Analytical Modeling of Current-Voltage Photovoltaic Performance: An Easy Approach to Solar Panel Behavior

: In this paper, we propose very simple analytical methodologies for modeling the behavior of photovoltaic (solar cells/panels) using a one-diode/two-resistor (1-D/2-R) equivalent circuit. A value of a = 1 for the ideality factor is shown to be very reasonable for the different photovoltaic technologies studied here. The solutions to the analytical equations of this model are simpliﬁed using easy mathematical expressions deﬁned for the Lambert W-function. The deﬁnition of these mathematical expressions was based on a large dataset related to solar cells and panels obtained from the available academic literature. These simpliﬁed approaches were successfully used to extract the parameters from explicit methods for analyzing the behavior of solar cells/panels, where the exact solutions depend on the Lambert W-function. Finally, a case study was carried out that consisted of ﬁtting the aforementioned models to the behavior (that is, the I-V curve) of two solar panels from the UPMSat-1 satellite. The results show a fairly high level of accuracy for the proposed methodologies.


Introduction
Based on the installation of power plants over the last few decades, it can be seen that photovoltaic energy has emerged as a very important factor in policies relating to renewable energy sources [1][2][3][4][5][6][7]. As an energy source, solar panels have a significant competitive edge over other renewable energy sources due to their potential for dual-use at either the industrial or the domestic scale. Both possibilities have provided an impetus for the increasing growth in demand for photovoltaic systems [5,[8][9][10][11].
Photovoltaic energy has also been demonstrated to be crucial for stand-alone power systems such as glow-in-the-dark lettering and illuminated signs on highways. One of the first such stand-alone industrial applications of photovoltaic technologies was in power generation in spacecraft. According to Rauschenbach, "The first solar cell array that successfully operated in space was launched on 17 March 1958, on board Vanguard I, the second U.S. earth satellite" [12].
Modeling the performance of power sub-systems for space missions is essential at the predesign stage. At these early configuration stages of a space mission, simple, fast simulations are required that are as accurate as possible. This trend has also been amplified by the increasing importance of Concurrent Design (CD) in industrial processes. According to the European Cooperation for Space Standardization (ECSS), CD is especially important

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The extraction of the parameters appearing in this equation, which define the performance of the photovoltaic device at a certain temperature and irradiance level, and; • The calculation of the output current as a function of the output voltage, or vice versa, since this equation is implicit.
There are multiple ways to fit the equation for the photovoltaic equivalent circuit models to experimental data [20][21][22][23][24][25][26][27][28][29]. All of these approaches can essentially be grouped into two different types: Numerical (in which the equation is fitted to a large dataset that represents the I-V curve of the photovoltaic device) and analytical (in which the parameters of the equation are extracted based on three points of the I-V curve: short circuit, maximum power, and open voltage).
Possible solutions to the aforementioned problems include explicit methods of photovoltaic modeling, which are based on explicit equations with parameters defined based on the characteristic points of the I-V curve. Some of these methods also require the use of the Lambert W-function to define the parameters [56].
In the present work, we explore two important aspects of the analytical approach to the one diode/two resistor (1-D/2-R) equivalent circuit model for solar panels: • Selection of a proper ideality factor for extraction of the model parameters based on analytical methods, as this forms the cornerstone for extracting the other four parameters [57], and; • Development of a simplified approach to the Lambert W-function, which is required in this analytical methodology in order to: Obtain the value of the first parameter of the 1-D/2-R equivalent circuit model, i.e., the value of the resistance of the series-connected resistor, and; Solve the implicit equation of this model to derive the output current in relation to the output voltage.
Our simplified approach to the Lambert W-function is also proven to be a relatively powerful tool for extracting the parameters of some of the most accurate explicit equations for photovoltaic modeling that can be found in the literature: the models of Kalmarlar and Haneefa [58,59] and Das [60].
The aim of this paper is to define very simple methodologies to measure the performance of photovoltaic devices (solar cells/panels), thus that these can be implemented as part of more complex simulations. One example of this type of simulation is the coupled thermo-electrical modeling of space systems with ESATAN © [61]. It should also be emphasized that these methodologies may be useful tools for professionals within small and medium-sized enterprises in the solar energy sector to allow them to estimate the performance of the systems they design. Additionally, the increasing use of photovoltaic generation in small grids [62,63] could increase the worth of the easy approximations to solar panels performance included in this paper. This paper is organized as follows. A simple model of solar panels is described in Section 2, and simplified equations for solving the 1-D/2-R equivalent circuit model and the Lambert W-function are given in Section 3. A case study is also described and solved in Section 3, using the procedures described in the preceding sub-sections. Finally, conclusions are presented in Section 4.

