# Parameter Estimation of the Bishop Photovoltaic Model Using a Genetic Algorithm
## 4 Results
### Estimation Stages
#### 4.2.2 Estimation of the Bishop Model Parameters  
This second estimation stage was carried out for the two cells with different technologies previously described (Cell 1 and Cell 2) to observe variations in the second quadrant parameters. In this case, in addition to the search ranges defined in Table 2, the ranges of parameters \(a\), \(V_{br}\) and \(m\) were also defined. Those additional ranges were obtained by analyzing the impact of each parameter on the I-V curve in the second quadrant. This procedure was carried out using simulations, verifying the values of which allowed us to reconstruct the I-V curve close to the experimental ranges obtained from the real cells. Those new parameter ranges are detailed in Table 8.  
The objective function was the same adopted in the previous stage, i.e., Equation (5); the set of constraints indicated in Equations (6)-(13) also held, which are detailed in Tables 2 and 8. As in the previous stage, the PSO algorithm was used to find the best set of parameters of the GA, similar to the tuning process presented in Section 4.2.1. In this case, the results were \(p=14\), \(iter_{max}=9709\) and \(iter_{max}=1261\). Then, using those GA parameters, the estimation of the Bishop model was repeated 100 times, obtaining the parameters' mean and standard deviations reported in Table 9. Moreover, Table 10 presents the sets of parameters providing the best and worst estimations obtained with the 100 repetitions. In this estimation process, the highest relative deviations were found in the same parameters: \(I_{sat}\), \(V_{br}\), \(m\) and \(a\). For Cell 1, those maximum variations were 13.3%, 23.2%, 33.3% and 22.22%, respectively, while, for Cell 2, those were 24.4%, 17.5%, 24.5% and 32.5%, respectively; the other parameters reported variations under 5%. The mean RMSE for Cell 1 (mc-SI), evaluated as the objective function, was lower than the value of Cell 2; however, the deviations were higher in Cell 2 (pc-SI). Table 11 reports the accuracy metrics evaluated for the complete Bishop model estimation. Comparing the results presented in Tables 7 and 11 confirm the large accuracy increment achieved with this second estimation stage, in comparison with the adoption of classical Bishop parameters found in the literature. In fact, the RMSE, MAPE and MBE values were reduced between one and two orders or magnitude. \begin{table}
\begin{tabular}{c c c} \hline
**Accuracy Measures** & **Cell 1 (mc-SI)** & **Cell 2 (pc-SI)** \\ \hline RMSE & \(0.15\pm 1.3\%\) & \(0.024\pm 8.3\%\) \\ MAPE & \(0.15\pm 2\%\) & \(0.02\pm 10\%\) \\ MBE & \(-8\times 10^{-2}\pm 2.5\%\) & \(9\times 10^{-3}\pm 33.33\%\) \\ \hline \end{tabular}
\end{table} 
Table 7: Accuracy measures for estimation with the SDM model and Bishop parameters (mean \(\pm\) standard deviation). \begin{table}
\begin{tabular}{c c c c} \hline
**Limits** & \(a\) & \(V_{br}\) & \(m\) \\ \hline Min & \(2\times 10^{-3}\) & \(-50\) V & 2 \\ Max & \(15\times 10^{-3}\) & \(-20\) V & 6 \\ \hline \end{tabular}
\end{table} 
Table 8: Ranges for parameter estimation in the second quadrant. Finally, Figures 13a and 14a compare the estimated I-V curves of Cell 1 and Cell 2 with the experimental data, respectively. Those comparison put into evidence the satisfactory performance of the Bishop model, at the second quadrant, with the parameters estimated using the proposed GA. Thus, this approach provides a much more accurate solution than that obtained using the Bishop parameters reported in [18] for those real PV cells. Moreover, Figures 13b and 14b report a representative sample of the convergence path for the estimation of Cells 1 and 2 parameters, respectively. In both figures, we can observe the consistent decrement of the O.F. as the number of iterations increases, which, after 7000 iterations, forces all solution paths to reach a similar value to the O.F., thus providing a similar accuracy in the second-quadrant reproduction.