## 4 Simulation Results  
In this section, a simulation of power frequency control with distributed energy source demonstrates the effectiveness and advantages of our proposed control method. The subsystem frequencies are controlled by our proposed controller, which can save more costs under the consideration of communication and system state. Meanwhile, a comparison is carried out to verify the benefit on the control of power system. The simulation is performed in MATLAB 2010b. We consider the electrical power network shown in Figure 4. Three subsystems are in the power system. It assumes that the composition of three electric power subsystems is same, which is illustrated in Figure 2. There are gas turbine generators, wind power generations and battery arrays in the system. Power supply is done to the electric power demand with these power generating machines. The gas turbine, wind power and battery power output are controllable. The parameters of each subsystem are given in Table 1.  
The maximal norm of disturbance assumes to be 0.1, and the weight \(e\) for communication cost is set to 0.1. The contour map of cost upper bound with respect to exponential stability parameter \(\beta\) and input-to-state stability parameter \(\gamma\) is illustrated in Figure 5, and the relationship between \(\beta\) and \(\tau_{min}\) is shown in Figure 6.  
According to the cost upper bound contour map, if the parameter \(\beta\) is selected to be very low (lower than 0.02), then the effect of parameter \(\gamma\) is not significant. Meanwhile, it shows that the faster convergence rate (larger \(\beta\)) makes the physical cost (state cost) lower. However, according to the relationship between the \(\tau_{min}\) and \(\beta\) illustrated in Figure 6, large \(\beta\) may bring a higher communication cost, because the controller has to reduce the triggering interval to satisfy higher physical cost requirements (faster rate of convergence). Thus, a tradeoff should be made to reduce the total cost. After solving the optimization problem described in (21) with MATLAB Optimization ToolBox, \(\beta\) and \(\gamma\) are obtained as 0.12 and 0.11, respectively. In addition, the feedback gain \(K\) and Lyapunov function parameter \(P\) are also calculated by the robust control design algorithm and LMI toolbox. The initial \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters (S)**} & \multirow{2}{*}{**Symbols (Unit) (S)**} & \multicolumn{3}{c}{**Values**} \\ \cline{3-5}  & & **Subsystem 1** & **Subsystem 2** & **Subsystem 3** \\ \hline Inertia constant & \(M\) (puMw s/Hz) & 0.20 & 0.14 & 0.16 \\ Damping constant & \(D\) (puMw/Hz) & 0.26 & 0.26 & 0.23 \\ Governor constant & \(T_{\delta}\) (s) & 0.20 & 0.20 & 0.12 \\ Gas turbine constant & \(T_{d}\) (s) & 5.0 & 4.5 & 5.0 \\ Regulation constant & \(R_{g}\) (Hz/pu Mw) & 2.5 & 2.5 & 1.5 \\ Synchronizing constant & \(T_{ij}\) (pu Mw) & 0.50 & 0.5 & 0.5 \\ TBC gain & \(K_{i}\) & 0.1 & 0.08 & 0.1 \\ Frequency bias & \(B_{i}\) (Mw/Hz) & 0.1 & 0.1 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter set. Figure 4: Power system structure for simulation  
Figure 8: The frequency deviation \(\Delta f\). Figure 7: The output power deviation of each subsystem \(P_{out}\). Figure 9: The power output deviation of the gas turbine generator \(P_{g}\). At last, they are in the same magnitude. Above all, under the same exponential convergence rate \(\beta\) and input-to-state stability parameter \(\gamma\) requirement, the convergence rate or the physical cost seems almost same and satisfies the performance. The proposed method is effective in the frequency control application of power system. However, the communication should be considered in our proposed control algorithm. Therefore the comparison of communication cost is illustrated. As the sampling rate of classical control method is fixed, it causes the communication cost of the classical control method to be much higher than for the STC method [49; 50].