Chaotic chimp-mountain gazelle optimized FOPID control for frequency regulation in islanded airport microgrids with heterogeneous energy systems

Abstract

Ensuring stable frequency regulation in islanded airport microgrids is a challenging task owing to the intermittency of renewable energy sources, unpredictable load variations, and nonlinear dynamics. Conventional optimization techniques often struggle with premature convergence and sub-optimal controller tuning, leading to a poor transient response and inadequate frequency stabilization. These challenges necessitate an advanced optimization strategy that can efficiently handle dynamic airport environments while ensuring enhanced frequency stability. To address these issues, this study proposes a Chaotic Chimp Mountain Gazelle Optimizer (CCMGO) algorithm. The CCMGO algorithm integrates the exploration capabilities of the Chimp Optimization Algorithm (ChOA) with the fast convergence of the Mountain Gazelle Optimizer (MGO), which is further enhanced by chaotic mapping to improve search diversity and avoid local optima. The effectiveness of the proposed CCMGO optimized dynamic controller was evaluated under various load perturbation scenarios, including impulse, step-ramp, and stochastic disturbances. The system considered here is a multi-source airport model integrating wave, wind, solar, biogas turbines, battery energy storage systems, ultra-capacitors, and electric vehicles. Simulation results demonstrates that the CCMGO optimized fractional order proportional-integral-derivative controller exhibits better performances compared to the conventional genetic algorithm and particle swarm optimization based controllers, as well as contemporary metaheuristic algorithms like grey wolf optimizer and whale optimization algorithm. The proposed methodology achieves notable reductions in frequency deviation, shorter settling time, and enhanced transient response characteristics.

Introduction

Basics

The increasing demand for resilient and sustainable energy systems has led to the widespread adoption of microgrids, particularly in critical infrastructures such as airports. Microgrids enable local power generation and can operate independently of the main grid, providing a reliable solution in the event of power outages and grid instability. However, maintaining a stable power frequency in an isolated microgrid poses a major challenge, particularly for systems with a diverse mix of generation sources. Frequency deviations can cause instability and affect sensitive equipment, making frequency control critical for microgrid operations. In isolated airport microgrids, integrating heterogeneous generation sources such as solar, wind, and wave energy creates a highly dynamic environment. Although these renewable sources are essential for sustainability, their intermittent nature complicates frequency control. ESSs have emerged as key elements for ensuring stability and reliability; they support primary frequency control and balance power generation fluctuations by responding rapidly to demand and supply imbalances. Primary frequency regulation aims to rapidly limit and stabilize deviations following a disturbance, effectively arresting further drift and securing system stability within acceptable limits. Finer adjustments to bring the frequency closer to the nominal value are managed by secondary control layers. Despite ESSs’ advantages, effective frequency control in microgrids requires more than storage capacity. An optimized control approach is necessary to balance the complexities of different generation types and their response times. We propose a hybrid optimization technique to improve primary frequency control in an islanded airport microgrid. This advanced control strategy enables the microgrid to dynamically adjust its response in real time, allowing smooth integration of renewable energy and efficient management of ESS resources. The hybrid optimization approach combines multiple optimization techniques to fine-tune control parameters, leading to more accurate frequency regulation, reduced deviations, and enhanced overall performance. Integrating an energy storage system into this optimized control strategy provides a robust and adaptable solution to challenges faced by island microgrids with heterogeneous energy generation. The proposed solution significantly improves the operational efficiency and reliability of island microgrids, making them suitable for airport applications where continuous power is critical.

Literature survey

The significance of effective frequency control in airport microgrid is crucial. This ensures safe and efficient operation of electrical equipment, minimizes downtime, and contributes to a greener industry. As renewable energy technologies advance, islanded microgrids, such as airport power systems, will play a key role in demonstrating the feasibility of integrating these sources into complex environments. This literature underscores the importance of LFC in maintaining power system stability, particularly with the increasing prevalence of microgrids and RESs. LFC are vital for sustaining system stability and reliability in modern power systems, especially with the growing mixing of RESs and decentralized generation. Latif et al.¹ provided a comprehensive overview of controllers and soft computing techniques for regulated load frequency management in both traditional and renewable energy-based power systems, highlighting the challenges posed by the variability of RESs. Traditional LFC strategies use classical controllers enhanced by optimization algorithms. El-Fergany and El-Hameed² employed a social-spider optimizer for efficient frequency controllers in autonomous two-area hybrid microgrid systems. Similarly, Latif et al.³ evaluated water cycle algorithm-optimized non-integer controllers in isolated two-area interconnected microgrids with wind power and plug-in hybrid electric vehicles. Daraz et al.⁴ proposed a modified PID controller for AGC of multi-source interconnected power systems using a fitness-dependent optimizer algorithm, demonstrating improved system performance. Ćalasan et al.⁵ employed a chaotic optimization approach to design controllers for automatic frequency control in hybrid generator units with different interconnection structures. Advancements in control strategies have led to the development of sophisticated controllers and optimization algorithms for LFC. Irudayaraj et al⁶ applied Matignon’s theorem for stability analysis in hybrid power systems using an atom-search optimized fractional-order PID (FOPID) controller. Daraz et al.⁷ introduced an improved fitness-dependent optimizer-based FOI-PD controller for AGC in multi-source interconnected power systems within deregulated environments. Roy et al.⁸ conducted small-signal stability analysis of hybrid power systems with a quasi-oppositional sine cosine algorithm-optimized fractional-order PID controller. The grasshopper optimization algorithm has been effectively used in LFC design. Nayak et al.⁹ optimized a multistage controller using this algorithm for AGC in power systems equipped with FACTS devices. Magzoub and Alquthami¹⁰ employed simulated annealing for the optimal design of AGC in interconnected two-area power systems using a hybrid PID-fuzzy controller. Krishna et al.¹¹ proposed a rank-sum-weight method for systematic determination of weights in controller tuning for AGC, enhancing the tuning process’s efficiency. MPC has emerged as a powerful tool for LFC. Gulzar et al.¹² designed an adaptive MPC for load frequency control in hybrid power systems, achieving significant improvements in control performance. Rajaguru and Annapoorani¹³ introduced a virtual synchronous generator-based superconducting magnetic energy storage unit for the LFC of microgrids, optimized using the African vulture optimization algorithm. Aftab et al. ¹⁴ proposed an optimized cascaded controller for frequency regulation of energy storage-integrated microgrids, considering communication delays to enhance reliability. The integration of ESS is pivotal in modern LFC schemes. Liu et al. ¹⁵ explored LFC for renewable energy sources in isolated power systems by introducing large-scale PV systems and storage batteries. Parol et al. ¹⁶ focused on optimal power and energy management in low voltage microgrids using evolutionary algorithms and energy storage solutions. Oshnoei et al. ¹⁷ proposed a disturbance observer and tube-based MPC for electric vehicles to aid frequency regulation in isolated power grids. The role of ESS was further highlighted in studies by Jufri et al. ¹⁸, who developed an optimal battery energy storage dispatch strategy for small-scale isolated hybrid renewable energy systems with different load profiles. Almasoudi et al. ¹⁹ presents a nonlinear coordination strategy for frequency regulation in hybrid power systems that integrate renewable energy sources (RES) and fuel cells. Mi et al. ²⁰ addressed frequency control in wind-diesel systems using hybrid energy storage, demonstrating the effectiveness of combining different storage technologies. DR and load management strategies are integral to modern LFC approaches. Barik and Das ²¹ investigated load–frequency regulation in demand response-supported bio-renewable cogeneration-based hybrid microgrids using quasi-oppositional selfish-herd optimization. Latif et al. ²² proposed a novel coordinated load frequency control strategy for an independent three-area microgrid system, incorporating a combination of diverse energy storage units and DC link integration. The study utilized a BOA-optimized PFOID controller to enhance system performance and stability. Almasoudi et al. ¹⁹ investigated a nonlinear coordination strategy between renewable energy sources and fuel cells for frequency regulation in hybrid power systems. The approach emphasizes the effective interaction between different energy sources to maintain system balance under dynamic conditions. Mohammed et al. ²³ examined AGC in future multi-source power systems with high renewable penetration and electric vehicles, using the Egyptian power system in 2035 as a case study. Wang et al. ²⁴ designed a controlled load damping factor controller for systems with significant renewable energy integration. Advanced intelligent control methods have been explored to address the complexities of modern power systems. Xi et al. ²⁵ developed a multistep unified reinforcement learning method for AGC in multiarea interconnected power grids. Magdy et al. ²⁶ developed a superconducting magnetic energy storage (SMES)-based PID controller for improving frequency stability in a hybrid power system with high wind power penetration. Their study demonstrated the effectiveness of the controller in mitigating frequency deviations caused by wind power fluctuations. Barik and Das ²⁷ introduced a grasshopper optimization algorithm for frequency control in an isolated renewable microgrid comprising solar photovoltaic, biogas, and biodiesel generators. Their findings highlight the efficiency of the algorithm in maintaining stable frequency dynamics under variable load conditions. Latif et al. ²⁸ extended their research by evaluating the performance of a demand response-supported dual-stage PIFOD-(1 + PI) controller optimized using the YSGA technique. The controller was applied to a wind-tidal-biodiesel-based two-area interconnected microgrid, demonstrating its capability to enhance frequency response and system stability. Moschos and Parisses ²⁹ combined frequency and voltage control in two-area multi-source interconnected microgrids using a two-degree-of-freedom tilted integral derivative with a fractional order (2DOF-TIDμ) controller. Daraz et al. ³⁰ addressed load frequency stabilization in distinct hybrid conventional and renewable power systems incorporating electric vehicles and capacitive energy storage. Specific case .studies have provided practical insights into the application of LFC strategies. Choudhary et al. ³¹ developed a grasshopper optimization-based robust power/frequency regulator for shipboard microgrids, demonstrating the adaptability of LFC techniques in maritime applications. Lü et al. ³² reviewed energy management in hybrid electric vehicles, emphasizing energy optimization in fuel cell hybrid power systems using genetic algorithms. Future trends in LFC involve innovative approaches and the incorporation of advanced technologies. Huy et al. ³³ proposed real-time power scheduling for isolated microgrids with renewable energy and ESS via a supervised-learning-based strategy, enhancing responsiveness and efficiency. Mohamed et al. ³⁴ introduced a hybrid cheetah particle swarm optimization-based hierarchical control for multiple microgrids, showcasing the potential of hybrid optimization algorithms. Khalil et al. ³⁵ developed a novel multi-objective tuning formula for LFC in isolated low inertia microgrids incorporating PV, wind, fuel cells, and battery energy storage systems. Luo et al. ³⁶ presented an optimal adaptive decentralized under-frequency load shedding strategy for islanded smart distribution networks, considering wind power uncertainty, which is crucial for maintaining stability in the face of unpredictable renewable generation. Elkasem et al. ³⁷ proposed a strategy for frequency regulation in hybrid renewable power grids using LFC and redox flow batteries, highlighting advancements in energy storage technologies. Recent literature emphasizes advanced optimization and control techniques for stabilizing the frequency and voltage in microgrids and interconnected power systems. Studies have applied innovative optimizers, such as the Barnacle Mating Optimizer ³⁸, fuzzy fractional-order controllers leveraging tidal turbines ³⁹, and dual-degree branched type-2 fuzzy control ⁴⁰ to improve frequency regulation. Others addressed power quality through particle swarm optimization for distributed generation and UPQC placement ⁴¹ and heuristic methods for DSTATCOM insertion ⁴². MPC-based strategies, including virtual inertia integration, have also enhanced frequency regulation performance ^43,44. Advanced approaches using chaos-based algorithms and deep learning address cyber-security concerns in deregulated power systems ⁴⁵. Finally, finite-time control schemes ⁴⁶ and hierarchical deep learning methods ⁴⁷ provided robust voltage and frequency management in networked microgrids. Pushkarna et al. (2022) ⁴⁸ proposed an analytical method for optimal Type-IV DG placement in an unbalanced distribution system to minimize power losses, while Pushkarna et al. (2024) ⁴⁹ enhanced this by using a PSO-based optimization with UPQC for improved voltage stability, and Kumar et al. (2024) ⁵⁰ introduced a heuristic approach integrating DSTATCOMs to mitigate harmonics and voltage sag for better power quality.

