New Accurate Model to Estimate Maximum Power of Photovoltaic Modules and Arrays

Transcription

1 New Accurate Model to Estimate Maximum Power of Photovoltaic Modules and Arrays M. Bencherif* and T. Benouaz Department of Physics, University of Tlemcen, B.p. 119, Tlemcen R. p , Algeria Received February 09, 2013; Accepted April 11, 2014 Abstract: Quantitative information regarding the maximum power point (MPP) of photovoltaic (PV) arrays is crucial for determining and controlling their operation. Accurate knowledge of the MPP for PV arrays is essential for the design and operation of grid-connected PV power systems under varied atmospheric conditions. The usual methods for tracking the MPP of PV arrays suffer from a serious problem that the MPP cannot be quickly acquired. Therefore, a simple and effective mathematical model to obtain the MP output in real time under all possible system conditions is indispensable to the development of a feasible PV generation system. We developed a new prediction model for directly estimating MPP for power tracking in PV arrays. The proposed model is a simple approach that takes the effect of solar cell resistances into consideration. The performance of the proposed model was evaluated at various temperatures and irradiation intensities. Keywords: single diode model, modeling maximum power, PV module, modeling maximum power point 1. Introduction The photovoltaic modules of a PV generation system convert solar energy into direct current (dc) electricity. However, the PV arrays exhibit the characteristics of an extremely non-linear current voltage (I V), which varies with array temperature and solar irradiation over time and complicates the locating of the MPP. For practical purposes it can be assumed that the power delivered by a photovoltaic generator that is connected to an MPPT is always the highest. If it is wished to study the behaviour of a PV generator in that situation, the most interesting aspect is to know the evolution of the maximum power point. Two of the authors that work with this topic, Jones and C.P Underwood [1], developed a model in (2002) that finds the maximum power of a photovoltaic *Corresponding author: moh.bencherif@gmail.com 60 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

2 module,.this model depends on the temperature of the cells, the solar irradiation and the fill factor. The model of the fill factor that we propose can be used to supplement the model developed by these authors. Jen-Cheng Wang, Yu-Li Su, Jyh-Cherng Shieh, Joe-Air Jiang [2] (2011) investigated the effectiveness of the proposed prediction MPP method for PV arrays subject to different irradiation intensities and temperatures.,the exposed model takes only the series resistance effect of the solar cells into consideration, and E. Skoplaki, J.A. Palyvos [3], (2009) exhibited many correlations expressing Tc, the PV cell temperature, as a function of weather variables such as the ambient temperature, Ta, and the local wind speed, as well as the solar radiation flux/irradiance, with material and system dependent properties as parameters, i.e, glazing-cover transmittance, plate absorptions, etc. The maximum power is a function of the efficiency of the PV modules. The model that we propose helps a direct prediction of the maximum power of PV arrays at various conditions and is expressed as the loss parameters of the PV modules and short circuit current and the open circuit voltage. The various parameters contained in the model can be easy evaluated in other conditions of temperature and irradiation intensity. The proposed model is verified through experiments carried out under various weather conditions and can also be applied to the method of MPPT and PV stability in the future. 2 The Standard Diode Model There are several mathematical models in the literature to describe photovoltaic cells, from simple to more complex models that account for different reverse saturation currents. The two-diode equations with the saturation currents Is1 and Is2 and with the diode factors A1 and A2 describe the diffusion and recombination characteristics of the charge carriers in the material itself and in the space-charge zone [4]. To simplify parameter adjustment, the two-diode model can be reduced to a one-diode model in which, according to the Shockley theory, recombination in the space-charge zone is neglected, so the second diode term is omitted [4]. An electrical circuit with a single diode (single exponential) is considered the equivalent photovoltaic cell in the present article. The basic model for a photovoltaic module is shown in figure 1. The current voltage (I V) characteristics of a photovoltaic module can be described with a single diode as [3-11]. V+ RsI V+ RsI I = Iph Is exp 1 a Rp (1) Where Iph is the light-generated current (A), it is the reverse saturation current of the p n diode (A). Rs is the series resistance of the cells (Ω). The series resistance Rs accounts for all voltage drops across the transport resistance of the solar cell, and Rs connects to a load or an inverter. Rsh is the shunt resistance of the cells in J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 61

