# Adaptive Integral Second-Order Sliding Mode Control Design for Load Frequency Control of Large-Scale Power System with Communication Delays

Authors: Anh-Tuan Tran; Bui Le Ngoc Minh; Phong Thanh Tran; Van Van Huynh; Van-Duc Phan; Viet-Thanh Pham; Tam Minh Nguyen
Journal: Complexity
 (2021)

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## Abstract

Nowadays, the power systems are getting more and more complicated because of the delays introduced by the communication networks. The existence of the delays usually leads to the degradation and/or instability of power system performance. On account of this point, the traditional load frequency control (LFC) approach for power system sketches a destabilizing impact and an unacceptable system performance. Therefore, this paper proposes a new LFC based on adaptive integral second-order sliding mode control (AISOSMC) approach for the large-scale power system with communication delays (LSPSwCD). First, a new linear matrix inequality is derived to ensure the stability of whole power systems using Lyapunov stability theory. Second, an AISOSMC law is designed to ensure the finite time reachability of the system states. To the best of our knowledge, this is the first time the AISOSMC is designed for LFC of the LSPSwCD. In addition, the report of testing results presents that the suggested LFC based on AISOSMC can not only decrease effectively the frequency variation but also make successfully less in mount of power oscillation/fluctuation in tie-line exchange.


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## Body

## 1. Introduction

In modern power networks, the frequency stability is one of the significant problems related to the large-scale power system (LSPS) with communication delays. The system has been more and more complicated for LFC due to matched or mismatched uncertainties, load variations, and time-delays [1–4]. Communications time-delay is necessary in the LSPS; it cannot ignore and remove in the practical LSPS. So, the time-delay must consider in the area control error (ACE) signal. Normally, the time-delay can appear in the state variables or in the input channel of LSPS. The input control signal is transported to the power plants through communication networks with delay time. The state delay-time is the essential reason to effect on delaying the basic elements of communication or transportation in the LSPS. The deviation of tie-line exchange power and frequency are combined and known as an ACE to confirm the control frequency aim for the LSPS. The balance in both the tie-line and frequency exchange power is assured when an ACE is controlled and limited. In the LSPS model, the time-delay of ACE from regulation station to control area of the system is important [4–6]. Moreover, the time-delay is known as time lag in the power system model. That is, a big challenge to design the control algorithm to deal with the time-delay in the LSPS. In ACE control signal, the time lag may occur and lead to oscillation and instability in the LSPS [6–9]. Therefore, it is absolutely necessary to consider and explore the LFC for the LSPSwCD. The main objectives of LFC for each control area are continuing to keep the nominal value of the system frequency in range of the standard value and to act in accordance with the scheduled active power interchange with neighboring control areas.In general, the previous LFC scheme such as PID control has been used to maintain tie-line power and frequency at schedule value which is one of the simplest controllers [10–13]. A suitable design of PID controller which is constructed on the direct synthesis method in frequency domain is given in [10]. In [11], a fractional-order PID control approach is built for the single-area LFC model using the Kharitonov’s theorem to eliminate steady state error. Robust PID controller-based stability boundary locus and Kharitonov’s theorem were used for LFC of the LSPS [12]. In [13], an ant lion algorithm is combined with PID to optimize the LFC loop parameters for improving the frequency regulation. The above approaches are suitable for application to design LFC of power network with nominal parameter and no uncertainty. However, the practical power network is always influenced or touched by an external factor of the uncertainties such as different load disturbances, which effect on the stability of power network. In order to overcome drawbacks and resolve the LFC problem effectively, advanced control techniques are developed for LFC design consisting of neural network and fuzzy control [14–16]. In [14], the LFC scheme for the LSPS was established and constructed on the adaptive fuzzy control method. A novel fuzzy PID control strategy with the fractional-order integrator and filtered derivative action was suggested to resolve automatic generation control for the power network [15]. In [16], the optimized operation of LFC is proposed for the LSPS with hybrid energy storage system.On the other hand, as a powerful robust control strategy, SMC has been successfully applied to a wide variety of practical systems [17–19]. The terminal SMC is designed for LFC with the renewable power networks [20]. In [21], the adaptive double integral SMC is proposed for LFC of the LSPS where the time-delay is not considered. The goal of Sarkar was to design an adaptive integral higher order SMC for LFC problems to guarantee the frequency variation [22]. In [23], this article indicated nonlinear SMC with parametric uncertainties for LFC utilization in the LSPS to vary the system damping characteristics under uncertainties and step load disturbances. The approach given in [24] developed the strategy of SMC by model order-reduction of the LFC approach of the microhydropower system. A new full order SMC scheme in the paper [25] is utilized to eliminate and avoid the singularity of derivative of terms with fractional power elements. A novel adaptive SMC method has been used and developed for the LFC of the LSPS [26]. The proportional and integral switching surface was established for LFC of LSPS where the chattering problem is existed due to first-order SMC used [27]. The SMC-based LFC of above approaches can reduce the tie-line power and frequency deviations to ensure the stability of power network. However, the performance of LSPS using control scheme given in [20–27] is achieved without considering the time-delay in the power grids. In addition, the existence of the delays usually leads to the degradation and/or instability of power network performance. In order to solve this problem, the control scheme given in [28–30] is proposed to regulate the frequency deviation of the LSPS in the presence of communication delays and sudden load change. In [31], the LFC of the LSPS with nonlinear perturbations and time-varying delays are proposed using SMC. In [32], the sliding mode LFC is designed for the LSPS with time-delay and load disturbance. The sliding mode LFC strategy is suggested for the LSPS with time-delay and significant wind power penetration [33]. However, these SMC approaches are developed based on first-order time derivative [31–33]. In addition, the first-order SMC provides low accuracies due to chattering phenomenon in the control input. Therefore, in this paper, the second-order SMC is proposed to solve this problem. Moreover, an adaptive control method is adopted to estimate the unknown upper bound of aggregated uncertainties. To the best of our knowledge, the adaptive integral second-order sliding mode control (AISOSMC) scheme has not been developed for LFC problem in the LSPSwCD so far. The major contributions of this paper are displayed as follows:The novel theory based on AISOSMC approach is offered for the LSPSwCDThe continuous control law is developed to deal with the influence of time-delay related to ACE on the LFC problem of the LSPSwCDA state feedback controller containing both present and delayed state information is designed to improve tolerable delay margin of the LSPSwCDA new LMI is established for proving the stability of whole PSPS based on Lyapunov stability theoryThe proposed LFC based on AISOSMC can not only decrease effectively the frequency variation but also make successfully less in mount of power oscillation/fluctuation in tie-line exchange


