# Further Improvement on Stability Results for Delayed Load Frequency Control System via Novel LKFs and Proportional-Integral Control Strategy

Authors: Xingyue Liu; Kaibo Shi; Lin Tang; Yiqian Tang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this work, novel stability results for load frequency control (LFC) system considering time-varying delays, nonlinearly perturbed load, and time-varying disturbance of system parameters are proposed by using proportional-integral control strategy. Considering the nonlinearly exogenous load disturbance and system parameters disturbance, an improved stability criterion in the form of linear matrix inequalities (LMIs) is derived by novel simple Lyapunov–Krasovskii functionals (LKFs). The delay-dependent matrix in quadratic term, cross terms of variables, and quadratic terms multiplied by 1st, 2nd, and 3rd degrees of scalar functions are included in the new simple LKF. Taking the single-area and two-area LFC system installed with proportional-integral (PI) controller as example, our results surpass the previous maximum allowable size of time delay. Meanwhile, the relationship between time delay varying rate, load disturbance degree, gains of PI controller, and delay margin of the LFC system is researched separately. The results can provide guidance to tune the PI controller for achieving maximum delay margin, in which the LFC system can withstand without losing stability. At last, the simulation results verify the effectiveness and superiority of the proposed stability criterion.


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## Body

## 1. Introduction

The usage of load frequency control (LFC) is widely in the power system, which restores the balance of the power system between load demand and generation supply [1–3]. The sudden change of load can bring threaten to the safe and economic operation of the power system. Thus, the analysis and research about LFC scheme of the power system are considered to be essential and important segment to guarantee the safe operation of the power system. The LFC scheme is also effectively applied in smart grid [4, 5].A wide and dedicated open communication network is needed to transmit control signals, measurements between remote RTUs, and the control center in the conventional LFC system [3, 6]. The usage of the communication network leads to a series of inevitable problems, like time delays [7–11], packet losses, and so on. Hence, many researchers focus on the time delay phenomenon of the power control system and the impact of network delay on the communication-based power system [12–16]. From the perspective of stability analysis, it is of significant to seek the maximum allowable network delay that the power system with LFC scheme can withstand without losing sable.Generally speaking, there are mainly two approaches to seek the upper bound of the time delay system, which have different restriction and conservatism. One is the direct approach, which is frequency domain method, such as tracing eigenvalue [17, 18], or cluster treatment of characteristic roots [19–21]. The advantage of these direct methods is that the accurate delay margin can be derived by calculating eigenvalues of the whole system. However, the disadvantage of direct methods is that they can only be applied to constant time delays situation. The other is time domain method, which is based on Lyapunov stability theory and linear matrix inequalities (LMIs) techniques [22–26]. Though the time domain methods are more conservatism than the frequency domain methods, the time domain methods can be applied in both constant and time-varying delays situation. So, the time domain methods based on Lyapunov stability theory are abroad exploited.The main effort to diminish the conservatism of time domain methods has focused on two aspects, methods of constructing L-K functional and analyzing techniques for bounding the derivatives of L-K functional with regard to time. The former includes delay-division functional, functional with matrices dependent on the time delays [27], functional with two-integral or triple-integral terms [28], and so on. The latter techniques include improved majorization technique, free weighting matrix method [29], integral inequality including Jesen inequality [23], Wirtinger inequality [30], auxiliary function-based integral inequality [31], and reciprocal convex technique [32]. More and more researchers focus on the delay-dependent criteria for stability of the power system because the delay-dependent criteria have less conservatism than delay-independent criteria [22, 33].Until now, there are a few excellent research studies about stability analysis of the delayed LFC systems. Ramakrishnan and Ray [34] focused on the delay-dependent robust stability problem for the LFC system with multiple time-invariant delays in H∞ framework. The model of the LFC system with both sampling and transmission delay was proposed in [35]. The stability analysis problem was considered in [3], which focused on the LFC system with electric vehicles and time delays. The novel linear operator inequality approach was proposed, and a delay-dependent stability criterion with less conservative was obtained. Unfortunately, the load disturbance and random disturbance of system parameters were not considered either in the stability analysis or simulation in the above literatures. It is worth noting that the bounded nonlinear load perturbation model was built without considering random disturbance of system parameters in literatures [22, 36, 37]. The augmented L-K functional including single and double integral terms was developed to analyze stability of the LFC system in these literatures. The novel LKF with delay-dependent matrix in quadratic term, cross terms of variables, and quadratic terms multiple by a higher degrees of scalar function proposed by this paper contains more information about state variables than the routine L-K functional adopted in [22, 36, 37], and the results in this paper are less conservative than those of [22, 36, 37].This work proposes a novel stability criterion for the time-varying delay LFC power system considering nonlinearly perturbed and random disturbance of system parameters. Considering unknown exogenous load disturbance and random disturbance of system parameters as constrained time-varying nonlinear function related to current and delayed state vectors, the delayed LFC system with PI controller can be described as a time-varying delay system. Then, the novel stability criterion of the LFC power system is expressed in the form of LMIs by a simple LKF and integral inequalities. The main opinion of constructing the new LKF is the application of delay-dependent matrix, cross terms of variables, and quadratic terms multiplied by 1st, 2nd, and 3rd degrees of scalar function. Moreover, the case studies are carried out taking the single-area and two-area LFC system installed with PI controller as example with simulation examples to verify the effectiveness and superiority of the proposed stability criteria.


## 2. System Description and Problem Formulation

The notations ofjth area for the LFC system are given in Table 1. The subscript j should be omitted for the single-area LFC system. For example, KPi can be changed to KP in the single-area LFC system. It is worth noting that ΔPtie=0 in the single-area LFC system.Table 1
Notations.
ΔfjDeviation of frequencyKPjProportional gainKIjIntegral gainACEjArea control errorΔLdjDisturbance of loadDjDamping constantMjGenerator inertia momentTtjTurbine time constantTgjTime-constantγjFrequency bias factorΔPtieTie-line power transferΔPmjGenerator outputΔPvjGenerator valve positionv^jSpeed dropRijSynchronize coefficientConsider the uncertainties in the power parameters of the actual LFC system. The state-space model with disturbance parameters and load disturbance of the multiarea LFC system (N area) is presented as follows:(1)x˙t=A+ΔAtxt+∑j=1NBj+ΔBjtxt−hjt+CΔLdt,xt=ϕt,t∈−maxhjt,0,where xt represents state vector. A and B represent the system matrices related to the current state and delayed state vectors. ΔAt=NFtD1 and ΔBt=NFtD2 represent perturbation parameters. Ft is a time-varying matrix, which satisfies FTtFt≤I. N, D1, and D2 are the constant matrix. hjt represents the network time-varying delay in j th area. C represents the known matrix related to the disturbance vector. The initial state can be described by a continuous function defined in t∈−maxhjt,0.


