Eel and Grouper Optimization-Based Fuzzy FOPI-TID^μ-PIDA Controller for Frequency Management of Smart Microgrids Under the Impact of Communication Delays and Cyberattacks

Abstract

In a smart microgrid (SMG) system that deals with unpredictable loads and incorporates fluctuating solar and wind energy, it is crucial to have an efficient method for controlling frequency in order to balance the power between generation and load. In the last decade, cyberattacks have become a growing menace, and SMG systems are commonly targeted by such attacks. This study proposes a framework for the frequency management of an SMG system using an innovative combination of a smart controller (i.e., the Fuzzy Logic Controller (FLC)) with three conventional cascaded controllers, including Fractional-Order PI (FOPI), Tilt Integral Fractional Derivative (TID^μ), and Proportional Integral Derivative Acceleration (PIDA). The recently released Eel and Grouper Optimization (EGO) algorithm is used to fine-tune the parameters of the proposed controller. This algorithm was inspired by how eels and groupers work together and find food in marine ecosystems. The Integral Time Squared Error (ITSE) of the frequency fluctuation (ΔF) around the nominal value is used as an objective function for the optimization process. A diesel engine generator (DEG), renewable sources such as wind turbine generators (WTGs), solar photovoltaics (PVs), and storage components such as flywheel energy storage systems (FESSs) and battery energy storage systems (BESSs) are all included in the SMG system. Additionally, electric vehicles (EVs) are also installed. In the beginning, the supremacy of the adopted EGO over the Gradient-Based Optimizer (GBO) and the Smell Agent Optimizer (SAO) can be witnessed by taking into consideration the optimization process of the recommended regulator’s parameters, in addition to the optimum design of the membership functions of the fuzzy logic controller by each of these distinct algorithms. The subsequent phase showcases the superiority of the proposed EGO-based FFOPI-TID^μ-PIDA structure compared to EGO-based conventional structures like PID and EGO-based intelligent structures such as Fuzzy PID (FPID) and Fuzzy PD-(1 + PI) (FPD-(1 + PI)); this is across diverse symmetry operating conditions and in the presence of various cyberattacks that result in a denial of service (DoS) and signal transmission delays. Based on the simulation results from the MATLAB/Simulink R2024b environment, the presented control methodology improves the dynamics of the SMG system by about 99.6% when compared to the other three control methodologies. The fitness function dropped to 0.00069 for the FFOPI-TID^μ-PIDA controller, which is about 200 times lower than the other controllers that were compared.

1. Introduction

1.1. Background and Motivation

Environmental concerns linked to the use of petroleum and coal in power generation have driven scientists to explore alternative energy sources, such as renewables. One promising solution is the smart microgrid (SMG), which offers enhanced performance when integrating distributed power plants [1,2]. However, renewable energy sources (RESs) introduce frequency control challenges, primarily due to the reduced net inertia in SMG networks [3]. Power imbalances between RESs and loads can lead to frequency instability, compromising the overall stability and reliability of the SMG structure [4]. To address these issues, recent studies have proposed advanced control strategies aimed at balancing generation and demand while ensuring stable power transfer across interconnected systems [5,6].

1.2. Literature Review and Research Gaps

Traditional controllers like PI and PID are structurally simple and effective in many cases but often underperform in complex, nonlinear environments [7]. To overcome these limitations, the TID controller was introduced, replacing the conventional proportional term with a tilting operator. Research shows that this slanted operator, defined by the $S^{-{\displaystyle \frac{1}{n}}}$ transfer function, allows TID controllers to outperform conventional I, PI, and PID controllers in microgrid frequency regulation [8]. Despite these improvements, limitations in traditional integral and derivative actions have prompted the use of fractional-order (FO) operators in control design [9,10]. FO operators offer additional tuning flexibility and an improved dynamic response by increasing the system’s degrees of freedom. Their effectiveness has been demonstrated across various applications, including practical implementations and industrial hardware [11,12]. In power systems, the fractional-order PID (FOPID) controller has been particularly effective in maintaining frequency stability, especially in contexts involving electric vehicle integration [13]. Achieving optimal performance with parallel control architectures remains challenging due to the interdependence of control actions, which affects the output signal and complicates tuning [14]. This can be mitigated through adaptive parameter adjustment. Alternatively, multistage control architectures have proven effective in addressing the drawbacks of parallel configurations and are widely applied in the load frequency control of standalone microgrids [14,15,16,17]. Combining FO operators with multistage architectures offers a promising approach by leveraging the benefits of fractional dynamics and hierarchical structures [17]. However, microgrid frequency regulation remains complex. FO operators introduce additional optimization variables, increasing the dimensionality and complexity of the design. Thus, while more control parameters may enhance flexibility, they do not necessarily ensure better performance and may even cause deviation from the optimal control trajectory.

In addition to the previously discussed controllers, various control strategies have been employed for frequency regulation in power systems. Fuzzy logic controllers have been widely used due to their adaptability across different operating conditions [18,19], though their reliance on expert knowledge and sensitivity to membership functions can be limiting [20]. Combined droop and sliding mode controllers have been explored for primary and secondary control layers [21]. Hybrid storage systems have also been integrated for voltage control and energy management during on/off-grid operation [22]. In [23], networked SMG systems are managed through voltage/frequency control and power sharing, while centralized observer-controlled electric vehicles help stabilize the network and mitigate load variation effects [24]. A cascade PD-(1 + I) regulator has been proposed for frequency control in SMGs using diverse energy storage devices (ESDs) [25], and a fuzzy type-2 controller based on an equilibrium approach was introduced in [26]. An adaptive control method was developed in [27] for managing frequency with solar and microturbine inputs. Efficient control over distributed energy sources reduces operational costs and improves performance [28]. The role of rapid-acting energy storage devices (RAESDs) in frequency regulation is discussed in [29]. Advanced techniques include equilibrium-optimizer-tuned multistage PID controllers [30], wind energy impact analysis [31], tilted PID control in hybrid systems [32], and a multi-stage PDF + (1 + PI) structure optimized using the grasshopper algorithm [33]. Additionally, [34] investigates the role of cascade regulator topologies and ESDs in frequency stability.

The literature presents a variety of mathematical models and intelligent control approaches for frequency management, primarily aimed at minimizing physical disturbances. However, cyber-attacks introduce cyber–physical disruptions that can trigger cascading failures in existing systems [35]. Therefore, it is essential to evaluate controllers under various cyber-attack scenarios. DoS attacks, in particular, demand attention to ensure the reliability and frequency stability of SMGs. This study focuses on a frequency-tolerant regulator capable of operating effectively despite communication delays or signals being denied to distributed energy resources. Recent regulator designs incorporate advanced optimization methods [36,37,38,39]. For instance, a PID controller using a modified Sine–Cosine Algorithm (SCA) was proposed in [40], while [41] introduced a PDF-PI regulator tuned by the Coyote Optimizer (CO). A fuzzy PD-(1 + I) controller tuned via the Mayfly Optimizer (MFO) was developed in [42]. FOPID controllers optimized using the Atom Search Algorithm (ASA) and Gray Wolf Optimizer (GWO) were suggested for hybrid energy systems [43,44]. Additionally, [45] proposed a regulator fine-tuned with the Salp Swarm Algorithm (SSA) for frequency regulation in a redox flow battery-based system.

1.3. Research Objectives and Main Contributions

An examination of the previous research reveals that many alternative methods with various control and optimization approaches have been proposed for the frequency regulation of diverse networks. However, not every system can be adequately addressed by a single strategy. As a result, this gives us the chance to investigate a different plan by establishing a novel controller framework and a more effective optimization method in order to produce a more effective frequency control strategy. Inspired by the superior effectiveness of fuzzy-logic and FO control methodologies, this study discusses frequency management by presenting a unique EGO-enhanced smart FFOPI-TID^μ-PIDA controller, which has not been used in frequency stability before. A thorough examination of the aforementioned literature indicates that several metaheuristic methods are employed in load frequency control research for the optimal adjustment of controller settings. EGO, a recently introduced metaheuristic method for global optimization in 2024 [46], has demonstrated superior efficacy compared to several algorithms, including SSA, SCA, GWO, the Fox algorithm, Multi-Verse Optimization (MVO), the Whale Optimizer (WOA), PSO, the Chimp Optimizer (ChOA), and Fitness-Dependent Optimization (FDO) across nineteen benchmark functions and various engineering challenges, such as in tension/compression springs, pressure vessels, piston levers, and car side impacts [46]; thus, it is employed for the frequency management of the suggested SMG system. The vital achievements of the work are documented as follows:

• This cutting-edge controller, known as fuzzy FOPI-TID^μ-PIDA (FFOPI-TID^μ-PIDA), integrates the characteristics of intelligent control (i.e., FLC) and traditional high-efficiency controllers (i.e., FOPI, TID^μ, and PIDA) within a singular framework for the first time in the frequency management of smart microgrids.
• Moreover, it employs the recently released EGO algorithm to enhance the suggested control structure (FFOPI-TID^μ-PIDA) by meticulously adjusting the settings of the FOPI-TID^μ-PIDA controller, selecting the optimal shape of the membership functions (MFs), and determining the ideal scaling inputs and output factors of the FLC.
• The superiority of the FFOPI-TID^μ-PIDA structure over traditional controllers such as PID and EGO-based intelligent structures like FPID [47] and FPD-(1 + PI) [48] is verified. Also, EGO showed an outstanding performance in a comparison scenario with the Gradient-Based Optimizer (GBO) [49] and Smell Agent Optimizer (SAO) [50].
• The efficacy of the proposed frequency management approach against different cyberattacks, such as DoS and communication delays, is examined.

