## 3 Problem Formulation  
The considered hybrid wind-diesel power system is schematically shown in Figure 1. The system comprises a diesel generator with a speed regulator, a wind turbine generator with a pitch angle controller, and battery energy storage. The electric power is produced by wind turbines, which are complemented by diesel generators and battery storage to compensate for variations in the active power. For the purposes of this analysis, first-order dynamical models for the wind turbine generator and battery energy storage are sufficient. The system under consideration has the following linearized model:  
\[\dot{x}(t) =Ax(t)+Bu(t)+Ew(t) \tag{3}\] \[u(t) =-Kx(t),\]  
where \(x(t)\in\mathbb{R}^{n_{11}}\) is the state, \(u(t)\in\mathbb{R}^{2}\) is the control input, \(w(t)\in\mathbb{R}^{2}\) is the external disturbance, \(x(t):=(x_{1},x_{2},\ldots,x_{11})\) with \(x_{1}(t)=\Delta f(t)\), \(x_{2}(t)=\Delta P_{g}(t)\), \(x_{3}(t)=\Delta\
u(t)\), \(x_{5}(t)=\Delta f_{w}(t)\), \(x_{6}(t)=\Delta P_{w}(t)\), \(x_{7}(t)=\Delta\beta(t)\), \(x_{9}(t)=\Delta P_{b}(t)\), \(x_{4}(t),x_{8}(t)\) are dummy states, and \(x_{10}(t),x_{11}(t)\) are augmented states for stability. The matrix \(K\) is the controller gain matrix to be designed, and plant matrices \(A,B\) are constant and have the following structure:  
Figure 1: Schematic of hybrid wind–diesel power system. \[A= \begin{bmatrix}\frac{-(1+K_{pt}K_{ig})}{I_{pt}}&\frac{K_{pt}}{I_{pt}}&0&0&\frac{K_ {pt}K_{ig}}{I_{pt}}&0&0&0&-\frac{K_{pt}}{I_{pt}}&0&0\\ 0&\frac{-1}{I_{44}}&\frac{1}{I_{44}}&0&0&0&0&0&0&0&0\\ \frac{-K_{41}T_{41}}{K_{41}^{2}t_{2}t_{3}}&0&\frac{-1}{t_{3}}&\frac{-K_{41}(T _{41}-T_{42})}{t_{2}t_{43}}&0&0&0&0&0&0\\ \frac{-1}{K_{41}^{2}t_{2}}&0&0&\frac{-1}{t_{2}}&0&0&0&0&0&0&0\\ \frac{K_{x}}{T_{x0}}&0&0&0&\frac{(K_{pt}-K_{ig}-1)}{T_{x0}}&\frac{1}{T_{x0}}&0 &0&0&0&0\\ 0&0&0&0&0&\frac{1}{T_{x0}}&\frac{K_{pK}K_{p3}}{T_{p3}}&0&0&0&0\\ 0&0&0&0&0&\frac{-1}{T_{p2}}&\frac{K_{p1}K_{p2}(1-T_{p2})}{T_{p2}}&0&0&0\\ 0&0&0&0&0&0&-1&0&0&0\\ \frac{1}{T_{b}}&0&0&0&0&0&0&0&\frac{-1}{T_{b}}&0&0\\ 1&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0\end{bmatrix}  \tag{4}\]  
\[B= \begin{bmatrix}0&0&\frac{K_{01}T_{41}}{t_{2}t_{3}}&\frac{1}{t_{2}}&0&0&0&0&0 &0\\ 0&0&0&0&0&0&\frac{K_{p1}K_{p2}T_{p1}}{T_{p2}}&1&0&0\end{bmatrix} \]  
\[E= \begin{bmatrix}\frac{-Kpt}{T_{pt}}&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&\frac{1}{T_{w}}&0&0&0&0&0\end{bmatrix}  \tag{5}\]  
Since the feedback information is sent to the controller over a digital channel, the controller can only access the value of \(x(t)\) at discrete-time instants \(t_{k},k\in\mathbb{N}\). Hence, considering the effect of the network, the plant model (3) is modified to  
\[\dot{x}(t) =Ax(t)+Bu(t)+Ew(t) \tag{6}\] \[u(t) =-Kx(t_{k})\hskip 28.452756 pt\forall t\in(t_{k},t_{k+1}],\]  
where the received state information \(x(t_{k})\) at the controller is kept constant between two transmissions, i.e., \(\forall t\in(t_{k},t_{k+1}]\), by means of the zero-order hold (ZOH) implementation. As such, we need to design both the feedback law and the triggering rule to generate the sequence of sampling times \(t_{k},k\in\mathbb{N}\). The latter will be designed using the event-triggered approach to reduce the amount of transmissions between the sensors and the controller. A major challenge in this context is to prevent the occurrence of Zeno phenomena while preserving the closed-loop stability. The networked control architecture of the hybrid power system is depicted in Figure 2. The full state measurement \(x(t)\) is received by the ETC mechanism to release the discrete-time value \(x(t_{k})\) at the next transmission instant \(t_{k},k\in\mathbb{N}\). Accordingly, the control input \(u(t_{k})\) is updated and it remains constant until the next update by means of ZOH implementation. We assume that the communication channel is noiseless and delay-free. **Problem statement**.