# Load Frequency Control for Interconnected Multi-area Power Systems with the Semi-Markov Jumping Parameter and Actuator Failure

Authors: Yanjun Jiang; Jinfeng Wang; Zeyang Bai; Zhengmou Ren; Xiaochen Sun; Guangliang Yu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, we study the load frequency control (LFC) problem for interconnected multiarea power systems (IMAPSs) with quantization and actuator failure. To effectively reduce the amount of data in the channel, input signals will be quantized before being transmitted from a controller to a system through the digital communication channel. To reveal the asynchronous phenomenon between the original plant and LFC with actuator failure, a hidden semi-Markov model is formulated. In addition, the stability of the jump system under network attack is discussed. On the basis of the Lyapunov theory, sufficient conditions are derived to ensure the stochastic stability of IMAPSs. Finally, the validity of the theoretical results is tested via a simulation example.


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## Body

## 1. Introduction

The power system is a complex nonlinear system, which has developed into a multiregional interconnected power system (PS) since the Industrial Revolution. To deal with the low-frequency little oscillations of interconnected PSs, the LFC was proposed in [1], which has been effectively applied to PSs [2–4]. According to the LFC technique, the frequency can be adjusted at a desired level, which guarantees the stability of entire PSs. Over the past few decades, researchers have proposed a number of techniques concerning with LFC, such as use of the integral control law [5], PI case [6], and PID case [7]. These approaches have been verified to improve the control performance of interconnected PSs [8, 9].In practical applications, the dynamic systems may undergo sudden changes in their parameters or structures due to component failures, sudden environmental changes, etc. In this case, Markov chains are widely adopted to model the variations in PS states. In [10], the Markov chain was employed to describe the random mutations of the discrete-time PS. In [11], the uncertain Markov chain was applied for the decentralized control of the PS. However, in most existing literature concerning Markov PSs, the residence time of Markov processes obeys a memory-free random distribution, in which the probability of the transition rate is time-independent. As signified in [12–14], compared with the conventional Markov chain, the semi-Markov chain is more general in approximating practical dynamics owing to its time-varying transition rate. Consequently, it is meaningful to study the LFC problem for PSs with semi-Markov jumping parameters, the so-called semi-Markov PSs. To the best of our knowledge, quite a few theoretical results have been applied to semi-Markov PSs due to their inherent difficulty, and this motivates this article.In reality, the signal is communicated through a limited bandwidth network [13, 15, 16]. Note that massive signals are transmitted via the limited network, which may lead to channel congestion, thus reducing the system performance. To overcome this shortcoming, we quantify the control inputs, in which quantization stands for the process of mapping the continuous values of a signal to a limited number of discrete values [17, 18]. By means of quantization, the amount of data in the channel and load of the channel can be effectively reduced. The quantizers can be roughly categorized into linear quantizers [19], and logarithm quantizers [20, 21]. The problem of stabilizing a continuous-time switched system affected by the time-varying delay and data quantization has been addressed in [22]. In the quantitative input multiarea, however, results of the power system are very few; in order to fill the gap, this article takes into account the interconnected more regional power system with quantitative input, and the quantification method is used as the basis of the design, making the obtained quantitative instruments with the crudest density and further reducing the burden of transmission. We would like to mention that, in practice, capturing system information is a tricky task. Therefore, the asynchrony between the system mode and the controller mode cannot be omitted. Nevertheless, the asynchronous control of the semi-Markov IMAPS has not been researched thoroughly, which partially motivates the current work.Motivated by the above considerations, this work considers the LFC problem of the IMAPS subject to quantization input. Different from the existing homogeneous Markov IMAPS, the semi-Markov chain is employed to describe the dynamic behavior of the IMAPS. Aiming to describe the asynchronous phenomenon between the original plant and LFC, the hidden semi-Markov model is formulated. By resorting to the Lyapunov theory, sufficient conditions are derived to ensure the stochastic stability of the resulting dynamic. In the end, one numerical example is inferred to show the correctness of the proposed method. The general structure is rendered as follows: the second section describes the asynchronous LFC of the semi-Markov PS with quantization form. In section3, sufficient conditions of random stability are given. A numerical example is given in section 4.Notations:diag∗ means a block-diagonal matrix; HeR=R+R⊤; P>0 means P is positive definite; Pr∗ implies occurrence probability; ε∗ indicates the mathematical expectation; ∗ signifies the Euclidean vector norm. ℑ signifies the identity matrix; A⊤ and A−1 stand for the transpose and inverse matrix, respectively.


