Fully Distributed Consensus Voltage Restoration Control Based on Dynamic Event-Triggered Mechanisms for Offshore Wind Power Systems

Abstract

Although voltage regulation in distributed generation (DG) in offshore wind power systems has been deeply studied, information on the global communication network structure and high communication bandwidths still remain key factors restricting the integration of a high proportion of new energy sources. In order to achieve precise voltage regulation while reducing communication, this paper proposes a fully distributed dynamic event-triggered secondary voltage restoration control for offshore wind power systems. Firstly, the nonlinear system model is switched to the linear second-order multi-agent system (MAS) by feedback linearization. This is a crucial step for the subsequent research. From there, a fully distributed voltage restoration control strategy is proposed, which utilizes adaptive event-triggered protocol. The proposed protocol operates independently of global information. Finally, the effectiveness of the control protocol is verified through simulation.

1. Introduction

Offshore wind power systems are usually small power systems composed of distributed fans, loads, and other devices, which have been used to realize independent power supply in remote offshore islands. They have been the key solution to solving energy depletion and environmental pollution [1,2,3,4]. Each wind turbine unit controls voltage, frequency, and active power in an independent manner with flexibility, reliability, and robustness [5,6]. Therefore, offshore wind power systems have received great social attention in recent years.

The process of information transmission and control play a vital role in offshore wind power systems. Scholars have proposed a hierarchical control structure for the coordinated control of distributed wind turbine power generation in offshore wind power systems. In particular, the control structure includes three levels: (1) primary control; (2) secondary control; (3) tertiary control. The primary control mainly maintains stable voltage and frequency in DG through droop control when the load changes. However, primary control can only keep the output level within an acceptable range, and deviates from its required reference value [7]. Thus, secondary control was proposed to eliminate the voltage and frequency deviations of the offshore wind power system caused by primary control or the distributed power source that is connected.

Distributed secondary control was considered as an alternative to the traditional centralized secondary control, because it only needs neighboring information and does not require the central controller [8]. The issue of secondary control could be converted into a consensus problem of either the first or second order through the application of feedback linearization methods. As a result, a consistency algorithm based on MASs has widely been used in offshore wind power systems [9]. A cooperative control strategy for the distribution network voltage based on deep reinforcement learning of MASs was proposed in the literature [10]. Although this method improved distribution network voltage stability and reduced network communication loss, its control method required global information. So, it was not fully distributed. A model construction method based on automated fusion of MAS information was proposed in the literature [11]. The advantage of this method is its ability to accurately utilize the information conveyed by the secondary control and improve the efficiency of information fusion. However, it does not consider the cost issue. An autonomous control method for distributed energy storage systems within smart grids based on MAS consensus theory was also proposed in the literature [12]. This method analyzed the advantages and disadvantages of multiple droop controls, and achieved efficient energy management under different droop control methods. However, it does not consider the effect of communication bandwidth.

Based on the U-Q droop characteristics, the degree of line voltage deviation was divided into four levels by the authors of [13]. Combined with the consensus algorithm, this control strategy achieved a good convergence speed and was able to effectively improve voltage fluctuations. However, it was unable to adjust the voltage timely. In study [14], an offshore wind power system was subjected to multi-objective optimization. The voltage regulation index and the formula for line losses were presented through the three-phase power flow calculation equations. However, this control strategy could not precisely control the global voltage due to the limitations of the objective function. The key to achieving the secondary control goal was communication between each DG. Distributed controllers based on the consistency of multi-agent systems were highly flexible, adaptive, and robust, and were an effective way to achieve this goal. A distributed secondary controller for ring power systems was proposed in the literature [15], which realized the voltage regulation and current distribution of power systems with the help of a MASs consistency algorithm.

