## 4 Application of the CrSA for Parameter Extraction of the Solar Module
### Formulation of the Objective Function  
The main goal of the extraction phase in determining coefficients for different PV models using OAs is to assess the efficacy of each parameter and determine the most ideal value for each parameter in the given context. This issue is under investigation. This strategy aims to reduce the discrepancy between estimated and measured current data by minimizing the error to the greatest extent possible. It is necessary to design and optimize the objective function to assess the suggested coefficients. The goal function is formulated to minimize the disparity between the estimated and measured current values to the greatest feasible degree. The existing literature focuses solely on achieving accuracy at the _MPP_. However, this study's objective function is designed to ensure accuracy at three different operating points: the \(OC\), \(SC\), and _MPP_.  
In order to create an accurate model, the minimizing of errors in these places is taken into consideration. For the SDiM, the errors at the \(OC\) (\(\varphi_{OC}\)), \(SC\) (\(\varphi_{SC}\)), and _MPP_ (\(\varphi_{MPP}\)) can be measured using Equations (11), (12), and (14). For the DDiM, this can be achieved using Equations (19)-(21). To extract the parameters of a solar PV module and obtain the optimal parameters, the objective function is formulated as net error \(\varphi\), which is the sum of the squares of \(\varphi_{OC}\), \(\varphi_{SC}\), and \(\varphi_{MPP}\). The steps to determine errors for a model with a single diode are as follows: (31)-(33).  
For the SDiM:  
Error at the \(OC\) point  
\[\varphi_{OC}=I_{sa}\bigg{[}e^{\{\frac{\delta(\
u_{OC})}{\rho_{MP}\
u_{1}}\}} -1\bigg{]}+\frac{\
u_{OC}}{\mathcal{R}_{sh}}-I_{ph} \tag{22}\]  
Error at the \(SC\) point  
\[\varphi_{SC}=I_{sc}+I_{sa}\bigg{[}e^{\{\frac{\delta(\
u_{OC}\mathcal{R}_{s})}{ \rho_{MP}\
u_{1}}\}}-1\bigg{]}+\frac{I_{sc}\mathcal{R}_{s}}{\mathcal{R}_{sh}}- I_{ph} \tag{23}\]  
Error at the _MPP_  
\[\varphi_{MPP}=I_{ph}-I_{sa}\bigg{[}e^{\{\frac{\delta(\
u_{MPP}+I_{MPP}\mathcal{ R}_{s})}{\rho_{MP}\
u_{1}}\}}-1\bigg{]}-\frac{V_{MPP}+I_{MPP}\mathcal{R}_{s}}{ \mathcal{R}_{sh}}-I_{MPP} \tag{24}\]Therefore, the objective function that has to be minimized for the SDiM is as follows  
\[\varphi=\varphi_{OC}^{2}+\varphi_{SC}^{2}+\varphi_{MPP}^{2} \tag{25}\]  
For the DDiM:  
Error at the \(OC\) point  
\[\varphi_{OC}=I_{sat1}\bigg{[}e^{\{\frac{q(i_{OC})}{P_{1}\sqrt{p_{1}}}\}}-1 \bigg{]}+I_{sat2}\bigg{[}e^{\{\frac{q(i_{OC})}{P_{2}\sqrt{p_{1}}}\}}-1\bigg{]} +\frac{\
u_{OC}}{\mathcal{R}_{sh}}-I_{ph} \tag{26}\]  
Error at the \(SC\) point  
\[\varphi_{SC}=I_{SC}+I_{sat1}\bigg{[}e^{\{\frac{q(I_{SC}\mathcal{R}_{s})}{P_{1} \sqrt{p_{1}}}\}}-1\bigg{]}+I_{sat2}\bigg{[}e^{\{\frac{q(I_{SC}\mathcal{R}_{s}) }{P_{2}\sqrt{p_{1}}}\}}-1\bigg{]}+\frac{I_{SC}\mathcal{R}_{s}}{\mathcal{R}_{sh }}-I_{ph} \tag{27}\]  
Error at the \(MPP\)  
\[\begin{split}\varphi_{MPP}=I_{ph}-I_{sat}\bigg{[}e^{\{\frac{q( \frac{q(i_{MPP}+IMPP\mathcal{R}_{s})}{P_{1}\sqrt{p_{1}}})}}{P_{1}\sqrt{p_{1}}} }-1\bigg{]}-I_{sat2}\bigg{[}e^{\{\frac{q(i_{MPP}+IMPP\mathcal{R}_{s})}{P_{2} \sqrt{p_{1}}}\}}-1\bigg{]}\\ -\frac{\
u_{MPP}+IMPP\mathcal{R}_{s}}{\mathcal{R}_{sh}}-I_{MPP} \end{split} \tag{28}\]  
Therefore, the objective function that has to be minimized for the DDiM is as follows  
\[\varphi=\varphi_{OC}^{2}+\varphi_{SC}^{2}+\varphi_{MPP}^{2} \tag{29}\]