# Parameter Estimation of the Bishop Photovoltaic Model Using a Genetic Algorithm
## 4 Results
### Estimation Stages  
#### 4.2.1 Estimation of the SDM Parameters  
In the first stage of the parameter estimation, the five parameters defining the behavior of the curve in the first quadrant (\(I_{ph}\), \(R_{s}\), \(I_{sh}\), \(I_{o}\) and \(\eta\)) were calculated. Thus, the ranges in Table 2 were selected as search ranges; those values were taken from the literature [41; 42; 43; 44; 45; 46; 47; 48; 49]. The photovoltaic current (\(I_{ph}\)) was adjusted to a range of \(\pm 5\%\) of the short-circuit current of the test (\(I_{sc}\)), this based on the fact that \(I_{sc}\) is caused by the generation and collection of light-generated carriers; therefore, \(I_{sc}\) and \(I_{ph}\) are very close values. The genetic algorithm was adjusted to obtain the best possible solution in terms of the objective function. This was achieved by implementing another optimization algorithm to obtain the GA parameters that best fit the minimization of the problem. In this case, we used the PSO algorithm proposed in [50; 51] to tune the following values: population size \(p\) (individuals or particles), with a search range of 2-100; number of iterations \(iter_{max}\) (maximum iterations allowed in the GA), with a search range of 1-10,000; and number of non-improvement iterations \(iter_{max}\) (this determines the point at which the algorithm stops performing convergence or minimizing the objective function), with a search range of 1-10,000. The results obtained from this tuning process were \(p=11\), \(iter_{max}=8761\) and \(iter_{max}=4061\). As part of the estimation process, the technique was evaluated 100 times per cell to observe metrics such as mean value and standard deviation for each estimated SDM parameter, which are reported in Table 3. The accuracy of the data was evaluated using the RMSE, since that metric was adopted as the objective function, obtaining small standard deviations for both cells, as reported in Table 4. The MAPE metric also showed a small error but with a higher standard deviation, especially for Cell 2, while the MBE metric exhibited very small values with high standard deviations. In any case, the three metrics had small values, which ensured a high accuracy on the I-V curves reproduction. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Limits** & \(n\) & \(R_{s}\) & \(R_{sh}\) & \(I_{o}\) & \(I_{ph}\) \\ \hline Min & 0.05 & \(1\times 10^{-5}\) & 5 & \(1\times 10^{-10}\) & 95\% \(\times\)\(I_{sc}\) \\ Max & 2 & 2 & 100 & \(1\times 10^{-7}\) & 105\% \(\times\)\(I_{sc}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Ranges for the estimation of parameters in the first quadrant. Figure 9: Experimental I–V curves of mc-SI and pc-SI cells. Analyzing the Cell 1 (mc-SI) parameters in Table 3, the highest relative deviations was found in \(I_{sat}\), with 23.3%. Concerning Cell 2 (pc-SI), the highest relative deviations were found in parameters \(I_{sat}\) and \(R_{slt}\), with 23.3% and 14%, respectively. Instead, the other parameters, for both cells, exhibited variations under 5%. The mean RMSE of Cell 1, evaluated as the objective function, was lower than the value obtained for Cell 2; however, the relative deviations for Cell 2 were lower than in the case of Cell 1. The best and worst solutions provided by the 100 estimations were also analyzed and compared based on the value of the objective function provided by the GA. Table 5 presents the sets of parameters that provide the best and worst performance for the SDM in each cell. Moreover, Figures 10a and 11a show the comparison between the I-V curves estimated with the best parameters and the experimental data for Cells 1 and 2, respectively. Those figures show a satisfactory reproduction of the I-V curve's first quadrant using the SDM parameters obtained with the proposed GA, where the most significant differences are presented in \(I_{ph}\) and \(R_{slt}\). In addition, Figures 10b and 11b show the convergence path of a sample of 10 repetitions of the estimation algorithm.