# A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads
## 2 Problem Formulation
### TCL Dynamics  
As stated in [14, 16, 25], for each TCL \(i\in\mathcal{N}\) at any time \(t\in\mathcal{T}\), the evolution of the temperature can be expressed as a first-order differential equation, such that,  
\[\frac{d\theta_{i}(t)}{dt}=-\frac{1}{C_{i}R_{i}}(P_{i}R_{i}W_{i}(t)+\theta_{i}( t)-\theta_{a,i}(t)), \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \(P_{i}\) & Cooling thermal power of TCL \(i\) \\ \(W_{i}(t)\) & Switch state of TCL \(i\) at instant \(t\) \\ \(\delta\) & Temperature deadband \\ \(T\) & Length of the time interval \(\mathcal{T}\) \\ \(T_{on}\) & Length of the “on” state in \(\mathcal{T}\) \\ \(\widehat{\theta}_{i}^{+}\) & Maximum energy demand in \(\mathcal{T}\) \\ \(\theta_{i}^{\prime}\) & Reference temperature of TCL user \(i\) \\ \(q_{i}\) & Reference energy demand of TCL user \(i\) in \(\mathcal{T}\) \\ \(j\) & Case of the switch state \(W_{i}(t_{k})\), \(j=1,2\) \\ \(b_{i}\) & Priority factor of the TCL user \(i\) \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 1: The framework of the energy trading process.  
where the notations are specified as below:  
* \(\theta_{i}(t)\) and \(\theta_{a,i}(t)\) represent respectively the internal temperature (\({}^{\circ}\)C) and the ambient temperature (\({}^{\circ}\)C) of TCL \(i\) at time \(t\).
* \(R_{i}\), \(C_{i}\) and \(P_{i}\) are thermal parameters which express the thermal resistance (kWh/\({}^{\circ}\)C), thermal capacitance (\({}^{\circ}\)C/kW) and cooling thermal power (kW) of TCL \(i\), respectively. For notational simplicity, we denote the thermal constant by \(\tau_{i}\), such that \(\tau_{i}\equiv R_{i}C_{i}\).
* The binary variable \(W_{i}(t)\in\{0,1\}\) represents the switch state of TCL \(i\) at instant \(t\).  
**Remark 1**.: _In (1), we consider the TCLs in different houses or offices where the evolution of the internal temperature is mainly effected by the heat exchange between inside and outside, hence there is no heat exchange among the TCLs [26]. Besides, Equation (1) is formulated for cooling TCLs such as air conditioners. Then, \(P_{i}\) in (1) is a positive constant._  
To avoid TCL \(i\) switching frequently around its set-point temperature \(\widehat{\theta}_{i}\), we adopt a temperature dead-band \(\delta\equiv|[\theta_{i}^{-},\theta_{i}^{+}]|\), where \(\theta_{i}^{-}\) and \(\theta_{i}^{+}\) are the lower and upper limit of the dead-band respectively, such that:  
\[\theta_{i}^{-}=\widehat{\theta}_{i}-\delta/2, \tag{2a}\] \[\theta_{i}^{+}=\widehat{\theta}_{i}+\delta/2. \tag{2b}\]  
Then, the switch state function in (1) is defined as follows [27; 28]:  
\[W_{i}(t+\triangle t)=\left\{\begin{array}{ll}0&\text{in case }\theta_{i}(t+ \triangle t)\leq\theta_{i}^{-}\\ 1&\text{in case }\theta_{i}(t+\triangle t)\geq\theta_{i}^{+}\\ W_{i}(t)&\text{otherwise}\end{array}\right. \tag{3}\]  
where \(\triangle t\) is an arbitrarily small time interval. The temperature evolution procedure is shown in Figure 2. \(T_{on,i}\) appeared in the figure is the time length that the \"on\" state of TCL \(i\) lasts during one time horizon \([t_{k},t_{k+1}]\).  
Figure 2: Temperature evolution procedure of thermostatically controlled loads (TCLs).