Mathematical Model for Determining the Performance Characteristics of Multi-Crystalline Photovoltaic Modules

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1 Proc. of the 9th WEA nt. onf. on Mathematical and omputational Methods in cience and Engineering, rinidad and obago, November -7, 7 79 Mathematical Model for Determining the Performance haracteristics of Multi-rystalline Photovoltaic Modules JUKUP MANOO, KAMAUZZAMAN OPAN, WAN AM WAN DAUD, MOHAMAD AGOU AND AZAM ZAHAM olar Energy esearch nstitute, Universiti Kebangsaan Malaysia, Bangi, elangor, Malaysia ksopian@vlsi.eng.ukm.my,wramli@vlsi.eng.ukm.my,azami@vlsi.eng.ukm.my Abstract: Mathematical models for predicting the performance characteristics of multi-crystalline photovoltaic (P) modules have been developed. n any prediction of the electric output of P systems, an understanding of its characteristics is of fundamental importance. hese characteristics are popularly expressed in the form of - curves that may be presented graphically; or using equations. here exists many equations for calculating the photogenerated current, saturated current o and a fitting-parameter a. he characteristics of P modules using simulations and s on multi-crystalline modules have been conducted. o achieve this, an al P system of kwp capacity was set-up comprising of modules, arranged in a string combination of 8 in series by in parallel. he current and voltage data were recorded at various solar radiation levels and module temperatures. his paper then presents and discusses a set of new equations pertaining to a,,, and the maximum power point. hese results were expressed using graphs and equations with constant values and compared with those obtained from the al set-up. his set of equations could be used to predict performances and maximum power points of a P module using temperature and solar radiation data only. Keywords: photovoltaic; P characteristic; maximum power point; mathematic model.. ntroduction Energy systems in the foreseeable future will tend to be environmental friendly; in which renewable energy resources offer a most promising option. hese renewable energy resources are not only clean but they are uniquely sustainable. One of the most promising renewable energy resources is solar energy. olar energy can be utilized as thermal energy, direct electricity, or a combination of both. Within a variety of renewable and sustainable energy technologies in progress, photovoltaic technology appears to be one of the most promising ways meeting the future energy demands as well as environmental issues. he olar Energy esearch nstitute (E) of University Kebangsaan Malaysia (UKM) has installed a kwp grid-interactive P system. he P system comprises modules, each string of 8 modules connected in series by modules in parallel. he grid-interactive inverter is equipped with a maximum power point tracker (P) and synchronizer. n order to predict the electric output of any P system, characteristics of the modules must be understood. he equations showing - and P- characteristic of P cells and modules have been established [,, ]. deally, a P module should always operate at the maximum power point, which is achieved by using a Maximum Power Point racker (P). o the results of output P electric will be always a maximum. he equation to predict maximum output has been others based on interpolation model []. his paper discusses a new set of equations on the curve fitting parameter (a), light current ( ), diode reserve saturation current ( ) and the maximum power point based on - and P- characteristics.

2 Proc. of the 9th WEA nt. onf. on Mathematical and omputational Methods in cience and Engineering, rinidad and obago, November -7, 7 8 able ummary of module characteristics. ype Maximum power output (P ) Maximum power voltage ( ) Maximum power current ( ) Open ircuit voltages ( O ) hort ircuit current ( ) oltage temperature coefficient (β ) urrent temperature coefficient (α ) ength Width Depth Multi-crystalline Wp.9 7. A. 7. A -. per o. A per o mm mm mm + s exp a At short circuit condition, the photogenerated current ( ) is equivalent to the short circuit current ( ). At open circuit condition, there is no current () and is relatively smaller compared to the exponential term. hus equation becomes O exp a At the maximum power condition with substitution equation () to () and neglecting the, the result is a ln + O he series resistance is independent of temperature and P modules with ε gap. e for silicon, and ε gap. e for gallium arsenide. ubscript is the erence condition.. - haracteristic of P. he P moduleequivalent circuit diagram is shown in Fig []. Here, the current balance can show the current voltage (-) characteristic equation of a P module. a a, G [, +, (, G, O. O,, gap. N exp ε a )] Figure he equivalent circuit for a P generator. Using Kirchoff s aw, the current balance is +. s D sh O exp a + sh f the shunt resistance ( sh ) is very much larger than the series resistance ( s ), the equation can be simplified as a, sc, O. oc d. d sc d. d, O, sc., O,, O + ε gap O (). N Parametric model, equation of light current ( ) uses α and β parameters as follows [] α ( + βc ) G

3 Proc. of the 9th WEA nt. onf. on Mathematical and omputational Methods in cience and Engineering, rinidad and obago, November -7, 7 8 And the diode saturation current ( ) is presented with M parameter a k c ε gap k ( + kc ) G M c exp () kc he result of the substitution of equations (7 to 9) into equation () is where k is Boltzman s constant (J/K). he interpolation model based on short circuit current ( ), open circuit voltage ( O ), maximum power point voltage ( ) and maximum power point current ( ). k k ( k ) G + exp k +. exp k s, exp,., O,, ln,, O,. Maximum Power Point of P deally, a P module should always operate at the maximum power point. n this condition the power is P, the current is and the voltage () is. he maximum power that can be obtained corresponds to the rectangle of maximum area under the - curve, or the maximum P- curve. he two parameters (D t ) and ( ) can be calculated Ai, et al. () uses interpolation model to with temperature of P and solar irradiation. evaluate P array, operated by maximum power point tracker (P). he equations of and G G in maximum power conditions are Δ α Δ + (), G, G, Δ β Δ + Δ + G (,.9log + ) β Δ Δ ell G, ell, () o the output current can be calculated with the equation below: + Δ Δ, exp O, + Δ Δ exp, Δ (), O, Many researchers have used equation () as standard equation, but for equations of a,, there are many variations, as such Kau, et al. (998); Beckman and Duffie (99), have used the equations of ownsend (equations to 9), Khouzam and Hoffman (99), have used the equation of parametric model (equation -). Parameter values, erence conditions and photovoltaic area are fixed, so those equations can be simplified as below:

