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

Abstract: Many diode-based equivalent circuits for simulating the electrical behaviour of photovoltaic (PV) cells and panels are reported in the scientific literature. Two-diode equivalent circuits, which require more complex procedures to calculate the seven model parameters, are less numerous. The model parameters are generally calculated using the data extracted from the datasheets issued by the PV panel manufactures and adopting simplifying hypotheses and numerical solving techniques. A criterion for rating both the usability and accuracy of two-diode models is proposed in this paper with the aim of supporting researchers and designers, working in the area of PV systems, to select and use a model that may be fit for purpose. 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. The analytical procedures, the simplifying hypotheses and the operative steps to calculate the parameters of the most famous two-diode equivalent circuits are exhaustively described in this paper. The accuracy of the models is tested by comparing the characteristics issued by the PV panel manufacturers with the current-voltage (I-V) curves, at constant solar irradiance and/or cell temperature, calculated with the analysed models with. The results of the study show that the two-diode models recently proposed reach accuracies that are comparable with the values derived from the one-diode models.


Introduction
Numerous analytical procedures for determining the model parameters of one and two diode equivalent circuits have been proposed .These models use a set of analytical relations derived from the performance data, usually provided by manufacturers, and arranged in an equation system whose solution is often made easier through the adoption of some simplifying hypotheses and/or iterative methods.Some authors have also faced the problem of the identification of the model parameters by means of alternative methods such as genetic algorithms, cluster analysis, Padè approximants, harmony search-based algorithms, Lambert W-function, reduced forms, evolutionary algorithms, artificial neural networks and small perturbations around the operating point [46][47][48][49][50][51][52][53][54][55][56][57][58][59].
The paper is organised along the lines of a previous study regarding simplified one-diode models for photovoltaic (PV) modules [60].The analytical procedures to extract the two-diode equivalent circuit parameters and the hypotheses assumed to simplify the mathematical computations are described.In order to verify the effectiveness and accuracy of the analysed models, the I-V characteristics calculated with the proposed procedures, are compared to the performance curves issued by the manufacturers of some silicon PV modules.The paper is organised as follows: Section 2