3 Figure 1 Equivalent circuit of the standard diode model. parallel with the diode of the solar cell model, representing the shunt paths that can occur in real solar cells across the surface, at pin holes in the p n junction, or at grain boundaries. A is the thermal voltage (V) depending on the cell temperature, which is defined as: a = ANsKT (2) q Where T is the cell temperature (K), K is Boltzmann s constant (J K -1 ), and q is the charge of the electron (C). Ns is the number of cells in the series, and A is the diode ideality factor. 3 Ideal Diode Model The equivalent circuit in figure 2 describes the behaviour at a fixed temperature and the solar radiation of the ideal diode model that assumes a null series resistance, infinity shunt resistance and ideality factor equal to 1. The current voltage (I V) characteristic of an ideal photovoltaic module is: V Io = Iph Iso exp 1 Vt (3) Where Iso is the reverse saturation current of the ideal diode and Vt is the thermal voltage (V) depending only on the cell temperature and the number Ns of the cells connected in series, which is given as: Vt = NsKT q (4) The power generated from the PV modules or PV arrays is: P= VI (5) 62 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

4 Figure 2 Equivalent circuit of the ideal diode model. 4 Theoretical Basis of the Proposed Method The effects of loss resistances on the I-V characteristic have been extensively studied in several works [12 16] although in different ways, both resistances contribute to the degradation of the I-V curve. In general, the current derived by the shunt resistance is very important in which part of the I-V curve runs from short circuit (Isc) to the vicinity of the maximum power point (Im). On the contrary, the voltage drop due to series resistance is greater at the voltages between the open circuit (Voc) and the maximum power (Pmax). In line with these observations, the point of maximum power lies in the transitional zone where there are higher effects of both resistances. Therefore, the two resistances contribute to a significant loss of the output power consequently the maximum power. Figures 3 shows the I-V curves of the polycrystalline module MSX110 plotted at standard test conditions (STC), depending on the values that their loss resistances take on. The curve presented by the continuous line in figure 3(a) shows the case of a null series resistance Rs. In that case, shunt resistance Rsh and an idealit, are equal to 1 (ideal PV module) and the curve in dotted line s is similar to that specified by the manufacturer, as it passes through the specific points It has been obtained by finding the values of Rs and Rsh, which are 0.7Ω, Ω respectively and a thermal voltage of (V). Figure 3(b) shows the effect of both resistances where their current - voltage characteristics are compared to ideal characteristics. Curve A shows the effect of series resistance on the I-V characteristic plotted with Rs = 0 Ω, Rsh = Ω and the thermal voltage a=2.174 (V). Curve (B) displays the effect of the shunt resistance on the I-V characteristic plotted with Rs = 0.7 Ω, Rsh = infinity and a thermal voltage a=2.174 (V). This figure shows that series resistance operates in the part of the I-V curve that runs from Pmax to the open circuit voltage (Voc). It also indicates that series resistance controls the position of the maximum power point; otherwise, it supervises the current and the voltage in this point. If the Rs value decreases, the voltage at the maximal power increases enormously and the maximum power point moves on the right; in contrary case, the voltage decreases considerably and the maximum power point moves on the left, while the maximum power current changes lightly, consequently the series resistance has an extensive effect on the voltage Vm of the maximum power so the maance controls the slope of I-V characteristic in at the open circuit conditions. In other words, Rsh acts on J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 63