Analysis: literature work study

The literature survey thoroughly explores LFC methodologies in power systems and microgrids, highlighting significant strides in integrating RES, ESS, and advanced control strategies optimized through various algorithms. This review aligns with the proposed research, focusing on enhancing primary frequency regulation in an islanded airport microgrid through nonlinear control methods governed by heterogeneous generation and assisted by energy storage elements. The survey covers traditional and advanced LFC strategies, RES and ESS integration, and optimization algorithms, providing a solid foundation for the proposed study. Several survey studies directly relate to the core aspects of the proposed research. Studies such as El-Fergany and El-Hameed ¹, Latif et al. ³, and Daraz et al. ⁸ explore optimized PID controllers in microgrids, pertinent to the comparative analysis of controllers (PID, cascaded PD-PID, FOPID, fractional-order PI-FOPID) in the proposed research. Table 1 analysis the state-of-the -art literature depicting the significance of the proposed research work. Furthermore, the exploration of advanced control strategies and optimization algorithms by Irudayaraj et al. ¹¹, Daraz et al. ¹⁴, and Roy et al. ¹⁶ paralleled the proposed use of the new CCMGO for controller optimization. These studies demonstrate the efficacy of using hybrid and intelligent optimization techniques to enhance the control performance in power systems, which supports the proposed methodology. The integration of diverse RES and ESS in microgrids is well documented in the literature survey through studies such as Liu et al. ⁹, Parol et al. ¹³, and Jufri et al. ¹⁸. These studies emphasize the importance of combining various renewable sources and storage solutions to improve grid stability and load frequency regulation, mirroring the proposed system architecture that includes wave energy, wind turbines, solar towers, photovoltaic energy, biogas turbines, microturbines, and diesel generators, along with BESS, ultra-capacitors, and EVs. The discussion on the role of ESS in frequency regulation is particularly relevant, as it underscores the benefits of ESS in compensating for generation-load imbalances and enhancing microgrid flexibility, which are the key objectives of the proposed research.

Table 1.

Analysis of state-of-the -art literature.

References	Controller stagey used				Penetration of RESs				Algorithms used	Additional advances incorporation in terms of advanced controller used	System Complexity in terms of renewable cogeneration	Proposed work
	TID	FOPID	PID	Cascaded PD-PID	Tidal power	Geothermal power plants	Bio-diesel	Wave energy
2	×	×	✔	×	×	×	×	×	SSO	Not Used	✔	Included
3	×	✔	×	×	×	×	✔	×	WCA	Fractional order used	✔	Included
4	×	×	✔	×	×	×	×	×	Dependent optimizer algorithm	Fractional order used	✔	Included
5	×	×	✔	×	×	×	×	×	COA	Fractional order used	✔	Included
8	×	✔	✔	×	×	×	×	×	QOSCA	Fractional order used	✔	Included
10	×	×	✔	×	×	×	×	×	HYBRID	Fractional order used	✔	Included
12	×	✔	✔	×	×	×	×	×	HYBRID	Fractional order used	✔	Included
14	×	×	×	✔	×	×	×	×	hBWT-LUS	Fractional order used	✔	Included
21	×	x	×	×	✔	✔	✔	✔	QSHO	Fractional order used	✔	Included
27	×	x	✔	×	×	×	×	×	×	Fractional order used	✔	Included
28	×	x	✔	×	×	×	×	×	CHIO	Fractional order used	✔	Included
29	✔	×	✔	×	×	×	×	×	COOT	Fractional order used	✔	Included
31	×	×	✔	×	×	×	×	×	GOA	×	✔	Included
34	×	×	×	×	×	×	×	×	HYCHOPSO	Droop control with PLL control technique	✔	Included
37	✔	×	×	×	×	×	×	×	COA	Fractional order used	✔	Included
49	×	×	×	×	×	×	×	×	ChOA	Not Used	✔	Included

Although the literature survey is comprehensive, it can be improved by including studies on frequency regulation in airport microgrids or similar critical infrastructures to highlight the unique challenges of these systems. Additionally, the survey could be strengthened by discussing chaotic maps and hybrid algorithms, such as the proposed CCMGO, in controller design. This provides a stronger theoretical foundation for the novel algorithm.

The research highlights nonlinear control and advanced controllers under dynamic loads, but the survey could benefit from more studies on nonlinear control strategies in microgrids with diverse generations and ESS. Comparing the performance of different controllers under similar conditions would also provide valuable benchmarks for the proposed controllers. Additionally, while ESS is discussed, more focus on ultra-capacitors and PHEVs would strengthen the survey, as their impact on frequency regulation is directly related to the proposed system.

Problem identification, motivation and the proposed approach

Despite advancements in LFC and integration of RES and ESS into microgrids, maintaining grid stability in isolated microgrids with heterogeneous generation remains a significant challenge. The operational dynamics of airport microgrids are unique, incorporating various RES like wave energy, wind turbines, solar towers, and photovoltaic systems, along with controlled generation units such as biogas turbines, microturbines, and diesel generators. Current research lacks comprehensive studies addressing frequency regulation intricacies in these environments, particularly regarding nonlinear control methods enhanced by innovative algorithms capable of managing diverse, fluctuating load conditions. Existing studies often focus on traditional control strategies or do not fully exploit hybrid optimization techniques to enhance primary frequency regulation in these critical infrastructures.

