# Observer-Based Sliding Mode Load Frequency Control of Power Systems under Deception Attack

Authors: Siwei Qiao; Xinghua Liu; Gaoxi Xiao; Shuzhi Sam Ge
Journal: Complexity
 (2021)

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## Abstract

This study investigates the observer-based sliding mode load frequency control for multiarea interconnected power systems under deception attack. By introducing the observer and combining it with the system state equation, the expression of the system error is obtained. A sliding mode surface is proposed to make sure the state of the systems to be stable. Then, the state equation of the system under sliding mode control is derived. The asymptotic stability of the whole system is proved by using the linear matrix inequality (LMI) technique and Lyapunov stability theory. Furthermore, a sliding mode control law is proposed to ensure that the attacked power system can reach a stable position. Numerical simulation results are presented to support the correctness of the results.


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## Body

## 1. Introduction

The developments of the power systems have greatly facilitated the human life. Ensuring stability and sustainability of power systems, as an increasingly important research area, has attracted extensive attentions [1–3]. One of the most popular goals of such studies is to ensure stable operation of the power system, to which the use of control methods helps provide a boost [4, 5].Sliding mode control, with its advantages in overcoming system uncertainties and achieving strong robustness against disturbances, has been widely used in a variety of fields [6–8]. System robustness is enhanced by combining dynamic surface control with the sliding mode [9–12]. It can be used in different systems such as nonlinear systems [13], Markovian jump systems [14], power systems, and so on. Here, we focus on the power system. The robust stability of adaptive event-triggered load frequency control (LFC) with sliding mode control (SMC) for multiarea power systems are investigated in [4]. The authors of [15] proposed a fractional-order sliding mode controller for effectively stabilizing a nonlinear power system in a fixed time. In [16], a highly robust observer sliding mode-based load frequency and tie-power control to compensate for primary frequency control of multiarea interconnected power systems were designed. While the power system can be kept stable by sliding mode control, the modern power system relies on the critical cyber infrastructure which makes it vulnerable to hostile cyber threats.Some robust control methods are applied to the LFC of the multiarea power system. It should be pointed out that in the analysis of multiarea power system problems, many studies ignore the attacks from outside the system. There are many types of cyber-attacks that can act on the power systems, among them include denial-of-service attack, deception attack, and data integrity attack [17–21]. A deception attack is to compromise the trustworthiness of data by manipulating the packets transmitted over the communication networks. When under a deception attack, the flow of information in the system will be changed [22, 23], which may destabilize or even crash the system. We pay special attention to the deception attack in this study. In [24], a new malicious data deception attack by considering a practical attacking situation was investigated. An active synchronous detection method was proposed in [25] to detect deception attack in microgrids without impeding system operations. In [26], the authors investigated the viability of machine learning methods in detecting cyber-attacks. We want to get the information that is already damaged, and the observer-dependent event-triggered scheme is used in this study.The event-trigger control works only when a certain set of event-trigger conditions are met. Using this method can not only make effective use of communication resources but also prolong the service time of the controller. This makes it be rapidly developed in the system control, especially in dealing with complex networked dynamic systems. A properly designed event-triggering scheme may also help us find out whether the system is under malicious attack. In [27], a dynamic event-triggered mechanism was proposed to modulate the transmission of data packets. The cooperative control problem has been investigated based on a novel event-triggered scheme in [28]. The random deception attack is well handled according to a memory-based event-triggered scheme proposed in [29]. In [30], a mathematical model concerning the event-triggered load frequency control under deception attack was developed. The collaborative design of both the event-trigger strategy and the adaptive robust controller is investigated in [31]. The authors in [32] investigated the observer-based predictive event-triggered load frequency control for a smart grid incorporating electric vehicles. Benefiting from the use of an event-triggered mechanism, the negative impacts of deception attack on information transmission can be effectively detected, based on which the control module may work in time to ensure the stable operation of the system.We design an observer-based sliding mode LFC scheme for controlling multiarea power systems under deception attack. First, by taking into account the characteristic of the deception attack, a model of the attacked power system is built. Then, an observer is proposed to estimate the state of the damaged system. Furthermore, the event-trigger is introduced to judge whether the normal operation of the system is compromised using the designed event-trigger conditions. The main contributions of this study can be generalized as follows:(1)
A novel sliding mode control was proposed for the multiarea power systems under deception attack. The proposed observe-based event-trigger sliding mode control helps save the limited network channel resources, so that the desired control can be performed successfully.(2)
The sliding mode load frequency problem in an interconnected power system with a deception attack was well-analyzed. Considering the designed sliding mode control surface, the equivalent control under the basic sliding mode control is obtained through analysis. The observer and the event-triggering scheme are introduced to analyze the power system under deception attack. Using the linear matrix inequality (LMI) method, the conditions of the system’s asymptotic stability are obtained. Finally, an appropriate sliding mode control law is designed to ensure the reachability of the system trajectory.The rest of this study is organized as follows. Section2 introduces the modeling for the multiarea power system. The switching surface designed for the ith area of the power system is given in Section 3. Then, based on the designed sliding mode controller, the H∞ problem of the power system, and control law is developed to ensure the dynamic quality of sliding mode surface. Some illustrative examples of a three-area interconnected power system are analyzed to demonstrate the feasibility of the proposed approach in Section 4. Finally, Section 5 concludes our discussion.


