Model of solar-cell array for terrestrial use

Solar Energy, Vol. 17, pp. 325-329. Pergamon Press 1975. Printed in Great Britain MODEL OF SOLAR-CELL ARRAY FOR TERRESTRIAL USE D. BIP, ANt and M. S. ERLICKI Faculty of Electrical Engineering, Technion--Israel Institute of Technology, Haifa, Israel (Received 17 October 1974; in revised form 20 July 1975) Abstract--Computer-aided analysis of conversion of solar energy into electricity under terrestrial conditions is discussed. The proposed approach permits determination of the supply system, for any load, on the basis of known climatic data. The working conditions of the system, and their influence on its cost, are analyzed. INTRODUCTION The modern world is becoming increasingly dependent on energy-consuming systems, and economic development is conditional on availability of adequate energy resources. The present study examined the possibilities of direct conversion of solar radiation into electricity for industrial and other uses. Its initial phase was devoted to identification of the basic elements of a solar-powered system, namely: solar-cell arrays, batteries, overcharge protection, and voltage-, current- and frequency-control devices. Next, a mathematical model was constructed (on the basis of local solar-radiation data) and computerized so as to yield the cost-optimized solution for a given location with given power requirement. The model was checked experimentally in the following terms: (i) Optimum solar cells. (ii) Optimum geometry and design of the arrays. (iii) Minimization of the effects of dirt and overcast on array output parameters. (iv) The suitable type of batteries. (v) Overcharge protection. GENERAL DESCRIPTION OF MODEL The flowchart in Fig. 1 comprises the following components, reading from the bottom: The consumer is a resistive or reactive load, character- ized by continuous or intermittent consumption of power under constant or variable conditions. Since the guiding principle is that the consumer must be practically provided for at all times, all other components are adapted to his requirements, represented by the total consumption per day (or per a longer working cycle) and by the relevant rated voltage (or voltage range). A converter (d.c./d.c. or d.c./a.c.) is an optional component required wherever a consumer calls for a.c. or high d.c. voltage. As it operates with the array and batteries installed in parallel, its influence is reflected in the required input voltage and in a slightly increased supply. For consumption W~, the required supply is W~ = W~bf, the conversion efficiency being 7/'< 1. The batteries provide storage for the converted d.c. ?Dr. D. Biran is with the Israeli D.F. Solar Radiation Solar- Cell Array Charge - Control Unit Batteries I Converter I Consumer (Load) Fig. l. Basic model of so]ar-cell based power supply. energy generated during sunshine periods. Each con- sumer is served by one battery or more, suitably connected in accordance with the discharge current dictated by the consumer's power and voltage require- ments. Usually cost considerations rule out nickel- cadmium in terrestrial arrays; ordinary lead units with storage capacity 100 kWh/m 3 are quite adequate. The charge-control unit is intended for keeping the batteries at near-maximum capacity, by arresting the charging process at peak voltage and triggering the recharging process at about 90%. The array comprises the requisite number of cells or modules (connected in series and parallel); a diode in series is provided for blocking reverse currents during periods of darkness. The power output of the array is governed by five factors: (a) Location. (b) Slant angle, relative to the horizon (the array faces south). 325 326 D. BIRAN and M. S. ERLICKI (c) Surface area. (d) Radiation conditions. (e) Serviceability of protective casing. The location is mainly dictated by that of the consumer. Within this limitation, the array must be installed so as to have no shadow cast on it at any time. The (seasonally variable )slant angle is a function of the geographical latitude of the location. The surface area is dictated by the consumer's requirements, by climatic conditions, by the charging efficiency of the batteries, etc. Year-round operation of the same array involves considerable seasonal fluctua- tions in input (summer, winter), and as the batteries are designed for overcharge irrespective of the season, charging time under these conditions is shorter in summer than in winter. A possible solution is seasonal disconnec- tion of part of the parallel branches of the array. Radiation