## 2. Problem Formulations

### 2.1. System Model

In this study, the dynamic model of the multiarea LFC is described as follows:(1)x˙t=Artxt+HrtΔPLt+Brtut,yt=Crtxt,where xit∈ℝ5, yit∈ℝ2, uit∈ℝ, and ΔPLit∈ℒ20,∞ represent the state, the output, the control input, and the disturbance, respectively, and(2)yit=ACEi∫0tACEisds⊤,yt=y1ty2t⋯yNt⊤,ut=u11u2t⋯uNt⊤,ΔPLt=ΔPL1tΔPL2t⋯ΔPLNt⊤,xit=Δfit,ΔPmit,ΔYit,ΔPtieit,∫0tACEisds⊤,xt=x1tx2t⋯xNt⊤,Birt=001Tgirt00⊤,Brt=diagB1rtB2rt⋯BNrt,Hirt=−1Mirt0000⊤,Hrt=diagH1rtH2rt⋯HNrtCirt=βirt001000001,Crt=diagC1rtC2rt⋯CNrt,Aiirt=−DirtMirt1Dirt0−1Dirt00−1Ttirt1Ttirt00−1RirtTgirt0−1Tgirt00∑j=1,j≠iN2πTijrt0000βirt0100.Aij=alsl,s=15 with a51=−2πTij, als=0, and A=Aijj,i=1N. Each ACE signal is described as a linear combination of the tie-line power exchange and frequency deviation, i.e., ACEi=βirtΔfit+Ptieit, where ΔPtiei refers to the net exchange of the tie-line power. The nomenclature of other parameters is shown in Table 1.Table 1
The physical meaning of parameters.
ParametersNomenclatureΔfitFrequency deviationΔPmitMechanical power output incrementΔYitValve position of turbineDirtMode-dependent damping coefficientTtirtTurbine time constantTgirtGovernor time constantβirtFrequency biasTijrtCoefficient between thei-th and j-th areaIn view of the uncertain fault time of the power system, the semi-Markov processrtt≥0∈S=1,2,…,S is adopted. For all k∈ℕ,(3)Rkthe Markov chainTkthe sojourn time, i.e.,Tk=tk−tk−1Gpthe probability distribution functionχpthe Markov chain.Then,Rk,rk is said to be a renewal process if rt=RNt where Nt=supk:t≥tk with(4)PrRk+1=q|Rk=p=Πpq,p≠q,0,p=q.Meanwhile, the probability distribution functionGp can be described as follow:(5)Gph=PrTk+1<h|rtk=p.According to the aforementioned observation, we have(6)Prrt+δ=q|rt=p=PrTk+1≤h+δ,Rk+1=q|Tk+1>h,Rk=p,p≠q,PrTk+1>h+δ,Rk+1=q|Tk+1>h,Rk=p,p=q.=πpqhδ+oδ,p≠q,1+πpqhδ+oδ,p=q.Hence, by simple calculation, the transition rate matrix can be defined byΨh=πpqhS×S, where(7)πpqh=limδ⟶0Prrt+δ=q|rt=pδ=Πpqχph1−Gph,p≠q,and πpph=−∑q=1,q≠pSπpqh<0.


### 2.2. Asynchronous Control Input with Quantized

As exhibited in Figure1, the control input ut is required to be quantized before sending it to the power system. Inspired by this fact, the logarithmic quantizer can be described as follows:(8)qut=q1u1tq2u2t⋯qιuιt,where the wth subquantizer qw⋅ satisfying qwuwt=−qw−uwt, w=1,2,…,ι. The set of the logarithmic quantization level can be described as(9)Uw=±θnw:±θnw=〉wnθ0w,n=±1±2,⋯∪±θ0w∪0,where 0<ϱwn<1 and θ0w>0 on behalf of the quantizer density and the initial quantization, respectively .And λω=1−ϱw/1+ϱw. Then, we define the subquantizer qωuωt as(10)qωuωt=θnω,ifθnω1+λω<uωt<θnω1−λω,0,ifuωt=0,−qω−uωt,ifuωt<0.Figure 1
The evolution of modesrt and φt.Then, we have that(11)qut=ut,where Λ=diagΛ1,Λ2,…,Λι, Λw⊤Λw≤ℑ.In this work, thei-th control area of an asynchronous PI controller is given by(12)uit,φt=KPφtiACEit+KIφti∫0tACEisds=Kφtiyit,where KPφti, and KIφti are the i -th proportional and integral gains, respectively, and Kφti=KPφtiKIφti. The variable φt that refers to the Markov chain belongs to the space ℳ=1,2,…,M, whose conditional probability matrix is inferred Γ=ρpmS×M with(13)Prφt=m|rt=p=ρpm,with ∑m∈ℳρpm=1. According to equations (11) and (12), the control signal uit,φt can be devised as follows:(14)uit,φt=aitℑi+ΛiKφtiyit,where ait∈0,1, and εait=a¯i.


