H-infinity intermittent sampling load frequency security control method for multi-area power system under deception attack

Abstract

The application discloses a kind of H∞ intermittent sampling load frequency security control method of multi-area power system under deception attack belongs to the technical field of power system safety control.In the application, intermittent control signal is only used in working interval, effectively save control cost, sampling control can save limited network bandwidth, and safety control effectively resists the influence of deception attack on multi-area power system.Firstly, according to the characteristics of intermittent control and sampling control, while considering the influence of deception attack, the dynamic model of intermittent sampling load frequency security control of multi-area power system under deception attack is established.Secondly, switching Lyapunov functional is proposed, and sufficient condition for the existence of intermittent sampling load frequency security controller is given in the framework of LMI, which guarantees the exponential stability of multi-area power system and meets H∞ performance.Finally, the effectiveness of the proposed H∞ intermittent sampling load frequency security controller is verified through simulation of three-area power system under deception attack.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of power system safety control, and particularly relates to an H∞ intermittent sampling load frequency safety control method for a multi-region power system under deception attack. BACKGROUND

[0002] In a multi-region power system, the key role of load frequency control (LFC) is to maintain frequency stability, match load demand, control power exchange, and quickly respond to load changes. In recent years, researchers have proposed some new improved schemes based on LFC methods, such as adaptive control, robust control, proportional differential control, and fuzzy logic control. However, the above control methods require continuous input of control signals and exert control, which will lead to an increase in control cost. At the same time, the operation of modern multi-region power systems relies on stable network space, and the open communication environment in the network space makes the system more vulnerable to network attack threats. Compared with traditional physical attacks, network attacks can also cause serious consequences such as information leakage and infrastructure interruption. Deception attack is a form of network attack that can destroy data integrity and reduce system performance. Therefore, in view of this situation, it is urgent to develop a new intermittent sampling safety control load frequency control method for a multi-region power system to meet the needs of actual work. SUMMARY

[0003] In order to solve the problems of the prior art, the application provides a new H∞ intermittent sampling load frequency safety control method for a multi-region power system.

[0004] In order to achieve the above purpose, the application adopts the following technical scheme:

[0005] A new H∞ intermittent sampling load frequency safety control method for a multi-region power system includes the following processes:

[0006] Establish a dynamic model of load frequency control of a multi-region power system;

[0007] Introduce an intermittent sampling control mechanism;

[0008] Establish a dynamic model of intermittent sampling load frequency control of a multi-region power system;

[0009] Consider deception attack;

[0010] Establish a dynamic model of intermittent sampling load frequency safety control of a multi-region power system under deception attack;

[0011] Based on Lyapunov functional theory, a sufficient condition for the existence of the intermittent sampling load frequency security controller is given, which can guarantee the exponential stability of the multi-area power system and meet the H∞ performance.

[0012] The application provides an H∞ intermittent sampling load frequency security control method for a multi-area power system under a deception attack. BRIEF DESCRIPTION OF DRAWINGS

[0013] The drawings constituting a part of the specification of the application are used to provide a further understanding of the application, and the illustrative embodiments of the application and the description thereof are used to explain the application, and do not constitute an improper limitation on the application.

[0014] Figure 1 The schematic diagram provided in the application.

[0015] Figure 2 The dynamic model schematic diagram of the load frequency control of the multi-area power system of the i-th region.

[0016] Figure 3 The intermittent sampling control mechanism schematic diagram.

[0017] Figure 4 The deception attack position schematic diagram.

[0018] Figure 5 The system state response schematic diagram of the first region.

[0019] Figure 6 The system state response schematic diagram of the second region.

[0020] Figure 7 The system state response schematic diagram of the third region.

[0021] Figure 8 The H∞ intermittent sampling security load frequency control signal schematic diagram. DETAILED DESCRIPTION

[0022] The application will be described in further detail below with reference to specific examples. The embodiments of the application are not limited to this.

