## 3 DEWH Load Scheduling under Different Robust Levels  
### The Bounded Uncertainties of the Hot Water Demand and Ambient Temperature  
In the aforementioned scheduling problem, the purpose of the DEWH is to maximize the economic benefits, and meanwhile keep the hot water temperature profiles within the comfort zones. While due to the randomness in the environment and human behaviors, the parameters in the DEWH load scheduling problem, such as the ambient temperature and the hot water demand, which cannot be predicted precisely. The uncertain parameters may lead to the violations on comfort zones and ineffectiveness of the original optimal schedule. Therefore, the impact of diverse levels of uncertain water demand and ambient temperature on hot water temperature in tank should be discussed in detail. According to the analysis in Reference [16; 22], it is reasonable to describe the uncertainties of the hot water demand and ambient temperature with uncertain-but-bounded parameters whose boundaries are always based on predicted values. The two uncertain-but-bounded parameters, water demand and ambient temperature, can be expressed as intervals by introducing two auxiliary parameters. Moreover, to control the degree of conservatism of the uncertain parameters, one pair of robust level indicators are predefined to divide the uncertainties of water demand and ambient temperature into different grades, which indicates the consumers' optimistic degree for the predicted uncertain-but-bounded parameters in household load scheduling. Given this background, the intervals under different levels of the uncertain water demand in \(n\)th time step is described as:  
\[d_{n}^{U}(a_{d},\Gamma_{d})=d_{n}^{\min}+a_{d}\Gamma_{d}\Delta d_{n} \tag{7}\]  
\[\Delta d_{n}=\frac{d_{n}^{\max}-d_{n}^{\min}}{N_{d}} \tag{8}\]  
\[a_{d}\in[0,1],\Gamma_{d}\in[0,N_{d}],\forall\Gamma_{d},N_{d}\in\mathrm{N}^{+} \tag{9}\]  
Similarly, the intervals under different levels of the uncertain ambient temperature in \(n\)th time step is described as:  
\[\theta_{e,n}^{U}(a_{\theta},\Gamma_{\theta})=\theta_{e,n}^{\max}+a_{\theta} \Gamma_{\theta}\Delta\theta_{e,n} \tag{10}\]  
\[\Delta\theta_{e,n}=\frac{\theta_{n}^{\min}-\theta_{n}^{\max}}{N_{\theta}} \tag{11}\]  
\[a_{\theta}\in[0,1],\Gamma_{\theta}\in[0,N_{\theta}],\forall\Gamma_{\theta},N_ {\theta}\in\mathrm{N}^{+} \tag{12}\]  
where \(a_{d}\) and \(a_{\theta}\) are the auxiliary parameters for intervals of water demand and ambient temperature respectively, making the uncertain parameters ranged in the intervals. Accordingly, \(\Gamma_{d}\) and \(\Gamma_{\theta}\) represent the robust levels for the two uncertain parameters. If the robust levels are equal to 0, it means that consumers are very confident and optimistic about the predicted values. In such situation, the intervalsare compressed into real numbers and the uncertain parameters lose the uncertain characteristic, if the robust levels are equal to \(N_{d}\) and \(N_{\theta}\), then it means that consumers are completely distrustful and pessimistic about the predicted values. In this case, the range of intervals reaches the maximum. When the robust levels are set between zero and maximum, diverse intervals under different levels of the uncertainties are created. In view of the analysis in Section 2, Equation (3) is the thermal dynamic model of a DEWH with deterministic parameters. If the uncertainties as shown in Equations (7) and (10) are considered, it is modified as:  
\[