H-infinity intermittent sampling load frequency safety control method for multi-region power system under spoofing attack

Abstract

The invention discloses an H-infinity intermittent sampling load frequency safety control method for a multi-region power system under spoofing attack, and belongs to the technical field of power system safety control. According to the method, the intermittent control signal is only used in the working interval, the control cost is effectively saved, the limited network bandwidth can be saved through sampling control, and the influence of spoofing attacks on the multi-area power system is effectively resisted through safety control. Firstly, according to the characteristics of intermittent control and sampling control, the influence of spoofing attack is considered at the same time, and a multi-region power system intermittent sampling load frequency safety control dynamic model under the spoofing attack is established. Secondly, a switching Lyapunov functional is provided, meanwhile, under the framework of an LMI, sufficient conditions for existence of an intermittent sampling load frequency safety controller are given, and the conditions guarantee that the index of a multi-area power system is stable and meet the H infinity performance; and finally, verifying the effectiveness of the H-infinity intermittent sampling load frequency safety controller through three-region power system simulation under spoofing attack.

Description

Technical Field

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

[0002] In multi-regional power systems, 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 improvement schemes based on the LFC method, such as adaptive control, robust control, proportional differential control, fuzzy logic control, etc. However, the above control methods require continuous input of control signals and exert control, which will lead to an increase in control costs. At the same time, the operation of modern multi-regional power systems depends on a stable cyberspace, and the open communication environment in the cyberspace makes the system more vulnerable to cyber attacks. Compared with traditional physical attacks, cyber attacks may also lead to serious consequences, such as information leakage and infrastructure disruption. Deception attacks are a form of cyber attacks 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 multi-regional power systems to meet the needs of practical work. Summary of the invention

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

[0004] In order to achieve the above object, the present invention adopts the following technical solution:

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

[0006] Establish a dynamic model for load frequency control of multi-regional power systems;

[0007] Introducing intermittent sampling control mechanism;

[0008] Establish a dynamic model for frequency control of intermittent sampling loads in multi-regional power systems;

[0009] Consider spoofing attacks;

[0010] Establish a dynamic model for the secure control of intermittent sampling load frequency in multi-regional power systems under deception attacks;

[0011] Based on Lyapunov functional theory, sufficient conditions for the existence of intermittent sampling load frequency security controller are given, which can ensure the exponential stability of multi-regional power systems and meet H∞ performance. According to this condition, the specific parameters of the intermittent sampling load frequency security controller are designed, and then the designed controller is applied to the multi-regional power system under deception attack.

[0012] The present invention provides a H∞ intermittent sampling load frequency safety control method for a multi-regional power system under a deception attack. In the present invention, the intermittent control signal is only used in the working interval, which can effectively save the control cost, the sampling control can save the limited network bandwidth, and the safety control can effectively resist the influence of the deception attack on the multi-regional power system. Firstly, according to the characteristics of intermittent control and sampling control, and considering the influence of the deception attack, a dynamic model of the intermittent sampling load frequency safety control of the multi-regional power system under the deception attack is established. Secondly, a switching Lyapunov functional is proposed, and under the framework of LMI, sufficient conditions for the existence of the intermittent sampling load frequency safety controller are given, which can ensure the exponential stability of the multi-regional power system and meet the H∞ performance. Finally, the effectiveness of the proposed H∞ intermittent sampling load frequency safety controller is verified through the simulation of the three-regional power system under the deception attack. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The accompanying drawings in the specification, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute improper limitations on the present invention.

[0014] Figure 1 A schematic diagram of the present invention.

[0015] Figure 2 Schematic diagram of the dynamic model of load frequency control in the multi-regional power system in the i-th region.

[0016] Figure 3 Schematic diagram of intermittent sampling control mechanism.

[0017] Figure 4 Schematic diagram of the deception attack location.

[0018] Figure 5 This is a schematic diagram of the system status response in the first area.

[0019] Figure 6 Schematic diagram of the system status response in the second area.

[0020] Figure 7 This is a schematic diagram of the system status response in the third area.

[0021] Figure 8 Schematic diagram of H∞ intermittent sampling safety load frequency control signal. DETAILED DESCRIPTION

[0022] The present invention is further described in detail below with reference to specific examples, but the embodiments of the present invention are not limited thereto.

