# Parameter Estimation of the Bishop Photovoltaic Model Using a Genetic Algorithm
## 4 Results
### Estimation Stages
#### 4.2.1 Estimation of the SDM Parameters
Both figures show that the algorithm started with different initial values but arrived at similar objective function (O.F.) values, which resulted in parameters with low deviations. \begin{table}
\begin{tabular}{c c c} \hline
**Accuracy Measures** & **Cell 1 (mc-SI)** & **Cell 2 (pc-SI)** \\ \hline RMSE & \(0.003\pm 6.6\%\) & \(0.008\pm 1.6\%\) \\ MAPE & \(0.014\pm 7.1\%\) & \(0.001\pm 70\%\) \\ MBE & \(-1.5\times 10^{-5}\pm 266\%\) & \(-2.8\times 10^{-6}\pm 1571\%\) \\ \hline \end{tabular}
\end{table} 
Table 4: Accuracy measures for estimation with the SDM model (mean \(\pm\) standard deviation). \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Cell 1 (mc-SI)** & **Cell 2 (pc-SI)** \\ \hline \(I_{ph}(A)\) & \(0.43\pm 0.16\%\) & \(0.32\pm 0.2\%\) \\ \(I_{sat}(A)\) & \(9\times 10^{-8}\pm 23.3\%\) & \(9\times 10^{-8}\pm 23.3\%\) \\ \(n\) & \(1.1\pm 2.7\%\) & \(1.2\pm 2.9\%\) \\ \(R_{s}(\Omega)\) & \(0.12\pm 3.3\%\) & \(0.08\pm 4.7\%\) \\ \(R_{slt}(\Omega)\) & \(15\pm 2\%\) & \(485\pm 14\%\) \\ O.F. & \(0.003\pm 7\%\) & \(0.008\pm 1.6\%\) \\ Time (s) & \(4.81\times 10^{3}\pm 199.4\%\) & \(1.34\times 10^{3}\pm 78\%\) \\ \hline \end{tabular}
\end{table} 
Table 3: Ranges of parameters estimated for the SDM model (mean \(\pm\) standard deviation). \begin{table}
\begin{tabular}{c c c c c} \hline
**Accuracy Measures** & **Cell 1 (mc-SI)** & **Cell 2 (pc-SI)** \\ \hline RMSE & \(0.003\pm 6.6\%\) & \(0.008\pm 1.6\%\) \\ MAPE & \(0.014\pm 7.1\%\) & \(0.001\pm 70\%\) \\ MBE & \(-1.5\times 10^{-5}\pm 266\%\) & \(-2.8\times 10^{-6}\pm 1571\%\) \\ \hline \end{tabular}
\end{table} 
Table 4: Accuracy measures for estimation with the SDM model (mean \(\pm\) standard deviation). In order to reproduce the I-V curve second quadrant, an exhaustive search in the literature was conducted to find Bishop model parameters used for PV cells, modules, or panels. Such a literature review confirmed that second-quadrant studies are commonly performed using parameters similar to those proposed in [18], with small variations; moreover, only few studies have reported the procedure followed to obtain those parameters. Table 6 lists some of the second-quadrant studies found in the literature, the values adopted for the second-quadrant parameters, the technology of the cell under study and the source from where the parameters were taken. Figure 11: Parameter estimation results for Cell 2 (pc-SI) using the SDM mode: (**a**) comparison of I-V curves using the best solution and (**b**) convergence results for a sample of 10 repetitions. Figure 10: Parameter estimation results for Cell 1 (mc-SI) using the SDM model: (**a**) comparison of I–V curves using the best solution and (**b**) convergence results for a sample of 10 repetitions. The values most commonly adopted in the literature for the second-quadrant parameters, observed in Table 6, are: \(a=2\times 10^{-3}\), \(m=3\) and \(V_{br}=-28\) V for Cell 1 and \(V_{br}=-26\) V for Cell 2. Using those values to estimate the behavior of both experimental cells (Cell 1 and Cell 2) led us to the results reported in Figure 12a,b, which shows a wrong reproduction of the second-quadrant behavior of real PV cells under real operation conditions. In fact, the RMSE of Cells 1 and 2 were 0.15 and 0.024, respectively, which are much higher than the RMSE obtained in the first quadrant. Table 7 reports the RMSE, MAPE and MBE metrics for those curve reproductions, where all the metrics grow consistently in comparison with the values achieved in Table 4 for the parameter estimation of the SDM model. Therefore, this test puts into evidence the need to also estimate the second-quadrant parameters instead of adopting the values reported in [18] for any condition. The following subsection addresses such a second estimation stage. Finally, Figure 12a,b shows that, although the Bishop model is used to represent the behavior of PV cells, its fitting is not consistent in the second quadrant if the parameters are not correct.