# High-Order Observer-Based Sliding Mode Control for the Isolated Microgrid with Cyber Attacks and Physical Uncertainties

Authors: Hao Wang; He Jiang; Yan Zhao; Huanxin Guan; Bo Hu; Shunjiang Wang
Journal: Complexity
 (2020)

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## Abstract

System security is essential for the operation of the island microgrid. However, the system security is generally threatened due to the presence of physical uncertainties and cyber attacks. In this article, a novel sliding mode load control strategy is proposed for the microgrid to mitigate cyber attacks and physical uncertainties. Firstly, a high-order disturbance observer (HODO) is designed to estimate the unmeasurable factors in the microgrid. Secondly, a HODO-based sliding mode control (SMC) strategy is proposed where the estimated value observed by the HODO is applied to the sliding mode surface and control law. It can better guarantee the security of the isolated microgrid. Then, the stability of the HODO-based SMC is demonstrated by Lyapunov stability theory. Finally, simulation results show that the proposed control strategy has excellent control performance.


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## Body

## 1. Introduction

The power system is regarded as a critical factor for economic development. With the rapid development of communication equipment, power system application, and energy management system [1–3], power systems merging primary and secondary systems are promoted to transform into the cyber-physical system (CPS). Consequently, the security of the cyber-physical power system has received widespread attention.CPS security includes the security of physical systems and cyber systems. Physical security is the security of the primary system, which can stabilize the system at scheduled operating point under physical uncertainties. Cyber security means the security of the secondary system which is vulnerable to cyber attacks [4–6]. The insecurity of the cyber-physical power system has a significant impact on the society. For example, in 2019, the primary system of Venezuelan power system became the target of an attack, resulting in a large-scale blackout. In 2003, the secondary system, computer network at the power plant, was hacked in Davis–Besse, USA. On the contrary, the security of the power grid can improve the utilization rate of clean energy power generation and enhance the reliability of the power grid [7, 8].Because of the integration of advanced measuring devices, application software, and renewable generations, the security of the power system is threatened by serious attacks [9–12]. Multitype intelligent analysis software relies on computers and communication networks, which make the system vulnerable to cyber attacks. Meanwhile, the parameters of physical equipment including generators, turbines, and transmission lines are uncertain. Currently, many critical techniques about cyber attacks, which are the major challenge in the CPS, were studied by scholars. There are some advanced resilient control technologies for cyber attacks [13–15], such as data intrusion attacks [16, 17], nontechnical loss fraud, time-delay attacks [18], and replay attacks. Liu and Li [19] proposed a load distribution attack model with the incomplete acquisition of power system information. In [20], a detection technique was studied for uncertain systems. When the power system was attacked, it can be detected and protected immediately. In [21], a control strategy to protect distributed time-delay power systems was proposed. The method of time-delay estimation was introduced to solve time-delay switch attacks. In recent years, the advanced SMC has been proposed to address the threat of cyber attacks and physical uncertainties [22–24]. In [25], SMC with the neural network observer was constructed, where the measured values were used for the control law, and it was proven to be superior in simulation. Mi et al. [26] proposed SMC based on the proportional-integral sliding mode surface, and this method proved that the microgrid can be immune to the attacks.In this paper, a control strategy is investigated. Firstly, a transformed dynamics system is established combining cyber attacks and physical uncertainties as a lumped attack. Secondly, the attack is measured by a high-order nonlinear observer where the attack and its derivatives are observed. Then, compared to the linear sliding surface, an improved sliding surface including the estimation value is proposed. By employing the estimation value, system states are forced to move to the sliding mode surface with the control law. Finally, simulation results on the isolated microgrid are carried out to verify the performance of the controller.The main contributions of the article are as follows:(1)
Considering the characteristic of the power system, the presented HODO can be used to measure the cyber attacks and physical uncertainties in the power system(2)
We construct the sliding mode surface and the control law based on the output of the HODO in the corresponding state space of the microgrid(3)
Using the proposed control strategy, the security of the microgrid will be significantly improved, especially the frequency indexThis paper is organized as follows: in Section2, the system structure of the microgrid and the dynamic equations are proposed. In Section 3, conventional SMC is illustrated. In Section 4, the control strategy is proposed. Firstly, HODO-based SMC is designed. Secondly, the stability is theoretically proved for the proposed method. The experimental simulation results are demonstrated in Section 5, while the work of this paper is summarized in Section 6.


