## 2 System Modelling
### Modelling of Appliance Consumption  
The household appliances in this research are classified based on their energy consumption patterns, as follows (See Table 2):  
* Non-shiftable appliances;
* Shiftable interruptible appliances;
* Fixed appliances.  
Non-shiftable appliances are designed to be easily utilized and cannot be stopped once the appliance is turned on. However, before operating, it might be moved from one time slot to another. Shiftable interruptible appliances may be turned off at any moment. They help save power costs when turned off during peak hours [54]. Fixed appliances are base appliances that are neither shiftable nor interruptible and require a continuous power supply. Non-shiftable appliances' power consumption may be divided into two modes,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Households** & **Category** & **Starting Time** & **Ending Time** & **Total Running Duration** & **Power Rating (kW)** \\ \hline Washing machine (WM) & Non-shiftable & 9 p.m. & 10 p.m. & 2 h & 0.38 \\ Vacuum cleaner (VC) & Shiftable & 1 p.m. & 3 p.m. & 0.50 h & 1.25 \\ Air conditioner (AC) & Shiftable & 1 p.m. & 10 p.m. & 9 h & 1.0 \\ Rice cooker (RIC) & Non-shiftable & 10 a.m. & 12 p.m. & 0.75 h & 0.80 \\ Humidifier (HU) & Shiftable & 6 a.m. & 12 a.m. & 4 h & 0.05 \\ Computer phones light (CPL) & Fixed & 6 a.m. & 6 p.m. & Full time-24 h & 1.20 \\ Oven (OV) & Fixed & 2 p.m. & 10 p.m. & 2 h & 2.0 \\ Hair-dryer (HD) & Shiftable & 7 a.m. & 8.30 a.m. & 30 min & 1.80 \\ Dishwasher (DW) & Non-shiftable & 8 a.m. & 11 p.m. & 3 h & 0.6 \\ Electrical water(EW) & Shiftable & 6 a.m. & 8 p.m. & 0.50 h & 1.5 \\ Robot (RO) & Shiftable & 11 a.m. & 2 p.m. & 2 h & 2.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Home appliances’ parameters.  
signifying a high and low consumption. Consequently, the price of the non-shiftable appliance power consumption at time \(t\) (\(C_{NSH}(t)\)) is calculated as follows (Equation (9)):  
\[C_{NSH}(t)=CT(t)\times Pr_{comp}(t)+\beta_{NSH}\times(Pr_{comp}^{mode}(t)-Pr_{NSH,max})^{2}+\hat{C}(t) \tag{9}\]  
The user can customize the appliances based on their comfort by specifying issues, such as the preferred duration, temperature, and interval. For non-shiftable appliances, the operating time can be delayed to the off-peak period; in this case, the dissatisfaction price \(\beta\) was indicated by the user. Thus, \(\beta_{NSH}\) for non-shiftable appliances is the updated operating time of non-shiftable appliances because said appliances cannot be interrupted once they start working. \(CT(t)\) is the electricity tariff at time \(t\), and \(Pr_{comp}^{mode(t)}\) is the power consumption of non-shiftable appliances during the mode at time \(t\), \(Pr_{NSH,max}\) is the power rating of the non-shiftable appliance. Moreover, \(\hat{C}\) indicates the predicted extra consumption cost at a specific time. Various requirements must be met before scheduling may begin. The following is how the cost of shiftable interruptible appliances is calculated (Equation (10)):  
\[C_{SH}(t)=CT(t)\times Pr_{comp}(t)+\beta_{SH}\times(S_{SH,updated}-S_{SH, scheduled})+\hat{C}(t) \tag{10}\]  
\(S_{SH,scheduled}\) is the scheduled start time of the appliance and \(S_{SH,updated}\) is the updated operating time. Moreover, \(Pr_{comp}(t)\) is the power consumption of a shiftable appliance at time \(t\). Additionally, the dissatisfaction price for shiftable appliances \(\beta_{SH}\) can be utilized to specify how quickly an appliance should complete its job; for example, minimizing the duration of electrical water or rescheduling the starting and ending time of a vacuum cleaner to clean the house before or after the peak hours. Finally, as a limitation, the appliance must complete its operation within the following twelve hours of being scheduled.