Voltage and frequency regulation in smart grids via a unique Fuzzy PIDD 2 controller optimized by Gradient-Based Optimization algorithm

This paper proposes a maiden intelligent controller design that consists of a Fuzzy Proportional– Integral–Derivative–Double Derivative (FPIDD 2 ) controller whose parameters are fine-tuned using the Gradient-Based Optimization algorithm (GBO). The proposed FPIDD 2 regulator is employed as a secondary regulator for stabilizing the combined voltage and frequency loops in a two-area interconnected power system. It has been shown that the GBO optimization algorithm outperforms other optimization strategies such as the Chimp Optimization Algorithm (ChOA), the Whale Optimization Algorithm (WOA), and the Gorilla Troops Optimization algorithm (GTO). The proposed FPIDD 2 controller is tested in a conventional two-area power system. Then, the investigation is expanded to a two-area hybrid system, with each area comprising a mix of traditional (thermal, gas, and hydraulic power plants) and renewable generation units (wind and solar power). Additionally, the proposed controller takes into account system nonlinearities (such as generation rate limitations, governor deadband, and communication time delays), system uncertainties, and load/renewables fluctuations. In the two tested systems, the dynamic responses of each system demonstrate that FPIDD 2 has a superior ability to attenuate the deviations in voltage and frequency in both areas of the system. In the studied conventional system, the proposed FPIDD 2 controller is compared with a PID controller tuned by the Multi-Objective Non-Linear Threshold Accepting Algorithm (MONLTA), which has been presented in the literature, and a Fuzzy PID (FPID) controller tuned by GBO. In the investigated hybrid system, the suggested FPIDD 2 regulator is compared to a GBO-tuned Integral Derivative-Tilted (ID-T) controller and FPID controller. As a fitness function (FF) for the GBO, the criteria of minimizing the integral time absolute error (ITAE) are applied. The results are presented in the form of MATLAB/SIMULINK time-domain simulations. © 2022TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).


Introduction
The simultaneous management of a synchronous generator's terminal voltage and area frequency can be considered one of the major hurdles that the engineering world encounters in the field of electrical power systems.The degradation of any of these characteristics has a tremendous effect on the life expectancy and performance of other power systems' operational equipment.Small load disruptions are dealt with by controlling devices placed in big complex power systems in order to maintain system voltage and frequency within defined limits.The generating power plants are always equipped with two operational loops in this regard.One of these loops is the load frequency control (LFC) loop, which regulates the frequency by lowering the gap between real power generation and load.The other is the automatic voltage regulator (AVR) loop, which is responsible for managing the system's reactive power and, as a result, the terminal voltage (Kalyan and Rao, 2021a).The combined AVR-LFC systems are required to assist the inter-area power generating systems' dependability, security, and performance.It has been demonstrated that certain interactions between the AVR loop and the LFC loop occur in response to dynamic perturbations (Saadat, 2011).This is because the AVR loops have a direct impact on the magnitude of the power generation voltage (Bingul and Karahan, 2018).Fosha and Elgerd (1970)  The rotor-swept area (m 2 ) β The pitch angle C P The power coefficient of the rotor blades V W The rated wind speed (m/s) r T The rotor radius λ The tip-speed ratio (TSR) λ I The intermittent TSR The output terminal voltage of area-1 (p.u) V out.2 The output terminal voltage of area-2 (p.u) ∆F 1 The frequency deviation in Area 1 (Hz) ∆F 2 The frequency deviation in Area 2 (Hz)

