Telechargé par Dominique Bonkoungou

Effect of Shading on Cds

Energy Conversion.
Vol. 14, pp. 61-71.
Pergamon Pres.?, 1975.
Printed in Great Britain
(ReceivedJuly 1973)
effects of shading on solar arrays in general has been analyzed using the specific Z-V characteristics of a CdS/Cu,S
solar cell. The major differences in the amount of power lost with various array configurations subjected to identical shading
conditions has been determined. The conditions under which localized heating or ‘hot spots’ occur has been treated and the criteria
for their avoidance has been precisely defined. The effects of protective diodes in reducing power loss and in preventing ‘hot spots’
has been included. Genera1 design rules to assist in obtaining optimal solar structures have been formulated and presented.
Loss of output power from an array of solar cells is
unavoidable if a portion of the array is shaded from
incident light. However, the amount of power lost
depends on the configuration used to connect the cells
[l-71. Some connection schemes with particular portions
shaded result in reverse voltage across individual cells.
These cells then absorb rather than deliver power and
severe localized heating can result [7, 81. Such ‘hot
spots’ can cause individual cel1 failure [9] and can
result in power loss from large sections of an array [7].
The addition of shunt diodes shorts out these reverse
voltages preventing ‘hot spots’ [2-71. Instead of reverse
voltages, recent studies have shown that other combinations of 41s and shading produce reverse currents
through individual cells [ 101. The resulting power
absorption is avoided by plating diodes in series with
the cells to black the reverse current. These reverse
current levels are low and no significant heating or
‘hot spot’ formation has been attributed to them.
In the references cited above, power loss studies have
been largely confined to particular cases. Few attempts
have been made to define rules for optimal arrays in
general. This previous work has dealt specifically with
silicon solar 41s intended for space applications where
considerations of weight, size, efficiency, but not tost,
are strong constraints. When tost becomes a major
concern, as in large scale terrestrial solar energy conversion, materials other than silicon become attractive
alternatives. In particular, thin film CdS/&S
heterojunction cells have the potential for reducing power
conversion costs by several orders of magnitude [l 1, 121
even though their efficiency is somewhat less than
that of silicon (5-8 per cent [l 1-131 compared to 11 per
cent [3]).
* Supported by a grant from the National Science Foundation.
t Present
Company, Palo Alto, California 94304, U.S.A.
$ Correspondence should be directed to this author. Electrical
of Delaware,
Delaware 19711, U.S.A.
5 Institute of Energy Conversion, University of Delaware,
Newark, Delaware 19722, U.S.A.
This paper investigates the effects of shading various
configurations in genera1 using CdS/Cu,S cel1 characteristics. Since the I-V properties of al1 solar 41s are
similar, implications for al1 cel1 types are obtained.
In particular, we have studied the precise shading
conditions under which reverse voltage and current
occur. Confìgurations less susceptible to ‘hot spots’
and excessive power loss have been found. We have
quantified the shading effects for the various array
contìgurations so that direct comparisons can be made
for series and parallel connections with and without
protective diodes. Finally, we have formulated genera1
design guidelines to aid in obtaining optimal solar array
2. Analysis of Shading
The effects of shading depend on the I-V characteristics of the solar cells, the type of biasing (e.g. maximum power, minimum load current fluctuation or
minimum load voltage fluctuation) and load used
(resistive, inductive, constant or variable) and the
connection scheme.
A. Measurement of cel1 characteristics
The I-V curves of a 3 x 3 in., CdS/Cu,S solar cel1
(No. 1-1: 6-5) manufactured by Clevite-Gould, were
measured under three conditions. First the cel1 was
exposed to a light intensity equivalent to the mean maximum solar value of 100 mW/cms. Next an opaque
cover was placed over one half the cel1 surface so that
only the uncovered portion was exposed to light. Finally
the I-V characteristics were measured in the dark. The
light source was a calibrated solar simulator of intensity
air mass one. Air mass one is the mean maximum
solar intensity at sea leve1 in contrast to air mass zero
which is the mean value in space. This simulator approximated the spectra1 distribution of sunlight and
was calibrated (& 2 per cent) with a standard cel1
flown from NASA’s Lewis Research Center. The resulting
reverse voltage characteristics are shown in Fig. 1.
Figure 2 gives the forward characteristics. Only in the
fourth quadrant is the I-Y product negative indicating
Flg. 1.The reverse voltage I-V clauam
of a CdS/Ca,S
solar cell ader fully shaded, half shaded and fally lllmalaated
maximum power available from a cel1 is by plotting
curves of constant power (IY = constant) on the cel1
characteristic [13]. This series of hyperbolas are shown
as the dashed lines in Fig. 2. For clarity only a few are
shown. The intersection of the cel1 curve with the highest
power hyperbola quickly gives the maximum power
(Pm) to be 220 mW at a current (Im> of 600 mA and a
voltage (Vm) of 0.36 V. The resistive load (&) that
allows this maximum power to be delivered is given by
Vm/Im and is equal to 0.6 Q. Since the characteristic
is not square, V,& is less than VoJsC. The ratio
is a measure of cel1 quality called the
fill factor F [14]. The squarer the characteristic, the
higher the fill factor. In the present case the F value is
67 per cent.
