## 2 Analysis of EV Controllability Considering Users' Travel Patterns
### The Establishment of Grid Connection Scenes of \(Ev\)  
The grid connection scenes of EVs have uncertainty on the temporal scale, mainly due to factors such as the commuting time and the daily mileage [24]. This paper takes the following five parameters to establish the grid connection scenes of EVs: the time leaving home, the time arriving at the workplace, the time leaving the workplace, the time arriving home, and the daily mileage of EVs. According to the 2011 Beijing Traffic Development Annual Report, the probability distributions of travel time of private EV owners are fitted [25]. The graphs of the original data are shown in Figures 12 and 13 in the Appendix A, and the results show that the time leaving home and leaving the workplace obey the Cauchy distribution, while the time arriving at the workplace and arriving home obey the normal distribution. The corresponding expressions of probability density functions \(f_{w,s}\), \(f_{e,s}\), \(f_{w,f}\), \(f_{e,s}\) are shown in Equations (1)-(4), and the distribution functions are \(F_{w,s}\), \(F_{e,s}\), \(F_{w,f}\), \(F_{e,f}\)[26].  
\[f_{w,s}(t_{1})=\left\{\begin{array}{ll}\frac{1}{1.2\pi[1+5(t_{1}-7.3)^{2}]} &0\leq t_{1}\leq 7.5\\ \frac{1}{1.05\pi[1+6(t_{1}-8.2)^{2}]}&7.5\leq t_{1}\leq 24\end{array}\right. \tag{1}\]  
\[f_{e,s}(t_{2})=\frac{1}{0.75\pi[1+3(t_{2}-17.2)^{2}]} \tag{2}\]  
\[f_{w,f}(t_{3})=\frac{1}{\sqrt{2\pi}\times 0.6}\exp[-\frac{(t_{3}-8.75)^{2}}{2 \times 0.6^{2}}] \tag{3}\]  
\[f_{e,f}(t_{4})=\frac{1}{\sqrt{2\pi}\times 0.8}\exp[-\frac{(t_{4}-18.2)^{2}}{2 \times 0.8^{2}}] \tag{4}\]  
where \(t_{1}\) is the time leaving home, \(t_{2}\) is the time leaving the workplace, \(t_{3}\) is the time arriving at the workplace, and \(t_{4}\) is the time arriving home.  
The daily mileage of EVs depend commuters' work and driving characteristics. Moreover, the driving characteristic of large-scale EVs is regular. Based on the results of 2009 National Household Travel Survey, it can be concluded that the daily mileage of EVs obeys the following normal distribution \(f_{d}(x)\)[27]:  
\[f_{d}(x)=\frac{1}{x\sigma_{d}\sqrt{2\pi}}\exp[-\frac{(\ln x-\mu_{d})^{2}}{2 \sigma_{d}{}^{2}}] \tag{5}\]where \(x\) is the daily mileage of EVs, \(\mu_{d}\) and \(\sigma_{d}\) are mean value and variance respectively, \(\mu_{d}=3.2\), \(\sigma_{d}=0.88\).  
Assuming that the five parameters of individual EVs shown in Equations (1)-(5) are mutually exclusive, EV users' travel patterns can be simulated by Monte Carlo method, i.e., grid connection scenes. If the scale of EVs in a certain area is \(NE\), the grid connection scenes of EV can be represented by the sampling matrix \(Q=[W_{s};E_{s};W_{f};E_{f};D]\). \(W_{s}\) is a \(1\times NE\)-dimensional matrix of the time leaving home, \(E_{s}\) is a \(1\times NE\)-dimensional matrix of the time leaving the workplace, \(W_{f}\) is a \(1\times NE\)-dimensional matrix of the time arriving at the workplace, \(E_{f}\) is a \(1\times NE\)-dimensional matrix of the time arriving home, and \(D\) is a \(1\times NE\)-dimensional matrix of daily mileage of EV users. The elements in the same column of the sampling matrix \(Q\) are the grid connection scenes of a single EV.