## 2. Problem Formulation

### 2.1. Model Description

The multiarea power system is a type of the complex control system. In this study, the control problem of a sliding mode load frequency control-based power system under deception attack is analyzed. Theith area of a power system under the deception attack model is shown in Figure 1, which consists of a generator, a governor, a turbine, and so on. The meanings of the notations in Figure 1 are given in Table 1.Figure 1
The model of theith area of the power system.Table 1
Definition of notations.
ΔfiFrequency deviationΔPtieiDeviation of tie-line power exchange between areasΔPmiMechanical output deviationΔPdiLoad deviationDiDamping coefficientMiInertia of the generatorΔPviThe deviation valueRiDroop coefficientTgiTime constant of the governoruiControl inputTchiTime constant of the turbineTijSynchronization coefficient between areasBased on the system model and information transfer, the dynamic equation of the system can be obtained as follows:(1)ΔP˙mit=−ΔPmitTchi+ΔPvitTchi,ΔP˙eit=ΔPtieit+βiΔfit,ΔP˙tieit=−2π∑j=1,i≠jnTijΔfit−Δfjt,ΔP˙vit=uitTgi−ΔfitRiTgi−ΔPvitTgi−ΔPeitTgi,Δf˙it=ΔPmitMi−ΔPditMi−DiΔfitMi−ΔPtieitMi.We can derive the state equation ofith area power system as(2)x˙t=Axt+But+Fwt,yt=Cxt,where wt is the disturbance of the system and defined as wt=ΔPdi, and wt is bounded and satisfies ΔPdi<hi, where hi represent the positive constants,(3)xit=x1Tt,x2Tt,…,xnTtT,ut=u1Tt,u2Tt,…,unTtT,yit=y1Tt,y2Tt,…,ynTtT,wit=ΔPd1t,ΔPd2t,…,ΔPdntT,xt=Δfi,ΔPmi,ΔPvi,ΔPei,ΔPtieiT,where A, B, C, and F are the parameter matrixes, A=Aijn×n, B=diagB1,…,Bi︸n, C=diagC1,…,Ci︸n, and F=diagF1,…,Fi︸n.(4)A=−DiMi1Mi00−1Mi0−1Tchi1Tchi00−1RiTgi0−1Tgi00βi0001−2π∑j=1,j≠inTij0000,B=001Tgi00T,F=0000−1MiT.


### 2.2. Event-Triggered Control

The data information of the power system is transmitted by the shared network to the corresponding sliding mode controller. In this section, a sliding mode observer-based load frequency controller for theith area power systems is given. An event-triggered scheme will be put forward taking into account the consideration of the deception attack:(5)y˜t=1−αtyt+αtεt,where εt is the deception attack from outside of the system and αt is an energy-bounded signal belonging to L2∈0,+∞. αt is a stochastic variable subject to a Bernoulli distribution satisfying(6)Pαt=1=Eαt≔α0,Pαt=0=1−Eαt≔1−α0.When the deception attack happens, the information transmitted will be changed, and the system equation of state can be expressed as(7)x˙t=Axt+But+Fwt,y˜t=1−αtCxt+αtεt.Using the observed information as the transmit information, to facilitate the analysis of the problem, the state of the observed information is defined asx^t; then,(8)x^˙t=Ax^t+But+Ly˜t−y^t,y^t=1−αtCx^t.To facilitate the detection of the deception attack on the system, the event-trigger is used in the system model (1). In the absence of deception attack, a common event-triggered condition [33] is given as(9)tk+1h=tkh+mind∈ℝ+dheTjkhϕejkh≥δtkhx^Ttkhϕx^tkh,where ϕ=ηCTC, η is a positive definite matrix, and ϕ=diagϕ1,…,ϕn is defined as a weighting matrix. We denote the information error as ejkh=x^jkh−x^tkh, jkh=tkh+dh,d∈ℝ0+. δ is a predefined scalar confined in the interval 0,1.The system error isΘt=xt−x^t. Then,(10)Θ˙t=AΘt+Fwt−Lαtεt+1−αtCΘt.


