Robust fault-tolerant control method for LFC system under FDI and ALD coupling effect

Abstract

The invention discloses a robust fault-tolerant control method for an LFC system under the FDI and ALD coupling effect. The robust fault-tolerant control method comprises the steps that S1, a multi-area load frequency control system model is constructed; s2, constructing a dynamic FDI attack model, dynamically simulating the FDI attack in the load frequency control system constructed in the S1, and defining a regional control error; s3, representing ALD characteristics under the switching of the multi-source communication link; s4, constructing a robust fault-tolerant controller, and enhancing the dynamic stability of the load frequency control system constructed in the S1 under the FDI attack constructed in the S2 and the ALD coupling action in the S3; and S5, solving the control gain based on the representation of the load frequency control system in the step S4. The robust control method is provided for solving the problem that dynamic performance of an LFC system is degraded due to the coupling effect of FDI attack and asynchronous leakage delay (ALD). According to the method, a dynamic attack model is constructed, an attack modulator (AM) is introduced to simulate a key control channel attack, and an ALD caused by multi-source link switching is embedded into a system feedback link.

Description

Technical Field

[0001] This application relates to the field of load frequency control system technology, and specifically to a robust fault-tolerant control method for an LFC system subjected to the coupling effect of FDI and ALD. Background Technology

[0002] Load frequency control (LFC) plays a crucial role in maintaining frequency stability and power balance in power systems, and is key to ensuring system safety and dynamic performance [1], [2]. Traditional LFC systems mainly rely on real-time adjustment of generator output to match power generation with load demand. This helps reduce frequency deviation and power exchange between stable regions. However, traditional LFC communication infrastructure usually relies on dedicated links with point-to-point or tree topology. Communication paths and equipment configurations are fixed and predefined [3]. Although this architecture has certain advantages in terms of physical isolation and basic stability, its rigid structure and closed communication protocols limit flexibility and compatibility. These limitations make it difficult to handle dynamic scenarios with frequent integration of distributed renewable energy. In addition, dedicated communication networks face limitations in bandwidth, interaction frequency, adaptive scheduling, etc., becoming a key bottleneck for the expansion and upgrading of traditional LFC systems.

[0003] As power systems evolve towards distributed, information-driven, and architecture-based systems, network-physical power systems (CPPS) have become the core infrastructure of the next-generation power grid [4]. CPPS integrates the physical layer and network layer to support the bidirectional flow of power and information. Sensors collect system status in real time and transmit the data to the control system through an open communication network. The controller analyzes the data and sends feedback signals to the actuators to achieve precise regulation and coordinated optimization of the power system [5]. The deployment of CPPS improves the visibility and controllability of the system, improves the efficiency of information exchange, and improves the control accuracy. However, it also brings some challenges, such as the complexity of link switching, network congestion, communication delay fluctuations, and potential network threats [6]. In view of the limitations of traditional load frequency control in fixed topology and closed protocols, negative LFC has become a promising research direction for power grid development. The LFC system based on the CPPS architecture integrates multi-link communication mechanisms, link switching strategies, and dynamic sensing control methods. These features show a strong potential to improve the robustness and adaptability of the system in complex and ever-changing environments.

[0004] In LFC systems, time delay mainly comes from signal transmission between the phasor measurement unit (PMU) and the management center, and between the management center and the controller, analog-to-digital conversion, input calculation, and GPS synchronization. Traditional research classifies LFC delay into discrete delay, distributed delay, neutral delay, and leakage delay[7],[8],[9],

[10] . Solutions for compensating for explicit transmission delay include robust fault-tolerant distributed PI control

[11] , resilient active compensation mechanisms

[12] , disturbance estimation techniques

[13] , and delay estimators

[14] . Leakage delay is a special type of delay, which refers to the time lag introduced by the negative feedback loop of the system, often referred to as the "forgetting period"

[15] . In control theory, leakage delay is characterized as instantaneous state feedback delay

[16] , and its stability analysis has been extensively studied in automatic control and neural networks

[17] .

[0005] With the rapid integration of distributed clean energy such as wind power and photovoltaic power generation, multi-regional power grid monitoring and dispatch must rely on multi-channel, multi-path parallel data acquisition, calculation and transmission mechanisms to achieve cross-regional, multi-node collaborative operation. Current LFC systems rely on a variety of heterogeneous communication paths, including optical fiber, power line carrier, copper cable, GSM / GPRS, WiMAX and WLAN, which usually operate in parallel. However, due to differences in access priorities, network congestion and protocol incompatibility, frequent exchanges or interconnections between these links may introduce non-stationary delay drift, which is prone to accumulate under high load or fault recovery conditions

[18] ,

[19] ,

[20] .

[0006] Existing literature often idealizes or ignores the unstructured packet-level delay caused by link switching. However, under real-world conditions of high load or post-fault, these delays may accumulate to a level that threatens the real-time performance and control stability of the system

[20] . The dynamic delay jitter introduced by multi-link switching and heterogeneous network interoperability effectively constitutes the time-varying disturbance component in the feedback loop. Therefore, in order to improve the robustness and fault tolerance of the control, the time-varying characteristics of link switching and leakage delay must be incorporated into a unified modeling framework. In addition, in order to improve the operational safety and dynamic stability of LFC system in complex communication environments, a fault-tolerant control strategy with integrated delay compensation needs to be designed.