[0023] The notations used in the present application are standard and are introduced here before proceeding. If the dimension of a matrix is not explicitly stated, it is assumed to be compatible with the algebraic operation. n denotes an n-dimensional identity matrix. n denotes an n-dimensional zero matrix. ||.|| denotes the Euclidean norm of a vector. L2[0,∞) describes the space of square integrable vector functions on [0,∞). denotes the set of natural numbers. n denotes the set of real numbers. denotes the n-dimensional Euclidean space, R n×m denotes the set of real numbers. denotes the set of n x m real matrices. col denotes column vector, diag denotes diagonal matrix. sup denotes the least upper bound of a set. For symmetric matrices A denotes the term caused by symmetry. I is the identity matrix. Define the matrix where A 11 , A 12 , …, A 1n , A 21 , …, A nn are arbitrary matrices, i, j = 1, 2, …, n, and such a matrix is defined as a special one-type matrix. Note that i, j are repeatable constants due to the high computational complexity of the present application.

[0024] As shown in Figure 1 , the present application provides a multi-region power system H∞ intermittent sampling load frequency security control method under deception attack, which comprises the following steps:

[0025] According to the dynamic equation describing the system state of each sub-region in the multi-region power system, the new power system model is constructed.

[0026] The framework of the i-th region of the multi-region power system load frequency control is as shown in Figure 2 The dynamic equation of the i-th sub-region is expressed as:

[0027]

[0028] The multi-region power system is composed of generators, water turbines and speed governors, wherein ΔP vi , ΔP mi , ΔP di and Δf irespectively represent valve position deviation, generator mechanical output deviation, load deviation and power system frequency deviation. R i , M i , D i , T chi , T gi are the speed droop, generator inertia, generator damping coefficient, turbine time constant and governor time constant of the i-th regional power system, respectively. ij is the tie-line synchronization coefficient between the i-th and j-th control area, ACE i is the linear combination of the frequency derivative Δf i and the tie-line power exchange ΔP tie-i , β i is the frequency bias factor. t is the time variable, and are the derivatives with respect to time t.

[0029] Let x(t) be the state vector of the multi-regional power system with n total regions, x i (t) = [Δf i (t) ΔP mi (t) ΔP vi (t) ΔP tie-i (t)] T is the state vector of the i-th regional power system, is the derivative of x(t) with respect to time t, y(t) be the output vector of the multi-regional power system with n total regions, y i (t) = ACE i (t) is the output vector of the i-th regional power system, u(t) be the input vector of the multi-regional power system with n total regions, u i (t) is the input vector of the i-th regional power system, w(t) be the disturbance variable of the multi-regional power system with n total regions, w i (t) = ΔP di (t) is the disturbance variable of the i-th regional power system. Then the mathematical model transformation function of the multi-regional power system can be obtained by converting from the dynamic equation (1.0), and the specific function expression is:

[0030]

[0031] wherein T ij = T ji , x(t) = [x1(t), x2(t), …, x i (t), …, x n (t)] T , B = diag{B1, B2, …, B i , …, B n}, u(t) = [u1(t), u2(t),...,u i n(t),...,u n n(t)] T , F = diag{F1,F2,...,F i n,...,F n n}, w(t) = [w1(t), w2(t),...,w i n(t),...,w n n(t)] T , C = diag{C1,C2,...,C i n,...,C n n}, y(t) = [y1(t), y2(t),...,y i n(t),...,y n n(t)] T , C i = [β i 0 0 1], n is the total number of regional power systems, at this time i = 1, 2,..., n, j = 1, 2,..., n, and i ≠ j.

[0032] As Figure 3 shown, the application introduces an intermittent sampling control mechanism. Specifically, the total time interval is divided into a series of disjoint time intervals 0 = t0 < t1 < t2 <... < t k < t k+1 ... satisfy Each control time interval [t k , t k+1 ) is composed of a rest interval [s k , t k+1 ) and a working interval [t k , s k ), s k is a time variable in the middle of the control interval [t k , t k+1 ], and the controller only works in the working interval. t k , t k+1 are the sampling time before and after, h k is the sampling period. Set two positive scalars h1 and h2, so that the working interval h k = s k -t k satisfies

[0033] Then under the intermittent sampling transmission scheme, the control signal can be expressed as:

[0034]

[0035] where the controller gain matrix

[0036] The system model of (1.1) is transformed using the control signal to obtain the closed-loop multi-area power system model, denoted as:

[0037]

[0038] The system output signal of the new power system is obtained by changing the control signal using the deception attack.