5 Figure 3 Effect of the loss parameters and the thermal voltage on MPP. the slope when the photovoltaic module works as the generator of current, in this region with a shunt resistance infinite the ideal curve is practically confused with the curves (B) in figure 3 (b). The curves (B) show the effect of shunt resistance on the current. If the value of Rsh decreases, the current decreases considerably (curve B). On the other hand, if the value of Rsh is significant, the current of the maximum power increases widely (curve C). Consequently the shunt resistance acts on the current Im, therefore, its incidence on the maximum power is considerable. Figures 3(c)-3(d) indicate that the effect of thermal voltage (ideality factor A) is significant on the maximum power point. In figure 3(c), the curves (A) and (B) present the ideal and the real I-V characteristics of the PV module MSX110 in standard test conditions respectively. Tracing different curves in various cases makes the comparison of the two curves; curve (C) shows the effect of the series resistance and thermal voltage (Rs=0Ω, A=1, a=ns KT/q). In this case the MPP is near the maximum power voltage of the ideal case. Curve (D) displays the effect of 64 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

6 shunt resistance and thermal voltage (Rsh= infinity Ω, A=1, a=ns KT/q). The MPP increases, i.e, the current Im and the voltage Vm increase simultaneously, the curve (E) depicts the effect of the thermal voltage only (Rs= 0.7 Ω, Rsh= Ω, A=1, a=ns KT/q). If the ideality factor of the diode decreases, the current and the voltage increases. In the contrary case the MPP falls abruptly. In figure 3(d), we observe that there is an increase in current Im and voltage Vm for an increase of 9% (only in relation to the value determined at standard test conditions, the percentage error made on Im, Vm and Pma. Ts 0.5%, 0.90% and 1.43% respectively). Indeed its reduction involves a voltage Vm drop and a waning of current Im, the percentage error due to this reduction is of the same order as its increase. The previous paragraph demonstrates that three parameters, Rs, Rsh and the thermal voltage or the temperature and the ideality factor of the diode, affect the output power and the maximum power. The single diode model of the PV modules or PV arrays is a non-linear model, with one operating point existing where the PV array produces the maximum power. The non-linear temperature dependent behaviors of PV modules or PV arrays have been successfully described by the p n junction recombination mechanism of semiconductors [17]. In order to perform maximum utilization efficiency, we developed a model to directly estimate the MPP by using the p n junction semiconductor theory. The maximum power point (MPP) is situated in the transition zone (AB) (figures 4) where there is the effect of both resistances. The tangents in this point to I-V characteristics of the PV modules in the ideal and real case seem to have very nearby directing coefficients with a very weak difference, which is located between 10 3 and according to the analysis of various PV modules. Figures 4 present the current - voltage characteristics and the tangents at these curves at the maximum power point of PV modules HR-185, MSX110, NA-F135 and SM 55 evaluated in the standard test conditions of which their data will be mentioned in the ensuing sections. Figures 4 show that the tangents are almost parallel in the transition zone (AB). Table 1 contains the absolute difference between the director coefficients of tangent lines to current- voltage characteristics of some modules at a maximum power point. These coefficients are characterized by the quotient I/V at MPP. Directing coefficients of both tangents are almost similar with insignificant differences; for the tested PV modules the difference between the directing coefficients of the tangents is about 10 3 which let both tangents at MPP almost parallel in the transitional zone (figure 4). This observation can be exploited in order to evaluate the current Im and voltage Vm of maximum power point and in consequence maximum power (MP) of any PV module. From this observation, we can write: di dv Ideal case di dv Real case (6) J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 65

7 Figure 4 Tangents at the maximum power points. The derivative of the current with respect to voltage V of eq. (6), we get: V+ RsI a Isexp + Iso V a Rsh exp Vt Vt Rs V + RsI Ideal a 1+ + RsIsexp Rsh a Real (7) At maximum power point, the following condition is met: P ( V, I) I I I = = I + = 0 = V V V V V (8) From eq. (8) the expression [Iso exp (V/Vt))/Vt] can be estimated by: Iso V Imo exp = Vt Vt Vmo (9) 66 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

