Dynamic Integral-Event-Triggered Control of Photovoltaic Microgrids with Multimodal Deception Attacks

Abstract

With the rapid development of smart grid technologies, communication networks have become the core infrastructure supporting control and energy optimization in microgrids. However, the excessive reliance of microgrid control on communication networks faces dual challenges: On one hand, the high-frequency information exchange under traditional periodic communication patterns causes severe waste of network resources; on the other hand, cyberattacks may cause information loss, abnormal delays, or data tampering, which can ultimately lead to system instability. To address these challenges, this paper investigates the secure dynamic integral event-triggered stabilization of photovoltaic microgrids under multimodal deception attacks. To address the communication resource constraints in photovoltaic (PV) microgrid systems, a dynamic integral-event-triggered scheme (DIETS) is proposed. This scheme employs average processing of historical state data to filter out redundant triggering events caused by noise or disturbances. Simultaneously, a time-varying triggering threshold function is designed by integrating real-time system states and historical information trends, enabling adaptive adjustment of dynamic triggering thresholds. In terms of cybersecurity, a secure control strategy against multi-modal deception attacks is incorporated to enhance system resilience. Subsequently, through the Lyapunov–Krasovskii functional and Bessel–Legendre inequality, collaborative design conditions for the controller gain and triggering matrix are formed as symmetric linear matrix inequalities to ensure system stability. The simulation results demonstrate that DIETS recorded only 99 triggering events, achieving a 55.2% reduction compared to the normal event-triggered scheme (ETS) and a 52.6% decrease relative to dynamic ETS, verifying the outstanding communication effectiveness of DIETS.

1. Introduction

Microgrids, as intelligent power systems integrating distributed energy resources, energy storage systems, and loads, offer core advantages in terms of enhancing the penetration of renewable energy, optimizing energy utilization efficiency, and strengthening the grid’s resilience to disturbances [1]. As renewable energy and energy storage systems are increasingly integrated into power networks, coupled with the emergence of novel electrical demands like data centers and electric vehicles, microgrids have become critical enablers for modern power systems transitioning toward decentralization, intelligence, and sustainability [2,3]. Among all renewable energy sources, particularly PV power generation has emerged as a leading source of energy today due to cost-effectiveness, high efficiency, environmental advantages and flexible deployment capabilities [4,5,6].

As essential components of modern power systems, microgrids maintain a bidirectional interactive relationship with the main grid, operating under grid-connected model and islanded model [7,8,9,10,11]. In grid-connected mode, the stable operation of microgrids heavily relies on coordinated interactions with the main grid and real-time state perception/control enabled by communication networks. Through the Point of Common Coupling (PCC), microgrids continuously exchange critical parameters with the main grid, including power flow, voltage phase, and frequency synchronization. This enables microgrids to make dynamic adjustments of local generation/storage outputs to mitigate the impacts of renewable energy fluctuations on the main grid. Concurrently, the main grid requires real-time access to microgrid data, such as generation forecasts, energy storage State of Charge (SOC), and load demand profiles to optimize system-wide dispatch strategies [12,13]. Therefore, ensuring communication network resource optimization and precise synchronization of critical information—while maintaining data integrity and confidentiality—constitutes a fundamental requirement for guaranteeing coordinated control and secure, stable system operation.

As microgrids and smart grids advance rapidly, communication networks have become the core infrastructure for distributed control and energy optimization scheduling. However, the deep integration of communication networks has also introduced multidimensional challenges, including cybersecurity threats, communication delays, inadequate real-time performance, and network reliability [14,15,16]. To address these challenges, researchers have proposed event-triggered control (ETC) methods [17,18,19,20]. Compared to traditional periodic control, ETC methods employ “on-demand communication” for data sampling and control updates, ensuring control performance while significantly reducing communication loads. Reference [21] proposes a distributed secondary control strategy incorporating consensus estimators and an event-triggered scheme (ETS), effectively achieving proportional current sharing and voltage regulation in single-bus DC microgrids while reducing communication traffic. To avoid Zeno behavior, reference [22] presents an improved event-triggered distributed secondary control strategy based on sampled data. The Lyapunov function method is employed to establish stability conditions for this control strategy under fixed and switching communication topologies, ensuring system stability. While traditional ETC methods based on fixed thresholds or periodic sampling can reduce communication frequency, their rigid trigger conditions often lead to redundant triggering and response delays.

Due to the distributed nature of microgrids, their control information and data are primarily transmitted through communication networks. Once the network is subjected to attacks, issues such as loss of control information, delays, or tampering may occur, potentially leading to microgrid instability or even collapse [23,24,25]. When a microgrid is subjected to False Data Injection (FDI) attacks, voltage and frequency fluctuations can be induced, compromising system stability [26]. In reference [27], a distributed dynamic event-triggered k-step attack observer is designed, and an elastic controller is developed based on the attack signals estimated by the observer. This method has been demonstrated to effectively ensure stable control of voltage and frequency under FDI attacks. During the operation of a microgrid, Denial of Service (DoS) attacks can cause communication interruptions and data loss, thereby undermining the real-time performance and consistency of the system and severely affecting its stability [28]. To address this issue, reference [29] proposed an event-triggered distributed consensus control method. A switching framework between communication intervals and attack intervals is considered, and sufficient conditions for the frequency and duration of DoS attacks are derived, ensuring system stability and secure current sharing. Additionally, deception attacks also pose a significant threat to microgrids. Attackers may forge or tamper with sensor data or control commands, misleading the system into making incorrect decisions and further exacerbating system instability [23,24,30]. To mitigate such attacks, reference [31] designed an improved attack-resilient event-triggered method. The attack signals are modeled as bounded deceptive false data, and the impact of attacks and disturbances is reduced through event-triggered functions and algorithms, achieving bounded consensus and effectively alleviating the threat of spoofing attacks to system stability. Therefore, considering the potential threats of communication network failures, control failures leading to system collapse, and economic losses, it is imperative to develop effective methods to enhance the fault tolerance capability and security of microgrids against network attacks, thereby ensuring stable operation and reliable power supply.

