A Criterion for Rating the Usability and Accuracy of the One-Diode Models for Photovoltaic Modules

Abstract: In selecting a mathematical model for simulating physical behaviours, it is important to reach an acceptable compromise between analytical complexity and achievable precision. With the aim of helping researchers and designers working in the area of photovoltaic systems to make a choice among the numerous diode-based models, a criterion for rating both the usability and accuracy of one-diode models is proposed in this paper. A three-level rating scale, which 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, is used. The proposed criterion is tested on some one-diode equivalent circuits whose analytical procedures, hypotheses and equations are minutely reviewed along with the operative steps to calculate the model parameters. To assess the achievable accuracy, the current-voltage (I-V) curves at constant solar irradiance and/or cell temperature obtained from the analysed models are compared to the characteristics issued by photovoltaic (PV) panel manufacturers and the differences of current and power are calculated. The results of the study highlight that, even if the five parameter equivalent circuits are suitable tools, different usability ratings and accuracies can be observed.


Introduction
In the field of photovoltaics, the diode-based equivalent circuits of photovoltaic (PV) cells and modules have been widely used because they allow the designer to optimize the system performance and maximize the effectiveness of the economic investment.The use of accurate models of the electrical behaviour of PV modules, which is obviously important to implement simulation tools, is also required to test the dynamic performances of the inverters equipped with the maximum power point tracker (MPPT).In order to get results, which would be confirmed by experimental measurements, the practical effectiveness of MPPT control algorithms was analysed using different diode-based equivalent circuits [1][2][3].It was observed that the study of the transient conditions, which are quite common for an MPPT, requires very reliable equations describing the behaviour of the PV array working far from the standard rating conditions [4].
Usability and accuracy, which have been already addressed by other authors [5], are important features that have to be carefully considered before deciding the mathematical model to be adopted to simulate the electrical behaviour of PV devices.The usability is mainly affected by the mathematical difficulties, which may be encountered in performing calculations, and the unavailability of the performance data used to evaluate the model parameters.Before starting working, it would be preferable to have a clear idea about the need of specific performance data, which may be not available Energies 2016, 9, 427 2 of 48 or difficult to extract from the issued datasheets, and the computation difficulties, which may require the use of mathematical tools ranging from simple algorithms to complex methods implemented in dedicated computational software.Also the accuracy is a relevant parameter, even though its achievable level may significantly depend on the physical characteristics of the modelled PV devices.
To better understand the origin of the diode-based equivalent circuits used for modelling PV devices it is useful to remember that, like a semiconductor diode, a PV cell consists of two layers of semiconductor material, usually silicon, differently doped, that are electrically connected to two metallic electrodes deposited on their outer surfaces.To better understand the origin of the diode-based equivalent circuits used for modelling PV devices it is useful to remember that a PV cell is a diode made with two layers of semiconductor material, usually silicon, differently doped, that are electrically connected to two metallic electrodes deposited on their outer surfaces.It is well-known [2,6] that the absorption of light in semiconductors can, under certain conditions, create an electric current due to the capability of the absorbed photons of converting fixed electrons in freely moving conduction electrons.
A silicon PV cell shares with semiconductor electronic devices, such as diodes and transistors, the same processing and manufacturing techniques used to create p-n junctions.An ideal PV cell behaves like an illuminated semiconductor diode whose I-V characteristic was described by Shockley [7] with the following equation: where I and V are the current and voltage, I L is the photocurrent generated by illumination, I 0 is the reverse saturation current of the diode, q is the electron charge (1.602 ˆ10 ´19 C), k is the Boltzmann constant (1.381 ˆ10 ´23 J/K), T is the junction temperature and diode ideality factor γ, in compliance with the traditional theory of semiconductors [8], is 1 for germanium and approximately 2 for silicon.Wolf [9] observed that in a PV cell the photocurrent is not generated by only one diode but it is the global effect of the presence of a multitude of elementary flanked diodes that are uniformly distributed throughout the surface that separates the two semiconductor slabs of the p-n junction.The assertion of Wolf arises from a significant difference between diodes and PV cells: unlike semiconductor diodes, the upper electrode of a PV cell is deposited with a discontinuous structure that embeds several metal elements (fingers), whose shape and size are chosen with the aim of maximizing the absorbing surface and minimizing the contact resistance between fingers and silicon.In a real diode the electrodes face each other and the carriers of electricity flow through the silicon slabs following linear paths, which are perpendicular to the junction.In contrast, into a PV cell, because of the discrete shape of the upper electrode, the carriers of electricity follow curved paths toward the fingers that collect them (Figure 1).
Energies 2016, 9, 427 2 of 47 difficulties, which may require the use of mathematical tools ranging from simple algorithms to complex methods implemented in dedicated computational software.Also the accuracy is a relevant parameter, even though its achievable level may significantly depend on the physical characteristics of the modelled PV devices.
To better understand the origin of the diode-based equivalent circuits used for modelling PV devices it is useful to remember that, like a semiconductor diode, a PV cell consists of two layers of semiconductor material, usually silicon, differently doped, that are electrically connected to two metallic electrodes deposited on their outer surfaces.To better understand the origin of the diode-based equivalent circuits used for modelling PV devices it is useful to remember that a PV cell is a diode made with two layers of semiconductor material, usually silicon, differently doped, that are electrically connected to two metallic electrodes deposited on their outer surfaces.It is well-known [2,6] that the absorption of light in semiconductors can, under certain conditions, create an electric current due to the capability of the absorbed photons of converting fixed electrons in freely moving conduction electrons.
