# Load Frequency Control for Power Systems with Actuator Faults within a Finite-Time Interval

Authors: Haifeng Qiu; Liguo Weng; Bin Yu; Yanghui Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper is concerned with the issue of finite-timeH∞ load frequency control for power systems with actuator faults. Concerning various disturbances, the actuator fault is modeled by a homogeneous Markov chain. The aperiodic sampling data controller is designed to alleviate the conservatism of attained results. Based on a new piecewise Lyapunov functional, some novel sufficient criteria are established, and the resulting power system is stochastic finite-time bounded. Finally, a single-area power system is adjusted to verify the effectiveness of the attained results.


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## Body

## 1. Introduction

Load frequency control (LFC), as an integral part of automatic generation control in power systems, has been adopted to regulate the frequency deviation and tie-line power exchanges [1–3]. Added by the LFC strategy, the high-quality electric energy can be maintained over a certain range [4]. In general, constant frequency deviation may lead to unreliable frequency devices, transmission lines overload, etc. Meanwhile, owing to the large size of the power grid, it raises the difficulty in frequency control. Therefore, it is a tough task to design suitable frequency control law. In practical applications, the loads are unexpected and unmeasurable, which indirectly regulate the system frequency. Accordingly, through the LFC strategy, the system performance can be guaranteed without affecting the generation capacity or frequency deviation. Up to now, the research on the LFC for power system gradually becomes a hot topic [5–7].In networked control systems, various faults can be encountered due to the long-term utilization of components [8–10]. Note that the actuator faults are the source of instability and performance deterioration. To overcome the above shortage and improve the dependability, a great deal of attention has been shifted to actuator faults, and plenty of results have emerged [11, 12]. However, the actuator faults are assumed to be time-unchanged, which limits the potential applications. As stated in [13], the so-called failure probability is common in the reliability industry, where failure rates can be governed by the Markov switching chain [14–16]. Despite the significant achievement has attained, no suitable attention has been devoted to the power systems.On the other hand, Lyapunov asymptotic stability is most common in the literature, where asymptotic behavior can be expected over the infinite-time domain. Nevertheless, in reality, the desirable transient performance is very important in many physical systems, which causes the inapplicability of the Lyapunov stability. Following this trend, finite-time stability (FTS) has been studied [17–19], which concerns the dynamic behavior within bound over a fixed time interval instead of an asymptotical case. As is well known that FTS is different from the Lyapunov case, it gives more solutions of transient performance control. Owing to the merits of the FTS, many valuable achievements have been made over the past years [20]. However, to our knowledge, most of the previous results are assumed that the data communication keeps continuous between sensors and controllers. In the fields of sampled-data control law, this assumption is not accurate. In general, with respect to the demand of actual systems, the sampler may encounter component aging, data losses, etc. [21–23]. These shortages may lead to unreliable periodic sampling. Fortunately, the aperiodic sampled-data control strategy is presented [24, 25], which can efficiently deal with the aforementioned issues. However, the finite-time aperiodic sampled-data control for power systems remains unsettled, not mentioned to the LFC, which motivates us for this study.Inspired by the above observations, we focus on the finite-timeH∞ load frequency control for power systems with actuator faults over the finite-time interval in this study. The main contributions can be summarized as follows: (1) different from the previous studies, to fully describe the randomly occurring actuator fault, the actuator fault is characterized by a homogeneous Markov chain. (2) To better characterize the actual demands of practical dynamics, a generalized framework of the actuator constraint is considered. (3) Apart from the traditional Lyapunov asymptotic stability, this study exploits the FTS for power systems and focuses on the finite-time control issue. By resorting to the piecewise Lyapunov theory, some novel results over the finite-time interval are reached. Finally, a numerical example is manifested to reveal the validity of the gained results.The remainder of this study is listed as follows. Section2 provides a description of the problem. Section 3 presents the main results, and the simulation validation is exhibited in Section 4. Section 5 concludes the study.


### 1.1. Notations

The notations of this paper are standard.⋅ means the Euclidean norm. ℝ indicates a set of n-dimensional matrix. E refers to the mathematical expectation. λmaxA/λminA means the largest/smallest eigenvalue of matrix A. Pr· means the occurrence probability. diag· represents a block-diagonal matrix.


## 1.1. Notations

The notations of this paper are standard.⋅ means the Euclidean norm. ℝ indicates a set of n-dimensional matrix. E refers to the mathematical expectation. λmaxA/λminA means the largest/smallest eigenvalue of matrix A. Pr· means the occurrence probability. diag· represents a block-diagonal matrix.


