# Robust Optimization for Household Load Scheduling with Uncertain Parameters
## 2 Mathematical Model
### Constraints
#### 2.3.1 Loads with Uncertain Parameters  
The loads with uncertain parameters widely exist in houses, such as the air conditioner (\(AC\)), water heater (WH), refrigerator, etc. As stated in Section 2.1, this paper takes the \(AC\) and WH as the typical examples to be studied, which are associated the uncertainties in the outdoor temperature and hot water use. As for an \(AC\), due to the heat exchange between the indoor and outdoor environment, the indoor temperature is closely related to the uncertain outdoor temperature, thus being an uncertain variable as well. This relationship can be described as the following Equation (5), which is derived based on the recursive formula presented in [24]:  
\[ \begin{split}\tilde{\theta}_{room,i+1}=&\sum\limits _{n=1}^{i}R\cdot p_{AC}^{r}(1-e^{\frac{-\Delta t}{RC}})\cdot(e^{\frac{-\Delta t }{RC}})^{i-n}\cdot x_{AC,n}+\sum\limits_{n=1}^{i}(1-e^{\frac{-\Delta t}{RC}}) \cdot(e^{\frac{-\Delta t}{RC}})^{i-n}\cdot\tilde{\theta}_{out,n}\\ +&\theta_{room,1}\cdot(e^{\frac{-\Delta t}{RC}})^{i }\quad\quad\forall i\in\{1...N\}\end{split}  \tag{5}\]  
where \(\tilde{\theta}_{room,i}\) and \(\tilde{\theta}_{out,i}\) are the uncertain indoor and outdoor temperatures at the \(i\)th time step, respectively; \(x_{AC,i}\) represents the working status of the \(AC\) (0--off status, 1--heating status and \(-1\)--cooling status); the constants \(R\) and \(C\) are the equivalent heat resistance and capacity;\(p_{AC}^{r}\) represents the rated power of the \(AC\). It should be noted that \(\theta_{room,1}\) is the initial indoor temperature, which is given as an initial condition before scheduling. The main constraint associated with the \(AC\) is the thermal comfort constraint. That is, the uncertain indoor temperature has to be maintained in a preset range at each time step:  
\[\theta_{room}^{\min}\leq\tilde{\theta}_{room,i+1}\leq\theta_{room}^{\max} \quad\quad\forall i\in\{1\dots N\} \tag{6}\]  
\[P_{AC,i}=|x_{AC,i}|\cdot p_{AC}^{r} \tag{7}\]  
where \(\theta_{room}^{\min}\) and \(\theta_{room}^{\max}\) are the lower and upper boundaries of the preset comfort range. Equation (7) shows the relationship among the power consumption, the working status and the rated power of the \(AC\). As for a WH, the decrease of the water temperature in tank results from the heat exchange with the environment and the hot water usage. Due to the excellent insulation, the heat loss in the process of the heat exchange can be ignored [25]. Then the main cause of the temperature dropping is the hot water usage and the followed supplementary cold water. Based on the energy conservation law, the water temperature in tank can be calculated as follows:  
\[\tilde{\theta}_{water,i+1}=\left\{ \begin{array}{ll}\frac{x_{EWH,i}\hat{p}}{ CM}+(1-\frac{\tilde{d}}{M})\cdot(\theta_{water,1}-\theta_{cold})+\theta_{cold},&i=1 \\ \sum\limits_{n=1}^{i-1}[\prod\limits_{k\sim n+1}^{i}(1-\frac{\tilde{d}}{M})] \cdot\frac{x_{EWH,i}\hat{p}}{CM}+\frac{x_{EWH,i}\hat{p}}{CM}+\prod\limits_{n=1 }^{i}(1-\frac{\tilde{d}}{M})\cdot(\theta_{water,1}-\theta_{cold})+\theta_{cold},&i=2\cdots N\end{array} \right. \tag{8}\]  
where \(\tilde{\theta}_{water,i+1}\) is the uncertain water temperature in tank at the \(i\)th time step; \(\tilde{d}_{i}\) is uncertain hot water usage; \(x_{EWH,i}\) is the heating power; the constant \(\rho\) is the transfer coefficient between J and kWh, taking the value of 3.6 \(\times\) 10\({}^{6}\); the constant C is the specific heat capacity of water, being 4.2 \(\times\) 10\({}^{3}\); \(M\) is the mass of the water in full tank; and \(\theta_{water,1}\) is the initial water temperature in tank, which is also given before scheduling. Similar to \(AC\), the constraint of the WH is also about thermal comfort:  
\[\theta_{water}^{\min}\leq\tilde{\theta}_{water,i+1}\leq\theta_{water}^{\max} \quad\forall i\in\{1...N\} \tag{9}\]  
\[P_{EWH,i}=x_{EWH,i} \tag{10}\]  
where \(\theta_{water}^{\min}\) and \(\theta_{water}^{\max}\) are the lower and upper thresholds which are preset by customers.