## 5 Analysis of Behavior and Results
### Simulation Results at Autumn/Winter ToU Tariff
#### 5.2.2 Winter Day Scenario  
Winter day simulated scenario is implemented by taking under consideration the power consumption profile (Figure 8) for an average day of winter and the ToU tariff of autumn/winter (Figure 1b). At the first discharging zone the weights are set as: \(w_{i}\) = [2 2 1 1 1]. Moreover, at the second discharging zone are as \(w_{i}\) = [1 1 1 2 3 2 2 2 1 1 1]. In that context, during the first discharging zone the EMS will prioritize the load cover at the first two hours (8 a.m.-10 a.m.). In addition, during the second discharging zone, the EMS will prioritize the load cover from 6 p.m. to 11 p.m. with an extra emphasis to the load from 7 p.m. to 8 p.m. (where \(w_{i}\) = 3).  
The operation of the BESS at this average winter day is presented at the obtained Figure 16a,b and Figure 17. During the first discharging zone is noticed that from 8 a.m. to 10 a.m. the household's load is fully covered by the BESS while from 10 a.m. to 1 p.m. is covered partially by the BESS and by the grid. At the end of first discharging zone the battery stack's load is 19.23 Ah. In order to fully charge the stack at the first charging zone the needed current is 40.39 A. Nevertheless, due to the BESS \(C_{rate}\) limitation (0.33C), the BESS is charged with current I = 33 A. Therefore, at the end of the first charging zone the BESS is not fully charged (\(q0\) = 85.23 Ah). During the second discharging zone, the BESS targets to fully cover the load from 7 p.m. to 8 p.m. (where \(w_{i}\) = 3) by discharging the stack to the house's load with I = 33 A. However, due to the \(C_{rate}\) limitation is achieved to cover 1612 Wh of the overall 1750 Wh. The rest 138 Wh are covered from the grid. Additionally, the BESS partially covers the loads from 6 p.m. to 7 p.m. (supplied energy is 581.8 Wh), from 8 p.m. to 9 p.m. (supplied energy is 591.6 Wh), from 9 p.m. to 10 p.m. (supplied energy is 227.6.8 Wh) and from 11 p.m. to midnight (supplied energy is 11.1 Wh). From 1 p.m. to 6 p.m. and from midnight to 2 a.m. the household's load if fully covered by the grid.  
At the end of the second discharging zone it is noted that the BESS SOC is 10%, which implies that the minimum limit (\(SOC_{\min}\)) was reached. BESS's load (\(q0\)) was reduced by 75.23 Ah and the average discharging current was I = 6.82 A (\(C_{rate}=0.0682\)). This yields that the estimated capacity loss at the optimal discharge schedule is probably significantly lower than the corresponding set point (25e) which was calculated for DOD = 80% and \(C_{rate}=0.33\). Hence, the optimizer could possibly permit further discharging of the BESS. Nonetheless, for safety reasons, further discharging was prevented from constraint (25d). At the second charging zone the BESS is charged with current I = 15 A in order to fully charge the battery stack (Figure 10b).  
Calculating the daily energy cost for the household with and without the BESS, it was found that the cost is 2.351 \(\upepsilon\) with the BESS and 2.579 \(\upepsilon\) without the BESS. This implies that the daily profit is 0.228 \(\upepsilon\) and for the whole winter would be 20.75 \(\upepsilon\). The increased profit during a typical winter day compared to a spring/summer day is an outcome of the corresponding ToU tariff which permits two full charge/discharge cycles per day instead of one. The estimated capacity loss for the first charge/discharge cycle of the day is 0.0859 Ah while the set point is 0.0984 Ah. In addition, for the second cycle of the day, the capacity loss is 0.0681 Ah and the corresponding set point is 0.0659 Ah. Finally, the estimated capacity loss for the whole day is 0.154 Ah while the capacity loss set point is 0.1643 Ah.