## 2 Domestic Electric Vehicle Charging Loads Modelling
### Users' Driving Habits and Preference of Charging Electric Vehicles  
As two critical parameters, users' daily return time from the last trip and daily travelling distance per vehicle need to be analyzed carefully, because they are directly relevant to users' driving habits and preference of charging electric vehicles. Figure 1 shows the statistical data of the probability distribution of domestic users' daily return time from the last trip, which came from the NHTS and was published by the Department of Transportation of U.S. [18]. Figure 1: Probability distribution of domestic users’ daily return time from the last trip; (**a**) For one year; (**b**) For different seasons. Figure 1a reveals that the probability distribution of domestic users' daily return time is similar to the normal distribution. Therefore, the probability density function of domestic users' daily return time from the last trip can be written as follows:  
\[f_{end}(t_{r})=\left\{\begin{array}{ll}\frac{1}{\sigma_{end}\sqrt{2\pi}}e^{ [-\frac{(t_{r}+24-u_{end})^{2}}{2\sigma_{end}^{2}}]}&0<t_{r}\leq u_{end}-12 \\ \frac{1}{\sigma_{end}\sqrt{2\pi}}e^{[-\frac{(t_{r}-t_{end})^{2}}{2\sigma_{ end}^{2}}]}&u_{end}-12<t_{r}\leq 24\right. \tag{1}\]  
In (1), \(f_{end}(t_{r})\) is the probability density function of daily return time from the last trip, \(\sigma_{end}\) represents the standard deviation, \(\mu_{end}\) is the mathematical expectation, and \(t_{r}\) is daily return time from the last trip. In addition, according to Figure 1b, users' daily return time from the last trip varies greatly according to the seasons. It is shown that compared to the summer, domestic users finish their last trips much earlier in winter. Therefore, in this paper, it is the first time that seasonal factors are taken into consideration when modelling the probability density function of domestic users' daily return time from the last trip. By implementing MATLAB (R2016b, MathWorks company, Nettie, MA, USA) Simulation, the standard deviations and mathematical expectations of users' return time from the last trip in different seasons can be acquired. Table 1 summarizes the simulation results of these two parameters in different seasons. As another important factor that can affect users' driving habits and preference of charging electric vehicles, user's daily travelling distance needs to be analyzed. Figure 2 shows the probability distribution of domestic users' daily travelling distance per vehicle (in mile). Figure 2a reveals that both of the probability distribution and the cumulative probability distribution of domestic users' daily travelling distance are similar to exponential distribution compared with that of logarithmic normal probability distribution. Therefore, in this paper, it is the first time that exponential distribution is used to represent the original data which was published by the Department of Transportation of U.S. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Spring** & **Summer** & **Autumn** & **Winter** \\ \hline \(\mu_{end}\) & 17:48 1 & 18:00 1 & 17:26 1 & 17:10 1 \\ \(\sigma_{end}\) & 3.60 & 3.59 & 3.60 & 3.62 \\ \hline \end{tabular}
\end{table} 
Table 1: The standard deviations and mathematical expectations of users’ return time in different seasons. Figure 2: _cont_. In Figure 2b, simulation intervals of daily travelling distance increases from 5 to 10 miles, and simulation results show that the error is significantly reduced when applying exponential distribution to represent the original data. This is because stochastic volatility can be reduced if simulation intervals are increased. In addition, Figure 2b shows that simulation results are accurate by implementing exponential distribution to represent the probability distribution of travelling distance for different seasons. Therefore, in this paper, the exponential distribution model is used to represent the probability distribution of daily travelling distance.