1.4. Organization of Paper

The next sections of this document are structured as follows: Section 2 delineates the recommended frequency response configuration for the SMG system and its constituent parts. Section 3 delineates the architecture of the regulator and the mathematical description of the optimization problem utilized. Section 4 provides a comprehensive examination of the EGO mathematical structure and its implementation inside the SMG system. Section 5 evaluates the simulation outcomes for system performance over various load fluctuation scenarios and cyberattacks, including a comparative analysis of different controllers and optimization approaches. Section 6 delineates the conclusions of this study. Section 7 illustrates the limitations of our work, along with the future investigations.

2. The Investigated SMG System

The smart microgrid (SMG) system shown in Figure 1 is utilized for the design and study of the recommended controller. Electrical vehicles, aqua electrolyzers, fuel cells, diesel engine generators, wind turbine generators, battery energy storage systems, flywheel energy storage systems, and solar photovoltaics are all components of the SG system. The SMG system uses a single centralized regulator for all of its controlled sources. This leads to a control system that is easier to manage (with fewer control parameters) and requires less maintenance. We implement rate constraints for all configurable sources to enhance the system’s realism. The following subsections illustrate the modelling process of the SMG system studied, as seen in Figure 1. Moreover, all values of the SMG system parameters studied can be found in

Table 1. The studied SMG system parameters.

Unit	Gain (K)	Time Constant (T)
PV	1	1.8
EV	1	1
Wind	1	1.5
Diesel Engine	0.003	2
Aqua Electrolyzer	0.002	0.5
Flywheel Energy Storage	−0.01	0.1
Battery Energy Storage	−0.003	0.1
Fuel Cell	0.01	4

.

2.1. Modeling of the PV Unit

The power generation of the photovoltaic (PV) unit is denoted by the following [48]:

$$P_{PV}=\zeta \cdot A\cdot \psi \left[1-0.005\left(T_a+25\right)\right]$$

(1)

where $\zeta$ denotes the conversion efficiency of the photovoltaic system (assumed to be 10%), $A$ represents the size of the photovoltaic array (assumed to be 4084 m²), and $\psi$ signifies the amount of solar radiation in kW/m². The photovoltaic system transfer function can be described as follows:

$$G_{PV}\left(s\right)={\displaystyle \frac{K_{PV}}{1+{sT}_{PV}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{PV}}{\mathsf{\Delta }\Psi }}$$

(2)

2.2. Modeling of the Wind Unit

The wind turbine is characterized by the nondimensional curves of the power coefficient Cp, which are presented as an expression of the blade pitch angle $\beta =0.1745$ and the tip speed ratio $\lambda$. The curves can be mathematically represented as follows [9]:

$$C_p=(0.44-0.0167\beta )sin\left[{\displaystyle \frac{\pi (\lambda -3)}{15-0.3\beta }}\right]-0.0184(\lambda -3)\beta$$

(3)

In this context, $\lambda$ denotes the ratio of the blade tip velocity of the wind turbine to the wind velocity and can be defined as follows:

$$\lambda ={\displaystyle \frac{R_B\cdot {\omega }_B}{V_{wind}}}$$

(4)

where $R_B=23.5 \mathrm{m}$ denotes the blades’ radius in the wind turbine, ${\omega }_B=3.14 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ represents the speed of rotation and $V_{wind}$ depicts the wind speed. The mechanical power output of the wind unit is expressed as follows:

$$P_{WP}={\displaystyle \frac{1}{2}}\rho \cdot A_r\cdot C_p\cdot V_{wind}^3$$

(5)

where $\rho$ denotes an air density of $1.25 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and $A_r$ represents the blades’ swept area of $1735 {\mathrm{m}}^2$. The transfer function used to simply model the wind unit is as follows:

$$G_{WT{G}_i}\left(s\right)={\displaystyle \frac{K_{wind}}{{1+sT}_{wind}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{WTG}}{\mathsf{\Delta }{P}_{WP}}}$$

(6)

where $i=1,2,3$.

2.3. Modelling of the Aqua Electrolyzer (AE) and Fuel Cells (FCs)

The AE generates hydrogen for the fuel cell by utilizing a portion of the electricity produced from renewable sources such as wind and solar energy. The behavior of the AE is characterized by the transfer function shown in Equation (7) [9], utilizing ($1-K_n$) of the total power from wind and photovoltaic sources to generate hydrogen, which is subsequently employed by two fuel cells to provide supplementary power to the grid.

$$G_{AE}\left(s\right)={\displaystyle \frac{K_{AE}}{{1+sT}_{AE}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{AE}}{(\mathsf{\Delta }{P}_{WTG}+\mathsf{\Delta }{P}_{PV})(1-K_n)}}$$

(7)

where $K_n$ represents a constant with a value of $0.6$. In a similar way, the FC behavior can also mathematically be represented as a first-order transfer function as follows:

$$G_{F{C}_i}\left(s\right)={\displaystyle \frac{K_{FC}}{{1+sT}_{FC}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{F{C}_i}}{\mathsf{\Delta }{P}_{AE}}}$$

(8)

where $i=1, 2$.

2.4. Modelling of the Energy Storage System (ESS)

The control strategy of the SMG system encompasses the Battery Energy Storage System (BESS), Flywheel Energy Storage System (FESS), and Electric Vehicle (EV). They are regulated by the signal derived from the controller output. These units operate as the source or load according to the requirements of the SMG system for frequency regulation. The transfer function of each unit can be expressed as follows [48]:

$$G_{EV}\left(s\right)={\displaystyle \frac{K_{EV}}{{1+sT}_{EV}}}$$

(9)

$$G_{BESS}\left(s\right)={\displaystyle \frac{K_{BESS}}{{1+sT}_{BESS}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{BESS}}{\mathsf{\Delta }U}}$$

(10)

$$G_{FESS}\left(s\right)={\displaystyle \frac{K_{FESS}}{{1+sT}_{FESS}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{FESS}}{\mathsf{\Delta }U}}$$

(11)

The rate constraints nonlinearities that have been considered for the EES are considered as $\left|{\dot{P}}_{EV}\right|<0.01$, $\left|P_{BESS}\right|<0.2$ and $\left|P_{FESS}\right|<0.9$.

2.5. Modelling of the Diesel Generator (DG)

The DG can make up for the shortfall in power and keep the gap between generation and load demand as little as possible. The DG behaviour characteristics can be simply expressed as a first-order transfer function [48]. However, the DG’s nonlinearity has been taken into account by considering $\left|{\dot{P}}_{DG}\right|<0.01$ and $0<\left|P_{DG}\right|<0.45$

$$G_{DG}\left(s\right)={\displaystyle \frac{K_{DG}}{{1+sT}_{DG}}}={\displaystyle \frac{\mathsf{\Delta }{P}_{DG}}{\mathsf{\Delta }U}}$$

(12)

2.6. Modelling of the Power System

Any alteration in the input to a power network causes an associated fluctuation in frequency, which in turn causes power fluctuation. The power system has the following transfer function:

$$G_{PS}\left(s\right)={\displaystyle \frac{1}{D+sM}}={\displaystyle \frac{\mathsf{\Delta }F}{\mathsf{\Delta }P}}$$

(13)

Throughout the current research, $M$ is the equivalent inertia constant of the SMG system, and $D$ is the damping constant. The nominal values for $M$ are $0.4$ and $0.03$ for $D$.

3. The Suggested FFOPI-TID^μ-PIDA Controller Structure and Problem Formulation

PID regulators are usually employed in control systems due to their simple structure, cost-effectiveness, and applicability in linear systems. Nevertheless, these conventional frameworks are typically inadequate for nonlinear systems. Conversely, Fuzzy Logic Control (FLC) is adaptable, straightforward to understand, and facile to implement. The control system efficacy may be enhanced by the use of a fuzzy logic controller-based PID (FPID) controller. An integral action is required in FPID to eradicate the steady-state inaccuracy. However, decelerating the system reaction leads to an increase in the integral gain. To resolve these conflicts, a multistage framework was developed. In light of the aforementioned, a fuzzy FOPI-TID^μ-PIDA, also known as FFOPI-TID^μ-PIDA, was developed to enhance the frequency performance of the examined SMG system, which comprises multiple green sources, in response to abrupt load variations, fluctuations in renewable energy sources, and diverse cyber-attack scenarios. The structure of the proposed FFOPI-TID^μ-PIDA controller can be seen in Figure 2. This section is separated into the following three subsections to explain the configuration of the suggested regulator’s conventional and intelligent parts, along with the mathematical formulation of the studied problem.