### 2.3. Sliding Mode Design

To make the system working stable, the sliding mode control method is introduced. We construct the following sliding mode surface function:(11)σt=Gx^t−∫0tGA+BKx^τdτ,where G is a constant matrix, and it is selected to ensure that GB stays nonsingular, 0≤τt≤h+τmax,τmax>0.We all know that the switch surface function should satisfy(12)σt=0,tσ˙t=0.The derivative of the sliding surface is(13)σ˙t=Gx^˙t−GAx^t−GBKx^t.According to (8) and (13), we have(14)σ˙t=GAx^t+But+Ly˜t−y^t−GAx^t−GBKx^t=−GBKx^t+GBut+GLy˜t−y^t.Noticing the function (12), we have(15)−GBKx^t+GBut+GLy˜t−y^t=0.Thus, we get the equivalent controller:(16)ueq=Kx^t−GB−1GLy˜t−y^t.Then, the controlled system and the error system are reformed as(17)x^˙t=A+BKx^t+L−BGB−1GLαtεt+1−αtCΘt,Θ˙t=AΘt+Fwt−Lαtεt+1−αtCΘt.


## 2.1. Model Description

The multiarea power system is a type of the complex control system. In this study, the control problem of a sliding mode load frequency control-based power system under deception attack is analyzed. Theith area of a power system under the deception attack model is shown in Figure 1, which consists of a generator, a governor, a turbine, and so on. The meanings of the notations in Figure 1 are given in Table 1.Figure 1
The model of theith area of the power system.Table 1
Definition of notations.
ΔfiFrequency deviationΔPtieiDeviation of tie-line power exchange between areasΔPmiMechanical output deviationΔPdiLoad deviationDiDamping coefficientMiInertia of the generatorΔPviThe deviation valueRiDroop coefficientTgiTime constant of the governoruiControl inputTchiTime constant of the turbineTijSynchronization coefficient between areasBased on the system model and information transfer, the dynamic equation of the system can be obtained as follows:(1)ΔP˙mit=−ΔPmitTchi+ΔPvitTchi,ΔP˙eit=ΔPtieit+βiΔfit,ΔP˙tieit=−2π∑j=1,i≠jnTijΔfit−Δfjt,ΔP˙vit=uitTgi−ΔfitRiTgi−ΔPvitTgi−ΔPeitTgi,Δf˙it=ΔPmitMi−ΔPditMi−DiΔfitMi−ΔPtieitMi.We can derive the state equation ofith area power system as(2)x˙t=Axt+But+Fwt,yt=Cxt,where wt is the disturbance of the system and defined as wt=ΔPdi, and wt is bounded and satisfies ΔPdi<hi, where hi represent the positive constants,(3)xit=x1Tt,x2Tt,…,xnTtT,ut=u1Tt,u2Tt,…,unTtT,yit=y1Tt,y2Tt,…,ynTtT,wit=ΔPd1t,ΔPd2t,…,ΔPdntT,xt=Δfi,ΔPmi,ΔPvi,ΔPei,ΔPtieiT,where A, B, C, and F are the parameter matrixes, A=Aijn×n, B=diagB1,…,Bi︸n, C=diagC1,…,Ci︸n, and F=diagF1,…,Fi︸n.(4)A=−DiMi1Mi00−1Mi0−1Tchi1Tchi00−1RiTgi0−1Tgi00βi0001−2π∑j=1,j≠inTij0000,B=001Tgi00T,F=0000−1MiT.


