Assessment of the Usability and Accuracy of the Simpliﬁed One-Diode Models for Photovoltaic Modules

: Models for photovoltaic (PV) cells and panels, based on the diode equivalent circuit, have been widely used because they are effective tools for system design. Many authors have presented simpliﬁed one-diode models whose three or four parameters are calculated using the data extracted from the datasheets issued by PV panel manufactures and adopting some simplifying hypotheses and numerical solving techniques. Sometimes it may be difﬁcult to make a choice among so many models. To help researchers and designers working in the area of photovoltaic systems in selecting the model that is ﬁt for purpose, a criterion for rating both the usability and accuracy of simpliﬁed one-diode models is proposed in this paper. The paper minutely describes the adopted hypotheses, analytical procedures and operative steps to calculate the parameters of the most famous simpliﬁed one-diode equivalent circuits. To test the achievable accuracy of the models, a comparison between the characteristics of some commercial PV modules issued by PV panel manufacturers and the calculated current-voltage ( I-V ) curves, at constant solar irradiance and/or cell temperature, is carried out. The study shows that, even if different usability ratings and accuracies are observed, the simpliﬁed one-diode models can be considered very effective tools. W/m , cell T ref ◦ and at AM


Introduction
A mathematical model used to simulate physical behaviours of PV modules needs a compromise between analytical complexity and achievable precision [1].
The one-diode model is a simplified version of the two-diode model proposed by Wolf [2] in order to represent the physical structure of a PV cell.As Wolf observed, the photocurrent in a PV cell is not generated by only one illuminated diode, but it is rather the global effect of the presence of a multitude of elementary flanked diodes that are uniformly distributed throughout the surface that separates the two slabs of the semiconductor junction.For this reason, a PV cell should be realistically approximated with a distributed constant electric circuit containing a multitude of elementary lumped components such as current generators, diodes and electrical resistances.Because such an equivalent circuit would be too complex to use, a simplified equivalent circuit was adopted.The circuit, which is depicted in Figure 1, contains only one pair of diodes with reverse saturation currents I 01 and I 02 , a current generator and two resistors R s and R sh , which take account of dissipative effects and parasitic currents within the PV panel.The second diode was added to consider the effect of the carrier recombination in the depletion region.The two-diode equivalent circuit of a PV module is described by the equation: 01 02 1 1 where, following the traditional theory, the photocurrent IL depends on the solar irradiance, the diode saturation currents I01 and I02 are affected by the cell temperature, n1 = a1Ncsk/q and n2 = a2Ncsk/q are the diode quality factors, a1 and a2 are the diode shape factors, Ncs is the number of cells of the panel that are connected in series.The values of Rs, Rsh, I01 and I02 variously affect the I-V characteristic of the PV panel [3].Because the evaluation of the parameters contained in the two-diode equivalent circuit is a complex problem, the one-diode equivalent circuit, depicted in Figure 2, was also considered.Many authors have proposed analytical procedures for determining the model parameters on the basis of the performance data usually provided by manufacturers .The identification of the parameters contained in the diode-based equivalent circuits has been also tackled exploring the possibility of using different procedures such as Lambert W-function, evolutionary algorithms, Padè approximants, genetic algorithms, cluster analysis, artificial neural networks, harmony search-based algorithms, small perturbations around the operating point and reduced forms [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63].Other authors have investigated some simplified versions of the one-diode equivalent circuit in order to obtain an adequate representation of the PV panel characteristics by means of a reduced number of model parameters.A large amount of simplified one-diode models, obtained by changing the used set of performance data, the adopted hypotheses and the analytical procedures for evaluating the model parameters, have been presented [64][65][66][67][68][69][70][71][72].
The selection of the model fit for purpose may be a difficult task that should carefully consider both the strong points and weaknesses of the examined method.Besides the achievable precision, each model has a different usability, as it needs specific performance data, which may be not available or difficult to extract from the available datasheets.The model also presents computation The second diode was added to consider the effect of the carrier recombination in the depletion region.The two-diode equivalent circuit of a PV module is described by the equation: where, following the traditional theory, the photocurrent I L depends on the solar irradiance, the diode saturation currents I 01 and I 02 are affected by the cell temperature, n 1 = a 1 N cs k/q and n 2 = a 2 N cs k/q are the diode quality factors, a 1 and a 2 are the diode shape factors, N cs is the number of cells of the panel that are connected in series.The values of R s , R sh , I 01 and I 02 variously affect the I-V characteristic of the PV panel [3].Because the evaluation of the parameters contained in the two-diode equivalent circuit is a complex problem, the one-diode equivalent circuit, depicted in Figure 2, was also considered.