Motivated by the aforementioned observations, this article studies the dynamic integral-event-triggered control problem for PV microgrids subject to multimodal deception attacks. The contributions are concluded as follows:

• A DIETS is proposed to mitigate redundant communication caused by noise and disturbances. Unlike normal ETSs that rely on instantaneous states, DIETS employs the mean of historical states to mitigate false triggering events induced by noise or disturbances, and designs an adaptive triggering threshold function that dynamically adjusts the threshold by integrating the system’s historical state information.
• A secure controller against multimodal deception attacks is developed. By introducing a Bernoulli stochastic process to characterize the time-varying characteristics of multimodal deception attacks and combining it with DIETS to achieve dynamic suppression of multimodal deceptive signals, the system’s resilience was effectively enhanced.
• Bessel–Legendre-based inequality is introduced to handle the integral term induced by the historical state and related Lyapunov–Krasovskii functional. Then, the conservatism of stability criteria obtained by this technique is reduced compared to the existing approximation method based on the Simpson rule.

The remainder of this paper is structured as follows: Section 2 describes the model formulation of PV microgrids and presents the proposed DIETS. Section 3 develops the main stability analysis framework using Lyapunov–Krasovskii functionals and Bessel–Legendre inequality techniques. Section 4 provides comprehensive simulation results validating the communication efficiency and security performance of the proposed method. Section 5 discusses the limitations and considerations. Finally, Section 6 concludes the paper with remarks on contributions.

2. Preliminaries

Notation: Let ${\mathbb{R}}^n$ denote the n-dimensional Euclidean space, and ${\mathbb{R}}^{n\times m}$ represent the set of all $n\times m$ real matrices. The notation $He\left(X\right)=X+X^T$ is used, where $X^T$ is the transpose of matrix X. The operator ⊗ denotes the Kronecker product. $I_a$ represents the $a\times a$ identity matrix. The expectation operator is denoted by $\mathbb{E}\{·\}$.

2.1. Single-Phase Inverters in PV Microgrids

In modern microgrids, distributed energy resources, such as PV systems, tidal turbines, and energy storage, are integrated into the grid through power electronic inverters. As essential nodes for energy conversion, the dynamic characteristics of inverters directly influence the overall behavior of the microgrid. Therefore, the state information of inverters can be utilized as the state information of the PV microgrid. The core function of the Boost circuit is to elevate the low voltage output from the photovoltaic array to the bus voltage required by the inverter. The relationship between its output voltage and duty cycle is borrowed from [32] and given as follows:

$$V_b={\displaystyle \frac{V_S}{1-D_b}},$$

(1)

where $V_b$ represents the boosted voltage, $D_b$ is the duty cycle of the Boost circuit, and $V_S$ denotes the voltage of the microgrid storage.

Since the inverter’s output voltage is governed by the control signal u, the resulting AC voltage $V_{in}$ is defined as follows:

$$V_{in}=u{V}_b={\displaystyle \frac{u{V}_S}{1-D_b}},$$

(2)

The application of Kirchhoff’s Voltage Law (KVL) to the loop in Figure 1 can be expressed as

$$\begin{array}{cc}\hfill V_{in}& =i_{Lf}{R}_f+L_f{\displaystyle \frac{d{i}_{Lf}}{dt}}+v_{Cf}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill v_{Cf}& ={\displaystyle \frac{v_0}{R_L}}{R}_o+L_o{\displaystyle \frac{d{i}_o}{dt}}+v_o\hfill \end{array}$$

(4)

where $i_{Lf}$ is the current through the filter inductor $L_f$, $R_f$ represents the equivalent series resistance of the inductor, and $v_{Cf}$ is the voltage across the filter capacitor $C_f$, $v_o$ is the output voltage across the load, $R_L$ is the load of the distributed generation unit, $L_o$ is the inductance of the transmission line, and $R_o$ represents the resistance of the transmission line.

Combine (2) and (3), the rate of change in the inductor current is set by the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{i}_{Lf}}{dt}}={\displaystyle \frac{1}{L_f}}\left({\displaystyle \frac{u\left(t\right)V_S}{1-D_b}}-R_f{i}_{Lf}-v_{Cf}\right),\end{array}$$

(5)

where the control signal u is chosen to be $u\left(t\right)$.

The load voltage $v_o$, influenced by the filter inductor and load resistance, can be expressed as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_o}{dt}}& ={\displaystyle \frac{R_L}{L_o}}\left(v_{Cf}-{\displaystyle \frac{R_o+R_L}{R_L}}{v}_o\right),\hfill \end{array}$$

(6)

where $v_o$ is the output voltage across the load, $R_L$ is the load resistance, $L_o$ is the output inductance, and $R_o$ represents the equivalent series resistance in the output path.

According to Kirchhoff’s Current Law (KCL), the charging/discharging process of the capacitor $C_f$ is determined by the difference between the inductor current and the load current:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{Cf}}{dt}}& ={\displaystyle \frac{1}{C_f}}\left(i_{Lf}-{\displaystyle \frac{v_o}{R_L}}\right).\hfill \end{array}$$

(7)

By defining the global state vector $\chi \left(t\right)=[i_{L_f},v_0,v_{C_f}]$ and control input $u\left(t\right)$, the overall system dynamics is given by the following:

$$\left\{\begin{array}{c}\dot{\chi }\left(t\right)=Q\chi \left(t\right)+Ju\left(t\right),\hfill \\ y\left(t\right)=E\chi \left(t\right),\hfill \end{array}\right.$$