conditions are determined from monthly maps for the location of the installation. Each region is represented by a reference locality, for which are given the daily global radiation (a multiannual average) and the number n~ of consecutive days in the month in question on which the radiation does not exceed a given minimum level by more than 25% or 10%. These factors, 1,25 and 1.1 (increments C in Tables 1 and 2), will be utilized for estimation of the dimension of the solar cells array. In Israel, for example, the reference locality for year-round operation is chosen on the basis of the December map (representing the worst conditions for the array). For intermittent (seasonal) operation, the approp- riate seasonal maps must be used. Serviceability o/ protective casing. The transparent protective casing of the array (made of plastic, perspex, glass, etc.) must be cleaned regularly as accumulating dirt Type % A 2 Table 2. Array data Solar Cells C K ~nth Slant Angle ,, , Cost (degrees) Total Area Number (IL) (cm2) in parallel L I0 0 Dec. 55.2 2400 12 2900 i Dec. 55.2 1120 5 1949 2 Dec. 55.2 960 4 1829 3 Dec. 55.2 800 4 1709 Dee. 55.2 800 4 1709 Dee. 55.2 800 4 1709 1.25 ol Dec. 55,2 2080 9 2660 1 Dec. 55,2 1280 5 2069 2 Dee. 55.2 1120 5 1949 3 Dec. 55.2 960 4 1829 4 Dee. 55.2 960 4 1829 5 Dec. 55.2 800 ~ 1709 i.i0 0 Oct. 50.0 960 4 1820 0 Dee. 55.0 2400 12 2900 0 March 35.0 1120 5 1940 0 May 30.0 800 4 1700 i Oct. 50.0 800 4 1709 i Dec. 55.0 1120 5 1949 i March 35.0 800 4 1709 i May 30.0 640 3 1589 2 Oct, 50.0 800 4 1709 2 Dec. 55.0 960 h 1829 2 March 35.0 640 3 1589 2 May 30;0 640 3 1589 3 Oct. 50.0 640 3 1589 3 Dee. 55.0 800 4 1709 3 March 35.0 640 3 1589 3 May 30.0 480 2 i469 4 Oct. 50.0 640 3 1589 h Dee. 55.0 800 4 1709 4 March 35.0 640 3 1589 4 M~y 30.0 480 2 1469 5 Oct. 50.0 640 3 1589 5 Dec. 55.0 800 4 1709 5 March 35.0 640 3 ] 1589 5 May 30.0 480 2 ] 1469 reduces the collecting capacity of the array. An antistatic coating is useful in this respect. MATHEMATICAL MODEL--UNDERLYING PRINCIPLES The proposed model comprises specific system data, memory data (mainly referring to the climate and to the cells and batteries, and readily adjustable in accordance with changes in any of the above), and the program. The output presents the solutions for the array (size and slant Table 1. Summary of results. Location: Zvital; latitude 32 °, reference locality: Beit Dagon. Daily energy consumption: Basic solar cell area---4 cm ; 64 cells in series 20.0 Wh at rated voltage-24V. 2 Battery Data Type C K Solsr Cell Area ' Cost of Syst~ (em 2 ) Manufacturer (IL) Quantity Capacity Voltage Unit Cost Coae Name (Ah) (V) A I l.lC 0 2400 6S4 ~ 4 6 50 2900 1 1120 6S7 2 45 12 109 1949 2 960 6S7 2 45 12 109 1829 3 800 6S7 2 45 12 109 1709 h 800 6S7 2 45 12 109 1709 5 800 6S7 ~ 2 45 12 109 1709 1,25 0 2080 6S4 ~ 4 4 6 50 2660 ] 1280 6S7 ~ 2 45 12 109 2089 2 1120 6S7 m 2 45 12 109 1949 3 960 6S7 ~ 2 45 12 109 1829 4 960 6S7 ~ 2 45 12 109 1829 I.I0 800 6s7 ~ ~ 2 45 12 lO9 1709 A 2 ~ 2400 6S4 ~ 4 4 6 50 29O0 i 1120 6S7 ~ 2 45 12 109 1949 2 960 6S7 2 45 12 109 1829 3 800 6S7 2 45 12 109 1709 4 800 6s7 ~ 2 45 12 109 1709 5 800 6s7 i 2 45 12 109 17o9 o 2080 684 4 4 6 50 2680 1"251' 1 1280 6S7 2 45 12 109 2069 2 1120 6S7 2 45 12 109 1949 3 960 6s7 2 45 12 109 1829 4 960 687 ~ 2 45 12 109 1829 5 800 6S7 ~ 2 45 12 109 1709 A 3 l.iC 0 2400 684 ~ 4 4 6 50 2900 i 1120 6S7 2 ~5 12 109 19~9 2 960 6s7 ~ 2 45 12 109 1829 3 800 6s7 ~ 2 45 12 109 17o9 4 800 6s7 ~ 2 45 12 109 17o9 5 800 6S7 ~ 2 45 12 109 1709 1.25 o 2o80 6s4 4 4 6 50 266o 1 1280 6S7 2 45 12 lO9 2089 2 1120 6s7 2 45 12 lO9 1949 3 960 6s7 2 45 12 lO9 1829 Model of solar-cell array for terrestrial use angle) and the batteries, their seasonal variation (dates of resetting) and the corresponding system costs. System data Two important parameters of the array are the daily consumption W, (the amount of energy per operation cycle) of the system, and its rated voltage. In the case of systems calling for a.c. or high d.c. voltage and necessitating a converter, the input voltage of the latter must be given. Three types of systems are taken into account: Type A~. Slant angle set for year-round operation, according to the "worst" month. Type As. Slant angle re-set four times during the year, according to the "worst" month in each season. As fluctuation is stronger in winter than in summer, the "seasons" are unequal; optimal performance has been obtained by re-setting in March, May, October and December. It seems that these months of re-setting may be accepted for the Northern Hemisphere. Type A3. Slant angle re-set monthly. The type parameter is optional. If it is given, the computation is confined to it; otherwise, all three types are covered by the program. 