## 2.2. Event-Triggered Control

The data information of the power system is transmitted by the shared network to the corresponding sliding mode controller. In this section, a sliding mode observer-based load frequency controller for theith area power systems is given. An event-triggered scheme will be put forward taking into account the consideration of the deception attack:(5)y˜t=1−αtyt+αtεt,where εt is the deception attack from outside of the system and αt is an energy-bounded signal belonging to L2∈0,+∞. αt is a stochastic variable subject to a Bernoulli distribution satisfying(6)Pαt=1=Eαt≔α0,Pαt=0=1−Eαt≔1−α0.When the deception attack happens, the information transmitted will be changed, and the system equation of state can be expressed as(7)x˙t=Axt+But+Fwt,y˜t=1−αtCxt+αtεt.Using the observed information as the transmit information, to facilitate the analysis of the problem, the state of the observed information is defined asx^t; then,(8)x^˙t=Ax^t+But+Ly˜t−y^t,y^t=1−αtCx^t.To facilitate the detection of the deception attack on the system, the event-trigger is used in the system model (1). In the absence of deception attack, a common event-triggered condition [33] is given as(9)tk+1h=tkh+mind∈ℝ+dheTjkhϕejkh≥δtkhx^Ttkhϕx^tkh,where ϕ=ηCTC, η is a positive definite matrix, and ϕ=diagϕ1,…,ϕn is defined as a weighting matrix. We denote the information error as ejkh=x^jkh−x^tkh, jkh=tkh+dh,d∈ℝ0+. δ is a predefined scalar confined in the interval 0,1.The system error isΘt=xt−x^t. Then,(10)Θ˙t=AΘt+Fwt−Lαtεt+1−αtCΘt.


## 2.3. Sliding Mode Design

To make the system working stable, the sliding mode control method is introduced. We construct the following sliding mode surface function:(11)σt=Gx^t−∫0tGA+BKx^τdτ,where G is a constant matrix, and it is selected to ensure that GB stays nonsingular, 0≤τt≤h+τmax,τmax>0.We all know that the switch surface function should satisfy(12)σt=0,tσ˙t=0.The derivative of the sliding surface is(13)σ˙t=Gx^˙t−GAx^t−GBKx^t.According to (8) and (13), we have(14)σ˙t=GAx^t+But+Ly˜t−y^t−GAx^t−GBKx^t=−GBKx^t+GBut+GLy˜t−y^t.Noticing the function (12), we have(15)−GBKx^t+GBut+GLy˜t−y^t=0.Thus, we get the equivalent controller:(16)ueq=Kx^t−GB−1GLy˜t−y^t.Then, the controlled system and the error system are reformed as(17)x^˙t=A+BKx^t+L−BGB−1GLαtεt+1−αtCΘt,Θ˙t=AΘt+Fwt−Lαtεt+1−αtCΘt.


## 3. Main Results

In this section, sufficient conditions are established for the closed-loop system (15) to be globally asymptotically stable with an L-∞ norm bound. Before presenting the main results, the following useful lemma is introduced.Definition 1.
Considering the power system (14), if wt=0 and εt=0, there exist a positive definite continuous first-order partial derivative Vx^t and negative definite EV˙x^t. One can conclude that the system is asymptotically stable. Besides, under the zero initial condition of x^t=0,t∈−τ,0, any nonzero wt,εt∈L20,+∞. The disturbance rejection level of H∞ is given as γ, and we have the following condition: Ey^t2≤γEwt2+εt2.Lemma 1.
For any real vectorsu and v and symmetric positive matrix Q with compatible dimensions, the following inequality holds:(18)uTv+vTu≤uTQu+vTQ−1v.


