Resilient distributed control of islanded microgrids under hybrid attacks

In this paper, a resilient control strategy is proposed to improve the stability of frequency and voltage recovery for the islanded microgrid (MG) under hybrid cyber attacks. To deal with the common false data injection attacks (FDI) and denial of service attacks (DoS) in MGs, the proposed resilient control strategy utilizes the observers to accurately estimate the potential FDI signals on both the sensors and actuators of each distributed generation unit (DG) and reconstruct the unavailable states in the system to enhance the system’s ability actively. The ultimate uniform boundedness (UUB) of the system under hybrid cyber attacks is proved by the Lyapunov stability theory. Finally, an islanded MG system is established in MATLAB/SIMULINK, and multiple scenarios are simulated to verify the effectiveness of the method.

1 Introduction

With the development of smart grid technologies, MGs are evolving into typical cyber-physical systems (CPS) by integrating more and more renewable energy sources, monitoring devices, communication facilities, and control systems. MGs can operate in both grid-connected mode and islanded mode (Albarakati et al., 2022). In islanding mode, due to the lack of support from the main grid, MGs are sensitive to disturbances such as power fluctuations, load switching, and cyber attacks, which can cause system instability or even large-scale blackouts. To improve system operation performance, MGs widely adopt the hierarchical control structure in islanded mode (Kang et al., 2018). The primary layer is the lowest control layer in MGs, including voltage and current control loops. In the primary control, each DG only uses its own measurements, so the control layer will cause steady-state errors in frequency and voltage. The secondary layer is responsible for voltage/frequency recovery and active/reactive power sharing between DGs (Dörfler et al., 2015). The tertiary layer is responsible for implementing economic dispatching, operation dispatching, and power flow between MGs and the main grid in the grid-connected mode (Liu et al., 2023).

Centralized and distributed control strategies are common approaches to achieving secondary control objectives in AC MGs. In recent years, centralized algorithms have been increasingly replaced by distributed schemes for secondary controllers since distributed techniques offer better robustness (no single point of failure) and scalability (Feng and Ma, 2022; Yang et al., 2022). Distributed consensus algorithms based on multiagent systems (MAS) have been widely studied in the secondary control of MGs (Ma et al., 2017). Since MAS-controlled AC MGs are typical CPS, researchers mainly concentrated on challenges related to unmodeled dynamics (Shotorbani et al., 2017; Hu et al., 2023), physical disturbances (Hu and Bhowmick, 2020), communication failures (Shahab et al., 2019), stability analysis (Majumder et al., 2009; Zhang and Fan, 2022), high integration of DGs (Sharma et al., 2018), economical and reliable operation of MGs (Albarakati et al., 2022; Liu et al., 2023), and synchronization problems in grid-connected mode (Hossain et al., 2019; Elshenawy et al., 2022), etc.

In addition to the objective and common problems mentioned above, MG CPS is very vulnerable to extreme human-made attacks such as cyber attacks. For example, in 2015, the power grid of Ukraine suffered from the coordinated attack of multiple types of cyber-attacks, leading to large-scale power failure (Dibaji et al., 2019). Common cyber-attacks include disruption attacks and deception attacks, etc (Dibaji et al., 2019; Chen et al., 2019). Among them, DoS and FDI attacks are considered to be the most prevalent and destructive forms of attacks (Uddin et al., 2023). Specifically, FDI attacks are typically deception attacks that inject or modify real data from sensors, actuators, and communication links in the secondary layer to degrade performance (Beg et al., 2017; Ding et al., 2018). DoS attacks can prevent data transmission in communication links (Lian et al., 2022). In a closed-loop feedback control system that uses distributed algorithms, such as MGs, FDI and DoS attacks have always been regarded as persistent security threats (Tan et al., 2020). In Zhang et al. (2006), the authors have concluded that the IEC 61850 protocol and its extensions are recommended when modeling the communication infrastructure for the MG. However, IEC 61850 and TCP/IP protocols are still potentially threatened by FDI attacks, which are tested by the remote monitoring based adaptive LTC controller (Wang and Yang, 2018) and a TCP/IP-based networked dc servo system (Yu et al., 2019). Industrial control systems security mechanisms such as industrial firewalls and white lists cannot effectively deal with these threats. If the cyber attacks on the MG’s communication system are not detected and corrective actions are not implemented in time, the DGs in the MG will be unable to supply the required power, which deviate from the system operating point and endanger the security and stability (Wang et al., 2023). Furthermore, note that FDI and DoS attacks may occur concurrently, resulting in a more devastating impact and lower detection probability, seriously jeopardizing the system’s security and stability (Liu et al., 2019; Tadepalli and Pullaguram, 2022). Therefore, it is crucial to investigate the resilient distributed control problems that arise from the simultaneous presence of DoS and FDI attacks.

