# Frequency Regulation of Nonlinear Power Systems using Neural Network Observer-Based Optimized Resilient Controller

Authors: Vivek Patel; Dipayan Guha; Shubhi Purwar
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

This study introduces a resilient frequency controller for nonlinear interconnected power systems to counteract endogenous/exogenous system disturbances. A neural network-based observer (NNO) is intended to estimate lumped system disturbances, such as unmodelled dynamics and unknown disturbances. The estimated NNO’s output is incorporated with a second-order sliding mode controller (SOSMC) to minimize chattering in the control effort and improve the nominal performance of the undertaken plant. The design parameters of SOSMC have been optimally identified by applying Harris hawk optimization (HHO), exercising integral error-based objective function. HHO has demonstrated superior tuning capabilities than other well-known optimization methodologies in terms of convergence rate and transient measurements of system outputs. The asymptotic convergence of estimated error and overall stability of the system has been established employing the Lyapunov argument. System outputs are compared with the results reported in the literature to validate the efficacy of the proposed resilient frequency controller. Presented results showcase the mastery of the applied NNO-based SOSMC over its counterparts in weaker chattering, fast disturbance rejection, and a high degree of robustness against endogenous/exogenous disturbances.


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## Body

## 1. Introduction

Due to the continuum shrinking of conventional fossil fuel resources, escalating fuel prices, geographical constraints, global warming, and increasing load demand, low-inertia inverter-based renewable energy resources (RERs) that are integrated into the existing power grid have been progressively increased. Among the RERs, wind power generator (WPG) is widely utilized due to their high energy conversion efficiency and relatively cheaper than photovoltaic system [1]. In the future, wind energy is predicted to meet a large portion of global energy needs. Several huge offshore wind farms have already been built and are connected to the electrical grid. Wind energy farms are predicted to produce 40% of all energy in the United States by 2030.Grid frequency stability and power system operation grow increasingly difficult as large-scale wind farms are developed. The frequency stability in the region is impacted by the randomness and nondispatchable characteristics of wind power [2]. WPGs contribute little or nothing to frequency stability. Due to the use of power electronic converters, modern wind farms are mostly isolated from grid frequency and do not contribute to power system inertia. Only synchronously connected rotating generators and motors give apparent inertia and help to maintain frequency stability. Hence, large penetration of variable renewable energy reduces the power system’s apparent inertia, which can cause undesired frequency variation, voltage instability, low-grade power quality, and reliability obstruction. Moreover, unpredictable load variations may cause the frequency to droop from its theoretical value (50 Hz/60 Hz). Excessive frequency droop may cause instability/blackout of the entire system. To preserve frequency stability under extensive penetration of WPGs, the design and realization of resilient frequency controllers are in high demand [2, 3].


### 1.1. Litrature Survey

To alleviate frequency oscillations caused by sudden load variations and/or RERs transient, various control methodologies have been reported in the state of the art, including metaheuristic-based classical controllers [4–6], robust controllers [7], fractional-order aided controllers [8–10], intelligent controllers [11, 12], and so on. According to a study of the existing literature, the performance of conventional controllers in frequency regulation of power systems with/without RERs incorporated has been widely used [2, 3]. An output feedback-based optimal control for in frequency regulation of reduced power systems model has been discussed in reference [13]. In the above-cited works, traditional controllers considerably improve system dynamics. However, its performance deteriorates with parametric uncertainty, unmodelled dynamics, and unknown/uncertain external perturbations. Fuzzy-aided controllers may overcome the above-said limitations and increases stability degree. However, the design of a fuzzy-aided controller requires high training data set, a suitable selection of input control parameters (e.g., membership function and scaling factors), and is computationally intensive [6].Robust control algorithms are necessary to achieve good dynamic performance throughout various operating conditions and severe loading fluctuation. The sliding mode controller (SMC) has appeared as one of the most powerful robust controllers for uncertain nonlinear interconnected power systems (NIPSs) due to its inherent robust operation against structured vulnerabilities and considerable improvement in the system dynamics [14, 15]. In frequency regulation of power systems with or without RPRs, the SMC has received much attention. In references [16–18], SMC’s ability to provide quick frequency response was highlighted, leaving high-frequency oscillation (chattering) in controller’s output. In reference [19], nonlinear SMLFC for power systems is presented. A disturbance-observer based SMLFC is discussed in references [20, 21]. Considering the above-said troubled situation, the SMC’s performance has been improved by integrating linear/nonlinear disturbance observers [21, 22]. Sliding mode observer-based SMLFC has been presented in reference [23].According to the above study, a frequency controller is developed for a rationalised power system that includes or excludes renewable energy inputs. After that, the performance of the controller is examined, considering system’s nonlinearities (governor dead band (GDB) and generation rate constraints (GRC)) as well as parametric uncertainties. The research mentioned above did not consider the system’s nonlinearities and uncertainties while building the system model. Chattering is generally acknowledged to be a major impediment to the traditional SMC. A literature assessment also reveals that a trade-off between chattering and robustness is required to increase SMC’s performance [24, 25].Disturbance compensation was used by power system researchers to reduce the impact of system’s nonlinearities and uncertainties. Thus, the planning a state and disturbance estimator with an effective approximation is a fundamental for achieving a superior control structure. The neural network is the most used estimation technique of the system uncertainties/dynamics, despite its excellent approximation analysis and adaptive learning capabilities. In reference [26], the performance of the neural network-aided integral SMC (NN-ISMC) was examined, and it was found to be more effective than conventional SMC in improving chattering tasks [25, 26]. In reference [27], NN-based nonlinear SMC has been applied for frequency regulation of a power system. NN was used to estimate unmatched plant uncertainties. In reference [28], NN-based fractional-order ISMC is described. However, the drawback of this literature was the controller parameter was taken by trial-error method, and the chattering was not eliminated in controller output. A resilient frequency controller has been attempted to counterbalance power-frequency oscillations of the studied power system, taking into account the benefits of neural networks and second-order sliding mode controllers (SOSMCs).


### 1.2. Motivations

Given the preceding discussion, the following attributes encourage the examination of the present work:(i)
To deliver a stable and consistent electrical power to end-users, the analysis of a realistic (including parametric uncertainty and nonlinearities) nonlinear interconnected power systems (NIPSs) with renewable energy resources integrated may be carried out.(ii)
Estimation of plant uncertainty is needed for enriching the controller performance.(iii)
The traditional SMC suffers from a chattering problem. Therefore, implementing a resilient frequency controller for stable power systems operation with the minimum chattering is imperative.However, to tackle the frequency regulation problem, a neural network observer (NNO) and second-order SMC (SOSMC) are fused due to their accuracy and extensive application. The NNO is best suited to predict system variables reliably, even in any power system vulnerability. The NNO may also get approximate realizations of plant dynamics in frequency regulation support based on its input-output history when subjected to uncertain nature constrained renewable power integration. In SOSMC, a control input’s derivative is used to perform a second-order derivative of the sliding variable. The control input’s derivative is discontinuous. However, the control input’s integration is continuous, making the control input less chattering. As a result, wear and tear issues in the governor valve actuator in the steam turbine servo system are reduced.