### 3.1. Stability Analysis

Theorem 1.
For the given conditions1>α0≥0, system (5) is asymptotically stable for the given positive constants γ and real symmetric matrices Q>0,W>0 satisfying the following matrix inequality:(19)ϖ11ϖ12ϖ12Tϖ22<0,where(20)ϖ11=ϖa−ϖb,ϖa=a7∗7,ϖb=ΓTΓ,a11=symXA+BK+Q,a13=1−α0XLC,a17=α0XL,a22=δjkhϕ,a25=−δjkhϕ,a33=symXA+W−1−α0XLC,a36=XF,a37=−α0XL,a55=δjkh−1ϕ,a66=a77=−γ2I,Γ=1−α0C01∗6,ϖ12=XB0000001−α0CTLTX003×103×103×100α0LTX,ϖ22=−GB000−1−α0X000−α0X.Proof.
Propose a Lyapunov functional(21)Vt=Vx^t+VΘt,where(22)Vx^t=x^TtXx^t+∫t−τttx^TvQx^vdv,VΘt=ΘTtXΘt+∫t−τttΘTvWΘvdv.
Calculating the derivative of (21), we have(23)V˙t=V˙x^t+V˙Θt,V˙x^t=2x^TtXx^˙t+x^TtQx^t,V˙Θt=2ΘTtXΘ˙t+ΘTtWΘt.
The expectation of (23) is(24)EV˙t=EV˙x^t+V˙Θt,for(25)EV˙x^t=2x^TtXA+BKx^t+2α0x^TtXL−BGB−1GLεt+21−α0x^TtXL−BGB−1GLCΘt+2ΘTtXFωt+x^TtQx^t,EV˙Θt=2ΘTtXAΘt−2α0ΘTtXLεt−21−α0ΘTtXLCΘt+ΘTtWΘt+eTjkhϕejkh−eTjkhϕejkh.
Notice that there exist some complex items in (24). According to Lemma 1, these complex items can be handled like(26)−2α0x^TtXBGB−1GLεt≤α0x^TtXBGB−1Gx^t+α0εTtLTXLεt.
Also, we have(27)−21−α0x^TtXBGB−1GLCΘt≤1−α0x^TtXBGB−1Gx^t+1−α0×ΘTtCTLTXLCΘt.
Take the event-triggered condition into considering(28)eTjkhϕejkh<δjkhx^t−τt−ejkhTϕx^t−τt−ejkh.
For convenience, we define a matrix as follows:(29)ξt=colx^Ttx^Tt−τtΘTtΘTt−τteTjkhωtεTt.
Then,(30)EVt+Ey^Tty^t−γ2EωTtωt+εTtεt≤ξTtϖ11−ϖ12ϖ22−1ϖ12Tξt.
Using the Schur theorem given in [34], (30) is concluded as(31)EVt≤γ2EωTtωt−y^Tty^t+εTtεt.
Integrating the deduction above, we have(32)EV+∞−EV0≤E∫0+∞−y^Tty^t+γ2ωTtωt+εTtεtdt.
Under zero initial condition, we can further obtain(33)Ey^t2≤γEωt2+Eεt2.
Whenωt=0 and εt=0, the inequality is concluded as(34)EV˙t≤−Ey^Tty^t<0.
Then, there exists a positive scalarϵ>0 that makes the following inequality holds(35)EV˙t≤−ϵx^t2.
Based on the proposed control scheme, the multiarea power system is to be robustly stable with an optimal attenuation levelγ. It completes the proof.Corollary 1.
System (5) is asymptotically stable for the given conditions α0≥0, if there exist matrixes Q>0,W>0, appropriate known positive number γ>0, real symmetric matrixes χ>0, and an arbitrary matrix Y satisfying the following LMI:(36)ϖ˜11ϖ˜12ϖ˜13∗ϖ˜2201∗3∗∗−I<0,where(37)ϖ˜11=a˜7∗7,a˜11=symAχ+BY+χQχ,a˜13=1−α0LCχ,a˜17=α0L,a˜22=δjkhϕ,a˜25=−δjkhϕ,a˜33=symAχ+χWχ−1−α0LCχ,a˜36=F,a˜37=−α0L,a˜55=δjkh−1ϕ,a˜66=a˜77=−γ2I,ϖ˜13=1−α0Cχ01∗6T,ϖ˜12=B0000001−α0χTCTNT003×103×103×100α0LT,ϖ˜22=−GB000−1−α0χ000−α0χ.Proof.
Defineχ=X−1 and Y=Kχ, pre and postmultiplying both sides of inequality (16) with diagχ,I,χ,I,I,I,I,I,χ,χ.We can notice that the matrix in Theorem2 cannot be solved directly. The reason is the nonlinear terms LCχ, χWχ, and χQχ. To deal with the nonlinear term LCχ in Theorem 2, we have added the following theorem, so that the nonlinear problem is solvable. Besides, χWχ and χQχ are defined as χWχ=W˜ and χQχ=Q˜, respectively.Theorem 2.
Based on Corollary1, for the given full rank matrix M and any proper matrix N with appropriate dimensions, (24) holds, and the following LMI:(38)ϖ˜11′ϖ˜12′ϖ˜13∗ϖ˜2201∗3∗∗−I<0,ξICTMT−XTCT∗−I<0,where ξ is approaching to 0, ϖ˜11′ and ϖ˜12′ are obtained from ϖ˜11 and ϖ˜12 by replacing LCχ with NC and replacing χWχ and χQχ with W˜ and Q˜ respectively, and the system is asymptotically stable with observe gains L=NM−1. It completes the proof.Remark 1.
Based on the proposed sliding mode load frequency control scheme, Theorem2 could guarantee the attacked multiarea power system to be robustly stable with an optimal attenuation level γ. In Theorem 2, the nonlinear coupling terms has been replaced by processable terms.