In previous works, resilient control strategies for islanded MGs under FDI and DoS attacks are generally studied separately. Several detection, identification, and mitigation schemes have been proposed for DGs under FDI attacks in MGs. In Peng et al. (2019), Abhinav et al. (2017), the influence of FDI attacks is alleviated by detecting the attacked DG and correcting the confidence factor of the communication links. Although these methods can resist various attacks on communication links, additional assumptions about the connectivity of the communication topology and the number of compromised DGs are usually required. Recently, several artificial intelligence algorithms, including reinforcement learning and unsupervised learning, have been used to detect and mitigate cyber attacks (Tian et al., 2022; Ramotsoela et al., 2023). However, the effectiveness of these data-driven schemes depends on sufficient historical data sets, which may have challenges such as a high amount of calculation, poor compatibility, and limited resistance to parameter changes. In addition, estimating attack signals or reconfiguring the system states by designing the state observers is also an essential technique to enhance system resilience. In Afshari et al. (2020), deception attack, dynamic interference, and uncertainty are regarded as bounded disturbances and an adaptive control is introduced to compensate for the influence of disturbances. In Jiang et al. (2021), Mohiuddin and Qi (2021), Shi et al. (2021), Barzegari et al. (2022), Muktiadji et al. (2022), Zhou et al. (2023), the distributed observer frameworks are used to estimate the real states of the system, respectively. In Wang et al. (2022a), Wang et al. (2022b), Zhan et al. (2023), the virtual layer-based compensation strategies are proposed to mitigate FDI attacks. For DoS attacks, a resilient controller based on sampling and communication mechanisms is employed to enhance the DC MGs against DoS attacks by rigorously limiting the frequency and duration of attacks (Lian et al., 2021). In Xu and Ma (2020), Li et al. (2022), Liu and Che (2022), distributed control schemes based on security event triggering mechanisms are proposed to ensure the secondary control objectives and secure economic dispatch of MG under DoS attacks, which establish a connection between the triggering mechanism and DoS attacks.

To summarize, the aforementioned methods mainly focus on the effects of MGs under FDI or DoS attacks, and few studies have investigated the resilient control of MGs that are simultaneously subjected to FDI and DoS attacks. Currently, to resist the influence of various cyber attacks, (Wang et al., 2023), proposes an adaptive control strategy to deal with the bounded sensor FDI attacks and tolerate the intermittent DoS attacks on the communication links. In Liu et al. (2021), the authors develop an adaptive gain-based control scheme to cope with intermittent DoS and impulse signal FDI attacks. However, these methods either do not address both sensor and actuator attacks in a unified framework or assume that these FDI signals are regarded as bounded natural disturbances, etc. Furthermore, the control strategies in Wang et al. (2023), Liu et al. (2021) do not consider the impact of DoS attacks on the controller, i.e., the control inputs are forced to zero when the communication links under DoS attacks. Although these studies have some effect, the resilience under attacks is not satisfactory. Therefore, the main goal of this paper is to develop a resilient distributed method to effectively address both FDI and DoS attacks in MGs. The core challenges that need to be addressed to achieve this goal include:

(1) How to develop an effective distributed resilient method to eliminate or mitigate the effects of attacks when FDI signals are unknown and even unbounded, and sensors and actuators of numerous DGs may be attacked simultaneously?

(2) How to cleverly design the resilient method to avoid the divergence or even infinite growth of the system states when MGs subjected to hybrid attacks, while ensuring the realization of secondary control objectives?

As a result, it is essential to consider designing a general and resilient framework to resist the impact of hybris attacks. Compared with the existing research, the main contributions can be summarized as follows:

1) Compared to most recent works that focus solely on the impact of single cyber attacks on islanded MGs, this paper further proposes a distributed control framework incorporating resilience against simultaneous sensor, actuator, and DoS attacks for MG systems. It is more practical for island MGs in complex working environments.

2) The proposed resilient control strategy utilizes the observer to accurately estimate the potential FDI signals on both the sensors and actuators of each DG and is capable of accommodating unbounded FDI attacks on any number of DGs, thus enhancing its robustness against potential attacks. During DoS attacks, the open-loop observer is used to reconstruct the unavailable neighbor DG state to enhance the system’s ability to resist DoS attacks actively.

3) The system’s stability is demonstrated using the Lyapunov function, which shows that the system can achieve UUB when subjected to hybrid cyber attacks.

2 Modeling of AC MGs and cyber attacks

2.1 Notations and graph theory

In this paper, $A\otimes B$ denotes the Kronecker product of matrix A and B. $I_N\in {\mathbb{R}}^{N\times N}$ denotes the Nth order identity matrix. The operator $\text{diag}\left\{•\right\}$ creates the diagonal matrix using elements in brackets. The communication network among DGs can be modeled as MAS, usually represented by undirected graphs. An undirected graph is generally denoted by $\mathcal{G}\left(\mathcal{V},\mathcal{E},\mathcal{A}\right)$, where $\mathcal{V}=\left\{\mathrm{1,2},...,\mathcal{N}\right\}$ denotes the set of nodes, including $M$ follower nodes and $N$ leader nodes. Define $F\in \left\{\mathrm{1,2},...,M\right\}$ and $\mathrm{\Omega }=\left\{M+1,M+2,...,N\right\}$. $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ denotes the adjacency matrix. If node i and node j can communicate with each other, then $a_{ij}>0,j\in F$, otherwise, $a_{ij}=0$. $N_i=\left\{j\left|\left(j,i\right)\in \mathcal{E}\right.\right\}$ denotes the sets of the neighboring nodes of node i. Laplacian matrix is $L=\mathcal{D}-\mathcal{A}$, where $\mathcal{D}=\text{diag}\left\{d_i\right\}\in {\mathbb{R}}^{N\times N}$ denotes the in-degree matrix, with $d_i={\sum }_{j\in N_i}{a}_{ij}$. The Laplace matrix can be chunked according to the number of leader nodes $\left[L_1{L}_2;0_{\left(N-M\right)\times M}{0}_{\left(N-M\right)\times \left(N-M\right)}\right]$, where $L_1\in {\mathbb{R}}^{M\times M}$ and $L_2\in {\mathbb{R}}^{M\times \left(N-M\right)}$ represent the Laplace matrix between followers and between leader and followers, respectively.