## 2. Model of the Cyber-Physical Power System

In this paper, a typical cyber-physical system composed of the power system and controller is considered. The matrix form of the cyber-physical power system is expressed as follows:(1)x˙t=Anxt+Bnut+FnΔPd,y=C¯xt,where(2)xt=Δft,ΔPgt,ΔXgtT,An=−1TPKPTP00−1TT1TT−1RTg0−1Tg,Bn=001TgT,Fn=−KPTP00T,C¯=100T,where xt is the system state vector; An,Bn, and Fn are system matrices; Δft,ΔPgt, and ΔXgt are the deviations of frequency, power output, and governor valve position, respectively; Tp, TT, and Tg are the time constants of the power system, turbine, and governor, respectively; Kp denotes the power system, R is speed drop; and ut and ΔPdt denote the control vector and the cyber attacks, respectively. The formulated cyber-physical power system is similar to that of the literature [25].Considering the physical uncertainties of the system dynamic model, equation (1) is written as(3)x˙t=A′+ΔAxt+B′+ΔBut+F′+ΔFΔPd,y=C¯xt,where A′, B′, and F′ are the determined physical system and ΔA, ΔB, and ΔF denote the uncertainties of the physical system. Equation (4) is the detailed representation of system dynamics (3):(4)Δf˙t=−1Tp+k11Δft+KpTp+k12ΔPgt−KpTp+k13ΔPdt,ΔP˙gt=−1TT+k21ΔPgt+1TT+k22ΔXgt,ΔX˙gt=−1RTg+k31Δft+−1Tg+k32ΔXgt+1Tg+k33ut,where kmn denotes physical uncertainties.The system dynamic model with cyber attacks and physical uncertainties can be represented as follows:(5)Δf˙t=−1TpΔft+KpTpΔPgt+d1x,t,ΔP˙gt=−1TTΔPgt+1TTΔXgt+d2x,t,ΔX˙gt=−1RTgΔft−1TgΔXgt+1Tgut+d3x,u,t,where(6)d1x,t=k11Δft+k12ΔPgt−KpTp+k13ΔPdt,d2x,t=k21ΔPgt+k22ΔXgt,d3x,u,t=k31Δft+k32ΔXgt+k33ut.The matrix form of system dynamic model (5) is(7)x˙t=Axt+But+Fdd,y=C¯xt,where A=An,B=Bn,Fd=100010001, and d=d1,d2,d3T.Assumption 1.
PairA is observable.
In order to design the observer conveniently, let us transform system dynamics (5) using the transformation matrix. The structure is(8)η˙1=η2+d11,η˙2=η3+d12,η˙3=C¯A3T−1η+C¯A2Bu+d13,where η=Txt and T=C¯,C¯A,C¯A2T.
The aforementioned system is represented as(9)η˙=A¯η+B¯ut+F¯dd,where A¯=TAT−1, B¯=TB, F¯d=TFd, and d11,d12,d13T=TFdd.
For the transformed dynamic equation (9), the following assumption is necessary.Assumption 2.
The attacksd1i are continuous, and their higher-order derivative with respect to time satisfies(10)d1iqx,tdtq≤χ,q=0,1,2,3,…,r,i=1,2,3,where χ is a positive number.Remark 2.
Using a linear nonsingular transformation, system dynamic model (5) can be transformed into system (9). It should be noted that system (9) facilitates the design of HODO-based SMC. Meanwhile, η is equivalent to xt in simulation analysis.


## 3. Conventional SMC Design

In the microgrid, the conventional SMC was proposed to ensure system security through secondary frequency regulation of the generator, which adjusts the system to the normal working range with the attacks.The design of the SMC is composed of two processes: firstly, to design a sliding surface; secondly, to design the control law. The designed sliding surface drives system states to the desired equilibrium asymptotically and remain on it. The system state can be driven to the sliding surface by the designed control law after sufficient time.


### 3.1. Linear SMC

Based on system dynamic (8), the linear SMC is designed as(11)s1=C1η=∑i=13ciηi,where C1=c1,c2,c3 are constants, and c3=1. ci meets that the polynomial c1+c2p+c3p2, which is Hurwitz, such that the eigenvalues of the polynomial are less than zero.According to the literature [27], the reaching condition is chosen as s1⋅s˙1<0. The equality reaching condition is selected as follows:(12)s˙1=−kd1s1−kε1signs1,where kd1 and kε1 are positive numbers and sign⋅ is the sign function.The control law is designed based on (8), (11), and (12), which drives the system state to the sliding surface:(13)ut=−C1B¯C1A¯η+C1χ+kd1s+kε1signs.


