policy and energy allocation laws. Directing policy, the solution of **P3**, decides which outlet that the PEV registered at the $k$-th entry point should be charged in. Distributed energy allocation laws, implemented by each charging station, are solutions of **P4**$\sim$**P6**.

### Directing Policy

After every PEV uploads their energy demands via entry points, and every charging station uploads their battery state, the central controller implements the following Algorithm \[alg\_1\].

$E_k(t),H_i(t)\eta^-$ The decision variable $W_{k,ij}(t)$ Define two sets: $\textbf{Y}\!\!=\!\!\{k|E_k(t)\!>\!0\}$ and $\textbf{Z}\!\!=\!\!\{(i,j)|Q_{ij}\!=\!0\}$ to include the nonempty entry points and empty outlets in time slot $t$ respectively. Let all $W_{k,ij}(t)=0$. Choose the entry point having the largest demand $E_k(t)$ from **Y**. Search for an empty outlet $(i,j)$ among **Z** to get the maximum of $E_k(t)+H_i(t)\eta^-$. Let $W_{k,ij}(t)=1$, remove $k$ from **Y** and $(i,j)$ from **Z**.

### Charging Rate of PEV

For every charging outlet, the charging rate is determined by solving problems **P4(a)** and **P4(b)**, case by case.

$a)$ For outlet that is non-empty ($Q_{ij}(t)>0$) at the beginning of $t$: The charging rate depends on the demand queue state and battery state.

$b)$ For outlet that is empty ($Q_{ij}(t)=0$) at the beginning of $t$: The charging rate depends on the coming demand in 