[0039] The system output signal is sampled to obtain the sampled output signal and the control signal under the deception attack; the attack location is as shown in Figure 4 From Figure 4 , it can be seen that the deception attack first attacks the sensor signal, and then sends the data after being attacked to the controller. The control signal uses an intermittent sampling mechanism for transmission. Deception attacks can tamper with data to affect the performance of the system, and even cause system instability. Deception attacks completely replace the original data by maliciously attacking the signal or add the signal to the original data, thereby destroying data transmission. This study assumes that the attacker will completely replace the original data, and the malicious attack signal can be modeled as v(t k ), which is related to the original transmitted data and satisfies the following settings.

[0040] Setting 2: The deception attack is bounded, i.e., the attack signal has the following bounded condition:

[0041] ‖v(t k )‖2≤‖Hx(t k )‖2

[0042] where H is a constant matrix artificially given to describe the strength of the upper bound of the attack.

[0043] Under the influence of the deception attack, the actual control input signal u(t) of the new multi-area power system can be represented as:

[0044]

[0045] The system model of (1.2) is transformed to obtain the closed-loop multi-area power system model, denoted as:

[0046]

[0047] The purpose of the present application is to develop an H∞ intermittent sampling load frequency safety controller to ensure that the closed-loop multi-area power system is exponentially stable under deception attacks while satisfying H∞ performance. Specifically, the controller design satisfies the following requirements.

[0048] (1) The closed-loop multi-area power system is exponentially stable under deception attacks with disturbance w(t) = 0.

[0049] (1) The inequality γ||w(t)||2≥||y(t)||2 holds for a given scalar γ > 0 and any non-zero disturbance

[0050] Subsequently, the following lemma is proposed to support the stability analysis of the closed-loop multi-area power system under deception attacks.

[0051] Lemma 1: For any matrix N > 0, constants p and q satisfy q > p, and the function m is an arbitrary constant, the following inequality holds:

[0052]

[0053] The time-dependent switching LF constructed by the present application is as follows:

[0054]

[0055] where

[0056] V P (t) = x T (t)Px(t)

[0057]

[0058] V X (t) = (s k -t)(t-t k )x T (t k )Xx(t k )

[0059]

[0060] where the matrix and

[0061]

[0062] The matrix

[0063] The sufficient condition for the closed-loop multi-area power system to be exponentially stable under deception attacks with disturbance w(t) = 0 can be represented by Theorem 1: ​

[0064] Theorem 1: Given scalars a > 0, β > 0, θ > 0, h2≥ hi > 0, where a, β and satisfy If there exists any matrix

[0065] and matrices R > 0, P > 0, X > 0 satisfy the following linear matrix inequalities (LMIs):

[0066]

[0067] Ξ1< 0 (1.7)

[0068] Π(h k ) > 0 (1.8)

[0069] where the special one-type matrix Γ1= [Γ ij ] 5n×5n , Δ(h k ) = [Δ ij ] 5n×5n , Λ(h k ) = [Λ ij ] 6n×6n , Ξ1= [Ξ ij ] 2n×2n , Π(h k ) = [Π(h k ) ij ] 3n×3n , other matrices Γ 12 = S1- S2- Y2+ P1 T BK, Γ 13 = P - Y3- P1 T + A T P2, Γ 14 = -S3, Γ 15 = P1 T B, Γ 23 = -Y3+ (BK) T P2, Γ 24 = -S4, Δ 23 = h k (-S1+ S2), Δ 24 = a h k S4, Δ 33 = h k R, Λ 16 = hk Y1 T ,Λ 22 =-h k X+(αh2 / 2)h k X, Π 12 =h k (-S1+S2),Π 13 =h k S3, Π 23 =h k S4, I is the identity matrix and the rest of the blocks are zero matrices. Then the closed-loop multi-area power system under the deception attack with w(t) = 0 is exponentially stable.

