# Advanced Sliding Mode Observer Design for Load Frequency Control of Multiarea Multisource Power Systems

Authors: Van Van Huynh; Phong Thanh Tran; Tien Minh Nguyen; Van-Duc Phan; Viet-Thanh Pham
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

Recently, to balance the increased electricity demands and total generated power, the multiarea power system (MAPS) has been introduced with multipower sources such as gas, nuclear, hydro, and thermal, which will impact the load frequency control (LFC). Therefore, the LFC of the two-area gas-hydro-thermal power system (TAGHTPS) is introduced by applying the single-phase sliding mode control-based state observer (SPSMCBSO). In this scheme, the TAGHTPS is the first model that considers the uncertainties of the parameters in the state and the interconnected matrix. Second, the state observer is employed to estimate the state variables for the feedback control. Third, the SPSMCBSO is developed to modify the basic sliding mode control to improve the performance of TAGHTPS in terms of overshoot and settling time. In addition, the SPSMCBSO is established to rely fully on the state observer so that the difficulty in the state variable measurement is solved. Fourth, the TAGHTPS stability analysis is performed using a new linear matrix inequality (LMI) scheme—Lyapunov stability theory. Lastly, the simulation results are shown and compared to recently established classical control methods to validate the SPSMCBSO choice of application for the LFC of the multiarea multisource power system (MAMSPS).


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## Body

## 1. Introduction

The main aim of power system (PS) control is always to balance the total net generated power and the electrical load. This can be viewed when the frequency is kept at the permissible level. However, if the industry’s load-dependent frequency increases, it will impact the PS frequency and cause changes between total net power and load demand. Matching total generation power with load is accomplished with load frequency control (LFC) [1]. Over the years, researchers have applied classical control methods to the LFC of PS. To achieve the classical scheme, the PS is modeled and represented in the transfer function. In review, the model allows only for the maximum of two inputs, which are frequency error and load disturbance, to be monitored. Furthermore, classical control schemes such as proportional-integral (PI) and proportional-integral-differential (PID) methods use control gains to adjust the parameters associated with the PS LFC after measuring the frequency error and load disturbance. The PI and PID schemes have been widely used for the industrial LFC of perturbed PS. As the electricity demand increases day by day, the PS begins to grow in size, shape, and complexity, which leads to the LFC of PS becoming more complicated. In reality, large PSs such as multiarea power system (MAPS), which includes many generating sets in each area, are characterized with long frequency transient time delay, area control error (ACE), parameter uncertainties, subsystem parameter deviation, random load disturbance, nonlinearity problem, tie-line power flow control problem, etc. These characteristics have an impact on the LFC of the MAPS. Therefore, the MAPS modeling requires these characteristics to be considered. However, many existing PS models for the LFC are not suitable to handle the above characteristics because it only allows for two inputs to be monitored. Recently, control engineers have solved the above problem by representing the PS model in state-space form, which allows for multi-inputs using the modern control. Meanwhile, some methods such as intelligent control (i.e., fuzzy logic), optimal control (i.e., particle swarm optimization (PSO)), adaptive technique, and observer scheme combined with classical PI and PID have been developed to study the LFC of MAPS. In [2], the fuzzy logic technique was applied to select the PI algorithm for the LFC of two PS areas following the frequency change. Based on load demand, the proposed control scheme using the PI parameters was updated online using fuzzy logic rules. Indirect adaptive fuzzy logic control combined with classical PI has been applied to track unknown parameters for the MAPS LFC, and the results have been shown to be superior compared with the classical control method [3]. In the case where some parameters of the MAPS are difficult to access, the observer scheme was employed. The Luenberger observer was the first to be used in which the PS is reconstructed to estimate the system state variables for the LFC of the original MAPS [4]. Observer PI was developed to study the LFC of the single area PS and was compared with the basic Luenberger observer to validate its superiority [4]. In practice, the LFC is required to be highly robust against large disturbances. Hence, variable structure control (VSC) was developed for the LFC of MAPS. The sliding mode control (SMC) scheme is the most popular of the VSC. Their designs follow the selection of the sliding surface and the construction of the switching law and control law that implies that variables must be brought to the surface and remain therein in the finite reaching time [5–9]. The SMC is very important because of its robustness and resistance to large disturbances. The SMC has already been studied for the LFC of MAPS [10–20]. Over time, several methods have been combined with the MAPS SMC to handle the LFC problem of MAPS. A novel adaptive technique based on SMC was utilized for MAPS LFC under step load disturbance [21]. On the other hand, the SMC via the observer has been applied for the LFC of MAPS where some state variables of the system were difficult to access, and the SMC was designed to fully depend on the observer [21]. The SMC-based integral output feedback control was developed for the MAPS LFC against certain disturbances [22]. The single-phase SMC was designed to modify the basic SMC so that variable trajectories get to the surface without reaching time and remain there for all time, making it highly robust for the LFC of MAPS. Recently, single-phase SMC via observer was newly established for the LFC of MAPS under the influence of a step load and a random load disturbance [23]. This method is further used for the LFC of New England 39-bus system under random load disturbance [23]. In [19], the controller parameter is determined using grey wolf optimization and particle swarm optimization approaches to get an ideal outcome in a sliding mode controller for frequency management in an interconnected power system. Because of the discontinuous control component in [20], the traditional integral SMC suffers the flaw of chattering. However, the above conventional SMC methods were applied for the LFC of MAPS consisting of thermal plants or hydro alone in each area without considering the mismatched disturbance and load variation. In reality, MAPS consists of the combination of many generators in each area, such as nuclear, hydro, gas, and thermal, which can be referred to as multiarea multisource power system (MAMSPS). In studies, only a few works have been done to study the LFC of the MSMAPS. A concept and implementation of a structured generation-based PID control using the bacterial foraging algorithm (BFA) had been utilized for the LFC of the two-area gas-hydro-thermal power system (TAGHTPS) after the step load change [24]. For automatic generation control, a novel teaching learning-based optimization (TLBO) algorithm with 2-degree-of-freedom of proportional-integral-derivative (2-DOF PID) controller has also been developed for LFC to improve the dynamic of TAGHTPS under load disturbance [25]. In addition, proportional-integral-derivative (PID) structured regulators of the optimized generation control (OGC) strategy for interconnected two control zones of diverse-source power systems are designed using a new artificial intelligence (AI) technique known as the Jaya algorithm [26]. These works were done only to study the LFC of the TAGHTPS under load disturbances, and the TAGHTPS is modeled without considering the impact of parameter and interconnection uncertainties in the system state matrix.Aside from that, the SMC has been utilized in conjunction with a state estimator to observe the MAPS disturbance to enhance the system’s performance by eliminating chattering [27–31]. In [30], a reduced-order disturbance observer based on SMC is used in a hybrid power system to decrease frequency deviation. Based on system conditions and expected disturbance, an adaptive super-twisting SMC is constructed. In [31], the nonlinear disturbance observer evaluates the mismatch between electrical power and mechanical power, which is subsequently used in the controller design to adjust for the disturbance. The disturbance observer is utilized in conjunction with the suggested fractional order three-degree-of-freedom tilt integral derivative controller in [32] to effectively predict the wind velocity’s uncertain profile and enrich the control law.These controllers were created utilizing a reduced observer controller or a nonlinear disturbance observer to notice load changes and maintain nominal frequency if all system state variables are measured. However, if some MAPS state variables are not measurable or impossible to measure, this cannot be guaranteed for the actual implementation of these above controllers. As a result, the design of an LFC based on a novel SMC where the state observer is entirely integrated into the sliding surface and an SPSMCBSO is employed to overcome the concerns above is motivated. Furthermore, the selection of switching strategies and sliding surfaces is critical. The switching strategy is used to shift the system states and keep them converged at a certain sliding surface. As a result, a novel single-phase sliding surface is built. The single-phase switching surface ensures robustness at the reaching stage without reaching time.In summary, we model the TAGHTS taking into account the impact of parameter and interconnection uncertainties on the state matrix and proposed the SPSMCBSO to study the LFC of the TAGHTPS under load disturbance, deviation of subsystem parameters, and impact of parameter uncertainties, which is simple and less stressful to implement.Meanwhile, this is the first time that the SPSMCBSO is used for the LFC of TAGHTPS, which is validated compared with the recently established above classical methods. In this work, the major contributions are stated as follows:(i)
The SPSMCBSO was designed to rely entirely on the state observer, making it particularly effective for the MAMSPS LFC, where some variables are difficult to obtain.(ii)
The novel controller is established to modify the basic SMC such that the order of the PS making it highly robust against disturbance is different than that of the basic SMC, which depends on the reaching time.(iii)
The Lyapunov stability theory-based novel linear matrix inequality (LMI) approach is used to theoretically show whole-system stabilization.(iv)
In comparison with recent LFC approaches [24–26, 30–32], the novel SMC through single-phase sliding surface does not require reaching time, ensuring greater system performance in terms of settling time and overshoot under the matched or mismatched disturbance and load variation.