Energies 2016, 9, 1019 2 of 40 The second diode was added to consider the effect of the carrier recombination in the depletion region.The two-diode equivalent circuit of a PV module is described by the equation: 01 02 1 1 where, following the traditional theory, the photocurrent IL depends on the solar irradiance, the diode saturation currents I01 and I02 are affected by the cell temperature, n1 = a1Ncsk/q and n2 = a2Ncsk/q are the diode quality factors, a1 and a2 are the diode shape factors, Ncs is the number of cells of the panel that are connected in series.The values of Rs, Rsh, I01 and I02 variously affect the I-V characteristic of the PV panel [3].Because the evaluation of the parameters contained in the two-diode equivalent circuit is a complex problem, the one-diode equivalent circuit, depicted in Figure 2, was also considered.Many authors have proposed analytical procedures for determining the model parameters on the basis of the performance data usually provided by manufacturers .The identification of the parameters contained in the diode-based equivalent circuits has been also tackled exploring the possibility of using different procedures such as Lambert W-function, evolutionary algorithms, Padè approximants, genetic algorithms, cluster analysis, artificial neural networks, harmony search-based algorithms, small perturbations around the operating point and reduced forms [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63].Other authors have investigated some simplified versions of the one-diode equivalent circuit in order to obtain an adequate representation of the PV panel characteristics by means of a reduced number of model parameters.A large amount of simplified one-diode models, obtained by changing the used set of performance data, the adopted hypotheses and the analytical procedures for evaluating the model parameters, have been presented [64][65][66][67][68][69][70][71][72].
The selection of the model fit for purpose may be a difficult task that should carefully consider both the strong points and weaknesses of the examined method.Besides the achievable precision, each model has a different usability, as it needs specific performance data, which may be not available or difficult to extract from the available datasheets.The model also presents computation One-diode equivalent circuit for a PV panel.
Many authors have proposed analytical procedures for determining the model parameters on the basis of the performance data usually provided by manufacturers .The identification of the parameters contained in the diode-based equivalent circuits has been also tackled exploring the possibility of using different procedures such as Lambert W-function, evolutionary algorithms, Padè approximants, genetic algorithms, cluster analysis, artificial neural networks, harmony search-based algorithms, small perturbations around the operating point and reduced forms [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63].Other authors have investigated some simplified versions of the one-diode equivalent circuit in order to obtain an adequate representation of the PV panel characteristics by means of a reduced number of model parameters.A large amount of simplified one-diode models, obtained by changing the used set of performance data, the adopted hypotheses and the analytical procedures for evaluating the model parameters, have been presented [64][65][66][67][68][69][70][71][72].
The selection of the model fit for purpose may be a difficult task that should carefully consider both the strong points and weaknesses of the examined method.Besides the achievable precision, each model has a different usability, as it needs specific performance data, which may be not available or difficult to extract from the available datasheets.The model also presents computation difficulties, which may require the use of mathematical tools ranging from simple algorithms to complex methods Energies 2016, 9, 1019 3 of 41 implemented in dedicated computational software.The usability is a qualitative parameter, whereas the accuracy achievable by a model requires a quantitative assessment.In order to select the simplified one-diode model which represents the best compromise between analytical complexity and expected accuracy, it is necessary to perform a complex synthesis of both qualitative and quantitative features.The criterion proposed in this paper, which is used to rate the performances of some of the most famous simplified one-diode models, can help researchers and designers, working in the area of photovoltaic systems, to select the model fit for purpose.
The paper is organised along the lines of a previous study regarding the one-diode models for PV modules [1].The criterion adopts a three-level rating scale that considers the ease of finding the data used by the analytical procedure, the simplicity of the mathematical tools needed to perform calculations and the accuracy achieved in calculating the current and power.Section 2 presents the simplified one-diode model and the effects of the series resistance on the shape of the I-V curves; Section 3 lists chronologically the most famous simplified one-diode equivalent circuits along with the used performance data, the required mathematical tools and the operative steps to obtain the model parameters.In Section 4, the accuracy of the tested simplified one-diode models is evaluated by calculating the I-V characteristics of some PV modules and comparing them with the performance curves issued by manufacturers.A criterion for rating the usability and accuracy of the analysed one-diode models is presented in Section 5.The minute descriptions of the mathematical procedures used to get the explicit or implicit expressions necessary to calculate the model parameters are listed in the Appendix A; such a review also contains the sequence of operative steps to easily calculate the model parameters.

