## 5 Results, Discussions and Comparison
### Case 1: Single and Double Diode Model (RTC France Company)  
The proposed algorithm is applied first to extract the electrical intrinsic parameters values for single and double diode models of a 57-mm-diameter commercial (RTC France) silicon solar cell under 1000 W/m\({}^{2}\) at 33 \({}^{\circ}\)C. The extracted parameters are compared with those found by: Biogeography-Based Optimization algorithm with Mutation strategies (BBO-M) [68], Levenberg-Marquardt algorithm combined with Simulated Annealing (LMSA) [47], Artificial Bee Swarm Optimization algorithm [48], Artificial Bee Colony optimization (ABC) [49], hybrid Nelder-Mead and Modified Particle Swarm Optimization (NM-MPSO) [50], Repaired Adaptive Differential Evolution (RADE) [59], Chaotic  
Figure 4: A pseudocode for the considered firefly algorithm implementation. Furthermore, to understand the quality of the curve fit between Firefly algorithm values and experimental data, the results are compared to other algorithms. The compared statistical analysis for each model is presented in Tables 7 and 8. The compared statistical criteria indicates that the Firefly algorithm ranks the overall lowest values for relative error (RE), median absolute error (MAE), residual sum of squares (SSE), and root mean square error (RMSE). The statistical errors are used to show the performance with the definitions as follows:  
\[RE=\frac{I_{\text{measured}}-I_{\text{estimated}}}{I_{\text{measured}}} \tag{17}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multirow{2}{*}{\(\mathbf{V_{Exp}\) (V)} & \multirow{2}{*}{\(\mathbf{I_{Exp}\) (A)} & \(\mathbf{I_{Calculated}\) (A)} & \multicolumn{3}{c}{**Individual Absolute Error (IAE)**} \\ \cline{5-8}  & & & & **FA** & **RADE** & **NM-MPSO** \\ \hline
1 & \(-0.2057\) & 0.7640 & 0.76404800 & 4.7990 \(\times\) 10\({}^{-5}\) & 9.2680 \(\times\) 10\({}^{-5}\) & 2.3000 \(\times\) 10\({}^{-5}\  
\[MAE=\sum_{i=1}^{m}\frac{|I_{\text{estimated}}-I_{\text{measured}}|}{m} \tag{18}\]  
\[SSE=\sum_{i=1}^{m}(I_{\text{measured}}-I_{\text{estimated}})^{2} \tag{19}\]  
\[RMSE=\sqrt{\frac{1}{m}\sum_{i=1}^{m}(I_{\text{measured}}-I_{\text{estimated}}) ^{2}} \tag{20}\]  
From Tables 7 and 8, we observe that the five and seven electrical parameters identified by the Firefly algorithm are close and more accurate than those found by all other compared algorithms. The performance of the proposed algorithm provides the lowest values for the statistical criteria, IAE, RMSE, SSE and MAE when compared to the other methods. Therefore, the Firefly algorithm is ranked first in achieving the lowest IAE, RMSE, SSE and MAE, while the Repaired Adaptive Differential Evolution algorithm and hybrid Nelder-Mead and Modified Particle Swarm Optimization (NM-MPSO) are ranked second and third, respectively. Therefore, the optimal parameters identified by the proposed Firefly Algorithm are very accurate because they are close to the real parameters of the system. The individual absolute error (IAE) and the relative error RE for each measurement using optimal values founded by the Firefly algorithm are  
The current-voltage and power-voltage characteristics resulting from extracted parameters by the Firefly algorithm along with experimental data are compared to estimated data to investigate the quality of the identified parameters. This is illustrated in Figures 7 and 8. The two figures show the reconstructed single diode model is in good agreement with experimental data and are very close to each other. Figure 9 shows the compared extracted current-voltage characteristics of the mono-crystalline for single and double diode, RTC France Company. The calculated current by extracted parameters  
Figure 8: Experimental power-voltage data compared with estimated data of the mono-crystalline silicon solar cell single double diode, RTC France Company. Figure 6: Relative Error (RE) plots for single and double diode for Mono-crystalline silicon solar cell, RTC France Company.