### 2.1. LFC Modeling

In order to simplify the calculation, it supposes thathjt=ht, which means the multiple delays are supposed to be equal and shown as a unified time-varying delay ht. So, the state-space model of multiple-area can be presented as follows:(2)x˙t=A+ΔAtxt+B+ΔBtxt−ht+CΔLdt0≤ht≤h,−μ≤h˙t≤μ,xt=ϕt,t∈−h,0,where B=∑i=1nBi and ΔBt=∑i=1nΔBit. h and μ represent the upper bound of time-varying delay and the delay derivative separately. The constraints of h>0 and 0<μ<1 can make time-varying delay systems well-posed because the fast time-varying delay will cause a series of problems, like causality, minimality, inconsistency, and so on.Furthermore, consideringΔAt=NFtD1 and ΔBt=NFtD2, the state-space model of multiple-area can be rewritten as follows:(3)x˙t=Axt+Bxt−ht+CΔLdt+NFtD1xt+D2xt−ht0≤ht≤h,−μ≤h˙t≤μ,xt=ϕt,t∈−h,0.For a typical two-area LFC system, the dynamic model is shown in Figure1. e−sht is network time-varying delay notation. The state vectors, load disturbance vectors, and system matrices are presented as follows:(4)xt=Δf1tΔPm1tΔPv1t∫ACE1tdtΔP12tΔf2tΔPm2tΔPv2t∫ACE2tdtT,A=−D1M11M100−1M100000−1Tt11Tt1000000−1v^1Tt10−1Tt1000000γ1000100002πR120000−2πR1200000001M2−D2M21M200000000−1Tt21Tt2000000−1v^2Tt20−1Tt200000−1γ2000,B=000000000000000000−γ1KP1Tg100−KI1Tg1−KP1Tg100000000000000000000000000000000000000000000KP2Tg2−γ2KP2Tg200−KI2Tg2000000000,C=−1M1000000000000−1M20000TΔLdt=ΔLd1tΔLd2tTFt=F1tF2tT.Figure 1
Dynamic model of two-areas LFC scheme and single-area without dotted lines.For thej th area, j=1,2, the notations are presented in Table 1.The dynamic model of the single-area LFC system is shown in Figure1 without dotted lines. The state vector, disturbance vector, and system matrices of the single-area LFC system are reduced as follows:(5)xt=ΔftΔPmtΔPvt∫ACEtdsTA=−DM1M000−1Tt1Tt0−1v^Tg0−1Tg0γ000,B=00000000−γKPTg00−KITg0000,ΔLdt=ΔLdtTFt=FtT,C=−1M000T.Remark 1.
When there areNN>1 subareas connected by tie-lines forming a multiarea power system, the dotted line will be copied by N times. The decentralized control strategy is applied in the multiarea power system, which means every control area is independent and has its own LFC center to maintain the balance of generation and load. As a result, the interactions between i th area and other areas, ∑j=1,j≠iNTijΔfj, are treated as disturbances for i th area. Then, define xit=colΔfTΔPmiTΔPvi∫ACEiTΔPline−iT, where (4) is the expanded form of xit in two-area power system. And the system matrices of multiareas are listed as follows:(6)A=A11A12…A1NA21A22…A2N⋮⋮⋱⋮AN1AN2…ANN,Aii=−DiMi1Mi00−1Mi0−1Tchi1Tchi00−1RiTgi0−1Tgi00βi00012πΣk=1,k≠iNTik0000,Aik=000000000000000000002πTik0000,B=diagB1,B2,…,BN,Bi=001Tgi00T,C=diagC1,C2,…,CN,Ci=βi000100010,F=diagF1,F2,…,FN,Fi=−1Mi0000T,ΔLdt=digΔLd1t…ΔLdit.


### 2.2. Disturbance Parameters and Load Disturbance Model of LFC System

This paper models the unknown disturbance of power system parameters asNFtD1xt+D2xt−ht=Npt,xt,xt−ht, which is nonlinear function related to current and delayed state vector. Owing to the restriction of FtTFt≤I, the following inequalities can be obtained:(7)pTt,xt,xt−htpt,xt,xt−ht=D1xt+D2xt−htTFTtFtD1xt+D2xt−ht≤D1xt+D2xt−htTD1xt+D2xt−ht.Similarly, the load disturbance is considered as time-varying nonlinear function with current and delayed state vector asCΔLdt=gt,xt,xt−ht, which satisfy the following norm-bounded constraints:(8)gt,xt,xt−ht≤ρxt+τxt−ht,(9)gTt,xt,xt−htgt,xt,xt−ht≤ρ2xTtGTGxt+τ2xTt−htHTHxt−ht.where ρ≥0 and τ≥0 represent known scalars. G and H represent known constant matrices with appropriate dimensions. ρ, τ, G, and H bound the magnitude of time-varying load disturbance of power system together.


## 2.1. LFC Modeling

In order to simplify the calculation, it supposes thathjt=ht, which means the multiple delays are supposed to be equal and shown as a unified time-varying delay ht. So, the state-space model of multiple-area can be presented as follows:(2)x˙t=A+ΔAtxt+B+ΔBtxt−ht+CΔLdt0≤ht≤h,−μ≤h˙t≤μ,xt=ϕt,t∈−h,0,where B=∑i=1nBi and ΔBt=∑i=1nΔBit. h and μ represent the upper bound of time-varying delay and the delay derivative separately. The constraints of h>0 and 0<μ<1 can make time-varying delay systems well-posed because the fast time-varying delay will cause a series of problems, like causality, minimality, inconsistency, and so on.Furthermore, consideringΔAt=NFtD1 and ΔBt=NFtD2, the state-space model of multiple-area can be rewritten as follows:(3)x˙t=Axt+Bxt−ht+CΔLdt+NFtD1xt+D2xt−ht0≤ht≤h,−μ≤h˙t≤μ,xt=ϕt,t∈−h,0.For a typical two-area LFC system, the dynamic model is shown in Figure1. e−sht is network time-varying delay notation. The state vectors, load disturbance vectors, and system matrices are presented as follows:(4)xt=Δf1tΔPm1tΔPv1t∫ACE1tdtΔP12tΔf2tΔPm2tΔPv2t∫ACE2tdtT,A=−D1M11M100−1M100000−1Tt11Tt1000000−1v^1Tt10−1Tt1000000γ1000100002πR120000−2πR1200000001M2−D2M21M200000000−1Tt21Tt2000000−1v^2Tt20−1Tt200000−1γ2000,B=000000000000000000−γ1KP1Tg100−KI1Tg1−KP1Tg100000000000000000000000000000000000000000000KP2Tg2−γ2KP2Tg200−KI2Tg2000000000,C=−1M1000000000000−1M20000TΔLdt=ΔLd1tΔLd2tTFt=F1tF2tT.Figure 1
Dynamic model of two-areas LFC scheme and single-area without dotted lines.For thej th area, j=1,2, the notations are presented in Table 1.The dynamic model of the single-area LFC system is shown in Figure1 without dotted lines. The state vector, disturbance vector, and system matrices of the single-area LFC system are reduced as follows:(5)xt=ΔftΔPmtΔPvt∫ACEtdsTA=−DM1M000−1Tt1Tt0−1v^Tg0−1Tg0γ000,B=00000000−γKPTg00−KITg0000,ΔLdt=ΔLdtTFt=FtT,C=−1M000T.Remark 1.
When there areNN>1 subareas connected by tie-lines forming a multiarea power system, the dotted line will be copied by N times. The decentralized control strategy is applied in the multiarea power system, which means every control area is independent and has its own LFC center to maintain the balance of generation and load. As a result, the interactions between i th area and other areas, ∑j=1,j≠iNTijΔfj, are treated as disturbances for i th area. Then, define xit=colΔfTΔPmiTΔPvi∫ACEiTΔPline−iT, where (4) is the expanded form of xit in two-area power system. And the system matrices of multiareas are listed as follows:(6)A=A11A12…A1NA21A22…A2N⋮⋮⋱⋮AN1AN2…ANN,Aii=−DiMi1Mi00−1Mi0−1Tchi1Tchi00−1RiTgi0−1Tgi00βi00012πΣk=1,k≠iNTik0000,Aik=000000000000000000002πTik0000,B=diagB1,B2,…,BN,Bi=001Tgi00T,C=diagC1,C2,…,CN,Ci=βi000100010,F=diagF1,F2,…,FN,Fi=−1Mi0000T,ΔLdt=digΔLd1t…ΔLdit.


