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

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). erefore, 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. ird, 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).


Introduction
e main aim of power system (PS) control is always to balance the total net generated power and the electrical load.
is 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. e 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. ese characteristics have an impact on the LFC of the MAPS. erefore, 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. e 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. e sliding mode control (SMC) scheme is the most popular of the VSC. eir 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][6][7][8][9]. e SMC is very important because of its robustness and resistance to large disturbances. e SMC has already been studied for the LFC of MAPS [10][11][12][13][14][15][16][17][18][19][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]. e SMC-based integral output feedback control was developed for the MAPS LFC against certain disturbances [22]. e 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]. is 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 generationbased PID control using the bacterial foraging algorithm (BFA) had been utilized for the LFC of the two-area gashydro-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]. ese 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][28][29][30][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. e disturbance observer is utilized in conjunction with the suggested fractional order three-degree-offreedom tilt integral derivative controller in [32] to effectively predict the wind velocity's uncertain profile and enrich the control law. ese 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. e 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. e 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) e 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) e 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) e 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][25][26][30][31][32], the novel SMC through singlephase 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.

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. e MAPS consists of many generating sets such as nuclear, gas, hydro, and gas plant in each area. is 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][25][26]. e gas plant can adapt to the demand for random loads, enabling it to fit the LFC system. erefore, 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. Taking into account the impact of the interconnection matrix and load disturbance, the PS model is constructed in the differential equation as follows: International Transactions on Electrical Energy Systems (   where ΔP pt i is the change in thermal turbine speed changer position (p.u.MW), ΔP Gh i is change in hydro-turbine speed changer position (p.u.MW), ΔP Gg i is change in gas turbine speed changer position (p.u.MW), ΔP D i is total incremental charge in the local load of the control area (p.u.MW), Δf i andΔf j are incremental change in frequency of each control area (Hz), and ΔP tie ij is incremental change in actual tie-line power flow from control areas 1 to 2 (p.u.MW), and ΔACE i 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: and the above equation is the state-space form of the TAGHTPS. Here, andF i are the system matrices given as follows: International Transactions on Electrical Energy Systems 5 International Transactions on Electrical Energy Systems In practical interconnected MAMSPS, changes in operating points constantly influence the fluctuating sources of load.
is factor can be considered as parameter uncertainties. Introducing this factor, system (14) can be rewritten as follows: (16) where Σ i (x i , t) and Ξ ij (x j , t) are time-varying parameter uncertainties andB i ψ i (x i , t) is the input disturbance. In other words, the aggregate uncertainty is therefore given as follows: erefore, the new dynamic model can be expressed as follows: International Transactions on Electrical Energy Systems 7 where Φ i (x i , 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 u i (t) is very important and it is based on the choice of the control engineers. Several techniques have been designed for u i (t) as seen in the literature. Meanwhile, to design the novel u i (t), we first make the following assumptions and recall the lemmas as follows.
where c i 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: Lemma 2. [22,33]: for a given inequality: where

Design of the Power System State Estimator
where Γ i is the observer gain, x i (t) is the estimation of x i (t), y i 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: Taking the derivative of the error x i , we have the following: e state error tends to zero depending on the eigenvalue of (A i − Γ i C i ).
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. e 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 matrices A i and B i and the lack of the initial error.

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][17][18][19]. e SMC design follows the selection of a sliding surface (SS) and the construction of a switching law and an equivalent control law [5][6][7][8][9][10][11][12][13][14][15]. e 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: where the matrix Μ i is selected to promise that the matrix Μ i B i is nonsingular. e design matrix Λ i ∈ R m i ×n i is chosen satisfying the nonlinearity condition.
If we take derivative of η i [x i (t)] with respect to time, we have the following: 8 International Transactions on Electrical Energy Systems As _ η i (t) � η i (t) � 0, then we can derive the equivalent control as follows: By closing the loop system, we substitute (27) into (18): To observe the MAMSPS (18), we combine (23) and (28) in the following 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. where Proof . of eorem 1: Lyapunov's function [6,34] is selected as follows: where Π i > 0 and Π i > 0 satisfy (30) for i � 1, 2, . . . , L. en, taking the derivative of time, we have the following: International Transactions on Electrical Energy Systems Introducing Lemma 1 into equation (32), we get the following: we where Furthermore, by the Schur complement of [33], the LMI is similar to the following: Combining (36) and (37), we get the following: International Transactions on Electrical Energy Systems is the constant value and the eigenvalue ρ min (Ψ i ) > 0. e term i c e − 2δ i t will be approaching zero when the time is approaching infinite. Hence, , which shows that the system is asymptotic stable.

