Distributed adaptive resilient control for AC microgrids against false data injection attacks on state information

1 Introduction

With the increasing penetration of distributed renewable generation, energy storage systems, and diverse loads, conventional AC microgrids are facing growing limitations in maintaining power balance, accommodating renewable energy, and providing reserve support (Chen et al., 2021). Interconnecting multiple AC microgrids through a flexible DC network enables the complementary source and load characteristics across different regions to be utilized, thereby facilitating inter-microgrid power exchange and coordinated operation (Wu et al., 2026). Multi-terminal AC/DC converters allow flexible regulation of power transmission while reducing direct electrical interactions among individual AC microgrids (Liu et al., 2023). This configuration enables each microgrid to maintain relatively independent voltage and frequency control while participating in energy exchange, thereby enhancing operational flexibility, energy utilization efficiency, and disturbance support capability (Chen et al., 2026; Wang et al., 2026).

However, the stable operation of DC-interconnected AC microgrids depends strongly on extensive data acquisition, frequent information exchange, and real-time control (Jiang et al., 2022; Zhou et al., 2023). A false data injection attack (FDIA) is a type of data integrity attack that manipulates measurement data or superimposes malicious signals on legitimate measurements (Jamali et al., 2024). As a result, the measured state variables, such as voltage, frequency, and power, may deviate from their actual values, causing controllers to generate erroneous control actions and potentially destabilizing the closed-loop system. From the perspective of attackers, the large number of distributed sensors, communication links, and AC/DC converter control interfaces creates multiple and dispersed attack entry points. Moreover, distributed secondary control continuously relies on local and neighboring state information for coordinated control. Carefully designed false data may evade the conventional bad data detection (BDD) mechanisms and propagate across the system through inter-node information exchange. In low-inertia systems, with a high penetration of power electronic devices, local state corruption may rapidly develop into voltage and frequency limit violations, severe overloads of converters and lines, and cross-regional disturbance propagation. In extreme cases, these effects may result in loss of synchronism among interconnected microgrids (Xiao et al., 2026).

Therefore, it is essential to investigate defense and control strategies against FDIAs in microgrids. From the perspective of defenders, existing studies can be generally classified into two categories: proactive prevention and post-attack recovery.

The proactive defense strategies in cyberattack defense aim to reduce the stealthiness and effectiveness of FDIAs by explicitly accounting for system uncertainties, optimizing the allocation of defense resources, and enhancing anomalous data identification. Giraldo et al. (2022) developed a decentralized moving target defense mechanism by combining data replication with randomized link-based transmission, thereby concealing attack paths and facilitating anomalous data identification. In Liu et al. (2022), the controller gains of converters are dynamically perturbed to invalidate the system model attackers acquire in advance. To balance defense effectiveness against operational costs, Lakshminarayana et al. (2021) formulate the system parameter perturbation as an attacker-defender game, while Lakshminarayana and Yau (2021) further optimize the allocation of limited defense resources through cost-benefit analysis. Considering the diffusion of malicious data across communication networks, Bi et al. (2022) established a game-theoretic model of FDIA and propagation to determine strategies for critical node protection and malicious data removal. To detect stealthy attacks, Karanfil et al. (2023) proposed a cyber-physical collaborative detection framework that integrates network operating states with the physical measurements of microgrids. In contrast, Su et al. (2024) employ a multi-graph attention network to extract the spatial, temporal, and topological features of measurement data, thereby improving FDIA identification performance. However, proactive defense methods generally require additional communication, measurement, or control resources, resulting in high implementation costs. Moreover, the effectiveness of data- or model-driven methods is highly dependent on algorithmic accuracy and data quality, making it difficult to meet the real-time prevention and mitigation requirements of microgrids.