## 2. Mathematical Model of the Interconnected Multiarea Multisource Power Network

In this section, the block chart of PS is presented. Dynamic models of power systems are generally nonlinear. The MAPS consists of many generating sets such as nuclear, gas, hydro, and gas plant in each area. This can be viewed as the MAMS. However, the nuclear plant is known as a base load system and does not apply to the LFC of MAPS [24–26]. The gas plant can adapt to the demand for random loads, enabling it to fit the LFC system. Therefore, in this section, we consider TAGHTPS in each area, as shown in Figure 1. Area control error (ACE) is computed as the power error from the linear combination of the power error of the link network and the system frequency errors.Figure 1
LFC block diagram of a TAMSPS.Taking into account the impact of the interconnection matrix and load disturbance, the PS model is constructed in the differential equation as follows:(1)Δf˙i=−ΔfiTPSi+ΔPptiKPSiαi1TPSi+ΔPGhiKPSiαi2TPSi+ΔPGgiKPSiαi3TPSi−KPSiaijTPSiΔPtieij−KPSiTPSiΔPDi,(2)ΔP˙pti=ΔPGtiTTi−ΔPptiTTi,(3)ΔP˙Gti=−ΔPGtiTRi+ΔXEtiTRi−ΔXEtiKRiTSGi−ΔfiKRiTSGiRi1+ΔACEiKRiTSGi+Ui1KRiTSGi,(4)ΔX˙Eti=−ΔXEtiTSGi−ΔfiTSGiRi1+ΔACEiTSGi+Ui1TSGi,(5)ΔP˙Ghi=2ΔPRhiTWi−2ΔPGhiTWi−2ΔXEhiTRHi+2ΔPRhiTRHi+2ΔXEhiTRSiTRHiTGHi+2ΔfiTRSiRi2TRHiTGHi−2ΔACEiTRSiTRHiTGHi−2Ui2TRSiTRHiTGHi,(6)ΔP˙Rhi=ΔXEhiTRHi−ΔPRhiTRHi−ΔXEhiTRSiTRHiTGHi−ΔfiTRSiRi2TRHiTGHi+ΔACEiTRSiTRHiTGHi+Ui2TRSiTRHiTGHi,(7)ΔX˙Ehi=−ΔXEhiTGHi−ΔfiTGHiRi2+ΔACEiTGHi+Ui2TGHi,(8)ΔP˙Ggi=−ΔPGgiTCDi+ΔPRgiTCDi,,(9)ΔP˙Rgi=−ΔPRgiTFi+ΔXVgiTFi+ΔXVgiTCRiYGiTFi−ΔXEgiTCRiYGiTFi+ΔXEgicgiXGiTCRibgiYGiTFi+ΔfiXGiTCRiRi3bgiYGiTFi−ΔACEiXGiTCRibgiYGiTFi−Ui3XGiTCRibgiYGiTFi,(10)ΔP˙Vgi=−ΔPVgiYGi+ΔXEgiYGi−ΔXEgicgiXGibgiYGi−ΔfiXGiRi3bgiYGi+ΔACEiXGibgiYGi+Ui3XGibgiYGi,(11)ΔX˙Egi=−ΔXEgicgibgi−ΔfibgiRi3+ΔACEibgi+Ui3bgi,(12)ΔAC˙Ei=BiΔfi+aijΔPtieij,(13)ΔP˙tieij=∑j=1j≠iL2πTijΔfi−Δfj,where ΔPpti is the change in thermal turbine speed changer position (p.u.MW), ΔPGhi is change in hydro-turbine speed changer position (p.u.MW), ΔPGgi is change in gas turbine speed changer position (p.u.MW), ΔPDi is total incremental charge in the local load of the control area (p.u.MW), Δfi andΔfj are incremental change in frequency of each control area (Hz), and ΔPtieij is incremental change in actual tie-line power flow from control areas 1 to 2 (p.u.MW), and ΔACEi is the area control error. By using the dynamic equations from (1)–(13) the ith area of the PS state-space model is given in (14) as follows:(14)x˙it=Aixit+Biuit+∑j=1j≠iLHij+ΔHijxjt+FiΔPdit,and the above equation is the state-space form of the TAGHTPS. Here, xit=ΔfiΔPptiΔPGtiΔXEtiΔPGhiΔPRhiΔXEhiΔPGgiΔPRgiΔPVgiΔXEgiΔACEiΔPtieijT is the state vector, xjt is the interconnected system state vector of xit, uit is the control vector, and ΔPdit is the disturbance. Ai, Bi, Hij, andFi are the system matrices given as follows:(15)Ai=Ai1Ai2Ai3Ai4,Ai1=−1TPSiKPSiαi1TPSi00KPSiαi2TPSi000−1TTi1TTi0000−KRiTSGiRi10−1TRi1TRi−KRiTSGi000−1TSGiRi100−1TSGi0002TRSiRi2TRHiTGHi000−2TWi2TWi+2TRHi2TRSiTRHiTGHi−2TRHi−TRSiRi2TRHiTGHi0000−1TRHi1TRHi−TRSiTRHiTGHi−1TGHiRi2000001TGHi,Ai2=KPSiαi3TPSi0000−KPSiaijTPSi000000000000000000000000000000000000,Ai3=0000000XGiTCRiRi3bgiYGiTFi000000−XGiRi3bgiYGi000000−1Ri3bgi000000Bi000000∑j=1j≠iL−2πTij000000,Ai4=−1TCDi1TCDi00000−1TFi1TFi+TCRiYGiTFicgiXGiTCRibgiYGiTFi−TCRiYGiTFi0000−1YGi1YGi−cgiXGibgiYGi00000−cgibGi0000000aij000000,Bi=00KRiTSGi1TSGi0000000000000−2TRSiTRHiTGHiTRSiTRHiTGHi1TGHi00000000000000−XGiTCRibgiYGiTFiXGibgiYGi1bgi00T,Fi=−KPSiTPSi000000000000T,Hij=000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000−2πTij000000000000.In