[0007] In LFC systems, Fake Data Injection (FDI) and Denial-of-Service (DoS) attacks are considered the most destructive types of network attacks. FDI attacks aim to inject malicious forged data into communication links to disrupt measurement information and reduce system performance. DoS attacks occupy network bandwidth by injecting malicious requests, leading to communication interruptions and latency accumulation, thereby weakening the stability and availability of the control system.

[21] ,

[22] ,

[23] , . According to

[24] , frequency measurement and tie-line power data are the most vulnerable targets of FDI attacks.

[25] ,

[26] discuss typical FDI attack strategies, including bias attacks, harmonic attacks and hybrid attacks, while

[27] further expands the impact analysis under various attack scenarios. Although previous studies have attempted to enhance the system's defense against FDI attacks from the perspective of detection and estimation, the design of robust fault-tolerant controllers for FDI attack environments is still insufficient

[28] . Some works, such as

[29] ,

[30] , have proposed robust control strategies against FDI attacks. However, there is limited research on the robustness and fault tolerance of intelligent load frequency control systems under multi-source uncertain environments, such as asynchronous leak delay (ALD) and intelligent attack behavior.

[0008] In summary, existing LFC systems have shortcomings in link switching and time delay disturbance modeling, making it difficult to meet the control requirements of multi-source heterogeneous and dynamically evolving environments in smart grids. In particular, when facing non-stationary feedback caused by the combined effects of ALD and FID, traditional control strategies are unable to effectively guarantee the stability and security of the system. Summary of the Invention

[0009] The purpose of this application is to provide a robust fault-tolerant control method for LFC systems subjected to the coupling effect of FDI and ALD. The specific technical solution is as follows:

[0010] A robust fault-tolerant control method for an LFC system subjected to the coupling effect of FDI and ALD includes: S1, constructing a multi-region load frequency control system model; S2, constructing a dynamic FDI attack model to dynamically simulate the load frequency control system constructed in S1 subjected to an FDI attack and defining the regional control error; S3, characterizing the ALD characteristics under multi-source communication link switching; S4, constructing a robust fault-tolerant controller to enhance the dynamic stability of the load frequency control system constructed in S1 under the coupling effect of the FDI attack in S2 and the ALD in S3; S5, solving for the control gain based on the representation of the load frequency control system in S4.

[0011] S1 includes:

[0012] S1.1. Based on the logical relationship between the transfer function and variables, the load frequency control system is... The time-domain mathematical model for each region is expressed as follows:

[0013]

[0014] in, For frequency deviation, For the power output of solar power plants, For the output power of the wind farm, For turbo output, For load disturbance, The time constant of the speed controller, For solar energy systems, For wind turbines, For turbines, Let be the system's equivalent inertial constant. The equivalent damping coefficient of the system is... This is the frequency offset coefficient;

[0015] S1.2, Definition:

[0016]

[0017] S1.3, based on S1.1 and S1.2, the multi-region load frequency control system model is represented as follows:

[0018] ,

[0019] in,

[0020] ,

[0021] ,

[0022] ,

[0023] .

[0024] S2 includes:

[0025] S2.1 Construct a dynamic FDI attack model, represented as follows:

[0026] ,

[0027] in, The actual signal received by the system; The actual transmitted signal; I is the identity matrix, whose dimensions are... The number of rows is the same; This is an FDI attack modulator whose dimension is compatible with I, and whose value changes over time;

[0028] S2.2, a dynamic FDI attack model built based on S2.1, when When the matrix is ​​a scalar, the corresponding scaling attack mode is represented as:

[0029] ,

[0030] ,

[0031] in, For a univariate bounded function, when When K is a zero matrix, it indicates that the system cannot receive control signals, and the FDI attack evolves into a denial-of-service attack;

[0032] S2.3, a dynamic FDI attack model built based on S2.1, when When the matrix is ​​nonscalar, the corresponding replay attack, harmonic attack, and bias attack are represented as follows:

[0033] ,

[0034] ,

[0035] in, It is a matrix function that determines the type of attack;

[0036] S2.4. Divide the measurement signals in the load frequency control system constructed in S1 that are vulnerable to FDI attacks into frequency error signals. and tie-line power error signal The measured values ​​of damage to the load frequency control system under FDI attack are expressed as follows:

[0037]

[0038] in, This represents the actual measured value of the frequency deviation under an FDI attack; This represents the actual measured value of the tie-line power deviation under FDI attack. and These are the bounded spurious data signals injected into the frequency signal and the tie-line power signal, respectively.

[0039] S2.5, The area control error is expressed as:

[0040] ,

[0041] Based on S2.1-S2.4, the FDI attack is defined as follows: The regional control error of each region is expressed as follows:

[0042] ,

[0043] in, .

[0044] S3 includes:

[0045] S3.1 Construct a hybrid leakage delay model, introducing a Bernoulli random variable to adjust the activation mechanism of different delay components, expressed as:

[0046] ,

[0047] in, It has an inherent time delay; To measure random delays caused by asynchrony and communication link switching; and All are bounded values ​​and satisfy... , , ; and Let be a Bernoulli random variable, taking values... or Let represent whether the delay is activated, and its probability satisfies . and ;

[0048] S3.2. Due to the unavoidable inherent delay, Always take as ,therefore ;when Also taken as When, it indicates the presence of a random delay; when Pick When , it indicates that there is no random delay; based on this, the time delay term can be rewritten as:

[0049] .

