## 3 DEWH Load Scheduling under Different Robust Levels
### The Scheduling Problem under Different Robust Levels  
To improve the flexibility of energy production and consumption, various time-varying pricing mechanisms are implemented around the world. In this paper, RTP price is considered as the time-varying pricing mechanism used in the analysis, as its retail electric prices update frequently to take the variations in the cost of the power supply into account [23].  
When considering the uncertainties, the optimization problem of DEWH load scheduling is still to minimize the electricity bill but subjects to comfort constraints with different robust levels. Therefore, the scheduling problem can be rebuilt as follow:  
\[\min\sum\limits_{n=1}^{N}p_{n}X_{n}P_{\Delta t} \tag{14}\]  
\[\theta^{down}\leq\theta_{n+1}^{U}(X_{n},a_{d},a_{\theta},\Gamma_{d},\Gamma_{ \theta})\leq\theta^{up} \tag{15}\]  
\[X_{n}\in\{0,1\},\ a_{d}\in[0,1],\ a_{\theta}\in[0,1] \tag{16}\]  
As the hot water temperature in tank in each step is involved with the uncertain auxiliary parameters \(a_{d}\) and \(a_{\theta}\), the optimization problem formulated by (14-16), obviously, could not be solved directly. To address this uncertain optimization problem, it is necessary to obtain the bounds of uncertain hot water temperature in tank. Hence, there is  
\[\theta_{n+1}^{U}(X_{n},a_{d},a_{\theta},\Gamma_{d},\Gamma_{\theta})=\left[ \theta_{n+1}^{\min}(X_{n},a_{d},a_{\theta},\Gamma_{d},\Gamma_{\theta}),\theta_ {n+1}^{\max}(X_{n},a_{d},a_{\theta},\Gamma_{d},\Gamma_{\theta})\right] \tag{17}\]  
The hot water temperature fluctuates in its interval, while the auxiliary parameters change. In order to figure out the constraints with uncertainties, the boundaries of the interval should be calculated. Appendix B fortunately indicates that  
\[\left\{\begin{array}{ll}\theta_{n+1}^{\min}(X_{n},a_{d},a_{\theta},\Gamma_{ d},\Gamma_{\theta})=\theta_{n+1}^{U}(X_{n},1,1,\Gamma_{d},\Gamma_{\theta})\\ \theta_{n+1}^{\max}(X_{n},a_{d},a_{\theta},\Gamma_{d},\Gamma_{\theta})=\theta_ {n+1}^{U}(X_{n},0,0,\Gamma_{d},\Gamma_{\theta})\end{array}\right. \tag{18}\]  
With such background, the constraint (15) could be transformed into  
\[\theta^{down}\leq\left[\theta_{n+1}^{U}(X_{n},1,1,\Gamma_{d},\Gamma_{\theta}), \theta_{n+1}^{U}(X_{n},0,0,\Gamma_{d},\Gamma_{\theta})\right]\leq\theta^{up} \tag{19}\]Besides, to make this optimization problem solvable, Equation (19), which is an uncertain inequality constraint, is transformed into two deterministic constraints, which is given by  
\[\theta_{n+1}^{U}(X_{n},0,0,\Gamma_{d},\Gamma_{\theta})\leq\theta^{up},\ \theta_{n+1}^{U}(X_{n},1,1,\Gamma_{d},\Gamma_{\theta})\geq\theta^{down} \tag{20}\]  
Based on the above analysis, the optimization problem of DEWH load scheduling with different robust levels could be formulated as integer linear programming, of which the objective function is shown in Equation (4) and the constraints are the Equations (7)-(13), (16) and (20).  
So far, the optimization problem has been transformed to an integer linear programming problem, and many existing methods, tools, and commercial software are available to obtain the optimal solution. Among the abundant methods, the heuristic-based evolutionary algorithm like particle swarm optimization (PSO) [21], model predictive method [24], and commercial software CPLEX (Version 12.6.3.0, IBM, New York, USA, 2016) [25] and MOSEK(Version 7.1.0.63, ApS, Copenhagen, Denmark, 2016) [26] are some of the representative ones. Especially, IBM ILOG CPLEX(r) Optimizer is used in this paper to solve the problem considering its strong capability.