The Simplified One-Diode Equivalent Circuit
The one-diode model depicted in Figure 2 is described by the well-known equation: where diode quality factor n = aN cs k/q and a is the diode shape factor.Despite its simplicity, the one-diode model adequately reproduces the I-V characteristic at standard rating conditions (SRC)-irradiance G ref = 1000 W/m 2 , cell temperature T ref = 25 • C and average solar spectrum at AM 1.5-of most of the modern and efficient crystalline PV modules.Because of their small series resistance and great shunt resistance, crystalline PV modules show a good fill factor and, consequently, an I-V characteristic with a very sharp bend.The model is based on parameters I L , I 0 , n, R s and R sh whose calculation generally requires the solution of an equation system containing five independent relations obtained from Equation (2) or from its derivative.The mathematical difficulties encountered in the simultaneous solution of the involved implicit transcendent equations have suggested solving the problem by introducing some simplifications in the one-diode equivalent model.The four-parameter model depicted in Figure 3, in which resistance R sh is set equal to infinity, has been often proposed.
difficulties, which may require the use of mathematical tools ranging from simple algorithms to complex methods implemented in dedicated computational software.The usability is a qualitative parameter, whereas the accuracy achievable by a model requires a quantitative assessment.In order to select the simplified one-diode model which represents the best compromise between analytical complexity and expected accuracy, it is necessary to perform a complex synthesis of both qualitative and quantitative features.The criterion proposed in this paper, which is used to rate the performances of some of the most famous simplified one-diode models, can help researchers and designers, working in the area of photovoltaic systems, to select the model fit for purpose.The paper is organised along the lines of a previous study regarding the one-diode models for PV modules [1].The criterion adopts a three-level rating scale that considers the ease of finding the data used by the analytical procedure, the simplicity of the mathematical tools needed to perform calculations and the accuracy achieved in calculating the current and power.Section 2 presents the simplified one-diode model and the effects of the series resistance on the shape of the I-V curves; Section 3 lists chronologically the most famous simplified one-diode equivalent circuits along with the used performance data, the required mathematical tools and the operative steps to obtain the model parameters.In Section 4, the accuracy of the tested simplified one-diode models is evaluated by calculating the I-V characteristics of some PV modules and comparing them with the performance curves issued by manufacturers.A criterion for rating the usability and accuracy of the analysed one-diode models is presented in Section 5.The minute descriptions of the mathematical procedures used to get the explicit or implicit expressions necessary to calculate the model parameters are listed in the Appendix A; such a review also contains the sequence of operative steps to easily calculate the model parameters.

The Simplified One-Diode Equivalent Circuit
The one-diode model depicted in Figure 2 is described by the well-known equation: where diode quality factor n = aNcsk/q and a is the diode shape factor.Despite its simplicity, the one-diode model adequately reproduces the I-V characteristic at standard rating conditions (SRC)-irradiance Gref = 1000 W/m 2 , cell temperature Tref = 25 °C and average solar spectrum at AM 1.5-of most of the modern and efficient crystalline PV modules.Because of their small series resistance and great shunt resistance, crystalline PV modules show a good fill factor and, consequently, an I-V characteristic with a very sharp bend.The model is based on parameters IL, I0, n, Rs and Rsh whose calculation generally requires the solution of an equation system containing five independent relations obtained from Equation (2) or from its derivative.The mathematical difficulties encountered in the simultaneous solution of the involved implicit transcendent equations have suggested solving the problem by introducing some simplifications in the one-diode equivalent model.The four-parameter model depicted in Figure 3, in which resistance Rsh is set equal to infinity, has been often proposed.The four-parameter model is governed by the following equation: As shown in Figure 4, series resistance R s impacts the shape of the I-V characteristic close and beyond the MPP, which is approximately set on the "knee" of the curve.The four-parameter model is governed by the following equation: As shown in Figure 4, series resistance Rs impacts the shape of the I-V characteristic close and beyond the MPP, which is approximately set on the "knee" of the curve.At a constant value of the solar irradiance, if the series resistance is lowered, the internal dissipation of energy is reduced and the panel becomes more efficient; the MPP will slide towards right and the "knee" will be sharper because the value of the open circuit voltage is not affected by the series resistance. .Conversely, smoother curves and greater values of Rs characterize the PV cells that are made with more energy dissipative materials and/or present higher electrical connection resistances.The analytical procedures proposed to calculate the four-parameter model generally require the following input data, which are usually available in the manufacturer datasheets: Sometimes, the number of series connected PV cells, or the derivative of the I-V curve at the MPP are also required.Because of the presence of current I in both terms of transcendent Equation (3), the solution of the four-equation system, which is necessary to calculate the model parameters, cannot be obtained by means of exact mathematical methods.To solve the problem, both approximate forms of the equations and numerical solving techniques have been used.

Usability of the Simplified One-Diode Models
Numerous procedures to calculate the parameters of the simplified one-diode model have been proposed.Some of these models were presented about 25 years ago.Nevertheless, they are still considered as effective and accurate as the most recent simplified one-diode models.Townsend [64] analysed several models of PV cells and panels and proposed a four-parameter model to be the most appropriate one for assessing the long-term performance of direct-coupled PV systems.In order to define the values of the model parameters, Townsend described three different procedures that in this paper are called Townsend n.1, Townsend n.2 and Townsend n.3, respectively.Other models were proposed by Duffie et al. [65], Xiao et al. [66], Ulapane et al. [67], Saloux et al. [68], Mahmoud et At a constant value of the solar irradiance, if the series resistance is lowered, the internal dissipation of energy is reduced and the panel becomes more efficient; the MPP will slide towards right and the "knee" will be sharper because the value of the open circuit voltage is not affected by the series resistance.Conversely, smoother curves and greater values of R s characterize the PV cells that are made with more energy dissipative materials and/or present higher electrical connection resistances.The analytical procedures proposed to calculate the four-parameter model generally require the following input data, which are usually available in the manufacturer datasheets: • open circuit voltage temperature coefficient µ V,oc and short circuit current temperature coefficient µ I,sc .
Sometimes, the number of series connected PV cells, or the derivative of the I-V curve at the MPP are also required.Because of the presence of current I in both terms of transcendent Equation (3), the solution of the four-equation system, which is necessary to calculate the model parameters, cannot be obtained by means of exact mathematical methods.To solve the problem, both approximate forms of the equations and numerical solving techniques have been used.