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: where θ i is the positive scalar and u i (t) 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.
e SSWRP and the controller are given by equations (24) and (39), respectively. en, the variable state trajectories of system (19) reach the single-phase sliding surface and lie on it for all time.

Remark 2. e concept of a single-phase SMC is centered on the robustness of motion throughout the state space. e 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 eorem 2: e Lyapunov function [6,34] is given as follows: Using the time derivative of V 1 , we obtain the following: Substituting equation (26) into equation (41), we have the following: According to equation (42), property ‖AB‖ ≤ ‖A‖‖B‖ and [‖Μ j ‖‖H ji ‖ ‖x i (t)‖], and it generates: By substituting u i (t) into equation (43), we achieve the following: e derivative of Lyapunov's function (40) is less than zero. erefore, the reachability proof is achieved.
Following that, Figure 2 depicts the flowchart of the proposed observer single-phase sliding mode control technique.

12
International Transactions on Electrical Energy Systems 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.

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. e simulation results of the investigation are compared with recent results of the control method in [24][25][26][30][31][32] as follows.

Simulation 1.
e classical control methods are commonly used for the LFC of MAMSPS under step load disturbance.
e 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]. e 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. e 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 ±0.001 Hz, but 8s settling time with the novel approach is comparatively lower than the settling time in [24,[30][31][32]. [24,[30][31][32], the novel proposed SPSMCBSO controller for the LFC of MAMSPS is more robust and responsive to load disturbances than the earlier technique. e load disturbance is clearly visible, and the system is quickly restored to steady state with smaller overshoots.

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]. e frequency variation in both areas is given in Figure 6. e 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. is means that the proposed system is a superior option for the MAMSPS LFC since it is easier to execute and less stressful.

Case 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. erefore, we assume the electricity demands as a random load disturbance applied to each area of the TAGHTPS as shown in Figure 8. e subsystem parameters are assumed to be nominal as in case 1 of this simulation. e frequency oscillation in both areas is shown in Figure 9 to Figure 10, while TLPD is displayed in Figure 11. e 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. [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.

Simulation 3 Case 3.
e 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]. e 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]. e frequency variation in area 1 is shown in Figure 12, while in area 2, the frequency deviation is shown in Figure 13. e 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.
is further implies that the proposed scheme is a better choice for the MAMSPS LFC, which is very simple and less stressful to implement.

Case 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 Figure 15, we treat electrical needs as a random load disturbance applied to each section of the TAGHTPS. Figures 16 and 17 depict the frequency oscillation in both locations, whereas Figure 18

International Transactions on Electrical Energy Systems
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. erefore, in this case, we simulate the TAGHTPS response under random load disturbance as shown in Figure 19 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 have ΔA 1 � ΔA 2 � ΔA 11 ΔA 12 , where: e frequency fluctuation in both areas is given in Figure 20, while TLPD is in Figure 21. e 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.        e TAGHTPS response has been simulated against the step load disturbance and random load changes to compare the optimal controller given in [26]. e results were seen better; however, the LFC schemes are required to be robust against wide range of disturbances. erefore, the proposed singlephase 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.

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. e transfer function model of test system is available in Figure 1 with i � 4. e comparative transient responses of test simulation 4 after 1% step load disturbance are depicted in Figures 22 and 23. e optimal controller parameters obtained by the proposed approach are presented from the frequency vitiation of four areas in Figure 22. e fluctuations in tie-line power are clearly shown in Figure 23.
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.
e 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. erefore, the designed controller is robust and effective to control the matching and mismatched parameter uncertainties of interconnected multiarea systems.

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-hydrothermal power system (TAGHTPS) model is chosen. Furthermore, the uncertainty of the state and interconnected parameters is considered for the TAGHTPS model. e proof of the stability of TAGHTPS is established by a new linear matrix inequality via the Lyapunov theory. e 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. erefore, the proposed SPSMCBSO is very useful for the LFC of MAMSPS.

B. LMI's Positive Matrices
By solving LMI (28), it is easy to verify that conditions in eorem 1 are satisfied with positive matrices:

ACE:
Area control error DSPSMCS: Decentralized single-phase sliding mode control scheme DE: Differential evolution MAPS: Multiarea power system MAMSPS: Multiarea multisource power system LMI: Linear matrix inequality LFC: Load frequency control TAGHTPS: Two-area gas-hydro-thermal power system SPSMCBSO: Single-phase sliding mode control-based state observer PS: Power system PI: Proportional-integral PID: Proportional-integral-differential PSO: Particle swarm optimization VSC: Variable structure control SMC: Sliding mode control OHS: Harmonic search FOTID: Fractional order tilt integral derivative PFA: Pathfinder algorithm RFB: Redox flow battery TLPD: Tie-line power deviation.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.