[0070] Proof: First, it is worth noting that we have

[0071]

[0072] Therefore, V(t) is continuous in time since Next, when t ∈ [t k ,s k ), the derivative of the time-dependent switching LF gives

[0073]

[0074] V X (t) is amplified by the following inequality:

[0075]

[0076] To solve the integral term in inequality (1.9), we use Lemma 1 to obtain the following result:

[0077]

[0078] where the function

[0079] By considering the setup of the deceptive attack, we can derive the following inequality condition

[0080] x T (t k )H T Hx(t k )-v(t k )v(t k )≥0

[0081] Considering the Newton-Leibnitz formula, for any matrices Y1, Y2, Y3, the following relation holds:

[0082]

[0083] When t∈[t k ,s k ), according to the multi-area power system under false data attacks, for any matrix P1, P2, it gets

[0084]

[0085] Substituting (1.10) - (1.14) into (1.9), we get

[0086]

[0087] where m(t)}.

[0088] Applying the Schur lemma and considering (1.5) and (1.6), we get

[0089]

[0090] It is worth noting that even if the matrix S is indefinite, V P (t) + V S (t) is positive, which is guaranteed by inequality (1.8). Specifically, we have

[0091]

[0092] where Φ(h k ) = ((s k -t) / h k )Π(h k )+((t-t k ) / h k )Π(0), Π(h k ) > 0,

[0093] The conditions R > 0 and X > 0 guarantee that V X (t) and V R (t) are positive definite, respectively, without requiring that V P (t) + V S (t) is positive definite. Therefore, V(t) > 0 holds. From we get

[0094]

[0095] When t∈[s k ,t k+1 ), taking the derivative of the time-dependent switching LF gives

[0096]

[0097] Similarly to (1.14), when t ∈ [s k ,t k+1 ), according to the multi-area power system under deception attacks, for any matrices Q1, Q2, we have

[0098]

[0099] Substituting (1.18) into (1.17), we obtain

[0100]

[0101] where

[0102] According to (1.7), we obtain

[0103]

[0104] Therefore, due to the positive definiteness of V(t), we obtain

[0105]

[0106] By referring to (1.16) and (1.20) and the continuity of V(t), for t ∈ [t k ,t k+1 ], the following inequality can be determined to hold:

[0107]

[0108] Given V(t) ≥ V P (t) + V S (t) and (1.16), we can derive a positive scalar ε > 0 that exists such that the following inequality holds:

[0109] V(t) ≥ ε‖x(t)‖ 2 (1.22)

[0110] According to (1.21), (1.22), and V(t0) = x T (t0)Px(t0), we can obtain

[0111]

[0112] In summary, the closed-loop multi-area power system under deception attacks with disturbance w(t) = 0 is exponentially stable. Next, under zero initial conditions, a sufficient condition for the inequality to hold for a given γ > 0 and any non-zero disturbance can be represented by Theorem 2.

[0113] Theorem 2: Given any scalars a > 0, β > 0, γ > 0, h2≥ hi > 0, where a, β and satisfy If there exist any matrices Y1, Y2, Y3, P1, P2, Q1, Q2, S1, S2, S3, S4, S5, and matrices R > 0, P > 0, X > 0 satisfying the following LMI:

[0114]

[0115] where Γ 1 = Γ1+ Δ(h k ), Θ 1 = Θ1+ Λ(h k ),

[0116]

[0117] Proof: When t ∈ [t k , s k ), according to the multi-zone power system under the deception attack, for any matrices P1, P2, we have

[0118]

[0119] By the Schur complement lemma, we can deduce from (1.9)-(1.13) and (1.27) that

[0120]

[0121] where ξ4(t) = col{ξ1(t), w(t)} and ξ5(t) = col{ξ2(t), w(t)}.

