# A Sampling Load Frequency Control Scheme for Power Systems with Time Delays

Authors: R. Sriraman; Jihad A. Younis; C. P. Lim; P. Hammachukiattikul; G. Rajchakit; N. Boonsatit
Journal: Complexity
 (2022)

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## Abstract

In this study, we investigate the effectiveness of a robust sampled-dataH∞ load frequency control (LFC) scheme for power systems with randomly occurring time-varying delays. By using the input-delay technique, the sampled-data LFC model is reformulated as a continuous time-delay representation. Then, Bernoulli-distributed white noise sequences are used to describe randomly occurring time-varying delays in the sampled-data LFC model. Some less conservative conditions are achieved by utilizing the Lyapunov–Krasovskii functional (LKF) and employing Jensen inequality and reciprocal convex combination lemma to ensure the considered power system has mean-square asymptotic stability under the designed control scheme. The derived results are based on linear matrix inequalities (LMIs) that can readily be solved using the MATLAB LMI toolbox. The criteria obtained are used to analyze the upper bounds for time delays, and a comparison study to validate the efficacy of the presented method is presented.


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## Body

## 1. Introduction

In recent years, research on power systems has advanced tremendously due to increased public demands for electricity, which promotes the incorporation of renewable energy into power systems [1, 2]. When a power system suffers from load disturbance or fluctuation, the system operating point can change, and deviations in the frequency of the system and the planned power exchanges can influence system performance in multiple contexts. Therefore, LFC has been proposed to ensure successful power system operations by maintaining an equilibrium among power supply and demand, thereby restoring the frequency of power systems to a proper range. Recently, a number of research studies have concentrated on the development of suitable controllers based on LFC for power systems [3–5]. In this respect, LFC schemes including proportional-integral (PI) control [2, 3], sliding mode control [6, 7], adaptive control [8], robust control [9, 10], sampled-data control [11], event-triggered control [12–14], and H∞ control [15] have been well defined in the literature.However, most LFC controllers operate in the continuous case, while practical controllers are often operated using a sampling period in the discrete case. In this context, the above-mentioned controllers do not always perform optimally in practice [16]. In power systems, control error signals are first received by sensors and sent to controllers via zero-order holders (ZOHs), before being transmitted to actuators through ZOHs [17]. This sensor-controller-actuator (SCA) technique has a discrete link design, implying that the power system can be considered a continuous-discrete sampled-data model. In recent years, many studies have introduced sampled-data control designs for LFC-based systems [18–20]. Based on the LMI theory, discrete-time multivariable PID controllers were discussed in reference [19]. The study in reference [20] devised an input-delay strategy to convert a sampled-data model to a continuous model, while the studies in references [21, 22] revealed useful additional results for LFC-based power systems.In real-world applications, random abrupt events are a common issue in sampled-data models due to environmental changes. Furthermore, owing to the appearance of time delays in dynamic systems, data transmission is frequently affected by random delays from sensors and remote receivers, which can significantly slow down the transmission of information [23, 24]. Therefore, investigations into sampled-data control systems with random delays have recently increased [11, 25, 26]. As an example, random delays are important for the application of the H∞ algorithm performance, as reported in reference [25]. In reference [26], type-2 fuzzy systems with probabilistic delays and actuator failures were explored under nonfragile sampled-data control. A novel method for the finite-time stability for sampled-data control systems with random delays was proposed in reference [11].On the other hand, many physical systems, including power systems, are susceptible to uncertainty under various conditions. It is essential to investigate power systems with uncertain parameters. There have been a few studies on robust LFC schemes for power systems [27–31]. In reference [29], the study examined LFC design for delayed power systems integrating a robust decentralized PI control. Applying Riccati equation, the study in reference [30] introduced a robust LFC method for one-area power systems. In reference [31], the potential of a physical system comprehension for robust controller designs in power systems was examined. On the other hand, the classic H∞ control theory defines a control rule that yields the minimal value of the measured performance under the assumption of zero initial values. In this regard, the delay-dependent H∞ LFC scheme for power systems has recently received significant interest from researchers [32–35]. For instance, in references [36, 37], based on the Lyapunov functional theory, some sufficient conditions are attained for H∞ performance of uncertain systems, which gives the minimum value of the measured performance. In reference [32], the issue of robust H∞ LFC methods for delayed multiarea power systems was examined. In reference [33], decentralized H∞ LFC strategy for multiarea power systems including communication uncertainties was studied. Some relevant studies have explored the issue of H∞ LFC methods for power systems with communication delays [34, 35]. To the best of our knowledge, the robust sampled-data H∞ LFC problem for power systems with randomly occurring time-varying delays has never been completely addressed. As such, there is still potential for further investigation and development in the stability and stabilization analysis of power systems by addressing the robust sampled-data H∞ LFC scheme, which motivated the present study.On the basis of the above discussions, our primary goal in this study is to address the problem of robust sampled-data LFC for stability and stabilization of power systems with randomly occurring time delays by tackling the robust sampled-dataH∞ LFC scheme. The following are the major contributions of our research:(i)
Different from the traditional analyses, this article presents the robust sampled-dataH∞ LFC for power systems with randomly occurring time-varying delays. By taking the probability distribution characteristic of communication delays into consideration for LFC design, the power systems with PI controllers are modelled as stochastic time-delay systems. Moreover, the input-delay strategy converts the sampled-data model into a continuous time-delay representation.(ii)
The LMI-based mean-square asymptotic stability and stabilization criteria for power systems are established under the designed sampled-dataH∞ LFC scheme by constructing an appropriate LKF and employing Jensen inequality and the reciprocal convex combination inequality.(iii)
Two examples are offered to demonstrate that the presence of maximum allowable upper bounds of time delays by our approach is much better than the most recent results. A comparison study is also performed, demonstrating the low computational efficiency of the obtained criteria.This study is organized as follows: the designed sampled-data LFC scheme for power systems is formally specified in Section2. In Section 3, the sufficient criteria for stability and stabilization of the considered model are presented. In Section 4, the application of the robust sampled-data H∞ LFC scheme to power systems with uncertainties is discussed. The case studies in Section 5 show the potential of the criteria presented. Conclusions are given in Section 6.Notations. In the following sections, the superscripts ″T″ and ″−1″ indicate matrix transposition and inverse, respectively. I is any matrix with identity. Any matrix Q>0Q<0 denotes the positive definite (negative definite) matrix. The block diagonal matrix is represented by diag…, while ℝn and ℝn×n denote the Euclidian n space and n×n real matrices, respectively. In a symmetric matrix, ∗ denotes symmetric terms.


## 2. Dynamic Model of Sampled-Data-Based LFC for Power System

In this section, the LFC scheme using sampled-data control and randomly occurring time-varying delays is detailed.


### 2.1. One-Area LFC Model

A block structure for a one-area LFC-based model with communication delays is shown in Figure1, which can be represented in the following formula:(1)xˇ.t=Aˇxˇt+ℬˇut+Dˇwt,yˇt=Cˇxˇt.Figure 1
Dynamic model of one-area LFC scheme.The notations of model (1) are standard and they are available in Appendix.The area control errorACE is clearly illustrated in Figure 1 as the model output with no net tie-line power exchange. It follows that yˇt=ACE=β△f, where β>0 is the frequency bias factor. Furthermore, ACE is utilized as a control input to develop the PI-based controller that follows(2)ut=−KPACE−KI∫ACE,where ∫ACE is the integration of ACE, while bot integral and proportional gains are denoted as KI and KP, respectively.