This research addresses the urgent need to improve airport microgrid dependability and resilience, given growing environmental challenges and the global shift to renewable energy. Reliable high-quality power supply is crucial for smooth airport operations. Incorporating various RES and ESS introduces power generation stability challenges, complicating frequency regulation. Addressing this issue is vital for maintaining operational efficiency and safety. The development of sophisticated optimization algorithms, like CCMGO, offers potential to enhance controller performance. This advancement contributes to control systems engineering progress and supports incorporating sustainable energy into critical infrastructure while preserving grid stability.

The suggested methodology involves examining a standalone airport microgrid with diverse renewable and traditional power sources, complemented by energy storage systems (BESS, UC, and EVs). The research will evaluate multiple control techniques, including PID, cascaded PD-PID, FOPID, and FOPI-FOPID controllers, under various load scenarios (impulse load, step-ramp load, and random load fluctuations). To enhance these controllers, a hybrid optimization algorithm called CCMGO will be created, combining MGO’s exploratory capabilities with chaotic maps and ChOA components. This algorithm aims to improve solution diversity and avoid early convergence for better optimization outcomes. Simulations will assess the CCMGO-optimized controllers’ performance in maintaining grid stability and responding to dynamic operating conditions typical of airport microgrids.

Key contribution

1. A complex isolated airport microgrid model is designed including sustainable sources such as AWEC, WTG, PV, ST, DEG, MTPG, BTGU, BESS, UC, and EV.

2. This study applies to the proposed CCMGO algorithm to solve the frequency regulation problem in airport microgrids involving the concept of chaotic and CHOA algorithms.

3. The superiority of the CCMGO algorithm with the design of PID, cascaded PD-PID, FOPID, and FOPI-FOPID controllers for LFC analysis in airport microgrids is established.

4. Extensive simulations were conducted to test how the optimized controllers perform under various load conditions, including impulse positive and negative, impulse positive, step-ramp changes, and random load variations.

The paper is organized to systematically explore the modelling and control of a power system, beginning with the complete description of the test system’s components in Sect. "Test system modelled and investigated", followed by the design and methodology of various controllers in Sect. "Modelling of controller: controller design and methodology", and the formulation of an optimization problem aimed at improving LFC in Sect. Problem formulation: Defining LFC objective function. Section “Implemented algorithm description” introduces the new chaotic chimp mountain gazelle optimization algorithm and presents the simulation results in Sect. “Simulation results and analysis” that demonstrate its effectiveness. The study concludes in Sects. “Limitations and constraints of this study” and Sects. “Conclusions and scopes of future work” by acknowledging its limitations and suggesting future research directions to further enhance the algorithm’s application in microgrid control.

Test system modelled and investigated

Basic description of the model

This study investigates LFC performance in an airport power system model. The test system integrates wave energy conversion, wind turbine generator, PV, and solar tower power unit as independent sources of generation, whereas biogas turbine generator, micro-hydro turbine generator, and biodiesel engine driven generator are used for controlled generation operating in standby mode. In addition, EV, UC and BESS were used in the studied system as a battery storage solution ³⁵.

The airport system with a bio renewable energy cycle is depicted in the perception model shown in Fig. 1. Figure 2 illustrates a simplified dynamic model of the airport power network. This model captures the dynamic interactions between various components of the network, such as generation, energy storage, and load dynamics. Figure 3 shows how renewable cogeneration units are interconnected with the modelled system, emphasizing the mechanism by which fluctuations in renewable output are mitigated. Figure 4 presents the test system through a transfer function based diagram, where each component behaviour is modelled using parameters such as gains and time constants. This approach provides a precise quantitative description of the network dynamic response. Finally, Fig. 5 details the overall network model and demonstrates its application to the airport system. This figure integrates the individual models into a comprehensive framework that highlights the interactions among RESs, cogeneration units, and control strategies.

Fig. 1.

Perception model of the airport network with bio-renewable energy cycle ²¹.

Fig. 2.

Simplified dynamic model of airport network including RESs ²¹.

Fig. 3.

Interconnection of airport renewable cogeneration with AGC implementation ²⁶.

Fig. 4.

Transfer function based modelled diagram of the test system under study ¹⁹.

Fig. 5.

Network model and its implementation to the airport power system.

Basic description of the model: autonomous generation

AWEC: independent power generation

AWEC is a method for generating electricity from the ocean waves by using the floater vertical motion caused by waves in the ocean. To convert machine driven wave energy into electric energy, AWS is connected to a permanent magnet alternator. It is possible to install this WECS at depths greater than 25 m. The AWEC’s dynamic velocity and force can be expressed as follows.

1

2

The irregular sinusoidal wave force of AWEC is expressed as.

3

The linear first order transfer function of AWEC (ignoring all nonlinearities) can be written as follows ²².

4

WTG: independent power generation

Wind power is one of the proven RESs which has huge potential. A wind turbine generator is a device that converts wind energy into electric power. The extractable output power of the wind turbine generator is expressed as.

5

The power coefficient of WTG is.

6

Where as

After simplifying, the transfer function of WTG is given as follow ²¹.

7

PV: independent power generation

Solar panels convert sunlight directly into electricity by exciting electrons in a semiconductor material. The amount of electricity generated depends on the temperature and sunlight level. To track more power from solar panels, PV system often use MPPT method. The transfer function of a PV model is follows ¹⁹.

8

ST: independent power generation

In a central receiver system, also known as a solar tower, numerous curved mirrors called heliostats concentrate sunlight onto a receiver positioned at the tower summit. This receiver, capable of reaching temperatures between 500 and 850 °C, employs fluids such as steam, heated air, or liquefied salt to transfer heat to a steam generator. Subsequently, the generated steam drives a turbine generator, producing electricity.

9

The net electrical power from ST is described by.

10

For simplicity, the fluid in the solar receiver tube was assumed to be homogeneous. As a result, the state equation describing the outlet temperature of the receiver tube can be formulated as.

11

The fluid velocity (Fo ) of the receiver tube can be expressed as.

12

Finally, the state equation of ST receiver can be represented as.

13

By linearizing the above, we could estimate the TF model of ST considering a thermal generator ²².

14

Basic description of the model: controlled generation

BDEG units: controlled generation

Biodiesel units, known as BDEG, can utilize fuel produced from the transesterification of waste cooking oils or suitable energy crops with diesel like properties. Biodiesel can be used either as a mixture or in its pure form. The linearized model of the BDEG ¹⁹ is approximated by considering the engine and inlet valve operations.

15

MTPG: controlled generation

A microturbine power generator is a compact, high speed turbine that produces 25—200 kW of power. It is quiet, has low emissions, and can use different fuels, making it ideal for homes and businesses. The power it generates is given by the following formula:

16

The linearised transfer function of MTPG ²¹ is expressed as follows.

17

BGTG unit: controlled generation

Biodegradable waste and animal waste of the community can be converted into biogas, which is then stored and effectively utilized in BGTG systems to produce electricity. Equation (18) represents the linearized BGTG model, which incorporates the operations of biogas inlet valve, combustor, and turbine²⁷.

18

EV: controlled generation

EVs contribute to system stability by mitigating fluctuations. They accomplish this by acting as a load during charging and as a power source during discharging. The first order transfer function of EV model is as follows ¹⁹.

19

UC: back-up device

Ultracapacitors store and release energy quickly. They are reliable, long-lasting, and work well in cold weather. The linear transfer function model of ultra capacitors ⁸ is shown in (20).

20

BESS: back-up device

BESS helps keep the system frequency stable by either storing energy or supplying power as needed. The BEES first order transfer function ¹⁹ is as follows.

21

Modelling of controller: controller design and methodology

PID controller: external controller

PID controller act in a parallel combination of proportional, integral, and derivative parameters. The controller’s input is ACE. Based on the error signal, the required control signal is produced. The transfer function of PID controller is represented by equation (22) and is shown in Fig. 6(a) ¹⁰.

22

Fig. 6.

Implemented controller structures: (a) PID, (b) cascaded PD-PID (c) FOPI-FOPID and (d) FOPID.

PD-PID controllers: external controller

A PD-PID controller is a cascaded controller that combines PD control for fast response and damping with PID control for steady-state error elimination. The transfer function of PD-PID controller is represented by Eq. (23), as shown in Fig. 6(b).