## 2. Problem Formulations

Block diagram of single-area LFC power model is exhibited in Figure1 [6]. Accordingly, the dynamic equation of power model can be listed as follows:(1)δ¯˙t=A¯δ¯t+B¯uFt+F¯ωt,y¯t=C¯δ¯t,where(2)δ¯t=ΔfΔPmΔPυT,ωt=ΔPd,C¯=β00,A¯=−DM1M00−1Tch1Tch−1RTg0−1Tg,B¯=001TgT,F¯=−1M00T,and the system parameters are expressed in Table 1.Figure 1
The structure of single-area power system.Table 1
Physical meaning.
ParametersPhysical meaningΔPυGoverner valve positionΔfDeviations of frequencyΔPmMechanical output of generatorβFrequency bias factorΔPdLoad disturbanceMMoments of inertia of the generatorRSpeed dropTdTime constant of the governorDDamping coefficient of generatorTchTime constant of turbineIn single-area, the area control error (ACE) is interpreted asyt=βΔf due to the unaccessiblity of the tie-line power exchange. In reality, the actuator faults cannot be neglected for long-term utilization of components, which can be expressed as(3)uFt=αrtut,where αrt=diagα1rt,α2rt,…,αfrt, and each element αmrt∈0,1m=1,2,…,f. More specifically, rt,t≥0 is identified as a right-continuous Markov chain taking values over a set S=1,2,…,S with generator Π=πpqS×S, and its transition probabilities are inferred as(4)Prrt+Δt=q|rt=p=πpqΔt+oΔt,ifq≠p,1+πpqΔt+oΔt,ifq=p,where Δt>0 and limΔt⟶0oΔt/Δt=0, for q≠p and πpp=−∑q≠pπpq for each p∈S.Taking the ACE as the desired controller input of LFC, the output of the proportional-integral (PI) controller is asserted as(5)ut=−KPy¯t−KI∫y¯t,where KP and KI signify the proportional and integral gains of the area, respectively.Letδt=δ¯Tt∫0ty¯TsdsT, yt=y¯Tt∫0ty¯TsdsT, the power model (1) is reformulated as(6)δ˙t=Aδt+Bαrtut+Fωt,yt=Cδt,where(7)A=−DM1M000−1Tch1Tch0−1RTg0−1Tg0β000,B=001Tg0,F=−1M000,C=β0000001.The purpose of this study is to solve the output feedback control for power system (6) with data sampling. Therefore, the sampling sequence attained at a set of time instants. Added by the data sampling technique, only the measured signal ytk can be released to the controller. Specifically, the sampling instants are represented as(8)0=t0<t1<⋯<tk<tk+1<⋯<limk⟶∞tk=∞.In light of periodic sampling instants, in this study, we consider the aperiodic sampling case. Following this trend, the sampling intervaltk,tk+1 is time-varying with the upper sampling period. Thus, one defines 0<tk+1−tk≜τk≤τ. Based on the input delay technique, we have that tk=t−t−tk≜t−τt with τ˙t=1 for t≠tk. Summarizing the above discussion, we have 0≤τt<τk≤τ.LettingK=KpKI, the PI-based sampled data LFC can be designed as(9)ut=Kytk,t∈tk,tk+1.Substituting (3) and (9) into (6), the closed loop power system can be governed by(10)δ˙t=Aδt+BαrtKCδtk+Fωt,t∈tk,tk+1.Before further derivation, some important contents are stated as follows.Assumption 1 (see [18]).
The external disturbanceωt belongs to L2t0,T, and it is assumed that there exists a parameter ω¯>0 such that ∫t0TωTtωtdt≤ω¯.Assumption 2.
From the viewpoint of the physical limitation of actuators in power systems, the control torque is assumed to meet(11)ust≤umax,s=1,2,…,nu.Definition 1 (see [26]).
Given parametersc1>0, c2>0, time interval T>0, and matrix R>0, the closed loop power system (10) is called stochastic finite-time stability (SFTS) with respect to c1,c2,T,R and ωt=0, if inequality EδTt0Rδt0≤c1⇒EδTtRδt≤c2 holds for ∀t∈t0,T.Definition 2 (see [26]).
Given parametersc1>0, c2>0, ω¯>0, time interval T>0, and matrix R>0, the closed loop power system (10) is called stochastic finite-time boundedness (SFTB) with respect to c1,c2,T,R,ω¯, if inequality EδTt0Rδt0≤c1⇒EδTtRδt≤c2 holds for ∀t∈t0,T.
The object of this study is to design sampled-data-based controller (9) such that(1)
The closed-loop power system (10) is SFTS with ωt=0.(2)
When disturbancesωt≠0, the power system (10) is called SFTB with H∞ performance index γ such that(12)Eyt22<γ2ωt22.Lemma 1 (see [27]).
For any vectorsℏ1t and ℏ2t, scalar ι∈0,1, a matrix A, and symmetric matrices B and C, the following inequality holds:(13)−1ιℏ1TtBℏ1t−11−ιℏ2TtCℏ2t≤−ℏ1tℏ2tTBA∗Cℏ1tℏ2t.