### 2.2. Sampled-Data LFC Scheme

In the continuous-time mode, state vectors are employed directly to generate the control signal. However, the sampled-data LFC scheme can only handle discrete measurements of state vectors in discrete time. Meanwhile, sampled output measurementsyˇt can only be used in the sampled-data control loop in the LFC scheme. As a result, we only use the measurement yˇtk at sampling instant tk. It is obvious that in Figure 2, network-induced delays affect the communication network, which is defined as dtk=dtksc+dtkca≤dM<+∞, where dM is constant. The sampling instants tkk=0,1,2,… are supposed to satisfy(3)0=t0<t1<t2<⋯<tk<⋯<limk⟶∞tk=+∞.Figure 2
Dynamic model of one-area LFC scheme over the communication network and the transmission delay from SCA.The sampling intervaldk=tk+1−tk is set to satisfy 0<dk=tk+1−tk=dM, where dM denotes the largest upper bound of dk. Then, taking account of the influence of sampling and communication network delays, the possible LFC control signal yˇt can be articulated as follows:(4)yˇt=yˇtk,t∈tk,tk+1.Then, the sampled-data LFC scheme in the network system is drawn out according to equations (2) and (4) as follows:(5)ut=utk=−KPyˇtk−KI∫yˇtkdt.By setting the following new vectors of virtual state and output measurementxt=△f△Pm△Pv∫ACET and yt=ACE∫ACET and incorporating equations (1) and (5), the sampled-data LFC model can be obtained for t∈tk,tk+1 as follows:(6)x.t=Axt−ℬxtk+Dwt,yt=Cxt.The expression of model (6) is given in Appendix. Then, we define dt=t−tk for t∈tk,tk+1. As such, we can denote the sampling instant as tk=t−t−tk=t−dt. It is supposed that dt≤tk+1−tk=dk≤dM for all tk. Based on the technique of input delay [20], we can further define the sampled-data control in equation (6) as follows:(7)ut=−Kxt−dt.It is noteworthy that time delays often occur in a probabilistic form in many practical control systems. Thus, the impact of random delays in this study is important. Therefore,dt is expected to satisfy the following two assumptions:A1. Consider the probability distribution of the time-varying delay dt that takes values in the range 0,d1 or d1,d2 and define the two sets as follows:(8)Π1=t:dt∈0,d1,Π2=t:dt∈d1,d2.We also consider the mapping functions as follows:(9)d1t=dt,fort∈Π1,0,forelse,d2t=dt,fort∈Π2,d1,forelse.It has been inferred that ift∈Π1 appears in the case of dt∈0,d1, that is, dt=d1t, and t∈Π2 appears in the case of dt∈d1,d2, that is, dt=d2t. New time-varying delays are d1t and d2t, such that 0≤d1t≤d1 and d1<d2t≤d2. A stochastic variable σt is defined as follows:(10)σt=1,fort∈Π1,0,fort∈Π2.A2. The variable σt is a Bernoulli-distributed sequence with the following properties: Probσt=1=Eσt=σ0 and Probσt=0=1−Eσt=1−σ0, where 0≤σ0≤1 and Eσt is the expectation of σt.Remark 1.
In practice, power systems often need broad open communication networks to supply relevant information. The operations of these networks are subject to undesirable events, including latent faults, packet loss, time delays, and others. Such nonlinear disturbances can appear in random ways as a result of different environmental conditions. As a result, the stochastic variableσt is used to represent this randomly occurring phenomenon.
By incorporatingd1t, d2t, σt, and equation (7) into model (6), we have(11)x.t=Axt−σtℬKCxt−d1t−1−σtℬKCxt−d2t+Dwt,yt=Cxt.
The initial condition of model (11) is given by xt=ψt,t∈−d2,0, where ψt is continuous on −d2,0.
The results presented below need the following definition and lemmas.Definition 1.
Given a prescribed scalarη>0, model (11) is considered asymptotically stable with H∞ performance if the conditions below hold [38]:(i)
Closed-loop model (11) with wt=0 is asymptotically stable.(ii)
For every nonzerowt∈ℒ20,∞ with a prescribed η>0, the following inequality is true under the zero initial condition: Eyt2≤ηEwt2.Lemma 1.
Given matrixO=OT>0, two matrices X1,X2∈ℝn×m, positive integers n and m, scalar θ∈0,1, and any vector ζ∈ℝm denote the function Ξθ,O with the following form [39]:(12)Ξθ,O=1θζTX1TOX1ζ+11−θζTX2TOX2ζ.There exists a matrixY∈ℝn×m satisfying OYYTO>0, then(13)minθ∈0,1Ξθ,O≥X1ζX2ζTOYYTOX1ζX2ζ.Lemma 2.
Given matrixO=OT>0, the following inequality is true for all continuously differentiable function xt in ξ1,ξ2∈ℝn as follows [40]:(14)−ξ2−ξ1∫t−ξ2t−ξ1xTαOxαdα≤−∫t−ξ2t−ξ1xαdαTO∫t−ξ2t−ξ1xαdα.Lemma 3.
LetΞ=ΞT,ℋ and E be real matrices, ℱt satisfies ℱTtℱt≤I. Then, Ξ+ℋℱtE+ℋℱtET<0, iff there exists a scalar ε>0 such that Ξ+ε−1ℋℋT+εETE<0 or equivalently [41]:(15)ΞℋεΕ∗−εI0∗∗−εI<0.In this study, the main aim is to obtain LMI-based sufficient criteria and ensure model (11) has mean-square stability and stabilization under the designed sampled-data robust H∞ LFC scheme.Problem 1.
The following conditions are derived, in order to attain the aim of this study:(i)
Delay-dependent LKF with full model information is constructed to obtain the stability and stabilization criteria of model (11) based on Definition 1.(ii)
Sufficient gain matrices for controlK=KPKI are calculated from the solution of LMIs to ensure stabilization of model (11) via the robust sampled-data H∞ LFC scheme.


## 2.1. One-Area LFC Model

A block structure for a one-area LFC-based model with communication delays is shown in Figure1, which can be represented in the following formula:(1)xˇ.t=Aˇxˇt+ℬˇut+Dˇwt,yˇt=Cˇxˇt.Figure 1
Dynamic model of one-area LFC scheme.The notations of model (1) are standard and they are available in Appendix.The area control errorACE is clearly illustrated in Figure 1 as the model output with no net tie-line power exchange. It follows that yˇt=ACE=β△f, where β>0 is the frequency bias factor. Furthermore, ACE is utilized as a control input to develop the PI-based controller that follows(2)ut=−KPACE−KI∫ACE,where ∫ACE is the integration of ACE, while bot integral and proportional gains are denoted as KI and KP, respectively.