## 2. Mathematical Model of a Large-Scale Power System with Communication Delays (LSPSwCD)

In this part, we model the LSPSwCD. Figure1 is the block diagram of the ith area of the LSPS with communication delay [28–30]. The model of the ith area system is composed of a governor, a nonreheat turbine, and a generator. The output of the generator is the frequency error. The tie-line power error is linearly proportional to the integration of frequency error. The linear combination of frequency error and the tie-line power error is the area control error (ACE). In addition, the time-delay of ACE signal is considered in the power network.Figure 1
The structure ofith area of the LSPSwCD.Nevertheless, when the load variation with minor change occurs during its conventional process, then the mathematical model of power network can be linearized near the stable operating point. Therefore, the dynamic equations of the aboveith area system are expressed as follows:(1)Δf˙it=−1TpiΔfit+KPiTpiΔPmit−KPiTpiΔPdit−KPi2πTpi∑i=1,j≠iNTs,ijΔδit−Δδjt,ΔP˙mit=−1TtiΔPmit+1TtiΔXgit,ΔX˙git=−1TgiRiΔfit−1TgiΔXgit−1TgiΔEit−τi+1Tgiuit,ΔE˙it=KBiKEiΔfit+KEi2π∑i=1,j≠iNTs,ijΔδit−Δδjt,Δδ˙it=2πΔfit,with i=1 to N and N is denoted as the number of areas, where Δfit and Δfjt are the frequency variation of the ith area and the jth area, ΔPmit is the variation in governor output command, ΔXgit is the governor valve position of each area, Δδit and Δδjt are the changes of rotor angle deviations of the ith area and the jth area, ΔPdit is the incremental change in local load of each area, Ts,ij is the tie-line power coefficient between the ith area and the jth area, Tgi is the time constant of governor, Tti is the turbine time constant, and Tpi is the time constant in power network, respectively. Kpi, Ri, KEi, and KBi are power network gain, droop coefficient of individual area, speed regulation coefficient, and frequency bias factor. ΔEit−τi is the area control error with the time-delay and uit is the control input.In state-space form, the system state variables are used as(2)zit=ΔfitΔPmitΔXgitΔEitΔδitT.So, the LSPSwCD described by Figure1 can be written and expressed in state-space representation as follows:(3)z˙it=Aizit+Dizit−τi+Biuit+∑j=1;j≠iNHijzjt+FiΔPdit,where i=1,2,…,N and N is denoted as the number of areas.The state-space matrixAi,Bi,Di,Fi,Hij in mathematical model is given as follows:(4)Ai=−1TPiKPiTPi00−KPi2πTPi∑i=1,j≠iNTs,ij0−1TTi1TTi00−1RiTGi0−1TGi00KBiKEi000KEi2π∑i=1,j≠iNTs,ij2π0000,Di=0000000000000−1TGi00000000000,Hij=0000−KPi2πTPi∑i=1,j≠iNTs,ij00000000000000KEi2π∑i=1,j≠iNTs,ij00000,Bi=001TGi00;Fi=−KPiTPi0000.In applied LSPSwCD, the operating point fluctuates continually induced by the fluctuating resource and load disturbance. In addition, by considering CDs element, the dynamic model of LSPSwCD with the uncertainties and parameter variations in equation (3) are further redefined as(5)z˙it=Ai+ΔAizi,tzit+Di+ΔDizi,t−τizit−τi+Biuit+ξizi,t+∑j=1;j≠iNHij+ΔHijzj,tzjt+FiΔPdit=Aizit+Dizit−τi+Biuit+∑j=1;j≠iNHijzjt+wizi,t,where Ai,Di,Bi,andHij are the system matrices with nominal value, ΔAizi,t, ΔHijzj,t, and ΔDizi,t−τi are the parameter uncertainties, and Biξizi,t is the disturbance input signal. The lumped uncertainty wizi,t is defined as follows:(6)wizi,t=ΔAizi,tzit+ΔDizi,t−τizit−τi+Biξizi,t+∑i=1;j≠iNΔHijzjt+FiΔPdit.Assumption 1.
The lumped uncertaintieswizi,t and the differential of w˙izi,t are bounded, i.e., there exist known scalars γi and ∂i such that wizi,t≤γi and w˙izi,t≤∂i, where . is the matrix norm.Assumption 2.
The time-delay state vector must satisfy the conditionzit−τi≤zimax, zimax=maxzimax, where . is the matrix norm.
In order to prove the system stability, we recall some lemmas.Lemma 1.
(see [34]). Let X and Y be actual matrices with appropriate dimension, then, for any scalar μ>0, the sequent matrix inequality obtains(7)XTY+YTX≤μXTX+μ−1YTY.Lemma 2.
(see [35]). The following matrix inequality(8)PzΓzΓTzQz>0,where Pz=PTz, Qz=QTz, Γz depends affinely on z, is equivalent to Qz>0 and Pz−ΓzQ−1zΓTz>0.Lemma 3.
(see [35]). Assume that x∈Rn, y∈Rn, N∈Rn×n, and N is the positive definite matrix. Then, the inequality(9)xTNy+yTNx≤1εxTNx+εyTNy,holds for all ε>0.


