Load Frequency Regulation for Thermal Units Integrated with Renewable Energy Sources and Energy Storage Systems

Abstract

This paper presents an advanced load frequency regulation strategy for interconnected thermal power systems integrated with renewable energy sources and energy storage systems. A two-area non-reheat thermal power system is investigated, where photovoltaic generation is incorporated in Area 1 and wind turbine generation in Area 2 to assess the impact of renewable penetration on system dynamics and frequency stability. To improve the dynamic response under varying operating conditions, a novel multi-stage TDn(1+PIDn) controller is proposed. The TDn stage enhances transient shaping, while the PIDn stage provides superior damping and steady-state accuracy. The controller parameters are optimally tuned using the pied kingfisher optimizer (PKO) and compared with particle swarm and grey wolf-based optimizers. Furthermore, vanadium redox flow batteries, superconducting magnetic energy storage, and hydrogen–air fuel cells are integrated into the hybrid system to mitigate frequency oscillations caused by renewable intermittency. Offline simulations and real-time validation using the OPAL-RT OP4512 simulator are conducted under different dynamic scenarios. The obtained results demonstrate that the proposed PKO-TDn(1+PIDn) method achieves the best transient performance, for example, reducing the $∆F_1$ overshoot, undershoot and settling time by 28%, 15% and 7.5%, respectively, relative to its closest-performing counterpart while attaining the minimum ITAE value of 0.037309. Consistent improvements are observed across the key performance metrics, confirming the robustness and effectiveness of the scheme for modern hybrid power systems.

1. Introduction

The growing penetration of renewable energy sources (RESs), including photovoltaic (PV) and wind turbine (WT) systems, into conventional power grids introduces significant challenges for load frequency control (LFC) [1,2,3]. The fluctuations associated with renewable generation, coupled with the dynamic behavior of thermal power plants, disturb system frequency and tie-line power [4]. This increases the risk of instability and reduces overall operational efficiency [5]. To ensure that frequency and tie-line power remain within the allowed boundaries, especially during system disturbances, LFC becomes essential. Consequently, LFC continues to be a critical area of focus for researchers aiming to enhance the stability and reliability of renewable-integrated power systems [6,7,8]. Energy storage systems (ESSs) are employed to mitigate deviations in system frequency and tie-line power. They absorb excess power when production is high and release it when production drops, thereby reducing sudden fluctuations and improving frequency regulation and keeping the grid stable [9].

1.1. Literature Review

Over the past decades, numerous power system (PS) designs have been developed to address the challenge of maintaining frequency stability. Prior studies [10,11] primarily examined LFC within single-area systems, whereas subsequent studies [12,13,14] extended the analysis to multi-area systems that incorporate nonlinear dynamics. Additional research efforts [15,16] highlighted issues related to the deregulated electricity network. To mitigate the complexities of frequency regulation in PS, a wide spectrum of control methodologies has been explored. These include model predictive control (MPC) [17], artificial intelligence (AI)-based approaches [18], robust control approaches (RCAs) [19], and fuzzy logic control (FLC) [20,21]. Despite such advancements, the proportional–integral–derivative (PID) controller remains the most extensively adopted solution in academic literature due to its low cost and common availability.

A variety of optimization strategies have been applied to the LFC problem, ultimately giving rise to a PID-based design methodology that has demonstrated reliable performance in practice [22,23]. To manage the challenges associated with complex control structures, researchers have proposed the use of new smart algorithms for tuning controller parameters. For instance, Sekyere et al. in [24] developed a PSO-variant tuning algorithm for PID controllers in systems integrated with RESs and validated its effectiveness across multiple load disturbance scenarios. Gopi et al. in [25] explored rat swarm optimization (RSO) to tune PID parameters in a single-area thermal system, demonstrating faster settling times and minimal frequency error compared to conventional methods. Ali, Aly & Little in [26] introduced a GA-fuzzy logic self-tuning PID controller for LFC in hybrid renewable systems, tuning via multiple heuristic methods and reporting superior transient stability.

Barakat in [27] proposed a FOPID-FOPI control strategy for interconnected PSs, where the tuning of controller parameters was carried out using a chaos game optimization (CGO) algorithm to enhance load frequency regulation performance. Çelik et al. in [28] proposed a cascade (1+PD)-PID controller for load frequency regulation, showing that it outperformed conventional PID by reducing overshoot, improving settling time, and providing greater robustness under diverse operating conditions. Sivalingam et al. in [29] developed a hybrid stochastic fractal search and local unimodal sampling method to tune a multistage PDF-(1+PI) controller, achieving superior frequency stabilization and damping compared with classical techniques. Dash et al. in [30] applied the bat algorithm (BA) to optimize a PD-PID cascade controller for multi-area thermal systems, achieving faster stabilization and reduced frequency deviations compared with classical methods.

Pawan and Anil in [31] highlighted that the wild horse optimizer (WHO) of the FTIDF-(1+I) controller offered superior performance compared with conventional TIDF and FTIDF controllers, particularly in addressing nonlinearities and uncertainties in microgrids. Sekyere et al. in [32] extended ANFIS control by integrating fractional-order dynamics into multi-area systems, improving both frequency regulation and system robustness under nonlinear disturbances.

Optimization techniques serve as key tools for tuning controller parameters and boosting system performance. These include the equilibrium optimization algorithm (EOA) [33], walrus optimization algorithm (WOA) [34], honey badger algorithm (HBA) for two-stage adaptive PID/NN design for microgrid over-frequency control [35], PI-TDF tuned salp swarm algorithm (SSA) [36], multi-objective mantis search algorithm (MOMSA) [37], golden jackal optimization (GJO) [38], grey wolf optimization (GWO) [39], and particle swarm optimization (PSO) [40].

Based on the surveyed literature, certain studies have been particularly valuable in shaping the methodological choices and theoretical orientation of this research. These works offered essential perspectives that were integrated into the design and execution of the current study. For instance, Daraz et al. in [41] examined frequency stabilization in interconnected diverse PS by integrating RESs with ESSs. Their results illustrate how the coordinated deployment of these technologies enhances overall system stability. Jabari et al. in [42] introduced a multistage TDn(1+PI) controller whose parameters were tuned using the bio-dynamic grasshopper optimization algorithm (BDGOA) for LFC in a two-area PV–reheat thermal system; they show that the BDGOA-tuned TDn(1+PI) mitigates frequency and tie-line oscillations and achieves faster settling than PI/PID-family baselines under nonlinearities such as time delay and governor deadband. Jabari et al. in [43] presented a multistage TDn(1+PIDn) controller, whose parameters were optimized using the diligent crow search algorithm (DCSA), for efficient pressure regulation in nonlinear shell-and-tube steam condensers. Their results show that the DCSA-TDn(1+PIDn) outperforms PI, PID, FOPID, and PI-PDn schemes in terms of faster settling, lower overshoot, and reduced steady-state error.

Ojha & Maddela in [44] applied the brown bear optimization algorithm (BOA) to tune both conventional PID controllers and cascaded PI-PDN structures for LFC in a two-area renewable-integrated PS. Their results show that the BOA-tuned controllers deliver enhanced capability to suppress frequency deviations, minimize tie-line power swings, and improve dynamic response compared to PSO and GWO benchmarks under a variety of renewable generation scenarios. Khan et al. in [45] proposed a PIλ(1+PDF) controller for LFC coordinated with hybrid RESs (PV, WT), ESSs (super magnetic energy storage (SMES), vanadium redox flow battery (VRFB), hydrogen aqua electrolyzer fuel cell (HAFC)), high-voltage direct current (HVDC) transmission line, IPFC-FACTS and biodiesel generator across different scenarios on a four-area PS with zebra optimization algorithm (ZOA)-based tuning, which yielded markedly lower overshoot/undershoot and shorter settling times than PID/FOPID under high-RES variability. Raj and Shankar in [46] proposed a cascaded fractional-order integral-derivative and tilt FID-T controller for LFC in a two-area PS with RESs and electric vehicle (EV) participation, tuned its gains via a modified quasi-opposition reptile search algorithm (QORSA), and validated practicality on an OPAL-RT hardware-in-the-loop platform, reporting improved damping and faster settling versus conventional techniques.

This study applies the pied kingfisher optimizer (PKO), a recently introduced metaheuristic optimization technique, to optimize the parameters of a newly developed multi-stage TDn(1+PIDn) controller. PKO models collective search patterns and cooperative prey-acquisition tactics exhibited by pied kingfishers [47].

The algorithm is divided into three key phases: (1) perching and hovering for prey exploration and diversification, (2) diving for prey exploitation and intensification, and (3) fostering symbiotic relations. The algorithm harmonizes exploration and exploitation, promoting efficient search-space coverage and reducing vulnerability to local optima.