The Two-Diode Equivalent Circuit
In the two-diode model, which is depicted in Figure 1, a second diode is added to consider the effect of the carrier recombination in the depletion region.The equivalent circuit contains seven parameters, which are photocurrent IL, diode reverse saturation currents I01 and I02, series resistance Rs, shunt resistance Rsh, and diode quality factors n1 = a1Ncsk/q and n2 = a2Ncsk/q in which a1 and a2 are the diode shape factors, 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 two-diode model is described by the well-known equation: where, following the traditional theory, photocurrent IL depends on the solar irradiance and diode currents I01 and I02 are affected by the cell temperature.Due to the large number of parameters used, the two-diode model is supposed to be fit to adequately represent any I-V characteristic, regardless of the shape peculiarities due to the different production technology of the simulated PV panels.Actually, because the production technology affects the shape of the I-V characteristics, crystalline silicon and thin-film PV cells and modules have very different performance curves.As depicted in Figure 2, in which the range-scaled I-V characteristics at the standard rating conditions (SRC)-irradiance Gref = 1000 W/m 2 , cell temperature Tref = 25 °C and average solar spectrum at AM 1.5-of some types of PV modules are compared, the crystalline PV modules show an I-V characteristic with a very sharp bent, whereas the thin-film modules are generally characterized by smoother I-V curves.The two-diode model is described by the well-known equation: where, following the traditional theory, photocurrent I L depends on the solar irradiance and diode currents I 01 and I 02 are affected by the cell temperature.Due to the large number of parameters used, the two-diode model is supposed to be fit to adequately represent any I-V characteristic, regardless of the shape peculiarities due to the different production technology of the simulated PV panels.Actually, because the production technology affects the shape of the I-V characteristics, crystalline silicon and thin-film PV cells and modules have very different performance curves.As depicted in Figure 2, in which the range-scaled I-V characteristics 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-of some types of PV modules are compared, the crystalline PV modules show an I-V characteristic with a very sharp bent, whereas the thin-film modules are generally characterized by smoother I-V curves.Different techniques are used to make crystalline and thin-film PV modules.Mono-crystalline and polycrystalline PV cells are made of wafers sawed from silicon ingots obtained by means of a method of crystal growth or from molten silicon, which is carefully cooled and solidified.Conversely, the material of thin-film PV modules is deposited onto a substrate, or onto previously deposited layers, by means of various chemical and/or physical methods.The slopes of the I-V curves of Figure 2 near the open circuit point (0, 1) confirm the fact that the high quality silicon slabs of polycrystalline modules dissipate less energy than the materials used to make amorphous or triple junction PV panels.The values of R s , R sh , n 1 , n 2 , I 01 and I 02 variously affect the I-V characteristic of the PV panel [61].
Energies 2017, 10, 564 3 of 32 The series and shunt resistances, whose effects are shown in Figures 3 and 4, take account of dissipative phenomena and parasitic currents within the PV panel.Different techniques are used to make crystalline and thin-film PV modules.Mono-crystalline and polycrystalline PV cells are made of wafers sawed from silicon ingots obtained by means of a method of crystal growth or from molten silicon, which is carefully cooled and solidified.Conversely, the material of thin-film PV modules is deposited onto a substrate, or onto previously deposited layers, by means of various chemical and/or physical methods.The slopes of the I-V curves of Figure 2 near the open circuit point (0, 1) confirm the fact that the high quality silicon slabs of polycrystalline modules dissipate less energy than the materials used to make amorphous or triple junction PV panels.The values of Rs, Rsh, n1, n2, I01 and I02 variously affect the I-V characteristic of the PV panel [61].The series and shunt resistances, whose effects are shown in Figures 3 and 4, take account of dissipative phenomena and parasitic currents within the PV panel.
Range-scaled I-V characteristics of crystalline and thin-film PV panels at the SRC.Different techniques are used to make crystalline and thin-film PV modules.Mono-crystalline and polycrystalline PV cells are made of wafers sawed from silicon ingots obtained by means of a method of crystal growth or from molten silicon, which is carefully cooled and solidified.Conversely, the material of thin-film PV modules is deposited onto a substrate, or onto previously deposited layers, by means of various chemical and/or physical methods.The slopes of the I-V curves of Figure 2 near the open circuit point (0, 1) confirm the fact that the high quality silicon slabs of polycrystalline modules dissipate less energy than the materials used to make amorphous or triple junction PV panels.The values of Rs, Rsh, n1, n2, I01 and I02 variously affect the I-V characteristic of the PV panel [61].The series and shunt resistances, whose effects are shown in Figures 3 and 4, take account of dissipative phenomena and parasitic currents within the PV panel.
Effects of the series resistance on the I-V characteristic.
Energies 2017, 10, 564 4 of 33 The series resistance impacts the shape of the I-V characteristic close and beyond the maximum power point (MPP), which is approximately set on the "knee" of the curve; the shunt resistance modifies the I-V curve for values of the voltage that are smaller than the MPP voltage.As depicted in Figure 5, the presence of the second diode saturation current modifies the curvature of the I-V characteristic close the MPP.The series resistance impacts the shape of the I-V characteristic close and beyond the maximum power point (MPP), which is approximately set on the "knee" of the curve; the shunt resistance modifies the I-V curve for values of the voltage that are smaller than the MPP voltage.As depicted in Figure 5, the presence of the second diode saturation current modifies the curvature of the I-V characteristic close the MPP.The series resistance impacts the shape of the I-V characteristic close and beyond the maximum power point (MPP), which is approximately set on the "knee" of the curve; the shunt resistance modifies the I-V curve for values of the voltage that are smaller than the MPP voltage.As depicted in Figure 5, the presence of the second diode saturation current modifies the curvature of the I-V characteristic close the MPP.At a constant value of the solar irradiance, the position of the MPP is lowered if Rs is increased, Rsh is reduced and I02 is much greater than I01.As a consequence, a small value of the filling factor is reached.Such a peculiarity characterizes thin-film PV modules that, for this reason, usually result less energy efficient than the crystalline silicon PV panels.
The parameters of the two-diode models are generally calculated using the following data which are usually available in the manufacturer datasheets:  At a constant value of the solar irradiance, the position of the MPP is lowered if R s is increased, R sh is reduced and I 02 is much greater than I 01 .As a consequence, a small value of the filling factor is reached.Such a peculiarity characterizes thin-film PV modules that, for this reason, usually result less energy efficient than the crystalline silicon PV panels.
The parameters of the two-diode models are generally calculated using the following 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 .
Some procedures also require the number of series connected PV cells, or the derivative of the I-V curve calculated at the short circuit and open circuit points.Due to the presence of current I in both terms of transcendent Equation (1), exact mathematical methods cannot be used to solve the seven-equation system, which is necessary to calculate the model parameters.Both approximate forms of the equations and numerical solving techniques have been used to solve the problem.