## 3. Adaptive Integral Second-Order Sliding Mode Control (AISOSMC) Design for LFC of Large-Scale Power System with Communication Delays (LSPSwCD)

In this section, the AISOSMC method is developed for LFC of the LSPSwCD under mismatched parameter uncertainties and load disturbances. To solve this problem, we work step by step to design and implement the new controller approach. Firstly, the integral sliding surface (ISS) is represented for LSPSwCD to assure that the whole system is asymptotically stable. Secondly, the decentralized adaptive integral second-order sliding mode control law (DAISOSMCL) is designed to force the system trajectories to the sliding manifold and keep it there for after.


### 3.1. Stability Analysis of a LSPSwCD in Sliding Mode Dynamics

In detail, we first begin to propose and build an ISS for a LSPS:(10)σizit=Eizit−∫0tEiAi−BiTiziτdτ,where Ei is the constant matrix and Ti is the design matrix, matrix Ei is designed to guarantee that matrix EiBi is nonsingular and matrix Ti is chosen via pole assignment such that the eigenvalues of matrix Ai−BiTi are always less than zero.If we recognize and differentiateσizit with respect to time combined with (3), then(11)σ˙izit=EiAizit+Dizit−τi+Biuit+∑j=1;j≠iNHijzjt+wizi,t−EiAi−BiTizit.So, the settingσizit=σ˙izit=0; the equivalent control is rewritten by(12)uieqt=−EiBi−1EiAizit+EiDizit−τi+∑j=1;j≠iNEiHijzjt+Eiwizi,t−EiAi−BiTizit.Substitutinguit with uieqt into the LSPSwCD yields the sliding motion:(13)z˙it=Ai−BiTizit+I−BiEiBi−1EiDizit−τi+∑j=1;j≠iNI−BiEiBi−1EiHijzjt+I−BiEiBi−1Eiwizi,t.The introduction of the following theorem makes a condition that the AISOSMC dynamic equation (11) is asymptotically stable.Theorem 1.
The sliding motion (13) is asymptotically stable if and only if there includes symmetric positive definite matrix Qi,Pi, i=1,2,…,N, and positive scalars q,φi. and βj such that the following LMIs hold:(14)Ωi+∑j=1j≠iNβj−1HjiTHji0DiTI−BiEiBi−1EiTQiI−BiEiBi−1Ei0−Pi00I−BiEiBi−1EiDi0−qQi−10I−BiEiBi−1EiTQi00−φi−1<0,where Ωi=Ai−BiTiTQi+QiAi−BiTi+Qi+Pi.Proof.
To study and analyze stability of the sliding motion (13), we use the Lyapunov function as follows:(15)V=∑i=1NziTtQizit+∫t−τitziTsPizisds,where Qi,Pi>0 satisfies (14). Then, taking the time derivative of (15) and using equation (13), we obtain(16)V˙=∑i=1NziTtAi−BiTiTQi+QiAi−BiTizit+ziTt−τiDiTI−BiEiBi−1EiTQizit+ziTtQiI−BiEiBi−1EiDizit−τi+∑j=1;j≠iNzjTtHijTI−BiEiBi−1EiTQizit+∑j=1;j≠iNziTtQiI−BiEiBi−1EiHijzjt+wiTzi,tI−BiEiBi−1EiTQizit+ziTtQiI−BiEiBi−1Eiwizi,t+ziTtPizit−ziTt−τiPizit−τi.
To apply Lemma1 in equation (16), we obtain(17)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTizit+ziTt−τiDitI−BiEiBi−1EiTQizit+ziTtQiI−BiEiBi−1EiDizit−τi+∑j=1;j≠iNziTtβi−1QiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+∑j=1;j≠iNzjTtβiHijTHijzjt+ziTtPizit−ziTt−τiPizit−τi+χi−1ziTtQiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+χiwiTzi,twizi,t.
Using Lemma3 and equation (17), we get(18)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTizit+ziTt−τiDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit−τi+ziTtQizit+∑j=1;j≠iNziTtβi−1QiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+∑j=1;j≠iNzjTtβiHijTHijzjt+ziTtPizit−ziTt−τiPizit−τi+χi−1ziTtQiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+χiwiTzi,twizi,t.
Since∑i=1N∑j=1,j≠iNzjTtβi−1HijTHijzjt=∑i=1N∑j=1,j≠iNziTtβj−1HjiTHjizit, we achieve that(19)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTi+Qi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+∑j=1;j≠iNziTtβjHjiTHjizit+χiwiTzi,twizi,t+ziTt−τiDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit−τi+ziTtPizit−ziTt−τiPizit−τi,where φi=χi−1+N−1βi−1.
The matrixDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDi is the semipositive definite. Since the zit for i=1,2,…,N are independent of each other. Then, from equation (32) of paper [36], the following is true:(20)Vz1t−τ1,z2t−τ2,z3t−τ3,…,zNt−τN≤qVz1t,z2t,z3t,…,zNt,for q>1 is equivalent to(21)∑i=1NziTt−τiDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit−τi≤q∑i=1NziTtDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit.
Then, we can get the following equation:(22)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTi+Qi+Pi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQi+qDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit+∑j=1;j≠iNziTtβjHjiTHjizit−ziTt−τiPizit−τi+χiwiTzi,twizi,t.
Based on Assumption1, the following equation can be achieved:(23)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTi+Qi+Pi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQi+qDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit+∑j=1;j≠iNziTtβjHjiTHjizit−ziTt−τiPizit−τi+ιi,where ιi=χiγi2.
Define the augmented vector(24)Ψit=ziTtziTt−τiT,V˙≤ΨiTtΘiΨit+ιi.
From Lemma2 and LMI (14), we get(25)Θi=−Ξi00−Pi>0,where(26)Ξi=Ai−BiTiTQi+QiAi−BiTi+Qi+Pi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQi+qDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDi+∑j=1,j≠iNβj−1HjiTHji.
According to equations (23) and (25), we obtain(27)V˙≤∑i=1N−λminΘiΨit2+ιi,where the constant value ιi and the eigenvalue λminΘi>0. Therefore, V˙<0 is achieved with Ψit>ιi/λminΘi. Hence, the sliding motion of system (13) is asymptotically stable.Remark 1.
The adaptive integral second-order sliding mode control design is composed of the hitting phase and the sliding phase. The proposed controller is used to force the system state trajectories to sliding phase and keep the system state trajectories on it thereafter. If the disturbance and uncertainty satisfy the matching conditionrankBi,wizi,t=rankBi, the system in the sliding mode is invariant to disturbance and uncertainty. The stability of system under matched condition is easier than mismatched condition. The proposed controller can compensate for disturbance and uncertainty directly under matched condition. Therefore, the stability of the system in the sliding mode under mismatched condition rankBi,wizi,t≠rankBi has been considered and proved using LMI technique based on Lyapunov stability theory.