## 2.2. Disturbance Parameters and Load Disturbance Model of LFC System

This paper models the unknown disturbance of power system parameters asNFtD1xt+D2xt−ht=Npt,xt,xt−ht, which is nonlinear function related to current and delayed state vector. Owing to the restriction of FtTFt≤I, the following inequalities can be obtained:(7)pTt,xt,xt−htpt,xt,xt−ht=D1xt+D2xt−htTFTtFtD1xt+D2xt−ht≤D1xt+D2xt−htTD1xt+D2xt−ht.Similarly, the load disturbance is considered as time-varying nonlinear function with current and delayed state vector asCΔLdt=gt,xt,xt−ht, which satisfy the following norm-bounded constraints:(8)gt,xt,xt−ht≤ρxt+τxt−ht,(9)gTt,xt,xt−htgt,xt,xt−ht≤ρ2xTtGTGxt+τ2xTt−htHTHxt−ht.where ρ≥0 and τ≥0 represent known scalars. G and H represent known constant matrices with appropriate dimensions. ρ, τ, G, and H bound the magnitude of time-varying load disturbance of power system together.


## 3. Main Results

Before deriving the main results, Lemma1 should be introduced. Lemma 1 bounds the integral of quadratic function multiplied by a 1st/2nd degree scalar function [26].Lemma 1.
For a positive definite matrixM with appropriate dimensions and a vector fs, the following inequalities can be obtained for all scalar valued function κs≥0,s∈t1,t2:(10)−∫t1t2κsfTsMfsds≤α1εtTF1TM−1F1εt+2εtTF1T∫t1t2κsfsds,−∫t1t2κ2sfTsMfsds≤α2εtTF2TM−1F2εt+2εtTF2T∫t1t2κsfsds.where α1=∫t1t2κs and α2=t2−t1. And F1 and F2 represent appropriate dimensional matrices and εt represents arbitrary vector. Then, we can obtain the delay-dependent robust stability criterion of system (3).Theorem 1.
Given constantsh and μ, the considered system (3) is asymptotically stable if there exist positive matrices P1,P2∈ℝ4n×4n, Q1,Q2,Q3∈ℝ+2n×2n, L1,L2∈ℝ+n×n, and any matrices K1,K4∈ℝ2n×9n and K2,K3,K5,K6∈ℝn×9n satisfying the following LMIs:(11)P1+htP2>0Ωβ1hK1ThK2ThK3T⋆−hQ300⋆⋆−L10⋆⋆⋆−h3L2<0,Ωβ2hK1ThK2ThK3T⋆−hQ300⋆⋆−L10⋆⋆⋆−h3L2<0,Ωβ3hK3ThK4ThK5T⋆−hQ300⋆⋆−L10⋆⋆⋆−h3L2<0,Ωβ4hK3ThK4ThK5T⋆−hQ300⋆⋆−L10⋆⋆⋆−h3L2<0,where β1,β2,β3,andβ4 represent ht=0 and h˙t=−μ, ht=0 and h˙t=μ, ht=h and h˙t=−μ, and ht=h and h˙t=μ, the four situations, respectively. Moreover, the details about other notations are listed in Appendix.Proof.
The Lyapunov functionals are adopted as follows:(12)Vt=∫t−httΞ1Tt,sQ1Ξ1t,s+Ξ2TsQ1Ξ2sds+∫t−ht−htΞ1Tt,sQ2Ξ1t,s+Ξ2TsQ2Ξ2sds+ξTtP1+htP2ξt+∫t−hth−t+sΞ2TsQ3Ξ2s+x˙Tsh−t+s2L1+h−t+s3L2x˙sds,where ξt=xTt,xTt−ht,xTt−h,∫t−htxTsdsT,Ξ1t,s=xTt,xTsT,Ξ2s=xTs,x˙TsT.
Then, defineζt∈ℝ9n×1 as ζt=xTtxTt−htxTt−h∫t−ht−htxTsds∫t−httxTsdsx˙Tt−htx˙Tt−h,gTt,xt,xt−htpTt,xt,xt−htT; ζTtekT represents the k th vector of ζt, e.g., ζTte1T=xTt.
Computing the derivative ofVt along the trajectories of system (3):(13)V˙t=2ξTtP1+htP2ξ˙t+2∫t−httΞ1Tt,sQ1Ξ˙1Tt,sds+2∫t−ht−htΞ1Tt,sQ2Ξ˙1t,sds+h˙tξTtP2ξt+Ξ1Tt,tQ1Ξ1t,t+1−h˙tΞ1Tt,t−htQ2−Q1Ξ1t,t−ht+Ξ2TtQ1+hQ3Ξ2t+h2x˙TtL1x˙t+1−h˙tΞ2Tt−htQ2−Q1Ξ2t−ht−Ξ1Tt,t−hQ2Ξ1t,t−h−Ξ2Tt−hQ2Ξ2t−h+h3x˙TtL3x˙t−∫t−htΞ2TsQ3Ξ2sds−2∫t−hth−t+sx˙TsL1x˙sds−3∫t−hth−t+s2x˙TsL2x˙sds=ζTtHeG1TP1+htP2G2+G3TQ1G4+G5TQ2G6+Πζt+fat.
The sum of all integral terms is expressed byfat:(14)fat=−∫t−htΞ2TsQ3Ξ2s+2h−t+sx˙TsL1x˙s+3h−t+s2x˙TsL2x˙sds.
Owing to−h−t+s≤−ht−t+s,∀s∈t−ht,t, the following inequalities can be obtained:(15)fat≤−∫t−ht−htΞ2TsQ3Ξ2s+2ht−t+sx˙TsL1x˙s+3ht−t+s2x˙TsL2x˙sds−∫t−httΞ2TsQ3Ξ2s+2ht−t+sx˙TsL1x˙s+3ht−t+s2x˙TsL2x˙sds=Υt.
