# Security-Based Asynchronous Control for Discrete-Time Markov Switching Power Systems⋆

Authors: Weifeng Xu; Bin Yu; Liguo Weng; Jianguo Zhou
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper addresses the asynchronous control problem for power systems subject to abrupt variations and cyber-attacks. In the sequel, the transient faults of circuit breakers can be described as the Markov process. In light of these situations, the power systems are transmitted to discrete-time Markov switching systems. Meanwhile, the deception attacks with time-varying delays in dispatchers are regulated by a Markov process. The controller and dispatcher are mode-dependent and their modes are nonsynchronous with those of power systems, which are modeled by the hidden Markov models. On the basis of the deception attacks, sufficient conditions are presented to guarantee the stochastic mean-square stability of the closed-loop dynamic. Finally, the proposed control design strategy is testified via a simulation result.


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## Body

## 1. Introduction

As a type of complex nonlinear system, power systems have gained considerable interest due to their spontaneous oscillation characters and high penetration [1–3]. Over the past decades, many significant methods have been devoted to power systems, such as state estimation [4–8]. For the purpose of stabilizing interconnected power systems, many efforts have been devoted to exploring the mismatch between load demand and total power generation. Following this character, many techniques are forwarded to maintain the frequency balance including the load frequency control, state feedback control [9–11]. Therefore, how to keep the frequency deviation of power systems within a certain range remains a hot topic. For instance, to overcome the low/zero inertia, stability has been studied for power systems with fluctuation and intermittency in [12]. In [13], the load frequency control technique has been utilized to balance the power exchanges among different areas. In [14], the network-induced time delay has been considered in supplying high-quality electric energy. In [15], the fuzzy-dependent power system stabilizer with uncertain factors has been investigated.In reality, owing to many unexpected factors such as component faults, external disturbances, and unknown attacks, the power systems always experience random variations in parameters/structures, which lead to the resulting operation changes and performance degeneration [16, 17]. Markov switching is identified as an effective tool in modeling the aforementioned conditions [18]. Note that Markov switching is ubiquitous, which has been applied in many physical situations, such as tunnel-diode-circuit-model and complex networks [18–20]. However, to our knowledge, little attention has been given to Markov switching power systems (MSPSs) except for [21–23]. In [21], the random switching of power systems can be modeled as MSPSs. Lately, the continuous-time interconnected multiarea MSPSs with load frequency control are considered in [22]. In [23], the discrete-time MSPSs are studied with a hidden Markov model. Nevertheless, in contrast with the fruitful achievements of power systems, MSPSs have not gained suitable attention.On the other hand, many valuable results have been reported on the networked power systems subjected to many network-induced factors, such as communication delays, packet losses, quantization effects, and event-triggered protocols [24–26]. By comparison, potential cyber-attacks may destroy the stability of power systems via a shared communication network. In light of the different attack ways, cyber-attacks can be summarized into three categories: denial-of-service [27], repeat attack [28, 29], and deception attacks [30–33]. In more detail, the former aims at destroying the channel of signal exchange, and repeat attacks inject historic data into the network affect the performance, while deception attacks attempt to inject false data into the communication network to destroy the data trustworthiness. On the basis of these cyber-attacks, the conventional control law becomes untrusted. It is of significance to be concerned with cyber-attacks. Among these cyber-attacks, deception attacks are common in practice. For instance, in [34], the unified power systems against random-occurring deception attacks have been studied. In [35], on the basis of credibility, the multiarea power systems with deception attacks have been well concerned. Nevertheless, the security issues of power systems with regard to cyber-attacks have not been adequately explored, such as random occurring deception attacks with time-varying delays, not to mention power systems against Markov switching.Through expounds of the above discussion, this work shall consider the nonsynchronous controller design issue for MSPSs with cyber-attacks. The main contributions are listed below: (1) a generalized MSPS is established, which covers asynchronous dispatcher, asynchronous controller, and deception attacks, simultaneously. (2) Deception attacks with time-varying delays are taken into consideration. The probability of each time-varying delay is different, and random-occurring deception attacks are described by a sequence of stochastic variables induced by a new Markov process. (3) A more general scenario is that the asynchronous phenomena among system mode, controller mode, and dispatcher mode are well revealed and the hidden Markov model technique is applied. Finally, the effectiveness of the gained methodology is verified via an illustrated example.Notations: the notations in this paper is standard.ℕ symbols the set of all non-negative integers. diag⋯ means a block-diagonal matrix. HeZ=Z+Z⊤. ⋅ implies the Euclidean norm of a vector. ℝn signifies the n dimensional real space. E⋅ represents the mathematical expectation. ∗ describes the symmetric term.


