lemma shows that the stepsize can be used to tradeoff convergence accuracy and speed. Due to space limitation, the detailed proof can be found in [@Boyd].

$H_i(t),c(t),\eta^-$ $D_i^d(t)$ Let all $\lambda_l=\lambda_{max}$. Compute new values of $\overrightarrow{\lambda}$ through (24). $D_i^d(n,t)=0$ $D_i^d(n,t)\!\!=\!\!\left( \sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right) -U_{i}\left( t\right)\right) ^{+}$ $n=n+1$ $D_i^d(t)=D_i^d(n,t)$

The whole algorithm for the problem $\widetilde\textbf{P1}$ is given in Fig. \[fig\_3\]. At the beginning of time slot $t$, every entry point uploads the energy demand arriving at them, and every charging station uploads the states of demand queues and batteries, for the central controller to guide different PEVs to charge at different charging outlets. At the same time each node in the distribution network computes the Lagrangian multipliers as the overload indication through (\[eq\_25\]) and broadcasts them to downstream charging stations. When receiving the Lagrange multipliers, each charging station calculates the charging rate, the utilization of renewable energy, and the energy consumption from the power grid.

Performance Analysis
====================

Stability of battery dynamics and demand queues should be guaranteed in the above algorithms. Also the performance gap between the proposed algorithm and the optimal one in solving **P1** is given.

Stability of Queues
-------------------

\[thm\_1\] All demand queues at every charging outlet 