Solar Cells – Silicon Wafer-Based Technologies
216

Irradiance
(W/m
2
)
Temperature
(°C)
I
ph
(A) I
0
(A) a Rs(Ω) Rp(Ω)
46 1006.70 33.05 2.8445 7.7494×10
-7
1.2922 1.3580 132.0408
47 1014.20 33.20 2.8548 9.1903×10
-7
1.3001 1.3490 162.1626
48 1014.90 33.95 2.8757 7.7393×10
-7
1.2816 1.3650 135.7198
49 599.50 44.10 2.0093 3.2553×10
-6
1.3443 1.3202 193.5047
50 756.85 50.55 2.2630 2.0566×10
-6
1.2958 1.3584 98.4074
51 776.20 50.35 2.3183 9.4350×10

-7
1.2435 1.2222 202.9395
62 455.65 37.60 1.7262 2.9317×10
-7
1.2605 1.2122 202.2739
63 602.50 38.40 2.0061 2.2729×10
-7
1.2318 1.2910 179.7304
64 706.90 38.45 2.1841 6.2885×10
-7
1.3100 1.2227 176.9047
65 705.40 36.60 2.1762 3.4172×10
-7
1.2607 1.2898 185.8031
66 703.90 38.70 2.1727 4.4171×10
-7
1.2803 1.2778 178.6681
67 780.75 37.00 2.2865 2.7213×10
-7
1.2499 1.2911 155.5827
68 777.75 36.40 2.2661 6.4822×10
-7
1.3257 1.2351 196.1866
69 777.00 35.80 2.2597 3.5896×10
-7
1.2797 1.2661 180.5390
70 886.60 44.45 2.4968 5.9216×10
-7
1.2747 1.2546 153.3574
71 879.15 44.25 2.4217 1.7378×10

-7
1.2774 1.2940 218.7826
82 615.20 36.50 2.0152 4.5464×10
-7
1.2507 1.3113 168.6899
83 648.75 37.90 2.0960 4.6946×10
-7
1.2710 1.2501 148.6441
84 778.50 35.70 2.3769 3.8760×10
-7
1.2713 1.2666 160.5721
85 836.70 25.00 2.4144 2.1683×10
-7
1.2840 1.2112 228.6814
86 850.10 25.40 2.4656 1.6939×10
-7
1.2639 1.2282 180.5302
87 839.65 23.15 2.4409 2.2484×10
-7
1.2938 1.1793 183.7797
Evaluation the Accuracy of One-Diode and Two-Diode
Models for a Solar Panel Based Open-Air Climate Measurements
217

Irradiance
(W/m
2
)
Temperature
(°C)

-6
1.4494 0.8720 323.0246
97 876.20 35.40 2.4382 5.6696×10
-7
1.2564 1.3492 157.0273
98 873.25 36.45 2.4151 1.3653×10
-6
1.3337 1.3058 180.0039
99 453.40 34.10 1.6490 1.1006×10
-6
1.3337 1.1999 245.1651
100 617.40 38.50 2.0113 3.5727×10
-7
1.2650 1.2431 213.8478
101 620.40 37.40 2.0119 4.5098×10
-7
1.2847 1.2074 196.3093
102 453.40 37.00 1.6437 1.0425×10
-6
1.3602 1.1132 275.7352
103 678.60 14.75 1.8721 1.7176×10
-7
1.3306 1.1480 837.2890
104 718.10 13.15 2.0527 7.0015×10
-8
1.2647 1.2034 427.2372
105 615.20 33.10 2.0934 1.15866×10
-7
1.2124 1.2524 113.7532
106 589.10 33.55 1.9420 3.2678×10

Temperature
(°C)
I
ph
(A) I
01
(A) a
1
I
02
(A) a
2
Rs (Ω) Rp(Ω)
1 644.30 22.95 1.9043 3.0432×10
-8
1.1883 1.3697×10
-7
1.3197 1.2341 294.5317
2 657.70 24.00 1.9446 2.0588×10
-8
1.1240 2.0918×10
-7
1.5696 1.3141 254.2053
3 662.18 24.50 1.9536 1.3177×10
-7
1.3606 4.0695×10
-7
1.3606 1.1380 318.7178
4 665.16 25.20 1.9729 2.9981×10
-8