3.1. The Optimum FOPI-TID^μ-PIDA Controller (Conventional Part)

This section delineates the arrangement of the conventional part of our recommended controller, comprising three series regulators that have been utilized independently in the literature to address frequency stability concerns. Nevertheless, the integration of these controllers has significantly enhanced their collective capabilities. This configuration achieves three main objectives: a robust capacity to mitigate disturbances, the ability to monitor set points with dynamic sensitivity, and the assurance of stability inside the closed-loop control system. The following subsections highlight the specific structure of each controller.

3.1.1. FOPI Controller

Fractional-order controllers are based on fractional calculus. In recent years, there has been increasing interest in control engineering with the development of the FOPID controller. Fractional calculus encompasses derivatives and integrals of an arbitrary real order. The predominant definitions of the fractional–integral differential operator in the literature were established by Riemann–Liouville and Caputo [51].

Riemann–Liouville Definition: A function’s $a^{th}$ order derivative, $h\left(t\right)$, is defined in the following way according to Riemann-Liouville:

$$\begin{array}{c}{D}_{t_0}^{a}h\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-a)}}{\displaystyle \frac{d^n}{d{t}^n}}\left({\int }_{t_0}^t {\displaystyle \frac{h\left(\tau \right)}{(t-\tau {)}^{1-(n-a)}}}d\tau \right)\\ \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} n-1<a\le n,n\in \mathbb{N}. \end{array}$$

(14)

Caputo definition: A function’s $a^{th}$ order derivative, $h\left(t\right)$, is defined in the following way according to Caputo:

$$D_{t_0}^{t}h(t)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(n-a)}}{\int }_{t_0}^t {\displaystyle \frac{h^n\left(\tau \right)}{(t-\tau {)}^{1-(n-a)}}}d\tau ,n-1<a<n\\ {\displaystyle \frac{d^n}{d{t}^n}}h(t),a=n\end{array}\right.$$

(15)

where $\mathsf{\Gamma }\left(z\right)$ represents the Gamma function. Although fractional calculus is theoretically defined through the Riemann–Liouville and Caputo formulations (as shown in Equations (14) and (15)), these definitions are often difficult to implement directly in control design and simulation due to their non-local, memory-dependent structure. To overcome this, we employ the Oustaloup recursive approximation in the Laplace domain, as shown in Equations (16)–(19) [52]. This method approximates $s^a$, the Laplace transform of a fractional-order derivative, using a rational transfer function that is implementable using standard integer-order system tools. Equation (16) expresses the approximation of $s^a$ as a product of first-order filters. Moreover, Equations (17)–(19) define the approximation parameters over a frequency range $[{\omega }_b,{\omega }_h]$, ensuring a good match in the desired bandwidth. The Laplace transform of the $a^{th}$ derivative is estimated with the following procedure:

$$s^{\mathrm{a}}\approx K\prod _{k=-N}^N {\displaystyle \frac{s+{\omega }_k^{\prime }}{s+{\omega }_k}}$$

(16)

$$K={\omega }_h^{\mathrm{a}}$$

(17)

$${\omega }_k^{\prime }={\omega }_b{\left({\displaystyle \frac{{\omega }_h}{{\omega }_b}}\right)}^{{\displaystyle \frac{k+N+(1-a)/2}{2N+1}}}$$

(18)

$${\omega }_k={\omega }_b{\left({\displaystyle \frac{{\omega }_h}{{\omega }_b}}\right)}^{{\displaystyle \frac{k+N+(1+a)/2}{2N+1}}}$$

(19)

Within the permitted frequency range of $[{\omega }_b,{\omega }_h]$, $N$ denotes the approximation order in this case. It is important to point out that the corelation $s^a=s^n{s}^{\gamma }$ is valid when $a\ge 1$. In this case, $n=a-\gamma$ represents the integer component of $a$, and $s^a$ is determined via the Oustaloup approximation Equation (16). For the mathematical calculation of any fractional terms, the FOMCON toolbox [53] is employed. It is worth noting that the Oustaloup recursive approximation offers several advantages over other approaches. It provides a computationally efficient and numerically stable way to approximate fractional operators over a defined frequency range. It also integrates easily into classical control frameworks and simulation environments such as MATLAB/Simulink. Compared to time-domain numerical methods or the direct implementation of the Riemann–Liouville and Caputo definitions, this frequency-domain approach is more practical for real-time and embedded applications, especially in engineering problems like the issue studied in this paper (LFC). Thus, the use of Laplace-domain approximation via Oustaloup’s method offers a balanced trade-off between theoretical rigor and practical feasibility.

The FOPI controller, a type of fractional-order controller, extends the regular PI controller and bases its operation on fractional calculus. The introduction of an extra parameter $\lambda$, termed the fractional operator, to the integral action of PI augments the design freedom. Figure 3 depicts the arrangement of the FOPI controller. Equation (20) provides the transfer function of the FOPI regulator [54].

$$C_{FOPI}\left(s\right)=KP1+{\displaystyle \frac{KI1}{s^{\lambda }}}$$

(20)

where $KP1$ and $KI1$ represent the proportional and integral action of the FOPI controller, respectively. Moreover, u(t) is the output signal from the FLC, and ΔU₁ is the first control law, which serves as the input to the second part of the FOPI-TID^μ-PIDA controller (i.e., the TID^μ component).

3.1.2. TID^μ Controller

The TID^μ controller is derived by transforming the derivative component of the TID regulator into a fractional form (i.e., adding the fractional operator $\mu$ to the derivative action). The integral component is not transformed into a fractional form. Furthermore, in contrast to the numerous advancements achieved in FOPID in prior research, fewer efforts have been devoted to TID. Despite this, the influence of the tilt compensator in the TID regulator cannot be disregarded. Furthermore, the TID^μ regulator has demonstrated its efficacy in modern frequency management research, as evidenced in Ref. [55]. Figure 4 illustrates the configuration of the TID^μ regulator. Equation (21) delineates the transfer function of the TID^μ regulator [55].

$$C_{{TID}^{\mu }}\left(s\right)=KT{\displaystyle \frac{1}{s^{\frac{1}{n}}}}+KI2{\displaystyle \frac{1}{s}}+KD1{\cdot s}^{\mu }\left({\displaystyle \frac{N1}{N1+s}}\right)$$

(21)

where $KT, KI2, KD1, n$ and $N1$ stand for the tilt, integral and derivative actions, as well as the tilt order and derivative filter coefficient of the TID^μ controller, respectively. Moreover, ΔU₁ is the output signal from the previous part (i.e., the FOPI component), and ΔU₂ is the second control law, which serves as the input to the third part of the FOPI-TID^μ-PIDA controller (i.e., the PIDA component).

3.1.3. PIDA Controller

The PIDA regulator can be considered as an enhancement of the conventional PID regulator, integrating an extra derivative (acceleration) component. This leads to enhanced dynamic reaction and superior disturbance rejection, particularly in systems necessitating precision control [18]. This controller possesses various advantages over PID, including an enhanced performance for extremely fluctuating systems, improved stability in converter-dominated systems, and the incorporation of an acceleration coefficient, which mitigates overshoot and enhances tracking precision. Furthermore, power electronic converters exhibit nonlinearities and rapid dynamics, necessitating superior control beyond PID. The incorporation of renewable energy sources, such as solar and wind, results in frequent load fluctuations and system disruptions. Consequently, PIDA can respond more rapidly than PID and assist in alleviating the rapid transients induced by changes in renewable energy sources [56]. PIDA possesses a greater phase margin than PID, indicating its capacity to endure more substantial communication delays prior to the onset of instability. The structure of PIDA mitigates delay-induced fluctuations, rendering it appropriate for smart grids and linked microgrids [57]. Moreover, the PIDA controller has proven its effectiveness in present-day frequency stability studies, as indicated in Ref. [18]. Figure 5 depicts the architecture of the PIDA controller. Equation (22) specifies the transfer function of the PIDA regulator [18].

$$C_{PIDA}\left(s\right)=KP2+KI3{\displaystyle \frac{1}{s}}+KD2\cdot s\left({\displaystyle \frac{N_2}{N_2+s}}\right)+KA\cdot s^2\left({\displaystyle \frac{1}{(N_2+s)(N_3+s)}}\right)$$

(22)

where $KP2, KI3, KD2, KA, N2$ and $N3$ stand for the proportional, integral, derivative and acceleration actions, as well as the derivatives’ filter coefficients of the PIDA controller, respectively. Moreover, ΔU₂ is the output signal from the previous part (i.e., the TID^μ component), and ΔU is the final (total) control law.

Finally, it is worth noting that the total transfer function of the conventional part of the proposed controller can be mathematically expressed as follows:

$$C_{FOPI-{TID}^{\mu }-PIDA}\left(s\right)=C_{FOPI}\left(s\right)\cdot C_{{TID}^{\mu }}\left(s\right)\cdot C_{PIDA}\left(s\right)$$

(23)

where the input signal to the previous transfer function is u(t) and the output signal is ΔU.