2.2 Dynamic modeling of MGs

In this paper, a typical inverter-based islanded MG is taken as the research object, and the control framework of a grid-forming converter is shown in Figure 1. In the grid-forming converters, the islanded MG includes two parts: 1) physical layer, including inverter-based DGs and their controllers, LC filter, coupling inductor (Pogaku et al., 2007), transmission lines, and loads; 2) the control layer, including the hierarchical control structure; 3) communication layer, which usually contains only communication links composed of directed or undirected graphs.

FIGURE 1

FIGURE 1. The MG layers schematic, including the physical, control, and communication layers. The control layer gives the schematic diagram of the proposed resilient control framework.

The secondary control is usually achieved by introducing additional regulation terms for voltage/frequency consensus as well as power sharing, as follows (Hennane et al., 2023):

${\omega }_i={\omega }_{n,i}-m_{P,i}{P}_i(1)$

${v_o}_i=V_{n,i}-n_{Q,i}{Q}_i(2)$

where ${\omega }_i$ is the output angular frequency of ith DG. ${v_o}_i$ is the capacitor voltage magnitude and equal to d-axis capacitor voltage ${v_o}_{di}$ in the dq reference frame. ${\omega }_{n,i}$ and $V_{n,i}$ are the nominal angular frequency and voltage set points, respectively. $P_i$ and $Q_i$ are the active power and reactive power of ith DG. $m_{P,i}$ and $n_{Q,i}$ are droop coefficients.

In MG dynamic modeling and control system design, the model of the MG system can safely ignore the fast switching model in Dehkordi et al. (2016), so the nonlinear dynamics model of the grid-forming DGs in the d-q reference system can be expressed as.

$\left\{\begin{array}{c}{\dot{x}}_i=f_i\left(x_i\right)+k_i\left(x_i\right)D_i+g_i\left(x_i\right)u_i\\ y_{i1}=v_{odi}=h_{i1}\left(x_i\right),y_{i2}={\omega }_i=h_{i2}\left(x_i\right)\end{array}\right.(3)$

where $x_i$ is the state vector of i th DG. The detailed expressions of $x_i$, $u_i$, $f_i\left(x_i\right)$, $g_i\left(x_i\right)$, $k_i\left(x_i\right)$, $h_{i1}\left(x_i\right)$, $h_{i2}\left(x_i\right)$ can be found in Yang et al. (2020).

Since the dynamic characteristics of the current and voltage controllers are much faster than the primary controller, the dynamic characteristics of the current and voltage controllers can be ignored in the study of the secondary controller. Therefore, the secondary control problem is transformed into a tracking control problem using the input-output feedback linearization technique. For the secondary voltage control of MGs, let $F_i\left(x_i\right)=f_i\left(x_i\right)+k_i\left(x_i\right)D_i$, then the voltage dynamics of each DG is modeled as:

${\ddot{y}}_{i1}=L_{F_i}^2{h}_{i1}\left(x_i\right)+L_{g_i}{L}_{F_i}{h}_{i1}\left(x_i\right)u_{i1}(4)$

where $L_{F_i}{h}_{i1}\left(x_i\right)$ is the Lie derivative of the scalar function $h_{i1}\left(x_i\right)$ along $F_i\left(x_i\right)$ (Xiao and Dong, 2020). Define the auxiliary control signal as $v_i$, the dynamics of Eq. 4 can be considered as a second-order linear system ${\ddot{y}}_{i1}=v_i$. Therefore, the control input is designed as $u_{i1}={\left(L_{g_i}{L}_{F_i}{h}_i\left(x_i\right)\right)}^{-1}\left(-L_{F_i}^2{h}_i\left(x_i\right)+v_i\right)$, and Eq. 4 can be written in the following compact form:

${\dot{y}}_{Vi}=A{y}_{Vi}+B{u}_{Vi},i\in F(5)$

where $y_{Vi}={\left[y_{i1},y_{i\mathrm{1,2}}\right]}^T={\left[v_{oi},{\dot{v}}_{oi}\right]}^T$, $A=\left[01;00\right]$, $B={\left[01\right]}^T$. The dynamic model of the leader is considered as ${\dot{y}}_l=A{y}_l,l\in \mathrm{\Omega }$, where $y_l={\left[v_{\text{ref},l}{\dot{v}}_{\text{ref},l}\right]}^T$.

Similar to the voltage dynamic model, according to Eq. 3, the system frequency dynamics can be obtained using feedback linearization as:

${\dot{y}}_{i2}=L_{F_i}^2{h}_{i2}\left(x_i\right)+L_{g_i}{L}_{F_i}{h}_{i2}\left(x_i\right)u_{i2}(6)$

The above equation can be written in the following form:

${\dot{y}}_{\omega i}={\dot{\omega }}_i={\dot{\omega }}_{ni}-m_{P,i}{\dot{P}}_i(7)$

2.3 Modeling of FDI attacks

Complex FDI attacks can modify real data in sensors, actuators, and communication links simultaneously. In this paper, we consider the case of unknown cyber attacks that occur in sensors and actuators in a MG system as shown in Figure 1. The sensor FDI corrupts the measured states or output signals, while the actuator attacks modify the control signals generated by the secondary controller. Therefore, the following typical FDI attack model is used for the islanded MG system:

${\overline{y}}_i=y_i+{\delta }_i^s(8)$

${\overline{u}}_i=u_i+{\delta }_i^a(9)$

where ${\delta }_i^s$ and ${\delta }_i^a$ are FDI signals injected by the sensor and actuator attacker under secondary voltage or frequency control, respectively. This means that the system’s true states and control signals are unknown, and only corrupted signals can be measured and used.