1.0648 7.7917×10
-7
1.6272 1.2959 306.1568
11 575.00 17.40 1.7003 2.2433×10
-8
1.1666 2.1302×10
-7
1.5016 1.2351 268.6247
12 601.00 18.10 1.7728 2.8641×10
-8
1.2150 1.8419×10
-7
1.3616 1.1607 275.4929

Solar Cells – Silicon Wafer-Based Technologies
218

Irradiance
(W/m
2
)
Temperature
(°C)
I
ph
(A) I
01
(A) a
1
I

-8
1.1918 1.8355×10
-7
1.3790 1.1295 254.8772
19 557.80 21.00 1.6153 2.9673×10
-8
1.2465 2.6445×10
-7
1.3220 1.1547 356.6941
20 548.80 22.00 1.5912 2.9889×10
-8
1.1972 1.6498×10
-7
1.3094 1.1999 289.9760
21 524.25 21.50 1.5337 2.4210×10
-8
1.1624 3.3020×10
-7
1.4010 1.1874 396.9221
22 517.50 20.65 1.4707 2.9247×10
-8
1.1909 2.4577×10
-7
1.3534 1.1742 395.9226
23 533.15 19.85 1.5767 2.9454×10
-8
1.1883 1.9557×10
-7
1.3592 1.1708 611.9569
24 946.25 40.85 2.6531 2.5017×10

-7
1.3334 1.5355×10
-6
1.3554 1.2526 350.9833
31 914.95 21.95 2.5641 2.9959×10
-8
1.2141 2.4595×10
-7
1.2967 1.3091 195.5702
32 917.95 23.85 2.5827 5.4386×10
-8
1.3181 5.0320×10
-7
1.3275 1.2698 199.2378
33 923.20 27.00 2.6221 9.0452×10
-9
1.0543 1.2920×10
-6
1.4846 1.3512 137.6304
34 1004.50 34.60 2.8311 6.8694×10
-8
1.1262 9.0237×10
-7
1.3614 1.4110 152.6305
35 1004.50 35.15 2.8410 6.4888×10
-8
1.1394 8.3690×10
-7
1.3241 1.3880 142.3507
36 994.75 34.25 2.8144 1.1388×10

-6
1.4692 3.4374×10
-6
1.4690 1.1719 217.3854
43 637.55 35.85 2.1516 9.4216×10
-8
1.3134 1.1090×10
-6
1.3134 1.3044 184.2421
44 406.40 34.10 1.5958 2.9422×10
-7
1.3599 1.3358×10
-6
1.3611 1.1440 237.6602
45 412.35 33.00 1.6175 1.2768×10
-6
1.4603 2.4944×10
-6
1.4605 1.0597 258.8879
46 1006.70 33.05 2.8442 5.6206×10
-8
1.1902 6.3607×10
-7
1.3014 1.3825 134.8388
47 1014.20 33.20 2.8506 2.0098×10
-7
1.3564 1.5479×10
-6
1.3583 1.3247 197.6830
48 1014.90 33.95 2.8735 1.1218×10

-7
1.3287 1.8279×10
-6
1.3287 1.3952 158.5665
Evaluation the Accuracy of One-Diode and Two-Diode
Models for a Solar Panel Based Open-Air Climate Measurements
219

Irradiance
(W/m
2
)
Temperature
(°C)
I
ph
(A) I
01
(A) a
1
I
02
(A) a
2
Rs (Ω) Rp(Ω)
55 392.25 45.35 1.4567 6.3763×10
-7
1.3514 2.1097×10
-6
1.3685 1.2832 449.1823

62 455.65 37.60 1.7239 3.0959×10
-8
1.1392 4.4560×10
-7
1.3674 1.2573 216.5685
63 602.50 38.40 2.0035 3.1412 ×10
-8
1.1813 4.8461×10
-7
1.3212 1.2244 180.2013
64 706.90 38.45 2.1800 5.4562×10
-8
1.3504 9.3287×10
-7
1.3503 1.2132 194.9991
65 705.40 36.60 2.1695 2.9715×10
-8
1.3401 8.3436×10
-7
1.3401 1.2273 210.6548
66 703.90 38.70 2.1708 2.9240×10
-8
1.3187 6.6092×10
-7
1.3187 1.2416 177.3929
67 780.75 37.00 2.2875 1.2985×10
-8
1.0604 1.8326×10
-6
1.6166 1.3705 151.6619