23

Fractional order concept

The concept of fractional order (FO) calculus originated from extending the traditional integral order (IO) calculus to include non-integer orders. Among the various FO integro-differential definitions, the most commonly used are based on the formulation of Rieman-Liouville (RL), Grunwald–Letnikov, and Caputo. Scientists often utilize RL fractional-order integration and derivatives, which are mathematically expressed in Eqs. (24) - (25)⁸.

24

25

Where is the Euler gamma function, n is an integer such that n-1 < α < n and α is proper fraction.

FOPI-FOPID controller: external controller

FOPI and FOPID controllers are sophisticated variants of traditional PI and PID controllers that utilize fractional calculus principles. These advanced controllers employ non-integer-order derivatives and integrals, in contrast to the integer order operation of conventional controllers. This approach provides greater tuning flexibility and improved control, especially in complex systems that continue memory effects, including time delays and fractional dynamics.

The FOPI controller improves the standard PI controller by incorporating the integration effect of fractional order, leading to more accurate adjustments and increased resilience to faults and uncertainties. The FOPID controller expands the PID controller by enabling fractures of both integral and differential components. This additional parameter provides excellent control over the dynamic system, providing improved stability margins and frequency response characteristics. These controllers are particularly valuable when to deal with nonlinear behaviour and temporal delays.

The FOPID controller offers excellent performance, especially in systems with complex dynamics, but the prerequisites for fractional order processes make practical applications mathematically intensive. However, the adaptability and improved control capabilities of the FOPI and FOPID controllers make them a critical device for modern control applications, with great advantages over traditional control systems. (refer Fig. 6(c)).

26

FOPID controller: external controller

FOPID Controller is a more advanced version of the classic PID, involving fractional calculus to provide enhanced control flexibility and robustness. It is expressed as PI^λD^μ, with λ and μ being the fractional orders of integration and differentiation, respectively. The transfer function of the FOPID controller is shown in equation (27) and is shown in Fig. 6(d) ³.

27

Problem formulation: defining LFC objective function

Mathematical problem formulation in AGC studies, especially when optimizing control parameters to improve system stability and performance, typically includes specifying an objective function that embodies the desired optimization criteria.

Objective: defining optimization goals and objective function

ITAE is a significant objective function which punishes long-term errors and promotes quick convergence of target points, smooth responses and grid stability. The ITAE objective function is expressed mathematically in (28) ⁴.

28

Constrained optimization: defining the boundaries of constrained

(a) The PID controller gains are the constraints for the optimization task. The boundaries of these parameters are bounded as given in (29).

29

(b) FOPID controller gains are the constraints for the optimization task. The boundaries of these parameters are given in (30).

30

(c) PD-PID controller gains are the constraints for the modelling optimization task. The boundaries of these parameters are given in (31)

31

(d) FOPI-FOPID controller gains are the constraints for the modelling optimization task. The limits of these parameters are given in (32)

32

These parameters represent the proportional, integral, derivative gains, and fractional-order coefficients (λ, µ).

Measure of performance: defining performance indices

Incorporating diverse performance indices into the AGC optimization process ensures both effective grid stability and efficient control actions in the face of RESs variability. Through a detailed comparative analysis of IAE, ITSE, ISE, and ITAE, this study aims to identify optimal control strategies that balance rapid disturbance response, long-term error minimization, and overall system performance enhancement. Choosing the right performance index is crucial for optimization. IAE measures the overall control accuracy, ITSE prioritizes rapid stabilization, and ISE emphasizes minimizing large deviations. Together, these indices enable a comprehensive optimization strategy, guiding the selection of AGC solutions that balance swift disturbance recovery, long-term stability, and precise control, thus improving grid reliability and performance in the presence of RES variability.

In the studied models, IAE, ITSE, and ISE values are taken as the performance indices. The mathematical expressions ^4,5 for these three indices are defined in order in Eqs. (33) (34) (35).

33

34

35

Here, is the time of simulation, is the deviation of frequency.

Implemented algorithm description

MGO algorithm

Introduced in 2022, MGO is an advanced optimization algorithm that takes cues from the complex social dynamics of mountain gazelles. These dynamics encompass various groups, including maternity herds, bachelor’s male herds, and lone territorial males ⁵¹. Scientists have utilized these natural patterns to create a mathematical framework that enhances the optimization process. MGO algorithm capitalizes on key behaviours exhibited by mountain gazelles, such as territorial instincts, social hierarchies, migratory patterns, and swift movements. By integrating aspects such as interactions within bachelor and maternity herds, as well as the foraging movements of territorial males, the algorithm replicates how virtual gazelles explore and interact within the solution space. This novel approach enables MGO to effectively tackle and optimize complex problems by employing the adaptive and dynamic strategies observed in mountain gazelle behaviour. The structure of the algorithm facilitates the efficient exploration and exploitation of the search space, making it a powerful tool in optimization research.

Concept of algorithm: inspiration

A species of gazelle known as the mountain gazelle is indigenous to the Arabian Peninsula and surrounding areas. Although it has a broad distribution, its population is relatively small. Robinia tree species play a crucial role in their habitat. During the late Holocene, the mountain gazelle’s range contracted due to increasing temperatures, allowing Gazella bennettii, which is more adapted to warmer environments to occupy its former territory. Mountain gazelles exhibit highly territorial behavior with considerable space between individual domains. They organize themselves into three distinct groups: herds consisting of mothers and their offspring, groups of young males, and solitary males. Mature males often engage in frequent but less violent territorial disputes compared to conflicts over females. Young males tend to use their horns more aggressively than older territory holding males. These gazelles are migratory travel over 120 km in search of food, and are incredibly fast runners, capable of sprinting several hundred meters at speeds up to 80 km/h. herd of mountain gazelles (refer Fig. 7).

Fig. 7.

Proposed MGO implementation procedure.

Mathematical formulation

(a) Territorial Solitary Males (TSM):

When male gazelles reach adulthood, they establish solitary territories and defend themselves. The position of the best global solution (the adult male) is updated using a mathematical model that considers random factors and the positions of other males in bachelor’s herds ⁵¹.

36

Where ‘malegazelle’ represents the position of the best global solution, ‘ri1’ and ‘ri2’ are random integers, ‘BH’ is the bachelor herd coefficient, ‘X(t)’ is the current position, and ‘Cofr’ is a randomly selected coefficient vector.

(b) Maternity Herds (MH)

Maternity herds are crucial for the birth of strong male gazelles. This behavior is modelled to ensure the creation of robust solutions by simulating the interactions within maternity herds and the influence of dominant males ⁵¹.

37

Here, ‘Cof1, r’ and ‘Cof3, r’ are random coefficient vectors, ‘ri3’ and ‘ri4’ are random integers, and ‘Xrand’ is a randomly selected position.

(c) Bachelor’s male herds (BMH):

As young males mature, they attempt to establish their territories, leading to conflicts with established males. This is modelled to represent the competition among solutions to improve their positions ⁵¹.

38

where ‘D’ is calculated using the current positions and ‘malegazelle’ is the best solution.

(d) Migration to Search for Food (MSF):

Gazelles constantly migrate in search of food, covering large distances. This behavior is modelled to enhance exploration capabilities, allowing the algorithm to search diverse areas of the solution space ⁵¹.

39

Where ‘ub’ and ‘lb’ are the upper and lower bounds of the problem, and ‘r₇’ is a random number.

CHOA algorithm

Chimpanzees, highly social great apes closely related to humans, live in dynamic fission–fusion societies where group size and composition fluctuate ⁵². This social structure is mirrored in the CHOA, where independent groups explore the search space using diverse strategies and leverage individual strengths to solve complex problem. Hunting occurred during the exploration and exploitation stages. Calculate the driving and pursuing prey. For driving and following prey, (40) and (41) are mathematical models ⁵².

40

41

The vectors a and c are examined in accordance with Eqs. (42) and (43), respectively.

42

43

Drivers, barriers, and chaser chimpanzees sometimes hunt, but attackers usually do so. The best prey location is unknown in the search space. It is hypothesized that the first attacker, the most ideal solution, and the driver, barrier, and chaser chimpanzees have a better understanding of the possible prey location to mathematically describe their behaviour. Thus, the other chimpanzees locations were adjusted to match those of these better chimps, helping them to improve their search techniques. The relationship mentioned is represented by Eqs. (44) to (46) ⁵².

44

45

46

In the final stage, the chimps abandon their roles and become chaotic. This is modelled by a 50% chance of using chaotic movement instead of the normal updating process. This helps the algorithm escape local optima and explore new solutions. The search process involved chimps with different roles, adjusting their positions based on the estimated prey location. The parameters are adaptively tuned for faster convergence. The ‘f’ value decreases over time, encouraging exploration initially and exploitation later. The mathematical model is expressed by Eq. (47) ⁵².

