a	appliance
t	time
$\tilde{t}$	internal time of appliances (${\mathsf{\Delta }}_{\tilde{t}}={\mathsf{\Delta }}_t$)
$ts$	time slot
${\mathsf{\Delta }}_t$	length of each time t
C	Controllable
D	set of similar days
$\mathcal{L}$	set of appliances
T	number of uniform time slots
$\mathcal{T}$	execution period
$NO$	Normally Operated
$Sc$	Scheduable
$Sh$	Shiftable
$CBE$	Comfort Based Elastic
$EBE$	Energy Based Elastic
$FTB$	Fixed and Task Based
$MCC$	Main Controller Controllable
$UnC$	Uncontrollable
$USh$	Unshiftable
$USc$	Unscheduable
$USC$	User Controllable
$dUSC$	Delayable User Controllable
$ndUSC$	Non-delayable User Controllable
${\mathit{U}}_D^a$	utilization matrix of an appliance $a\in \mathcal{L}$ for the set of the similar days D
${\tilde{P}}^a$	power profile vector of an appliance $a\in \mathcal{L}$ (W)
$P_d^a\left(t\right)$	power consumption of an appliance $a\in \mathcal{L}$ at a t of a day $d\in D$ (W)
$T_m^a$	measurement duration of an appliance $a\in \mathcal{L}$
$t_s^a$	starting time of an appliance $a\in \mathcal{L}$
${\mathcal{L}}_{\mathcal{UC}}$	set of user-controllable appliances
${\mathcal{L}}_{\mathcal{MC}}$	set of MC-controllable appliances
$P_{\mathcal{UC}}^a\left(t\right)$	power consumption of a user-controllable appliance $a\in {\mathcal{L}}_{\mathcal{UC}}$ at a t (W)
$T_e^a$	one execution period of an appliance $a\in {\mathcal{L}}_{\mathcal{UC}}$
$P_{\mathcal{MC}}^a\left(t\right)$	power consumption of a MC-controllable appliance $a\in {\mathcal{L}}_{\mathcal{MC}}$ at a t (W)
$T_o^a$	operation period of an appliance $a\in {\mathcal{L}}_{\mathcal{MC}}$
$T{S}_D^a$	set of starting times of an appliance a on the similar days D
$N{C}_D^a\left(t_s^a\right)$	number of $t_s^a$ to be chosen as starting time of an appliance $a\in {\mathcal{L}}_{\mathcal{MC}}$
$P{r}_D^a\left(t_s^a\right)$	probability of $t_s^a$ to be chosen as starting time of an appliance $a\in {\mathcal{L}}_{\mathcal{MC}}$
${P{r}_L}_d^a\left(t_s^a\right)$	priority level of $t_s^a$ to be chosen as starting time of an appliance $a\in {\mathcal{L}}_{\mathcal{MC}}$
$T_{s_{interval}}^{a_j}$	specific starting time interval for each of MC-controllable appliance
	$a_j\in {\mathcal{L}}_{\mathcal{MC}}$
$P_{d_{exp}}^a\left(t\right)$	expected power consumption of an appliance $a\in {\mathcal{L}}_{\mathcal{UC}}$ at a t of a day
	$d\in D$ (W)
$P_{d_{exp}}^{{\mathcal{L}}_{UC}}\left(t\right)$	expected power consumption of all USC appliances at a t of a day
	$d\in D$ (W)
$P_d^{{\mathcal{L}}_{MC}}\left(t\right)$	power consumption of a MC-controllable appliance $a\in {\mathcal{L}}_{\mathcal{MC}}$ at a t of a
	day $d\in D$ (W)
$P_d^{{\mathcal{L}}_{MC}}\left(t\right)$	power consumption of all MC-controllable appliances at a t of a day
	$d\in D$ (W)
$P_d^{\mathcal{L}}\left(t\right)$	total power consumption of all appliances at a t of a day $d\in D$ (W)
$P_{lim}$	power limit (W)
$T_s^{{\mathcal{L}}_{MC}}$	a starting time combinations of MC-controllable appliances, i.e.,
	$t_s^{a_1},t_s^{a_2},\dots ,t_s^{a_{|L_{MC}|}}$
$T_{s_i}^{{\mathcal{L}}_{MC}}$	ith possible combination of starting times of MC-controllable appliances
$T_{s_{optimal}}^{{\mathcal{L}}_{MC}}$	optimal combination of starting times
$C_d^{{\mathcal{L}}_{MC}}\left(T_s^{{\mathcal{L}}_{MC}}\right)$	total electricity cost of MC-controllable appliances for $T_s^{{\mathcal{L}}_{MC}}$ ($)
$I_d\left(T_s^{{\mathcal{L}}_{MC}}\right)$	power limit indicator for $T_{\mathcal{MC}}^{\mathcal{L}}s$
${P{r}_L}_d^{{\mathcal{L}}_{MC}}\left(T_s^{{\mathcal{L}}_{MC}}\right)$	total priority level of MC-controllable appliances for $T_{\mathcal{MC}}^{\mathcal{L}}s$
$C_{dscaled}^{{\mathcal{L}}_{MC}}\left(T_s^{{\mathcal{L}}_{MC}}\right)$	scaled total electricity cost for $T_{\mathcal{MC}}^{\mathcal{L}}s$
$I_{dscaled}\left(T_s^{{\mathcal{L}}_{MC}}\right)$	scaled power indicator value for $T_{\mathcal{MC}}^{\mathcal{L}}s$
${\widehat{P{r}_L}}_d^{{\mathcal{L}}_{MC}}$	ideal priority level
${\widehat{C}}_d^{{\mathcal{L}}_{MC}}$	ideal total electricity cost
${\widehat{I}}_d$	ideal power limit indicator
$w_C$	weight of the total electricity cost
$w_I$	weight of the power limit indicator
$w_{Pr}$	weight of the total priority level
$d_i$	output of distance function for ith possible combination of starting times of MC-controllable appliances