Using Lemma1 to estimate Υt:(16)Υt≤ζTth−htK1TQ3−1K1+2K1Te4,e2−e3T+h−ht2K2TL1−1K2+4K2Th−hte2−e4T+3h−htK3TL2−1K3+6K3Th−hte2−e4T+htK4TQ3−1K4+2K4Te5,e1−e2T+h2tK5TL1−1K5+4K5Thte1−e5T+3htK6TL2−1K6+6K6Thte1−e5Tζt.
The following inequality holds for anyς≥0 from (9) and (7), respectively:(17)−ςg.Tg.+ςρ2xTtGTGxt+ςτ2xTt−htHTHxt−ht≥0,−p.Tp+D1xt+D2xt−htTD1xt+D2xt−ht≥0.
They can be expressed by the augmented state vectorζt, respectively:(18)ζTte8T−ςIe8+e1Tςρ2GTGe1+e2Tςτ2HTHe2ζt≥0,ζTte9T−Ie9+e1TD1TD1e1+e1TD1TD2e2+e2TD2TD1e1+e2TD2TD2e2ζt≥0.
By substituting (16) into (13) and adding the inequality (18), the following inequality can be obtained:(19)V˙t≤ζTtHeG1TPG2+G3TQ1G4+G5TQ2G6+H1+H2+H3+Π+Λ+Ψζt,where G1,G2,G3,G4,G5,G6,H1,H2,H3,Λ,Π have been defined in Appendix, and(20)Ψ=h−htK1TQ3−1K1+h−ht2K2TL1−1K2+3h−htK3TL2−1K3+htK4TQ3−1K4+htH5TL1−1K5+3htK6TL2−1K6.
Finally, considering0≤ht≤h and −μ<h˙t<μ, inequality (9) is obtained by Schur complement, which ensures the following inequality:(21)V˙t≤ζTtHeG1TPG2+G3TQ1G4+G5TQ2G6+H1+H2+H3+Π+Λ+Ψζt,<−ϵζt2.∃ϵ>0,≤−ϵ^xt2.∃ϵ^>0.
Hence, the system is asymptotically stable. The proof is finished.Remark 2.
The condition of positive matricesP1,P2∈ℝ4n×4n, Q1,Q2,Q3∈ℝ+2n×2n, and L1,L2∈ℝ+n×n in Theorem 1 can guarantee the adopted Lyapunov functionals Vt being positive. Thus, the positive definite of Vt and negative definite of Vt in time domain −h,0 can guarantee the asymptotic stability of system (3) in −h,0.Remark 3.
Our work has investigated the stability condition of LFC system (3) under the limitation 0≤ht≤h and −μ≤h˙t≤μ. The limitation conditions are 0<ht<h,−μ<h˙t<μ. That is to say, the stability condition should be satisfied under the four situations, β1:ht=h1,h˙t=−μ, β2:ht=h1,h˙t=μ, β3:ht=h2,h˙t=−μ, and β4:ht=h2,h˙t=μ. V˙t<0 must be guaranteed under the four situations. Thus, inequality (9) under β1−β4 in Theorem 1 must be satisfied simultaneously for system asymptotic stability.Remark 4.
The main contribution of this section is to propose a novel Theorem1 in the form of LMIs which can derive a less conservative result of maximum allowable network delay for LFC system (3). The LKF constructed by literature [36] is Vt=xTtPxt+∫t−τttxTsQ1xsds+∫t−τtxTsQ2xsds+τ∫−τ0∫t+θtx˙TsRx˙sdsdθ. Compared with that, the LKF proposed by this paper has four improvement points. (i) The augmented variable xTt,xTt−ht,xTt−h,∫t−htxTsdsT, rather than xt in [36], which can reduce the conservatism by increasing the link between variables. (ii) The time-delay dependent matrix P1+htP2 is introduced in quadratic term, rather than constant matrix P in [36], which can reduce the conservatism by introducing more messages about time-varying delay. (iii) The cross terms of variable xt,xs,xs,x˙s in Ξ1t,s,Ξ2t,s also can reduce the conservatism by increasing the link between variables. (iv) The 4th, 5th, and 6th terms of the novel LKF multiplied by 1st, 2nd, and 3rd degrees of a scalar function h−t+s. The increasing of h−t+s by one signifies the increasing the number of integral by one. Compared with the LKF of literatures [22, 36, 37], we increase the quadratic terms multiplied by 2nd and 3rd degrees of the scalar function h−t+s, which may reduce the conservatism compared with literatures [22, 36, 37].Remark 5.
If there is a small disturbance or no disturbed load in the LFC system, the delay-dependent robust stability criterion can be developed from Theorem1 by setting ρ=0,τ=0. If there is no or very small disturbance in LFC system parameters, set N=0.