47

Briefly, ChOA searches begin with chimpanzees and possible solutions. These chimpanzees were randomly assigned to attackers, barricades, chasers, and drivers. Group-based "f" coefficient updates were used for each chimp. In each iteration, the attacker, barrier, chaser, and driver chimpanzees assess their prey’s position. Each potential solution modifies its prey distance based on this estimate. The "c" and "m" vectors are adaptively modified to accelerate convergence and prevent local optima. To improve exploitation and prey capture, "f" is progressively lowered from 2.5 to zero. Candidate solutions split and explore various search space areas if |a|> 1. If the inequality is not fulfilled, the potential solutions converge on the prey.

Chaotic concept

The use of a chaotic map instead of a random process has become increasingly popular in optimization. Random-based optimization approaches may use disorderly motion instead of stochastic motion for unpredictability ⁵³. If it is difficult to construct a general rule for a given class of systems, experimental studies show that chaotic signals outperform random signals for the most part. Using chaotic features in optimization helps avoid local minima better than stochastic optimization. Random based algorithms enable unpleasant situations or solutions with predetermined probabilities. A chaos based optimization strategy seeks order in a chaotic system to escape a local optimum. Owing to its sensitivity to the initial condition and quasi stochasticity, chaos is valuable in optimization. Thus, the output of a search strategy may vary significantly depending on its beginning position. If the correct starting point is selected and the search is performed inside a finite sphere around it, all potential outcomes may be found. The equation for the same is shown in Eq. (48).

48

The value of is taken as 4. The flowchart of the (refer Fig. 8).

Fig. 8.

Flowchart of the proposed CCMGO algorithm.

Proposed algorithm: CCMGO

CCMGO is an amalgamated algorithm that combines the strengths of CHOA and MGO with chaotic maps. CCMGO uses MGO capacity to imitate mountain gazelle social characteristics such as territoriality, social hierarchy, and migration to explore and utilize the search area. It uses the CHOA specific hunting functions and adaptive techniques to boost convergence and search. Chaotic maps provide unpredictability and assist the algorithm in escaping local optima, encouraging exploration and variety in solution searches. Combining these strong components, CCMGO aspires to exceed existing methods in accuracy, efficiency, and resilience for complicated optimization tasks.

Simulation results and analysis

This study focuses on the comparative assessment of PID, cascaded PD-PID, FOPI-FOPID, and FOPID control strategies using the CCMGO algorithm with the ITAE as the performance index. The controllers were evaluated based on their dynamic performance, effectiveness in transient responses, ability to integrate energy storage devices, and control over generation units. The optimization process was implemented in MATLAB R2024a, while the system was simulated in Simulink using the FOMCON toolbox. Initially, the CCMGO algorithm was compared with GA, PSO, GWO, and WOA for optimizing the FOPID controller under positive impulse and negative impulse load disturbance. The results, presented in Table 2, demonstrate lower ITAE values and superior performance metrics, indicating enhanced frequency regulation, reduced overshoot, and smoother operation. CCMGO outperformed PSO, GA, GWO, and WOA due to its unique integration of chaotic dynamics and hybrid metaheuristic strategies. By combining chaotic maps to enhance population diversity and prevent premature convergence, CCMGO integrates the social hierarchy of chimp optimization with the agile exploration mechanisms inspired by mountain gazelle behaviour. This hybrid framework enables adaptive balancing of exploration and exploitation phases, dynamically adjusting search intensity based on fitness feedback. Unlike PSO and GA, which suffer from rigid parameterization and diversity loss in high-dimensional spaces, or GWO/WOA, which rely on fixed social structures, CCMGO’s chaotic perturbations and multi-strategy collaboration allow it to navigate the non-convex, fractional-order search space of FOPID controllers more effectively. After demonstrating the efficacy of CCMGO, a variety of operational scenarios were analyzed to assess frequency stability under dynamic conditions.

Table 2.

Comparative analysis with GA and PSO.

Algorithm	ITAE (Lower is Better)	IAE	ITSE	ISE
PSO	13.9083	0.1622	0.2943	0.0035
GA	16.6077	0.4153	0.6108	0.0088
GWO	13.68	0.9985	0.3078	0.0738
WOA	14.0470	0.7920	0.1965	0.0394
CCMGO (Proposed)	13.385	0.003	0.142	0.293

Scenario A:Analysing test system performance under positive impulse and negative impulse load disturbance.

Scenario B:Assess test system functioning under positive impulse load disturbance.

Scenario C:Examine the test system funsctioning under step ramp combination-based load disturbance.

Scenario D:Evaluate test system functioning under stochastic random load perturbation

Scenario A: Analyzing test system performance under positive and negative impulse load disturbances

The model examined here for evaluation is already shown in Figs 1,2,3,4,5. This scheme is similar to PID, cascaded PD-PID, FOPI-FOPID and FOPID controllers. The valuation is executed together with the load profile applied. The most optimal values obtainable for all the controller attributes are attained via CCMGO by means of ITAE. The finest probable values are manifested in Table 3, and potent outcomes are attained as shown in Figs. 9,10,11,12,13

Table 3.

Optimized PID controller gains for test system scenario A.

Techniques	Kp1	Ki1	Kd1	Kp2	Ki2	Kd2	λ 1	λ 2	µ
CCMGOFOPI-FOPID [Studied]	0.010	0.010	-	0.010	0.010	0.010	0.010	0.010	0.876
CCMGO PD-PID [Studied]	0.001	-	0.001	0.001	0.001	0.001	-	-	-
CCMGO PID [Studied]	-	-	-	2.000	1.600	2.000	-	-	-
CCMGO FOPID [Proposed]	-	-	-	0.900	0.900	0.900	0.900	-	0.700

Figure 9 shows the inputs applied to the independent power generation, such as the wave input condition, wind input condition, solar tower, and solar radiation condition. Figure 10 shows the independent power generation such as AWEC, WTG, PV, and STPG. Figure 11 shows the LFC response of the system in terms of frequency deviation, load demand, controller output response, and generation load demand response. Figure 12 shows the profile of energy storage systems, i.e., for EV, BESS, and UC. Figure 13 shows the controlled power generation i.e., DEG, BTGU, and MTPG. In the outcomes of Fig. 11, it is noticeable that the improvement can be observed in the amount of frequency deviation in the presence of FOPID controller. The controlled power generation output of various sources using the centralized controller provides LFC responses are satisfactory.

Fig. 9.

Applied inputs to the independent power generation (refer Scenario A): (a) Wave Input, (b) Wind Input (c) Solar radiation and (d) PV Radiation.

Fig. 10.

Independent power generation (refer Scenario A): (a)Wave power generation, (b) Wind power generation, (c) Solar power generation (d) PV power generation.

Fig. 11.

LFC response profile (Refer Scenario A) (a) Frequency deviation, (b) Load demand. (c) Controller output and (d) Generation load demand.

Fig. 12.

Profile of the energy storage system (refer Scenario A): (a) EV, (b) BESS and (c)UC.

Fig. 13.

Controlled power generation (refer Scenario A): (a) DEG, (b) BTG, and (c) MTPG.

The data provided in Table 4 compares the performance of four control strategies optimized by CCMGO PD-PID, CCMGO PID, CCMGO FOPI—FOPID, and the proposed CCMGO FOPID under Scenario A. The performance is evaluated based on ITAE, IAE, ITSE, and ISE, where lower values generally indicate better control system performance.

Table 4.

Comparative measures of performance indices for test system Scenario A.

Techniques	FOD	Performance indices
	ITAE	IAE	ITSE	ISE
CCMGO FOPI-FOPID [Studied]	677.324	19.074	188.059	5.278
CCMGO PD-PID [Studied]	669.192	18.623	183.817	5.031
CCMGO PID [Studied]	37.816	1.792	1.094	0.088
CCMGO FOPID [Proposed]	7.998	0.565	0.115	0.030

Additionally, the computational time required for each controller was also considered (refer Table 5). The results demonstrated that the CCMGO FOPID algorithm significantly outperformed the other controllers in terms of all performance indices. It achieves an ITAE of 7.998, an IAE of 0.565, an ITSE of 0.115, and an ISE of 0.030. These values are considerably lower than those of the studied one, highlighting superior performance of the CCMGO FOPID in minimizing error and achieving faster settling times. The CCMGO FOPID algorithm also showed competitive performance, particularly in terms of ITAE (7.998) and IAE (0.565), indicating its effectiveness in reducing errors.

Table 5.

Computational time for test system in scenario A.