A silicon PV cell shares with semiconductor electronic devices, such as diodes and transistors, the same processing and manufacturing techniques used to create p-n junctions.An ideal PV cell behaves like an illuminated semiconductor diode whose I-V characteristic was described by Shockley [7] with the following equation: where I and V are the current and voltage, IL is the photocurrent generated by illumination, I0 is the reverse saturation current of the diode, q is the electron charge (1.602 × 10 −19 C), k is the Boltzmann constant (1.381 × 10 −23 J/K), T is the junction temperature and diode ideality factor γ, in compliance with the traditional theory of semiconductors [8], is 1 for germanium and approximately 2 for silicon.Wolf [9] observed that in a PV cell the photocurrent is not generated by only one diode but it is the global effect of the presence of a multitude of elementary flanked diodes that are uniformly distributed throughout the surface that separates the two semiconductor slabs of the p-n junction.The assertion of Wolf arises from a significant difference between diodes and PV cells: unlike semiconductor diodes, the upper electrode of a PV cell is deposited with a discontinuous structure that embeds several metal elements (fingers), whose shape and size are chosen with the aim of maximizing the absorbing surface and minimizing the contact resistance between fingers and silicon.In a real diode the electrodes face each other and the carriers of electricity flow through the silicon slabs following linear paths, which are perpendicular to the junction.In contrast, into a PV cell, because of the discrete shape of the upper electrode, the carriers of electricity follow curved paths toward the fingers that collect them (Figure 1).As a result of the unequal length of the current paths, the carriers of electricity flow through different thicknesses of semiconductor and different electrical resistances are thus opposed.As consequence, each elementary portion of the p-n junction has a different electrical behaviour and, As a result of the unequal length of the current paths, the carriers of electricity flow through different thicknesses of semiconductor and different electrical resistances are thus opposed.As consequence, each elementary portion of the p-n junction has a different electrical behaviour and, in turn, a different I-V characteristic.In order to get a realistic representation, a PV cell may be Energies 2016, 9, 427 3 of 48 approximated with the distribute constants electric circuit of Figure 2, which contains a multitude of elementary lumped components composed of a current generator and a diode.Numerous electrical resistances take account of various dissipative effects.Major contributions to the internal series resistance come from the sheet resistance of the p-layer, the bulk resistance of the n-layer and the resistance between the semiconductor layers and the metallic contacts.
Energies 2016, 9, 427 3 of 47 in turn, a different I-V characteristic.In order to get a realistic representation, a PV cell may be approximated with the distribute constants electric circuit of Figure 2, which contains a multitude of elementary lumped components composed of a current generator and a diode.Numerous electrical resistances take account of various dissipative effects.Major contributions to the internal series resistance come from the sheet resistance of the p-layer, the bulk resistance of the n-layer and the resistance between the semiconductor layers and the metallic contacts.The elementary diodes are inter-connected by resistors Rp and Rn, which represent the transverse distributed resistances of p-layer and n-layer, respectively; resistors Rc are included to consider the contact resistance between the semiconductor and the fingers, or the back contact.Because such equivalent circuit would be too complex to be used, simplified equivalent circuits, which contain one or two diodes, a current generator and two resistors, were considered.
Because of the amount of combinations that can be obtained by changing the used set of performance data, the adopted hypotheses and the analytical procedures for evaluating the model parameters, a great number of one-diode models have been reported in the scientific literature.The selection of the model fit for purpose may be a difficult task, which cannot disregard some important aspects, such as: • the kind and availability of the performance data used by the model; • the reliability of the hypotheses on which the model is based; • the procedure followed to obtain the expressions used to calculate the model parameters; • the mathematical methods, tools and/or computer routines required to solve the equation system; • the robustness and stability of the mathematical approach; • the achievable precision of the model.
The features mentioned above affect the model effectiveness and change its usability rating and accuracy level.The usability is a qualitative parameter, whereas the accuracy achievable by a model requires a quantitative assessment.An aware choice of the one-diode model, which should be the best compromise between analytical complexity and expected accuracy, would require the capability of performing the complex synthesis of both qualitative and quantitative features.In order to help researchers and designers, working in the area of photovoltaic systems, to select the model fit for purpose, the usability and accuracy of some of the most famous one-diode models are overviewed and rated in this study.The paper is organised as follows: Section 2 presents the The elementary diodes are inter-connected by resistors R p and R n , which represent the transverse distributed resistances of p-layer and n-layer, respectively; resistors R c are included to consider the contact resistance between the semiconductor and the fingers, or the back contact.Because such equivalent circuit would be too complex to be used, simplified equivalent circuits, which contain one or two diodes, a current generator and two resistors, were considered.
Because of the amount of combinations that can be obtained by changing the used set of performance data, the adopted hypotheses and the analytical procedures for evaluating the model parameters, a great number of one-diode models have been reported in the scientific literature.The selection of the model fit for purpose may be a difficult task, which cannot disregard some important aspects, such as:

