## 3 Mathematical Modeling of PV Models
### Single-Diode Model  
The SDiM is recognized as a fundamental circuit type that is regularly utilized and built upon physical principles. The SDiM is commonly used because of its ease of integration and thoroughness in characterizing the intrinsic characteristics of stationary solar cells.  
Figure 1a illustrates the single-diode model used to represent a solar cell. The photon current (\(I_{ph}\)) generated in the solar cell as a result of solar irradiance is transferred to the load after being extracted. The current flowing through a diode is known as (\(I_{ph}\)). In addition, the circuit model can be adjusted according to the modifications illustrated in Figure 1b. These modifications take into account the combined resistance of contacts, metal grids, and \(p\) and \(n\) layers, represented by the series resistance (\(\mathcal{R}_{s}\)), as well as the leakage current of the \(p\)-\(n\) junction, which is accounted for by the shunt resistance (\(\mathcal{R}_{sh}\)).  
Through the application of [PERSON]'s current law (KCL), the relationship between the output current (\(I_{L}\)), \(I_{ph}\), diode current (\(I_{di}\)), and shunt-resistor current (\(I_{s}\)) can be expressed as  
\[I_{L}=I_{ph}-I_{di}-I_{s} \tag{7}\]  
The mathematical equations for \(I_{di}\) and \(I_{s}\) are expressed as  
\[I_{di}=I_{sa}\left[e^{\{\frac{(\varphi+I_{L}\mathcal{R}_{s})}{\rho\varphi t} \}}-1\right] \tag{8}\]  
\[I_{s}=\frac{\
u+I_{L}\mathcal{R}_{s}}{\mathcal{R}_{sh}} \tag{9}\]  
Through the utilization of Equations (8) and (9), Equation (7) undergoes modification as  
\[I_{L}=I_{ph}-I_{sa}\left[e^{\{\frac{(\varphi+I_{L}\mathcal{R}_{s})}{\rho\varphi t }\}}-1\right]-\frac{\
u+I_{L}\mathcal{R}_{s}}{\mathcal{R}_{sh}} \tag{10}\]where \(I_{sa}\) denotes the saturation current, \(\rho\) signifies the diode quality factor, \(\
u\) represents the output voltage, \(\tau\) represents the temperature of the \(p\)-\(n\) junction (in kelvin), \(q\) represents the electron charge, \(\eta\) denotes the number of solar cells in series, and \(\kappa\) stands for the Boltzmann constant.  
Equation (10) indicates that the parameters \(I_{L}\), \(I_{sa}\), \(\mathcal{R}_{sh,r}\), \(\mathcal{R}_{s}\), and \(\rho\) need to be recovered from the SDiM.  
Under open circuit (\(OC\)) circumstances (\(I_{L}=0\); \(\
u=\
u_{\text{OC}}\)), Equation (8) is changed to  
\[I_{ph}=I_{sa}\bigg{[}e^{\frac{\{\frac{\{\theta(\
u_{\text{OC}})}}{\rho\eta\tau} }}{\rho\eta\tau}}-1\bigg{]}+\frac{\
u_{\text{OC}}}{\mathcal{R}_{sh}} \tag{11}\]  
Equation (10) is changed under short circuit (\(SC\)) circumstances (\(I_{L}=I_{\text{SC}}\); \(\
u=0\)) to  
\[I_{ph}=I_{sc}+I_{sa}\bigg{[}e^{\{\frac{\{I_{sa}\mathcal{R}_{s}\mathcal{R}_{s} }}{\rho\eta\tau}}}-1\bigg{]}+\frac{I_{sc}\mathcal{R}_{s}}{\mathcal{R}_{sh}} \tag{12}\]  
Through utilizing Equations (9) and (10), the expression for \(I_{sa}\) is obtained as  
\[I_{sa}=\frac{I_{sc}+\frac{I_{sa}\mathcal{R}_{s}}{\mathcal{R}_{sh}}-\frac{\
u_ {\text{OC}}}{\mathcal{R}_{sh}}}{e^{\{\frac{\
u_{\text{OC}}}{\rho\eta\tau}\}} }-e^{\{\frac{\{I_{sa}\mathcal{R}_{s}\mathcal{R}_{s}}}{\rho\eta\tau}\}} \tag{13}\]  
At the maximum power point (_MPP_) (\(I_{L}=I_{MPP}\); \(\
u=\
u_{MPP}\)), Equation (8) is written as  
\[I_{MPP}=I_{ph}-I_{sa}\bigg{[}e^{\{\frac{\{\eta(\
u_{MPP}+I_{MPP}R_{s})}}{\rho \eta\tau}\}}-1\bigg{]}-\frac{V_{MPP}+I_{MPP}\mathcal{R}_{s}}{\mathcal{R}_{sh}} \tag{14}\]