### 2.3. Model Transformation

Letrt=p and φt=m, substituting equations (14) into (1), the closed-loop IMAPS is formulated as follows:(15)x˙it=Apiixit+∑j=1,j≠iNApijxjt+a¯iBpiℑi+ΛiKpmiCpixit+HpiΔPLit,yit=Cpixit.For analysis convenience, based on the compatible matrixB^pi, we can obtain a full rank matrix B¯pi=BpiB^pi. We define Xit=B¯pi−1xit, we have that(16)X.it=A¯piiXit+∑j=1,j≠iNA¯pijXjt+a¯iℑi+ΛiK¯miCpiXit+H¯piΔPLit,yit=C¯piXit,where A¯pij=B¯pi−1ApijB¯pi, K¯m=Kmi⊤0⊤, C¯pi=CpiB¯pi, and H¯pi=B¯pi−1Hpi. It is worth noting that linear transformation B¯pi is invertible. Thus, the overall IMAPS can be inferred as(17)X.t=A¯pXt+a¯ℑ+ΛK¯mC¯pXt+H¯pΔPLt,yt=C¯pXt,C¯pXt,where a¯=diagNa¯i, ℑ+Λ=diagNℑi+Λi, C¯p=diagNC¯pi, H¯pi=diagNH¯pi, and(18)A¯p=A¯p11A¯p12⋯A¯p1NA¯p21A¯p22⋯A¯p2N⋮⋮⋱⋮A¯pN1A¯pN2⋯A¯pNN.Definition 1.
([23]) The interconnected power system with ΔPLt=0 is called stochastic stability if the following equation holds:(19)ε∫0∞Xt2dt|rt0,φt0<∞.
Under the zero initial condition, the system withΔPLt∈ℒ20,∞ and γ>0 , the H∞ performance index is satisfied:(20)ε∫0∞yTtyt−γ2PLTtPLt<0.Lemma 1.
([21]) For the given matrix Ω1 and matrices Ω2 and Ω3 with appropriate dimensions, if inequality Ω1+HeΩ2ΛΩ3<0 holds for all Λ≤ℑ, for any scalar s1, such that Ω1+s1Ω3⊤Ω3+s1−1Ω2⊤Ω2<0.


## 2.1. System Model

In this study, the dynamic model of the multiarea LFC is described as follows:(1)x˙t=Artxt+HrtΔPLt+Brtut,yt=Crtxt,where xit∈ℝ5, yit∈ℝ2, uit∈ℝ, and ΔPLit∈ℒ20,∞ represent the state, the output, the control input, and the disturbance, respectively, and(2)yit=ACEi∫0tACEisds⊤,yt=y1ty2t⋯yNt⊤,ut=u11u2t⋯uNt⊤,ΔPLt=ΔPL1tΔPL2t⋯ΔPLNt⊤,xit=Δfit,ΔPmit,ΔYit,ΔPtieit,∫0tACEisds⊤,xt=x1tx2t⋯xNt⊤,Birt=001Tgirt00⊤,Brt=diagB1rtB2rt⋯BNrt,Hirt=−1Mirt0000⊤,Hrt=diagH1rtH2rt⋯HNrtCirt=βirt001000001,Crt=diagC1rtC2rt⋯CNrt,Aiirt=−DirtMirt1Dirt0−1Dirt00−1Ttirt1Ttirt00−1RirtTgirt0−1Tgirt00∑j=1,j≠iN2πTijrt0000βirt0100.Aij=alsl,s=15 with a51=−2πTij, als=0, and A=Aijj,i=1N. Each ACE signal is described as a linear combination of the tie-line power exchange and frequency deviation, i.e., ACEi=βirtΔfit+Ptieit, where ΔPtiei refers to the net exchange of the tie-line power. The nomenclature of other parameters is shown in Table 1.Table 1
The physical meaning of parameters.
ParametersNomenclatureΔfitFrequency deviationΔPmitMechanical power output incrementΔYitValve position of turbineDirtMode-dependent damping coefficientTtirtTurbine time constantTgirtGovernor time constantβirtFrequency biasTijrtCoefficient between thei-th and j-th areaIn view of the uncertain fault time of the power system, the semi-Markov processrtt≥0∈S=1,2,…,S is adopted. For all k∈ℕ,(3)Rkthe Markov chainTkthe sojourn time, i.e.,Tk=tk−tk−1Gpthe probability distribution functionχpthe Markov chain.Then,Rk,rk is said to be a renewal process if rt=RNt where Nt=supk:t≥tk with(4)PrRk+1=q|Rk=p=Πpq,p≠q,0,p=q.Meanwhile, the probability distribution functionGp can be described as follow:(5)Gph=PrTk+1<h|rtk=p.According to the aforementioned observation, we have(6)Prrt+δ=q|rt=p=PrTk+1≤h+δ,Rk+1=q|Tk+1>h,Rk=p,p≠q,PrTk+1>h+δ,Rk+1=q|Tk+1>h,Rk=p,p=q.=πpqhδ+oδ,p≠q,1+πpqhδ+oδ,p=q.Hence, by simple calculation, the transition rate matrix can be defined byΨh=πpqhS×S, where(7)πpqh=limδ⟶0Prrt+δ=q|rt=pδ=Πpqχph1−Gph,p≠q,and πpph=−∑q=1,q≠pSπpqh<0.