### 3.2. First-Order SMC Design

In order to guarantee the reachability of state variables to the ISS (10), the decentralized first-order integral SMC law is designed as follows [18]:(28)uiFOSMCt=−EiBi−1EiBiTizit+EiDizit−τi+∑j=1;j≠iNEjHjizit+Eiγ¯i+ε˜isatσizit,where ε˜i>0 and(29)satσizit=1,σizit>1,σizit,When−1≤σizit≤1,−1,σizit<−1.Remark 2.
The first-order SMC can be used to study LFC of power system under matched uncertainties. However, the parametric uncertainties not usually satisfy the matched condition in real power network. Consequently, some main constraints are necessary to design the first-order SMC to compensate the uncertainties, which can guarantee the convergence in nominal frequency and the system stability but the system trajectories cannot reach to origin point. Therefore, the second-order ISS has been used as the following part to force the system trajectory to equivalent point and to make better the transient performance.


### 3.3. Decentralized Adaptive Integral Second-Order Sliding Mode Control Law (DAISOSMCL) Design

In this step, the DAISOSMCL is developed for the LSPSwCD to reduce the frequency deviation. The main purpose of the proposed control scheme is to effect on the second-order derivative of the sliding variablesσizit. By using the discontinuous control signal u˙it, it is simple to make σizit and σ˙izit converge to zero. So, the input control signal uit of LSPSwCD can get by integrating the discontinuous signal u˙it to make continuous signal uit. Therefore, the DAISOSMCL approach removes some undesired frequency oscillations in the control signal of LSPSwCD.We define and establish the sliding manifold (SMd)Gizit as(30)Gizit=σ˙izit+εiσzit,(31)G˙izit=σ¨izit+εiσ˙izit,where εi>0 is a positive constant, according to equation (30); the equation (31) can be redefined as(32)G˙izit=EiAiz˙it+Diz˙it−τi+Biu˙it+∑j=1;j≠iNHijz˙jt+w˙izi,t−EiAi−BiTiz˙it+εiσ˙izit.Based on the definition of sliding surface and SMd, the continuous DAISOSMCL for LFC of a LSPSwCD is given as follows:(33)u˙it=u˙iSOSMCt+u˙iadtt,where(34)u˙iSOSMCt=−EiBi−1EiBiTiz˙it+EiDiz˙t−τi+∑j=1;j≠iNEjHjiz˙it+εiσ˙izit+Ei∂i+ε¯iGizitGizit,u˙iadtt=−EiBi−1I∂^itEi+p¯i∂i24,i=1,2,…,N,where(35)∂^˙it=pi−p¯i∂^it+Ei,i=1,2,…,N,in which pi and p¯i are the positive constants.Then, we have the main result which is presented as follows.Theorem 2.
Consider the closed loop of the power systems with the DAISOSMCL (33). Then, every solution trajectory of system state is directed towards the SMd Gizit, and once the trajectory hits the SMd Gizit, it remains on the sliding manifold thereafter.Proof.
The Lyapunov function is introduced as follows:(36)V¯t=∑i=1NGizit+0.5pi∂˜i2,where ∂˜it=∂i−∂^it.
So, taking the derivative ofV¯t, we have(37)V¯˙=∑i=1NGiTzitGizitG˙izit−∂˜it∂^˙itpi=∑i=1NGiTzitGizitEiAiz˙it+Diz˙t−τi+Biu˙it+∑j=1j≠iNHijz˙jt+w˙izi,t−EiAi−BiTiz˙it+εiσ˙izit−∑i=1N∂˜it∂^˙itpi.
According to equation (37) and property AB≤AB, we have(38)V¯˙=∑i=1NEiAiz˙it+Diz˙t−τi+Biu˙it+∑j=1j≠iNHijz˙jt+w˙izi,t+∑i=1NGiTzitGizitEiBiu˙it−∑i=1N∂˜it∂^˙itpi≤∑i=1NEiBiTiz˙it+EiDiz˙it−τi+δiσ˙izit+∑j=1j≠iNEiHijz˙it+Eiw˙izi,t+∑i=1NGiTzitGizitEiBiu˙it−∑i=1N∂˜it∂^˙itpi.
Using Assumption1, we achieve(39)V¯˙≤∑i=1NEiBiTiz˙it+EiDiz˙it−τi+Ei∂i+∑j=1j≠iNEjHjiz˙it+εiσ˙izit+∑i=1NGiTzitGizitEiBiu˙it−∑i=1N∂˜it∂^˙itpi.
Using the DAISOSMCL (33) yields(40)V¯˙≤∑i=1N−ε¯i−∂^itEi−p¯i∂i24+Ei∂i−∑i=1N∂iEi−∑i=1Np¯i∂^i2t−p¯i∂i∂^it−∂^itEi=∑i=1N−ε¯i−p¯i∂i24−∑i=1Np¯i∂^i2t−p¯i∂i∂^it=∑i=1N−ε¯i−p¯i∂i24−∑i=1Np¯i∂^It−∂i2−p¯i∂i24=∑i=1N−ε¯i−∑i=1Np¯i∂^It−∂i2.
The above inequality implies that the system trajectories of the LSPSwCD (3) reach the SMd Gizit and keep it for later.Remark 3.
The SMC is capable to reach the bounded system stability and is able to maintain within a range to converge zero, and then the power system is notably stable. Equation (10) indicates that the integral term ∫0tEiAi−BiTiziτdτ is only reflected in the proposed DAISOSMCL (33). Therefore, the control law (33) is to improve the performance of steady state error in comparison with the traditional integral SMC.