[0122] Similarly to (1.27), when t ∈ [s k , t k+1 ], according to the multi-zone power system under the deception attack, for any matrices Q1, Q2, we have

[0123]

[0124] By the Schur complement lemma, from (1.17) and (1.29) we can deduce that

[0125]

[0126] where ξ6(t) = col{ξ3(t), w(t)}.

[0127] Then, under zero initial conditions, we can derive

[0128]

[0129] Therefore, by integrating both sides of (1.28) and (1.30) from 0 to t k+1 we obtain from the above equation

[0130]

[0131] Taking the limit as t→∞, we can derive from the above equation

[0132]

[0133] for all and the initial condition is zero.

[0134] However, the above Theorem 2 cannot be directly used to calculate the controller gain matrix K by LMI toolbox. Next, we will introduce a method to calculate the controller gain matrix K, which is used to convert Theorem 2 into Theorem 3 that can calculate the controller gain matrix K.

[0135] Theorem 3: Given scalars α>0, β>0, γ>0, h2≥h1>0, where α, β and satisfy If there exist any matrix

[0136] and any positive definite matrix satisfy the following LMI:

[0137]

[0138] where matrix Special matrix of type one Other matrix

[0139]

[0140]

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] Further, the control gain matrix is given by

[0147] Proof: Define

[0148]

[0149] and define P2 = aP1, Q1 = bP1, Q2 = cP1,

[0150] It is obvious that the results of inequalities (1.31)-(1.34) in Theorem 3 can be obtained by left multiplying and right multiplying Y i , i = 1, 2, 3, 4, respectively.

[0151] Since the LMIs (1.31)-(1.34) in Theorem 3 include the tuning parameters α > 0, β > 0, γ > 0, a, b, c, it is essential to determine suitable values of these parameters for solving the LMI. In this paper, the Latin hypercube sampling method is used to find suitable values of α > 0, β > 0, γ > 0, a, b, c. Unlike the grid search method, the Latin hypercube sampling method provides good global parameter search and is an easy-to-use design technique and popular experimental technique with low cost and easy implementation.

[0152] The effects of the present application can be further illustrated by the following simulation experiment. Consider a simulation example of a three-zone power system under a deception attack to illustrate the practicality of the proposed H ∞ interval sampling safe load frequency control method.

[0153] Set the deception attack v(t k ) = [-tanh(H1x1(t k )) ; -tanh(H2x2(t k )) ; -tanh(H3x3(t k ))], where H = diag{H1, H2, H3}, H1 = 0.1C1, H2 = 0.2C2, H3 = 0.3C3. Note that v(t k ) satisfies the condition ‖v(t k )‖2≤‖Hx(t k )‖2.

[0154] The parameters of a region of a multi-zone power system under a general deception attack: M1 = 10, β1 = 21.0, D1 = 1.0, R1 = 0.05, T​g1 = 0.1, T ch1 = 0.3; two-region parameters: M2= 12, β2= 21.5, D2= 1.5, R2= 0.05, T g2 = 0.17, T ch2 = 0.4; three-region parameters: M3= 12, β3= 21.8, D3= 1.8, R3= 0.05, T g3 = 0.1, T ch3 = 0.35; tie-line synchronization coefficient T 12 = 0.2, T 13 = 0.2, T 23 = 0.2. Meanwhile, the sampling period h k = h1= h2= 0.05, the time ratio of the rest interval in the total interval is Other parameters can be obtained by the Latin hypercube sampling method, and the parameter values are as follows: a = 4, β = 3.01, a = 0.1, b = 0.2, c = 0.3, γ = 10. Then we use the LMI toolbox in Matlab to solve theorem 3, and the corresponding controller gain matrix is:

[0155]

[0156] Let the disturbance ω(t) be:

[0157]