8 Substituting eq. (9) into eq. (7), we obtain: Rs Vmo a 1+ Imo + Vm RsIm exp = Rsh Rsh Is. (10) a Vmo RsImo Applying the short circuit conditions to eq. (1), Iph can be obtained by: Rs IscRs Iph = Isc 1+ + Is exp. (11) Rsh a In the same way, the open circuit conditions lead to an equation for Iph: Voc IscRs Iph = + Is exp 1. Rsh a (12) Most of the times, Iph is simplified by: Rs Iph = Isc 1+ Rsh (13) Substituting eq. (11) into eq. (12) leads to an equation for Is: Rs Voc Isc 1+ Rsh Rsh Is = Voc IscRs exp exp a a. (14) Since, exp (Voc/a)>>> exp (Isc Rs/a), eq. (14) is simplified to: Rs Voc Voc Is = Isc 1+ exp. (15) Rsh Rsh a If we assume that Voc/Rsh < Isc (1+Rs/Rsh) the last eq. (15) is reduced to the following relation: Rs Voc Is Isc 1+ exp. Rsh a (16) J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 67

9 The voltage of the maximum power point deduced from eq. (10) and eq. (16) is given by: Rs Vmo a 1+ Imo = + Rsh Rsh Vm Voc RsIm alog. (17) Rs ( Vmo RsImo) Isc 1+ Rsh Substituting eq. (17)) into eq. (1) leads to an equation for Im: Rs Vmo a 1+ Imo + Rsh Rsh Voc alog Rs Vmo Rs a + ( ) + 1 Imo Vmo RsImo Isc 1 Im = Rsh Rsh Rsh Iph. (18) Vmo RsImo Rsh Eq. (18) of current Im can be simplified to: Rs Vmo a 1+ Imo + Rsh Rsh Voc alog Rs Vmo ( Vmo RsImo) Isc a 1+ Imo Im = Rsh Rsh Isc Vmo RsImo Rsh (19) Substituting eq. (19) into eq. (17) leads to an equation for Vm: Rs Vmo Rs Vmo a 1+ Imo a 1+ Imo Rs Rp Rp Rp Rp Vm = Voc alog ( ) Rs Isc Rsh Vmo RsImo Isc Vmo RsImo (20) After some unquestionable approximations, eq. (20) can be shortened to: 2Rs aimo aimo Vm = 1+ Voc + alog Rs Isc Rsh ( ) Vmo RsImo Isc Vmo RsIm o (21) Equations (19) and (21) predict that the current Im and voltage Vm depend only on the loss parameters Rs, Rsh and the quality factor of the diode and the temperature Tc of the modules as well as the short circuit current Isc, the open circuit voltage Voc and the current Imo and voltage Vmo of maximum power point, in the 68 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

10 ideal case. In order to determine the current Imo and the voltage Vmo of an ideal module we resort to the fill factor equation of an ideal cell (ideal module) operating in the same conditions of temperature and irradiation intensity of a real cell or solar modules given by the accurate green model, which is as follows [18]. Where the entity Vc is given by [18]: ( Vc ) Vc log FFo = Vc + 1. (22) Vc = Voc Vt. (23) The maximum power point of an ideal cell, ideal modules or an ideal PV array can be calculated exactly by the following relations that we propose using eq. (23): ao = Vc +1. (24) ao bo =. (25) ao + 1 bo ( ) Imo= Isc 1 ao. (26) Imo Vmo = Voc + Vtlog 1. (27) Isc Vmo = FFoIscVoc Im o (28) 5 Maximum Power Model (MPM) The output current and voltage of PV arrays are directly affected by many factors such as resistances, irradiation intensity and temperature. For this study, we developed a prediction model for directly estimating the MPP of PV arrays which takes the resistance effect of the solar cells into consideration. The maximum power can be evaluated by the usual following eq. (29): Pmax = ImVm (29) J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 69