74 749.45 38.95 2.2701 8.8746×10
-8
1.3359 9.5496×10
-7
1.3522 1.1952 238.7603
75 746.45 38.70 2.2769 3.4376×10
-8
1.3022 6.5867×10
-7
1.3174 1.2048 173.1026
76 604.75 45.95 2.0155 2.1057×10
-7
1.2517 1.4359×10
-6
1.2742 1.3825 166.8131
77 987.30 48.80 2.7327 3.8961×10
-6
1.4222 5.1035×10
-6
1.4221 1.3555 150.1287
78 981.05 50.00 2.7015 2.1267×10
-6
1.3779 4.0844×10
-6
1.3779 1.4000 164.3968
79 519.00 33.70 1.7890 3.6873×10
-7
1.4160 1.7341×10
-6
1.4160 1.1666 388.5935

86 850.10 25.40 2.4609 3.0968×10
-8
1.3033 2.4821×10
-7
1.3033 1.2173 195.0512
87 839.65 23.15 2.4396 2.5111×10
-
1.1490 1.9083×10
-7
1.4551 1.2452 169.0755
88 838.16 23.05 2.4322 2.4870×10
-8
1.1509 2.0781×10
-7
1.4440 1.2393 172.6510
89 844.15 23.35 2.4427 1.2633×10
-8
1.0985 1.0025×10
-9
1.7022 1.3270 152.1720
90 781.50 20.80 2.2524 9.4105×10
-9
1.1001 1.4127×10
-6
1.7709 1.2685 219.9114
91 775.50 20.45 2.2298 1.6292×10
-8
1.1488 4.3632×10
-7
1.4821 1.1884 260.3549

(W/m
2
)
Temperature
(°C)
I
ph
(A) I
01
(A) a
1
I
02
(A) a
2
Rs (Ω) Rp(Ω)
97 876.20 35.40 2.4400 5.9112×10
-8
1.1237 6.9247×10
-7
1.3634 1.3776 148.9166
98 873.25 36.45 2.4181 5.7254×10
-8
1.1237 6.8324×10
-7
1.3591 1.3861 154.3058
99 453.40 34.10 1.6455 4.0638×10
-7
1.3908 1.5872×10
-6

1.3848 1.1947 116.7962
106 589.10 33.55 1.9408 2.9259×10
-8
1.1168 3.8318×10
-7
1.4000 1.3141 242.0028
107 649.50 37.85 2.1087 3.3872×10
-8
1.1031 7.3445×10
-7
1.4255 1.3455 152.8020
108 648.05 37.90 2.0881 9.7714×10
-7
1.4012 2.0786×10
-6
1.4040 1.1869 209.7529
109 653.95 38.15 2.0926 2.2252×10
-7
1.3486 1.6059×10
-6
1.3486 1.2456 228.6922
110 665.20 39.20 2.1417 2.5978×10
-7
1.3349 1.3822×10
-6
1.3349 1.2873 185.4866
111 947.05 42.55 2.6777 4.5174×10
-7
1.3483 1.7329×10
-6

corresponding parameters. The Fill Factor is described by Equation (7) [1].

mp mp
oc sc
VI
FF
VI

(7)
In continue dependency of the models parameters over environmental conditions is
expressed. Figures 15, 16 and 17 show appropriate sheets fitted on the distribution data (i.e.
some of one-diode model parameters) drawn by MATLAB (thin plate smoothing splint
fitting). Dependency of the model parameters could be seen from the figures. It could be
easily seen that the relation between I
ph
and irradiance is approximately increasing linear
and its dependency with temperature is also the same behavior. Other commentaries could
be expressed for other model parameters. Thin plate smoothing splint fitting could be also
carried out for two-diode model.
Evaluation the Accuracy of One-Diode and Two-Diode
Models for a Solar Panel Based Open-Air Climate Measurements
221 (a)
sc
(A) V
oc
(V) Fill Factor (FF)
Measurements
1-diode
model
2-diode
model
Measurements
1-
diode
model
2-
diode
model
Measurements
1-diode
model
2-diode
model
Measurements
1-
diode
model
2-
diode
model
33 31.8064 31.8 31.9203 2.5993 2.5929 2.5908 19.9972 19.9972 19.9972 0.6119 0.6134 0.6161
63 25.1194 25.1201 25.0915 1.9918 1.9884 1.9866 19.6316 19.6316 19.6316 0.6424 0.6427 0.6434

