# A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads  
[PERSON]  
1 School of Automation, Beijing Institute of Technology (BIT), Beijing 100081, China; [EMAIL_ADDRESS]  
[PERSON]  
2 Automatic Control Laboratory, Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology (ETH), Zurich 8092, Switzerland 2*  
In order to coordinate TCLs, model formulation of TCLs should be illustrated. Based upon the dynamics of TCL [12; 13], two different aggregated TCL models were proposed to mitigate the imbalance of the power gird, say homogeneous model [14] and heterogeneous one [15]. In [5; 16], modeling and control of the aggregated TCLs were studied aiming at different goals. However, the preference of each TCL user is not reflected in these works, which is an important indicator to describe the comfort level of TCL users. As stated in [17; 18; 19; 20], the discomfort functions can be defined to reflect the discomfort level w.r.t the energy demand of TCL users. In this paper, we propose a discomfort function with respect to the dynamics of each TCL user. We study the coordination of TCLs in a typical office or residential building. An electric energy control center (EECC), as a coordinator, is equipped to play the role of buying energy from the wholesale market and selling it to TCL users. Then, an energy trading process occurs between EECC and the TCL users, such that, EECC determines a selling price to maximize the utility benefits and each TCL user adjusts its set-point temperature to maximize their own profits with respect to the selling price from EECC. Considering the dynamics of TCLs, we build a relationship between the set-point temperature of TCLs and the energy demand to reach this temperature. Based upon the above relationship, the energy trading process between EECC and TCL users can be formulated w.r.t the energy demand of TCLs. Moreover, since the decisions between EECC and the TCL users are interacted, we apply a Stackelberg game which is an effective method in power systems [11; 21]. Specifically, in this paper, a one-leader \(N\)-follower Stackelberg game is established such that EECC serves as a leader and the TCL users are the \(N\) followers. We show that the Stackelberg equilibrium exists and is unique, which can be achieved by a backward induction method [22]. Above all, the main contributions of this work can be summarized as below:  
* We study the demand response of the TCLs to control their set-point temperatures by considering the tradeoff between the electricity payment and TCL user's comfort preference;
* According to the dynamics of TCLs, we set up the relationship between the energy demand and set-point temperature. Besides, we formulate the dissatisfaction function to represent the discomfort level of the set-point temperature;
* Based upon the interaction between EECC and TCL users, we formulate the specific energy trading process as a one-leader N-follower Stackelberg game;
* We show the existence and uniqueness of the equilibrium for the underlying Stackelberg games, and develop a DR algorithm based on the Backward Induction method to achieve the equilibrium. The reminder of the paper is organized as follows. In Section 2, we specify the relationship between the energy demand and the set-point temperature and formulate the energy trading process as a DR problem under the RTP scheme. In Section 3, a one-leader \(N\)-follower Stackelberg game is established and the existence and uniqueness of the Stackelberg equilibrium is observed. Section 4 presents numerical simulations for the proposed method. In Section 5, we provide a conclusion for the developed work. The key variables and parameters used in this paper are listed in Table 1.