Usability of the Simplified One-Diode Models
Numerous procedures to calculate the parameters of the simplified one-diode model have been proposed.Some of these models were presented about 25 years ago.Nevertheless, they are still considered as effective and accurate as the most recent simplified one-diode models.Townsend [64] analysed several models of PV cells and panels and proposed a four-parameter model to be the most appropriate one for assessing the long-term performance of direct-coupled PV systems.In order to define the values of the model parameters, Townsend described three different procedures that in this paper are called Townsend n.1, Townsend n.2 and Townsend n.3, respectively.Other models were proposed by Duffie et al. [65], Xiao et al. [66], Ulapane et al. [67], Saloux et al. [68], Mahmoud et al. [69,70], Cristaldi et al. [71] and Averbukh et al. [72].The model parameters are always evaluated by solving an equation system that represents the information related to the physical properties of the PV panels and/or geometrical conditions concerning the I-V characteristics.
The usability of a procedure may be significantly lowered by the difficulties encountered in using it.For this reason to assess the usability rating it is necessary to explore the complete sequence of operative steps that permit to calculate the model parameters.Sometimes the solution of the equation system is obtained adopting some simplifying hypotheses and iterative procedures.Some models evaluate the parameters on the basis of a similar set of information, but do not adopt the same simplifying hypotheses and/or use different relations to describe the dependence on the cell temperature and/or the solar irradiance.A synthetic description of the used information, simplifying hypotheses and solving techniques is contained in the paper and the analytical procedures to calculate the model parameters are minutely described in the Appendix A. Because the simplifying hypotheses are quite realistic, the values of the model parameters obtained by the approximate procedure may result almost equal to the values calculated without recourse to mathematical simplifications.Such an occurrence has been observed for some of the analysed models and consequently produces very similar I-V curves whose comparison may be considered trivial.Nevertheless, for the sake of completeness, the comparison of these models is also presented in this paper.

Townsend n.1 Model
The Townsend n.1 model [64] uses the following information: which are described by the four independent equations listed in the Appendix A. No simplifying hypothesis is assumed and the equation system is solved by means of the Newton-Raphson method.

Townsend n.2 Model
The Townsend n.2 model [64] is based on the same information of the Townsend n.1 model: (4) derivative of power at the MPP [∂P/∂V = 0 at The following hypotheses, which are usually verified for a PV module, are assumed: and the model parameters can be calculated using the explicit equations described in the Appendix A.

Townsend n.3 Model
For the Townsend n.3 model [64] the following information is used: The same hypotheses of Equation ( 4) are assumed.Due to the presence of implicit forms, the equation system is solved with the iterative procedure described in the Appendix A.

Duffie and Beckman Model
The Duffie and Beckman model [65] is based on the same information used by the Townsend n.3 model: Due to the adoption of the hypotheses described in Equation ( 4), the model does not require any iterative procedure and parameters I L,ref , I 0,ref , R s and n can be calculated with the explicit equations listed in the Appendix A.

Xiao, Dunford and Capel Model
Xiao et al. [66] presented a four-parameter model whose parameters are calculated on the basis of the following information: Assuming the following hypothesis: 3.6.Ulapane, Dhanapala, Wickramasinghe, Abeyratne, Rathnayake and Binduhewa Model The model proposed by Ulapane et al. [67] uses the same information and hypothesis adopted by Xiao et al. ( The model parameters are calculated with the iterative procedure described in the Appendix A. A different approach is used to describe the physical behaviour of the PV panel for conditions far from the SRC.

Saloux, Teyssedou and Sorin Model
A three-parameter model, in which resistance R s is set equal to zero, was proposed by Saloux at al. [68].The model uses the following simplified analytical equation: Energies 2016, 9, 1019 7 of 41 The model parameters are calculated imposing that the following points belong to the I-V curve: (1) short circuit point [I = I sc,ref ; Assuming the following hypotheses: the parameters of the model can be easily calculated with the explicit equations listed in the Appendix A.