### 1.3. Contributions

As a result of the abovementioned aspects, the significant contributions in this study are listed below.(i)
Design and access the SOSMC’s frequency control performance in a nonlinear interconnected power system (NIPS).(ii)
The SOSMC’s impacts are evaluated using a power system model that includes both thermal and wind power plants.(iii)
The impact of power system’s nonlinearities such as GDB and GRC on controller performance has been examined.(iv)
NNO is designed and used to estimate system states and lumped unknown/uncertain plant mismatches, and an improved control law/effort is generated based on this estimation.(v)
Harris hawk optimization (HHO) is applied to tune the parameter of SOSMC and compared with salp swarm algorithm (SSA), particle swarm optimization (PSO), deferential evaluation (DE), and crow search algorithm (CSA). The tuning competence of HHO algorithm has been measured in terms of convergence rate with PSO, DE, CSA, and SSA under identical simulation environment.(vi)
A comparison analysis was conducted to verify the superiority of the applied optimized SOSMC over the NN-ISMC [26] in terms of system performance, chattering, and robustness against system mismatches.


### 1.4. Organization of the Study

The rest of the study is laid out as follows. Section2 presents the modelling of the analysed NIPS with a wind power system. Section 3 discusses the implementation and analysis of NNO-based SOSMC. Section 4 contains the simulation outputs and a comparison discussion. The conclusion of the current work is described in Section 5.


## 1.1. Litrature Survey

To alleviate frequency oscillations caused by sudden load variations and/or RERs transient, various control methodologies have been reported in the state of the art, including metaheuristic-based classical controllers [4–6], robust controllers [7], fractional-order aided controllers [8–10], intelligent controllers [11, 12], and so on. According to a study of the existing literature, the performance of conventional controllers in frequency regulation of power systems with/without RERs incorporated has been widely used [2, 3]. An output feedback-based optimal control for in frequency regulation of reduced power systems model has been discussed in reference [13]. In the above-cited works, traditional controllers considerably improve system dynamics. However, its performance deteriorates with parametric uncertainty, unmodelled dynamics, and unknown/uncertain external perturbations. Fuzzy-aided controllers may overcome the above-said limitations and increases stability degree. However, the design of a fuzzy-aided controller requires high training data set, a suitable selection of input control parameters (e.g., membership function and scaling factors), and is computationally intensive [6].Robust control algorithms are necessary to achieve good dynamic performance throughout various operating conditions and severe loading fluctuation. The sliding mode controller (SMC) has appeared as one of the most powerful robust controllers for uncertain nonlinear interconnected power systems (NIPSs) due to its inherent robust operation against structured vulnerabilities and considerable improvement in the system dynamics [14, 15]. In frequency regulation of power systems with or without RPRs, the SMC has received much attention. In references [16–18], SMC’s ability to provide quick frequency response was highlighted, leaving high-frequency oscillation (chattering) in controller’s output. In reference [19], nonlinear SMLFC for power systems is presented. A disturbance-observer based SMLFC is discussed in references [20, 21]. Considering the above-said troubled situation, the SMC’s performance has been improved by integrating linear/nonlinear disturbance observers [21, 22]. Sliding mode observer-based SMLFC has been presented in reference [23].According to the above study, a frequency controller is developed for a rationalised power system that includes or excludes renewable energy inputs. After that, the performance of the controller is examined, considering system’s nonlinearities (governor dead band (GDB) and generation rate constraints (GRC)) as well as parametric uncertainties. The research mentioned above did not consider the system’s nonlinearities and uncertainties while building the system model. Chattering is generally acknowledged to be a major impediment to the traditional SMC. A literature assessment also reveals that a trade-off between chattering and robustness is required to increase SMC’s performance [24, 25].Disturbance compensation was used by power system researchers to reduce the impact of system’s nonlinearities and uncertainties. Thus, the planning a state and disturbance estimator with an effective approximation is a fundamental for achieving a superior control structure. The neural network is the most used estimation technique of the system uncertainties/dynamics, despite its excellent approximation analysis and adaptive learning capabilities. In reference [26], the performance of the neural network-aided integral SMC (NN-ISMC) was examined, and it was found to be more effective than conventional SMC in improving chattering tasks [25, 26]. In reference [27], NN-based nonlinear SMC has been applied for frequency regulation of a power system. NN was used to estimate unmatched plant uncertainties. In reference [28], NN-based fractional-order ISMC is described. However, the drawback of this literature was the controller parameter was taken by trial-error method, and the chattering was not eliminated in controller output. A resilient frequency controller has been attempted to counterbalance power-frequency oscillations of the studied power system, taking into account the benefits of neural networks and second-order sliding mode controllers (SOSMCs).


## 1.2. Motivations

Given the preceding discussion, the following attributes encourage the examination of the present work:(i)
To deliver a stable and consistent electrical power to end-users, the analysis of a realistic (including parametric uncertainty and nonlinearities) nonlinear interconnected power systems (NIPSs) with renewable energy resources integrated may be carried out.(ii)
Estimation of plant uncertainty is needed for enriching the controller performance.(iii)
The traditional SMC suffers from a chattering problem. Therefore, implementing a resilient frequency controller for stable power systems operation with the minimum chattering is imperative.However, to tackle the frequency regulation problem, a neural network observer (NNO) and second-order SMC (SOSMC) are fused due to their accuracy and extensive application. The NNO is best suited to predict system variables reliably, even in any power system vulnerability. The NNO may also get approximate realizations of plant dynamics in frequency regulation support based on its input-output history when subjected to uncertain nature constrained renewable power integration. In SOSMC, a control input’s derivative is used to perform a second-order derivative of the sliding variable. The control input’s derivative is discontinuous. However, the control input’s integration is continuous, making the control input less chattering. As a result, wear and tear issues in the governor valve actuator in the steam turbine servo system are reduced.


## 1.3. Contributions

As a result of the abovementioned aspects, the significant contributions in this study are listed below.(i)
Design and access the SOSMC’s frequency control performance in a nonlinear interconnected power system (NIPS).(ii)
The SOSMC’s impacts are evaluated using a power system model that includes both thermal and wind power plants.(iii)
The impact of power system’s nonlinearities such as GDB and GRC on controller performance has been examined.(iv)
NNO is designed and used to estimate system states and lumped unknown/uncertain plant mismatches, and an improved control law/effort is generated based on this estimation.(v)
Harris hawk optimization (HHO) is applied to tune the parameter of SOSMC and compared with salp swarm algorithm (SSA), particle swarm optimization (PSO), deferential evaluation (DE), and crow search algorithm (CSA). The tuning competence of HHO algorithm has been measured in terms of convergence rate with PSO, DE, CSA, and SSA under identical simulation environment.(vi)
A comparison analysis was conducted to verify the superiority of the applied optimized SOSMC over the NN-ISMC [26] in terms of system performance, chattering, and robustness against system mismatches.