practical interconnected MAMSPS, changes in operating points constantly influence the fluctuating sources of load. This factor can be considered as parameter uncertainties. Introducing this factor, system (14) can be rewritten as follows:(16)x˙it=Ai+Σixi,txit+Biuit+ψixi,t+∑j=1j≠iLHij+Ξijxj,txjt+FiΔPdit,where Σixi,t and Ξijxj,t are time-varying parameter uncertainties andBiψixi,t is the input disturbance. In other words, the aggregate uncertainty is therefore given as follows:(17)Φixi,t=Σixi,txit+Biψixi,t+∑j=1j≠iLΞijxjt+FiΔPdit.Therefore, the new dynamic model can be expressed as follows:(18)x˙it=Aixit+Biuit+∑j=1j≠iLHijxjt+Φixi,tyi=Cixi,where Φixi,t is the aggregated disturbance that represents the uncertainties of the matched and mismatched parameters. If we consider the state-space model (17), the designing of the controller uit is very important and it is based on the choice of the control engineers. Several techniques have been designed for uit as seen in the literature. Meanwhile, to design the novel uit, we first make the following assumptions and recall the lemmas as follows.Assumption 1.
The system matrix (Ai,Bi) is controllable and (Ai,Ci) is observable.Assumption 2.
It is assumed that the load disturbanceΦixi,t is bounded, such that Φixi,t≤γi, where γi is the known scalar and . is the matrix norm.Lemma 1.
[22, 33]: if X and Y are real matrix of suitable dimension, then, for any scalar μ>0, the following matrix inequality holds:(19)XTY+YTX≤μXTX+μ−1YTY.Lemma 2.
[22, 33]: for a given inequality:(20)QxSxSxTRx<0,where Qx=QxT and Rx=RxT such that Sx depends on the affinity on x; therefore, Rx<0 and Qx−SxRx−1SxT<0.


## 3. Design of the Power System State Estimator

Few works have applied observer techniques to solve the LFC of PS where some state variables are uneasy to solve. The state observer was invented to estimate the state variables of the system and reject disturbance by reconstructing the original PS model. The advantage is that the reconstructed PS model is nearly the same as the original PS model. The LFC of MAMSPS where some state variables are difficult to access can benefit from the observer scheme. For this reason, we apply the observer method to reconstruct the model of the original TAGHTPS (18) as follows:(21)x^˙it=Aix^it+Biuit+∑j=1j≠iLHijx^jt+Γiyi−y^iy^i=Cix^i,where Γi is the observer gain, x^it is the estimation of xit, yi is the output vector, and y^i is the state observer output, respectively. It can be calculated using the pole placement method. Next, we examine the state error dynamic where the state error is given as follows:(22)x˜it=xit−x^it.Taking the derivative of the errorx˜i, we have the following:(23)x˜˙i=Ai−ΓiCix˜i+∑j=1j≠iLHijx˜j+Φixi,t.The state error tends to zero depending on the eigenvalue ofAi−ΓiCi.Remark 1.
In this scheme, it is called a full-order state observer when the state observer observes all state variables of the system, regardless of whether some state variables are valuable for direct measurement. The mathematical model of the observer is basically the same as that of the plant, except that we include an additional term that includes the estimation error to compensate for inaccuracies in matricesAi and Bi and the lack of the initial error.


