station’s $j$-th outlet in time slot $t$
  $Q_{ij}(t)$                                Demand queue of the $j$-th outlet of the $i$-th charging station at the beginning of time slot $t$
  $\eta^+,\eta^-$                            Charging efficiency, discharging efficiency
  $H_i(t)$                                   Shift version of $B_i(t)$
  $\overrightarrow{\theta(t)}$               State of system queues at the beginning of time slot $t$
  $\mathcal L(\overrightarrow{\theta(t)})$   Lyapunov function
  $\triangle(\overrightarrow{\theta(t)})$    Lyapunov drift

  : Mathematical notations[]{data-label="tab:1"}

System Models
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This section introduces the system model of a distribution network, which includes several charging stations, uncontrollable non-PEV loads, and random request of PEVs.

Distribution Network and Charging Stations
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In this paper, a distribution network (Fig. 1), composed of multiple transformers and lines, is considered. The set of transformers and lines is $\textbf{L}\!=\!\{1,2,...,L\}$. Every transformer and transmission line has its power capacity, determined by the type of conductor, the highest temperature permitted, and other physical and environmental factors, and the capacity of the $l$-th ($l\!\in\!\textbf{L}$) transformer/line is denoted by $P_l$. The considered system is modeled as a slotted one and it is referred to the $t$-th interval $\![tT, (t+1)T\!)$, where $T$ is the length of a slot. In the following $T$ is normalized to unit for easier presentation.

![Distribution network.[]{data-label="fig_1"}](DisNet.eps){width="90mm"}

Let $i\!\in\!\textbf{I}\!=\!\{1,2,\cdots,N\}$ denote the $i$-th charging station and $j\!\in\!\textbf{J}_i\!=\!\{1,2,...,J_i\}$ 