accuracy of the models, output characteristics of the solar panel provided from simulation
results were compared with the data provided from experimental results. The comparison
showed that the results from simulation are compatible with data form measurement for
both models and the both proposed models have the same accuracy in the measurement
range of environmental conditions approximately. Finally, it was shown that all parameters
of the both models have dependency on environmental conditions which they were
extracted by thin plates smoothing splint fitting. Extracting mathematical expression for
dependency of the each parameter of the models over environmental conditions will carry
out in our future research.

Solar Cells – Silicon Wafer-Based Technologies
226
7. Appendix
Equations (8-12) state the one-diode model nonlinear equations for a solar panel. Five
unknown parameters;
ph 0 s
I,I,n,R and
p
R should be specified.

sc s
T
IR
V
sc s
sc ph 0
p
IR
II Ie
R


xxs
T
VIR
V
xxs
xph0
p
VIR
II Ie
R


  (11)

xx xx s
T
VIR
V
xx xx s
xx ph 0
p
VIR
IIIe
R


  (12)
Therefore, the five aforementioned nonlinear equations must be solved to define the model.
Newton’s method is chosen to solve the equations which its foundation is based on















3
45
44444
12345
55555
12345
f
xx
fffff
xxxxx
fffff
xxxxx








To solve the equations, a starting point
0ph0Tsp
x[I,I,V,R,R]

must be determined and
both matrixes
R&J
are also examined at that point. Then x

is described based on the Eq.
(13) and consequently Eq. (14) states the new estimation for the root of the equations.

kk k
Jx R


 (13)
Evaluation the Accuracy of One-Diode and Two-Diode
Models for a Solar Panel Based Open-Air Climate Measurements
227

new old
xx x


Spain, June 2007

De Soto, W.; Klein, S.A. & Beckman, W.A. (2006). Improvement and validation of a model
for photovoltaic array performance,
Elsevier, Solar Energy, Vol. 80, No. 1, (June
2005), pp. 78–88, doi:10.1016/j.solener.2005.06.010
Celik, A.N.; Acikgoz, N. (2007). Modeling and experimental verification of the operating
current of mono-crystalline photovoltaic modules using four- and five-parameter
models,
Elsevier, Applied Energy, Vol. 84, No. 1, (June 2006), pp. 1–15,
doi:10.1016/j.apenergy.2006.04.007
Chenni, R.; Makhlouf, M.; Kerbache, T. & Bouzid, A. (2007). A detailed modeling method for
photovoltaic cells,
Elsevier, Energy, Vol. 32, No. 9, (Decembere 2006), pp. 1724–1730,
dio:10.1016/j.energy.2006.12.006

Gow, J.A. & Manning, C.D. (1999). Development of a Photovoltaic Array Model for Use in
Power-Electronic Simulation Studies,
IEE proceeding, Electrical Power Applications,
Vol. 146, No. 2, (September 1998), pp. 193-200, doi:10.1049/ip-epa:19990116
Merbah, M.H.; Belhamel, M.; Tobias, I. & Ruiz, J.M. (2005). Extraction and analysis of solar
cell parameters from the illuminated current-voltage curve,
Elsevier, Solar Energy
Material and Solar Cells, Vol. 87, No. 1-4, (July 2004), pp. 225-233,
doi:10.1016/j.solmat.2004.07.019
Xiao, W.; Dunford, W. & Capel, A. (2004). A novel modeling method for photovoltaic cells,
35
th
IEEE Power Electronic Specialists Conference, ISBN: 0-7803-8399-0, Germany, June
2004