The post-attack recovery control aims to maintain closed-loop stability and restore system states to their prescribed operating values after malicious data enter measurement, communication, or control channels. Recent research has progressed along three main directions. First, model-based methods employ observers to reconstruct unavailable states or attack signals and then compensate for their effects. Hu et al. (2023) designed a piecewise observer and a resilient load-frequency controller for FDIAs. Zuo et al. (2023) developed a fully distributed framework for unbounded and correlated attacks on sensing, actuation, and communication channels. Zhou et al. (2024) combined distributed voltage observation with resilient secondary control under unbounded FDIAs. Jamali et al. (2024) addressed frequency synchronization and active power sharing under state-dependent FDIAs. Second, communication-efficient approaches reduce the information exchange required for recovery. Annavaram et al. (2024) proposed an event-triggered distributed control for voltage restoration and current sharing. Zhang et al. (2024) integrated distributed attack detection with event-triggered recovery learning to correct compromised control inputs. Third, data-driven techniques improve adaptability to uncertain models and changing conditions. Xia et al. (2024) used transfer learning to reconstruct corrupted interaction signals under different operating conditions. Beikbabaei et al. (2024) combined long short-term memory (LSTM)-based detection and bidirectional long short-term memory (BiLSTM)-based mitigation with model-free coordinated inverter control. Lu et al. (2025) developed a digital-twin-assisted observer for attack-vector estimation and measurement reconstruction under converter-model uncertainties. Mukherjee et al. (2025) proposed vertical federated reinforcement learning for networked microgrids and validated the recovery policies through real-time experiments. However, model-based methods often depend on the attack detectability and model accuracy, while data-driven approaches require representative training data and substantial computation. Fully distributed online recovery of both frequency and voltage under random and unbounded attacks remains insufficiently studied.

To address the challenges above, this article proposes a distributed adaptive resilient control strategy for AC microgrids subject to FDIAs on state information. The main contributions are summarized as follows:

• By introducing the distributed consensus-based control framework, rapid correction of local operating points is achieved, thereby ensuring coordinated operation among multi-terminal AC microgrids.
• By introducing the adaptive resilient controller, coordinated mitigation of FDIAs in frequency and voltage channels is achieved, thereby preventing local state deviations from causing system instability.
• By introducing a lightweight control model with limited information exchange between neighboring terminals, decreased control and communication requirements are achieved, thereby reducing the impact of FDIAs on system stability and limiting the risk of successful attacks.

The remainder of this article is organized as follows. In Section 2, the cyber-physical architecture of DC-interconnected AC microgrids is introduced, and the principles and limitations of conventional consensus-based distributed control under FDIAs are analyzed. In Section 3, the distributed secondary frequency and voltage control framework is established, and the proposed distributed adaptive resilient controllers are developed. In Section 4, the effectiveness and adaptability of the proposed strategy are validated through simulation studies under frequency, voltage, and coordinated attacks. The main conclusions are summarized in Section 5.

2 Cyber-physical system of AC microgrids and basics of consensus-based distributed control method

In this section, the physical topology and communication network of DC-interconnected AC microgrids are first established. The fundamental principle of consensus-based distributed control is introduced, and its main limitations under FDIAs are analyzed.

2.1 Cyber-physical system architecture of AC microgrids

A typical cyber-physical system (CPS) of AC microgrids interconnected via a flexible DC network is illustrated in Figure 1. At the physical layer, multiple AC microgrids are connected to a DC network through hybrid modular multilevel converters (MMCs). On the DC side, a low-voltage ring network enables flexible interconnection among multiple AC regions while supplying various types of loads. A virtual measurement unit (VMU) is deployed at each MMC terminal. Corresponding to the topology of the physical system, the cyber layer primarily comprises distributed control agents (DCAs) and communication links. Each MMC is equipped with a DCA that performs real-time control of its local optimal operating point based on locally measured data and information obtained from neighboring agents.

Based on the CPS architecture described above, the DCAs in the cyber layer are identified as the system components most vulnerable to FDIAs. As the computational core of distributed control and the hub of data processing, a DCA constitutes a particularly attractive target for adversaries. Even a localized disturbance can readily propagate through information exchange and induce system instability, thereby enabling an attack at relatively low cost. Given the low probability of simultaneous attacks at multiple locations, this study focuses on the scenario in which an adversary maliciously injects a random, unbounded false signal into the input channel of a DCA controller terminal.

