An Analysis of a Photovoltaic Panel Model

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1 An Analysis of a Photovoltaic Panel Model Comparison Between Measurements and Analytical Models Ciprian Nemes, Florin Munteanu Faculty of Electrical Engineering Technical University of Iasi Iasi, Romania cnemes@ee.tuiasi.ro Abstract This paper focuses on an analysis of accuracy degree of a photovoltaic panel model. Therefore, a brief review of onediode model having in view the main analytical equations of this model is presented in the paper. Based on this model and its required parameters evaluated based on the datasheet of panel, various simulation studies have been carried out in order to evaluate the influence of irradiance and temperature on the current-voltage and power-voltage characteristics curves. A comparison between values of the one-diode model and those obtained from real measurements, having in view the operational parameters of photovoltaic panel, has been also conducted in the paper. Keywords- solar enery; photovoltaic characteristics; modeling I. INTRODUCTION During the last decades, a growing interest in renewable energy sources has been observed. The photovoltaic systems are ones of the suitable renewable energy sources whose utilization becomes more common due to its nature. A photovoltaic system directly converts the sun radiation in electricity using the elementary elements, namely the photovoltaic cells. The photovoltaic cells are basically semiconductor diodes exposed to solar radiation. These photovoltaic cells may be interconnected in series and parallel in order to form photovoltaic panels, characterised by a large values of output voltage and current. The mechanism to obtain current from photovoltaic cells is called the photoelectric effect and consists in an energy transfer from photons that reach on the surface of semiconductor material to electrons, generating in such way, a continuous electrical current. Photovoltaic cells are made of various types of semiconductors materials and using different manufacturing techniques. The silicon and thin film photovoltaic cells are currently the main technologies used in commercial photovoltaic applications [, ], thus, some issues of each of them are summarized as follows: Mono-crystalline silicon photovoltaic cells are the most efficient ones, the efficiency of these cells being around -8%. Instead, the manufacturing process is more expensive and must be carefully conducted at higher temperatures; Polycrystalline silicon photovoltaic cells have a good performance relative to the price. These cells have a lower efficiency than mono-crystalline cells, around -%, but the manufacturing process is less expensive than for the mono-crystalline cells and in addition they can be manufactured from waste electronics; The amorphous silicon photovoltaic cells require a simpler and cheaper manufacturing process than previous ones, where the amorphous silicon (a non-crystalline form) is deposited as thin-films, at low temperatures, on different substrates. Instead, these cells have a lower efficiency, around -8%, but, with a good behavior under diffuse component of solar radiation. Other new photovoltaic cells are made of copper indium gallium (di)selenide (CIGS) or cadmium telluride (CdTe). These materials are also thin film deposited, they having the same advantages as the silicon thin film photovoltaic cells but with a better efficiency, around -%. The performance of photovoltaic systems is usually determined by its electrical behavior for various environmental and load conditions. The modeling and simulation of photovoltaic system could be used to predict the electrical performance in various working conditions. Different photovoltaic systems have different electrical performances, thus the models used to describe the electrical behaviors are also different. Accurate prediction of the electrical behavior of photovoltaic panel needs to have comprehensive and precise models for whole system, especially for their photovoltaic panels that compose the system. The main idea of this paper is to evaluate the accuracy of the one-diode model for a real photovoltaic panel, based on the comparison between results of analytical model and real data measurements. In this order, the paper has been organized as follows. Section presents a brief review of one-diode model having in view the main analytical equations of the model. Then, the review of photovoltaic model is followed in section by the modelling and simulation of a COMITINI- photovoltaic panel, the results of simulation being compared with those obtained from measurements in section. Furthermore, in section, these results are involved in statistical analyses related to evaluate the accuracy degree of one-diode model. Finally, based on results of comparisons, the main conclusions are given in section.

