## 3 FR Capacity of EV Cluster
### Real-Time Prediction of EVA's FR Capacity  
As a kind of transportation, EV's traffic characteristics cannot be ignored. It is possible for EV owners to violate dispatching instructions and leaving the grid before the declared departure time, resulting in the inaccurate prediction of EV's FR capacity. When the number of EVs violating the FR dispatching plan reaches a certain level, the FR task allocated to the EVA from the grid cannot be effectively accomplished.  
Therefore, in order to solve this problem, the concept of credibility is introduced into the prediction of EV's FR capacity [22]. If an EV obeys the declared plan strictly, leaving the grid according to the declared time, the credibility of this EV is regarded as high; thus, its FR capacity will be fully included in the FR capacity of EVA. However, if an EV is used to suddenly leaving the grid during the declared residence time, the credibility of this EV is regarded as low, and its FR capacity will be included only partially in EVA's FR capacity with the proportion based on the value of credibility. The definition of credibility is shown in (12):  
\[\rho_{i}=\frac{\overline{T}_{i}^{a}}{\overline{T}_{i}^{d}}=\left\{\begin{array}[ ]{ll}1&\rho_{up}\leq\frac{\overline{T}_{i}^{d}}{\overline{T}_{i}^{d}}\\ \frac{\overline{T}_{i}^{d}}{\overline{T}_{i}^{d}}&\rho_{down}\leq\frac{ \overline{T}_{i}^{d}}{\overline{T}_{i}^{d}}<\rho_{up}\\ 0&\frac{\overline{T}_{i}^{d}}{\overline{T}_{i}^{d}}<\rho_{down}\end{array}\right. \tag{12}\]  
where \(\overline{T}_{i}^{d}\) is the historical average actual FR participation time of the EV; \(\overline{T}_{i}^{d}\) is historical average declared FR participation time of the EV; \(\rho_{up}\) is the upper threshold of credibility. When the credibility is too high/low, the value is set to be 1/0.  
Taking the credibility of EVs into account, the FR capacity of EVA at time \(j\) can be predicted in real-time by (13)-(15):  
\[R_{up,j}^{ev}=\sum_{i=1}^{N_{j}^{dc}}\left(m\cdot\rho_{i}\right) \tag{13}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Group** & **Mode** & **Description** \\ \hline \multirow{3}{*}{Discharging group} & \multirow{3}{*}{1} & The EV is included in discharging group. If \(SOC_{i,j}>SOC_{\text{min}}\), the EV remains in discharging group and discharges for FR. \\ \cline{2-3}  & & The EV is included in the charging group. If \(SOC_{i,j}\geq SOC_{\text{max}}\), the EV switches into the discharging group and discharges for FR. \\ \hline \multirow{3}{*}{Charging group} & \multirow{3}{*}{3} & The EV is included in the charging group. If \(SOC_{i,j}<SOC_{\text{max}}\), the EV remains in charging group and charges for FR. \\ \cline{2-3}  & & The EV is included in discharging group. If \(SOC_{i,j}\leq SOC_{\text{min}}\), the EV switches into charging group and charges for FR. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Grouping strategy for EVs (electric vehicles) in FR (frequency regulation) service group.  
\[R_{down,j}^{ev}=\sum_{i=1}^{N_{j}^{c}}\left(-m\cdot\rho_{i}\right) \tag{14}\]  
\[R_{j}^{ev}=\left\{\begin{array}{ll}R_{up,j}^{ev}&\Delta f<0\\ R_{down,j}^{ev}&\Delta f>0\\ 0&\Delta f=0\end{array}\right. \tag{15}\]  
where \(R_{up,j}^{ev}\) is EVA's real-time predicted capacity for up-regulation; \(R_{down,j}^{ev}\) is EVA's real-time predicted capacity for down-regulation; \(m\) is the limit of EV's charging/discharging power; \(R_{j}^{ev}\) is the real-time predicted FR capacity of EVA; \(\Delta f\) is the system frequency deviation.