## 4 Simulation Results
### The Sscheduling Problem under Different Robust Levels  
When consumers believe that the forecast values are highly convincing so that the uncertainties can be ignored, the robust level pair takes the value of (0, 0). Thus, the water demand and ambient temperature lose the characteristic of uncertainty and become real numbers. In this case, the optimal schedule is deemed to be solved without uncertainties, called \"deterministic schedule\" for convenience. However, in practice, the uncertainties of hot water demand and ambient temperature actually give rise to huge fluctuations of the water temperature in tank, so it is possible that the huge fluctuations may make the water temperature out of the threshold when applying the deterministic schedule.  
Therefore, it is necessary to make a sensitivity analysis to figure out the influence of these uncertainties. As stated in Section 3.1, the robust parameters somehow indicate the different uncertainty levels of the parameters, therefore that are used here to generate the various scenarios that are used to verify the deterministic schedule. Specifically, the robust parameters are both set from zero to ten, creating 121 scenarios of different water demand and ambient temperature values. While still applying the deterministic schedule, the actual hot water temperatures in the 121 scenarios are shown in Figure 4.  
In Figure 4, the red line represents the upper boundary of the actual water temperature in tank. However, the 121 different sorts of blue lines represent different lower boundaries, while robust level pairs are set from (0, 0) to (10, 10). Specially, the special line colored black is exactly the lower boundary of water temperature under the highest level of uncertainties, where the values of two robust levels both are 10. As shown in Figure 4, from 8 a.m. to 12 p.m., most blue lines are lower than the consumers' lower threshold line, which means that the thermal comfort constraints are violated. This fact demonstrates that the deterministic schedule cannot satisfy the consumers' thermal comfort requirement under most levels of uncertainties and in most times of a day. The minimum water temperature is down to 48.53 \({}^{\circ}\)C, and the difference between the minimum value and the threshold reaches 6.47 \({}^{\circ}\)C. Such a great temperature deviation is unacceptable for common people.  
To evaluate the degrees of the violation of the constraints under crossed impact of the two parameters, average differences of water temperature between the lower boundaries and threshold are shown in Figure 5. Obviously, with the uncertainty levels of the two parameters increasing, the average temperature difference rises more and more seriously. Also, it is easy to find that the robust level, \(\Gamma_{d}\), brings much more dangerous degree of violation than the robust level \(\Gamma_{\theta}\). That is to say,  
Figure 4: The actual water temperature under 121 scenarios applying the deterministic schedule.  
the hot water demand will create more negative drop on the water temperature than the ambient temperature. In one word, the aforementioned the DEWH load scheduling model with different robust levels is indispensable for the consumers to obtain robust solutions.