### 3.2. Proportional-Integral SMC

The proportional-integral SMC is presented in this section. The proportional-integral sliding surface is selected as(14)s2=C1η−∫0tC1A¯−C1B¯Kηdt,where matrix K is designed as λC1A¯−C1B¯K<0.Similar to (12), we have(15)s˙2=−kd2s2−kε2signs2.The control law is designed as follows:(16)ut=−C1B¯C1B¯Kη+C1χ+kd2s2+kε2signs2.However, there are two obvious drawbacks including large overshoot and the lack of estimation for the attack.


## 3.1. Linear SMC

Based on system dynamic (8), the linear SMC is designed as(11)s1=C1η=∑i=13ciηi,where C1=c1,c2,c3 are constants, and c3=1. ci meets that the polynomial c1+c2p+c3p2, which is Hurwitz, such that the eigenvalues of the polynomial are less than zero.According to the literature [27], the reaching condition is chosen as s1⋅s˙1<0. The equality reaching condition is selected as follows:(12)s˙1=−kd1s1−kε1signs1,where kd1 and kε1 are positive numbers and sign⋅ is the sign function.The control law is designed based on (8), (11), and (12), which drives the system state to the sliding surface:(13)ut=−C1B¯C1A¯η+C1χ+kd1s+kε1signs.


## 3.2. Proportional-Integral SMC

The proportional-integral SMC is presented in this section. The proportional-integral sliding surface is selected as(14)s2=C1η−∫0tC1A¯−C1B¯Kηdt,where matrix K is designed as λC1A¯−C1B¯K<0.Similar to (12), we have(15)s˙2=−kd2s2−kε2signs2.The control law is designed as follows:(16)ut=−C1B¯C1B¯Kη+C1χ+kd2s2+kε2signs2.However, there are two obvious drawbacks including large overshoot and the lack of estimation for the attack.


## 4. Methodology

### 4.1. High-Order Observer for Cyber Attacks and Physical Uncertainties

In this section, an observer is proposed to estimate the attack [28]. In Figure 1, physical attacks kmn appear in the governing system, turbine, and power system. Meanwhile, cyber attacks ΔPdt corrupt the power system. When the system is attacked, the boundaries of the undetectable attack will be directly used in SMC without the HODO. Thus, the control is conservative. The proposed control strategy where the HODO can accomplish the detection of unknown attacks compensates this shortcoming to make the controller output more accurate.Figure 1
Control system with the observer.The HODO can estimate the attacks for system (9) as follows:(17)d^11q−1d^12q−1d^13q−1=p11qp12qp13q+Lqη1η2η3,(18)p˙11qp˙12qp˙13q=−LqA¯η+B¯ut+F¯dd^+d^11qd^12qd^13q,q=1,2,3,…,r−1,(19)d^11r−1d^12r−1d^13r−1=p11rp12rp13r+Lqη1η2η3,(20)p˙11rp˙12rp˙13r=−LrA¯η+B¯ut+F¯dd^,where Lq=l11q000l12q000l13q, q=1,2,3,…,r, are constant matrices which are necessary to select Lq>0 for the stability of the HODO; d˜1iq−1 and d^1rr−1 are estimations of d1iq−1 and d1rr−1, respectively, and p1iq are auxiliary variables i=1,2,3;q=1,2,3,…,n.The estimation errors are defined as(21)e˜=d˜11,d˜12,d˜13,d˙˜11,d˙˜12,d˙˜13,…,d˙˜11r−1,d˙˜12r−1,d˙˜13r−1,where(22)d˜1i=d1i−d^1i,d˙˜1i=d˙1i−d˙^1i,i=1,2,3,d˙˜1iq−1=dq−1d1idtq−1−d˙^1iq−1,i=1,2,3;q=3,4,5,…,n,and d˜1iq−1 is the error in the estimation of d1iq−1. From (8), (17), and (18), it follows that(23)d^˙11q−1d^˙12q−1d^˙13q−1=Lqd˜11d˜12d˜13+d^11qd^12qd^13q,q=1,2,3,…,r−1.Obviously,d˙11q−1,d˙12q−1,d˙13q−1T=d˙11q,d˙12q,d˙13qT. Subtracting both sides of equation (23) from d˙11q,d˙12q,d˙13qT yields(24)d˙11q−1−d^˙11q−1d˙12q−1−d^˙12q−1d˙13q−1−d^˙13q−1=−Lqd˜11d˜12d˜13+d11q−d^11qd12q−d^12qd13q−d^13q,(25)d˜˙11q−1d˜˙12q−1d˜˙13q−1=−Lqd˜11d˜12d˜13+d˜11qd˜12qd˜13q.From (19) and (20), we can get(26)d˜˙11r−1d˜˙12r−1d˜˙13r−1=−Lrd˜11d˜12d˜13+d11rd12rd13r.Differentiating (25) and using (26) give(27)d˜11rd˜12rd˜13r=−∑q=1rLqdr−qd˜11dtr−qdr−qd˜12dtr−qdr−qd˜13dtr−q+drd11dtrdrd12dtrdrd13dtr.Sinced11qx,t/dtq,d12qx,t/dtq,d13qx,t/dtqT is bounded as Assumption 2, the stability of estimation errors depends on the selection of matrices Lq. The HODO error dynamics can be expressed in the matrix form as(28)e˜˙=Dpe˜+Eω,where(29)Dr=−L1I30⋯0−L20I3⋯0⋮⋮⋮⋮⋮−LP−100⋯I3−LP0000,E=00⋮0I2,ω=drd11dtrdrd12dtrdrd13dtr,I3=100010001.In equation (28), the estimation error vector e˜ is illuminated in (21). The derivatives of all vectors in the estimation error vector can be calculated from (25) and (26). From (28), obviously, we can choose appropriate matrix Lqq=1,2,3,…,r such that the eigenvalues of Dr can be placed arbitrarily. Assume that Lq are designed to guarantee the eigenvalues of Dr less than zero. A positive symmetric matrix can be selected as follows:(30)DrTP+PDr=−Q.Define a Lyapunov functional, andλmin is the smallest eigenvalue; then,(31)Ve˜=e˜TPe˜.Substituting (28) into the derivative of Ve˜ becomes(32)V˙e˜=e˜˙TPe˜+e˜TPe˜˙=e˜˙TDpTP+PDpe˜+2e˜TPEω≤−e˜TQe˜+2PEe˜ω≤−λmine˜2+2PEe˜ω=−e˜λmine˜−2PEω.Consequently, for (8), (25), and (26), after sufficiently long time, the norm of the estimation error is ultimately bounded by(33)e˜≤2PEωλmin.When the error state trajectory enters into the closed ball centered ate˜=0 with radius φ and the smallest eigenvalue λmin>0, the Lyapunov function satisfies V˙e˜<0. It implies that the estimation error system is stable. The bound of the estimation error can be lowered by the appropriate choice of the parameter Lqq=1,2,3,…,r.