[0050] S4 includes:

[0051] S4.1. Based on the regional control error representation in S2, when the load frequency control system is subjected to an FDI attack, it is represented as follows:

[0052] ,

[0053] definition:

[0054]

[0055] S4.2, Based on a proportional-integral controller, the first The control law for each region is represented as follows:

[0056] ,

[0057] in, They represent the first The proportional coefficient and integral coefficient of each area controller;

[0058] S4.3 Substituting S4.2 into S4.1 yields a multi-region system with random leakage time delay, expressed as:

[0059] ,

[0060] in, , ;

[0061] S4.4. Ensure the performance estimation of the multi-zone load frequency control system based on the new constraints, using the multi-zone load frequency control system representation in S4.3, for a given positive scalar. , , , , and the given matrix If a positive definite matrix exists , , and If the following inequalities are satisfied, then the multi-zone load frequency control system in S4.3 will... Disturbance attenuation level The following is called mean-square stability, and the specific inequality is:

[0062] ,

[0063] in,

[0064]

[0065] S5 includes:

[0066] The performance estimation of the multi-zone load frequency control system is ensured based on the new constraints, and on the multi-zone load frequency control system represented in S4, for a given positive scalar... , , , , and the given matrix If a matrix of suitable dimension exists... and positive definite matrix , , and If the following inequalities are satisfied, then the multi-region load frequency control system represented in S4 is in Disturbance attenuation level The following is called mean-square stability, and the specific inequality is:

[0067] ,

[0068] in,

[0069]

[0070] Based on this, the control gain is expressed as:

[0071]

[0072] in, As an indicator vector, and They represent and The pseudo-inverse matrix makes and Become an identity matrix of appropriate dimension .

[0073] The beneficial effects of this application are as follows: To address the problem of dynamic performance degradation in LFC systems caused by the coupling effect of FDI attacks and asynchronous leakage delay (ALD), a robust control method is proposed. This method constructs a dynamic attack model and introduces an attack modulator (AM) to simulate attacks on critical control channels, while embedding ALD caused by multi-source link switching into the system feedback loop. Stability analysis is conducted using an improved Lyapunov-Krasovskii function, and the effectiveness of the proposed method in terms of disturbance rejection, stability, and engineering adaptability is verified through dual-region power system simulation. A dynamic FDI attack model for critical control channels is constructed. By introducing an attack modulator (AM) to adjust the attack intensity and timing in real time, this model can accurately simulate the actual attack process. A systematic method for characterizing asynchronous leakage delay (ALD) caused by multi-source communication link switching is proposed, and this delay characteristic is embedded in the control loop, thereby fully reflecting the feedback lag characteristics under complex communication environments. Based on the improved Lyapunov–Krasovskii function, a robust control method is proposed to maintain system stability and improve dynamic response under uncertain environments of FDI and ALD coupling. Attached Figure Description

[0074] Figure 1 For the first in this application A schematic diagram of LFCs showing regions subjected to FDI attacks and ALD attacks;

[0075] Figure 2 This is a schematic diagram of a multi-regional power grid under an FDI attack as described in this application;

[0076] Figure 3 This is a schematic diagram illustrating the frequency deviation under different attack modes in this application;

[0077] Figure 4 This is a schematic diagram of the area control error under different attack modes in this application;

[0078] Figure 5 This is a schematic diagram of the frequency deviation and area control error induced by the attack in this application;

[0079] Figure 6 This is a schematic diagram of the area control error of the LFC system under hybrid attack in this application;

[0080] Figure 7 This is a schematic diagram of the LFC frequency deviation under hybrid attacks in this application;

[0081] Figure 8 This is a schematic diagram of the mixed time delay with random uncertainty in this application.

[0082] Figure 9 This is a schematic diagram of frequency deviation under different time delays in this application;

[0083] Figure 10 This is a schematic diagram of the regional control error under different time delays in the application. Detailed Implementation

[0084] To make the objectives, technical solutions, and advantages of this application clearer, the application will be further described in detail below with reference to specific embodiments and accompanying drawings. It should be understood that these descriptions are merely exemplary and not intended to limit the scope of this application. Furthermore, descriptions of well-known structures and technologies are omitted in the following description to avoid unnecessarily obscuring the concepts of this application.

[0085] like Figure 1-10 As shown:

[0086] A robust fault-tolerant control method for LFC systems subjected to the coupling effects of FDI and ALD includes:

[0087] S1. Construct a multi-region load frequency control system model. Specifically, this includes:

[0088] S1.1. Based on the logical relationship between the transfer function and variables, the load frequency control system is... The time-domain mathematical model for each region is expressed as follows:

[0089]

[0090] in, For frequency deviation, For the power output of solar power plants, For the output power of the wind farm, For turbo output, For load disturbance, The time constant of the speed controller, For solar energy systems, For wind turbines, For turbines, Let be the system's equivalent inertial constant. The equivalent damping coefficient of the system is... This is the frequency offset coefficient;

[0091] S1.2, Definition:

[0092]

[0093] S1.3, based on S1.1 and S1.2, the multi-region load frequency control system model is represented as follows:

[0094] ,

[0095] in,

[0096] ,

[0097] ,

[0098] ,

[0099] .