1.2. Research Gap and Motivation

Although notable progress has been made in advancing LFC strategies for hybrid PSs, several unresolved challenges persist. Most previous studies have considered PV, WT, and thermal power plants separately, without fully addressing their complex interactions in hybrid systems. In such hybrid settings, RES-related fluctuations often perturb system frequency and tie-line power. Furthermore, optimization methods such as PSO and GWO, though widely employed for controller tuning, exhibit inherent shortcomings. In recent LFC tuning studies, PSO has been shown to remain liable to local optima with limited search accuracy [48], while GWO has been reported to stall at local optima on several benchmark functions and to converge markedly more slowly than newer optimizers and to reduce solution accuracy [49,50]. Their susceptibility to slow convergence rates and limited exploration capability often constrains their effectiveness in handling the nonlinearities and uncertainties present in hybrid systems, including governor deadband effects, boiler dynamics, and time delays. Additionally, while multi-stage and fractional-order controllers have shown potential to strengthen robustness and improve transient behavior, their application within PV-WT-thermal hybrid systems has not been comprehensively studied. In particular, the ability of such advanced controllers to simultaneously deliver stable transient and steady-state responses under nonlinear operating conditions and variable system parameters remains underexplored. PKO, by contrast, was benchmarked in its original work against thirty-nine functions and eight engineering problems and showed a stronger balance between exploration and exploitation [47].

Motivated by the identified gaps, this research applies a recently introduced PKO-based optimization technique to tune a multi-stage TDn(1+PIDn) controller for LFC in a two-area hybrid PS comprising thermal, PV, and WT, supported by an ESS. The adoption of PKO ensures faster convergence and robust global search capabilities, thereby avoiding early trapping in local solutions and improving controller tuning efficiency compared to conventional algorithms. The proposed TDn(1+PIDn) controller enhances control flexibility by combining multi-stage and fractional-order features with the well-established PID structure, leading to improved transient response, reduced frequency deviations, and superior steady-state accuracy. Furthermore, the inclusion of PV and WT increases renewable penetration, while the ESS mitigates fluctuations in renewable output, provides rapid frequency support, and strengthens overall grid resilience. By addressing these challenges and leveraging the combined advantages of advanced optimization, robust control design, and energy storage integration, this study aims to deliver a comprehensive solution that significantly enhances stability, reliability, and practical applicability of modern hybrid power grids under nonlinear and uncertain operating conditions. The main findings from these recent studies are summarized in

Table 1. Summary of literature review and proposed methodology on LFC.

Article	Controller	Optimization Algorithm	No. of Areas	RES	ESS	Key Findings
[10]	MPC-LQG	COA	1-area and 2-area (thermal non-reheat)	WT	-	COA-MPC-LQG delivers lower-frequency/tie-line deviations and faster settling under load disturbances and varying wind penetration, outperforming integral/MPC and PSO/GWO/ALO-tuned counterparts
[12]	I-SMC	GTO	1-area (microgrid Wind-diesel)	WT	-	GTO-I-SMC improves frequency regulation in a wind–diesel autonomous microgrid, yielding smaller frequency deviations, shorter settling times, and lower integral error indices
[21]	Fuzzy PID	GSA	2-area (thermal reheat)	-	-	GSA-tuned PID-Fuzzy for a two-area reheat thermal system yields smaller frequency/tie-line deviations and lower integral error indices, outperforming PSO- and ABC-tuned counterparts under severe load disturbances
[27]	FOPID-FOPI	CGO	2-area (thermal non-reheat, thermal reheat-hydro-gas) and 3-area (thermal-thermal-hydro)	-	-	GO-FOPID-FOPI decreases deviation magnitudes and faster return to steady state across four IPS test systems: two-area non-reheat (±GDB), two-area reheater-hydro-gas multi-unit, and three-area hydrothermal with GRC, outperforming recent soft-computing controllers under modeled nonlinearities
[30]	PD-PID	BA	3-area (thermal reheat)	-	-	BA-PD–PID on a multi-area thermal system with single-reheat units and GRC delivers smaller frequency/tie-line deviations and quicker recovery than PI/PID, BA tuning outperforms GA/BF alternatives
[31]	FTIDF-(1+I)	WHO	1-area (microgrid solar-wind-diesel)	PV-WT	FESS	WHO-FTIDF-(1+I) for an islanded microgrid (PV + WTG + diesel + flywheel ESU) achieves lower-frequency excursions and faster recovery than conventional alternatives under RES variability and load changes
[32]	FO-ANFIS	ADIWACO	3-area (thermal reheat)	PV-WT	-	FO-ANFIS (2-input & 3-input) for three-area RES system, trained from FOPI-FOPIDN (ADIWACO-tuned) data; 3-input FO-ANFIS achieves lower settling in area-1 with minimal over/undershoot, and near-zero tie-line overshoot/undershoot. Robust under GDB, delay and parameter variations (lowest integral error indices across scenarios besting FOPI-FOPIDN and IO-ANFIS variants)
[36]	PI-TDF	SSA	1-area (diesel-solar-wind-fuel cell-aqua electrolyzer)	PV-WT	BESS-FESS-UC	SSA-PI-TDF for a hybrid power system achieves lower frequency excursions and error indices than SSA-tuned PI/PID/TID/TIDF/P-TIDF comparators; validation under GRC nonlinearity, subsystem switching (BESS/FESS/diesel on/off), and parameter variations shows robust, less-sensitive performance
[37]	PID	MOMSA	2-area (thermal non-reheat)	PV-WT	EV	MOMSA-PID for an interconnected microgrid with RESs + EVs markedly attenuates frequency oscillations and tie-line power fluctuations, outperforming PSO/MPA/GA, with faster transient response and improved stability across scenarios
[38]	PID	GJO	2-area (thermal-hydro-nuclear, thermal-gas)	PV-WT	SMES	GJO-PID on a two-area multi-source system: thermal, hydro, nuclear, gas, wind, and PV delivers smaller frequency & tie-line excursions versus PSO, WOA, SSA, ALO, Harris’ hawks, and TLBO, robustness is verified under variable step loads, parameter variations, and nonlinearities (GRC, GDB), stability margins are verified by frequency-response analysis, with fewer tuning iterations
[41]	FOTIDD2	MOA	2-area (marine bio-diesel-sea wave energy)	PV-WT	BESS-CES	MOA-FOTIDD2 on a two-area diverse (sea-wave, PV, wind, biodiesel, BESS, and CES) with communication delay achieves lower frequency and tie-line excursions, faster settling, and improved damping than PID baselines, while MOA consistently surpasses SCA/FOA/GWO in tuning effectiveness and robustness
[42]	TDn(1+PI)	BDGOA	2-area (PV-thermal reheat)	PV	-	The BDGOA-TDn(1+PI) proposed design retains stability and tracking quality under delays and uncertainty and outperforms standard controllers in reducing frequency and tie-line power fluctuations
[48]	PID	GOA-PSO	1-area (thermal non-reheat-solar-wind-EV) and 2-area (thermal non-reheat)	PV-WT	EV	Standalone PSO exhibits premature convergence and local optima entrapment in renewable-integrated LFC; the proposed GOA-PSO hybrid exploits GOA’s exploratory strength to escape local optima, achieving marked reductions in overshoot, undershoot, and settling time over conventional PSO-PID
[49]	FOPID	MPO	2-area and 4-area (thermal reheat-hydro-wind)	WT	RFB-HAFC	Conventional metaheuristics suffer limited population diversity leading to premature convergence; the proposed MPO avoids trapping in local optima taking advantage over GWO through memory-based election and random exploration strategies, reducing ITAE by 8.023% and 20.071% in two- and four-microgrid systems, respectively
[50]	FOPID	ASIA	3-area (thermal non-reheat-wind-hydro)	WT	-	GA, GWO, SCIA, and ASIA are benchmarked comparatively for PID/FOPID tuning in three-area LFC; ISE-based GA/SCIA and ITSE-based GWO/ASIA-tuned FOPID controllers deliver superior regulation under parameter perturbations and varying load conditions; ASIA/SCIA are highlighted against GA/GWO
Proposed	TDn(1+PIDn)	PKO	2-area (thermal non-reheat)	PV-WT	VRFB-SMES-HAFC	PKO-TDn(1+PIDn) on a two-area system (area 1: thermal non-reheat, PV, VRFB, SMES, area 2: thermal non-reheat, WT, VRFB, HAFC) achieves lower frequency and tie-line excursions, faster settling, and improved damping than PID baselines, while PKO consistently surpasses GWO/PSO in tuning effectiveness and robustness; validating the results with OPAL-RT

.

