## 4 Simulation Results and Discussion
### Statistical Analysis  
In order to verify the effectiveness of the proposed method, we examined the optimal scheduling of the controllable loads obtained by the optimization in case 1\(\sim\)case 4 how robust it is for uncertainties. Monte Carlo simulation is conducted for 1000 scenarios in order to obtain a statistically certain cost. Figure 12 indicate that the power flow when the scheduling of controllable loads obtained by the optimization in case 1\(\sim\)case 4 was operated for uncertain PV output. In case 1, it is obviously observed that power flow deviates from the bandwidth in the daytime affected by the PV output as shown in Figure 12. The deviation of power flow from the bandwidth in case 2 and case 3 is reduced in comparison with case 1 by considering uncertainties or re-forecast and re-plan as can be seen from Figure 12,c. The deviation of power flow in case 4 is further improved in comparison with case 2 and case 3 by incorporating the scenario-based approach, re-forecast and re-plan into the optimization process. Figure 11: Simulation results with considering re-forecast, re-plan and uncertainties on a fair day (Case 4). Figure 13 and Table 4 show distribution and histogram of the operational cost and the expected cost obtained by Monte Carlo simulation for 1000 scenarios which are derived by adding possible forecast error based on the normal distribution to the forecasted PV output. The verification results indicate that the expected value of operational cost for case 1 in which both re-forecast, re-plan and uncertainties are not considered, is the most expensive as depicted in Figure 13a (482 Yen/day). Furthermore, in case 2 which uncertainties are only considered, the cost is 456 [Yen/day] as depicted in Figure 13b. In case 3 which re-forecast and re-plan are only considered, the cost is 426 [Yen/day] as depicted in Figure 13c. The cheapest cost is depicted in case 4 which re-forecast, re-plan and uncertainties are considered as shown in Figure 13d (412 Yen/day). More discussion for the simulation results between case 1 and case 4, is as follows. if the real PV output would come out exactly as the forecasted one, the operational cost is minimized by operating the optimal scheduling of controllable load obtained by the optimization in even case 1. It means that the power flow as shown in Figure 8c exists in the low position of the bandwidth in Region A so that power consumption is minimized based on set the objective function. However, real PV output often deviates from the forecasted one. Thus, the controllable loads should be scheduled while considering the uncertainties in advance. That is the power flow shown in Figure 11c exists in the center within the bandwidth in the day time in preparation for the capability in which the forecast error would be relatively higher throughout the day. The frequency of the state of charge on the battery for 1000 scenarios in case 1 and case 4 are depicted in Figure 14a,b, respectively. The state of charge on the battery exists at almost 80% from 10:00 to 24:00 in case 1 which re-forecast, re-plan and uncertainties are not considered since the battery acts at the maximum of its capacity without consideration of the forecast error of PV output as shown in Figure 14a. On the other hand, in case 4 which re-forecast, re-plan and uncertainties are considered, the state of charge on that battery exists at almost 40\(\sim\)60% from 10:00 to 24:00 in preparation for the forecast error of PV output as shown in Figure 14b. These results imply that the capacity of the battery may be even cut down by introducing the proposed method. Figure 12: Statistical analysis result in regards to power flow in the smart home in each case.