that any feasible solution of **P1** is also a feasible solution of $\widetilde \textbf{P1}$ since $\widetilde \textbf{P1}$ is less constrained than **P1**. Therefore, $P_{re}^*\leq P^*$. We will solve $\widetilde \textbf{P1}$ first. Later we will prove the performance gap between the proposed algorithm and the optimal solution to **P1**. More interestingly, the proposed algorithm for $\widetilde\textbf{P1}$ also satisfies the constraints of (\[eq\_5\]).

Problem Decomposition and Solution
==================================

One challenge in solving $\widetilde \textbf{P1}$ is due to the fact that the supply, demand and price are all stochastic processes. If all statistical information is known, it can be solved offline. However, the solution to this problem should be achieved by real-time implementations. Firstly, some transformation of constraint (1) shall be done to write it into a deterministic form.

Transformation of Chance Constraint
-----------------------------------

Assume the distribution of uncontrollable load $N_l(t)$ is symmetric about its mean. Then there is $$\label{eq_8}\begin{split}
Pr[N_l(t)\!\!\geq\!\! P_l\!\!-\!\!\!\!\!\!\displaystyle \sum_{i:X_{li}=1}\!\!\!\!D_i^d(t)]\!\!
&=\!\!\frac{1}{2}Pr[|N_l(t)\!\!-\!\!\textbf{E}\{\!N_l(t)\!\}|\!\!\geq\!\! P_l\!\!-\!\!\!\!\!\sum_{i:X_{li}=1}\!\!\!\!D_i^d(t)\!\!-\!\!\textbf{E}\{N_l(t)\}]\\
&\leq \!\frac{1}{2} \frac{\sigma_{N_l}^2}{[P_l-\!\!\displaystyle\sum_{i:X_{li}=1}D_i^d(t)\!-\!\textbf{E}\{N_l(t)\}]^2}\leq \epsilon,
\end{split}$$where $\sigma_{N_l}$ is the standard deviation of $N_l(t)$. As in [@Tarasak], Chebyshev’s inequality is given below: $$\begin{small}
Pr[|x|\geq a]\leq{\sigma_x^2}/{a^2}
\end{small}$$ for a zero-mean random variable $x$, and $\sigma_x$ is the standard deviation of $x$. Therefore, the constraint in (\[eq\_8\]) is transferred to $$\label{eq_9}
\begin{small}