1.3. Contribution and Paper Organization

This research provides several important contributions to the topic of LFC in PS:

• Methodological contribution: a multi-stage TDn(1+PIDn) controller is developed for LFC of a two-area non-reheat thermal system, where the TDn stage shapes the fast transient, while the (1+PIDn) stage delivers damping and steady-state accuracy, giving faster recovery than single-stage controllers.
• Optimization contribution: a recently introduced PKO algorithm is applied to tune this controller.
• Influence of nonlinearity sources: a systematic examination of generation rate
• constraints (GRC), governor dead band (GDB), and communication time delays (CTD) quantifies their effect on frequency regulation performance, offering essential insights into their function in LFC problem.
• Renewable energy integration: the effect of PV (Area 1) and WT (Area 2) penetration on frequency and tie-line dynamics is quantified across uniform and random RES profiles and different load cases.
• Energy storage integration: the measured benefit of VRFB/SMES (Area 1) and VRFB/HAFC (Area 2) is quantified across all scenarios and showed the improvement in system’s performance that exposed to fluctuations in frequency due to the intermittent nature of solar and wind.
• Validation of the aforementioned cases with OPAL-RT simulator: the PS is experimentally analyzed through the OPAL-RT simulator to ensure the validation of numerical simulation by integrating the fidelity of physical simulation with the adaptability of the numerical simulation.

This paper is organized as follows: Section 2 presents the dynamic model of test system. Section 3 presents the proposed TDn(1+PIDn) controller. Section 4 provides the optimization technique, PKO and objective function. Section 5 shows the results. Lastly, Section 6 presents the conclusions.

2. Dynamic Model of Test System

2.1. Two-Area Power System

A two-area interconnected non-reheat thermal–thermal PS (case 1) is adopted, as illustrated in Figure 1. This configuration is widely used in [51,52,53] for the study of the dynamic behavior of interconnected systems under both steady-state and transient operating scenarios. Each area has generating units with a total installed capacity of 2000 MW, operating at a nominal base load of 1000 MW. Both areas incorporate conventional control components, including a speed governor, a non-reheat steam turbine, and PS units. Nominal values of the model parameters are from [51,52] and are presented in

Table 2. Nominal values of system parameters.

Parameters	Values	Parameters	Values
F	60 Hz	a12	−1
Pr1 = Pr2	2000 MW	B1 = B2	0.425 pu MW/Hz
PD	1000 MW	R1	2.4 Hz/pu MW
GRC	±0.05 or ±0.025	KP1 = KP2	120
TP1 = TP2	20 s	TT1 = TT2	0.3 s
T12	0.545	TG1 = TG2	0.08 s

. The nonlinear turbine model shown in Figure 2, which incorporates a GRC with limits of α = ±0.05 pu/s and α = ±0.025 pu/s, replaces the linear non-reheat turbine transfer function of Figure 1 to account for the physical ramp-rate limitation of the steam turbine under realistic operating conditions. The GDB represents a nonlinear characteristic of the governor system, defining a range of frequency deviations within which no corrective valve action is initiated. This behavior originates from mechanical valve-positioning tolerances and intentional filtering mechanisms designed to limit unnecessary actuator movements. Consequently, the governor response exhibits an inherent delay that may increase phase lag and reduce the effectiveness of LFC. To account for this practical nonlinearity, the GDB is incorporated into the governor model through a modified transfer function, expressed as $(\left(-0.2/\pi \right)s+0.8)/(T_{G}s+1)$, which replaces the linear governor transfer function of Figure 1, allowing for a more realistic representation of valve dynamics under small frequency variations [27,54,55]. In addition, CTD is a characteristic of modern wide-area LFC systems due to the use of communication networks.

This delay is associated with the transmission of measured system variables to the control center and the subsequent delivery of control signals to generation units. Following a load disturbance, these communication lags affect the timely generation and execution of corrective control actions, thereby affecting the performance of the PS. To account for this practical nonlinearity, the CTD is incorporated into the overall plant model after the controller in Figure 1 using the exponential function $e^{-s\tau }$, where $\tau$ denotes the communication delay time, considered to be 0.1 s. This representation enables a more realistic assessment of system dynamics and controller performance under networked operating conditions [55].

RESs are added to the PS for environmental and economic aspects; therefore, PV in area 1 and WT in area 2 (case 2) are adopted, as illustrated in Figure 3. To mitigate the fluctuations resulting from RESs, ESSs are added to PS; therefore, VRFB and SMES in area 1 and VRFB and HAFC in area 2 (case 3) are adopted, as illustrated in Figure 4. The transfer function of the governor $G_G\left(s\right)$, turbine $G_T\left(s\right)$, and generator $G_P\left(s\right)$ are represented as follows [56]:

$$G_G\left(s\right)={\displaystyle \frac{\Delta P_V\left(s\right)}{\Delta P_G\left(s\right)}}={\displaystyle \frac{1}{1+s{T}_G}}.$$

(1)

The speed governor has two inputs, $\Delta P_{ref}\left(s\right)$ and $\Delta F,$ with one output, $\Delta P_G\left(s\right),$ given by:

$$\Delta P_G\left(s\right)=\Delta P_{ref}\left(s\right)-{\displaystyle \frac{1}{R}}\Delta F \left(s\right),$$

(2)

$$G_T\left(s\right)={\displaystyle \frac{\Delta P_T\left(s\right)}{\Delta P_V\left(s\right)}}={\displaystyle \frac{1}{1+s{T}_T}},$$

(3)

$$G_P\left(s\right)={\displaystyle \frac{K_P}{1+s{T}_P}},$$

(4)

where $K_P=1/D$ and $T_P=2H/FD$.

$D$ is the load frequency dependency parameter given in (5).

$$D={\displaystyle \frac{Nominal load \left(P_D\right)}{Nominal frequency \left(F\right)}},$$

(5)

where $T_G$ is the generator time constant, $T_T$ is the turbine time constant, $K_P$ is the power system gain constant, $T_P$ is the power system time constant, $P_D$ is the nominal load, $H$ is the inertia constant of the generator, and $F$ is the nominal frequency.

The generator has two inputs, $\Delta P_T\left(s\right)$ and $\Delta P_D\left(s\right),$ with one output, $\Delta F\left(s\right),$ given by:

$$\Delta F\left(s\right)=G_P\left(s\right)\left[\Delta P_T\left(s\right)-\Delta P_D\left(s\right)\right].$$

(6)

The calculation of area control error (ACE) values for both areas is given in (7) and (8).

$${ACE}_1=B_1∆F_1+\Delta P_{tie},$$

(7)

$${ACE}_2=B_2∆F_2+\Delta P_{tie}.$$

(8)

2.2. Renewable Energy Sources

2.2.1. PV Model

Uniform Model

The dynamic behavior of PV can be represented using a first-order transfer function, which effectively captures the system’s transient response to variations in irradiance.

Under the assumption of constant ambient temperature, the generated electrical power is linearly correlated with the incident solar radiation. Consequently, the PV system can be modeled as follows [44]:

$$G_{PV}\left(s\right)={\displaystyle \frac{K_{PV}}{1+s{T}_{PV}}},$$

(9)

where $K_{PV}$ denotes the photovoltaic system gain constant, and $T_{PV}$ represents the photovoltaic system time constant reflecting the PV module’s response delay.

This simplified linear model enables analytical control design and stability analysis of the PV-integrated PS.

Table 3. Values of uniform model of PV and WT.

Parameters	Values	Parameters	Values
KPV	1	KWT	1
TPV	1.3 s	TWT	1.5 s

shows the values of these parameters [44].

Random Model

PV exhibits power variability owing to fluctuations in solar radiation and ambient temperature.

To accurately emulate this dynamic behavior in simulation studies, random power models are typically adopted to represent the temporal variability in PV output. PV generation, as depicted in Figure 5, employs a stochastic model adapted from [45] corresponding to a 6 MW solar facility. The dynamic behavior of the PV plant can be expressed through a first-order transfer function [45], where $\Delta P_{Solar}$ characterizes the variance in PV output power.

$$\Delta P_{PV}={\displaystyle \frac{K_{PV}}{1+s{T}_{PV}}}\left(\Delta P_{Solar}\right).$$

(10)

2.2.2. WT Model

Uniform Model

The dynamic performance of WT can be approximated using a first-order transfer function that characterizes the system’s response to variations in wind speed. Under the assumption of uniform air density and negligible mechanical losses, the electrical power generated by the turbine is directly proportional to the cube of the effective wind velocity. Accordingly, the transfer function of the WT can be represented as follows [44]:

$$G_{WT}\left(s\right)={\displaystyle \frac{K_{WT}}{1+s{T}_{WT}}},$$

(11)

where $K_{WT}$ denotes the wind system gain constant, and $T_{WT}$ represents the wind system time constant reflecting the WT’s response delay. This simplified linear model enables analytical control design and stability analysis of WT-integrated PS. Table 3 shows the values of these parameters [44].

Random Model

With the growing urgency to address global energy sustainability, RESs have gained significant attention, with wind power emerging as a major contributor to modern power systems. In this study, a 5 MW offshore WT model developed by the National Renewable Energy Laboratory (NREL) under the Wind Energy Technologies Programme of the U.S. Department of Energy was utilized. This model is widely recognized as a benchmark for offshore WT research, design optimization, and performance evaluation across diverse operating conditions. The 5 MW capacity represents the industry standard for large-scale offshore turbines, ensuring compatibility with international modeling frameworks and cost-efficiency analyses. Due to the substantial infrastructure and installation costs associated with deep-water deployment, turbines in this category generally possess ratings of 5 MW or higher to achieve favorable economic performance [57,58,59].

