## 3 Residential Flexible Load Models and DR Strategies
### EWH Load  
The temperature of hot water in the EWH changes reciprocally within a certain temperature range up and down the setting temperature point. Based on the equivalent thermal parameter model, the differential equation of the EWH can be constructed [19], as shown in Equation (3). \[C\frac{\text{d}T_{\text{EWH},w}(t)}{\text{d}t}=Q_{\text{EWH}}(t)-\frac{A\Big{(}T _{\text{EWH},w}(t)-T_{\text{EWH},\rho}\Big{)}}{R}-\rho c_{\rho}\Big{(}T_{\text{ EWH},w}(t)-T_{\text{EWH},i}\Big{)}WO(t) \tag{3}\]  
where \(A\) is the surface area of water storage tank, m\({}^{2}\); \(\rho\) is water density, kg/m\({}^{3}\); \(c_{p}\) is specific heat capacity of water, J/(kg \({}^{\circ}\)C); \(R\) is equivalent thermal resistance of water storage tank, \({}^{\circ}\)C/W; \(C\) is equivalent heat capacity, J/\({}^{\circ}\)C; \(R\) and \(C\) can be calculated by the 24-h inherent energy consumption coefficient \(\varepsilon\)[24] and volume \(V\) of the EWH; \(Q_{\text{EWH}}\) is the heating power, W; \(T_{\text{EWH},w}\) is water temperature in the EWH, \({}^{\circ}\)C; \(T_{\text{EWH},\rho}\) is temperature outside the EWH, \({}^{\circ}\)C; \(T_{\text{EWH},i}\) is cold water temperature, \({}^{\circ}\)C; \(WO\) is water consumption of user, m\({}^{3}\)/s. When the EWH is running, the thermostat controller is used to keep the water temperature constant, and the user determines the setting temperature \(T_{\text{EWH}}^{\text{set}}\). During a certain period of time, the working state of the EWH can be divided into two types depending on whether the hot water is tapped. The EWH remains heated while the hot water is tapped. Otherwise, the EWH only needs to keep the water temperature constant. Figure 4 shows the relationship between the running state of the EWH and the water temperature in the storage tank. The EWH has good thermal energy storage characteristics and can participate in DR by changing the setting temperature value and heating in advance. After analysis, it is found that changing the setting temperature value has little effect on the power operation of the EWH, while heating in advance can well achieve the effect of load transfer. Therefore, the EWH participates in DR by heating in advance, as shown in Figure 5. Before and after DR, the EWH electricity consumption change value of user \(i\) is \(Q_{\text{EWH},i,j}^{\text{DR}}\) in period \(j\). Establishing higher temperatures will increase thermal losses and reduce the efficiency of the EWH. This section takes an EWH as the research object and analyzes the effect of increasing the setting temperature value on the efficiency. The volume \(V\) of the EWH is set to 80 L, \(Q_{\text{EWH}}\) is set to 2 kW, \(T_{\text{EWH},\rho}\) and \(T_{\text{EWH},i}\) are set to 20 \({}^{\circ}\)C and 15 \({}^{\circ}\)C, \(\varepsilon\)[24] is set to 0.8. The setting temperature value is initially set at 55 \({}^{\circ}\)C, and it is raised to 70 \({}^{\circ}\)C from 1:00 a.m. to 5:00 a.m., which can achieve the effect of load transfer. Before and after DR, the operation state of the EWH and water temperature change curve are shown in Figure 6.  
Figure 4: The relationship between the running state of the electric water heater (EWH) and the water temperature in the storage tank. In two scenarios, the electricity consumption of the EWH can be calculated, as shown in Table 1. Due to the increase of thermal losses, the electricity consumption also responds to the increase. After DR, the electricity consumption in all day of the EWH is 6.80 kWh, which is 2.6% more than before DR. However, during the low electricity consumption period from 1:00 a.m. to 5:00 a.m., the load increased by 1.67 kwh, which is 25.2% of electricity consumption in all day. For residents, although the electricity consumption increases, the compensation for residents participating in DR is far greater than the increase of electricity cost. The increase of thermal losses has little influence on the overall income of residents, so it can be ignored.