### 4.2. SMC Based on the HODO

When the system is attacked, SMC is an effective control strategy to guarantee the security of the system. However, the shortcoming is that the conventional SMC method would bring some adverse effects such as introducing overshoot. In this section, a HODO-based SMC is presented. It should be noted that SMC and HODO are designed, respectively.To be immune to attacks, a linear sliding surface based on the HODO is applied to improve the stability of the power system.The sliding surface is selected as follows:(34)s∗=∑i=13ciηi+c2d^11+d˙^11+d^12.Theorem 1.
Using HODO (17)–(20) and the designed controller law (35), all the states of (9) are ultimately bounded; therefore, the closed-loop system is asymptotically stable around equilibrium with the following control law:(35)u=−1CA2Bc1η2+d^11+c2η3+d^12+C¯A3T−1η+d^13+ς,where(36)ς=kds∗+kεsigns∗,kε=c1+c2l111+l112+c2+l121+4e˜1+3χ+θ,and kd and θ are positive constants.Proof.
Construct a Lyapunov candidate function as(37)Vs∗=12s∗2.
It obviously elicits(38)V˙s∗=s∗s˙∗.
From (9) and (34), it follows that(39)s˙∗=c1η˙1+c2η˙2+η˙3+c2d^˙11+d˙^˙11+d^˙12−ζ.
Inserting (35) into (39), we obtain(40)s˙∗=c1+c2l111+l112d˜11+c2+l121d˜12+d˜13+d˙^11+d˙^12+d¨^11−ζ.
According to (24), we have(41)d^˙11=l111d˜11+d˙^11d^˙12=l121d˜12+d˙^12d^˙13=l131d˜13+d˙^13d˙^˙11=l112d˜11+d¨^11d˙^˙12=l122d˜12+d¨^12d˙^˙13=l132d˜13+d¨^13.
Then, substituting (41) to (40), we have(42)s˙∗=c1+c2l111+l112d˜11+c2+l121d˜12+d˜13+d˙11−d˙˜11+d˙12−d˙˜12+d¨11−d¨˜11−ζ.
Substituting (42) into (38), it follows that(43)V˙s∗=s∗s˙∗≤−kεs∗−kds∗2+c1+c2l111+l112+c2+l121+4e˜1s∗+3χs∗<−θs∗−kds∗2<0.
According to (21), e˜1=d˜11,d˜12,d˜13,d˙˜11,d˙˜12,d¨˜11 is bounded as follows:(44)e˜1<e˜≤2PEωλm.
This completes the proof.
It should be noted thatCA2B is invertible. The control block diagram of the proposed HODO-based SMC is shown in Figure 2.
Figure2 briefly illuminates the steps of the HODO-based SMC. Firstly, the system dynamics is obtained, and coordinate transformation is performed. Secondly, the transformed coordinate is applied in the HODO. Thirdly, the estimated values calculated by the HODO work on the sliding mode surface. Finally, the calculated control law obtained from the sliding mode surface has an effect on the power system and the HODO.Figure 2
Block diagram of the system with the proposed SMC.


