# A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads
## 2 Problem Formulation
### Energy Trading Process  
As shown in Figure 1, EECC first collects the energy from the wholesale market under the market price P, and then sells the energy to TCL users at a broadcasted price \(p\). Each TCL user adjusts its set-point temperature based on the broadcasted price from EECC. Suppose that EECC and TCL users are strategic players, and all of them make decisions by optimizing their individual objectives. Next we will introduce the preference of EECC and TCL users. For TCL user \(i\in\mathcal{N}\), determining \(\widehat{\theta}_{i}\) is equivalent to determining \(u_{i}\) as we have a relationship between them in (11). Hence, TCL user can optimize its energy demand by minimizing its individual cost, which contains the electricity payment and the cost associated with its discomfort level. The individual cost of the \(i\)-th TCL user with respect to \(u_{i}\) is given in the following:  
\[C_{i}(u_{i};p)\triangleq pu_{i}+\omega d_{i}(u_{i}), \tag{12}\]  
wherein the first term represents the electricity payment and the second is the dissatisfaction cost, and \(\omega\) denotes a weighting coefficient concerning the importance of the TCL user's discomfort during \(\mathcal{T}\). For a rational TCL user, its discomfort level continuously decreases with the reduction of the set-point temperature. By (11), the dissatisfaction cost is a function of \(u_{i}\), say \(d_{i}(u_{i})\). At time \(t_{k}\), before choosing the set-point temperature \(\widehat{\theta}_{i}\), each TCL user has a reference temperature, denoted by \(\theta^{\prime}_{i}\), representing its comfortable temperature. Then, the corresponding reference demand, denoted by \(q_{i}\), can be computed by (11) such that:  
\[q_{i}= \begin{cases}0,&\text{in case }\theta^{\prime}_{i}>\widehat{\theta}^{+}_{i, j}\\ f_{i}(\theta^{\prime}_{i}),&\text{in case }\widehat{\theta}^{-}_{i,j}\leq \theta^{\prime}_{i}\leq\widehat{\theta}^{+}_{i,j}\\ \beta^{+}_{i},&\text{in case }\theta^{\prime}_{i}<\widehat{\theta}^{-}_{i,j} \end{cases}  \tag{13}\]  
where \(\widehat{\theta}^{+}_{i,j}\) and \(\widehat{\theta}^{-}_{i,j}\) represent the \(i\)-th TCL user's maximum and minimum set-point temperature in Case \(j\) respectively, with \(j=\{1,2\}\). **Remark 3**. : _In (13), the reference temperature \(\widehat{\theta}^{\prime}_{i}\) is the threshold value of the comfortable temperature, which is related to each TCL user's preference and external environment. It can be recognized as a criterion of the comfort level of TCL users._  
**Remark 4**. : _By (11), the expression of \(f_{i}(\widehat{\theta}_{i})\) is distinct in different cases. Then, by the feasible set of energy demand in (6), we have \(\widehat{\theta}^{+}_{i,j}\) and \(\widehat{\theta}^{-}_{i,j}\) are related to the case \(j\) for all \(i\in\mathcal{N}\)._  
As specified in [18; 19; 20], the dissatisfaction cost function \(d_{i}(u_{i})\) is continuous and has the following properties:  
\[d_{i}(u_{i}) \begin{cases}>0,&\text{in case }u_{i}<q_{i}\\ =0,&\text{in case }u_{i}=q_{i}\\ <0,&\text{in case }u_{i}>q_{i}\end{cases} \]For \(u_{i}<q_{i}\), i.e., \(\widehat{\theta}_{i}>\theta_{i}^{r}\), the TCL user is dissatisfied with the current temperature and the discomfort level will increase rapidly as \(\widehat{\theta}_{i}\) (demand \(u_{i}\)) is away from the reference temperature \(\theta_{i}^{r}\) (demand \(q_{i}\)). For \(u_{i}>q_{i}\), i.e., \(\widehat{\theta}_{i}<\theta_{i}^{r}\), the TCL user is satisfied with the current temperature, but the comfort level will not increase infinitely and change slowly as \(\widehat{\theta}_{i}\) is away from the reference temperature \(\theta_{i}^{r}\). Based on the above properties, we apply the dissatisfaction cost function \(d_{i}(u_{i})\) as the following form [18]:  
\[d_{i}(u_{i})\triangleq e^{b_{i}(1-u_{i}/q_{i})}-1, \tag{14}\]  
with \(b_{i}>0\), where \(b_{i}\) represents the priority factor of TCL user \(i\). For EECC, it can obtain benefits by buying energy from the market and selling it to TCL users.