An arbitrary number N of identical cells can be
connected in series. If each cel1 is biased at its maximum
power point, a total output voltage of NV, is obtained
but the output current remains Im. Thus the load resistance is NR, and the maximum power output is
NPm. If the N cells are not identical, the composite
I-V characteristic can be obtained by adding the voltage
across each cel1 for each current value [7]
V&O =
Fig. 2. The forwardvoltage I-V clmactetics of a C!dS/Ca,S
cmditbm. Dashed liaea iadicate the locas of points of amtaat
power. The load line iadica~ the re&tive load for maximam
power generation. In the fìrst and third quadrants, the
product is positive and power is absorbed by the cell.
For full illtination,
the short circuit current (&)
was 700 mA and the open circuit voltage (Vac) was
0.47 V. These curves are the basis for al1 the calculations
that follow.
It is of interest to note that no detectable cel1 damage
resulted from forcing up to 1.10 A of third quadrant
current through the cel1 for short periods of time.
By this we mean that the fourth quadrant characteristic
remained unchanged within the measurement accuracy
(_t 2 per cent). Thi‘s is in contrast to results found with
silicon cells. Jett and Miller [7] reported that breakdown
and irreversible damage occurred to silicon devices at
shaded cell, third quadrant current levels approximately
equal to the full illumination, short circuit current
B. Bìasing for maximum power
One of the simplest methods
of determining
where V{(I) is the voltage across the ith cel1 which is a
single valued function of 1. Knowing the composite
I-V characteristic, the load resistance for maximum
power and the maximum power available can be determined, as in the case of a single cell.
Alternately M identical cells biased for maximum
power can be connected in parallel. From the duality
theorem the output voltage is V,, the load current is
MIm, the load resistance is RmIM and the maximum
available power is MP,. Also if the M cells are not
identical, the composite I-V characteristic can be
obtained by adding the current of each cel1 for each
voltage value
E la(V),
where la(v) is the current from the ith cel1 which is
single valued function of V. The maximum power
parameters again can be obtained from this composite
curve as with a single cell.
C. Efects of shding
In realistic applications of solar cells, the load could
be expected to change from time to time. Such random
variations are difficult to treat analytically. Since it is
desirable to obtain as much power as possible from the
cells, we have chosen to treat the case of solar arrays
connected to constant resistive loads equal to that
needed for maximum power output when there is no
shading. Correctly designed systems should not have
loads fluctuating signitìcantly from this value.
(1) Series conjìgurution. For N identical cells connected in series and one cel1 completely shaded, Kirch-
Fig. 3. The ratio of the power los.9to the unsbded power output
a3 8 fmction of slmdiag for the simple series conmction witb
and without ideal shunt diodes and lor the theoretical minimum
power loss case.
hoff’s voltage law requires
(N - l)(Vr - I&&) + (VS - IR??&)= 0,
where VZ and VS are the voltages across the illuminated
and shaded cells respectively. Figures 1 and 2 were used
to obtain plots of (VZ - Iz&) VSIZ and of ( Va - 18Rm)
VS Is, where Ir and Is are the current of the illuminated
and shaded cells respectively. These were used to obtain
trial and error solutions to Equation (3) so that
Ir = Is = 1 for N values between 2 and 50.
Since N = 2 represents 50 per cent shading, N = 3
represents 33.3 per cent shading, etc. these results have
been plotted in Fig. 3 (solid curve) as the ratio PL/Po VS
per cent shading. Here PL is the power lost with shading
and PO is the initial unshaded power output. The power
values were obtained from
[(N -
1) Vz-I$$]
value the reverse voltage across the shaded cel1 is
- 5.1 V. This means that the maximum power dissipated
by the shaded cel1in a series string under worst conditions
is 3.06 W.
This reverse voltage and power absorption can be
completely eliminated if an ideal diode is placed in
parallel with each series 1~11, as shown in Fig. 4(a)
[2, 31. Shunt diodes short out any reverse voltage and
shunt current around shaded cells. The positive voltage
of illuminated cells reverse bias their diodes so that
unshaded cells are unatfected. Kirchhoff’s law for this
case is the same as Equation (3) except that now V. is
zero so that
Po = NV,,&
PL = NV,&
Fig. 4. Array co~tions
for (a) the simple series amay witb
shunt diodes and @) the simple parallel army witb series diodes.
1)Vz + Vs]l.
Results for shading between 50 and 100 per cent were
determined by considering 10 cells in series and 6-9
cells shaded to give the 60-90 per cent data points.
For this, equation (3) was modified for multiple cel1
shading and solved by trial and error. Note that power
loss is essentially 100 per cent if shading is 12.5 per cent
or more because shaded cel1 current (Fig. 1) is negligible
unless a voltage greater than - 2.0 V is applied. This
required eight or more illuminated cells (12.5 per cent
or less shading).
As shading decreases to 2.8 per cent the power loss
drops to 35 per cent and the load current rises to approx. 480 mA. At this value the shaded cel1 drops
- 4.75 V so that the shaded cel1 absorbs 2.28 W instead
of delivering the 0.22 W available when it was unshaded.
This power dissipation heats the shaded cel1 and produces
the so-called ‘hot spot’ [8]. Note that the shape of the
reverse characteristic strongly affects this power loss [5].
Cells with lower reverse currents result in greater power
loss for a given current value.