Assumption 1. Sensor attack signals ${\delta }_i^s$ and actuator attacks ${\delta }_i^a$ can be unknown and unbounded, but their time derivatives $\left|{\dot{\delta }}_i^s\right|$ and $\left|{\dot{\delta }}_i^a\right|$ satisfy the bounded condition.

Remarks 1:. Malicious signals are often modeled in the papers as random disturbances such as constants, steps, sines, and noise (Afshari et al., 2020; Barzegari et al., 2022). However, unlike random disturbances such as power fluctuations or noise, FDI signals should not be regarded as unintentionally caused bounded signals. Moreover, the attacker carefully designs cyber attacks and does not need information about the system dynamics to perform FDI. Therefore, inspired by Wang et al. (2022b), this paper argues that malicious cyber attacks can be unbounded time-varying signals, i.e., the amplitude of a FDI signal is allowed to be sufficiently large, but as long as its derivatives are constrained, the malicious signals will not grow at an infinitely fast rate. Due to these aspects, the designs available in Afshari et al. (2020), Barzegari et al. (2022) may not be directly applicable. Additionally, the model is universal for the problem of frequency and voltage restoration in MGs with bounded disturbances (attacks, disturbances, uncertainties, etc.).

2.4 Modeling of DoS attacks

The MG system based on the MAS distributed control strategy is shown in Figure 2. The communication network of the MG is usually not sufficiently resistant to DoS attacks. Once the DG fails to receive the states sent by all its neighbors, the control objective of the system will not be accomplished. Consider a class of non-periodic DoS attacks and assume that one DoS attack over a period of time can be denoted as $\left[t_n^{\text{on}},t_n^{\text{off}}\right)$. Therefore, the set of times when the system suffers a DoS attack during $\left[0,t\right]$ is ${\mathrm{\Xi }}_a\left(0,t\right)=\left\{\left.\underset{n\mathbb{\in }\mathbb{N}}{\cup }[t_n^{\text{on}},t_n^{\text{off}}\right)\right\}\cap \left[0,t\right]$. Similarly, the set of times when the system communicates normally is ${\mathrm{\Xi }}_s\left(0,t\right)=\left[0,t\right]\backslash {\mathrm{\Xi }}_a\left(0,t\right)$. Due to the energy constraints of the attacker, the following common assumption is introduced in this paper.

FIGURE 2

FIGURE 2. Framework of the resilient secondary controller.

Assumption 2. There exist constants ${\tau }_a>1$ and ${\mathrm{\Xi }}_0>0$ during the period $\left[0,t\right]$, such that $\left|{\mathrm{\Xi }}_a\left(0,t\right)\right|\le {\mathrm{\Xi }}_0+t/{\tau }_a$ is satisfied.

Remark 2. Assumption 2 limits the duration of DoS attacks considering the limited attack resources and the self-healing mechanisms of real systems, which are widely used in Xu and Ma (2020)).

3 Distributed resilient voltage control under hybrid attack

This section proposes a resilient control scheme for hybrid network attacks in secondary control of MGs. The proposed resilient control framework is shown in Figure 2, and the UUB stability of the whole system is guaranteed.

3.1 Design of resilient voltage controller

The real states of MGs is usually not available due to FDI attacks. Therefore, this paper first introduces state observers to reconstruct the real states of the system and estimate the sensor and actuator FDI signals. To compensate for the effect of FDI on MGs, the following state observers are designed:

${\dot{\widehat{y}}}_{Vi}\left(t\right)=A{\widehat{y}}_{Vi}\left(t\right)+B{u}_{Vi}\left(t\right)+M_1{e}_{Vi}\left(t\right)(10)$

${\dot{\widehat{\delta }}}_{Vi}^s\left(t\right)=-\left(M_1+M_2\right)e_{Vi}\left(t\right)(11)$

where $e_{Vi}\left(t\right)={\overline{y}}_{Vi}\left(t\right)-{\widehat{y}}_{Vi}\left(t\right)-{\widehat{\delta }}_{Vi}^s$. ${\widehat{y}}_{Vi}\left(t\right)$ is the estimated value of the true state $y_{Vi}$, and ${\widehat{\delta }}_{Vi}^s\left(t\right)$ is the estimated value of the sensor FDI signal ${\delta }_{Vi}^s\left(t\right)$. $M_1$ and $M_2$ are the observation gain. The actuator attack signal ${\widehat{\delta }}_{Vi}^a$, which is used to estimate and compensate for the actuator attack signal ${\delta }_{Vi}^a$, is designed in the following form:

${\widehat{\delta }}_{Vi}^a\left(t\right)=\frac{B^T{Q}_1{e}_{Vi}\left(t\right){\chi }_i{\left(t\right)}^2}{‖e_{Vi}^T\left(t\right)Q_{1}B‖{\chi }_i\left(t\right)+{\rho }_i\left(t\right)}(12)$

where ${\rho }_i\left(t\right)>0$ is a uniform continuous function and satisfies $\underset{t\to +\infty }{\lim }{\int }_{t_0}^t{\rho }_i\left(s\right)ds\le {\overline{\rho }}_i<+\infty$. $Q_1$ is the control gain. ${\chi }_i\left(t\right)$ is the adaptive parameter satisfying the following adaptive regulation law:

${\dot{\chi }}_i\left(t\right)={\mu }_i‖e_{Vi}^T\left(t\right)Q_{1}B‖(13)$

where ${\mu }_i$ is a positive constant.

