outlet in the $i$-th charging station. It is reasonable that each charging outlet can only serve one entry point and only one outlet can be connected to each entry point simultaneously. Therefore, $W_{k,ij}(t)$ satisfies $$\label{eq_2}
\begin{small}
\sum\nolimits_{i=1}^{I}\sum\nolimits_{j=1}^{J_i}\!W_{k,ij}(t)\!\leq\!1 ~~(\forall k\!\in\!\textbf{K}),
\end{small}$$ $$\label{eq_3}
\begin{small}
\sum\nolimits_{k=1}^K\!W_{k,ij}(t)\!\leq\!1 ~~(\forall i\!\in\!\textbf{I}, j\!\in\!\textbf{J}_i).
\end{small}$$

The dynamic operation of this system is stated here. At the beginning of every time slot, the central controller collects the energy demand at each entry point and some information of each charging station before broadcasting the guidance information. At the same time, every node computes the overload indication according to current PEV loads and uncontrollable loads imposed on it. Finally, each charging station computes other control variables $D_i^d(t)$ and $R_{i}(t)$ in a distributed way based on the local information and the overload indication.

Battery Dynamics and Demand Queue
---------------------------------

For each charging station, its battery’s energy state satisfies the following iterating equation: $$\label{eq_4}\begin{small}
B_{i}\!\left(t+1\!\right)\!=\!B_{i}(t)\!-\!\eta^{-}\!\left(\!\left(\!\sum\limits_{j=1}^{J_{i}}\!r_{ij}(t)\!-\!U_{i}
(t)\!\right)^{+}\!-\!D_{i}^{d}(t)\!\right)\!+\!\eta^{+}\!R_{i}(t),
\end{small}$$where $0\!<\!\eta^{+}\!\leq\!1$ and $\eta ^{-}\!\geq\!1$ represent the charging and discharging efficiency respectively, and $\eta ^{-}\left(\!\left(\sum\limits_{j=1}^{J_{i}}r_{ij}(t)\!-\!U_{i}(t)\!\right)^{+}\!-\!D_{i}^{d}(t)\!\right)$ denotes the amount of energy that the $i$-th charging station extracts from its battery. The following inequality stands for every battery: $$\label{eq_5}\begin{small}
0\!\leq\!\eta ^{-}\!\left(\!\left(\!\sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right)\!-\!U_{i}\left( t\right)\!\right) ^{+}\!-\!D_{i}^{d}\left( t\right) \right)\!\leq\!B_i(t)\!\leq\!B_{i,max},
\end{small}$$which means charging 