∆P tie
The tie-line power deviation (p.u)K P Proportional gain of the PIDD 2 controller K I Integral gain of the PIDD 2 controller K D1 , K D2 Derivative gains of the PIDD 2 controller N 1 , N 2 The coefficients of the filters K T Tilted gain of the ID-T controller n Fractional component of the ID-T controller K 1 , K 2 Scaling factors of the fuzzy controller inputs et al., 2018;Hakimuddin et al., 2020;Barisal and Mishra, 2018;Rao, 2020)/PIDD 2 (Kalyan and Suresh, 2021) controllers have been extensively used for many types of power systems with multiple areas due to their ease of implementation.In addition, the authors used the Chemical Reaction-based Particle Swarm Optimizer (CRPSO) (Mohanty and Hota, 2018), hybrid Bacterial Foraging-Particle Swarm Optimizer (BFOA-PSO) (Dhillon et al., 2016), Grey Wolf Optimizer (GWO) (Guha et al., 2016), Flower Pollination Algorithm (FPA) (Madasu et al., 2018), Genetic Algorithm (GA) (Hakimuddin et al., 2020), Improved Particle Swarm Optimizer (IPSO) (Barisal and Mishra, 2018), Differential Evolution (DE) (Rao, 2020), and Grey Wolf Optimizer (GWO) (Kalyan and Suresh, 2021).The authors in Mohanty and Hota (2018), Guha et al. (2016) and Madasu et al. (2018) investigated a twoarea system containing hydro-thermal producing units, while the authors in Hakimuddin et al. (2020), Barisal and Mishra (2018), Rao (2020) and Kalyan and Suresh (2021) worked on a multi-area interconnected power system containing also conventional units such as thermal, hydro and gas units.Furthermore, because controllers provide extra optimizing parameters in fractional order (FO), they are gaining popularity, particularly in LFC schemes, and are largely approved by academics (Tungadio and Sun, 2019).Also, from the FOCs family, a tilt integral derivative (TID) controller has recently been employed to overcome LFC difficulties.As a result, various research (Topno and Chanana, 2018;Elmelegi et al., 2021) recommended the TID controller as a solution to LFC difficulties.Ahmed et al. (2022) proposed a modified version of TID called ID-T whose parameters are finetuned by Archimedes optimization algorithm (AOA).The particle swarm optimizer (PSO) (Al-Hinai et al., 2021), Firefly Algorithm combined with Pattern Search (FAPS) (Rajesh et al., 2019), Wild Horse Optimizer (WHO) (Khudhair et al., 2022), Imperialist Competitive Algorithm (ICA) (Yogendra, 2018), and others have been used in previous studies.The authors in the above-mentioned papers focused primarily on system frequency stabilization, focusing solely on the LFC problem and ignoring the AVR coupling.Separately, there is a vast number of studies on LFC and AVR.Researchers are now focusing on doing work with integrated LFC-AVR models; however, they are limited in some ways.By inserting a damper winding into the synchronous generator rotor of the AVR loop, Ref. Gupta et al. (2014) examined a combined LFC-AVR model and studied the controlled response, although the research was restricted to one area.A combined LFC-AVR study of a standalone thermal power unit controlled by a proportional-integral (PI) controller is shown in Ref. Rakhshani et al. (2009). Ref. Vijaya Chandrakala andBalamurugan (2016) applied an LFC-AVR combination study to the multi-regional systems with conventional hydrothermal power plants, all under the assistance of classic PID control based on Simulated Annealing (SA).The authors of Rajbongshi and Saikia (2017a) investigated the application of coupled LFC-AVR model to a system consisting of three areas controlled by an F-based regulator welltuned using the Lightning Search Algorithm (LSA), however only diesel and thermal power plants were inserted in each respective area.The authors in Refs.Lal and Barisal (2019) and Kalyan and Rao (2021b) used a multi-area system involving various generating power plants in their study of LFC-AVR combination, even though the study was confined to the use of traditional PID utilizing the Moth Flame Optimization technique (MFO) (Lal and Barisal, 2019) and the hybrid Differential Evolution-Artificial Electric Field Algorithm (DE-AEFA) (Kalyan and Rao, 2021b), respectively.Authors in Gupta et al. (2016) presented a hybrid controller that utilizes the benefits of neural networks and rapid traversal filters in order to investigate the interaction between the LFC loop and AVR loop in a one-area power system.For a two-area power system, Ref. Shyama et al. (2012) introduces an AVR-LFC solution that relies on a Fuzzy Gain Scheduled Proportional-Integral (FGSPI) structure.This method surpassed a traditional Proportional-Integral (PI) controller in a comparison that employed many performance metrics.When considering convergence stability, implementation simplicity, and computing efficacy, the FGSPI technique outperformed the others.The PID controller parameters of an AVR-LFC combination of a one-area power system are determined using a Particle Swarm Optimization (PSO) approach in Soundarrajan et al. (2010).Refs.Morsali and Esmaeili (2020), Rajbongshi and Saikia (2019), Rajbongshi et al. (2018) and Kalyan et al. (2022) introduces more research inquiries into the connection between the AVR-LFC regulatory methods.Regardless of the diversity of PID control systems, the performance and robustness of such solutions were tested by the uncertainties in the dynamics of the power production units.Some recent research studies have focused on customizing and developing unique optimization processes that can survive large levels of uncertainty in dynamical parameters and consequently obtain superior results (Nahas et al., 2019;Nahas and Nourelfath, 2014;Nahas et al., 2021).In Refs.Nahas and Nourelfath (2014), Nahas et al. (2021), Ekinci and Hekimoğlu (2019) and Ranjan et al. (2021) heuristics such as the Nonlinear Threshold Accepting Algorithm (NLTA), Multi-Objective Nonlinear Threshold Accepting Algorithm (MONLTA), improved kidney-inspired algorithm (IKA), and Mine Blast Algorithm (MBA) have been developed to enhance the settings of Proportional-Integral-Derivative (PID) controller For AVR control systems.Refs.Chatterjee andMukherjee (2016), Ortiz-Quisbert et al. (2018) and Modabbernia et al. (2020) proposes other ways relied on a teaching-learning controller, fractional-order PID regulator, and H∞ control strategy with µ-analysis.For multi-area interconnected power systems, the problem at hand includes interacting loops of AVR-LFC regulators.For individual AVR or LFC, typical solutions are implemented using either analytical or heuristic techniques.These, on the other hand, do not strike a balance between competing objectives (e.g., regulating voltage to the desired level or regulating frequency owing to load changes), nor do they handle the physical interconnectivity between the various power system areas (Kumar et al., 2021).
This inspired the developers of this study to construct a newly smart-based fuzzy PIDD 2 regulator for an LFC-AVR combination model for a dual-area system containing numerous conventional and renewable generating units in order to concurrently stabilize voltage and frequency.Until recently, writers have solely concentrated on the installation of conventional and FO-based controllers in the LFC-AVR combined analysis stream.Because the combined LFC-AVR system is more sophisticated, Standard controllers cannot be used anymore, particularly for strong disturbances.In both the LFC and AVR loops, fuzzy PIDD 2 fine-tuned using the GBO algorithm was used as a secondary regulatory strategy.