[0100] S2. Construct a dynamic FDI attack model to dynamically simulate an FDI attack on the load frequency control system built in S1, and define the regional control error. Specifically, this includes:

[0101] S2.1 Construct a dynamic FDI attack model, represented as follows:

[0102] ,

[0103] in, The actual signal received by the system; The actual transmitted signal; I is the identity matrix, whose dimensions are... The number of rows is the same; This is an FDI attack modulator whose dimension is compatible with I, and whose value changes over time;

[0104] S2.2, a dynamic FDI attack model built based on S2.1, when When the matrix is ​​a scalar, the corresponding scaling attack mode is represented as:

[0105] ,

[0106] ,

[0107] in, For a univariate bounded function, when When K is a zero matrix, it indicates that the system cannot receive control signals, and the FDI attack evolves into a denial-of-service attack;

[0108] S2.3, a dynamic FDI attack model built based on S2.1, when When the matrix is ​​nonscalar, the corresponding replay attack, harmonic attack, and bias attack are represented as follows:

[0109] ,

[0110] ,

[0111] in, It is a matrix function that determines the type of attack;

[0112] S2.4. Divide the measurement signals in the load frequency control system constructed in S1 that are vulnerable to FDI attacks into frequency error signals. and tie-line power error signal The measured values ​​of damage to the load frequency control system under FDI attack are expressed as follows:

[0113]

[0114] in, This represents the actual measured value of the frequency deviation under an FDI attack; This represents the actual measured value of the tie-line power deviation under FDI attack. and These are the bounded spurious data signals injected into the frequency signal and the tie-line power signal, respectively.

[0115] S2.5, The area control error is expressed as:

[0116] ,

[0117] Based on S2.1-S2.4, the FDI attack is defined as follows: The regional control error of each region is expressed as follows:

[0118] ,

[0119] in, .

[0120] S3 characterizes the ALD (Asynchronous Leakage Delay) characteristics under multi-source communication link switching. In the operation of networked control systems and distributed power systems, asynchronous leakage delay has become a crucial factor that must be considered. Unlike traditional synchronous communication, modern industrial control uses asynchronous transmission mechanisms for multi-source links to improve system reliability. However, this structure inevitably introduces asynchronous time-varying delays in the leakage term, meaning that non-uniform and uncertain delays occur in the leakage links of feedback or error signals. Existing research shows that such delays directly affect system stability and performance. If not modeled and compensated for, they may lead to the loss of asymptotic stability of the closed-loop system, or even oscillations or instability under small disturbances.

[0121] From a theoretical perspective, ignoring leakage delays can indeed lead to distortion or invalidation of stability criteria. Many Lyapunov proofs based on no-delay or only constant-delay conditions may no longer hold true when faced with time-varying or random leakage delays. Existing literature, focusing on a class of nonlinear difference systems with time-varying leakage delays, demonstrates that by explicitly incorporating the leakage delay into the model and comprehensively utilizing fixed-point theorems, Lyapunov–Krasovskii functionals, and model transformation techniques, a stability criterion expressed in linear matrix inequalities (LMIs) can be derived, depending on the upper bound of the delay and its derivative. This method effectively advances theoretical analysis from sufficient conditions under the "no-delay assumption" to more conservative sufficient conditions that accurately characterize the effects of delays. Existing research results show that even for the case of constant leakage delays, the obtained criterion is less conservative than some recent publications. This strongly demonstrates that considering leakage delays can provide a more reliable quantitative basis for the selection of control gain, thereby avoiding erroneous safety judgments in engineering design.

[0122] At the transient and performance level, neglecting asynchronous leakage delays significantly degrades the system's dynamic response indicators, directly impacting grid dispatch and operational safety. Specifically, this manifests as increased ACE peaks after disturbances, prolonged frequency overshoot and recovery time, and increased oscillation amplitude and duration. These all directly amplify the frequency of reserve capacity activation and dispatch costs. Engineering simulations and field tests show that the interaction between communication delays and leakage items can cause even well-designed PI / AGC controllers to exhibit significant overshoot in real-world grids and require longer stabilization times, leading to frequency deviations exceeding dispatch tolerances and triggering more emergency measures or market / performance penalties. The National Renewable Energy Laboratory (NREL) and numerous engineering studies have indicated that under distributed energy resources (DERs)-dominated or hybrid communication conditions, delays have a more significant impact on controller performance than before; even neglected occasional delays can push the system into unfavorable operating ranges, resulting in significant economic and operational costs.

[0123] In terms of security and attack resistance, ignoring asynchronous leakage delays amplifies the success rate of both passive and active attacks, creating serious systemic risks. The Time-Delay Switch (TDS) attack, proposed and verified in academic work, exploits this characteristic of "path-activated delay": attackers selectively inject or switch delays in critical feedback paths, causing the controller to gradually lose stability or performance without triggering traditional fault alarms, resulting in long-term ACE offsets, frequency recovery failures, and even the spread of oscillations in neighboring regions. Existing journal work has demonstrated the actual damage paths and simulation results of TDS-type attacks in LFCs, further showing that without incorporating these delay patterns into the threat model, it is impossible to design effective detectors or compensators. From an engineering perspective, this means that unmodeled asynchronous leakage delays are both unintentional reliability flaws and potential attack surfaces, and must be considered simultaneously in design and auditing.

[0124] In summary, ignoring asynchronous leakage delays can trigger a chain reaction across multiple domains within the system, potentially escalating into systemic risks. Literature and engineering case studies demonstrate that only by identifying and quantifying the sources of delay can these risks be mitigated.

[0125] These risks can only be reduced to an acceptable level by explicitly introducing path- or event-activated delay terms into the model and employing delay-dependent stability criteria and compensation or detection mechanisms in the control design. Specifically, this includes:

[0126] S3.1 Construct a hybrid leakage delay model, introducing a Bernoulli random variable to adjust the activation mechanism of different delay components, expressed as:

[0127] ,

[0128] in, It has an inherent time delay; To measure random delays caused by asynchrony and communication link switching; and All are bounded values ​​and satisfy... , , ; and Let be a Bernoulli random variable, taking values... or Let represent whether the delay is activated, and its probability satisfies . and ;

[0129] S3.2. Due to the unavoidable inherent delay, Always take as ,therefore ;when Also taken as When, it indicates the presence of a random delay; when Pick When , it indicates that there is no random delay; based on this, the time delay term can be rewritten as:

[0130] .

