## 3 Results
### 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  
Figure 9: (**a**) Negative branch of [PERSON]’s W-function, W-1, calculated for points \(x\) (Equation (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 [PERSON]’s W-function, \(W_{-1}\), calculated for points \(x\) (Equation (33)) corresponding to the explicit models proposed by [PERSON] and [PERSON], based on data for solar cells/panels. Our approximate expression for \(W_{-1}\) (Equations (35) and (36)) is also shown. 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). 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:  
* The best fit of the 1-D/2-R equivalent circuit model (see Table 6);
* The analytical fit for \(a=1\) (Equations (5-12)), and our approximations to the Lambert W-function (Equations (26)-(28),(31,32));
* The explicit models proposed by [PERSON] and [PERSON], [PERSON] (Equations (14)-(20), (35,36)), and [PERSON] (Equations (21) and (22)), see Table 7. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Solar Panel** & **Fitting \(I_{pp}\) [A]** & \(I_{0}\) [A] & \(a\) & \(R_{s}\) [\(\Omega\)] & \(R_{sh}\) [\(\Omega\)] & \(\xi\) \\ \hline UPM-5 & Num. & 1.4314 & 1.0495.10\({}^{-09}\) & 1.105 & 1.0368 & 4.3761\(\cdot 10^{-03}\) & 1.80\(\cdot 10^{-03}\) \\  & An. & 1.4323 & 1.0855\(\cdot 10^{-10}\) & 1 & 1.2045 & 1.3023\(\cdot 10^{+03}\) & 5.63\(\cdot 10^{-03}\) \\ UPM-6 & Num. & 1.4295 & 1.0285\(\cdot 10^{-09}\) & 0.902 & 0.8483 & 9.9115\(\cdot 10^{+02}\) & 9.50\(\cdot 10^{-03}\) \\  & An. & 1.4238 & 7.3448\(\cdot 10^{-09}\) & 1 & 0.7207 & 1.3524\(\cdot 10^{+03}\) & 1.70\(\cdot 10^{-02}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters for the 1-D/2-R equivalent circuit model (where Num. represents the numerical best fit, and An. represents the analytical fit with \(a=1\)) for the UPM-5 and UPM-6 solar panels of the UPMSat-1 satellite. \(\xi\) is the non-dimensional RMSE for the experimental data. Figure 10: The UPMSat-1 satellite; launched in 1995, this was the tenth university-developed space mission in history [93]. (**a**) Satellite during the integration in the Ariane IV-40 launcher (V75 flight). (**b**) Exploded view drawing. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Solar Panel** & **Technology** & \(n\) & \(T\) [\({}^{\circ}\)C] & \(I_{sc}\) [A] & \(I_{mp}\) [A] & \(V_{mp}\) [V] & \(V_{oc}\) [V] \\ \hline UPM-5 & Si & 51 & 25 & 1.431 & 1.329 & 25.139 & 30.513 \\ UPM-6 & Ga-As & 64 & 25 & 1.423 & 1.318 & 25.806 & 31.351 \\ \hline \hline \end{tabular}
\end{table} 
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). Figure 11: \(I\)-\(V\) experimental curve (left axis) for the UPM-5 solar panel. Curves for the 1-D/2-R equivalent circuit model (obtained using two procedures: best fit to the parameters obtained numerically and analytical extraction) and the explicit methods proposed by [PERSON] and [PERSON], [PERSON] and [PERSON] and [PERSON] are shown. The differences in the current compared to the experimental data are indicated on the right axis.