*2.2. Electric Vehicle Charging Model*

The probability of an individual EV traveling a distance *d* can be represented by the logarithmic normal distribution function [20]

$$h(d,\mu,\sigma) = \frac{1}{d\sqrt{2\pi\sigma^2}}e^{-\frac{\left(\ln d - \mu\right)^2}{2\sigma^2}}\tag{2}$$

where μ and σ are the mean and standard deviation, respectively.

According to the traveling distance *d* of the EV, the remaining capacity can be calculated:

$$SOC = (1 - \frac{d}{d\_m}) \times 100\% \tag{3}$$

where *dm* is the maximum travel distance.

According to [21], the charging start time can be modeled as a normal distribution function with specified parameters, and we represent the time when the EV starts charging as a set.

$$T\_{start}^{k} = \begin{bmatrix} \underline{T\_{start}^{k}} \, \overline{T\_{start}^{k}} \end{bmatrix} \tag{4}$$

where *T<sup>k</sup> start* is the charging start time of the *<sup>n</sup>*th EV, and *<sup>T</sup><sup>k</sup> start* and *<sup>T</sup><sup>k</sup> start* are the lower and upper limits, respectively.

In this study, we considered that all EV batteries have the same capacity *E* and the same charging power *Pc*. Thus, the charging end time *T<sup>k</sup> end* can be obtained:

$$T\_{cud}^k = T\_{start}^k + \frac{(1 - SOC^k) \times E}{P\_\varepsilon} \tag{5}$$

The total charging power of EVs at each moment is the sum of the charging power of an individual EV:

$$PEV\_t^G = \sum\_{k=1}^K PEV\_{k,t}^G \tag{6}$$

where *PEV<sup>G</sup> <sup>k</sup>*,*<sup>t</sup>* is the charging power of the *k*th EV at period *t*. In addition, *K* is the number of EVs dispatched.

The uncertainty of the charging power of the EV is still expressed in the form of a set:

$$\begin{aligned} PEV\_t^G &= \overline{PEV\_t} + PEV\_t\\ \text{s.t.} & \underline{PEV\_t} \le PEV\_t \le \overline{PEV\_t} \end{aligned} \tag{7}$$

where *PEV<sup>G</sup> <sup>t</sup>* is the total charging power of EVs at period *t*, *PEVt* is the forecasted value of the charging power of EVs at period *t*, *PEV*ˆ *<sup>t</sup>* is the deviation, and *PEV*ˆ *<sup>t</sup>*, *PEV*ˆ *<sup>t</sup>* are its lower and upper limits, respectively.