## 1.4. Organization of the Study

The rest of the study is laid out as follows. Section2 presents the modelling of the analysed NIPS with a wind power system. Section 3 discusses the implementation and analysis of NNO-based SOSMC. Section 4 contains the simulation outputs and a comparison discussion. The conclusion of the current work is described in Section 5.


## 2. System Modeling

To quantify the mastery of the proposed NNO-SOSMC, a thermal power plant having wind power plant integrated has been undertaken to perform the present investigation. Figure1 depicts the linearized model of the studied test system. To carry out the investigation on the realistic domain, detailed mathematical models of WTG, including wind aerodynamics, coupling shaft, and generator, have been derived and considered for the assessment of the proposed controllers’ performance. The GRC and GDB as physical constraints are involved in the model to assess the proposed controller’s efficacy in coping with physical constraints. The dynamical model of the studied system is represented by the following differential equations:(1)Δfi⋅t=−1TpiΔfit+KpiTpiΔPgit−KpiTpiΔPijt−KpiTpiΔPditΔpti⋅t=−ΔPtit+ΔXgit1Tti+n2Tgi−ΔfitR−ΔEit+ut1TtiΔXgi⋅t=−1TgiΔXgit+1Tgin1−n2Tgi−ΔfitRi−ΔEit+utΔEi⋅t=KeiBiΔfit+KeiΔPtieijtΔPtieij⋅t=2πTi,jΔfit−Δfjtwherei,j=1,2,i≠j.where Δfi,ΔPti,ΔPtieij,ΔXgi, and ΔEi are the changes in area frequency, nonreheat turbine output, tie-line interchange power, governor valve position, and integral control effort, respectively; ΔPdit and Pw are the deviation in the load demand and uncertain output power of WTG, respectively; Tgi,Tpi, and Tti are the time constants of speed governor, power system, and turbine,respectively; T12 is the time constant of tie-line. Table 1 provides the nominal values of the thermal power system.Figure 1
Two-area nonlinear interconnected power system (test system 1).Table 1
Thermal power system parameters.
Kp1,Kp2120Tt1,Tt20.3B1,B20.425Ki1,Ki21.2n10.8Tp1,Tp220Tg1,Tg20.08T120.05R1,R22.4n2−0.2/π


### 2.1. Wind Turbine Generator (WTG) Model

A variable speed WTG system model, as presented in Figure1, is obtained to quantify the performance of proposed NNO-SOSMC against the uncertain/unknown wind power output. The mechanical power output of wind aerodynamic is given by the following equation:(2)Pm=12ρAbVw3CPλ,βwhere,CP=c1c2λk−c3β−c4exp−c5λk+c6λ1λk=1λ+0.08β−0.03β3+1,where ρ is the air density (kg/m3), Vw is the incoming wind speed (m/s2), Ab is the WT blade swept area (m2), and Cp is the power coefficient [19, 25].The model of coupling shaft is given by the following equations:(3)ω•tt=−Bt+BshItωtt+BshItωgt−δtgtIt+1ItTr,ω•gt=Bsh2Igωtt−Bg+BshIgωgt+δtgtNgIg−1IgTgtδ•tgt=Kshωtt−ωgtNg,,where Tg and Tr are the torque on generator and rotor side, respectively; Ig and It are the inertia of generator and rotor side, respectively; Bg and Bt are the rotor damping coefficients of generator, and turbine,respectively; Bsh and Ksh are the damping and spring constant of the shaft,respectively; Ng is the gear-train ratio; ωg and ωt are the generator speed and rotor speed, respectively; δtg is the shaft stiffness of the two-mass model. An asynchronous generator has been described by a first-order differential equation, as given in equation (4). The generator power Pw=Tg×ωg is deliberated as a net WTG’s power output [20, 28].(4)Tg•=−1τgTg+uwτg,where τg is the generator’s time constant.The state-space model of the investigated power system, including dynamic variation in system parameter, is obtained as in the following equation:(5)xt·=A1xt+B1ut+Bd1ΔPt,where A1=A+ΔA,B1=B+ΔB,Bd1=Bd+ΔBdxt=Δf1ΔPt1ΔGg1ΔE1Δf2ΔPt2ΔGg2ΔE2ΔPtie12ΔωtΔωrΔδtrΔTrT∈ℝ13×1ut=u1u2uw∈ℝ3×1ΔPt=ΔPd1ΔPd2PwTgT∈ℝ4×1, xt is the system states, ut is the states, and ΔPt is the disturbance input of the system. Including the parameter uncertainties, equation (5) can be written as follows:(6)xt·=A+ΔAxt+B+ΔBut+Bd+ΔBdΔPtxt•=Axt+But+BdΔPt+ΔAxt+ΔBut+ΔBdΔPt,where A∈ℝ13×13 is the system matrix, B∈ℝ13×13is the input matrix, Bd∈ℝ13×4 is the disturbance matrix, computed at nominal value, and ΔAxt, ΔBut, and ΔBdΔPt show the parametric uncertainties.GRC bounds the maximum generated power. The steam turbine’s output with GRC can be described [25] as follows:(7)ΔPti−1Tti∫ψdtαt<−ψ,1Tti∫αtdtαt≤−ψ,1Tti∫ψdtαt>ψ.Therefore,αt=ΔGgit−ΔPtitandψ>0. Considering GRC, equation (6) is modifiedas follows:(8)xt·=Axt+But+BdΔPt+ΔAxt+ΔBut+ΔBdΔPtφt.where φt in equation (8) denotes the impact of GRC. Therefore, the state-space model of the studied power system is described as follows:(9)xt•=Axt+But+dtwheredt=ΔAxt+ΔBut+ΔBdΔPt+ϕt+BdΔPt,where dt symbolizes the lumped unknown plant uncertainties and nonlinearities, and A, B, and H matrices are the matrices computed for test system 1 described in Appendix. The proposed control technique for the system (equation (9)) is presented in the next section.


