# A Multi-Objective Hybrid BESSA Optimization Scheme for Parameter Extraction from PV Modules
## 2 Analysis and Modeling of Analytical PV Cell Introduction  
### Mathematical Modelling of the PV Module  
Accurate forecasting of PV module performance under different weather conditions only depends upon the degree of exactness in the modeling of the photovoltaic(PV) cell, which again depends upon correctness in the values of the extracted parameters.  
The current(I)-voltage(V) characteristics of a PV cell [1] during dark operating conditions are governed by (1).  
\[I_{\Delta}=I_{\
u}\left(e^{\
u/\beta\pi\
u}-1\right) \tag{1}\]  
In (1), the diode current and reverse saturation current are designated as \(I_{\Delta}\) and \(I_{\
u}\), respectively. The symbol \(c\) isthe charge of an electron in coulombs (\(1.602\times 10^{-19}\)\(C\)) and \(V\) and \(\delta\) signify output voltage and diode ideality factor, respectively. Diode ideality factor (\(\delta\)) depends on the quality of the semiconductor material used and has a value between 1 and 2 [1]. In the above equation, \(k\) represents the Boltzmann constant. The value of \(k\) is \(1.381\times 10^{-23}\) J/K. \(T^{\prime}\) represents junction temperature.  
Photon illumination of the PV module produces photon current \(I_{\phi}\). Thus, an ideal PV cell is considered as one which has a current source with an intensity of \(I_{\phi}\) and is in parallel connection with the diode. The PV characteristics [4] of a diode are explained via the Shockley equation. The Shockley equation is mathematically expressed by (2).  
\[I_{\Delta}=I_{\phi}-I_{\
u}\left(e^{\
u/\beta\pi\
u}-1\right) \tag{2}\]All of the analysis discussed above is for the ideal case. The effect of electrodes is not taken into account. However, practically, the current conducted from each elementary diode moves across the semiconductor slabs using different paths which have varying resistances and voltage drops. Thus, the electrical equivalent of the PV cell is envisaged as a current source connected in parallel with a diode and shunt resistance (\(\mathcal{R}_{g}\)) and in series with resistance(\(\mathcal{R}_{S}\)).  
The constructional defects and dissipative effects [4] are taken into account by these resistances. The constructional defects and dissipative effects cause a flow of parasitic current in the PV panel.  
Five parameters of the single-diode model are represented in Figure 1. Here, \(I_{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\  
\[V_{OC\_ref}(G,T)=V_{OC\_SOC}+\sigma_{OC}(T-T_{SOC}) \tag{12}\]  
\[P_{MPP\_ref}(G,T)=\frac{G}{G_{SOC}}[P_{MPP\_SOC}+\sigma_{MPP}(T-T_{SOC})] \tag{13}\]  
In the above equations, \(I_{SC\_SOC}\), \(V_{OC\_SOC}\), and \(P_{MPP\_SOC}\) represent short circuit current, open circuit voltage, and power at MPP, respectively, in SOC. The terms \(\sigma_{SC}\), \(\sigma_{OC}\) and \(\sigma_{MPP}\) represent the temperature coefficient at short circuit current, open circuit voltage, and maximum power point, respectively. The following terms' information, \(V_{OC\_ref}\), \(I_{SC\_ref}\), \(P_{MPP\_ref}(G,T)\), \(\sigma_{OC}\), \(\sigma_{SC}\), and \(\sigma_{MPP}\), are provided by the manufacturer's data sheet.