With increases in electricity consumption demand, offshore wind power systems are becoming increasingly complex [16,17]. The growing number of wind turbines and load demands have placing a significant strain on the communication infrastructure of the grid. Adding an event-triggered mechanism could effectively solve this problem, which has made it a hot research topic in recent years [18]. A newly distributed iterative event-triggered control scheme was proposed in [19] to synchronize the voltages of DGs to their rated values and provide optimal load sharing for their economic operation. A distributed event-triggered protocol was proposed in [20] to solve the secondary voltage recovery problem with a fully distributed control scheme. It is worth pointing out that the traditional, static event-triggered mechanism mentioned above could not further reduce unnecessary communication, and a better event-triggered mechanism needs to be proposed. In recent years, dynamic event-triggered mechanisms have been recognized as a more efficient way to alleviate communication pressure. Dynamic event-triggered mechanisms are more flexible and efficient than traditional, static event-triggered mechanisms, as has been proven in [21]. A dynamic event-triggered mechanism in secondary controllers was introduced in [22], which adjusted the thresholds of triggered conditions through internal dynamic variables. However, this triggered mechanism only optimized the event-triggered conditions. A distributed controller with higher precision that incorporated the dynamic event-triggered mechanism needed to be proposed so that it could reduce the number of triggered events and extend the communication intervals while ensuring that the system would still have a faster convergence rate and higher stability. A secondary robust frequency controller based on the dynamic event-triggered mechanism was proposed in [23] to overcome the uncertainties of renewable energy sources. This method extended communication intervals while maintaining the stability of the secondary frequency control. However, a scheme for secondary voltage control was not proposed in [23]. A distributed resilience restoration scheduling method for offshore wind transmission and distribution networks was proposed in [24]. This method considered the power quality of offshore wind farms under severe weather conditions such as typhoons and lightning strikes. However, it did not consider the problem of limited communication bandwidth in harsh environments. A frequency control strategy under non-periodic intermittent control based on wind energy was proposed in [25]. Although this method was effective in reducing the communication bandwidth, it could not guarantee the stability of the bus voltage. A brief description of the existing literature is summarized in

Table 1. Brief description of existing literature.

Year	Paper	Focus
2020	Zhang et al. [7]	Use droop control to maintain voltage stabilization within a certain range
2021	Liu et al. [15]	Secondary control of a toroidal DC microgrid
2024	Yang et al. [25]	Automatic power generation strategy with wind energy under nonperiodic intermittent control
2025	Yin et al. [10]	Cooperative control of voltage zoning in distribution networks based on deep reinforcement learning with multiple intelligences
2025	Cui et al. [11]	Smart Grid MASs Information Automation Fusion Model Construction
2025	Zhang et al. [12]	Autonomous Control Method for Distributed Energy Storage System within Smart Grid Based on Multi-Intelligent Body Coherence Theory

.

To sum up, this paper proposes a secondary coordinated control strategy for offshore wind power systems based on a dynamic event-triggered mechanism. The strategy effectively compensates for the voltage deviations resulting from primary control and load fluctuations. The main contributions of this paper are as follows:

(1)

In contrast to existing works [26,27,28,29,30,31], which rely on global information from the Laplacian matrix, this paper presents a novel consensus protocol for MASs that operates independently of global information, achieving a fully distributed approach.

(2)

Compared with previous research [32,33,34], this paper introduces a new, sufficient condition for leader-following consensus, offering greater feasibility and applicability to a broader range of linear MASs.

(3)

Compared to traditional, static event-triggered mechanisms, this paper proposes a dynamic event-triggered control scheme for offshore wind power systems. Unlike existing control approaches, the proposed scheme achieves faster convergence. By analyzing the voltage characteristics of the offshore wind power system, dynamic triggering conditions related to the system state are introduced to effectively reduce the communication burden and simplify system complexity.

The description of the symbols used in this article is shown in

Table 2. Description of symbols.

Notations	Definitions
$D{G}_s$	Distributed generations
$MASs$	Multi-agent systems
$N$	The number of MASs
$‖ \cdot ‖$	2-norm of vectors (matrices)
$t_{i|k}$	The k-th trigger moment of i-th MAS
$\mathrm{Re}{\lambda }_i$	Real part of the ith eigenvalue
${ℝ}^{m\times n}$	m × n order matrix
$X^T$	The transpose of the matrix
$X^{-1}$	The inverse of the matrix
$col(X)$	Column vector
$\otimes$	Kronecker product

.