Usability of the Two-Diode Models
Some procedures to calculate the parameters of the two-diode model have been proposed.Early models for PV cells and panels, which were presented by Chan et al. [40], Enebish et al. [41] and Hovinen [42], were conceived to calculate the I-V characteristic at certain values of solar irradiance and cell temperature, which can be the SRC or any others.Some models, able to give a complete representation of the performance curves for any condition different from the SRC, were proposed by Ishaque et al. [43], Gupta et al. [44] and Hejri et al. [45].Such recent models face the complex problem of the analytical solution of the involved equations by assuming some simplifying hypotheses and/or reducing the number of independent parameters.

Chan and Phang Model
Chan et al. [40] used Equation (1) to represent the I-V characteristic of a PV solar cell at the SRC.To make the calculated curve coincide with an experimental characteristic, the following information was considered: (1) shape factor a 1 = 1; (2) shape factor a 2 = 2; Moreover, as described in the Appendix A, some exponential terms containing the parameter R s are substituted with their respective power series.Using the first two terms, or the first three terms, of the power series, the equation that describes the derivative of current at the open circuit point can be approximated with a quadratic form, or a cubic form, respectively.Depending on the use of the quadratic or cubic form, two models were presented, which in this paper are named Chan et al. n.1 and Chan et al. n.2 models, respectively.The model parameters can be calculated with the explicit equations listed in the Appendix A. A new set of model parameters should be calculated for any generic value of solar irradiance and/or cell temperature.

Enebish, Agchbayar, Dorjkhand, Baatar and Ylemj Model
The determination of a solar cell characteristic at the SRC was presented by Enebish et al. [41] who proposed a double diode model based on the following information: (1) shape factor a 1 = 1; (2) shape factor a 2 = 2; The above information is used to write an equation system that is solved using the Newton-Raphson technique.Because the convergence of the procedure strongly depends on the initial values of I L,ref , I 01,ref , I 02,ref , R s , and R sh , the use of some relations described in the appendix was suggested.The model was only used to calculate the I-V characteristics at the SRC.

Hovinen Model
Hovinen [42] used the following information to calculate the parameters of the two-diode equivalent circuit:

Ishaque, Salam and Taheri Model
An improved modelling approach for the two-diode model was proposed by Ishaque et al. [43].The model is based on the following information: (1) shape factor a 1 = 1; (2) shape factor a 2 ≥ 1.
In which n = a 1 N cs k/q and p = a 1 + a 2 , photocurrent I L,ref at the SRC and shunt resistance R sh can be calculated with the iterative procedure described in the Appendix A.

Gupta, Tiwari, Fozdar and Chandna Model
Gupta et al. [44] based on the following information the analytical procedure to calculate the parameters of a two-diode model of photovoltaic modules suitable for the use in simulation studies: (1) shape factor a 1 = 1; (2) shape factor The two-diode equation is transformed in the following form: in which coefficients K 1 , K 2 and K 3 are calculated with the equations listed in the Appendix A.

Hejri, Mokhtari, Azizian, Ghandhari and Söder Model
Hejri et al. [45] proposed a procedure for the extraction of the parameters of the two-diode equivalent model.A set of approximate analytical solutions for the model parameters, which can be used as initial conditions for the numerical solutions based on the Newton-Raphson method, were also proposed.The model is based on the following information: (1) shape factor a 1 = 1; (2) shape factor a 2 = 2; the model parameters are expressed the equations listed in the appendix, which are solved with the Newton-Raphson method.

Summary of the Information Used by the Models
In order to better appreciate the analogies and differences between the various models, the sets of information, hypotheses and solving techniques, on which the analysed procedures are based, are summarised in in Table 1.

Simplif.
Hypoth.Despite the fact that the same pieces of information are often shared, each model has a particular capability to reproduce the I-V characteristics because of the different mathematical approaches used, which can be very simple or require the implementation of iterative routines and the use of specific mathematical methods, are adopted.

