# A Multi-Objective Hybrid BESSA Optimization Scheme for Parameter Extraction from PV Modules
## 6 Parameter-Setting of the Optimization Technique
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Function**} & \multicolumn{2}{c}{**Range**} & \multicolumn{1}{c}{\(f_{min}\)} \\ \hline \(f_{14}(p)=\left(\frac{1}{500}+\sum\limits_{j=1}^{25}\frac{1}{i+\frac{2}{i+2}(p_{i }-a_{j})}\right)^{-1}\) & \(\left[-65,65\right]^{2}\) & \(1\) \\ \(f_{15}(p)=\sum\limits_{j=1}^{i}\left[a_{i}-\frac{p_{i}(b_{j}^{2}+b_{j}p_{j}+a_{ j})}{b_{j}^{2}+b_{j}p_{j}+a_{j}}\right]^{2}\) & \(\left[-5,5\right]^{4}\) & \(0.00030\) \\ \(f_{16}=4p_{1}^{2}-2.1p_{1}^{4}=\frac{1}{3}p+p_{1}y_{2}-4p_{2}^{2}+4p_{2}^{4}\) & \(\left[-5,5\right]^{2}\) & \(-1.0316\) \\ \(f_{17}(p)=\left(p_{2}-\frac{51}{4p}p_{1}+\frac{5}{4p}p_{1}-6\right)^{2}+10 \left(1-\frac{1}{8\pi}\right)cos(p_{1})+10\) & \(\left[-5,5\right]^{2}\) & \(0.398\) \\ \(f_{18}(p)=\left[1+(p_{1}+p_{2}+1)^{2}\left(19-14p_{1}+3p_{1}^{2}-14p_{2}+6p_{1 }p_{2}+3p_{2}^{2}\right)\right]\) & \(\times\) & \(\left[-2,2\right]^{2}\) & \(3\) \\ \(\left[30+(2p_{1}-3p_{2})^{2}\times\left(18-32p_{1}+12p_{1}^{2}+48p_{2}-36p_{1 }p_{2}+27p_{2}^{2}\right)\right]\) & \(\times\) & \(\left[1,3\right]^{3}\) & \(-3.86\) \\ \(f_{20}(p)=-\sum\limits_{j=1}^{4}c_{i}exp(-\sum\limits_{j=1}^{6}A_{ij}\left(p_{j }-m_{ij}\right)^{2})\) & \(\left[0,1\right]^{6}\) & \(-3.32\) \\ \(f_{21}(p)=-\sum\limits_{j=1}^{5}\left[(P-A_{i})(P-A_{i})^{T}+c_{i}\right]^{-1}\) & \(\left[0,10\right]^{4}\) & \(-10.1532\) \\ \(f_{22}(p)=-\sum\limits_{i=1}^{7}\left[(P-A_{i})(P-A_{i})^{T}+c_{i}\right]^{-1}\) & \(\left[0,10\right]^{4}\) & \(-10.4028\) \\ \(f_{23}(p)=-\sum\limits_{i=1}^{10}\left[(P-A_{i})(P-A_{i})^{T}+c_{i}\right]^{-1}\) & \(\left[0,10\right]^{4}\) & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fixed dimension multimodal benchmark functions. Multi-Objective Parameter Extraction Scheme and Problem Formulation Based on the Hybrid BESSA Technique  
A multi-objective parameter extraction scheme based on the Hybrid BESSA approach was devised for the simultaneous extraction of the parameters of a photovoltaic module from a combined objective function. The optimal combinations of three operating parameters were then used for the mathematical modeling of the photovoltaic module. The accuracy of the PV modeling was judged based on the ISE of power (\(\epsilon_{\text{P}}\)), ISE of voltage (\(\epsilon_{\text{V}}\)), and ISE of current(\(\epsilon_{\text{I}}\)) values obtained from simulation via the BESSA and BES techniques. For any value of irradiance and temperature, the corresponding open  
Figure 5: Convergence profile for function \(p_{8}\). Figure 6: Convergence profile for function \(p_{13}\). Figure 7: Convergence profile for function \(p_{14}\). circuit voltage, short circuit current, and maximum power point can be obtained using Equations (7)-(9), whereas the respective reference short circuit current, reference open circuit voltage, and reference maximum power can be obtained using Equations (11)-(13). The difference between reference values of the voltage, current, and power (i.e., \(V_{OC\_ref}\), \(I_{SC\_ref}\), and \(P_{MPP\_ref}\), respectively) and the actual values of the voltage, current, and power (i.e., \(V_{0},\ I_{SC}\), and \(P_{MPP}\), respectively) gives the voltage error (E\({}_{V}\)), current error (E\({}_{I}\)), and power error (E\({}_{p}\)), as demonstrated in Figure 9.  
A multi-objective optimization problem statement is framed with the problem of computing the optimum combination of three PV parameter values that include the shunt(\(\mathcal{R}_{g}\)) and series resistance (\(\mathcal{R}_{s}\)) and diode ideality factor, which is represented by \(\delta\), with respect to obtaining the percentage error of the ISE of power (\(\epsilon_{\mathrm{P}}\)),the percentage error of the ISE of voltage (\(\epsilon_{\mathrm{V}}\)), and the percentage error of the ISE of current (\(\epsilon_{I}\)).The combined objective was to minimize the ISE of power (\(\epsilon_{\mathrm{P}}\)), ISE of voltage (\(\epsilon_{\mathrm{V}}\)), and ISE of current (\(\epsilon_{\mathrm{I}}\)) from (31) to (36) and the combined fitness function (37).