# Analytical Modeling of the Maximum Power Point with Series Resistance
## 5 Numerical Results  
In this section, the accuracy of Equation (19) is tested. For this, the focus is on \(V_{\text{mpp}}\) and \(P_{\text{mpp}}\), given by \(P_{\text{mpp}}=V_{\text{mpp}}i(V_{\text{mpp}})\), with \(i\) being given by Equation (7). The label \"mod\" is used to denote the values of \(V_{\text{mpp}}\) and \(P_{\text{mpp}}\) obtained from Equation (19). The label \"ref\" is used to denote the reference values to which \(V_{\text{mpp}}^{\text{mod}}\) and \(P_{\text{mpp}}^{\text{mod}}\) are compared. The \"ref\" quantities are obtained by numerically solving Equation (8). The accuracy of Equations (17) and (18) is also tested, and the label \"app\" is used to denote these. For all numerical calculations, the AM 1.5G spectrum was assumed. The numerical single-diode model used to test the accuracy of Equation (19) was fed with cell parameters corresponding to six different technologies found in the latest installment of the solar cell efficiency tables [19]. The parameters corresponding to these cells are summarized in Table 1. As a seventh case, the derived model was also tested against a numerical single-diode model using \(E_{\text{g}}=1.125\) eV and \(\text{ERE}=10^{-4}\). These values are typical for silicon cells. This case is therefore referred to as a numerically modeled silicon cell. All cells were assumed to be at a temperature of 300 K. Figure 2 displays (a) \(V_{\text{mpp}}\), (b) \(i_{\text{mpp}}\), and (c) \(P_{\text{mpp}}\) as a function of the series resistance. Additionally, the corresponding current-voltage characteristic (Figure 2d) is shown for several values of the series resistance. All curves correspond to the numerically modeled silicon cell described above. In Figure 2d, dotted lines show how the maximum power point changes with increasing series resistance. The red dotted line was obtained by solving Equation (8) numerically, for multiple values of the series resistance, and evaluating Equation (7) with the obtained \(V_{\text{mpp}}\) values. The purple dotted line was obtained from Equation (15). The values for \(r_{\text{L}}\) and \(r_{\text{max}}\) (Figure 2a-c) and their correspondent values of \(V_{\text{mpp}}\) (Figure 2d) are displayed with black vertical dashed lines. From Figure 2a-c, it can be seen that the values calculated with the new analytical model were in good agreement with the numerical reference model for \(r\leq r_{\text{L}}\). For \(r\geq r_{\text{L}}\), \(V_{\text{mpp}}^{\text{mod}}\) appears to be underestimated (Figure 2a) and \(i_{\text{mpp}}^{\text{mod}}\) overestimated (Figure 2b). As a result, \(P_{\text{mpp}}^{\text{mod}}\) overlaps well with \(P_{\text{mpp}}^{\text{ref}}\) (Figure 2c). Finally, for \(r\geq r_{\text{max}}\), \(V_{\text{mpp}}^{\text{mod}}\) appears to increase with increasing series resistance. The accuracy of the expression for \(V_{\text{mpp}}^{\text{mod}}\), Equation (19), decreases with increasing resistance. This can be seen in Figure 3, where \(V_{\text{mpp}}\) is plotted as a function of the series resistance for the six technologies presented in Table 1. The dashed lines correspond to the reference values, \(V_{\text{mpp}}^{\text{ref}}\) and the continuous lines to \(V_{\text{mpp}}^{\text{mod}}\). The points corresponding to a series resistance equal to \(r_{\text{L}}\) are marked with crosses. Focusing on the graphs representing the CIGS cell, the mismatch between \(V_{\text{mpp}}^{\text{ref}}\) and \(V_{\text{mpp}}^{\text{mod}}\) becomes noticeable for \(r\geq r_{\text{L}}\). For additional comparison, the \(V_{\text{mpp}}\) calculated from Equation (3) (i.e., without series resistance) for the GaAs cell is displayed in Figure 3. This is represented by the blue straight line with zero slope. At \(r=0\)\(\Omega\cdot\text{cm}^{2}\), \(V_{\text{mpp}}^{\text{ref}}\), \(V_{\text{mpp}}^{\text{mod}}\), and Equation (3) overlap, but as soon as \(r\) starts increasing, Equation (19) predicts the value of \(V_{\text{mpp}}^{\text{ref}}\) with higher accuracy.