The power system, a complex infrastructure with numerous components and devices, relies heavily on synchronous generators as core elements in energy conversion processes (Liu et al., 2025a; Liu et al., 2025b). Systems composed of multiple generators have garnered attention due to their distributed nature, allowing effective coordination, control, and interconnection (Abdollahi et al., 2021; Zhao et al., 2021; Hu et al., 2022; Kenyon et al., 2023; Liu et al., 2025a). Actuator failures during operation, leading to potential system instability and equipment damage, pose a significant challenge (Xu et al., 2023). Addressing these faults becomes imperative for enhancing power system reliability and security, emphasizing the need for fault-tolerant control schemes (Ren et al., 2015). For large-scale distributed generation systems, hierarchical distributed model predictive control (Kong et al., 2019) and asynchronous distributed power control for hybrid microgrids have been proposed. These approaches aim to achieve distributed control, inspired by collaborative control principles (Hu et al., 2024; Zhang et al., 2023; Rostampour et al., 2019; Hu et al., 2025). However, nonlinear interactions among synchronous generators, characterized by adverse coupling, present challenges in controller design for power systems.

In recent years, scholars have dedicated substantial attention to researching fault-tolerant control methods for power systems afflicted by actuator faults to uphold system stability. Notably, a collaborative fault-tolerant control protocol was devised by Huang et al. (2021) to address actuator faults in autonomous AC power systems. Furthermore, in Afshari et al. (2020), Huang et al. proposed a fault-tolerant control scheme tailored for active DC power systems, with a primary objective of ensuring the stability of system voltage regulation. Additionally, a distributed fault-tolerant control protocol focusing on frequency-voltage synchronization in power systems was presented in Shahab et al. (2020). This protocol adeptly regulates both voltage and frequency, ensuring precise sharing of active power ratios.

To tackle actuator faults in power systems, as discussed in Dehkordi and Moussavi (2020), a fault-tolerant control protocol grounded in distributed consensus was developed, employing a sliding mode control strategy. Moreover, in Hu and Bhowmick (2020), a fully distributed consensus-based secondary robust voltage restoration control protocol was crafted specifically for islanded microgrids. The primary aim was to enhance the overall reliability and flexibility of microgrid operations. In another innovative approach, Huang et al. proposed a novel fault-tolerant control protocol in Huang et al. (2023). This protocol, based on elastic adaptive consensus methods, sought to mitigate the adverse effects of actuator faults on power system stability, particularly in regulating frequency and voltage. However, actuator faults in synchronous generators often remain in an unknown state, posing a significant challenge. Conventional methods struggle to achieve safe and stable control of power systems comprising synchronous generators due to the coupling effects between these generators. This inherent complexity necessitates the exploration of more sophisticated control strategies and fault-tolerant mechanisms to ensure the reliability and stability of power systems.

In recent years, researchers have made significant strides in designing control protocols for nonlinear multi-agent systems, employing innovative backstepping methods. Ma et al. (2023) developed a consistent control strategy for multi-agent systems, achieving effective leader-following consensus control. Chen et al. (2020), guided by consensus theory, explored the capability of multi-agent systems to recover collaborative performance after encountering faults or uncertainties, specifically addressing unknown faults on communication links. Liu et al. (2023) proposed a game-based backstepping control method for high-order nonlinear multi-agent systems with unknown dynamics and input saturation. In another approach, Zhao et al. (2023) integrated adaptive radial basis function neural networks with backstepping techniques, presenting an adaptive output-feedback approach for constructing consensus tracking control protocols and adaptive laws. However, it’s crucial to note that studies such as (Ma et al., 2023; Chen et al., 2020; Liu et al., 2023; Zhao et al., 2023) are theoretical in nature and may lack consideration of practical contexts. Consequently, the integration of these strategies into power systems comprising synchronous generators assumes significant importance. Moreover, nonlinear actuators may introduce oscillations, thereby amplifying uncertainty within the power system. To ensure stability, it becomes imperative to formulate appropriate control strategies and develop effective optimal actuator control schemes, particularly to achieve distributed voltage-frequency restoration with optimal control. This motivation underscores the focus and relevance of the present study.

Inspired by the studies mentioned above, this article proposes a collaborative fault-tolerant control solution for power systems equipped with generators featuring actuators. By leveraging the backstepping method and Lyapunov function stability theory, we demonstrate the boundedness of all signals in the closed-loop system, ensuring the secure and stable operation of the power system. The key contributions of this work are outlined as follows:

In alignment with the consensus theory of multi-agent systems, this study extends the backstepping method and distributed consensus control algorithm to power systems with synchronous generators affected by actuator faults. It introduces a novel adaptive distributed collaborative fault-tolerant control protocol for the first time. The research demonstrates the feasibility of this theoretical approach in the context of power systems composed of synchronous generators;

The subsequent sections of this paper are organized as follows: Section 2 presents the basic model, problem formulation, and fundamental assumptions and definitions. Section 3 introduces the design of the distributed adaptive collaborative fault-tolerant control protocol and adaptive parameters. The stability analysis proof is provided in Section 4. Simulation experiments are presented in Section 4. Finally, Section 5 concludes this paper.