2 II. MODEL FOR PHOTOVOLTAIC PANEL OUTPUT POWER Ones of the most important characteristics of photovoltaic cells are the nonlinear characteristics of current-voltage (I-V) and power-voltage (P-V) curves, which indicate the features of the current and output power versus voltage values, as shown in Fig.. Between I-V and P-V characteristics there is a reciprocal dependence, the output power from a photovoltaic cell could be estimated from its current-voltage curve. Power (W) Current (A) A photovoltaic panel is a set of cells connected in series in order to form strings, which, in turn, are connected in parallel in order to form a panel. The one-diode model could be extended for a photovoltaic panel [], having in view the number of cells in the panel connected in series (N s ) and the number of strings connected in parallel (N p ), as follows: q( U I( Ns / N p ) Rs ) N kat U I( Ns / N p) Rs I N pil N pi e s ( N p / Ns) Rp The photovoltaic current (I L ) of the photovoltaic panel could be evaluated considering the values of series and parallel resistances (R s,r p ), the short-circuit current (I sc ), and also the influence of the solar radiation (G) and the temperature (T) in accordance with the coefficient of current (K I ), as shown following relationship: () Voltage (V) Figure. Characteristics I-V and P-V curves for a photovoltaic panel. Various electrical models are available in the literature in order to model the I-V and P-V curves of photovoltaic cells. The I-V characteristic of a photovoltaic cell has an exponential characteristic similar to that of a diode, thus the models that are mostly used to express the electrical behaviour of a photovoltaic cell are one-diode and two-diode models [-]. In this paper, a one-diode model is considered in order to model the photovoltaic panel characteristics. The equivalent circuit of photovoltaic cell is a current source in parallel with a single diode and a parallel resistance, and also connected in series with a series resistance, as is depicted in Fig.. The parallel resistance indicates the leakage currents in the diode, while the series resistance represents the losses to the connections contacts. I L D I D Figure. Single-diode equivalent circuit for a photovoltaic cell. The electrical behaviour of a solar cell can be represented by the one-diode electrical model [,], its current-voltage characteristic being expressed by the following relationship: I IL I R p R s I U q( U IRs ) kat U IR e s Rp () where I and U are the photovoltaic cell output current and voltage, I L and I are the photovoltaic and saturation currents, q is the charge of an electron (. -9 C), k is the Boltzmann constant (.8 - JK - ), T is the absolute temperature (in Kelvin degrees), A is the diode quality factor, and R s and R p are the equivalent series and parallel resistances of the photovoltaic cell. I I L STC KI T G GSTC L, () where I L,STC is the photovoltaic current generated under Standard Test Conditions (STC), which take into account the influence of series and parallel resistances of the cell, the nominal value under STC being assumed that I L, STC ( Rs Rp ) Rp Isc [,]. The STC indicate an irradiance level of G STC = W/m², with the reference air mass coefficient A.M. and a cell junction temperature of C (T STC =98. K). The saturation current I has been considered dependent only on the temperature and could be expressed as follow: T qe g I I, exp () STC T ka T T STC STC where I,STC is the saturation current evaluated under STC and E g is bandgap energy (equal to. ev). Unfortunately, some of the parameters required for adjusting photovoltaic panel model cannot be found in the manufacturer s datasheets. Manufacturers of photovoltaic panels provide some information regarded by the main three important points of the power-voltage characteristic (the opencircuit point, the short-circuit point and the maximum power point) and few experimental data about electrical and thermal characteristics. All photovoltaic panel datasheets bring basically the following information: the nominal open-circuit voltage, the nominal short-circuit current, the maximum output power (MPP), the voltage at the MPP, the current at the MPP, the open-circuit voltage-temperature coefficient and the shortcircuit current-temperature coefficient, all these information being always provided with reference to STC. Therefore, the corresponding models parameters, namely the series resistance and parallel resistance, the diode saturation and photovoltaic currents have to be determined in order to find the analytical expression of the electrical model. This evaluation has to be carefully performed having in view that some parameters are variables with the solar radiation and with the cell junction temperature.