Modeling of Photovoltaic Systems
The most widely used method of modeling the performance of a solar cell/panel (based on its I-V curve, where I is the output current and V the output voltage) is an equivalent circuit based on one current source, one diode, and two resistors (series and shunt resistors), as shown in Figure 1. The equation that describes the performance of this model is as follows [57,[64][65][66]: where I pv is the photocurrent, I 0 is the reverse saturation current of the diode, R sh is the resistance of the shunt resistor, R s is the resistance of the series-connected resistor, and V T is the thermal voltage: In the above equation, κ = 1.38064852·10 −23 m 2 kg s −2 K −1 is the Boltzmann constant, T is the temperature expressed in K, and q = 1.60217662·10 −19 C is the electron charge. In Equation (1), a is the ideality factor of the diode (in principle, the effect of temperature on this parameter can be left aside [64]), and n is the number of series-connected cells in the solar panel (obviously, n = 1 when studying the performance of a single cell).
The first problem of modeling a photovoltaic device with the 1-D/2-R equivalent circuit model lies in extracting the five parameters of the model: I pv , I 0 , a, R s , and R sh . Depending on the available information, which might be either:

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The I-V curve with an enough large number of points, or; • The three characteristic points of the I-V curve (short circuit output current, I sc , open circuit output voltage, V oc , and output current and voltage levels at the Maximum Power Point (MPP), I mp and V mp , It is possible to extract these five parameters either by numerically fitting Equation (1) to the I-V curve, or by using the three characteristic points, which in practice represent four conditions [67]: the first three of which force the equation to match the points (0, I sc ), (V mp , I mp ) and (V oc , 0), and the fourth is related to the maximum power condition: Modeling the performance of photovoltaic systems with the equivalent circuit described here can be very advantageous, as these circuits somehow preserve the physical processes of a pair formed of a current source and a p-n junction [68]. The effect of the series-connected resistor, R s , is mainly associated with power losses in the solder bonds and interconnections between cells, whereas the effect of the shunt resistor, R sh , is associated with current leakages across the p-n junction [69,70]. In recent years, several reviews of the different procedures and techniques for the parameter extraction problem related to the 1-D/2-R model have been published [20,21,23,24,67,[71][72][73]. The present work focuses on an analytical approach based on the use of information from the characteristic points to calculate the five parameters in Equation (1). The following equations can be derived [57]: Three problems arise at this point: • A sufficiently accurate estimation of the ideality factor a is required; • Equation (4) is an implicit mathematical expression for solving R s , and an iterative process is, therefore, required to extract this parameter; and • Equation (1), which defines the performance of the solar cell/panel, is also an implicit expression. As a consequence, once all the parameters of this equation have been extracted, an additional iterative process will be required to solve it (i.e., to derive the value of the output current, I, for a given value of the output voltage level, V).
Fortunately, the ideality factor can be estimated by taking a value within the range from a = 1 to a = 1.5 [74,75], and the value of the resistance of the series-connected resistor, R s , can be derived from [64]: where W −1 is the negative branch of the Lambert W-function (see Section 2.1). The variables A, B, C, and D are defined as: However, the problem of solving the implicit equation in (1) still remains after the extraction of parameters a and R s . Authors such as Peng et al. [50] have shown that this equation can also be solved using the Lambert W-function: where W 0 is the positive branch of the Lambert W-function (see Section 2.1).