4 Proc. of the 9th WEA nt. onf. on Mathematical and omputational Methods in cience and Engineering, rinidad and obago, November -7, P P he result of the substitution equations ( and 8) into equation () is urrent (Amper)..7. P- Power (Watt) Δ + a ln a Δ Δ Δ Δ + a ln a O oltage (olt) Figure ypical - and P- curves for a P module. n the maximum power point condition occurs maximum value of *, or at P- curve the P is the zenith or the reverse point, so the differential (dp/): dp d( ) + d + d Based on equation () and the is small compared to the exponential term, the - characteristic of the P at maximum power condition is 7 a ln 978 W/m, 8. o K 8 W/m,. o K Δ a ln 78 W/m, 9. o K 7 W/m,. o K Δ () d d W/m a Δ a, 7.7 o K (7) Δ 8 W/m,. o K Δ (8) oltage (olt) Δ a Δ a ln a + Δ P. esult and Discussions Data of the correlation between current () and voltage () of a photovoltaic module, the result of s in some variations of temperatures and solar radiations, is used for the suitable values evaluation of a, and. he optimization values a, and used the Hooke-Jeeves method, with minimization of sum square of error from - and with equation () data by trial and error of a, and values. he comparison of and data of the P module characteristic (Figure ), shows that data not so different with data. ) urrent (Amper 8 W/m, 8. o K

5 Proc. of the 9th WEA nt. onf. on Mathematical and omputational Methods in cience and Engineering, rinidad and obago, November -7, 7 8 Figure - characteristic of P module, and data. Figure Experimental and of (equation ) at varying temperature. he optimizations of k to k used Hooke-Jeeves and Golden-ection methods with minimization of E among a, and s and with equation (7-9). he result of these optimizations are k., k., k.77e-8, k and k o (Amper) O k exp(-k / ) k k a.... a k. k. emperature of P ( o K)..9.8 Figure Experimental and of (equation) at varying temperature..7. emperature ( o K) Figure Experimental and of a (equation) at varying temperature. Figures,, and show that the al data are not so different with the calculated data. he - curve as the result of predictions with equation () at fixed temperature and several radiation levels is shown at Figure 7; and at fixed radiation level and several temperature is shown at Figure 8. hese findings agree with Beckman and Duffie (99), in that a module operating at fixed temperature and several radiation levels, the short circuit voltage ( O ) increases logarithmically and the short circuit current ( ) is nearly proportional to the incident radiation. At fixed radiation level, increasing temperature leads to decreased open circuit voltage and slightly increased short circuit current. he conclusion, that equations of mathematic model in this research are qualified. he constants in this research are special for multicrystalline silicon. 7 k (+k. )G k. k (Amper) emperature ( o K) Figure 7 - characteristic and maximum power point of a P module at temperature of o and solar radiations of,,,, and 7 W/m.

6 Proc. of the 9th WEA nt. onf. on Mathematical and omputational Methods in cience and Engineering, rinidad and obago, November -7, Okt 8 (Watt/m ) P (Watt) Figure 8 - characteristic and maximum power point of a P module at temperatures of,,, and o, with solar radiation of W/m. Equations () and () are used to decide maximum power point, with values of a,, from equations (7), (8), (9) and constants (k to k ) that have been known. he solution of equation () can be solved by numerical methods (Newton- aphson, for example) so it will be understood maximum current ( ) value, then maximum voltage can be calculated with equation (), and maximum power whith equation (). From this, it can be concluded too that the equations model in this research will be able to predict maximum power point of P module, only by temperature of P and solar radiation data. he prediction results of maximum power point (P) of P module in several temperature and solar radiation levels are shown in Figures 7 and 8. (Amper) 8 Okt 7: 9: : : :8 9: ime mp mp Figure 9 and of a P module on 8 October. 8 8 (olt) olar adiation Pmp 7: 9: : : :8 9: ime Figure Maximum power of a P module on 8 October.. onclusions he equations of curve fitting parameter (a), photogenerated current ( ), and diode reserve saturation current ( ) have been simplified. his paper serves mathematic equations a,, and equation to calculate maximum power point of P module, with constants k to k. he equations of mathematic model in this research are simple and qualified, and can predict performance and maximum power point of P module, only by temperature of P and solar radiation data. he constants in this research are only for multicrystalline silicon. eferences [] Duffie J.A.and Beckman W.A.; olar Engineering of hermal Processes., nd, Wiley nterscience, New York; 99. [] Khauzam K. and Hoffman K.; eal ime imulation Photovoltaic Modules. n J. olar Energy, 99; ; -. [] ucumo M., osa A. D., Ferraro., Kaliakatsos D., Marinelli.; Performance analysis of a kw grid-connected photovoltaic plant. n J. enewable Energy, ; xx:. [] Ai, B., Yang, H., hen, H. and iao, X. omputer-aided design of P/wind hybrid system. n J. enewable Energy, ; 8 : 9-. -