3.2. The Optimum FLC Controller (Intelligent Part)

The intelligent component of the suggested FFOPI-TID^μ-PIDA regulator is the FLC, which has been integrated into the series configuration of the FOPI-TID^μ-PIDA regulator to enhance its capability and performance. The FLC has four primary parts, as seen in Figure 6: the fuzzification interface, the knowledge base, the decision-making unit, and the defuzzification interface. The FLC possesses two input signals: the error (e(t)) and its derivative (de(t)). It also has one output signal (u(t)). More information regarding the FLC components and how they have been manipulated to benefit our study may be found in the following subsections.

3.2.1. Fuzzification Interface

Fuzzification is the process via which a precise value is converted into a fuzzy value by leveraging details from the database. Various types of curves are found in widely employed areas of the fuzzification process, specifically Gaussian, triangular, and trapezoidal membership functions (MFs). In our study, triangular membership functions were chosen for both inputs due to their computational simplicity, ease of implementation, and symmetrical representation of uncertainty. These functions are commonly preferred in control applications where real-time implementation and fast computation are required [18,19]. Additionally, triangular MFs offer a good balance between accuracy and efficiency, making them particularly suitable for systems like load frequency control (LFC), where response speed is essential [31,44]. In this context, the inputs for the fuzzification process consist of the error in the system’s frequency response and its derivative, each multiplied by a scaling factor ($K1$ and $K2$, respectively) prior to their incorporation into the Fuzzy Logic Controller (FLC), to ensure that their values fall within a suitable range for fuzzy inference. Furthermore, the membership functions of inputs range from −50 to 50, which was determined based on the expected dynamic behaviour of the system and empirical tuning to adequately cover the operational range of these signals. Five linguistic variables are employed. The variables are represented by the symbols VN, TN, Z, TP, and VP, which correspond to Vast Negative, Tiny Negative, Zero, Tiny Positive, and Vast Positive, respectively. These terms provide sufficiently fine granularity to describe the state of the system while keeping the rule base manageable in size. To further refine the performance, EGO algorithm was employed to optimize the position and shape parameters of the MFs. Rather than relying solely on heuristics or fixed membership function placements, the EGO algorithm fine-tuned the centers and widths of the MFs to yield an optimal control performance. This optimization ensures that the MFs are not only well-distributed but also adaptively shaped based on the system’s dynamics, thereby enhancing the FLC’s effectiveness. The final, optimized fuzzy sets are shown in Figure 7, which illustrates the resulting MF configurations for both inputs after EGO-based optimization.

3.2.2. Knowledge Base

The knowledge base allows the fuzzification of inputs and delineates the manner in which the inference system will implement the rules to ascertain the fuzzy control output. The knowledge base is made up of two primary parts: the rule base and the data base. The rule base is a collection of IF-THEN rules that explain how input and output variables are related to one another.

Table 2. The employed fuzzy rule base set.

e(t)	de(t)
	VN	TN	Z	TP	VP
VN	VN	VN	TN	TN	Z
TN	VN	TN	TN	Z	TP
Z	TN	TN	Z	TP	TP
TP	TN	Z	TP	TP	VP
VP	Z	TP	TP	VP	VP

shows the rule base set that was employed for this research. The database outlines the membership functions (MFs) for the fuzzy sets that correspond to each variable. It also establishes the shape (triangular, trapezoidal, Gaussian, etc.) and range of the fuzzy sets.

3.2.3. Decision Making-Unit (DMU)

The DMU is in charge of dealing with fuzzy inputs and executing the rule base, which is derived from expert knowledge or empirical data, in order to identify the fuzzy output. It is often referred to as the Inference Engine or the Fuzzy Inference System (FIS). The functions of the DMU can be summarized as follows:

• Employing the Fuzzy Rule Base: Applies established fuzzy IF-THEN rules to ascertain the impact of input fuzzy sets on the output.
• Executing Fuzzy Inference: Employs inference techniques such as Mamdani or Sugeno to obtain fuzzy output values from fuzzy inputs. Our investigation employed a Mamdani fuzzy interface that employs minimum (AND) and maximum (OR) operators for rule evaluation assessment.
• Integrating Fuzzy Rules: Consolidates the outputs of all relevant rules to provide a definitive fuzzy output set.

3.2.4. Defuzzification Interface

Defuzzification is the process of converting the fuzzy output from the DMU into a single crisp value (i.e., a meaningful control action u(t)). This signal serves as the input for the following part (FOPI-TID^μ-PIDA controller) of the presented methodology. Some of the most frequent defuzzification strategies are the center of gravity (COG), mean of maximum, maximal criteria, and weighted average approach. In this case, we employ the COG approach, which calculates the center of mass of the fuzzy output distribution, since it generates a smooth and balanced output. Furthermore, membership functions are crucial in the defuzzification process, as they characterize the interpretation and conversion of fuzzy output into a definitive control action. Similar to the input membership functions, we employed the EGO method to optimize the shapes of the output membership functions and the output scaling factor ($K3$) to attain optimal configurations that produce the most favourable results. The configuration of the output membership functions is the same as the input regarding shape, number, and range. Figure 8 depicts the EGO-based optimum designs of the membership functions for the fuzzy output. Moreover, the surface of the entire control process of the FLC is illustrated in Figure 9.

3.3. The Mathematical Formulation of the Problem

The purpose of the assessment function is critical for identifying errors in any system. It serves as a cognitive channel between the system and the optimization procedure. Minimizing the fitness function (FF) will facilitate the determination of optimal parameters for the FFOPI-TID^μ-PIDA controller using the Eel and Grouper Optimization (EGO) method. Integral Absolute Error, Integral Time Absolute Error, Integral Square Error, and Integral Time Square Error are some of the most well-known fitness functions in the field of frequency stability. These functions are also known as IAE, ITAE, ISE, and ITSE, respectively. All of these functions are important criteria for evaluating errors based on a variety of formulas. In comparison to other standards, the ITSE metrics exhibit high sensitivity, maintain durability, and produce fewer disturbances and overshoots, rendering them valuable criteria for control system development [58]. The below equation represents the ITSE criterion utilized for error computation:

$$FF=ITSE={\int }_0^{T_s}t\cdot ∆F^2 dt$$

(24)

In Equation (24), $\mathsf{\Delta }F$ signifies the system frequency fluctuation from the nominal value and $T_s$ indicates the simulation time interval. The gains of the suggested FFOPI-TID^μ-PIDA regulator are constrained in accordance with Equation (25):

$$\left\{\begin{array}{ccc}{K1}_{LB}\le K1\le {K1}_{UB}& {K2}_{LB}\le K2\le {K2}_{UB}& {K3}_{LB}\le K3\le {K3}_{UB}\\ {KP1}_{LB}\le KP1\le {KP1}_{UB}& {KI1}_{LB}\le KI1\le {KI1}_{UB}& {\lambda }_{LB}\le \lambda \le {\lambda }_{UB}\\ {KT}_{LB}\le KT\le {KT}_{UB}& {KI2}_{LB}\le KI2\le {KI2}_{UB}& {KD1}_{LB}\le KD1\le {KD1}_{UB}\\ {N1}_{LB}\le N1\le {N1}_{UB}& n_{LB}\le n\le n_{UB}& {\mu }_{LB}\le \mu \le {\mu }_{UB}\\ {KP2}_{LB}\le KP2\le {KP2}_{UB}& {KI3}_{LB} \le KI3\le {KI3}_{UB}& {KD2}_{LB}\le KD2\le {KD2}_{UB}\\ {KA}_{LB}\le KA\le {KA}_{UB}& {N2}_{LB}\le N2\le {N2}_{UB}& {N3}_{LB}\le N3\le {N3}_{UB}\end{array}\right.$$

(25)

where $LB$ and $UB$ denote the lower and upper bounds, respectively. The suggested control method has 18 gains to be optimized using the EGO algorithm.

Table 3. The proposed controller parameters’ search space range.

The FFOPI-TIDμ-PIDA Controller	FLC		FOPI		TIDμ		PIDA
	Parameter	Range	Parameter	Range	Parameter	Range	Parameter	Range
	$K1$	0 → 3	$KP1$	0 → 1	$KT$	0 → 5	$KP2$	0 → 5
	$K2$	0 → 3	$KI1$	0 → 5	$KI2$	0 → 5	$KI3$	0 → 2
	$K3$	0 → 3	$\lambda$	0 → 1	$KD1$	0 → 1	$KD2$	0 → 2
					$N1$	100 → 500	$KA$	0 → 2
					$n$	1 → 10	$N2$	100 → 500
					$\mu$	0 → 1	$N3$	100 → 500

delineates the search space range for each parameter during the optimization procedure.

4. Comprehensive Description of the EGO Mathematical Structure

This section introduces the Eel and Grouper Optimization (EGO) algorithm. Then, we give a detailed presentation of the EGO approach together with its mathematical framework.