[0131] S4. Construct a robust fault-tolerant controller to enhance the dynamic stability of the load frequency control system built in S1 under the coupling effects of the FDI attack in S2 and the ALD in S3. Specifically, this includes:

[0132] S4.1. Based on the regional control error representation in S2, when the load frequency control system is subjected to an FDI attack, it is represented as follows:

[0133] ,

[0134] definition:

[0135]

[0136] S4.2, Based on a proportional-integral controller, the first The control law for each region is represented as follows:

[0137] ,

[0138] in, They represent the first The proportional coefficient and integral coefficient of each area controller;

[0139] S4.3 Substituting S4.2 into S4.1 yields a multi-region system with random leakage time delay, expressed as:

[0140] ,

[0141] in, , ;

[0142] S4.4. Ensure the performance estimation of the multi-zone load frequency control system based on the new constraints, using the multi-zone load frequency control system representation in S4.3, for a given positive scalar. , , , , and the given matrix If a positive definite matrix exists , , and If the following inequalities are satisfied, then the multi-zone load frequency control system in S4.3 will... Disturbance attenuation level The following is called mean-square stability, and the specific inequality is:

[0143] ,

[0144] in,

[0145]

[0146] To demonstrate the stability of the multi-region load frequency control system represented in S4, LKFs are chosen, specifically as follows:

[0147] ,

[0148] in,

[0149]

[0150] So, The time derivative is shown below:

[0151]

[0152] in, .

[0153] set up and It is any scalar. For vector functions and any positive definite matrix The following inequalities are satisfied:

[0154] ,

[0155] Using this integral inequality to scale time derivative The integral term in the equation. Therefore, the following inequality holds:

[0156] ,

[0157] in,

[0158]

[0159] in,

[0160]

[0161] Then, consider the derivative properties of asynchronous leakage delay. We obtain the following inequality:

[0162] ,

[0163] in,

[0164] .

[0165] Furthermore, the multi-region load frequency control system represented in S4, given any invertible symmetric matrix... This makes the following equation true:

[0166] ,

[0167] in,

[0168] ;

[0169] Based on this formula, the following inequality is obtained:

[0170] ,

[0171] in,

[0172]

[0173] Based on the above conditions, if the inequality holds, and where... If this also holds true, then:

[0174] ,

[0175] Therefore, the following inequality holds:

[0176] .

[0177] Therefore, the multi-region load frequency control system represented in S4 satisfies Performance standards are used to obtain proof.

[0178] S5. Based on the representation of the load frequency control system in S4, solve for the control gain. Specifically, this includes:

[0179] The performance estimation of the multi-zone load frequency control system is ensured based on the new constraints, and on the multi-zone load frequency control system represented in S4, for a given positive scalar... , , , , and the given matrix If a matrix of suitable dimension exists... and positive definite matrix , , and If the following inequalities are satisfied, then the multi-region load frequency control system represented in S4 is in Disturbance attenuation level The following is called mean-square stability, and the specific inequality is:

[0180] ,

[0181] in,

[0182]

[0183] Based on this, the control gain is expressed as:

[0184]

[0185] in, As an indicator vector, and They represent and The pseudo-inverse matrix makes and Become an identity matrix of appropriate dimension .

[0186] To make this application easier to understand, the following explanation is based on practical applications.

[0187] This embodiment conducted experiments on a multi-source power grid in two regions, such as... Figure 4 As shown in Table 1, this model is built upon previous research, and key parameters have been extracted and presented.

[0188] Table 1: The multi-source power grid parameter values ​​for the two regions are as follows:

[0189]

[0190] This article considers Performance, for a given different There will be different control gains. Theoretically, The smaller the value, the better the control effect and the stronger the system's anti-interference ability. However, this is not always the case. The smaller the value, the better. For a given system, there exists a relatively optimal value. This value is used to achieve a relatively optimal control gain. Therefore, this paper sets it to... Within the range, and according to the given Values ​​and Theorem 2 are used to calculate the controller gain, bringing it closer to the optimal value. For a given delay... , , , , And the system parameters in Table 1. According to Theorem 2, The calculation results are as follows:

[0191] Table 2: Control gain values ​​for different performance indicators are as follows:

[0192]

[0193] In the subsequent simulation experiments, we will first select the relatively optimal control gain from Table 2 for simulation experiments.

[0194] In practical LFC systems, attackers typically employ time-segmented FDI attacks to reduce costs and enhance stealth. This section will simulate the segmented attack strategy to verify the effectiveness of the control strategy designed in this paper. Specifically, based on equation (2), the variables in equation (12) of the two-zone LFC system are... In Perform a sine wave attack. The amplitude of the sine wave is 0.01, and the frequency step is 0.2Hz. Set the attack duration to 10 seconds, initiating three attack phases at 0 seconds, 35 seconds, and 70 seconds respectively. The scaling attack formula is: ,in Scaling factor This represents the selection matrix, which determines the location of the target perturbation and the subset of system states used to construct the attack.