Contribution of the paper
This research offers an intelligent-based fuzzy PIDD 2 controller for frequency and voltage stability in various systems (i.e., conventional and hybrid) with significant RESs penetration, considering a variety of load patterns, system uncertainties, and nonlinearities.The major contributions of the work are described as follows concerning the latest research on related threads: (a) The proposal of a maiden robust controller combining the benefits of fuzzy and PIDD 2 controllers (FPIDD 2 ) for simultaneous voltage and frequency stabilization of conventional and hybrid two-area interconnected power systems.(b) The use of the innovative and efficient optimization approach GBO to determine the finest settings of the presented controller.(c) Comparisons to the performance of other, more sophisticated algorithms (such as ChOA Khishe and Mosavi, 2020, WOA Mirjalili and Lewis, 2016, and GTO Abdollahzadeh et al., 2021) are used to prove the GBO's superiority.(d) To analyze the system stability status, multiple problems were taken into account, including the significant penetration of RESs in both areas, various load perturbation patterns, communication time delay, and time-varying desired output voltage.(e) The superiority of fuzzy PIDD 2 was demonstrated through performance comparisons with classic PID (Nahas et al., 2021), ID-T (Ahmed et al., 2022), and intelligent fuzzy PID (Tasnin and Saikia, 2018) controllers.(f) The consideration of many different cases, such as tuning an isolated LFC and AVR systems, and tuning the combined LFC-AVR system.
The following is how the rest of the paper is structured: The architecture of the investigated systems is shown in Section 2. The suggested controller structure and the optimization utilized for tuning its parameters are presented in Section 3, while Section 4 shows the results of the simulation for conventional and hybrid systems with different cases and their discussion.The paper's conclusion is found in Section 5.

Investigated system modeling
The proposed controller is tested on two systems: a basic conventional two-area power system and a hybrid system, which are described in detail in the following subsections.

The configuration of the conventional power system
The power system shown in Fig. 1 consists of two areas with equal generation capacity.Fig. 1 shows in a simple form how the AVR loop is connected to the LFC loop for each area.Controlling the output voltages of the generating units is the responsibility of the AVR loop system, whereas the LFC loop system's focus is on regulating frequency variations brought on by active load disturbances (Gozde and Taplamacioglu, 2011;Gaing, 2004).The variations in the active load that occur in one region are not just reflected in the frequency variations that occur in that region; rather, they also serve as a source of disturbance for the other regions that are linked.In addition, Fig. 1 clarifies how the two areas are connected using a tie-line, which is also the channel via which power is transferred from one area to another.The total coupling of the two-area power system may be thought of as a graphical system, with the synchronization coefficient C12 serving as a representation of the graph weights (Nahas et al., 2021).This synchronization-like system depicts a consensus process for each region, where the states are the frequency-deviation values that reflect the various parts of the power system.The entire synchronization speed is determined by the synchronization coefficient.In the sections that follow, each part of the dynamical structure of the combined AVR-LFC scheme is explained in more detail.

Automatic voltage regulation
The AVR system aim is to minimize reactive power losses caused by voltage mismatches between targeted voltages and the exciter terminal voltage V e .Variations in a generator's reactive power load cause changes in the terminal voltage V g .As shown in Fig. 2, this voltage is detected using a single-phase potential transformer (i.e., voltage V s ) and then compared to a desired reference voltage V Ref .The error signal is amplified (i.e., voltage V a ) and used to control the exciter's field and hence the exciter's terminal voltage.This causes changes in the generator's field current and, as a result, changes in the induced emf (Nahas et al., 2021).Table 1 shows the detailed mathematical modeling of the AVR unit.

Load frequency control
As depicted in Fig. 1, an LFC-based power system includes a rotating mass, governor, turbine, and load demands.The LFC structure's major objective is to split dynamic load changes among the various generators, maintain uniform frequency operating values, and govern tie-line exchange schedules (Hasanien and El-Fergany, 2019).The variations in frequency ∆F for each location reflect changes in the generator's rotor angle ∆δ.The frequency shift is detected, and the error signal is amplified and regulated before being utilized to create a real-power correction ∆P g (Saadat, 2011).The prime mover is commanded to produce a torque variation via the real-power correction.Three basic differential equations are used to describe the LFC dynamical scheme, demonstrating the relationship between the Governor (a physical actuation system), Turbine, and Generator/Load units (Saadat, 2011;Hasanien and El-Fergany, 2019).These equations can be mathematically modeled for a single-area power system as shown in Table 2.The governor's dynamical behavior ∆X g demonstrates that the signal u activates the governor's valve openings when it is accompanied by a negative feedback loop with a gain of 1/R, as seen in Fig. 1.The governor actuation function is based on the LFC control gains and the speed regulation term R, with the linked load frequency deviations serving as inputs.As a result, the frequency fluctuation ∆F caused by load variations within each power system area is regulated by the LFC control signal u, which is the controller's output signal.

Combined AVR-LFC power system
The AVR and LFC systems' modest dynamical coupling allowed for independent control schemes for the voltage and frequency variables in each area.But, the AVR system's activities cause a terminal voltage change, which has a considerable impact on real power generation (Rajbongshi and Saikia, 2017b).As a result, the automatic voltage regulator has an immediate and significant effect on the load frequency control loop.Fig. 2 depicts the AVR loop with coupling coefficients.This coupling scheme describes Governor speed regulation parameters the relationship between minor changes in stator emf on real electric power (K 2 ), small changes in rotor angle on terminal voltage (K 3 ), the effect of tiny rotor angle changes on stator emf (K 4 ), and the effect of small stator emf changes on rotor angle (K 5 ) (Nahas et al., 2021).The values of the coupling coefficients are taken as: 4, and K 5 = 0.5.The nominal value of the synchronization coefficient (C 12 ) between the two areas is given as: C 12 = 0.545.

