Resilient Event-Triggered H_∞ Control for a Class of LFC Systems Subject to Deception Attacks

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper considers an integrated power grid model, where the joint impact of ETTS and fraudulent interference is captured in a unified analytical framework to mitigate the effects of random attacks aimed at misleading the public environment.

Remarks

1. Formula 3 describes the LFC system using the state space method. The authors should define this method.

2. Insert the meaning of Fig. 1 and Fig. 2 into the text.

3. What is AMSS? Write the meaning of the abbreviation and explain it!

4. Explain the application of Schur's complement in Lemma 1, Section 3!

5. You should revise the conclusion! The abstract states that this paper investigates a strategy for an event-driven load frequency controller for smart grids, including electric vehicles. The conclusion does not mention this. What is a sufficient condition for stabilization? Add plans for future research development!

6. Citing more literature sources would be beneficial. Let them be more modern than those already added.

Author Response

Summary on the Revision

Resilient Event-Triggered H∞ Control for a Class of LFC Systems Subject to Deception Attacks

Yunfan Wang, Zesheng Xi, Bo Zhang, Tao Zhang and Chuan He

We would like to thank the editor, and the reviewers for their careful reading, valuable comments and suggestions. In this response letter, we present a point-by-point summary of editor’s and the reviewers’ comments, and our modification are colored blue in the revision.  Please revise the formal response in the attached PDF file, as some formulas cannot be displayed correctly in the form fields.

 

Responses to Reviewer 1

This paper considers an integrated power grid model, where the joint impact of ETTS and fraudulent interference is captured in a unified analytical framework to mitigate the effects of random attacks aimed at misleading the public environment.

Response: Thank you for your comments. We have revised the manuscript carefully according to your comments.

 

Comment 1: Formula 3 describes the LFC system using the state space method. The authors should define this method.

Response: Thank you for your comments. We have now clarified the definition of the state- space method in the revision. Please see Definition 1:

Definition 1: The state-pace method is a modeling method describing system dynamics with a set of differential or difference equations, which are especially suitable for analyzing multi-input multi-output (MIMO) systems.

 

Comment 2:  Insert the meaning of Figure 1 and Figure 2 into the text.

Response: Thanks for your comments. We have added the meaning of Figure 1 and Figure 2 into the revision.

• Basedon the information flow and transfer functions illustrated, the entire control loop involves the following modules, including EVs, governors, turbines, and generators.  The controller generates output signals that regulate the power frequency deviation of the gen- erator through two control subloops. In the first path, the controller output is distributed to the governor via a regulation gain. The signal then passes through the governor dynamics and the governor droop characteristics to produce a speed control signal, which is further processed by the turbine dynamics to generate the turbine output power.  In the another subloop, the controller output is  allocated to  the EVs module,  leading to  the incremen- tal change in the EVs’ output power.  Taking external disturbance into account, the total generation power can be calculated from the turbine and the EVs unit.

 

• Theinterval between t1h and t2h can be divided into five sub-intervals according to the designed principle.  Consequently, by means of the virtual partition method proposed in

[21], for any given time period between two consecutive triggering instants tkh and tk+1h, after considering the respective time delays at the triggering instant from the sensor to the controller, it can be expressed by the following subperiods:

 

 

 

Comment 3:  What is AMSS? Write the meaning of the abbreviation and explain it!

Response: Thank you for your comments. The abbreviation AMSS refers to Asymptotically Mean-Square Stability.  Specially, AMSS refers to a probabilistic concept of system stability, ensuring that the expected value of the square of the system state converges to zero when time tends to infinity. Mathematically, a stochastic system is said to be AMSS if

 

where x k  denotes the system state at instant k.  The stability metric is suitable for systems subject to random attacks, noise, or packet dropouts.  We have revised this manuscript to clearly define this term. Please see

Definition 2:  (Asymptotically mean-square stability (AMSS)) Mathematically, a stochastic system is said to be AMSS if

lim E {jj x kj j 2} = 0,

where x k  denotes the system state at instant k.