Accuracy of the Simplified Two-Diode Models
The accuracy of the analysed two-diode models was verified using the various procedures to calculate the I-V characteristics extracted from the manufacturer datasheets.For the sake of brevity, only the I-V characteristics of two PV modules based on different production technologies were used, although such an approach cannot be considered exhaustive because the results are significantly affected by the particular shape of the considered I-V curves.In any case, the purpose of this paper is not indicate the best or the worst among the analysed models, but only to evaluate the range of predictable precision in order to calibrate the criterion.The performance data of the simulated PV modules are listed in Table 2.
Table 2. Performance data of the simulated PV panels.

Panel
Type N cs V oc,ref (V) To evaluate the differences between the calculated and the experimental data, numerous points were extracted from the I-V characteristics issued by the manufacturers, considering both the constant solar irradiance and the constant cell temperature curves.The graphical procedure described in [26] was used to calculate R sho and R so , which correspond to the reciprocal of slopes of the I-V curve in correspondence of the short circuit and open circuit.Tables 3 and 4 list the values of the parameters obtained using the procedures of the analysed models.The values of Tables 3 and 4                       It can be generally observed in Figures 8-13 that the models result less accurate for values of voltage greater than the MPP voltage.Moreover it seems that the analysed models are more precise if they are used to evaluate the I-V characteristics of the Kyocera PV panel.This may be due to the different shape of the issued I-V curves; actually, the I-V characteristics of the Sanyo PV module show sharper "knees" close to the MPP.The Hejri et al. and the Ishaque et al. models adequately reproduce the issued I-V characteristics of the Kyocera PV panel at the SRC, whereas they are less effective for the Sanyo PV module; the curves calculated with the Gupta et al. model at the SRC are rather different from the issued I-V characteristics.Such occurrences contrast with the fact that the two-diode models should be particularly able to represent the I-V characteristics regardless the shape of the simulated Energies 2017, 10, 564 12 of 32 curves.In this regard, it must be highlighted that none of the analysed models take full advantage of the seven independent parameters of the two-diode equivalent circuit.It easy to verify that, if constant values for a 1 and a 2 are arbitrarily assumed, as was made by all the analysed procedures, the number of independent parameters is reduced from seven to five.Moreover, if it is set I 02 = I 01 , as it was proposed by Ishaque et al., the number of independent parameters is further lowered to four.Only three independent parameters are used by the Gupta et al. model, who set a fixed ratio of I 02 to I 01 and neglected the shunt resistance.A lucky guess of the values of a 1 and a 2 , and the fact that the system of equations is solved without recourse to mathematical simplifications, are probably the reasons why the Enebish et al. model better reproduce the I-V characteristic of the simulated PV panels.
To quantify the accuracy of the analysed models, the mean absolute difference (MAD) for current and power was calculated with the following expressions: 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: MD(P) = MAX V iss,j I calc,j − V iss,j I iss,j 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.Considering the solar irradiance variation, for the Kyocera PV panel the smallest MAD(I)s range from 0.053 to 0.109 A; the smallest MAD(P)s vary from 1.620 to 3.298 W. For the Sanyo PV module the smallest MAD(I)s vary between 0.073 and 0.277 A. The smallest MAD(P)s are in the range from 2.590 to 10.669 W. The greatest MAD(I)s for the Kyocera PV panel vary from 0.078 to 0.299 A; the greatest MAD(P)s range from 2.103 to 9.931 W. For the Sanyo PV module the greatest MAD(I)s are contained in the range from 0.171 to 0.376 A. The greatest MAD(P)s vary from 5.467 to 14.897 W. Table 6.Mean absolute current and power differences between the calculated and the issued I-V characteristics at irradiance G = 1000 W/m 2 .