4.1. Idea and Concept

Groupers (Plectropomus pessuliferus) are large predatory fish that patrol coral reefs in open waters, while moray eels (Gymnothorax javanicus) are nocturnal hunters that weave their way through the complex crevices of the reef. On their own, each is a formidable predator, but when they join forces, their hunting efficiency increases dramatically and the chances of prey escaping drop significantly. If a target seeks shelter within the reef’s nooks and crannies, the moray eel can flush it out—or consume it directly. If the prey flees into open water, it becomes easy pickings for the grouper.

What makes this partnership remarkable is that it represents one of the only known cases—outside of humans—where two unrelated species engage in cooperative hunting. Even more fascinating, groupers have been observed to actively recruit moray eels by performing a distinctive head-shaking signal, inviting them to join the hunt. This behaviour has been likened to a human bringing a hunting dog along to flush out game, highlighting the strategic intelligence and adaptability of these marine predators. The collaboration between groupers and moray eels is a rare and compelling example of interspecies coordination in the wild.

The hunting strategies of these predators varies greatly. The grouper’s hunting style makes victims vulnerable at sea, whereas the moray’s assault mode prevents hiding in holes. For 38 min, two couples maintained a separation that was 1 to 3 times the body length of one grouper [46]. Groupers frequently employ visual cues to engage Moray eels in cooperative search. While swimming, the groupers notify a nearby eel to take it to the prey’s hiding place. The groupers perform a headstand above the prey’s hiding location in the fissure while shaking their heads vigorously. The eel explores the crevices after attaching the grouper. Additionally, hunger correlates with groupers’ signalling rate [46].

According to [59], interspecific participation hunting requires gulping down complete prey, as seen in eels and groupers. Although interspecific predation may enhance mammalian predators’ hunting efficiency, it is seldom observed. Cooperation between eels and groupers falls within the mutualism category of animal interactions. Additionally, hunting is a notable social activity among these fish. Ref. [59] outlines the main processes used by groupers and eels to hunt:

• Grouper tracking target.
• Moray eels are signaled by groupers to hunt together and seek hiding locations.
• Hunger impacts grouper signalling.
• Attack prey together.
• Partnership helps both sides.

Next, we mathematically model the eel and grouper cooperative hunting approach to develop the suggested EGO algorithm.

4.2. Building the Mathematical Model

In this part, the mathematical framework that underpins the recommended EGO approach is detailed; this involves the technique used by groupers to track and locate prey (exploration), communication with moray eels, and then attacking the prey together (exploitation).

4.2.1. Exploration Phase (Grouper’s Tracking Technique for Locating Prey)

The grouper fish optimally forages according to the proximity of its prey. The initial position of the moray eel is arbitrarily selected from the initial population. To enhance exploration capabilities and encourage divergence from local optima, we introduce two control coefficients, $C_1$ and $C_2$. These coefficients are assigned random values that either exceed 1 or fall below −1, intentionally designed to push the search agent away from currently known optimal solutions. This repulsive mechanism serves to expand the search space and reduce the risk of premature convergence. During the exploration phase of the algorithm, instead of exploiting the best-known solution, a randomly chosen search agent is used to guide the update of another agent’s position. This stochastic update mechanism strengthens the global search ability by preventing overfitting to local optima and encouraging a more diverse search trajectory across the solution space. The behaviour of the algorithm in this phase is mathematically described by the following model:

$${\overrightarrow{X}}_i^{t+1}={\overrightarrow{X}}_{rand }+\overrightarrow{C_1}\cdot \left|{\overrightarrow{X}}_i^t-\overrightarrow{C_2}\cdot {\overrightarrow{X}}_{rand }\right|$$

(26)

$${\overrightarrow{XG}}_i^{t+1}={\overrightarrow{X}}_i^{t+1}, if fitness \left({\overrightarrow{X}}_i^{t+1}\right)>{\overrightarrow{XG}}_i^t$$

(27)

$$\overrightarrow{C_1}=2a\times r_1-a$$

(28)

$$\overrightarrow{C_2}=2r_2$$

(29)

where $t$ denotes the present iteration in the $i^{th}$ size and $X_{rand}$ signifies the arbitrary location vector. The optimal candidate from the earlier iteration is denoted by the position of vector $XG$. $C_1$ and $C_2$ represent the coefficient vectors, $r_1$ and $r_2$ generate a randomized value within the range of zero to unity, and · signifies element-wise multiplication. It is essential to acknowledge that if a superior solution is available, $XG$ should be modified in every iteration. The grouper is in motion following its hunt.

The phenomenon of a shrinking encircling action is induced by reducing variable $a$ in Equations (30) and (31). Variable $a$ exhibits a declining pattern as the number of repeats increases. $C_1$ is a stochastic variable within the interval $[-a, a]$, whereas $C_2$ is a stochastic variable within the interval $[0, 2]$, with $a$ diminishing from $2$ to $0$ over the course of iterations. The range of variance for $C_1$ is similarly diminished by variables $a$. According to Equation (29), $C_2$ is a coefficient of $r_2$. In Equation (32), $rand$ represents an arbitrary value within the interval of zero to unity.

$$\begin{gathered} a=2-2\times \left({\displaystyle \frac{t}{Ma{x}_{iteration}}}\right) \\ {r}_1,r_2 is a random number between \mathrm{0,1} \end{gathered}$$

(30)

$$r_3=(a-2)\times r_1+2$$

(31)

$$r_4=100\times rand$$

(32)

4.2.2. Communicating with Moray Eels

Groupers communicate their attacking locations and insights by head bobbing. Groupers, especially those who are famished and have recently lost their prey to a reef break, approach the closest moray eel in its diurnal concealment and perform an unusual head-shaking dance while fluttering its dorsal fin. The position of each eel relative to the grouper is renewed numerically utilizing Equation (33). The degree of grouper hunger influences signaling in this formula. Elevating the degree of starving in grouper fish enhances the likelihood of persuading the moray eel. In Equation (34), the $starvation\_rate$ is a value ranging from zero to one hundred that escalates over time (number of repeats). $r_4$ can further be computed using Equation (32). If the eel is unassociated, its location is going to shift to a new arbitrary point.

$${\overrightarrow{XE}}_i^t=C_2\cdot {\overrightarrow{XG}}_i^t, if r_4\le starvation\_rate$$

(33)

$$starvation\_rate=100\times {\displaystyle \frac{t}{Max\_iteration}}$$

(34)

4.2.3. Exploitation (Attacking the Prey Together)

Moray eels pursue their prey by traversing the reef, whereas groupers use the open waters next to coral reefs to encircle their targets. Researchers have found that when hunting together, these two species are five times more effective than when hunting alone. In 58% of observed cases, moray eels first emerged from their hiding places and then both predators moved together across the reef. The probability of prey being caught by either the grouper or the eel is equal. To simulate this cooperative behaviour, there is a 50% chance that a search agent’s position will be updated using either the grouper or the eel hunting model. The parameter $r_3$ is calculated using Equation (31). The subsequent is the mathematical structure:

$$X_1=e^{b{r}_3}\cdot sin\left(2\pi r_3\right)\cdot {\overrightarrow{C}}_1\left|{\overrightarrow{XE}}_i^t-{\overrightarrow{XP}}_i^t\right|+{\overrightarrow{XE}}_i^t$$

(35)

$$X_2={\overrightarrow{XG}}_i^t+{\overrightarrow{C}}_1\cdot \left|{\overrightarrow{XG}}_i^t-{\overrightarrow{XP}}_i^t\right|$$

(36)

$${\overrightarrow{X}}_i^{t+1}=\left\{\begin{array}{c}{\displaystyle \frac{0.8X_1+0.2X_2}{2}}, if p<0.5\\ {\displaystyle \frac{0.2X_1+0.8X_2}{2}}, if p\ge 0.5\end{array}\right.$$

(37)

As shown in Figure 10, the undulating update localization method begins by measuring the distance between the moray eel at coordinates $(XE,YE)$ and the prey at $(XP,YP)$. A rotational equation is then applied to simulate their positions, capturing the natural, wave-like undulating movement of the eel as it approaches its target. This motion is modeled mathematically in Equation (35). Due to the periodic characteristics of the sine function used in this model, the algorithm allows a smooth and continuous transition between successive positions, ensuring that the space between them is thoroughly explored and no potential solution is overlooked.

In contrast, groupers detect the location of prey and immediately move toward it. When the exact position of the target within the solution space is unknown, the EGO algorithm assumes that the best-known solution so far is either optimal or close to the global optimum. Based on this assumption, all groupers update their positions to converge toward this most promising solution. This directed movement behaviour is captured in Equation (36), reinforcing exploitation in the search process.

Finally, as presented in Equation (37), the algorithm preserves the two best solutions—one from the moray eel and one from the grouper—and guides the remaining search agents to relocate based on the average of these top-performing solutions. The influence of each agent is controlled through empirically derived weighting factors, designed to reflect the natural balance observed in cooperative hunting behaviour. When the probability $p<0.5$, greater significance is assigned to the moray eel’s position $X_1$, with an influence factor of $0.8$. Conversely, if $p\ge 0.5$, the grouper’s position $X_2$ takes precedence with the same weight, directing the movement of the rest of the agents. This dynamic adjustment helps maintain a balance between exploration and exploitation throughout the optimization process.