## 3.1. Stability Analysis of a LSPSwCD in Sliding Mode Dynamics

In detail, we first begin to propose and build an ISS for a LSPS:(10)σizit=Eizit−∫0tEiAi−BiTiziτdτ,where Ei is the constant matrix and Ti is the design matrix, matrix Ei is designed to guarantee that matrix EiBi is nonsingular and matrix Ti is chosen via pole assignment such that the eigenvalues of matrix Ai−BiTi are always less than zero.If we recognize and differentiateσizit with respect to time combined with (3), then(11)σ˙izit=EiAizit+Dizit−τi+Biuit+∑j=1;j≠iNHijzjt+wizi,t−EiAi−BiTizit.So, the settingσizit=σ˙izit=0; the equivalent control is rewritten by(12)uieqt=−EiBi−1EiAizit+EiDizit−τi+∑j=1;j≠iNEiHijzjt+Eiwizi,t−EiAi−BiTizit.Substitutinguit with uieqt into the LSPSwCD yields the sliding motion:(13)z˙it=Ai−BiTizit+I−BiEiBi−1EiDizit−τi+∑j=1;j≠iNI−BiEiBi−1EiHijzjt+I−BiEiBi−1Eiwizi,t.The introduction of the following theorem makes a condition that the AISOSMC dynamic equation (11) is asymptotically stable.Theorem 1.
The sliding motion (13) is asymptotically stable if and only if there includes symmetric positive definite matrix Qi,Pi, i=1,2,…,N, and positive scalars q,φi. and βj such that the following LMIs hold:(14)Ωi+∑j=1j≠iNβj−1HjiTHji0DiTI−BiEiBi−1EiTQiI−BiEiBi−1Ei0−Pi00I−BiEiBi−1EiDi0−qQi−10I−BiEiBi−1EiTQi00−φi−1<0,where Ωi=Ai−BiTiTQi+QiAi−BiTi+Qi+Pi.Proof.
To study and analyze stability of the sliding motion (13), we use the Lyapunov function as follows:(15)V=∑i=1NziTtQizit+∫t−τitziTsPizisds,where Qi,Pi>0 satisfies (14). Then, taking the time derivative of (15) and using equation (13), we obtain(16)V˙=∑i=1NziTtAi−BiTiTQi+QiAi−BiTizit+ziTt−τiDiTI−BiEiBi−1EiTQizit+ziTtQiI−BiEiBi−1EiDizit−τi+∑j=1;j≠iNzjTtHijTI−BiEiBi−1EiTQizit+∑j=1;j≠iNziTtQiI−BiEiBi−1EiHijzjt+wiTzi,tI−BiEiBi−1EiTQizit+ziTtQiI−BiEiBi−1Eiwizi,t+ziTtPizit−ziTt−τiPizit−τi.
To apply Lemma1 in equation (16), we obtain(17)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTizit+ziTt−τiDitI−BiEiBi−1EiTQizit+ziTtQiI−BiEiBi−1EiDizit−τi+∑j=1;j≠iNziTtβi−1QiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+∑j=1;j≠iNzjTtβiHijTHijzjt+ziTtPizit−ziTt−τiPizit−τi+χi−1ziTtQiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+χiwiTzi,twizi,t.
Using Lemma3 and equation (17), we get(18)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTizit+ziTt−τiDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit−τi+ziTtQizit+∑j=1;j≠iNziTtβi−1QiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+∑j=1;j≠iNzjTtβiHijTHijzjt+ziTtPizit−ziTt−τiPizit−τi+χi−1ziTtQiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+χiwiTzi,twizi,t.
Since∑i=1N∑j=1,j≠iNzjTtβi−1HijTHijzjt=∑i=1N∑j=1,j≠iNziTtβj−1HjiTHjizit, we achieve that(19)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTi+Qi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQizit+∑j=1;j≠iNziTtβjHjiTHjizit+χiwiTzi,twizi,t+ziTt−τiDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit−τi+ziTtPizit−ziTt−τiPizit−τi,where φi=χi−1+N−1βi−1.
The matrixDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDi is the semipositive definite. Since the zit for i=1,2,…,N are independent of each other. Then, from equation (32) of paper [36], the following is true:(20)Vz1t−τ1,z2t−τ2,z3t−τ3,…,zNt−τN≤qVz1t,z2t,z3t,…,zNt,for q>1 is equivalent to(21)∑i=1NziTt−τiDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit−τi≤q∑i=1NziTtDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit.
Then, we can get the following equation:(22)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTi+Qi+Pi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQi+qDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit+∑j=1;j≠iNziTtβjHjiTHjizit−ziTt−τiPizit−τi+χiwiTzi,twizi,t.
Based on Assumption1, the following equation can be achieved:(23)V˙≤∑i=1NziTtAi−BiTiTQi+QiAi−BiTi+Qi+Pi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQi+qDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDizit+∑j=1;j≠iNziTtβjHjiTHjizit−ziTt−τiPizit−τi+ιi,where ιi=χiγi2.
Define the augmented vector(24)Ψit=ziTtziTt−τiT,V˙≤ΨiTtΘiΨit+ιi.
From Lemma2 and LMI (14), we get(25)Θi=−Ξi00−Pi>0,where(26)Ξi=Ai−BiTiTQi+QiAi−BiTi+Qi+Pi+φiQiI−BiEiBi−1EiI−BiEiBi−1EiTQi+qDiTI−BiEiBi−1EiTQiI−BiEiBi−1EiDi+∑j=1,j≠iNβj−1HjiTHji.
According to equations (23) and (25), we obtain(27)V˙≤∑i=1N−λminΘiΨit2+ιi,where the constant value ιi and the eigenvalue λminΘi>0. Therefore, V˙<0 is achieved with Ψit>ιi/λminΘi. Hence, the sliding motion of system (13) is asymptotically stable.Remark 1.
The adaptive integral second-order sliding mode control design is composed of the hitting phase and the sliding phase. The proposed controller is used to force the system state trajectories to sliding phase and keep the system state trajectories on it thereafter. If the disturbance and uncertainty satisfy the matching conditionrankBi,wizi,t=rankBi, the system in the sliding mode is invariant to disturbance and uncertainty. The stability of system under matched condition is easier than mismatched condition. The proposed controller can compensate for disturbance and uncertainty directly under matched condition. Therefore, the stability of the system in the sliding mode under mismatched condition rankBi,wizi,t≠rankBi has been considered and proved using LMI technique based on Lyapunov stability theory.


## 3.2. First-Order SMC Design

In order to guarantee the reachability of state variables to the ISS (10), the decentralized first-order integral SMC law is designed as follows [18]:(28)uiFOSMCt=−EiBi−1EiBiTizit+EiDizit−τi+∑j=1;j≠iNEjHjizit+Eiγ¯i+ε˜isatσizit,where ε˜i>0 and(29)satσizit=1,σizit>1,σizit,When−1≤σizit≤1,−1,σizit<−1.Remark 2.
The first-order SMC can be used to study LFC of power system under matched uncertainties. However, the parametric uncertainties not usually satisfy the matched condition in real power network. Consequently, some main constraints are necessary to design the first-order SMC to compensate the uncertainties, which can guarantee the convergence in nominal frequency and the system stability but the system trajectories cannot reach to origin point. Therefore, the second-order ISS has been used as the following part to force the system trajectory to equivalent point and to make better the transient performance.