‚
the kind and availability of the performance data used by the model; ‚ the reliability of the hypotheses on which the model is based; ‚ the procedure followed to obtain the expressions used to calculate the model parameters; ‚ the mathematical methods, tools and/or computer routines required to solve the equation system; ‚ the robustness and stability of the mathematical approach; ‚ the achievable precision of the model.
The features mentioned above affect the model effectiveness and change its usability rating and accuracy level.The usability is a qualitative parameter, whereas the accuracy achievable by a model requires a quantitative assessment.An aware choice of the one-diode model, which should be the best compromise between analytical complexity and expected accuracy, would require the capability of performing the complex synthesis of both qualitative and quantitative features.In order to help researchers and designers, working in the area of photovoltaic systems, to select the model fit for purpose, the usability and accuracy of some of the most famous one-diode models are overviewed and rated in this study.The paper is organised as follows: Section 2 presents the five-parameter equivalent circuit.To assess the usability rating the analytical procedures of some one-diode models are synthetically reviewed in Section 3 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 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.Moreover, the model, which represents the ideal compromise between the usability and accuracy is highlighted.The appendix lists the detailed descriptions of the procedures used by the ranked models and the explicit or implicit expressions necessary to evaluate the model parameters; such a review also contains the sequence of operative steps to easily calculate the model parameters.

The One-Diode Equivalent Circuit
The one-diode equivalent circuit depicted in Figure 3 is characterized by five parameters, which are photocurrent I L , diode reverse saturation current I 0 , series resistance R s , shunt resistance R sh , and diode quality factor n = aN cs k/q, in which a is the shape factor, N cs is the number of cells of the panel that are connected in series, q is the electron charge (1.602 ˆ10 ´19 C) and k is the Boltzmann constant (1.381 ˆ10 ´23 J/K).
Energies 2016, 9, 427 4 of 47 five-parameter equivalent circuit.To assess the usability rating the analytical procedures of some one-diode models are synthetically reviewed in Section 3 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 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.Moreover, the model, which represents the ideal compromise between the usability and accuracy is highlighted.The appendix lists the detailed descriptions of the procedures used by the ranked models and the explicit or implicit expressions necessary to evaluate the model parameters; such a review also contains the sequence of operative steps to easily calculate the model parameters.

The One-Diode Equivalent Circuit
The one-diode equivalent circuit depicted in Figure 3 is characterized by five parameters, which are photocurrent IL, diode reverse saturation current I0, series resistance Rs, shunt resistance Rsh, and diode quality factor n = aNcsk/q, in which a is the shape factor, Ncs is the number of cells of the panel that are connected in series, q is the electron charge (1.602 × 10 −19 C) and k is the Boltzmann constant (1.381 × 10 −23 J/K).The one-diode model is described by the well-known equation: where T is the cell temperature.Following the traditional theory, the photocurrent depends on the solar irradiance and the diode reverse saturation current is affected by the cell temperature.The values of Rs, Rsh, n and I0 variously affect the I-V characteristic of the PV panel [83].The series and shunt resistances take account of dissipative phenomena and parasitic currents within the PV panel.At a constant value of the solar irradiance, the internal dissipation of energy is reduced if the series resistance is lowered and/or the shunt resistance is increased.As a consequence, the panel becomes more efficient because the maximum power point (MPP) slides towards right and the I-V curve becomes sharper.Thin-film PV cells and modules, which present significant values of Rs and Rsh due to the use of materials that are more energy dissipative than the mono or poly crystalline silicon, are usually characterized by smooth I-V curves.
The analytical procedures proposed to calculate the five-parameter model generally require the following data, some of them are provided by the manufacturer datasheets: The one-diode model is described by the well-known equation: where T is the cell temperature.Following the traditional theory, the photocurrent depends on the solar irradiance and the diode reverse saturation current is affected by the cell temperature.The values of R s , R sh , n and I 0 variously affect the I-V characteristic of the PV panel [83].The series and shunt resistances take account of dissipative phenomena and parasitic currents within the PV panel.At a constant value of the solar irradiance, the internal dissipation of energy is reduced if the series resistance is lowered and/or the shunt resistance is increased.As a consequence, the panel becomes more efficient because the maximum power point (MPP) slides towards right and the I-V curve becomes sharper.Thin-film PV cells and modules, which present significant values of R s and R sh due to the use of materials that are more energy dissipative than the mono or poly crystalline silicon, are usually characterized by smooth I-V curves.
The analytical procedures proposed to calculate the five-parameter model generally require the following data, some of them are provided by the manufacturer datasheets: Because of the presence of current I in both terms of transcendent Equation ( 2), the solution of the five-equation system, which is necessary to calculate the model parameters, cannot be faced by means of exact mathematical methods.For this reason, both numerical calculation techniques and approximate forms of the equations were adopted.

Usability of the One-Diode Models
The models proposed by Hadj Arab et al. [24] [44] were selected in order to assess the effectiveness of the proposed criterion.The models are based on some fundamental equations that, for the first time, Kennerud [84], Phang et al. [85] and de Blas et al. [23] described even more than 30 years ago.Unfortunately, such early models were not conceived to allow a complete representation of the I-V characteristic for values of the solar irradiance and cell temperature different from the SRC.
To assess the usability rating of a procedure, which may be significantly lowered by the difficulties encountered in using it, it is necessary to explore the complete sequence of operative steps that permit to reach the wished results.Some of the analysed procedures evaluate the model parameters on the basis of similar information, but they adopt different simplifying hypotheses to solve the equations and/or do not use the same 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 following paragraphs; the analytical procedures to calculate the model parameters are minutely described in the appendix.

Hadj Arab, Chenlo and Benghanem Model (Link 4)
The model Hadj Arab et al. [20] uses the following information: The model parameters can be calculated using the explicit equations described in the appendix.

