R~h~.

References [1] R.M. Turfler, T.J. Lambarski, K.E. Bradwell and C.B. Raggers, Technique lbr aggregating cells in series and parallel+ Proc. 14th IEEE Photovolatic Specialist Conf.. San Diego, CA (1980) 528. [2] L.J. Goldstine and G.R. Case. PVSS - A photovoltaic system simulation program user manual, Sandla Laboratories+ SAND 77-0814, 1977. [3] T.J. Lambarski, F. Malmberg, E. Melick+ R.M. Turfler and G. Semmens, Photovoltic Transient Analysis Program User Guide, BMD corporation, 1978. [4] T.J. Watkine and E.L. Burgess, The effect of solar cell parameter variation on array power output, Proc. 13th IEEE Photovoltaic Specialist Conf.+ Washington DC (1978) 1061. [5] R. Lari, An experimental study of series combination of Solar Cells+ Proc. 13th Photovoltaic Specialist Conf., Washington DC (1978) 1080. [6] M. Shechter, J. Applebaum and G. Yekutieli+ Parameters of solar cell array, IEEE Trans. Electron Devices 30 (1983) 616. [7] A. Kapoor, V.K. Sharma and K.N. Tripathi, General expression tor the fill factor of a real homojunction solar cell using a single exponential model, Phys. Star Sol.(a) 136 (1993) 261. [8] S. Sokolic, D. Krizaj and S. Amon, Lumped series resistance of solar cell as a result of distributed sheet resistance, Solid State Electronics 36 (1993) 623.