As N approaches itinity (shading approaches 0 per
cent), (VS - IRm) < (N - l)( VZ - IRm) and Equation
(3) requires that 1 be equal to V./R, = Im. At this
must be satisfied. This was solved directly by plotting
the load line R = NRm/(N - 1) on the I-V characteristic (Fig. 2) and obtaining VZ and 1 from the intersection of the curves. The results are shown by the
dashed line curve in Fig. 3 as the ratio PLIPo VS percentage shading where Po is the same as Equation (4)
PL = NV,,&
- (N -
As can be seen, shunt diodes drastically reduce power
loss for low percentage shading. With 50 per cent shading
the loss is 65 per cent compared to 100 per cent without
diodes. For 10 per cent shading, PL/Po is 11 per cent
rather than 96 per cent without diodes, a factor of 8.8
reduction in lost power. As percentage shading decreases
further, this reduction factor grows smaller and eventually approaches one at zero shading. One might conclude
that diodes are unnecessary for large arrays where
per cent shading should remain smal1 but the diodes
prevent ‘hot spots’. As percentage shading decreases
the ‘hot spot’ heating grows worse eventually approaching
a constant value asymptotically (of 3.06 W per shaded
cel1 in our case) for very large arrays with very small
percentage shading.
Real diodes of course have some forward bias voltage
drop so that the actual shunt diode reduction in power
loss is not as great as shown in Fig. 3 for ideal diodes.
This real voltage drop is nevertheless much less than the
- 5.0 V of the example discussed for cells without
diodes. Comparing power absorption at 580 mA, a
diode with 0.2 V drop would dissipate 96 mW rather
than the 2.28 W occurring with the unshunted cell, a
factor of 24 reduction even with real diodes. Since
germanium diodes have approximately a factor of three
less forward voltage drop than silicon (0.2 compared
to 0.7 V typically), germanium should provide the best
shunt diode protection.
A calculation of the theoretically minimum power
loss possible when one cel1 is shaded provides a useful
standard for evaluating various connection schemes.
Power loss to the shaded cel1 is avoided by shunting it
with an ideal diode. To obtain the ideal minimization,
the other unshaded cells should be maintained at their
maximum power points. This is accomplished by reducing
the load resistance to a value of (N - l)R,. This then
gives the lowest power as, using Equation (5)
PL = NV&,
- (N -
l)VmZ, = VmIm,
so that
_ =_,1
Po N
under the best possible circumstances. This is shown as
the dotted line in Fig. 3. As percentage shading drops
from 50 to 10 per cent, the minimum power loss decreases from 50 to 10 per cent compared to the 65-11 per
cent decrease with shunted cells. This means that a
variable load could produce an additional factor of
1.3 reduction in power loss for 50 per cent shading
and a factor of 1.1 at 10 per cent shading. This ratio
of real to theoretical minimum loss becomes less than
1.01 for shading below 6 per cent. Hence a variable
load cannot reduce power loss appreciably for low
shading percentages.
(2) Parallel conjîguration. With M identical cells
connected in parallel, Section 2B gives the load resistance,
voltage and current for maximum power output with
no shading. With one cel1 shaded Kirchhoff’s law again
Notice that Equation (10) can be obtained from Equation
(3) by replacing N by M, Vl by ZI and I by V. This is
essentially the well-known duality theorem. Plots of
(11 - VZ/&) VS VZ and of (Is - V,I&J VS VS were
obtained from Figs. 1 and 2 and trial and error solutions
to Equation (10) were found so that VZ = VS = V for
M values between 2 and 50. These results are plotted
as the solid curve in Fig. 5 for shading levels between
0 and 50 per cent. The values for 50-100 per cent shading
were obtained for 10 cells in parallel by modifying
Equation (10) to apply for shading 6-9 cells.
The power loss varies from 69 to 12 per cent as percentage shading decreases from 60 to 10 per cent. This
contrasts sharply with the series case which had 100 per
cent power loss for 50 per cent shading that only decreased to 96 per cent at 10 per cent shading (Fig. 3).
Obviously, simple parallel connection results in much
less power loss with shading than simple series connection.
There is some power absorption by a shaded cel1 in
parallel with other illuminated ones. The unshaded
ones maintain a positive voltage across the shaded
cel1 (except for P configuration ‘hot spot’ formation
which is discussed later). Figure 2 shows that such
positive voltage results in some current flow in the
shaded cel1 and biases it in the power absorbing fitst
quadrant. This partially shorts out the photocurrent
generated in the illuminated cells. In contrast to the
series case, the shaded cel1 voltage and current both
remain low. With VS around 0.3 V, Is is less than 25 mA
(Fig. 2) so that less than 7.5 mW is absorbed compared
to absorption as high as 2.28 W calculated for the
series case. Resultant heating is insignificant in comparison. The forward characteristics do not strongly
affect this power loss unless the cel1 becomes a low
resistance short circuit.
As M approaches infinity, percentage shading approaches zero and the operating current and voltage
Fig. 5. The ratio of the power loss to the uushaded power output as a
function of shading for the simple parallel connection with and without
ideal series diodes and for the theoretical miaimmn power loss case.
Effect of Shding on CdS/CuaS Solar Cells and Optimd Solar Army Design
approach their values before shading (similar to N + co
for series cordìguration). For this worst condition case,
the shaded cell dissipates 12.6 mw, a factor of 17.5 less
than its unshaded power production. This contrasts
sharply with the series case where shaded power loss
for the worst case was over an order of magnitude larger
than the unshaded power production.
This smal1 loss can be eliminated by the addition of
ideal diodes in series with each cel1 and oriented so
that only forward current can flow as seen in Fig. 4(b).