## 3. Main Results

Theorem 1.
For given parametersρ>0, c1>0, c2>0, T>0, ω¯>0, umax, and matrix R, the closed-loop power system (10) is called SFTB with respect to c1,c2,ω¯,T,R, if there exists matrix Pp>0, N1>0, N3>0, Q1>0, and Q2>0, such that ∀p∈S(14)Qf≥0,f=1,2,(15)Θ1<0,(16)Θ2τpTℋpT∗−τQ1<0,(17)ψc1+γ2ω¯<e−ρTminp∈SλminP˜pc2,(18)−usmax2CTKsT∗−Rc2<0,s=1,2,…,nu.where(19)Θi=Θ11iΘ12iΘ13Θ14iΘ15∗Θ22iH8⊤Θ24iH10∗∗Θ3300∗∗∗Θ44iT2⊤F∗∗∗∗−γ2I,Q1=N1U∗N1,Q2=Q2V∗Q2,Θ111=∑q=1SπpqPq−M1+Q1−Q2+T1TA+ATT1−ρPp−τρM1−H1−H1T,Θ112=∑q=1SπpqPq−M1+Q1−Q2+T1TA+ATT1−ρPp−H1−H1T,Θ121=Pp+M1−M2−N2+T1TBαpKC−τρ−M1+M2+H6T−H2,Θ122=Pp+M1−M2−N2+T1TBαpKC+H6T−H2,Θ13=−H3+V,Θ141=τM1−T1T+ATT2−H4,Θ142=−T1T+ATT2−H4,Θ15=T1TF−H5,Θ221=−M1+M2+M2T+N2+N2T−τρM1−M2−M2T+H7+H7T,Θ222=−M1+M2+M2T+N2+N2T−τN3+H7+H7T,Θ241=−M1T+τM2T+N2T+T2TBαKCT+H9,Θ242=−M1T+T2TBαKCT+H9,Θ33=−Q1−Q2,Θ441=τN1+τ2Q2−T2T−T2,Θ442=τ2Q2−T2T−T2,Θ441=τN1+τ2Q2−T2T−T2,Θ442=τ2Q2−T2T−T2,ψ≜maxp∈SλmaxP˜p+λmaxQ˜1+λmaxQ˜2,P˜p=R−1/2QpR−1/2,Q˜s=R−1/2QsR−1/2,s=1,2.Proof.
Establishing a Lyapunov functional asVt,rt=∑i=14Vit,rt, where(20)V1t,rt=δTtPrtδt,V2t,rt=τ−t−tkϑ1Ttℳϑ1t,V3t,rt=τ−t−tk∫tkteρt−sϑ2Ts,tkNϑ2s,tkds,V4t,rt=∫t−τteρt−sδTsQ1δsds+τ∫−τ0∫t+θteρt−sδ˙TsQ2δ˙sdsdθ,where(21)ϑ1t=δTtδTtkT,ϑ2s,tk=δ˙TsδTtkT,ℳ=M1−M1+M2∗M1−M2−M2T,N=N1N2∗N3.
The weak infinitesimal operatorℒ· can be inferred as(22)ℒVt,rt=∂Vt,rt∂t+δ˙Tt∂Vt,rt∂t+∑q=∈SπpqPqVt,rt.
Applying the operatorℒ· along the power system (10), which yields(23)ℒV1t,rt=δTtPpδ˙t+δ˙TtPδt+δTt∑q=1SπpqPqδt,(24)ℒV2t,rt=τ−t−tkϑ1Ttℳδ˙Tt0T+τ−t−tkδ˙t0Tℳϑ1t−ϑ1Ttℳϑ1t,(25)ℒV3t,rt≤−∫tktδ˙TsN1δ˙sds−2δTtU2δtk+δTtkN2+N2T−t−tkN3δtk+τ−t−tkδ˙TtN1δ˙t+2τ−t−tkδ˙TtN2δtk+ρV3t,rt,(26)ℒV4t,rt≤δTtQ1δt−δTt−τQ1δt−τ+τ2δ˙TtQ2δ˙t−τ∫t−τtδ˙TsQ2δ˙sds+ρV4t,rt.
Based on Lemma1, the following inequality can be devised:(27)−∫tktδ˙TsN1δ˙sds≤−1t−tkξTtW1TQ1W1ξt,(28)−τ∫t−τtδ˙TsQ2δ˙sds≤−ξTtW2TQ2W2ξt,where(29)ξTt=δTtδTtkδTt−τδ˙tωTt,W1=I00000−I000,W2=I000000−I00.
It is well known that for any matricesℋ, one gets(30)1t−tkQ1W1−t−tkℋTQ1−1Q1W1−t−tkℋ≥0.
The aforementioned condition can be rewritten as(31)−1t−tkW1TQ1W1≤−ℋTW1−W1Tℋ+t−tkℋTQ1−1ℋ.
On the other hand, for any matricesT1 and T2, it is clear that(32)0=2δTtT1T+δ˙TtT2T−δ˙t+Aδt+BαpKCδtk+Fωt.
Substituting (23)–(32) into (20), it can be deduced that(33)EℒVt,rt−ρVt,rt−γ2ωTtωt≤ξTtΘt+t−tkℋTQ1−1ℋξt,t∈tk,tk+1,where(34)Θt=Θ11Θ12Θ13Θ14Θ15∗Θ22H8TΘ24H10∗∗Θ3300∗∗∗Θ44T2TF∗∗∗∗−γ2I,Θ11=∑q=1SπpqPq−M1+Q1−Q2+T1TA+ATT1−ρPp−τ−t−tkρM1−H1−H1T,Θ12=Pp+M1−M2−N2+T1TBαpKC−τ−t−tkρ−M1+M2+H6T−H2,Θ13=−H3+V,Θ14=τ−t−tkM1−T1T+ATT2−H4,Θ15=T1TF−H5,Θ22=−M1+M2+M2T+N2+N2T−t−tkN3−τ−t−tkρM1−M2−M2T+H7+H7T,Θ24=−M1T+τ−t−tkM2T+N2T+T2TBαKCT+H9,Θ33=−Q1−Q1,Θ44=τ−t−tkN1+τ2Q2−T2T−T2.
Note that (33) is a convex combination of t−tk and τ−t−tk, in accordance with Schur complement; one can deduce that Θt+t−tkℋTQ1−1ℋ<0 if and only if (15) and (16) hold. Therefore, one can see that(35)EℒVt,rt<ρVt,rt+γ2ωTtωt,t∈tk,tk+1.