The configuration of the hybrid power system
In this work, the problem of the combined LFC-AVR in relevance to electrical power grids is discussed by researching dual-area interconnected hybrid power systems.The investigated power grid consists of two interconnected areas in which area.1 has a thermal unit, hydropower unit, gas unit, and PV unit, and area.2 has the same conventional units as the area.1,but instead of the PV unit, a wind unit is inserted into area.2,as presented in Fig. 3.In this system, the conventional units have 2000 MW of rated power for each area, of which the thermal power unit provides 1000 MW, accounting for the majority of the electrical power share, afterwards we have the hydropower unit, which provides 500 MW, and the gas turbine, which supplies 240 MW to the whole output.And, the renewable units have 120 MW of rated power, of which the PV power unit supplies 50 MW, and the wind power unit provides 70 MW with a nominal load of 1740 MW for each area.Refs.Morsali et al. (2018) andSinghParmar et al. (2012) provide more information about the system under study and its parameters.Additionally, the system nonlinearities are taken into account, as are the power system's physical restrictions, such as the GRC and GDB of the thermal power plants, in which the GRC (generation rate constraint) of the hydropower station is 270% p.u/min = (0.045 p.u MW/s) and 360% p.u/min = (0.06 p.u MW/s), respectively for both rising and decreasing rates and the GRC for the thermal unit is set at 10% p.u/min (0.0017 p.u MW/s) for rising and decreasing rates.The transfer functions included in the considered power system have been listed in Table 3, and their configurations are made clear in Table 4.

The setup of PV generation model
Fig. 4 depicts how the Photovoltaic (PV) model may be constructed using the professional software MATLAB/SIMULINK (R2020a).The output power generated by the model is equivalent to the generated output power provided by an actual PV plant.In addition, about 50 MW of the PV model's output power permeates the first area of the examined power system.Here, Utilizing the white-noise block found inside the R2020a version of the MATLAB program allows for the production of random output fluctuations, which are then multiplied by the typical output power of an actual PV plant.The power generated by the proposed PV model may be calculated using Eq. ( 1) (Khamari et al., 2020).Fig. 5 presents the random power output generated by the PV model.

The setup of wind generation model
In order to apply the simplified model of wind generation power in order to share its power in the second area of the power system that is being studied, the expert software MAT-LAB/SIMULINK program (R2020a) is being utilized.The power created by the following wind power model behaves exactly the same as the electricity generated by actual wind farms, hence the model is quite accurate.This is accomplished with the use of a white-noise block, which is employed for making an arbitrary speed form and is then multiplied by the speed of the wind, as seen in Fig. 6 (Elkasem et al., 2021).The irregular output power of 93 wind units is depicted in Fig. 7, where each wind unit produces 0.75 MW of power.The value of the power generated by the wind farm that was investigated is around 70 MW.Following is a set of equations that may be used to describe the output power that was captured from the wind model (Elkasem et al., 2021): (2) In Eq. ( 2), P W denotes the wind turbine output power, ρ denotes the air density in kg/m 3 , A T denotes the swept area by the rotor in m 2 , V W denotes the wind's nominal speed in m/s, and C P denotes the rotor's blade parameter.In Eq. ( 3), C 1 to C 7 denote the parameters of the turbine, β denotes the pitch angle of the blade,

Transient droop compensation
λ T denotes the optimal tip speed ratio (TSR), and λ I denotes the intermittent tip speed ratio.In Eq. ( 4), r T denotes the radius of the rotor.The nominal wind generation coefficients are listed in Table 5.

Gradient-based optimization algorithm
Ahmadianfar et al. ( 2020) proposed the GBO method, which is among the most recent metaheuristic optimization techniques.GBO's operation is made up of two concepts: gradient-based Newton's phenomenon and two trajectories for diversification and intensified search.The Gradient Search Rule (GSR) trajectory facilitates exploration and accelerates convergence to get the optimum fitness in the search space.The Local Escaping Operator (LEO) trajectory, on the other hand, serves the primary purpose of allowing the GBO to flee from the local solution.A graphic representation of the GBO algorithm's operation is presented in Fig. 8.In order to facilitate comprehension, the GBO algorithm's working mechanism has been broken down into the steps listed below.

Initialization
Individually, the GBO creates a main population from a uniform arbitrary distribution.In a population with N trajectories, each agent is given the term "trajectory'', and the population is free to explore a domain with D dimensions.The following mathematical statement represents the initialization procedure.
where X min and X max are the decision variable X boundaries, and rand (0, 1) represents an arbitrary value within the range [0 1].

Gradient search rule (GSR)
During the optimization process, the GSR in the algorithm is in charge of delivering unpredictable behavior that boosts the algorithm's diversion power.Based on the GSR, direction of movement (DM), and beginning location of the search agent, the location of the trajectory (x m n ) may be computed using Eq. ( 7).
where, where m denotes the present iteration number, M represents the maximum number of iterations, β min is 0.2, β max is 1.2, and ε denotes a number between [0, 0.1].ρ 2 may be computed as follows: opensr2, r3, and r4 (r1 and rand(1 : N) is an N-dimensional arbitrary number.The step in Eq. ( 15) denotes the step size that is described by x best and x m r1 .By swapping the site of the best trajectory (x best ) with the current trajectory (x m n ) found in Eq. ( 6), the new trajectory (X 2 m n ) can be described as: where, the new solution at the next iteration (x m+1 n ), established on the locations X 1 m n , X 2 m n , and the current location (X m n ), can be described as:

Local escaping operator (LEO)
Incorporating the LEO into GBO improves the algorithm's ability to resolve difficult optimization issues.Helps the algorithm break out of the ''local optimality'' trap.The LEO employs a number of solutions to arrive at the best solution (X m LEO ), including the best location (x best ), the locations X 1 m n and X 2 m n , two arbitrary locations x m r1 and x m r2 , and a contemporaneous arbitrarily produced location (x m k ).The following methodology is used to construct the answer X m LEO .
if rand < pr if rand < 0.5 where f 1 is an even arbitrary number, f 2 is an arbitrary number in a normal distribution with a zero mean and a unity standard deviation, pr is the probability, and u 1 , u 2 , and u 3 are three arbitrary values, which are given as: where rand denotes an arbitrary value within the domain [0 1], and µ 1 denotes a value within the domain [0 1].We can rewrite the above-mentioned equations as: ) ) where L 1 represents a binary parameter, whose value is 0 or 1.If the value of the parameter µ 1 gets smaller than 0.5, the value of L 1 becomes 1, otherwise, it becomes 0. To describe the solution x m k in Eq. ( 6), The strategy may be expressed mathematically as follows: (29) The best objective functions obtained by the four algorithms.