The effectiveness of wind energy generation primarily depends on the transformation of wind’s kinetic energy into electrical energy. This process starts with the turbine blades capturing the kinetic energy of the wind and converting it into mechanical energy at the rotor shaft. Mechanical energy is subsequently transformed into electrical energy through the generator unit. The total aerodynamic power extracted from the wind can be calculated from (12)–(15) [60,61].

$$P_{Wind}={\displaystyle \frac{1}{2}}.\rho .A_T.V^3.C_P\left({\lambda }_T,\beta \right),$$

(12)

$$C_P\left({\lambda }_T,\beta \right)=C_1\left({\displaystyle \frac{C_2}{{\lambda }_1}}-C_3\beta -C_4{\beta }^2\right)e^{-{\displaystyle \frac{C_5}{{\lambda }_1}}}+C_6{\lambda }_T,$$

(13)

$${\displaystyle \frac{1}{{\lambda }_1}}=\left({\displaystyle \frac{1}{{\lambda }_T+0.08\beta }}-{\displaystyle \frac{0.035}{{\beta }^3+1}}\right),$$

(14)

$${\lambda }_T={\displaystyle \frac{V_{TP}}{V}}={\displaystyle \frac{{R\omega }_r}{V}},$$

(15)

where $P_{Wind}$ represents the wind turbine power, $A_T$ is the swept area of the turbine, $V$ is the wind speed, $\rho$ is the air density, $C_P$ is the turbine performance coefficient, $\mathsf{\beta }$ is the blade pitch angle, ${\lambda }_T$ is the tip speed ratio, ${\lambda }_1$ is the effective tip speed ratio, $V_{TP}$ is the tip speed, $R$ is the blade radius, and ${\omega }_r$ is the rotor speed. $C_P$ represents a nonlinear function that depends on both the aerodynamic configuration of the turbine blades and the operational conditions of WT. It characterizes the conversion efficiency by which the kinetic energy of the wind is transformed into mechanical energy.

The coefficients ${\mathrm{C}}_1$ through ${\mathrm{C}}_6$ are determined by the blade’s shape and its aerodynamic properties. Notably, the turbine attains its maximum power output when the blade pitch angle $\mathsf{\beta }$ equals zero degrees, corresponding to the optimal aerodynamic alignment.

The technical specifications of the 5 MW NREL wind turbine model are taken from [45,60,62,63]. The radius of the swept area (R₁) is 61.5 m, while the rotor diameter (D) is 126 m and the hub diameter (D_H) is 3 m. The rotor operates at a speed (ω_r) of 12.1 rpm. The cut-in wind speed (V_in) is 3 m/s, whereas the rated wind speed (V) reaches 11.4 m/s. The air density (ρ) is taken as 1.225 kg/m³. In addition, the optimal tip-speed ratio (λ_T) is 7.55. The power coefficient constants (C₁–C₆) are given as 0.517, 116, 0.4, 5, 21, and 0.0068, respectively.

The maximum power of the NREL- 5 MW WT can be evaluated using (16)–(18).

$$C_{P\_max}=C_P\left({\lambda }_T\right)=0.517\left({\displaystyle \frac{116}{{\lambda }_1}}-5\right)e^{-{\displaystyle \frac{21}{{\lambda }_1}}}+0.0068{\lambda }_T,$$

(16)

$${\displaystyle \frac{1}{{\lambda }_1}}=\left({\displaystyle \frac{1}{{\lambda }_T}}-0.035\right),$$

(17)

$$P_{Wind\_max}={\displaystyle \frac{1}{2}}.\rho .A_T.V^3.C_{P\_max}.$$

(18)

A model of fluctuating WT for frequency control is shown in Figure 6. The random wind power is obtained by multiplying the wind speed standard deviation by the random output fluctuation from the white noise block [23,64,65]. Figure 7 shows the WT model of the NREL 5 MW with maximum power generation [62].

2.3. Energy Storage Systems

2.3.1. Vanadium Redox Flow Battery Model

VRFBs emerge as an encouraging large-scale energy storage system within renewable power systems owing to their high energy density, long cycle life, and rapid response features.

These attributes make VRFBs particularly suitable for mitigating fluctuations and enhancing overall grid reliability and stability. The transfer function of the VRFB employed for load frequency control is illustrated in Figure 8. In this model, K_RFB denotes the vanadium redox flow battery gain constant, while T_dRFB is the vanadium redox flow battery time delay constant, and T_cRFB is the resetting vanadium redox flow battery time constant.

Table 4. Values of ESS parameters.

Parameters	Values	Parameters	Values
KRFB	1	TdRFB	0 s
TcRFB	0.3 s	KSMES	0.2035
TSMES	0.03 s	T1	0.2333 s
T2	0.016 s	T3	0.7087 s
T4	0.2481 s	Kn	0.6
Kae	0.002	Kfc	0.01
Tae	0.5 s	Tfc	4 s

shows the values of these parameters. This transfer function-based formulation provides an analytically tractable means of characterizing the electrochemical–dynamic behavior of the battery within load frequency control frameworks [66]. It reproduces the dominant charge and discharge response relevant to frequency support, but it does not capture the accurate electrochemical behavior or the battery’s state of charge. Richer electrochemical and state-of-charge estimation methods, of the kind surveyed in [67], would support a more detailed handling over long regulation terms.

2.3.2. Super Magnetic Energy Storage Model

SMES systems are instrumental in maintaining frequency stability within modern power grids. These units regulate system dynamics by supplying or absorbing electrical power as needed by the grid, thereby mitigating frequency deviations and enhancing overall reliability. The transfer function representation of the SMES, as demonstrated in Figure 9, is developed using a two-stage lead–lag compensator structure [68]. In this model, T₁, T₂, T₃, and T₄ denote the lead-lag blocks time constants. The parameter K_SMES is the super magnetic energy storage gain constant, and T_SMES defines the super magnetic energy storage time constant. Table 4 shows the values of these parameters.

2.3.3. Hydrogen Aqua Electrolyzer Fuel Cell Model

Hydrogen-based energy conversion systems are increasingly recognized as key enablers of sustainable power generation, offering a viable pathway to reduce reliance on fossil fuels.

Among these systems, the integration of HAFC into hybrid PS has proven effective in enhancing grid frequency regulation and power quality. In such arrangements, surplus renewable electricity is converted into hydrogen through electrolysis, stored for extended periods, and later reconverted to electrical energy by the fuel cell. This closed-loop process enables more flexible utilization of renewable resources and contributes to system reliability. Beyond its operational role, the HAFC architecture delivers multiple advantages, including reduction in greenhouse gas emissions, improved energy recovery, and reduced maintenance costs, with reported round-trip efficiencies typically ranging between 20% and 40% [69].

Within a hybrid renewable framework, the power generated by RESs is dynamically balanced through coordinated operation of the electrolyzer and fuel cell components to sustain the overall system equilibrium. To capture the HAFC’s dynamic behavior, the coupled electrolyzer–fuel cell structure can be reasonably approximated using a first-order transfer function model.

In this study, power fluctuation is used as the input disturbance driving the HAFC response, as shown in Figure 10 [70]. The transfer function of HAFC is expressed in (19).

$$\Delta P_{HAFC}=\left[\left({\displaystyle \frac{K_{fc}}{T_{fc}.s+1}}·{\displaystyle \frac{K_{ae}}{T_{ae}.s+1}}·{\displaystyle \frac{{1-K}_n}{1}}\right)+K_n\right]·\Delta F,$$

(19)

where $K_{fc}$ and $T_{fc}$ define the gain and time constants of fuel cell, and $K_{ae}$ and $T_{ae}$ correspond to the gain and time constants of aqua electrolyzer. $K_n$ is a conversion factor that equals

$$K_n={\displaystyle \frac{P_{H_2}}{P_{WT}+P_{PV}}},$$

(20)

and is assumed to be 0.6, where $P_{H_2}$ defines the power output from the hydrogen fuel cell, $P_{WT}$ is the power generated from WT, and $P_{PV}$ is the power generated from PV.

The HAFC is captured by a coupled first-order electrolyzer–fuel-cell model that reproduces the fast, bounded power exchange relevant to the LFC problem. This representation omits the reversible voltage loss and the recovery that follows it; both affect efficiency and durability over longer terms [71]. Additionally, this representation does not resolve the internal thermal field of the stack. In a physical PEMFC, the speed at which power can be ramped is tied to the temperature distribution across the membrane and to the effectiveness of the cooling circuit; so, a rapid regulation command may be followed by a short thermal-inertia lag before the electrical output fully settles. The recent review of liquid-cooled PEMFC thermal management by Song et al. [72] sets out these effects and the associated control challenges in detail. Coupling the electrical model with a thermal-management sub-model would give a more faithful account of the achievable ramp rate and of the long-term efficiency and service life of the stack.