327 localities). Temperature must be taken into consideration in view of the effect of overheating on cell efficiency. Assuming that both radiation and heating are at peak around noon, 14.00 was taken as the reference time. Experience having shown that with the ambient tempera- ture above 25°C, cell temperature may exceed it by 20°C; this last figure was taken as the maximum correction for April-October, as against 15°C for November-March. Cell data are referred to 25°C, with the increment AT determined as the excess of the working temperature over it. Battery data These fall into two categories: identil~cationparameters (name/number of manufacture, and cost per unit), and working parameters. The latter include: (i) Capacity--referred to a discharge period of 20 hr as the closest to the 24-hr operation cycle. For shorter periods (say 10 hr) the capacity must be reduced by about 5%. (ii) Voltage--fixed at 6 or 12 V (with series combina- tions, or a converter, provided if necessary). (iii) Charging e1~ciency--fixed at 0.75 (the usual level for lead-acid batteries). Memory data The climatic parameters in this group are: (i) The sun's inclination 4~, varying over an annual cycle and used here in monthly averages. (ii) The average and minimum daily radiation R~ and R2 in Wh/cm 2. R~ and R2 are taken from multiannual measurements. As noted earlier, there are two alternatives for the minimum: multiplied by the factors 1.1 or 1.25 respec- tively (increment C in Tables 1 and 2). The factors were introduced in order to avoid reliance on accidental non-recurrent minima. The lower factor is obviously safer, but the corresponding array would be larger and more expensive. Other increments may also be consi- dered in an economic analysis. (iii) The daily sunshine period D, also varying over an annual cycle. Disregarding local atmospheric conditions, the radiation pattern over the period is as shown schematically in Fig. 2. The part of the period correspond- ing to effective radiation is represented by a factor a, determined experimentally as 0.6 for October-February, and 0.7 for March-September. (iv) Temperature (monthly averages for reference T R tPe Effective Radiation -I-D/~ a.D riod of Fig. 2. Sunshine radiation as function of tirne. Array data Unlike the battery data (which allow for a variety of makes), these are based on a single manufacturer (chosen as optimum in terms of parameters and cost) and must be modified if the cell is to be changed. The cost comprises that of the cell (or submodule) proper, plus the appropriate fi'actions for the materials and labour required for its installation in the array. The parameters (provided by the manufacturer) are: (i) Cell or submodule size S (area in cm2). (ii) Cell or submodule voltage VL under optimal load--0.34 to 0.40 V for silicon cells, depending on quality, radiation level and load. (iii) Conversion efficiency "0r--taken as constant re- gardless of diurnal fluctuations, and referred to a radiation basis of 100 mW/cm 2 (approximately corresponding to the noontime level). The assumption is correct within +-10%. (iv) Diode voltage drop----the only diode parameter included in the input for considerations of convenience. (Other parameters obtained in the output). Computer program The flowchart of the main computer program used is shown in Fig. 3. The program starts with the read-in of all climatic, battery and solar-cell data into the memory. Input is effected by entering a punched card coded for: (a) The location, latitude and reference locality of the system. (b) The nominal voltage of the load. (c) The daily energy consumption (in Wh). The steps are as follows: (1) Computation of the required charging voltage (which depends on the type of batteries in use--lead-acid 328 D. BIRAN and M. S. ERLICKI jar Computation of [ Fixed Slant Angle (December) Read - in Data Computation of Charging Voltage Computation of r Number of Cells in Series + Computation of Four-Season Optimal S ant Angles Computation of Work Cycle Computation of Total Cycle Radiation Energy --]A, Computation of Monthly Slant Angles Computation of Average Daily Radiation Energy Computation of Corrected Radiation Data ? Computation of Solar Cell Arrays I Determination of Determination of I Area Us ing Four-Season Optimal Data of December Areas Computation of Number of Cells in Parallel of Batteries Capacity /w,i,e: NO Battery No / Found Computation of The System