De Soto, Klein and Beckman Model (Link 5)
The model of De Soto et al. [25] is based on the following information: No simplifying hypothesis is assumed.To simultaneously solve the system of five equations described in the appendix, De Soto et al. use a non-linear equation solver, such as Engineering Equation Solver (EES) [86].Laudani et al. proposed the use of closed forms to find the parameters of the De Soto et al. model [87].

Sera, Teodorescu and Rodriguez Model (Link 6)
The model proposed by Sera et al. [26] calculates the model parameters by means of following information: The following hypotheses are assumed: Due to the presence of implicit forms, the equation system is solved with the nine-step procedure described in the appendix.

Villalva, Gazoli and Filho Model (Link 8)
Villalva et al. [27] propose a model based on the following information: The following hypotheses are assumed: and the nine-step iterative method described in the appendix is used to calculate the model parameters.

Lo Brano, Orioli, Ciulla and Di Gangi Model (Link 9)
The model Lo Brano et al. [29] uses the following information: (1) short circuit point [I = I sc,ref ; The model adopts a modified version of Equation (2): where α G = G/G ref denotes the ratio between the generic solar irradiance and the solar irradiance at the SRC and K is a thermal correction factor similar to the curve correction factor described by the IEC891.Obviously, at the SRC, it is α G = 1, T = T ref and Equation ( 6) becomes equal to the traditional five-parameter Equation (2).R s,ref and R sh,ref are the series and shunt resistances at the SRC, respectively.Due to presence of term α G in Equation ( 6), it is also assumed that the series and shunt resistance are inversely affected by the solar irradiance: No simplifying hypothesis is assumed and the model parameters are calculated by means of the numerical iterative procedure, based on ten steps, described in the appendix.

Seddaoui, Rahmani, Kessal and Chauder Model (Link 10)
The Seddaoui et al. model [35] is based on the following information: The main difference with the Hadj Arab et al. model consists in a different way to calculate the I-V characteristics for conditions far from the SRC.The model parameters are calculated using the explicit equations described in the appendix.No simplifying hypothesis is used and the model parameters are calculated solving some implicit equations by means of the iterative procedure based on twelve steps described in the appendix.

Yetayew and Jyothsna Model (Link 12)
Yetayew et al. [52] propose a model based on the following information: No simplifying hypothesis is assumed and the equation system described in the appendix is solved by means of the Newton-Raphson method.It is known [88] that the Newton-Raphson technique may result instable because of the wide range of variation of the model parameters.To overcome such a difficulty, the definition of initial values of the model parameters, which should differ from the correct values less than an order of magnitude, is required.Such a task, which is not easy, may make difficult to use the Newton-Raphson method.

Orioli and Di Gangi Model (Link 13)
The model proposed by Orioli et al. [44] adopts the following information: and assumes the following hypotheses: To calculate the model parameters, the equations, which are based on the modified form used by Lo Brano et al., are solved using the seven-step iterative procedure described in the appendix.

Features Affecting the Model Usability
In order to better appreciate the analogies and differences between the various models, the sets of used information, hypotheses and solving techniques, on which the analysed procedures are based, are summarised in in Table 1.
The set of information used by a one-diode model may affect the usability rating of its analytical procedure.Actually, many performance data can be easily found, because they are always listed in tabular form in the PV module datasheets, whereas some data can be extracted only if a complete set of I-V curves is provided by the manufacturers.Also, the required mathematical tools, which include simple algorithms, iterative routines, mathematical methods or dedicated computer software, may have a significant impact on the procedure usability.