Test functions
The best objective function obtained 27.13 28.94 25.64 20.14 x 2 i 6.3 × 10 −33 4.6 × 10 −13 6.9 × 10 −49 1.9 × 10 −126  29) can be re-written in a simplified expression as: where L 2 represents a binary parameter, whose value is 0 or 1.If the value of the parameter µ 2 gets smaller than 0.5, the value of L 2 becomes 1, elsewise, it becomes 0. Fig. 9 (Ahmadianfar et al., 2020) provides an overview of the pseudo-code for the GBO algorithm.The flowchart of the approach that was employed to achieve the optimum gains of the suggested FPIDD 2 regulator in this work can be seen in Fig. 10.
Finally, six commonly used benchmark functions from previous research (Khishe and Mosavi, 2020;Abdollahzadeh et al., 2021;Ahmadianfar et al., 2020) are used to comprehensively evaluate the performance of the GBO algorithm.The effectiveness of GBO is evaluated by comparing it with three other metatheuristic algorithms (GTO, ChOA and WHO).30 iterations of each optimization procedure are performed for each test function.A maximum number of 500 iterations was allowed, and the population size was set to 50.Table 6 summarizes the best objective functions obtained by the four algorithms.Fig. 11 shows GBO convergence curves and other algorithms that have been realized in the tested benchmark functions.The results highlight the advantage of GBO over competing algorithms.

Controller structure and suggested solution methodology
The current section discusses the design of the intelligent fuzzy PIDD 2 controller to handle the LFC problem, given that traditional controllers have certain flaws in coping with the uncertainties of the system derived from prior research.The suggested controller mingles the benefits of Fuzzy logic and PIDD 2 controllers, which ultimately allows for superior load frequency regulation.The PIDD 2 controller setup is exactly the same as that of the PID controller, with the addition of a second derivative action to the standard PID components.Fig. 12.a denotes the structure of PIDD 2 regulator.Eq. ( 32) (Kalyan and Suresh, 2021) is a representation of the PIDD 2 controller's transfer function.
where K P , K I , (K D1 , K D2 ) and (N 1 , N 2 ) represent the proportional gain, integral gain, derivative gains, and filter coefficients, respectively, of the PIDD 2 controller.
Additionally, a fuzzy logic controller, also known as an FLC, may be attached to the PIDD 2 regulator in order to enhance both its functionality and its effectiveness.However, the effectiveness of fuzzy logic regulators is highly dependent on the membership functions (MFs) and the convoluted development of a proper fuzzy rule base interface system.Fig. 12.b depicts the configuration of the FPIDD 2 controller used for the combined LFC and AVR investigation where the error input (E) and the change of error (CE) serve as inputs to the fuzzy controller.Scaling factors are represented by the gains K 1 and K 2 .Following is a brief overview of the key procedures involved in deploying an FLC (Cam et al., 2017;Bhateshvar et al., 2017): The first phase is called ''fuzzification'', and it involves the FLC converting E and CE into linguistic variables.For the sake of clarity, simplicity in real-time processing and the need for lowlevel memory (Tasnin and Saikia, 2018;Yakout et al., 2021), the inputs and outputs of the FLCs in this study are all triangular membership functions as shown in Fig. 13.Regarding the inputs and output, there are five linguistic variables employed, including LN, SN, Z, SP, and LP, denoting large negative, small negative, zero, small positive, and large positive.It is clear that the membership functions of both inputs and output are located in the interval [−40, 40].The implementation of the rule base constitutes the second phase of the process.Table 7 displays the results of the FLC's application of fuzzy rules to the linguistic variables derived from the fuzzification process; the FIS utilized in this case is Mamdani (Yakout et al., 2021).It is up to the designer's skill level    to determine the scope and nature of the FLC's underlying rule set.Each system relies on its own set of rules to get the best possible outcome.
Reaching the final phase, which is called defuzzification.The defuzzification procedure takes in linguistic variables as inputs, and the output of the FIS system is itself a linguistic variable.additionally, the defuzzification procedure transforms these variables to crisp variables.In this work, the fuzzy output Control Law (CL) was calculated using the defuzzification procedure's center of gravity approach.The CL signal is passed to PIDD 2 controller.Fig. 14 depicts the control surface that represents the FLC's input-output correlation.It is a tough challenge to determine the optimal input and output values and shapes for FLC.The primary objective of the suggested controller is to lessen frequency deviation (∆F 1 , ∆F 2 ) and lower tie-line power deviations (∆P tie−line ), caused by system uncertainties, in the LFC loop, and improve the dynamic response of the voltage in both areas by lowering the voltage deviations (∆V 1 , ∆V 2 ).This may be accomplished by fine-tuning the FPIDD 2 controller's settings In this study, the integral time absolute error, often known as ITAE, was chosen to serve as the objective function for evaluating the controller's performance.According to Eq. ( 33) (Kalyan, 2021), ITAE is expected to be the method that is most successful in significantly lowering response overshoots and settling time in the combined LFC-AVR problem .
where T sim represents simulation time.The parameters of the controller are regulated as follows: For all following Cases, the lower bounds are set [0, 0, 0, 0, 100, 100, 0, 0] while the higher bounds are set [4, 4, 4, 2, 500, 500,   4, 4].The frequency limits are assumed to be ±1 Hz, the tieline power limits are taken as ±10%, and the voltage limits are assumed to be ±5% (Nahas et al., 2019).The simulation's results and observations will be presented in the next section under a wide variety of different operating circumstances.