## 2.1. Wind Turbine Generator (WTG) Model

A variable speed WTG system model, as presented in Figure1, is obtained to quantify the performance of proposed NNO-SOSMC against the uncertain/unknown wind power output. The mechanical power output of wind aerodynamic is given by the following equation:(2)Pm=12ρAbVw3CPλ,βwhere,CP=c1c2λk−c3β−c4exp−c5λk+c6λ1λk=1λ+0.08β−0.03β3+1,where ρ is the air density (kg/m3), Vw is the incoming wind speed (m/s2), Ab is the WT blade swept area (m2), and Cp is the power coefficient [19, 25].The model of coupling shaft is given by the following equations:(3)ω•tt=−Bt+BshItωtt+BshItωgt−δtgtIt+1ItTr,ω•gt=Bsh2Igωtt−Bg+BshIgωgt+δtgtNgIg−1IgTgtδ•tgt=Kshωtt−ωgtNg,,where Tg and Tr are the torque on generator and rotor side, respectively; Ig and It are the inertia of generator and rotor side, respectively; Bg and Bt are the rotor damping coefficients of generator, and turbine,respectively; Bsh and Ksh are the damping and spring constant of the shaft,respectively; Ng is the gear-train ratio; ωg and ωt are the generator speed and rotor speed, respectively; δtg is the shaft stiffness of the two-mass model. An asynchronous generator has been described by a first-order differential equation, as given in equation (4). The generator power Pw=Tg×ωg is deliberated as a net WTG’s power output [20, 28].(4)Tg•=−1τgTg+uwτg,where τg is the generator’s time constant.The state-space model of the investigated power system, including dynamic variation in system parameter, is obtained as in the following equation:(5)xt·=A1xt+B1ut+Bd1ΔPt,where A1=A+ΔA,B1=B+ΔB,Bd1=Bd+ΔBdxt=Δf1ΔPt1ΔGg1ΔE1Δf2ΔPt2ΔGg2ΔE2ΔPtie12ΔωtΔωrΔδtrΔTrT∈ℝ13×1ut=u1u2uw∈ℝ3×1ΔPt=ΔPd1ΔPd2PwTgT∈ℝ4×1, xt is the system states, ut is the states, and ΔPt is the disturbance input of the system. Including the parameter uncertainties, equation (5) can be written as follows:(6)xt·=A+ΔAxt+B+ΔBut+Bd+ΔBdΔPtxt•=Axt+But+BdΔPt+ΔAxt+ΔBut+ΔBdΔPt,where A∈ℝ13×13 is the system matrix, B∈ℝ13×13is the input matrix, Bd∈ℝ13×4 is the disturbance matrix, computed at nominal value, and ΔAxt, ΔBut, and ΔBdΔPt show the parametric uncertainties.GRC bounds the maximum generated power. The steam turbine’s output with GRC can be described [25] as follows:(7)ΔPti−1Tti∫ψdtαt<−ψ,1Tti∫αtdtαt≤−ψ,1Tti∫ψdtαt>ψ.Therefore,αt=ΔGgit−ΔPtitandψ>0. Considering GRC, equation (6) is modifiedas follows:(8)xt·=Axt+But+BdΔPt+ΔAxt+ΔBut+ΔBdΔPtφt.where φt in equation (8) denotes the impact of GRC. Therefore, the state-space model of the studied power system is described as follows:(9)xt•=Axt+But+dtwheredt=ΔAxt+ΔBut+ΔBdΔPt+ϕt+BdΔPt,where dt symbolizes the lumped unknown plant uncertainties and nonlinearities, and A, B, and H matrices are the matrices computed for test system 1 described in Appendix. The proposed control technique for the system (equation (9)) is presented in the next section.


## 3. Control Strategy

A neural network-based observer (NNO) is developed in this section to effectively estimate limped plant uncertainty [29–32]; later, the estimated output is incorporated with second-order SMC for increasing speed of response and minimizing chattering. The design of the proposed method is not distinctive and can be realized on four-area power system. The schematic diagram of proposed control structure is shown in Figure 2.Figure 2
Schematic diagram of neural network observer with a controller.


### 3.1. Neural Network Observer

The Chebyshev neural network is a useful tool for estimating vulnerabilities with a basic topological structure that fills the gap between system stability and the uncertain boundary value. The following recursive formula can be used to define Chebyshev polynomials equation (10) [29, 30].(10)Pj+1=2xPjx−Pj−1x,P0x=1,where Pjx is the Chebyshev polynomial and j is the polynomial order.The structure of neural network (NN) is revealed in Figure3. The output of NN is described in the following equation:(11)d¯x=wTϕx,u+εx,where εx is the reconstruction error and εx≤ε1 is bounded.w is the ideal weight of NN.Figure 3
Structure of NN.The NN observer (NNO) for the system (equation (9)) is defined as follows:(12)x^˙t=Ax^t+But+d^t+Lyt−y^ty^t=Cx^t.where x^ is the estimated state variable and y^ is the estimated output. The observer gain L∈ℝn×m is chosen such that the matrix (A − LC) is a Hurwitz matrix [28]. Therefore, matrix (A, C) should be observable.Assumption 1.
The ideal weightwF≤wm, where ⋅F represents the Frobenius norm.
The NN estimate ofd¯x is described by using the following equation:(13)d¯^x=w^Tϕx,t,where w^ is the estimated weight w, ϕx=1Pjx1…Pjx13T, and j = 2.
The NNO error dynamics is represented by the following equation:(14)et=xt−x^t.
Using equations (9) and (12), the error dynamics of NNO is represented as follows:(15)et·=A−LCet+wTϕx,u−ϕx^,u+εx+w˜Tϕx^,u,where w˜=w−w^,andAL=A−LC, and the tuning law of weight w^ is given in equation (31).


### 3.2. Design of Second-Order Sliding Mode Controller (SOSMC)

Sliding mode control (SMC) is identified as an effective, robust control technique due to its (i) fast dynamic response and (ii) insensitivity to variations in plant parameters and external disturbance. SOSMC is developed in this section to reduce chattering at the SMC output while maintaining the system’s robustness. The main inadequacy of SMC is the chattering, which may cause damage to the final control element and makes the system more vulnerable to instability. Hence, it is required to quickly reduce/eliminate the chattering at the controller output effectively. Using a SOSMC can help reduce chattering and increase the overall robustness of a closed-loop control system. SOSMC design comprises two basic steps, similar to traditional SMC design: choosing a sliding surface and developing an effective control law. SOSMC has been designed based on the assumptions listed below [20, 27].Assumption 2.
The pair (A, B) is fully state controllable.Assumption 3.
d˙t≤d¯,d¯>0, where ⋅ indicates the standard vector norm, dt is continuously differentiable function and bounded; that is, d¯ is unknown but bounded positive constant.
Let the first-order sliding surface be(16)σt=α1x^t−∫0tα1A−α1Bα2x^τdτ,where α1 and α2 are the controller design parameters with appropriate dimension, the matrix α2 is to be selected to make sure the eigenvalues of ReA−Bα2<0, and α1B≠0.
The prime objective of the SOSMC is to act on second-order derivative of the sliding surfaceσt rather than the first derivative as in standard-sliding mode, also retaining the advantage of the SMC, and SOSMC also has the advantage of eliminating the chattering effect [24]. The basic equality condition to design SOSMC is σt=σt•=0.
To design the SOSMC, the first and second-order time derivatives of sliding surface (equation (16)) are obtained, as in equations (17) and (18), respectively:(17)σt·=α1x^t·−α1A−α1Bα2x^t=0.(18)σt··=α1x^t··−α1A−α1Bα2x^t•=0.
For designing SOSMC, the sliding function is chosen as shown in the following equation:(19)σnt=σt•+ϒσt,ϒ>0,where σnt is the second-order sliding surface. Differentiating equation (19) yields(20)σnt·=σt··+ϒσt·.
Substituting equations (17) and (19) into equation (20) gives(21)σn⋅t=α1x^⋅t−α1A−α1Bα2x^⋅t+ϒα1x^⋅t−α1A−α1Bα2x^torσn⋅t=α1Ax^⋅t+Bu⋅t+d⋅t−α1A−α1Bα2x^⋅t+ϒσ⋅t.
The equality hitting condition (σntσnt·<0) is assumed for designing the controller [20]:(22)σnt·=−γ1σnt−γ2sgnσnt.
From equations (21) and (22), we get(23)ut·=−α2xt·−α1B−1α1d¯t+ϒσt·+γ1σnt+γ2sgnσnt.
It is impractical to know the exact boundary valued¯t of the lumped plant uncertainty. The controller is designed under the assumption that the uncertainties are unknown boundary. To refine the control effort, NNO is applied to estimate the upper bound of uncertain/unknown lumped plant uncertainty d¯t.
Equation (23) is modified by using the Chebyshev-based NNO approximation:(24)ut·=−α2xt·−α1B−1α1d¯^t+ϒσt·+γ1σnt+γ2sgnσntut=∫0t−α2xt·−α1B−1α1d¯^t+ϒσt·+γ1σnt+γ2sgnσntdτ.
Equation (24) represents the control law of the proposed SOSMC integrated with NNO (NNO-SOSMC). The controller parameter is optimized by Harris hawk algorithm, which is given in Section 4. The stability analysis of the proposed controller is given in the next subsection.