The proposed control architecture, depicted in Figure 1, employs a backstepping-based fuzzy adaptive strategy where each local controller calculates tracking errors z i , 1 and generates fault-tolerant signals u i through distributed coordination. Virtual control inputs ς i , 1 and final control protocols u i are subsequently designed for the power system.

To simplify the subsequent backstepping design process, the adaptive parameter is defined as θ i , h = max ‖ W i , h ‖ 2 , h = 1,2 , (4) where W i , h shows the ideal weight vector. Simultaneously, we introduce θ ̂ i , h as the estimate of θ i , h . Additionally, θ ̃ i , h = θ ̂ i , h − θ i , h is defined as the estimate error. Next, define the coordinate transformation as follows. z i , 2 = y i 1,2 − ς i , 1 , (5) where ς i , 1 is virtual control protocol. Subsequently, the genuine adaptive voltage restoration consensus controller is formulated in Step 2.

Step 1

Choose the candidate Lyapunov function as V i , 1 = z i , 1 2 2 + θ ̃ i , 1 2 2 γ i , 1 , where γ i , 1 > 0 represents the design parameter.

Then, based on (Equation 3), we can obtain V ̇ i , 1 = z i , 1 b i + d i f i 1,1 z i , 2 + f i 1,1 ς i , 1 + μ i 1 − b i f r v r , t − ∑ j = 1 N a i , j f j 1,1 y j 1,2 + μ j 1 + 1 γ i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 . (6)

According to Young’s inequality, we have − z i , 1 b i f r v r , t ≤ z i , 1 b i f r v r tanh z i , 1 b i f r v r ϑ i , 1 + 0.2785 ϑ i , 1 z i , 1 b i + d i f i 1,1 z i , 2 ≤ 1 2 z i , 1 2 b i + d i 2 f i 1,1 2 + 1 2 z i , 2 2 furthermore, we can obtain V ̇ i , 1 ≤ z i , 1 b i + d i f i 1,1 ς i , 1 + μ i 1 + Γ i , 1 x i + 1 2 z i , 2 2 + 0.2785 ϑ i , 1 + 1 γ i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 , (7) where Γ i , 1 x i , 1 = b i f r v r tanh z i , 1 b i f r v r ϑ i , 1 + 1 2 z i , 1 b i + d i 2 f i 1,1 2 − ∑ j = 1 N a i , j f j 1,1 y j 1,2 + μ j 1 . then, for a given accuracy parameter π i > 0 , we have Γ i , 1 x i , 1 = W i , 1 T Φ i , 1 x i , 1 + π i . furthermore, according to Φ T ( x ) Φ ( x ) < 1 and Young’s inequality, we can obtain z i , 1 Γ i , 1 x i , 1 ≤ 1 2 a ̃ i , 1 2 z i , 1 2 θ i , 1 + 1 2 a ̃ i , 1 2 + 1 2 z i , 1 2 + 1 2 π i , 1 2 , (8) where a ̃ i , 1 > 0 is a positive constant. Substituting (Equation 8) into (Equation 7), we have V ̇ i , 1 ≤ z i , 1 b i + d i f i 1,1 ς i , 1 + μ i 1 + 1 2 z i , 1 + 1 2 a ̃ i , 1 2 z i , 1 θ i , 1 + 1 2 z i , 2 2 + 1 2 a ̃ i , 1 2 + 0.2785 ϑ i , 1 + 1 2 π i , 1 2 + 1 γ i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 . (9) then, the virtual controller ς i , 1 is designed as ς i , 1 = − 1 b i + d i f i 1,1 z i , 1 c i , 1 + 1 2 − b i + d i μ i 1 − z i , 1 2 a ̃ i , 1 2 θ ̂ i , 1 , where c i . 1 > 0 is the design parameter.

The adaptive law is defined as θ ̂ ̇ i , 1 = − γ i , 1 2 a ̃ i , 1 2 z i , 1 2 − ω i , 1 θ ̂ i , 1 , (10) where ω i , 1 > 0 is the adaptation decay coefficient.