(As is traditional, a forward photocurrent is plotted as
a negative value in Figs. 1 and 2.) Application of Kirchhoff’s voltage law gives the same result as Equation (10)
except that now I8 is zero. This can be rewritten as
This was solved directly by plotting the load line
R=R,(Ml)/M on the 1-V characteristic for the
various M values. As before this expression was modified
for multiple cel1 shading to obtain values for shading
between 50 and 100 per cent. The diode reduction in
power loss was so small that the results, either with or
without diodes, are shown by the same solid line in
Fig. 5. This is due to the shaded ccll’s smal1 current
value (less than 35 mA) compared to the illuminated
cel1 current (over 600 mA). The cel1 voltages for both
cases are comparable.
The power loss that occurs in real series diodes makes
their use impractical. For a forward cc11 current of
600 mA through an illuminated cel1 and diode, a forward
voltage drop of 0.2 V across the diode would drop
120 mW at each diode. This is approximately half the
power generated by the cell. Such 50 per cent power loss
with no shading at al1 is of course unacceptable.
The theoretical minimum power loss for M cells in
parallel would be obtained with ideal diodes in series
with each cel1 and a variable load that would change
from R,,,/M to R,,,/(M - 1) when one cel1 is shaded.
This would maintain maximum power output from the
remaining cclls and give the theoretical minimum power
loss ratio as
This minimum loss is shown as the dashed line in
Fig. 5.
Connecting cells in parallal is equivalent to increasing
the surface area of a single cell. Thus the results of
Fig. 5 apply to partial shading of a single cel1 as wel1
as parallel cells. In particular, Fig. 5 should correctly
describe power loss due to shading by the conducting
grid placed over the front of solar cells. Actual measurement on a CdS/Cu,S cel1 with the grid shaded area
reduced from 15 to 5 per cent showed a 20 per cent
increase in output power [15]. This agrees with the
solid curve of Fig. 5 to within 3.5 per cent.
(3) Combination series and parallel arrays. The large
decrease of power loss with shading for simple parallel
strings of cells rather than series strings make the former
Fig. 6. Large array cxmnection schemes for (a) the basically
parallel or P array witb shunt diodw and (b) the basically series
or s array witb shunt diodes.
configuration attractive. However, the voltage output
for this arrangement (approximately 0.36 V) is too
low for practica1 applications. Series connection must
be used to obtain useful voltage output levels. If enough
cells are available to provide a given voltage, the question
arises as to how to best connect in additional cclls.
Two possibilities immediately present themselves and
are shown in Fig. 6. The primarily parallel arrangement
of Fig. 6(a) is called the P coníìguration, and the more
series layout of Fig. 6(b) is termed the S cotiguration.
A series of specific examples have been taken to
analyze the characteristics of these two connection
schemes. Arrays of N cells high (2 I N I 10) and
M cells wide (2 I M 2 10) are the basis of this analysis.
The N cel1 height is connected in series to give a desired
voltage and the M cel1 width is connected in either the
P and S coníìgurations to give a desired current. The
amount of power lost when one cell was shaded was
calculated for a constant resistive load. By extending
the results of Section 2B, the load resistance for maximum
power output is NR,/M for both coníigurations with
no shading. This load value was used for each case
considered which gave an unshaded output power of
(a) P conzguration. For the P configuration,
a single cel1 lowers the current output from the parallel
section in which it is located. Since this section is in
series with the other (N - 1) non-shaded sections, the
total current is decreased. Kirchhoff’s voltage law
requires that
(N - 1)Vr + Ve - MII(NR,/M)
= 0,
where VS is the voltage across the shaded cel1 and the
(M - 1) fully illuminated cells in parallel with it. The
voltage Vz and current 11 are the voltage and current
of al1 the other illuminated cells in the remaining
sections. From Kirchhoff’s current law, the total current
(1t) through each parallel section must be equal so
1t = MIZ = (M - 1>z;+ I,,
where & is the current through the shaded cel1 and 1;
is the current through each of the illuminated cells in
parallel with the shaded one.
Trial and error solutions to Equations (13 and 14)
were found in the following manner. An arbitrary
value of It was chosen somewhat less than its unshaded
value. Next, values of Iz and 1; were determined from
Equation (14) assuming Is was zero. Corresponding
values of VZ and V8 were then found directly from
Figs. 1 and 2. This VS value and the shaded I-V curves
then specified a non-zero Is. Therefore, 1; was reduced
so that Equation (14) was satisfied (using this non-zero
Is), and a new for VS was thus specified. Using this
latter value, the total voltage Vt was found from the
sum (N- 1)Vz + Vs. If the correct values had been
found, the ratio Vt/It equaled the load resistance NR,/M.
If not, a new value of It was chosen, and the whole
process repeated. Since Is is so small, more than two
iterations were not usually required to obtain the
solution within f 2 per cent.
Figures 7 and 8 show the results of these calculations
plotted as PL/Po VS shading. Figure 7 gives the power
loss with width shading for various constant height
shading values. Figure 8 gives the loss VSheight shading
for various constant widths. Shading greater than
50 per cent was not considered. The dashed line shown
in both figures separates regions where ‘hot spots’
occur from regions where they do not form. Note that
low percentage width shading of a P conflguration
does not result in ‘hot spots’. Addition of shunt diodes
(Fig. 6a) eliminates the ‘hot spots’ at large shading by
shorting out reverse voltages. Ideal shunt diodes thus
clamp the power loss at the point where the heating
just begins and significantly reduce the loss of power
at high shading. The power loss with diodes is shown
in Figs. 9 and 10. Notice that diodes do not reduce
power loss if width shading is sufficiently small. Hence
diodes should not be required if the P array can be
made so large that a given shading area wil1 be below
those percentages indicated in Fig. 7.