Accuracy of the One-Diode Models
In order to assess the accuracy of the proposed procedures, a comparison between the tested one-diode models was made using the I-V characteristics extracted from manufacturer datasheets by reading the coordinates of a large number of points on each curve.For the sake of brevity, only two PV modules, based on different production technologies, were considered.Obviously, because the results are strongly affected by the particular shape of the considered I-V characteristics, a more reliable assessment may require the use of a greater number of PV modules.Actually, the aim of this accuracy assessment is not ranking the best or the worst among the analysed models, but only reckoning the range of predictable precision in order to calibrate the proposed criterion.The data of the simulated PV modules are listed in Table 2. To calculate parameter K, these models use the values of voltage and current at the MPP for G = 1000 W/m 2 and T = 75 ˝C, which are 22.50 V and 8.35 A, for the Kyocera PV panel, and 28.89 V and 7.13 A, for the Sanyo PV module.In order to get a reliable comparison between the calculated and the experimental data, numerous points were extracted from the I-V characteristics issued by the manufacturer, considering both the constant solar irradiance and the constant cell temperature curves.To calculate R sho and R so , the reciprocal of slopes of the I-V curve in correspondence of the short circuit and open circuit points were extracted from the issued I-V characteristics following the graphical procedure described in [44].Table 3 and Table 4 list the values of the parameters evaluated with the analysed models.
The values of Table 3 and Table 4 were used to calculate the I-V characteristics of the selected PV panels.In Figure 4, Figure 5, Figure 6, Figure 7, Figure 8     The values of Tables 3 and 4 were used to calculate the I-V characteristics of the selected PV panels.In Figures 4-9                            It can be generally observed in Figures 4-15 that most of the models result less accurate for values of voltage greater than the MPP voltage.Moreover, it can be inferred that the 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.It can be also observed that models that use similar values of the parameters listed in Tables 3 and 4 yield different I-V curves for values of the solar irradiance and the cell temperature far from the SRC because different approaches were adopted to describe the effects of the solar irradiance and cell temperature.To quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated with the following expressions:   It can be generally observed in Figures 4-15 that most of the models result less accurate for values of voltage greater than the MPP voltage.Moreover, it can be inferred that the 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.It can be also observed that models that use similar values of the parameters listed in Tables 3 and 4 yield different I-V curves for values of the solar irradiance and the cell temperature far from the SRC because different approaches were adopted to describe the effects of the solar irradiance and cell temperature.To quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated with the following expressions:  It can be generally observed in Figures 4-15 that most of the models result less accurate for values of voltage greater than the MPP voltage.Moreover, it can be inferred that the 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.It can be also observed that models that use similar values of the parameters listed in Table 3 and Table 4 yield different I-V curves for values of the solar irradiance and the cell temperature far from the SRC because different approaches were adopted to describe the effects of the solar irradiance and cell temperature.To quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated with the following expressions: ˇˇI calc,j ´Iiss,j ˇˇ(11) ˇˇV iss,j I calc,j ´Viss,j I iss,j ˇˇ( 12) Energies 2016, 9, 427 16 of 48 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: MDpIq " MAX " I calc,j ´Iiss,j ı MDpPq " MAX " V iss,j I calc,j ´Viss,j I iss,j ı Tables 5 and 6 list the MAD(I)s and MAD(P)s for the Kyocera KD245GH-4FB2 and Sanyo HIT-240 HDE4 PV panels.For these models differences varying between 0.724 A and 1.125 A were calculated.Table 9 and Table 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 .The smallest MD(I)s for the Kyocera PV module at constant solar irradiance arise from the Lo Brano et al.  11-14 list the MD(P)s calculated for the analysed PV modules.Table 11.Maximum power differences between the calculated and the issued I-V characteristics of Kyocera KD245GH-4FB2, at temperature T = 25 ˝C.15 and 16, the percentage ratios of MAD(I) to the current at the issued MPP, and of MAD(P) to the rated maximum power, are listed.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 0.40% to 0.77% of the current at the MPP; the greatest MAD(I)s vary from 1.65% to 7.96%.The smallest MAD(I)s for the Sanyo PV module are in the range 0.22% to 0.61% of the current at the MPP; the greatest MAD(I)s range from 3.71% to 5.11%.The smallest MAD(P)s range from 0.44% to 0.72% of the rated maximum power for the Kyocera PV panel; the greatest MAD(P)s vary from 1.79% to 7.43%.For the Sanyo PV module the smallest MAD(P)s are in the range 0.20% to 0.67% of the rated maximum power; the greatest MAD(P)s vary from 3.74% to 5.54%.

Rating of the Usability and Accuracy of the One-Diode Models
As it was previously observed, the usability is a qualitative parameter whereas the accuracy level is quantitatively described.Even though it is not simple to find an index able to globally represent both the usability and accuracy of the considered analytical procedure, an attempt to get a concise description of the model performances, is necessary in order to define the rating criterion.The approach proposed, which is based on a three-level rating scale, 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: ‚ low, when the analytical procedure requires the use of dedicated computational software.
In order to condense the precision data, the results of the accuracy assessment are summarized in Table 17, where the average ratios of MAD(I) to the rated current at the MPP, and of MAD(P) to the rated maximum power, extracted from Table 15 and Table 16, are listed.It can be observed that the global accuracy listed in Table 17, which is calculated averaging the accuracies evaluated for the Kyocera and Sanyo PV panels, ranges from 0.53% to 3.27%.Such range of variation was divided in three equal intervals, which were used to qualitatively describe the accuracy of the analysed models:

‚
high, for values of the mean difference in the subrange 0.52% to 1.39%; ‚ medium, for values of the mean difference in the subrange 1.39% to 2.26%; ‚ low, for values of the mean difference in the subrange 2.26% to 3.14%.
Table 18 lists the rating of the ease of finding data, simplicity of mathematical tools, and accuracy in calculating the current and power, based on the three-level rating scale previously described.
As it can be observed in tackle a medium level of mathematical difficulty, the data finding results more elaborate and a low level of accuracy in the calculated current and power was obtained.As it was predictable, the choice of the one-diode model requires a wise compromise between usability and accuracy, which may prefer the usability to the accuracy, the accuracy to the usability, or try to find an acceptable balance between such features.No model achieved the highest rating for all the considered features.On the basis of the practical application of the proposed criterion, a medium degree of mathematical difficulty and a high level of current and power accuracy was provided by Orioli et al. model despite the major difficulty in the data finding.The information, hypotheses and solving techniques required by the model represent the best possible compromise for researchers and designers to calculate precise and reliable parameters.