### 3.3. Stability Analysis

Assumption 4.
wTϕx−d¯≤ξ, ξ is a positive constant.Theorem 1.
Take Assumption1–3 into account. Considering the system (equation (9)), sliding surface (equation (19)), and control law equation (24), if the system is stable, then weight update law for NN has the following form:(25)w^·=Τk1ϕσnT−kwσnΤw^,where Τ is a positive constant. The weight error w˜ and sliding surface σn are constrained by(26)σn>kwwm2/4−γ2−α1ξγ1min,w˜F>wm2+kwwm24−γ2−α1ξ.Proof.
Take into account the following Lyapunov function candidate:(27)V=12σnTσn+12w˜TΤ−1w˜,where w˜=w−w^ and w˜˙=−w^˙.
Differentiating equation (27)(28)V˙=σnTσn·−w˜TΤ−1w^˙.
Using equations (21) and (28)(29)V·=σnTα1x^t··−α1A−α1Bα2x^t·+ϒα1x^t·−α1A−α1Bα2x^t−trw˜TΓ−1w˜·V·=σnTα1Ax^t·+Bx^t·+dt·−α1A−α1Bα2x^t·+ϒα1x^t·−α1A−α1Bα2x^t−trw˜TΓ−1w˜·V·=σnTα1But·+dt·+α1Bα2x^·t+ϒσt·−trw˜TΓ−1w˜·.
Substituting the control law equation (24) into equation (29)(30)Vt·=σnTt−α1dt^−α1Bα2x^t·−ϒσnt·−γ1σnt−γ2sgnσnt+α1dt^+α1Bα2x^t·+ϒσt·−trw˜TΤ−1w^·Vt•=σnTt−α1dt^−dt^−γ1σnt−γ2sgnσnt−trw˜TΤ−1w^·.
Substituting equation (25) into equation (30)(31)V·≤σnTt−α1wTϕ−ξ−w^Tϕ−γ1σnt−γ2sgnσnt−trw˜TΤ−1α1ΤϕσnT−kwσnΤw^.
It is noted thatσnTtsignσnt=σn in equation (31). Owing to σnTtγ1σnt=σnγ1minσn, γ1min is the minimum eigenvalue of γ1. Equation (31) can be rewritten as follows:(32)V·≤−σnγ1minσn−γ2σn−α1ξσn+α1w˜TϕσnT−trw˜Tα1ϕσnT−kwσnw^V·≤−σnγ1minσn−μσn−α1ξσn+trkwσnw˜Tw^.
The inequality is applied as given in the following equation:(33)trw˜Tw−w˜≤wFw˜F−w˜F2.
We get(34)V·≤−σnγ1minσn+γ2+α1ξ+kww˜Fw˜F−wmV·≤−σnγ1minσn+γ2+α1ξ+kww˜F−wm22−kwwm24,σn>kwwm2/4−γ2−α1ξγ1min,w˜F>wm2+kwwm24−γ2−α1ξ.
As a result,V· (equation (34)) is negative outside of a compact set, demonstrating that it is uniformly ultimately bounded (UUB) of both σn and w˜F.
Harris hawk optimization (HHO) is applied to tune the parameter of SOSMC, which is given in next section.


## 3.1. Neural Network Observer

The Chebyshev neural network is a useful tool for estimating vulnerabilities with a basic topological structure that fills the gap between system stability and the uncertain boundary value. The following recursive formula can be used to define Chebyshev polynomials equation (10) [29, 30].(10)Pj+1=2xPjx−Pj−1x,P0x=1,where Pjx is the Chebyshev polynomial and j is the polynomial order.The structure of neural network (NN) is revealed in Figure3. The output of NN is described in the following equation:(11)d¯x=wTϕx,u+εx,where εx is the reconstruction error and εx≤ε1 is bounded.w is the ideal weight of NN.Figure 3
Structure of NN.The NN observer (NNO) for the system (equation (9)) is defined as follows:(12)x^˙t=Ax^t+But+d^t+Lyt−y^ty^t=Cx^t.where x^ is the estimated state variable and y^ is the estimated output. The observer gain L∈ℝn×m is chosen such that the matrix (A − LC) is a Hurwitz matrix [28]. Therefore, matrix (A, C) should be observable.Assumption 1.
The ideal weightwF≤wm, where ⋅F represents the Frobenius norm.
The NN estimate ofd¯x is described by using the following equation:(13)d¯^x=w^Tϕx,t,where w^ is the estimated weight w, ϕx=1Pjx1…Pjx13T, and j = 2.
The NNO error dynamics is represented by the following equation:(14)et=xt−x^t.
Using equations (9) and (12), the error dynamics of NNO is represented as follows:(15)et·=A−LCet+wTϕx,u−ϕx^,u+εx+w˜Tϕx^,u,where w˜=w−w^,andAL=A−LC, and the tuning law of weight w^ is given in equation (31).