Fig. 9. The power loss ratio as a function of width shading of a
P array with ideal shunt diodes for various, constant height
shadmg values.
Fig. 7. The power loss ratio as a function of width shading of a
P array without shunt diodes for various, constant height shading
values, dashed liie separates ‘hot spot’ and ‘no hot spot’ @ons.
10 %
Fig. 10. The power loss ratio as a function of height shading of a
P array with ideal shunt diodes for various, constant width
shading values.
Fig. 8. The power loss ratio as a function of height shadmg
of a P array without shunt diodes for various, constant width
shading values. Dashed lme separates ‘hot spot’ and ‘no hot
spot’ regions.
Since cells in parallel are equivalent to a single cel1
with equivalent area, the results for percent width
shading applies even for partial shading of cells. However, the results for height shading apply only for shading
multiples of complete cells. This occurs because partial
cel1 shading is equivalent to width shading even though
the shadow may extend vertically rather than horizontally
across the cell.
of shadiag oa cds/cus
solar ceb
(b) S confgurution. For the S contìguration, shading
a single cell lowers the output voltage across the series
section in which it is located. Since this section is in
parallel with the other (M - 1) non-shaded sections,
the total voltage is decreased. To satisfy Kirchhoff’s
law we must require that
(M - 1>11+ 1. - NVz(M/NRm) = 0.
Again this is the dual equation for Equation
Likewise, the dual for Equation (14) is
Vt = NVZ = (N - 1>v; + VS,
where VS and V; are the voltages across the shaded cel1
and across each of the illuminated cells in series with
the shaded one, respectively. Similar to the P contìguration, trial and error solutions were found in the following
manner. An arbitrary value of the total voltage (Vt) was
chosen less than its value before shading. Thus VZ = Vt/N
for (il4 - 1) the illuminated sections. Using Fig. 2, the
corresponding value of IZ was then determined. Since
the series sections are in parallel, the total voltage in
the shaded section (N - 1)V; + V8, must also equal
Vt. Specifying & determmes V8 and V; from Figs. 1
and 2. Hence various values of le were tried until a value
was found which gave corresponding V, and V; amplitudes correctly adding to Vt. This Is determination then
and optimal solar Array Design
allowed the total current 18 to be obtained from the
sum (N- l)lz + Is. If the correct solution had been
found, the ratio V& equaled the load resistance NR,JM.
If not, a new value of Vt was chosen and the process
repeated. In many cases Is was zero. This greatly simplified the calculations so that few iterations were usually
The results are shown in Figs. 11 and 12. The dashed
line in these graphs again separates regions where ‘hot
spots’ occur from regions where they do not. As with
P arrays, ‘hot spots’ only occur for the larger width
shading percentages. However, with height shading, the
situation is exactly reversed. Smal1 shading causes cel1
heating that only disappears when enough of series
section is shaded to reduce its current to zero. Since no
single cells are connected in parallel, these S array
calculations apply only for shading multiples of complete
cells (no partial cel1 shading) for width as wel1 as height.
Hence S arrays are very susceptible to ‘hot spot’ problems if N (the height) is large enough that shading a
complete single cel1 is less than 12.5 per cent (i of the
20 % EIGHT
13. The power loss ratio as a fanction of width shadiag of
an S array with ideal shant diodes for various constant height
shading values.
ll. The power los3 ratio as 8 tbaction of width shading
of au S array without shunt diodes for various constant height
shadiug values. Dashed line separates ‘hot spot’ and ‘IIOhot
spot’ regions.
Fig. 14. The power loss ratio as a fanction of height shading of
an S array with ideal shrmt diodes lor various constant width
shading values.
Fig. 12. The power loss ratio as 8 t%nction of height shading
of an S array without shunt diodes for various constaat width
shadiag values. Dashed Iiae separates ‘hot spot’ and ‘DOhot
Of course ideal shunt diodes (Fig. 6b) prevent ‘hot
spots’ and significantly reduce power loss as shown in
Figs. 13 and 14. This reduction is most dramatic for
height shading (see Figs. 12 and 14) although it is stil1
quite significant for width shading (Figs. 11 and 13).
3. ‘Hot Spot’ Development
It was pointed out earlier that localized heating or
‘hot spots’ occur in the shaded cel1 in a series string.
Consider an S array with N cells in series and M sections
in parallel resistively biased for maximum power at no
shading. If K cells in a single series string are shaded,
‘hot spots’ can form if the remaining (N - K) unshaded
cells force current through the shaded ones. To prevent
this, we must require that K be restricted so that the
total voltage developed in the shaded string is less
than the load voltage VL. This can be expressed as
KV, + (N - K)v; < NVZ,
where V, and V; are the voltages developed in the
partially shaded string across the shaded and unshaded
cells respectively and where VZ is the voltage across
each cel1 in the unshaded strings so that NV1 = V..