## 4.1. High-Order Observer for Cyber Attacks and Physical Uncertainties

In this section, an observer is proposed to estimate the attack [28]. In Figure 1, physical attacks kmn appear in the governing system, turbine, and power system. Meanwhile, cyber attacks ΔPdt corrupt the power system. When the system is attacked, the boundaries of the undetectable attack will be directly used in SMC without the HODO. Thus, the control is conservative. The proposed control strategy where the HODO can accomplish the detection of unknown attacks compensates this shortcoming to make the controller output more accurate.Figure 1
Control system with the observer.The HODO can estimate the attacks for system (9) as follows:(17)d^11q−1d^12q−1d^13q−1=p11qp12qp13q+Lqη1η2η3,(18)p˙11qp˙12qp˙13q=−LqA¯η+B¯ut+F¯dd^+d^11qd^12qd^13q,q=1,2,3,…,r−1,(19)d^11r−1d^12r−1d^13r−1=p11rp12rp13r+Lqη1η2η3,(20)p˙11rp˙12rp˙13r=−LrA¯η+B¯ut+F¯dd^,where Lq=l11q000l12q000l13q, q=1,2,3,…,r, are constant matrices which are necessary to select Lq>0 for the stability of the HODO; d˜1iq−1 and d^1rr−1 are estimations of d1iq−1 and d1rr−1, respectively, and p1iq are auxiliary variables i=1,2,3;q=1,2,3,…,n.The estimation errors are defined as(21)e˜=d˜11,d˜12,d˜13,d˙˜11,d˙˜12,d˙˜13,…,d˙˜11r−1,d˙˜12r−1,d˙˜13r−1,where(22)d˜1i=d1i−d^1i,d˙˜1i=d˙1i−d˙^1i,i=1,2,3,d˙˜1iq−1=dq−1d1idtq−1−d˙^1iq−1,i=1,2,3;q=3,4,5,…,n,and d˜1iq−1 is the error in the estimation of d1iq−1. From (8), (17), and (18), it follows that(23)d^˙11q−1d^˙12q−1d^˙13q−1=Lqd˜11d˜12d˜13+d^11qd^12qd^13q,q=1,2,3,…,r−1.Obviously,d˙11q−1,d˙12q−1,d˙13q−1T=d˙11q,d˙12q,d˙13qT. Subtracting both sides of equation (23) from d˙11q,d˙12q,d˙13qT yields(24)d˙11q−1−d^˙11q−1d˙12q−1−d^˙12q−1d˙13q−1−d^˙13q−1=−Lqd˜11d˜12d˜13+d11q−d^11qd12q−d^12qd13q−d^13q,(25)d˜˙11q−1d˜˙12q−1d˜˙13q−1=−Lqd˜11d˜12d˜13+d˜11qd˜12qd˜13q.From (19) and (20), we can get(26)d˜˙11r−1d˜˙12r−1d˜˙13r−1=−Lrd˜11d˜12d˜13+d11rd12rd13r.Differentiating (25) and using (26) give(27)d˜11rd˜12rd˜13r=−∑q=1rLqdr−qd˜11dtr−qdr−qd˜12dtr−qdr−qd˜13dtr−q+drd11dtrdrd12dtrdrd13dtr.Sinced11qx,t/dtq,d12qx,t/dtq,d13qx,t/dtqT is bounded as Assumption 2, the stability of estimation errors depends on the selection of matrices Lq. The HODO error dynamics can be expressed in the matrix form as(28)e˜˙=Dpe˜+Eω,where(29)Dr=−L1I30⋯0−L20I3⋯0⋮⋮⋮⋮⋮−LP−100⋯I3−LP0000,E=00⋮0I2,ω=drd11dtrdrd12dtrdrd13dtr,I3=100010001.In equation (28), the estimation error vector e˜ is illuminated in (21). The derivatives of all vectors in the estimation error vector can be calculated from (25) and (26). From (28), obviously, we can choose appropriate matrix Lqq=1,2,3,…,r such that the eigenvalues of Dr can be placed arbitrarily. Assume that Lq are designed to guarantee the eigenvalues of Dr less than zero. A positive symmetric matrix can be selected as follows:(30)DrTP+PDr=−Q.Define a Lyapunov functional, andλmin is the smallest eigenvalue; then,(31)Ve˜=e˜TPe˜.Substituting (28) into the derivative of Ve˜ becomes(32)V˙e˜=e˜˙TPe˜+e˜TPe˜˙=e˜˙TDpTP+PDpe˜+2e˜TPEω≤−e˜TQe˜+2PEe˜ω≤−λmine˜2+2PEe˜ω=−e˜λmine˜−2PEω.Consequently, for (8), (25), and (26), after sufficiently long time, the norm of the estimation error is ultimately bounded by(33)e˜≤2PEωλmin.When the error state trajectory enters into the closed ball centered ate˜=0 with radius φ and the smallest eigenvalue λmin>0, the Lyapunov function satisfies V˙e˜<0. It implies that the estimation error system is stable. The bound of the estimation error can be lowered by the appropriate choice of the parameter Lqq=1,2,3,…,r.


