## 2 System Modelling
### Modelling of TOU and RTP Pricing  
Different power tariffs (e.g., ToU, CPP, and RTP.) have been introduced under the DR program to minimize electricity use during on-peak hours. The DR method incentivises load transfer from on-peak to off-peak hours, where costs are lower during off-peak hours. Prices increase when the demand for power rises during on-peak hours, and the extra demand is satisfied by expensive utility tariffs. Several factors, including the generation cost, determine the electricity price per unit. The key factors influencing electricity prices are distribution, transmission, and generation costs. In addition, prices are greatly impacted by the utility and consumer demand. The utility is a service provider that regulates power rates on a seasonal (ToU), weekly, daily, or hourly basis (RTP). RTP and ToU are popular pricing tariffs across the world. However, the RTP fluctuates in response to the demand and wholesale price. This variation may cause the customer some discomfort. The ToU tariff is divided into three blocks: on-peak, shoulder-peak, and off-peak hours, with pricing remaining constant throughout the season during these periods. Utilities determine pricing rates for each block of ToU after monitoring the consumers' past energy consumption behaviour. The adjusted ToU tariff is based on extra generation costs and user load demand during particular hours. Changes are implemented per hour unit for on-peak and shoulder-peak hours. The formal expression of the adjusted ToU price signal (\(C_{ToU}\)) is given as (Equation (23)) [58]:  
\[C_{ToU}(t)= \begin{cases}ToU\;\;if\;\;t_{off}\\ C^{+}\;\;else\end{cases}  \tag{23}\]  
A \(t_{off}\) indicates the off-peak hours and \(C^{+}\) denotes the extra cost. Utilities set the pricing tariffs based on demand, with electricity costs set higher during on-peak hours and cheaperduring off-peak hours. Off-peak hours are when households and companies consume the least amount of power. The RTP price is an hour-ahead, day-ahead, and week-ahead price. The day-ahead and week-ahead pricing approaches are based on prior power usage, while the RTP hourly price is the daily prediction of demand and cost. It is simple to program electrical appliances for day-ahead and week-ahead pricing systems. However, hourly price scheduling is rather challenging because the price fluctuates between on-peak and off-peak hours. Load forecasting can be employed to address this issue [59]. However, shiftable appliances are adjusted from on-peak to off-peak prices. Vacuum cleaners, humidifiers, and hairdryers are regarded as shiftable appliances. The total hourly power consumed by all shiftable appliances (SH) in a day is determined as follows:  
\[ \begin{split} C_{RTP,SH}(t)=\Sigma_{SH}^{t}(P_{sh}\times m_{sh}(t) )\\ C_{RTP,SH}(t)=\Sigma_{t=1}^{T}\Sigma_{SH}^{t}(P_{sh}\times m_{sh}(t) )\end{split}  \tag{24}\]  
\(T\) = 24 and \(P_{sh}\) signifies the power rating for each shiftable appliance. An \(m_{sh}(t)\) denotes the hourly status of each shiftable appliance (\(ON\) or \(OFF\)). Each appliance's state for the next period is indicated as follows:  
\[\hat{m}_{sh}(t)= \begin{cases}1\;\;if\;\;\hat{m}_{sh}\;\;is\;\;ON\\ 0\;\;if\;\;\hat{m}_{sh}\;\;is\;\;OFF\end{cases}  \tag{25}\]  
The second appliance type is non-shiftable, meaning the appliances are not interrupted or shifted. These types of appliances may not help with load scheduling or energy management. Non-shiftable appliances must follow a preset execution pattern. This study considers dishwashers and rice cookers as non-shiftable appliances. Equation (26) calculates the hourly total power consumption (kWh) in a day for non-shiftable appliances (\(C_{RTP,NSH}(t)\)). \[ \begin{split} C_{RTP,NSH}(t)=\Sigma_{NSH}^{t}(P_{nsh}\times m_{ nsh}(t))\\ C_{RTP,NSH}(t)=\Sigma_{t=1}^{T}\Sigma_{NSH}^{t}(P_{nsh}\times m_{ nsh}(t))\end{split}  \tag{26}\]  
where \(P_{nsh}\) is the rating of power of non-shiftable appliances and \(m_{nsh}\) indicates the status of each non-shiftable appliance (\(ON\) or \(OFF\)). \[\hat{m}_{nsh}(t)=