Numerical simulation and analysis
In this part of the paper, the dynamic performance of the previously presented power systems (i.e., conventional and hybrid) is evaluated under various cases: (1) Conventional power system studied cases • Tuning the AVR system individually.
• Tuning the LFC system individually.
• Tuning of combined LFC-AVR system.
All the previous cases are studied under nominal and disturbed system parameters.
• Applying communication time delay (CTD) to the controller output.
• Applying random load perturbations (RLP) in area-1 as well as a time-varying desired output voltage.
• Varying the parameters of the system.
RESs penetration is taken into account in all hybrid power system studied cases.
In the conventional system that has been analyzed, the PID controller that has been tuned using MONLTA (Nahas et al., 2021) and the FPID controller that has been tuned using GBO are compared with the newly suggested FPIDD 2 regulator.The suggested FPIDD 2 regulator is evaluated in the context of the examined hybrid system and contrasted with a GBO-tuned ID-T controller and an FPID controller.
In the following subsections, each system's dynamic performance has been studied, and every case has been discussed in detail.The optimization processes with the GBO are evaluated with a maximum iteration of 50 and a number of search agents of 20.

The conventional system cases
Firstly, the superiority of the GBO algorithm is proved by comparing it with other recent algorithms such as ChOA (Khishe and Mosavi, 2020), WOA (Mirjalili and Lewis, 2016) and GTO (Abdollahzadeh et al., 2021).The comparison is carried out on the basis of fine-tuning the proposed controller parameters to enhance the voltage and frequency stability of the combined LFC-AVR twoarea power system under normal case in which the nominal values for the parameters of the combined LFC-AVR are considered in area-1, while a disturbed parameters are considered in area-2.The disturbed AVR system's parameters are T a = 0.1, K a = 10, T e = 0.6, K e = 1.5, T n = 1.5, K n = 1.5, T s = 0.01 and K s = 1, whereas the disturbed LFC system's parameters are K l = 100, T l = 10, T t = 0.15, T g = 0.12, and R = 1.2.An SLP of 2% is applied in both areas.The synchronization coefficient is taken as 0.545.The convergence curve for the four techniques is seen in Fig. 15.By evaluating the efficacy of the GBO method to that of ChOA, WOA, and GTO, it is possible to prove that the GBO algorithm has outstanding convergence over other algorithms.The resultant ITAE from the GBO algorithm is 0.03.

Case I: Tuning the AVR system individually
In this case, under the umbrella of two conditions for the AVR system's parameters (i.e., nominal and disturbed), The performance of the proposed GBO-optimized FPIDD 2 controller is compared to the performance of different controllers, such as the GBO-optimized FPID and the conventional PID controller optimized by the MONLTA technique (Nahas et al., 2021).
(a) Under nominal parameters In this part, Individually, the AVR system is investigated using the nominal parameters listed in Table 1.The step response of the AVR system's output terminal voltage is depicted in Fig. 16, demonstrating the efficiency of the suggested controller over the other comparative controllers.Fig. 16 depicts a quick voltage rise to the reference value with a small overshooting amplitude and settling time.Table 8 shows the settings of the three comparative controllers, while Table 9 shows the dynamic AVR system performance, which clearly shows that the ITAE obtained by the

Table 8
The Optimum settings of the controllers in Case I under nominal case.

Controller
Controller parameters

(b) Under disturbed parameters
Using a slight fluctuation in AVR system parameters around the nominal values, the influence of AVR dynamical system uncertainty on the quality of optimum settings determined by the optimization methodologies is depicted.The disturbed AVR system's parameters are T a = 0.1, K a = 10, T e = 0.6, K e = 1.5, T n = 1.5, K n = 1.5, T s = 0.01 and K s = 1.It is shown in Fig. 17 that the AVR system's output voltage step response is more efficient than the output voltage of the other controllers.As shown in Table 10, the dynamic AVR system performance reveals

Table 11
The Optimum settings of the controllers in Case II under nominal case.

Controller
Controller parameters that the proposed controller's reaction is unaffected by changes in any of the system parameters.The suggested controller's ITAE is nearly identical to the nominal case.

Case II: Tuning the LFC system individually
Like the AVR system, the LFC system has been investigated individually under two different conditions (nominal and disturbed) in order to validate the superiority of the suggested controller (FPIDD 2 ) over the other comparative controllers (i.e., MONLTA-tuned PID and GBO-tuned FPID).
(a) Under nominal parameters The LFC system is studied in this section, considering the nominal parameters indicated in Table 2. Fig. 18 is a representation of the frequency variation caused by 10% SLP injection into the LFC system.This figure depicts the capability of the proposed controller to provide appropriate system stability and considerably lessen the impact of system fluctuation compared to the other controllers.The optimum setups of the three comparative controllers are detailed in Table 11, and the performance of the dynamic LFC system is presented in Table 12.It is abundantly obvious from this table that the recommended regulator achieved the lowest overshoot, undershoot, settling time, and ITAE values.
(b) Under disturbed parameters Herein, a disturbed version of the LFC system is investigated for the purpose of testing the sensitivities of three comparative controllers to changes in system parameters in order to demonstrate the superiority of the controller that has been presented (FPIDD 2 ).The disturbed LFC system's parameters are K l = 100, T l = 10, T t = 0.15, T g = 0.12, and R = 1.2.Fig. 19 demonstrates that the frequency deviation that occurs under the management of the recommended regulator is nearly identical to that which occurs in the normal case.Furthermore, in contrast to other regulators, the ITAE that occurs under the suggested regulator's control is the lowest.The performance of the dynamic LFC system is displayed in Table 13, and it demonstrates that the reaction of the suggested controller is nearly unaffected by the changes in any of the system parameters.There has been no alteration in the maximum overshooting or undershooting, but there has been a tiny rise in the settling time, which has pushed the ITAE to 4.5 × 10 −5 .