Explicit Equations/Models as Alternative to the 1-D/2-R Equivalent Circuit Model
The difficulties associated with solving the 1-D/2-R model implicit equation drove researchers to develop explicit equations for modeling the performance of solar cells/ panels [58,59,[76][77][78][79][80][81][82]. These photovoltaic models give the output current as a function of the output voltage by using simple mathematical models or equations that can be easily solved without the need for advanced mathematical tools. Of these, three models appear to be fairly accurate in relation to the I-V curve. The model proposed by Kalmarkar and Haneefa [58,59] is: where [56]: and: the model proposed by Das [60] is: where [56]: while the model of Pindado and Cubas [56,80] is: where: In the graph in Figure 2, these models are fitted to the experimental I-V curve for an RTC solar cell, taken from the well-known work by Easwarakhanthan et al. [83]. The 1-D/2-R equivalent circuit model is also fitted to these data, and it can be seen that the accuracy of the explicit models is quite high, as stated in [56,81]. The problem with using some of these models again lies in the need to solve the Lambert W-function, which requires some mathematical ability.

The Lambert W-Function
From the methodologies described above, it is clear that in order to work with the equations of the implicit 1-D/2-R model and some of the explicit models, some knowledge of the Lambert W-function is required. The Lambert W-function, W(z) (plotted in Figure 3), is defined as: In the above equation, z is a complex number. If a real variable x is considered, the Lambert function is then defined within the range [−1/e, ∞]. It should also be noted that this function gives a double value within the range [−1/e, 0]. Two different branches, called the positive and negative branches, are normally defined for this function, as follows: This function is a useful tool for solving equations that involve exponentials since if X = Yexp(Y), then Y = W(X). However, as stated above, a certain level of expertise is required to solve these [84,85]. Barry et al. developed an interesting explicit approach, although this may not be sufficiently direct to obtain a solution [86].

Results
In this section, a method that facilitates the use of the 1-D/2-R equivalent circuit model is presented. It is organized into three sub-sections, as follows: • Sub-Section 3.1 describes the problem of selecting a suitable value for the ideality factor, a, in order to obtain the values for the rest of the parameters. • Very simple equations for the Lambert W-function, for use in solving the equations described in the section above, are included in Sub-Section 3.2.

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Finally, a case study in which our methodologies are applied to the modeling of spacecraft solar panels is included in Sub-Section 3.3.

On the Best Value for the Ideality Factor
In a paper published in the 2nd International Conference on Renewable Energy Research and Applications (ICRERA 2013), our research group demonstrated a hyperbolic relationship between the non-dimensional RMSE, ξ, of the 1-D/2-R equivalent circuit model fitted to the I-V curve of a solar cell, and the value of the ideality factor, a [87]. The non-dimensional (or normalized) RMSE is defined as: where N is the number of point of the I-V curve, I is the current obtained from modeling the corresponding solar cell/panel, and I ref is the measured current associated with each one of those points. The aforementioned relationship between ξ and a is plotted in Figure 4. It can be seen that the curve has a hyperbolic shape, with a minimum value of the nondimensional RMSE at a = 1.18. This hyperbolic shape was also confirmed in a recent work by Elkholy and Abou El-Ela [88]. The normalized RMSE can be used to compare the accuracy of different models applied to different photovoltaic technologies. However, it should also be taken into account that variations in temperature and irradiance could somehow affect the accuracy of the model's accuracy. In the present work, the current at the short circuit from the reference I-V curve has been selected as it was in previous works such as [57]. This way to normalize the RMSE has been used by other authors [89]. However, it was not the only one, as the average current value from the measured dataset that represents the I-V curve [88,90], or the difference between the current at two points [91], has also been proposed as current values to normalize the RMSE. The value of the ideality factor, a, was studied with the aim of providing future researchers with an accurate but much simpler fitting of the 1-D/2-R equivalent circuit model. To carry out this analysis, the I-V curves from four solar cells and three solar panels were selected from previous works [56,81], as shown in Table 1. The parameters extracted by fitting Equation (1) to these I-V curves using Matlab © are given in Table 2. These fits were used as a reference for the values calculated analytically since it could be reasonably assumed that they were the most accurate ones for the I-V curves. It should also be remarked that the equations for the 1-D/2-R equivalent circuit model resulting from these fits may not exactly meet the requirements for the short circuit, MPP, and open-circuit conditions, as they represent the best fits not only to the three characteristic points but to a large number of additional points. Table 1. Solar cell/panel I-V curves used in the present work (where n is the number of series-connected cells for each of these photovoltaic devices; T is the temperature; and the characteristic points are I sc , I mp , V mp , and V oc ).

