## 4 Simulation Results
### IMGO for the PV Parameter Estimation Problem  
The Double Diode (DD) model is an extension of the single diode which shows better accuracy. This model has an additional diode, as shown in Figure 18, to account for recombination processes in the space-charge region. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Energy Losses/Day (kWday)** & **Improvement** \\ \hline Proposed IMGO & 10,495.75376 & \\ MGO & 10,582.90161 & 0.82\% \\ ASBA & 10,554.05 & 0.55\% \\ DA & 10,653.42 & 1.48\% \\ NGO & 10,660.27 & 1.54\% \\ COA & 10,715.77 & 2.05\% \\ OMA & 10,798.92 & 2.81\% \\ OOA & 10,964.46 & 4.27\% \\ SBA & 11,369.16 & 7.68\% \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Comparisons of the proposed IMGO and the reported results for energy loss/day (kWday) minimization. Figure 17: Box and whisker plot of MGO, IMGO, DE, PSO, GA, and MVO for energy loss minimization. Figure 18: Schematic diagram of DD-PV model. Where: \(I_{Ph}\) is the irradiation light current. \(R_{sh}\) and \(R_{ss}\) are the parallel and series resistances. \(\eta_{1}\) and \(\eta_{2}\) are the ideality factor regarding the two diodes. Id1 and Id2 are the diodes reverse saturation currents. The primary aim of this application is to accurately determine the optimal I-V characteristics of the PV model by identifying the most precise representation that aligns closely with the experimentally recorded data. Achieving this requires extracting a set of parameters that not only reflect the physical and electrical properties of the solar cell or module but also ensure a high degree of fidelity in replicating its behavior under various operating conditions. To quantify the accuracy of the extracted parameters and evaluate how well the modeled I-V characteristics correspond to the measured experimental data, the root mean square error (RMSE) is utilized as the performance metric. RMSE serves as a robust indicator of the deviation between the observed and modeled values, effectively capturing the extent of discrepancies across all data points. The formulation of RMSE, as used in this study, is given by the following equation:  
\[RMSE=\sqrt{\frac{1}{MP}\left(\sum_{L=1}^{MP}\left(I_{calc}^{L}\left(V_{t,obs} ^{L}x\right)-I_{obs}^{L}\right)^{2}\right)} \tag{31}\]  
Here, _MP_ represents the total number of measured data points, while \(I_{obs}\) and \(V_{obs}\) denote the observed current and voltage values, respectively. The variable \(x\) signifies the candidate solution during the optimization process and encapsulates the potential PV characteristics being evaluated. The equivalent circuit design of the DD-PV model is illustrated in Figure 18. Using Kirchhoff's Current Law (KCL), the I-V characteristic curve of the DD model can be mathematically expressed as follows:  
\[I=I_{ph}+\frac{V_{t}}{R_{sh}}-\frac{IR_{ss}}{R_{sh}}-I_{S1}\left[\exp^{\left( \frac{IR_{ss}+V_{t}}{V_{th}}\right)}-1\right]-I_{S2}\left[\exp^{\left(\frac{IR _{ss}+V_{t}}{V_{th}}\right)}-1\right] \tag{32}\]  
In this context, \(V_{t}\) represents the terminal voltage of the PV cell. The thermal voltage (_Vth_) of the PV cell is an important parameter and can be computed using Equation (33). \[V_{th}=K_{BZ}\times\frac{T}{q_{c}} \tag{33}\]  
This equation incorporates the following fundamental constants and operating conditions: \(T\) is the cell temperature measured in Kelvin, and \(K_{BZ}\) is Boltzmann's constant (1.3806503 \(\times\) 10\({}^{-23}\) J/K). The Shockley diode equation also employs \(qc\), the electron's charge magnitude (1.60217646 \(\times\) 10\({}^{-19}\) C), to account for the charge carriers' behavior within the cell. For the application, the commercial silicon solar R.T.C. France cell is considered under a sun radiance of 1000 W/m\({}^{2}\) and runs at 33 \({}^{\circ}\)C. It has an open circuit voltage of 0.5727 V and a short-circuit current of 0.7605 A. Moreover, the maximum point voltage and current of R.T.C. France are 0.4590 V and 0.6755 A, respectively.