2. Modeling Framework

2.1. Graph Theory

In this paper, each DG is considered an agent in MASs. The communication topology between the agents can be represented by an undirected graph, $G=\{T,E,A\}$, where $T=\{1,\cdots ,N\}$ denotes N nodes in the system. Denote the communication link between nodes by the set of edges as $E=T\times T$. $\mathit{A}=[a_{ij}]\in {ℝ}^{N\times N}$ denotes the adjacency matrix of the system. For the i-th node, $a_{ii}=0$. If $(T_j,T_i)\in E$, $a_{ij}=1$, else $a_{ij}=0$. Definitional degree matrix $\mathit{D}=\mathrm{diag}\{d_1,d_2,\cdots ,d_N\}$, where the diagonal elements are defined as $d_i={\displaystyle \sum _{j=1,j\ne i}^N}{a}_{ij}$. Define the Laplace matrix as $L=D-A$. Define the drawdown matrix as $B=\mathrm{diag}\left\{b_i\right\}\in {ℝ}^{N\times N}$. Consider an offshore wind power system where a virtual voltage reference value node exists to provide a reference voltage for DG units. If the $\mathrm{DG}i$ is able to receive the reference value from the virtual voltage reference value node, then $b_i=1$; otherwise, $b_i=0$. It is important to note that since the transmitted information is undirected, if $\mathrm{DG}j$ is a neighbor node of $\mathrm{DG}i$, then $a_{ij}=a_{ji}=1$; otherwise, $a_{ij}=a_{ji}=0$. A graph $G$ is considered connected if a path exists at any two nodes in the graph.

2.2. Structural Modeling of Offshore Wind Power Systems

The structure of the offshore wind power system is shown in Figure 1. Each unit of the offshore wind power system consists of a wind turbine, a droop controller, a voltage controller, a current controller, and a power calculation module, the structure of which is shown in Figure 2. Current controllers, voltage controllers and droop controllers constitute the primary control in hierarchical control. Voltage controllers are given in [35] with specific control laws. Droop control adjusts the voltage of each DG output according to the droop coefficient.

Droop control is generally a localized control scheme. Assuming that the line is purely inductive, which means that the output reactive power is related to the voltage magnitude, then the specific control equation of the droop controller can be expressed as:

$$\begin{array}{c}{V}_{ref}=V^{\#}-nQ\end{array}$$

(1)

where $\begin{array}{c}{V}^{\#}\end{array}$ is the output voltage amplitude of ${\mathrm{DG}}_i$, $\begin{array}{c}{V}_{ref}\end{array}$ is the voltage reference for ${\mathrm{DG}}_i$, $\begin{array}{c}Q\end{array}$ is the reactive power output value of ${\mathrm{DG}}_i$, and $\begin{array}{c}n\end{array}$ is the reactive power droop factor of ${\mathrm{DG}}_i$.

After the current controller and voltage controller, the large signal model of each distributed power supply can be represented as:

$$\left\{\begin{array}{l}{\dot{x}}_i=f_i(x_i)+k_i(x_i){\Phi }_i+g_i(x_i)u_i\hfill \\ y_i=h_i(x_i)\hfill \end{array}\right.$$

(2)

where x is a state variable of the 13th order. ${\Phi }_i,{\mathrm{f}}_1{,\mathrm{g}}_1$ are elaborated in the literature [36]. We then obtain the second order derivation of the output of the above equation of state:

$${\ddot{y}}_i=L_{F_i}^2{h}_i+L_{g_i}{L}_{F_i}{h}_i{u}_i$$

(3)

where

$$F_i(x_i)=f_i(x_i)+k_i(x_i){\Phi }_i$$

(4)

$L_{F_i}{h}_i$ is the Lie derivative. We define $L_{f_i}^2{h}_i+L_{g_i}{L}_{F_i}{h}_i{u}_i=v_i$. Then, Equations (2) and (4) can be presented as:

$${\ddot{y}}_i=v_i$$

(5)

The control signal u is represented as:

$$u_i={\left(L_{g_i}{L}_{F_i}{h}_i\right)}^{-1}\left(-L_{F_i}^2{h}_i+v_i\right)$$

(6)

After feedback linearization, the system dynamics equation, including N DGs, can be written as:

$$\left\{\begin{array}{l}{\dot{y}}_i\equiv y_{i,1}\hfill \\ {\dot{y}}_{i,1}=v_i,i=1,\cdots ,N\hfill \end{array}\right.$$

(7)

The dynamic model of the $D{G}_i$ after feedback linearization can be described as follows:

$${\dot{x}}_i(t)=A{x}_i(t)+B{u}_i(t),i=1,\dots ,N.$$

(8)

where $x_i={[v_{odi},{\dot{v}}_{odi}]}^T,A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],B={[0,1]}^T$.

The virtual voltage leader can be expressed as:

$$\begin{array}{c}{\dot{x}}_0(t)=A{x}_0(t)\end{array}$$

(9)

where $x_0={\left[\begin{array}{c}{v}_{ref},0\end{array}\right]}^T$.