Solar Cell/Panel
Technology The 1-D/2-R equivalent circuit model was also analytically fitted to the aforementioned I-V curves. Using specific values for the ideality factor, a, and the characteristic points of the curve, I sc , V mp , I mp and V oc , the different parameters of the model were found using the following procedure: • R s was calculated using Equation (8) to (12); • R sh was calculated using Equation (5); • I pv was calculated using Equation (6); • I 0 was calculated using Equation (7).
Once all the parameters were known, the I-V curve was found by obtaining the values of the current, I, for each value of the voltage, V, using Equation (13). The ideality factor was initially varied from a = 0.5 to a = 1.8 for each photovoltaic device (although this range was extended for some devices), and the non-dimensional RMSE, x, was calculated in each case. The results are plotted in Figure 5. Table 3 shows the values of the parameter a at which the minimum value of x is reached, with the rest of the parameters for the 1-D/2-R equivalent circuit.  Some important conclusions can be drawn based on these results, as follows: • The analytical methodology suggested in the present work may give unacceptable results when the ideality factor exceeds a certain value, as complex numbers start to emerge in the calculations.

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One of the parameters obtained analytically for the best fit in terms of the nondimensional RMSE has no physical meaning: this is the negative value of the resistance of the shunt resistor shown in Table 3, which is obtained for the Photowatt PWP 201 solar panel.

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The best analytical fit gave remarkably low values of a in two cases: the Azur Space 3G30C solar cell, and the Selex Galileo SPVS X5 solar panel (composed of five Azur Space 3G28C series-connected cells), as their values were outside the range [1,1.5] suggested by Villalva et al. [74,75].
In order to compare both types of fit, the values of the ideality factor that gave the lowest values of x for the analytical fittings, a bf-a , were plotted against the values obtained from the numerical fittings, a bf-n , in Figure 6. It can be seen that the correlation between the fits is high, with the exception of only two devices: the Azur Space 3G30C solar cell, and the Selex Galileo SPVS X5 solar panel (which are both based on the same photovoltaic technology, as shown in Table 1). However, this correlation does not allow us to identify any rule that would suggest a reasonable value for the ideality factor. We, therefore, plotted the average values of the non-dimensional RMSE, ξ av , obtained from the analytical fits versus the ideality factor, a. It can be seen from Figure 6 that the minimum value of ξ av was obtained at a = 0.98. As a consequence, the value suggested for all devices is a = 1.   Table 4) for all of the photovoltaic devices. Although the differences may appear high, it should be taken into account that photovoltaic systems are normally designed to operate at the MPP, which is exactly reproduced by all of the analytical approximations for solving the parameter extraction of the 1-D/2-R equivalent circuit model. In addition, we reviewed the results from several works published over the last five years on parameter extraction for 1-D/2-R equivalent circuit models, based on the same experimental data from the RTC France solar cell and the Photowatt PWP 201 solar panel. The values obtained in these works for the non-dimensional RMSE for the fits were lower than the results from our method, with average values of ξ = 1.20·10 −03 (RTC France solar cell) and ξ = 4.03·10 −03 (Photowatt PWP 201 solar panel). Although these more accurate values were based on numerical rather than analytical procedures, it should be pointed out that numerical approaches do not always reach a more accurate solution [92].  Table 2), an analytical fit (see Table 3), and an analytical fit with a = 1 (see Table 4). Table 4. Analytically fitted parameters for 1-D/2-R equivalent circuit models, with a = 1, for I-V curves corresponding to RTC France, TNJ Spectrolab, ZTJ Emcore, and Azrur Space 3G30C solar cells, and Photowatt PWP 201, Kyocera KC200GT-2, and Selex Galileo SPVS X5 solar panels (where ξ is the non-dimensional RMSE for the fit).