[0195] like Figure 3 The simulation shows that the system frequency deviates significantly from the nominal value within each attack interval, with regions 1 and 2 exhibiting different dynamic response characteristics. Sine attacks induce typical periodic disturbance behavior, particularly during the 35s-45s and 70s-80s attack phases, where the frequency deviation exhibits clear damped oscillatory convergence characteristics. This indicates that although the disturbance causes short-term fluctuations, the system can effectively suppress oscillations and eventually recover to steady state due to the zero-mean characteristic of the sinusoidal attack and the control strategy proposed in this paper. In contrast, scaling attacks cause more severe steady-state frequency drift. After each attack activation, the frequency deviation does not converge to the reference value but continues to deviate, indicating that the system faces a greater stability challenge when facing such multiplicative disturbances. However, simulation results also show that under the intervention of the proposed control strategy, the drift is effectively suppressed within a certain range, reflecting the controller's anti-interference capability. Simulation results demonstrate that the robust control strategy proposed in this paper has good oscillation suppression performance against sinusoidal attacks and can effectively reduce the impact of dynamic disturbances. Furthermore, although the steady-state drift caused by scaling attacks is not completely eliminated, it can effectively limit the amplitude and velocity of the deviation. This demonstrates that designing a control strategy with generalized robustness and dynamic adaptive capability is crucial for ensuring the stable operation of power system frequencies in the face of segmented fake data injection attacks.

[0196] Under the same segmented fake data injection attack configuration Figure 4 The dynamic response characteristics of the two-zone intelligent coordinated load frequency converter system (ACE) were evaluated. Sine wave attacks and multiplicative scaling attacks were applied to three predefined attack windows. Simulation results are based on the proposed coordinated robust control strategy. System elasticity analysis shows that the sine wave attack in zone 1 induces typical damped oscillatory convergence. ACE exhibits alternating sign decay, particularly during the 35s-45s period, with an amplitude decay rate reaching 62.3%. This verifies the controller's dynamic suppression capability against periodic disturbances. Conversely, the scaling attack causes steady-state frequency drift, with ACE continuously accumulating positive deviations during the attack period, and the largest drift slope occurring in the 70-80 window, highlighting the long-term instability effect of multiplicative disturbances. The response in zone 2 further reveals the system's heterogeneity: the scaling attack triggers step-like transient convergence within 0-10s, indicating that the discrete adjustment mechanism of proportional error injection has asymptotic compensation characteristics; while the sine wave attack of the same period is rapidly suppressed, with the decay time constant of its high-frequency oscillation component being less than 2s. Subsequent attack windows (35-45 seconds, 70-80 seconds) reproduced different reaction patterns. Sine attacks maintained the convergence of the damped oscillations, while scaling attacks continued to induce steady-state biases. In summary, this control strategy is robust to dynamic attacks, but limited by static gain uncertainty, it can only partially mitigate the accumulated static error caused by scaling attacks. This conclusion underscores the necessity of introducing steady-state error feedforward compensation in multi-region LFC architectures to enhance the system's asymptotic stability against steady-state spurious data injection.

[0197] To evaluate the long-term stability of the system and the controller's continuous anti-interference capability, this section extends the FDI interval to the entire simulation interval based on segmented attacks. Experiments compare system simulations of two typical interference scenarios: proportional-to-proportional attacks and periodic sinusoidal attacks. The focus is on studying the time-domain response characteristics of the system's frequency deviation and regional control error, evaluating the robustness and dynamic stability of the control system.

[0198] In scaled attack scenarios, frequency deviation and ACE response are as follows: Figure 5As shown in the upper left and upper right corners, the system exhibits significant transient disturbances at the start of the attack, with region 2 showing a larger frequency deviation amplitude, indicating that this region is more sensitive to disturbances. However, due to the intervention of the designed intelligent coordinated control strategy, especially its closed-loop adjustment mechanism based on state observation and local interactive information fusion, the system is able to quickly adjust to the disturbance. Within approximately 15 seconds, the frequency and ACE signal gradually recover to near zero, demonstrating the system's good transient regulation performance and steady-state convergence capability. This type of attack does not introduce changes in frequency components, only producing linear interference in amplitude, falling into the category of low-frequency, slowly varying interference. This intelligent control strategy effectively suppresses proportional attacks by estimating the interference intensity online and dynamically adjusting the control gain, giving the system strong convergence and anti-interference capabilities.

[0199] In contrast, the interference from periodic sinusoidal attacks is more severe, and the system response is as follows: Figure 5 As shown in the lower left and lower right corners, due to the persistence and periodicity of the attack signal, the frequency deviation and ACE signal oscillate continuously throughout the simulation time domain, failing to achieve complete attenuation, and the system falls into a state of obvious periodic dynamic response. This high-frequency disturbance not only disrupts the frequency balance adjustment process but also generates a cooperative propagation effect between regions through the power coupling path of the interconnecting lines, amplifying the dynamic uncertainty and oscillation risk of the system. Nevertheless, the control strategy proposed in this paper effectively limits the frequency range and amplitude of the oscillation, achieving forced convergence of the system under continuous attacks. In particular, after 20 seconds, the system oscillation tends to stabilize, verifying that the designed control system has strong fault-tolerant adjustment capability and strong adaptive capability in the face of periodic disturbances.