2.2 Basic principle of the consensus-based distributed control method

In the consensus-based distributed control method (CDCM), the sparse communication network among DCAs in a microgrid system can be modeled as a directed graph. The DCAs are represented as nodes, while the communication links between them are represented as directed edges. The directed graph is defined as ρ = v , e , A, where *v* = {v ₁, v ₂, … , v _*N*} is the set of nodes, e ∈ v × v is the set of edges, and A = A i j ∈ R N × N is the weighted adjacency matrix. Let (v _*j*, v _*i*) denote a directed edge from the jth node to the ith node and let A _*ij* denote its associated weight. If (v _*j*, v _*i*)∈*e*, the ith node can receive information from the jth node, and A _*ij*>0; otherwise, A _*ij* = 0. The set of neighboring nodes from which the ith node receives information is defined as *N* _***i*** = {j|(v _*j*, v _*i*)∈*e}. The in-degree matrix of the directed graph is defined as D* = diag{d _*i*}, where d i = ∑ j ∈ N i A i j. Accordingly, the graph Laplacian is defined as *L = D − A*. A directed path from the ith node to the jth node consists of a sequence of directed edges and can be represented as v i , v m , v m , v n , . . . , v l , v j. A directed graph is said to contain a directed spanning tree if it has a root node from which every other node in the graph is reachable through a directed path.

Let x _*i* denote the state information of the ith node in the cyber layer. Under the framework of CDCM, each node updates its state x _*i* using only its own information and that received from its neighboring nodes. Given a consensus reference value x _ref, the tracking synchronization control law is formulated as Equation 1:where G _*i* denotes the pinning gain associated with the edge connecting the ith node to the consensus reference. If the communication network contains a directed spanning tree rooted at x _ref, the state information of all the nodes can converge to the consensus reference value.

x ˙ i t = − ∑ j ∈ N i A i j x i t − x j t − G i x i t − x ref , (1)

The conventional CDCM enables each DCA to coordinate its state using only local and neighboring information, offering low communication overhead and high scalability. However, it assumes that exchanged measurements and control signals are trustworthy. Under FDIAs, corrupted data can propagate through the communication network, drive agents toward a false equilibrium, and overwhelm fixed-gain consensus feedback. The limitations motivate the distributed adaptive resilient control method (DARCM) developed in this article.

3 Proposed distributed adaptive resilient control strategy for AC microgrids under FDIAs

In this section, the architecture and model of primary control and distributed secondary frequency and voltage control for the AC microgrids are first established. Second, distributed resilient frequency and voltage controllers are incorporated into the control framework to mitigate the impact of FDIAs through adaptive corrective control.

3.1 Consensus-based distributed control of frequency and voltage for AC microgrids

3.1.1 Primary droop control

The primary control layer consists of a droop controller, a voltage controller, and a current controller. Under droop control, the real-time power output of the converter is first obtained from the VMU. The voltage reference is then determined according to the droop characteristics. Subsequently, PWM signals are generated through the outer voltage and inner current control loops. The active power-frequency and reactive power-voltage droop control equations are expressed using Equation 2:where m _*i* and n _*i* denote the frequency and voltage droop coefficients of the ith converter, respectively. P _N*i* is the rated active power of the ith converter. ω _N is the rated frequency. ω _*i* is the frequency of the microgrid associated with the ith converter. P _*i* and Q _*i* are the actual active power and reactive power output of the ith converter, respectively. U ₀ is the open-circuit voltage. U i * is the voltage reference of the ith converter.

ω i = ω N − m i P i − P N i U i * = U 0 − n i Q i , (2)

3.1.2 Distributed secondary frequency control

The droop control inherently results in steady-state deviations. Therefore, the distributed secondary controller is introduced; its control mechanism is illustrated in Figure 2. The distributed secondary frequency control law is formulated as Equation 3:where Ω i denotes the output of the distributed secondary frequency controller.where c ω i is the frequency control parameter. g _*i* denotes the fixed coupling gain from the leader node to the ith node, and ω _ref is the system frequency reference. To ensure that all converters in the system share the load in proportion to their rated capacities, the droop coefficients are generally selected to be inversely proportional to their respective capacities. When m 1 P 1 = m 2 P 2 = . . . = m N P N is satisfied, all converters share the load proportionally to their rated capacities.

ω i = ω N − m i P i − P N i + Ω i , (3)

Ω ˙ i = c ω i ∑ j ∈ N i A i j ω j − ω i + g i ω ref − ω i + ∑ j ∈ N i A i j m j P j − m i P i , (4)

3.1.3 Distributed secondary voltage control

The distributed secondary voltage control aims to regulate the average voltage of all the nodes to its rated value while achieving accurate reactive power sharing among all converters.