By integrating the both sides of (35) from tk to t and simple derivation, it yields(36)EVt,p<eρt−tkEVtk,p+∫tkteρt−sγ2ωTsωsds≤eρt−tk−1EVtk−1,p+eρt−tk−1∫tk−1tkeρtk−sγ2ωTsωsds+∫tkteρt−sγ2ωTsωsds≤⋯≤et−t0EVt0,p+eρt−t1∫t0t1eρt1−sγ2ωTsωsds+⋯+∫tkteρt−sγ2ωTsωsds≤eρt−t0EVt0,p+∫t0tγ2ωTsωsds≤eρTEVt0,p+γ2ω¯.
Recalling the Lyapunov functional (20), we can get(37)EVt,rt≥λminP˜pδTtRδt,(38)EVt0,rt≤maxp∈SλmaxP˜p+λmaxQ1+λmaxQ2δTt0Rδt0.
Substituting (37) and (38) into (36), we can obtain(39)infp∈SλminP˜pEδTtRδt<eρTψc1+γ2ω¯.
In light of (17), it can be concluded from (39) that EδTtRδt<c2. Thus, from Definition 2, we have to derive that power system (10) is SFTB over the time interval t0,T.
In the following, the actuator constraints (18) will be discussed. In light of (9), one has(40)usFt≤ust=KsCδtk2≤KsCR−1/22×R1/2δtk2=KsCR−1CTKsTδTtkRδtk≤c2KsCR−1CTKsT.
Recalling Assumption2, it yields(41)c2KsCR−1CTKsT<usmax2,s=1,2,…,nu.
According to Schur complement, (18) can be guaranteed by (41), which completes the proof of Theorem 1.Theorem 2.
For given parametersρ>0, c1>0, c2>0, T>0, ω¯>0, umax, and matrix R, the closed-loop power system (10) is called SFTB with respect to c1,c2,ω¯,T,R and meet an H∞ performance index γ¯=γeρT, if there exists matrix Pp>0, N1>0, N3>0, Q1>0, Q2>0, and matrices X, Y with suitable dimensions, such that ∀p∈S(42)Θ¯1Z1Z2∗−ε1X+XT0∗∗−ε2X+XT<0,(43)Θ¯2τℋTZ1Z2∗−τQ1−100∗∗−ε1X+XT0∗∗∗−ε2X+XT<0,(44)eρTψ¯c1+eρTγ2ω¯<λ1c2,(45)λ1R<Pp<λ2R,Q1<λ3R,Q2<λ4R,(46)−usmax2YCI−X∗−Rc2ε3CTYT∗∗−ε3X+XT<0,s=1,2,…,nu,where(47)Θi=Θ¯11iΘ12iΘ13Θ14iΘ15∗Θ22iH8TΘ24iH10∗∗Θ3300∗∗∗Θ44iT2TF∗∗∗∗−γ2I,ψ¯≜λ2+τλ3+τ2λ4,Θ¯111=∑q=1SπpqPq−M1+Q1−Q2+T1TA+ATT1−ρPp−τρM1−H1−H1T−CTC,Θ¯112=∑q=1SπpqPq−M1+Q1−Q2+T1TA+ATT1−ρPp−H1−H1T−CTC,Z1=αT1TB−BXTεYC000T,Z2=YC00αT2TB−BXT0T.
The controller gain is deduced as(48)K=X−1Y.Proof.
By resorting to the same Lyapunov functional as displayed in (20), and adopting the similar derivations in the proof of Theorem 1, for t∈tk,tk+1, the following inequality can be realized:(49)EℒVt,rt<ρVt,rt−yTtyt+γ2ωTtωt,t∈tk,tk+1.
Integrating the both sides of (49) from tk to t, it yields(50)EVt,p<eρt−tkVtk,p+∫tkteρt−s−yTsys+γ2ωTsωsds,t∈tk,tk+1.
Clearly, (50) can be rewritten as(51)EVt,p<eρt−t0Vt0,p+∫t0teρt−s−yTsys+γ2ωTsωsds.
Under the zero-initial conditions, one getsVt0,p=0. It follows from (51) that(52)∫t0teρt−syTsysds<∫t0teρt−sγ2ωTsωsds.
Since1≤eρt−s≤eρt and t∈t0,T, (52) signifies(53)E∫t0tyTsysds<γ¯2∫t0tωTsωsds,with γ¯=eρTγ. Therefore, the H∞ performance index γ¯ is ensured.
Meanwhile, in light of Lemma1 of [28], conditions (42) and (43) can be easily attained. Similarly, (46) can be ensured. The proof is completed.