(8)

where

$$Q=\left[\begin{array}{ccc}-{\displaystyle \frac{R_f}{L_f}}& 0& -{\displaystyle \frac{1}{L_f}}\\ 0& -{\displaystyle \frac{R_o+R_L}{L_o}}& {\displaystyle \frac{R_L}{L_o}}\\ {\displaystyle \frac{1}{C_f}}& -{\displaystyle \frac{1}{C_f{R}_L}}& 0\end{array}\right],J=\left[\begin{array}{c}{\displaystyle \frac{V_S}{L_f(1-D_b)}}\\ 0\\ 0\end{array}\right],E=\left[\begin{array}{ccc}1& 1& 0\end{array}\right].$$

(9)

2.2. Dynamic Integral-Event-Triggered Scheme

By considering the limitation of communication bandwidth, the following DIETS based on the average value of memory outputs are presented to improve network resource utilization, which is given as

$$t_{k+1}=\underset{t}{sup}\left\{t\ge t_k|{ϵ}^T\left(t\right)\mathsf{\Phi }ϵ\left(t\right)\le \delta \left(t\right){\tilde{y}}^T\left(t\right)\mathsf{\Phi }\tilde{y}\left(t\right)\right\},$$

(10)

where $t_k$ and $t_{k+1}$ are the latest and next triggering times, $\tilde{y}\left(t\right)\triangleq {\displaystyle \frac{1}{h}}{\int }_{-h}^{0}y(t+s)ds$ characterizes the averaged historical state data and $ϵ\left(t\right)=\tilde{y}\left(t\right)-y\left(t_k\right)$ represents the discrepancy between the integrated state and its triggered value, $\mathsf{\Phi }>0$ is a weighting matrix and $h>0$ is the time span of the historical memory interval. The dynamic triggering threshold $\delta \left(t\right)$ is defined as $\delta \left(t\right)={\delta }_m-({\delta }_m-{\delta }_M)e^{-∥\tilde{y}\left(t\right)∥}$, with $\delta \left(t\right)\in [{\delta }_m,{\delta }_M](0<{\delta }_m\le {\delta }_M\le 1)$.

Remark 1.

The proposed DIETS can smooth system state variations and reduce the frequent triggering caused by instantaneous fluctuations or noises by averaging historical data. Compared to traditional ETSs [17,18] , relying on instantaneous states and fixed triggering threshold, DIETS demonstrates superior performance in complex environments, enhancing control accuracy, system stability, and robustness while mitigating the impact of random disturbances.

Remark 2.

Notably, when the time interval h is reduced to zero and the triggering threshold is fixed, the DIETS degenerates to

$$\begin{array}{c}\hfill t_{k+1}=\underset{t}{sup}\left\{t\ge t_k\mid {\widetilde{ϵ}}^T\left(t\right)\mathsf{\Phi }\widetilde{ϵ}\left(t\right)\le \delta y{\left(t\right)}^T\mathsf{\Phi }y\left(t\right)\right\},\end{array}$$

(11)

where $\widetilde{ϵ}\left(t\right)=y\left(t\right)-y\left(t_k\right)$ is the instantaneous error. This generalization demonstrates that the DIETS covers some existing normal ETSs [17,18] as specific cases.

2.3. Closed-Loop Event-Triggered Control System

Due to the vulnerability of the communication network, the control signal is highly possible to be attacked by malicious adversaries. Then, the actual secure static output controller under multimodal deception attacks is modeled as follows:

$$\begin{array}{cc}\hfill u\left(t\right)& =K(y\left(t_k\right)+\sum _{g=1}^d{\alpha }_g\left(t\right){\sigma }_g\left(y\left(t\right)\right))\hfill \\ \hfill & =K\left(E{\int }_{-h}^0{\displaystyle \frac{1}{h}}\chi (t+s)ds-ϵ\left(t\right)\right)+K\beta \left(t\right)\sum _{g=1}^d{\alpha }_g\left(t\right){\sigma }_g\left(y\left(t\right)\right),\hfill \end{array}$$

(12)

where $\beta \left(t\right)$ and ${\alpha }_g\left(t\right)$ are mode-dependent stochastic Bernoulli variables, meaning that the multimodel deception attack happens or not, and different modes of attack happen or not. Specifically, when $\beta \left(t\right)=1$, it signifies that the control signal is injected by a deception attack. Conversely, when $\beta \left(t\right)=0$, it denotes that the communication network operates normally. In addition, ${\alpha }_1\left(t\right),\dots ,{\alpha }_d\left(t\right)(g=1,\dots ,d)$ are independent Bernoulli variables and satisfy the following constraints:

$$\begin{array}{c}\hfill E\left\{\beta \left(t\right)\right\}=\overline{\beta },E\left\{{\alpha }_g\left(t\right)\right\}={\overline{\alpha }}_g,\sum _{g=1}^d{\alpha }_g\left(t\right)=1,\sum _{g=1}^d{\overline{\alpha }}_g=1,{\overline{\alpha }}_g\le 1.\end{array}$$

(13)

The deception attack function ${\sigma }_g\left(t\right)$ meets the next inequality:

$$\begin{array}{c}\hfill \parallel {\sigma }_g{\left(y\left(t\right)\right)\parallel }_2\le {\parallel {\mathcal{D}}_{g}y\left(t\right)\parallel }_2,\end{array}$$

(14)

where ${\mathcal{D}}_g$ are constant matrices with compatible dimensions.

For simplicity, we define the following:

$$\begin{array}{c}\hfill \begin{array}{c}\alpha \left(t\right)=\left[\begin{array}{ccc}{\alpha }_1\left(t\right)\hfill & \cdots \hfill & {\alpha }_d\left(t\right)\hfill \end{array}\right],\sigma \left(y\left(t\right)\right)={\left[\begin{array}{ccc}{\sigma }_1^T\left(y\left(t\right)\right)\hfill & \cdots \hfill & {\sigma }_d^T\left(y\left(t\right)\right)\hfill \end{array}\right]}^T.\hfill \end{array}\end{array}$$

(15)

From (12) and (15), the secure static output controller is rewritten as

$$u\left(t\right)=K\left(E{\int }_{-h}^0{\displaystyle \frac{1}{h}}\chi (t+s)ds-ϵ\left(t\right)\right)+K\beta \left(t\right)\alpha \left(t\right)\sigma \left(y\left(t\right)\right).$$

(16)

Remark 3.