[0023] The notation used in this invention is standard and is introduced here before proceeding. If the dimensions of a matrix are not explicitly stated, it is assumed to be compatible with algebraic operations. n Represents an n-dimensional identity matrix. n represents an n-dimensional zero matrix. ||.|| represents the Euclidean norm of a vector. L2[0,∞) describes the space of square-integrable vector functions on [0,∞). Represents the set of natural numbers. n and Both represent n-dimensional Euclidean space, R n×m and represents a set of n×m real matrices. col represents a column vector, diag represents a diagonal matrix. sup represents the minimum upper bound of a set. For a symmetric matrix A<0 and C<0 * refers to the term caused by symmetry. I is the identity matrix. Define the matrix Among them A 11 , A 12 ,…,A 1n , A 21 ,…,A nn is an arbitrary matrix, in which case i, j = 1, 2, ..., n, and such a matrix is ​​defined as a special type I matrix. Note that due to the high computational complexity of the present invention, i, j are reproducibly defined constants.

[0024] like Figure 1 As shown, the present invention provides a method for securely controlling H∞ intermittent sampling load frequency in a multi-regional power system under a deception attack, including:

[0025] The novel power system model is constructed according to the dynamic equations describing the system state of each sub-region in the multi-region power system.

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

[0027]

[0028] The multi-regional power system consists of generators, turbines and speed regulators, where ΔP vi , ΔP mi , ΔP di and Δf iThey represent valve position deviation, generator mechanical output deviation, load deviation and power system frequency deviation respectively. i 、M i , D i , T chi , T gi are the speed reduction of the power system in the i-th region, the generator moment of inertia, the generator damping coefficient, the turbine time constant, and the governor time constant. ij is the synchronization coefficient of the tie line between the i-th and j-th control areas, ACE i is the frequency derivative Δf i Power exchange with tie line ΔP tie-i The linear combination of i is the frequency offset factor. t is the time variable, and is the derivative with respect to time t.

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

[0030]

[0031] in 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 0 1], n is the total number of regions in the multi-region power system, where i = 1, 2, ..., n, j = 1, 2, ..., n, and i ≠ j.

[0032] like Figure 3 As shown, the present invention introduces an intermittent sampling control mechanism. Specifically, the total time interval Divide into a series of disjoint time intervals Where 0=t0<t1<t2…<t k <t k+1 …satisfy Each control time interval [t k ,t k+1 ) respectively by the rest area [s k ,t k+1 ) and working range [t k ,s k ) composition, s k is the control interval [t k ,t k+1 ]Intermediate time variable, the controller only works in the working range. k , t k+1 are the two sampling moments before and after, respectively, h k is the sampling period. Set two positive scalars h1 and h2 so that the working interval h k =s k -t k satisfy

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

[0034]

[0035] The controller gain matrix

[0036] The closed-loop multi-region power system model is obtained by converting the system model of (1.1) using the control signal, which is expressed as:

[0037]

[0038] The multi-regional power system is attacked by using a deception attack to change the control signal to obtain a system output signal of the new power system.

[0039] The system output signal is sampled to obtain the sampled output signal and control signal under the deception attack; the attack occurs at the position such as Figure 4 As shown, from Figure 4 It can be seen that the deception 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. Deception attacks can tamper with data to affect system performance and even cause system instability. Deception attacks completely replace the original data with malicious attack signals or attach signals to the original data, thereby destroying data transmission. This study assumes that the attacker will completely replace the original data. The malicious attack signal can be modeled as v(t k ), the attack signal is related to the originally sent data and meets the following settings.

[0040] Setting 2: The deception attack is bounded, that is, the attack signal has the following bounded conditions:

[0041] ‖v(t k )‖2≤‖Hx(t k )‖2

[0042] Among them, H is an artificially given constant matrix, which is used to describe the strength of the upper bound of the attack.

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

[0044]

[0045] Then, the system model of (1.2) is transformed to obtain the closed-loop multi-regional power system model, which is expressed as:

[0046]

[0047] The purpose of this invention is to develop an H∞ intermittent sampling load frequency security controller to ensure that the closed-loop multi-region power system is exponentially stable under spoofing attacks while meeting the H∞ performance. Specifically, the controller is designed to meet the following requirements.

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

[0049] (1) Under zero initial conditions, the inequality γ||w(t)||2≥||y(t)||2 holds for a given scalar γ>0 and any non-zero disturbance. Established.