## 2.2. Sampled-Data LFC Scheme

In the continuous-time mode, state vectors are employed directly to generate the control signal. However, the sampled-data LFC scheme can only handle discrete measurements of state vectors in discrete time. Meanwhile, sampled output measurementsyˇt can only be used in the sampled-data control loop in the LFC scheme. As a result, we only use the measurement yˇtk at sampling instant tk. It is obvious that in Figure 2, network-induced delays affect the communication network, which is defined as dtk=dtksc+dtkca≤dM<+∞, where dM is constant. The sampling instants tkk=0,1,2,… are supposed to satisfy(3)0=t0<t1<t2<⋯<tk<⋯<limk⟶∞tk=+∞.Figure 2
Dynamic model of one-area LFC scheme over the communication network and the transmission delay from SCA.The sampling intervaldk=tk+1−tk is set to satisfy 0<dk=tk+1−tk=dM, where dM denotes the largest upper bound of dk. Then, taking account of the influence of sampling and communication network delays, the possible LFC control signal yˇt can be articulated as follows:(4)yˇt=yˇtk,t∈tk,tk+1.Then, the sampled-data LFC scheme in the network system is drawn out according to equations (2) and (4) as follows:(5)ut=utk=−KPyˇtk−KI∫yˇtkdt.By setting the following new vectors of virtual state and output measurementxt=△f△Pm△Pv∫ACET and yt=ACE∫ACET and incorporating equations (1) and (5), the sampled-data LFC model can be obtained for t∈tk,tk+1 as follows:(6)x.t=Axt−ℬxtk+Dwt,yt=Cxt.The expression of model (6) is given in Appendix. Then, we define dt=t−tk for t∈tk,tk+1. As such, we can denote the sampling instant as tk=t−t−tk=t−dt. It is supposed that dt≤tk+1−tk=dk≤dM for all tk. Based on the technique of input delay [20], we can further define the sampled-data control in equation (6) as follows:(7)ut=−Kxt−dt.It is noteworthy that time delays often occur in a probabilistic form in many practical control systems. Thus, the impact of random delays in this study is important. Therefore,dt is expected to satisfy the following two assumptions:A1. Consider the probability distribution of the time-varying delay dt that takes values in the range 0,d1 or d1,d2 and define the two sets as follows:(8)Π1=t:dt∈0,d1,Π2=t:dt∈d1,d2.We also consider the mapping functions as follows:(9)d1t=dt,fort∈Π1,0,forelse,d2t=dt,fort∈Π2,d1,forelse.It has been inferred that ift∈Π1 appears in the case of dt∈0,d1, that is, dt=d1t, and t∈Π2 appears in the case of dt∈d1,d2, that is, dt=d2t. New time-varying delays are d1t and d2t, such that 0≤d1t≤d1 and d1<d2t≤d2. A stochastic variable σt is defined as follows:(10)σt=1,fort∈Π1,0,fort∈Π2.A2. The variable σt is a Bernoulli-distributed sequence with the following properties: Probσt=1=Eσt=σ0 and Probσt=0=1−Eσt=1−σ0, where 0≤σ0≤1 and Eσt is the expectation of σt.Remark 1.
In practice, power systems often need broad open communication networks to supply relevant information. The operations of these networks are subject to undesirable events, including latent faults, packet loss, time delays, and others. Such nonlinear disturbances can appear in random ways as a result of different environmental conditions. As a result, the stochastic variableσt is used to represent this randomly occurring phenomenon.
By incorporatingd1t, d2t, σt, and equation (7) into model (6), we have(11)x.t=Axt−σtℬKCxt−d1t−1−σtℬKCxt−d2t+Dwt,yt=Cxt.
The initial condition of model (11) is given by xt=ψt,t∈−d2,0, where ψt is continuous on −d2,0.
The results presented below need the following definition and lemmas.Definition 1.
Given a prescribed scalarη>0, model (11) is considered asymptotically stable with H∞ performance if the conditions below hold [38]:(i)
Closed-loop model (11) with wt=0 is asymptotically stable.(ii)
For every nonzerowt∈ℒ20,∞ with a prescribed η>0, the following inequality is true under the zero initial condition: Eyt2≤ηEwt2.Lemma 1.
Given matrixO=OT>0, two matrices X1,X2∈ℝn×m, positive integers n and m, scalar θ∈0,1, and any vector ζ∈ℝm denote the function Ξθ,O with the following form [39]:(12)Ξθ,O=1θζTX1TOX1ζ+11−θζTX2TOX2ζ.There exists a matrixY∈ℝn×m satisfying OYYTO>0, then(13)minθ∈0,1Ξθ,O≥X1ζX2ζTOYYTOX1ζX2ζ.Lemma 2.
Given matrixO=OT>0, the following inequality is true for all continuously differentiable function xt in ξ1,ξ2∈ℝn as follows [40]:(14)−ξ2−ξ1∫t−ξ2t−ξ1xTαOxαdα≤−∫t−ξ2t−ξ1xαdαTO∫t−ξ2t−ξ1xαdα.Lemma 3.
LetΞ=ΞT,ℋ and E be real matrices, ℱt satisfies ℱTtℱt≤I. Then, Ξ+ℋℱtE+ℋℱtET<0, iff there exists a scalar ε>0 such that Ξ+ε−1ℋℋT+εETE<0 or equivalently [41]:(15)ΞℋεΕ∗−εI0∗∗−εI<0.In this study, the main aim is to obtain LMI-based sufficient criteria and ensure model (11) has mean-square stability and stabilization under the designed sampled-data robust H∞ LFC scheme.Problem 1.
The following conditions are derived, in order to attain the aim of this study:(i)
Delay-dependent LKF with full model information is constructed to obtain the stability and stabilization criteria of model (11) based on Definition 1.(ii)
Sufficient gain matrices for controlK=KPKI are calculated from the solution of LMIs to ensure stabilization of model (11) via the robust sampled-data H∞ LFC scheme.


## 3. Design of Sampled-DataH∞ LFC Scheme

We present a delay-dependent stability analysis in Section3.1 and a stabilization analysis in Section 3.2 for model (11) using Jensen integral inequality, reciprocally convex combination lemma, and LMI methodology. The following notations simplify the remaining presentation:(16)ζt=colxt,xt−d1t,xt−d1,xt−d2t,xt−d2,∫t−d1t−d1txαdα,∫t−d1ttxαdα,∫t−d2t−d2txαdα,∫t−d2ttxαdα,x.t,wt,∫t−d1ttx˙αdα=xt−xt−d1t,∫t−d1t−d1tx˙αdα=xt−d1t−xt−d1,∫t−d2ttx˙αdα=xt−xt−d2t,∫t−d2t−d2tx˙αdα=xt−d2t−xt−d2.