At the point where current just begins to flow in the
shaded string, Figs. 1 and 2 show that Vf is equal to
the open circuit voltage V,, and VS is - 2.0 V. Let V*
denote this latter negative voltage when current just
begins to flow. Then shaded cel1 current cannot flow
and ‘hot spots’ cannot develop as long as
KV” + (N - K)Vo, I NVZ.
Rearranging terms this can be expressed as
1 - vz/voc
N - 1 - V*/Voc’
Since V* is negative, the right hand side of this expression
is always less than one.
The worst possible case occurs when M is one (simple
series string). In this case VZ is zero and K/N must
be greater than 1/(1 - V*/Voc). Since Vac is 0.47 V,
this requires fractional shading greater than 0.19. Not
allowing any shaded cel1 current is a conservative
criterion for ‘hot spot’ avoidance and so gives a higher
minimum shading percentage than the 12.5 per cent
determined earlier by graphical means. Some shaded
cel1 current does flow with shading between 12.5 and
19 per cent, but it is so smal1 that insignificant heating
Since we have considered .the worst case, we can say
that no shaded cel1 current wil1 flow in any S array
(M > 1) as long as K/N is more than 0.19. Somewhat
less conservatively, we can say that no significant heating
occurs as long as fraction shading is above 0.125. One
should remember that even though fractions are used,
these criteria apply only for completely shading multiples
of single cells (no partial cel1 shading).
In contrast to this, the largest possible value VZ can
have (under best conditions) occurs when M approaches
infìnity. Then shading a single series string has negligible
effect on the load current so that the load voltage
remains NV,. This requires VZ = Vm. Hence the upper
and lower bounds for the right hand side of Equation
(19) are
1 - v,/v,, < 1 - vz/voc <
1 - v*/voc - 1 - v*/v0e - 1 -
Fig. 15. The miniium height shading of an S array without
shunt diodes required to avoid ‘hot spots’ as a function of
width shadimg.
For our case with V, = 0.36 V, this means that shaded
cel1 ‘hot spots’ always occur with S arrays if height
shading (KIN) is less than 4.46 per cent and never
occur if height shading is greater than 19 per cent. For
M values between one and inflnity, the minimum height
shading to avoid ‘hot spots’ are shown in Fig. 15. For
convenience the M values have been presented as per
cent width shading (i.e. M = 1 is 100 per cent, M = 2
is 50 per cent, M = 3 is 33.3 per cent, etc.). Again these
height and width percentages apply only for multiples
of completely shaded celis. This plot also applies if the
horizontal and vertical axes are relabeled maximum
width shading and height shading respectively. Then
it gives the largest allowable width shading without
‘hot spots’ as a function of height shading.
When P arrays are considered, a second mode of
‘hot spot’ formation can occur, but now on fully illuminated cells [2-71. This is explained by the reverse voltage
characteristics shown in Fig. 1. Observe that an illuminated cel1 very rapidly goes to high negative voltage
values for current levels in excess of its short circuit
current Zse. Such conditions develop in a P array when
a single parallel section is partially shaded. The shaded
cel1 currents fa11to near zero so that almost al1 the load
current must flow through the remaining unshaded
cells. If there are enough unshaded sections in series,
they can force so much current through the shaded
section that short circuit current levels are exceeded.
Since these illuminated cel1 currents are much higher
than for shaded cells at the same negative voltage,
several times more power loss can be developed across
a cel1 for this P array mode than for S array cel1 heating.
At a - 4.0 V cel1 potential, unshaded cel1 current is
1.15 A (Fig. 1) compared to only 0.2 A for a shaded
cell. Hence power lost to the unshaded cel1 at this
voltage is 5.75 times as large as that of the shaded cel1
(4.6 compared to 0.8 W) and is a factor of 21 times
greater than the 0.22 W per cel1 delivered when there is
no shading.
Consider a P array with M cells in parallel and N
parallel sections in series connected to a maximum
power resistive load. If K cells in one parallel section are
completely shaded, the total current through each
section must stil1 be equal so that
KZa + (M - K)Z; = MZz,
where Z8 and Zi are currents in the partially shaded
vertically or horizontally across the cell. Therefore, the
per cent heights of Fig. 16 apply only for multiples of
complete cel1 heights.
Again the horizontal and vertical axes can be respectively relabeled as minimum height shading and width
shading to give the largest allowable height shading
without ‘hot spots’ as a function of shading width.
is satisfied. At the point where power loss just begins This is equivalent to saying that operating the array
to develop, Z8 and the shaded cel1 voltage both pass in regions below this curve avoids cel1 heating. This
through zero (see Figs. 1 and 2). Hence no negative safe area under the curve is somewhat less than the
voltage can occur as long as
safe area above the curve for S arrays (Fig. 15). However,
the P array is safe for both smal1 width and smal1 height
(23) shading. The S array is only safe for smal1 width shading.
It always gives cel1 heating at smal1 height shading
The worst case condition occurs when Zz takes on its
values. These ‘hot spots’ can of course be eliminated
maximum value as N approaches itinity. Then shading
with shunt diodes. However, this requires a diode for
a single parallel section has negligible effect on load
each cel1 of an S array, potentially a very large number.
voltage so that a total load current of MZm continues
In contrast P arrays with a given shading area wil1
to flow. Since Zm = 600 mA and Zgc= 700 mA in our
not develop ‘hot spots’ if large parallel sections (or
case, K/M must be less than 0.143.
equivalently large area single cells) are used so that
Best possible conditions are realized when N is one.
fractional width shading stays below the limits given
Then ZZ is zero since there are no unshaded sections.
in Fig. 16.
