Algorithm 1 AS-REMS algorithm.

Input:${\mathcal{L}}_{MC}$, ${\mathcal{L}}_{UC}$, d, $T_{s_{interval}}^{a_i}$,  ${\mathit{U}}_D^a$, ${\tilde{P}}^a$, $P_{lim}$, $Tariff$. Step 1. Determine the set of possible starting times $T{S}_D^{a_i}$ of each $a_i\in {\mathcal{L}}_{MC}$ such that:       If $T_{s_{interval}}^{a_i}=empty$             Find $T{S}_D^{a_i}$ via Equation (6)       Else             $T{S}_D^{a_i}=T_{s_{interval}}^{a_i}$             ${P{r}_L}_d^{a_i}\left(t_s^{a_i}\right)=0$       End If Step 2. Determine the set of possible combination of possible starting times MC-controllable appliances $T{S}_D^{{\mathcal{L}}_{MC}}$ as in Equation (16) Step 3. Determine corresponding parameters values for each $T_{s_j}^{{\mathcal{L}}_{MC}}\in T{S}_D^{{\mathcal{L}}_{MC}}$            ${P{r}_L}_d^{{\mathcal{L}}_{MC}}\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)$ via Equation (9)            $C_d^{{\mathcal{L}}_{MC}}\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)$ via Equation (10)            $I_d\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)$ via Equation (15) Step 4. Normalize the corresponding parameter values for each $T_{s_j}^{{\mathcal{L}}_{MC}}\in T{S}_D^{{\mathcal{L}}_{MC}}$ as           ${P{r}_L}_d^{{\mathcal{L}}_{MC}}\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)\to {P{r}_L}_d^{{\mathcal{L}}_{MC}}\left(T_{s_i}^{{\mathcal{L}}_{MC}}\right)$           $C_d^{{\mathcal{L}}_{MC}}\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)\to C_{dscaled}^{{\mathcal{L}}_{MC}}\left(T_{s_i}^{{\mathcal{L}}_{MC}}\right)$           $I_d\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)\to I_{dscaled}\left(T_{s_j}^{{\mathcal{L}}_{MC}}\right)$ Step 5. Call BFCP Algorithm with ${P{r}_L}_d^{{\mathcal{L}}_{MC}}\left(T_s^{{\mathcal{L}}_{MC}}\right)$, $I_{dscaled}\left(T_s^{{\mathcal{L}}_{MC}}\right)$, $C_{dscaled}^{{\mathcal{L}}_{MC}}\left(T_s^{{\mathcal{L}}_{MC}}\right)$ Output: $T_{s_{optimal}}^{{\mathcal{L}}_{MC}}$