## 2.2. Asynchronous Control Input with Quantized

As exhibited in Figure1, the control input ut is required to be quantized before sending it to the power system. Inspired by this fact, the logarithmic quantizer can be described as follows:(8)qut=q1u1tq2u2t⋯qιuιt,where the wth subquantizer qw⋅ satisfying qwuwt=−qw−uwt, w=1,2,…,ι. The set of the logarithmic quantization level can be described as(9)Uw=±θnw:±θnw=〉wnθ0w,n=±1±2,⋯∪±θ0w∪0,where 0<ϱwn<1 and θ0w>0 on behalf of the quantizer density and the initial quantization, respectively .And λω=1−ϱw/1+ϱw. Then, we define the subquantizer qωuωt as(10)qωuωt=θnω,ifθnω1+λω<uωt<θnω1−λω,0,ifuωt=0,−qω−uωt,ifuωt<0.Figure 1
The evolution of modesrt and φt.Then, we have that(11)qut=ut,where Λ=diagΛ1,Λ2,…,Λι, Λw⊤Λw≤ℑ.In this work, thei-th control area of an asynchronous PI controller is given by(12)uit,φt=KPφtiACEit+KIφti∫0tACEisds=Kφtiyit,where KPφti, and KIφti are the i -th proportional and integral gains, respectively, and Kφti=KPφtiKIφti. The variable φt that refers to the Markov chain belongs to the space ℳ=1,2,…,M, whose conditional probability matrix is inferred Γ=ρpmS×M with(13)Prφt=m|rt=p=ρpm,with ∑m∈ℳρpm=1. According to equations (11) and (12), the control signal uit,φt can be devised as follows:(14)uit,φt=aitℑi+ΛiKφtiyit,where ait∈0,1, and εait=a¯i.


## 2.3. Model Transformation

Letrt=p and φt=m, substituting equations (14) into (1), the closed-loop IMAPS is formulated as follows:(15)x˙it=Apiixit+∑j=1,j≠iNApijxjt+a¯iBpiℑi+ΛiKpmiCpixit+HpiΔPLit,yit=Cpixit.For analysis convenience, based on the compatible matrixB^pi, we can obtain a full rank matrix B¯pi=BpiB^pi. We define Xit=B¯pi−1xit, we have that(16)X.it=A¯piiXit+∑j=1,j≠iNA¯pijXjt+a¯iℑi+ΛiK¯miCpiXit+H¯piΔPLit,yit=C¯piXit,where A¯pij=B¯pi−1ApijB¯pi, K¯m=Kmi⊤0⊤, C¯pi=CpiB¯pi, and H¯pi=B¯pi−1Hpi. It is worth noting that linear transformation B¯pi is invertible. Thus, the overall IMAPS can be inferred as(17)X.t=A¯pXt+a¯ℑ+ΛK¯mC¯pXt+H¯pΔPLt,yt=C¯pXt,C¯pXt,where a¯=diagNa¯i, ℑ+Λ=diagNℑi+Λi, C¯p=diagNC¯pi, H¯pi=diagNH¯pi, and(18)A¯p=A¯p11A¯p12⋯A¯p1NA¯p21A¯p22⋯A¯p2N⋮⋮⋱⋮A¯pN1A¯pN2⋯A¯pNN.Definition 1.
([23]) The interconnected power system with ΔPLt=0 is called stochastic stability if the following equation holds:(19)ε∫0∞Xt2dt|rt0,φt0<∞.
Under the zero initial condition, the system withΔPLt∈ℒ20,∞ and γ>0 , the H∞ performance index is satisfied:(20)ε∫0∞yTtyt−γ2PLTtPLt<0.Lemma 1.
([21]) For the given matrix Ω1 and matrices Ω2 and Ω3 with appropriate dimensions, if inequality Ω1+HeΩ2ΛΩ3<0 holds for all Λ≤ℑ, for any scalar s1, such that Ω1+s1Ω3⊤Ω3+s1−1Ω2⊤Ω2<0.