If the number of attack mode is $d=1$, the considered multimodal deception attacks in (12) includes only one attack mode, which reduces to some existing single mode deception attack in many existing results. In addition, without considering the multimodal deception attack $\sigma \left(t\right)$, the proposed secure static output controller (16) degrades to

$$\begin{array}{c}\hfill u\left(t\right)=K\left(E{\int }_{-h}^0{\displaystyle \frac{1}{h}}\chi (t+s)ds-ϵ\left(t\right)\right),\end{array}$$

(17)

which is viewed as the standard static output controller.

To simplify the analysis, we define an auxiliary function $\psi \left(s\right)\triangleq {\displaystyle \frac{1}{h}}$ and let $\tilde{\psi }\left(s\right)={\left[\begin{array}{c}{\psi }_0\left(s\right)\dots {\psi }_i\left(s\right)\dots {\psi }_{\kappa }\left(s\right)\end{array}\right]}^T$ be a vector composed of Legendre polynomials defined on $s\in [-h,0]$, ${\psi }_0\left(s\right)\triangleq \psi \left(s\right)$, $i=0,1,\dots ,\kappa$ with $\kappa \in {\mathbb{R}}^{n\times n}$.

Defining $\varphi \left(s\right)=\tilde{\psi }\left(s\right)\otimes I$, $\mathcal{I}=\left[\begin{array}{cc}I& 0_{3,3\kappa }\end{array}\right]$ and ${\mathbb{X}}_{\chi }\left(t\right)={\int }_{-h}^0\varphi \left(s\right)\chi (t+s)ds$, one obtains

$$\begin{array}{cc}\hfill & {\int }_{-h}^0\psi \left(s\right)\chi (t+s)ds=\mathcal{I}{\mathbb{X}}_{\chi }\left(t\right).\hfill \end{array}$$

(18)

By combining (12) and (18), the closed-loop event-triggered PV microgrid system is established as follows:

$$\begin{array}{c}\hfill \dot{\chi }\left(t\right)=Q\chi \left(t\right)+JK\beta \left(t\right)\alpha \left(t\right)\sigma \left(y\left(t\right)\right)+JK\left(E\mathcal{I}{\mathbb{X}}_{\chi }\left(t\right)-ϵ\left(t\right)\right).\end{array}$$

(19)

Lemma 1

([33]). For a matrix $\mathcal{Z}>0\in {\mathbb{R}}^{n\times n}$, let $\tilde{\psi }\left(s\right)={\left[\begin{array}{c}{\psi }_0\left(s\right)\dots {\psi }_i\left(s\right)\dots {\psi }_{\kappa }\left(s\right)\end{array}\right]}^T$ be a vector composed of Legendre polynomials defined on $s\in [-h,0]$, ${\psi }_0\left(s\right)\triangleq \psi \left(s\right)$, $i=0,1,\dots ,\kappa$ with $\kappa \in {\mathbb{R}}^{n\times n}$, satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{d\tilde{\psi }\left(s\right)}{ds}}={\displaystyle \frac{\mathcal{W}}{h}}\tilde{\psi }\left(s\right),\end{array}$$

(20)

where $\mathcal{W}\in {\mathbb{R}}^{(\kappa +1)\times (\kappa +1)}$ is a lower-triangular matrix with entries ${\mathcal{W}}_{ij}=(2j+1)(1-{(-1)}^{i+j})$ for $i\ge j$. Then, one has

$$\begin{array}{cc}\hfill & {\int }_{-h}^0{\chi }^T\left(s\right)\mathcal{Z}\chi \left(s\right)ds\ge {\mathbb{X}}_{\chi }^T\left(t\right)(\mathsf{\Omega }\otimes \mathcal{Z}){\mathbb{X}}_{\chi }\left(t\right),\hfill \end{array}$$

(21)

where ${\mathsf{\Omega }}^{-1}={\int }_{-h}^0\tilde{\psi }\left(s\right){\tilde{\psi }}^T\left(s\right)ds$.

3. Main Results

To simplify the presentation, the following matrix is defined as

$$\begin{array}{cc}\hfill & {\mathbb{M}}_a\triangleq \left\{\begin{array}{cc}\left[\begin{array}{ccc}{0}_{3,3(a-1)}& I& 0_{3,3(\kappa +1)+d+1}\end{array}\right],\hfill & a=1,\dots ,3\\ \left[\begin{array}{ccc}{0}_{3(\kappa +1),9}& I_{3(\kappa +1)}& 0_{3(\kappa +1),d+1}\end{array}\right],\hfill & a=4+\kappa \\ \left[\begin{array}{ccc}{0}_{d,3(4+\kappa )}& I_d& 0_{d,1}\end{array}\right],\hfill & a=5+\kappa \\ \left[\begin{array}{cc}{0}_{3,3(\kappa +4+d)}& I\end{array}\right].\hfill & a=6+\kappa \end{array}\right.\hfill \end{array}$$

Next, sufficient stability conditions are obtained in Theorem 1 and a set of co-design conditions for deriving the output controller and the triggering matrix are proposed in Theorem 2.

3.1. Stability Analysis

Theorem 1.