## 4. Case Studies

In this section, the typical single-area and two-area LFC systems are carried out to analyze the delay margins based on Theorem1. The parameters of the real two-area LFC system are listed as Table 2. For verifying the less conservatism of our proposed method, the parameters of the two-area LFC system given in [37] are presented in Table 2 for comparison purposes. M, D, γ, Tg, Tt, and v^ represent the inertia moments of generator, the generator damping coefficient, the frequency bias factor, time constant of the governor, the turbine time constant, and the speed drop separately. Delay margins with respect to different value of ρ, τ, gains of PI controller are calculated for constant and time-varying time delays. Then, simulation studies based on MATLAB/Simulink are shown to investigate the effect of time delay on control performance and verify the improvement and effectiveness of the Theorem 1 proposed by this paper.Table 2
Parameters of two-area LFC systems.
ParameterTt/sTg/sv^DγM/sT12Area 10.30.10.051.021100.1986Area 20.40.170.051.521.512


### 4.1. Theoretical Delay Margins

#### 4.1.1. Single-Area LFC System

The parameters of Area 1 in Table2 are adopted as single-area LFC system parameters.SettingKP as very small, we investigate the effect of Integral (I) controller on delay margin under three situations of load disturbance (ρ=0,τ=0; ρ=0,τ=0.025; and ρ=0.025,τ=0.025) for both constant and time-varying delays. Suppose G and H in (9) and D1 and D2 in (8) as 0.1In, where n represents the dimension of the system state vector xt. It can be seen that n=4 in the single-area LFC system from (5). Theorem 1 is adopted to calculate the delay margins. The results are shown in Table 3 and Figures 2–4.Table 3
Delay marginh∝KI under three situations of load disturbance (single-area LFC).
KI0.050.10.20.40.61ρ=0τ=0μ=032.8915.7527.823.82132.2140.9824μ=0.525.17810.8185.8122.8131.51230.5812ρ=0,τ=0.025μ=026.12412.4536.5123.5412.1120.8721μ=0.524.9819.7184.7742.5721.4170.4882ρ=0.025,τ=0.025μ=025.51612.0115.9183.2111.8120.743μ=0.523.8287.5213.5142.0121.2280.334Figure 2
Delay marginh∝KI for single-area with ρ=0,τ=0.Figure 3
Delay marginh∝KI for single-area with ρ=0,τ=0.025 .Figure 4
Delay marginh∝KI for single-area with ρ=0.025,τ=0.025.The following conclusions can be obtained from Table3 and Figures 2–4:(i)
The increasing ofρ and τ represents that the load fluctuation becomes more pronounced. With the increase in ρ and τ, the delay margin that PI controller can withstand without decrease in losing stability under the same value of μ and KI. The effect of load disturbance on delay-dependent stability of the LFC system can be obtained.(ii)
It can be seen from Figures2–4 that the increasing of μ causes the decreasing of delay margin h with the same value of KI under the three situations of load disturbance (ρ=0,τ=0; ρ=0,τ=0.025; ρ=0.025,τ=0.025). The effect of time delay changing rate on delay-dependent stability of the LFC system can be obtained.(iii)
Simultaneously, it can be seen from Figures2–4 that maximum allowable network delay decreases with the increasing of KI with the same value of μ under the three situations of load fluctuation.As for the nominal system conditionρ=0,τ=0 with constant μ=0 and time-varying delay μ=0.9, the maximum allowable network delays are listed in Table 4 compared with the other results from literatures [22, 36, 37]. It can be seen that the results calculated by Theorem 1 are less conservative than those of [22, 36, 37], owing to the progressiveness of Lyapunov functional and Lemma 1 adopted by this paper. Therefore, when compared with [22, 36, 37], the results proposed by this paper are more realistic.Table 4
Delay marginh by Theorem 1 for single-area LFC with ϱ=0,τ=0.
KP000000.10.10.10.10.1KI0.050.20.40.610.050.20.40.61μ=0Theorem132.897.823.82132.2140.982432.987.893.832.291.12[22]27.926.693.121.910.8827.036.943.292.020.96[36]27.926.693.121.910.8827.056.943.292.020.96[37]30.917.333.382.040.9231.617.793.612.191.01μ=0.9Theorem127.876.543.021.820.7822.846.643.141.860.79[22]20.454.591.811.0110.4817.394.671.851.050.48[36]26.376.252.851.680.7420.255.942.871.750.74[37]27.266.432.911.710.75226.593.111.840.75Results of the maximum allowable network delays with regard to gains of PI controllerKP,KI are listed in Table 5 and Figure 5 with constant delays μ=0 and no load disturbance ρ=0,τ=0 and in Table 6 and Figure 6 with time-varying delays μ=0.9 and load disturbance ρ=0.025,τ=0.025, separately. Results demonstrate that smaller h is obtained under the bigger KP,KI, and a relatively larger h is acquired for the smaller KP,KI. h is reduced with the increasing of KP when KI≤0.05. For KI≥0.1, h increases at first and then decreases with the increasing of KP. This results implies that for a given KI, there is a optimal value of KP which can acquire the maximum h, which can give guidance to tune PI controller to achieve maximum delay margins. Moreover, it can be found that h of the time-varying delays is smaller and decreases faster than that of constant delays.Table 5
Delay marginh∝KP,KI for single-area LFC with μ=0,ρ=0,τ=0.
hKPKI00.050.10.20.40.610.0532.8931.7230.9828.31226.42425.12420.8180.116.4314.3416.8915.2410.245.712.140.159.238.829.518.924.141.560.750.27.826.237.897.8993.121.520.540.43.82132.4133.834.213.021.420.320.62.2141.8122.293.313.011.020.1210.98240.8131.122.322.480.830.08Figure 5
h∝KP,KI (single-area with μ=0,ρ=0,τ=0).Table 6
Delay marginh∝KP,KI for single-area LFC with μ=0.9,ρ=0.025,τ=0.025.
hKPKI00.050.10.20.40.610.0522.54121.03119.38312.5412.842.60.5310.111.83012.16311.1647.8022.8232.540.5270.158.138.1538.0636.1422.8042.520.5190.26.5847.7026.4725.2342.7842.110.5110.43.6123.9233.8423.4362.6842.010.50.62.0112.0482.0521.9431.670.980.481.00.8210.7270.7410.7990.680.540.44Figure 6
h∝KP,KI (single-area with μ=0.9,ρ=0.025,τ=0.025).


#### 4.1.2. Two-Area LFC System

Maximum allowable network delayh is calculated by Theorem 1 for the two-area LFC system listed in Table 2 by simplifying the multiple delay as a single delay like (3) and gains of PI controller for two areas as same (KPi=KP, KIi=KI). In the two-area LFC system, both the values of G, H, D1, and D2 are taken as 0.1In, where n=9.In two-area LFC system case, for more clearly comparisons with literatures [22, 36, 37], the results are summarized in Table 7 with the condition of ρ=0,τ=0 under constant μ=0 and time-varying delay μ=0.5. Similarity, the listed results are less conservative than those of [22, 36, 37]. Also, the results of Table 7 show the effect of PI controller gains on the delay margins.Table 7
Delay marginh by Theorem 1 for two-area LFC with ϱ=0,τ=0.
KP00000.10.10.1KI0.20.40.610.20.40.6μ=0Theorem17.513.542.10.827.813.522.31[22]6.631.740.576.883.171.86[36]6.631.740.576.883.171.86[37]7.233.241.860.587.673.472.01μ=0.5Theorem16.522.921.840.526.812.921.82[22]5.552.361.180.225.352.551.3[36]6.142.681.40.356.342.831.51[37]6.412.811.540.416.752.841.53Also, the results of delay margin with respect to gains of PI controller are listed in Table8 and Figure 7 with constant delay μ=0 and no load fluctuation ρ=0,τ=0 and in Table 9 and Figure 8 with time-varying delay μ=0.9 and load fluctuation ρ=0.025,τ=0.025, respectively. The delay margin changing trend with regard to gains of PI controller of the two-area LFC system is consistent with that of the single-area LFC system. Hence, the details can be omitted here.Table 8
Delay marginh∝KP,KI for two-area LFC with μ=0,ϱ=0,τ=0.
hKPKI00.050.10.20.40.610.0528.44228.38727.52126.1921.01215.3280.5640.114.69914.82114.25013.78211.0128.3320.4640.159.2349.5249.2628.7728.0145.0210.4440.27.517.827.936.8725.8234.6210.4330.43.543.723.813.922.922.010.3840.62.12.242.412.582.241.890.34210.810.920.981.020.920.810.227Figure 7
h∝KP,KI (two-area LFC with μ=0,ρ=0,τ=0).Table 9
Delay marginh∝KP,KI for two-area LFC with μ=0.9,ϱ=0.025,τ=0.025.
hKPKI00.050.10.20.40.610.0524.98223.98223.01221.47514.4640.9240.3280.112.75413.07112.6311.4217.7450.9120.320.157.9178.278.0127.2594.9050.8230.3120.26.556.977.026.784.620.790.3040.42.4722.6722.8542.5411.8240.620.2720.61.922.012.342.481.020.520.2410.320.340.380.40.410.250.172Figure 8
h∝KP,KI (two-area LFC with μ=0.9,ρ=0.025,τ=0.025).


### 4.2. Simulation Verification

The models of the single-area and two-area LFC system are built in MATLAB/Simulink separately. The simulation results verify the above calculation results. The state vectorsΔf are selected as the observation since the load fluctuation directly affects the frequency of the LFC system.


#### 4.2.1. Single-Area LFC System

It can be seen from Table4 that the maximum allowable network delay calculated by Theorem 1 is h=6.64 under the condition of ρ=τ=0,μ=0.9,KP=0.1,KI=0.2. The time-varying delay ht is supposed to be a sine function ht=3.32+3.32sin0.27t. Figure 9 shows that the state vector Δf converges asymptotically to stable point for h=6.64; however, the delay margin calculated by [22] is 4.67, calculated by [36] is 5.94, and calculated by [37] is 6.59. Moreover, if h exceeds beyond 6.64, the system becomes unstable as shown in Figure 10 for h=7s.Figure 9
Frequency responses withh=6.64μ=0.9 of single-area.Figure 10
Frequency responses withh=7 of single-area.