Further, the following definition and assumptions need to be clarified.

Definition 1.

If the system consists of N agents and $x_i(t)$ denotes the MASs’ state values, then the basic definition of the system realizing state consistency is that, for any initial state, the following condition is satisfied:

$$\underset{t\to \infty }{\lim }\parallel x_i(t)-x_j(t)\parallel =0$$

(10)

Assumption 1.

The topologies of systems (8) and (9) have directed generation, with the leader as the root node.

Assumption 2.

(A,B) is stabilizable.

3. Fully Distributed Secondary Voltage Control Based on Adaptive Dynamic Event-Triggered Mechanisms

3.1. Dynamic Event Trigger Controller Design

This section designs adaptive dynamic event-triggered consistency controls based on MASs. The sufficiency of the proposed control protocol is then demonstrated, and it is worth noting that Zeno’s behavior can be avoided.

Using the consistency protocol, the DG state voltage error function is defined as follows:

$$\begin{array}{c}{\mathcal{Z}}_i(t)={\displaystyle \sum _{j=1}^N}{a}_{ij}(x_i(t)-x_j(t))+g_i(x_i(t)-x_0(t))\end{array}$$

(11)

Considering the latest triggered instant $t_{i|k}$, the measurement error ${\mathcal{G}}_i(t)$ can be expressed as follows:

$${\mathcal{G}}_i(t)={\mathcal{Z}}_i(t_{i|k})-{\mathcal{Z}}_i(t)$$

(12)

where $t_{i|0}=0$. All triggered sequences for the secondary voltage recovery controller of ${\mathrm{DG}}_i$ are $t_{i|0},t_{i|1},t_{i|2},\cdots t_{i|\mathrm{N}}$; triggering depends on the following conditions:

$$\begin{array}{c}{t}_{i|k+1}=\inf \{t:t>t_{i|k}\hspace{1em}\mathrm{and}\hspace{1em}{\mathcal{T}}_i(t)⩾0\}\end{array}$$

(13)

The triggered function, ${\mathcal{T}}_i(t)$, is denoted as:

$${\mathcal{T}}_i(t)={\mathcal{C}}_i(t){\mathcal{G}}_i^T(t)F{\mathcal{G}}_i(t)-\eta {\mathcal{C}}_i(t){\mathcal{Z}}_i^T(t)F{\mathcal{Z}}_i(t)-\beta e^{-\gamma t}$$

(14)

where $\beta ,\eta ,\gamma$ are positive parameters to be designed. ${\mathcal{C}}_i(t)$ is updated to:

$${\dot{\mathcal{C}}}_i(t)=\alpha {\mathcal{Z}}_i^T(t_{i|k})F{\mathcal{Z}}_i(t_{i|k}),\hspace{1em}t\in [t_{i|k},t_{i|k+1})$$

(15)

where $F$ is the controller gain matrix of the corresponding dimension, to be defined later. $\alpha$ is a positive constant.

Consider the voltage recovery objective, which is $\underset{t\to \infty }{\lim }{v}_{odi}\left(t\right)-v_{ref}(t)=0$ $i=1,\cdots ,N$. By combining the event-triggered control, the voltage controller of the ${\mathrm{DG}}_i$ is designed as follows:

$$u_i(t)=-{\mathcal{C}}_i(t)K{\mathcal{Z}}_i(t_{i|k}),\hspace{1em}t\in [t_{i|k},t_{i|k+1})$$

(16)

where $K\in {ℝ}^{m\times n}$ is the controller gain matrix.

Notably, the consensus protocol was designed and implemented using only local information from each agent and its neighboring agents. Unlike the traditional, static event-triggered mechanism, this paper introduces dynamic variables ${\mathcal{C}}_i(t)$ and ${\mathcal{T}}_i(t)$ in the event-triggered function. The event-triggered threshold can be adjusted according to the deviation of the system state, so that the system can be regulated more rationally.

Lemma 1.

If there exists a spanning tree with the leader as the root in the undirected graph $G$, then there is $Re{\lambda }_i(H)>0$ holds for all entries ${\lambda }_i(H)\in \lambda (H)$, where $H=L+B$.

Lemma 2.

There exists a symmetric positive definite matrix $P_H>0$ such that $P_{H}H+H^T{P}_H>0$ holds, provided that a matrix $H$ satisfies the condition $Re{\lambda }_i(H)>0$.

Theorem 1.