PV Panel
Absolute Mean Difference Temperature ( At constant solar irradiance, the smallest MAD(I)s for the Kyocera PV panel range from 0.059 to 0.124 A MD(I)s; the smallest MAD(P)s vary from 1.931 to 3.383 W. For the Sanyo PV module the smallest MAD(I)s vary between 0.143 and 0.228 A. The smallest MAD(P)s vary between 4.216 and 8.005 W. For the Kyocera PV module, the greatest MAD(I)s are contained in the range from 0.272 to 0.669 A. The greatest MAD(P)s vary between 8.924 and 19.517 W. The greatest MAD(I)s for the Sanyo PV panel vary from 0.366 to 0.478 A. The greatest MAD(P)s range from 13.513 to 16.587 W. In Tables 7  and 8 the values of the percentage ratio MD(I)/I mp,ref for the analysed panels, calculated considering the I-V curves at a constant cell temperature of 25 • C, are listed.are in the range from 3.383% to 12.349%, the greatest vary between 4.919% and 18.035%.Tables 9 and 10 list the values of the percentage ratio MD(I)/I mp,ref calculated for Kyocera KD245GH-4FB2 and Sanyo HIT-240 HDE4 PV panels at a constant solar irradiance of 1000W/m 2 .Table 8.Maximum current differences between the calculated and the issued I-V characteristics of Sanyo HIT-240 HDE4, at temperature T = 25 The smallest percentage values of MD(I)/I mp,ref for the Kyocera PV module at constant solar irradiance range from 2.321% to 4.763%; the greatest percentage values of MD(I)I mp,ref vary between 10.656% and 20.437%.For the Sanyo PV panel the smallest percentage values of MD(I)/I mp,ref vary from −4.047% to 8.479%; the greatest are contained in the range from 17.829% to 18.360%.Tables 11-14 show the values of the percentage ratio MD(P)/V mp,ref I mp,ref 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. For the Kyocera PV panel, the smallest percentage values of MD(P)/V mp,ref I mp,ref at constant cell temperature vary from 1.462% to 4.064%.The greatest percentage values of MD(P)/V mp,ref I mp,ref are in the range 2.449% to 13.368%.For the Sanyo PV module, the smallest percentage values of MD(P)/V mp,ref I mp,ref at constant temperature vary from 3.553% to 13.812%; the greatest range 5.344% to 20.777%.

Parameters at the
Considering the MD(P)/V mp,ref I mp,ref at constant solar irradiance, the smallest percentage values for the Kyocera PV panel range from 2.536% to 4.151%; the greatest vary between 12.518% and 20.235%.The smallest percentage values of MD(P)/V mp,ref I mp,ref for the Sanyo PV module are in the range from −4.319% to 9.435%; the greatest vary from 17.803% to 20.777%.Tables 15 and 16 list the percentage ratios of MAD(I) to the current at the issued MPP and of MAD(P) to the rated maximum power.
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 indicated in the last column.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 .For the Kyocera PV panel the smallest MAD(I)s range from 0.64% to 1.51% of the current at the MPP; the greatest MAD(I)s vary from 0.95% to 8.49%.The smallest MAD(I)s for the Sanyo PV module are in the range 1.08% to 4.09% of the current at the MPP; the greatest MAD(I)s range from 2.53% to 7.06%.The smallest MAD(P)s range from 0.66% to 1.34% of the rated maximum power for the Kyocera PV panel; the greatest MAD(P)s vary from 0.86% to 7.96%.For the Sanyo PV module the smallest MAD(P)s are in the range 1.08% to 4.44% of the rated maximum power; the greatest MAD(P)s vary from 2.27% to 6.90%.

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 criterion based on a three-level rating scale described in [60] was adopted.The 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: • high, when only tabular data are required; • medium, when the data have to be extracted by reading the I-V characteristics; • low, when the derivative of the I-V curves are required.
The simplicity of the used mathematical tools is considered: • high, if only simple calculations are necessary; • medium, if an iterative procedure is used; • low, when the analytical procedure requires the use of dedicated computational software.
Table 17 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 15 and 16.The global accuracy listed in Table 17, which is calculated averaging the accuracies evaluated for the Kyocera and Sanyo PV panels, varies between 2.27% and 3.88%.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 2.27% to 2.81%; • medium, for values of the mean difference in the subrange 2.81% to 3.34%; • low, for values of the mean difference in the subrange 3.34% to 3.88%.
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.In order to assess the suitability of adopting two-diode models instead of one-diode models, a comparison with the performances of the best known diode-based models was carried out considering the I-V characteristics of the same PV panels.Table 19 lists the usability and accuracy ratings of the one-diode models ranked in [60,62] along with the ones of the two-diode models analysed in the present paper.To make a consistent comparison, the accuracy was rated on the basis of the smallest and the greatest mean differences calculated for all the analysed models.According to such minimum and maximum values, the following accuracy subranges were defined:

•
high, for values of the mean difference in the subrange 0.53% to 1.91%; • medium, for values of the mean difference in the subrange 1.91% to 3.30%; • low, for values of the mean difference in the subrange 3.30% to 4.68%.
It can be observed that the analysed two-diode models reach values of the accuracy comparable with the precision of the simplified one-diode models.Such result is not surprising because, as it was previously pointed out, only a part of the seven parameters of the two-diode models are obtained from the equations that describe the relevant proprieties of the I-V curves.Actually, the Hejri

Conclusions
In order to rate the usability of the two-diode models for PV cells and panels, the analytical procedures to evaluate the model parameters and the hypotheses, which were adopted to simplify calculations, were described in detail.Using the data extracted from the datasheets issued by the manufactures of two different types of PV modules, the I-V curves at constant cell temperature and solar irradiance were calculated by means of the analysed models.In order to test the model accuracies, the calculated I-V curves were compared with the issued I-V characteristics.The maximum difference and the mean absolute difference between the calculated values of current and the numerous values of current extracted from the issued I-V characteristics were considered; also the maximum difference and the mean absolute difference for the generated power were evaluated.
The achieved accuracy obviously depends on the used model and the considered I-V curve.For the most effective two-diode equivalent circuits, the calculated current differences averagely vary between 0.64% and 1.51% of the current at the MPP, for the poly-crystalline Kyocera KD245GH-4FB2 PV panel.The values of the power difference averagely range from 0.66% to 1.34% of the rated maximum power.For the Sanyo HIT-240 HDE4 PV module smaller accuracies were generally observed.The current differences averagely vary from 1.08% to 4.09% of the current at the MPP.The power accuracies averagely range from 1.08% and 4.44% of the rated maximum power.The accuracies of the less effective models averagely reach 8.49% of the current at the MMP and 7.96% of the rated maximum power for the Kyocera PV panel, whereas average differences of 7.06% of the current at the MMP and of 6.90% of the rated maximum power were observed for the Sanyo PV module.
It is not a trivial matter to identify the most usable and accurate model because no model reaches the highest ratings for all the features considered by the adopted criterion.Among the previously analysed models, the Ishaque et al. model is the most accurate and has a medium degree of mathematical difficulty.If the model comparison is extended to the one-diode based models ranked in [60,62], the best ratings among the simplified one-diode models are given to the Townsend n.2 model, the Saloux et al. model and the Mahmoud et al. n.2 model, which present the same degree of ease of data finding, mathematical simplicity and current and power accuracy; the Orioli et al. model reaches the best rating among the five-parameter models.The analysed two-diode models do not confirm their supposed capability to yield very accurate results.The lack of effectiveness is probably due to the fact that the proposed analytical procedures arbitrarily fix some of the seven parameters of the two-diode model with the consequence of wasting the opportunities given by the presence of a wider number of model parameters.
in which n = N cs k/q.Assuming the following hypotheses, the equations can be approximated in order to simplify the evaluation of the model parameters: Using the first two terms of Equations (A13) and (A14), Equation (A12) can be approximated with the following quadratic form: whereas, if the first three terms of Equations (A13) and (A14) are used, a cubic form can be obtained:

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

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

Energies 2017, 10 , 564 3 of 33 Figure 2 .
Figure 2. Range-scaled I-V characteristics of crystalline and thin-film PV panels at the SRC.

Figure 3 .
Figure 3. Effects of the series resistance on the I-V characteristic.

Figure 2 .
Figure 2. Range-scaled I-V characteristics of crystalline and thin-film PV panels at the SRC.

Figure 3 .
Figure 3. Effects of the series resistance on the I-V characteristic.

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

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

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

Figure 5 .
Figure 5. Effects of the saturation currents on the I-V characteristic.

I 8 Figure 5 .
Figure 5. Effects of the saturation currents on the I-V characteristic.