3 III. MODELING OF A REAL PHOTOVOLTAIC PANEL The mathematical model from previous section has been implemented in Matlab package software, in order to evaluate the unknown parameters of the one-diode model. The previous relationships do not allow directly calculating the unknown parameters as an analytical expression, they being solved using a numerical iterative method. The photovoltaic panel under study is a COMITINI- photovoltaic panel, manufactured based on mono-crystalline silicon cells. Table reports the main technical data of the photovoltaic panel at the standard test conditions, based on its datasheets available on [7]. TABLE I. TECHNICAL DATA OF THE COMITINI PANEL COMITINI - Mono-crystalline silicon P max W* I sc. A V oc. V*. V mp. V I mp.7 A Coefficient of current K Isc. %/C Coefficient of voltage K Voc -. %/C Coefficient of power K Pmax -. %/C Temperature of Operation - 8 C Nominal operating cell temperature. C Fill factor 7 The objective of this section is to find the value of series and parallel resistances and also the photovoltaic and saturation currents, based on the datasheet of panel. Series and parallel resistances required in the model have been stated constant and their dependencies on solar radiation and temperature conditions have been neglected. Instead, the photovoltaic and saturation currents have been considered as variables which depend on the solar radiation and temperature. An iterative technique initially tried has been used in this paper, with good performances, the values of series and parallel resistances, evaluated in STC, have been found R s =. Ω and R p = 9.79 Ω, while the photovoltaic and saturation currents have been found I L =. A and I =. -9 A, values that are in concordance with those found in standard books and in literature [8,9]. For any photovoltaic panel, the output electrical characteristics depend on the values of temperature and the solar radiation from the site where the photovoltaic panel is placed. This dependence has been simulated and investigated using the analytical model in order to analyse the electrical behaviour in different environmental conditions. The currentvoltage and power-voltage characteristics under various solar radiation values (between and W/m ), but with same temperature value (C) are shown in Figs. and. As can be seen, the output current and power increase with the solar radiation increase. Instead, the output current and power decrease with temperature. Figs. and show the current and output power characteristics versus voltage, for same value of solar radiation ( W/m ) and various temperature values (between - and C). Current [A] 7 G= W/m G= W/m G=8 W/m G= W/m G= W/m I-V curve for T= o C Figure. The photovoltaic panel I-V curves for various irradiation values. Power [W] 8 8 P-V curve for T= o C G= W/m G= W/m G=8 W/m G= W/m G= W/m Figure. The photovoltaic panel P-V curves for various irradiation values. Current [A] I-V curve for G= W/m T= - o C T= o C T= o C T= o C Figure. The photovoltaic panel I-V curves for various temperature values. Power [W] 8 8 P-V curve for G= W/m T= - o C T= o C T= o C T= o C Figure. The photovoltaic panel P-V curves for various temperature values.