## 3.3. Decentralized Adaptive Integral Second-Order Sliding Mode Control Law (DAISOSMCL) Design

In this step, the DAISOSMCL is developed for the LSPSwCD to reduce the frequency deviation. The main purpose of the proposed control scheme is to effect on the second-order derivative of the sliding variablesσizit. By using the discontinuous control signal u˙it, it is simple to make σizit and σ˙izit converge to zero. So, the input control signal uit of LSPSwCD can get by integrating the discontinuous signal u˙it to make continuous signal uit. Therefore, the DAISOSMCL approach removes some undesired frequency oscillations in the control signal of LSPSwCD.We define and establish the sliding manifold (SMd)Gizit as(30)Gizit=σ˙izit+εiσzit,(31)G˙izit=σ¨izit+εiσ˙izit,where εi>0 is a positive constant, according to equation (30); the equation (31) can be redefined as(32)G˙izit=EiAiz˙it+Diz˙it−τi+Biu˙it+∑j=1;j≠iNHijz˙jt+w˙izi,t−EiAi−BiTiz˙it+εiσ˙izit.Based on the definition of sliding surface and SMd, the continuous DAISOSMCL for LFC of a LSPSwCD is given as follows:(33)u˙it=u˙iSOSMCt+u˙iadtt,where(34)u˙iSOSMCt=−EiBi−1EiBiTiz˙it+EiDiz˙t−τi+∑j=1;j≠iNEjHjiz˙it+εiσ˙izit+Ei∂i+ε¯iGizitGizit,u˙iadtt=−EiBi−1I∂^itEi+p¯i∂i24,i=1,2,…,N,where(35)∂^˙it=pi−p¯i∂^it+Ei,i=1,2,…,N,in which pi and p¯i are the positive constants.Then, we have the main result which is presented as follows.Theorem 2.
Consider the closed loop of the power systems with the DAISOSMCL (33). Then, every solution trajectory of system state is directed towards the SMd Gizit, and once the trajectory hits the SMd Gizit, it remains on the sliding manifold thereafter.Proof.
The Lyapunov function is introduced as follows:(36)V¯t=∑i=1NGizit+0.5pi∂˜i2,where ∂˜it=∂i−∂^it.
So, taking the derivative ofV¯t, we have(37)V¯˙=∑i=1NGiTzitGizitG˙izit−∂˜it∂^˙itpi=∑i=1NGiTzitGizitEiAiz˙it+Diz˙t−τi+Biu˙it+∑j=1j≠iNHijz˙jt+w˙izi,t−EiAi−BiTiz˙it+εiσ˙izit−∑i=1N∂˜it∂^˙itpi.
According to equation (37) and property AB≤AB, we have(38)V¯˙=∑i=1NEiAiz˙it+Diz˙t−τi+Biu˙it+∑j=1j≠iNHijz˙jt+w˙izi,t+∑i=1NGiTzitGizitEiBiu˙it−∑i=1N∂˜it∂^˙itpi≤∑i=1NEiBiTiz˙it+EiDiz˙it−τi+δiσ˙izit+∑j=1j≠iNEiHijz˙it+Eiw˙izi,t+∑i=1NGiTzitGizitEiBiu˙it−∑i=1N∂˜it∂^˙itpi.
Using Assumption1, we achieve(39)V¯˙≤∑i=1NEiBiTiz˙it+EiDiz˙it−τi+Ei∂i+∑j=1j≠iNEjHjiz˙it+εiσ˙izit+∑i=1NGiTzitGizitEiBiu˙it−∑i=1N∂˜it∂^˙itpi.
Using the DAISOSMCL (33) yields(40)V¯˙≤∑i=1N−ε¯i−∂^itEi−p¯i∂i24+Ei∂i−∑i=1N∂iEi−∑i=1Np¯i∂^i2t−p¯i∂i∂^it−∂^itEi=∑i=1N−ε¯i−p¯i∂i24−∑i=1Np¯i∂^i2t−p¯i∂i∂^it=∑i=1N−ε¯i−p¯i∂i24−∑i=1Np¯i∂^It−∂i2−p¯i∂i24=∑i=1N−ε¯i−∑i=1Np¯i∂^It−∂i2.
The above inequality implies that the system trajectories of the LSPSwCD (3) reach the SMd Gizit and keep it for later.Remark 3.
The SMC is capable to reach the bounded system stability and is able to maintain within a range to converge zero, and then the power system is notably stable. Equation (10) indicates that the integral term ∫0tEiAi−BiTiziτdτ is only reflected in the proposed DAISOSMCL (33). Therefore, the control law (33) is to improve the performance of steady state error in comparison with the traditional integral SMC.