Consider the MASs composed of Equations (8) and (9) under the event-based adaptive control protocol given in (16). The control gains $F$ and $K$ are designed as $F=PB{B}^{T}P,K=B^{T}P$. MAS consensus can be achieved for $\beta >0,0<\eta <1,\gamma >0$ if assumption 1 is satisfied. P represents the positive definite solution to the control algebraic Riccati equation (ARE).

$$PA+A^{T}P-PB{B}^{T}P+Q=0$$

(17)

Proof. 

Then, using (8) and (9), (11) and (12), (15) and (16), we derive the trajectories of the system:

$${\dot{x}}_i=A{x}_i-c_{i}BK({\mathcal{Z}}_i+{\mathcal{G}}_i),\hspace{1em}i\in N_i$$

(18)

Equation (18) can be further expressed as:

$$\dot{x}=(I\otimes A)x-(C\otimes BK)(\mathcal{G}+\mathcal{Z})$$

(19)

where $\mathcal{G}=col({\mathcal{G}}_1,{\mathcal{G}}_2,\cdots ,{\mathcal{G}}_N)$. $\mathcal{Z}=col({\mathcal{Z}}_1,{\mathcal{Z}}_2,\cdots ,{\mathcal{Z}}_N)$. $x=col(x_1,x_2,\cdots ,x_N)$. $\otimes$ denotes the Kronecker product.

We derive the trajectories of the error system as ${\epsilon }_i=x_i-x_0$. ${\dot{\epsilon }}_i$ can be expressed as follows:

$$\dot{\epsilon }=\dot{x}-{\dot{x}}_0=(I\otimes A)\epsilon -(C\otimes BK)(\mathcal{Z}+\mathcal{G})$$

(20)

It follows directly that $\underset{t\to 0}{\lim }{x}_i(t)-x_0(t)=0$ holds if and only if $\underset{t\to 0}{\lim }\epsilon (t)=0$.

We then construct the Liapunov function [37]:

$$V={\epsilon }^T(H\otimes P)\epsilon +\sum _{i\in N_i}{\displaystyle \frac{1-\eta }{4\alpha (1+\eta )}}{({\mathcal{C}}_i-\overline{\mathcal{C}})}^2$$

(21)

where H = L + G. $\overline{\mathcal{C}}$ can be expressed as:

$$\overline{\mathcal{C}}={\displaystyle \frac{2(1+\eta )(1+\delta )}{{\lambda }_1\delta (1-\eta )(1-\eta -\eta \delta )}},0<\delta <{\displaystyle \frac{1-\eta }{\eta }}$$

(22)

Observe that $q=(H\otimes I)\epsilon$ holds due to $(L\otimes I)x_0=0$. The time derivative of $V$ along $\epsilon$ is:

$$\begin{array}{ll}\dot{V}& =2{\epsilon }^T(H\otimes P)\dot{\epsilon }+{\displaystyle \sum _{i\in N_i}}{\displaystyle \frac{1-\eta }{2\alpha (1+\eta )}}({\mathcal{C}}_i-\overline{\mathcal{C}}){\dot{\mathcal{C}}}_i\\ & =2{\epsilon }^T(H\otimes PA)\epsilon -2{\mathcal{Z}}^T(C\otimes F)\mathcal{Z}-2{\mathcal{Z}}^T(C\otimes F)e+\\ & {\displaystyle \sum _{i\in N_i}}{\displaystyle \frac{1-\eta }{2(1+\eta )}}({\mathcal{C}}_i-\overline{\mathcal{C}})\parallel K{\mathcal{Z}}_i(t_k^i){\parallel }^2\end{array}$$

(23)

The following three inequalities hold [38]:

$$-2{\mathcal{Z}}^T(C\otimes F)\mathcal{G}⩽{\mathcal{Z}}^T(C\otimes F)\mathcal{Z}+{\mathcal{G}}^T(C\otimes F)\mathcal{G}$$

(24)

$$\sum _{i\in N_i}{\displaystyle \frac{1-\eta }{2(1+\eta )}}{c}_i\parallel K{\mathcal{Z}}_i(t_{i|k}){\parallel }^2⩽{\displaystyle \frac{1-\eta }{1+\eta }}{\mathcal{Z}}^T(C\otimes F)\mathcal{Z}+{\displaystyle \frac{1-\eta }{1+\eta }}{\mathcal{G}}^T(C\otimes F)\mathcal{G}$$