Conclusions
A criterion for rating the usability and accuracy performances of diode-based equivalent circuits was tested on some one-diode models.In order to define the criterion an accurate examination of the used analytical procedures along with the comparison between calculated results and reference data was carried out.The procedures adopted by the tested models were minutely described and analysed along with the used performance data, simplifying hypotheses and mathematical methods needed to calculate the model parameters.Most of the performance data can be easily found, because they are always listed in the PV module datasheets, whereas some data can be extracted only if a complete set of I-V curves is provided by the manufacturers.Moreover, mathematical tools with different degrees of complexity, which include simple algorithms, iterative routines, mathematical methods or dedicated computer software, may be necessary to calculate the model parameters.
The tested models were implemented to calculate the I-V curves, at constant cell temperature and solar irradiance, of two different types of PV modules.The accuracy achievable with the one-diode models was assessed by comparing the calculated I-V curves with the I-V characteristics issued by manufactures and evaluating the maximum difference and the mean absolute difference between the calculated and issued values of current and power.Depending on the used model, the most effective one-diode equivalent circuits yielded for the poly-crystalline Kyocera KD245GH-4FB2 PV panel values of the current difference that averagely range from 0.40% to 0.77% of the current at the MPP.The values of the power difference averagely vary between 0.44% and 0.72% of the rated maximum power.For the Sanyo HIT-240 HDE4 PV module greater accuracies were generally observed.The current differences averagely vary from 0.22% to 0.61% of the current at the MPP.The power accuracies averagely range from 0.20% to 0.67% of the rated maximum power.The accuracies of the less effective models averagely reach 7.96% of the current at the MPP and 7.43% of the rated maximum power for the Kyocera PV panel, whereas average differences of 5.11% of the current at the MPP and of 5.54% of the rated maximum power were observed for the Sanyo PV module.
In order to rate both the usability and accuracy of the considered analytical procedure, a three-level rating scale (high, medium and low) was defined considering some significant features, such as the ease of finding the performance 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 of the analysed PV modules.No model achieved the highest rating for all the considered features.As it was predictable, the selection of the one-diode model requires that researchers and PV designers, who are the only persons aware of the peculiarities of the problem to be solved, reach a suitable compromise between analytical complexity and expected accuracy.In our opinion, even though the presented criterion is obviously debatable and other approaches may be used to rate the usability and accuracy of the one-diode models, the information provided in this paper may be useful to make more aware choices and support the users in implementing the selected model. in order to be substituted in Equations (A5) and (A9), respectively.After some manipulations.Equations (A5)-(A7) and (A10) can be rewritten in the following forms:

˙(A21)
The model parameters are calculated by means of explicit Equations (A17)-(A21).Phang et al.only calculate the I-V characteristic at the SRC.

A3. de Blas et al. Model
The same five pieces of information adopted by Phang et

Figure 1 .
Figure 1.Schematic electrical charges flow paths (a) in a diode and (b) in a photovoltaic (PV) cell.

Figure 1 .
Figure 1.Schematic electrical charges flow paths (a) in a diode and (b) in a photovoltaic (PV) cell.

Figure 3 .
Figure 3. One-diode equivalent circuit for a PV panel.

•Figure 3 .
Figure 3. One-diode equivalent circuit for a PV panel.

48 ( 4 )
) derivative of current at the short circuit point [BI/BV = ´1/R sho at I = I sc,ref ; V = 0]; derivative of current at the open circuit point [BI/BV = ´1/R so at I = 0

( 1 )
short circuit point [I = I sc,ref ; V = 0]; (2) open circuit point [I = 0; V = V oc,ref ]; (3) derivative of current at the short circuit point [BI/BV = ´1/R sho at I = I sc,ref ; V = 0]; (4) derivative of current at the open circuit point [BI/BV = ´1/R so at I = 0; V = V oc,ref ]; (5) MPP [I = I mp,ref ; V = V mp,ref ].The following hypotheses, also adopted by Phang et al. and Hadj Arab et al., are assumed: e V oc,re f nT re f ąą e I sc,re f Rs nT re f
IP Notes: SCP: Short Circuit Point; OCP: Open Circuit Point; MPP: Maximum Power Point; DSCP: Derivative of I at the SCP; DOCP: Derivative of I at the OCP; DMPP: Derivative of power at the MPP; OCP *: OCP at condition ‰ SRC; SC: Simple Calculation; NES: Non-linear Equation Solver; IP: Iterative Procedure; NRM: Newton-Raphson Method.
and Figure 9 the I-V curves evaluated at T = 25 ˝C with the models of Hadj Arab et al., De Soto et al., Sera et al., Villalva et al., Lo Brano et al., Seddaoui et al., Siddique et al., Yetayew et al. and Orioli et al. are compared with the characteristics issued by manufacturers.
the I-V curves evaluated at T = 25 °C with the models of Hadj Arab et al., De Soto et al., Sera et al., Villalva et al., Lo Brano et al., Seddaoui et al., Siddique et al., Yetayew et al. and Orioli et al. are compared with the characteristics issued by manufacturers.

Figure 4 .
Figure 4. Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 4 .
Figure 4. Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 ˝C and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 4 .
Figure 4. Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 5 .
Figure 5.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Villalva et al., the Yetayew et al. and the Seddaoui et al. models.

Figure 5 . 47 Figure 6 .
Figure 5.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 ˝C and the characteristics calculated by means of the Villalva et al., the Yetayew et al. and the Seddaoui et al. models.Energies 2016, 9, 427 11 of 47

Figure 7 .
Figure 7.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 6 . 47 Figure 6 .
Figure 6.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 ˝C and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 7 .
Figure 7.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 7 .
Figure 7.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 ˝C and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 8 .
Figure 8.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Villalva et al., the Yetayew et al. and the Seddaoui et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 8 . 47 Figure 8 .
Figure 8.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 ˝C and the characteristics calculated by means of the Villalva et al., the Yetayew et al. and the Seddaoui et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 9 .
Figure 9.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 ˝C and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

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

Figure 10 . 47 Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

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

Figure 11 .Figure 12 .
Figure 11.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Villalva et al., the Yetayew et al. and the Seddaoui et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 12 . 47 Figure 12 .
Figure 12.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at G = 1000 W/m 2 and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = W/m 2 and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Sera et al., the Orioli et al. and the De Soto et al. models.

