## 6 Simulation
### Simulation Results and Discussions  
The simulation results on a sunny day of case 1 and 3 are shown in Figures 4 and 5. Figures 4a and 5a show the electric power [kW] on the time axis. In these Figures, \(P_{DC}\) [kW] indicates the charge/discharge power of battery determined by TS. If \(P_{DC}\) is positive, the battery will charge electric power. \(P_{PV}\) [kW] is the power output of PV, \(P_{PL}\) [kWh] is the power consumption of load, \(P_{HP}\) [kWh] is the power consumption of the heat pump system, \(P_{BUY}\) [kWh] is the purchasing electric power consumption, \(P_{SALE}\) [kWh] is selling electric power. Figures 4b and 5b show the running energy capacity of battery \(C_{B}\) [kWh], and state of charge \(\eta_{CB}\) [%]. In Figures 4c and 5c, the left side vertical axis indicates the inlet temperature of hot-water storage tank \(T_{T}\) [\({}^{\circ}\mathrm{C}\)], the right side vertical axis indicates the heat quantity \(Q\) [MJ], and the horizontal axis indicates hour. In these Figures, \(Q_{t}\) [MJ] is heat quantity in tank, \(Q_{tu}\) [MJ] is used heat quantity from tank. As shown in Figures 4 and 5, all simulation results are satisfied with constraint conditions as described in Section 3. Figure 3: Forecasting method. It is also confirmed by comparing the Figures 4 and 5, that the result of the proposed re-planning operation in case 3 is similar to case 1. Additionally, the battery charges the electric power at night time at an inexpensive electric rate, and provides the charged power to the heat pump system in a sunless period (19:00). More detail regarding simulation results about rainy days is beyond the scope of this research. However, Table 2 shows the calculation results of operational cost, electric power selling, electric power buying, generated electric power, electric power load, heat load, electric power loss and forecast errors for all cases on a rainy day. The lower line in Table 2 represents the mean absolute value (and the percentage values) for solar radiation forecast errors. The percentage values of solar radiation forecast errors are calculated by dividing the total errors by total solar radiation for 1 day. Table 3 shows the calculation results of operational cost for all cases on a sunny day. These simulation results show that  
Figure 4: Simulation results (sunny day, case 1: no forecast errors). (**a**) Optimized result; (**b**) Battery; (**c**) Temperature of tank and used heat quantity. the daily operational cost is minimized by the proposed method on two solar radiation patterns. This also means that the benefit of the installed battery is confirmed by these results. The calculation results of cases 1-3 are similar to the simulation results as described in Figures 4 and 5. It should be noted that weather varies with the seasons. To understand more about simulation results, Table 4 represents the mean generated electric power of PV, absolute values of forecast errors and the percentages every month in 2005. It can be confirmed that case 3 is proposed to be superior to case 2. Whereas the simulation results of operational cost for case 3 are very similar to case 1, the results of electric power loss are higher even than case 2. This reason for this is clear because the objective function in this research is to minimize the operational cost, as described in Section 3. So, if we change the objective function, the simulation results will be different. Otherwise, the difference of the operation time between cases 2 and 3 causes the electric power loss due to the charge/discharge electric power of battery. Therefore, considering further research that is not explored in this article, our future research  
Figure 5: Simulation results (sunny day, case 3: re-planning operation). (**a**) Optimized result; (**b**) Battery; (**c**) Temperature of tank and used heat quantity.