[0200] On the other hand, we propose combining two different attack models into a hybrid attack and a progressive penetration attack mode. The experimental design verifies the robustness of our proposed controller against multimodal attacks. Attacks combining harmonics and scaling are implemented at three time points: 0 seconds, 20 seconds, and 45 seconds. The signal is divided into three intervals, with the duration of each interval gradually increasing, forming a variable step-size feature to simulate the dynamic adjustment strategy of a real attacker. This segmented design detects the system response through a shorter initial attack and amplifies the interference effect through a longer subsequent attack, improving attack efficiency. During the simulation cycle, the stealth of this attack lies in its non-uniform perturbation pattern, which can bypass conventional anomaly detection mechanisms. Dynamically adjusting the attack strength makes it difficult for the controller to respond effectively to achieve the optimal destructive effect.

[0201] Figure 6Experimental results for frequency error and ACE of the LFC system under mixed attack conditions are presented. It can be seen that under the control of the controller, the system exhibits good stability and robustness; within approximately 20 seconds after the initial disturbance, the ACE of regions 1 and 2 recovers to the initial equilibrium state. However, the FDI attack injected starting at age 40 causes the ACE to reach significant peaks, approximately 0.3 and 0.25 respectively, reflecting the asymmetric impact of the attack on different regions. The controller, through dynamic adjustment, allows the ACE to converge to a near-zero steady-state range within 20 seconds after the attack, highlighting its crucial role in maintaining the system's frequency and power balance. As the duration of the FDI attack increases, the adjustment complexity of the controller also increases accordingly, leading to a gradual increase in the time for the system to return to equilibrium. These ACE changes are also reflected in the system's frequency deviation dynamics, further confirming the asymmetric impact of the attack and the effectiveness of the controller's adjustment. A crease trend in the first equilibrium time is also shown.

[0202] Simulation results of frequency deviation changes in region 1 and region 2 under hybrid attack are as follows: Figure 7 As shown, the results indicate that initial equilibrium is reached approximately 30 seconds after the initial disturbance. However, the FDI attack starting at 40 seconds causes peak frequency deviations of approximately 0.15 Hz and 0.2 Hz, respectively, indicating more severe interference in region 2. Through precise intervention, the controller suppresses the deviation to a near-zero stable range within approximately 70 seconds, with a recovery time of approximately 30 seconds, ensuring frequency stability. The increasing first equilibrium time may be related to the distortion of the frequency control signal caused by the attack, which increases the difficulty of dynamic system adjustment, particularly the deviation amplification effect in region 2, exacerbating the response delay. In summary, the distortion of the frequency control signal and the deviation amplification effect in region 2 increase the difficulty of system adjustment, resulting in a longer equilibrium time.

[0203] This embodiment addresses the scenario of FDI attacks and asynchronous leakage delays. We focus on simulating the performance of the LFC system in the presence of FDI attacks and random, uncertain leakage delays. Since both regions of the system may experience random leakage delays, we investigate a more challenging scenario where both regions exhibit uncertainty simultaneously. We examine the coupling effect of FDI attacks and mixed delays to verify the robustness and effectiveness of the proposed controller under this dual challenge. Time delays are as follows: Figure 8 As shown, the random time delay is active for 50% of the entire simulation time interval.

[0204] Delay structure model such as Figure 8 As shown, this model consists of two time-varying delay components. Inherent transmission delay The value varies randomly within the boundary interval [0.04s, 8.00s], and its lower bound is defined. Upper Realm To better evaluate the control strategy's ability to handle time delays, a random time delay was set to have only two states: active or standby. The maximum delay was 2 seconds when active and 0 seconds when standby, with activation occurring within four randomly distributed time periods. (Mixed delay) The synthesis delay range is [0.04s, 10.00s].

[0205] Frequency deviation response curve Figure 9 The results show that the hybrid delay has a particularly significant impact on the transient frequency modulation process of the system. Under the influence of the inherent delay, the frequency deviation fluctuates after the disturbance but quickly approaches zero. However, when a random delay is added, the curve shows a stronger initial offset, with the frequency deviation amplitude increasing in the first 10 seconds and the convergence speed slowing down. Especially during the initial activation, the overshoot of the red curve is significantly higher than that of the case with the inherent delay alone, indicating that the random delay has an amplifying effect on the dynamic characteristics of the system. In addition, the distribution of the gray activation region further illustrates the intermittent effect of the uncertain delay, which poses a challenge to the stability of the frequency recovery process. Nevertheless, under the action of the controller, the system frequency eventually gradually recovers to near zero, indicating that the strategy has strong adaptability and robustness in steady-state performance. Overall, the hybrid delay mainly affects the dynamic quality of the system, manifested as an increase in oscillation amplitude and a decrease in convergence speed, but does not affect its long-term stability, providing validation for control applications in real-world delay environments.

[0206] Figure 10 The figures show a comparison of the dynamic behavior of ACE under the influence of mixed delays. The results indicate that the activation of random delays significantly affects the transient behavior of the system. Particularly in the initial activation phase, the ACE curve exhibits a significant amplification effect on oscillation amplitude, indicating that sudden delay changes weaken the system's ability to quickly suppress disturbances. Comparing the two curves, with only the inherent time delay, the system converges quickly and remains near zero. However, with the introduction of mixed delays, the ACE curve exhibits shift and amplitude superposition in the early stages, leading to a prolonged transient recovery time. Nevertheless, over time, the ACE recovers to a relatively small steady-state error range in both cases, demonstrating a certain degree of resilience to random delay disturbances. Comparisons between different regions also show that the effect of mixed delays on the initial response is more pronounced, manifested as an increase in ACE oscillation amplitude and a slowdown in convergence speed. This suggests that the delay of random activation poses a potential challenge to the coordinated control of multi-region systems, but the steady-state performance remains stable, demonstrating the robustness of the control strategy.