3. Proposed Multi-Stage Controller

The proposed TDn(1+PIDn) control strategy is designed to enhance the dynamic performance of interconnected PS in LFC applications [43]. Accurate frequency regulation is crucial to maintaining the stability and reliability of modern grids under varying load profiles and renewable energy variations. By integrating a nonlinear tilt-derivative component with a filtered PID block, the controller achieves improved stability, reduced overshoot, and faster transient response under varying operating conditions. The coordinated interaction between both control stages enables superior robustness against external disturbances and parameter uncertainties.

3.1. TDn Controller

The TDn section represents an advanced derivative element that incorporates a nonlinear tilt function and an n-filter. Unlike conventional derivative terms that rely solely on the rate of change of the error signal, the TDn introduces nonlinear dynamics into the derivative path. This modification enhances the controller’s responsiveness to rapid frequency deviations in the PS, enabling faster corrective actions and improved damping of transient oscillations. Moreover, the filter effectively mitigates the noise amplification problem often observed in conventional derivative terms, thereby reinforcing the overall robustness and stability of the load frequency control system under fluctuating grid conditions.

3.2. PIDn Controller

The PIDn section integrates proportional, integral, and derivative control actions, each enhanced with an n-filter to suppress noise and maintain stability. In load frequency control applications, this nonlinear configuration enables adaptive tuning under diverse operating conditions. It ensures effective mitigation of transient frequency deviations, accurate steady-state regulation, and overall improvement in system reliability and power balance.

3.3. Interaction Between TDn and PIDn

The coordinated operation of the TDn and PIDn sections enables effective control of both transient and steady-state frequency deviations. The TDn part ensures rapid stabilization following load disturbances and RES fluctuations, while the PIDn section refines the response to maintain frequency near its nominal value. Their combined action strengthens system stability and robustness against uncertainties in load frequency control applications.

3.4. Multi-Stage Structure

The proposed TDn(1+PIDn) controller introduces a multi-stage configuration that markedly enhances the stability and dynamic response of interconnected PS under load and RES fluctuations. This architecture allows for progressive refinement of control actions across distinct operating phases, enabling precise frequency regulation and improved adaptability to nonlinear system dynamics. The initial control phase focuses on rapid frequency restoration, mitigating transient deviations and suppressing oscillations resulting from sudden load or RES disturbances. Subsequently, the intermediate correction phase ensures smooth transitional behavior by addressing moderate deviations and nonlinearities within the PS. The final stage provides high-accuracy steady-state regulation, reducing residual frequency errors and maintaining long-term system stability even under persistent disturbances.

By integrating these stages, the TDn(1+PIDn) scheme achieves an optimal compromise between responsiveness and robustness. Its adaptive characteristics allow for consistent performance across a wide range of operating conditions, outperforming conventional controllers such as PID, the common configuration. This advanced design not only strengthens the frequency restoration capability but also improves tie-line power regulation, making the proposed controller particularly suitable for modern hybrid PS where renewable integration introduces strong nonlinearities and dynamic uncertainties. The block diagram of the proposed controller is shown in Figure 11.

The open-loop transfer function for the first-stage controller is as follows:

$$G_{TDn}\left(s\right)={\displaystyle \frac{\Delta U_1\left(s\right)}{\Delta F\left(s\right)}}={\displaystyle \frac{K_T}{s^{\frac{1}{n}}}}+\left[K_D\ast \left({\displaystyle \frac{Ns}{N+s}}\right)\right].$$

(21)

The open-loop transfer function of the second stage controller is given in (22).

$$G_{(1+PIDn)}\left(s\right)={\displaystyle \frac{\Delta U\left(s\right)}{\Delta U_1\left(s\right)}}=1+\left[K_P+{\displaystyle \frac{K_I}{s}}+{\displaystyle \frac{K_{DD}NNs}{s+NN}}\right].$$

(22)

In order to explain the open-loop representation of the proposed controller, (23) can be formulated as follows:

$$G_{TDn(1+PIDn)}\left(s\right)={\displaystyle \frac{\Delta U\left(s\right)}{\Delta F\left(s\right)}}=\left[{\displaystyle \frac{K_T}{s^{\frac{1}{n}}}}+K_D\ast \left({\displaystyle \frac{Ns}{N+s}}\right)\right]\ast \left[1+\left(K_P+{\displaystyle \frac{K_I}{s}}+{\displaystyle \frac{K_{DD}NNs}{s+NN}}\right)\right],$$

(23)

where $K_D$, $K_{DD}$, $K_P$, and $K_I$ are the derivative, proportional and integral gains of controller, while $N$, $NN$, and $n$ are the controller filters.

3.5. Frequency-Domain Interpretation of the Proposed Controller

To move beyond empirical demonstration and provide a mechanistic explanation for the performance gains achieved by the proposed structure, the open-loop characteristics of each stage are examined in the frequency domain. The TDn stage operates through two complementary elements: the tilt element $K_T/s^(1/\mathrm{n})$ introduces a fractional-order gain slope at low frequencies, enhancing attenuation of slow-varying disturbances such as sustained load perturbations and gradual renewable power fluctuations, while the filtered derivative element $K_D\ast \left(Ns/(N+s\right))$ injects phase advancement in the mid-frequency band, enabling rapid suppression of transient deviations before they propagate further; the filter $N$ simultaneously prevents amplification of high-frequency measurement noise. The succeeding (1+PIDn) stage operates on the pre-shaped signal to accomplish steady-state and bandwidth objectives: the integral action eliminates residual steady-state error, the proportional gain governs the closed-loop bandwidth, and the filtered derivative contributes additional damping without reintroducing the noise attenuated upstream, while the direct unity feed-through path preserves the frequency content already shaped by the TDn stage. The structural consequence of this two-stage arrangement is that fast transient shaping and precise steady-state correction are assigned to dedicated stages operating in complementary frequency bands, a functional separation that a single conventional PID structure cannot achieve simultaneously.

Table 5. Optimal ITAE and parameters of proposed and other controllers.

Algorithm- Controller	ITAE	Area	Parameters
			${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN	λ	μ	PW	DW
PKO-TDn(1+PIDn)	0.0373	Area 1	1.96	0.627	0.992	1.006	1.966	0	2	286	-	-	-	-
		Area 2	0	1.999	0.694	1000	2	1.985	1.99	464	-	-	-	-
GWO-TDn(1+PIDn)	0.0544	Area 1	2	0.237	0.001	117.9	0.182	2	0.26	10.8	-	-	-	-
		Area 2	1.54	0.686	0.419	11.96	0.920	0.417	1.94	786	-	-	-	-
PSO-TDn(1+PIDn)	0.05550	Area 1	2	0.047	0	947.5	0.309	2	0.37	85.5	-	-	-	-
		Area 2	2	1.370	0.427	727.5	1.657	0.368	0.08	406	-	-	-	-
PKO-PID	0.12696	Area 1	-	0.394	-	-	1.084	2	-	-	-	-	-	-
		Area 2	-	0.543	-	-	1.997	0.057	-	-	-	-	-	-
GWO-PID	0.12701	Area 1	-	0.399	-	-	1.086	2	-	-	-	-	-	-
		Area 2	-	0.562	-	-	2	0.246	-	-	-	-	-	-
PSO-PID	0.12702	Area 1	-	0.402	-	-	1.092	2	-	-	-	-	-	-
		Area 2	-	0.561	-	-	1.981	0.156	-	-	-	-	-	-
DSA-FOPID [73]	0.0778	Area 1	-	0.467	-	-	2.033	2.999	-	-	1.0007	1.0517	-	-
		Area 2	-	NA	-	-	NA	NA	-	-	NA	NA	-	-
WOA-tuned 2DOF TIDF [74]	0.0734	Area 1	3.88	1.856	0.105	227.8	-	3.963	-	-	-	-	1.243	0.423
		Area 2	NA	NA	NA	NA	-	NA	-	-	-	-	NA	NA
WOA-tuned TIDF [74]	0.1167	Area 1	1.40	0.377	0.111	230.9	-	1.877	-	-	-	-	-	-
		Area 2	NA	NA	NA	NA	-	NA	-	-	-	-	-	-
EPSDE-PID [39]	0.1497	Area 1	-	0.388	-	-	0.859	1.773	-	-	-	-	-	-
		Area 2	-	1.011	-	-	1.041	0.165	-	-	-	-	-	-
CLPSO-PID [39]	0.1569	Area 1	-	0.384	-	-	1.014	1.705	-	-	-	-	-	-
		Area 2	-	0.583	-	-	1.720	0.428	-	-	-	-	-	-
HPSO-PS tuned fuzzy PI [52]	0.1438	Area 1	-	-	-	-	0.985	0.559	-	-	-	-	-	-
		Area 2	-	-	-	-	0.933	0.720	-	-	-	-	-	-
PS-tuned fuzzy PI [52]	0.6334	Area 1	-	-	-	-	0.731	0.647	-	-	-	-	-	-
		Area 2	-	-	-	-	0.450	0.547	-	-	-	-	-	-
PSO-tuned fuzzy PI [52]	0.4470	Area 1	-	-	-	-	0.508	0.510	-	-	-	-	-	-
		Area 2	-	-	-	-	0.817	0.794	-	-	-	-	-	-

Note: NA indicates that the corresponding source does not report parameters for Area 2.

summarizes the optimal parameters and ITAE values of the proposed and benchmark controllers, all obtained on the identical two-area non-reheat thermal test system under the same parameters and conditions. Beyond PID-family controllers, the comparison includes advanced fractional-order and multi-degree-of-freedom designs from recent research, specifically a DSA-tuned FOPID and WOA-tuned 2DOF-TIDF and TIDF, and other optimization techniques with fuzzy PI controllers. Additionally, a framework of the hybrid PS, the proposed controller in two-area system, the optimization method, and the objective function studied in this research is shown in Figure 12.

