## 3 Residential Flexible Load Models and DR Strategies  
### HVAC Load  
A thermodynamic model of the building to which the HVAC belongs is constructed using equivalent thermal parameters based on circuit simulation [17]. The differential equation form of the equivalent thermodynamic model of the HVAC can be derived in Equation (1). \[\frac{dT_{\text{HVAC},i}(t)}{dt}=\frac{1}{R_{1}C_{a}}\Big{(}T_{\text{HVAC},o}(t )-T_{\text{HVAC},i}(t)\Big{)}-\frac{Q_{\text{HVAC}}^{c/h}(t)}{C_{a}} \tag{1}\]  
where \(T_{\text{HVAC},i}(t)\) is the indoor temperature at time \(t\), \({}^{\circ}\)C; \(T_{\text{HVAC},o}(t)\) is the outdoor temperature at time \(t\), \({}^{\circ}\)C; \(R_{1}\) is the indoor equivalent thermal resistance, \({}^{\circ}\)C/W; \(C_{a}\) is the indoor equivalent heat capacity, J/\({}^{\circ}\)C; \(Q_{\text{HVAC}}^{c/h}(t)\) is the cooling or heating energy of the HVAC at time \(t\), W.  
Assume that the outdoor temperature is \(T_{\text{HVAC},o}(t_{k})\) and remains constant during the time period \([t_{k},t_{k+1}]\). The indoor temperature at time \(t_{k+1}\) can be obtained by solving Equation (1) with \(T_{\text{HVAC},i}(t_{k})\) as the initial value. The user determines the indoor setting temperature \(T_{\text{HVAC}}^{\text{set}}\) according to comfort level, and the relationship between the operating state of the HVAC and indoor temperature is shown in Figure 2. The upper and lower limits of temperature are \(T_{\text{HVAC}}^{\text{max}}\) and \(T_{\text{HVAC}}^{\text{min}}\), \(\delta_{\text{HVAC}}\) is the parameter that characterizes the control precision of a constant temperature controller. Since the short-term change of the temperature in the house will not make the human body have a greater perception, taking refrigeration as an example, the DR strategy of the HVAC is mainly to increase the setting temperature within the period of DR to achieve the purpose of decrease load. Figure 3 shows the operating state and indoor temperature changes of the HVAC after changing the setting temperature. It can be found that the total operating time of the HVAC decreases significantly after increasing the setting temperature. Figure 1: The framework of the load aggregator (LA) bidding. The demand response project is carried out in a period of time. In the case of refrigeration, the user changes the setting temperature at the beginning of the demand response. At the end of the demand response, the user restores the original setting temperature. The reduced electricity consumption of the HVAC of the user \(i\) during the period \(j\) is obtained by Equation (2), which is used as the calculation index of the LA's incentive compensation to users. \[Q_{\text{HVAC},i,j}^{\text{DR}}=Q_{\text{HVAC},i,j}^{pre}-Q_{\text{HVAC},i,j}^{ post} \tag{2}\]  
where \(Q_{\text{HVAC},i,j}^{pre}\) is the electricity consumption of the HVAC of the user \(i\) before DR, kWh. \(Q_{\text{HVAC},i,j}^{pre}\) can be predicted by the user's historical loads. \(Q_{\text{HVAC},i,j}^{post}\) is electricity consumption of the HVAC after DR, kWh, which is the actual electricity consumption of the user during the period \(j\). \(Q_{\text{HVAC},i,j}^{\text{DR}}\) is the difference of electricity consumption of the HVAC before and after DR in period \(j\), kWh. In addition, a large number of HVAC loads change the setting temperature at the beginning of the demand response, which will lead to sharp load changes. At the end of the demand response period to restore the original setting temperature, there will be load rebound phenomenon. Therefore, in this paper, the LA adopts the method of rotation control to determine the demand response period and the change of temperature setting value for each HVAC user, so as to avoid the phenomenon of sudden increase and decrease of load [18]. Figure 3: Before and after DR, the relationship between the operating state of the HVAC and the change of indoor temperature. Figure 2: The relationship between the operating state of the heating ventilation and air conditioning (HVAC) and the change of indoor temperature.