## 2. Problem Formulations

As sketched in Figure1, a type of single-machine infinite bus (SMIB) through tie line is explored in the current study. From the SMIB, we can observe the dynamic behavior of large interconnected power systems. Following this trend, the basic components of SMIB power systems (SMIBPSs) are expressed in Figure 2. Accordingly, the following formula can be established:(1)δ˙=ω0ω,ω˙=Tm−TeM,Tdo′E˙q′=Efd−xd+xexd′+xeEq′+xd+xd′xd′+xeVcosδ,TEE˙fd=KEEref−KEVt+KEu−Efd,where δ, ω, Eq′, and Efd refer to the generator angle, generator speed, q-axis voltage, and generator voltage, respectively. Meanwhile, other physical meanings are summarized in Table 1.Figure 1
Signal model of SMIBPSs.Figure 2
Basic components of SMIBPSs with faults.Table 1
The parameters of power systems.
xd′Generatord-axis transient reactancexd,xqd−q axis synchronous reactanceTe,TmElectrical, mechanical torquekE,TEExciter gain and time constantVInfinite busbar voltageVtTerminal voltageP,QReal and reactive power loadingsMInertia coefficientTdo′Open circuitd-axis transient time constantuStabilizing signalxeExternal reactanceΔwSpeed deviationk1,…,k6Linearized model constants of the synchronous machineIn view of the aforementioned observation, the fourth-order state-space model of SMIBPS is formulated as(2)x˙t=Axt+But,where(3)xt=ΔδΔωΔEq′ΔEfd,A=0ω000−k1M0−k2M0−k4Tdo′0−1k3Tdo′−1Tdo′−k5kETE0−k6kETE−1TE,B=000kETE.Similar to the work of [10], Q and P, respectively, represent the reactive and real power loading, whose functions are presented by the parameters kll=1,…,6. More specifically, the values of parameters in Table 1 are given by xq=1.55pu, xd=1.6pu, xd′=0.32pu, Tdo′=6s, M=10s, xe=0.4pu, ω0=50πrad/s, TE=0.05s, kE=50, and Ll=0.8l=1,2.By resorting to a discretization periodT, the discrete-time SMIBPS (2) can be established as(4)xk+1=Axk+Buk,where A=eAT and B=∫0TeAtBdt.In light of the unreliability of the network medium, the abrupt variations of SMIBPS cannot be avoided. To model the variation of SMIBPS in a suitable way, a stochastic variableσk takes values within a space ℳ=1,…,ℳ, is presented to depict the Markov switching SMIBPSs as follows:(5)xk+1=Aσkxk+Bσkuk,k=0,1,…where σk refers to a Markov chain, whose transition probability matrix (TPM) Π=πmnℳ×ℳ is inferred as(6)πmn=Prσk+1=n|σkm,where 0≤πmn≤1 and ∑n=1ℳπmn=1 for all m,n∈ℳ=1,2,…,ℳ.Notice that the data are transmitted to controllers via an unencrypted communication network, which is always being attacked on the sensor-to-controller channels. It is well-known that deception attacks are commonly encountered, which launch some deception signals to destroy the information authenticity ofxk. Thus, as depicted in Figure 1, we consider the random occurring deception attacks in power systems, which damage/destroy the performance to the data integrity. Therefore, the real system information is modeled as(7)x¯k=xk+αεkkζk,where ζk=−xk+fxk−τεkk. τεkk stand for the jump-mode-dependent time-varying delay of deception attacks, and τεkk∈τ1,τ2. τ1 and τ2 are two constants, which satisfies 0≤τ1≤2. fxk=fx1k−τεkk,fx2k−τεkk,…,fxnk−τεkk⊤ being the nonlinear function of deception attacks subject to random occurring time-varying delays. αεkk implies the Bernoulli variable, in which αεkk=1 and αεkk=0 signify the transmission channel with and without attack. It yields(8)Eαεkk=1=α¯εk,Eαεkk=0=1−α¯εk,Eαεkk−α¯εk2=α¯εk1−α¯εk=α^εk2.In particularly,εk is a Markov process having values over a set S=1,…,S, whose TPM Ω=ψmsℳs with(9)ψms=Prεk=s|σk=m,where 0≤ψms≤1 and ∑s=1Sψms=1 for any m∈ℳ and s∈S.Accordingly, the actual system information can be reformulated as(10)x¯k=1−αεkkxk+αεkkfxk−τεkk.Remark 1.
In contrast to the reporting literature with mode-independent deception attacks [30–33], the deception attacks have the Markov behavior, and the attack indication scalar αεkk is presented to describe the dynamic behavior. Meanwhile, in the current study, time-varying delays against Markov behavior are considered in the deception attacks, which covers the existing deception attacks as special cases [30–33].Assumption 1.
(see [36]). The embedded function fxk−τεkk, which is adopted to restrain deception attacks with random occurring time-varying delays, satisfies the following condition:(11)fxk−τεkk2≤Qxk−τεkk2,whereQ indicates a known matrix implying an upper bound of embedded function fxk−τεkk.It is noteworthy that the resulting mode information determines the controller design and effects the performance. With respect to the mode information that cannot be observed when an attack occurs, in the current study, an asynchronous controller was developed as follows:(12)uk=Kθkx¯k,where Kθk being the controller gains. Stochastic variable θk implies a Markov chain having values in a space D=1,2,…,D, whose TPM Ξ=φmdℳD with(13)φmd=Prθk=d|σk=m,where 0≤φmd≤1 and ∑d=1Dφmd=1 for any m∈ℳ and d∈D.Letσk=m∈ℳ, εk=s∈S, and θk=d∈D, combining (5), (10), and (12), we have(14)xk+1=Amsdxk+α¯sBmKdfxk−τsk+αsk−α¯sBmKd−xk+fxk−τsk,where Amsd=A+1−α¯sBmKd.Remark 2.
Actually, the mode information of the SMIBPSs is difficult to achieve due to many factors, including higher costs and time-wasting. In order to describe the dynamic behavior of random occurring deception attacks and control laws, the hidden Markov models are adopted to model these asynchronous phenomena. More specifically, we get the two-independent conditional probabilities as follows:(15)Prεk=s,θk=d|σk=m=Prεk=s|σk=m×Prθk=d|σk=m=ψmsφmd.For the convenience of presentation, the following definition is recalled:Definition 1.
(see [37]). The closed-loop dynamic (14) is stochastically mean-square stable (SMSS), if under any initial condition αt0, φt0∈ℒ, and υt0∈ℳ, such that(16)E∑0∞xk2|xk0,σk0<∞.


