## 6 Case Study  
In this paper, a residential consumer load was comprised of multiple appliances with an ESS, and PV generation was considered for the implementation of the proposed methodology. We considered 12 household appliances in this study. Seven base loads were considered, which could complete their operating time in single or multiple cycles. These cycles may be consecutive or split within the consumer's preferred time span. For example, an electrical car can be charged in any consecutive 3 h from 18:00 to 06:00. In addition, it can also be charged in two separate cycles summing to 3 h of charging, e.g., 20:00 to 22:00 and 05:00 to 06:00. Another option may be to charge the car in three disconnected hours, e.g., 19:00 to 20:00, 23:00 to 24:00, and 04:00 to 05:00 can be possible hours that are not continuous but sum to an operating duration of 3 h. Such different possible combinations are tested by the proposed algorithm and the optimal operating schedule of the appliance was generated. An additional operating constraint for non-deferrable appliances, such as interior lighting, was that their operation could not be shifted, while deferrable loads could be shifted in terms of their allocated time. Three deferrable and two non-deferrable appliances were considered in this study. Different consumer load parameters with respect to their operating preferences are given in Table 4.  
The solar irradiance data used in the study was collected from [PERSON] [48] and characteristics of this data was modeled by using the beta-probability-distribution-based model to generate daily solar irradiance patterns. The daily solar irradiance patterns of original dataset and scenarios generated after the beta distribution modeling are presented in Figure 5. In this study, the generated solar irradiance data was used for calculating the PV generation by randomly selecting a pattern for each run of the simulation. The output power of the PV generation was calculated using Equations (10)-(12). Limits and other capacity restrictions of the different system components, such as the PV, grid, and battery, are given in Table 5. Two different pricing schemes--RTP and CPP--were considered in this study. Both schemes had the same prices of energy except from 08:00 to 11:00, where the CPP scheme charged more than RTP. The peak price in both schemes was observed between 10:00 to 11:00, as shown in Figure 6. Furthermore, it was assumed that the tariff for the H2G energy export was half of the pricing charged in the energy import from the G2H [27]. Three different scenarios considering PV generation, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Appliance** \\ **Classification** \\ \end{tabular} & **Appliance Name** & **Power Rating (kW)** & **Starting Time, \(\lambda\) (h)** & \begin{tabular}{c} **Finishing Time,** \\ \(\beta\) **(h)** \\ \end{tabular} & \begin{tabular}{c} **Length of** \\ **Operation Time (h)** \\ \end{tabular} \\ \hline \multirow{4}{*}{Base load appliances} & Microwave & 1.7 & 6 & 10 & 1 \\  & Cooker hub & 3 & 6 & 10 & 1 \\  & Vacuum cleaner & 1.2 & 9 & 17 & 1 \\  & Cooker oven & 5 & 18 & 20 & 1 \\  & Laptop & 0.1 & 18 & 24 & 2 \\  & Deskip & 0.3 & 18 & 24 & 3 \\  & Electrical car & 3.5 & 18 & 8 & 3 \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Deferrable \\ appliances \\ \end{tabular} } & Washing machine & 1.5 & 9 & 12 & 2 \\  & Dish washer & 1.5 & 9 & 17 & 2 \\  & Spin dryer & 2.5 & 13 & 18 & 1 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Non-deferrable \\ appliances \\ \end{tabular} } & Refrigiator & 0.3 & 1 & 24 & 24 \\  & Interior lighting & 0.84 & 18 & 24 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Consumer load parameters [35]. \begin{table}
\begin{tabular}{c c} \hline \hline \begin{tabular}{c} **Parameters** \\ \end{tabular}  & **Value** \\ \hline \