For given scalars ${\delta }_M$, h, $\overline{\beta }$, $\overline{\alpha }$, and controller gain K, the system (19) is asymptotically stable in a mean-square sense under the DIETS (10), if there exist symmetric matrices P, $\mathcal{S}>0$, $\mathcal{G}>0$, $\mathsf{\Phi }>0$, and matrices W, $\mathcal{F}$ such that

$$\begin{array}{cc}\hfill & \mathcal{P}>0,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & \mathsf{\Pi }+{\mathsf{\Theta }}_5<0,\hfill \end{array}$$

(23)

where

$$\begin{array}{cc}\hfill & \mathcal{P}=P+diag\{0_4,(\mathsf{\Omega }\otimes U)\},\widehat{\mathcal{W}}=\mathcal{W}\otimes I_4,\hfill \\ \hfill & \mathsf{\Pi }=\mathsf{\Lambda }+He\left(\mathcal{W}\mathcal{G}\right),\mathcal{W}={({\rho }_{1}W{\mathbb{M}}_1+{\rho }_{2}W{\mathbb{M}}_2)}^T,\hfill \\ \hfill & \mathcal{G}=Q{\mathbb{M}}_2+JK\overline{\beta }\overline{\alpha }{\mathbb{M}}_5+JKE{\mathbb{M}}_4-JK{\mathbb{M}}_6-{\mathbb{M}}_1,\hfill \\ \hfill & \mathsf{\Lambda }={\mathsf{\Theta }}_1+{\mathsf{\Theta }}_2+{\mathsf{\Theta }}_3+{\mathsf{\Theta }}_4,\hfill \\ \hfill & {\mathsf{\Theta }}_1=He\left(M^{T}PR\right),\hfill \\ \hfill & {\mathsf{\Theta }}_2={\mathbb{M}}_2^T(\mathcal{S}+h\mathcal{G}){\mathbb{M}}_2-{\mathbb{M}}_3^T\mathcal{S}{\mathbb{M}}_3-{\mathbb{M}}_4^T(\mathsf{\Omega }\otimes \mathcal{S}){\mathbb{M}}_4,\hfill \\ \hfill & {\mathsf{\Theta }}_3=-{\mathbb{M}}_5^T\mathcal{F}{\mathbb{M}}_5,\mathcal{D}={\left[\begin{array}{ccc}{\mathcal{D}}_1^T& \dots & {\mathcal{D}}_d^T\end{array}\right]}^T,\hfill \\ \hfill & {\mathsf{\Theta }}_4={\delta }_M{\mathbb{M}}_4^T{\mathcal{I}}^T{E}^T\mathsf{\Phi }E\mathcal{I}{\mathbb{M}}_4-{\mathbb{M}}_6^T\mathsf{\Phi }{\mathbb{M}}_6,\hfill \\ \hfill & {\mathsf{\Theta }}_5={\mathbb{M}}_2^T\left(E^T{\mathcal{D}}^T\mathcal{D}E\right){\mathbb{M}}_2,\hfill \\ \hfill & {\mathbb{M}}_h=\left[\begin{array}{c}{\mathbb{M}}_4\\ ⋮\\ {\mathbb{M}}_{4+\kappa }\end{array}\right],R=\left[\begin{array}{c}{\mathbb{M}}_1\\ \mathcal{W}\left(0\right){\mathbb{M}}_2-\mathcal{W}(-h){\mathbb{M}}_3-\widehat{\mathcal{W}}{\mathbb{M}}_h\end{array}\right],M=\left[\begin{array}{c}{\mathbb{M}}_2\\ {\mathbb{M}}_h\end{array}\right].\hfill \end{array}$$

Proof. 

The following Lyapunov–Krasovskii functional is selected as

$$\begin{array}{c}\hfill V\left(t\right)={\theta }^T\left(t\right)P\theta \left(t\right)+{\int }_{t-h}^t{\chi }^T\left(s\right)[\mathcal{S}+(s-t+h)\mathcal{G}]\chi \left(s\right)ds,\end{array}$$

(24)

where $\theta \left(t\right)=\left[\begin{array}{c}\chi \left(t\right)\\ {\mathbb{X}}_{\chi }\left(t\right)\end{array}\right]$. The first term ${\theta }^T\left(t\right)P\theta \left(t\right)$, where $\theta \left(t\right)=\left[\begin{array}{c}\chi \left(t\right)\\ {\mathbb{X}}_{\chi }\left(t\right)\end{array}\right]$ incorporates both the current system state $\chi \left(t\right)$ and the historical state ${\mathbb{X}}_{\chi }\left(t\right)={\int }_{-h}^0\varphi \left(s\right)\chi (t+s)ds$. The second term ${\int }_{t-h}^t{\chi }^T\left(s\right)\left[S+(s-t+h)\mathcal{G}\right]\chi \left(s\right)ds$ handles the integral of historical state correlations over the memory interval h.

By using Lemma 1 we can have

$$\begin{array}{cc}\hfill & {\int }_{-h}^0{\chi }^T(t+s)\mathcal{S}\chi (t+s)ds\ge {\mathbb{X}}_{\chi }^T\left(t\right)(\mathsf{\Omega }\otimes \mathcal{S}){\mathbb{X}}_{\chi }\left(t\right).\hfill \end{array}$$

(25)

Based on (24) and (25), the following result can be derived

$$\begin{array}{c}\hfill V\left(t\right)\ge {\theta }^T\left(t\right)\mathcal{P}\theta \left(t\right)+{\int }_{-h}^0{\chi }^T(t+s)\mathcal{G}\chi (t+s)ds.\end{array}$$

(26)

Therefore, $V\left(t\right)>0$ is guaranteed by $\mathcal{G}>0$, $\mathcal{P}>0$.