## 3. Main Results

In the current section, the SMSS criteria for the closed-loop dynamic (14) and controller design method will be established in Theorem 1 and Theorem 2, respectively.Theorem 1.
For given scalarsτ2≥τ1≥0, andα¯ss∈S., and gain matrixKd, the closed-loop dynamic (14) is SMSS, if there exist matricesPm>0m∈ℳ,R>0, such that for anym,n∈ℳ,s∈S,d∈D,(17)∑s=1Sψms∑d=1DφmdUmsd−1<Pm−1,(18)Σ^msd=Υmsd1+Υmsd2⊤PmΥmsd2+α^s2Υmsd3⊤PmΥmsd3<0,where(19)Υmsd1=diag−Umsd−1+τ2−τ1+1R,−R+α¯sQ⊤Q,−α¯sI,Υmsd2=Am+1−α¯sBmKd0α¯sBmKd,Υmsd3=−BmKd0BmKd.Proof.
Let us construct the following Lyapunov functional:(20)Vxk,σk=∑ι=13Vιαt,φt,υt,where(21)V1xk,σk=x⊤kPσk−1xk,V2xk,σk=∑δ=k−τskk−1x⊤δRxδ,V3xk,σk=∑l=−τ2+2−τ1+1∑δ=k+l−1k−1x⊤δRxδ.
Calculating the derivation ofVxk,σk along the trajectories (14), yields(22)EΔVk=EVxk+1,σk+1=n|xk,σk=m−Vxk,σk=Ex⊤k+1Pmxk+1−x⊤kPm−1xk=Amsdxk+α¯sBmKdfk−τsk⊤∑s=1Sψms∑dDφmdPm×Amsdxk+α¯sBmKdfk−τsk+α^s2BmKd−xk+fk−τsk⊤∑s=1Sψms∑dDφmdPm×BmKd−xk+fk−τsk−x⊤kPm−1xk.(23)EΔV2k=EV2xk+1,σk+1−V2xk,σk≤x⊤kRxk−x⊤k−τskRxk−τsk+∑k+1−τ2k−τ1x⊤δRxδ.(24)EΔV3k=EV3xk+1,σk+1−V3xk,σk≤τ2−τ1x⊤kRxk−∑k+1−τ2k−τ1x⊤δRxδ.
Recalling Assumption1, the following formula can be acquired:(25)−α¯sx⊤k−τskQ⊤Qxk−τsk+α¯sf⊤k−τskfk−τsk≤0.
If follows from (20)–(25), it can be obtained that(26)EΔVk=EΔV1k+EΔV2k+EΔV3k=Eη⊤k∑s=1Sψms∑d=1DφmdΣ^msdηk−x⊤kPm−1xk≤Eη⊤k∑s=1Sψms∑d=1DφmdΣ¯msdηk−x⊤kPm−1xk=Ex⊤k∑s=1Sψms∑d=1DφmdUmsd−1−Pm−1xk,where η⊤k=x⊤kx⊤k−τskf⊤xk−τsk, Σ¯msdmsd=diagUmsd−1,0,0.
In light of condition (17), the above inequality equivalents to(27)EΔVk≤−λmin−Σ¯msdη⊤kηk≤−υx⊤kxk,where υ=infλmin−Σ¯msd. Obviously, it yielded EΔVk<0, which means(28)E∑k=0∞xk2≤1υEVx0,σ0−EVx∞,σ∞≤1υEVx0,σ0<∞.
Based on Definition1, the closed-loop dynamic (14) achieves SMSS, which completes the proof.□Theorem 2.
