Voltage angle between bus $i$, $k$   Radian
  $\alpha$                                    Coef. of $t_k$                       -
  $\beta$                                     Coef. of $p_k/i_n$                   -
  $\mu_k$                                     Coef. of $d_{j,k}$                   -
  $\eta_k$                                    Coef. of $z_{j,k}$                   -
  $\gamma_k$                                  Coef. of $r_{j,k}$                   -
  $\lambda_k$                                 Coef. of $g_{j,k}$                   -
  $\delta_k$                                  Coef. of $m_{j,k}$                   -

  : Parameter Description

In this paper, we postulate the problem of EV charging with an oligopolistic market structure that has multiple charging service providers (oligopolists). The service providers aim to maximize their expected utility while satisfying the QoS constraints by selecting optimal station placements.

In particular, we consider the case of three service providers that offer three EV charging services [@specification], namely, Level 1, Level 2, and Level 3 (see Table \[ta1\] for details). Level 1 and Level 2 are AC charging. Level 3 charging is DC Fast charging. Let $\mathcal{O}=\{1,2,3\}$ denote the set of charging service providers. Moreover, we assume that service provider 1 offers Level 1 charging, service provider 2 offers Level 2 charging, and service provider 3 offers Level 3 charging. The three charging levels represent three charging services, which have different charging voltages and currents, charging times, and charging experiences. In economic terms, they are imperfect substitutes to each other. In our model, we 