## 2 Modeling of Wind Curtailment and EV Charging
### Mitigation of the Wind Power Variation  
The simplest measure of wind power variation is a step change [17] in the average power between two consecutive times. For a time step \(\tau\), the maximum upward and downward variations are defined as follows:  
\[uv_{AW}=\max(p_{AW,\ t+\tau}-p_{AW,t}) \tag{4}\]  
\[dv_{AW}=-\min(p_{AW,\ t+\tau}-p_{AW,t}) \tag{5}\]  
where \(uv_{AW}\) and \(dv_{AW}\) are the maximum upward and downward variations of wind power, respectively; \(p_{AW,t}\) and \(p_{AW,t+\tau}\) represent the wind power output at the initial time and after a time \(\tau\), respectively.  
The wind power variation is subject to a different distribution depending on the given initial wind output [1]. The maximum variations in wind power that are obtained from the historical wind power data from Jeju in Korea are shown in Figure 2. With this dependency, the maximum upward and downward variations in wind power are defined as functions of the initial wind power.  
Figure 1: Wind power curtailment using the absolute power limit.  
With a given available wind power \(p_{AW}\), wind generation positively varies up to \(p_{AW}+uv_{AW}(p_{AW})\) when there is no limitation. However, when wind curtailment is executed, the wind power varies up to the lower value between the two absolute power limits of \(l_{W}\) and \(p_{AW}+uv_{AW}(p_{AW})\). The maximum upward variation (\(w_{W}\)) of the wind generation can be therefore expressed as follows:  
\[uv_{W}(p_{AW})=\min(p_{AW}+uv_{AW}(p_{AW}),\;l_{W})-p_{W} \tag{6}\]  
where \(w_{AW}(p_{AW})\) is the maximum upward variation without wind curtailment, which is a function of available wind power.  
On the other hand, the downward variation in wind power is reduced because of the restriction of the initial wind power to be below the absolute power limit. With a given available wind power of \(p_{AW}\), wind generation negatively varies up to \(p_{AW}-dv_{AW}(p_{AW})\) when there is no limitation. When wind power curtailment is executed, initial wind power is restricted to be the lower value between the absolute power limit \(l_{W}\) and the available wind power \(p_{AW}\), and this wind generation can vary up to \(p_{AW}-dv_{AW}(p_{AW})\). The maximum downward variation (\(dv_{W}\)) in the wind generation can be therefore expressed as follows:  
\[dv_{W}(p_{AW})=p_{W}-\;\min(p_{AW}-\;dv_{AW}(p_{AW}),\;l_{W}) \tag{7}\]  
where \(dv_{AW}(p_{AW})\) is the maximum downward variation without wind curtailment in terms of available wind power.  
Figure 2: Maximum short-term variations in wind power with varying initial wind power levels: (**a**) upward variation and (**b**) downward variation.  
Figure 3 shows an example of maximum variation in wind power generation with different absolute power limits. With an absolute power limit within the available generation range as shown in Figure 3a, the maximum upward variation is considerably decreased, while it cannot be expected that the maximum downward variation will decrease. In case an absolute power limit is significantly lower than the available generation range as shown in Figure 3b, it is expected that the wind power forecast error will be approximately zero because almost all the possible wind generations will be restricted by the absolute power limit. It can be also expected that the wind power stays constant without any variation in both the directions.