In a three-dimensional space, Figure 11 illustrates the rationale behind Equation (26). The position of the searching agent $(\overrightarrow{XG},\overrightarrow{YG},\overrightarrow{ZG})$ can be modified according to the coordinates of the optimal attainable outcome $(\overrightarrow{XP},\overrightarrow{YP},\overrightarrow{ZP})$. By utilizing the values of the $\overrightarrow{C_1}$ and $\overrightarrow{C_2}$ vectors, many positions surrounding the most efficient agent may be reached from the present place. It is important to observe that the arbitrary vectors $\overrightarrow{r_1}$, $\overrightarrow{r_2}$, and $\overrightarrow{r_3}$ can reach any location inside the search region delineated by the key points illustrated in Figure 11. Equation (26) allows any particle to adjust its position near the current optimal solution, emulating prey pursuit in a three-dimensional space.

An identical concept may be applied in an n-dimensional solution space, where search agents navigate within hypercubes surrounding the current optimal solution. As discussed in the previous section, the grouper and the moray eel work in coordination to pursue prey, reflecting a unique form of cooperative hunting. While it is theoretically possible to incorporate crossover operations and other evolutionary techniques into the EGO algorithm to more precisely simulate the natural behaviors of these two predators, our approach deliberately avoids adding such complexities. Instead, we aim to minimize the number of heuristics and reduce the dependency on multiple input parameters. This results in a more streamlined and computationally efficient version of the EGO algorithm, without compromising its core exploratory and exploitative capabilities.

Figure 12 demonstrates how Equation (35) regulates the amplitude of the sine function through successive iterations. As the optimization process advances, the range of the sine wave becomes progressively narrower, effectively refining the search space over time. The figure illustrates that the EGO algorithm performs a fine-tuned local search when the sine function values remain within the standard range of $[-1, 1]$, allowing for precise exploitation near potential optima. Conversely, when the sine values extend into broader ranges such as $[-2,-1]$ and $[1, 2]$, the algorithm shifts toward a more exploratory mode, probing less-visited regions of the solution space.

Figure 13 and Equation (37) illustrate the positional dynamics of the search agents within the solution space, which are influenced by the estimated location of the prey. The optimal position is assumed to lie randomly within a circular region defined by the positions of the moray eel and the grouper, representing the area of potential convergence. In one scenario, the grouper actively estimates the prey’s location and moves toward it in a direct manner, while the eel randomly adjusts its position within the vicinity of the grouper. Through this cooperative behaviour, both agents approach the prey using different yet complementary strategies.

Based on these interactions, the EGO algorithm can now be formulated using the mathematical models introduced in the previous subsections. These models define the behaviour and movement rules of the search agents. Figure 14 presents the overall flowchart of the EGO algorithm, outlining each step from initialization to convergence.

5. Results and Discussions

This section presents a comprehensive evaluation of the SMG control framework under a wide range of operational scenarios. These scenarios encompass various load perturbations (including single-step and multi-step disturbances), variability in renewable energy input, and cyber threats such as Denial-of-Service (DoS) attacks and communication latency issues. The proposed control strategy, based on EGO-tuned FFOPI-TID^μ-PIDA, is assessed through a two-phase analysis.

In the first phase, the optimization proficiency of the EGO algorithm is critically examined by benchmarking it against two alternative metaheuristic techniques (i.e., GBO and SAO). This comparison focuses on the parameter tuning of the proposed controller as well as the optimum design of the membership functions of the fuzzy logic controller by each of these distinct algorithms.

The second phase involves a comparative performance analysis of the proposed EGO-based FFOPI-TID^μ-PIDA controller against both traditional and intelligent control structures, all optimized via EGO. These include the classical PID controller, as well as more advanced fuzzy-based schemes such as FPID and FPD-(1 + PI). All simulation experiments were carried out in the MATLAB/Simulink R2024b environment using the ODE45 solver with a variable time step, a maximum step size of $1\times {10}^{-4}$, and executed under in normal simulation mode.

5.1. Assessments of Different Optimizations Used for Tuning the Proposed Control

Initially, the efficacy of the EGO optimization technique is determined by a series of tuning trials including several optimization techniques. In order to ascertain the superiority of the EGO optimization strategy, it is compared to contemporary methods such as SAO and GBO. In order to ensure a fair comparison, all the strategies studied employed the same optimization process settings, which included 100 iterations, 30 search agents, and 30 independent runs. The numerical values the EGO parameters settings used in this study are illustrated in

Table 4. The parameter setting of the suggested EGO.

Parameter	Symbol	Value
Population size	$N$	30
Maximum iterations number	$Ma{x}_{iteration}$	100
arbitrary vectors	$r_1, r_2=[0:1]$	30
Coefficient vector	$C_2=[0:2]$	1000
Grouper attraction coefficient	$\beta$	1.6
Damping ratio	$\lambda$	0.9

. The optimal gains for the proposed controller, achieved through three different methods, are summarized in

Table 5. FFOPI-TID^μ-PIDA optimum parameters using different optimization techniques.

Optimization	Controller Part	Parameters	Value	ITSE
SAO		$K1$	2.368	0.368
		$K2$	0.214
		$K3$	1.452
	FOPI	$KP1$	0.562
		$KI1$	1.356
		$\lambda$	0.893
	TIDμ	$KT$	2.579
		$KI2$	0.214
		$KD1$	0.12
		$N1$	356
		$n$	1.436
		$\mu$	0.769
	PIDA	$KP2$	4.325
		$KI3$	1.528
		$KD2$	0.115
		$KA$	1.658
		$N2$	429
		$N3$	500
GBO	FLC	$K1$	0.986	0.0045
		$K2$	1.121
		$K3$	1.432
	FOPI	$KP1$	0.234
		$KI1$	3.769
		$\lambda$	0.653
	TIDμ	$KT$	3.338
		$KI2$	4.682
		$KD1$	0.085
		$N1$	467
		$n$	8.671
		$\mu$	0.293
	PIDA	$KP2$	1.864
		$KI3$	0.259
		$KD2$	1.245
		$KA$	0.368
		$N2$	500
		$N3$	243
EGO	FLC	$K1$	1.789	0.00069
		$K2$	0.593
		$K3$	0.864
	FOPI	$KP1$	0.379
		$KI1$	2.722
		$\lambda$	0.984
	TIDμ	$KT$	1.561
		$KI2$	3.158
		$KD1$	0.022
		$N1$	482
		$n$	9.727
		$\mu$	0.253
	PIDA	$KP2$	2.579
		$KI3$	0.122
		$KD2$	0.011
		$KA$	0.116
		$N2$	500
		$N3$	479

. This study compares the convergence behaviour of EGO, SAO, and GBO, as depicted in Figure 15. Figure 15 clearly indicates that EGO exhibits a superior convergence performance in comparison to SAO and GBO, hence validating its greater optimization capabilities. EGO achieves the minimum fitness function, namely the Integral Time Squared Error (ITSE), with a value of 0.00069. EGO achieves a 99.8% improvement in the ITSE index compared to GBO, and an 84.7% improvement compared to SAO.

5.2. Assessments of Different Controllers Used to Enhance System Performance

In this part and after the demonstration of the efficiency of EGO in the previous subsection, EGO is applied to optimize the settings of all compared controllers (i.e., PID, FPID, FPD-(1 + PI), and FFOPI-TID^μ-PIDA). The optimum settings of these controllers’ coefficients are provided in

Table 6. Optimum controllers’ parameters using EGO optimization technique.

Controller									ITSE
PID									1.041
$KP$			$KI$			$KD$
19.976			19.515			0.516
FPID									0.461
$K1$	$K2$		$KP$		$KI$		$KD$
1.996	0.011		15.731		15.951		3.978
FPD-(1 + PI)									0.1616
$K1$	$K2$		$KP1$	$KD$		$KP2$	$KI$
1.987	0.196		3.059	0.078		1.248	1.208
FFOPI-TIDμ-PIDA (Proposed)									0.00069
$K1$	$K2$	$K3$	$KP1$	$KI1$	$\lambda$	$KT$	$KI2$	$KD1$
1.789	0.593	0.864	0.379	2.722	0.984	1.561	3.158	0.022
$N1$	$n$	$\mu$	$KP2$	$KI3$	$KD2$	$KA$	$N2$	$N3$
482	9.727	0.253	2.579	0.122	0.011	0.116	500	479

. We conducted this test, exposing the system to 0.1 pu of step load variation (ΔPL) and 0.15 pu of constant renewable powers (ΔRPs), to demonstrate the outstanding performance of the proposed FFOPI-TID^μ-PIDA controller. Figure 16 illustrates the system’s frequency response in this particular situation, in which one can clearly see that FFOPI-TID^μ-PIDA outperforms PID, FPID, and FPD-(1 + PI). Furthermore, the control output signals corresponding to the different controllers are illustrated in Figure 17.

Table 7. Characteristics of the frequency deviation response of the system using different controllers.

