## 1 Introduction
### The Problem of Determining the I-V Curve Maximum Power Point (MPP)  
We can conclude from the above results that analytical methods need the characteristic points of the _I-V_ curve in order to extract the parameters of a photovoltaic system (cell or panel) equivalent circuit. The advantage of using these points lies in the small amount of information needed, which is normally provided by the manufacturer. However, if experimental data are the source to calculate the equivalent circuit of a solar cell/panel using an analytical method, the main problem turns to be the accurate determination of the _I-V_ curve's maximum power point (MPP). Unlike the other two characteristic points (short-circuit and open-circuit), whose determination is relatively simple, the MPP correct estimation requires multiple measurements at different voltage levels around this particular point. This fact weakens one of the main advantages of the analytical methods, as quite large resources are required to determine the MPP accurately. This is, in fact, one of the most relevant problems in the solar energy industry: how to make the photovoltaic devices work at the maximum extractable power despite the changing conditions (temperature, solar radiation, loads... ). Different maximum power point tracking (MPPT) techniques have been developed in the past decades to track continuously the maximum power point and force the photovoltaic device to work as close as possible. Among the MPPT methods described in the available literature (see a quite complete review of the MPPT methods in the recent works by [PERSON] and [PERSON] [11], [PERSON] et al. [12], [PERSON] et al. [13], and [PERSON] et al. [14]), it is possible to distinguish between offline and online methods [15]. In offline methods, the maximum power point is calculated using information regarding the panel _I-V_ curve. One of the main advantages of these methods is the possibility to change the operating voltage \begin{table}
\begin{tabular}{c c c} \hline
**Characteristic Data** & **R.T.C. France** & **Photowatt PWP201** \\ \hline \(I_{\text{ac}}\) (A) & 0.7603 & 1.0300 \\ \(V_{\text{ac}}\) (V) & 0.5728 & 16.778 \\ \(V_{\text{np}}\) (V) & 0.4507 & 12.649 \\ \(I_{\text{np}}\) (A) & 0.6894 & 0.9120 \\ \(R_{\text{n}0}\) (\(\Omega\)) & 246.80 \({}^{*}\) & 689.13 \({}^{*}\) \\ \(K_{\text{n}0}\) (\(\Omega\)) & 0.0907 \({}^{*}\) & 2.5193 \({}^{*}\) \\ \(T\) (K) & 306.15 & 318.15 \\ \(N\) & 1 & 36 \\ \hline \end{tabular} \({}^{*}\) Estimated from data in [4].  
\end{table} 
Table 1: Characteristic data from R.T.C. France solar cell and Photowatt PWP201 solar panel. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Model** & \(R_{s}\) **(\(\Omega\))** & \(R_{\text{sh}}\) **(\(\Omega\))** & \(I_{\text{pp}}\) **(A)** & \(I_{\text{OI}}\) **(A)** & \(a_{1}\) & \(I_{\text{02}}\) **(A)** & \(a_{2}\) & \(\xi\) \\ \hline
1D & - & - & 0.7603 & \(1.12\times 10^{-5}\) & 1.9509 & - & - & \(2.25\times 10^{-2}\) \\
1D/1R & 0.0233 & - & 0.7603 & \(2.07\times 10^{-6}\) & 1.6944 & - & - & \(6.84\times 10^{-3}\) \\
1D/2R-1 & 0.0481 & 28.931 & 0.7616 & \(4.14\times 10^{-8}\) & 1.3 \({}^{*}\) & - & - & \(5.95\times 10^{-3}\) \\
1D/2R-2 & 0.0261 & 246.77 & 0.7604 & \(1.44\times 10^{-6}\) & 1.6478 & - & - & \(5.48\times 10^{-3}\) \\
2D/2R & 0.0450 & 246.76 & 0.7604 & \(1.54\times 10^{-9}\) & 1.1087 & \(5.15\times 10^{-6}\) & 2.0 \({}^{*}\) & \(2.50\times 10^{-3}\) \\ \hline \end{tabular} \({}^{*}\) Selected as reasonable values for calculations.  
\end{table} 
Table 2: Equivalent circuit parameters (see Figure 1, and Equations (1)–(4)) and normalized root mean square error (RMSE), \(\xi\), in relation to the R.T.C. France solar cell.