(25)

$$-\parallel K{\mathcal{Z}}_i(t_{i|k}){\parallel }^2⩽{\displaystyle \frac{\delta \beta }{{\mathcal{C}}_i}}{e}^{-\gamma t}-{\displaystyle \frac{\delta (1-\eta -\eta \delta )}{1+\delta }}\parallel K{\mathcal{Z}}_i{\parallel }^2$$

(26)

Combining Equations (23) and (24), the $\dot{V}$ can be expressed as:

$$\dot{V}⩽2{\epsilon }^T(H\otimes PA)\epsilon -{\mathcal{Z}}^T(C\otimes F)\mathcal{Z}+{\mathcal{G}}^T(C\otimes F)\mathcal{G}+\sum _{i\in N_i}{\displaystyle \frac{1-\eta }{2(1+\eta )}}({\mathcal{C}}_i-\overline{\mathcal{C}})\parallel K{\mathcal{Z}}_i(t_k^i){\parallel }^2$$

(27)

Combining Equations (23)–(25), the $\dot{V}$ can be expressed as:

$$\dot{V}⩽2{\epsilon }^T(H\otimes PA)\epsilon -{\displaystyle \frac{2\eta }{1+\eta }}{\mathcal{Z}}^T(C\otimes F)\mathcal{Z}+{\displaystyle \frac{2}{1+\eta }}{\mathcal{G}}^T(C\otimes F)\mathcal{G}-\sum _{i\in N_i}{\displaystyle \frac{\overline{\mathcal{C}}(1-\eta )}{2(1+\eta )}}\parallel K{\mathcal{Z}}_i(t_{i|k}){\parallel }^2$$

(28)

It is worthwhile to note that in Equation (14), ${\mathcal{C}}_i(t){\mathcal{G}}_i^T(t)F{\mathcal{G}}_i(t)<\eta {\mathcal{C}}_i(t){\mathcal{Z}}_i^T(t)$. $F{\mathcal{Z}}_i(t)-\beta e^{-\gamma t}$. That is:

$${\mathcal{G}}^T(C\otimes F)\mathcal{G}⩽\eta {\mathcal{Z}}^T(C\otimes F)\mathcal{Z}+N\beta e^{-\gamma t}$$

(29)

The $\dot{V}$ can further be expressed as:

$$\dot{V}⩽2{\epsilon }^T(H\otimes PA)\epsilon -\sum _{i\in N_i}{\displaystyle \frac{\overline{\mathcal{C}}(1-\eta )}{2(1+\eta )}}\parallel K{\mathcal{Z}}_i(t_{i|k}){\parallel }^2+{\displaystyle \frac{2N\beta }{1+\eta }}{e}^{-\gamma t}$$

(30)

By applying inequality (26), one can derive the following result:

$$\begin{array}{ll}\dot{V}⩽& 2{\epsilon }^T(H\otimes PA)\epsilon -{\displaystyle \frac{\overline{\mathcal{C}}(1-\eta )\delta (1-\gamma -\eta \delta )}{2(1+\eta )(1+\delta )}}\\ & {\displaystyle \sum _{i\in N_i}}\parallel K{\mathcal{Z}}_i{\parallel }^2+{\displaystyle \sum _{i\in N_i}}{\displaystyle \frac{\overline{\mathcal{C}}(1-\eta )\delta \beta }{2(1+\eta ){\mathcal{C}}_i}}{e}^{-\gamma t}+{\displaystyle \frac{2N\beta }{1+\eta }}{e}^{-\gamma t}\end{array}$$

(31)

By simplifying,

$$\dot{V}⩽2{\epsilon }^T(H\otimes PA)\epsilon -\overline{\mathcal{C}}\varphi {\epsilon }^T(H^2\otimes F)\epsilon +\phi e^{-\gamma t}$$

(32)

where $\varphi ={\displaystyle \frac{(1-\eta )\delta (1-\eta -\eta \delta )}{2(1+\eta )(1+\delta )}}$, and $\phi ={\displaystyle \frac{N\overline{\mathcal{C}}(1-\eta )\delta \beta }{2(1+\eta )\min {\mathcal{C}}_i(0)}}+{\displaystyle \frac{2N\beta }{1+\eta }}$. Since the range of $\delta$ is $0<\delta <\begin{array}{c}{\displaystyle \frac{1-\eta }{\eta }}\end{array}$, $\varphi ,\phi ⩾0$.