Lambert W-Function Simplified Equations for Solar Cell/Panel Modeling
A thorough review of the available literature from between 2000 and 2020 was carried out to find relevant data on photovoltaic devices, including the five parameters of the 1-D/2-R equivalent circuit model (I pv , I 0 , a, R s , and R sh ), the characteristic points of the I-V curve (I sc , V mp , I mp , and V oc ), the temperature of the cells, T, which is related to the performance curve, and the number of series-corrected cells, n. Information was found for 90 photovoltaic devices, most of which were solar panels.
The positive branch of the Lambert W-function needs to be solved for in Equation (13), and must be evaluated at certain points x for a given value of the output voltage, V (within the range [0, V oc ]): x = f I pv , I 0 , a, R s , R sh , n, T, V .
After post-processing the data from the 90 solar cells/panels, it was clear that Equation (13) refers to the right-side section of the Lambert's W-function positive branch, W 0 + . The following expressions were proposed:  10]. (28) The error for these equations was below 0.15% (Equation (26)), 0.16% (Equation (27)), and 0.08% (Equation (28)). For larger values of x, we recommend using the approximation proposed by Barry et al. [86] (see Appendix A). In Figure 8, the positive branch of Lambert's W-function, W 0 + , calculated at the points x (Equation (25)) corresponding to the data for the solar cells/panels, evaluated at V = 0 and V = V oc , is shown together with Equation (26) to (28). Depending on the magnitude of x (that is, depending on how closely it approaches 0 + ), W 0 + can be reasonably estimated as: for points x (Equation (25)) corresponding to the 1-D/2-R equivalent circuit model, based on the data for the solar cells/panels. Our approximation for W 0 + (Equation (26) to (28)) is also shown.
The negative branch of the Lambert W-function, W −1 , needs to be calculated at: when estimating the series resistance, R s , for the 1-D/2-R equivalent circuit model (Equation (8) to (12)). Since the above equation for the different solar cells/panels from the database gives values of x within the range [−2.77·10 −3 , −10 −35 ], two expressions were defined: The error for these equations was below 0.42% (Equation (31)) and 0.02% (Equation (32)) (see Figure 9). Equation (32) can be applied for values of up to x = −10 −50 , with errors of below 0.07%.  (30)) corresponding to the series resistance, R s , of the 1-D/2-R equivalent circuit model, based on data for solar cells/panels. Our approximate expression for W −1 (Equations (31) and (32)) is also shown. (b) Negative branch of Lambert's W-function, W −1 , calculated for points x (Equation (33)) corresponding to the explicit models proposed by Kalmarkar and Haneefa and Das, based on data for solar cells/panels. Our approximate expression for W −1 (Equations (35) and (36)) is also shown.
Finally, if the explicit models proposed by Kalmarkar and Haneefa, and Das are used, W -1 needs to be calculated at points x defined by the characteristic points of the I-V curve (I sc , V mp , I mp , and V oc (see Equations (16) and (19)): x = f I sc , I mp , V oc , V mp (33) as shown in Figure 9. Based on the post-processed data for the 90 solar cells/panels, the values of x calculated using these explicit methods were within the range [−0.304, −0.1].
The following equation for W -1 was initially suggested, which was characterized by an error of less than 1.6%22 within this range: In order to extend both the range of validity and the accuracy of the above equation, a splitting into two sub-branches at the point where the curvature of the function changes its sign (x = −0.27), the following equations were suggested: The error for the above equations was below 0.23% (Equation (35)) and 0.02% (Equation (36)) (see Figure 9). It should also be emphasized that caution needs to be used with regard to Equation (35), as it gives complex solutions when x→−1/e + (that is, when x ∈ [−1/e, −0.36785]).