 

Comment 4:  Explain the application of Schur’s complement in Lemma 1, Section 3!

Response: Thank you for your comments.

• TheSchur complement Lemma is applied a matrix inequality involving block matrices into an equivalent Linear Matrix Inequality (LMI) form, which converts complex matrix conditions into tractable formats for convex optimization. Specially, consider a symmetric block matrix with the following form:

 

 

when C < 0, such inequality is equivalent for

A - BC-1 BT < 0.

This equivalence is called Schur’s complement Lemma.

 

• Inthis manuscript, Eq.(23) is transformed into an LMI form of Eq.(16) by means of Lemma 1, which facilitates the subsequent controller synthesis and parameter optimization utilizing convex optimization techniques.

By utilizing Lemma 1 from (16), one has

Γ 11 - Γ21(T)Γ21 Γ21 ≤ 0.

 

Comment 5:  You should revise the conclusion! The abstract states that this paper investigates a strategy for an event-driven load frequency controller for smart grids, including electric vehicles. The conclusion does not mention this. What is a sufficient condition for stabilization? Add plans for future research development!

Response: Thank you for your comments. The conclusion has been revised according to your comments.

This study presents an event-triggered H∞  load frequency control (LFC) strategy for power systems, particularly incorporating electric vehicles (EVs) . By employing an event-based commu- nication scheme, the data transmission and the network burden can be reduced and alleviated, while maintaining satisfactory control performance in contrast to  conventional time-triggered approaches.  Specially, it compensates for the detrimental effects of random deception attacks through the resilient controller.  Sufficient conditions for the AMSS of the closed-loop system has been obtained by utilizing the Lyapunov theory in terms of LMIs.  The simulation results illustrate the feasibility and availability of the proposed method for power systems under limited bandwidth and cyber threats.  Future work involves exploring distributed and full decentralized event-triggered LFC schemes, as well as data-driven and model-free control methods for resilient controller design in large-scale power networks [42].

 

Comment 6:  Citing more literature sources would be beneficial. Let them be more modern than those already added.

Response: Thank you for your comments. In this revision, we have updated and enriched the references by including several recent works published in the past five years. Please see:

• LiuM.; Teng F.; Zhang Z.; Ge P.; Sun M.; Deng R. Enhancing cyber-resiliency of DER- based smart grid: A survey. IEEE Trans. Smart Grid. 2024, 15, 4998-5030
• Wang,H.; Chen, S.; Li, M.; Zhu, C.; Wang Z. Demand-driven charging strategy-based distributed routing optimization under traffic restrictions in internet of electric vehicles. IEEE Internet Things J. 2024, 11, 35917-35927.
• Zhou,S.; Deng, C.; Fan, S.; Wang, B.; Che W.-W. Resilient distributed Nash equilibrium control for nonlinear MASs under DoS attacks. IEEE Trans. Cybern. 2025, 55, 2316-2326.
• Liu,X.; Yang G.-H. Optimal intermittent deception attacks with energy constraints for cyber-physical systems. IEEE Trans. Man Cybern. B Cybern. 2024, 54, 5889-5900.
• XiaoZ.; Jiang Y.; Yao Z.; He Z.; Jiang Y.; Li Y. A hybrid data-driven power loss mini- mization method of dual-active bridge converters. IEEE Trans. Power Electron. 2024, 39, 5820-5832.

