## 2 Analysis of EV Controllability Considering Users' Travel Patterns
### The Controllable Capacity Forecast of Large-Scale EVs  
The establishment of grid connection scenes of EV is considered as the basic data for large-scale EVs' controllable capacity forecasting. According to the driving characteristics of EV users on weekdays, it is generally believed that the users will charge the batteries uncoordinatedly during the working period or in the time at home. It is supposed that the morning rush hour is \([W_{s,i},W_{f,i}]\) and the evening rush hour is \([E_{s,i},E_{f,i}]\). The charging period division of EV\(i\) on working days is shown in Figure 1.  
During the driving period, the private EVs are not in a dispatchable state, thus the controllable capacity is set to be zero. EVs can receive the dispatching orders from electricity grid at any time during the users' working hours at the workplace and rest hours at home, and select an appropriate charging strategy, according to their own demands. The number of idle EVs in the time period \(\Delta T\) of the above two can be expressed by Equation (6):  
\[ \begin{split}&\Omega_{free}((k_{free}+1)\Delta T)=\\ &\left\{\begin{array}{l}NE\times\sum\limits_{k_{1}=0}^{NT_{1}-1 }[F_{w,f}(k_{1}\Delta T)-F_{e,s}(k_{1}\Delta T)]\ \ \ \ \ if\ free=1\\ \sum\limits_{k_{2}=0}^{NT_{2}-1}[F_{e,f}(k_{2}\Delta T)-F_{w,s}(k_{2}\Delta T )]\ \ \ \ \ if\ free=2\end{array}\right.\end{split}  \tag{6}\]  
where \(free=1\) and \(free=2\) represent that the EV is in the working hours at the workplace and during rest hours at home, respectively; \(NT_{1}\) is the total number of dispatching periods included in the working hours; \(NT_{2}\) is the total number of dispatching periods included in the rest hours; \(\Delta T\) is the time length of each dispatching period, \(k_{1}=0,1,2,\ldots,NT_{1}-1\), \(k_{2}=0,1,2,\ldots,NT_{2}-1\). This paper divides 24 h a day into 96 time periods, and each period is 15 min. However, with driving and charging demands, if the EV needs to be in a controllable state, not only does the EV need to be idle during the dispatching periods, but it also needs to be in a sufficient state of charge (SOC) for the user demand required before leaving the grid. The sequence diagram of the charging scenarios for judging whether the individual EVs are in a controllable state is shown in Figure 2. \(t_{a,i}\) is the time when EV \(i\) is connected to the grid, \(t_{r,i}\) is the leaving time from the grid for EV \(i\), \(t_{e,i}\) is the ending time when EV \(i\) charges the battery to the user's required SOC, \(t_{p,i}\) and \(t_{q,i}\) represent the start time of the \(p-th\) and \(q-th\) dispatching of electricity grid for EV \(i\), \(t_{p+\Delta T,i}\) and \(t_{q+\Delta T,i}\) represent  
Figure 1: The charging period division of electric vehicles (EVs) \(i\) on working days. the stop time of the \(p-th\) and \(q-th\) dispatching of electricity grid for EV \(i\), and \(t_{d,i}\) is the leaving time when EV participates one dispatching period, \(SOC_{s,i}\) is the state of charge when EV \(i\) is connected to the grid, \(SOC_{r,i}\) is the required state of charge when EV \(i\) leaves the grid, \(SOC_{p,i}\) and \(SOC_{q,i}\) are the initial state of charge of the \(p-th\) and \(q-th\) dispatching of electricity grid for EV \(i\), respectively. In this figure, the black curve is the scenario where the EV is charged with constant power after connecting to the grid. The red curve indicates that the EV accepts the dispatching order at \(t_{p,i}\) during the charging period and temporarily leaves the grid, and then reconnects to electricity grid for charging after the dispatching. Meanwhile, the blue curve shows the scenario in which the EV completes two dispatching orders of the grid. In addition, the EV is considered to be uncontrollable when the EV is charged with full power or the amount of charged energy for EV user demand is not satisfied before leaving the grid, i.e., the EV keeps charging during idle periods without reacting to the dispatch signal that interrupts its charging process.