### 3.1. Sampled-Data LFC-BasedH∞ Stability Analysis

This subsection focuses on obtaining the sufficient criteria for establishing mean-square asymptotic stability of model (11), which is stated in Theorem 1.Theorem 1.
Under given control gainK with H∞ performance index η, model (11) is mean-square asymptotically stable for given positive scalars d1, d2, and η, if there exist matrices P, Qi, ℛi, Si, Yii=1,2, and ℳ, which satisfy the following criteria:(17)P>0,Qi>0,ℛi>0,Si>0,i=1,2,(18)SiYiYiTSi≥0,i=1,2,(19)Ξ11×11<0,where Ξ1,1=Q1+Q2+d12ℛ1+d22ℛ2−S1−S2+ℳA+ℳAT+CTC, Ξ1,2=S1−Y1−σ0ℳℬKK, Ξ1,3=Y1, Ξ1,4=S2−Y2−1−σ0ℳℬKK, Ξ1,5=Y2, Ξ1,10=C−ℳ+ℳT, Ξ1,11=ℳA, Ξ2,2=−S1−S1+Y1T+Y1, Ξ2,3=S1−Y1, Ξ2,10=−σ0ℳℬKKT, Ξ3,3=−Q1−S1, Ξ4,4=−S2−S2+Y2T+Y2, Ξ4,5=S2−Y2, Ξ4,10=−1−σ0ℳℬKKT, Ξ5,5=−Q2−S2, Ξ6,6=−ℛ1, Ξ7,7=−ℛ1, Ξ8,8=−ℛ2, Ξ9,9=−ℛ2, Ξ10,10=d12S1+d22S2−ℳ−ℳT, Ξ10,11=ℳA, and Ξ11,11=−η2I.Proof.
Construct the LKF candidate as follows:Vxt=V1xt+V2xt+V3xt+V4xt, where(20)V1xt=xTtCxt,V2xt=∑i=12∫t−ditxTαQixαdα,V3xt=∑i=12di∫t−dit∫αtxTβℛixβdβdα,V4xt=∑i=12di∫t−dit∫αtx.TβSix.βdβdα.
Obtaining the derivatives ofVxt and taking the mathematical expectation, we have(21)EV.1xt=E2xTtPx.t,(22)EV.2xt=E∑i=12xTtQixt−xTt−diQixt−di,(23)EV.3xt=E∑i=12xTtdi2ℛixt−∑i=12di∫t−ditxTαℛixαdα,(24)EV.4xt=E∑i=12x.Ttdi2Six.t−∑i=12di∫t−ditx.TαSix.αdα.
According to A1, the integral term in equation (23) can be employed by utilizing Lemma 2 in the form of(25)−∑i=12di∫t−dit−ditxTαℛixαdα≤−∑i=12∫t−dit−ditxαdαTℛi∫t−dit−ditxαdα,(26)−∑i=12di∫t−dittxTαℛixαdα≤−∑i=12∫t−dittxαdαTℛi∫t−dittxαdα.
According to A1, the integral term in equation (24) can be employed by utilizing Lemma 2 in the form of(27)−∑i=12di∫t−ditx.TαSix.αdα=−∑i=12didi−dit∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12didit∫t−dittx.αdαTSi∫t−dittx.αdα=−∑i=12∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12ditdi−dit∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12∫t−dittx.αdαTSi∫t−dittx.αdα−∑i=12di−ditdit∫t−dittx.αdαTSi∫t−dittx.αdα.
IfSiYiYiTSi≥0,i=1,2 by Lemma 1, the following inequality is true:(28)∑i=12ditdi−dit∫t−dit−ditx.αdαdi−ditdit∫t−dittx.αdαTSiYiYiTSi×∑i=12ditdi−dit∫t−dit−ditx.αdαdi−ditdit∫t−dittx.αdα≥0,which implies(29)−∑i=12ditdi−dit∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12di−ditdit∫t−dittx.αdαTSi∫t−dittx.αdα≤−∑i=12∫t−dit−ditx.αdαTYi∫t−dittx.αdα−∑i=12∫t−dittx.αdαTYiT∫t−dit−ditx.αdα.
From equations (27) and (29), we can obtain that(30)−∑i=12di∫t−ditx˙TαSix˙αdα≤−∑i=12∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12∫t−dittx.αdαTSi∫t−dittx.αdα−∑i=12∫t−dit−ditx.αdαTYi∫t−dittx.αdα−∑i=12∫t−dittx.αdαTYiT∫t−dit−ditx.αdα.
In addition, for any matrixℳ with a suitable size, the following inequalities are valid:(31)0=2xt+x.tℳ−x.t+Axt−σ0ℬKCxt−d1t−1−σ0ℬKCxt−d2t+Dwt.
From equations (21)–(31), we can deduce that for all nonzero wt∈ℒ20,∞,(32)EV.xt+yTtyt−η2wTtwt≤EζTtΞ11×11ζt.
Under zero conditions, we haveV0=0 and V∞≥0. Integrating both sides of equation (30) yields Eyt2≤ηEwt2 for every wt∈ℒ20,+∞. As such, model (11) is mean-square asymptotically stable under Definition 1. The proof is completed.


### 3.2. Sampled-Data LFC-BasedH∞ Stabilization Analysis

According to the stability results developed in Section3.1, we focus on obtaining stabilization results through the use of sampled-data LFC scheme for model (11), which is stated in Theorem 2.Theorem 2.
Under sampled-data LFC withH∞ performance index η, model (11) is mean-square asymptotically stable for given positive scalars d1, d2, η, and ς⟶0, if there exist matrices Pˇ, Qˇi, ℛˇi, Sˇi, Yˇii=1,2, ℒ, W, and Z, which satisfy the following criteria:(33)Pˇ>0,Qˇi>0,ℛˇi>0,Sˇi>0,i=1,2,(34)SˇiYˇiYˇiTSˇi≥0,i=1,2,(35)Ξ˜11×11Υ¯∗−I<0,(36)−ςIWC−CℒT∗−I<0,where Ξ˜1,1=Qˇ1+Qˇ2+d1ℛˇ1+d2ℛˇ2−Sˇ1−Sˇ2+Aℒ+AℒT, Ξ˜1,2=Sˇ1−Yˇ1−σ0ℬZC, Ξ˜1,3=Yˇ1, Ξ˜1,4=Sˇ2−Yˇ2−1−σ0ℬZC, Ξ˜1,5=Yˇ2, Ξ˜1,10=Pˇ−ℒ+ℒAT, Ξ˜1,11=D, Ξ˜2,2=−Sˇ1−Sˇ1+Yˇ1+Yˇ1T, Ξ˜2,3=Sˇ1−Yˇ1, Ξ˜2,10=−σ0ℬZCT, Ξ˜3,3=−Qˇ1−Sˇ1, Ξ˜4,4=−Sˇ2−Sˇ2+Yˇ2+Yˇ2T, Ξ˜4,5=Sˇ2−Yˇ2, Ξ˜4,10=−1−σ0ℬZCT, Ξ˜5,5=−Qˇ2−Sˇ2, Ξ˜6,6=−ℛˇ1, Ξ˜7,7=−ℛˇ1, Ξ˜8,8=−ℛˇ2, Ξ˜9,9=−ℛˇ2, Ξ˜10,10=d12Sˇ1+d22Sˇ2−ℒ−ℒT, Ξ˜10,11=D, Ξ˜11,11=−η2I, Υ˜=Aℒ01×10T, and K=ZW−1.Proof.
We can define the change of variables in order to achieve the controller gain as follows:ℒ=ℳ−1, Pˇ=ℒℒ, Qˇ1=ℒQ1ℒ, Qˇ2=ℒQ2ℒ, ℛˇ1=ℒℛ1ℒ, ℛˇ2=ℒℛ2ℒ, Sˇ1=ℒS1ℒ, Sˇ2=ℒS2ℒ, Yˇ1=ℒY1ℒ, and Yˇ2=ℒY2ℒ. Post and premultiplying equations (18) and (19) by diagℒ,ℒ and diagℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,I, we can obtain that(37)Ξ¯11×11<0,where Ξ¯1,1=Qˇ1+Qˇ2+d1ℛˇ1+d2ℛˇ2−Sˇ1−Sˇ2+Aℒ+AℒT+ℒCTCℒ, Ξ¯1,2=Sˇ1−Yˇ1−σ0ℒKC, Ξ¯1,3=Yˇ1, Ξ¯1,4=Sˇ2−Yˇ2−1−σ0ℒBKC, Ξ¯1,5=Yˇ2, Ξ¯1,10=Pˇ−ℒ+ℒCT, Ξ¯1,11=D, Ξ¯2,2=−Sˇ1−Sˇ1+Yˇ1+Yˇ1T, Ξ¯2,3=Sˇ1−Yˇ1, Ξ¯2,10=−σ0ℒBKCT, Ξ¯3,3=−Qˇ1−Sˇ1, Ξ¯4,4=−Sˇ2−Sˇ2+Yˇ2+Yˇ2T, Ξ¯4,5=Sˇ2−Yˇ2, Ξ¯4,10=−1−σ0ℒBKCT, Ξ¯5,5=−Qˇ2−Sˇ2, Ξ¯6,6=−ℛˇ1, Ξ¯7,7=−ℛˇ1, Ξ¯8,8=−ℛˇ2, Ξ¯9,9=−ℛˇ2, Ξ¯10,10=d12Sˇ1+d22Sˇ2−ℒ−ℒT, Ξ¯10,11=D, and Ξ¯11,11=−η2I, Because of the existence of nonlinear terms in equation (37) and C is not invertible, let Y=KCℒ, then YC−1ℒ−1=K is not applicable to directly find K. Therefore, by defining ZC=KCℒ, WC=Cℒ and ZW−1=K.
By using Schur complement, equation (37) is equivalent to equation (35) and the nonlinear functions are changed to W problem as in reference [42]. Moreover, we can formulate inequality (36) for every small scalar ς and use Schur complement in WC−CℒTWC−Cℒ=0. Here, ℒ>0 and W is an invertible matrix. The proof is completed.