## 3. Main Results

Theorem 1.
For given scalarsγ>0and η>0 and the matrix Mp, the IMAPS is stochastic stability with preset performance, such that(21)Ξ¯pm=Ξpm11+C¯p⊤C¯p∗∗Ξpm21−HeηM⊤∗H¯p⊤M⊤ηH¯p⊤M⊤−γ2<0,where(22)Ξpm11=∑q=1SΠ¯Pq+∑m=1MρpmHeM⊤A¯p+a¯ℑ+ΛK¯mC¯p,Ξpm21=−M+Pp⊤+∑m=1MρpmηM⊤A¯p+a¯ℑ+ΛK¯mC¯p,Π¯pq=επpqh=∫0∞πpqhχphdh.Proof.
Establish the semi-Markov-based Lyapunov function as follows:(23)VXT,rt,φt=X⊤tPrtXt.
It follows that(24)εℒVXt,rt,φt=limδ⟶01δε∑q≠pπpqhδX⊤t+δPqXt+δ+1+πpphδX⊤t+δPpXt+δ+X⊤tPpXt=X⊤t∑q=1SΠ¯pqPqXt+HeX.⊤tPpX⊤t.
On the basis of equation (17), for any proper matrix M, such that(25)0=2X⊤tM⊤+ηX.tM⊤×−X.t+A¯pXt+a¯ℑ+ΛK¯mC¯pXt+H¯pΔPLt.
It follows from equations (24) and (25) that(26)εℒVXt,rt,φt=X⊤t∑q=1SΠ¯pqPqXt+HeX.⊤tPpX⊤t+2X⊤tM⊤+ηX.tM⊤×−X.t+A¯pXt+a¯ℑ+ΛK¯mC¯pXt+H¯pΔPLt=ζ⊤tΞpmζt,where ζ⊤t=ξ⊤tΔP⊤t, ξ⊤t=X⊤tX.⊤tΠ¯pq=επpqh=∫0∞πpqhχphdh and(27)Ξpm=Ξpm11∗∗Ξpm21−HeηM⊤∗H¯p⊤M⊤ηH¯p⊤M⊤0.
In case ofΔPLt=0, we have εℒVXt,rt,φt≤ξ⊤tΞpm1ξ⊤t, by equation (21), we have Ξpm1<0, when Ξpm1=Ξpm11∗Ξpm21−HeηM⊤, Ξpm11=∑q=1SΠ¯Pq+∑m=1MρpmHeM⊤A¯p+a¯ℑ+ΛK¯mC¯p, Ξpm21=−M+Pp⊤+∑m=1MρpmηM⊤A¯p+a¯ℑ+ΛK¯mC¯p. Subsequently, we have(28)εℒVXt,rt,φt≤−λεXt2∣Xt0,rt0,φt0,where λ=λmin−Ξpm1, we can further have(29)ε∫0∞Xt2|Xt0,rt0,φt0≤1λVXt0,rt0,φt0<∞.
Furthermore, forΔPLt≠0, according to equation (24), it yields(30)ℒV+y⊤tyt−γ2ΔPL⊤tΔPLt≤ζ⊤tΞ¯pmζt.
Note thatΞ¯pm<0, from which one can obtain(31)∫0∞ℒV+z⊤tzt−γ2ΔPL⊤tΔPLtdt<0,which indicates(32)∫0∞z⊤tzt−γ2ΔPL⊤tΔPLtdt<0,this completes the proof.Theorem 2.
For given scalarsγ>0, η>0, and s1>0, system equation (13) is stochastic stability with the preset H∞ performance index γ, if there exist matrices Mand K¯m, such that(33)Ξ¯pm11∗∗∗∗∗∗Ξ¯pm21−HeηM⊤∗∗∗∗∗H¯p⊤M⊤ηH¯p⊤M⊤−γ2ℑ∗∗∗∗C¯p00−ℑ∗∗∗θp⊤000θp11∗∗θp111⊤ηθp111⊤000−Hea^s¯1∗M⊤ηM⊤0000−Hea^s1<0,where s¯1=s1−1 , a^=ηa¯−1(34)Ξ¯pm11=Π¯ppPp+HeM⊤A¯p+∑m=1MρpmHea¯YKmC¯p,Ξ¯pm21=−M+Pp⊤+∑m=1Mρpma¯ηYKmC¯pθp11=−diagP1P2⋯Pp−1Pp+1⋯PS,θp111=ρp1C¯⊤K¯1⊤ρp2C¯⊤K¯2⊤⋯ρpMC¯⊤K¯M⊤θp=Π¯p1P1Π¯p2P2⋯Π¯pp−1Pp−1Π¯pp+1Pp+1⋯Π¯pSPS,Y=diagY1,Y2,…,YN,Yi=diagℑp,0.
Furthermore, controller gain matrices are given byK¯m=M⊤−1YKm .Proof.
LetYKm=M⊤K¯m, with the purpose of structure of the matrix K¯m. In detail,(35)K¯mi=Mi⊤−1YiKmi==M1i−100M2i−1ℑp000Kmi.
According to Lemma1 and by using the Schur complement in equation (21)(10) the proof is completed.