To achieve frequency and voltage restoration and active power and reactive power sharing among DGs, we design the auxiliary control inputs $u_{Vi}\left(t\right)$ based on consensus protocol.

$u_{Vi}\left(t\right)=c_i\left(t\right)K\widehat{\epsilon }\left(t\right)(14)$

${\dot{c}}_i\left(t\right)=\text{Pro}{\mathrm{j}}_{\left[{\underset{\_}{c}}_i,{\overline{c}}_i\right]}\left\{c_i\left(t\right)\right\}=\left\{\begin{array}{l}0,\text{if\hspace{0.17em}}{c}_i\left(t\right)={\overline{c}}_i,{\mathrm{\Phi }}_i>0\text{or\hspace{0.17em}}{c}_i\left(t\right)={\underset{\_}{c}}_i{,\mathrm{\Phi }}_i<0\\ {\kappa }_i{\mathrm{\Phi }}_i,\text{others}\end{array}\right.(15)$

where ${\widehat{\epsilon }}_i\left(t\right)={\sum }_{j=1}^M{a}_{ij}\left({\widehat{y}}_{Vi}\left(t\right)-{\widehat{y}}_{Vj}\left(t\right)\right)+{\sum }_{l=M+1}^N{a}_{il}\left({\widehat{y}}_{Vi}\left(t\right)-y_{Vl}\left(t\right)\right)$ is the local tracking error. $c_i\left(t\right)\in \left[{\underset{\_}{c}}_i,{\overline{c}}_i\right]$ is the adaptive control parameter. ${\mathrm{\Phi }}_i=-{\beta }_i{c}_i\left(t\right)+{\widehat{\epsilon }}_i^T\left(t\right)\mathrm{\Gamma }{\widehat{\epsilon }}_i\left(t\right)$, where ${\beta }_i$ and ${\kappa }_i$ are positive constants. $K$ and $\mathrm{\Gamma }$ are the feedback gain matrix to be designed.

When the communication link (i, j) suffers from DoS attacks, the states sent by the j th DG during DoS cannot be received by the i th DG. Therefore, to mitigate or cope with the impact of DoS attacks on the secondary controller as much as possible, inspired by the literature (Xiao and Dong, 2020), the following open-loop observer is introduced to reconstruct the unavailable state of the MG system during DoS as follows

$\left.{\dot{\stackrel{\sim }{y}}}_{Vj}\left(t\right)=A{\stackrel{\sim }{y}}_{Vj}\left(t\right),t\in [t_n^{\text{on}},t_n^{\text{off}}\right)(16)$

where ${\stackrel{\sim }{y}}_{Vj}\left(t_n^{on}\right)={\widehat{y}}_{Vj}\left(t_n^{on}\right)$, denotes the last successful data exchange of the i th DG with the j th DG before the DoS attacks. Therefore, the estimated local tracking error during DoS is ${\stackrel{\sim }{\epsilon }}_i\left(t\right)={\sum }_{j=1}^M{a}_{ij}\left({\widehat{y}}_{Vi}\left(t\right)-{\stackrel{\sim }{y}}_{Vj}\left(t\right)\right)+{\sum }_{l=M+1}^N{a}_{il}\left({\widehat{y}}_{Vi}\left(t\right)-y_{Vl}\left(t\right)\right)$, and ${\stackrel{\sim }{\mathrm{\Phi }}}_i=-{\beta }_i{c}_i\left(t\right)+{\stackrel{\sim }{\epsilon }}_i^T\left(t\right)\mathrm{\Gamma }{\stackrel{\sim }{\epsilon }}_i\left(t\right)$.

Remark 3. Proj{·} denotes the projection operator, which has been used to design fully distributed controllers and fault-tolerant controls (Xiao and Dong, 2020). Since it is difficult for the open-loop observer to recover the lost data perfectly, introducing the projection operator and the adaptive controller constrain the tracking error during DoS, preventing the system’s consensus error from diverging or even being unbounded. In addition, the open-loop estimator is activated only during the attack. If a particular DG’s states are unavailable, the neighboring DGs are aware of the attacks. The last estimate is kept until the communication state of the MG returns to normal.

3.2 Stability analysis

Before moving forward, the estimated errors of the FDI signals are defined as ${\stackrel{\sim }{\delta }}_{Vi}^s\left(t\right)={\delta }_{Vi}^s\left(t\right)-{\widehat{\delta }}_{Vi}^s\left(t\right)$ and ${\stackrel{\sim }{\delta }}_{Vi}^a\left(t\right)={\delta }_{Vi}^a\left(t\right)-{\widehat{\delta }}_{Vi}^a\left(t\right)$, then

${\dot{e}}_{Vi}\left(t\right)=\left(A+M_2\right)e_{Vi}\left(t\right)+B{\stackrel{\sim }{\delta }}_{Vi}^a\left(t\right)-A{\stackrel{\sim }{\delta }}_{Vi}^s\left(t\right)+{\dot{\delta }}_{Vi}^s\left(t\right)(17)$

${\dot{\stackrel{\sim }{\delta }}}_{Vi}^s\left(t\right)=\left(M_1+M_2\right)e_{Vi}\left(t\right)+{\dot{\delta }}_{Vi}^s\left(t\right)(18)$

Let ${\vartheta }_i\left(t\right)=\text{col}\left\{e_{Vi}\left(t\right),{\stackrel{\sim }{\delta }}_{Vi}^s\left(t\right)\right\}$, then

${\dot{\vartheta }}_i\left(t\right)=\overline{A}{\vartheta }_i\left(t\right)+\overline{B}{\stackrel{\sim }{\delta }}_{Vi}^a\left(t\right)+{\Delta }_i\left(t\right)(19)$

where $\overline{A}=\left[A+M_2,-A;M_1+M_2,0\right]$, $\overline{B}=\left[B;0\right]$, ${\Delta }_i\left(t\right)=\text{col}\left\{{\dot{\delta }}_{Vi}^s\left(t\right),{\dot{\delta }}_{Vi}^s\left(t\right)\right\}$.