There is an orthogonal matrix $M$ satisfying:

$$M^{-1}HM=M^{T}HM=\Lambda =diag({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N)$$

(33)

where ${\lambda }_i\ge 0$ is the eigenvalue of matrix $H$.

Define $J(t)=(M^T\otimes I)\epsilon (t)$. Equation (32) can be written as:

$$\begin{array}{ll}\dot{V}& ⩽2J^T(\Lambda \otimes PA)J-\overline{\mathcal{C}}\varphi J^T({\Lambda }^2\otimes F)J+\phi e^{-\gamma t}\\ & ={\displaystyle \sum _{i=1}^N}{\lambda }_i{J}_i^T[A^{T}P+PA-\overline{\mathcal{C}}\varphi {\lambda }_{i}PB{B}^{T}P]J_i+\phi e^{-\gamma t}\\ & ⩽{\displaystyle \sum _{i=1}^N}{\lambda }_i{J}_i^T(A^{T}P+PA-PB{B}^{T}P)J_i+\phi e^{-\gamma t}\\ & =-{\displaystyle \sum _{i=1}^N}{\lambda }_i{J}_i^{T}Q{J}_i+\phi e^{-\gamma t}\end{array}$$

(34)

From the ARE in Equation (17), it follows that Q > 0, which indicates that $\begin{array}{c}\underset{t\to \infty }{\lim }\dot{V}(t)⩽0\end{array}$ through Equation (34). Therefore, we conclude that $\underset{t\to \infty }{\lim }\epsilon (t)\to 0$. The proof is complete. □

The control protocol proposed in this paper does not rely on global network communication, and is fully distributed and scalable.

3.2. No Zeno Behavior Proof

Theorem 2.

Consider the MASs consisting of (8) and (9) with the event-based adaptive control protocol of (16). The Zeno phenomenon can be avoided in this paper.

Proof. 

Derivation of ${\mathcal{G}}_i^T(t)F{\mathcal{G}}_i(t)$ at $\left[t_{i|k},t_{i|k+1}\right)$:

$$\begin{array}{ll}{D}^{+}({\mathcal{G}}_i^T(t)F{\mathcal{G}}_i(t))& =D^{+}\parallel K{\mathcal{G}}_i{\parallel }^2⩽\parallel 2{\mathcal{G}}_i^T{K}^{T}K{\dot{\mathcal{G}}}_i\parallel \\ & =\parallel 2{\mathcal{G}}_i^T{K}^{T}K{\dot{\mathcal{Z}}}_i\parallel =2{\mathcal{G}}_i^T{K}^{T}K{\displaystyle \sum _{i=1}^N}{a}_{ij}(A(x_i-x_j)\\ & +B(u_i-u_j))+g_i(A{x}_i+B{u}_i)\parallel ⩽\mathbb{M}\end{array}$$

(35)

where $M={\max }_i\sup t\in [t_{i|k},t_{i|k+1})\parallel {\displaystyle \sum _{i=1}^N}{a}_{ij}(A(x_i-x_j)+B(u_i-u_j))+g_i(A{x}_i+B{u}_i)\parallel$.

The next event time instant is ${\mathcal{T}}_i(t)$ = 0:

$$\parallel K{e}_i{\parallel }^2⩽\mathbb{M}(t-t_k^i)$$

(36)

$$\begin{array}{c}{c}_i\parallel K{\mathcal{G}}_i(t_{ik+1}){\parallel }^2=\eta c_i\parallel K{\mathcal{Z}}_i(t_{ik+1}){\parallel }^2+\beta e^{-\gamma t_{ik+1}}\end{array}$$

(37)

According to Equations (36) and (37):

$$\begin{array}{c}\beta c_i^{-1}\mathcal{G}⩽\Vert K{\mathcal{G}}_i(t_{ik+1}){\Vert }^2⩽\mathbb{M}(t_{ik+1}-t_{ik})\end{array}$$

(38)

The inequality (38) shows that $\tau >0$ holds for any finite horizon. Consequently, the Zeno behavior for any agent $i$ is fundamentally eliminated. The proof is complete. □

4. Simulation Results

To verify the feasibility of the above proposed secondary control of the offshore wind power system under a dynamic event-triggered mechanism, the voltage simulation results are provided in this section. The offshore wind power system model with 4 $DG$ and a rated output voltage amplitude of 380 V is constructed in MATLAB R2023b, which is shown in Figure 3. The communication topology is shown in Figure 4. The electrical and control parameters are shown in

Table 3. Table of electrical and control parameters for offshore wind power systems.