Based on the state estimation of voltage, the average voltage of all the nodes can be estimated using Equation 5:where U ¯ i denotes the average voltage of all nodes estimated by the ith node. k _*Ui* is a control parameter.

U i ¯ · = U ˙ i * + k U i ∑ j ∈ N i A i j U j ¯ − U i ¯ , (5)

The distributed secondary voltage control law is expressed as Equation 6:where h _*i* denotes the output of the secondary voltage controller and is calculated using Equation 7:where c _*Ui* is the voltage control parameter and U _ref is the voltage reference, which is set to the rated voltage in this article.

U i * = U 0 − n i Q i + h i , (6)

h ˙ i = c U i g i U ref − U ¯ i + ∑ j ∈ N i A i j n j Q j − n i Q i , (7)

When the communication network contains at least one spanning tree, the proposed control procedure guarantees that the average voltage converges to its rated value. Meanwhile, the reactive power outputs of the converters are shared in proportion to their respective droop coefficients, that is, n 1 Q 1 = n 2 Q 2 = . . . = n N Q N.

3.2 Design of fully distributed adaptive resilient controllers

3.2.1 Cyberattack model for secondary control

At the cyber layer, the FDIAs are modeled as additive malicious signal injections into the output channels of the distributed secondary controllers. Specifically, as illustrated in Figure 2, the attack signal is superimposed on the nominal frequency or voltage regulation signal generated from the corresponding state information, before the resulting compromised signal enters the subsequent stage of primary control.

Let ξ i = Ω ˙ i and u i = h ˙ i. The cyberattack model can be formulated as Equation 8:where δ i denotes the malicious signal injected into the secondary controller. ξ i F and u i F represent the compromised output of the secondary frequency and voltage controllers, respectively. Both μ i and κ i are binary variables. μ i = 1 indicates that the frequency controller is under attack, whereas μ i = 0 indicates that no attack is launched against it. A similar formulation applies to attacks on the voltage controller.

ξ i F = ξ i + μ i δ i u i F = u i + κ i δ i , (8)

In this article, the FDIA signal δ i may take either an unbounded form or an arbitrarily bounded form. The FDIA signal at any time can be expressed as Equation 9:where a, b, and c are constants. The term a denotes a bounded constant attack signal. f(t) represents an unbounded function. g(t) represents a bounded non-constant function.

δ i = a + b f t + c g t , (9)

3.2.2 Distributed resilient frequency controller

To eliminate the adverse effects of FDIAs on frequency control, a distributed adaptive resilient frequency controller is developed in this section, as illustrated in Figure 2A. A compensation term Δ ^ i is incorporated into the intermediate output of the distributed secondary frequency controller. Accordingly, the original output is corrected and updated to k _*i*, as expressed in Equation 10.

k i = ξ i + Δ ^ i = c ω i ∑ j ∈ N i A i j ω j − ω i + g i ω ref − ω i + ∑ j ∈ N i A i j m j P j − m i P i + Δ ^ i . (10)

The adaptive compensation term Δ ^ i is defined by Equations 11, 12. where ϑ i is an adaptive updated variable. The parameters α i and γ i are positive constants. A continuous function e − α i t is introduced to ensure smooth control action.

Δ ^ i = ξ i ϑ i ξ i + e − α i t , (11)

ϑ ˙ i = γ i ξ i , (12)

To facilitate the subsequent theoretical analysis, the distributed secondary frequency control law in Equation 4 can be reformulated as follows:

ξ i = Ω ˙ i = c ω i ∑ j ∈ N i A i j ω j + m j P j − ω i + m i P i + g i ω ref + m i P i − ω i + m i P i . (13)

Let Θ i = ω i + m i P i, and Θ ref = ω ref + m * P *. m ^* and P ^* are the droop coefficient and steady-state active power output of each converter, respectively. Accordingly, Equation 13 can be reformulated as follows:

Θ ˙ i = c ω i ∑ j ∈ N i A i j Θ j − Θ i + g i Θ r e f − Θ i = c ω i − d i + g i Θ i + ∑ j ∈ N i A ij Θ j + g i Θ r e f . (14)

Equation 14 can be expressed in the global form given in Equation 15.where *G* = diag(g _*i*), and ε = Θ − 1 N Θ ref is defined as the global cooperative regulation error.