## 3.1. Sampled-Data LFC-BasedH∞ Stability Analysis

This subsection focuses on obtaining the sufficient criteria for establishing mean-square asymptotic stability of model (11), which is stated in Theorem 1.Theorem 1.
Under given control gainK with H∞ performance index η, model (11) is mean-square asymptotically stable for given positive scalars d1, d2, and η, if there exist matrices P, Qi, ℛi, Si, Yii=1,2, and ℳ, which satisfy the following criteria:(17)P>0,Qi>0,ℛi>0,Si>0,i=1,2,(18)SiYiYiTSi≥0,i=1,2,(19)Ξ11×11<0,where Ξ1,1=Q1+Q2+d12ℛ1+d22ℛ2−S1−S2+ℳA+ℳAT+CTC, Ξ1,2=S1−Y1−σ0ℳℬKK, Ξ1,3=Y1, Ξ1,4=S2−Y2−1−σ0ℳℬKK, Ξ1,5=Y2, Ξ1,10=C−ℳ+ℳT, Ξ1,11=ℳA, Ξ2,2=−S1−S1+Y1T+Y1, Ξ2,3=S1−Y1, Ξ2,10=−σ0ℳℬKKT, Ξ3,3=−Q1−S1, Ξ4,4=−S2−S2+Y2T+Y2, Ξ4,5=S2−Y2, Ξ4,10=−1−σ0ℳℬKKT, Ξ5,5=−Q2−S2, Ξ6,6=−ℛ1, Ξ7,7=−ℛ1, Ξ8,8=−ℛ2, Ξ9,9=−ℛ2, Ξ10,10=d12S1+d22S2−ℳ−ℳT, Ξ10,11=ℳA, and Ξ11,11=−η2I.Proof.
Construct the LKF candidate as follows:Vxt=V1xt+V2xt+V3xt+V4xt, where(20)V1xt=xTtCxt,V2xt=∑i=12∫t−ditxTαQixαdα,V3xt=∑i=12di∫t−dit∫αtxTβℛixβdβdα,V4xt=∑i=12di∫t−dit∫αtx.TβSix.βdβdα.
Obtaining the derivatives ofVxt and taking the mathematical expectation, we have(21)EV.1xt=E2xTtPx.t,(22)EV.2xt=E∑i=12xTtQixt−xTt−diQixt−di,(23)EV.3xt=E∑i=12xTtdi2ℛixt−∑i=12di∫t−ditxTαℛixαdα,(24)EV.4xt=E∑i=12x.Ttdi2Six.t−∑i=12di∫t−ditx.TαSix.αdα.
According to A1, the integral term in equation (23) can be employed by utilizing Lemma 2 in the form of(25)−∑i=12di∫t−dit−ditxTαℛixαdα≤−∑i=12∫t−dit−ditxαdαTℛi∫t−dit−ditxαdα,(26)−∑i=12di∫t−dittxTαℛixαdα≤−∑i=12∫t−dittxαdαTℛi∫t−dittxαdα.
According to A1, the integral term in equation (24) can be employed by utilizing Lemma 2 in the form of(27)−∑i=12di∫t−ditx.TαSix.αdα=−∑i=12didi−dit∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12didit∫t−dittx.αdαTSi∫t−dittx.αdα=−∑i=12∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12ditdi−dit∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12∫t−dittx.αdαTSi∫t−dittx.αdα−∑i=12di−ditdit∫t−dittx.αdαTSi∫t−dittx.αdα.
IfSiYiYiTSi≥0,i=1,2 by Lemma 1, the following inequality is true:(28)∑i=12ditdi−dit∫t−dit−ditx.αdαdi−ditdit∫t−dittx.αdαTSiYiYiTSi×∑i=12ditdi−dit∫t−dit−ditx.αdαdi−ditdit∫t−dittx.αdα≥0,which implies(29)−∑i=12ditdi−dit∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12di−ditdit∫t−dittx.αdαTSi∫t−dittx.αdα≤−∑i=12∫t−dit−ditx.αdαTYi∫t−dittx.αdα−∑i=12∫t−dittx.αdαTYiT∫t−dit−ditx.αdα.
From equations (27) and (29), we can obtain that(30)−∑i=12di∫t−ditx˙TαSix˙αdα≤−∑i=12∫t−dit−ditx.αdαTSi∫t−dit−ditx.αdα−∑i=12∫t−dittx.αdαTSi∫t−dittx.αdα−∑i=12∫t−dit−ditx.αdαTYi∫t−dittx.αdα−∑i=12∫t−dittx.αdαTYiT∫t−dit−ditx.αdα.
In addition, for any matrixℳ with a suitable size, the following inequalities are valid:(31)0=2xt+x.tℳ−x.t+Axt−σ0ℬKCxt−d1t−1−σ0ℬKCxt−d2t+Dwt.
From equations (21)–(31), we can deduce that for all nonzero wt∈ℒ20,∞,(32)EV.xt+yTtyt−η2wTtwt≤EζTtΞ11×11ζt.
Under zero conditions, we haveV0=0 and V∞≥0. Integrating both sides of equation (30) yields Eyt2≤ηEwt2 for every wt∈ℒ20,+∞. As such, model (11) is mean-square asymptotically stable under Definition 1. The proof is completed.