## 4. Integral Single-Phase Sliding Surface Design

In practice, the LFC scheme is required to be highly robust against certain disturbances to achieve MAMSPS stability. Over the years, the SMC has been applied to attenuate various disturbances for the LFC of MAPS [12, 16–19]. The SMC design follows the selection of a sliding surface (SS) and the construction of a switching law and an equivalent control law [5–15]. The reachability of system state variable trajectories to the SS using the basic SMC depends on a finite reach time. However, the PS with a long transient time might cause a drawback to the basic SMC. With this information, we propose that the SPSMCBSO and the sliding surface without reaching phase (SSWRP) are given as follows:(24)ηix^it=Μix^it−∫0tΜiAi−BiΛix^itdτ−Μix^i0e−δit,where the matrix Μi is selected to promise that the matrix ΜiBi is nonsingular. The design matrix Λi∈Rmi×ni is chosen satisfying the nonlinearity condition.(25)ReλmaxAi−BiΛi<0.If we take derivative ofηix^it with respect to time, we have the following:(26)η˙ix^it=ΜiAix^it+ΜiBiuit+∑j=1j≠iLΜiHijx^jt+ΜiΓiyi−y^i−ΜiAi−BiΛix^it+δiΜix^i0e−δit.Asη˙it=ηit=0, then we can derive the equivalent control as follows:(27)uieqt=−ΜiBi−1ΜiAix^it+∑j=1j≠iLΜiHijx^jt+ΜiΓiyi−y^i−ΜiAi−BiΛix^it+δiΜix^i0e−δit=−ΜiBi−1ΜiBiΛix^it+ΜiΓiyi−y^i+δiΜix^i0e−δit+∑j=1j≠iLΜiHijx^jt.By closing the loop system, we substitute (27) into (18):(28)x˙it=Ai−BiΛixit+BiΛi−BiΜiBi−1ΜiΓiCix˜it+∑j=1j≠iLHij−BiΜiBi−1ΜiHijxjt+∑j=1j≠iLBiΜiBi−1ΜiHijx˜jt+Φixi,t−δiBiΜiBi−1Μix^i0e−δit.To observe the MAMSPS (18), we combine (23) and (28) in the following equation:(29)x˙ix˜˙i=Ai−BiΛiΘi0Ai−ΓiCixix˜i+∑j=1j≠iLHij−ϒiHijϒiHij0Hijxjx˜j+Φixi,tΦixi,t+Νie−δit0,where Θi=BiΛi−BiΜiBi−1ΜiΓiCi, Νi=−δiBiΜiBi−1Μix^i0, and ϒi=BiΜiBi−1Μi.Equation (29) is the dynamic system of the MAMSPS. Hence, we analyze the stability of (29) via the new LMI given in (30), which is accompanied by the theorem as stated.Theorem 1.
Equation (29) is asymptotic stable if the symmetric positive definite matrices Πi and Π¯i, where i=1,2,…,L, and the positive scalars λi, ρi, and γ^i are assumed so that the below new LMI holds:(30)ΧiΠiΘiΠiΠiNi0ΘiTΠiΧ¯i00Π¯iΠi0−λi−100NiTΠi00−γ^i−100Π¯i00−ρi−1<0,where Χi=ΠiAi−BiΛi+Ai−BiΛiTΠi+∑j=1j≠iLλ¯jHji−ϒjHjiTHji−ϒjHji and Χ¯i=Π¯iAi−ΓiCi+Ai−ΓiCiTΠ¯i+∑j=1j≠iLλ˜j−1HjiTHji+∑j=1j≠iLλ^jϒjHjiTϒjHji.Proof .
of Theorem1: Lyapunov’s function [6, 34] is selected as follows:(31)V=∑i=1Lxix˜iTΠi00Π¯ixix˜i,where Πi>0 and Π¯i>0 satisfy (30) for i=1,2,…,L. Then, taking the derivative of time, we have the following:(32)V˙=∑i=1Lx˙ix˜˙iTΠi00Π¯ixix˜i+∑i=1Lxix˜iTΠi00Π¯ix˙ix˜˙i=∑i=1Lxix˜iTΠiAi−BiΛi+Ai−BiΛiTΠiΠiΘiΘiTΠiΠ¯iAi−ΓiCi+Ai−ΓiCiTΠ¯ixix˜i+∑i=1LxiTΠiΝie−δit+e−δitTNiTΠixi+∑i=1L∑j=1j≠iLxjTHij−ϒiHijTΠixi+xiTΠiHij−ϒiHijxj+∑i=1L∑j=1j≠iLx˜jTϒiHijTΠixi+xiTΠiϒiHijx˜j+∑i=1L∑j=1j≠iLx˜iTΠ¯iHijx˜j+x˜jTHijTΠ¯ix˜i+∑i=1Lx˜iTΠ¯iΦi+xiTΠiΦi+ΦiTΠixi+ΦiTΠ¯ix˜i.
Introducing Lemma1 into equation (32), we get the following:(33)V˙≤∑i=1Lxix˜iTΠiAi−BiΛi+Ai−BiΛiTΠiΠiΘiΘiTΠiΠ¯iAi−ΓiCi+Ai−ΓiCiTΠ¯ixix˜i+∑i=1L∑j=1j≠iLλ¯ixjTHij−ϒiHijTHij−ϒiHijxj+∑i=1L∑j=1j≠iLλ¯i−1xiTΠiΠixi+λ^ix˜jTϒiHijTϒiHijx˜j+∑i=1L∑j=1j≠iLλ^i−1xiTΠiΠixi+λ˜ix˜iTΠ¯iΠ¯ix˜i+λ˜i−1x˜jTHijTHijx˜j+∑i=1Lγ¯ix˜iTΠ¯iΠ¯ix˜i+γ¯i−1ΦiTΦi+γ˜ixiTΠiΠixi+γ˜i−1ΦiTΦi+∑i=1Lγ^i−1xiTΠiΝiNiTΠixi+γ^ie−δitTe−δit.
Since(34)∑i=1L∑j=1j≠iLλ¯ixjTHij−ϒiHijTHij−ϒiHijxj=∑i=1L∑j=1j≠iLλ¯jxiTHji−ϒjHjiTHji−ϒjHjixi∑i=1L∑j=1j≠iLλ^ix˜jTϒiHijTϒiHijx˜j=∑i=1L∑j=1j≠iLλ^jx˜iTϒiHijTϒjHjix˜i,(35)∑i=1L∑j=1j≠iLλ˜i−1x˜jTHijTHijx˜j=∑i=1L∑j=1j≠iLλ˜j−1x˜iTHjiTHjix˜i,
we(36)V˙≤∑i=1Lxix˜iTΠiAi−BiΛi+Ai−BiΛiTΠiΠiΘiΘiTΠiΠ¯iAi−ΓiCi+Ai−ΓiCiTΠ¯ixix˜i+∑i=1L∑j=1j≠iLλ¯jxiTHji−ϒjHjiTHji−ϒjHjixi+∑i=1L∑j=1j≠iLλ¯i−1xiTΠiΠixi+λ^jx˜iTϒjHjiTϒjHjix˜i+∑i=1L∑j=1j≠iLλ^i−1xiTΠiΠixi+λ˜ix˜iTΠ¯iiΠ¯iix˜i+λ˜j−1x˜iTHjiTHjix˜i+∑i=1Lγ¯ix˜iTΠ¯iΠ¯ix˜i+γ˜ixiTΠiΠixi+γ¯i−1ΦiTΦi+γ˜i−1ΦiTΦi+∑i=1Lγ^i−1xiTΠiΝiNiTΠixi+γ^ie−δitTe−δit≤∑i=1Lxix˜iTΩiΠiΘiΘiTΠiΩ¯ixix˜i+∑i=1Lγ¯i−1+γ˜i−1γi2+∑i=1Lγ^ie−δitTe−δit.where λi=N−1λ¯i−1+λ^i−1+γ˜i and ρi=λ˜iN−1+γ¯i, Ωi=ΠiAi−BiΛi+Ai−BiΛiTΠi+γ^i−1ΠiΝiNiTΠi
+λiΠiΠi+∑j=1j≠iLλ¯jHji−ϒjHjiTHji−ϒjHji and Ω¯i=Π¯iAi−ΓiCi+Ai−ΓiCiTΠ¯i+ρiΠ¯iΠ¯i+∑j=1j≠iLλ˜j−1HjiTHji+∑j=1j≠iLλ^jϒjHjiTϒjHji.
Furthermore, by the Schur complement of [33], the LMI is similar to the following:(37)Ψi=ΩiΠiΘiΘiTΠiΩ¯i<0.
Combining (36) and (37), we get the following:(38)V˙≤∑i=1L−ρminΨixix˜i2+ζi+γ^ie−2δit,where ζi=∑i=1Lγ¯i−1+γ˜i−1γi2 is the constant value and the eigenvalue ρminΨi>0. The term γ^ie−2δit will be approaching zero when the time is approaching infinite. Hence, V˙<0 is derived with xix˜i>ζi/ρminΨi, which shows that the system is asymptotic stable.