[0158] And the given general initial state of the system is x0= [x 10 x 20 x 30 ] T , where x 10 = [0.020.04-0.05-0.09] T , x 20 = [-0.020.05-0.05-0.04] T , x 30 = [0.040.04-0.015-0.05] T . Now, we apply the H ∞ interval sampling load frequency safety controller with the above controller gain to the three-region power system under deception attack to obtain the simulation results. Figures 5 to 7 The system state responses of the first to third regions are shown in Fig. 6. The H ∞ interval sampling safety load frequency control signal is shown in Fig. 7. Figure 8 ​

Claims

1. A method for H∞ intermittent sampling load frequency security control in a multi-regional power system under deception attack, I n Represents an n-dimensional identity matrix; 0 n represents an n-dimensional zero matrix; ||.|| represents the Euclidean norm of a vector; It describes the space of square-integrable vector functions on [0,∞); R represents the set of natural numbers; n and Both represent n-dimensional Euclidean space, R n×m and Let A denote the set of n×m real matrices; col denotes column vectors; diag denotes diagonal matrices; sup denotes the smallest upper bound of a set; for a symmetric matrix where A < 0 and C < 0... *Refers to terms resulting from symmetry; I is the identity matrix; define the matrix. Where A 11 A 12 ,…,A 1n A 21 ,…,A nn Let i be any matrix, where i,j = 1, 2, ..., n; The method specifically includes: A new power system model is constructed based on the dynamic equations describing the system state of each sub-region in a multi-region power system; The dynamic equation for the i-th region of a multi-region power system load frequency control is expressed as: A multi-regional power system consists of generators, turbines, and governors, where ΔP vi ΔP mi ΔP di and Δf i These represent the valve position deviation, generator mechanical output deviation, load deviation, and power system frequency deviation for the i-th region, respectively; Δf j The frequency deviation of the power system in region j is shown; R i M i D i T chi T gi These represent the speed reduction of the power system in region i, the generator moment of inertia, the generator damping coefficient, the turbine time constant, and the governor time constant, respectively; T ij ACE is the synchronization coefficient of the tie line between the i-th and j-th control areas. i The frequency derivative Δf i Power exchange ΔP with tie line tie-i A linear combination of β i t is the frequency bias factor; t is the time variable. and Let be the derivative with respect to time t; Define x(t) as the state vector of a multi-region power system with a total number of regions n. i (t)=[Δf i (t)ΔP mi (t)ΔP vi (t)ΔP tie-i (t)] T Let be the state vector of the power system in region i. Let x(t) be the derivative of x(t) with respect to time t, and y(t) be the output vector of a multi-regional power system with a total number of regions n. i (t)=ACE i u(t) is the output vector of the power system in the i-th region, and u(t) is the input vector of the multi-region power system with a total of n regions. i w(t) is the input vector of the power system in the i-th region, and w(t) is the disturbance variable of the multi-region power system with a total of n regions. i (t)=ΔP di (t) represents the disturbance variable of the power system in the i-th region; then, the transformation function of the mathematical model of the multi-region power system can be obtained by transforming the dynamic equation (1.0), and the specific function expression is as follows: where A=[A ij ] n×n , T ij =T ji x(t) = [x1(t), x2(t), ..., x i (t),…,x n (t)] T B = diag{B1, B2, ..., B i ,…,B n }, u(t)=[u1(t),u2(t),…,u i (t), ...,u n (t)] T F = diag{F1, F2, ..., F i ,…,F n }, w(t)=[w1(t),w2(t),…,w i (t),…,w n (t)] T C = diag{C1, C2, ..., C i ,…,C n }, y(t)=[y1(t),y2(t),…,y i (t),…,y n (t)] T , C i =[β i 0 01], n is the total number of regions in the multi-regional power system, where i = 1, 2, ..., n, j = 1, 2, ..., n, and