4 IV. SIMULATION AND RESULTS The main aim of this section is to validate and to investigate the accuracy of the one-diode model associate to a COMITINI- photovoltaic panel. In this order, the I-V and P-V characteristics curves, obtained through analytical model, have been compared with those from real measurements. The experimental data used in this paper has been recorded in the Laboratory for Renewable Hydrogen (IDRILab), Faculty of Electrical Engineering, University of Catania, where a COMITINI- photovoltaic panel is installed on a photovoltaic system []. The photovoltaic system uses an automatic tracker system, which permanently orientates the panel on the sun direction, thus its surface is always perpendicular to the solar rays. The photovoltaic system contains also two acquisition systems that allow to simultaneously recording the environmental conditions and electrical parameters. The first system allows recording the solar radiation and photovoltaic panel temperature, and others parameters concerning to the photovoltaic panel position (azimuth and tilt angles). The acquisition system is managed by a software, the data being recorder in real time. For comparison, the solar radiation and temperature values have been also recorded with a portable analyzer (IV Photovoltaic Panel Analyzer), the acquisition software and portable analyser being presented in Fig. 7. The measurements have been carried out in different days and different environmental conditions, thus, different photovoltaic panel curves have been recorded for different solar radiation and temperature values. From these curves have been eliminated those characterised by an inconstant solar radiation, where the radiation has been changed during measurements interval. To assess the accuracy of the models, several extracted I-V and P-V characteristic curves have been grouped for various radiation intervals and temperatures, in order to coverage a wide range of environmental condition. Therefore, some sets of I-V and P-V curves have been extracted from measurement data and involved in a comparison process. For instance, Figs. 9 and show the I-V and P-V characteristics, drawn based on the analytical model (continuous line) and from measurements data (scatter points), both curves being drawn in the same diagrams. Current [A] I-V curve for G= W/m and T=. o C Figure 9. Comparison between I-V characteristic curves obtained from analytical model and real data measurements (G= W/m, T=.C). P-V curve for G= W/m and T=. o C Figure 7. The I-V and P-V characteristics of COMITINI- photovoltaic panel. The second acquisition system allows recording the current and output power values versus voltage of the photovoltaic panel. The photovoltaic panel has been connected to an electric load, and current-voltage and power-voltage characteristics curves have been drawn over each seconds time interval, using registered data points, Fig. 8 shown these curves. Power [W] 8 Figure. Comparison between P-V characteristic curves obtained from analytical model and real data measurements (G= W/m, T=.C). Figure 8. The I-V and P-V characteristics of COMITINI- photovoltaic panel. There have been founded similar features between I-V and P-V curves obtained from one-diode analytical model and those obtained from real measurements. As can be seen, the analytical model results are few underestimated, especially for lower values of voltage, but it can be stated that the analytical model gives fairly close results for a preliminary evaluation of electrical behaviour of photovoltaic panels, the accuracy degree of model being evaluated in following section.

5 V. COMPARISON BETWEEN MEASUREMENTS AND ANALYTICAL MODEL The degree of accuracy of one-diode model is evaluated by two statistical tests, mean bias error (MBE) and root mean square error (RMSE), these statistical tests being widely used tests in assessing the performance of the analytical models. To obtain dimensionless statistical indicators, the MBE and RMSE have been normalized to average of measured values, the percentage of relative errors being calculated with following expressions: Calculated power [W] 8 8 corrcoef=.9879 N xi xˆ i N MBE (%) i () N xˆ i N i N xi xˆ i N i RMSE (%) () N xˆ i N i where xi and xˆ i are the i th values from analytical model and measurement database, respectively. The MBE provides information about the model s performance, a lower MBE value being desirable. Positive values indicate overestimated values, while negative values indicate underestimated values. The RMSE is always positive, and a lower value is desirable, too. RMSE test provides also the information on the performance of the model considering the deviation between the calculated values and the desired values. All values used in previous equations refer to output current and output power of the photovoltaic panel. The comparisons are made for different sets of data, for various moments characterised by different solar radiation and temperature values, the set of data containing the current and power values, calculated and measured for same voltage values. Figs. and show the correlation between calculated and measured values of current (with a correlation coefficient of.989) and power (with a correlation coefficient of.9879), respectively. As can be seen, it has been obtained higher values for the correlation coefficients. Calculated current [A] corrcoef=.989 Measured current [A] Figure. Corelation between calculated and measured values of current (=.989). 8 Measured power [W] Figure. Corelation between calculated and measured values of power (=.9879). The mean bias and root mean square errors have been computed based on the calculated and measured output current and power values for various case study (characterised by different solar radiation and temperature values), the percentage of the relative errors being calculated and reported in Table II. TABLE II. Cases G= W/m T=.C G=7 W/m T=9C G=88 W/m T=.C G= W/m T=.C G= W/m T=C PERCENTAGE MBE AND RMSE FOR DIFERENT SETS OF DATA I-V curve P-V curve MBE (%) RMSE (%) MBE (%) RMSE (%) As can be seen in Table II, the percentage of MBE for all under evaluated cases varies from.7% to 7.7%, while RMSE varies from.% to 9.7%. The first case has registered the highest error values of all cases. It can be observed that the MBE, for all considered cases, are lower than %, instead the RMSE has higher values. The analysis of the errors shows that the calculated values from analytical model are in agreement with those obtained from measurements, especially for the environmental conditions close to solar radiation around to W/m and temperature around to C, which corresponds to STC. For lower values of solar radiation, the errors have higher values, especially the RMSE that indicate a higher deviation of calculated values around to those obtained from measurements. Furthermore, the errors indicate that the calculated values are underestimated estimated relative to measured values, for all cases, the highest degree of errors being registered for lower radiation values. As a conclusion, it can be stated that this model provide fairly results to real ones, especially for those conditions close to STC, thus the previous model can be used for a preliminary evaluation of the electrical characteristics.