Mahmoud, Xiao and Zeineldin n.1 Model
Mahmoud et al. [69] presented a three-parameter model based on Equation ( 6) whose parameters were calculated using the same information adopted by Saloux et al. ( Numerical methods are used to solve the equations listed in the Appendix A. A different approach is adopted to describe the physical behaviour of the PV panel for conditions far from the SRC.

Cristaldi, Faifer, Rossi and Toscani Model
Cristaldi et al. [71] proposed a four-parameter model based on the following information: The following hypotheses are adopted: and the model parameters are calculated by means of the analytical procedure based on the explicit equations listed in the Appendix A.

Averbukh, Lineykin and Kuperman Model
The model proposed by Averbukh et al. [72] uses on the same information used by Cristaldi et al. ( (4) derivative of power at the MPP [∂P/∂I = 0 at Instead of using Equation (3), the following equivalent expression is adopted: The equations listed in the Appendix A, which describe the short circuit, open circuit, maximum power points and the derivative of power at the MPP, are normalized using six per-unit dimensionless parameters and solved by means of a modern dedicate software.
In order to state the equivalent circuit representation and calculate the model parameters, the procedure described in the Appendix A is adopted.

Summary of the Information Used by the Models
In order to better appreciate the analogies and differences between the various models, the used information, hypotheses and solving tools are summarised in Table 1.It can be observed that the same information is often shared among different models.For this reason one may suppose that these models should be quite similar and yield the same results.Actually, because different simplifying hypotheses, solving techniques and relations to evaluate the PV panel performance curves at conditions different from SRC are adopted, each model has a particular capability to reproduce the I-V characteristics by means of mathematical approaches, which can be very simple or require the implementation of iterative routines and the use of specific mathematical methods and computer software.

Accuracy of the Simplified One-Diode Models
With the aim of verifying the accuracy of the analysed procedures, a comparison between the simplified one-diode models was made using the I-V characteristics extracted from the manufacturer datasheets.For the sake of brevity only two PV modules, based on different technologies, were considered.Obviously, even using a greater number of PV modules, the comparison would never be exhaustive because the results are strongly affected by the particular shape of the considered I-V characteristics.Moreover, the purpose of this paper is not ranking the best or the worst among the analysed models, but only defining the range of predictable precision in order to calibrate the criterion.The performance data of the simulated PV modules are listed in Table 2. Considering both the constant solar irradiance and the constant cell temperature curves, numerous points were extracted from the I-V characteristics issued by the manufacturers in order to get a reliable comparison between the calculated and the measured data.Tables 3 and 4 list the values of the parameters evaluated with the analysed models.3 and 4 were used to calculate the I-V characteristics of the selected PV panels.The Townsend n.Observing Figures 5-16 it can be generally deduced that all models result less accurate for voltage values greater than the MPP voltage.Moreover, it can be also noted that the simplified one-diode models are more precise if they are used to evaluate the I-V characteristics of the Kyocera PV panel.This occurrence may be due to the different shape of the I-V curves used to compare the analysed models.Actually, the I-V characteristics of the Sanyo module generally show sharper "knees" close to the MPP, probably due to the used heterojunction with intrinsic thin layer (HIT) technology.Moreover, it can be observed that models that use similar values of the parameters listed in Tables 2 and 3 yield different I-V curves for values of the solar irradiance and the cell temperature far from the SRC; this condition is obviously due to the different approaches adopted to describe the effects of the solar irradiance and the cell temperature.In this respect, the models of Xiao et al.Mahmoud et al. n.1 and of Averbukh et al. seem to be less accurate.In order to quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated using the following expressions:  Observing Figures 5-16 it can be generally deduced that all models result less accurate for voltage values greater than the MPP voltage.Moreover, it can be also noted that the simplified one-diode models are more precise if they are used to evaluate the I-V characteristics of the Kyocera PV panel.This occurrence may be due to the different shape of the I-V curves used to compare the analysed models.Actually, the I-V characteristics of the Sanyo module generally show sharper "knees" close to the MPP, probably due to the used heterojunction with intrinsic thin layer (HIT) technology.Moreover, it can be observed that models that use similar values of the parameters listed in Tables 2 and 3 yield different I-V curves for values of the solar irradiance and the cell temperature far from the SRC; this condition is obviously due to the different approaches adopted to describe the effects of the solar irradiance and the cell temperature.In this respect, the models of Xiao et al.Mahmoud et al. n.1 and of Averbukh et al. seem to be less accurate.In order to quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated using the following expressions: Observing Figures 5-16 it can be generally deduced that all models result less accurate for voltage values greater than the MPP voltage.Moreover, it can be also noted that the simplified one-diode models are more precise if they are used to evaluate the I-V characteristics of the Kyocera PV panel.This occurrence may be due to the different shape of the I-V curves used to compare the analysed models.Actually, the I-V characteristics of the Sanyo module generally show sharper "knees" close to the MPP, probably due to the used heterojunction with intrinsic thin layer (HIT) technology.Moreover, it can be observed that models that use similar values of the parameters listed in Tables 2 and 3 yield different I-V curves for values of the solar irradiance and the cell temperature far from the SRC; this condition is obviously due to the different approaches adopted to describe the effects of the solar irradiance and the cell temperature.In this respect, the models of Xiao et al.Mahmoud et al. n.1 and of Averbukh et al. seem to be less accurate.In order to quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated using the following expressions: Energies 2016, 9, 1019 15 of 41 V iss,j I calc,j − V iss,j I iss,j (12) in which V iss,j and I iss,j are the voltage and current of the j-th point extracted from the I-V characteristics issued by manufacturers, I calc,j is the value of the current calculated in correspondence of V iss,j and N is the number of extracted points.Moreover, in order to assess the range of dispersion of the results, also the maximum difference (MD) for current and power was evaluated using the following relations: In Tables 5 and 6, the percentage ratios of MAD(I) to the current at the issued MPP, and of MAD(P) to the rated maximum power, are listed.The underline represents the highest value.
Table 6.Percentage ratio of MAD(P) to the rated maximum power.The underline represents the highest value of percentage ratio of MAD(P).
Energies 2016, 9, 1019 In the last column the average values of the ratios of MAD(I) to the current at the issued MPP, and of MAD(P) to the rated maximum power, calculated for all I-V curves, are listed.For the Kyocera PV panel the smallest MAD(I)s range from 0.97% to 2.71% of the current at the MPP; the greatest MAD(I)s vary from 3.94% to 15.03%.The smallest MAD(I)s for the Sanyo PV module are in the range 0.92% to 3.18% of the current at the MPP; the greatest MAD(I)s range from 4.19% to 13.13%.The smallest MAD(P)s range from 0.97% to 2.49% of the rated maximum power for the Kyocera PV panel; the greatest MAD(P)s vary from 4.35% to 14.26%.For the Sanyo PV module the smallest MAD(P)s are in the range 0.93% to 3.05% of the rated maximum power; the greatest MAD(P)s vary from 4.64% to 13.25%.In Tables 7 and 8 the values of MD(I) and MD(P) for the analysed panels, calculated considering the I-V curves at a constant cell temperature of 25 • C, are listed.The underline represents the highest value of maximum current differences for each irradiance.
Energies 2016, 9, 1019 Table 8.Maximum current differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, at temperature T = 25 The underline represents the highest value of maximum current differences for each irradiance.
Considering the I-V curves at constant temperature of the Kyocera PV panel, the Townsend n.2 model seems to be the most accurate; the MD(I)s vary from 0.161 A to 0.259 A. The greatest current differences, which are contained in the range from 0.906 A to 1.379 A, are observed for the Duffie et   The underline represents the highest value of maximum current differences for each temperature.The underline represents the highest value of maximum current differences for each temperature.
The smallest MD(I)s for the Kyocera PV module at constant solar irradiance are obtained by adopting the Townsend n.The underline represents the highest value of maximum current differences for each irradiance.
Table 12.Maximum power differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, at temperature T = 25   The underline represents the highest value of maximum current differences for each temperature.The underline represents the highest value of maximum current differences for each temperature.
Considering the MD(P)s at constant solar irradiance, the smallest values for the Kyocera PV panel are obtained by means of the Townsend n.