### 3.2. Control Law Design

For sliding mode control theory, a proper sliding control law is of great importance. The trajectory of the power system shall be reachable to the sliding surface in a limited time under the supplied controller. For the controlled multiarea power system (7), the corresponding reachability condition for the ith area can be described by the following theorem.Theorem 3.
For the considered sliding mode surface (9), a proper control law is given as(39)ut=Kx^t−1−αtGB−1GLCΘt−hiGB−1σit−hiGB−1GFsgnσit,where hi is a positive constant, and sgnσit is the switching function:(40)sgnσit=−1,ifσit<0,0,ifσit=0,1,ifσit>0.Proof.
The Lyapunov function is constructed as(41)VL=12σiTtσit.
From (14), (39), and (41), the derivative of VLt is(42)V˙L=12σ˙iTtσit+12σiTtσ˙it=σiTtσ˙it=σit−GBKx^+GBut+1−αtGLCΘt=σit−GBKx^+1−αtGLCΘt+GBKx^t−1−αtGB−1GLCΘt−hiGB−1σit−hiGB−1GFsgnσit=−hiσit2−hiσitGFsgnσit<−hiσit2−hiσitGF<0.
Therefore, one can conclude that the system can be kept stable by the designed control law (39).Remark 2.
The variable structure control has the advantages of fast response, being insensitive to parameter changes and disturbances. The way of sliding mode control can be designed and independent of plant parameters and disturbances. This method can be simple implemented in physical. The main disadvantage, however, is that the imperfect of all kinds of components of the actual system may contribute to causing the control switching delay. Specifically, when the state trajectory reaches the sliding surface, it is difficult to slide strictly along the sliding surface towards the equilibrium point. Instead, it may pass back and forth on both sides of the sliding surface, resulting in chattering, the error of the operation in MATLAB. Besides, the use of sliding mode control will certainly cause some buffeting phenomena.