Controllers used	Computational time (s)
CCMGO PID [Studied]	632
CCMGO PD-PID [Studied]	618
CCMGO FOPI-FOPID [Studied]	628
CCMGO FOPID [Proposed]	603

In terms of computational efficiency, CCMGO PID requires a computational time of 632 s, CCMGO PD-PID requires a computational time of 618 s, CCMGO FOPI-FOPID requires a computational time of 628 s, and CCMGO FOPID requires a computational time of 603 s. In this CCMGO FOPID, less computational time was required. With the above information, the proposed CCMGO based FOPID emerges as an effective controller for the test system under Scenario A, showing superior performance in minimizing errors and achieving faster settling times. While it requires a slightly higher computational time compared to CCMGO PD-PID, the trade-off is worthwhile given the substantial improvement in the control system performance. The CCMGO FOPID algorithm is also a viable option, particularly if computational efficiency is the primary concern. The choice between CCMGO FOPI-FOPID and CCMGO FOPID ultimately depends on the specific requirements and priorities of the application.

Table 6 represents the Eigen value analysis for the CCMGO FOPID for Scenario A, Scenario B, Scenario C and Scenario D. The eigenvalues confirm that all system poles lie in the left half of the complex plane, ensuring system stability under different load conditions.

Table 6.

Eigen value analysis.

Scenario A	Scenario B	Scenario C	Scenario C
1.0e + 03 *	1.0e + 03 *	1.0e + 03 *	1.0e + 03 *
-1.4071 + 0.0000i	-1.0230 + 0.0000i	-1.4637 + 0.0000i	-1.1608 + 0.0000i
-1.0000 + 0.0000i	-1.0000 + 0.0000i	-1.0000 + 0.0000i	-1.0000 + 0.0000i
-0.1054 + 0.4647i	-0.6031 + 0.0000i	-0.0930 + 0.5310i	-0.5370 + 0.0000i
-0.1054—0.4647i	-0.3033 + 0.0000i	-0.0930—0.5310i	-0.0293 + 0.3026i
-0.3947 + 0.0000i	-0.1622 + 0.0000i	-0.3772 + 0.0000i	-0.0293—0.3026i
-0.3033 + 0.0000i	-0.0109 + 0.1172i	-0.3033 + 0.0000i	-0.3033 + 0.0000i
-0.1037 + 0.0000i	-0.0109—0.1172i	-0.1023 + 0.0000i	-0.1357 + 0.0000i
-0.0864 + 0.0000i	-0.0864 + 0.0000i	-0.0864 + 0.0000i	-0.0864 + 0.0000i
-0.0294 + 0.0000i	-0.0424 + 0.0000i	-0.0297 + 0.0000i	-0.0378 + 0.0000i
-0.0246 + 0.0000i	-0.0246 + 0.0000i	-0.0246 + 0.0000i	-0.0246 + 0.0000i
-0.0200 + 0.0000i	-0.0200 + 0.0000i	-0.0200 + 0.0000i	-0.0200 + 0.0000i
-0.0125 + 0.0000i	-0.0121 + 0.0000i	-0.0125 + 0.0000i	-0.0125 + 0.0000i
-0.0100 + 0.0000i	-0.0125 + 0.0000i	-0.0100 + 0.0000i	-0.0111 + 0.0000i
-0.0088 + 0.0000i	-0.0100 + 0.0000i	-0.0093 + 0.0000i	-0.0100 + 0.0000i
-0.0071 + 0.0000i	-0.0071 + 0.0000i	-0.0071 + 0.0000i	-0.0071 + 0.0000i
-0.0050 + 0.0000i	-0.0050 + 0.0000i	-0.0050 + 0.0000i	-0.0050 + 0.0000i
-0.0043 + 0.0000i	-0.0043 + 0.0000i	-0.0043 + 0.0000i	-0.0043 + 0.0000i
-0.0033 + 0.0000i	-0.0035 + 0.0000i	-0.0033 + 0.0000i	-0.0032 + 0.0000i
-0.0027 + 0.0000i	-0.0033 + 0.0000i	-0.0031 + 0.0000i	-0.0033 + 0.0000i
-0.0025 + 0.0000i	-0.0025 + 0.0000i	-0.0025 + 0.0000i	-0.0025 + 0.0000i
-0.0021 + 0.0000i	-0.0021 + 0.0000i	-0.0021 + 0.0000i	-0.0021 + 0.0000i
-0.0008 + 0.0003i	-0.0012 + 0.0000i	-0.0010 + 0.0003i	-0.0012 + 0.0000i
-0.0008—0.0003i	-0.0011 + 0.0000i	-0.0010—0.0003i	-0.0008 + 0.0002i
-0.0012 + 0.0000i	-0.0009 + 0.0000i	-0.0012 + 0.0000i	-0.0008—0.0002i
-0.0011 + 0.0000i	-0.0010 + 0.0000i	-0.0011 + 0.0000i	-0.0011 + 0.0000i
-0.0010 + 0.0000i	-0.0007 + 0.0000i	-0.0010 + 0.0000i	-0.0010 + 0.0000i
-0.0004 + 0.0001i	-0.0003 + 0.0001i	-0.0004 + 0.0001i	-0.0004 + 0.0001i
-0.0004—0.0001i	-0.0003—0.0001i	-0.0004—0.0001i	-0.0004—0.0001i
-0.0001 + 0.0000i	-0.0001 + 0.0000i	-0.0001 + 0.0000i	-0.0001 + 0.0000i
-0.0001—0.0000i	-0.0001 + 0.0000i	-0.0001 + 0.0000i	-0.0001 + 0.0000i
-0.0010 + 0.0000i	-0.0010 + 0.0000i	-0.0010 + 0.0000i	-0.0010 + 0.0000i
-0.0125 + 0.0000i	-0.0125 + 0.0000i	-0.0125 + 0.0000i	-0.0125 + 0.0000i
-0.0003 + 0.0000i	-0.0003 + 0.0000i	-0.0003 + 0.0000i	-0.0003 + 0.0000i
-0.0008 + 0.0000i	-0.0008 + 0.0000i	-0.0008 + 0.0000i	-0.0008 + 0.0000i
-0.0014 + 0.0000i	-0.0014 + 0.0000i	-0.0014 + 0.0000i	-0.0014 + 0.0000i
-0.0007 + 0.0000i	-0.0007 + 0.0000i	-0.0007 + 0.0000i	-0.0007 + 0.0000i
-0.0003 + 0.0000i	-0.0003 + 0.0000i	-0.0003 + 0.0000i	-0.0003 + 0.0000i
-0.0003 + 0.0000i	-0.0003 + 0.0000i	-0.0003 + 0.0000i	-0.0003 + 0.0000i
-0.0020 + 0.0000i	-0.0020 + 0.0000i	-0.0020 + 0.0000i	-0.0020 + 0.0000i
-0.0020 + 0.0000i	-0.0020 + 0.0000i	-0.0020 + 0.0000i	-0.0020 + 0.0000i

Scenario B: assess the test system functioning under a positive impulse load disturbance

For this scenario, the optimal values of the controller’s attributes attained via the algorithms are recorded in Table 7. The aapplied inputs to the independent power generation are shown in Fig. 14 and the corresponding power generation is shown on Fig. 15. Figure 16 illustrates the dynamic behavior of the system in response to changes in demand. It shows the frequency deviation and fluctuations in the load demand. Additionally, it presents the controller output and generation load demand response, providing insights into the control actions taken to regulate the system. Figure 17 focuses on the energy storage system, depicting the profiles of EV, BESS, and UC. This figure shows the profile of power generation patterns, highlighting their role in supporting LFC systems. Figure 18 visualizes controlled power generation from various sources, including DEG, BTGU, and MTPG. This figure demonstrates how the power output from these sources is adjusted in response to load changes and frequency deviations, thereby contributing to the overall stability and balance of the system. Exhaustive inspection of Figs. 14,15,16,17,18 imitates that the potent outcome via the CCMGO augmented subordinate controller offers enhanced dynamics with diminished levels of oscillations.

Table 7.

Optimized PID controller gains for test system in Scenario B.

Techniques	Kp1	Ki1	Kd1	Kp2	Ki2	Kd2	λ 1	λ 2	µ
CCMGOPD-PID [Studied]	0.001	-	0.001	0.001	0.001	0.001	-	-	-
CCMGO FOPI-FOPID [Studied]	0.010	0.010	-	0.010	0.010	0.010	0.010	0.010	0.900
CCMGO PID [Studied]	-	-	-	1.000	0.737	1.000	-	-	-
CCMGO FOPID [Proposed]	-	-	-	0.900	0.900	0.899	0.900	-	0.242

Fig. 14.