Rating of the Usability and Accuracy of the Simplified One-Diode Models
In order to rate the usability and accuracy of the analysed models, the same approach used in [1] was adopted.The rating criterion is based on a three-level rating scale that takes into consideration the following features: • the ease of finding the performance data used by the analytical procedure; • the simplicity of the mathematical tools needed to perform calculations; • the accuracy achieved in calculating the current and power of the analysed PV modules.
The ease of finding the input data is assumed: The simplicity of the used mathematical tools is considered: • high, if only simple calculations are necessary; • medium, if an iterative procedure, which not necessarily requires dedicated computational software, is used; • low, when the analytical procedure requires the use of numerical solvers or complex mathematical methods usually implemented in dedicated computational software.
Table 15 lists the average ratios of MAD(I) to the rated current at the MPP, and of MAD(P) to the rated maximum power, extracted from Tables 5 and 6.
Table 15.Average ratios of MAD(I) to the rated current at the MPP andof MAD(P) to the rated maximum power.It can be observed that the global accuracy listed in Table 15, which is calculated averaging the accuracies evaluated for the Kyocera and Sanyo PV panels, ranges from 2.34% to 4.68%.Such range of variation was divided in three equal intervals, which were used to qualitatively describe the accuracy of the analysed models:

Figure 1 .
Figure 1.Two-diode equivalent circuit for a PV panel.

Figure 2 .
Figure 2. One-diode equivalent circuit for a PV panel.

Figure 1 .
Figure 1.Two-diode equivalent circuit for a PV panel.

Figure 1 .
Figure 1.Two-diode equivalent circuit for a PV panel.

Figure 2 .
Figure 2. One-diode equivalent circuit for a PV panel.

