# Robust Optimization for Household Load Scheduling with Uncertain Parameters
## Simulation Result  
### Simulation Design  
In this paper, a day-ahead household load scheduling (from 0 a.m. to 12 p.m.) is presented. The time length of each step is set as one hour, and that is to say, the total number of time steps in the scheduling horizon is 24. The power output of the photovoltaic generation with 10 kWh capacity and the power consumption of uncontrollable loads are taken from [19], as shown in Figure 1a. The real-time price for buying electricity is easily taken from [14], shown in Figure 1b, and the price for selling electricity is assumed to be a fixed tariff, being 0.34 yuan/kWh. The operational parameters of interruptible loads, uninterruptible loads, and the energy storage device are listed in Tables 1-3.  
Taken from [13] and [27], the heat resistance and capacity for the air conditioner are 18 \({}^{\circ}\)C/kW and 0.525 kWh/\({}^{\circ}\)C, and the mass capacity of the tank is 100 L. Additionally, the rated power of the air conditioner and water heater are 1.8 kW and 3.6 kW, respectively. The preset temperature allowable ranges as comfort constraints of the air conditioner and water heater are [16 \({}^{\circ}\)C, 24 \({}^{\circ}\)C] and [37 \({}^{\circ}\)C, 53 \({}^{\circ}\)C], respectively. Note that the air conditioner is working in the heating mode to warm the room.  
According to the statistics in [17], the uncertainty set of the outdoor temperature (the possible range of the outdoor temperature) for the next day is shown in Figure 2a. As for the hot water demand, it is assumed that extra hot water usage occurs at each time step from 6 a.m. to 11 a.m., and thus the corresponding uncertainty set is assumed as shown in Figure 2b. Given the above settings, based on the theory of robust optimization, the robust parameter \(\Gamma_{EWH,i}\) should follow:  
\[\begin{split}\Gamma_{EWH,i}&\in\{0\},i\in[1,6]\\ \Gamma_{EWH,i}&\in[0,i],i\in[7,11]\\ \Gamma_{EWH,i}&\in[0,11],i\in[12,24]\end{split} \tag{48}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Appliance** & \(b_{\text{UL}}\) & \(e_{\text{UL}}\) & \(l_{\text{UL}}\) (h) & \(p_{\text{UL}}^{r}\) (KWh) \\ \hline Cloth Washer (CW) & 7:00 & 17:00 & 3 & 1 \\ Electrical Vehicle (EV) & 0:00 & 8:00 & 4 & 2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of interruptible loads.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\eta_{ch}\) & 0.95 & \(p_{\text{max}}^{ch}\) (kW) & 2 & \(SOC_{\text{min}}\) & 0.1 \\ \(\eta_{dch}\) & 0.95 & \(p_{\text{max}}^{ch}\) (kW) & 2 & \(SOC_{\text{max}}\) & 0.9 \\ \(\varepsilon\) (kWh/h) & 0.004 & \(SOC_{ini}\) & 0.5 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the energy storage device.  
Figure 1: (a) Power output of the photovoltaic generation and power consumption of the uncontrollable loads for the next day; and (b) the real-time price for the next day.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Appliance** & \(b_{\text{UL}}\) & \(e_{\text{UL}}\) & \(l_{\text{UL}}\) (h) & \(p_{\text{UL}}^{r}\) (KWh) \\ \hline Cloth Washer (CW) & 7:00 & 17:00 & 3 & 1 \\ Electrical Vehicle (EV) & 0:00 & 8:00 & 4 & 2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of uninterruptible loads.