Controller	Max.OS (Hz)	Max.US (Hz)	Set-Time (Sec.)
PID	0.354	0.04853	22.01
FPID	0.2145	0.04767	14.1
FPD-(1 + PI)	0.1532	0.04672	13.8
FFOPI-TIDμ-PIDA	0.0221	0.0382	6.314

presents the key attributes of the system’s frequency response that were examined when employing several controllers, including Max.OS, Max.US, and Set-Time. The FFOPI-TID^μ-PIDA controller obtains optimal values for Max.OS, Max.US, and Set-Time, namely 0.0221 Hz, 0.0382 Hz, and 6.314 Sec, respectively. In Table 6, the EGO-based FFOPI-TID^μ-PIDA controller shows the smallest fitness function (0.00069) compared to EGO-based PID, FPID, and FPD-(1 + PI) controllers, highlighting the effectiveness of FFOPI-TID^μ-PIDA in this design. The dynamic system responses using the EGO-tuned PID, FPID, FPD-(1 + PI), and FFOPI-TID^μ-PIDA controllers show that FFOPI-TID^μ-PIDA Max.OS is 7 times better than FPD-(1 + PI), 10 times better than FPID, and 16 times better than PID. The subsequent subsections present additional scenarios in which fluctuations in load and power from renewable sources occur, utilizing the optimal parameters of the controllers derived in this specific case (Table 6).

5.2.1. Case 1: Load Variation with Constant Renewable Powers

In this scenario, only load variation (ΔPL) is considered, as shown in Figure 18, while renewable powers (ΔRPs) remain constant. The dynamic system responses using the EGO-tuned PID, FPID, FPD-(1 + PI), and FFOPI-TID^μ-PIDA controllers are presented in Figure 19. One can note from Figure 19 that the performance of the FFOPI-TID^μ-PIDA regulator surpasses that of the PID, FPID, and FPD-(1 + PI) regulators. The power provided by controllable plants under the EGO-tuned FFOPI-TID^μ-PIDA controller during the initial 30 s, when the power imbalance is positive (ΔRPs > ΔPL), is absorbed by controllable sources to reduce the imbalance. Between 30 and 60 s, when the power generation is insufficient to meet the load demand, these energy sources step in to supply the deficit. During this interval, these sources inject additional power into the grid to maintain stability and ensure a continuous supply of electricity. Finally, between 60 and 90 s, when the power imbalance returns to zero, controllable sources neither supply nor absorb power.

Table 8. Characteristics of the frequency deviation response of the system using different controllers under the impact of Case 1.

Controller	Max.OS (Hz)	Max.US (Hz)	Set-Time (Sec.)	St.St. Error (Hz)	ITSE
PID	0.3472	0.07958	22	0.0006	1.919
FPID	0.2111	0.07837	13.22	0.0003	0.612
FPD-(1 + PI)	0.1644	0.07667	11.4	0.0003	0.211
FFOPI-TIDμ-PIDA	0.1533	0.05997	7.117	0.0001	0.019

presents the key attributes of the system’s frequency response that were examined when employing several controllers, including Max.OS, Max.US, Set-Time, steady-state error (St.St. Error) and ITSE for Case 1. The FFOPI-TID^μ-PIDA controller obtains optimal values for Max.OS, Max.US, St.St. Error and ITSE, namely 0.1533 Hz, 0.05997 Hz, 7.117 Sec., 0.0001 Hz, and 0.019, respectively.

5.2.2. Case 2: Variations in Solar Irradiance, Wind Speed and Load

For this situation, we take into account the fluctuations in load (ΔPL), solar power (ΔPV), and wind power (ΔPW), as depicted in Figure 20. Figure 21 displays the system frequency responses obtained from the EGO-tuned PID, FPID, FPD-(1 + PI), and FFOPI-TID^μ-PIDA controllers. The FFOPI-TID^μ-PIDA regulator shows a superior performance compared to the PID, FPID, and FPD-(1 + PI) controllers in terms of system performance metrics, including Max.OS, Max.US, and Set-Time. When the amount of renewable power generation (ΔRPs) is greater than the amount of load demand (ΔPL), controlled sources absorb the excess power. Conversely, when the load demand is greater than the renewable generation, these sources send energy to the system in order to maintain balance.

Table 9. Characteristics of the frequency deviation response of the system using different controllers under the impact of Case 2.

Controller	Max.OS (Hz)	Max.US (Hz)	Set-Time (Sec.)	St.St. Error (Hz)	ITSE
PID	0.6481	0.3146	20.71	0.15	23
FPID	0.4515	0.306	15.98	0.1	10.15
FPD-(1 + PI)	0.5307	0.3012	14.41	0.07	6.168
FFOPI-TIDμ-PIDA	0.3913	0.07728	10.76	0.001	0.498

displays the vital features of the system’s frequency response that were analyzed while using various controllers, such as Max.OS, Max.US, Set-Time, St. St. Error and ITSE for Case 2. The FFOPI-TID^μ-PIDA controller establishes that the best values for Max.OS, Max.US, Set-Time, St. St. Error and ITSE are 0.3913 Hz, 0.07728 Hz, 10.76 Sec., 0.001 Hz, and 0.498, respectively. When comparing FFOPI-TID^μ-PIDA to FPD-(1 + PI), FPID, and PID, the frequency performance is enhanced in terms of Max.US by 74.3%, 74.7%, and 75.4%, respectively. Additionally, in terms of Set-Time, there is an improvement of 25.3%, 32.8%, and 48%, respectively.

5.2.3. Case 3: Various Cyber-Attack Situations

In this scenario, the system’s response is evaluated under various cyberattack circumstances, such as Denial of Service (DoS) and data transmission delays. These cyberattacks can significantly impact the communication between distributed energy resources and the controller, leading to disturbances in frequency regulation. The system’s dynamic performance with EGO-tuned PID, FPID, FPD-(1 + PI), and FFOPI-TID^μ-PIDA controllers is analyzed under these cyberattack conditions. The outcomes demonstrate that the FFOPI-TID^μ-PIDA regulator efficiency is higher than the PID, FPID, and FPD-(1 + PI) regulators, providing better resilience to the effects of cyberattacks. The FFOPI-TID^μ-PIDA controller maintains system stability and ensures reliable frequency control even when subjected to signal disruptions and delays. Various performance metrics, such as integral errors, maximum overshoot, and maximum undershoot, indicate that the FFOPI-TID^μ-PIDA controller achieves a better overall performance, when contrasted to the PID, FPID, and FPD-(1 + PI) controllers during cyberattack scenarios. This proves the resilience of the FFOPI-TID^μ-PIDA regulator in maintaining grid stability despite communication challenges.

(A)

Case 3A: DoS Attack Against Electric Vehicle (EV)

In the present scenario, the electric vehicle unit is entirely withdrawn from the system, but the other components remain unchanged. Figure 22 provides a clear illustration of the outstanding performance of the FFOPI-TID^μ-PIDA controller. An analysis of multiple measures, including undershoot, indicates that the FFOPI-TID^μ-PIDA controller surpasses the PID, FPID, and FPD-(1 + PI) regulators. The recommended controller attains a frequency deviation of 0.0921 Hz, while the PID, FPID, and FPD-(1 + PI) controllers yield deviations of 0.36 Hz, 0.353 Hz, and 0.345 Hz, respectively. FFOPI-TID^μ-PIDA stabilizes within 8 s, exhibiting oscillations around zero with a range of −0.005 to 0.005. In contrast, PID, FPID, and FPD-(1 + PI) oscillate around zero with ranges of −0.2 to 0.16, −0.13 to 0.101, and −0.07 to 0.07, respectively.

(B)

Case 3B: DoS Attack Against EV and Renewable Units

In this specific situation, the electric vehicle (EV) unit and the renewable units (solar and wind) are totally taken out of the system, but the other parts of the system are kept as they are. Figure 23 provides an obvious visualization of the outstanding performance of the FFOPI-TID^μ-PIDA regulator. The FFOPI-TID^μ-PIDA regulator performs better than the PID, FPID, and FPD-(1 + PI) controllers, according to the outcomes of several metrics, including overshoot. The proposed controller obtains a deviation in the frequency response of 0.152 Hz, while PID, FPID, and FPD-(1 + PI) achieve shifts in the frequency of 0.295 Hz, 0.183 Hz, and 0.176 Hz, respectively. In contrast to PID, which has a settling time of 17 s, FPID, which has a settling time of 12 s, and FPD-(1 + PI), which has a settling time of 10 s, the FFOPI-TID^μ-PIDA demonstrates the most promising settling time of 6 s.

(C)

Case 3C: DoS Attack Against EV and FESS Units

In this scenario, the Electric Vehicle (EV) and Flywheel Energy Storage System (FESS) are completely removed from the system, while all other components remain unchanged. The superior performance of the FFOPI-TID^μ-PIDA controller is clearly demonstrated in Figure 24. A comparative analysis of various performance metrics, such as undershoot, highlights the efficacy of the presented regulator in relation to the other control strategies, including PID, FPID, and FPD-(1 + PI). Specifically, the FFOPI-TID^μ-PIDA controller achieves an undershoot value of 0.101 Hz, significantly lower than the values obtained with the PID (0.359 Hz), FPID (0.351 Hz), and FPD-(1 + PI) (0.345 Hz) controllers. Furthermore, the FFOPI-TID^μ-PIDA controller reaches a steady-state condition within 7 s, exhibiting minimal oscillations around zero, with an oscillation range of −0.0041 to 0.0045. In contrast, the PID, FPID, and FPD-(1 + PI) controllers exhibit more pronounced oscillations around zero, with respective ranges of −0.251 to 0.21, −0.16 to 0.12, and −0.091 to 0.092. These results underscore the enhanced stability and dynamics of the presented FFOPI-TID^μ-PIDA regulator compared to conventional control approaches.