## 4.2. SMC Based on the HODO

When the system is attacked, SMC is an effective control strategy to guarantee the security of the system. However, the shortcoming is that the conventional SMC method would bring some adverse effects such as introducing overshoot. In this section, a HODO-based SMC is presented. It should be noted that SMC and HODO are designed, respectively.To be immune to attacks, a linear sliding surface based on the HODO is applied to improve the stability of the power system.The sliding surface is selected as follows:(34)s∗=∑i=13ciηi+c2d^11+d˙^11+d^12.Theorem 1.
Using HODO (17)–(20) and the designed controller law (35), all the states of (9) are ultimately bounded; therefore, the closed-loop system is asymptotically stable around equilibrium with the following control law:(35)u=−1CA2Bc1η2+d^11+c2η3+d^12+C¯A3T−1η+d^13+ς,where(36)ς=kds∗+kεsigns∗,kε=c1+c2l111+l112+c2+l121+4e˜1+3χ+θ,and kd and θ are positive constants.Proof.
Construct a Lyapunov candidate function as(37)Vs∗=12s∗2.
It obviously elicits(38)V˙s∗=s∗s˙∗.
From (9) and (34), it follows that(39)s˙∗=c1η˙1+c2η˙2+η˙3+c2d^˙11+d˙^˙11+d^˙12−ζ.
Inserting (35) into (39), we obtain(40)s˙∗=c1+c2l111+l112d˜11+c2+l121d˜12+d˜13+d˙^11+d˙^12+d¨^11−ζ.
According to (24), we have(41)d^˙11=l111d˜11+d˙^11d^˙12=l121d˜12+d˙^12d^˙13=l131d˜13+d˙^13d˙^˙11=l112d˜11+d¨^11d˙^˙12=l122d˜12+d¨^12d˙^˙13=l132d˜13+d¨^13.
Then, substituting (41) to (40), we have(42)s˙∗=c1+c2l111+l112d˜11+c2+l121d˜12+d˜13+d˙11−d˙˜11+d˙12−d˙˜12+d¨11−d¨˜11−ζ.
Substituting (42) into (38), it follows that(43)V˙s∗=s∗s˙∗≤−kεs∗−kds∗2+c1+c2l111+l112+c2+l121+4e˜1s∗+3χs∗<−θs∗−kds∗2<0.
According to (21), e˜1=d˜11,d˜12,d˜13,d˙˜11,d˙˜12,d¨˜11 is bounded as follows:(44)e˜1<e˜≤2PEωλm.
This completes the proof.
It should be noted thatCA2B is invertible. The control block diagram of the proposed HODO-based SMC is shown in Figure 2.
Figure2 briefly illuminates the steps of the HODO-based SMC. Firstly, the system dynamics is obtained, and coordinate transformation is performed. Secondly, the transformed coordinate is applied in the HODO. Thirdly, the estimated values calculated by the HODO work on the sliding mode surface. Finally, the calculated control law obtained from the sliding mode surface has an effect on the power system and the HODO.Figure 2
Block diagram of the system with the proposed SMC.