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

[0051] Lemma 1: For any matrix N>0, the 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 form constructed by the present invention is as follows:

[0054]

[0055] in

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

[0057]

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

[0059]

[0060] The matrix and

[0061]

[0062] matrix

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

[0064] Theorem 1: Given scalars α>0, β>0, θ>0, h2≥h1>0, Among them, α, β and satisfy If there is any matrix

[0065] And the matrix R>0, P>0, X>0 satisfies the following linear matrix inequality (LMI):

[0066]

[0067] Ξ1<0 (1.7)

[0068] Π(h k )>0 (1.8)

[0069] Among them, the special 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 =αh 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, is the unit matrix, and the remaining blocks are all zero matrices. Then the closed-loop multi-area power system is exponentially stable under the deception attack with interference w(t)=0.

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

[0071]

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

[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] The function

[0079] By considering the setting of 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-Leibniz formula, for any matrices Y1, Y2, Y3, the following relationship holds:

[0082]

[0083] When t∈[t k ,s k ), according to the multi-regional power system under deception attack, for any matrix P1, P2, it is obtained

[0084]

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

[0086]

[0087] in m(t)}.

[0088] Applying Schur's lemma and considering (1.5) and (1.6), we obtain

[0089]

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

[0091]

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

[0093] The conditions R>0 and X>0 guarantee V X (t) and V R (t) is positive definite, and V is not required P (t)+V S (t) is positive definite. Therefore, V(t)>0 holds. We get

[0094]

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

[0096]

[0097] Similar to (1.14), when t∈[s k ,t k+1 ), according to the multi-regional power system under deception attack, for any matrix Q1, Q2, we have

[0098]

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

[0100]

[0101] in

[0102] According to (1.7), we have

[0103]

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

[0105]

[0106] By referring to equations (1.16) and (1.20) and the continuity of V(t), for t∈[t k ,t k+1 ), we can determine that the following inequality holds:

[0107]

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

[0109] V(t)≥ε‖x(t)‖ 2 (1.22)

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

[0111]

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

[0113] Theorem 2: Given any scalar α>0, β>0, γ>0, h2≥h1>0, Among them, 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 that satisfy 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-regional power system under deception attack, for any matrix P1, P2, it is obtained

[0118]

[0119] Using Schur's complement lemma, we can infer from equations (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] Similar to (1.27), when t∈[s k ,t k+1 ), according to the multi-regional power system under deception attack, for any matrix Q1, Q2, we have

[0123]

[0124] Using Schur's complement lemma, we can infer from equations (1.17) and (1.29) that

[0125]

[0126] Where ξ6(t)=col{ξ3(t),w(t)}.

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

[0128]

[0129] Therefore, by extending both sides of (1.28) and (1.30) from 0 to t k+1 Integrating, we can get:

[0130]

[0131] Let t→∞, from the above formula we can deduce

[0132]

[0133] For all And the initial condition is zero.

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

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

[0136] And any positive definite matrix Meet the following LMIs:

[0137]

[0138] The matrix Special Type 1 Matrix Other matrices

[0139]

[0140]

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] Furthermore, the control gain matrix is ​​given by Given.

[0147] Proof: Definition

[0148]

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

[0150] Obviously, the results of inequalities (1.31)-(1.34) in Theorem 3 can be obtained by multiplying inequalities (1.23)-(1.26) in Theorem 2 by and right multiply Υ i Get, where i = 1, 2, 3, 4.

[0151] Since the LMIs (1.31)-(1.34) in Theorem 3 include tuning parameters α>0, β>0, γ>0, a, b, c, determining the appropriate values ​​of these parameters is crucial to solving the LMI. In this paper, the Latin Hypercube Sampling method is used to find the appropriate values ​​of α>0, β>0, γ>0, a, b, c. Different from the grid search method, the Latin Hypercube Sampling method provides a good global parameter search and is an easy-to-use design technique and a popular experimental technique with low cost and easy implementation.

[0152] The effect of the present invention can be further illustrated by the following simulation experiment. Consider a simulation example of a three-region power system under a deception attack to illustrate the proposed H ∞ Practicality of intermittent sampling safety load frequency control method.