## 5. Total Output Feedback Sliding Mode Controller Design

In this segment, we design the decentralized single-phase SMC scheme (DSPSMCS) for the LFC of the MAMSPS (18) as follows:(39)uit=−ΜiBi−1ΜiBiΛix^it+δiΜix^i0e−δit+ΜiΓiyi−y^i+∑j=1j≠iLΜjHjix^it+θiηix^tηix^it,i=1,2,...,L,where θi is the positive scalar and uit is the decentralized single-phase SMC scheme. In this subsection, the system state variable reachability proof is also derived with the Lyapunov function accompanying the theorem postulated below.Theorem 2.
The SSWRP and the controller are given by equations (24) and (39), respectively. Then, the variable state trajectories of system (19) reach the single-phase sliding surface and lie on it for all time.Remark 2.
The concept of a single-phase SMC is centered on the robustness of motion throughout the state space. The dimension of the state space is equal to the order of the motion equation in sliding mode. As a result, the resilience of complex interconnected power systems may be guaranteed throughout the system’s full response, beginning with the initial time instance.Proof .
of Theorem 2: The Lyapunov function [6, 34] is given as follows:(40)V1=∑i=1Lηix^it.
Using the time derivative ofV1, we obtain the following:(41)V˙1=∑i=1LηiTx^itηix^itη˙ix^it.
Substituting equation (26) into equation (41), we have the following:(42)V˙1=∑i=1LηiTx^itηix^itΜiAix^it−ΜiAi−BiΛix^it+δiΜix^i0e−δit+ΜiΓiyi−y^i+∑i=1LηiTx^itηix^itΜiBiuit+∑i=1L∑j=1j≠iLηiTx^itηix^itΜiHijx^jt.
According to equation (42), property AB≤AB and ∑i=1L∑j=1j≠iLΜiHijx^jt=∑i=1L∑j=1LΜjHjix^it, and it generates:(43)V˙1=∑i=1LΜiBiΛix^it+δiΜix^i0e−δit+∑i=1LΜiΓiyi−y^i+∑i=1L∑j=1j≠iLΜjHjix^it+∑i=1LηiTx^itηix^itΜiBiuit.
By substitutinguit into equation (43), we achieve the following:(44)V˙1≤−∑i=1Lθi<0.
The derivative of Lyapunov’s function (40) is less than zero. Therefore, the reachability proof is achieved.
Following that, Figure2 depicts the flowchart of the proposed observer single-phase sliding mode control technique.Figure 2
Flowchart of the proposed observer single-phase sliding mode-based load frequency control.Remark 3.
Paper [6, 34] shows the LFC’s stability in a power system while employing the LMI approach. In the LMI equations, however, the aforementioned method requires the discovery of four positive matrices. As a result, the suggested method only needs to locate two positive matrices in LMI equations, making finding a feasible solution easier.


## 6. Simulation Results and Discussions

In this session, to test the efficiency and robustness of the proposed control strategy, the various cases in four simulations are implemented to prove the performance.In this session, to test the efficiency and robustness of the proposed control strategy, various cases in three simulations are implemented to prove the performance of the suggested single-phase sliding mode control-based state observer (SPSMCBSO) for the LFC of the two areas gas-hydro-thermal power system (TAGHTPS) under step load disturbance, random load disturbance, and parameter uncertainties. The simulation results of the investigation are compared with recent results of the control method in [24–26, 30–32] as follows.


### 6.1. Simulation 1

The classical control methods are commonly used for the LFC of MAMSPS under step load disturbance. The recent application of bacterial foraging algorithm (BFA) for the design and implementation of generation-based PID structured automatic generation control algorithm was developed to investigate the LFC of TAGHTPS [24]. In this case, we test the proposed controller of PS under 1% to 2% increase in step load change with the nominal parameters kept in the same manner [24]. The frequency deviation in both areas is displayed in Figure 3 to Figure 4, and the tie-line power deviation (TLPD) is shown in Figure 5. The TAGHTPS performance with settling time and overshoot is compared rightly with that seen in [24]. It can be seen that both controllers produced smaller frequency overshoot in other words keeping the operating frequency with the permissible level ±0.001Hz, but 8s settling time with the novel approach is comparatively lower than the settling time in [24, 30–32].Figure 3
Frequency deviation (Hz) with 1% to 2% step load in area 1.Figure 4
Frequency deviation (Hz) with 1% to 2% step load in area 2.Figure 5
Tie-line power variation (p.u.MW).Remark 4.
With this comparison [24, 30–32], the novel proposed SPSMCBSO controller for the LFC of MAMSPS is more robust and responsive to load disturbances than the earlier technique. The load disturbance is clearly visible, and the system is quickly restored to steady state with smaller overshoots.