Case Study: The Solar Panels of the UPMSat-1
The general characteristics from the I-V curves for the UPM-5 and UPM-6 solar panels for UPMSat-1 (see Figure 10) are shown in Tables 1, 2 and 5 (Appendix B). These panels are based on two different technologies: Si (UPM-5) and Ga-As (UPM-6). In the latter, the number of series-connected cells n was 32; however, since they were based on double-junction technology, this number must be multiplied by two, unlike in the 1-D/2-R equivalent circuit model (the Selex Galileo SPVS X5, formed from five Azur Space triple-junction technology series-connected solar cells, where n = 15, as shown in Table 1). Figure 10. The UPMSat-1 satellite; launched in 1995, this was the tenth university-developed space mission in history [93].  Table 5. Characteristics of the measured I-V curves corresponding to the UPM-5 and UPM-6 solar panels of UPMSat-1 (see Figure 10).

Solar Panel Technology n T [ • C] I sc [A] I mp [A] V mp [V] V oc [V]
UPM-5 The I-V curves for the UPM-5 and UPM-6 solar panels are plotted in Figures 11 and 12, respectively, with several other curves corresponding to:

Solar
The differences were relatively small, with the maximum always located in a reasonably small range between the MPP and the open voltage point. The values of the non-dimensional RMSE, ξ, for the fits are plotted in Figure 13. The results show reasonably good agreement with the mathematical approaches proposed in the present work.

Conclusions
In this study, we have reviewed analytical approaches for extracting the five parameters of the 1-D/2-R equivalent circuit model from the characteristic points of the I-V curve (short circuit, MPP, and open circuit points). Five different problems were identified, as follows: • An initial estimation of the ideality factor, a, is required.

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The equation for the value of the resistance of the series-connected resistor, Rs, is an implicit expression, meaning that either an iterative process or the Lambert W-function is required. • When all the parameters for the 1-D/2-R equivalent circuit model have been extracted, an implicit equation must be solved (or the Lambert W-function must be used) to derive the value of the output current for a given output voltage.

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The use of the Lambert W-function requires some numerical and calculation resources and skills.

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The use of explicit models rather than the 1-D/2-R equivalent circuit model, in order to avoid the problems described above, may not be possible, as some of them require the Lambert W-function to derive their parameters, based on the characteristic points of the I-V curve.
I-V curves from seven different solar cells/panels were used to test our proposed approach. I-V curves from two of the solar panels of the UPMSat-1 spacecraft were also analyzed as a case study.
The most important conclusions of this work are as follows: • A value of a = 1 for the 1-D/2-R equivalent circuit model was shown to be reasonable for most photovoltaic technologies.

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The analytical procedure for extracting the parameters for the 1-D/2-R equivalent circuit model may give values for the resistance of one resistor (or even both) that are negative. However, this does not affect the results (i.e., the modeled performance of the photovoltaic device).

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The Lambert W-function can be simplified for use in modeling the performance of photovoltaic devices. Accurate simplified versions of the Lambert W-function are proposed here for three cases, depending on the specific need: (i) calculation of Rs; (ii) calculation of the output current using the equation for the 1-D/2-R equivalent circuit; or (iii) calculation of the parameters for certain explicit models. • Explicit models are also accurate alternatives to the 1-D/2-R equivalent circuit model.
Our approach was carefully verified in a case study in which the fits of the1-D/2-R equivalent circuit model and the explicit methods to the measured data (I-V curve) for two solar panels from the UPMSat-1 satellite were compared with the results of the proposed method.
Finally, it should be highlighted that the results from this work open up new possibilities for coupled calculations, both for the performance of photovoltaic systems and in other disciplines such as thermodynamics and power distribution in grids. The simple but accurate solutions to the 1-D/2-R equivalent circuit model and the explicit methods described here can easily be implemented in software packages such as ESATAN © .