i ≠ j; Total time interval Divided into a series of non-overlapping time intervals [t] k ,t k+1 ], Where 0 = t0 < t1 < t2 ... < t k <t k+1 ···satisfy Each control time interval [t] k ,t k+1 ) are respectively located in the rest area [s k ,t k+1 ) and work interval [t k ,s k Composed of ) s k For the control interval [t] k ,t k+1 Intermediate time variable; the controller only operates within its designated working interval; t k , t k+1 These are the two sampling times, h. k The sampling period is defined; two positive scalars h1 and h2 are set such that the working interval h k =s k -t k Satisfying 0 < h1 ≤ h k ≤h2, In the intermittent sampling transmission scheme, the control signal is represented as: Where the controller gain matrix Using this control signal, the system model in (1.1) is transformed to obtain a closed-loop multi-region power system model, which is expressed as: The multi-regional power system was attacked using a deception attack to change the control signal and obtain the system output signal of a new power system. The spoofing attack first attacks the sensor signal, and then sends the attacked data to the controller. The control signal is transmitted using an intermittent sampling mechanism. The attacker is set to completely replace the original data; the malicious attack signal is modeled as v(t) k The attack signal is related to the previously sent data and meets the following settings; Setting 2: Deception attacks are bounded, meaning the attack signal is bounded under the following conditions: ‖v(t k )‖2≤‖Hx(t k )‖2 Where H is a constant matrix given by the user to describe the strength of the upper bound of the attack; Under the influence of a deception attack, the actual control input signal u(t) of the new multi-regional power system is expressed as: Then, the system model in (1.3) is transformed to obtain a closed-loop multi-region power system model, which is expressed as: The controller design must meet the following requirements; (1) The closed-loop multi-regional power system is exponentially stable under a deceptive attack that interferes with w(t) = 0; (1) Under zero initial conditions, the inequality γ||w(t)||2≥||y(t)||2 holds true for a given scalar γ>0 and any non-zero disturbance. Established; Subsequently, the following lemma is proposed to support the stability analysis of closed-loop multi-region power systems under deception attacks; Lemma 1: For any matrix N > 0, the constants p and q satisfy q > p, and the function For any constant m, the following inequalities hold: The constructed time-dependent switching LF form is as follows: in V P (t)=x T (t)Px(t) V X (t)=(s k -t)(t-t k )x T (t k )Xx(t k ) Where the matrix and matrix A sufficient condition for a closed-loop multi-regional power system to be exponentially stable under a deceptive attack with disturbance w(t) = 0 can be expressed by Theorem 1: Theorem 1: Given scalars α > 0, β > 0, θ > 0, h2 ≥ h1 > 0, Where α, β and satisfy If any matrix exists And the matrices R > 0, P > 0, X > 0 satisfy the following linear matrix inequality (LMI): Ξ1<0 (1.7) P(h k )>0 (1.8) Where the special type I matrix Γ1=[Γ ij ] 5n×5n ,Δ(h k )=[Δ ij ] 5n×5n ,Λ(h k ) = [Λ ij ] 6n×6n Ξ1=[Ξ ij ] 2n×2n , Π(h k )=[Π(h k ) ij ] 3n×3n Other matrices Γ 12 =S1-S2-Y2+P1 T BK, Γ 13 =P-Y3-P1 T +A T P2, Γ 14 =-S3,Γ 15 =P1 T B, Γ 23 = -Y3 + (BK) T P2, Γ 24 =-S4, Γ 55 =-I, Δ 23 =h k (-S1+S2), Δ 24 =αh k S4, Δ 33 =h k R, Λ 16 =h k Y1 T Λ 22 =-h k X+(αh2 / 2)h k X, Π 12 =h k (-S1+S2), Π 13 =h k S3. Π 23 =h k S4, If the matrix is ​​an identity matrix and the remaining squares are all zero matrices, then the closed-loop multi-region power system is exponentially stable under a deceptive attack with interference w(t) = 0. Proof: First, it is worth noting that there is Therefore, V(t) is continuous in time, because Next, when t∈[t k ,s k When ), taking the derivative of the time-dependent switching LF yields: in The derivative of V(t) is expressed as denoted by . V X (t) is amplified