Taking the derivative of $V\left(t\right)$ leads to

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & 2{\theta }^T\left(t\right)P\dot{\theta }\left(t\right)+{\chi }^T\left(t\right)(\mathcal{S}+h\mathcal{G})\chi \left(t\right)\hfill \\ \hfill & -{\chi }^T(t-h)\mathcal{S}\chi (t-h)-{\int }_{-h}^0{\chi }^T(t+s)\mathcal{G}\chi (t+s)ds.\hfill \end{array}$$

(27)

The integral term ${\int }_{-h}^0{\chi }^T(t+s)\mathcal{G}\chi (t+s)ds$ in (27) can be relaxed via Lemma 1 as follows:

$$-{\int }_{-h}^0{\chi }^T(t+s)\mathcal{G}\chi (t+s)ds\le -{\mathbb{X}}_{\chi }^T\left(t\right)(\mathsf{\Omega }\otimes \mathcal{G}){\mathbb{X}}_{\chi }\left(t\right).$$

(28)

According to (14) and assuming $0<\mathcal{F}<I$, one can obtain

$$\begin{array}{c}\hfill {\chi }^T\left(t\right)E^T{\mathcal{D}}^T\mathcal{D}E\chi \left(t\right)-{\sigma }^T\left(y\left(t\right)\right)\mathcal{F}\sigma \left(y\left(t\right)\right)\ge 0.\end{array}$$

(29)

Define $\zeta \left(t\right)={\left[{\dot{\chi }}^T\left(t\right){\chi }^T\left(t\right){\chi }^T(t-h){\mathbb{X}}_{\chi }^T\left(t\right){\sigma }^T\left(y\left(t\right)\right){ϵ}^T\left(t\right)\right]}^T.$

Subsequently, a zero term is introduced as

$$\begin{array}{c}\hfill (Q{\mathbb{M}}_2+JK\beta \left(t\right)\alpha \left(t\right){\mathbb{M}}_5+JKE{\mathbb{M}}_4-JK{\mathbb{M}}_6-{\mathbb{M}}_1)\zeta \left(t\right)=0.\end{array}$$

(30)

In terms of (20), we have

$${\dot{\mathbb{X}}}_{\chi }\left(t\right)=\varphi \left(0\right)\chi \left(t\right)-\varphi (-h)\chi (t-h)-\widehat{\mathcal{W}}{\mathbb{X}}_{\chi }\left(t\right).$$

(31)

Then, $\theta \left(t\right)$ and $\dot{\theta }\left(t\right)$ can also be expressed as

$$\begin{array}{cc}\hfill & \theta \left(t\right)=M\zeta \left(t\right),\dot{\theta }\left(t\right)=R\zeta \left(t\right).\hfill \end{array}$$

(32)

From (28), (29) and (32), (27) is ensured by

$$\begin{array}{c}\hfill {\zeta }^T\left(t\right)\Gamma \zeta \left(t\right)<0.\end{array}$$

(33)

From (30) and the construction of $\mathcal{W}={({\rho }_{1}W{\mathbb{G}}_1+{\rho }_{2}W{\mathbb{G}}_2)}^T$, one can derive

$$\begin{array}{c}\hfill {\zeta }^T\left(t\right)He\left(\mathcal{W}(Q{\mathbb{M}}_2+JK\beta \left(t\right)\alpha \left(t\right){\mathbb{M}}_5+JKE{\mathbb{M}}_4-JK{\mathbb{M}}_6-{\mathbb{M}}_1)\right)\zeta \left(t\right)=0.\end{array}$$

(34)

Taking the mathematical expectation of (34) results in

$$\begin{array}{c}\hfill \mathbb{E}\left\{{\zeta }^T\left(t\right)He\left(\mathcal{W}\mathcal{G}\right)\zeta \left(t\right)\right\}=0.\end{array}$$

(35)

Based on $\chi \left(t\right)=\mathcal{I}{\mathbb{X}}_{\chi }\left(t\right)=\mathcal{I}{\mathbb{M}}_4\zeta \left(t\right)$, $ϵ\left(t\right)={\mathbb{M}}_7\zeta \left(t\right)$ and the triggering condition (10), one has

$${\delta }_M{\chi }^T\left(t\right)E^T\mathsf{\Phi }E\chi \left(t\right)-{ϵ}^T\left(t\right)\mathsf{\Phi }ϵ\left(t\right)={\zeta }^T\left(t\right){\mathsf{\Theta }}_4\zeta \left(t\right)>0.$$

(36)

By combining (36) and (33), it yields

$$\begin{array}{c}\hfill \mathbb{E}\left\{{\zeta }^T\left(t\right)\Gamma \zeta \left(t\right)\right\}<0.\end{array}$$

(37)

Inserting (34) into (37) leads to

$$\begin{array}{c}\hfill \mathbb{E}\left\{{\zeta }^T\left(t\right)\mathsf{\Pi }\zeta \left(t\right)\right\}<0,\end{array}$$

(38)

which is equivalent to (23). □

Remark 4.

By constructing the block-diagonal matrix structure $\mathcal{P}=P+\mathrm{diag}\{0_4,(\mathsf{\Omega }\otimes R)\}$ , the stability condition (22) only requires the composite matrix $\mathcal{P}$ to be positive definite, rather than enforcing the positive definiteness of P alone. By adopting this strategy, the conservatism of stability assessments is notably mitigated, as it allows for a more flexible and less restrictive condition to ensure system stability.

3.2. Secure Controller Design

Theorem 2.