11 We pose Rs Vmo a 1+ Imo Rp Rp X = ( Vmo RsImo) (30) Substituting eq (19) and (21) in eq (29) the obtained relation of maximum power is given in eq (31). X Voc + alog 2Rs a aimo Rs X Pmax = VocIsc log ( ) ( ) Isc X 1 Isc Rsh Voc Vmo RsImo Isc Voc Isc RshIsc (31) The output power Pmax generated from the PV arrays is expressed as a function of the current Isc and the voltage Voc and the loss parameters Rs, Rsh and thermal voltage a. The eqs (19) and (21) demonstrate that resistances have a critical effect on the output power. The parameters appearing in eq. (31) depend exclusively of temperature and solar irradiation. 6 Fill Factor The fill factor describes the squareness of the I V curve. It is defined as: Pmax ImVm FF = = IscVoc IscVco (32) Where Im and Vm are the maximum power point current and voltage, respectively. This ratio is due to the physical constraints on diode quality, the practical limit to the fill factor is less than the ideal value of 1.The behavior of a real diode will deviate from the ideal, primarily as a result of recombination occurring at the junction. From the eq (31), we extract the fill factor FF given in eq. (33). X Voc + alog 2Rs a aimo Rs X FF = log ( ) ( ) Isc X 1 Isc Rsh Voc Vmo RsImo Isc Voc Isc RshIsc (33) The values of Voc and Isc are obtained from the datasheets and Rs, Rsh and thermal voltage can be obtained from the datasheets by using the method edited by M.Bencherif and A. Chermitti [19] or another method. While the values under various situations can be calculated using the temperature coefficients β of open 70 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

12 circuit voltage Voc and temperature coefficient α of short circuit current Isc, on the other hand, the series and shunt resistances can be determined by the edited method of M.Bencherif and A. Chermitti [19] and Valerio Lo Brano, Aldo orioli and Giuseppina Ciulla [20]. The proposed model is a new and simple approach with a low calculation burden, which can be used to directly determine the MPP of PV arrays. This direct prediction method could be further applied to the MPPT algorithm for any kind of PV array. It is expected that by using the proposed method we can achieve a maximum power output in real time and substantially increase the output power of PV arrays in a solar generation system. In other conditions than the standard test conditions, the short circuit Isc, open circuit voltage Voc and loss parameters can be obtained by the following relations. The short circuit Isc is calculated by [21, 23]: ( ) G (, ) = ref + a( ref ) Isc Tc G Icc Tc T G ref (34) The open circuit voltage Voc can be computed by [21, 23]: G Voc( Tc, G) = Vocref + bδ T + alog Gref (35) The thermal voltage a at a cells temperature Tc can be evaluated by [24]: ( ) = atc a ref Tc T ref (36) The series and the shunt resistance can be estimated with [19, 20]: Rs Rsh G = ref ref G G Rs = ref ref G Rsh (37) (38) Where G is the irradiance, Gref is the reference irradiance 1000 W/m2, AM 1.5, and Rsref and Rshref are the series and shunt resistance calculated at standard test conditions. J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 71

13 7 Results and Discussion In order to evaluate the performances of the suggested method, we resorted to the data published by Engin Karatepe, Mutlu Boztepe and Metin Çolak [25] and of the experimental values which we published [21]. 7.1 Basic Performance Evaluation The basic performance evaluation of the proposed models given in eqs (19), (21) and (31) is made with the issued data by Engin Karatepe, Mutlu Boztepe and Metin Çolak [25] of the mono-crystalline PV module SM55, its electrical specification are listed in table 1. The edited specific points Isc, Vco, Im, Vm and the loss parameters Rs, Rsh and ideality factor A were determined in various irradiations intensities and two different temperatures. Table 3 summarizes the irradiation intensities and temperature dependence of the edited specific points and the loss parameters. The specific points and the loss parameters in table 3 were used to predict the performance of the proposed models. The obtained results will be compared to the edited current and voltage of maximum power and maximum power [25]. The power voltage curves of the solar module assessed with different loss parameters and short circuit current Isc and open circuit voltage Voc under different irradiation intensities and temperatures are examined by substituting the related parameters into eqs.(19),(21),and (31) of the direct prediction models. Figure 5 show the edited maximum power point, which is presented on the curves by a star, and the evaluated maximum power point obtained with the proposed models marked on the curves by a circle. Figs 5(a) 5(b) display the simulated power curves pass almost by all evaluated maximum power points for all the tested cases. Table 1 Absolute difference between the director coefficients. HR-185 MSX110 NA-F135 SM Table 2 Electrical specification of SM 55 PV module (AM 1.5, 1000 W/m2, 25 o C). Maximum power Pmax 54.81W Open circuit voltage (Voc) 21.7 V Short circuit current (Isc) 3.45 A Operating voltage at maximum power (Vm) 17.4 V Operating current at maximum power (Im) 3.15 A 72 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