#### 4.2.2. Two-Area LFC System

At first, whenht=0, the state-space model of two-area LFC system becomes x˙t=A+ΔA+B+ΔBxt+CΔLdt. Figure 11 shows that the delay-free system is asymptotically stable. It can be seen from Table 9 that the maximum allowable delay calculated by Theorem 1 is h=6.78 under the condition of ρ=τ=0.025,μ=0.9,KP=KI=0.2. Then, it can be seen from Figure 11 that the two-area LFC system is asymptotically stable under h=6.78. Moreover, if h exceeds beyond 6.78, the system becomes unstable as shown in Figure 12 for h=7s.Figure 11
Frequency responses withh=6.81μ=0.5 and without delay of two areas.Figure 12
Frequency responses withh=7 of two areas.


## 4.1. Theoretical Delay Margins

### 4.1.1. Single-Area LFC System

The parameters of Area 1 in Table2 are adopted as single-area LFC system parameters.SettingKP as very small, we investigate the effect of Integral (I) controller on delay margin under three situations of load disturbance (ρ=0,τ=0; ρ=0,τ=0.025; and ρ=0.025,τ=0.025) for both constant and time-varying delays. Suppose G and H in (9) and D1 and D2 in (8) as 0.1In, where n represents the dimension of the system state vector xt. It can be seen that n=4 in the single-area LFC system from (5). Theorem 1 is adopted to calculate the delay margins. The results are shown in Table 3 and Figures 2–4.Table 3
Delay marginh∝KI under three situations of load disturbance (single-area LFC).
KI0.050.10.20.40.61ρ=0τ=0μ=032.8915.7527.823.82132.2140.9824μ=0.525.17810.8185.8122.8131.51230.5812ρ=0,τ=0.025μ=026.12412.4536.5123.5412.1120.8721μ=0.524.9819.7184.7742.5721.4170.4882ρ=0.025,τ=0.025μ=025.51612.0115.9183.2111.8120.743μ=0.523.8287.5213.5142.0121.2280.334Figure 2
Delay marginh∝KI for single-area with ρ=0,τ=0.Figure 3
Delay marginh∝KI for single-area with ρ=0,τ=0.025 .Figure 4
Delay marginh∝KI for single-area with ρ=0.025,τ=0.025.The following conclusions can be obtained from Table3 and Figures 2–4:(i)
The increasing ofρ and τ represents that the load fluctuation becomes more pronounced. With the increase in ρ and τ, the delay margin that PI controller can withstand without decrease in losing stability under the same value of μ and KI. The effect of load disturbance on delay-dependent stability of the LFC system can be obtained.(ii)
It can be seen from Figures2–4 that the increasing of μ causes the decreasing of delay margin h with the same value of KI under the three situations of load disturbance (ρ=0,τ=0; ρ=0,τ=0.025; ρ=0.025,τ=0.025). The effect of time delay changing rate on delay-dependent stability of the LFC system can be obtained.(iii)
Simultaneously, it can be seen from Figures2–4 that maximum allowable network delay decreases with the increasing of KI with the same value of μ under the three situations of load fluctuation.As for the nominal system conditionρ=0,τ=0 with constant μ=0 and time-varying delay μ=0.9, the maximum allowable network delays are listed in Table 4 compared with the other results from literatures [22, 36, 37]. It can be seen that the results calculated by Theorem 1 are less conservative than those of [22, 36, 37], owing to the progressiveness of Lyapunov functional and Lemma 1 adopted by this paper. Therefore, when compared with [22, 36, 37], the results proposed by this paper are more realistic.Table 4
Delay marginh by Theorem 1 for single-area LFC with ϱ=0,τ=0.
KP000000.10.10.10.10.1KI0.050.20.40.610.050.20.40.61μ=0Theorem132.897.823.82132.2140.982432.987.893.832.291.12[22]27.926.693.121.910.8827.036.943.292.020.96[36]27.926.693.121.910.8827.056.943.292.020.96[37]30.917.333.382.040.9231.617.793.612.191.01μ=0.9Theorem127.876.543.021.820.7822.846.643.141.860.79[22]20.454.591.811.0110.4817.394.671.851.050.48[36]26.376.252.851.680.7420.255.942.871.750.74[37]27.266.432.911.710.75226.593.111.840.75Results of the maximum allowable network delays with regard to gains of PI controllerKP,KI are listed in Table 5 and Figure 5 with constant delays μ=0 and no load disturbance ρ=0,τ=0 and in Table 6 and Figure 6 with time-varying delays μ=0.9 and load disturbance ρ=0.025,τ=0.025, separately. Results demonstrate that smaller h is obtained under the bigger KP,KI, and a relatively larger h is acquired for the smaller KP,KI. h is reduced with the increasing of KP when KI≤0.05. For KI≥0.1, h increases at first and then decreases with the increasing of KP. This results implies that for a given KI, there is a optimal value of KP which can acquire the maximum h, which can give guidance to tune PI controller to achieve maximum delay margins. Moreover, it can be found that h of the time-varying delays is smaller and decreases faster than that of constant delays.Table 5
Delay marginh∝KP,KI for single-area LFC with μ=0,ρ=0,τ=0.
hKPKI00.050.10.20.40.610.0532.8931.7230.9828.31226.42425.12420.8180.116.4314.3416.8915.2410.245.712.140.159.238.829.518.924.141.560.750.27.826.237.897.8993.121.520.540.43.82132.4133.834.213.021.420.320.62.2141.8122.293.313.011.020.1210.98240.8131.122.322.480.830.08Figure 5
h∝KP,KI (single-area with μ=0,ρ=0,τ=0).Table 6
Delay marginh∝KP,KI for single-area LFC with μ=0.9,ρ=0.025,τ=0.025.
hKPKI00.050.10.20.40.610.0522.54121.03119.38312.5412.842.60.5310.111.83012.16311.1647.8022.8232.540.5270.158.138.1538.0636.1422.8042.520.5190.26.5847.7026.4725.2342.7842.110.5110.43.6123.9233.8423.4362.6842.010.50.62.0112.0482.0521.9431.670.980.481.00.8210.7270.7410.7990.680.540.44Figure 6
h∝KP,KI (single-area with μ=0.9,ρ=0.025,τ=0.025).