For given scalars ${\delta }_M$, h, $\overline{\beta }$, $\overline{\alpha }$, ${\rho }_1$ and ${\rho }_2$, the system (19) is asymptotically stable in the mean-square sense under the DIETS (10), if there exist symmetric matrices $\tilde{P}$, $\tilde{\mathcal{S}}>0$, $\tilde{\mathcal{G}}>0$, $\tilde{\mathsf{\Phi }}>0$ and matrices X and $\mathcal{K}$ such that

$$\begin{array}{cc}\hfill & \tilde{\mathcal{P}}>0,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}\tilde{\mathsf{\Pi }}& {\mathbb{M}}_2^{T}X{\left(\mathcal{D}E\right)}^T\\ \ast & -I\end{array}\right]<0,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}-\gamma I& \ast \\ EX-HE& -I\end{array}\right]<0,\hfill \end{array}$$

(41)

where

$$\begin{array}{cc}\hfill & \tilde{\mathcal{P}}=\tilde{P}+diag\{0_4,(\mathsf{\Omega }\otimes \tilde{\mathcal{S}})\},\hfill \\ \hfill & \tilde{\mathsf{\Pi }}=\tilde{\mathsf{\Lambda }}+He\left(\tilde{\mathcal{W}}\tilde{\mathcal{G}}\right),\tilde{\mathcal{W}}={({\rho }_1{\mathbb{M}}_1+{\rho }_2{\mathbb{M}}_2)}^T,\hfill \\ \hfill & \tilde{\mathcal{G}}=QX{\mathbb{M}}_2+J\mathcal{K}\overline{\beta }\overline{\alpha }{\mathbb{M}}_5+J\mathcal{K}E{\mathbb{M}}_4-J\mathcal{K}{\mathbb{M}}_6-X{\mathbb{M}}_1,\hfill \\ \hfill & \tilde{\mathsf{\Lambda }}={\tilde{\mathsf{\Theta }}}_1+{\tilde{\mathsf{\Theta }}}_2+{\tilde{\mathsf{\Theta }}}_3+{\tilde{\mathsf{\Theta }}}_4,\hfill \\ \hfill & {\tilde{\mathsf{\Theta }}}_1=2He\left(M^T\tilde{P}R\right),\hfill \\ \hfill & {\tilde{\mathsf{\Theta }}}_2={\mathbb{M}}_2^T(\tilde{\mathcal{S}}+h\tilde{\mathcal{G}}){\mathbb{M}}_2-{\mathbb{M}}_3^T\tilde{\mathcal{S}}{\mathbb{M}}_3-{\mathbb{M}}_4^T(\mathsf{\Omega }\otimes \tilde{S}){\mathbb{M}}_4,\hfill \\ \hfill & {\tilde{\mathsf{\Theta }}}_3=-{\mathbb{M}}_5^T\tilde{\mathcal{F}}{\mathbb{M}}_5,\hfill \\ \hfill & {\tilde{\mathsf{\Theta }}}_4=\delta {\mathbb{M}}_4^T{\mathcal{I}}^T{E}^T\tilde{\mathsf{\Phi }}E\mathcal{I}{\mathbb{M}}_4-{\mathbb{M}}_6^T\tilde{\mathsf{\Phi }}{\mathbb{M}}_6.\hfill \end{array}$$

Therefore, the controller gain is solved by $K=\mathcal{K}{H}^{-1}$.

Proof. 

By applying the Schur complement technique to Equation (23), it leads to

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}\mathsf{\Pi }& {\mathbb{M}}_2^T{\left(\mathcal{D}E\right)}^T\\ \ast & -I\end{array}\right]<0.\hfill \end{array}$$

(42)

Define $\tilde{\mathcal{G}}=X^T\mathcal{G}X$, $\tilde{\mathcal{S}}=X^T\mathcal{S}X$, $\tilde{\mathsf{\Phi }}=H^T\mathsf{\Phi }H$, $\tilde{\mathcal{F}}=H^T\mathcal{F}H$, $\tilde{P}={(I_{\kappa +1}\otimes X)}^{T}P(I_{\kappa +1}\otimes X)$ and $X=W^{-1}$. To handle the nonlinear term $JKEX$ in (23), the equality condition $EX=HE$ is adopted.

Based on the above definitions, and pre- and post-multiplying (42) from both sides with matrix $\mathcal{X}=diag\{X,X,X,I_{\kappa +1}\otimes X,H,H\}$ and its transpose ${\mathcal{X}}^T$, one has

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}\widehat{\mathsf{\Pi }}& {\mathbb{M}}_2^{T}X{\left(\mathcal{D}E\right)}^T\\ \ast & -I\end{array}\right]<0,\hfill \end{array}$$

(43)

which equals the condition (40). Following a similar way, the condition (39) can be deduced easily.

Then, it results in ${(EX-HE)}^T(EX-HE)=0.$ By applying the Schur complement lemma, the following matrix inequality can be derived as follows:

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}-\gamma I& \ast \\ EX-HE& -I\end{array}\right]<0,\hfill \end{array}$$

(44)

where $\gamma >0$ is a sufficiently small scalar. Thus, the proof is fulfilled. □

To sum up, the control design process is shown in Figure 2.

4. Simulation Results

The parameter values of the considered PV microgrid system are chosen as

Table 1. The parameter values of PV microgrid.

$V_{dc}$	65 V	$D_b$	$0.8375$	$L_0$	1.2 mH	$L_f$	18 mH
$C_f$	220 $\mathsf{\mu }$F	$R_0$	$0.25\mathsf{\Omega }$	$R_f$	$0.33\mathsf{\Omega }$	$R_L$	$29.32\mathsf{\Omega }$

[32].

The simulations are implemented in MATLAB/Simulink with a step size of 0.02 s. Specifically, the system dynamic equations were established through the State-Space module, while the event-triggered mechanism, secure controller, and multimodal deception attacks module were constructed via individual programming in the Matlab Function module. The multimodal deception attack with two different modes is modeled as ${\sigma }_1\left(\left(y\left(t\right)\right)\right)=2.5sin\left(y\left(t\right)\right)$ with ${\overline{\alpha }}_1=0.2,{\mathcal{D}}_1=2.5$ and ${\sigma }_2\left(\left(y\left(t\right)\right)\right)=1.5tanh\left(y\left(t\right)\right)$ with ${\overline{\alpha }}_2=0.8,{\mathcal{D}}_2=1.5$. The auxiliary function $\varphi \left(s\right)$ is set to the standard Legendre polynomials of degree $\kappa =1$. The other parameters are selected as ${\delta }_m=0.01$, ${\delta }_M=0.2$, $h=0.08$, ${\rho }_1=0.5$, ${\rho }_2=1$. By solving the conditions in Theorem 2, one can obtain the controller gain and triggering matrix as