## 3.2. Design of Second-Order Sliding Mode Controller (SOSMC)

Sliding mode control (SMC) is identified as an effective, robust control technique due to its (i) fast dynamic response and (ii) insensitivity to variations in plant parameters and external disturbance. SOSMC is developed in this section to reduce chattering at the SMC output while maintaining the system’s robustness. The main inadequacy of SMC is the chattering, which may cause damage to the final control element and makes the system more vulnerable to instability. Hence, it is required to quickly reduce/eliminate the chattering at the controller output effectively. Using a SOSMC can help reduce chattering and increase the overall robustness of a closed-loop control system. SOSMC design comprises two basic steps, similar to traditional SMC design: choosing a sliding surface and developing an effective control law. SOSMC has been designed based on the assumptions listed below [20, 27].Assumption 2.
The pair (A, B) is fully state controllable.Assumption 3.
d˙t≤d¯,d¯>0, where ⋅ indicates the standard vector norm, dt is continuously differentiable function and bounded; that is, d¯ is unknown but bounded positive constant.
Let the first-order sliding surface be(16)σt=α1x^t−∫0tα1A−α1Bα2x^τdτ,where α1 and α2 are the controller design parameters with appropriate dimension, the matrix α2 is to be selected to make sure the eigenvalues of ReA−Bα2<0, and α1B≠0.
The prime objective of the SOSMC is to act on second-order derivative of the sliding surfaceσt rather than the first derivative as in standard-sliding mode, also retaining the advantage of the SMC, and SOSMC also has the advantage of eliminating the chattering effect [24]. The basic equality condition to design SOSMC is σt=σt•=0.
To design the SOSMC, the first and second-order time derivatives of sliding surface (equation (16)) are obtained, as in equations (17) and (18), respectively:(17)σt·=α1x^t·−α1A−α1Bα2x^t=0.(18)σt··=α1x^t··−α1A−α1Bα2x^t•=0.
For designing SOSMC, the sliding function is chosen as shown in the following equation:(19)σnt=σt•+ϒσt,ϒ>0,where σnt is the second-order sliding surface. Differentiating equation (19) yields(20)σnt·=σt··+ϒσt·.
Substituting equations (17) and (19) into equation (20) gives(21)σn⋅t=α1x^⋅t−α1A−α1Bα2x^⋅t+ϒα1x^⋅t−α1A−α1Bα2x^torσn⋅t=α1Ax^⋅t+Bu⋅t+d⋅t−α1A−α1Bα2x^⋅t+ϒσ⋅t.
The equality hitting condition (σntσnt·<0) is assumed for designing the controller [20]:(22)σnt·=−γ1σnt−γ2sgnσnt.
From equations (21) and (22), we get(23)ut·=−α2xt·−α1B−1α1d¯t+ϒσt·+γ1σnt+γ2sgnσnt.
It is impractical to know the exact boundary valued¯t of the lumped plant uncertainty. The controller is designed under the assumption that the uncertainties are unknown boundary. To refine the control effort, NNO is applied to estimate the upper bound of uncertain/unknown lumped plant uncertainty d¯t.
Equation (23) is modified by using the Chebyshev-based NNO approximation:(24)ut·=−α2xt·−α1B−1α1d¯^t+ϒσt·+γ1σnt+γ2sgnσntut=∫0t−α2xt·−α1B−1α1d¯^t+ϒσt·+γ1σnt+γ2sgnσntdτ.
Equation (24) represents the control law of the proposed SOSMC integrated with NNO (NNO-SOSMC). The controller parameter is optimized by Harris hawk algorithm, which is given in Section 4. The stability analysis of the proposed controller is given in the next subsection.


## 3.3. Stability Analysis

Assumption 4.
wTϕx−d¯≤ξ, ξ is a positive constant.Theorem 1.
Take Assumption1–3 into account. Considering the system (equation (9)), sliding surface (equation (19)), and control law equation (24), if the system is stable, then weight update law for NN has the following form:(25)w^·=Τk1ϕσnT−kwσnΤw^,where Τ is a positive constant. The weight error w˜ and sliding surface σn are constrained by(26)σn>kwwm2/4−γ2−α1ξγ1min,w˜F>wm2+kwwm24−γ2−α1ξ.Proof.
Take into account the following Lyapunov function candidate:(27)V=12σnTσn+12w˜TΤ−1w˜,where w˜=w−w^ and w˜˙=−w^˙.
Differentiating equation (27)(28)V˙=σnTσn·−w˜TΤ−1w^˙.
Using equations (21) and (28)(29)V·=σnTα1x^t··−α1A−α1Bα2x^t·+ϒα1x^t·−α1A−α1Bα2x^t−trw˜TΓ−1w˜·V·=σnTα1Ax^t·+Bx^t·+dt·−α1A−α1Bα2x^t·+ϒα1x^t·−α1A−α1Bα2x^t−trw˜TΓ−1w˜·V·=σnTα1But·+dt·+α1Bα2x^·t+ϒσt·−trw˜TΓ−1w˜·.
Substituting the control law equation (24) into equation (29)(30)Vt·=σnTt−α1dt^−α1Bα2x^t·−ϒσnt·−γ1σnt−γ2sgnσnt+α1dt^+α1Bα2x^t·+ϒσt·−trw˜TΤ−1w^·Vt•=σnTt−α1dt^−dt^−γ1σnt−γ2sgnσnt−trw˜TΤ−1w^·.
Substituting equation (25) into equation (30)(31)V·≤σnTt−α1wTϕ−ξ−w^Tϕ−γ1σnt−γ2sgnσnt−trw˜TΤ−1α1ΤϕσnT−kwσnΤw^.
It is noted thatσnTtsignσnt=σn in equation (31). Owing to σnTtγ1σnt=σnγ1minσn, γ1min is the minimum eigenvalue of γ1. Equation (31) can be rewritten as follows:(32)V·≤−σnγ1minσn−γ2σn−α1ξσn+α1w˜TϕσnT−trw˜Tα1ϕσnT−kwσnw^V·≤−σnγ1minσn−μσn−α1ξσn+trkwσnw˜Tw^.
The inequality is applied as given in the following equation:(33)trw˜Tw−w˜≤wFw˜F−w˜F2.
We get(34)V·≤−σnγ1minσn+γ2+α1ξ+kww˜Fw˜F−wmV·≤−σnγ1minσn+γ2+α1ξ+kww˜F−wm22−kwwm24,σn>kwwm2/4−γ2−α1ξγ1min,w˜F>wm2+kwwm24−γ2−α1ξ.
As a result,V· (equation (34)) is negative outside of a compact set, demonstrating that it is uniformly ultimately bounded (UUB) of both σn and w˜F.
Harris hawk optimization (HHO) is applied to tune the parameter of SOSMC, which is given in next section.