For given scalarsτ2≥τ1≥0, andα¯ss∈S, and gain matrixK¯d, the closed-loop dynamic (14) is SMSS, if there exist matricesPm>0m∈ℳ.,Umsd>0m,n∈ℳ,s∈S,d∈D.,R¯>0, and matrixYdd∈Dsuch that for anym,n∈ℳ,s∈S,d∈D,(29)−PmΦmsd∗diag−Um11,…,−Umsd,…−UmSD<0,where(30)Φmsd=ψm1φm1Pm…ψmsφmdPm⋯ψmSφmDPm,Υ⟶msd1=diagUmsd−symYd+τ2−τ1+1R¯d,−R¯d,α¯sI−symYd,Υ⟶msd2=ϖmΥ⟶msd2,1⊤ϖmΥ⟶msd2,2⊤Υ⟶msd2,3,Υ⟶3=diagΥ3,1,Υ3,1,−I,Υ⟶msd2,1=AmYd+1−α¯sBmK¯d0α¯sBmK¯d,Υ⟶msd2,2=−BmK¯d0BmK¯d,Υ⟶msd2,3=0αsQYd0⊤,ϖm=πm1Iπm2I⋯πmℳI,Υ3,1=diag−P1,−P2,…,−Pℳ.
Furthermore, the controller gains are achieved as(31)Kd=K¯dYd−1,∀d∈D.Proof. Firstly, premultiplying and postmultiplying the condition (29) by term diagPm−1,I,…,I and its transpose, we can get that(32)−Pm−1Φ¯msd∗diag−Um11,…,−Umsd,⋯−UmSD<0,where(33)Φ¯msd=ψm1φm1I⋯ψmsφmdI⋯ψmSφmDI.According to Schur complement, one can derive (33) is equivalent to (17).Next, by means of Schur complement, it can be easily obtained from (17)as follows:(34)Υ¯msd1ϖmΥmsd2∗Υ⟶3<0,where(35)Υ¯msd1=diag−Umsd−1+τ2−τ1+1R,−R,−α¯sI,ϖmΥmsd2⊤ϖmΥmsd3⊤Υmsd4,Υ¯msd2=Υmsd4=αsQ00⊤.Premultiplying and postmultiplying (35) by diagYd,Yd,Yd,I,I,…,I and its transpose, respectively. It yields(36)Υ^msd1Υ^msd2∗Υ⟶m3<0,where(37)Υ^msd1=diag−YdUmsd−1Yd⊤+τ2−τ1+1R¯d,−R¯d,−Ydα¯sYd⊤,Υ^msd2=ϖmΥ^msd2,1⊤ϖmΥ^msd2,2⊤Υ⟶msd2,3,Υ^msd2,1=AmYd+1−α¯sBmKdYd0α¯sBmKdYd,Υ^msd2,2=−BmKdYd0BmKdYd,R¯d=YdRYd⊤.On the other hand, with respect toUmsd>0, it can be derived that(38)Umsd−YdUmsd−1Umsd−Yd⊤≥0,which equivalents to(39)Umsd−symYd≥−YdUmsd−1Yd⊤.Similarly, in light ofα¯s≥0, one can also obtain(40)α¯sI−symYd≥−Ydα¯sIYd⊤.Substituting (40) and (41) into (37) and utilizing Schur complement, (30) can be guaranteed. This completes the proof.□