( 3 )
short circuit point (I = I sc,ref ; V = 0); (4) open circuit point (I = 0; V = V oc,ref ); (5) MPP (I = I mp,ref ; V = V mp,ref ); (6) derivative of current at the short circuit point (∂I/∂V = −1/R sho at I = I sc,ref ; V = 0); (7) derivative of current at the open circuit point (∂I/∂V = −1/R so at I = 0; V = V oc,ref ).In order to simplify the evaluation of the model parameters, the following hypotheses are assumed: e V oc,re f nT re f >> e I sc,re f Rs nT re f , e V oc,re f 2nT re f >> e I sc,re f Rs

32 ( 1 )
shape factor a 1 = 1; (2) shape factor a 2 = 2; (3) short circuit point (I = I sc,ref ; V = 0); (4) open circuit point (I = 0; V = V oc,ref ); (5) MPP (I = I mp,ref ; V = V mp,ref ); (6) derivative of current at the short circuit point (∂I/∂V = −1/R sho at I = I sc,ref ; V = 0); (7) derivative of power at the MPP (∂P/∂V = 0; V = V mp,ref ).As described in the Appendix A, from the information used, parameters I 01,ref , I 02,ref , R sh , and I L,ref can be calculated by means of an iterative procedure.Hovinen did not use the model to calculate the I-V characteristics for values of solar irradiance and cell temperature different from the SRC.

( 3 )
short circuit point (I = I sc,ref ; V = 0); (4) open circuit point (I = 0; V = V oc,ref ); (5) MPP (I = I mp,ref ; V = V mp,ref ); (6) derivative of current at the short circuit point (∂I/∂V = −1/R sho at I = I sc,ref ; V = 0); (7) derivative of power at the MPP (∂P/∂V = 0; V = V mp,ref ).Adopting the following hypotheses: e V oc,re f nT re f >> e I sc,re f Rs nT re f , e V oc,re f 2nT re f >> e I sc,re f Rs 2nT re f (6) I 01,re f nT re f e I sc,re f Rs nT re f NRM SCP: Short Circuit Point; OCP: Open Circuit Point; MPP: Maximum Power Point; DSCP: Derivative of I at SCP; DOCP: Derivative of I at OCP; DMPP: Derivative of power at MPP; SC: Simple Calculation; IP: Iterative Procedure; NRM: Newton-Raphson Method; Simplif.Hypoth.: Simplifying Hypotheses; Mathem.Tools: Mathematical Tools.

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

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

2 Figure 6 . 33 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 Chan et al. models.

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

2 Figure 7 . 33 Figure 6 .
Figure 7.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 • C and the characteristics calculated by means of the Enebish et al. and the Hovinen models.

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

Figure 8 . 2 Figure 8 .Figure 9 .
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 Chan et al. models.

Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 9 . 33 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 Enebish et al. and the Hovinen models.

Figure 10 .
Figure 10.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 °C and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 10 . 33 Figure 11 .
Figure 10.Comparison between the issued I-V characteristics of Kyocera KD245GH-4FB2 at T = 25 • C and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.Energies 2017, 10, 564 11 of 33

Figure 11 .
Figure 11.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 • C and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 11 .
Figure 11.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

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 Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4at G = 1000 W/m 2 and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

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 Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 11 .
Figure 11.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4 at T = 25 °C and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

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 Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4at G = 1000 W/m 2 and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

Figure 13 .
Figure 13.Comparison between the issued I-V characteristics of Sanyo HIT-240 HDE4at G = 1000 W/m 2 and the characteristics calculated by means of the Hejri et al., the Gupta et al. and the Ishaque et al. models.

I− 1 − 1 −
mp,re f = I L,re f − I 01,re f e V mp,re f +I mp,re f Rs nT re f I 02,re f e V mp,re f +I mp,re f Rs 2nT re f − V mp,re f +I mp,re f R s R sh