[0207] The embodiments described above are merely illustrative of specific implementations of the present invention, and while the descriptions are detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A robust fault-tolerant control method for an LFC system subjected to the coupling effect of FDI and ALD, characterized in that, include: S1. Construct a multi-regional load frequency control system model; S2. Construct a dynamic FDI attack model to dynamically simulate the FDI attack on the load frequency control system constructed in S1, and define the area control error. S3, characterizing ALD features under multi-source communication link switching; S4. Construct a robust fault-tolerant controller to enhance the dynamic stability of the load frequency control system constructed in S1 under the coupling effect of the FDI attack constructed in S2 and the ALD in S3. S5. Based on the representation of the load frequency control system in S4, solve for the control gain.

2. The robust fault-tolerant control method for LFC systems subjected to the coupling effect of FDI and ALD as described in claim 1, characterized in that, S1 includes: S1.

1. Based on the logical relationship between the transfer function and variables, the load frequency control system is... The time-domain mathematical model for each region is expressed as follows: in, For frequency deviation, For the power output of solar power plants, For the output power of the wind farm, For turbo output, For load disturbance, The time constant of the speed controller, For solar energy systems, For wind turbines, For turbines, Let be the system's equivalent inertial constant. The equivalent damping coefficient of the system is... This is the frequency offset coefficient; S1.2, Definition: S1.

3. Based on S1.1 and S1.2, the multi-region load frequency control system model is represented as follows: ， in, ， ， ， 。 3. The robust fault-tolerant control method for LFC systems subjected to the coupling effect of FDI and ALD as described in claim 2, characterized in that, S2 includes: S2.1 Construct a dynamic FDI attack model, represented as follows: ， in, The actual signal received by the system; The actual transmitted signal; I is the identity matrix, whose dimensions are... The number of rows is the same; This is an FDI attack modulator whose dimension is compatible with I, and whose value changes over time; S2.2, Based on the dynamic FDI attack model constructed in S2.1, when When the matrix is ​​a scalar, the corresponding scaling attack mode is represented as: ， ， in, For a univariate bounded function, when When K is a zero matrix, it indicates that the system cannot receive control signals, and the FDI attack evolves into a denial-of-service attack; S2.3, Based on the dynamic FDI attack model constructed in S2.1, when When the matrix is ​​nonscalar, the corresponding replay attack, harmonic attack, and bias attack are represented as follows: ， ， in, It is a matrix function that determines the type of attack; S2.

4. Divide the measurement signals in the load frequency control system constructed in S1 that are vulnerable to FDI attacks into frequency error signals. and tie-line power error signal The measured values ​​of damage to the load frequency control system under FDI attack are expressed as follows: in, This represents the actual measured value of the frequency deviation under an FDI attack; This represents the actual measured value of the tie-line power deviation under FDI attack. and These are the bounded spurious data signals injected into the frequency signal and the tie-line power signal, respectively. S2.5, The area control error is expressed as: ， Based on S2.1-S2.4, the first step under an FDI attack is defined. The regional control error of each region is expressed as follows: ， in, .

4. The robust fault-tolerant control method for LFC systems subjected to the coupling effect of FDI and ALD as described in claim 3, characterized in that, S3 includes: S3.1 Construct a hybrid leakage delay model, introducing a Bernoulli random variable to adjust the activation mechanism of different delay components, expressed as: ， in, It is due to inherent time delay; To measure random delays caused by asynchrony and communication link switching; and All are bounded values ​​and satisfy... , , ; and Let be a Bernoulli random variable, taking values... or Let represent whether the delay is activated, and its probability satisfies . and ; S3.

2. Due to the unavoidable inherent delay, Always take as ,therefore ;when Also taken as When, it indicates the presence of a random delay; when Pick When , it indicates that there is no random delay; based on this, the time delay term can be rewritten as: 。 5. The robust fault-tolerant control method for LFC systems subjected to the coupling effect of FDI and ALD as described in claim 4, characterized in that, S4 includes: S4.1 Based on the regional control error representation in S2, when the load frequency control system is subjected to an FDI attack, it is represented as follows: ， definition: S4.2, Based on a proportional-integral controller, the first The control law for each region is represented as follows: ， in, They represent the first The proportional coefficient and integral coefficient of each area controller; S4.3 Substituting S4.2 into S4.1, we obtain a multi-region system with random leakage time delay, expressed as: ， in, , ; S4.

4. Ensure the performance estimation of the multi-zone load frequency control system according to the new constraints, based on the multi-zone load frequency control system representation in S4.3, for a given positive scalar , , , , and the given matrix If a positive definite matrix exists , , and If the following inequality is satisfied, then the multi-region load frequency control system in S4.3 will... Disturbance attenuation level The following is called mean-square stability, and the specific inequality is: ， in, 6. The robust fault-tolerant control method for LFC systems subjected to the coupling effect of FDI and ALD as described in claim 5, characterized in that, S5 includes: The performance estimation of the multi-zone load frequency control system is ensured based on the new constraints, and on the multi-zone load frequency control system represented in S4, for a given positive scalar. , , , , and the given matrix If a matrix of suitable dimension exists... and positive definite matrix , , and If the following inequalities are satisfied, then the multi-region load frequency control system represented in S4 is in Disturbance attenuation level The following is called mean-square stability, and the specific inequality is: ， in, Based on this, the control gain is expressed as: in, As an indicator vector, and They represent and The pseudo-inverse matrix makes and Become an identity matrix of appropriate dimension .

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