## 5 Simulation and Result Analysis  
Taking the four-area interconnected power system as research object in this paper and each control area can be distinguished by setting different values to various parameter variables. In the case of Control Area 1, the parameter settings are shown in Table 1.  
The electrical simulations of interconnected power system for four control areas area carried out through Matlab 2012a/Simulink in following three different cases:  
* Case 1: Traditional LFC, i.e., power system without the HESS. * Case 2: Power system with the HESS and external PID controller [14; 34]. * Case 3: Power system with the HESS and external H\({}_{\infty}\) robust state feedback controller. Set the simulation time to 100 s. To test the robust stability and the suppression ability of the control system under load disturbance, load interference is added to the power system in Control Area 1, as shown in Figure 7, which consists of step signal, ramp signal and sinusoidal signal [35]. It is worth pointing out that, when Area 1 is disturbed by load interference, the load frequency of power systems in the other three control areas will be affected through the tie-lines between each area. Take the Area 1 as an example. In the process of H\({}_{\infty}\) robust state feedback controller design, the weighting coefficients and performance indices are given as Table 2. Combining Section 4, the state feedback control coefficient matrix of Area 1 can be obtained by Matlab as \(K_{1}=[-10.8625,\,25.5883\)\(5.2364,\,-8.8031,\,-7.6875,\,-9.2341]\) and coefficient matrices for other three control areas can be easily obtained by using the method above. For Case 3, Figure 8 shows the control signal \(u_{1}\), \(u_{2}\), \(u_{3}\), \(u_{4}\) for each control area from the H\({}_{\infty}\) state feedback controller designed in this paper. These control signals denote the power instruction transmitted to the HESS from H\({}_{\infty}\) controller. More attention should be paid to the magnitude order of the axis because they are different in each figure. It can be noticed that the orders of magnitude in Figure 8b-d are really small, respectively, \(10^{-5}\), \(10^{-5}\) and \(10^{-6}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(T_{gi}\) **(s)** & \(T_{ri}\) **(s)** & \(T_{ri}\) **(s)** & \(R_{i}\) **(Hz/p.u.Mw)** & \(T_{1j(j=1,2,3,4)}\) **(s)** \\ \hline
0.08 & 4.2 & 0.3 & 2.4 & 0.0707 \\ \hline \(K_{p1}\) **(s)** & \(T_{p1}\) **(s)** & \(c_{1}\) & \(\beta_{1}\) **(Hz/p.u.Mw)** & \(K_{AF1}\) \\ \hline
120 & 20 & 0.35 & 0.425 & 1.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in Control Area 1 of the interconnected power system. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \((i\)=1) & \(q_{i1}\) & \(q_{i2}\) & \(q_{i3}\) & \(q_{i4}\) & \(q_{i5}\) & \(q_{i6}\) & \(\rho_{1}\) & \(\gamma_{1}^{2}\) \\ \hline Area 1 & 0.0125 & 0.01 & 0.001 & 0.005 & 0.001 & 0.013 & 0.09 & 0.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of controller design for Control Area 1. Figure 7: The external load disturbance added to Control Area 1. Figure 9 shows the load frequency deviation \(\Delta f_{i}(i=1,2,3,4)\) of each power system under load disturbance in Case 3. When the load disturbance of Figure 8 arises in Control Area 1, load frequencies of each power system will be subjected to varying degrees of influence. Under the condition of LFC with the HESS and external H\({}_{\infty}\) robust state feedback controller, in Figure 8 we can see that the load frequency \(f_{i}(i=1,2,3,4)\) restores stability rapidly in seconds and \(\Delta f_{i}(i=1,2,3,4)\) was guaranteed to be within \(\pm 0.0006\) Hz since the maximum interference is almost 10 Kw. The result comparison of PID controller and H\({}_{\infty}\) state-feedback controller is shown in Figure 10 under the condition of LFC with the HESS. Figure 8: Control signals from controller of four areas in Case 3: (a) control signal \(u_{1}\); (b) control signal \(u_{2}\); (c) control signal \(u_{3}\); and (d) control signal \(u_{4}\).