is then charged into the battery[^5]. Let $R_{i}\!\left(t\!\right)$ be the current charging rate to the battery and thus $R_{i}\!\left( t\!\right)\!\in\!\left[0,\min\!\left\{R_{i}^{\max},\!\left(U_{i}\!\left(t\!\right)\!-\!\sum\limits_{j=1}^{J_{i}}\!r_{ij}\!\left( t\!\right)\!\right)^{+}\!\right\}\!\right]$, where $\!\left( x\!\right) ^{+}\!\hat{=}\!\max\!\left\{ 0,x\!\right\}$. Otherwise an amount of $\left(\sum\limits_{j=1}^{J_{i}}r_{ij}\left(t\right)-U_{i}\left(t\right)\right)^{+}$ load will be served either by the battery or by the distribution networks depending on the energy availability in battery and usage cost of power grid. Let $D_{i}^{d}\left(t\right)\!\in\!\left[0,D^{d,\max}\!\right] $ denote the amount of energy withdrawn from the distribution network and $\!\left(\sum\limits_{j=1}^{J_{i}}r_{ij}\!\left(t\!\right)\!-\!U_{i}\!\left(t\!\right)\!\right)^{+}\!\!-\!\!D_{i}^{d}\!\left(t\!\right)$ the amount of energy supplied by the battery. To ensure that the distribution network can sustain the charging load independently, we assume $D^{d,\max}\!>\!r_{i,\max }$. Thus the total loads imposed on the $l$-th node is $\sum_{i:X_{li}=1}\!D_i^d(t)\!+\!N_l(t)$.

If no overload is permitted, $\sum\nolimits_{i:X_{li}=1}\!D_i^d(t)\!+\!N_l(t)\!\leq\!P_l$ should stand for every time slot $t$. Since the uncontrollable loads are time varying and each node always allows slight overload for the safety reason [@Ardakanian], the above inequality can be slacked to the “chance constraint”: $$\label{eq_1}
\begin{small}
Pr[\sum\nolimits_{i:X_{li}=1}\!D_i^d(t)\!+\!N_l(t)\!>\!P_l]\!\leq\!\epsilon,
\end{small}$$where $\epsilon$ is chosen to ensure that the node is safe even with a little overload.

At the end of this subsection, let $c(t)$ denote the power price in time slot $t$ in the system, and it is also bounded by $[C_{min},C_{max}]$.

PEVs’ Request and Guidance
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In 