## 3 Results
### On the Best Value for the Ideality Factor  
In a paper published in the 2 nd International Conference on Renewable Energy Research and Applications (ICRERA 2013), our research group demonstrated a hyperbolic relationship between the non-dimensional RMSE, \(\xi\), 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:  
\[\xi=\frac{\text{RMSE}}{I_{sc}}=\frac{1}{I_{sc}}\sqrt{\frac{1}{N}\sum_{i=1}^{N }\left(I-I_{ref}\right)^{2}}, \tag{24}\]  
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 \(\xi\) and \(a\) is plotted in Figure 4. It can be seen that the curve has a hyperbolic shape, with a minimum value of the non-dimensional RMSE at \(a=1.18\). This hyperbolic shape was also confirmed in a recent work by [PERSON] and [PERSON] [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. Figure 3: The Lambert W-function, with its two different branches. 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(r) 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. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Solar Cell/Panel** & **Technology** & \(n\) & \(T\) [\({}^{\circ}\)C] & \(I_{Sc}\) [A] & \(I_{mp}\) [A] & \(V_{mp}\) [V] & \(V_{oc}\) [V] \\ \hline RTC France 1 & Si & 1 & 33 & 0.7605 & 0.6894 & 0.4507 & 0.5727 \\ TNJ Spectrolab 2 & GaInP2/GaAs/Ge & 3 & 28 & 0.5259 & 0.4969 & 2.273 & 2.592 \\ ZTJ Emcore 2 & InGaP/InGaAs/Ge & 3 & 28 & 0.4634 & 0.4424 & 2.398 & 2.726 \\ Azur Space 3G30C 3 & GaInP/GaAs/Ge & 3 & 28 & 0.5270 & 0.5023 & 2.468 & 2.711 \\ Photowatt PWP 201 1 & Si & 36 & 45 & 1.032 & 0.9255 & 12.493 & 16.778 \\ Kyocera KC200 GT-2 2 & Si polycrystalline & 54 & 25 & 8.182 & 7.605 & 26.90 & 32.92 \\ Selex Galileo SPVS X5 4 & GaInP/GaAs/Ge & 15 & 20 & 0.5029 & 0.4783 & 12.406 & 13.603 \\ \hline \end{tabular}
\end{table} 
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 \(V_{oc}\)).