Furthermore, Equation 9 can be rewritten as V ̇ i , 1 ≤ − c i , 1 z i , 1 2 + 1 2 z i , 2 2 + 1 2 a ̃ i , 1 2 + 0.2785 ϑ i , 1 + 1 2 π i , 1 2 − ω i , 1 γ i , 1 θ ̃ i , 1 θ ̂ i , 1 . (11)

Step 2

According to Equation 5, we can obtain z ̇ i , 1 = χ i + f i 1,2 u i f − ς ̇ i , 1 . then, choose the candidate Lyapunov function as V i , 2 = V i , 1 + 1 2 z i , 2 2 + 1 2 γ i , 2 θ ̃ i , 2 2 , where γ i , 2 > 0 represents the design parameter.

Furthermore, V ̇ i , 2 ≤ V ̇ i , 1 + z i , 2 χ i + f i 1,2 u i f − ∂ ς i , 1 ∂ y i 1,1 f i 1,1 y i 1,2 + μ i 1 − ∂ ς i , 1 ∂ y r f r v r − ∑ j ∈ n i ∂ ς j , 1 ∂ y j 1,1 f j 1,1 y j 1,2 + μ j 1 − ∂ ς j , 1 ∂ θ ̂ i , 1 θ ̂ ̇ i , 1 + 1 γ i , 2 θ ̃ i , 2 θ ̂ ̇ i , 2 . (12)

Based on Young’s inequality, we get − z i , 2 ∂ ς j , 1 ∂ y r f r v r ≤ z i , 2 ∂ ς j , 1 ∂ y r f r v r tanh z i , 2 ∂ ς j , 1 ∂ y r f r v r ϑ i , 2 + 0.2785 ϑ i , 1 . (13)

Subsequently, in accordance with (Equation 2), (Equations 11–13) is reformulated as V ̇ i , 2 ≤ − c i , 1 z i , 1 2 − ω i , 1 γ i , 1 θ ̃ i , 1 θ ̂ i , 1 + 1 2 z i , 2 2 + 0.2785 ϑ i , 1 + 0.2785 ϑ i , 2 + 1 2 π i , 1 2 + 1 γ i , 2 θ ̃ i , 2 θ ̂ ̇ i , 2 + z i , 2 u i + Γ i , 2 x i , 2 + χ i , where Γ i , 2 x i , 2 = z i , 2 ∂ ς j , 1 ∂ y r f r v r tanh z i , 2 ∂ ς j , 1 ∂ y r f r v r ϑ i , 2 − ∑ j ∈ n i ∂ ς j , 1 ∂ y j 1,1 f j 1,1 y j 1,2 + μ j 1 − ∂ ς j , 1 ∂ θ ̂ i , 1 θ ̂ ̇ i , 1 + ϖ t − ∂ ς i , 1 ∂ y i 1,1 f i 1,1 y i 1,2 + μ i 1 .

Using the same approach as in Step 1, we obtain z i , 2 Γ i , 2 x i , 2 ≤ 1 2 a ̃ i , 2 2 z i , 2 2 θ i , 2 + 1 2 a ̃ i , 2 2 + 1 2 z i , 2 2 + 1 2 π i , 2 2 .

The distributed voltage collaborative fault-tolerant control protocol and parameter adaptation rates are designed as u i = − c i , 2 + 1 z i , 2 − χ i − 1 2 a ̃ i , 2 2 z i , 2 θ ̂ i , 2 , (14) and θ ̂ ̇ i , 2 = − γ i , 2 2 a ̃ i , 2 2 z i , 2 2 − ω i , 2 θ ̂ i , 2 , (15) where c i , 2 > 0 and ω i , 2 > 0 are design constants.

Furthermore, we can obtain V ̇ i , 2 ≤ − ∑ j = 1 2 c i , j z i , j 2 − ∑ j = 1 2 ω i , j γ i , j θ ̃ i , j θ ̂ i , j + 1 2 ∑ j = 1 2 a ̃ i , j 2 + 0.2785 ∑ j = 1 2 ϑ i , j 2 + 1 2 ∑ j = 1 2 π i , j 2 . (16)

By using Young’s inequality, Equation 16 is rewritten as V ̇ i , 2 ≤ − ∑ j = 1 2 c i , j z i , j 2 − ∑ j = 1 2 ω i , j γ i , j θ ̃ i , j 2 + Δ i , where Δ i = ∑ j = 1 2 ω i , j 2 γ i , j θ i , j 2 + 1 2 ∑ j = 1 2 a ̃ i , j 2 + 0.2785 ∑ j = 1 2 ϑ i , j 2 + 1 2 ∑ j = 1 2 π i , j 2 .

Define N i = min { c i , j ≥ 0 , j = 1,2 } , we have V ̇ i , 2 ≤ − N i V i , 2 + Δ i . (17)

Based on the above discussion, the following theorem is proposed, summarizing the current research and conducting a stability analysis.