This then gives the upper as wel1 as the lower bounds
This beneficial property obtained with large area
for the right hand side of Equation (23) as
parallel sections contradicts conclusions drawn by Jett
(1 - Zm/Z8C)
< (1 - 11/18C) < 1.0.
(24) and Miller [7]. These authors compared the cel1 heating
that results from shading a single cel1 in two conflguraThis means that ‘hot spots’ never form with P arrays
tions, a simple series array of 154 cells and a P array
for width shading (KIM) less than 14.3 per cent and
with 4 cells in parallel and 154 sections in series. Less
in some cases do not form even with 100 per cent width
heating occurred with the simple series connection
shading. The maximum values that K/M (in per cent
since shading a complete cel1 greatly reduced the current
width) can have without ‘hot spot’ formation are shown
output. Thus they concluded that the 1 x 154 array
in Fig. 16 VS per cent height shading. Since cells in
had shading characteristics superior to the 4 x 154
parallel are equivalent to a single cel1 of equivalent
array. What they failed to realize was that shading
surface area, the per cent width shading (of Fig. 16)
25 per cent of one cel1 in the 1 x 154 array would have
applies even for fractional shading of single cells. In
been just as bad as shading one out of four in the
particular this applies to single cells connected in a
4 x 154 array. The current densities in the unshaded
simple series string. Then it is perhaps clearer to express
portions in parallel with the shaded portion would
Equation (23) in its equivalent form
exceed J8, for both cases. The localized heating would
fi (1- h/J8C),
(25) then have been similar.
Note what Figs. 15 and 16 imply about shading
where JI is the current density in the illuminated cells
pattems. If a long thin shadow is tast across an array,
in series with the partially shaded cell, Jee is the short
these diagrams show that no cel1 heating would occur
circuit current density, and f is the fraction of the cel1
if the length of the shadow fel1 along the height of
area shaded. Partially shading a cel1 is equivalent to
either the P or the S array but that they would form
width shading regardless of whether the shadow falls
if the length fel1 along the width.
section flowing through the shaded and ilhuninated
cells respectively and where ZI is the current flow through
each cel1 in unshaded sections. To prevent ‘hot spots’
we must require that Zi be less than ZsCso that
4. Optimal Array Design
ahlmt diodes lkquimd to avoid ‘hot spots’ as a flmction of
heisht sbadiog.
The array configuration which performs best depends
on the criteria used to specify optimal and on the design
constraints imposed. If least power loss with shading
and avoidance of ‘hot spots’ are the criteria, then the
simple parallel array (Fig. 4(b) without diodes) gives
the best performance. Under no shading conditions
can cell heating occur with this cor&u.ration,
power loss with shading (Fig. 3) is near the theoretical
minimum. Although the power loss is slightly less with
a simple series array shunted with ideal diodes, real
diodes drop so much power (0.2 V forward voltage
drop compared to V’ = 0.36 V) that actual series
arrays would have larger losses. Unfortunately, for
CdS/Cu$S (and Si) solar cells, the simple parallel array
gives too low an output voltage for practica1 applications
(less than 05 V). If other materials were used to construct a higher voltage solar cel1 (say above 5 V) the
band gap would have to be so high that the majority
of the solar spectrum would be incapable of exciting
the cell. Extremely low efficiency would then result.
If, in addition to requiring low power loss and ‘hot
spot’ avoidance, the constraints of higher output voltage
and no protective diodes are imposed, either the P or
S arrays must be used to obtain the voltage. As far as
power loss with shading is concerned, the P array is
superior under al1 circumstances. As concerns ‘hot
spots’, smal1 percentage shading does not produce
localized heating with the P array but does give it with
the S array. At larger shading values, ‘hot spots’ develop
with the P connection and disappear for the S connection. However, this large S shading is such that it
completely cuts off current flow from partially shaded
series strings including fully illuminated cells. Since
array placement where large shading occurs is inherently bad design, the P array should perform best
under most circumstances. This is particularly true if
large parallel sections can be constructed so that maximum shading can be held below that indicated in
Fig. 16.
Removing the restriction on diodes and retaining
the criteria of low loss, higher voltage and no ‘hot
spots’ leads to P and S arrays with shunt diodes. Series
diodes are impractical as discussed earlier. Depending
on the exact shading conditions, sometimes the P array
gives lower losses and sometimes the S array does.
However, the differente between them remains small.
With ideal diode protection, there is no chance of
‘hot spot’ development in either case. With real diodes,
reverse voltages up to - 0.2 V (with germanium diodes)
develop so that some heating (approx. 140 mw) is
possible across fully illuminated cells in the P array.
With the S array, shaded cel1 current is insignificant
at - 0.2 V so that no heating occurs. Significantly
fewer diodes are required with the P configuration
(Fig. 6) and thus this feature makes it somewhat more
attractive. Furthermore, the P array is particularly
good when shading percentages can be kept low. Then
it performs just as wel1 without diodes as with diodes
and would hence be the logical, simpler design choice.
Under few conditions does the S array without
diodes give adequate performance. If shading is large
enough to prevent cel1 heating, it also effectively cuts
off current flow from part of the array. If shading is
lower, ‘hot spots’ form. It always gives higher losses
than the other conflgurations. Such a connection scheme
is only suitable if no shading or large shading is involved.