Cost J t Table I: I Summary J Lta ! ~A3 Determination of Monthly Areas Fig. 3. Flowchart of the computer program. or nickel-cadmium) and of the number of solar cells or submodules to be connected in series. (2) Computation of the slant angle at which the solar cell array must be installed. This slant angle is determined for the worst conditions for each of the three types of systems (A~, A2, A3) as discussed above, and used subsequently for correcting the radiation data of the location, retrieved from the memory. (3) Computation of the working cycle of the system. A compromise is sought between the required solar cell area and the battery capacity. The solar cell array being the most expensive element in the system, the most economical combination is the smallest possible array with a larger battery (which is usually inexpensive). Accordingly, the working cycle is increased from a given minimum n~ to n~ + K; n~--the number of consecutive days on which the radiation energy is minimal; this number is included in the radiation data related to the location for each month separately. K in this program ranges from 0 to 5. (4) Retrieval of all radiation data for the location from the memory. These consist of the minimum global "tin the present study, a total of 102 array areas were calculated: 6 for type A, (yearly system), 24 for type A2 (four-season system) and 72 for type A3 (monthly system), with 6 values of K assumed for each type. radiation E, the average global radiation/~ for each month of the year, and the number n~. The radiation energy per cycle is then computed as follows (factor C equals 1.1 or 1.25 is taken into account): Ecy~to = E × ni × C +/~ x K. The average daily radiation energy is given by: Ear = Ecycle nl +K subject to correction for the slant angle (the radiation data being referred to a horizontal plane). (5) Computation of the array area f, using the solar cell or submodule data (size and efficiency), the corrected solar radiation average for each working cycle, the efficiency of the batteries used, and the energy required per daily cycle (plus that dissipated in the diode). An additional useful correction accounts for ambient temper- ature variations and the charging current efficiency, which depends on the diurnal radiation distribution. In the case of A2, an optimizing subroutine is used for each "season", with a view to the minimum possible area yielding the maximum energy throughout the "season" in question. Using the number of solar cells in series for different Model of solar-cell array for terrestrial use 329 areas, the computer determines the number of cells or submodules for parallel connection. (6) Computation of the battery capacity needed for the solar cell areas, the appropriate configuration of the available batteries (according to the type used--24, 12 or 6 V) and the number needed for each system. (7) Computation of the system cost--the sum of all items in the installation. COMPUTER OUTPUT AND ENGINEERING DECISION The computer output comprises a summary of results (Table 1) and the solar cell array data (Table 2). The summary, which provides the information for the engineering decision based on lowest cost, operational requirements and reliability (factor C)--is in two parts, the first part (identical for all possible types of systems) incorporating the basic input data of the system require- ments and fixed output data such as the number of cells in series, the charging current, and the basic solar cell dimensions retrieved from the memory. The table lists for each type A~, each C (10 or 25%), and each K (0 -< K -< 5), the following data: solar cell array area (cm2), the battery data (name and number of manufacturer, quantity, capacity, nominal voltage and unit cost), and the cost of the system. For each A~, Cand K, the main data refer to the worst case for the given type. By this means, the engineer chooses the type and the final system according to the appropriate C and K. Having made this decision, he turns to Table 2 for the corresponding array data: the month or months of re-setting, the slant angle, the area, the number of cells in parallel, and finally the cost. The seasonal data enable a system intended for a limited period to be designed economically, instead of on a year-round basis. CONCLUSION The proposed model permits a high degree of flexibility in terms of loads. The data used in the memory may be adapted for any location for which climatic data are available. The program also allows for improvements in future technology and performance without basic modifi- cations.