# A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads
## Appendix A Proof of Lemma 2  
According to the value of market price \(\mathsf{P}\), we have the following two cases.  
**Case 1**: \(\mathsf{P}\leq\min_{i\in\mathcal{N}}\left\{\frac{\omega b_{i}}{q_{i}}e^{b_{i}( 1-\beta_{i}^{+}/q_{i})}\right\}\)  
Based on the boundary conditions in (21), the value of the leader's strategy \(p\) in Case 1 can be divided into two subcases.  
**Case (IA)**: \(\mathsf{P}\leq p\leq\min_{i\in\mathcal{N}}\left\{\frac{\omega b_{i}}{q_{i}}e^ {b_{i}(1-\beta_{i}^{+}/q_{i})}\right\}\)  
By (21), we have \(u_{i}^{*}(p)=\beta_{i}^{+},\forall i\in\mathcal{N}\). Then by (17), we obtain the leader's optimal strategy \(p^{*}\) in the following:  
\[p^{*}=\operatorname*{arg\,max}_{p\in\mathcal{P}_{1}}(p-\mathsf{P})\sum_{i=1} ^{N}\beta_{i}^{+}-\omega\sum_{i=1}^{N}d_{i}(\beta_{i}^{+}), \tag{22}\]  
where \(\mathcal{P}_{1}\equiv\left\{p|\mathsf{P}\leq p\leq\min_{i\in\mathcal{N}}\left\{ \frac{\omega b_{i}}{q_{i}}e^{b_{i}(1-\beta_{i}^{+}/q_{i})}\right\}\right\}\).  
**Case (IB)**: \(\min_{i\in\mathcal{N}}\left\{\frac{\omega b_{i}}{q_{i}}e^{b_{i}(1-\beta_{i}^ {+}/q_{i})}\right\}<p<p_{max}\)  
We denote the feasible set of \(p\) in Case (1B) by \(\mathcal{P}_{2}\), such that,  
\[\mathcal{P}_{2}\triangleq\left\{p|\min_{i\in\mathcal{N}}\left\{\frac{\omega b _{i}}{q_{i,j}}e^{b_{i}(1-\beta_{i}^{+}/q_{i})}\right\}<p<p_{max}\right\}. \tag{23}\]  
In addition, we specify three sets \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\) and \(\mathcal{N}_{3}\), such that,  
\[\mathcal{N}_{1}\triangleq\left\{m|u_{m}^{*}(p)=\beta_{m}^{+},m \in\mathcal{N}\right\}, \tag{24}\] \[\mathcal{N}_{2}\triangleq\left\{n|u_{n}^{*}(p)=q_{n}-\frac{q_{n}} {b_{n}}\ln\frac{pq_{n}}{\omega b_{n}},n\in\mathcal{N}\right\},\] (25) \[\mathcal{N}_{3}\triangleq\left\{l|u_{l}^{*}(p)=0,l\in\mathcal{N} \right\}. \tag{26}\]  
By Lemma 1 and (23), we have \(\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}\cup\mathcal{N}_{3}\) and \(\mathcal{N}_{2}\
eq\emptyset\).  
Then, together with (17), we obtain that,  
\[\begin{array}{ll}\max_{p\in\mathcal{P}_{2}}S_{E}(p;\mathbf{u}^{*}(p))&=\max_{p\in \mathcal{P}_{2}}\left\{(p-\mathsf{P})\left(\sum_{m\in\mathcal{N}_{1}}\beta_{m}^ {+}+\sum_{n\in\mathcal{N}_{2}}\left(q_{n}-\frac{q_{n}}{b_{n}}\ln\frac{pq_{n}}{ \omega b_{n}}\right)\right)\right.\\ &\left.-\omega\left(\sum_{m\in\mathcal{N}_{1}}d_{m}(\beta_{m}^{ +})+\sum_{n\in\mathcal{N}_{2}}\left(\frac{pq_{n}}{\omega b_{n}}-1\right)+\sum _{l\in\mathcal{N}_{3}}d_{l}(0)\right)\right\}\end{array} \tag{18}\]  
Take the second derivative of the utility function \(S_{E}(p;\mathbf{u}^{*}(p))\) with respect to \(p\), we have,  
\[\frac{\partial^{2}}{\partial p^{2}}S_{E}(p;\mathbf{u}^{*}(p))=\sum_{n\in\mathcal{N }_{2}}\left(-\frac{q_{n}}{pb_{n}}-\frac{\mathsf{P}}{p^{2}b_{n}}\right)<0.\]  
Hence, the optimization problem (18) has a unique optimal strategy \(p^{*}\).  
**Case 2**: \(\min_{i\in\mathcal{N}}\left\{\frac{\omega b_{i}}{q_{i}}e^{b_{i}(1-\beta_{i}^ {+}/q_{i})}\right\}<\mathsf{P}<p_{max}\)  
By (15), we have \(p\in[\mathsf{P},p_{max})\). In addition, by Lemma 1, TCL users will have different optimal strategies \(\beta_{i}^{+},q_{i}-\frac{q_{i}}{b_{i}}\ln\frac{q_{i}p}{\omega b_{i}}\), 0. Similar with **Case (1B)**, there exists a unique strategy of the optimization problem (18).  
In sum, consider \(p\in[\mathsf{P},p_{max})\), there exists a unique optimal strategy \(p^{*}\) in (22).