## 4 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. * 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. * The use of the Lambert W-function requires some numerical and calculation resources and skills. * 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. * 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). * 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. Figure 13: Non-dimensional RMSE, \(\zeta\), for the numerical fit of the 1-D/2-R equivalent circuit model, and the analytical fit with the proposed method, for the UPM-5 and UPM-6 solar panels of UPMSat-1. Values derived from the explicit methods studied here are also shown. 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(r). Project administration, S.P., J.C., E.R.-M.; methodology, S.P. ; conceptualization, S.P. ; formalism analysis, S.P. ; supervision S.P., J.C., E.R.-M.; investigation, J.M.A., D.A.-C., J.-E., M.S.-P., R.J.-P., A.G.-E.; software, J.M.A., D.A.-C., J.C.-E., M.S.-P., R.J.-P., A.G.-E.; validation, visualization, J.M.A., D.A.-C., J.C.-E., M.S.-P., R.J.-P., A.G.-E.; writing--original draft, J.M.A., D.A.-C., E.R.-M., J.C.-E., M.S.-P., R.J.-P., A.G.-E.; writing--review and editing, S.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.