14 Table 3 Published specific points in various levels of irradiation and temperatures. Operating conditions PV module equivalent circuit parameters G [W/m 2 ] T [oc] A Rs [Ω] Rsh[ Ω] Voc [V] Isc [A] Vm [V] Im [A] Pm [W] J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 73

15 Figure 5 Power curve in conditions and maximum power in different conditions. In other words, the simulated and edited maximum power points are practically confused; this means that the currents and voltages of maximum power provided by the exposed model in relation to the published values are really similar with an insignificant difference (see subsequent tables 4 and 5). These figures show a good agreement between the edited and calculated data; on the other hand this confirms that the mathematical models proposed offer an irreproachable estimation. These results verify that the maximum power currents of the PV arrays and MP can also be accurately estimated with the proposed direct prediction models. The proposed models in eq. (19) and eq. (21) can be used to inspect the maximum power currents and voltage of a PV array and thus, can also estimate their values with a high accuracy by the proposed direct prediction model. Table 4 contains the evaluated operating current Im and voltage Vm of maximum power and maximum power examined with different irradiances level and temperatures. The computed values are compared to the edited values of current Im and voltage Vm and maximum power [25], these values show a small difference between them, the obtained results are a good approval to the edited data. Some little inaccuracies still occur for the maximum power current and voltage and maximum power, with a small absolute difference and absolute percentage prediction error (see table 5 and 6) of the current and voltage of maximum power compared to the edited values. To prove what was claimed, the maximum power of the panels was obtained on the basis of the model described here. We can observe that the calculated results with the edited data in table 3 show that maximum power, current Im and voltage Vm are pretty similar. These differences (table 5) are due to the errors committed on the main parameters Rs, Rsh and thermal voltage of the PV module. We can see that the values estimated are not far to the edited values. In addition, two prediction performance indices are used to measure the prediction performance of the proposed models, and they are defined as follows: (1) Prediction 74 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

16 Table 4 The current Im and the voltage Vm and maximum power evaluated with different irradiances level and temperatures. Operating conditions Calculated Vm and Im Edited Pmax Calculated Maximum power G [W/m 2 ] T [ o C] Vm [V] Im [A] Pmax [W] Pmax [W] Table 5 Absolute prediction error between issued and calculated at different illuminations and temperatures. Operating conditions Prediction error G [W/m 2 ] T [ o C] ΔVm [V] Δ Im [A] ΔPmax [W] Table 6 Absolute percentage prediction error between published and calculated at different illuminations and temperatures. Operating conditions Percentage Prediction error % G [W/m 2 ] T [ o C] ΔVm ΔIm ΔPmax J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 75