## 3.1. Stability Analysis

Theorem 1.
For the given conditions1>α0≥0, system (5) is asymptotically stable for the given positive constants γ and real symmetric matrices Q>0,W>0 satisfying the following matrix inequality:(19)ϖ11ϖ12ϖ12Tϖ22<0,where(20)ϖ11=ϖa−ϖb,ϖa=a7∗7,ϖb=ΓTΓ,a11=symXA+BK+Q,a13=1−α0XLC,a17=α0XL,a22=δjkhϕ,a25=−δjkhϕ,a33=symXA+W−1−α0XLC,a36=XF,a37=−α0XL,a55=δjkh−1ϕ,a66=a77=−γ2I,Γ=1−α0C01∗6,ϖ12=XB0000001−α0CTLTX003×103×103×100α0LTX,ϖ22=−GB000−1−α0X000−α0X.Proof.
Propose a Lyapunov functional(21)Vt=Vx^t+VΘt,where(22)Vx^t=x^TtXx^t+∫t−τttx^TvQx^vdv,VΘt=ΘTtXΘt+∫t−τttΘTvWΘvdv.
Calculating the derivative of (21), we have(23)V˙t=V˙x^t+V˙Θt,V˙x^t=2x^TtXx^˙t+x^TtQx^t,V˙Θt=2ΘTtXΘ˙t+ΘTtWΘt.
The expectation of (23) is(24)EV˙t=EV˙x^t+V˙Θt,for(25)EV˙x^t=2x^TtXA+BKx^t+2α0x^TtXL−BGB−1GLεt+21−α0x^TtXL−BGB−1GLCΘt+2ΘTtXFωt+x^TtQx^t,EV˙Θt=2ΘTtXAΘt−2α0ΘTtXLεt−21−α0ΘTtXLCΘt+ΘTtWΘt+eTjkhϕejkh−eTjkhϕejkh.
Notice that there exist some complex items in (24). According to Lemma 1, these complex items can be handled like(26)−2α0x^TtXBGB−1GLεt≤α0x^TtXBGB−1Gx^t+α0εTtLTXLεt.
Also, we have(27)−21−α0x^TtXBGB−1GLCΘt≤1−α0x^TtXBGB−1Gx^t+1−α0×ΘTtCTLTXLCΘt.
Take the event-triggered condition into considering(28)eTjkhϕejkh<δjkhx^t−τt−ejkhTϕx^t−τt−ejkh.
For convenience, we define a matrix as follows:(29)ξt=colx^Ttx^Tt−τtΘTtΘTt−τteTjkhωtεTt.
Then,(30)EVt+Ey^Tty^t−γ2EωTtωt+εTtεt≤ξTtϖ11−ϖ12ϖ22−1ϖ12Tξt.
Using the Schur theorem given in [34], (30) is concluded as(31)EVt≤γ2EωTtωt−y^Tty^t+εTtεt.
Integrating the deduction above, we have(32)EV+∞−EV0≤E∫0+∞−y^Tty^t+γ2ωTtωt+εTtεtdt.
Under zero initial condition, we can further obtain(33)Ey^t2≤γEωt2+Eεt2.
Whenωt=0 and εt=0, the inequality is concluded as(34)EV˙t≤−Ey^Tty^t<0.
Then, there exists a positive scalarϵ>0 that makes the following inequality holds(35)EV˙t≤−ϵx^t2.
Based on the proposed control scheme, the multiarea power system is to be robustly stable with an optimal attenuation levelγ. It completes the proof.Corollary 1.
System (5) is asymptotically stable for the given conditions α0≥0, if there exist matrixes Q>0,W>0, appropriate known positive number γ>0, real symmetric matrixes χ>0, and an arbitrary matrix Y satisfying the following LMI:(36)ϖ˜11ϖ˜12ϖ˜13∗ϖ˜2201∗3∗∗−I<0,where(37)ϖ˜11=a˜7∗7,a˜11=symAχ+BY+χQχ,a˜13=1−α0LCχ,a˜17=α0L,a˜22=δjkhϕ,a˜25=−δjkhϕ,a˜33=symAχ+χWχ−1−α0LCχ,a˜36=F,a˜37=−α0L,a˜55=δjkh−1ϕ,a˜66=a˜77=−γ2I,ϖ˜13=1−α0Cχ01∗6T,ϖ˜12=B0000001−α0χTCTNT003×103×103×100α0LT,ϖ˜22=−GB000−1−α0χ000−α0χ.Proof.
Defineχ=X−1 and Y=Kχ, pre and postmultiplying both sides of inequality (16) with diagχ,I,χ,I,I,I,I,I,χ,χ.We can notice that the matrix in Theorem2 cannot be solved directly. The reason is the nonlinear terms LCχ, χWχ, and χQχ. To deal with the nonlinear term LCχ in Theorem 2, we have added the following theorem, so that the nonlinear problem is solvable. Besides, χWχ and χQχ are defined as χWχ=W˜ and χQχ=Q˜, respectively.Theorem 2.
Based on Corollary1, for the given full rank matrix M and any proper matrix N with appropriate dimensions, (24) holds, and the following LMI:(38)ϖ˜11′ϖ˜12′ϖ˜13∗ϖ˜2201∗3∗∗−I<0,ξICTMT−XTCT∗−I<0,where ξ is approaching to 0, ϖ˜11′ and ϖ˜12′ are obtained from ϖ˜11 and ϖ˜12 by replacing LCχ with NC and replacing χWχ and χQχ with W˜ and Q˜ respectively, and the system is asymptotically stable with observe gains L=NM−1. It completes the proof.Remark 1.
Based on the proposed sliding mode load frequency control scheme, Theorem2 could guarantee the attacked multiarea power system to be robustly stable with an optimal attenuation level γ. In Theorem 2, the nonlinear coupling terms has been replaced by processable terms.