## 3.2. Sampled-Data LFC-BasedH∞ Stabilization Analysis

According to the stability results developed in Section3.1, we focus on obtaining stabilization results through the use of sampled-data LFC scheme for model (11), which is stated in Theorem 2.Theorem 2.
Under sampled-data LFC withH∞ performance index η, model (11) is mean-square asymptotically stable for given positive scalars d1, d2, η, and ς⟶0, if there exist matrices Pˇ, Qˇi, ℛˇi, Sˇi, Yˇii=1,2, ℒ, W, and Z, which satisfy the following criteria:(33)Pˇ>0,Qˇi>0,ℛˇi>0,Sˇi>0,i=1,2,(34)SˇiYˇiYˇiTSˇi≥0,i=1,2,(35)Ξ˜11×11Υ¯∗−I<0,(36)−ςIWC−CℒT∗−I<0,where Ξ˜1,1=Qˇ1+Qˇ2+d1ℛˇ1+d2ℛˇ2−Sˇ1−Sˇ2+Aℒ+AℒT, Ξ˜1,2=Sˇ1−Yˇ1−σ0ℬZC, Ξ˜1,3=Yˇ1, Ξ˜1,4=Sˇ2−Yˇ2−1−σ0ℬZC, Ξ˜1,5=Yˇ2, Ξ˜1,10=Pˇ−ℒ+ℒAT, Ξ˜1,11=D, Ξ˜2,2=−Sˇ1−Sˇ1+Yˇ1+Yˇ1T, Ξ˜2,3=Sˇ1−Yˇ1, Ξ˜2,10=−σ0ℬZCT, Ξ˜3,3=−Qˇ1−Sˇ1, Ξ˜4,4=−Sˇ2−Sˇ2+Yˇ2+Yˇ2T, Ξ˜4,5=Sˇ2−Yˇ2, Ξ˜4,10=−1−σ0ℬZCT, Ξ˜5,5=−Qˇ2−Sˇ2, Ξ˜6,6=−ℛˇ1, Ξ˜7,7=−ℛˇ1, Ξ˜8,8=−ℛˇ2, Ξ˜9,9=−ℛˇ2, Ξ˜10,10=d12Sˇ1+d22Sˇ2−ℒ−ℒT, Ξ˜10,11=D, Ξ˜11,11=−η2I, Υ˜=Aℒ01×10T, and K=ZW−1.Proof.
We can define the change of variables in order to achieve the controller gain as follows:ℒ=ℳ−1, Pˇ=ℒℒ, Qˇ1=ℒQ1ℒ, Qˇ2=ℒQ2ℒ, ℛˇ1=ℒℛ1ℒ, ℛˇ2=ℒℛ2ℒ, Sˇ1=ℒS1ℒ, Sˇ2=ℒS2ℒ, Yˇ1=ℒY1ℒ, and Yˇ2=ℒY2ℒ. Post and premultiplying equations (18) and (19) by diagℒ,ℒ and diagℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,I, we can obtain that(37)Ξ¯11×11<0,where Ξ¯1,1=Qˇ1+Qˇ2+d1ℛˇ1+d2ℛˇ2−Sˇ1−Sˇ2+Aℒ+AℒT+ℒCTCℒ, Ξ¯1,2=Sˇ1−Yˇ1−σ0ℒKC, Ξ¯1,3=Yˇ1, Ξ¯1,4=Sˇ2−Yˇ2−1−σ0ℒBKC, Ξ¯1,5=Yˇ2, Ξ¯1,10=Pˇ−ℒ+ℒCT, Ξ¯1,11=D, Ξ¯2,2=−Sˇ1−Sˇ1+Yˇ1+Yˇ1T, Ξ¯2,3=Sˇ1−Yˇ1, Ξ¯2,10=−σ0ℒBKCT, Ξ¯3,3=−Qˇ1−Sˇ1, Ξ¯4,4=−Sˇ2−Sˇ2+Yˇ2+Yˇ2T, Ξ¯4,5=Sˇ2−Yˇ2, Ξ¯4,10=−1−σ0ℒBKCT, Ξ¯5,5=−Qˇ2−Sˇ2, Ξ¯6,6=−ℛˇ1, Ξ¯7,7=−ℛˇ1, Ξ¯8,8=−ℛˇ2, Ξ¯9,9=−ℛˇ2, Ξ¯10,10=d12Sˇ1+d22Sˇ2−ℒ−ℒT, Ξ¯10,11=D, and Ξ¯11,11=−η2I, Because of the existence of nonlinear terms in equation (37) and C is not invertible, let Y=KCℒ, then YC−1ℒ−1=K is not applicable to directly find K. Therefore, by defining ZC=KCℒ, WC=Cℒ and ZW−1=K.
By using Schur complement, equation (37) is equivalent to equation (35) and the nonlinear functions are changed to W problem as in reference [42]. Moreover, we can formulate inequality (36) for every small scalar ς and use Schur complement in WC−CℒTWC−Cℒ=0. Here, ℒ>0 and W is an invertible matrix. The proof is completed.


## 4. Design of Robust Sampled-DataH∞ LFC Scheme

In practical, uncertainties are commonly encountered in modelling of practical systems due to environmental imperfections and changes. As such, this study takes into account norm-bounded uncertainty, and we can describe the power system as follows:(38)x.t=A+△Atxt−σtℬKCxt−d1t−1−σtℬKCxt−d2t+Dwt,yt=Cxt,where △At is the parametric uncertainty satisfying △At=ℋℱtE, where ℋ and E are known matrices, and ℱt is time-varying unknown matrix that satisfies ℱtTℱt≤I.Theorem 3.
Under sampled-data LFC withH∞ performance index η, model (38) is mean-square robust asymptotically stable for given positive scalars d1, d2, η, and ς⟶0, if there exist matrices Pˇ, Qˇi, ℛˇi, Sˇi, Yˇii=1,2, ℒ, W, and Z, which satisfy the following criteria:(39)Pˇ>0,Qˇi>0,ℛˇi>0,Sˇi>0,i=1,2,(40)SˇiYˇiYˇiTSˇi≥0,i=1,2,(41)Ξ˜11×11ε¯Λ¯1Λ¯2Υ¯∗−ε¯I00∗∗−ε¯I0000−I<0,(42)−ςIWC−CℒT∗−I<0,where Ξ˜11×11 and Υ¯ are given in Theorem 2, and Λ¯1=ℋT01×8ℋT0T, Λ2=EℒT01×10, ε¯=ε−1, and K=ZW−1.Proof.
First, replacingA by A+△At in equation (19), we have(43)Ξ11×11+Λ1ℱtΛ2+Λ1ℱtΛ2T,
It follows from Lemma3, there exists a scalar ε>0, such that(44)Ξ11×11+1εΛ1Λ1T+εΛ2TΛ2<0,where Λ1=ℳℋT01×8ℳℋT0T and Λ2=E01×10, and applying Lemma 3, it is easy to obtain(45)Ξ11×11Λ1εΛ2∗−εI0∗∗−εI<0.
Post- and premultiplying equation (45) by diagℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,ℒ,I,ε−1I,ε−1I and by applying a similar procedure of Theorem 2 through the use of Schur complement lemma, equation (41) can be obtained. The proof is completed.Remark 2.
In comparison with prior publications [21, 22], the proposed method is novel for model (1), which not only covers the model with sampled-data control but also considers random time-varying delays in the controller implementation that is characterized by Bernoulli-distributed sequences. It has many advantages over regular power system analysis.
Forσt=1, since only one delay interval with 0≤d1t≤d1 exists, model (11) reduces to(46)x.t=Axt+Adxt−d1t+Dwt,yt=Cxt.Corollary 1.
Model (46) with wt=0 is asymptotically stable for given positive scalars d1, if there exist matrices P, Q1, ℛ1, S1, Y1, which satisfy the following criteria:(47)P>0,Q1>0,ℛ1>0,S1>0,S1Y1Y1TS1≥0,Γ=Γ1,1∗∗∗∗∗Γ2,1Γ2,2∗∗∗∗Y1TΓ3,2Γ3,3∗∗∗000−ℛ1∗∗0000−ℛ1∗d1ATS1d1AdTS1000−S1<0,where Γ1,1=Q1+d12ℛ1−S1+PA+PPT, Γ2,1=S1T−Y1T+AdTPT, Ξ2,2=−S1−S1+Y1+Y1T, Ξ3,2=S1T−Y1T, and Ξ3,3=−Q1−S1.Proof.
Choosing the same LKF as in Theorem1 with i=1 and computing the derivative of Vxt, we have(48)V.xt=2xTtPAxt+Adxt−dt+xTtQ1xt−xTt−d1Q1xt−d1+d12xTtℛ1xt−d1∫t−d1txTαℛ1xαdα+d12Axt+Adxt−dtTS1Axt+Adxt−dt−d1∫t−d1tx.TαS1x.αdα.
By applying Lemma2 and using Theorem 1, the remaining proof can be easily obtained.Remark 3.
The following constraint optimization issue is presented to solve the stability criterion in Corollary1:
Optimization problem is(49)minP,Q1,ℛ1,S1d1,s.t(50)S1Y1Y1TS1≥0,P=PT>0,Q1=Q1T>0,ℛ1=ℛ1T>0,S1=S1T>0,Γ<0.
The convex optimization issues are be easily solved by using certain numerical packages [39–43].Remark 4.
There have been a number of recent studies concentrating on developing suitable LFC-based controllers for power systems. For example, the sliding mode LFC for hybrid power system is based on disturbance observer [8], an adaptive control scheme for LFC of multiarea power systems [9], robust H∞ LFC of delayed multi-area power systems with stochastic disturbances [10, 37, 41], decentralized H∞ LFC for multi area power systems with communication uncertainties [11, 42], and resilient reliable H∞ LFC of power system with random gain fluctuations [44]. However, no work has yet been reported regarding the topic of robust sampled-data LFC schemes for power systems with randomly occurring time-varying delays. So, in order to fill these gaps, we introduced new sufficient conditions for ensuring mean-square asymptotic stability and stabilization criteria for power systems by addressing robust sampling of data H∞ LFC schemes. Thus, the main results of this study are novel and distinct from those of the existing literature.