### 4.1.2. Two-Area LFC System

Maximum allowable network delayh is calculated by Theorem 1 for the two-area LFC system listed in Table 2 by simplifying the multiple delay as a single delay like (3) and gains of PI controller for two areas as same (KPi=KP, KIi=KI). In the two-area LFC system, both the values of G, H, D1, and D2 are taken as 0.1In, where n=9.In two-area LFC system case, for more clearly comparisons with literatures [22, 36, 37], the results are summarized in Table 7 with the condition of ρ=0,τ=0 under constant μ=0 and time-varying delay μ=0.5. Similarity, the listed results are less conservative than those of [22, 36, 37]. Also, the results of Table 7 show the effect of PI controller gains on the delay margins.Table 7
Delay marginh by Theorem 1 for two-area LFC with ϱ=0,τ=0.
KP00000.10.10.1KI0.20.40.610.20.40.6μ=0Theorem17.513.542.10.827.813.522.31[22]6.631.740.576.883.171.86[36]6.631.740.576.883.171.86[37]7.233.241.860.587.673.472.01μ=0.5Theorem16.522.921.840.526.812.921.82[22]5.552.361.180.225.352.551.3[36]6.142.681.40.356.342.831.51[37]6.412.811.540.416.752.841.53Also, the results of delay margin with respect to gains of PI controller are listed in Table8 and Figure 7 with constant delay μ=0 and no load fluctuation ρ=0,τ=0 and in Table 9 and Figure 8 with time-varying delay μ=0.9 and load fluctuation ρ=0.025,τ=0.025, respectively. The delay margin changing trend with regard to gains of PI controller of the two-area LFC system is consistent with that of the single-area LFC system. Hence, the details can be omitted here.Table 8
Delay marginh∝KP,KI for two-area LFC with μ=0,ϱ=0,τ=0.
hKPKI00.050.10.20.40.610.0528.44228.38727.52126.1921.01215.3280.5640.114.69914.82114.25013.78211.0128.3320.4640.159.2349.5249.2628.7728.0145.0210.4440.27.517.827.936.8725.8234.6210.4330.43.543.723.813.922.922.010.3840.62.12.242.412.582.241.890.34210.810.920.981.020.920.810.227Figure 7
h∝KP,KI (two-area LFC with μ=0,ρ=0,τ=0).Table 9
Delay marginh∝KP,KI for two-area LFC with μ=0.9,ϱ=0.025,τ=0.025.
hKPKI00.050.10.20.40.610.0524.98223.98223.01221.47514.4640.9240.3280.112.75413.07112.6311.4217.7450.9120.320.157.9178.278.0127.2594.9050.8230.3120.26.556.977.026.784.620.790.3040.42.4722.6722.8542.5411.8240.620.2720.61.922.012.342.481.020.520.2410.320.340.380.40.410.250.172Figure 8
h∝KP,KI (two-area LFC with μ=0.9,ρ=0.025,τ=0.025).


## 4.1.1. Single-Area LFC System

The parameters of Area 1 in Table2 are adopted as single-area LFC system parameters.SettingKP as very small, we investigate the effect of Integral (I) controller on delay margin under three situations of load disturbance (ρ=0,τ=0; ρ=0,τ=0.025; and ρ=0.025,τ=0.025) for both constant and time-varying delays. Suppose G and H in (9) and D1 and D2 in (8) as 0.1In, where n represents the dimension of the system state vector xt. It can be seen that n=4 in the single-area LFC system from (5). Theorem 1 is adopted to calculate the delay margins. The results are shown in Table 3 and Figures 2–4.Table 3
Delay marginh∝KI under three situations of load disturbance (single-area LFC).
KI0.050.10.20.40.61ρ=0τ=0μ=032.8915.7527.823.82132.2140.9824μ=0.525.17810.8185.8122.8131.51230.5812ρ=0,τ=0.025μ=026.12412.4536.5123.5412.1120.8721μ=0.524.9819.7184.7742.5721.4170.4882ρ=0.025,τ=0.025μ=025.51612.0115.9183.2111.8120.743μ=0.523.8287.5213.5142.0121.2280.334Figure 2
Delay marginh∝KI for single-area with ρ=0,τ=0.Figure 3
Delay marginh∝KI for single-area with ρ=0,τ=0.025 .Figure 4
Delay marginh∝KI for single-area with ρ=0.025,τ=0.025.The following conclusions can be obtained from Table3 and Figures 2–4:(i)
The increasing ofρ and τ represents that the load fluctuation becomes more pronounced. With the increase in ρ and τ, the delay margin that PI controller can withstand without decrease in losing stability under the same value of μ and KI. The effect of load disturbance on delay-dependent stability of the LFC system can be obtained.(ii)
It can be seen from Figures2–4 that the increasing of μ causes the decreasing of delay margin h with the same value of KI under the three situations of load disturbance (ρ=0,τ=0; ρ=0,τ=0.025; ρ=0.025,τ=0.025). The effect of time delay changing rate on delay-dependent stability of the LFC system can be obtained.(iii)
Simultaneously, it can be seen from Figures2–4 that maximum allowable network delay decreases with the increasing of KI with the same value of μ under the three situations of load fluctuation.As for the nominal system conditionρ=0,τ=0 with constant μ=0 and time-varying delay μ=0.9, the maximum allowable network delays are listed in Table 4 compared with the other results from literatures [22, 36, 37]. It can be seen that the results calculated by Theorem 1 are less conservative than those of [22, 36, 37], owing to the progressiveness of Lyapunov functional and Lemma 1 adopted by this paper. Therefore, when compared with [22, 36, 37], the results proposed by this paper are more realistic.Table 4
Delay marginh by Theorem 1 for single-area LFC with ϱ=0,τ=0.
KP000000.10.10.10.10.1KI0.050.20.40.610.050.20.40.61μ=0Theorem132.897.823.82132.2140.982432.987.893.832.291.12[22]27.926.693.121.910.8827.036.943.292.020.96[36]27.926.693.121.910.8827.056.943.292.020.96[37]30.917.333.382.040.9231.617.793.612.191.01μ=0.9Theorem127.876.543.021.820.7822.846.643.141.860.79[22]20.454.591.811.0110.4817.394.671.851.050.48[36]26.376.252.851.680.7420.255.942.871.750.74[37]27.266.432.911.710.75226.593.111.840.75Results of the maximum allowable network delays with regard to gains of PI controllerKP,KI are listed in Table 5 and Figure 5 with constant delays μ=0 and no load disturbance ρ=0,τ=0 and in Table 6 and Figure 6 with time-varying delays μ=0.9 and load disturbance ρ=0.025,τ=0.025, separately. Results demonstrate that smaller h is obtained under the bigger KP,KI, and a relatively larger h is acquired for the smaller KP,KI. h is reduced with the increasing of KP when KI≤0.05. For KI≥0.1, h increases at first and then decreases with the increasing of KP. This results implies that for a given KI, there is a optimal value of KP which can acquire the maximum h, which can give guidance to tune PI controller to achieve maximum delay margins. Moreover, it can be found that h of the time-varying delays is smaller and decreases faster than that of constant delays.Table 5
Delay marginh∝KP,KI for single-area LFC with μ=0,ρ=0,τ=0.
hKPKI00.050.10.20.40.610.0532.8931.7230.9828.31226.42425.12420.8180.116.4314.3416.8915.2410.245.712.140.159.238.829.518.924.141.560.750.27.826.237.897.8993.121.520.540.43.82132.4133.834.213.021.420.320.62.2141.8122.293.313.011.020.1210.98240.8131.122.322.480.830.08Figure 5
h∝KP,KI (single-area with μ=0,ρ=0,τ=0).Table 6
Delay marginh∝KP,KI for single-area LFC with μ=0.9,ρ=0.025,τ=0.025.
hKPKI00.050.10.20.40.610.0522.54121.03119.38312.5412.842.60.5310.111.83012.16311.1647.8022.8232.540.5270.158.138.1538.0636.1422.8042.520.5190.26.5847.7026.4725.2342.7842.110.5110.43.6123.9233.8423.4362.6842.010.50.62.0112.0482.0521.9431.670.980.481.00.8210.7270.7410.7990.680.540.44Figure 6
h∝KP,KI (single-area with μ=0.9,ρ=0.025,τ=0.025).


