## 3 Results
### 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 \(R_{sl}\)), the characteristic points of the \(I\)-\(V\) curve (\(I_{sc}\), \(V_{mp}\), \(I_{mp}\), and \(V_{\alpha\alpha}\)), 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, \(V_{\alpha\alpha}\)]):  
\[x=f\big{(}I_{pv},I_{0},a,R_{s},R_{sh},n,T,V\big{)}. \tag{25}\]  
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:  
\[W_{0}^{+}(x)=x\exp\left(0.71116x^{2}-0.98639x\right),\ x\in\left[2\cdot 10^{-1 6},2\cdot 10^{-1}\right], \tag{26}\]  
\[W_{0}^{+}(x)=-1.6579+0.1396\Big{(}2.9179\cdot 10^{5}-(x-22.8345)^{4}\Big{)}^{0.2 5},\ x\in[0.2,1.2], \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Solar Cell/Panel** & \(I_{pv}\) **[A]** & \(I_{0}\) **[A]** & \(a\) & \(R_{s}\) **[\(\Omega\)]** & \(R_{sl}\) **[\(\Omega\)]** & \(\xi\) \\ \hline RTC France & 7.638\(\cdot 10^{-01}\) & 2.721\(\cdot 10^{-10}\) & 1 & 6.912\(\cdot 10^{-02}\) & 1.615\(\cdot 10^{+01}\) & 1.77\(\cdot 10^{-02}\) \\ TNJ Spectrolab & 5.262\(\cdot 10^{-01}\) & 1.786\(\cdot 10^{-15}\) & 1 & 1.078\(\cdot 10^{-01}\) & 1.895\(\cdot 10^{+02}\) & 5.08\(\cdot 10^{-03}\) \\ ZT) Emcore & 4.636\(\cdot 10^{-01}\) & 2.835\(\cdot 10^{-16}\) & 1 & 1.348\(\cdot 10^{-01}\) & 3.723\(\cdot 10^{+02}\) & 9.51\(\cdot 10^{-03}\) \\ Azur Space 3G30C & 5.269\(\cdot 10^{-01}\) & 3.881\(\cdot 10^{-16}\) & 1 & \(-6.068\cdot 10^{-02}\) & 2.643\(\cdot 10^{+02}\) & 3.38\(\cdot 10^{-02}\) \\ Photowatt PWP 201 & 1.036\(\cdot 10^{+00}\) & 4.163\(\cdot 10^{-08}\) & 1 & 1.982\(\cdot 10^{+00}\) & 5.430\(\cdot 10^{+02}\) & 1.76\(\cdot 10^{-02}\) \\ Kyocera KC200 GT-2 & 8.195\(\cdot 10^{+00}\) & 3.950\(\cdot 10^{-10}\) & 1 & 2.522\(\cdot 10^{-01}\) & 1.638\(\cdot 10^{+02}\) & 6.71\(\cdot 10^{-03}\) \\ Selex Galileo SPVS X5 & 5.028\(\cdot 10^{-01}\) & 1.261\(\cdot 10^{-16}\) & 1 & \(-3.091\cdot 10^{-01}\) & 1.185\(\cdot 10^{+03}\) & 3.96\(\cdot 10^{-02}\) \\ \hline \end{tabular}
\end{table} 
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 Azur Space 3G30C solar cells, and Photowatt PWP 201, Kyocera KC200 GT-2, and Selex Galileo SPVS X5 solar panels (where \(\xi\) is the non-dimensional RMSE for the fit). Figure 7: Values for the non-dimensional RMSE, \(x\), obtained from a numerical fit of the 1-D/2-R equivalent circuit model to the studied \(I\)-\(V\) curves (see Table 2), an analytical fit (see Table 3), and an analytical fit with \(a=1\) (see Table 4). \[W_{0}^{+}(x)=-1.2216+3.4724\cdot 10^{-2}\Big{(}1.7091\cdot 10^{8}-(x-114.146)^{4} \Big{)}^{0.25},\ x\in[1.2,10]. \tag{28}\]  
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 [PERSON] et al. [86] (see Appendix A). In Figure 8, the positive branch of [PERSON]'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 \(V=V_{\infty}\), is shown together with Equation (26) to (28).