### 6.2. Simulation 2

Case 1.
Again, the TAGHTPS with a recently developed teaching learning-based optimization (TLBO) algorithm with 2-degree-of-freedom of proportional-integral-derivative (2-DOF PID) controller was simulated with 0.01 p.u.MW load disturbance in each area and nominal parameters are given as demonstrated in [25]. To analyze the performance of PS, we propose SPSMCBSO and then simulate the TAGHTPS response with the proposed approach in the same manner as in [25]. The frequency variation in both areas is given in Figure 6. The TLPD is presented in Figure 7 accordingly. Both frequency overshoots in every area seem to be better, but the 7 s settling time with the novel approach is very low compared with 13 s settling time seen in [25]. Table 1 compares the two controllers in detail. This means that the proposed system is a superior option for the MAMSPS LFC since it is easier to execute and less stressful.Figure 6
Frequency deviation (Hz) with 1% step load in area 1 and area 2.Figure 7
Tie-line power variation (p.u.MW).Table 1
Setting timeTs(s) and maximum overshoot MOS(Hz) comparison.
ControllerThe proposed methodThe recent method in [25]ParameterTssMOS (Hz)TssMOS (Hz)Δf170.004130.019Δf270.0012130.017Case 2.
As industrial activities continue to increase, the electricity demands from neighboring industries, hospitals, homes, and other kinds of load also increase. On the other hand, the MAMSPSs are required to meet the demands with frequency kept at the permissible level. Therefore, we assume the electricity demands as a random load disturbance applied to each area of the TAGHTPS as shown in Figure8. The subsystem parameters are assumed to be nominal as in case 1 of this simulation.
The frequency oscillation in both areas is shown in Figure9 to Figure 10, while TLPD is displayed in Figure 11. The frequency response in Figures 9 and 10 has improved under random load changes. Again, the frequency is kept at tolerable level during operation, therefore validating the proposed SPSMCBSO for the LFC of MAMSPS.Figure 8
Random load variation.Figure 9
Frequency variation (Hz) in area 1.Figure 10
Frequency variation (Hz) in area 2.Figure 11
Change in tie-line power (p.u.MW).Remark 5.
TAGHTPS was simulated with only step load disturbance from demands of customers as seen in [25]. However, in reality, demands from load change daily, so this demand is assumed to be a random load change, which is subjected to the PS. Under the random load condition of the TAGHTPS, the proposed SPSMCBSO proves to be very useful for the stability of the power system, making it better for the LFC of MAMSPS.


### 6.3. Simulation 3

Case 3.
The TAGHTPS response was simulated again using the Jaya method to develop PID structured regulators for the optimized generation control (OGC) strategy, using a step load disturbance of 1% and the subsystem parameter from [26]. The above applied the metaheuristic algorithm to optimize the control parameter search for PS. However, implementing this strategy for the LFC of MAMSPS might be challenging as well. In review, the metaheuristic approach is quite complex and time-consuming too. In other words, we again proposed the SPSMCBSO, which is simpler to implement. In this case, we simulate the TAGHTPS response with the proposed approach under a similar condition to the one observed in [26]. The frequency variation in area 1 is shown in Figure 12, while in area 2, the frequency deviation is shown in Figure 13. The TLPD is given in Figure 14, respectively. Both controllers are seen with better overshoot, but 5 s settling time with the proposed approach again is comparatively lower than the 8 s settling time with the controller in [26]. Table 2 shows a detailed comparison of both controllers. This further implies that the proposed scheme is a better choice for the MAMSPS LFC, which is very simple and less stressful to implement.Figure 12
Frequency variation (Hz) in area 1.Figure 13
Frequency variation (Hz) in area 2.Figure 14
Change in tie-line power (p.u.MW).Table 2
Setting timeTs(s) and maximum overshoot MOS(Hz) comparison.
ControllerThe proposed methodThe recent method in [26]ParameterTssMOS (Hz)TssMOS (Hz)Δf150.002580.007Δf26.250.001580.003Case 4.
As industrial activity grows, so does the need for power from surrounding enterprises, hospitals, households, and other types of loads, MAMSPS, on the other hand, must meet the needs with a frequency that is within acceptable limits. As a result, as illustrated in Figure15, we treat electrical needs as a random load disturbance applied to each section of the TAGHTPS.
Figures16 and 17 depict the frequency oscillation in both locations, whereas Figure 18 depicts the TLPD. Under random load variations, the frequency response in Figures 16 and 17 has improved. The frequency is maintained at a reasonable level during operation, validating the proposed SPSMCBSO for the MAMSPS LFC.Figure 15
Random load variation.Figure 16
Frequency variation (Hz) in area 1.Figure 17
Frequency variation (Hz) in area 2.Figure 18
Change in tie-line power (p.u.MW).Case 5.
To be more realistic in achieving a better LFC of MAMSPS, it is required for the controller to be robust against certain disturbances such as random load change, parameter uncertainties, and subsystem parameter deviation. Therefore, in this case, we simulate the TAGHTPS response under random load disturbance as shown in Figure19 and ±20 deviation in subsystem parameters. We also assume mismatched uncertainties in the system state matrix as a result of the change in valve positions of the TAGHTPS represented in the cosine function given as follows.
We haveΔA1=ΔA2=ΔA11ΔA12, where:(45)ΔA11=000000000000004cost0000000.46cost00cost000cost0000cost0cost00000000000000000000cost000000cost000000000001.6cost000000000000000ΔA12=000000000000000000000000000000000000000006cost000000000cost00000.2costcost00000000000000000000
The frequency fluctuation in both areas is given in Figure20, while TLPD is in Figure 21. The proposed SPSMCBSO maintained higher robustness by rejecting the various disturbances and keeping the frequency at the operating permissible point with the TLPD properly managed as well.Figure 19
Variation in random load.Figure 20
Frequency deviations (Hz) in area 1 and area 2.Figure 21
Changes in tie-line power (p.u.MW).Remark 6.
The TAGHTPS response has been simulated against the step load disturbance and random load changes to compare the optimal controller given in [26]. The results were seen better; however, the LFC schemes are required to be robust against wide range of disturbances. Therefore, the proposed single-phase sliding mode control-based state observer (SPSMCBSO) scheme is simulated under parameter uncertainties, subsystem parameter deviations, random load disturbance, and step load change and it proves to be robust by rejecting these disturbances and maintaining the TAGHTPS stability.