17 error: ΔIm= absolute (Im (direct-prediction model)- Im(edited)), ΔVm= absolute (Vm (direct-prediction model)- Vm(edited) )and ΔPmax= absolute (Pmax (direct- prediction model)- Pmax (edited)) and (2) Percentage prediction error in (%):ΔIm%= 100x absolute(prediction error/ Im(edited)), ΔVm%= 100x absolute( Prediction error/ Vm(edited) ) and ΔPmax %= 100x absolute( Prediction error/ Pmax (edited)). The test results for these two performance indices are also included in tables 5 and 6. For the analyzed panel, table 6 itemizes the absolute percentage prediction error of current Im, voltage Vm and maximum power between the issued and the computed values. The greatest absolute percentage prediction error values in different test conditions of the chosen module SM55 are around %, %, and % for the current Im, voltage Vm and maximum power, respectively. The models show a little underestimation or overestimation of maximum power current and voltage and maximum power (table 5), which will lead to more moderate results and wiser predictions when the models are used to simulate the behavior of a PV system and to evaluate the benefit of economic investments. The maximum absolute percentage prediction error for the current Im, the voltage Vm, and MP does not exceed 0.5%. To achieve this study, we use some experimental published data of different modules that we carried out at the laboratory with M.Bencherif and A.Chermitti [21]; these data have been picked with the intention of making this study as general as possible with different data and technologies. The modules are mono- crystalline, Hareon model HR-185; polycrystalline, Solarex model MSX110; Sharp model NA-F135, which have a maximum power of 185W±10%, 110W±5%, 158.9%±10%, at standard test conditions respectively. Table 7 groups their manufacturer-edited data at standard test conditions and the experimental data at the test conditions of various temperatures and irradiation intensities. Figure 6 show the current voltage characteristic of the panels where the maximum power points are marked on each curve of each panel, the maximum power point determined by mean of the exposed model are practically confused with the manufacturer s data. This proves that the proposed model is very accurate to use for evaluating the maximum power point current, voltage and maximum power. Table 8 list the estimated values of maximum power current and voltage, MP, the prediction and percentage prediction error between the obtained values computed with the edited parameters by using the proposed models cited in eq. (19), eq. (21) and eq. (31) of current Im, voltage Vm and maximum power to the values given by the manufacturers and edited experimental data. The results of prediction error of the current Im and maximum power of the modules HR-185, MSX110 and NA-F135 at standard test conditions are A, A and A and W, W and W respectively. In the same way the prediction errors of voltage Vm are V, V and V, taken in the precedent order. The greatest absolute difference does not exceed A and V for the current Im and voltage Vm respectively. The absolute difference is very weak and almost insignificant. The percentage 76 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

18 Table 7 Manufacturer s and experimental edited data [19]. PV panels G [W/m 2 ] T [ o C] Icc Vco Im Vm Rs Rsh a Pmax HR Infinity Infinity MSX NA-F J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 77

19 Figure 6 I-V characteristics and maximum power points of the three PV modules in standard test conditions. prediction errors shown in table 8 are all lower than 0.5% for the current Im and voltage Vm and maximum power (MP); its calculated values are still below the ceiling of 1%. These results show that the proposed model offers higher degrees of approximation for the calculation of the current Im, voltage Vm and the maximum power. The maximum power can be calculated with a high exactitude using the developed models given in eq. (31). Figure 7 shows the current voltage characteristic plotted in conditions of temperature and irradiations intensities using the edited experimental data of the PV modules. By substituting the main related data into eqs.(19),(21),and (31) of the direct prediction models, the MPP of the different solar modules for each loss parameters were estimated, as indicated by the labeled circle and edited MPP is labeled with star in Figure 7. Examining figure 7 we can observe that the performance of the developed models of current, voltage of maximum power point (MPP) and maximum power (MP) of PV arrays indicate that the evaluated MPP is perfectly determined with an accurate approximation and is more far from the authorized limit of 1%. The small prediction error and percentage prediction error for the current and voltage at MPP of the PV arrays indicate that the proposed method provides pretty good prediction performance. Figure 7 shows the I V characteristic curves of the PV arrays plotted with experimental data and the MPP calculated by eq. (31) where, the obtained results are listed in table 8. In these test cases, different temperatures and irradiation intensities are used. The circle symbols show the estimated MPP, the solid lines indicate the experimental V I characteristics and the star symbol indicates the edited MPP of the PV arrays under different temperatures and irradiation intensities. The calculated maximum power is a good agreement with the published experimental data in different conditions of temperatures and irradiations (see table 8). 78 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