Theorem 1

For a power system with actuator faults, based on the backstepping method and multi-agent system collaborative control theory, the fuzzy adaptive collaborative fault-tolerant control protocol (Equation 14) and the adaptive laws (Equations 6, 15) are constructed to ensure the safe and stable operation of the power system, with all signals being bounded. Furthermore, for any ε ̄ > 0 , by appropriately adjusting the design parameters, the following inequality holds. lim t → ∞ y − y ̄ r ⩽ ε ̄ . (18)

Remark 3

Under partial connectivity, voltage stability maintains within local subgraphs through ‖ y i − y r , i ‖ ≤ Δ i N i . Global consensus requires the directed spanning tree condition in Assumption 1.

Proof. Choose the global Lyapunov function as V = ∑ i = 1 n i V i , j .

According to (Equation 17), we get V ̇ ≤ − N V + Δ , where N = min { N i , i = 1 , … , n i } and Δ = ∑ i = 1 n i Δ i .

According to the definition of V and Lemma 3, it indicates that the tracking error is bounded. Then, by choosing appropriate parameters, we can obtain Δ N = ε ̄ 2 2 ξ H + B 2 . (19)

Considering (Equation 19) and Lemma 3, we obtain (Equation 18). Therefore, the proposed adaptive collaborative fault-tolerant control protocol ensures the boundedness of all signals within the closed-loop system. This concludes the proof.

Remark 4

The UUB stability achieved contrasts with the ISS in due to differing fault-handling approaches. The matrix pencil method attains ISS via algebraic decoupling of known faults, while our adaptive strategy (Equations 4, 10, 15) estimates unknown bounds θ i , h , introducing Lyapunov errors θ ̃ i , h that necessitate UUB. This trade-off eliminates prior fault knowledge requirements at the cost of weaker guarantees.

The convergence rate of the multi-agent tracking control is governed by: N = min c i , k , ω i , k γ i , k λ min H + B − 1 where λ min ( ( H + B ) − 1 ) denotes the minimum eigenvalue of the inverse graph Laplacian. The tracking error bound satisfies: ‖ z i , 1 t ‖ ≤ κ e − N t ‖ z i , 1 0 ‖ with κ > 0 being a scaling constant. This quantitative relationship demonstrates how the communication topology impacts the convergence performance of the distributed control system.

In this section, we conducted simulations using Matlab software to validate the feasibility of the proposed method. The simulations were designed for the distributed adaptive collaborative fault-tolerant control protocol, which is based on multi-agent system collaborative control theory and the backstepping method within a power system.

In the simulation, the parameters were chosen as follows. c 1,1 = c 2,1 = c 2,2 = c 3,2 = 1.7 , c 3,1 = c 5,1 = c 4,2 = 1.3 , c 4,1 = c 1,2 = c 5,2 = 2.4 , τ ̃ i , j = 0.05 (initial value for a ̃ i , j ), χ i , j = 1.2 , ω i , j = 7 . Additionally, the initial conditions were set as: v a ( 0 ) = [ 0 0 ] T , i a ( 0 ) = [ 0 0 ] T , v r ( 0 ) = 0 , θ ̂ i , j = [ 0 0 ] T .

By implementing the developed distributed adaptive cooperative capacity in a power system composed of synchronous generating units, the simulation results are depicted in Figure 2. As shown in Figure 2a, the proposed adaptive distributed collaborative fault-tolerant control protocol effectively achieves the safe and stable control of the power system. Additionally, Figure 2b indicates that the controller of the power system is ultimately consensus stable.

The simulation results clearly demonstrate that, in the event of actuator failure, all voltage outputs from synchronous generators within the power system successfully follow the prescribed voltage reference values. Consequently, through the utilization of the back-stepping method, the adaptive collaborative fault-tolerant control protocol presented in this paper adeptly achieves secure and stable control of the power system, which is comprised of synchronous generators experiencing actuator failure. Employing the proposed method effectively addresses the voltage recovery challenge in the power system, ensuring consistent recovery and maintaining overall system stability. Importantly, this method guarantees the boundedness of all signals involved.

In this paper, a novel adaptive collaborative fault-tolerant control protocol is proposed for a power system consisting of synchronous generators affected by actuator faults. The protocol leverages the consistency theory of multi-agent systems to enhance fault tolerance in the power system. This method effectively overcomes the coupling effects among diverse synchronous generator units, thereby guaranteeing the stable operation of the power system. Moreover, it enhances the system’s resilience to interference and ensures that asymptotic errors and all signals within the system remain bounded. To substantiate the efficacy of the proposed control strategy, a simulation example is provided.