### 6.4. Simulation 4

To investigate the computational efficacy of SPSMCBSO scheme, the study is extended to a complex and realistic system, namely four-area multisource power system. Area 1, area 2, area 3, and area 4 consist of thermal-hydro-gas plant in each control area. The transfer function model of test system is available in Figure1 with i = 4. The comparative transient responses of test simulation 4 after 1% step load disturbance are depicted in Figures 22 and 23. The optimal controller parameters obtained by the proposed approach are presented from the frequency vitiation of four areas in Figure 22. The fluctuations in tie-line power are clearly shown in Figure 23.Figure 22
Frequency variation (Hz) in area 1, area 2, area 3, and area 4.Figure 23
Changes in tie-line power (p.u.MW).In detail, the typical transient specifications in terms of peak undershoot and settling time of system oscillations are noted to manage and keep the frequency at the operating permissible point with the changes in the tie-line properly managed and the proposed control scheme.Remark 7.
A SPSMCBSO controller is designed for load frequency control of four-area interconnected power systems. The proposed single-phase surface and the designed decentralized SMC can reduce the overshoot and improve the response speed and can also limit the deviation of frequency to zero. Therefore, the designed controller is robust and effective to control the matching and mismatched parameter uncertainties of interconnected multiarea systems.Remark 8.
From simulation 1 to simulation 4, a sensitivity analysis is undertaken to evaluate the benefit of the proposed SPSMCBSO scheme-based LFC. The settling times and minimum under/overshoot value under normal and varied conditions for the concerned power system are offered in each case. It is clearly viewed that performance values are changing within the acceptable limits and almost equal to the respective values obtained at the nominal condition. Hence, it may be concluded that the proposed controller gains are insensitive to the matched and mismatched disturbance and the parameter variations and perform satisfactorily under the wide change in the loading condition and system parameters compared with [24–26, 30–32].


## 6.1. Simulation 1

The classical control methods are commonly used for the LFC of MAMSPS under step load disturbance. The recent application of bacterial foraging algorithm (BFA) for the design and implementation of generation-based PID structured automatic generation control algorithm was developed to investigate the LFC of TAGHTPS [24]. In this case, we test the proposed controller of PS under 1% to 2% increase in step load change with the nominal parameters kept in the same manner [24]. The frequency deviation in both areas is displayed in Figure 3 to Figure 4, and the tie-line power deviation (TLPD) is shown in Figure 5. The TAGHTPS performance with settling time and overshoot is compared rightly with that seen in [24]. It can be seen that both controllers produced smaller frequency overshoot in other words keeping the operating frequency with the permissible level ±0.001Hz, but 8s settling time with the novel approach is comparatively lower than the settling time in [24, 30–32].Figure 3
Frequency deviation (Hz) with 1% to 2% step load in area 1.Figure 4
Frequency deviation (Hz) with 1% to 2% step load in area 2.Figure 5
Tie-line power variation (p.u.MW).Remark 4.
With this comparison [24, 30–32], the novel proposed SPSMCBSO controller for the LFC of MAMSPS is more robust and responsive to load disturbances than the earlier technique. The load disturbance is clearly visible, and the system is quickly restored to steady state with smaller overshoots.


## 6.2. Simulation 2

Case 1.
Again, the TAGHTPS with a recently developed teaching learning-based optimization (TLBO) algorithm with 2-degree-of-freedom of proportional-integral-derivative (2-DOF PID) controller was simulated with 0.01 p.u.MW load disturbance in each area and nominal parameters are given as demonstrated in [25]. To analyze the performance of PS, we propose SPSMCBSO and then simulate the TAGHTPS response with the proposed approach in the same manner as in [25]. The frequency variation in both areas is given in Figure 6. The TLPD is presented in Figure 7 accordingly. Both frequency overshoots in every area seem to be better, but the 7 s settling time with the novel approach is very low compared with 13 s settling time seen in [25]. Table 1 compares the two controllers in detail. This means that the proposed system is a superior option for the MAMSPS LFC since it is easier to execute and less stressful.Figure 6
Frequency deviation (Hz) with 1% step load in area 1 and area 2.Figure 7
Tie-line power variation (p.u.MW).Table 1
Setting timeTs(s) and maximum overshoot MOS(Hz) comparison.
ControllerThe proposed methodThe recent method in [25]ParameterTssMOS (Hz)TssMOS (Hz)Δf170.004130.019Δf270.0012130.017Case 2.
As industrial activities continue to increase, the electricity demands from neighboring industries, hospitals, homes, and other kinds of load also increase. On the other hand, the MAMSPSs are required to meet the demands with frequency kept at the permissible level. Therefore, we assume the electricity demands as a random load disturbance applied to each area of the TAGHTPS as shown in Figure8. The subsystem parameters are assumed to be nominal as in case 1 of this simulation.
The frequency oscillation in both areas is shown in Figure9 to Figure 10, while TLPD is displayed in Figure 11. The frequency response in Figures 9 and 10 has improved under random load changes. Again, the frequency is kept at tolerable level during operation, therefore validating the proposed SPSMCBSO for the LFC of MAMSPS.Figure 8
Random load variation.Figure 9
Frequency variation (Hz) in area 1.Figure 10
Frequency variation (Hz) in area 2.Figure 11
Change in tie-line power (p.u.MW).Remark 5.
TAGHTPS was simulated with only step load disturbance from demands of customers as seen in [25]. However, in reality, demands from load change daily, so this demand is assumed to be a random load change, which is subjected to the PS. Under the random load condition of the TAGHTPS, the proposed SPSMCBSO proves to be very useful for the stability of the power system, making it better for the LFC of MAMSPS.