through simulation and real case studies, in order to investigate the solutions which were
thought to minimize the effects of the mismatch.
2. Series/parallel mismatch in the I-V characteristic
Firstly, it will be worthy to explain the I-V mismatch in general for the solar cells, making a
classification in series and parallel mismatch. In the first case the effect of the different short-
circuit current (and maximum power point current) of each solar cell is that the total I-V
characteristic of a string of series-connected cells can be constructed summing the voltage of
each cell at the same current value, fixed by the worst element of the string. This means that
the string I-V curve is strongly limited by the short-circuit current of the bad cell, and
consequently the total output power is much less than the sum of each cell maximum
power. This phenomenon is more relevant in the case of shading than in presence of
production tolerance. It will be shown that the bad cell does not perform as an open circuit,
but like a low resistance (a few ohms or a few tens of ohms), becoming a load for the other
solar cells. In particular, it is subject to an inverse voltage and it dissipates power, then if the
power dissipation is too high, it will be possible the formation of some “hot spots”, with

Solar Cells – Silicon Wafer-Based Technologies

230
degradation and early aging of the solar cell. Furthermore, if the inverse voltage applied to a
shaded cell exceeds its breakdown value, it could be destroyed. The worst situation is with
the string in short-circuit, when all the voltage of the irradiated cells is applied to the shaded
ones. It is clear that the most dangerous case occurs if the shaded cell is only one, while the
experience shows that usually with two shaded cells the heating is still acceptable.
The solution adopted worldwide for this problem is the by-pass diode in anti-parallel
connection with a group of solar cell for each module. In this way, the output power
decreases only of the contribution of the group of bad cells and the inverse voltage is limited
by the diode.
In the case of parallel of strings, it is the voltage mismatch which becomes important. The
total I-V characteristic can be constructed summing the current of each string at the same

S
gSd
K
gd







(2)
where S() is the absolute spectral response of a silicon cell (A/W) and g() the irradiance
spectrum (W/m
2
m).
A suitable software, which calculates the global radiation spectrum on a selected tilted
plane, has been used. Apart from month, day and time, the input parameters are
meteorological and geographical data: global and diffuse irradiance on horizontal plane
(W/m
2
), ambient temperature (°C), relative humidity (%), atmospheric pressure (Pa);
latitude and longitude. Among the output parameters, it is important the global irradiance
spectrum (on the tilted plane) versus wavelength. By the spectral response of a typical
mono-crystalline silicon cell, it is possible to calculate K
S
. As an example, Figure 1 shows the
quantities S(), g
1
() and g


): 24 February
0
300
600
900
1200
1500
1800
2100
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Wavelength (m)
Solar spectrum (W/m
2

m)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Spectral response (A/W)
S()
g
1
()
g


m]

i2
(

)

i1
()

Fig. 2. Comparison of spectral current density in winter and summer.
The rated power of the PV devices is defined at Standard Test Conditions (STC),
corresponding to the solar spectrum at noon in the spring/autumn equinox, with clear sky.
This global irradiance (G
STC
= 1000 W/m
2
) is also referred as Air Mass (AM) equal to 1.5.
Then, considering the non linear diode, on the one hand, the first equivalent circuit is based
on a single exponential model for the P-N junction, in which the reverse saturation current
o
I and quality factor of junction m are the diode parameters to be determined:

1
j
c
qV
mkT
jo

jj
cc
qV qV
mkT mkT
jo o
II e I e







(4)
The model with a single exponential is used in this chapter (Fig. 3). In this one, the series
resistance
s
R accounts for the voltage drop in bulk semiconductor, electrodes and contacts,
and the shunt resistance
sh
R represents the lost current in surface paths.
Thus, five parameters are sufficient to determine the behaviour of the solar cell, namely, the
current source
p
h
I
, the saturation current
o
I , the junction quality factor m, the series
resistance

j
D
R
s
I
I
sh
R
sh
V
j
V

Fig. 3. Equivalent circuit of solar cell with one exponential.
Finally, the dependence on the solar irradiance
G(t) and on the cell temperature T
c
(t) is
explained for the ideal PV current
I
ph
and the reverse saturation current I
0
in the following
expression:


