## 2 Demand Response Modeling for ACL
### Constraints for the DR of the ACL  
In the process of load reduction, the power company should consider consumer comfort, making sure that the indoor temperature is within a comfortable range. It is assumed that the lowest and highest indoor temperatures that the consumers are comfortable with are \(T_{\text{min}}\) and \(T_{\text{max}}\), respectively; then  
\[T_{\text{min}}\leq T_{in}\leq T_{\text{max}} \tag{7}\]  
Because it is difficult to collect the real-time indoor temperature, the current study adopts the equivalent thermal parameter model of the ACL, transferring the temperature constraints in Equation (7) into a response time constraint. During load control periods \(x\), we assume that the averagerated power of the ACL is \(\overline{P}\) and that the outdoor temperature in the control cycle is a constant, \(T_{out,x}\). When a consumer turns off the air conditioner, the indoor temperature will increase as time goes by, until it reaches the highest temperature constraint \(T_{\text{max}}\). When a consumer turns on the air conditioner, the temperature will decrease as time passes by, until it drops to the lowest temperature constraint. Thus, Equations (1) and (2) can be used to calculate the longest air conditioner off-time period \(t_{\textit{off}}\) and the corresponding longest on-time period \(t_{\textit{on}}\) from the following:  
\[T_{\text{max}}=T_{out,x}-(T_{out}-T_{\text{min}})e^{-t_{\textit{off}}/RC} \tag{8}\]  
\[T_{\text{min}}=T_{out,x}-COP\cdot\overline{P}/A-(T_{out}-COP\cdot\overline{P}/A -T_{\text{max}})e^{-t_{\textit{on}}/RC} \tag{9}\]  
Among the various control methods for the DR of the ACL, this paper adopts the state-queuing method [31], providing the computational method for air-conditioning load shedding.  
Assuming that there are \(N_{AC}\) types of ACLs in the DR program in a specific area, the \(n\)th-type ACL is selected and divided into \(t_{c}\) groups, and each group is controlled in turn (shown in Figure 2). If the state time interval \(\Delta t\) is 1 min, then at each moment, there are \(t_{\textit{on}}\) groups of ACLs that are _on_ and \(t_{\textit{off}}\) groups of ACLs that are _off_ in the temperature interval [\(T_{\text{min}}\), \(T_{\text{max}}\)]. The total number of controlling groups/states is equal to the sum of \(t_{\textit{on}}\) and \(t_{\textit{off}}\).  
Thus, during load control periods \(x\), the largest load reduction of the \(n\)th-type ACL \(\overline{P}_{AC,n,x}\) is as follows:  
\[\overline{P}_{AC,n,x}=\frac{t_{\textit{off}}}{t_{c}}\cdot N_{n}\overline{P} \tag{10}\]  
where \(N_{n}\) refers to the total number of the \(n\)th-type ACLs in the DR program. According to Equations (8) and (9), \(t_{\textit{off}}\) and \(t_{\textit{on}}\) can be calculated; then \(\overline{P}_{AC,n,x}\) can be calculated as follows:  
\[\overline{P}_{AC,n,x}=\frac{\ln\left(\frac{T_{out,x}-T_{\text{max}}}{T_{out,x} -T_{\text{min}}}\right)}{\ln\left(\frac{T_{out,x}-T_{\text{max}}}{T_{out,x}-T _{\text{min}}}\right)+\ln\left(\frac{T_{out,x}-\overline{P}/A-T_{\text{min}}} {T_{out,x}-\overline{P}/A-T_{\text{max}}}\right)}\cdot\overline{P_{n}} \tag{11}\]  
The constraints for ACLs are that the load shedding \(P_{AC,n,t}\) of the \(n\)th ACL during load control periods \(x\) should not be higher than the largest load shedding:  
\[P_{AC,n,x}\leq\overline{P}_{AC,n,x} \tag{12}\]  
The power of ACLs is normally determined by the outdoor temperature and the set value of the indoor temperature. This study aims to reduce the peak loads by controlling the usage of ACLs. During peak-load periods, the system sends a load reduction signal to the smart controllers that are  
Figure 2: The state of the air-conditioning load (ACL).  
connected with the air conditioners. Using the state-queuing method, the smart controllers cyclically control the on/off status of the air conditioners to decrease the ACL and therefore the peak loads.