20 Table 8 Estimated Im, Vm, MP, Absolute prediction and Absolute percentage prediction error. Operating conditions Estimated values Absolute prediction error Absolute percentage prediction error PV panels G T W/m 2 [ o C] Im Vm Pmax Δ Im Δ Vm ΔPmax Δ Im% Δ Vm% Δ Pmax% HR MSX NA-F J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 79

21 Table 8 shows a negligible difference between the computed values using the maximum power model of eq (31) and the edited values. In conclusion, the results indicate that the proposed models can provide a more accurate estimation of the MPPs of PV modules under environmental variations while obtaining the maximum power output for practical applications. 7.2 Performance Evaluation of Fill Factor The model of PV module fill factor exposed in eq. (33) was evaluated in different conditions of irradiance intensities and temperatures with the edited data listed in table 3. Their computed values and the absolute relative errors are reported in tables 9 and 10. The results show a small absolute difference between the computed fill factor with the published data and estimated values with eq. (33), the greatest absolute percentage prediction error is % and all calculated values don t reach 0.33%. Figure 8 (a)-8(b) depict the edited and evaluated fill factor using eq. (33), these figures show a good agreement of the appreciated values with eq (33) and the issued data, that prove the accuracy of this mathematical model. The analysis of the obtained results of the performance evaluation of the exposed models describing the dependence of the maximum power (MP) and fill factor (FF) of the main loss parameters of PV modules in different conditions of temperature and irradiance. These results demonstrate that the developed models deliver an accurate approximation of the maximum power and fill factor with a highest absolute relative error below 0.33%. Figure 7 I-V Characteristics and maximum power points of the three PV modules in different conditions. 80 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

22 Table 9 Comparison between the edited and the calculated Fill Factor with the Eqs (33). Operating G [W/m 2 ] conditions T [ o C] Edited FF Calculated FF Table 10 Prediction error and percentage prediction error. Operating conditions Absolute difference G [W/m 2 ] T [ o C] ΔFF Relative Error ΔFF % J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC 81

23 Figure 8 Comparison between computed and edited fill factor. 8 Conclusion A new, simple, accurate, and low calculation burden method for estimating the MPP of PV arrays is presented. In this study, we investigate the effectiveness of the proposed direct-prediction MPP model for PV arrays subject to different irradiation intensities and temperatures. Based on the p n junction recombination mechanism, we are able to simply estimate the MPP of PV arrays with our prediction models to ensure that the maximum utilization efficiency and maximum power output of PV arrays can be achieved by impractical applications. The model of MPP has a proportional relationship between the product of the short-circuit current and the open circuit voltage and the fill factor given in eq. (33), which is a function of the loss of parameters of PV arrays and the current and voltage of maximum power of an ideal PV arrays. The simulated values MPP compared to the edited data predicted values are more accurate. This holds true even if the irradiation intensity or temperature changes. The effectiveness of the proposed method is verified through experiments carried out under various weather conditions and can also be applied to the method of MPPT and PV stability in future. The developed model to estimate the MPP of PV arrays according to the obtained results given in tables 4 and 8 offer a high degree of accuracy by using a normal scientific calculator, Hence, the programming and computational cost of the suggested equations of the models of the proposed method are minimal. In all the analyzed cases in this study, the results showed that the errors made on MPP of the different modules in various conditions, they are negligible and bearable. The greatest error observed of the maximal power of the tested modules was never higher than 0.33% (see tables 4 and 8). Moreover, this model can be used to determine the main loss parameters Rs, Rsh and thermal voltage of PV module, the developed model constitute an additional equation which can be added to the 82 J. Sustainable Energy Eng., Vol. 2, No. 1, May Scrivener Publishing LLC

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