Let ${\widehat{y}}_F\left(t\right)=\left[{\widehat{y}}_{V1}\left(t\right),...,{\widehat{y}}_{VM}\left(t\right)\right]$, ${\widehat{y}}_L\left(t\right)=\left[y_{V\left(M+1\right)}\left(t\right),...,y_{VN}\left(t\right)\right]$, $\widehat{\epsilon }\left(t\right)=\left[{\widehat{\epsilon }}_1\left(t\right),...,{\widehat{\epsilon }}_M\left(t\right)\right]$, $\stackrel{\sim }{\epsilon }\left(t\right)=\left[{\stackrel{\sim }{\epsilon }}_1\left(t\right),...,{\stackrel{\sim }{\epsilon }}_M\left(t\right)\right]$, $e_V\left(t\right)=\left[e_{V1}\left(t\right),...,e_{VM}\left(t\right)\right]$, $\stackrel{⌢}{c}\left(t\right)=\text{diag}\left\{c_1\left(t\right),...,c_M\left(t\right)\right\}$. Then the global containment error is ${\theta }_V\left(t\right)={\widehat{y}}_F\left(t\right)-L_1^{-1}{L}_2{y}_L\left(t\right)$, where ${\theta }_V\left(t\right)=L_1\widehat{\epsilon }\left(t\right)$. According to (10)–(16), there are the following switching dynamics

$\left\{\begin{array}{c}{\dot{\theta }}_V\left(t\right)=A{\theta }_V\left(t\right)+\left(\widehat{c}\left(t\right)\otimes BK\right)\widehat{\epsilon }\left(t\right)+M_1{e}_V\left(t\right),t\in [t_{n-1}^{\text{off}},t_n^{\text{on}})\\ {\dot{\theta }}_V\left(t\right)=A{\theta }_V\left(t\right)+\left(\stackrel{\mathrm{ˇ}}{c}\left(t\right)\otimes BK\right)\widehat{\epsilon }\left(t\right)+M_1{e}_V\left(t\right),t\in [t_n^{\text{on}},t_n^{\text{off}})\end{array}\right.(20)$

Next, the stability analysis of MGs under hybrid attack is discussed. The main results are as follows.

Theorem 1. Assuming that the communication topology $\mathcal{G}$ is a strongly connected graph, for given positive constants ${\mathrm{\gamma }}_{\mathrm{i}}\left(\mathrm{i}=\mathrm{1,2,3,4}\right)$ and ${\mathrm{\varpi }}_1,{\mathrm{\varpi }}_2$, if there exist a positive constant ${\mathrm{\tau }}_{\mathrm{a}}$ and positive definite matrices $\mathrm{P}$ and $\mathrm{Q}$, such that (20)–(24) hold, then the controllers (10)–(16) can ensure that the voltage consensus errors of all DGs can achieve UUB under hybrid attacks.

$Q=\left[\begin{array}{cc}{Q}_1& 0\\ 0& Q_2\end{array}\right]>0(21)$

such that

$-{\mathrm{\Omega }}_2=A^{T}P+PA-\eta PB{B}^{T}P+{\gamma }_3{I}_M<0(22)$

$-{\mathrm{\Omega }}_1={\overline{A}}^{T}Q+Q\overline{A}+{\gamma }_1{Q}^{T}Q+\mathrm{\Psi }<0(23)$

$A^{T}P+PA-\left({\gamma }_4+{\stackrel{\sim }{\gamma }}_4\right)P+{\gamma }_3{I}_M<0(24)$

$\frac{{\varpi }_1+{\varpi }_2}{{\varpi }_1}-{\tau }_a<0(25)$

where ${\varpi }_1=\min \left\{{\lambda }_{\min }\left({\mathrm{\Omega }}_2\right)/{\lambda }_{\max }\left(Q\right),{\lambda }_{\min }\left({\mathrm{\Omega }}_1\right)/{\lambda }_{\max }\left(P\right),{\beta }_i{\kappa }_i\right\}$, ${\varpi }_2=\max \left\{-{\lambda }_{\min }\left({\mathrm{\Omega }}_2\right)/{\lambda }_{\max }\left(Q\right),{\gamma }_4,-{\beta }_i{\kappa }_i\right\}$, $\mathrm{\Psi }=\text{diag}\left\{\left[\left(1/{\gamma }_2\right){M_1}^T{P}^2{M}_1,0\right]\right\}$.

The block diagram for the proposed resilient control scheme is depicted in Figure 2. Proof of stability for the proposed scheme is also presented in the Supplementary Appendix SA1.

