can derive the average probability that s/he will choose a certain charging station by taking the expectation over the unobservable utilities defined in Equation (\[cdf\]). The probability that the $n$-th EV owner will choose the $j$-th charging station of service provider $k$ is [@cite11]

$$\begin{aligned}
\Phi_{j,k}^n&=\mathbf{Prob}\left(\overline{U}_{j,k}^n+\epsilon_{j,k}^n > \overline{U}_{i,l}^n + \epsilon_{i,l}^n, \forall i \neq j, \textrm{or } l \neq k \right)\\
&=\int_{-\infty}^{+\infty}F_{j,k}(\overline{U}_{j,k}^n-\overline{U}_{1,1}^n+\epsilon_{j,k}^n,\overline{U}_{j,k}^n-\overline{U}_{2,1}^n+\epsilon_{j,k}^n,\\
&\cdots,\epsilon_{j,k}^n,\cdots,\overline{U}_{j,k}^n-\overline{U}_{L,3}^n+\epsilon_{j,k}^n)d\epsilon_{j,k}^n,
\end{aligned}$$

where $F_{j,k}$ denotes the derivative of $F$ with respect to $\epsilon_{j,k}^n$, i.e. $F_{j,k}=\partial F/\partial \epsilon_{j,k}^n$. Evaluating this integral with the above assumptions, we obtain

$$\Phi_{j,k}^n=\frac{e^{\overline{U}_{j,k}^n/\sigma_k}\left(\sum_{l=1}^Le^{\overline{U}_{l,k}^n/\sigma_k}\right)^{\sigma_k-1}}
{\sum_{t=1}^3\left(\sum_{l=1}^Le^{\overline{U}_{l,t}^n/\sigma_t}\right)^{\sigma_t}}.$$

Charging Demand Estimation
--------------------------

Once the EV owners’ choice probability is computed, we can predict the charging demand at any charging station. Let $q_n\;(n=1,2,\cdots,N)$ denote the total electricity (measured in kWh) that the $n$-th EV owner purchases from the charging station. Further, let $q_n$ be a random variable uniformly distributed in the range $[Q_a, Q_b]$, where $Q_a$ and $Q_b$ are, respectively, the lower and upper limit of charging demand for all EVs. The total predicted charging demand of charging station $j$ of service provider $k$ is modeled as $$\psi_{j,k}=\sum_{n=1}^Nq_n\Phi_{j,k}^n.$$

All coefficients in the nested logit model can be estimated and calibrated from preference survey data. The nested logit model enables us 