1 298
ph T c
SC STC








(6)
where I
SC|STC
is the short-circuit current evaluated at STC (T
STC
= 25°C = 298 K),

T
is the
temperature coefficient of I
ph
, E
g
is the energy gap and k is the Boltzmann constant. The cell
temperature is evaluated by considering a linear dependence on the ambient temperature T
a

Non-Idealities in the I-V
Characteristic of the PV Generators: Manufacturing Mismatch and Shading Effect

233
and the irradiance G, according to the NOCT definition valid for modules installed in
mounting structures which allow the natural air circulation (maximum wind speed equal to

consider a 35 W
p
rated power PV module of 36 solar cells in poly-crystalline silicon, with a
short circuit current of 2.4 A in STC. Figure 4 shows the I-V curves of:
a.
36 cells totally irradiated;
b.
35 cells totally irradiated;
c.
1 completely shaded cell;
d.
36 cells with 1 shaded cell.

0
0,5
1
1,5
2
2,5
3
-40 -30 -20 -10 0 10 20 30
Current (A)
Voltage (V)
I-V curves at STC
a)
b
)
c)
d)
P

c
decrease, namely
the working conditions of the PV module are less dangerous for the solar cells.

Solar Cells – Silicon Wafer-Based Technologies

234

N
c
μ P
c
[W] U
c
[V]
1 0.11 24 18
2 0.06 4.3 9.2
3 0.04 1.8 6.1
4 0.03 1 4.4
18 0 0 0
36 0 0 0
Table 1. Normalized power of the PV module μ, dissipated power P
c
and inverse voltage U
c

on shaded solar cell, under STC, depending on the number of shaded solar cells.
3. Manufacturing I-V mismatch
Considering at first the mismatch among PV modules due to production tolerance, a first
study is presented in the paper (Abete et al., 1998) in which an experimental set up has been

2
(I) –U
1
(I). In the dual circuit (“parallel type”) the
Non-Idealities in the I-V
Characteristic of the PV Generators: Manufacturing Mismatch and Shading Effect

235
PV modules are parallel connected: in case of mismatch, the voltage output measurement
of the unbalanced bridge, for each voltage value, is directly proportional to the difference
of the module’s currents I. This I vs. voltage U represents the voltage difference
characteristics I
2
(U) – I
1
(U).
Fig. 5 and Fig. 6 show the series and parallel bridge measuring circuits. Each bridge has two
active branches constituted by two modules, PV
1
(reference) and PV
2
(testing), which are
subject to the same irradiance G and cell temperature T. The other two branches of each
bridge are two equal resistors, R
s
with high resistance in Fig. 5 and R
p
with low resistance in
Fig. 6, such as to have a negligible loading effect on the I(U) characteristics of PV
1

PV
1
(ref.)
u
0
PR
u
2
= K i
2
u
1
= K i
1

C
R
s
R
s
+
+
PV
1
(ref.)
PV
2
(testing)
u
2

0
of the unbalanced
bridge measures the difference I of the two modules currents by I = U
0
(1/R
p
+2/R
0
)
with R
0
input resistance of the instrument which measures the voltage output U
0
.
Therefore, the measurement of the voltage difference U vs. the current I gives the
difference curve of the series connected modules; the measurement of the current difference
I vs. the voltage U gives the difference curve of the parallel connected modules. For
mismatch assessment, besides the difference of open circuit voltages U
oc
and of short
circuit currents I
sc
, it is profitable, in the maximum power point P
M
= (I
M
,U
M
) of the
reference module, to know the following parameters:

can be assumed as “mismatch parameters”.
The measuring signals of the circuits in Fig. 5 and Fig. 6 (K current probe constant), with a
suitable sampling rate (10-100 kSa/s), are digitized by an Automatic Data Acquisition
System (ADAS). This ADAS processes the signals for providing current-voltage curves of
the PV modules, the difference characteristics and the mismatch parameters. These
experimental results, concerning series and parallel connected polycrystalline silicon
modules, are shown respectively in Fig. 7 and Fig. 8. In Fig. 7 the testing module I(U
2
) curve
extends as far as the second quadrant, while the reference module I(U
1
) curve does not run
through all the first quadrant. This proves that the short circuit currents of the two modules
are different and consequently the testing module can operate as a load of the reference
module. In Fig. 8, likewise, the testing module I
2
(U) curve extends as far as the fourth
quadrant, while the reference module I
1
(U) curve does not run through all the first
quadrant. This proves that the open circuit voltages of the two modules are different and
thus the testing one can operate as a load. Once the power reduction are P
MI
and P
MU
are
measured, it is possible to choose the connection of the modules in the array to achieve the
optimum performance. Finally, the presented circuits can be profitably employed in
manufacturer quality control and customer acceptance testing.