## 4. Results and Discussion

Based on the interconnected time-delay power network, the four cases are offered to prove the strength of the suggested AISOSMC approach under the required conditions such as the different load disturbances and the parameter variations. A LSPSwCD is analyzed to explain the robustness and effectiveness of the suggested AISOSMC approach. The parameters of LSPSwCD were given in [22] as shown Table 1.Table 1
The parameters of three-area power system.
AreasTPiKPiTTiTGiRiKEiKBiKij120120.00.300.0802.410.00.410.550225112.50.330.0722.79.00.370.650320115.00.350.0702.57.10.400.545Case 1.
The presented LFC based on AISOSMC is used to compare with the traditional LFC approach under the same condition in [22]. The proposed LFC based on AISOSMC has been examined with different step load disturbance effect on the thermal power plant with nominal parameter conditions. We have considered the system parameters of matched uncertainties. In this case, the load disturbances are ΔPd1=0.02,ΔPd2=0.015,andΔPd3=0.01 (p.u.MW) at t1=0 in three areas of the LSPS. The frequency deviations of Δf1,Δf2,andΔf3 for the LSPS with delay-time as τi=0.1s (i=1,2,3) are displayed in Figures 2–4 which are the tie-line power deviation and control input signal for three-area power networks. It is easy to realize that the transient responses achieved in the proposed LFC based on AISOSMC is faster in the settling time, but it has less magnitude of overshoot percentage with the proposed recent controller in [22].Figure 2
Frequency variation of the system.Figure 3
Tie-line variation of the system.Figure 4
Control signal of the system.Remark 4.
Due to Figures2 to 4, the testing simulation results in this part are appointed in Table 2. In particular, the report of results can show in an effective comparison while the time-delay communication is considered for the large-scale power network. Therefore, the system performance of the suggested AISOSMC is well balanced, and frequency variation is zero after 1s.Table 2
Comparison between the proposed AISOSMC method with the previous method in [22].
Kinds of controllerThe AISOSMC approachPrevious load frequency controller in [22]ParametersTs (s)Max.O. S (pu)Ts (s)Max.O. S (pu)Δf11−3.8×10−32−3.8×10−3Δf21−2.1×10−32−2.1×10−3Δf31−1.9×10−32−2.0×10−3Case 2.
As the same matched parameter uncertainty of the three areas is used in [22], the cosine function around the nominal operation point is used to verify the usefulness and robustness of the suggested controller to load disturbance. The step load disturbances in LSPSwCD are chosen as ΔPd1=0.01 p.u.MW at t1=0s and ΔPd2=0.015 p.u.MW at t2=0 and ΔPd3=0.02 p.u.MW at t3=0.
ΔA1=0000000000−2.26cost2cost−2.604cost3cost00000000000 and the matched uncertainty among subsystems are assumed the same as [22] and ΔA1=ΔA2=ΔA3.
Figure5 indicates that the proposed LFC based on AISOSMC to three areas with CDs, τi=0.1; the frequency deviation reaching to zero is about 2 s much smaller than the comparative LFC used in [22] within 10 s. Figure 6 plots the control signal of the system under matched uncertainty. The tie-line power variations of three areas with LFC based on AISOSMC always are kept with maximum value as 4.7 × 10−3 p.u.MW in Figure 7.Figure 5
Frequency deviation of the system under matched uncertainty.Figure 6
Control signal of the system under matched uncertainty.Figure 7
Tie-line deviation of the system under matched uncertainty.Remark 5.
In this configuration, the influents of time-delay signals are considered to compare with the simulation result in [22], the DAISOSMCL based on the offered switching surface can not only make better in the response speed but also upgrade the transient performance to decrease the overshoot percentage. So, the designed control method is powerful and strong enough to regulate and control the matched parameter uncertainties of multiarea interconnected time-delay grids.Case 3.