(D)

Case 3D: Cyberattacks on Communication Infrastructure and DoS Attacks Targeting the EV

In this specific scenario, the EV unit is entirely removed from the system, while all other components remain unchanged. Additionally, a modification is introduced in the signal transmission latency, increasing it from 10 ms to 20 ms. Despite these changes, the exceptional performance of the FFOPI-TID^μ-PIDA controller remains evident, as illustrated in Figure 25. A detailed comparative analysis reveals that the suggested regulator outperforms the other compared regulators, including PID, FPID, and FPD-(1 + PI), across multiple performance metrics, with a particular emphasis on undershoot. The FFOPI-TIDμ-PIDA controller demonstrates a significantly lower undershoot value of 0.0584 Hz, whereas the PID, FPID, and FPD-(1 + PI) controllers exhibit higher undershoot values of 0.361 Hz, 0.354 Hz, and 0.347 Hz, respectively. This substantial reduction in undershoot highlights the superior disturbance rejection capability of the proposed controller. Moreover, the FFOPI-TID^μ-PIDA controller reaches a steady-state condition within 9 s, further emphasizing its rapid response characteristics. While the controller maintains oscillations around zero, these oscillations are minimal, with a narrow range of −0.0098 to 0.0073. In contrast, the PID, FPID, and FPD-(1 + PI) controllers exhibit significantly larger oscillation ranges of −0.199 to 0.16, −0.14 to 0.101, and −0.082 to 0.077, respectively.

These findings confirm that the FFOPI-TID^μ-PIDA controller not only enhances the transient response by reducing the overshoot and settling time but also improves the steady-state performance by minimizing oscillatory behaviour. The ability of the proposed controller to maintain stability and robustness under increased signal transmission latency further reinforces its effectiveness in dynamic and complex system conditions. Consequently, the FFOPI-TID^μ-PIDA controller emerges as a highly reliable solution for improving system performance compared to existing control strategies.

A summary of the frequency response characteristics for the various cyberattack scenarios examined in Case 3 is presented in Figure 26. The comparative analysis highlights the superior performance of the proposed FFOPI-TID^μ-PIDA controller compared to the other control strategies, including PID, FPID, and FPD-(1 + PI). Under different cyberattack conditions, the FFOPI-TID^μ-PIDA controller consistently demonstrates enhanced robustness and resilience, effectively mitigating the adverse effects of cyber threats on system stability and performance.

For instance, in Case 3C, the FFOPI-TID^μ-PIDA controller achieves a significantly lower ITSE index value of 1.257, representing a substantial improvement over the conventional controllers. Specifically, the proposed controller reduces the ITSE value by a factor of 31.7 compared to the PID controller, 13.9 compared to the FPID controller, and 7.3 compared to the FPD-(1 + PI) controller. This notable reduction in ITSE underscores the superior capability of the FFOPI-TID^μ-PIDA controller in minimizing error accumulation over time, thereby enhancing system reliability and performance in the presence of cyber threats. These results further validate the effectiveness of the FFOPI-TID^μ-PIDA controller in safeguarding system operation against cyberattacks by ensuring rapid error correction and improved dynamic response. The ability of the proposed controller to maintain a superior performance under adverse conditions highlights its potential for practical implementation in cyber-resilient control systems.

6. Conclusions

This study introduced an advanced frequency management framework for smart microgrid (SMG) systems, integrating a smart FLC with three cascaded conventional controllers—FOPI, TID^μ, and PIDA. The control parameters were optimally tuned using the recently developed Eel and Grouper Optimizer (EGO), which draws inspiration from the collaborative foraging strategies observed between eels and groupers in natural marine environments. This bio-inspired metaheuristic was employed to fine-tune both the controller gains and the fuzzy logic membership functions, with the aim of achieving superior system performance. To evaluate the effectiveness of the EGO algorithm, its results were benchmarked against those obtained using other state-of-the-art optimization techniques, specifically the Gradient-Based Optimization (GBO) and the Smell Agent Optimizer (SAO). Through these comparative analyses, EGO consistently demonstrated better optimization capability. The Integral of Time Squared Error (ITSE) of frequency deviations was adopted as the objective function, ensuring that the tuning process prioritized both fast response and minimal oscillation, thereby enhancing the overall dynamic performance of the system.

The proposed EGO-based FFOPI-TID^μ-PIDA controller was extensively tested under various operating conditions, including the presence of highly fluctuating renewable energy sources, uncertain load variations, and various cyberattacks such as denial of service (DoS) and signal transmission delays. The comparative performance assessment against conventional and intelligent control strategies—namely, EGO-based PID, Fuzzy PID (FPID), and Fuzzy PD-(1 + PI) (FPD-(1 + PI))—demonstrated the clear advantages of the proposed approach in enhancing frequency stability, reducing transient deviations, and improving system resilience under adverse conditions. For instance, in the common case, by exposing the system to a step load variation of 0.1 pu and a constant renewable power variation of 0.15 pu, the proposed controller improved the Max.OS by 93.8% compared to the PID controller, 89.7% compared to the FPID controller, and 85.6% compared to the FPD-(1 + PI) controller. In Case 1, when series variations were applied to the load while maintaining constant renewable power variation, the proposed FFOPI-TID^μ-PIDA controller enhanced the frequency response performance, resulting in an improvement in the fitness function by 99%, 97%, and 91% compared to the PID, FPID, and FPD-(1 + PI) controllers, respectively. Furthermore, in Case 2, where series variations in solar irradiance, wind speed, and load were introduced, the proposed controller yielded enhancements in the Max.US by 74.3%, 74.7%, and 75.4% compared to the PID, FPID, and FPD-(1 + PI) controllers, respectively.

Consequently, the results confirmed that the FFOPI-TID^μ-PIDA controller significantly enhanced the frequency regulation capabilities of the SMG system, achieving an overall performance improvement of approximately 99.6% in comparison to the competing control strategies. Notably, the fitness function value dropped to 0.00069, which is nearly 200 times lower than that obtained using the other compared controllers, underscoring the exceptional accuracy and effectiveness of the proposed framework. Additionally, the controller exhibited superior dynamic performance by reducing overshoot, undershoot, and settling time, thereby ensuring rapid and stable frequency regulation even in the presence of cyber threats and communication delays.

The findings of our study emphasize the potential use of the EGO-based FFOPI-TID^μ-PIDA controller as an effective and cyber-resilient solution for frequency regulation in SMG systems. By ensuring enhanced robustness, rapid response, and improved stability under uncertain and hostile conditions, our suggested controller represents a promising advancement in modern power system control strategies.

7. Limitations and Future Aspects

Despite the promising results, this study has certain limitations that warrant further investigation. First, the proposed controller was tested under a specific set of operating conditions and cyberattack scenarios. However, real-world power grids may encounter more complex and unpredictable disturbances, requiring further validation under diverse real-time scenarios, including large-scale grid integration and multiple attack vectors. Additionally, this study assumes known and fixed system parameters; however, in real-world applications, parameter uncertainties can significantly degrade controller performance. Furthermore, while specific cyberattack scenarios have been considered, the study does not cover the full spectrum of potential threats, such as advanced persistent threats, coordinated multi-point attacks, or zero-day vulnerabilities. These sophisticated cyber threats pose serious risks to system stability and highlight the need for the further development of adaptive and resilient control strategies capable of detecting and responding to novel and evolving attacks. Furthermore, the computational complexity associated with implementing the EGO algorithm, especially difficulties in tuning due to the numerous parameters of the proposed FFOPI-TID^μ-PIDA controller, may pose challenges in systems with stringent processing constraints, necessitating the exploration of lightweight optimization techniques or hardware-based acceleration methods.

Future studies may concentrate on expanding this framework to encompass extensive power networks and distributed energy resources (DER) systems, incorporating supplementary cyber–physical security protocols, and investigating adaptive or real-time optimization methodologies to augment its practical use. Moreover, future research could focus on enhancing the controller’s robustness against parameter variations by incorporating uncertainty modeling techniques such as interval analysis, Monte Carlo simulations, or robust control design (e.g., H-infinity, μ-synthesis). Furthermore, the integration of artificial intelligence (AI)-based predictive control algorithms, reinforcement learning, or hybrid optimization methods could enhance the controller’s responsiveness to dynamic grid conditions. Ultimately, employing real-time simulators such as OPAL-RT or dSpace to evaluate the suggested control approach within a practical SMG testbed will provide insights into its efficacy in real-world applications and the potential challenges that may arise during implementation.