2.0
-20 -15 -10 -5 0510 15 20

U
U
1
U
2
Non-Idealities in the I-V
Characteristic of the PV Generators: Manufacturing Mismatch and Shading Effect

237 Fig. 8. Experimental results with parallel connected polycrystalline silicon modules.
3.2 Manufacturing I-V mismatch and reverse currents in large Photovoltaic arrays
As an example of the consequences of the production tolerance in large PV plants, a brief
summary of a study on this topic is reported here. This work has dealt with the current-
voltage mismatch consequent to the production tolerance as a typical factor of losses in
large photovoltaic plants (Spertino & Sumaili, 2009). The results have been simulated
extracting the parameters of the equivalent circuit of the solar cell for several PV modules
from flash reports provided by the manufacturers. The corresponding I-V characteristic of
every module has been used to evaluate the behavior of different strings and the interaction
among the strings connected for composing PV arrays. Two real crystalline silicon PV
systems of 2 MW and 20 kW have been studied. The simulation results have revealed that
the impact of the I-V mismatch is negligible with the usual tolerance, and the insertion of the
blocking diodes against reverse currents can be avoided with crystalline silicon technology.
On the other hand, the experimental results have shown a remarkable power deviation (3%-
4%) with respect to the rated power, mainly due to the lack of measurement uncertainty in
the manufacturer flash reports.

0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 2 4 6 8 10 12 14 16 18 20
Voltage [V]
Current [A]
I
1
I
2

I

I
M

Solar Cells – Silicon Wafer-Based Technologies

238
the shading on all the strings. Finally, in the simulation conditions the impact of the shading
losses on yearly basis is limited to 1-3%.
4.1 Analysis of some shading patterns
In order to establish some guidelines for minimising the shading effect in multi-rows
PV arrays, a comparison among different configurations of module connections is carried
out within simplifying assumptions, i.e., all the shaded modules are located only in a single





__
Ssh Ssh
P
SS
Nonestr Nallstr
N
NN

(8)
Obviously, the previous parameter N
Ssh
(all_str) ≥ 1 only if N
P
≤ N
S
.
In our study, the chosen arrays are two, the first one with usual number of modules per
string (N
S
= 16) and low number of parallel strings (N
P
= 4) concerns a decentralized
inverter (Figures 9 and 10), whereas the second one deals with a centralized inverter
(N
S
= 16, N

N
NN


(9)
with N
Ssh
(one_str) = 16 and N
Ssh
(all_str) = 4 corresponding to the maximum number of
shaded modules per string in this example. In Figure 9 in every string, even if there are both
shaded modules (four) and totally irradiated modules (twelve), it is assumed the same
temperature for uniformity reasons and this one is equal to the temperature of the totally
irradiated modules. Consequently, the I-V curve can be calculated.
Non-Idealities in the I-V
Characteristic of the PV Generators: Manufacturing Mismatch and Shading Effect

239
On the other hand, in the second array with 6.25%- shading amount the situations are 8
shaded modules in the same string (half a string in Conf. 4 of Fig. 12) vs. one shaded module for
every string (Conf. 3 of Fig. 11), i.e., the eq. (8) becomes





__
1
2
Ssh Ssh
+ +
+
+
+ ++
++
++
1
2
15
16
+ +
+
+
+ ++
++
++
1
2
15
16Fig. 10. Array (N
S
= 16, N
P
= 4) with shading patterns - Configuration 2


14
15
16

Fig. 11. Array (N
S
= 16, N
P
= 8) with shading patterns - Configuration 3

++
1
2
3
7
8
12
13
14
15
16
1
2
37
8
++
1
2
3
7