Case III: Tuning of combined LFC-AVR system
To confirm the exceptional performance of the recommended FPIDD 2 in the enhancement of voltage and frequency stability, this section makes use of the combined LFC-AVR scheme of the dual-area that was described earlier in Fig. 1 in which the nominal

Table 13
The LFC system dynamics as a consequence of Case II impact under disturbed case.Fig. 20 depicts the combined LFC-AVR system's performance under the effects of the preceding circumstances.The optimum configurations of the three controllers under comparison (PID, FPID, and FPIDD 2 ) are listed in Table 14, and the combined LFC-AVR system dynamics are provided in Table 15, in which the

Table 14
The Optimum settings of the controllers in Case III under the normal scenario.

Controller
Area.1 AVR LFC PID tuned by MONLTA (Nahas et al., 2021) PID tuned by MONLTA (Nahas et al., 2021) 16 summarizes the dynamic performance of the system in terms of ITAE values for output voltages, frequency deviations, and tie-line power.In comparison to the MONLTAtuned PID and the GBO-tuned FPID controllers, the GBO-tuned FPIDD 2 controller has the smallest rising and settling times.The suggested FPIDD 2 regulator reduces the frequency oscillations relatively fast, in addition to improved control quality, despite coping with rapid and gradual load fluctuations.The total ITAE value of the FPIDD 2 regulator tuned using the GBO approach was boosted by 94.79% with the MONLTA-based PID and 67.42% with GBO-based FPID controllers (see Fig. 23).

Case I: Applying an MSLP in area 1 with the consideration of RESs penetration
The examined hybrid system's combined LFC and AVR model was investigated using several regulators such as classic ID-T and intelligent-based fuzzy PID and PIDD 2 in both LFC and AVR loops via applying MSLP, as denoted in Fig. 25, in area-1.Additionally, a PV solar unit with a rating of 50 MW is inserted into area-1 after 250 s, and a wind farm unit with a rating of 70 MW is inserted into area-2 after 100 s.These RESs disturbances are previously presented in Figs. 3 and 5, respectively.The convergence curves of the three examined controllers are exhibited in Fig. 26, demonstrating the superiority of the suggested controller employing the GBO.The responses of the combined LFC-AVR system were analyzed in light of the criteria of maximum overshooting (MO) and undershooting (MU) for the LFC loop, and maximum overshoot magnitude (Mp), rising and settling times (τ r, τ s) for the AVR loop, as shown in Fig. 27.After studying the results in Fig. 27, we came to the conclusion that the FPIDD 2 minimized undershoots and overshoots much better than other controllers.Furthermore, the terminal voltages were swiftly reached with the GBO-tuned FPIDD 2 regulator.Table 17 lists the ID-T/FPID/FPIDD 2 controller parameters that were optimally obtained using the GBO approach.Table 18 shows that the responses approached the steady state faster with the FPIDD 2 controller compared with other controllers.As a result, the intelligent FPIDD 2 demonstrated   its superiority in managing the behavior of the complicated hybrid system of the LFC-AVR combination model.The ITAE value of the FPIDD 2 regulator tuned by the GBO technique was enhanced by 91.66% with the GBO-based ID-T and 56.96% with GBO-based FPID controllers.

Case II: Applying a CTD to the controller output with the consideration of RESs penetration
This case involves an endurance challenge during which RESs have been introduced into both areas of the power system that is being examined.The PV unit is inserted at 80 s while the wind  unit is inserted at 220 s.Additionally, the injection of 1% SLP to area-1 at 10 s and 5% SLP to area-2 at 150 s.Furthermore, to test the efficacy and robustness of the controllers, a 0.1 s CTD is added to the controllers' output.

Table 17
The Optimum settings of the controllers in Case I.

Controller
Area.1 AVR LFC  of the system in this case.Fig. 28 depicts the three controllers' convergence curves.Both the voltage response and the frequency variation of both areas of the power system network that were analyzed are depicted in Fig. 29, together with the flow of power in the tie-line.The behavior of the system has significantly wavered as a result of disruptions caused by RES sources and the use of a communication time delay.On the other hand, the FPIDD 2 regulator that was recommended can achieve enough stability for the system power network and considerably lessen the impact of system fluctuations.After analyzing the data in Fig. 29, we observed that the FPIDD 2 suppressed undershoots and overshoots in frequency deviations significantly better than other controllers.Furthermore, the GBO-tuned FPIDD 2 regulator quickly achieved the reference terminal voltage with a small peak overshoot and settling time.The ITAE index value of the FPIDD 2 regulator tuned by the GBO technique was enhanced by 39.09% with the GBO-based ID-T and 35.28% with the GBO-based FPID controllers.

Case III: Applying an RLP in area 1 as well as a time-varying desired output voltage with the consideration of RESs penetration
In this particular scenario, the researched system dynamics are analyzed in the presence of significant perturbations to corroborate the reliability and dominance of the FPIDD 2 controller that was recommended.To begin, random load perturbations, which are depicted in Fig. 30, are implemented in area-1, which may be portrayed by a series of industrial loads connected to a power system network.Moreover, the penetrations of RESs that are depicted in Figs. 5 and 7 are represented by the connection of the photovoltaic unit to area-1 and the wind unit to area-2 after time intervals of 250 and 100 s respectively.Furthermore, both areas require the time-varying reference voltages indicated in Fig. 31.Fig. 32 displays the system's response to this complex control case using multiple control strategies (i.e., FPIDD 2 , FPID, and ID-T controllers based on the GBO).Table 21 provides a summary of the system's dynamic performance in the form of ITAE values for output voltages and deviations in frequency and tie-line power.Compared to the ID-T and FPID controllers, the suggested FPIDD 2 controller tuned by the GBO technique has the quickest reaction and the greater ability to quickly obtain the target voltage with a smaller steady-state error.And from the perspective of frequency stability, the recommended FPIDD 2 controller dampens oscillations extremely quickly, with the lowest undershooting and overshooting, in addition to superior control quality, while dealing with rapid and gradual load variations.The total ITAE value of the FPIDD 2 regulator tuned by the GBO technique was enhanced by 90.9% with the GBO-based ID-T and 55.4% with GBO-based FPID controllers.

