of marginal losses [@PJM]. Let $p_k$ represent the retail charging price of provider $k\;(k=1,2,3)$, and $p_{-k}$ be the retail charging prices of the other two service providers except $k$. Let $c_{j,k}$ be the LMP of the $j$th charging station candidate of service provider $k$, and $\psi_{j,k}$ be the predicted charging demand at the $j$th charging station candidate of service provider $k$. The vector $S_k=[s_{1,k},s_{2,k},\cdots, s_{L,k}]^{\textrm{T}}$ represents the placement policy of service provider $k$, where $s_{j,k}\in\{0,1\}$ is an indicator with $s_{j,k}=1$ implying that service provider $k$ will place the $j$th charging station. Let $S_{-k}$ represent the placement policies of the other two service providers, $\theta_{j,k}$ be the placement cost of the $j$th charging station. The total profit of service provider $k$ is given by

$$\Pi_k=p_k\Psi_k^{\textrm{T}}S_k-\textrm{diag}[C_k]\Psi_k^{\textrm{T}}S_k
-\Theta_k^{\textrm{T}}S_k,$$

and the total revenue of service provider $k$ is, $$\label{revenue}
R_k=p_k\Psi_k^{\textrm{T}}S_k-\textrm{diag}[C_k]\Psi_k^{\textrm{T}}S_k,$$ where $\Psi_k=[\psi_{1,k},\psi_{2,k},\psi_{3,k},\cdots,\psi_{L,k}]^{\textrm{T}}$, $C_k=[c_{1,k},c_{2,k},\cdots,c_{L,k}]^{\textrm{T}}$ and $\Theta_k=[\theta_{1,k},\theta_{2,k},\cdots,\theta_{L,k}]^{\textrm{T}}$. The notation $\textrm{diag}[.]$ is an operator to create a diagonal matrix using the underlying vector, and $[.]^{\textrm{T}}$ is the transpose operation. In Equation (\[revenue\]), the total revenue from the sales is $p_k\Psi_k^{\textrm{T}}S_k$, the cost of purchasing electricity is $\textrm{diag}[C_k]\Psi_k^{\textrm{T}}S_k$, and the placement cost is $\Theta_k^{\textrm{T}}S_k$.

The Disturbance on Power Grid Due to EV Charging
------------------------------------------------

It is conceivable that 