6 VI. CONCLUSIONS Measurement and modelling of electrical behaviour of photovoltaic panels is an important factor for the performance evaluation and development of solar energy systems. Various electrical models are available in the literature in order to model the electrical behaviour of photovoltaic panels. In this paper, a one-diode model was considered in order to model the electrical behaviour of a real photovoltaic panel, namely a COMITINI- panel. The purpose of this paper was to evaluate the parameters of one-diode model based on datasheet of panel and to compare the current-voltage and power-voltage characteristics curves with those obtained from real measurements. The method for extracting the panel parameters from datasheet values was based on an iterative calculation, and the obtained values were used to check the accuracy of the model. Validation was performed by comparing the current-voltage and power-voltage curves drawn by the one-diode model with those curves recorded for a real photovoltaic panel, running in various environmental conditions. Based on results of comparisons, it can be stated that the analytical model gives fairly results to real values, especially for environmental conditions close to STC, thus the one-diode model can be used for a preliminary evaluation of the electrical characteristics of photovoltaic panel. ACKNOWLEDGMENT This paper was supported by the project PERFORM-ERA Postdoctoral Performance for Integration in the European Research Area (ID-79), financed by the European Social Fund and the Romanian Government. REFERENCES [] M.Gilbert, Renewable and Efficient Electric Power Systems, John Wiley and Sons Publishing, New Jersey,, pp. 9-. [] T.Markvart, L.Castaner, Practical Handbook of Photovoltaics: Fundamentals and Applications, Elsevier Science Ltd.,, pp [] A. Cheknane, S. Hikmat, H. Djeffal, B. Benyoucef, J.Charles, An equivalent circuit approach to organic solar cell modelling, Microelectronics Journal 9, 8, pp [] M. G. Villalva, J. R. Gazoli, E. R. Filho, Comprehensive Approach to Modeling and Simulation of Photovoltaic Arrays, IEEE Transactions on Power Electronics, vol., no., may 9, pp [] V. Quaschning, R. Hanitsch Numerical simulation of photovoltaic generators with shaded cells, th Universities Power Engineering Conference, Greenwich, Sept. -7, 99, pp [] Loteta, W. Ferrara,G. Bellia, F. Aleo, G. Gigliucci, G.M. Tina, Energy rating of photovoltaic systems comparison between measurement and model, rd International Conference on Clean Electrical Power, th - th June, Ischia, Italy, pp. -, DOI:.9/ICCEP..9. [7] [8] D. Sera, R. Teodorescu, P. Rodriguez, PV panel model based on datasheet values IEEE International Symposium on Industrial Electronics, 7. ISIE 7. pp. 9-9, --7-9/7/$. '7 IEEE. [9] M. Azab, Improved Circuit Model of Photovoltaic Array, International Journal of Electrical Power and Energy Systems Engineering -, 9, pp []