In this case, the load disturbances areΔPd1=0.01 p.u.MW at t1=0s, ΔPd2=0.02 p.u.MW at t2=0, and ΔPd3=0.03 p.u.MW at t3=0 in area 1, area 2, and area 3, respectively. We discuss about the impact of mismatch parameter uncertainty in the state matrix and values of the time-delay as τi=0.2s (i=1,2,3) for all the subsystems of LSPSwCD which are designed as follows:(41)ΔA1=0Δf1000sint000000costcost00000costcost0000,ΔA2=ΔA3=ΔA1.
The mismatch interconnected between subsystems is designed as follows:(42)ΔH12=00000.178cost00000000000000−0.296sint00000,ΔH12=ΔH23=ΔH31.
The frequency variation in LSPSwCD is displayed in Figure8, the control input signal is presented in Figure 9, and tie-line power variation is displayed in Figure 10. We observe the system performance, the transient response makes achievement by the proposed control scheme which reduces the amplitude of over/undershoot percentage. It is to be clear that the input control signal is to converge quickly and drive the tie-line power variation and frequency variation to zero. Therefore, the suggested controller carried out with better design, both in terms of reducing the over/undershoots and minimizing the settling time in comparison to [21, 22].Figure 8
Frequency deviation of the system under mismatched uncertainty.Figure 9
Tie-line deviation of the system under mismatched uncertainty.Figure 10
Control signal of the system under mismatched uncertainty.Remark 6.
In this case, though considering to the influence of the mismatched uncertainty of LSPSwCD, the performance of proposed controller still is kept powerful and stable. Therefore, the proposed ISOSMC is proved to be suitable for LSPSwCD. The approach given in [21, 22] cannot be applied for the LSPSwCD of this case.Case 4.
In this case, the nominal parameters of LSPSwCD are in Table1 and the same parameters with previous cases. However, in the practical power system, the parameters of LSPSwCD are always varied due to different operation conditions. Therefore, the parameters of LSPSwCD are selected to vary by ±20% synchronously from their normal values. The delay time is 0.2 s and load variation is as shown in Figure 11. The tie-line power and frequency deviation decay quickly as observed in Figures 12 and 13. The control signal has been presented in Figure 14. It is to be clear that the proposed control scheme is able to handle with the time-delay and random load disturbances. It is proved that the proposed DAISOSMCL has the good quality and achievement to reject disturbance with small control signal.Figure 11
Load variations of three-area power network.Figure 12
Frequency variation of the system under load variations and mismatched uncertainty.Figure 13
Tie-line power variation of the system under load variations and mismatched uncertainty.Figure 14
Control signal of the system under load variations and mismatched uncertainty.Remark 7.
In this approach, the structure of LSPSwCD is more general than the approach given in [28–33]. Also, the system responses in terms of convergence of frequency variation and power exchange error are improved by using the proposed DAISOSMCL. Therefore, the proposed design controller is suitable for LFC of LSPSwCD.


## 5. Conclusions

In this paper, the LFC based on decentralized adaptive integral second-order sliding mode control (DAISOSMC) has been developed for the large-scale power system with communication delays (LSPSwCD), load variations, and parameter uncertainties. It is shown that the proposed DAISOSMCL ensures the finite time reachability of the system states and moreover the dynamics of LSPSwCD in the sliding mode is asymptotically stable under certain conditions. To the best of our knowledge, this is the first time the DAISOSMC approach is designed for LFC of the LSPSwCD. The report of simulation results indicates that the proposed DAISOSMC approach can adequately make less the deviation of the tie-line power and frequency variation of LSPSwCD. The suggested control scheme is therefore proved to be more effective for the LSPSwCD implementation.



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