## 5. Simulation Results with SMC

In this section, the proposed control strategy is analyzed by several numerical simulations. As we all know, in the cyber-physical power system, frequency deviation is the most important security index. Consequently, the frequency deviation is mainly discussed in the simulation.In order to verify the effectiveness of the studied control strategy for the CPS, comparative results between the HODO-based SMC and the conventional SMC are simulated and analyzed. The system parameters for simulation are listed in Table1 [23]. In this simulation, the second- and the third-order observer are analyzed for the effect of this control strategy.Table 1
The parameters of power systems.
ParametersRKPTPTTTGValues2.7112.5250.330.072By using the HODO based on equations (17)–(20), the second- and third-order disturbance observers are expressed as follows.The second-order disturbance observer:(45)d^1i=p1i1+l1i1ηi,p˙1i1=−l1i1ηi+1+d^1i+d˙^1i,d˙^1i=p1i2+l1i2ηi,p˙1i2=l1i2ηi+1+d^1i,d^13=p131+l131η3,p˙131=−l131−77.4η1−42.8η2−17η3+189.4u+d^13+d˙^13,d˙^13=p132+l132η3,p˙132=l132−77.4η1−42.8η2−17η3+189.4u+d^13.The third-order disturbance observer:(46)d^13=p131+l131η3,p˙131=−l131−77.4η1−42.8η2−17η3+189.4u+d^13+d˙^13,d˙^13=p132+l132η3,p˙132=l132−77.4η1−42.8η2−17η3+189.4u+d^13+d¨^13,d¨^13=p133+l133η3,p˙133=l133−77.4η1−42.8η2−17η3+189.4u+d^13.The control parameters and the initial variables are selected as(47)c1=8,c2=3,c3=1,kd=kd1=kd2=100,kε=kε1=kε2=0.01,x0=0.1,0,0T.


### 5.1. Step Cyber Attack

In this case, a step cyber attack is applied without physical uncertainties. Cyber attack is executed to the microgrid, which is 0.1 pu. And the cyber attack is added at the initial time and ends at 5 s. The attack boundary isχ=0.1.The values are considered as follows:Parameter 1 (P1):l111=l121=l131=10000andl112=l122=l132=1500Parameter 2 (P2):l111=l121=l131=10000andl112=l122=l132=1000Parameter 3 (P3):l111=l121=l131=10000andl112=l122=l132=500Parameter 4 (P4):l111=l121=l131=10000andl112=l122=l132=200We can see that the proposed SMC can ensure system security (especially, frequency deviation). The simulation result is presented in Figure3. Moreover, it has a small overshoot compared with the conventional SMC at the initial time.Figure 3
Frequency deviation with the step load attack.An evaluation index based on frequency deviation is employed to demonstrate HODO’s control performance, which is(48)fD=∫0tΔfdt.Table2 shows the evaluation index of frequency deviation between the conventional SMC and the second-order disturbance observer with different parameters. It can be concluded from Table 2 that the adjustment ability of the second-order disturbance observer is superior to the conventional SMC. Furthermore, the selection of the parameter in the second-order disturbance observer has an improvement on the overshoot.Table 2
The evaluation index of frequency deviation.
Linear SMCProportional-integral SMCP1P2P3P4fD0.24230.59770.06610.06420.06250.0617


### 5.2. Random Cyber Attacks

In this case, the designed SMC with the second-order disturbance observer and third-order disturbance observer is tested in the microgrid. The random external attack is injected into the microgrid, which isΔPdt=0.2sint and ends at 15 s. The parameters of the disturbance observer are as follows:(a)
The second-order disturbance observer:(49)l111=l121=l131=500,l112=l122=l132=100.(b)
The third-order disturbance observer:(50)l111=l121=l131=500,l112=l122=l132=100,l113=l123=l133=25.The plot of the estimation valuesΔP^dt is shown in Figure 4. The disturbance can be tracked in 8 s, while the estimated value accurately estimates the reference disturbance after 10 s. The frequency deviation with the HODO-based SMC is presented in Figure 5.Using equation (48), we get(51)fD2=∫2022Δfdt=0.007,fD3=∫2022Δfdt=0.003,wherefD2 and fD3 represent the evaluation index of the second- and third-order disturbance observer.It can be concluded that it quickly converges to 0 with the third-order disturbance observer, compared with the effect of the second-order disturbance observer.Figure 4
Actual and estimated attack.Figure 5
Frequency deviation with the disturbance observer.