by the following inequality: To solve the integral term in inequality (1.9), we obtain the following result using Lemma 1: Where the function By considering the scenario of deceptive attacks, the following inequality conditions are derived. x T (t k )H T Hx(t k )-v(t k )v(t k )≥0 Considering the Newton-Leibniz formula, for any matrix Y1, Y2, Y3, the following relationship holds: When t∈[t k ,s k When, according to the multi-regional power system under deception attack, for any matrix P1, P2, it obtains Substituting (1.10)-(1.14) into (1.9) yields in Applying Schur's lemma and considering (1.5) and (1.6), we obtain Even if matrix S is indeterminate, V P (t)+V S The positivity of (t) is also guaranteed by inequality (1.8); specifically, we have among themΦ(h k )=((s k -t) / h k )Π(h k )+((tt k ) / h k )Π(0),Π(h k )>0, Conditions R > 0 and X > 0 respectively guarantee V X (t) and V R (t) is positive definite, so V is not required. P (t)+V S (t) is positive definite; therefore, V(t) > 0 holds; by get When t∈[s] k ,t k+1 When ), taking the derivative of the time-dependent switching LF yields... Similar to (1.14), when t∈[s] k ,t k+1 When, according to the multi-regional power system under deception attack, for any matrix Q1, Q2, we have Substituting (1.18) into (1.17), we get in According to (1.7), we get Therefore, due to the positive definiteness of V(t), we obtain By referencing equations (1.16) and (1.20) and the continuity of V(t), for t∈[t k ,t k+1 The following inequalities hold true: Given V(t)≥V P (t)+V S From (t) and equation (1.16), we obtain a positive scalar ε > 0, which holds under the following inequality: V(t)≥ε‖x(t)‖ 2 (1.22) According to equations (1.21), (1.22), and V(t0) = x T (t0)Px(t0), thus obtaining Where λ max (P) represents the maximum value of the eigenvalues ​​of matrix P; In summary, the closed-loop multi-region power system is exponentially stable under a deceptive attack with disturbance w(t) = 0. Furthermore, under zero initial conditions, the inequality γ||w(t)||2≥||y(t)||2 holds for a given γ > 0 and any non-zero disturbance. The sufficient condition for this to hold can be expressed by Theorem 2; Theorem 2: Given any scalars α > 0, β > 0, γ > 0, h2 ≥ h1 > 0, Where α, β and satisfy If there exist any matrices Y1, Y2, Y3, P1, P2, Q1, Q2, S1, S2, S3, S4, S5, and matrices R > 0, P > 0, X > 0 satisfying the following LMI: P(h k )>0(1.26) among themC 1 =Γ1+Δ(h k ),Θ 1 =Θ1+Λ(h k ), Proof: When t∈[t k ,s k When, according to the multi-regional power system under deception attack, for any matrix P1, P2, it obtains Using Schul's complement lemma, we can deduce from equations (1.9)-(1.13) and (1.27) Where ξ4(t)=col{ξ1(t),w(t)} and ξ5(t)=col{ξ2(t),w(t)}; Similar to (1.27), when t∈[s] k ,t k+1 When, according to the multi-regional power system under deception attack, for any matrix Q1, Q2, we have Using Schul's complement lemma, we can deduce from equations (1.17) and (1.29) that... Where ξ6(t) = col{ξ3(t), w(t)}; Then, under zero initial conditions, we derive Therefore, by changing both sides of (1.28) and (1.30) from 0 to t k+1 Integrating, we get the following from the above equation: Let t→∞, from the above equation we can deduce For all And the initial condition is zero; This paper introduces a method for calculating the controller gain matrix K, which transforms Theorem 2 into Theorem 3 for calculating the controller gain matrix K. Theorem 3: Given scalars α > 0, β > 0, γ > 0, h2 ≥ h1 > 0, Where α, β and satisfy If any matrix exists and any positive definite matrix The following LMIs must be met: Where the matrix Special Type I Matrix Other matrices Furthermore, the control gain matrix is ​​composed of Give; Proof: Definition And define P2 = aP1, Q1 = bP1, The results of inequalities (1.31)-(1.34) in Theorem 3 are multiplied on the left by inequalities (1.23)-(1.26) in Theorem 2 respectively. And right multiplication Υ i We obtain the following, where i = 1, 2, 3, 4.

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