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1. To what extent does the integrated model capture the complexities of real-world power grids, especially considering the potentially nonlinear and time-varying characteristics of electric vehicle loads and the dynamic nature of cyberattacks?
2. How is the effectiveness of the proposed ETTS quantified beyond simply reducing communication frequency, and are there specific metrics or case studies demonstrating its impact on control performance during different grid operating conditions?
3. Are the assumptions underlying the H∞ controller design (e.g., linearity, knowledge of disturbance bounds) realistic for power systems, and how sensitive is the controller's performance to deviations from these assumptions?
4. Does the proof of asymptotically mean-square stability adequately address the potential for instability due to the interplay of random deception attacks, event-triggered sampling, and communication delays under different operating condition for power systems?
5. How practical is the joint optimization of controller gain and triggering condition using linear matrix inequalities for large-scale power systems, considering the computational complexity and potential for scalability issues?
6. How representative are the selected case studies of real-world power grid scenarios, and do they adequately demonstrate the robustness and effectiveness of the proposed control strategy under a variety of attack patterns and operating conditions?
7. Is the comparison with existing methods sufficiently detailed and comprehensive, and does it clearly highlight the advantages of the proposed approach in terms of resilience, performance, and computational efficiency?
8. What are the main challenges associated with implementing the proposed control strategy in a real-world power grid environment, considering factors such as communication infrastructure limitations, sensor accuracy, and real-time computational constraints?

Author Response

Summary on the Revision

Resilient Event-Triggered H∞ Control for a Class of LFC Systems Subject to Deception Attacks

Yunfan Wang, Zesheng Xi, Bo Zhang, Tao Zhang and Chuan He

We would like to thank the editor, and the reviewers for their careful reading, valuable comments and suggestions. In this response letter, we present a point-by-point summary of editor’s and the reviewers’ comments, and our modification are colored blue in the revision. Please revise the formal response in the attached PDF file, as some formulas and images  cannot be displayed correctly in the form fields.

 

Responses to Reviewer 2

Comment 1:  To what extent does the integrated model capture the complexities of real-world power grids, especially considering the potentially nonlinear and time-varying characteristics of electric vehicle loads and the dynamic nature of cyberattacks?

Response:  Thanks for your comments.  The integrated model proposed in this manuscript possesses characteristics of real-world power systems, particularly incorporating EVs and cyber- attacks. According to Figure 1, the entire control loop involves the following modules, including EVs, governors, turbines, and generators. The controller generates output signals that regulate the power frequency deviation of the generator through two control subloops. In the first path, the controller output is distributed to the governor via a regulation gain. The signal then passes through the governor dynamics and the governor droop characteristics to produce a speed con- trol signal, which is further processed by the turbine dynamics to generate the turbine output power.

In the another subloop, the output signal from the controller is distributed to the EVs by a regulation gain. This module consists of a gain module Ke and dynamic response part  , followed by a droop control mechanism ρe to generate the output power incremental change of EVs Pe(s). When consider load disturbances Pl(s), the total generated power of the system is calculated by combining the output power from both control loops and the external disturbances. After the generator module, one has the frequency deviation. In contrast to traditional control structure, the proposed control scheme takes into signal transmission delays and random attacks in communication networks, which not only reduces the computation burden but also improves the system’s resiliency.

 

Comment 2: How is the effectiveness of the proposed ETTS quantified beyond simply reducing communication frequency, and are there specific metrics or case studies demonstrating its impact on control performance during different grid operating conditions?

Response: Thanks for your comments. Apart from evaluating the communication efficiency, several performance metrics to assess the control effectiveness of the proposed ETTS have been added under various grid operating conditions. Specifically:

• Thefrequency deviation, settling time, and steady-state curve of the system are analyzed to measure the transient and steady-state performance under both normal and attack scenarios. Figure 2 illustrates that the system realizes the AMSS by the proposed controller and tradition controller in the absence of deception attacks.

 

• Theproposed controller is tested under different level of load disturbances and random

 

deception attacks. For example
(ϖ(t) = { (	0.01,   0 < t ≤ 40, 0,     t > 40.

 

The comparative results verifies that the proposed ETTS can maintain system stability and performance compared to traditional time-triggered schemes, which is shown in Figure 3.

• Asillustrated in Figure 4, the proposed ETTS adaptively regulates the number of trans- mission in response to the frequency of deception attacks, thereby improving the overall control performance. In contrast to conventional communication schemes, it exhibits su- perior robustness and lower sensitivity.

 

Comment 3: Are the assumptions underlying the H∞ controller design (e.g., linearity, knowl- edge of disturbance bounds) realistic for power systems, and how sensitive is the controller’s performance to deviations from these assumptions?