4. Optimization Technique and Objective Function

4.1. Optimization Technique

The PKO is a recently introduced swarm-based metaheuristic inspired by the unique hunting strategies and cooperative interactions of pied kingfishers in their natural environment [47].

The algorithm is organized into three functional phases: (1) perching/hovering for prey (exploration/diversification), (2) diving for prey (exploitation/intensification), and (3) fostering symbiotic relations. These biological behaviors are mathematically modeled to navigate complex and high-dimensional search spaces effectively.

To validate its robustness and adaptability, the PKO has been benchmarked against thirty-nine test functions covering unimodal, multimodal, hybrid, and composite landscapes. Moreover, its capabilities are further demonstrated through application to eight real-world engineering design problems, including both constrained and unconstrained formulations.

The overall results emphasize PKO’s strong capability in handling complex optimization tasks involving difficult and diverse search landscapes. The algorithm shows a well-balanced approach between exploration and exploitation, enabling it to efficiently navigate the search space while minimizing the risk of becoming trapped in local optima.

As with many population-based optimization techniques, the PKO begins by randomly generating an initial set of candidate solutions across the search space. This initial population serves as the algorithm’s first exploration trial. The initialization of each solution is performed using (24).

$$X_{i,j}=LB+\left(UB-LB\right)\times rand, i=1, 2,\dots ,N and j=1, 2,\dots , Dim,$$

(24)

where $X_{i,j}$ refers to the location of the ith individual at the jth dimension, rand is a uniformly distributed random number in the range [0, 1], and $LB$ and $UB$ represent the lower and upper bounds of the search space, respectively. After generating the initial population, a fitness function is applied to measure the quality of each candidate based on its ability to address the given optimization task. Individuals with the best fitness values are selected to produce a new generation.

4.1.1. Exploration Phase

The exploration phase of PKO is inspired by the perching and hovering strategies observed in pied kingfishers during foraging. Field studies show that these birds alternate between launching attacks from stationary perches and from mid-air hovering, depending on environmental and situational factors. In the context of PKO, these natural behaviors guide how search agents explore the solution space. Each agent’s position is dynamically adjusted based on this biologically inspired foraging mechanism, and the positions are updated in (25).

$$X_i\left(t+1\right)=X_i\left(t\right)+\alpha \ast T\ast {(X}_j\left(t\right)-X_i\left(t\right)), i,j=1, 2,\dots ,N and j \ne i.$$

(25)

During the iterative update process, the position of a search agent in the next iteration is expressed as $X_i\left(t+1\right)$, while $X_i\left(t\right)$ represents its current location. The parameter α is computed as $2\mathrm{\ast }randn\left(1,Dim\right)-1,$ where $randn$ represents a random number from a normal distribution. $N$ denotes the population size, and $Dim$ signifies the dimensionality of the problem under consideration. The parameter $T$ is a key component in our approach, with its value adaptively changing according to the current behavioral strategy, either perching or hovering. The calculation of $T$ is specifically designed to align with the characteristics of each strategy, enabling efficient operation in different system modes.

Perching

Pied kingfishers are known to initiate hunting by perching on elevated natural or artificial structures such as trees, rocks, or utility lines, from which they scan nearby water bodies for signs of prey. Initially, they rely on monocular vision to survey a broad area and detect subtle movements on the water’s surface. Upon identifying a potential prey, they switch to binocular vision to accurately judge distance and lock onto the prey, allowing for precise and well-timed capture movements.

However, the crest angle of a pied kingfisher can also influence its hunting dynamics. The perching mechanism can be mathematically modeled as previously described in (25) and the $T$ parameter can be calculated as follows:

$$T=(\mathit{exp}\left(1\right)-\mathit{exp}{\left({\displaystyle \frac{t-1}{Ma{x}_{-}Iter}}\right)}^{{\displaystyle \frac{1}{BF}}})\ast \mathrm{c}\mathrm{o}\mathrm{s}(Cres{t}_{-}angles),$$

(26)

$$cres{t}_{-}angles=2\ast pi\ast rand.$$

(27)

The maximum number of iterations is specified by $Ma{x}_{-}Iter$. The constant value $BF$, beating factor, is set to 8, and $rand$ is a random value between 0 and 1.

Hovering

The pied kingfisher can hover in place due to its fast and powerful wing beats. Its wings, though relatively short and broad, are built to beat at a high frequency, keeping the bird steadily suspended in the air. Unlike many other birds, its long and pointed wings create enough lift to support this hovering behavior. The hovering strategy can be mathematically expressed as previously given in (25) and the $T$ parameter can be calculated as follows:

$$T=beatin{g}_{-}rate\ast \left({\displaystyle \frac{t^{\frac{1}{BF}}}{{Ma{x}_{-}Iter}^{\frac{1}{BF}}}}\right),$$

(28)

$$beatin{g}_{-}rate=rand\ast \left({\displaystyle \frac{PK{O}_{-}Fitness\left(j\right)}{PK{O}_{-}Fitness\left(i\right)}}\right).$$

(29)

The fitness of the ith and jth pied kingfishers is represented by $PK{O}_{-}Fitness\left(i\right)$ and $PK{O}_{-}Fitness\left(j\right)$, respectively. As it is a constant value, $BF$, the beating factor, is set to 8.

4.1.2. Exploitation Phase

The pied kingfisher exhibits exceptional precision in diving behavior. During hunting, the bird perches above water to locate prey, and then performs a rapid, high-angle dive to capture fish held in its beak with remarkable accuracy. This natural behavior embodies a balance between exploration (searching from elevation) and exploitation (targeted diving toward the prey), which is abstracted into the PKO algorithm’s search dynamics. The beak’s optimized shape, due to its strength and precision, have also influenced modern engineering designs such as reducing the noise of bullet trains and realizing efficient underwater vehicles, reinforcing the kingfisher’s relevance as a model for intelligent and adaptive optimization strategies. The mathematical representation is presented in (30)–(33):

$$X_i\left(t+1\right)=X_i\left(t\right)+HA\ast o\ast \alpha \ast \left(b-X_{best}\left(t\right) \right), i=1, 2,\dots ,N,$$

(30)

$$HA=rand\ast \left({\displaystyle \frac{PK{O}_{-}Fitness\left(i\right)}{Bes{t}_{-}Fitness}}\right),$$

(31)

$$o=exp{\left({\displaystyle \frac{-t}{Ma{x}_{-}Iter}}\right)}^2,$$

(32)

$$b=X_i\left(t\right)+o^2\ast randn\ast X_{best}\left(t\right),$$

(33)

where $Bes{t}_{-}Fitness$ represents the best fitness value obtained over all iterations, $\alpha$ is a control parameter, calculated as follows: $randn\left(1,Dim\right)-1$; $o$ and $HA$ denotes the hunting ability.

4.1.3. Commensalism Phase

The pied kingfisher exhibits adaptive hunting behavior influenced by environmental and ecological factors such as prey availability and competition. It often forms a commensal association with otters, benefiting from fish disturbed during otter activity. This cooperative interaction enhances its hunting efficiency without mutual harm. In PKO, this ecological relationship is abstracted as the commensalism or local escape phase, enabling candidates to share information and refine their positions through cooperative learning for improved local search efficiency. This behavior is mathematically presented as follows:

$$X_i\left(t+1\right)=\left\{\begin{array}{cc}{X}_m\left(t\right)+o.\alpha .\left|\right(X_i\left(t\right)-X_n\left(t\right)\left)\right| if rand>\hfill & (1-PE)\hfill \\ X_i\left(t\right)\hfill & otherwise\hfill \end{array}\right.,$$

(34)

$$i=1, 2,\dots ,N,$$

(35)

$$o=exp{\left({\displaystyle \frac{-t}{Ma{x}_{-}Iter}}\right)}^2,$$

(36)

$$\alpha =2\ast randn\left(1,Dim\right)-1,$$

(37)

$$PE={PE}_{max}-\left({PE}_{max}-{PE}_{min}\right)\ast \left({\displaystyle \frac{t}{Ma{x}_{-}Iter}}\right).$$

(38)

Two individuals, denoted as $X_m$ and $X_n$, are randomly selected from the population to model the local interaction behavior. The predatory efficiency ($PE$) parameter controls this interaction, bounded within ${PE}_{min}$ = 0 and ${PE}_{max}$ = 0.5, to regulate the adaptive balance between exploration and exploitation in the search process.

