# Robust Optimization for Household Load Scheduling with Uncertain Parameters
## 3. Robust Optimization for Household Load Scheduling
### Robust Counterpart Transformation
#### 3.2.2 Robust Counterpart Transformation of Uncertain Water Demand  
Similarly for the water heater, the linear robust counterparts for the constraints in Equations (8) and (9) with the uncertain water demand should be given. However, unlike the air conditioner, strong duality should be used to deduce the robust counterpart considering the multiple uncertainties (i.e., outdoor temperature and hot water use) involved. Except the uncertain parameters in matrix \(\mathbf{A}\), there is another one uncertain parameter which do not appear in matrix \(\mathbf{A}\) in each constraint. Thus, introducing the new variable \(x_{EWH,N+1}\ =\ 1\) makes this uncertain parameter classified into matrix \(\mathbf{A}\). To simplify the proof, set \(k=\rho/CM,r=(\theta_{water,1}-\theta_{cold})\ >0\). Then the matrix form for constraints is as follows:  
\[\mathbf{c}\leq\mathbf{\hat{A}}\cdot\left[\begin{array}{c}x_{EWH,1}\\ x_{EWH,2}\\ \vdots\\ x_{EWH,N+1}\end{array}\right]\leq\mathbf{b},\mathbf{\hat{A}}=\left[\begin{array} []{cccccc}k&0&\cdots&\cdots&0&(1-\frac{\tilde{q}_{i}}{\text{M}})\cdot r\\ k\cdot(1-\frac{\tilde{q}_{i}}{\text{M}})&k&0&\cdots&\vdots&\overset{2}{\prod}(1 -\frac{\tilde{q}_{i}}{\text{M}})\cdot r\\ \vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\ddots&0&\vdots\\ k\cdot\overset{N}{\prod}(1-\frac{\tilde{q}_{i}}{\text{M}})&k\cdot\overset{N}{ \prod}(1-\frac{\tilde{q}_{i}}{\text{M}})&\cdots&\cdots&k&\overset{N}{\prod}(1 -\frac{\tilde{q}_{i}}{\text{M}})\cdot r\end{array}\right] \tag{36}\]  
\[\mathbf{b}=\left[\begin{array}{cccccc}\theta_{water}^{\text{max}}-\theta_{ cold}&\theta_{water}^{\text{max}}-\theta_{cold}&\cdots&\cdots&\theta_{ water}^{\text{max}}-\theta_{cold}\\ \mathbf{c}=\left[\begin{array}{cccccc}\theta_{water}^{\text{min}}-\theta_{ cold}&\theta_{water}^{\text{min}}-\theta_{cold}&\cdots&\cdots&\theta_{water}^{\text{min}}- \theta_{cold}\end{array}\right]^{\mathsf{T}}\end{array}\right.\]  
The functions with uncertain variables in the coefficient matrix are all monotonically decreasing function, and \(\tilde{q}_{i}\in[\tilde{d}_{i},\tilde{d}_{i}+\hat{d}_{i}]\). Thus, the elements in the coefficient matrix are decreasing, along with the increase of \(\hat{d}_{i}\). Thus, \(\tilde{a}_{ij}\in[\overline{a}_{ij}-\tilde{a}_{ij},\overline{a}_{ij}],\overline {a}_{ij}\in\overline{\mathbf{A}},\tilde{a}_{ij}\in\mathbf{\hat{A}}\  
\[\mathbf{1}\leq\mathbf{X}\leq\mathbf{u},\mathbf{Z}=\{\mathbf{z}\Bigg{|}-1\leq z_{ ij}\leq 0,\forall j\in J_{i},\sum\limits_{j\in J_{i}}\left|z_{ij}\right|\leq\Gamma_{i} \},\Gamma_{i}\in[0,i],x_{AC,N+1}=1 \tag{41}\]  
Obviously, there is \(\max\limits_{z_{i}\in\mathbf{Z}_{i}}\{\sum\limits_{j\in J_{i}}\hat{a}_{ij}x_{ AC,j}z_{ij}\}=0\) in Equation (39). While Equation (40) cannot be directly solved which should apply the strong duality to transform. The final form of the linear robust counterparts is directly given as follows:  
\[\sum\limits_{j=1}^{N}\overline{a}_{ij}x_{AC,j}\leq b_{i}\ \ \ \ \ \forall i \tag{42}\]  
\[\sum\limits_{j=1}^{N}\overline{a}_{ij}x_{AC,j}-(\Gamma_{EWH,i}\cdot p_{i}+\sum \limits_{j\in I_{i}}q_{ij})\geq c_{i}\ \ \ \ \ \forall i \tag{43}\]  
\[p_{i}+q_{ij}\geq\hat{a}_{ij}y_{j}\ \ \ \ \ \forall i,\forall j\in J_{i} \tag{44}\]  
\[-y_{j}\leq x_{j}\leq y_{j}\ \ \ \ \ \forall j \tag{45}\]  
\[p_{i},q_{ij}\geq 0\ \ \ \ \ \forall i,\forall j\in J_{i} \tag{46}\]  
\[\overline{a}_{ij}\in\overline{\boldsymbol{A}},\hat{a}_{ij}\in\boldsymbol{A},b_{i}\in\boldsymbol{b},c_{i}\in\boldsymbol{c} \tag{47}\]  
where \(\Gamma_{EWH,i}\) is the parameter that controls the solution robustness for the water heater, which takes a value in the range [0,\(i\)].