eV
oc,re f nT re f >> e I sc,re f Rs nT re f , e V oc,re f 2nT re f >> e I sc,re f Rs2nT re f , R sh >> R s , R sho >> R s ,re f R s << V oc,re f (A7)Using I L,ref from Equation (A2) and assuming the hypotheses in Equations (A6) and (A7), Equations (A1)-(A5) can be rewritten as:I 01,re f e V oc,re f nT re f + I 02,re f e V oc,re f 2nT re f − I sc,re f + V oc,re f R sh = 0 (A8) I 01,re f e V oc,re f nT re f − e V mp,re f +I mp,re f Rs nT re f + I 02,re f e V oc,re f 2nT re f − e V mp,re f +I mp,re f Rs 2nT re f + V oc,re f −V mp,re f R sh − I mp,re f = 0 (A9) R sh = R sho (A10) (R so − R s ) I 01,re f nT re f e V oc,re f nT re f + I 02,re f 2nT re f e V oc,re f 2nT re f − 1 = 0 (A11)Extracting I 01,ref and I 02,ref from Equations (A8) and (A9), and using Equation (A10), the following expression, which only contains the unknown series resistance, can be obtained from Equation (A11):I sc,re f − I mp,re f − V mp,re f R sho − V oc,re f R sho − I sc,re f + 2nT re f R so −R s eV mp,re f s −V oc,re f get the solution of Equation (A12), the exponential terms containing parameter R s can be substituted with their respective power series:

−I
A16)Both Equations (A15) and (A16) can be easily solved by means of ordinary mathematical methods because the involved coefficients a and b only contain known quantities.Diode currents I 01,ref and I 02,ref can be calculated with the following equations obtained by solving Equations (A8) and (A11):I 01,re f = V oc,re f R sho − I sc,re f + 2nT re f R so − R s e 02,re f = I sc,re f − V oc,re f R sho − nT re f R so − R s e − V oc,re f 2nT re f (A18) (R sho − R s ) A71)one can conclude that R sho ≈ R sh and Equation (A71) can be used in the form: (R sh − R s ) I 01,re f nT re f e I sc,re f Rs nT re f + I 02,re f 2nT re f e I sc,re f Rs2nT re f + 1 R sh − 1 = 0 (A73)that avoids the graphical extraction of parameter R sho from the experimental I-V curve of the analysed PV panel.Because the derivative of the current is: regarding the maximum power:∂(P) ∂V V = V mp,re f I = I mp.re f = ∂(V I) ∂V V = V mp,re f I = I mp,re f = I mp,re f + V mp,re f ∂I ∂V V = V mp,re f I = I mp,re f = 0 (A75)it can be extracted the following form:∂I ∂V V = V mp,re f I = I mp,re f = − I mp,re f V mp,re f (A76) ref and short circuit current I sc,ref at the standard reporting conditions (SRC);

Table 1 .
Summary of the information and solving techniques used by the analysed models.

Table 3 .
Model parameters of Kyocera KD245GH-4FB2 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 7 .
Maximum current differences between the calculated and the issued I-V characteristics of Kyocera KD245GH-4FB2, at temperature T = 25 • C.

Parameters at the Maximum Difference Points Irradiance (W/m 2 ) 200 400 600 800 1000
Considering the I-V curves at constant temperature of the Kyocera PV panel, the smallest percentage values of MD(I)/I mp,ref vary from 1.324% to 3.670% and the greatest are contained in the range from 2.284% to 11.422%.The smallest percentage values of MD(I)/I mp,ref for the Sanyo PV module

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 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 .

Table 15 .
Percentage ratio of MAD(I) to the rated current at the MPP.

Table 16 .
Percentage ratio of MAD(P) to the rated maximum power.

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

Table 18 .
Usability and accuracy ratings of the analysed one-diode models.Hejri et al. model, the models require data that are easy to be found.The Gupta et al. model achieves a small accuracy and presents the greatest mathematical difficulties.The Ishaque et al. model, which is very accurate and has a medium degree of mathematical difficulty, may be considered the best option among the two-diode models.
et al. model is a five-parameter model because it arbitrarily sets the values of a 1 and a 2 .The Ishaque et al. model is a four-parameter model because it also fix I 02 = I 01 .The Gupta et al. model is a tree-parameter model because the values of a 1 , a 2 , R s and R sh are not obtained from calculations.As a consequence, it is quite logical that such incomplete seven-parameter models do not surpass the accuracy of the one-diode models.No model achieves the highest ratings for all the considered features.For this reason the choice of the best model requires a wise compromise between usability and accuracy.The Orioli et al. model, the Townsend n.2 model, the Saloux et al. model and the Mahmoud et al. n.2 model have the best global rating.The Orioli et al. model, which reaches a high precision, presents some mathematical difficulties; conversely, the parameters of the Townsend n.2 model, the Saloux et al. model and the Mahmoud et al. n.2 model can be easily calculated but these models are less precise.

Table 19 .
Usability and accuracy ratings of the analysed one-diode based models.