Applied inputs to the independent power generation (refer Scenario B): (a) Wave Input, (b) Wind Input (c) Solar radiation and (d) PV Radiation.

Fig. 15.

Independent power generation (refer Scenario B): (a)Wave power generation, (b) Wind power generation, (c) Solar power generation (d) PV power generation.

Fig. 16.

LFC response profile (Refer Scenario B) (a) Frequency deviation, (b) Load demand. (c) Controller output and (d) Generation load demand.

Fig. 17.

Profile of the energy storage system (refer Scenario B): (a) EV, (b) BESS and (c)UC.

Fig. 18.

Controlled power generation (refer Scenario B): (a) DEG, (b) BTG, and (c) MTPG

The data provided in Table 8 compares the performance of three control strategies optimized by CCMGO PD-PID, CCMGO PID, and the proposed CCMGO FOPID under Scenario B. The evaluation was based on various performance indices (i.e., ITAE, IAE, ITSE, and ISE) and computational time. The results indicate that CCMGO FOPID exhibits better performance in terms of ITAE (5.097) and IAE (0.245), suggesting a lower overall error and faster response compared to the other algorithms. However, this improved performance comes at the cost of increased computational time (1243 s). In terms of computational efficiency, CCMGO PID requires a computational time of 1278 s, CCMGO PD-PID requires a computational time of 1258 s, CCMGO FOPI-FOPID requires a computational time of 1266 s, and CCMGO FOPID requires a computational time of 1243 s. In this CCMGO FOPID, less computational time was required (refer Table 9).

Table 8.

Comparative measures of performance indices for test system in Scenario B.

Techniques	FOD	Performance indices
	ITAE	IAE	ITSE	ISE
CCMGO PD-PID [Studied]	730.118	19.964	219.168	5.789
CCMGO PD-PID [Studied]	669.044	18.875	183.321	5.160
CCMGO PID [Studied]	8.206	0.767	0.182	0.050
CCMGO FOPID [Proposed]	5.097	0.245	0.025	0.003

Table 9.

Study of computational time for test system Scenario B.

Algorithms	Computational time (s)
CCMGO PID [Studied]	1278
CCMGO PD-PID [Studied]	1258
CCMGO FOPI-FOPID [Studied]	1266
CCMGO FOPID [Proposed]	1243

Scenario C: examine the test system functioning under a step-ramp combination-based load disturbance

For this case study, the optimal values of controller attributes attained via the studied algorithms are recorded in Table 10. The applied inputs to the independent power generation are shown in Fig. 19 and the corresponding power generation is shown on Fig. 20. Figure 21 illustrates the dynamic behavior of the system in response to changes in demand for load. It shows the frequency deviation and fluctuations in the load demand. Additionally, it presents the controller output and generation load demand response, providing insights into the control actions taken to regulate the system. Figure 22 focuses on the energy storage system, depicting the profiles of EV, BESS, and UC. This figure reveals the profile of storage devices, highlighting their role in supporting LFC systems by providing ancillary services. Figure 23 visualizes controlled power generation from various sources, including DEG, BTGU, and MTPG. This figure demonstrates how the power output from these sources is adjusted in response to load changes and frequency deviations, thereby contributing to stability of the system. Exhaustive inspection of Figs.19,20,21,22,23 imitates that the potent outcome via the CCMGO-augmented subordinate controller offers enhanced dynamics with diminished levels of oscillations.

Table 10.

Optimized PID controller gains for test system in Scenario C.

Techniques	Kp1	Ki1	Kd1	Kp2	Ki2	Kd2	λ 1	λ 2	µ
CCMGO PD-PID [Studied]	0.001	-	0.001	0.001	0.001	0.496	-	-	-
CCMGO FOPI-FOPID [Studied]	0.010	0.010	-	0.010	0.010	0.010	0.010	0.010	0.891
CCMGO PID [Studied]	-	-	-	1.000	1.000	0.446	-	-	-
CCMGO FOPID [Proposed]	-	-	-	2.000	2.000	0.993	0.900	-	0.725

Fig. 19.

Applied inputs to the independent power generation (refer Scenario C): (a) Wave Input, (b) Wind Input (c) Solar radiation and (d) PV Radiation.

Fig. 20.

Independent power generation (refer Scenario C): (a)Wave power generation, (b) Wind power generation, (c) Solar power generation (d) PV power generation.

Fig. 21.

LFC response profile (Refer Scenario C) (a) Frequency deviation, (b) Load demand. (c) Controller output and (d) Generation load demand.

Fig. 22.

Profile of the energy storage system (refer Scenario C): (a) EV, (b) BESS and (c)UC.

Fig. 23.

Controlled power generation (refer Scenario C): (a) DEG, (b) BTGU, and (c) MTPG.

The data provided in Table 11 compares the performance of three control strategies optimized by CCMGO: CCMGO PD-PID, CCMGO PID, and the proposed CCMGO FOPID under Scenario C of a test system. The evaluation was based on various performance indices (ITAE, IAE, ITSE, and ISE) and computational time. The results show that CCMGO FOPID outperforms ITAE (20.367) and IAE (0.448), indicating a lower overall error and faster response. In terms of computational efficiency, CCMGO PID requires a computational time of 608 s, CCMGO PD-PID requires a computational time of 592 s, CCMGO FOPI-FOPID requires a computational time of 602 s, and CCMGO FOPID requires a computational time of 586 s. In this CCMGO FOPID, less computational time was required (refer Table 12).

Table 11.

Comparative measures of performance indices for test system Scenario C.

Techniques	FOD	Performance indices
	ITAE	IAE	ITSE	ISE
CCMGO PD-PID [Studied]	3207	44.5	1080	146
CCMGO FOPI-FOPID [Studied]	2929	41.9	911	12.9
CCMGO PID [Studied]	35.341	0.900	0.589	0.044
CCMGO FOPID [Proposed]	20.367	0.448	0.187	0.014

Table 12.

Computational time for test system in Scenario C.

Algorithms	Computational time (s)
CCMGO PID [Studied]	608
CCMGO PD-PID [Studied]	592
CCMGO FOPI-FOPID [Studied]	602
CCMGO FOPID [Proposed]	586

Scenario D: evaluate the test system functioning under stochastic random load perturbation

Fo this case, the optimal values of the controller attributes attained via the algorithms are recorded in Table 13. The applied inputs to the independent power generation are shown in Fig. 24 and the corresponding power generation is shown on Fig. 25. Figure 26 illustrates the dynamic behavior of the system in response to changes in load demand. It shows the frequency deviation and fluctuations in the load demand. Additionally, it presents the controller output and generation load demand response, providing insights into the control actions taken to regulate the system. Figure 27 focuses on the profiles of EV, BESS, and UC. This figure reveals their role in supporting LFC systems by providing ancillary services. Figure 28 visualizes the controlled power generation from various sources, including DEG, BTGU, and MTPG. This figure demonstrates how the power output from these sources is adjusted in response to load changes and frequency deviations, thereby contributing to the overall stability. The outcomes accomplished are shown in Figs.24,25,26,27,28 Exhaustive inspection of Figs. 24,25,26,27,28 imitates that potent outcome via CCMGO augmented subordinate controller offers enhanced dynamics with diminished levels of oscillations. The data provided in Table 14 compares the performance of three control strategies optimized by CCMGO: CCMGO PD-PID, CCMGO PID, and the proposed CCMGO FOPID under Scenario D of a test system. The evaluation was based on various performance indices (ITAE, IAE, ITSE, and ISE) and computational time. The results show that CCMGO FOPID outperforms ITAE (43.491) and IAE (0.264), indicating a lower overall error and faster response. In terms of computational efficiency, CCMGO PID requires a computational time of 928 s, CCMGO PD-PID requires a computational time of 916 s, CCMGO FOPI-FOPID requires a computational time of 924 s, and CCMGO FOPID requires a computational time of 901 s. In this CCMGO FOPID, less computational time was required (refer Table 15).

Table 13.

Optimized PID controller gains for the test system in scenario D.

Techniques	Kp1	Ki1	Kd1	Kp2	Ki2	Kd2	λ 1	λ 2	µ
CCMGO PID [Studied]	-	-	-	1.000	1.000	1.000	-	-	-
CCMGO FOPI-FOPID [Studied]	0.010	0.010	-	0.900	0.900	0.010	0.010	0.900	0.900
CCMGO PD-PID [Studied]	0.010	-	0.010	1.000	1.000	0.010	-	-	-
CCMGO FOPID [Proposed]	-	-	-	3.000	3.000	2.999	0.900	-	0.376

Fig. 24.

Applied inputs to the independent power generation (refer Scenario D): (a) Wave Input, (b) Wind Input (c) Solar radiation and (d) PV Radiation.