[0153] Set the spoofing 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] Parameters of one area of ​​a multi-area power system under a general deception attack: M1 = 10, β1 = 21.0, D1 = 1.0, R1 = 0.05, Tg1 =0.1, T ch1 =0.3; Region II 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; synchronization coefficient of the tie line between control areas T 12 =0.2, T 13 =0.2, T 23 =0.2. At the same time, the sampling period h is specified k =h1=h2=0.05, the time spent in the rest area accounts for Other parameters can be obtained by Latin hypercube sampling method, and the parameter values ​​are as follows: α = 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 given the system's general initial state 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 have H with the above controller gains ∞ The intermittent sampling load frequency security controller is applied to the three-area power system under deception attack and simulation results can be obtained. Figures 5 to 7 They are the system status responses of the first to third regions respectively. ∞ Intermittent sampling safety load frequency control signal such as Figure 8 shown.

Claims

1. A H∞ intermittent sampling load frequency security control method for multi-regional power systems under deception attacks, I n Represents an n-dimensional identity matrix; 0 n represents an n-dimensional zero matrix; ||.|| represents the Euclidean norm of the vector; L2[0,∞) describes the space of square-integrable vector functions on [0,∞); represents the set of natural numbers; R n and Both represent n-dimensional Euclidean space, R n×m and represents a set of n×m real matrices; col represents a column vector, diag represents a diagonal matrix; sup represents the minimum upper bound of a set; for symmetric matrices with A<0 and C<0 * refers to the term caused by symmetry; I is the identity matrix; define the matrix Among them A 11 , A 12 ,…,A 1n , A 21 ,…,A nn is an arbitrary matrix, in which case i,j=1,2,…,n. The method specifically includes: Constructing the novel power system model according to the dynamic equations describing the system state of each sub-region in the multi-region power system; The dynamic equation of the i-th region of the multi-region power system load frequency control is expressed as: The multi-regional power system consists of generators, turbines and speed regulators, where ΔP vi , ΔP mi , ΔP di and Δf i They represent valve position deviation, generator mechanical output deviation, load deviation and power system frequency deviation respectively; R i 、M i , D i , T chi , T gi are the speed reduction of the power system in the i-th region, the generator moment of inertia, the generator damping coefficient, the turbine time constant, and the governor time constant; T ij is the synchronization coefficient of the tie line between the i-th and j-th control areas, ACE i is the frequency derivative Δf i Power exchange with tie line ΔP tie-i The linear combination of i is the frequency offset factor; t is the time variable, and is 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, x i (t) = [Δf i (t)ΔP mi (t)ΔP vi (t)ΔP tie-i (t)] T is the state vector of the power system in the ith region, is the derivative of x(t) with respect to time t, y(t) is the output vector of the multi-region power system with a total number of regions n, and y i (t) = ACE i (t) is the output vector of the power system in the i-th region, u(t) is the input vector of the multi-region power system with a total number of regions n, and u i (t) is the input vector of the power system in the i-th region, w(t) is the disturbance variable of the multi-region power system with a total number of regions n, and w i (t) = ΔP di (t) is the disturbance variable of the power system in the ith region; then the deformation function of the mathematical model of the multi-regional power system can be obtained by transforming the dynamic equation (1.0). The specific function expression is: in 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 0 1], n is the total number of regions in the multi-region power system, in which case i = 1, 2, ..., n, j = 1, 2, ..., n, and i ≠ j; The total time interval Divide 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 ) respectively by the rest area [s k ,t k+1 ) and working range [t k ,s k ) composition, s k is the control interval [t k ,t k+1 ]Intermediate time variable, the controller only works in the working range; t k , t k+1 are the two sampling moments before and after, respectively, 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 0 k ≤h2, ​ Then, under the intermittent sampling transmission scheme, the control signal is expressed as: The controller gain matrix The closed-loop multi-region power system model is obtained by converting the system model of (1.1) using the control signal, which is expressed as: Attacking the multi-region power system by using a deception attack to change the control signal to obtain a system output signal of the 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; Assuming that the attacker will completely replace the original data, the malicious attack signal is modeled as v(t k ), the attack signal is related to the originally sent data and meets the