## 3 Mathematical Modeling of PV Models
### Double-Diode Model  
One limitation of the SDiM is its inability to consider the impact of ongoing recombination losses in the reduction region. In addressing this issue, the researchers introduced the DDiM, known for its ability to calculate recombination current losses in the depletion zone by incorporating an extra diode. The DDiM is thought to provide superior accuracy in comparison to the SDiM. The photoproduced current source in the DDiM is realized by employing a diode pair. The D1 diode generates the propagation current in the p-n junction, whereas the D2 diode handles the effects of recombination in the junction area related to space freightage.  
To enhance the precision of the I-V characteristics of the PV panel, it is advisable to utilize the DDiM instead of the SDiM. The DDiM takes into account the impact of recombination current loss in the depletion area resulting from the recombination carriers. The solar PV module-equivalent circuit, represented as a DDiM, is illustrated in Figure 2.  
The output current (\(I_{L}\)) can be expressed using KCL as  
\[I_{L}=I_{ph}-I_{d11}-I_{d22}-I_{s} \tag{15}\]  
where \(I_{d11}\) and \(I_{d22}\) represent diode currents, which can be expressed in terms of saturation currents \(I_{sa1}\) and \(I_{sa2}\), as well as diode quality factors \(\rho_{1}\) and \(\rho_{2}\), as shown:  
\[I_{d11}=I_{sa1}\bigg{[}e^{\{\frac{\{\theta(\
u+I_{l}\mathcal{R}_{s})}}{\rho_{1 }\eta\tau}\}}-1\bigg{]} \tag{16}\]  
\[I_{d22}=I_{sa2}\bigg{[}e^{\{\frac{\{\theta(\
u+I_{l}\mathcal{R}_{s})}}{\rho_{2 }\eta\tau}\}}-1\bigg{]} \tag{17}\]  
By employing Equations (9), (16), and (17), Equation (15) is altered as  
\[I_{L}=I_{ph}-I_{sa1}\bigg{[}e^{\{\frac{\{\theta(\
u+I_{l}\mathcal{R}_{s})}}{ \rho_{1}\eta\tau}\}}-1\bigg{]}-I_{sa2}\bigg{[}e^{\{\frac{\{\theta(\
u+I_{l} \mathcal{R}_{s})}}{\rho_{2}\eta\tau}\}}-1\bigg{]}-\frac{\
u+I_{L}\mathcal{R}_{ s}}{\mathcal{R}_{sh}} \tag{18}\]Equation (18) indicates that the following seven parameters need to be derived from the DDiM: \(I_{ph}\), \(I_{sa1}\), \(I_{sa2}\), \(\mathcal{R}_{s}\), \(\mathcal{R}_{sh}\), \(\rho_{1}\), and \(\rho_{2}\).  
Under \(OC\) circumstances (\(I_{L}=0\); \(\
u=\
u_{OC}\)), Equation (16) is changed as  
\[I_{ph}=I_{sal}\bigg{[}e^{\{\frac{\delta(\
u_{OC})}{\rho_{1}\
u_{1}\
u_{1}}\}}-1 \bigg{]}+I_{sa2}\bigg{[}e^{\{\frac{\delta(\
u_{OC})}{\rho_{2}\
u_{1}}\}}-1 \bigg{]}+\frac{\
u_{OC}}{\mathcal{R}_{sh}} \tag{19}\]  
Equation (18) is changed under \(SC\) circumstances (\(I_{L}=I_{SC}\); \(\
u=0\)) as  
\[I_{ph}=I_{SC}+I_{sal}\bigg{[}e^{\{\frac{\delta(I_{SC}\mathcal{R}_{s})}{\rho_{1 }\
u_{1}\
u_{1}}\}}-1\bigg{]}+I_{sa2}\bigg{[}e^{\{\frac{\delta(I_{OC}\mathcal{ R}_{s})}{\rho_{2}\
u_{1}}\}}-1\bigg{]}+\frac{I_{SC}\mathcal{R}_{s}}{ \mathcal{R}_{sh}} \tag{20}\]  
At the _MPP_ ( \(I_{L}=I_{MPP}\); \(\
u=\
u_{MPP}\)), Equation (18) is written as  
\[\begin{split} I_{MPP}=I_{ph}-I_{sa1}\bigg{[}& e^{\{\frac{\delta(\
u_{MPP}+I_{MPP}\mathcal{R}_{s})}{\rho_{1}\
u_{1}\
u_{1}}\}}-1 \bigg{]}-I_{sa2}\bigg{[}e^{\{\frac{\delta(\
u_{MPP}+I_{MPP}\mathcal{R}_{s})}{ \rho_{2}\
u_{1}}\}}-1\bigg{]}\\ &-\frac{\
u_{MPP}+I_{MPP}\mathcal{R}_{s}}{\mathcal{R}_{sh}}-I_{ MPP}\end{split} \tag{21}\]