Remarks 4. The main differences between this paper and related work are clarified in the following two aspects: i) Compared with most recent work that only considers a single cyber attack, this paper proposes a resilient control framework that resists both FDI and DoS attacks. ii) In Wang et al. (2023), Liu et al. (2021), although resilient control strategies are designed to resist FDI and DoS attacks, the ability of the system to actively resist DoS attacks has not been investigated, that is, when the communication network is subjected to DoS attacks, the control input is forced to zero. In addition, the type of FDI attacks is limited to bounded actuator attacks. On the contrary, this paper proposes a resilient control strategy that can effectively compensate for the unbounded FDI attacks of sensors and actuators and reconstruct the unavailable neighbor DG state to resist DoS attacks actively.

Remark 5. Similar to the resilient voltage control, the frequency consensus error can be UUB under hybrid attacks by applying similar schemes (10)–(16). The proof is similar to Theorem 1.

4 Simulation analysis

In order to verify the performance of the proposed resilient distributed control strategy, a 380V/50Hz islanded MG with four inverter-based DGs is taken as the test system, as shown in Figure 3. The simulation is analyzed in the MATLAB/Simulink platform. The parameters of load, transmission line, and DG controller can be found in Table 1. The effectiveness of the proposed control strategy is verified through several scenarios such as load fluctuation, plug-and-play (P&P) of DG, and hybrid cyber attacks.

FIGURE 3

FIGURE 3. Block diagram of an islanded MG.

TABLE 1

TABLE 1. Parameters of the islanded MG.

4.1 Distributed resilient control strategy without attacks

In this section, the performance of the proposed resilient distributed frequency and voltage controller is analyzed by simulation without considering any cyber attacks. The parameters of the controllers are chosen as ${\overline{c}}_i=20,{\underset{\_}{c}}_i=0$, ${\rho }_i\left(t\right)=e^{-0.2t}$, ${\mu }_i=20$, ${\beta }_i=2.0$, ${\kappa }_i=30,{\mathrm{\Xi }}_0=0.2$. In order to satisfy the conditions of Theorem 1, let $M_2=-{\alpha }_1{M}_1,Q_1={\alpha }_2{Q}_2$ (${\alpha }_1,{\alpha }_2$ are the positive constants), and then (20)–(24) are transformed into a linear matrix inequality (LMI) problem to solve. The detailed steps can be found in remark 4 in Deng and Che (2019).

The results are discussed in time sequence as below: i) at t = 1.0 s, the proposed resilient controller starts up, and the communication topology between DGs is shown in Figure 4, given the voltage reference value ${\left[v_{ref,1}{\dot{v}}_{ref,1}\right]}^T=\left[\mathrm{380,0}\right]$ V, ${\left[v_{ref,2}{\dot{v}}_{ref,2}\right]}^T=\left[\mathrm{390,0}\right]$ V, and the frequency reference value ${\omega }_{ref,1}={\omega }_{ref,2}$ $=100\pi$ rad. ii) at t =2.0 s, 50% of load L₂ is removed. iii) DG 4 is plugged out of and plugged into the MG at t = 3.0 s and t = 4.0 s, respectively. iv) at t = 5.0 s, load L₅ increases by 50%.

FIGURE 4

FIGURE 4. Communication topologies under normal situation and DoS attacks.

The simulation results are shown in Figure 5. When the proposed resilient control is adopted, the islanded MG frequency is restored to 50Hz and the output voltage of DGs returns to the range of 380∼390 V. The load fluctuation and P&P of DG4 occur in the MG system during 2.0∼6.0 s. Under these system disturbances, the frequency and active power can quickly maintain consistency and respond rapidly. However, due to factors such as MG line impedance mismatch, there is a deviation in the voltage and reactive power consensus. Figure 5D gives the reactive power dynamic characteristics of the system under load fluctuations. The reactive power deviation rises at t = 5 s as the MG’s load distribution becomes more unbalanced. During the P&P of DG4, the frequency, voltage, active power, and reactive power of MG remain stable within a reasonable range. The simulation results show that the proposed method will not influence the normal operation of the MG system.

FIGURE 5

FIGURE 5. Performance evaluation of the proposed method without cyber attacks: (A) frequency, (B) voltage, (C) active power, (D) reactive power.

4.2 Distributed resilient control strategies under hybrid attacks

In this subsection, the MG system is considered to suffer from both FDI attacks and DoS attacks. Also, the performance of the MG system with distributed controllers using traditional consensus algorithms is tested to show the impact of hybrid attacks. It will also serve as a benchmark for comparison with the proposed approach. The effectiveness of the proposed resilient control strategy in secondary frequency and voltage control is evaluated for the sensor and actuator attacks and intermittent interruptions of the communication network in Figure 4, respectively. The duration of the DoS attacks is designed to $\left|{\mathrm{\Xi }}_a\left(0,t\right)\right|=2.514$ s such that (25) satisfies.

Simulation events are set as follows: i) at t = 0 s, the MG operates in an islanded mode, and the DoS attacks persist and lead to two communication topologies as shown in Figure 4. ii) at t = 0.5 s, the proposed resilient controller is activated and an actuator is attacked at DG3 with ${\delta }_{Vi,1}^a=10\mathit{sin}\left(10t+5\right)$ V. iii) at t = 2.5 s, the actuator attack on DG3 is mitigated, but sensors are attacked at both DG1 and DG4 with signals ${\delta }_{Vi,1}^s=2\mathit{sin}\left(15t+4.5\right)$ V. iv) at t =4.5 s, the sensor attacks on DG1 and DG4 are mitigated. The sensor of DG2 is attacked by ${\delta }_{Vi,1}^s=2\left(t-3\right)$ V, and the actuator of DG3 is attacked by ${\delta }_{Vi}^a=5\left(t-3\right)$ V. The FDI signals persist in subsequent processes. v) at t =5.0 s, the load L₃ increases by 50%.