## 4. Harris Hawk Optimization (HHO)

The Harris hawk (HH) is one of the most intellectual and illustrious birds in nature with unique mass hunting skills in tracking, swirling, rinsing, and capturing potential animals (rabbits) in the group in search of food. The starting population here is supposed to be a crowd of hawks (solving optimization problems) trying to hunt the target rabbit with seven killing strategies from different directions. First, the leader hawk attempts to attack the prey. If the animal cannot be grabbed due to prey dynamism and escape behavior, switching tactics are performed to attack the escaped prey until another group will hit the escaped prey until caught. The key benefit of this tactic is that birds can chase pointed rabbits through the confusion and exhaustion of fleeing prey. Therefore, HHO indicates the exploration and exploitation stages, which are described in Figure2 [33–35].Step 1.
Exploration Phase. The perch of the leader hawks is determined by the location of family members and prey. This is expressed as a mathematical equation for the change in distance between the hawks and the prey, as defined in the following equation:(35)Zj+1=Zrandj−r1Zrandj−2r2Zj,p0≥0.5,Zrabbitj−Zmj−r3LB+r4UB−LB,p0<0.5,where Zj+1 is the updating vector of the Hawk’s position at the j + 1 iteration, Zrj is the position of the prey, Zj is the position vector of the hawks at the jth iteration, r1,r2,r3,r4,andp0 are random numbers between 0 and 1, LB and UB are the lower and upper bounds of variables, and Zraj and Zmj are the initial population assumed randomly.
The average position of each hawk is defined in the following equation:(36)Zi+1j=1N∑i=1NZij,where Zij is the present location of hawks, Zi+1j is the updated location, and the total number of hawks is represented by N.Step 2.
During the exploration phase, the hawks attempt to locate and hit the prey. Due to this, there may be substantial alternate withinside the energy (Ee) of the prey and it is given in the following equation:(37)Escaping Energy,Ee=2Eeo1−t0Γ0,where t0 is the present iteration, Γ0 is the maximum iteration count, and the starting energy (Eeo) varies randomly between −1 and 1 at each iteration. Ee≥1 shows prey’s leaping behavior and the hawks are searching for prey in different location, and Ee<1 suggests that the prey turns into exhausted.Step 3.
Exploitation Phase. At this point, the prey is attacked using a variety of strategies. When the prey is successfully fleeing from hawks, the probability of escaping is depicted as er, and when er<0.5 and er≥0.5, the prey is successfully and unsuccessful escaped, respectively. If the prey escaped when er≥0.5,Ee<0.5 and er≥0.5,Ee≥0.5, then a hard siege and soft siege occurs, respectively.Step 4.
Soft Siege (SS). Here, the rabbit possesses energy and attempts to escape by leaping and the hawks surround it softly, which is shown in the following equation:(38)Zj+1=ΔZj−EIZrabbitj−Zj,ΔZj=Zrabbitj−Zj.
The random jump of the rabbit is given byI=21−r5, where ΔZj is the difference between the position of consecutive iteration and r5 is the number n between 0 and 1.Step 5.
Hard Siege (HS). Here, the prey is completely exhausted and the hawks encircle it hardly. Equation (45) is used to update the positions, as shown in the following equation:(39)Xj+1=Xrabbitj−EeΔXj.Step 6.
Soft Siege with Consecutive Quick Dives. The rabbit still has energy and attempts to flee, which is depicted as Ee≥0.5 and er<0.5; therefore, a soft siege is necessary before the hawks’ surprise pounce. This is a more intelligent step than the one before it. The Levy flight (LF) idea was developed here for progressive quick dives of hawks to execute a soft siege, with the hawks evaluating the prey’s next move using the following equation:(40)Y=Zrabbitj−EIZrabbitj−Zmj.
Despite many efforts, the hawks evaluate each of their motions to the previous dive to decide whether it was a successful dive. It executes an uneven, sudden, and quick dive to reach the prey animal if the dive is not reasonable. We suppose the hawks dive based on LF-based patterns and use the following rule:(41)H=Y+S×LFM,where M is the dimension of the problem and S is the random vector of size 1 × M.
The levy flight function (LF) can be demonstrated, as shown in the following equation:(42)LFz=0.01×μ×ζυ1/δ∴ζ=τ1+δ×sinπδ/2τ1+δ/2×δ×2δ−1/21/δ.Here, μ and υ are accidental values in the range (0, 1) and δ is a constant supposed to be equal to 1.5. As a result, in the soft siege phase, the final updating rule for the hawk’s position is defined in the following equation:(43)Zj+1=YfY<fZj,HfH<fZj,where Y and H are calculated using equations (40) and (41).Step 7.
Hard Siege with Continuous Quick Dives. In this situation Ee<0.5, the prey animal loses energy and becomes fatigued. The hawks then employ a hard siege to close the distance between them and the prey, allowing them to kill the prey. The case updating rule is shown as follows:(44)Zj+1=YfY<fZj,HfH<fZj,(45)Y=Zrabbitj−EIZrabbitj−Zmj,(46)H=Y+S×LFM.
Y in equation (45) and H in equation (46) are the next positions for the new iteration until the prey is killed. The proposed approach is shown (Figure 4) in the form of HHO pseudocode [33].
A regulatory constraint (boundary) optimization problem can be used to describe the proposed frequency regulation problem. The optimum gains of NNO-SOSMC are explored, and by applying HHO, an integral time absolute error (ITAE)-based objective function is defined in equation (47). ITAE-based objective function is chosen for parameter optimization as it concurrently improves both transient and steady-state performance. Moreover, to carry out an unbiased comparative study, an identical objective function as defined in [36] is chosen in this work and defined in equation (47).
Similar objective function defined in [36] has been deliberated in equation (47). The optimized parameters of NNO-SOSMC are R,C,andϒ. Performing an empirical study, the above-said controller parameter boundary ranges are set as α1∈0.01,4, α2∈0.01,4, γ1∈0.01,2andγ2∈0.01,1. To carry out the simulation, maximum iteration and population size are considered 100 and 25, respectively. The control settings are listed in Table 2.(47)J=∫0tΔfi+Δfj+ΔPtieijtdt.
The simulation outputs of the proposed optimization technique (HHO) are compared with other well-known optimization methods, as given in the next section.Figure 4
Pseudocode of proposed HHO.Table 2
Controller design parameter.
α10.2890.6290.3010.1520.020.0630.0310.01880.5100000.1560.1180.1081.0180.0230.9410.1230.0421.23700000000000002.1090.7010.3290.026α21.0231.9410.7430.4140.5660.1810.02871.1881.23700000.5660.1810.0281.1881.0231.9410.74310.4121.23700000000000003.0890.8010.4290.026γ1diag0.10.21.01γ2diag0.120.220.01