Figure 4 .
Figure 4. Effects of the series resistance on the I-V characteristic.

Figure 4 .
Figure 4. Effects of the series resistance on the I-V characteristic.
) derivative of voltage at the open circuit point [∂V oc /∂T = µ V,oc at G = 1000 W/m 2 ].
,ref , I 0,ref , n and R s can be calculated with the iterative procedure described in the Appendix A.

3. 11 .
Mahmoud, Xiao and Zeineldin n.2 Model Mahmoud et al. [70] presented a procedure to automatically transform the five-parameter model into a four-parameter model, in which only the series resistance, or only the shunt resistance, is present.The equivalent circuit representation depends on the physical properties of the simulated PV panel.The model is based on the same information used by Cristaldi et al. and Averbukh et al. (1) short circuit point [I = I sc,ref ; V = 0]; (2) open circuit point [I = 0; V = V oc,ref ]; (3) MPP [I = I mp,ref ; V = V mp,ref ]; (4) derivative of power at the MPP [∂P/∂I = 0 at I = I mp,ref ; V = V mp,ref ]; and assumes the following hypotheses: e I sc,re f Rs nT re f 3 model was not considered because it perfectly corresponds to the Duffie et al. model.The Cristaldi et al. model was not taken into account because the I-V curves calculated with the model perfectly overlap the characteristics obtained from the Ulapane et al. model for all values of solar irradiance and cell temperature.Actually, the results are numerically indistinguishable because the only difference, which should make the Cristaldi et al. model a bit more imprecise, is due to the last two hypotheses described in Equation (8), which are thoroughly confirmed by real PV modules.In Figures 5-10 the I-V curves, evaluated at T = 25 • C, are compared with the characteristics issued by manufacturers.Energies 2016, 9, 1019 10 of 40 The Cristaldi et al. model was not taken into account because the I-V curves calculated with the model perfectly overlap the characteristics obtained from the Ulapane et al. model for all values of solar irradiance and cell temperature.Actually, the results are numerically indistinguishable because the only difference, which should make the Cristaldi et al. model a bit more imprecise, is due to the last two hypotheses described in Equation (8), which are thoroughly confirmed by real PV modules.In Figures 5-10 the I-V curves, evaluated at T = 25 °C, are compared with the characteristics issued by manufacturers.

Figure 5 .
Figure 5.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 6 .
Figure 6.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 5 .
Figure 5.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 • C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 5 .
Figure 5.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 6 .
Figure 6.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 6 .
Figure 6.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 • C and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 7 .
Figure 7.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 8 .
Figure 8.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Xiao et al., the Averbukh et al. and the Saloux et al. models.

Figure 7 . 40 Figure 7 .
Figure 7.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at T = 25 • C and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 8 .
Figure 8.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Xiao et al., the Averbukh et al. and the Saloux et al. models.

Figure 8 . 40 Figure 7 .
Figure 8.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 • C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 8 .
Figure 8.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Xiao et al., the Averbukh et al. and the Saloux et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = 25 • C and the characteristics calculated by means of the Xiao et al., the Averbukh et al. and the Saloux et al. models.

Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = °C and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 11 .
Figure 11.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = • C and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at T = °C and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 11 .
Figure 11.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 11 .
Figure 11.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 12 .
Figure 12.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 14 .
Figure 14.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 12 . 40 Figure 12 .
Figure 12.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 14 .
Figure 14.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 13 . 40 Figure 12 .
Figure 13.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of the Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

Figure 14 .
Figure 14.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 14 .
Figure 14.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Townsend n.1, the Townsend n.2 and the Ulapane et al. models.

Figure 15 .
Figure 15.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 16 .
Figure 16.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

2 Figure 15 . 40 Figure 15 .
Figure 15.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Xiao et al. the Averbukh et al. and the Saloux et al. models.

Figure 16 .
Figure 16.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.

2 Figure 16 .
Figure 16.Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Duffie et al. and the Mahmoud et al. models.
al. and the Xiao et al. model.The smallest MD(I)s result for the Sanyo PV module by using the Townsend n.2, the Duffie et al. the Saloux et al. and the Mahmoud et al. n.1 models.These differences are in the range −0.356 to 0.632 A. The greatest inaccuracies derive from the Xiao et al., the Saloux et al. and the Mahmoud et al. n.1 models.For these models differences varying between 0.888 A and 1.403A were calculated.Tables 9 and 10 list the MD(I)s calculated for Kyocera KD245GH-4FB2 and Sanyo HIT-240 HDE4 PV panels at a constant solar irradiance of 1000 W/m 2 .
2 and the Ulapane et al. models.Such differences range from 0.206 A to 0.775 A. The greatest inaccuracies derive from the Duffie et al. and the Averbukh et al. models, for which differences varying between 1.332 A and 3.205 A are noted.The Townsend n.2 model seems to be the most accurate for the Sanyo PV panel; the MD(I)s vary from 0.531 A to 0.610 A. The greatest current differences, which are contained in the range from 1.221 A to 2.429 A, are provided by the Averbukh et al. the Saloux et al. and the Mahmoud et al. n.1 models.Tables11-14show the effectiveness of the analysed models to predict the power generated by PV modules.
2, the Xiao et al. and the Ulapane et al. models.The differences range −5.84 W to 20.76 W. The greatest inaccuracies derive from the Duffie et al. and the Averbukh et al. models.Differences, varying between 46.61 W and 97.01 W, were calculated.Townsend n.2 model yields the smallest MD(P)s for the Sanyo PV module, which are in the range from 19.19 W to 24.26 W. The greatest values are obtained with the Averbukh et al., the Saloux et al. and the Mahmoud et al. n.1 models.Such differences vary from 49.94 W to 92.72 W.

