Energy Conversion. Vol. 14, pp. 61-71. Pergamon Pres.?, 1975. Printed in Great Britain Effect ofShading onCdS/Cu,S Solar Cells andOptimal Solar Array Design* MOHAMED SAYEDj-§ and LARRY PARTAINS (ReceivedJuly 1973) Abstract-The 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. 1.rntroduction 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 address: Microwave Division, Hewlett-Packard Company, Palo Alto, California 94304, U.S.A. $ Correspondence should be directed to this author. Electrical Engineering Department, University of Delaware, Newark, Delaware 19711, U.S.A. 5 Institute of Energy Conversion, University of Delaware, Newark, Delaware 19722, U.S.A. 61 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 structures. 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 62 VOLTAGE (VOLTS 1 Flg. 1.The reverse voltage I-V clauam of a CdS/Ca,S solar cell ader fully shaded, half shaded and fally lllmalaated COIldltlOUS. VOLTAGE ( VOLTS) 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 (V,&/Vo&) 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 = -800 i i I I //’ \ I Fig. 2. The forwardvoltage I-V clmactetics of a C!dS/Ca,S BolarcellImdertilllyshaded,halfshadedaadfallyillaaliaated cmditbm. Dashed liaea iadicate the locas of points of amtaat power. The load line iadica~ the re&tive load for maximam pOWW. 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 value. B. Bìasing for maximum power One of the simplest methods of determining the E i=l vt(G, (1) 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 MV) = E la(V), i=l (2) 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- 63 2 ,..’ I 10 20 x) 40 50 SH4DIffi so 70 80 90 100 1PERCENT) 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, (3) 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$$] =o, (4) and - 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 (N- Po = NV,,& PL = NV,& (bl 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. (5) 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) but PL = NV,,& - (N - 1)VzI. (9 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 64 MOHAMED SAYED and LARRY PARTAIN 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, (8) so that _ =_,1 Po N PL (9) 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 requires - (M-l)(II-;)+(I,-;)=O* (l”) 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 StWJBNG ( PERCENT) 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 65 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 (11) 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 PL _=_* Po 1 M (12) 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 WIDTH-hl (bl 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 MNP,,,. (a) P conzguration. For the P configuration, shading 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, (13) where VS is the voltage across the shaded cel1 and the (M - 1) fully illuminated cells in parallel with it. The MOHAMRD SAYRD and LARRY PARTAIN 66 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 that 1t = MIZ = (M - 1>z;+ I,, (14) 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. I I I I 20 1 20 WlDTH F 30 SIUUNG 40 50 1 PERCENT ) 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. 60 [ ao p-l L 10 60 40 r I 20 50 10 20 WlOTH 30 SHADING 40 50 ( PERCENT 1 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. J” % WIDTH 33 % WIDTH 20 % WIDTH L t 10 % 60 - J -II 50 % WIDTH HEIGHT Sl-!ADING (PERCENT 1 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. HEIGHT SHADING ( PERCENT 1 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. Exkt 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. (15) Again this is the dual equation for Equation Likewise, the dual for Equation (14) is Vt = NVZ = (N - 1>v; + VS, (13). (16) 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 67 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 required. 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 height). 80 1 50 % HEIGHT 33 % HEIGHT 20 % EIGHT 60- !! 10 40- 20 30 40 50 WDTH SHADING(PERCENT) & 20. v I 10 20 30 WlOTHWDING ( PEikNT 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. Fig. 5 % HEIGHT 50 ) 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. 2 60 !8 ; 40 Y 20 10 20 30 40 50 HffiHT SH4LMNG( PERCENT) 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. 10 20 30 40 50 I-EIGHT SHAOING(PERCENT) 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 spot’regioas. 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). 68 MOHAMFD SAYED and LARRY PARTAIN 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, (17) 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. (18) Rearranging terms this can be expressed as 1 - vz/voc 3 N - 1 - V*/Voc’ (19) 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 results. 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 - 1 v*/voc’ I I 20 40 60 WIDTH SHADING (PERCENT1 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, (20) J 80 (21) 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 m-KTsIz 8C> (22) shading to give the largest allowable height shading M-K 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 [email protected]~~ 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, and 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 70 MOHAMED SAYED and LARRY PARTAIN 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 io 10 ti io 90 5J+lDM ( PERCENT WiDTH 1 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. 25 % WlDTH SHADING 40 P WITH 10 20 HEIGHT DIODES 30 SHAIING 40 50 ( PERCENT) 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 71 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. [2] W, R. Baron and P. F. Virobik, Proceedings 4th Photovoltaic 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 1970, pp. 407-423. [6] G. Ravelli and C. Arduini, Proceedings of the International Colloquium on Soiar Ce& Toulouse, France ,July 1970, pp. 519-533. [7] P. L. Jett and J. L. Miller, Proceedings of the ZECEC, August 1971, pp. 889-900. [S] F. Blake and K. L. Hanson, Proceedings of the IECEC, September 1969, pp. 575-581. [9] H. S. Rauschenback and E. E. Maiden, Proceedings of the yl: i:fE Photovoltaic Specialists Conference, May 1972, pp. [lO] A. At& and J. Cerpart, Proceedings of the 9th IEEE Photovoltaic Speciaiists Conference, May 1972, pp. 206-216. [ll] K. W. Boer, 3rd Conference on Lurge Scale Soiar Energy Conversion for Terrestrial Use, Newark, Delaware, October 1971. [12] F. A. Shirland, 3rd Conference on Large Scale SoIar Energy Conversion for Terrestrial Use, Newark, Delaware, October 1971. [13] L. R. Shiozawa, F. Augustine. G. A. Sullivan, J. M. Smith. 111 and W. R. Cook, Aeiospaci Research Laboiatories Repor; No. ARL 69-0155. October 1969. [14] R. A. Crossley,~ G. T. Noel and M. Wolf, National Aeronautics and Space Administration Report No. AED R-3346, June 1968, page 1-1. 1151H. Hadley, Institute of Energy Conversion, University of Delaware, private communication.