## 5. Simulation Results

The control performance of NNO-based SOSMC under the LFC framework has been assessed for two- and four-area interconnected power systems. The robustness of the applied/suggested nonlinear controller against system uncertainties has been evaluated by assuming four different operating scenarios. Initially, the practicability of developed NNO in estimating system disturbance without SMC/SOSMC was studied for test system 1. To perform the simulation, the load perturbations in the areas are assumed toΔPd1=0.08p.u,ΔPd2=0.06p.u.Figure5(a) shows the load disturbance profile applied to area 1. The estimated output of NNO is also (ΔPd1=0.08p.u) plotted on the same figure for better comparison. It is worth noting that NNO efficiently estimates the actual load profile within 0.6 sec, which establish the proficiency of the developed NNO. The frequency deviation (FD) and tie-line power deviation (T-LPD) of studied test system 1 with/without NNO are depicted in Figures 5(b) and 5(c), respectively. The estimation error is given in Figure 5(b).Figure 5
(a) Actual and estimated disturbance. (b) Error in disturbance estimation. (c) FD in area 2. (d) T-LPD.
(a)(b)(c)(d)To measure the effectiveness of the developed control strategy, the system dynamics have been measured considering the following simulation environments:Case 1.
A step load perturbations are applied to the test system 1 in the presence of GRC nonlinearity and an extensive comparative study is performed with HHO and other well-known optimization methods.Case 2.
A multiple load perturbation is applied simultaneously to test system 1 and dynamic performance of system compare with the proposed HHO-NNO-SOSMC and NN-ISMC [26] controller.Case 3.
A multiple load perturbation and wind power perturbation are simultaneously applied to test system 1 and dynamic performance of system is compared with the proposed HHO-NNO-SOSMC and NN-ISMC controller.Case 4.
Lastly, the performance of controller is investigated under the four-area NIPS (test system 2).Case 5.
In this case, two step load perturbationsΔPd1=0.015p.u,andΔPd2=0.01p.u at t = 5sec are applied simultaneously to the test system 1 in the presence of GRC nonlinearity to carry out a fair comparison with the result of reference [26]. The optimized parameters of NNO-SOSMC and minimum objective function value (ITAE) are computed with different optimization techniques for a population size of 25 and 100 iterations, respectively. The convergence characteristic of HHO is obtained and is compared with Figure 6(a). It is noteworthy from Figure 6(a) that HHO attains the global minimum solution quickly and gives the least minimum objective function value compared to SSA, CSA, DE, and PSO.
The dynamic performances of test system 1 using HHO-tuned NN-SOSMC are illustrated in Figures6(b)–6(d). The outputs obtained with PSO based PID, NN-ISMC [26], and SSA/DE/CSA/PSO-tuned NNO-SOSMC are also plotted in Figures 6(b)–6(d) to quantify the effectiveness control methodology. Table 3 lists the time response measurements of Figures 6(b)–6(d).
It is apparent from Table3 and Figures 6(b)–6(d) that HHO-tuned NNO-SOSMC yields better system outputs than SSA, DE, CSA, and PSO-tuned NNO-SOSMC controllers in terms of settling time and peak magnitude of system oscillations. As a result, it is possible to conclude that HHO is capable of obtaining improved near-optimal frequency and power deviations for the investigated test system. In the following investigation, HHO is only considered to optimize the controller parameters. The control effort of the controllers is given in Figure 6(f), where NNO-SOSMC shows less chattering in comparison to NN-ISMC [26].Figure 6
(a) Convergence profile of applied optimization techniques. (b) FD in area 1. (c) FD in area 2. (d) T-LPD. (e) NN outputs. (f) Control inputs.
(a)(b)(c)(d)(e)(f)Table 3
Comparison of transient analysis of NIPS with different optimization techniques.
ControllersΔf1Δf2ΔPtieJSettling time (s)UndershootSettling time (s)UndershootSettling time (s)UndershootNN-ISMC [26]9.6460.01158.2660.00738.620.0036—PSO-PID9.910.01010.520.00310.020.0280.55HHO-NN-SOSMC6.6150.01057.5980.00668.140.0030.02PSO-NN-SOSMC10.2120.01219.8120.006710.120.00320.10DE-NN-SOSMC9.2410.013110.120.00789.510.00330.08CSA-NN-SOSMC10.1210.01429.890.00828.220.00400.42SSA-NN-SOSMC9.4520.01227.6210.00818.510.0030.05Bold faces show best results.Case 6.
In this case, a multiple load perturbation given in Figure7(a) is applied simultaneously to two-area NIPS with GRC and GDB nonlinearities. The dynamic performance NIPS after these perturbations is revealed in Figures 7(b)–7(d). The results show that the suggested HHO-NNO-SOSMC effectively reduces frequency oscillations in this scenario. Furthermore, the given controller provides a response with the least amount of peak overshoot and undershoot, as well as a short settling time. As a result, it is possible to conclude that HHO-NNO-SOSMC is superior to NN-ISMC [26]. The control effort and NN output of the controllers are shown in Figures 7(e) and 7(f), respectively.Figure 7
(a) Load profile. (b) FD in area 1. (c) FD in area 2. (d) T-LPD. (e) Output of NN. (f) Control inputs.
(a)(b)(c)(d)(e)(f)Case 7.
In this case, multiple load perturbation (Figure7(a)) and wind power perturbation (Figure 8(a)) are simultaneously applied to NIPS. The dynamic performance of NIPS after these perturbations with HHO-NNO-SOSMC and NN-ISMC controller is plotted in Figures 8(b)–8(d). The frequency deviation with HHO-NNO-SOSMC is quicker with less undershoot and overshoot than with NN-ISMC, as shown in Figures 8(b)–8(c). The control effort and NN output of the controllers are given in Figures 8(e) and 8(f), respectively.Figure 8
(a) Wind power disturbance pattern. (b) FD in area 1. (c) FD in area 2. (d) T-LPD. (e) Output of NN. (f) Control inputs.
(a)(b)(c)(d)(e)(f)Case 8.
The performance of controller is investigated under the four-area IPS, as shown in Figure9. In this case, multiple load perturbation (Figure 7(a)) in area 1 and area 2 and random wind power perturbation (Figure 8(a)) in area 1 are simultaneously applied to 4-area IPS. The dynamic performance of 4-area NIPS after these perturbations is plotted in Figures 10(a)–10(b). The obtained results of HHO-NN-SOSMC controller are compared reported work [26] to demonstrate the superiority of the HHO-NN-SOSMC controller. It is revealed from Figures 10(a)–10(b) that frequency deviation with HHO-NNO-SOSMC reaches the steady-state level with minimum undershoot and significantly faster as compared to other controllers.Figure 9
Four-area NIPS (test system 2).Figure 10
(a) FD in area-1. (b) FD in area-2. (c) T-LPD 12. (d) T-LPD 31.
(a)(b)(c)


## 6. Conclusion

This study presents the neural network observer (NNO)-based resilient frequency controller for nonlinear interconnected power systems. An NNO is developed to estimate unmodelled system dynamics and exogenous disturbances. An optimized second-order sliding mode control (SOSMC), augmented with the estimated output of NNO, is applied to regulate frequency outputs of the studied power systems against unknown and uncertain plant disturbances. The design parameters of SOSMC are obtained using Harris hawk optimization and compared with other well-known optimization techniques to measure its tuning competence. Simulation results infer that HHO-NNO-SOSMC has minimum integral time absolute error value (0.02) and settling time (6.615 s) compared to other control techniques for Case1. It also attenuates the chattering better than NNO-based integral SMC. In addition, the disturbance rejection ability and robust stability degree of the developed closed-loop control system improve significantly with the proposed controller.In the future, the controller performance will be evaluated considering communication delay in the system. Furthermore, advanced SMC’s variants may be developed to further reduce chattering while maintaining a closed-loop stability.



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