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

Figure 15 .
Figure 15.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 14 . 47 Figure 14 .
Figure 14.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Villalva et al., the Yetayew et al. and the Seddaoui et al. models.

Figure 15 .
Figure 15.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.

Figure 15 .
Figure 15.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at G = 1000 W/m 2 and the characteristics calculated by means of the Lo Brano et al., the Siddique et al. and the Hadj Arab et al. models.
from 0.041 A to 0.054 A can be obtained.In the same conditions, the MAD(P)s vary from 1.074 W to 1.769 W. For the Sanyo PV module, the smallest MAD(I)s, which occur when the Hadj Arab et al., the Lo Brano et al. and the Seddaoui et al. models are used, vary between 0.015 A and 0.041 A. The smallest MAD(P)s derived from the same models.These differences vary between 0.487 W and 1.601 W. The greatest MAD(I)s for the Kyocera PV panel, which are calculated with the Siddique et al. and the Yetayew et al. models, vary from 0.136 A to 0.253 A. The Yetayew et al. model yields the greatest MAD(P)s, which range from 4.387 W to 8.123 W. For the Sanyo PV module, the greatest MAD(I)s are noted for the De Soto et al., the Siddique et al. and the Yetayew et al. models.These differences are contained in the range from 0.251 A to 0.346 A. The greatest MAD(P)s, which belong to the same models, vary from 9.427 W to 13.321 W. At constant solar irradiance, if the Hadj Arab et al., the Lo Brano et al. and the Seddaoui et al. models are used for the Kyocera PV panel, MAD(I)s ranging from 0.042 A to 0.063 A can be observed.In the same conditions, the MAD(P)s vary from 1.333 W to 1.611 W. For the Sanyo PV module, the smallest MAD(I)s, which are obtained from the Hadj Arab et al., the Lo Brano et al. and the Seddaoui et al. models, vary between 0.021 A and 0.034 A. The smallest mean power differences are obtained from the same models.These differences vary between 0.828 W and 1.142 W. For the Kyocera PV module, the greatest MAD(I)s are generated by the Seddaoui et al. and the Siddique et al. models.These differences are contained in the range from 0.166 A to 0.655 A. The greatest MAD(P)s, which are obtained from the Seddaoui et al. and the Yetayew et al. models, vary between 6.625 W and 18.222 W. The greatest MAD(I)s for the Sanyo PV panel, which are obtained from the Sera et al. and the Siddique et al. models, vary from 0.273 A to 0.337 A. The Siddique et al. and the Yetayew et al. models yield the greatest MAD(P)s, which range from 8.991 W to 13.321 W of the issued rated maximum power.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.Considering the I-V curves at constant temperature for the Kyocera PV panel, the Lo Brano et al. model seems to be the most accurate; the MD(I)s vary from ´0.257A to 0.132 A. The greatest current differences, which are contained in the range ´0.574A to 0.636 A, are observed for the Siddique et al. and the Yetayew et al. models.The smallest MD(I)s are obtained for the Sanyo PV module from the Hadj Arab et al., the Lo Brano et al. and the Seddaoui et al. models; these differences are in the range ´0.040A to ´0.147 A. The greatest inaccuracies derive from the de Soto et al., the Siddique et al. and the Yetayew et al. models.
model.Such differences range from 0.132 A to 0.223 A. The greatest inaccuracies are provided by the Seddaoui et al. and the Siddique et al. models, for which differences varying between 0.481 A and 1.516A are observed.The Hadj Arab et al., the Lo Brano et al. and the Seddaoui et al. model seem to be the most accurate for the Sanyo PV panel; the MD(I)s vary from ´0.099A to ´0.145 A. The greatest current differences, which are contained in the range 0.858 A to 1.125 A, are observed for the Siddique et al. and the Yetayew et al. models.Table

