## 2 Modeling of Wind Curtailment and EV Charging
### Daily EV Charging Load  
The EV charging load is modeled as a random variable while considering both the randomness of the activation time for charging the battery as well as the initial state of charge (SoC) at the beginning of the charging process. Once an EV starts charging at time \(t_{cs}\) and continues to be charged until time \(t\), the battery SoC will be increased from the initial SoC with the rated charging power until the battery reaches full capacity. The battery SoC (\(S_{t}\)) at time \(t\) can be expressed with the initial SoC (\(S_{ini}\)) as follows:  
\[S_{t}=\ S_{ini}+(t-t_{cs})\cdot(\eta_{EV}\cdot P_{EV}^{r})/C_{EV} \tag{8}\]  
Figure 3: Maximum variation in wind power with different absolute power limits (\(l_{W}\)): (**a**) absolute power limit within the available wind generation range and (**b**) absolute power limit that is significantly lower than the available generation.  
where \(\eta_{EV}\) is the charging efficiency considering the efficiency of the battery itself and the charger; \(P^{r}_{EV}\) and \(C_{EV}\) are the rated powers of the charger and the battery capacity, respectively. The initial SoC can be derived from daily travel distance [18].  
Figure 4 shows a change in the probability density of the SoC by charging for one hour. After the EV charges for one hour, the SoC increases such that the SoC distribution is shifted toward a higher SoC region. As the charging duration increases, the probability of the EV being fully charged will increase.  
Given that an EV starts charging at time \(t_{cs}\), the probability of an event (\(C_{t}\)) in which the EV is not fully charged until time \(t\) is calculated as follows:  
\[\begin{array}{rcl}\mathrm{pr}(\,C_{t}|\,t_{cs})&=&\mathrm{pr}\big{(}S_{ini}+(t -t_{cs})\cdot(\eta_{EV}\cdot P^{r}_{EV})/C_{EV}\;<\;S^{\mathrm{max}}\big{)}\\ &=&F_{ini}\big{(}S^{\mathrm{max}}-(t-t_{cs})\cdot(\eta_{EV}\cdot P^{r}_{EV})/C_ {EV}\big{)}\;,\end{array} \tag{9}\]  
where \(F_{ini}\) is the CDF of the initial SoC. If an EV that starts charging at time \(t_{cs}\) (before time \(t\)) does not reach a maximum SoC until time \(t\), then the EV is still in charging mode at time \(t\). Based on the total probability theorem [16], the probability that an EV is in charging mode at time \(t\) is expressed as follows:  
\[\mathrm{pr}(C_{t})\;=\sum_{t_{cs}=1}^{N_{T}}\{pr(t_{cs})\cdot F_{ini}(S^{ \mathrm{max}}-(t-t_{cs})\cdot(\eta_{EV}\cdot P^{r}_{EV})/C_{EV})\} \tag{10}\]  
where \(\mathrm{pr}(t_{cs})\) is the probability that an EV starts charging at time \(t_{cs}\). From Bernoulli trials of \(C_{t}\), the number of EVs that will be in the charging state is subject to a binomial distribution of B(\(N_{EV}\)-\(\mathrm{pr}(C_{t})\)), which can be approximated as a normal distribution. Consequently, the total EV charging demand at time \(t\) is subject to a normal distribution with following mean and variance:  
\[\mu_{EV,t}=P^{r}_{EV}\cdot N_{EV}\cdot\mathrm{pr}(C_{t}) \tag{11}\]  
\[\sigma^{2}_{EV,t}=P^{r}_{EV}\cdot N^{EV}\cdot\mathrm{pr}(C_{t})\cdot\{1- \mathrm{pr}(C_{t})\} \tag{12}\]  
The EV charging load can be merged with non-EV load by summation of the two normal distributions, and this load is used in the generation scheduling procedure as described in the following section.