Case IV: Sensitivity analysis with the consideration of RESs penetration
This case examines the FPIDD 2 performance when system parameters are changed by ±50%.The first region had 0.01 p.u step load penetration after 10 s and the second had 0.03 p.u after 150 s.PV solar and wind turbines are interlinked at 80 and 220 s.Table 17 provides the FPIDD 2 parameters for this case.Table 22 summarizes the power system dynamics.Fig. 33 shows the dynamic power system responses to a ±50% setting change.
Based on the dynamical analysis results, it is probably fair to say that the FPIDD 2 regulator is resilient to variations in system model parameters and step load penetration.

Conclusion
Connecting the AVR loop with the LFC through the use of coupling coefficients allows for simultaneous attention to be paid to the stability of the voltage and frequency of the interconnected hybrid power system.In this research, the LFC and AVR loops were controlled by a GBO-tuned FPIDD 2 implemented as a secondary regulator.The dynamical study was widely performed on different two-area systems (conventional and hybrid).For the conventional system, the suggested FPIDD 2 proves to have great performance compared to the MONLTA-based PID and the GBObased FPID controllers under numerous cases (i.e., tuning the

Table 19
The Optimum settings of the controllers in Case II.

Controller
Area.1 AVR LFC    AVR and LFC systems individually or coupled) with nominal and disturbed system parameters.Additionally, the proposed FPIDD 2 regulator has demonstrated greater stability and robustness compared to the GBO-based ID-T/FPID controllers under the impact of different scenarios (i.e., the injection of multiple perturbations such as MSLP, RLP with time-varying desired output voltage, and the application of communication time delay to the controller output, and ±50% system parameters' variations considering RESs penetration for all scenarios) on the hybrid system.The responses of the combined LFC and AVR model demonstrate the superiority of FPIDD 2 over both the traditional ID-T and the intelligent FPID.Future research may investigate the impact of adding electric vehicles and energy storage devices to the hybrid system, and a four-area hybrid power system may be studied.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 3 .
Fig. 3.The investigated dynamic model consisting of a two-area hybrid power system with various sources.

Fig. 11 .
Fig. 11.The convergence curves of the four comparative algorithms achieved in some popular benchmark functions.

Fig. 16 .
Fig. 16.The output voltage step response considering nominal parameters of the AVR system.

Fig. 17 .
Fig. 17.The output voltage step response considering disturbed parameters of the AVR system.

Fig. 18 .
Fig. 18.The frequency deviation considering nominal parameters of the LFC system.

Fig. 19 .
Fig. 19.The frequency deviation considering disturbed parameters of the LFC system.
parameters of AVR and LFC systems are employed in area-1, whereas the disturbed parameters of AVR and LFC systems are applied in area-2.The investigation has been partitioned into two distinct scenarios, which are as follows:(a) Normal scenario In this scenario, unit-step references for the output terminalvoltages are used (V Ref .1 = V Ref .2= 1 p.u)and a 2% SLP is applied in both areas (∆P d1 = ∆P d2 = 0.02 p.u).The synchronization coefficient between the two areas equals 0.545 (C 12 = 0.545).
GBO-optimized FPIDD 2 has accomplished the lowest settling time for voltage and frequency responses and obtained an ITAE of 0.03 with an enhancement of 93.2% compared to the MONLTAoptimized PID controller and 77.78% compared to the GBO-tuned FPID controller.(b)Disturbed scenario This challenging scenario puts the three comparative controllers' resilience to the test since it considers time-varying required output voltages, random load perturbations, and timevarying synchronization coefficient as depicted in Figs.20 to 22. Fig.24demonstrates the combined LFC-AVR system's response to this complicated control situation employing different control methodologies (i.e., FPIDD 2 and FPID controllers based on the GBO, and PID controller based on theMONLTA Nahas et al.,

Fig. 26 .
Fig. 26.The three controllers' convergence curves as a consequence of Case I impact.

Fig. 28 .
Fig. 28.The three controllers' convergence curves as a consequence of Case II impact.
performed groundbreaking research in the area of LFC for power grid networks.Since that day, a large

Table 1
AVR system modeling and setting.
a , T a 10, 0.1 Gains and time constants of amplifier, exciter, generator and sensor.

Table 3
Transfer functions of the system model.

Table 4
The power system settings.
hyd , R g , R Th 2.4, 2.4, 2.4 Hz/MW Governor speed regulation parameters of thermal, hydro, and gas units PF hyd , PF g , PF Th 0.287, 0.138, 0.575 Participation factors of hydro, gas, and thermal units GRC with Hydro -(0.045 p.u MW/s) and (0.06 p.u MW/s.For both rising and decreasing rates), respectively GRC with Thermal -The GRC (generation rate constraint) for the thermal unit is set (0.0017 p.u MW/s) For rising and decreasing rates

Table 5
The wind power plant coefficients.

Table 7
The fuzzy Logic controller's rule base.

Table 10
The AVR system dynamics as a consequence of Case I impact under disturbed case.

Table 16
The combined LFC-AVR system dynamics as a consequence of Case III impact under the disturbed scenario represented by the ITAE index value.

Table 21
The system dynamics as a consequence of Case III impact represented by the ITAE index value.