## 2 Mathematical Modeling of EV/DC Fast Charging Station
### Modelling of Photovoltaic Generator  
PV power generation has numerous advantages if compared with other green sources; such as installation flexibility with minimum maintenance. PV integration in the distribution system is widely accepted due to universal sunlight availability, high modularity, easy maintainability, long life cycle and mobility, short time for design, incurring no fuel operation cost, environment-friendly, installation and start-up, and the ability for off-grid application [38]. PV array installation is easy and the continuous price reduction of PV modules makes it more attractive. Hence, PV array-based EV battery charging is adopted by many EV users [39]. Despite clear advantages, solar PV is un-dis patchable with fluctuating power output which results in voltage variation, degraded protection, leakage power flow, non-availability during the night and increased fault level [40; 41].  
An equivalent circuit of a solar cell consist of a single diode model is selected for this configuration. It consists of a current source in parallel with a diode with shunt and series resistance. An equivalent model of tandem solar cells is displayed in Figure 4 [17].  
The PV output is calculated with the help of a single diode model. The output current of PV primarily depends on temperature (\(T\)) and solar irradiance intensity (\(G\)). The output current of PV, i.e., \(I_{pv}\), is given in;  
\[I_{pv}=I_{ph}+I_{sat}\left(e^{q(\frac{V+I_{pv}*R_{s}}{NR_{pv}})}\right)-\frac {V+I_{pv}*R_{s}}{R_{sh}} \tag{1}\]  
\[I_{ph}\ =I_{pho}(1+k_{0}(T-300)) \tag{2}\]  
\[I_{sat}=k_{1}T^{3}e^{\frac{-qV_{g}}{kT}} \tag{3}\]  
Figure 4: Equivalent model of PV cell. Reprinted with permission from ref. [17]. Copyright 2021 from IEEE.  
\[I_{ph}-\text{solar induced current}\] \[I_{pho}-\text{solar induced current at 300 }^{\circ}\text{C}\] \[I_{sat}-\text{saturation current of the diode}\] \[V_{g}-\text{voltage applied to the terminals of diode}\] \[k_{0},k_{1}-\text{constants depend on the value of PV system}\] \[T_{PV}-\text{operating temperature}\] \[N-\text{diode quality factor}\] \[V=\frac{\left(\left(a\times K\times t\right)\right)}{q} \tag{4}\] \[V=\text{thermal voltage}\] \[q=\text{electron elementary charge (1.602 }\times 10^{-19}\text{ C)}\] \[a=\text{ideality factor (1 }<a<2)\] \[k=\text{Boltzmann constant (1.381 }\times 10^{-23}\text{ J/K)}\] \[R_{s}\text{ }and\text{ }R_{p}\text{ are series and shunt resistances, respectively.}\]  
The output power of the PV generator at time step t is calculated by the following equation [42];  
\[P_{pv}(t)=N_{pv}\times\eta_{pv}\times A_{pv}\times\frac{G_{i}(t)}{1000} \tag{5}\]  
where \(\eta_{pv}\) is the efficiency of the PV panels, \(A_{pv}\) is the area of each PV panel in m\({}^{2}\), \(G_{i}(t)\) is the aggregate solar irradiation incident on the tilted PV panels in W/m\({}^{2}\) at time step t, and \(N_{PV}\) is the optimum number of panels that are determined at each iteration of the optimal sizing procedure. The most crucial feature of the MPPT is its ability to track the MPP as quickly and efficiently as possible. These can be configured using a fixed or adaptive time step. The algorithm used for maximum tacking is used Perturb and Observe method. The yearly temperature variations and solar insolation are delineated in Figures 5 and 6.  
Figure 5: Temperature variations of a studied area over a year.