Figures 17 and 18 illustrate the points made above
for two specific cases with P and S arrays. Notice that
power loss is consistently less for the P array if no
diodes are used and is sometimes less if shunt diodes
are used. While no hard rules can be made that apply
Fig. 17. The power loss ratio as a function of width shading for
Pand S arrays with and without ideal shunt diodes for a constant
25 per cent height shading.
Fig. 18. The power loss ratio as a function of height shading for
P and S arrays with and without ideal shunt diodes for a constant
25 per cent width shading.
in every case, the shunt diode protected P array seems
somewhat more attractive in genera1 due to fewer
protective diodes being required and its generally low
power loss with shading. If cheap, inexpensive diodes
with low forward voltage drop could be easily and
reliably attached to each cell, the S array with shunt
diodes would be just as attractive. However, if large
enough arrays can be constructed to keep shading
percentages low, the P array without diodes would
certainly be the best and simplest array.
Array designs to guard against cel1 failure requires
special considerations. Among the many possible failure
modes, we here consider two, cel1 open circuit and
cel1 short circuit. A shorted cel1 has least effect on an
S array where the main result is just the loss of the
single cell’s output power (approximately). However,
with P arrays, shorting a single cel1 shorts out the power
supplied from a complete parallel section. If many cells
are in parallel, many times more power is lost than with
the S array.
If shorting is a problem but a P configuration is
desired a modified P form can be used. For this the
number of cells connected in parallel would be restricted
below a given fraction of the total number of cells in
the array. This fraction would be the maximum fractional
power loss that could be tolerated with the shorting of
a single cell. Several P arrays of the desired voltage
would be constructed under this constraint and con-
Effect of Shading on CdS/Cu,S Solar Celks and Optimal Solar Array Design
nected in parallel with each other until al1 available
cells were used. This then would give a modified P
array protected against cel1 shorting.
An open circuited cel1 has its strongest effect on S
arrays without protective diodes. Then the open circuit
effectively removes al1 the cells in a series string from
the array. Addition of shunt diodes shorts around the
open circuit and reduces the power loss to approximately
just that of a single 41. For a P array, with or without
diodes, an open cel1 has minimum effect, approximately
just a single cell’s power 10s~. Therefore, if diodes are
not used, a shorted cel1 has least effect on an S array
and an open circuit cel1 has least effect on a P array.
If diodes are used, power loss is near minimum with
an S array for both open and shorted 41s. Shunt diodes
have no beneficial effect on a P array with either shorted
or open 41s (as long as the current density in the
parallel cells remains below their short circuit value).
Cells with unequal I-V characteristics can have
strong effects on an S array. The cel1 with the lowest
current limits current through cells in series with it
and reduces power output. However, with P arrays
the result is different. Since cells in parallel are equivalent
to a single cel1 of large area, parallel P sections effectively
average the characteristics of individual cells. Connected
in series, the sections then have IJ characteristics that
are wel1 matched.
The result of non-uniform light illumination on a
cel1 array is to make the I-V characteristics different
for cells exposed to different light intensities. Hence
P arrays should perform better than S arrays under
such conditions. If parts of an array make different
angles with a light source, cells tumed most away from
the source effectively have lower light intensity. Therefore, solar arrays should lie in a single plane or in
parallel planes pointed directly at a light source for
best performance. If arrays have to be placed on angles
or curved surfaces, P arrays would be least sensitive to
the resulting non-uniform illumination.
5. Conclusions
The results of shading various solar array confìgurations have been quantified and presented in graphical
form. The conditions under which localized cel1 heating
or ‘hot spots’ occur have been precisely determined,
and the precautions necessary to avoid their formation
have been presented. Comparative studies have shown
the major dilferences between P and S arrays. Analysis
of optimal design considerations has shown that a P
array or a moditìed P array with shunt diodes has
optimal or near optimal performance characteristics
under many conditions including shading, cel1 failure,
non-uniform illumination and unequal I-V characteristics. However, this P connection does not have
minimum power loss under al1 shading conditions, and
a shunt diode protected S array does perform better
if both shorted and open circuited cells constitute a
significant problem.
[l] W. H. Evans, A. E. Mann, 1. Weiman and W. V. Wright,
AR.9 Space Power Systems Conference, Santa Monica, Cahfornia, September 1960.
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Specialists Conference, Cleveland, Ohio, June 1964.
[3\97:3.&Baron, Proceedings of the IECEC, August 1967, pp.
__. _-..
[4] W. Luft and J. R. Barton, Proceedings of the IECEC, August
1967, pp. 247-257.
[5] R. M. Diamond and E. D. Steele, Proceedings of the International ColIoquium on Solar Celis, Toulouse, France, July
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[6] G. Ravelli and C. Arduini, Proceedings of the International
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1971, pp. 889-900.
[S] F. Blake and K. L. Hanson, Proceedings of the IECEC,
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[9] H. S. Rauschenback and E. E. Maiden, Proceedings of the
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[ll] K. W. Boer, 3rd Conference on Lurge Scale Soiar Energy
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[12] F. A. Shirland, 3rd Conference on Large Scale SoIar Energy
Conversion for Terrestrial Use, Newark, Delaware, October
[13] L. R. Shiozawa, F. Augustine. G. A. Sullivan, J. M. Smith.
111 and W. R. Cook, Aeiospaci Research Laboiatories Repor;
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1151H. Hadley, Institute of Energy Conversion, University of
Delaware, private communication.