Response: Thanks for your comments. The assumptions underlying the H∞ controller design in this article, such as linear dynamics and known disturbance bounds, are both considered and regarded as reasonable approximations for power systems.

• Whilecertain nonlinearities such as saturation or dead zones are exhibited in real-world power systems, linearized modes provide accurate descriptions of system behavior for small

 

signal analysis and frequency regulation near an operating point. The proposed controller is designed according to such standard linear approximation.

• Inpractical systems, the disturbance bounds (e.g. communication noise and load fluctua- tion) can be inferred from historical data and physical constraints, which offers a realistic rule of deign for resilient controller.

 

• Asdepicted in Figure 3, when random deception attacks exist, simulation results demon- strates that the proposed resilient controller provides a significantly smoother and im- proved system response, even under moderate modelling inaccuracies.

 

Comment 4: Does the proof of asymptotically mean-square stability adequately address the potential for instability due to the interplay of random deception attacks, event-triggered sam- pling, and communication delays under different operating condition for power systems?

Response: Thank you for your comments. The proof of asymptotically mean-square stabili- ty (AMSS) in this study has been carried out when considering the complex interplay between random deception attacks, event-triggered communication scheme, and bounded communication delays. Particularly:

• Thedeception attacks are modeled as stochastic processes, the random variable λ(t) reg- ulates the randomness of the deception attacks. Particularly, when λ(t) = 1, indicating that the system is under attack. Conversely, when λ(t) = 0, the system runs normally, and it follows Bernoulli distribution with the following expression:

Pr {λ(t) = 1} = λ(¯),

Pr {λ(t) = 0} = 1 - λ(¯),

where λ(¯) ∈ [0, 1].

• The Lyapunov theory is employed in terms of linear matrix inequalities(LMIs) to rigorous- ly derive sufficient conditions for guaranteeing AMSS. Construct the following Lyapunov functional:

 

x˙(t) =Ax(t) - (1 - λ(t))BKC(x(t - µ(t)) - e(t)) - λ(t)BKψ(y(t kh)) + Dϖ(t),

where random deception attacks and event-triggered communication scheme are both con- sidered, which reflects realistic operating conditions of power systems.

 

• Thederived stability condition holds under bounded delay assumptions and the designed resilient controller keeps resilient to external disturbances.  In addition, the simulation results under various attack schemes validate that the system maintains AMSS (Figure 2 and 3), showing the effectiveness of the proposed methods.

 

Comment 5: How practical is the joint optimization of controller gain and triggering condi- tion using linear matrix inequalities for large-scale power systems, considering the computational complexity and potential for scalability issues?

Response: Thank you for your comments. In this investigation, the joint design of the output- feedback controller gain and the triggering condition are both formulated as linear matrix in- equalities (LMIs), as shown in Theorem 2 (Eqs. (26)-(28)). Then the AMSS and H∞ system performance in the presence of deception attacks and communication constraints are both en- sured. It is acknowledgeable that for large-scale systems, solving LMIs become computationally challengeable. Nevertheless, the following practical aspects should be highlighted:

• Theproposed joint optimization is performed offline during the controller design process. When the optimal gain and triggering parameter are gained, online implementation only demands judging the event-triggering condition and executing standard feedback control, incurring negligible real-time computational burden.

 

• Thestructure of the LMIs in our design can be exploited to reduce computational cost. For example, sparsity-preserving techniques and decomposition methods can be used for high-dimensional systems. Simulations illustrate that the optimization remains tractable when incorporating nonlinear dynamics, random deception attacks, and ETTS jointly.

 

• Extendingto large power systems requires more scalable methods. As mentioned in the part of Conclusion of this manuscript, a promising direction for future discussion centers around the development of distributed or decentralized event-triggered LFC strategies, which can distribute the computational burden into each sub-area.

 

In conclusion, while the joint LMI-based design does face potential scalability problems for large networks, the proposed approach stays practical for the concerned power systems and pro- vides the foundation for further extension to distributed optimization and synthesis frameworks.