## 6.3. Simulation 3

Case 3.
The TAGHTPS response was simulated again using the Jaya method to develop PID structured regulators for the optimized generation control (OGC) strategy, using a step load disturbance of 1% and the subsystem parameter from [26]. The above applied the metaheuristic algorithm to optimize the control parameter search for PS. However, implementing this strategy for the LFC of MAMSPS might be challenging as well. In review, the metaheuristic approach is quite complex and time-consuming too. In other words, we again proposed the SPSMCBSO, which is simpler to implement. In this case, we simulate the TAGHTPS response with the proposed approach under a similar condition to the one observed in [26]. The frequency variation in area 1 is shown in Figure 12, while in area 2, the frequency deviation is shown in Figure 13. The TLPD is given in Figure 14, respectively. Both controllers are seen with better overshoot, but 5 s settling time with the proposed approach again is comparatively lower than the 8 s settling time with the controller in [26]. Table 2 shows a detailed comparison of both controllers. This further implies that the proposed scheme is a better choice for the MAMSPS LFC, which is very simple and less stressful to implement.Figure 12
Frequency variation (Hz) in area 1.Figure 13
Frequency variation (Hz) in area 2.Figure 14
Change in tie-line power (p.u.MW).Table 2
Setting timeTs(s) and maximum overshoot MOS(Hz) comparison.
ControllerThe proposed methodThe recent method in [26]ParameterTssMOS (Hz)TssMOS (Hz)Δf150.002580.007Δf26.250.001580.003Case 4.
As industrial activity grows, so does the need for power from surrounding enterprises, hospitals, households, and other types of loads, MAMSPS, on the other hand, must meet the needs with a frequency that is within acceptable limits. As a result, as illustrated in Figure15, we treat electrical needs as a random load disturbance applied to each section of the TAGHTPS.
Figures16 and 17 depict the frequency oscillation in both locations, whereas Figure 18 depicts the TLPD. Under random load variations, the frequency response in Figures 16 and 17 has improved. The frequency is maintained at a reasonable level during operation, validating the proposed SPSMCBSO for the MAMSPS LFC.Figure 15
Random load variation.Figure 16
Frequency variation (Hz) in area 1.Figure 17
Frequency variation (Hz) in area 2.Figure 18
Change in tie-line power (p.u.MW).Case 5.
To be more realistic in achieving a better LFC of MAMSPS, it is required for the controller to be robust against certain disturbances such as random load change, parameter uncertainties, and subsystem parameter deviation. Therefore, in this case, we simulate the TAGHTPS response under random load disturbance as shown in Figure19 and ±20 deviation in subsystem parameters. We also assume mismatched uncertainties in the system state matrix as a result of the change in valve positions of the TAGHTPS represented in the cosine function given as follows.
We haveΔA1=ΔA2=ΔA11ΔA12, where:(45)ΔA11=000000000000004cost0000000.46cost00cost000cost0000cost0cost00000000000000000000cost000000cost000000000001.6cost000000000000000ΔA12=000000000000000000000000000000000000000006cost000000000cost00000.2costcost00000000000000000000
The frequency fluctuation in both areas is given in Figure20, while TLPD is in Figure 21. The proposed SPSMCBSO maintained higher robustness by rejecting the various disturbances and keeping the frequency at the operating permissible point with the TLPD properly managed as well.Figure 19
Variation in random load.Figure 20
Frequency deviations (Hz) in area 1 and area 2.Figure 21
Changes in tie-line power (p.u.MW).Remark 6.
The TAGHTPS response has been simulated against the step load disturbance and random load changes to compare the optimal controller given in [26]. The results were seen better; however, the LFC schemes are required to be robust against wide range of disturbances. Therefore, the proposed single-phase sliding mode control-based state observer (SPSMCBSO) scheme is simulated under parameter uncertainties, subsystem parameter deviations, random load disturbance, and step load change and it proves to be robust by rejecting these disturbances and maintaining the TAGHTPS stability.


## 6.4. Simulation 4

To investigate the computational efficacy of SPSMCBSO scheme, the study is extended to a complex and realistic system, namely four-area multisource power system. Area 1, area 2, area 3, and area 4 consist of thermal-hydro-gas plant in each control area. The transfer function model of test system is available in Figure1 with i = 4. The comparative transient responses of test simulation 4 after 1% step load disturbance are depicted in Figures 22 and 23. The optimal controller parameters obtained by the proposed approach are presented from the frequency vitiation of four areas in Figure 22. The fluctuations in tie-line power are clearly shown in Figure 23.Figure 22
Frequency variation (Hz) in area 1, area 2, area 3, and area 4.Figure 23
Changes in tie-line power (p.u.MW).In detail, the typical transient specifications in terms of peak undershoot and settling time of system oscillations are noted to manage and keep the frequency at the operating permissible point with the changes in the tie-line properly managed and the proposed control scheme.Remark 7.
A SPSMCBSO controller is designed for load frequency control of four-area interconnected power systems. The proposed single-phase surface and the designed decentralized SMC can reduce the overshoot and improve the response speed and can also limit the deviation of frequency to zero. Therefore, the designed controller is robust and effective to control the matching and mismatched parameter uncertainties of interconnected multiarea systems.Remark 8.
From simulation 1 to simulation 4, a sensitivity analysis is undertaken to evaluate the benefit of the proposed SPSMCBSO scheme-based LFC. The settling times and minimum under/overshoot value under normal and varied conditions for the concerned power system are offered in each case. It is clearly viewed that performance values are changing within the acceptable limits and almost equal to the respective values obtained at the nominal condition. Hence, it may be concluded that the proposed controller gains are insensitive to the matched and mismatched disturbance and the parameter variations and perform satisfactorily under the wide change in the loading condition and system parameters compared with [24–26, 30–32].


## 7. Conclusions

In this study, for the first time, the single-phase sliding mode control-based state observer (SPSMCBSO) is developed for the load frequency control (LFC) of the multiarea multisource power system (MAMSPS). To test the feasibility of the constructed SPSMCBSO, the two-area gas-hydro-thermal power system (TAGHTPS) model is chosen. Furthermore, the uncertainty of the state and interconnected parameters is considered for the TAGHTPS model. The proof of the stability of TAGHTPS is established by a new linear matrix inequality via the Lyapunov theory. The superiority of the SPSMCBSO is concentrated in the comparison of the simulation results with the results of some recent methods. It is evident that the performance improvement of the TAGHTPS with the proposed SPSMCBSO is better than that of the recently mentioned methods. Furthermore, the SPSMCBSO further demonstrated robustness and is not affected by subsystem parameter deviation, random load disturbance, and parameter uncertainty in state and interconnected matrix. Therefore, the proposed SPSMCBSO is very useful for the LFC of MAMSPS.



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*Source: 6587194-2022-03-16.xml*