# A Stackelberg Game Approach for Price Response Coordination of Thermostatically Controlled Loads
## 4 Simulation
### Homogeneous Case  
We first consider \(N=100\) homogeneous TCLs, and the parameters of the TCLs are specified in Table 2[32]. We set the weighting factor of the importance of the discomfort level as \(\omega=0.2\). Without loss of generality, assume that \(W_{i}(t_{k})=0\) with \(t_{k}=11:00\), for all \(i\in\mathcal{N}\), i.e., the switch state of each TCL is \"off\" at 11:00. As specified in Section 2.2, the \"off\" state implies that \(j=2\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Homogeneous TCL** & **Heterogeneous TCL** \\ \hline \(R\) & 2 °C/kW & 2 °C/kW \\ \(C\) & 5 kWh/°C & 6 kWh/°C \\ \(P\) & 11 kW & 14 kW \\ \(\beta^{+}\) & 2.75 kWh & 3.5 kWh \\ \(b\) & 1.1 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: TCL parameters. Figure 4: Ambient temperature and Reference temperature from 11:00 to 20:00 on a summer day. Figure 3: Market price data. We also consider the internal temperature \(\theta_{i}(t_{k})=27\,^{\circ}\mathrm{C}\) for all \(i\in\mathcal{N}\) and the temperature dead-band \(\delta=0.25\,^{\circ}\mathrm{C}\). Then, according to the reference temperature \(\theta_{i}^{r}\) of TCL user \(i\) shown in Figure 4, we obtain the reference demand energy \(q_{i}\) by (13), which is displayed by the blue dash-dot line in Figure 5. More specifically, taking one time horizon [11:00, 11:15] as an example and given \(\theta_{a,i}(t_{k})=31.2\,^{\circ}\mathrm{C}\), \(\theta_{i}(t_{k})=27\,^{\circ}\mathrm{C}\) and \(\theta_{i}^{r}(t_{k})=26\,^{\circ}\mathrm{C}\), we calculate that \(\widehat{\theta}_{i}^{-}=f_{i}^{(-1)}(\beta_{i}^{+})=26.56\,^{\circ}\mathrm{C}\) in Case 2. Then by (13), we have \(q_{i}(t_{k})=\beta_{i}^{+}=2.275\,\mathrm{kWh}\). By applying Algorithm 1, EECC implements the optimal price \(p^{*}\) w.r.t \(\mathbf{u}^{*}(p)\) by (22), which is displayed by the red line in Figure 6.  
The broadcast price \(p^{*}\) satisfies \(p^{*}\in\left(\frac{\omega b_{i}}{q_{i}}e^{(1-\beta_{i}^{+}/q_{i})},\frac{ \omega b_{i}}{q_{i}}e_{i}^{p}\right)\). Then, by (21), the optimal energy demand of each TCL user \(u_{i}^{*}\) increases as \(p^{*}\) decreases from 11:00 to 20:00, which is displayed in Figure 5.  
Subsequently, according to the relationship between the set-point temperature and the energy demand specified in Section 2.2, each TCL user adjusts its set-point temperature by \(\widehat{\theta}_{i}^{*}=f_{i}^{(-1)}\left(u_{i}^{*}\right)\), which is displayed by the red line in Figure 7.  
Figure 5: Optimal energy demand of the TCL. Figure 6: Broadcast price \(p^{*}\) from electric energy control center (EECC). Consider the time horizon [13:00, 13:15] as an example. Based upon the reference demand \(q_{i}=0.691\) kWh, the market price \(\mathrm{P}=0.12\,\mathrm{\SIUnitSymbolSkip}/\mathrm{kWh}\) and the optimal reaction curve \(u_{i}^{*}\left(p\right)=q_{i}-\frac{q_{i}}{b_{i}}\) in \(\frac{pq_{i}}{ab_{i}}\) given in Lemma 1, we obtain the optimal broadcast price \(p^{*}=0.219\,\mathrm{\SIUnitSymbolSkip}/\mathrm{kWh}\) by (22). Afterwards, TCL users observe the best strategy of EECC and compute their best strategies \(u_{i}^{*}(p^{*})\) by (21). The corresponding set-point temperature is \(\widehat{\theta}_{i}^{*}=25.98\,\mathrm{\SIUnitSymbolCelsius}\). Because of the existence of the dead-band \(\delta\), the internal temperature varies by (1) in [13:00, 13:15], and the switch state will change when the internal temperature hits the upper limit \(\widehat{\theta}_{i}^{*}+\delta/2=26.11\,\mathrm{\SIUnitSymbolCelsius}\). Moreover, after 13:00, for keeping the internal temperature around the reference temperature \(26\,\mathrm{\SIUnitSymbolCelsius}\), the switch state changes one time within each time horizon and the optimal set-point temperature stays around \(26\,\mathrm{\SIUnitSymbolCelsius}\), as illustrated in Figure 7. However, given the same ambient temperature \(\widehat{\theta}_{a,i}\), by (9) and (10), the associate reference energy demand \(q_{i}\) are distinct in different cases.