## 3 EVs Charging Station Load Modeling
### Probabilistic Estimation of EVs Charging Station Load  
One of the most commonly used numerical methods for handling the uncertainties is MCS. This method is used when the system is highly non-linear or it has a lot of uncertain variables [34]. Although batteries have memory and it is better to use the sequential MCS, which generates sequential samples of SoCs for numerous time intervals [35], but we have used non-sequential MCS to avoid complicating the simulation and time-consuming convergence. Therefore, this method has been used in the proposed model because of its many uncertain parameters. The proposed algorithm for the estimation of EVs charging station load is as follows: \begin{table}
\begin{tabular}{c c} \hline
**Value of Each Scenario** & **Probability of Each Scenario** \\ \hline \(\mu-2.5\sigma\) & 0.025 \\ \hline \(\mu-1.5\sigma\) & 0.13 \\ \hline \(\mu\) & 0.69 \\ \hline \(\mu+1.5\sigma\) & 0.13 \\ \hline \(\mu+2.5\sigma\) & 0.025 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Probability of scenarios in the considered normal PDF. Figure 4: The PDF for states of charge (SOCs) of EVs batteries. \begin{table}
\begin{tabular}{c c c c} \hline
**Class** & **Market Share** & \(MinC_{bat}\) (**kWh**) & \(MaxC_{bat}\) (**kWh**) \\ \hline Micro car & 20\% & 10 & 30 \\ \hline Economic car & 30\% & 30 & 60 \\ \hline Mid-size car & 30\% & 30 & 60 \\ \hline Light truck & 30\% & 60 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Specifications of different EVs. \(g=1\). 2. Set MCS counter M = 1. 3. \(j=1\). 4. Randomly generate samples for type of charger used by EV \(j\), type of EV \(j\), charging or discharging of EV \(j\), SoC of EV \(j\) and battery capacity of EV \(j\) all based on their distribution functions assigned in Section 3.1. 5. With \(m_{j}^{2}\) and \(m_{j}^{type}\) equal to 1, if \(m_{j}^{C/D}\) is equal to 1, then the charging time of EV \(j\) can be calculated via (3) with the rate of a level 2 charger (i.e., \(cl=2\)). Otherwise the discharging time of EV \(j\) can be calculated via (4) with \(cl=2\). If \(m_{j}^{type}\) is equal to 0, thus the EV \(j\) can only be charged with the charging time as (3). \[t_{c,g,j}^{cl} =\frac{(SoC_{max}-SoC_{j})\times Cap_{j}}{r_{c,g}^{cl}}\] (3) \[t_{c,g,j}^{cl} =\frac{(SoC_{j}-SoC_{min})\times Cap_{j}}{r_{d,g}^{cl}}\] (4) where \(t_{c,g,j}^{cl}\) and \(t_{d,g,j}^{cl}\) are the charging and discharging time of EV \(j\) with charger level \(cl\) which uses charger \(g\). \(t_{c,g}^{cl}\) and \(t_{d,g}^{cl}\) are the charging and discharging rates of charger level \(cl\) and charger \(g\). Furthermore, \(SoC_{max}\) and \(SoC_{min}\) are the maximum and minimum SoC for EV \(j\). The similar process occurs when \(m_{j}^{2}\) equal to 0 with the difference that the rate of charger level 3 would be considered in (3) and (4) (i.e., with \(cl=3\)). 6. The charging or discharging energy of EV \(j\) or \(E_{j}\) is calculated using (5). \[E_{j}=r_{c,g}^{cl}t_{c,g,j}^{cl}-r_{d,g}^{cl}t_{d,g,j}^{cl}\] (5)
7. If \(j<N_{EV},j=j+1\) and go back to 4; otherwise calculate the charger load using (6). \[\overline{P}_{t,g,MC}^{sta} =\frac{\sum\limits_{j=1}^{N_{EV}}E_{j}}{N_{T}}\] (6) where \(\overline{P}_{t,g,MC}^{sta}\) is the average consuming power of charger \(g\) in repetition M.
8. Evaluation of \(P_{t,g}^{sta}=(1/M)\sum\limits_{M}\overline{P}_{t,g,MC}^{sta}\). \(P_{g}^{sta}\) is the average consuming power of charger \(g\) in all repetitions. 9. Check if \(P_{t,g}^{sta}\) is converged then go to 10, else M = M + 1 and go to 3. 10. If \(g<N_{g},g=g+1\) and go back to 2; otherwise calculate the EVs charging station load through the sum of the calculated chargers loads using (7). \[P_{t}^{station}=\sum\limits_{g=1}^{N_{g}}p_{t,g}^{sta}\] (7)
11. Finish. The proposed MCS-based algorithm is presented in Figure 5. The convergence condition of the loop is considered as follows.