following settings; Setting 2: The deception attack is bounded, that is, the attack signal has the following bounded conditions: ‖v(t k )‖2≤‖Hx(t k )‖2 Among them, H is a constant matrix given by humans, which is used to describe the strength of the upper bound of the attack; Under the influence of deception attack, the actual control input signal u(t) of the new multi-area power system is expressed as: Then, the system model of (1.2) is transformed to obtain the closed-loop multi-regional power system model, which is expressed as: The controller is designed to meet the following requirements; (1) The closed-loop multi-area power system is exponentially stable under the deception attack with the disturbance w(t) = 0; (1) Under zero initial conditions, the inequality γ||w(t)||2≥||y(t)||2 holds 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-area power systems under spoofing attacks; Lemma 1: For any matrix N>0, the constants p and q satisfy q>p, and the function m is an arbitrary constant, the following inequality holds: 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 ) The matrix and matrix The sufficient condition for the closed-loop multi-area power system to be exponentially stable under a deception 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, Among them, α, β and satisfy If there is any matrix And the matrix R>0, P>0, X>0 satisfies the following linear matrix inequality (LMI): Ξ1<0(1.7) P(h k )>0(1.8) Among them, 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, is the unit matrix, and the remaining blocks are all zero matrices; then the closed-loop multi-area power system is exponentially stable under the deception attack with interference w(t) = 0; Proof: First, it is worth noting that Therefore, V(t) is continuous in time because Next, when t∈[t k ,s k ), the derivative of the time-dependent switching LF yields: V X (t) is amplified by the following inequality: In order to solve the integral term in inequality (1.9), Lemma 1 is used to obtain the following result: The function By considering the setting of deceptive attack, the following inequality condition is derived x T (t k )H T Hx(t k )-v(t k )v(t k )≥0 Considering the Newton-Leibniz formula, for any matrices Y1, Y2, Y3, the following relationship holds: When t∈[t k ,s k ), according to the multi-regional power system under deception attack, for any matrix P1, P2, it is obtained Substituting (1.10)-(1.14) into (1.9) we obtain in Applying Schur's lemma and considering (1.5) and (1.6), we obtain Even if the matrix S is indefinite, V P (t)+V S The positivity of (t) is also guaranteed by inequality (1.8); specifically, Among themΦ(h k )=((s k -t) / h k )Π(h k )+((tt k ) / h k )Π(0),Π(h k )>0, The conditions R>0 and X>0 guarantee V X (t) and V R (t) is positive definite, and 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 ), the time-dependent switching LF is derived to obtain Similar to (1.14), when t∈[s k ,t k+1 ), according to the multi-regional power system under deception attack, for any matrix Q1, Q2, we have Substituting (1.18) into (1.17), we obtain in According to (1.7), we get Therefore, due to the positive definiteness of V(t), we have By referring to equations (1.16) and (1.20) and the continuity of V(t), for t∈[t k ,t k+1 ), confirm that the following inequality holds: Given V(t)≥V P (t)+V S (t) and equation (1.16), we get a positive scalar ε>0, for which the following inequality exists: V(t)≥ε‖x(t)‖ 2 (1.22) According to equation (1.21), equation (1.22), and V(t0) = x T (t0)Px(t0), we get In summary, the closed-loop multi-area power system is exponentially stable under the deception attack with disturbance w(t) = 0; next, under zero initial conditions, the inequality For a given γ>0 and any non-zero interference The sufficient conditions for the validity of can be expressed by Theorem 2; Theorem 2: Given any scalar α>0, β>0, γ>0, h2≥h1>0, Among them, 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 that satisfy 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 ), according to the multi-regional power system under deception attack, for any matrix P1, P2, it is obtained Using Schur's complement lemma, we can infer from equations (1.9)-(1.13) and (1.27) that 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 ), according to the multi-regional power system under deception attack, for any matrix Q1, Q2, we have Using Schur's complement lemma, we can infer 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 extending both sides of (1.28) and (1.30) from 0 to t k+1 Integrating, we can get: Let t→∞, and from the above formula we can infer For all And the initial condition is zero; A method for calculating the controller gain matrix K is introduced to transform 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, Among them, α, β and satisfy If there is any matrix And any positive definite matrix Meet the following LMIs: The matrix Special Type 1 Matrix Other matrices Furthermore, the control gain matrix is ​​given by given; Proof: Definition And define P2 = aP1, Q1 = bP1, Q2 = cP1, The results of inequalities (1.31)-(1.34) in Theorem 3 are obtained by multiplying inequalities (1.23)-(1.26) in Theorem 2 by and right multiply Υ i Get, where i = 1, 2, 3, 4.

Images