## 4 Simulation Results
### The Trade-Off Between the Conservatism of Uncertainties and the Economy  
From the above analysis, it is obvious that the bill for consumers will be expensive when they are willing to cover high degree of conservatism of uncertainties. Consequently, acquiring one robust schedule that has not only the best economy, but also the highest level of robustness is a troublesome problem. First of all, to tackle this problem, we predefine two types of scenarios: scenario A is taking the conservatism of only one of the uncertain parameters into account, while scenario B considers the conservatism of both uncertain parameters.  
In scenario A, the marginal effect indexes (MEI) for the robust level \(\Gamma_{d}\) and \(\Gamma_{\theta}\) are defined in Equations (21) and (22), respectively, reflecting the average incremental cost per level of the single uncertain parameter (the water demand or ambient temperature). When calculating the MEI for one robust level, the other robust level should be assumed as a constant. The marginal effect index bars for the robust level \(\Gamma_{d}\) and \(\Gamma_{\theta}\) are plotted in Figure 8.  
\[\mathrm{MEI}_{\Gamma_{\theta}}(\Gamma_{d})=\frac{\mathrm{cost}(\Gamma_{d}, \Gamma_{\theta})-\mathrm{cost}(0,\Gamma_{\theta})}{\Gamma_{d}}\forall\Gamma_ {d}\in[1,10],\forall\Gamma_{\theta}\in[0,10] \tag{21}\]  
\[\mathrm{MEI}_{\Gamma_{d}}(\Gamma_{\theta})=\frac{\mathrm{cost}(\Gamma_{d}, \Gamma_{\theta})-\mathrm{cost}(\Gamma_{d},0)}{\Gamma_{\theta}}\forall\Gamma_ {\theta}\in[1,10],\forall\Gamma_{d}\in[0,10] \tag{22}\]  
By means of the MEI, consumers can choose a trade-off robust schedule that contains the most conservatism at the lowest cost when one of the robust level is clear. For example, assuming that the robust level \(\Gamma_{d}\) is clearly equal to be nine, but the other robust level \(\Gamma_{\theta}\) may be any integer in the interval [5,8]. This assumption is possible, meaning that consumers have complete confidence in the 9 th level of uncertainty of the hot water demand, and have hesitation between 5 th and 8 th level of the uncertainty of ambient temperature. Form the red bars shown in Figure 8b, the MEI has the lowest  
Figure 7: The electricity bill under different robust levels.  
value when \(\Gamma_{\theta}\) is equal to 6, indicating that it is the best balance between the conservatism of ambient temperature and the economy. Therefore, the robust schedules under the robust parameter pair (6, 9) is recommended to consumers.  
Figure 8: (**a**) The marginal effect index bars for the robust level \(\Gamma_{d}\) under different values of \(\Gamma_{\theta}\); (**b**) the marginal effect index bars for the robust level \(\Gamma_{\theta}\) under different values of \(\Gamma_{d}\).  
Similarly, the joint effect index (JEI) is defined in Equation (23) to handle the Scenario B. Differing from the single robust level situation in the Scenario A, the root mean square (RMS) of the two robust levels \(\Gamma_{d}\) and \(\Gamma_{\theta}\) stands for the per level of the joint impact caused by two uncertain parameters. Just like the MEI, the joint effect index plotted in Figure 9 offers the consumers the most balanced schedule as well. Due to the change of the two robust levels, the bars are only colored for the height of the bars.  
\[\mathrm{JEI}(\Gamma_{\theta},\Gamma_{d})=\frac{\mathrm{cost}(\Gamma_{\theta}, \Gamma_{d})-\mathrm{cost}(0,0)}{\sqrt{(\Gamma_{d}^{2}+\Gamma_{\theta}{}^{2})/ 2}}\psi\Gamma_{\theta}\in[1,10],\psi\Gamma_{d}\in[1,10] \tag{23}\]  
Taking another instance for demonstration, we assume that the two robust level \(\Gamma_{d}\) and \(\Gamma_{\theta}\) are a integer which is ranged in the interval [6,8]. The black frame in Figure 9 boxes the area where the consumers are not sure about both of the two robust parameters from 6 th to 8 th level. In this area, the robust level pair (7, 8) has the lowest value and should be offered for consumers.