minimize the time-average cost for serving PEVs by renewable energy with non-ideal storage and power networks that supply some uncontrollable loads at the same time.

2.  An online algorithm consisting of guidance of PEV charging and energy management at stations is proposed based on Lyapunov optimization and Lagrange dual decomposition techniques. The algorithm can result in provable performance with non-ideal storage and ensure the safety of the distribution network.

3.  Simulations with real data demonstrate that the asymptotic optimality and comparative waiting time can be achieved.

The rest of this paper is organized as follows: system models are introduced in Section ; problem formulation is given in Section ; an online algorithm is developed in Section ; mathematical proofs of algorithm performance can be found in Section ; simulations are demonstrated in Section ; at last, conclusion of this paper is given in Section . The key mathematical notations used in the paper are listed in Table 1.

  Notation                                   Physical interpretation
  ------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------
  $l,\textbf{L}$                             Index and set of transformers and lines (nodes), $\textbf{L}\!=\!\{1,2,...,L\}$
  $i,\textbf{I}$                             Index and set of charging stations, $\textbf{I}\!=\! \{1,2,3,...,N\}$
  $j,\textbf{J}_i$                           Index and set of charging outlets belonging to the $i$-th charging station, $\textbf{J}_i\!=\!\{1,2,...,J_i\}$
  $k,\textbf{K}$                             Index and set 