## 5. Case Studies

This section presents case studies including the MAUBs of time delays,H∞ performance index, and efficiency of proposed sampled-data LFC scheme. In addition, the simulation results are given to demonstrate how the control scheme supports the achievement of system performance.


### 5.1. The Case of a One-Area LFC-Based Power System

Table1 lists the parameters of one-area LFC-based power system taken from references [35, 45]. Correlations between gain of the sampled-data LFC scheme and effects of time delays on stability margins and H∞ performance based on the proposed results are discussed in this subsection.Table 1
Parameter values of one-area LFC-based model (1).
TgTchβℛDℳ0.10.321.00.051.010Case 1.
MAUBs: based on Theorem 1, an MAUB of d2 is calculated for model (11) without load disturbance. For different settings of σ0 and fixed KP=0, KI=0.05, d1=0.1, and η=6.0512, the MAUB values of d2 are given in Table 2.
Following that, the MAUB ofd1 is calculated and displayed in Table 3 for various gains of KP and KI, in order to demonstrate the merit of the results obtained. Furthermore, the results presented in reference [45] are provided for comparison, in order to account for the advantage of the method presented in Corollary 1. According to Tables 3 and 4, our proposed method in Corollary 1 can achieve better MAUBs than those in reference [45] with respect to identical control gain.
The results presented in Tables3 and 4 demonstrate that d1 can significantly larger when KP and KI are smaller, and that d1 is significantly lower for larger KP and KI. With a lower KI≤0.05, d1 drops as KP increases. With the growth of KI≥0.1, d1 initially increases and then decreases with the increase of KP. As such, there is a range of KP that can achieve the MAUBs of d1 for a given KI.Table 2
The MAUBs ofd2 for various settings of σ0 and fixedKP, KI, d1, and η.
Methodsσ00.20.40.60.8Theorem14.28744.09643.19253.0028Table 3
The MAUB ofd1 for various control gains in reference [45].
d1KIKP0.050.10.150.20.40.61.0027.9213.779.056.693.121.910.880.0527.8714.069.286.863.211.970.920.1027.0313.689.226.943.292.020.960.2025.1112.768.616.533.322.101.010.4020.3610.427.065.382.831.911.010.6014.617.475.153.952.131.470.821.00.540.530.530.520.480.430.34Table 4
The MAUB ofd1 for various control gains in Corollary 1.
d1KIKP0.050.10.150.20.40.61.0033.9215.7712.058.694.122.911.880.0531.8716.069.286.863.211.970.920.1028.0315.6810.227.944.292.821.160.2025.1114.768.616.533.322.101.100.4023.3612.428.066.383.832.911.080.6015.6110.476.154.954.133.471.021.00.840.720.640.610.570.550.52Case 2.
MinimumH∞Performance Index: the acceptable minimum H∞ performance index ηmin that uses Theorem 1 in this study is noted in Table 5 for various σ0, fixed maximum delay bounds d1=0.2 and d2=0.8, and given gains KP=0.05 and KI=0.1. It is found that the designed control strategy in the power system is effective for managing load disturbances.Table 5
The allowable minimumηmin for given values of d1, d2, KP, andKI and different σ0.
Methodsσ00.20.40.60.8Theorem14.20134.51086.05126.9512


### 5.2. Sampled-DataH∞ LFC Controller Design

To compute the control gain in this example, first letd1=0.1, d2=0.3, η=8.5, σ=0.8, and ς=0.01, then the inequalities (33)–(36) in Theorem 2 are solved, and the control gain is obtained, as presented in Table 6.Table 6
The control gainK for fixed d1, d2, η, σ0, and ς in Theorem 2.
d1d2σ0ςηK0.10.30.80.018.50.03090.1343Let us use the disturbance to graphically verify the given results:(51)wt=0.1pu,ift=10s,20s,30s0,otherwise.The time-varying delays for the simulation results are chosen asd1t=0.05sint+0.05 and d1t=0.10sint+0.20 satisfying d1=0.1 and d2=0.3. State trajectories of model (11) are displayed in Figures 3 and 4. These simulation results indicate that model (11) is stabilized with load disturbance and the presence of a stochastic variable and the control gain is listed in Table 6.Figure 3
Evaluation of△f in equation (11) in Theorem 2.Figure 4
Evaluation ofACE in equation (11) in Theorem 2.


### 5.3. Sampled-Data RobustH∞ LFC Controller Design

For this scenario, model (38) is considered with ℋ=0.20.20.20.2T, E=diag0.10.10.10.1, and ℱt=diagcostcostcostcost. In addition, we assume that d1=0.1, d2=0.3, η=6.5, σ0=0.4, and ς=0.01. Then, inequalities (39)–(42) in Theorem 3 are solved, and the control gain is obtained, as given in Table 7.Table 7
The control gainK for fixed d1, d2, η, ς, and σ0 in Theorem 3.
d1d2σ0ςηK0.10.30.40.018.50.03300.0275Delaysd1t and d2t and disturbance signal wt are selected to be the same as in above case. To show the effect of the established control design scheme, state trajectories of model (38) are depicted in Figures 5 and 6 under the proposed sampled-data robust LFC control gain in Table 7.Figure 5
Evaluation of△f in equation (38) in Theorem 3.Figure 6
Evaluation ofACE in equation (38) in Theorem 3.


### 5.4. Comparative Analysis

Case 3.
Calculation of MAUBd1: the benefit of the proposed criterion in Corollary 1 is proven numerically in the following example. Model (44) is considered to obtain the following equation:(52)A=−2.000−0.9,Ad=−1.00−1.0−1.
With the above example, the comparison results are analyzed in detail. The MAUB values ofd1 are shown in Figure 7. They are compared with those published in references [39, 43, 44, 46, 47] for the case of μ=0. The results in Figure 7 clearly reveal the advantages of the proposed method.Figure 7
Comparative analysis of MAUBd1 of Corollary 1 with references [39, 43, 44, 44, 46].Case 4.
Computational Efficiency: the results presented in this study have been achieved on the basis of the LMI methodology. The total number of decision variables in LMI leads to computational issues of the main results. Consequently, the goal of this study is to establish new stability criteria that are less conservative and have a small number of decision variables. To do this, we have leveraged the Jensen-based integral inequality and reciprocal convex combination methods to reduce conservatism without adding new variables to the derivation of the main results. In addition, we have calculated the number of decision variables and compared them with those in prior studies [39, 43, 44, 46, 47], as listed in Table 8. In reference [43], enhanced stability criteria based on novel LKFs were discussed, which required 79.5n2+4.5n in decision variables. With the derived results in Corollary 1, only a limited number of decision variables 3n2+2n are required. Therefore, it is evident that the stability criteria formulated in this study yields less conservatism with a smaller computational burden.Table 8
Comparative analysis of number of decision variables in Corollary1.
MethodsNumber of decision variables[39]3.5n2+2.5n[44]10.5n2+3.5n[47]21n2+6n[46]54.5n2+9.5n[43]79.5n2+4.5nCorollary13n2+2n