## 4. Numerical Example

In this section, to show the effectiveness of the attached methodology, a numerical example of the 3-area interconnected semi-Markov switching power system is presented. Suppose that the coefficients are selectedT12rt=0.2, T13rt=0.25, and T23rt=0.12, rt=1,2. Furthermore, for any i=1,2,3, set(36)B^pi=0.00100000.00100000.0010000.0010.001.Leta¯i=0.5, γ=7, η=0.2, and s1=0.4. Other nominal parameters are listed in Table 2 [3, 24].Table 2
Parameters of the IMAPS.
ModeMirtDirtTtirtTgirtβirtRirtArea 1r=11010.30.1210.05r=2101.10.350.1520.50.05Area 2r=11210.40.1721.50.05r=2101.50.40.1180.05Area 3r=1121.80.350.1221.80.05r=212.51.250.150.15230.05Specifically, the transition rate function can be writtenπpq=Πpqhk/ϵkhk−1, and the Weibull distribution density function is given by χph=k/ϵh/ϵk−1exp−h/ϵk, where k and ϵ indicate the parameters of the shape and scale, respectively. Noting that when p=1, we choose ϵ=1 and k=3. Otherwise, for p=2, we set ϵ=1 and k=4. Accordingly, the transition rate matrix can be expressed as(37)πpqh=−3h23h24h3−4h3.Thus, we have(38)Π¯pq=επh=−2.70822.70823.6763−3.6763.In order to better describe the asynchronous phenomenon, the condition probability matrix is set as followsΓ=0.60.4;0.50.5. According to Theorem 2, the asynchronous controller gains can be easily devised. We select the initial state x10=0.048−0.040.0650.024−0.55⊤, x20=0.045−0.090.0150.022−0.46⊤, x30=0.07−0.050.07−0.08−1.1⊤, and the load disturbance ΔPLi=0.005sink, i=1,2,3. Added by the above-derived gains, the mode evolution of rt and φt is shown in Figure 1. The trajectories of the state are plotted in Figure 2. Figure 3 thatdepict the state trajectories and the measured output. These simulation results verify the effectiveness of the proposed method.Figure 2
The state trajectoriesxiti=1,2,3.Figure 3
The measured outputyiti=1,2,3.


## 5. Conclusions

In this study, the problem of the asynchronous load frequency control problem for semi-Markov interconnected multi-area power systems with the quantization effect has been addressed. In this case, the system under consideration is modeled as the semi-Markov jump system. The merit of this work is to tackle the asynchronous phenomenon between the control and semi-Markov interconnected multi-area power systems. By designing an asynchronous controller with quantized form, the quantized closed-loop system has stochastic stability under the specified performance. At last, the effectiveness of the developed method has been tested by the simulation result.



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