•
high, when only tabular data are required (short circuit current, open circuit voltage, MPP current and voltage); • medium, when the data have to be extracted by reading the I-V characteristics (open circuit voltage at conditions different from the SRC); • low, when the derivative of the I-V curves are required.
Nomenclature a, a 1 , a 2 diode shape factors G solar irradiance (W/m 2 ) G ref solar irradiance at the SRC (1000 W/m 2 ) I current generated by the panel (A) I calc,j current of the j-th calculated point of the I-V characteristic (A) I iss,j current of the j-th point extracted from the issued I-V characteristic (A) I L photocurrent (A) I L,ref photocurrent (A) at the SRC (A) I mp,ref current in the maximum power point at the SRC (A) I sc short circuit current of the panel (A) I sc,ref short circuit current of the panel at the SRC (A) I 0 , I 01 , I 02 diode saturation current (A) I 0,ref diode saturation current at the SRC (A) k Boltzmann constant (J/K) n, n 1 , n 2 diode quality factors (V/K) N number of points extracted from the issued I-V characteristic N cs number of cells connected in series P power generated by the panel (W) q electron charge (C) R s series resistance (Ω) R sh shunt resistance (Ω) T temperature of the PV cell ( • K) T ref temperature of the PV panel at the SRC (25 • C-298.15• K) V voltage generated by the PV panel (V) V oc open circuit voltage of the PV panel (V) V oc,ref open circuit voltage of the PV panel at the SRC (V) V iss,j voltage of the j-th point extracted from the issued I-V characteristic (A) V mp,ref voltage in the maximum power point at the SRC (V) V T voltage in the maximum power point at the SRC (V) ε G bandgap energy of the material (eV) µ I,sc thermal coefficient of the short circuit current (A/ • C) µ V,oc thermal coefficient of the open circuit voltage (V/ • C) ref and short circuit current I sc,ref at the SRC; • voltage V mp,ref and current I mp,ref at the MPP at the SRC;

Table 1 .
Summary of the information and solving techniques used by the simplified one-diode models.

Table 2 .
Performance data of the simulated PV panels.

Table 3 .
Model parameters of the Kyocera KD245GH-4FB2 at the SRC.

Table 4 .
Model parameters of the Sanyo HIT-240 HDE4 at the SRC.For the analysed PV modules, the procedure proposed by the Mahmoud et al. n.2 model always generated an equivalent circuit representation in which only the series resistance is present.The values of Tables

Table 5 .
Percentage ratio of MAD(I) to the rated current at the MPP.

Table 7 .
Maximum current differences between the calculated and the issued I-V characteristics of Kyocera KD245GH-4FB2, at temperature T = 25 • C.

Table 9 .
Maximum current differences between the calculated and the issued I-V characteristics of Kyocera KD245GH-4FB2, at irradiance G = 1000 W/m 2 .

Table 10 .
Maximum current differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, at irradiance G = 1000 W/m 2 .

Table 11 .
Maximum power differences between the calculated and the issued I-V characteristics of Kyocera KD245GH-4FB2, at temperature T = 25 • C.

Table 12 .
Cont.The underline represents the highest value of maximum current differences for each irradiance.For the Kyocera PV panel, the smallest MD(P)s at constant cell temperature are again ascribed to the Townsend n.2 model that yields values varying from −5.84 W to 8.41 W. The greatest MD(P)s, which occur with the Xiao et al. and the Duffie et al. models, are in the range 32.15 W to 48.28 W. For the Sanyo PV module, the smallest MD(P)s at constant temperature, which vary from −14.46 W to 24.90 W, are obtained by means of the Townsend n.2, the Duffie et al. the Saloux et al. and the Mahmoud et al. n.1 models.The Xiao et al., the Saloux et al. and the Mahmoud et al. n.1 models yield the greatest inaccuracies, which vary from 35.80 W to 57.83 W.

Table 13 .
Maximum power differences between the calculated and the issued I-V characteristics of Kyocera KD245GH-4FB2, at irradiance G = 1000 W/m 2 .

Table 14 .
Maximum power differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, at irradiance G = 1000 W/m 2 .