## 4.1.2. Two-Area LFC System

Maximum allowable network delayh is calculated by Theorem 1 for the two-area LFC system listed in Table 2 by simplifying the multiple delay as a single delay like (3) and gains of PI controller for two areas as same (KPi=KP, KIi=KI). In the two-area LFC system, both the values of G, H, D1, and D2 are taken as 0.1In, where n=9.In two-area LFC system case, for more clearly comparisons with literatures [22, 36, 37], the results are summarized in Table 7 with the condition of ρ=0,τ=0 under constant μ=0 and time-varying delay μ=0.5. Similarity, the listed results are less conservative than those of [22, 36, 37]. Also, the results of Table 7 show the effect of PI controller gains on the delay margins.Table 7
Delay marginh by Theorem 1 for two-area LFC with ϱ=0,τ=0.
KP00000.10.10.1KI0.20.40.610.20.40.6μ=0Theorem17.513.542.10.827.813.522.31[22]6.631.740.576.883.171.86[36]6.631.740.576.883.171.86[37]7.233.241.860.587.673.472.01μ=0.5Theorem16.522.921.840.526.812.921.82[22]5.552.361.180.225.352.551.3[36]6.142.681.40.356.342.831.51[37]6.412.811.540.416.752.841.53Also, the results of delay margin with respect to gains of PI controller are listed in Table8 and Figure 7 with constant delay μ=0 and no load fluctuation ρ=0,τ=0 and in Table 9 and Figure 8 with time-varying delay μ=0.9 and load fluctuation ρ=0.025,τ=0.025, respectively. The delay margin changing trend with regard to gains of PI controller of the two-area LFC system is consistent with that of the single-area LFC system. Hence, the details can be omitted here.Table 8
Delay marginh∝KP,KI for two-area LFC with μ=0,ϱ=0,τ=0.
hKPKI00.050.10.20.40.610.0528.44228.38727.52126.1921.01215.3280.5640.114.69914.82114.25013.78211.0128.3320.4640.159.2349.5249.2628.7728.0145.0210.4440.27.517.827.936.8725.8234.6210.4330.43.543.723.813.922.922.010.3840.62.12.242.412.582.241.890.34210.810.920.981.020.920.810.227Figure 7
h∝KP,KI (two-area LFC with μ=0,ρ=0,τ=0).Table 9
Delay marginh∝KP,KI for two-area LFC with μ=0.9,ϱ=0.025,τ=0.025.
hKPKI00.050.10.20.40.610.0524.98223.98223.01221.47514.4640.9240.3280.112.75413.07112.6311.4217.7450.9120.320.157.9178.278.0127.2594.9050.8230.3120.26.556.977.026.784.620.790.3040.42.4722.6722.8542.5411.8240.620.2720.61.922.012.342.481.020.520.2410.320.340.380.40.410.250.172Figure 8
h∝KP,KI (two-area LFC with μ=0.9,ρ=0.025,τ=0.025).


## 4.2. Simulation Verification

The models of the single-area and two-area LFC system are built in MATLAB/Simulink separately. The simulation results verify the above calculation results. The state vectorsΔf are selected as the observation since the load fluctuation directly affects the frequency of the LFC system.


### 4.2.1. Single-Area LFC System

It can be seen from Table4 that the maximum allowable network delay calculated by Theorem 1 is h=6.64 under the condition of ρ=τ=0,μ=0.9,KP=0.1,KI=0.2. The time-varying delay ht is supposed to be a sine function ht=3.32+3.32sin0.27t. Figure 9 shows that the state vector Δf converges asymptotically to stable point for h=6.64; however, the delay margin calculated by [22] is 4.67, calculated by [36] is 5.94, and calculated by [37] is 6.59. Moreover, if h exceeds beyond 6.64, the system becomes unstable as shown in Figure 10 for h=7s.Figure 9
Frequency responses withh=6.64μ=0.9 of single-area.Figure 10
Frequency responses withh=7 of single-area.


### 4.2.2. Two-Area LFC System

At first, whenht=0, the state-space model of two-area LFC system becomes x˙t=A+ΔA+B+ΔBxt+CΔLdt. Figure 11 shows that the delay-free system is asymptotically stable. It can be seen from Table 9 that the maximum allowable delay calculated by Theorem 1 is h=6.78 under the condition of ρ=τ=0.025,μ=0.9,KP=KI=0.2. Then, it can be seen from Figure 11 that the two-area LFC system is asymptotically stable under h=6.78. Moreover, if h exceeds beyond 6.78, the system becomes unstable as shown in Figure 12 for h=7s.Figure 11
Frequency responses withh=6.81μ=0.5 and without delay of two areas.Figure 12
Frequency responses withh=7 of two areas.


## 4.2.1. Single-Area LFC System

It can be seen from Table4 that the maximum allowable network delay calculated by Theorem 1 is h=6.64 under the condition of ρ=τ=0,μ=0.9,KP=0.1,KI=0.2. The time-varying delay ht is supposed to be a sine function ht=3.32+3.32sin0.27t. Figure 9 shows that the state vector Δf converges asymptotically to stable point for h=6.64; however, the delay margin calculated by [22] is 4.67, calculated by [36] is 5.94, and calculated by [37] is 6.59. Moreover, if h exceeds beyond 6.64, the system becomes unstable as shown in Figure 10 for h=7s.Figure 9
Frequency responses withh=6.64μ=0.9 of single-area.Figure 10
Frequency responses withh=7 of single-area.


## 4.2.2. Two-Area LFC System

At first, whenht=0, the state-space model of two-area LFC system becomes x˙t=A+ΔA+B+ΔBxt+CΔLdt. Figure 11 shows that the delay-free system is asymptotically stable. It can be seen from Table 9 that the maximum allowable delay calculated by Theorem 1 is h=6.78 under the condition of ρ=τ=0.025,μ=0.9,KP=KI=0.2. Then, it can be seen from Figure 11 that the two-area LFC system is asymptotically stable under h=6.78. Moreover, if h exceeds beyond 6.78, the system becomes unstable as shown in Figure 12 for h=7s.Figure 11
Frequency responses withh=6.81μ=0.5 and without delay of two areas.Figure 12
Frequency responses withh=7 of two areas.


## 5. Conclusion

The stability criterion for the time-varying delay LFC system with nonlinearly load disturbance and random disturbance of system parameters has been investigated by adopting a novel simple LKF and integral inequality (Lemma1). Considering the exogenous load disturbance and random disturbance of system parameters as constrained nonlinearly function, a time-varying delay system is modeled to describe the delayed LFC system. The novel simple LKF includes delay-dependent matrix in quadratic term, cross terms of variables, and quadratic terms multiplied by 1st, 2nd, and 3rd degrees of scalar function. Lemma 1 is adopted to bound the quadratic terms multiplied 1st and 2nd degrees of scalar function emerging in the derivative of LKF. The results of the novel simple LKF contributing to reduce conservatism and the relationships between the time delay varying rate, load disturbance degree, the gains of PI controller, and delay margins are derived by the case studied. Meanwhile, simulation verifies the effectiveness and superiority of our results. Nevertheless, the security threats in actual communication network such as denial of service (DOS) attacks and deception attacks are not considered in our work. In the future, how to analyze the stability of the LFC system under the simultaneous existence of various network attack, such as DOS and deception attacks, will be the focus of our following work.



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*Source: 4798415-2022-05-09.xml*