## 4. Computational Experiments

In this example, a single-area power system is studied for simulating the derived results. Similar to [6], the parameters of system (1) are expressed in Table 2.Table 2
Physical meaning.
ParametersRDMsβTgsTchs0.051.01021.00.10.3Taking the actuator faults into consideration, which is characterized by a Markov process, the possible fault coefficients areα1=1, α2=0.8, and α3=0.5. Furthermore, the transition probability matrix among faults is given by(54)Π=0.20.50.30.80.10.10.30.30.4.On the other hand, other parameters are selected asτ=0.2, γ=0.6, ρ=0.2, c1=0.1, c2=1, ω¯=0.8, R=I4×4, and T=8. The control input ut is supposed to be constrained by ut≤umax=2. By solving the linear matrix inequalities of Theorem 2, the desired PI-type controller is derived as(55)K=−0.0002−0.0183.For graphically verifying the achieved results, we select the initial state disturbanceωt as(56)ωt=1.8sin4t,ift≤π,0,otherwise.Added by the aforementioned controller, the simulation results are plotted in Figures2–7. Figure 2 plots the simulated frequency, and Figure 3 displays the evolution of ACE. Meanwhile, the mode switching of actuator faults is shown in Figure 4, and control output is presented in Figure 5. Furthermore, with the disturbance given in Figure 6, the evolution of δTtRδt is expressed in Figure 7. One can be observed from Figure 7 that the state of closed-loop system stays in the prefixed region, which implies the resulting system is SFTB. Meanwhile, the input constraint is also satisfied.Figure 2
Evolution of frequency derivation.Figure 3
Evolution of ACE.Figure 4
The mode switching of actuator faults.Figure 5
Control output.Figure 6
The disturbanceωt.Figure 7
The evolution ofδTtRδt.


## 5. Conclusions

In this study, the finite-time LFC problem for power systems with actuator fault has been considered. To better reflect the actual demands of practical dynamics, a generalized framework of the actuator constraint has been studied. Given the randomly occurring actuator fault, a homogeneous Markov chain-based actuator fault has been studied. Together with the piecewise Lyapunov theory, sufficient conditions have been attained. In the end, a numerical example has been applied to verify the effectiveness of the developed results.



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