Figures 6A, B depict the dynamic characteristics of the proposed resilient control strategy in secondary voltage control. It can be found that the existence of DoS attacks will affect the control performance, resulting in system voltage and power fluctuations. However, if the duration of the DoS attacks satisfies (25), the system’s stability will not be threatened. Furthermore, the communication topology after DoS attacks is complete, which can guarantee system convergence under the proposed resilient approach. During 0.5∼6 s, the system suffers from sinusoidal form and linear growth FDI attacks, respectively, and the simulation results show that as long as the derivatives of the injected FDI signals are bounded [the growth rate of the malicious signals satisfies (30)], the resilient control strategy ensures that the system achieves UUB. The estimated values of the FDI attack signals are shown in Figure 7. It can be seen that the proposed control strategy can better estimate the FDI attack signals in the system, which will accurately and effectively alleviate the impact of FDI attacks on the MG. Besides, the effectiveness of the proposed method can be clearly demonstrated by comparing the results of Figure 6 and Figure 8.

FIGURE 6

FIGURE 6. The test results of the proposed resilient voltage control under hybrid attack: (A) voltage, (B) reactive power.

FIGURE 7

FIGURE 7. Sensor and actuator attack signals (solid lines) and their estimated values (dashed lines): (A) sensor FDI, (B) actuator FDI.

FIGURE 8

FIGURE 8. The test results of the traditional voltage consensus algorithms under hybrid attack: (A) voltage, (B) reactive power.

Figure 9 shows the performance of the proposed control strategy in secondary frequency control. In the case of load fluctuation and mixed network attack, the proposed method can still ensure the system’s normal operation. In addition, the proposed algorithm can effectively deal with bounded or unbounded sensor and actuator attacks. Compared with the results of the traditional consensus algorithm shown in Figure 10, the proposed strategy effectively suppresses the frequency and active power fluctuations caused by hybrid attacks in frequency control and performs better.

FIGURE 9

FIGURE 9. The test results of resilient frequency control under hybrid attack: (A) frequency, (B) active power.

FIGURE 10

FIGURE 10. The test results of the traditional frequency consensus algorithms under hybrid attack: (A) frequency, (B) active power.

4.3 Comparison with other resilient control methods

In this section, the proposed control method is compared with a cyber-resilient control approach in the latest work (Wang et al., 2023). The cyber-resilient control approach is based on an adaptive strategy to compensate for bounded FDI attacks on the secondary controller, and it can also tolerate DoS attacks on the communication link. Considering the characteristics of the cyber-resilient controller, the following modifications and clarifications are made in this paper: 1) the cyber-resilient controller only targets bounded actuator attacks, and it utilizes adaptive techniques to mitigate hybrid network attacks and does not reconstruct or estimate unavailable states and FDI attack signals in the system; 2) the cyber-resilient control approach is designed for secondary frequency control. This section only compares the frequency results of the MG system under hybrid attacks.

Figure 11 shows the performance of the cyber-resilient controller in the secondary frequency control. Compared with the resilient control strategy proposed in this paper, both of them have better performance under bounded actuator FDI signals and DoS attacks. During 0.5–2.5 s, both the cyber-resilient and the proposed controller can effectively resist the influence of the bounded sensor FDI signals, and the frequency and active power of the system only fluctuate or shift slightly. After t = 2.5 s, the system suffers from sensor FDI and linear growth FDI signals, respectively. Compared with the traditional consensus algorithm, the cyber-resilient controller can alleviate the impact of unbounded FDI signals but cannot effectively compensate for attacks. The system with the cyber-resilient controller is awful under unbounded FDI attacks. Therefore, the resilience strategy proposed in this paper can better resist the influence of multiple concurrent attacks and improve the stability of the islanded MG system under hybrid attacks.

FIGURE 11

FIGURE 11. Performance evaluation of cyber-resilient controller under hybrid attack: (A) frequency, (B) active power.

5 Conclusion

To deal with the frequency and voltage restoration problem of islanded MGs under hybrid FDI and DoS attacks, this paper proposes a resilient distributed control protocol to improve the robustness of frequency and voltage restoration. Using the state observers, the proposed control strategy can effectively estimate the potential FDI attack signals on each DG. The open-loop observer is utilized to reconstruct the neighbor information during DoS attacks, and the adaptive parameters are used to constrain the consensus error. In addition, the proposed control strategy can improve the system’s ability to resist unbounded FDI attacks on sensors and actuators and is more practical in real-world security applications. The stability of the system is proved by the Lyapunov function, which shows that the system can realize UUB under hybrid attacks. The simulation results verify that the proposed control method can effectively mitigate or cope with the impact of concurrent DoS and FDI attacks on MGs. Future work will further explore the coordinated control framework of the AC MG system under other attack scenarios, such as replay attacks, communication delays, and failures.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

YL: Writing–original draft, Writing–review and editing. ZD: Writing–original draft, Writing–review and editing. YC: Writing–original draft, Writing–review and editing. HZ: Writing–original draft, Writing–review and editing.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This research was supported by the State Key Laboratory of HVDC (Project No. SKLHVDC-2022-KF-14).

Conflict of interest

Author YC was employed by CSG Electric Power Research Institute China Southern Power Grid.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fenrg.2023.1320968/full#supplementary-material

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Nomenclature