### 5.3. Physical Uncertainties with Cyber Attacks

The stochastic step attack is executed to the microgrid (see Figure6). Adjust the uncertainty time constant TT, the governor time constant Tg, and the speed drop R to 0.38, 0.08, and 2.5 in 20 s, respectively. System uncertainties and cyber attacks occur simultaneously between 15 and 20 seconds. When physical uncertainties appear in the secondary system, the system variables of the microgrid are gradually stable within the limited time by using HODO-based SMC, which are shown in Figure 7.Figure 6
Stochastic attack.Figure 7
State deviation (frequency, power output, and governor valve position) of the microgrid.


## 5.1. Step Cyber Attack

In this case, a step cyber attack is applied without physical uncertainties. Cyber attack is executed to the microgrid, which is 0.1 pu. And the cyber attack is added at the initial time and ends at 5 s. The attack boundary isχ=0.1.The values are considered as follows:Parameter 1 (P1):l111=l121=l131=10000andl112=l122=l132=1500Parameter 2 (P2):l111=l121=l131=10000andl112=l122=l132=1000Parameter 3 (P3):l111=l121=l131=10000andl112=l122=l132=500Parameter 4 (P4):l111=l121=l131=10000andl112=l122=l132=200We can see that the proposed SMC can ensure system security (especially, frequency deviation). The simulation result is presented in Figure3. Moreover, it has a small overshoot compared with the conventional SMC at the initial time.Figure 3
Frequency deviation with the step load attack.An evaluation index based on frequency deviation is employed to demonstrate HODO’s control performance, which is(48)fD=∫0tΔfdt.Table2 shows the evaluation index of frequency deviation between the conventional SMC and the second-order disturbance observer with different parameters. It can be concluded from Table 2 that the adjustment ability of the second-order disturbance observer is superior to the conventional SMC. Furthermore, the selection of the parameter in the second-order disturbance observer has an improvement on the overshoot.Table 2
The evaluation index of frequency deviation.
Linear SMCProportional-integral SMCP1P2P3P4fD0.24230.59770.06610.06420.06250.0617


## 5.2. Random Cyber Attacks

In this case, the designed SMC with the second-order disturbance observer and third-order disturbance observer is tested in the microgrid. The random external attack is injected into the microgrid, which isΔPdt=0.2sint and ends at 15 s. The parameters of the disturbance observer are as follows:(a)
The second-order disturbance observer:(49)l111=l121=l131=500,l112=l122=l132=100.(b)
The third-order disturbance observer:(50)l111=l121=l131=500,l112=l122=l132=100,l113=l123=l133=25.The plot of the estimation valuesΔP^dt is shown in Figure 4. The disturbance can be tracked in 8 s, while the estimated value accurately estimates the reference disturbance after 10 s. The frequency deviation with the HODO-based SMC is presented in Figure 5.Using equation (48), we get(51)fD2=∫2022Δfdt=0.007,fD3=∫2022Δfdt=0.003,wherefD2 and fD3 represent the evaluation index of the second- and third-order disturbance observer.It can be concluded that it quickly converges to 0 with the third-order disturbance observer, compared with the effect of the second-order disturbance observer.Figure 4
Actual and estimated attack.Figure 5
Frequency deviation with the disturbance observer.


## 5.3. Physical Uncertainties with Cyber Attacks

The stochastic step attack is executed to the microgrid (see Figure6). Adjust the uncertainty time constant TT, the governor time constant Tg, and the speed drop R to 0.38, 0.08, and 2.5 in 20 s, respectively. System uncertainties and cyber attacks occur simultaneously between 15 and 20 seconds. When physical uncertainties appear in the secondary system, the system variables of the microgrid are gradually stable within the limited time by using HODO-based SMC, which are shown in Figure 7.Figure 6
Stochastic attack.Figure 7
State deviation (frequency, power output, and governor valve position) of the microgrid.


## 6. Conclusion

In this paper, a HODO-based SMC is employed to guarantee the security of the cyber-physical power system. Firstly, HODO is applied to measure cyber attacks and physical uncertainties with matching and unmatching. Secondly, the SMC with the estimated value obtained by the HODO effectively stabilizes the system, furthermore, as compared with the conventional SMC and the proposed control strategy, and the advantage of the HODO-based SMC is small overshoot. In future, further research will be extended to the power-interconnected and time-delay system.



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*Source: 6662638-2020-12-15.xml*