## 3.2. Control Law Design

For sliding mode control theory, a proper sliding control law is of great importance. The trajectory of the power system shall be reachable to the sliding surface in a limited time under the supplied controller. For the controlled multiarea power system (7), the corresponding reachability condition for the ith area can be described by the following theorem.Theorem 3.
For the considered sliding mode surface (9), a proper control law is given as(39)ut=Kx^t−1−αtGB−1GLCΘt−hiGB−1σit−hiGB−1GFsgnσit,where hi is a positive constant, and sgnσit is the switching function:(40)sgnσit=−1,ifσit<0,0,ifσit=0,1,ifσit>0.Proof.
The Lyapunov function is constructed as(41)VL=12σiTtσit.
From (14), (39), and (41), the derivative of VLt is(42)V˙L=12σ˙iTtσit+12σiTtσ˙it=σiTtσ˙it=σit−GBKx^+GBut+1−αtGLCΘt=σit−GBKx^+1−αtGLCΘt+GBKx^t−1−αtGB−1GLCΘt−hiGB−1σit−hiGB−1GFsgnσit=−hiσit2−hiσitGFsgnσit<−hiσit2−hiσitGF<0.
Therefore, one can conclude that the system can be kept stable by the designed control law (39).Remark 2.
The variable structure control has the advantages of fast response, being insensitive to parameter changes and disturbances. The way of sliding mode control can be designed and independent of plant parameters and disturbances. This method can be simple implemented in physical. The main disadvantage, however, is that the imperfect of all kinds of components of the actual system may contribute to causing the control switching delay. Specifically, when the state trajectory reaches the sliding surface, it is difficult to slide strictly along the sliding surface towards the equilibrium point. Instead, it may pass back and forth on both sides of the sliding surface, resulting in chattering, the error of the operation in MATLAB. Besides, the use of sliding mode control will certainly cause some buffeting phenomena.


## 4. Numerical Examples

In this section, numerical examples are provided to demonstrate the effectiveness of the proposed design scheme. Taking a three-area interconnected power system as an example, the simulation results verify the effectiveness and correctness of the proposed sliding mode control scheme. By analyzing the simulation diagram, one can conclude that the system can reach a stable state under the designed sliding mode control.Parameter values of the three-area interconnected power system are presented in Table2. We consider a three-area interconnected power system. By using the MATLAB LMI control toolbox, the stability condition could be verified effectively.Table 2
Parameters of the three-area interconnected power system.
D1, D2, D31, 1.5, 1.8Tch1, Tch2, Tch30.3, 0.17, 0.2Tg1, Tg2, Tg30.37, 0.4, 0.35R1, R2, R30.05, 0.05, 0.05Beta1, beta2, beta321, 21.5, 21.8Pd1, Pd2, Pd30.1, 0.1, 0.1γ1,γ2,γ3,2, 2, 2The controller gain and observe gain can be calculated as(43)K=−223.783−5.012−3.28091.8641.795−157.956−2.073−2.59767.5331.631−160.963−2.360−2.44869.5191.510,L=0.2721.219−0.2800.075−0.1160.108−0.04−0.2240.019−0.2950.207−0.017−0.2920.063−0.135T.Under the control scheme, the frequency derivative of the power system is shown in Figure2. It can be seen that with the designed control scheme, the frequency derivative value of the deception attacked power system gradually tends to a stable state. This demonstrates that the proposed controller can effectively reduce the frequency deviation of the system, improving the robustness of the attacked power system, that is, the power system can be stable under sliding mode control, although it was attacked.Figure 2
Frequency deviation of the power system.The designed sliding surface trajectory is shown in Figure3. It can be seen from the figure that the sliding surface can reach a stable state in a certain period and keep running. One can prove that the sliding mode which we designed can work normally and satisfies our purpose.Figure 3
Designed sliding mode surface.It can be seen from Figure4 that the power system under equivalent control can operate to a stable state over time. This means the power system can work in a stable state under the provided control method.Figure 4
Equivalent control of the sliding mode-controlled power system.To ensure that the system can reach the sliding surface in a limited time, the state trajectory of the sliding mode control law is shown in Figure5. According to the curve, the simulation results can easily show the effectiveness of the proposed sliding mode load frequency control scheme.Figure 5
State of sliding mode control law.We test on the case where something is random and the event-trigger frequency and probability are equally random. Figure6 shows the event interval during the deception attack. The disorder event is shown in Figure 6, conforming to our requirements for random attacks.Figure 6
Event interval during deception attack interval.


## 5. Conclusion

This study proposed a sliding mode load frequency control for multiarea power systems under deception attack. An observer-based event-triggered transmission scheme was proposed for multiarea power systems. To be specific, an observer has been provided to gain disturbed transmission information. An event-trigger scheme is added to judge the attack is occurred or not. On the basis of the above analysis, a sliding mode controller has been constructed to keep the stability of the multiarea power system which is injured by some attack. This study has considered a kind of attack. In fact, it is possible that several attacks influence the system at the same time. The modeling of attack is also a problem worthy of consideration. In addition, with more and more renewable energy access to the power grid, its inherent randomness and uncertainty should be paid attention to. It will be a research direction in the future to consider the random disturbance and distribution caused by renewable energy into the power system.



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*Source: 8092206-2021-10-04.xml*