## Simulation Studies
### Results and Discussion
#### 5.2.3 With HEMS  
In this scenario, there is HEMS employed in the house to schedule the appliances. All the appliances are scheduled using MPC strategy in this case. Figure 5c clearly depicts the benefits of using HEMS and presents the total power consumed by all the household appliances. In Figure 5c, it can be seen that for initial time steps, total power consumption goes beyond the capacity limit of 4 kW, which happens due to the high power consumption of water heater since initial water temperature in the tank is low. At later time steps, it can be seen that the maximum total power consumed by all the appliances stays around the capacity limit. The constraints on the capacity limit are allowed to violate to ensure the feasibility of the optimization problem (soft constraints).  
Figure 8 shows the operation schedule of the appliances in this case. It can be seen that, the non-thermal appliances are interrupted occasionally to satisfy the capacity constraints and minimize the electricity cost. For example, at the noon of the fourth day of the week, washing-machine and the dishwasher is not running rather they are shifted to run later in the day.  
The total electricity cost as per the time varying prices used in this study, is detailed in the Table 4. It should be noted that this includes the cost of running both thermal and non-thermal appliances for entire simulation period. In comparison to the case without HEMS and LQR controller for the thermal appliances, a reduction of 70% is achieved in total electricity cost by using the HEMS. Similarly, in comparison to the case without HEMS and MPC controller for the thermal appliances, the HEMS achieves a reduction of 45%.  
To see the effectiveness of HEMS in reducing the peak-load, the total violation of the capacity constraint over the simulation period and the maximum peak-load consumed by the appliances for all the cases was calculated, and is presented in Table 5. The total violation for the case when appliances are scheduled with HEMS is about 10 times lower than the case of no HEMS with MPC control for thermal appliances. Similarly, by using HEMS, the peak power consumption is reduced by 57% and 35% in comparison with the case of no HEMS with LQR control and MPC control for thermal appliances, respectively. The PAR for all the cases is detailed in the Table 6. It is observed that by using the HEMS, PAR is reduced by 35% and 24% in comparison to other two cases.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Case** & **Electricity Cost (€)** \\ \hline Without HEMS and LQR control & 15.14 \\ \hline Without HEMS and MPC control & 8.2005 \\ \hline With HEMS & 4.4934 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The total electricity cost for the entire simulation duration.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Overconsumption (kW)** & **Peak-Power Consumption (kW)** \\ \hline Without HEMS and LQR control & 172 & 12.98 \\ \hline Without HEMS and MPC control & 59.8947 & 7.3 \\ \hline With HEMS & 6.4062 & 4.7285 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Total violation of capacity constraint during the simulation period.  
Figure 6: The operation schedule of the appliances without HEMS and LQR control for the thermal appliances.  
Figure 7: The operation schedule of the appliances without HEMS and. MPC control for the thermal appliances.  
Figure 8: Operation schedule of the appliances with HEMS.