# A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads
## 4 Simulation
### Heterogeneous Case  
In general, the aggregated TCLs' switch state are different [14]. For the purpose of demonstration, we suppose that the total 100 TCLs are partitioned into two categories, say 50 TCLs are with \(W_{i}(t_{k})=1\) and another 50 TCLs with \(W_{i}(t_{k})=0\). As a sequence, the profile of the aggregated energy demand of the 100 TCLs is displayed by the black line in Figure 8.  
As observed in Figure 8, the fluctuations of individual TCLs are alleviated by the aggregated TCLs with different \(W_{i}(t_{k})\). Thus, we may induce the TCL users to adjust its set-point temperature, to mitigate the fluctuation of the power grid by broadcasting different prices to the groups of TCL1 and TCL2 respectively.  
Furthermore, because of the different characteristics of the TCL users, the reference temperature will change with respect to the variational external environment, such as the ambient temperature and the human actions in the room. Therefore, in Figure 9, we consider a scenario with variational reference temperature. EECC broadcasts price \(p^{*}\) (displayed by the red line) and TCL user \(i\) implements the set-point temperature \(\widehat{\theta}_{i}^{*}\) accordingly (displayed by the purple line) at each instant to maximize the utility benefit and minimize the individual cost of each TCL user.  
In reality, the TCLs' properties vary according to the different preferences of TCL users. Thus, besides the above study for homogeneous TCLs, here we also apply Algorithm 1 for the heterogeneous cases.  
We first consider different priority factors of TCL users. By (14), we obtain that the TCL user with higher \(b\) will have more discomfort when the set-point temperature exceeds the reference temperature. Therefore, the set-point temperature of TCL2 with \(b_{2}=1.2\) decreases faster than TCL1 with \(b_{1}=1.1\), which is displayed in Figure 10.  
Figure 7: Set-point temperature and Internal temperature of the TCLs.  
Figure 8: One hundred TCLs with the same \(W(t_{k})\) vs. 100 TCLs with different \(W(t_{k})\).  
Figure 10: TCLs with different priority factor \(b\).  
Figure 9: Variational reference temperature.  
In the future, unlike the model considered in the current work, we will extend our work by considering the heat exchanges among the TCLs which are interactive with each other. Besides, we would like to design a different electricity price scheme to satisfy different users' preferences and maximize the utility benefits.  
This paper is a result of the collaboration of all authors. [PERSON] and [PERSON] conceived and designed this work. [PERSON] performed the experiments. [PERSON] and [PERSON] wrote the paper.  
This research was funded by International S&T Cooperation Program of Beijing Institute of Technology grant number GZ2016065101.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} TCL & Thermostatically controlled loads \\ EECC & Electric energy control center \\ DR & Demand response \\ RTP & Real-time pricing \\ \end{tabular}