## 5 Results, Discussions and Comparison
### Case 3: Commercial Silicon Photovoltaic Module Photowatt-PWP 201  
The prototype of the Photowatt-PWP 201 photovoltaic module has six solar panels, two are connected in series and three photovoltaic panels are connected in parallel. The measured voltage and current are taken under 25 \({}^{\circ}\)C and 1000 W/m\({}^{2}\). In this case, 26-pair current-voltage measured values are the same as [29], which are derived from 36 polycrystalline silicon cells which are connected in series. The extracted optimal parameters values for the photovoltaic module by Firefly algorithm have been reported in Table 12. Moreover, the optimal parameters are compared with several other techniques: Newton-Raphson [29] Pattern Search (PS) [55], Genetic Algorithm (GA) [56] and Simulated Annealing algorithm (SA) [58] based on the same experimental data. The purpose of comparison is to validate the accuracy of the Firefly algorithm in the parameter extraction process with a short time of convergence. The quality of the results in the extracted parameters are used to calculate the theoretical current values and compared to experimental measurements as show in Table 13.  
The optimal value of the following statistical errors: individual absolute error (IAE), relative error (RE), root means square error (RMSE) and residual sum of squares (SSE), for each measurement using the Firefly algorithm and other parameter extraction techniques are given in Table 14. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Item** & **FA** & **Newton-Raphson** & **PS** & **GA** & **SA** & **NM-MPSO** \\ \hline \(I_{ph}\) (A) & 1.0306 & 1.0318 & 1.0313 & 1.0441 & 1.0331 & 1.0305 \\ \(I_{0}\) (\(\mu\)A) & 3.4802 & 3.2875 & 3.1756 & 3.4360 & 3.6642 & 3.6817 \\ \(a\) & 48.6551 & 48.4500 & 48.2889 & 48.5862 & 48.8211 & 48.8598 \\ \(R_{s}\) (\(\Omega\)) & 1.2014 & 1.2057 & 1.2053 & 1.1968 & 1.1989 & 1.1944 \\ \(R_{sh}\) (\(\Omega\)) & 971.1396 & 555.5556 & 714.2857 & 555.5556 & 833.3333 & 983.9970 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Optimal parameter values identified by FA for Photowatt-PWP 201 polycrystalline photovoltaic module single diode compared with other methods. Figure 11: Comparison between, (**a**) \(I\)-\(V\) and (**b**) \(P\)-\(V\) characteristics resulting from the experimental data, Q-N, Soma and FA. Table 14 proves that the Firefly algorithm has the lowest IAE, RMSE, SSE and MAE compared to other parameter extraction techniques such as, Newton-Raphson, Pattern Search (PS), Genetic Algorithm (GA) and Simulated Annealing algorithm (SA), since the Firefly algorithm found the minimum value of statistical analysis in parameter extraction for the photovoltaic module. The comparison between Newton-Raphson, Pattern Search (PS), Genetic Algorithm (GA) and Simulated Annealing algorithm (SA) and the proposed algorithm, with the optimal value of IAE for each measurement, is illustrated in Figure 12. This Figure shows that the FA algorithm has better performance than the other parameter extraction algorithms. The total IAE values for each measurement is also calculated and listed in Table 14. The total IAE value shown in Table 14 highlights that the FA has the lowest total IAE compared to other algorithms for the photovoltaic module. Table 14 and Figure 12 indicate that FA outperforms the compared algorithms for this parameter extraction problem.