## 5 Results, Discussions and Comparison
### Case 2: Organic Flexible Hydrogenated Amorphous Silicon a-Si:H Solar Cell  
The Firefly algorithm, based on a parameter estimation method is used to extract the eight optimal parameters of flexible dual junction amorphous silicon solar cell under standard sunlight, based on data obtained in light intensity of 1000 W/m\({}^{2}\) and at a temperature of 300 K. The experimental data are used from [48]; only the open circuit voltage \(V_{oc}\) and short circuit current \(I_{sc}\) are obtained. Moreover, the optimal parameters are compared with several other techniques based on the same experimental data. The extracted optimal parameters by Firefly algorithm have been reported in Table 9, compared with the Quasi-Newton method and Self-Organizing Migrating Algorithm. Since it is difficult to extract the flexible amorphous silicon solar cell circuit model parameters and the research is still comparatively rare, the Quasi-Newton (Q-N) method and Self-Organizing Migrating Algorithm (SOMA) [48] have been chosen for comparison because in [29; 48] they were demonstrated to provide good results for parameter extractions. To verify and validate the performance of the quality of the results, statistical analyses were carried out to measure the accuracy of the estimated parameters. The estimated current values are compared to experimental current by means of the following statistical errors: the individual absolute error (IAE), Standard deviation (SD), residual sum of squares (SSE), the root mean square error (RMSE) and the mean bias error (MBE) of the solar cell for each measurement, respectively. The statistical errors are used to compare term by term, the difference between estimated and experimental electric current. Generally, the lower these parameters, the more the efficiency of the model. Table 10 presents the current calculated for the Firefly algorithm and the individual absolute error, Table 11 summarizes the statistical errors for each measurement using the optimal values of \(x\) found by the Quasi-Newton method and Self-Organizing Migrating algorithm [48] compared with Firefly algorithm. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithm** & \(I_{ph}\) **(uA)** & \(d\) **(m)** & \(\mu\)**\(\tau_{off}\) (\(\frac{\mu\text{m}^{2}}{\text{m}^{2}}\))** & \(V_{bi}\) **(V)** & \(R_{s}\) **(fN)** & \(I_{0}\) **(A)** & \(a\)** & \(R_{sh}\) **(fN)** \\ \hline FA & 0.3167 & \(5.8065\times 10^{-8}\) & \(3.3306\times 10^{-5}\) & 0.9895 & 0.4242 & \(3.0691\times 10^{-14}\) & 2 & 13.4978 \\ Q-N & 0.3043 & \(5.8065\times 10^{-8}\) & \(4.8812\times 10^{-5}\) & 0.9759 & 0.4242 & \(3.0691\times 10^{-14}\) & 1.9998 & 11.9138 \\ SOMA & 0.3181 & \(4.9743\times 10^{-8}\) & \(3.3277\times 10^{-5}\) & 0.9963 & 0.4706 & \(3.0783\times 10^{-14}\) & 1.9931 & 13.9288 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison among different parameter extraction of flexible silicon a-Si:H solar cell. Figure 9: Compared experimental current-voltage and power-voltage of the mono-crystalline single diode silicon solar cell, RTC France Company. Figure 10 presents the compared individual absolute error of each measurement used for current and power of optimal value \(x\) found by Firefly algorithm compared with the Quasi-Newton method and Self-Organizing Migrating Algorithm. From Figure 10, Tables 10 and 11 we know that the Firefly algorithm and Self-Organizing Migration Algorithm have the lowest SD, RMSE, SSE and MBE values among these three compared methods. Furthermore, the Firefly algorithm has better performance than the Quasi-Newton method and Self-Organizing Migration presented in [48]. In order to illustrate the quality of the extracted optimal values \(x_{3}\) found by the Firefly algorithm, the extracted values of \(I_{ph}\), \(d_{i}\), \(\mu\tau\), \(V_{bi}\), \(R_{s}\), \(I_{0}\), \(a\) and \(R_{sh}\) are put into Equation (3), then the current-voltage and power-voltage characteristics of this model is reconstructed with 16 pairs of current-voltage.