 

Comment 6: How representative are the selected case studies of real-world power grid scenarios, and do they adequately demonstrate the robustness and effectiveness of the proposed control strategy under a variety of attack patterns and operating conditions?

 

Response:  Thank you for your comments.  The presented case studies in this manuscript closely characterize realistic power systems, and the resiliency and effectiveness of the proposed control scheme under several operating conditions are rigorously assessed.

• The considered system in the simulation part is an interconnected powernetwork with het- erogeneous generation dynamics and EVs participation, which is in line with the structure of modern power networks.

 

• Thesimulations include random deception attacks with deterministic probability, as well as load disturbances and communication delays, which reflects common vulnerabilities encountered in real-world power systems.

 

• Theresults are evaluated through key performance metrics such as frequency deviation, triggering rate, control effort, and resilience under attacks. The comparison with tradi- tional time-triggered schemes clearly emphasizes the superior resiliency and adaptability of the proposed event-triggered H∞ controller.

In summary, the concerned case studies are sufficiently representative of real power systems and effectively demonstrate the effectiveness of the proposed approach.

 

Comment 7: Is the comparison with existing methods sufficiently detailed and comprehensive, and does it clearly highlight the advantages of the proposed approach in terms of resilience, performance, and computational efficiency?

Response: Thank you for your comments. We fully agree that a detailed and comprehensive comparison with existing approaches is crucial to stress the advantage of the proposed control policy.  Therefore, the detailed explanation are given from the following aspects to addresses your concerns more clearly.

• Anevent-triggered H∞ LFC design method is carefully developed to jointly optimize the output-feedback controller gain along with the communication parameters.  Unlike conventional time-triggered controllers, the proposed H∞ LFC controller enhances control performance by effectively utilizing available information, even in the presence of deception attacks, achieving superior resilience compared to standard approaches.

 

• Asdepicted in Figure 3, when random deception attacks exist, simulation results demon- strate that the proposed resilient controller provides a significantly smoother and improved system response in contrast the traditional controller.

 

• Although thecontroller design includes offline LMIs optimization, the online implementa- tion is lightweight and suitable to extend to medium-to-large power systems.

 

In summary, the case studies are sufficiently representative of real power systems and effectively demonstrate the effectiveness of the proposed approach.

 

Comment 8: What are the main challenges associated with implementing the proposed control strategy in a real-world power grid environment, considering factors such as communication infrastructure limitations, sensor accuracy, and real-time computational constraints?

Response: Thank you for your comments. It is acknowledged that several challenges exist in the practical deployment of the proposed event-triggered H∞ controller, including communication infrastructure limitation, sensor inaccuracies, and real-time constraints, which are discussed as follows:

• Inspite of the proposed event-triggered transmission scheme greatly reduces communi- cation load, its employment indeed requires reliable data exchange between sensors and controllers.  In real-world power grids, communication networks may encounter packet losses, delays, and asynchronous updates, which could affect system performance. Conse- quently, bounded transmission delays and packet loss are both considered in this study, and asynchronous event-triggered strategies should be explored in future work.

 

• Inrealistic systems, sensor readings are inevitably affected by noise or bias errors. While the proposed controller design involves resiliency to bounded disturbances and modeling uncertainties, future studies include investigating advanced filtering methods for improving system performance in the presence of sensor inaccuracies.

 

• Whilethe controller gains and triggering parameters are calculated offline, real-time im- plementation still requires timely evaluation of triggering conditions and application of control conducts. However, the structure of the proposed control algorithm is lightweight and decentralized in nature, which is proper to be deployed in parallel and embedded implementations.

Figure 1: Block of the LFC system model

 

Figure 2: Simulation results in Case 1. Top: State trajectories of x(t) with Controller 2; Bottom: State trajectories of x(t) with Controller 1.

 

Figure 3: Simulation results in Case 2. Top: State trajectories of x(t) with Controller 2; Bottom: State trajectories of x(t) with Controller 1

 

Figure 4: The deception attack and triggering instant sequences

 

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

No further comments