$$\begin{array}{c}\hfill K=-17.45,\mathsf{\Phi }=7055.\end{array}$$

To validate the advantages of the proposed DIETS over normal ETS and dynamic ETS, it is assumed that the PV microgrid system is susceptible to abrupt disturbances and noise interference $f\left(t\right)=r\left(t\right)sin\left(2\pi t\right)e^{-0.5t}$ in measurement outputs, where $r\left(t\right)\in [-1,1]$ is a random variable. In this case, the simulation process focuses on comparing the triggering frequency of the proposed DIETS with traditional ETS and dynamic ETS to validate the advantages of DIETS in reducing communication load. The dynamic response curves of PV microgrid and the distribution of triggering instants are shown in Figure 3 and Figure 4. The dynamic triggering threshold is drawn in Figure 5. The number of triggering times ${\mathcal{N}}_1$ under our DIETS and normal ETS are recorded in

Table 2. Amount of triggered events ${\mathcal{N}}_1$ and average trigger interval ${\mathcal{L}}_1$ under our DIETS and normal ETS.

Method	${\mathcal{N}}_1$	${\mathcal{L}}_1$
DIETS	99	0.050
Dynamic ETS [19]	209	0.024
Normal ETS [17,18]	221	0.023

.

From Figure 3 and Figure 4, one can see that similar state responses under the proposed DIETS and traditional ETSs (dynamic\normal ETS) have been obtained. The simulation results presented in Table 2 reveal a notable advantage of DIETS in event-triggering efficiency. Specifically, the number of triggering events recorded for DIETS stands at 99, marking a reduction of $52.6\%$ compared to dynamic ETS and $55.2\%$ compared to normal ETS. The advantage of DIETS in event-triggering efficiency is further corroborated by the average trigger interval, where the average trigger interval for DIETS is $0.05$ s, significantly higher than the $0.024$ s of dynamic ETS and $0.023$ s of normal ETS. Thus, it can be observed that in the event of multimodal deception attacks on the PV microgrid, the DIETS effectively reduces some redundant events triggered by transient fluctuations induced by measurement noises and disturbances without decreasing much control performance.

Deception attacks refer to scenarios where attackers forge input signals to a control system, deceiving it into performing erroneous operations. The objective of such attacks is to mislead the control system, causing it to execute incorrect actions that ultimately harm the system. As illustrated in Figure 6, a comparison of response curves between a standard controller under attack and one operating normally reveals that the system suffering from deception attacks misjudges external environmental changes. This leads to the activation of faulty control strategies, rendering the control ineffective. Compared to traditional deception attack patterns, the multimodal deception attacks shown in Figure 7 demonstrate more sophisticated threat characteristics. The attack signals typically mimic legitimate data formats, evading detection by conventional security mechanisms.

To further demonstrate the effectiveness of the proposed secure static output controller when encountering multimodal deception attacks, the following section compares the standard static output controller (17) with our method. Figure 8 and Figure 9 present the corresponding system response curves and the distribution of triggering instants. The amount of triggering times ${\mathcal{N}}_2$ for our secure static output controller and standard static output controller are presented in

Table 3. Amount of triggered events ${\mathcal{N}}_2$ and average trigger interval ${\mathcal{L}}_2$ under proposed secure controller and standard controller.

Method	${\mathcal{N}}_2$	${\mathcal{L}}_2$
Our secure controller	99	0.050
Standard controller	119	0.042

.

In terms of Figure 8, better state responses under our secure static output controller considering deception attacks are generated than the standard static output controller without considering deception attacks. In addition, as shown in Figure 9 and Table 3, the standard control method generates triggering times of 119, while our secure control strategy reduces it to 99, achieving a $16.8\%$ reduction in communication resource.

5. Limitations Discussion

Although this paper proposes innovative solutions to address the challenges of communication efficiency and cybersecurity in photovoltaic microgrid control systems, there are still some limitations that require further exploration in subsequent work: First, the current research did not model or analyze the time-delay issues in communication networks, while in practical microgrid distributed control architectures, the coupling effect of communication delays and measurement noise may significantly affect system robustness. Future work will incorporate delay-dependent analysis into the event-triggered control framework, focusing on investigating the dynamic interaction mechanisms between delays and noise and their impact on closed-loop performance. Furthermore, the multimodal deception attack model proposed in this paper remains constrained by static attack patterns and prior assumptions. Future work will extend security defense mechanisms by integrating adaptive learning-based attack models to design more deceptive attack strategies. Additionally, we plan to conduct research on Load Frequency Control in large-scale power grids under random delays and cyberattacks, focusing on dynamic characteristic analysis and security control strategies. Finally, this paper primarily focuses on theoretical methodology design and has not yet fully considered hardware constraints and communication interference in practical microgrid deployments. Subsequent research will validate the engineering applicability of proposed strategies through hardware-in-the-loop experiments and explore multi-node scalability challenges.

6. Conclusions

This paper focuses on the complex challenges posed by limited communication resources and cybersecurity threats in PV microgrids. Compared to conventional dynamic ETS, this paper designs a novel dynamic threshold and triggering function based on historical state integration rather than relying on traditional instantaneous states. By utilizing historical state averaging and adaptive triggering thresholds, DIETS effectively reduces redundant triggering events caused by noise. With only 99 recorded triggering events, it achieves a $55.2\%$ reduction compared to normal ETS and a $52.6\%$ decrease relative to dynamic ETS, demonstrating the outstanding communication efficiency. Furthermore, a secure event-triggered static output controller was proposed to stabilize PV microgrid operations under multimodal deception attacks. Lastly, the simulations are implemented to illustrate the benefits of the developed strategy.