Description	Symbol	Values
DGi	$m_P$	$1.5\times {10}^{-5}$
	$n_Q$	$2\times {10}^{-4}$
	$R_C$	$0.35 \Omega$
	$L_C$	$1.847 \mathrm{mH}$
	$K_{PV}$	$0.2013$
	$K_{IV}$	$480$
	$K_{PC}$	$6.72$
	$K_{IC}$	$3360$
Lines	Line 1 & Line 3	Line 2
	$R_{l1}=R_{l2}=0.2 \Omega$	$R_{l1}=0.35 \Omega$
	$L_{l1}=L_{l2}=0.318 \mathrm{mH}$	$L_{l1}=1.847 \mathrm{mH}$
RL Loads	$P_1=P_2=50 \mathrm{kW}$	$Q_1=Q_2=12 \mathrm{kVAr}$

.

Consider systems (8) and (9), each of which has a leader as well as a system of four followers, with the topology of information transfer as shown in Figure 4. Choosing $\alpha =0.3,\beta =0.1,\eta =0.5,\gamma =0.05$, according to the theorem, the unknown gain matrix of the controller (15)–(18) is $K=\left[\begin{array}{cc}1& 1.7321\end{array}\right]$, $F=\left[\begin{array}{cc}1& 1.7321\\ 1.7321& 3\end{array}\right]$.

Case 1.

Response experiment for the proposed strategy.

In order to show the tracking performance of the proposed fully distributed event-triggering algorithm, we perform two sets of experiments for comparison. Figure 5 shows the simulation without using the event-triggered algorithm, and Figure 6 shows the simulation with the event-triggered algorithm. The secondary control algorithm is enabled at moment 0, and all supply voltage amplitudes return to the reference value within 10 s. By comparing Figure 5 and Figure 6, it can be concluded that the control strategy proposed in this paper significantly reduces communication bandwidth while maintaining the same level of control performance as the full communication approach. The triggering instant of each distributed power supply is shown in Figure 6c, and no Zeno phenomenon occurs.

Case 2.

Load switching test experiment.

The waveform response of load switching with the proposed strategy is shown in Figure 7. The load is increased to two times its original value at 20 s, and all supply voltage amplitudes return to the reference value within 8 s. From the figure, it can be seen that the load switching voltage amplitude stabilizes at the desired steady state after small fluctuations. This indicates that the system can remain stable after fluctuations after load switching using the proposed strategy. The triggering instant of each distributed power supply is shown in Figure 7c, and no Zeno phenomenon occurs.

Case 3.

Plug-and-play test experiments.

To validate the plug-and-play performance of the system, $D{G}_1-D{G}_3$ form a system at 0 s and $D{G}_4$ is connected to the system at 20 s. From Figure 8, it can be seen that $D{G}_1-D{G}_3$ can still be expected to form a complete system and still receive information from the virtual leader node. Therefore, $D{G}_1-D{G}_3$ can still realize the purpose of secondary coordinated control. When $D{G}_4$ is connected, the system can be stabilized within 5 s, after undergoing minor oscillations. The triggering instant of each distributed power supply is shown in Figure 8c, and no Zeno phenomenon occurs.

To clearly delineate the advancements of our approach, a detailed comparison with existing methods in the literature are shown in

Table 4. Comparison of the number of triggers under different trigger mechanisms.

Method Name	[39]	[40]	[41]	[This Paper]
Global information	Yes	Yes	Yes	No
Triggering number	High	High	Middle	Low

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5. Conclusions

Since the high communication bandwidth requirements and reliance on global topology information in previous studies have hindered large-scale consumption of offshore wind power energy, this paper has proposed a dynamic event-triggered coordinated control strategy for fully distributed offshore wind power systems, which has effectively combined adaptive control and event-triggered mechanisms. It has realized the coordinated control of DG voltage, while reducing communication burdens and saving resources, along with decreasing the number of communications. The validity of the proposed fully distributed dynamic event-triggered secondary control has been illustrated by three numerical examples of the offshore wind power system’s use during a response experiment, load switching, and plug-and-play conditions. The proposed strategy is efficient and reliable, which is of significance for the development and application of offshore wind power systems.

However, the communication network topology utilized in this paper was undirected, and network attack situations have not been taken into account. Future research will focus on resolving these two issues.