## 4. Numerical Examples

In the current section, to evaluate the efficiency of the attained methodology, the off-on jumping of circuit breakers subject to Markov switching that is associated with two modes, similar to [9, 11], the parameters are listed as follows:For Modem=1(41)A1=0.98339.368−0.01264−1.833×10−5−3.529×10−30.9833−2.655×10−3−5.542×10−66.39×10−3−2.936×10−20.9553.681×10−3−0.2263−1.253−11.310.5231,B1=−1.417×10−4−5.836×10−56.133×10−222.27⊤.For Modem=2(42)A2=0.98669.378−8.638×10−3−1.253×10−5−2.837×10−30.9866−1.812×10−3−3.878×10−62.054×10−3−1.243×10−20.94433.67×10−3−0.68943.545−16.250.5118,B2=−9.692×10−5−3.99×10−56.123×10−222.15⊤.As stated in [11], a Markov chain is adopted to describe the fault switching in power lines, the transition probabilities are illustrated in Table 2. From which, it is clear that Π=−0.070.93−0.20.8.Table 2
The transition probabilities.
System reliabilityMode 1Mode 2Model 1 (maximum)0.8260.174Model 2 (minimum)0.001160.99884Furthermore, the condition TPMΩ with two modes and condition TPM Ξ with three modes are selected as(43)Ω=0.40.60.50.5,Ξ=0.10.40.50.60.20.2.Meanwhile, choosingα1=0.5, α2=0.2, τ1=1, and τ2=2. The embedded function of deception attacks is chosen as(44)fxk−τεkk=−tanhϱ1x1k−τεkk−tanhϱ2x1k−τεkk−tanhϱ3x1k−τεkk−tanhϱ4x1k−τεkk,with ϱ1=10−4, ϱ2=5∗10−4, ϱ3=2∗10−4, and ϱ4=3∗10−4. From which, we get the upper bound Q=10−4×1,5,2,3. On the basis of conditions in Theorem 2, the controller gains can be computed as(45)K1=−0.05462.04860.0083−0.0010,K2=−0.04921.74930.0296−0.0013,K3=−0.04821.69550.0334−0.0013.The simulations are plotted in Figure3, which presents the uncontrolled curves of xk. The evolution of system mode σk, dispatcher mode εk, and controller mode θk are depicted in Figure 4. The random occurring deception attacks and evolution of time-varying delays τεk are described in Figures 5 and 6, respectively. Under the aforementioned control gains, the state trajectories of the closed-loop dynamic (14) is displayed in Figure 7, and the control input uk is exhibited in Figure 8. It can be seen from Figure 8 that the developed methodology works well.Figure 3
State trajectories of open-loop dynamic (14).Figure 4
Evolution of modes.Figure 5
Evolution ofαεkk.Figure 6
Evolution of time-varying delay.Figure 7
State trajectories of closed-loop dynamic (14).Figure 8
Curve of control input with deception attacks.Meanwhile, whenℳ=S=D=2 and TPMs Π=Ω=Ξ=−0.070.93−0.20.8, the asynchronous controller degrades into the synchronous case. Similarly, the controller gains can be acquired as(46)K1=−0.00820.37730.0391−0.0044,K2=−0.04591.9166−0.0047−0.0006.On the basis of the abovementioned controller gains, the state trajectories of closed-loop dynamic (14) are plotted in Figure 9 and the control input uk is shown in Figure 10. Clearly, it can be seen from Figures 9 and 10 that the ideal control situation (synchronous control) can achieve better performance.Figure 9
State trajectories of synchronous closed-loop dynamic (14).Figure 10
Curve of synchronous control input.On the other hand, to further verify the impact of the deception attacks, letαεkk=0, which signifies no deception attacks occur. Similarly, by solving the conditions of Theorem 2, the corresponding gains can be calculated as(47)K1=−0.05711.86930.3956−0.0407,K2=−0.05981.98240.3964−0.0453,K3=−0.05972.00210.3918−0.0458.Under the aforementioned control gains, the state trajectories of the closed-loop dynamic (14) are displayed in Figure 11 and the control input uk is exhibited in Figure 12. From the figures, it is clear that a better performance can be achieved without considering the deception attacks. From which, one concludes that deception attacks degrade the dynamic performance to a certain extent.Figure 11
State trajectories of closed-loop dynamic (14) without deception attacks.Figure 12
Curve of control input without deception attacks.


## 5. Conclusion

The problem of power systems being subject to abrupt variations and cyber-attacks has been addressed in the current study. Two hidden Markov models are expressed to characterize the asynchronous phenomena among system mode, dispatcher mode, and controller mode. Furthermore, unlike the existing cyber-attacks, the deception attacks with time-varying delays are regulated by a Markov process. Based on the Lyapunov theory, sufficient conditions are presented to guarantee the stochastic mean-square stability of the closed-loop dynamic. Finally, the proposed control design strategy is testified via a simulation result.



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