Θ ˙ = − diag c ω i L + G Θ − 1 N Θ ref , (15)

3.2.3 Distributed resilient voltage controller

To eliminate the adverse effects of FDIAs on voltage control, the distributed resilient voltage control model is developed based on a mechanism similar to that of the frequency controller, as illustrated in Figure 2B. The output of the secondary voltage controller is modified using Equation 16:where ϕ ^ i denotes the additional compensation term, which is calculated according to Equations 17, 18. where λ i is an adaptive updated variable. The parameters α i and σ i are positive constants.

S i = h ˙ i + ϕ ^ i = c U i g i U r e f − U ¯ i + ∑ j ∈ N i A i j n j Q j − n i Q i + ϕ ^ i , (16)

ϕ ^ i = u i λ i u i + e − α i t , (17)

λ ˙ i = σ i u i , (18)

4 Case study

In this section, the effectiveness of the proposed consensus-based distributed control strategy and adaptive resilient control strategy against FDIAs is validated. The proposed strategy is evaluated under multiple attack scenarios, including attacks targeting the frequency controller, attacks targeting the voltage controller, and coordinated attacks targeting both controllers. The attack mitigation performance under different parameter settings is analyzed.

The simulations are conducted on the four-terminal AC microgrids interconnected via a ring DC microgrid, as illustrated in Figure 1. The rated voltage and frequency of the AC side are 380 V and 50 Hz, respectively. The line impedances and initial load parameters at each node are listed in Table 1. The main controller parameters are set as c ω i = 10, c U i = 10, k U i = 0.025, and α i = 4 for all the microgrids. To evaluate the parameter sensitivity, γ i and σ i are varied according to the settings specified in Cases II–IV.

TABLE 1. Parameters of DC connected AC microgrids in Cases I–IV.

4.1 Case I: effectiveness verification of consensus-based distributed frequency and voltage control

In this case, the system is initially operated under only primary droop control. The distributed secondary frequency controller is enabled at 1 s, and the distributed secondary voltage controller is then enabled at 4 s. To validate the response to operating changes, the active load at terminal 1 is reduced by 4 kW, and its reactive load is increased by 8 kVar during 6–7 s. During 8–9 s, the active load at terminal 3 is reduced by 6 kW, and its reactive load is increased by 6 kVar.

Figure 3A shows that the terminal frequencies initially differ because of the local operating conditions. After the distributed secondary frequency control is activated, the four trajectories converge rapidly to approximately 49.91 Hz, with only a small transient. As the load-side active power demand decreases continuously, the resulting active power surplus causes the frequency at each terminal to increase linearly until a new operating point is reached. The results confirm that the distributed controller effectively removes the initial frequency differences and maintains coordinated frequency regulation using local and neighboring information. Also, as illustrated in Figure 3B, the activation of the distributed voltage controller causes the initially dispersed terminal voltages to converge rapidly to a common level. The increases in reactive load lead to linear voltage changes.

Therefore, Case I verifies the effectiveness of the distributed consensus mechanism under operating condition changes. The coordination relies on the integrity of the exchanged states and controller outputs. However, the persistent false signal may consequently enter the control loop and spread through information exchanges, establishing the need for the adaptive resilient compensation investigated in the following cases.

4.2 Case II: validation of resilient control against FDIAs targeting frequency controller

To evaluate the effectiveness of the proposed control strategy under severe attack conditions, δ i = 60 t is adopted as the unbounded attack signal injected into the voltage and/or frequency control channels in Cases II–IV. Once the attack is initiated, the magnitude of the injected signal increases linearly with time.

As shown in Figure 4, the unbounded FDIA is applied to the frequency controller of terminal 1 at 2 s while distributed frequency regulation remains enabled in this case. The proposed resilient frequency controller is then activated at 4 s. The load disturbance conditions are identical to those considered in Case I. Three different control coefficients are considered to evaluate their influence on the recovery process.