( 5 )
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 diode saturation current (A) I 0,ref diode saturation current at the SRC (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 diode saturation current (A) I 0,ref diode saturation current at the SRC (A) k Boltzmann constant (J/K) K thermal correction factor (Ω/ ˝C) n diode quality factor (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 so reciprocal of slope of the I-V characteristic for V = V oc and I = 0 (Ω) R s,ref series resistance at the SRC (Ω) R sh shunt resistance (Ω) R sho reciprocal of slope of the I-V characteristic for V = 0 and I = I sc (Ω) R sh,ref shunt resistance at the SRC (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) α G ratio of the current irradiance to the irradiance at SRC γ diode ideality factor 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) BP/BV = 0 at I = I mp,ref ; V = V mp,ref :I 0,re f nT re f ˜1 ´Rs I mp,re f V mp,re f ¸e V mp,re f `Imp,re f Rs nT re f `1 R sh ˜1 ´Rs I mp,re f V mp,re f ¸" I mp,re f V mp,re f (A8)Using the above five pieces of information, Equations (A4)-(A8), listed in the appendix, are written and solved using the Newton-Raphson technique in order to calculate the model parameters I L,ref , I 0,ref , R s , R sh and n.The model cannot be used for any value of solar irradiance and cell temperature because Kennerud described the operation of PV devices at only the SRC.A2.Phang et al.Model Phang et al. calculate the model parameters by means of the following information: (1) short circuit point (I = I sc,ref ; V = 0); (2) open circuit point (I = 0; V = V oc,ref ); (3) derivative of current at the short circuit point (BI/BV = ´1/R sho at I = I sc,ref ; V = 0); (4) derivative of current at the open circuit point (BI/BV = ´1/R so at I = 0; V = V oc,ref ); (5) MPP (I = I mp,ref ; V = V mp,ref ).The first four pieces of information are described by Equations (A4)-(A7); the fifth piece of information corresponds to the equation:I mp,re f " I L,re f ´I0,re f ˜e V mp,re f `Imp,re f Rs nT re f´1¸´V mp,re f `Imp,re f R s R sh (A9) The following expressions of I L,ref are obtained from Equations (A4) and (A5): I L,re f " I sc,re f `I0,re f ˜e I sc,re f Rs nT re f ´1¸`I sc,re f R s R sh (A10) I L,re f " I 0,re f ˜e V oc,re f nT re f ´1¸`V oc,re f R sh (A11) al. are used by de Blas et al. and the following hypotheses are assumed: A11) is substituted in Equation (A4), and the first of the hypotheses in Equation (A22) is considered, the diode reverse saturation current can be calculated as: I 0,re f " I sc,re f ´1 `Rs Equations (A11) and (A23) are used in Equation (A9), it is possible to write parameter n in the following form: n " V mp,re f `Imp,re f R s ´Voc,re f T re f ln " pI sc,re f ´Imp,re f qp1`Rs{R sh q´V mp,re f {R sh I sc,re f p1`R s {R sh q´V mp,re f {R sh  (A24) From Equation (A3) it is possible to extract the general form of the derivative of the current: to write the conditions regarding the derivatives in the short circuit and open circuit points at the SRC: BI BV ˇˇˇV " 0 I " I sc,re f " ´I0,ref nT re f e I sc,re f Rs nT re f `1 R sh 1 `Rs ˜I0,ref nT re f e I sc,re f Rs nT re f `1 R sh ¸" ´1 R sho (A26) V oc,ref and short circuit current I sc,ref at the 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; ‚ voltage V mp,ref and current I mp,ref in the MPP at the SRC; , De Soto et al. [25], Sera et al. [26], Villalva et al. [27], Lo Brano et al. [29], Seddaoui et al. [35], Siddique et al. [49], Yetayew et al. [52] and Orioli et al.

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

Table 2 .
Performance data of the simulated PV panels.The Lo Brano et al. and the Orioli et al. models also use the open voltage at G = 200 W/m 2 and T = 25 ˝C, which are 34.40V and 40.61 V per the Kyocera and the Sanyo PV modules, respectively.

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

Table 4 .
Model parameters of Sanyo HIT-240 HDE4 at the SRC.

Table 4 .
Model parameters of Sanyo HIT-240 HDE4 at the SRC.

Table 5 .
Mean absolute current and power differences between the calculated and the issued I-V characteristics at temperature T = 25 ˝C.

Table 6 .
Mean absolute current and power differences between the calculated andthe issued I-V characteristics at irradiance G = 1000 W/m 2 .
Considering the solar irradiance variation, if the Hadj Arab et al., the Villalva et al., the Lo Brano et al. and the Seddaoui et al. models are used for the Kyocera PV panel, MAD(I)s ranging

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

Table 8 .
Maximum current differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, 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 12 .
Maximum power differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, at temperature T = 25 ˝C.

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 .For the Kyocera PV panel, the smallest MD(P)s at constant cell temperature are again obtained with the Lo Brano et al. model that yields values varying from ´2.82 W to ´8.83 W. The greatest MD(P)s, which occur with the Siddique et al. and the Yetayew et al. models, are in the range ´20.40 W to 21.32 W. For the Sanyo PV module, the smallest MD(P)s at constant temperature, which vary from ´1.41 W to ´6.19 W, are generated by the Hadj Arab et al., the Lo Brano et al. and the Seddaoui et al. models.The De Soto et al., the Siddique et al. and the Yetayew et al. models yield the greatest inaccuracies, which vary from 28.21 W to 45.68 W. Considering the MD(P)s at constant solar irradiance, the smallest values for the Kyocera PV panel are obtained from the Lo Brano et al. model.The differences range from ´4.44 W to 5.79 W. The greatest inaccuracies are provide by the Siddique et al. and the Yetayew et al. models.Differences, varying between 16.60 W and 44.90 W, were calculated.The Hadj Arab et al., the Lo Brano et al., and the Seddaoui et al. models yield the smallest MD(P)s for the Sanyo PV module, which are in the range from ´4.22 W to ´5.52 W. The greatest values are due to the Siddiqui et al. and the Yetayew et al. models.Such differences vary from 32.76 W to 45.68 W. In Tables

Table 15 .
Percentage ratio of the mean absolute difference, MAD(I) to the rated current at the maximum power point.

Table 16 .
Percentage ratio of MAD(P) to the rated maximum power.
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, current and voltage of the 4th and 5th points); ‚ low, when the derivative of the I-V curves, at the short circuit and open circuit points, are required.

Table 17 .
Average ratios of MAD(I) to the rated current at the MPP andof MAD(P) to the rated maximum power.

Table 18 ,
some models, such as De Soto et al., Yetayew et al., Villalva et al., Siddique et al. have good level of usability, but reach a smaller accuracy.Adversely, some models, such as Hadj Arab et al., Sera et al. and Lo Brano et al. are more accurate, but their usability is lower.Although the solving technique required by Seddaoui et al. model allows users to

Table 18 .
Usability and accuracy rating of the analysed one-diode models.
re f ˜e V oc,re f nT re f ´e I sc,re f Rs nT re f ¸´I sc,re f ˆ1 A15) are significantly simplified and, after some manipulations, the model parameters can be calculated by means of the following explicit forms: