## 7 Results and Discussion  
The obtained results in the previous section are verified by simulation using the parameters given in [43], which leads to the following state space matrices. \[A = \begin{bmatrix}-1.4202&4.9957&0&0&1.3588&0&0&0&-4.9957&0&0\\ 0&-0.3333&0.3333&0&0&0&0&0&0&0&0\\ -2.560&0&-40.0&12.80&0&0&0&0&0&0\\ -0.10&0&0&-0.50&0&0&0&0&0&0\\ 0.0680&0&0&-0.3176&0.250&0&0&0&0&0\\ 0&0&0&0&0&-1.0&0.1120&0&0&0\\ 0&0&0&0&0&0&-24.3902&12.1951&0&0&0\\ 0&0&0&0&0&0&0&-1.0&0&0&0\\ 20.0&0&0&0&0&0&0&-20.0&0&0\\ 1.0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1.0&0&0&0&0&0\end{bmatrix}  \tag{38}\] \[B = \begin{bmatrix}0&0&12.80&0.50&0&0&0&0&0&0\\ 0&0&0&0&0&18.2927&1.0&0&0&0\end{bmatrix} ^{T}\] \[E = \begin{bmatrix}-4.9957&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0.25&0&0&0&0&0&0\end{bmatrix} ^{T}.\]  
We design the controller, as described in Section 4, where we set the control input matrix \(R=\text{diag}\left\{3,3\right\}\times 10^{4}\) and the state matrix \(Q=\text{diag}\left\{0.01,0.0,0.03,0,0,0,0,0.01,0.03\right\}\) by manual tuning, and we obtain  
\[K= \begin{bmatrix}0.1&1.4&0&0.3&1.1&3&0&0&0&0.6&0.2\\ 0&0.2&0&0&3.1&0.8&0&0&0&-0.1&1\end{bmatrix} \times 10^{-3}.\]  
Then, by using the MATLAB environment with the YALMIP toolbox and the SeDuMi solver [49], the following values are obtained from the solution of the LMI condition (13): \(\varepsilon=2.0835\times-5\), \(\gamma=1.0001\), \(\eta=31.6228\), \(\sigma=0.0046\) and \(\mathcal{T}(\gamma,\beta)=1.51\) s. Thus, all of the ETC parameters were established. Then, by using the hybrid dynamical systems MATLAB toolbox HyEQ [50], simulations of system (21) were executed with the arbitrary initial values \(x(t_{0})=(2,3,1,2,3,5,4,5,3,-2,-4)\), \(e(t_{0})=[0]_{11\times 1}\) and \(\tau(t_{0})=0\) for 120 s with random disturbances \(w\). The resulting average and lowest inter-transmission intervals are \(\tau_{\text{avg}}=6.1684\), \(\tau_{\min}=T=1.51\) s, respectively. The closed-loop response is shown in Figures 3-11 for the continuous-time, time-triggered, and event-triggered controls. The simulation results show that the system frequency \(f(t)\), the active power of the diesel generator \(P_{g}(t)\), the active power of wind generation \(P_{\text{up}}(t)\), the speed of governor valve \(\
u(t)\), the frequency response of wind generation \(f_{w}(t)\), the output response of the hydraulic pitch actuator \(\beta(t)\), and the active power of the battery all converge to the origin in a short time. Figure 10 shows the sampling-induced errors of the system states, where the sampling errors are reset to zero at each transmission instant \(t_{k},k\in\mathbb{N}\), according to (9). The generated transmission instants \(t_{k},k\in\mathbb{N}\) by the proposed ETC (12) are depicted in Figure 11, where the ETC generates inter-transmission times that are much larger than (more than 10 times) the sampling period \(T=1.51\) s, demonstrating the effectiveness of the ETC in reducing the feedback transmission amount. Moreover, the state response shows that the event-triggered controller provides almost the same response as the continuous-time and triggered implementations while the transmission amounts are reduced. In Figure 3, we see that the system frequency \(f(t)\) starts at an initial value of \(f(0)=2\) (Hz), increases to about 2.8 Hz, and then converges rapidly to the origin at around 15 s.  
The active power of the diesel generator \(P_{g}(t)\) starts at the initial value of \(P_{g}(0)=3\) (KW) and converges to the origin in less than 15 s, while no overshoot is exhibited in the response, as shown in Figure 4.  
Figure 5 shows the active power of wind generation \(P_{w}(t)\), which starts at \(P_{g}(0)=5\) (KW) and converges to the origin at a short time, about 7 s, while no overshoot is observed. The speed response of the governor valve \(\upsilon(t)\) is depicted in Figure 6, where it starts from the initial value \(\upsilon(0)=1\) (m/s) and converges to the origin with an observed overshoot that is less than 10%. Figure 4: Trajectory of the active power of the diesel generator.