The algorithm iterates through these steps until the maximum number of iterations is reached, continuously updating the best solution. Finally, once the termination criteria are met, the algorithm returns the optimal solution found.

Substantially, the PKO algorithm balances exploration and exploitation through the perching, hovering, diving phase, and commensalism phase, where the first two drive broad global search via dynamically scaled stochastic updates, the diving phase intensifies local refinement toward the optimal solution through a fitness-weighted step mechanism, and the commensalism phase actively prevents local optima entrapment by probabilistically relocating stagnating agents across the population. This structured transition from exploration to exploitation, governed by adaptive parameters that evolve with iteration progress, overcomes the premature convergence and local optima susceptibility inherent in PSO and GWO, enabling PKO to achieve superior convergence accuracy and solution quality in the high-dimensional, multimodal parameter search space characteristic of the LFC controller tuning problem addressed in this work. Figure 13 represents the flowchart of procedural stages of the proposed PKO.

4.2. Objective Function

The objective function (OF) serves as a quantitative measure for evaluating the dynamic performance of the PS and the effectiveness of the proposed method. It ensures that the controller output maintains optimal performance under varying conditions while minimizing steady-state error. In this study, the parameters of the TDn(1+PIDn) controller are optimized using the PKO algorithm to mitigate frequency and tie-line power oscillations in PS that include RESs and ESSs. Accordingly, ITAE is employed as the objective function, expressed as follows [75]:

$${OF}_{ITAE}={\int }_0^{t_{sim}}\left(\left|∆F_1\right|+\left|∆F_2\right|+\left|∆P_{tie}\right|\right).t.dt.$$

(39)

In the above Equation, $∆F_1$ and $∆F_2$ are the system frequency deviations in area 1 and area 2, respectively. $∆P_{tie}$ is the incremental change in tie line power, and $t$ and $t_{sim}$ are the present and simulation times.

To keep the comparison fair, PSO, GWO and PKO were run under matched conditions: population size (30), number of iterations (50), the same gain bounds from

Table 6. Limits of proposed controller parameters.

Gains	${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
Lower limit (min)	0	0	0	1	0	0	0	1
Upper limit (max)	2	2	1	1000	2	2	2	1000

, the same ITAE objective function in (39), and number of independent runs (20), with every run facing the same disturbance and renewable sequences and conditions. PSO used an inertia weight $w$ initialized at 1 and damped multiplicatively by a factor of 0.99 per iteration, together with a personal learning coefficient $c_1$ = 1.5 and a global learning coefficient $c_2$ = 2, and GWO used the convergence coefficient a decreasing linearly from 2 to 0 with iteration [76]. PKO’s own parameters, a beating factor ($BF$) of 8 and a predatory efficiency ($PE$) range from 0 to 0.5 for the commensalism phase, were taken directly from the original paper [47] and left unchanged; so, no technique received a tuning advantage. Because a single realization of a stochastic optimizer carries little weight on its own, each algorithm was executed over these twenty independent runs and its performance judged on the distribution of outcomes rather than on any one trajectory. The statistical analysis across the independent runs is reported in

Table 7. Statistical analysis using the proposed controller.

Algorithm	PSO	GWO	PKO
Best	0.055503	0.054415	0.037309
Worst	0.10953	0.056291	0.057297
Mean	0.0682428	0.0553071	0.0530751
Standard deviation	0.021565285	0.0005967	0.006091139

.

The superiority of the proposed PKO-TDn(1+PIDn) method in LFC problem is demonstrated through an extensive comparative analysis with several modern methods for the same test system with identical parameters and conditions, as summarized in Table 6. Table 5 shows that PKO-TDn(1+PIDn) achieves the lowest ITAE value (0.0373) among all evaluated methods, corresponding to reductions of approximately 31% and 33% relative to GWO and PSO with the same controller. The reduction reaches about 49% and 52% relative to WOA-tuned 2DOF TIDF and DSA-FOPID. Relative to the WOA-tuned TIDF, HPSO-PS-tuned fuzzy PI, EPSDE-PID, and CLPSO-PID methods, ITAE reductions are roughly 68%, 74%, 75%, and 76%, respectively. In addition, the proposed method outperforms PSO-tuned fuzzy PI and PS-tuned fuzzy PI methods by more than 91% and 94%, respectively. Consequently, the proposed method exhibits superior transient performance in terms of faster response, lower overshoot, and improved steady-state accuracy.

The controller parameters are optimized subject to predefined bounds to ensure system stability and practical realizability. The admissible limits of the controller parameters are summarized in Table 6. The design problem can be formulated as the following optimization problem.

Minimize ${OF}_{ITAE}$ subject to:

$${\theta }^{Min}\le \theta \le {\theta }^{Max},$$

(40)

where $\theta$ is vector of controller parameters, and ${\theta }^{Min}$, ${\theta }^{Max}$ are bounds for each controller parameter taken from Table 6.

The PKO algorithm was executed in many separate runs, with Table 7 displaying the best, worst, mean, and standard deviation of OF values achieved by different optimization algorithms for the same proposed controller.

The boxplot in Figure 14 of PKO, GWO, and PSO under the proposed controller shows a more compact interquartile range and lower median value for PKO compared with PSO, with GWO lying in between. Such a distribution suggests that PKO not only yields better median performance but also limits the occurrence of outliers, highlighting its capability to avoid poor local minima and maintain stable convergence characteristics.

The PKO algorithm is executed for a sufficient number of iterations to guarantee convergence toward an optimal solution.

The PKO technique is used to evaluate the effectiveness of the proposed controller. Figure 15 shows that PKO achieves the lowest OF values with a consistently superior convergence rate throughout the optimization process, whereas PSO and GWO stagnate prematurely at higher OF values, confirming PKO’s enhanced ability to avoid becoming trapped in local optima and identify more optimal controller parameters compared to its counterparts.

5. Results

In this section, the simulation results and their analysis are shown to assess the overall performance of the suggested controllers and algorithms. For this purpose, all cases are implemented in MATLAB/Simulink 2024b. Optimization algorithms such as PKO, GWO, and PSO are implemented as MATLAB m-files.

For clarity, the load disturbances, nonlinearity sources, renewable profiles, and energy storage types used throughout the results are collected in

Table 8. Summary of tested dynamic scenarios.

Case/Scenario	Load Disturbance	RES Profile	ESS Integration	Nonlinearity/ Uncertainty Source
C1-S1	Step +0.1 pu in Area 1 (t = 0)	-	-	-
C1-S2	Step +0.05 pu in Area 1 (t = 0)	-	-	GRC ±0.05 GDB CTD 0.1 s
C1-S3	Step +0.05 pu in Area 1 (t = 0)	-	-	GRC ±0.025 GDB CTD 0.1 s
C2-S1	Step +0.1 pu in Area 1 (t = 10 s)	Uniform PV in Area 1 Uniform WT in Area 2	-	UniformRES
C2-S2	Step +0.1 pu in Area 1 (t = 10 s)	Random PV in Area 1 Random WT in Area 2	-	RandomRES
C2-S3	Random in Area 1	Uniform PV in Area 1 Uniform WT in Area 2	-	UniformRES Randomload
C2-S4	Random in Area 1	Random PV in Area 1 Random WT in Area 2	-	RandomRES Randomload
C3-S1: C3-S4	As C2-S1: C2-S4	As C2-S1: C2-S4	VRFB and SMES in Area 1 VRFB and HAFC in Area 2	Uniformor RandomRES Randomload
C4 (Validation by OPAL-RT)	tested scenarios: C1-S1 C-S1 C3-S3	As case	As case	As case

.

The random PV/WT profiles are generated using band-limited white noise blocks with a fixed default seed applied uniformly to all controllers, ensuring a controlled and reproducible comparison on the same stochastic realization. The robustness of the proposed controller is evidenced by its consistent superiority across four random-profile scenario combinations spanning random renewable generation, and its simultaneous occurrence.