Before the attack, all terminal frequencies remain synchronized near 49.91 Hz. During 2–4 s, the injected signal induces a continuous decrease in the frequency at terminal 1, while the coordinated regulation mechanism produces frequency deviations of the opposite sign at the remaining terminals. The phenomenon demonstrates that the consensus-based distributed control without resilient compensation cannot reject an unbounded attack. After the resilient frequency controller is enabled, the frequency at each terminal exhibits a certain degree of recovery, moving away from its safety limit and thereby increasing the safety margin until a new steady state is reached.

It can be seen form Table 2 that the control coefficient γ clearly affects the recovery performance. The coefficient of 0.5 results in a slow corrective response and a noticeable frequency difference. Increasing the coefficient to 1 accelerates the recovery, while a value of 5 provides the fastest restoration after a short corrective transient. Under the proper coefficient, the terminal frequencies can respond coherently to the subsequent load variations. Therefore, the proposed DARCM can prevent the sustained frequency divergence and preserve coordinated regulation under an unbounded frequency attack.

TABLE 2. Quantitative performance comparison under frequency FDIAs.

4.3 Case III: validation of resilient control against FDIAs targeting voltage controller

In Case III, an unbounded FDIA is introduced to the voltage control channel of terminal 1 at 2 s, while the distributed voltage controller remains enabled. The resilient voltage controller is then activated at 3 s. The performance under the voltage control coefficients of 0.8, 1, and 1.5 is tested.

As shown in Figure 5, an attack targeting a single voltage controller may cause a substantial deviation in AC voltages, and if not promptly mitigated, it may eventually lead to voltage collapse and widespread power outages. Once the resilient voltage control strategy is enabled at 3 s, the rate of voltage increase is rapidly suppressed, and the voltage gradually returns to its steady state before the attack.

Based on the results under different configurations, a larger control coefficient improves both the peak suppression and recovery speed. According to Table 3, the coefficient of 0.8 produces the largest voltage peak and the longest recovery interval, whereas coefficients of 1 and 1.5 progressively reduce the peak and shorten the restoration process. Therefore, Case III confirms that the proposed resilient voltage controller suppresses the sustained voltage excursion caused by an unbounded attack, while the control coefficient determines the transient performance.

TABLE 3. Quantitative performance comparison under voltage FDIAs.

4.4 Case IV: validation of resilient control against coordinated frequency-voltage FDIAs

To validate the effectiveness of the proposed method under a more severe attack scenario, coordinated cyber-attacks targeting both the voltage and frequency control loops are considered in this case. The frequency-channel attack begins at 1.5 s, and resilient frequency control is activated at 2 s. The voltage attack begins at 4.5 s, followed by the enabling of resilient voltage control at 5 s. Two types of parameter settings are compared.

As can be readily observed from Figure 6, the DARCM enables coordinated mitigation and rapid recovery of both system frequency and voltage, further confirming the effectiveness of the proposed method. Moreover, as illustrated in Table 4, relatively large frequency and voltage control gains result in faster responses of the state variables while suppressing overshoot to a certain extent. These results demonstrate that the proposed adaptive control strategy maintains system resilience under multiple attack scenarios and possesses favorable scalability and adaptability to diverse operating conditions.

TABLE 4. Quantitative performance comparison under coordinated frequency–voltage FDIAs.

5 Conclusion

To enhance the cyber resilience of AC microgrids, a distributed adaptive resilient control strategy is developed against FDIAs targeting secondary-control state information. Based on the simulation results, the following conclusions can be drawn.

• The proposed control strategy enables rapid and coordinated restoration of frequency and voltage across multiple terminals under unbounded FDIAs while ensuring reliable load supply.
• Compared with standard consensus-based distributed secondary control, the proposed method provides online compensation of local operating points under diverse attack scenarios, thereby enhancing system resilience and transient disturbance rejection while exhibiting strong scalability.
• The proposed fully distributed architecture requires only local measurements and information exchanged with neighboring nodes, thereby limiting private information exposure and reducing the susceptibility to cyberattacks.

It is noted that this study mainly focuses on control-level mitigation of FDIAs in a simulation environment. Resilient control triggering under communication imperfections and experimental validation will be investigated in future work.

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Footnotes

Edited by: Shuai Lu, Southeast University, China

Reviewed by: Xin Wang, City University of Hong Kong, Hong Kong SAR, China

Yuchen Fang, Dalian University of Technology, China