[@cite11]. The main idea of nested logit model is that a consumer is a utility maximizer. The consumer will choose the product which brings him/her the maximum utility.

In our problem, the utility that the $n$-th EV owner can obtain from choosing charging station $j\;(j=1,2,\cdots,L)$ of service provider $k\;(k=1,2,3)$ is denoted as $U_{j,k}^n=\overline{U}_{j,k}^n+\epsilon_{j,k}^n$, where $\overline{U}_{j,k}^n$ is the observable utility and $\epsilon_{j,k}^n$ is the unobservable utility. The vector of unobservable utility $\epsilon^n=[\epsilon_{1,1}^n,\cdots,\epsilon_{L,1}^n,\epsilon_{1,2}^n,\cdots,\epsilon_{L,2}^n, \epsilon_{1,3}^n,\cdots,\epsilon_{L,3}^n]^{\textrm{T}}$ is assumed to have a generalized extreme value (GEV) distribution with cumulative distribution function (CDF) [@cite11]

$$\label{cdf}
F(\epsilon^n)=\exp\left(-\sum_{k=1}^3\left(\sum_{l=1}^Le^{-\epsilon_{l,k}^n/\sigma_k}\right)^{\sigma_k}\right),$$

where $\sigma_k$ is a measure of the degree of independence in the unobservable utility among the charging stations owned by service provider $k$. For the nested logit model, $\epsilon_{j,k}$ is correlated within each charging level, and uncorrelated across different charging levels.

We can decompose the observable utility $\overline{U}_{j,k}^n$ into two components—the utility of choosing service provider $k$ and the utility of choosing a charging station $j$. In addition, we assume home charging is the “outside good" in this market [@outside1]-[@outside2]. Thus, $\overline{U}_{j,k}^n$ for EV owner $n$ can be expressed as

$$\overline{U}_{j,k}^n=\overline{W}_{k}^n+\overline{V}_{j,k}^n,$$

where $\overline{W}_{k}^n$ is the observable utility of choosing service provider $k$ (choosing nest $k$), and $\overline{V}_{j,k}^n$ is the observable 