## 5.1. The Case of a One-Area LFC-Based Power System

Table1 lists the parameters of one-area LFC-based power system taken from references [35, 45]. Correlations between gain of the sampled-data LFC scheme and effects of time delays on stability margins and H∞ performance based on the proposed results are discussed in this subsection.Table 1
Parameter values of one-area LFC-based model (1).
TgTchβℛDℳ0.10.321.00.051.010Case 1.
MAUBs: based on Theorem 1, an MAUB of d2 is calculated for model (11) without load disturbance. For different settings of σ0 and fixed KP=0, KI=0.05, d1=0.1, and η=6.0512, the MAUB values of d2 are given in Table 2.
Following that, the MAUB ofd1 is calculated and displayed in Table 3 for various gains of KP and KI, in order to demonstrate the merit of the results obtained. Furthermore, the results presented in reference [45] are provided for comparison, in order to account for the advantage of the method presented in Corollary 1. According to Tables 3 and 4, our proposed method in Corollary 1 can achieve better MAUBs than those in reference [45] with respect to identical control gain.
The results presented in Tables3 and 4 demonstrate that d1 can significantly larger when KP and KI are smaller, and that d1 is significantly lower for larger KP and KI. With a lower KI≤0.05, d1 drops as KP increases. With the growth of KI≥0.1, d1 initially increases and then decreases with the increase of KP. As such, there is a range of KP that can achieve the MAUBs of d1 for a given KI.Table 2
The MAUBs ofd2 for various settings of σ0 and fixedKP, KI, d1, and η.
Methodsσ00.20.40.60.8Theorem14.28744.09643.19253.0028Table 3
The MAUB ofd1 for various control gains in reference [45].
d1KIKP0.050.10.150.20.40.61.0027.9213.779.056.693.121.910.880.0527.8714.069.286.863.211.970.920.1027.0313.689.226.943.292.020.960.2025.1112.768.616.533.322.101.010.4020.3610.427.065.382.831.911.010.6014.617.475.153.952.131.470.821.00.540.530.530.520.480.430.34Table 4
The MAUB ofd1 for various control gains in Corollary 1.
d1KIKP0.050.10.150.20.40.61.0033.9215.7712.058.694.122.911.880.0531.8716.069.286.863.211.970.920.1028.0315.6810.227.944.292.821.160.2025.1114.768.616.533.322.101.100.4023.3612.428.066.383.832.911.080.6015.6110.476.154.954.133.471.021.00.840.720.640.610.570.550.52Case 2.
MinimumH∞Performance Index: the acceptable minimum H∞ performance index ηmin that uses Theorem 1 in this study is noted in Table 5 for various σ0, fixed maximum delay bounds d1=0.2 and d2=0.8, and given gains KP=0.05 and KI=0.1. It is found that the designed control strategy in the power system is effective for managing load disturbances.Table 5
The allowable minimumηmin for given values of d1, d2, KP, andKI and different σ0.
Methodsσ00.20.40.60.8Theorem14.20134.51086.05126.9512


## 5.2. Sampled-DataH∞ LFC Controller Design

To compute the control gain in this example, first letd1=0.1, d2=0.3, η=8.5, σ=0.8, and ς=0.01, then the inequalities (33)–(36) in Theorem 2 are solved, and the control gain is obtained, as presented in Table 6.Table 6
The control gainK for fixed d1, d2, η, σ0, and ς in Theorem 2.
d1d2σ0ςηK0.10.30.80.018.50.03090.1343Let us use the disturbance to graphically verify the given results:(51)wt=0.1pu,ift=10s,20s,30s0,otherwise.The time-varying delays for the simulation results are chosen asd1t=0.05sint+0.05 and d1t=0.10sint+0.20 satisfying d1=0.1 and d2=0.3. State trajectories of model (11) are displayed in Figures 3 and 4. These simulation results indicate that model (11) is stabilized with load disturbance and the presence of a stochastic variable and the control gain is listed in Table 6.Figure 3
Evaluation of△f in equation (11) in Theorem 2.Figure 4
Evaluation ofACE in equation (11) in Theorem 2.


## 5.3. Sampled-Data RobustH∞ LFC Controller Design

For this scenario, model (38) is considered with ℋ=0.20.20.20.2T, E=diag0.10.10.10.1, and ℱt=diagcostcostcostcost. In addition, we assume that d1=0.1, d2=0.3, η=6.5, σ0=0.4, and ς=0.01. Then, inequalities (39)–(42) in Theorem 3 are solved, and the control gain is obtained, as given in Table 7.Table 7
The control gainK for fixed d1, d2, η, ς, and σ0 in Theorem 3.
d1d2σ0ςηK0.10.30.40.018.50.03300.0275Delaysd1t and d2t and disturbance signal wt are selected to be the same as in above case. To show the effect of the established control design scheme, state trajectories of model (38) are depicted in Figures 5 and 6 under the proposed sampled-data robust LFC control gain in Table 7.Figure 5
Evaluation of△f in equation (38) in Theorem 3.Figure 6
Evaluation ofACE in equation (38) in Theorem 3.


## 5.4. Comparative Analysis

Case 3.
Calculation of MAUBd1: the benefit of the proposed criterion in Corollary 1 is proven numerically in the following example. Model (44) is considered to obtain the following equation:(52)A=−2.000−0.9,Ad=−1.00−1.0−1.
With the above example, the comparison results are analyzed in detail. The MAUB values ofd1 are shown in Figure 7. They are compared with those published in references [39, 43, 44, 46, 47] for the case of μ=0. The results in Figure 7 clearly reveal the advantages of the proposed method.Figure 7
Comparative analysis of MAUBd1 of Corollary 1 with references [39, 43, 44, 44, 46].Case 4.
Computational Efficiency: the results presented in this study have been achieved on the basis of the LMI methodology. The total number of decision variables in LMI leads to computational issues of the main results. Consequently, the goal of this study is to establish new stability criteria that are less conservative and have a small number of decision variables. To do this, we have leveraged the Jensen-based integral inequality and reciprocal convex combination methods to reduce conservatism without adding new variables to the derivation of the main results. In addition, we have calculated the number of decision variables and compared them with those in prior studies [39, 43, 44, 46, 47], as listed in Table 8. In reference [43], enhanced stability criteria based on novel LKFs were discussed, which required 79.5n2+4.5n in decision variables. With the derived results in Corollary 1, only a limited number of decision variables 3n2+2n are required. Therefore, it is evident that the stability criteria formulated in this study yields less conservatism with a smaller computational burden.Table 8
Comparative analysis of number of decision variables in Corollary1.
MethodsNumber of decision variables[39]3.5n2+2.5n[44]10.5n2+3.5n[47]21n2+6n[46]54.5n2+9.5n[43]79.5n2+4.5nCorollary13n2+2n


## 6. Conclusion

In this article, robust sampled-dataH∞ LFC scheme for power systems with randomly occurring time-varying delays has been considered. To better reflect the actual demands of practical dynamics, a generalized framework of the robust sampled-data H∞ LFC scheme has been studied. By leveraging the input-delay technique, the sampled-data model is converted into a continuous representation. Bernoulli-distributed sequences are used to characterize random time-varying delays in the sampled-data LFC scheme. Less conservative conditions are achieved by utilizing the LKF and employing Jensen inequality and reciprocal convex combination lemma to ensure the considered power system is mean-square asymptotic stability under the designed control strategy. The results derived in this study are based on LMIs that can be easily solved using the MATLAB LMI toolbox. The criteria obtained have been used to analyze the upper bounds of time delays, and a comparison study has been presented to validate the efficacy of the designed control method.It is worth noting that Markov processes are widely used for modelling complex systems that undergo unpredictable changes. Therefore, LFC for power systems with Markov processes is essential. As a result, we intend to analyze observer-based sliding mode LFC of power systems under deception attacks using the proposed Markovian jump approaches in references [48, 49]. The corresponding results will be carried out in the near future.



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