\right)\!-\!H_{i}\left( t\right) R_{i}\left(
t\right) \eta ^{+}\right\} \\
&\mbox{s.t.}\,\,(9), r_{ij}\left( t\right)\!\in\!\left[ 0,r_{i,\max }\right], 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], \forall i\!\in\!\mathbf{I}.\notag\end{aligned}$$

Another subproblem can be extracted from **P2**, to decide the connections between PEV requests at different entry points and charging outlets, for $\forall Q_{ij}(t)=0$, shown as below: $$\begin{aligned}
\label{eq_17}
\textbf{P3:}~~&\max_{W_{k,ij}(t)}~~\sum\limits_{k=1}^{K}E_{k}\left( t\right) W_{k,ij}\left( t\right)
+H_{i}\left( t\right) \eta ^{-}\\
&\mbox{s.t.}\,\,(2), (3) \notag\end{aligned}$$which means that the algorithm should meet the largest amount of demand with the empty charging outlet whose battery’s residual energy is the most.

Notice that there is variable coupling due to the constraint (\[eq\_9\]) in **P2**. To decouple this constraint, Lagrange dual method will be employed in the next subsection.

Decoupling of Primal Problems
-----------------------------

Considering Lagrangian relaxation for **P2**, two functions are defined in the case of $Q_{ij}(t)>0$ and $Q_{ij}(t)=0$ respectively: $$\begin{split}
&F_t^1(\overrightarrow{r}\!,\!\overrightarrow{D^d}\!,\!\overrightarrow{R}\!,\!\overrightarrow{\lambda})\!\!=\!\!\sum_{i=1}^I\{\sum_{j=1}^{J_i}\!r_{ij}(t)[Q_{ij}(t)\!\!+\!\!H_i(t)]\!-\!\!
D_i^d(t)(H_i(t)\eta^-\!\!+\!\!Vc(t))\!\!-\!\!H_i(t)R_{i}(t)\eta^+\}\!\\&
+\!\!\sum_{l=1}^L\lambda_l(P_l\!-\!\!\!\!\!\!\sum_{i:X_{li}=1}\!\!\!\!D_i^d(t)\!\!-\!\!\textbf{E}\{N_l(t)\}\!\!-\!\!\frac{\sigma_{N_l}}{\sqrt{2\epsilon}}),
\end{split}$$ $$\begin{split}
&F_t^2(\!\overrightarrow{r}\!,\!\overrightarrow{D^d}\!,\!\overrightarrow{R}\!,\!\overrightarrow{\lambda})\!\!=\!\!\sum_{i=1}^I\!\{\sum_{j=1}^{J_i}\!r_{ij}(t)[\sum_{k=1}^K E_k(t)W_{k,ij}(t)\!\!+\!\!H_i(t)\eta^-]\!\!\\&
\!\!-\!\!(D_i^d(t)\!\!+\!\!D_i^b(t))(H_i(t)\eta^-\!\!+\!\!Vc(t))\!\!-\!\!H_i(t)R_{i}(t)\eta^+\}\!\!+\!\!
\sum_{l=1}^L\lambda_l(P_l\!\!-\!\!\sum_{i:X_{li}=1}D_i^d(t)\!\!-\!\!\textbf{E}\{N_l(t)\}\!\!-\!\!\frac{\sigma_{N_l}}{\sqrt{2\epsilon}}),
\end{split}$$where $\overrightarrow{r}=(r_{11}(t),...,r_{IJ_i}(t))$, $\overrightarrow{D^d}=(D_1^d(t),...,D_I^d(t))$, $\overrightarrow{R}=(R_1(t),...,R_I(t))$, and $\overrightarrow{\lambda}=(\lambda_1,...,\lambda_L)$.

The dual functions $\Gamma_t^1(\overrightarrow{\lambda})$ and $\Gamma_t^2(\overrightarrow{\lambda})$ are defined as the partial maximum of $F_t^1(\overrightarrow{r},\overrightarrow{D^d},\overrightarrow{R},\overrightarrow{\lambda})$ and $F_t^2(\overrightarrow{r},\overrightarrow{D^d},\overrightarrow{R},\overrightarrow{\lambda})$ with respect to $\overrightarrow{r}$, $\overrightarrow{D^d}$, $\overrightarrow{R}$: $$\Gamma_t^1(\overrightarrow{\lambda})=\max_{\overrightarrow{r},\overrightarrow{D^d},\overrightarrow{R}} F_t^1(\overrightarrow{r},\overrightarrow{D^d},\overrightarrow{R},\overrightarrow{\lambda}),~\Gamma_t^2(\overrightarrow{\lambda})=\max_{\overrightarrow{r},\overrightarrow{D^d},\overrightarrow{R}} F_t^2(\overrightarrow{r},\overrightarrow{D^d},\overrightarrow{R},\overrightarrow{\lambda})$$

s.t.  $D_{i}^{d}\left( t\right)\!\!\in\!\!\left[ 0,\left(
\sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right)\!-\!U_{i}\left( t\right) \right)
^{+}\!\right], R_{i}\left( t\right)\!\!\in\!\!\left[0,\min\!\left\{ R_{i}^{\max },\!\left(
U_{i}\left( t\right)\!-\!\sum\limits_{j=1}^{J_{i}}r_{ij}\left( t\right) \right)^{+}\right\}\!\right], (\forall i\!\in\!\textbf{I},j\!\in\!\textbf{J}_i)$

Now scrutinizing the dual 