# Parameter Extraction of Photovoltaic Cells and Panels Using a PID-Based Metaheuristic Algorithm
## 3 Results and Discussion
### mSi Cell
The alignment of the estimated and observed I-V and P-V characteristics generated by the PSA is shown in Figures 12 and 13, allowing a comparison of the measured and computed currents for both the SDMs and DDMs. Figure 14 shows the absolute errors, which are the differences between the observed and estimated currents for both models as determined by the algorithm. In the case of the DDM, the absolute errors are very low in comparison with the ones calculated with the SDM. The convergence curves of the PSA method for both models are shown in Figure 15, with a maximum iteration of 1500 and a population size of 100. The graphic depicts the iteration at which the algorithm starts producing the lowest RMSE result. Figure 12: mSi SDM and DDM I-V characteristics. Starting with iteration 220 at 66 population for the SDM, the PSA showed evidence of escaping the local minimum and nearing the ideal RMSE value. For the DDM, the algorithm began this process at iteration 100, although it took a significant jump to obtain the optimum RMSE at iteration 780 and 70 population. To provide additional insight into the algorithm's reliability, shaded regions corresponding to \(\pm\) standard deviation across iterations are included. These bands demonstrate the level of variability in RMSE values at each iteration, offering a visual measure of algorithm stability. Narrower bands indicate consistent behavior across optimization steps, affirming PSA's robustness in both SDMs and DDMs. An initial investigation was performed on both models with a fixed iteration number set at 100 and the size of the population ranging between 20 and 100 to determine the most effective combination of iteration size and population size for reaching the best RMSE. The lowest RMSE value was attained when the population size and the number of iterations were both adjusted to 100; therefore, this method clearly did not work. The study showed  
Figure 14: mSi SDM and DDM calculated absolute error. Figure 13: mSi SDM and DDM P–V characteristics. that the RMSE for the SDM was \(1.61\times 10^{-3}\) and the calculation time was \(0.4073\) s, whereas the RMSE for the DDM was \(1.3605\times 10^{-3}\) and the computation time was \(0.6340\) s.  
In the next experiment, the population size was held constant, but the number of iterations was increased by 200. The lowest RMSE value achievable by the method was thus determined. It took around \(2.2343\) s to obtain the optimum RMSE solution for the SDM with a population size of 50 and 1100 iterations. The DDM took around \(7.082\) s to calculate after 1500 iterations with a population size of 50. Since DDM requires the extraction of seven parameters, the search for the best RMSE takes more time, resulting in a higher number of iterations. Table 12 presents the results of the SDM study, while Table 13 presents the findings of the DDM analysis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of** & **Population** & **Number of** & **RMSE** & **Run Time(s)** \\ \cline{2-5}
**Runs** & **Size** & **Iterations** & **RMSE** & **Run Time(s)** \\ \hline \hline \multirow{8}{*}{5} & 20 & 100 & \(3.3469\times 10^{-3}\) & 0.0888 \\ \cline{2-5}  & 30 & 100 & \(2.4630\times 10^{-3}\) & 0.1277 \\ \cline{2-5}  & 50 & 100 & \(8.4297\times 10^{-4}\) & 0.2075 \\ \cline{2-5}  & 60 & 100 & \(3.2550\times 10^{-3}\) & 0.2474 \\ \cline{2-5}  & 80 & 100 & \(1.9829\times 10^{-3}\) & 0.3272 \\ \cline{2-5}  & 100 & 100 & \(1.6111\times 10^{-3}\) & 0.4074 \\ \cline{2-5}  & 50 & 300 & \(9.8442\times 10^{-4}\) & 0.6108 \\ \cline{2-5}  & 50 & 500 & \(7.4310\times 10^{-4}\) & 1.0200 \\ \cline{2-5}  & 50 & 700 & \(6.7488\times 10^{-4}\) & 1.4103 \\ \cline{2-5}  & 50 & 900 & \(5.9296\times 10^{-4}\) & 1.8384 \\ \cline{2-5}  & 50 & 1100 & \(5.6309\times 10^{-4}\) & 2.2344 \\ \cline{2-5}  & 50 & 1300 & \(5.6309\times 10^{-4}\) & 2.6551 \\ \cline{2-5}  & 50 & 1500 & \(5.6309\times 10^{-4}\) & 3.1299 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: mSi iteration vs. population size SDM case. Figure 15: mSi SDM and DDM convergence curve.