Fig. 25.

Independent power generation (refer Scenario D): (a)Wave power generation, (b) Wind power generation, (c) Solar power generation (d) PV power generation.

Fig. 26.

LFC response profile (Refer Scenario D) (a) Frequency deviation, (b) Load demand. (c) Controller output and (d) Generation load demand.

Fig. 27.

Profile of the energy storage system (refer Scenario D): (a) EV, (b) BESS and (c)UC.

Fig. 28.

Controlled power generation (refer Scenario D): (a) DEG, (b) BTGU, and (c) MTPG.

Table 14.

Comparative measures of performance indices for test system Scenario D.

Techniques	FOD	Performance indices
	ITAE	IAE	ITSE	ISE
CCMGO PID [Studied]	145.566	0.909	1.610	0.009
CCMGO FOPI-FOPID [Studied]	116.830	0.703	0.641	0.003
CCMGO PD-PID [Studied]	96.029	0.580	0.553	0.003
CCMGO FOPID [Proposed]	43.491	0.264	0.227	0.001

Table 15.

Computational time for test system in Scenario D.

Algorithms	Computational time (s)
CCMGO PID [Studied]	928
CCMGO PD-PID [Studied]	916
CCMGO FOPI-FOPID [Studied]	924
CCMGO FOPID [Proposed]	901

Limitations and constraints of this study

The study uses a shipboard microgrid with simple power system models, which may not completely depict the power system complexity and problems. In addition, the analysis ignores system dynamics, assumes ideal communication, and assumes already system parameter information. Although simulation provides insights, limited experimental validation may not completely represent real-world implementation challenges and uncertainties. Furthermore, while this research addresses the shipboard microgrid AGC issue and performs well in power system simulations, its efficacy in different configurations, operating circumstances, and optimization issues may require further study. Finally, although the study evaluates the CCMGO algorithm in power system models, a more comprehensive analysis could compare it to other state-of-the-art optimization algorithms used in AGC or related optimization problems to better identify its strengths and weaknesses.

Conclusions and scopes of future work

This study significantly improves the frequency stability of an airport power system during the transition from conventional to renewable cogeneration. Using an enhanced LFC controller and regulated BESS and UC, a resilient structure was provided to sustain the frequency of the power system throughout this transition. CCMGO, which combines ChOA and MGO, is a robust optimization tool for optimizing PID, cascaded PD-PID, FOPI-FOPID, and the proposed FO-PID controllers for parallel DGs, and it outperforms the other controllers. The suggested controller also interacts with regulated generating units to stabilize the system, especially during the high penetration of renewable energy. This paper shows that the algorithm tunes these controllers to improve the microgrid stability. In island mode, the BESS, UC, and PHEV effectively share power with autonomous power production, improving peak demand power availability. This study supports the use of CCMGO algorithm for microgrid frequency control.

Future work will focus on enhancing CCMGO algorithm to optimize power sharing, frequency stability, and dynamic response under varying operational conditions, especially in shipboard power systems. Further, the study can be extended to its scalability for larger microgrids and diverse grid topologies, considering location specific climate conditions and seasonal variability. Additionally, the future study will explore advanced adaptive control methods for broader performance comparisons and improved grid management.

List of symbols

K_AWEC.T_AWEC

Gain and time constant of the AWEC unit (1,0.3S)

K_WT, T_WT

Time constant and gain of the WTG unit (1,1.5 s)

ρ, A_r, λ, β, C_P, V_W, R_b and W_b

Air density, blade-swept area, tip-speed ratio and blade pitch angle, power coefficient, speed of the wind, radius of blades and turbine nominal speed of WTG (1.25 kg/m3 ,1735m2, -23.5 and 3.14.)

V_W, F_AW, x and m_ft

Floater and generator translator velocity, wave forces, translator displacement and total mass (-)

F,ω_AW

Amplitude and angular frequency of wave forces (-)

β_G, β_ω

Damping constant of generator, Damping constant of AWEC (N s/m) (-)

k_C

Spring constant (-)

K_PV, T_PV

Time constant and gain of PV (1.1.8 s)

K_RF, K_RV, K_G, K_T

Gain values of refocus, receiver, governor, and turbine of ST (1, 1, 1, 1)

T_RF, T_RV, T_G, T_T

Time constant of refocus, receiver, governor, and turbine of ST (1.33, 4, 0.08, 1 s)

A,I,ƞH

: Heliostats area, incident solar radiation and constant parameter (-)

K_VA, T_VA, K_BE, T_BE

Valve gain, valve actuator delay, engine gain and time constant of BDEG unit respectively (1, 0.05 s,1.0.0.5 s)

K_BD, K_BG

Participation factor of BDEG and BGTG units respectively (0.5)

K_VP, K_FA

Valve positioner and fuel system actuator gain of MTPG (1, 1)

R₁, R₂, R₃

Droop constants of BGTG, MTPG, &BDEG (Hz/p.u. MW) (2.0)

P_T, P_CP

Output power of turbine and consumed power of compressor (-)

T_VP, T_FA

Valve positioner and fuel system actuator time constant of MTPG (0.8 s, 0.3 s)

K_BT, T_CR, T_BG, X_c, Y_c, b_B, T_BT

Turbine gain, combustion reaction delay, biogas delay, lead time, lag time, valve actuator and discharge time constants of BGTG unit, respectively (1, 0.01 s, 0.23 s, 0.6, 1 s, 0.05, 0.2 s)

K_EV, T_EV

Time constant and gain of the EV (1.0,1.0 s)

K_BESS, T_BESS

Time constant and gain of the BESS (-0.03,0.1 s)

K_UC, T_UC

Time constant and gain of ultra capacitor (-0.7,0.9 s)

Abbreviations

CCMGO

Chaotic chimp-mountain gazelle optimizer

COA

Chimp optimization algorithm

PSO

Particle swarm optimization

LFC

Load frequency control

AGC

Automatic generation control

DGTC

Dynamic gain-tuning control

YSGA

Yin-yang selfish genetic algorithm

BOA

Biogeography-based optimization

GOA

Grasshopper optimization

MPC

Model predictive control

AWS

Archimedes wave swings

AWEC

Archimedes wave energy conversion

PV

Photo-voltaic

BDEDG

Biodiesel engine driven generator

MTPG

Micro turbine power generator

UC

Ultra capacitor

EV

Electric vehicle

ESS

Energy Storage systems

BESS

Battery Energy Storage system

ACE

Area control error

IAE

Integral of absolute error

ITAE

Integral of time-weighted absolute error

FOPID

Fractional-order Proportional plus integral and derivative

PD-PID

Cascaded Proportional derivative - Proportional plus integral and derivative

Frequency deviation

TSM

Territorial solitary males

BHM

Bachelor male herds

MGO

Mountain Gazelle Optimizer

GA

Genetic algorithms

RESs

Renewable energy sources

BESS

Battery energy storage systems

PID

Proportional plus integral derivative

FOPID

Fractional order proportional plus integral and derivative

QSHO

Quasi-opposition selfish-herd optimization

WCA

Water cycle algorithm

DR

Demand response

ChOA

Chimp optimization algorithm

LFR

Linear Fresnel reflector

WTG

Wind turbine generator

ST

Solar tower

BGTG

Biogas turbine generator unit

PTC

Parabolic trough collector

WECS

Wave energy converters

CCGT

Combined cycle gas turbine

FACTS

Flexible AC transmission system

PID

Proportional plus integral derivative

ICμG

Interconnected micro grid

ISE

Integral of square error

ITSE

Integral of time-weighted square error

K_p, K_i, K_d, λ, and µ

Parameter of FOPID controller

K_p, K_i, K_d

Parameter of PID controller

TF

Transfer function

MH

Maternity herds

MSF

Migration to search for food

Author contributions

Odelu P. and Chandan Kumar Shiva designed the simulation work for the study. Odelu P. and Chandan Kumar Shiva wrote most of the paper’s content. Sachidananda Sen, B. Vedik and Chandra Sekhar Reddy write the remaining part of the paper and checked the grammar. All authors read and approved the manuscript.

Funding

The authors did not receive specific funding for this work from any funding agency. This research is solely the authors’ own.

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the study.

Declarations

Competing interests

The authors declare no competing interests.

Ethics Approval

The authors affirm adherence to accepted ethical standards for original studies.

Informed consent

All authors agree with the manuscript’s content and have followed all relevant instructions provided by the journal’s rules, regulations, and editors. All authors have approved the manuscript for submission to this journal and are equally in agreement with its content.

AI Declaration

AI tools were used to enhance the readability and language of the research article but were not employed to replace key tasks that should be performed by the authors, such as interpreting data or conducting scientific calculations.