Each area is regulated by an independent TDn(1+PIDn) block carrying eight optimized parameters, the five gains, $K_T$, $K_D$, $K_P$, $K_I$ and $K_{DD}$, the two measurement-filter coefficients, $N$ and $NN$, and the fractional filter order, $1/n$, so that sixteen parameters are tuned in total for the two-area plant. The additional parameters of the proposed controller do not translate into a heavy implementation. Once tuned, the controller reduces to a fixed low-order linear block, and its real-time simulation on the OPAL-RT produced no overruns, confirming a small online cost. The number of tunable gains is comparable to the cascade and FOPID designs already listed in Table 5, and the tuning itself is performed automatically by the optimizer within the bounds of Table 6. Set against this limited cost, the accuracy gain is clear.

5.1. Case 1—Scenario 1: Sudden Load Increase in Area 1

In this scenario, a +0.1 pu $\Delta P_D$ is applied in area 1 at t = 0. The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded for system performance assessment.

As shown in Figure 16 and

Table 9. ITAE and parameters of controllers with optimizations in case 1 scenario 1.

Algorithm Controller	ITAE	Area	Parameters
			${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
PKO-TDn(1+PIDn)	0.037309	Area 1	1.9674	0.627478	0.99274406	1.0066812	1.9662028	0	2	285.75433
		Area 2	0	1.9999704	0.69481776	1000	2	1.9854747	1.9989607	463.98808
GWO-TDn(1+PIDn)	0.054415	Area 1	2	0.2375788	0.000853708	117.9092	0.1828841	2	0.2639371	10.74717
		Area 2	1.5497	0.68634	0.4195544	11.96116	0.9200438	0.4173851	1.942215	785.5651
PSO-TDn(1+PIDn)	0.055503	Area 1	2	0.04735223	0	947.5054	0.3098349	2	0.3760524	85.46221
		Area 2	2	1.370524	0.4270762	727.5113	1.657196	0.3687319	0.0822369	406.3412
PKO-PID	0.12696	Area 1	-	0.39474	-	-	1.084	2	-	-
		Area 2	-	0.54378	-	-	1.997	0.057774	-	-
GWO-PID	0.12701	Area 1	-	0.399	-	-	1.0869	2	-	-
		Area 2	-	0.5627	-	-	2	0.24698	-	-
PSO-PID	0.12702	Area 1	-	0.40273	-	-	1.092	2	-	-
		Area 2	-	0.56175	-	-	1.9812	0.15687	-	-

and

Table 10. Comparative performance analysis in case 1 scenario 1.

Algorithm Controller	${∆\mathit{F}}_1$			${∆\mathit{F}}_2$			${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)
PKO-TDn(1+PIDn)	−0.0624	0.001027	4.573	−0.01679	0.0001482	3.303	−0.00729	0.0000485	3.302
GWO-TDn(1+PIDn)	−0.0734	0.001428	4.604	−0.02670	0.0000059	4.103	−0.01147	0.0000062	4.292
PSO-TDn(1+PIDn)	−0.0744	0.002759	4.944	−0.02900	0.0000029	3.896	−0.01173	0.0000110	4.296
PKO-PID	−0.1137	0.006389	6.208	−0.06175	0.0000753	3.938	−0.02243	0.0000236	4.403
GWO-PID	−0.1131	0.005735	6.267	−0.06136	0.0000781	3.997	−0.02232	0.0000239	4.397
PSO-PID	−0.1130	0.005238	6.253	−0.06119	0.0000795	4.222	−0.02215	0.0000247	4.395

, the PKO-TDn(1+PIDn) controller delivers the strongest overall transient performance as follows: (1) the ITAE reduction from 0.0544 (GWO) and 0.0555 (PSO) to 0.0373 corresponds to 31% and 33% improvement for the multi-stage controller, respectively, and nearly 71% improvement vs. the PID family. (2) For $∆F_1$, undershoot is reduced by 15–16% compared with GWO-TDn(1+PIDn) and PSO-TDn(1+PIDn), and by about 45% relative to PID methods. (3) Overshoot decreases by 28–63% relative to GWO-TDn(1+PIDn) and PSO-TDn(1+PIDn) and 80–84% vs. PID. Its settling time is faster by 7.5% vs. PSO-TDn(1+PIDn) and 26–27% vs. PID. For $∆F_2$, PKO-TDn(1+PIDn) lowers undershoot by 37–42% compared with the other TDn(1+PIDn) designs and by nearly 73% vs. PID, achieving 15–20% shorter settling times. A similar trend appears in $∆P_{tie}$, where undershoot improves by 36–38% relative to other TDn(1+PIDn) and 67% relative to PID, with settling time reduced by 23–25%.

The $∆F_2$ and $∆P_{tie}$ overshoot values for PKO-TDn(1+PIDn) (1.482 × 10⁻⁴ Hz and 4.85 × 10⁻⁵ pu, respectively) are somewhat higher than the near-zero values recorded for GWO-TDn(1+PIDn) and PSO-TDn(1+PIDn) (of the order of 10⁻⁶ Hz and pu), but all three remain extremely small in absolute terms; for PKO-TDn(1+PIDn) itself, the overshoot on each signal is under 1% of the corresponding undershoot; so, the response is, for practical purposes, overshoot-free. Such near-zero values are reported for completeness; the physically meaningful basis for comparison is the undershoot, settling time and ITAE, where PKO-TDn(1+PIDn) shows a consistent advantage across all three signals. Overall, PKO-TDn(1+PIDn) provides the most strongly damped and fastest response among all configurations. The sensitivity analysis shown in

Table 11. Sensitivity analysis using the proposed method in case 1 scenario 1.

System Parameters	Change (%)	ITAE	${∆\mathit{F}}_1$			${∆\mathit{F}}_2$			${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
			US	OS	${\mathit{T}}_{\mathit{S}}$ (s)	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)
$K_P$	+25	0.03704	−0.0685	0.000984	4.4	−0.01661	0.000155	3.2	−0.00716	0.000038	3.25
	−25	0.03769	−0.055	0.001255	4.8	−0.01661	0.000135	3.4	−0.00731	0.000067	3.35
$T_P$	+25	0.03764	−0.0571	0.001198	4.7	−0.01672	0.000137	3.5	−0.00733	0.000063	3.4
	−25	0.03707	−0.0703	0.001049	4.5	−0.0165	0.000157	3.2	−0.00711	0.000035	3.22
$T_{12}$	+25	0.03461	−0.0615	0.00055	4.4	−0.0193	0.000168	3.15	−0.00837	0.0000576	3.21
	−25	0.04304	−0.0635	0.001762	4.75	−0.0137	0.000105	3.55	−0.00596	0.0000305	3.46

confirms the strong robustness of the proposed method against realistic power system parameter uncertainty. Under ±25% variation in $K_P$ and $T_P$, the controller maintained near-nominal performance, with ITAE deviating by no more than 1.76% from the nominal case and all undershoot, overshoot, and settling-time indices for $∆F_1$, $∆F_2$, and $∆P_{tie}$ remaining consistent. This demonstrates the controller’s excellent insensitivity to uncertainties in equivalent generation-side inertia and damping representation. The controller also responded favorably to strengthening of the tie-line coupling ($T_{12}$ + 25%), where ITAE improved by 6.56% relative to nominal, indicating that the design can exploit stronger inter-area synchronization to further enhance frequency regulation.

Even under the more challenging condition of a weakened tie-line ($T_{12}$ − 25%), the controller preserved stable operation and bounded dynamic performance across all indices, without divergence or instability, underscoring the reliability of the PKO-optimized gains across a wide operating envelope.

5.2. Case 1—Scenario 2: Sudden Load Increase in Area 1 with GRC ±0.05, GDB, and CTD 0.1 s in Both Areas

In this scenario, a +0.05 pu $\Delta P_D$ is applied in area 1 at t = 0. The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded for system performance assessment.

5.3. Case 1—Scenario 3: Sudden Load Increase in Area 1 with GRC ±0.025, GDB, and CTD 0.1 s in Both Areas

In this scenario, the setup is the same as in scenario 2, but with a tighter GRC. The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded for system performance assessment and compared with scenario 2.

According to Figure 17 and

Table 12. Comparison of ITAE and parameters of the proposed method between case 1 scenario 2 and scenario 3.

Scenario	Algorithm Controller	ITAE	Area	Parameters
				${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
2	PKO-TDn(1+PIDn)	1.0633	Area 1	0.3834	0	0.99737	222.6333	0.5336	0	0	929.609
			Area 2	0.0287	0	0.85564	524.225	1.9691	0.05651	0.1418	239.805
3	PKO-TDn(1+PIDn)	1.4269	Area 1	0.6132	2	0.00675	802.2939	0.6001	0.56383	0.35423	527.522
			Area 2	0.0584	0.26	0	197.2155	0.0843	0.11329	0.80599	607.188

and

Table 13. Comparative performance analysis between case 1 scenario 2 and scenario 3.

Scenario	Algorithm Controller	${∆\mathit{F}}_1$			${∆\mathit{F}}_2$			${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
		US	OS	${\mathit{T}}_{\mathit{S}}$ (s)	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)	US	OS	${\mathit{T}}_{\mathit{S}}$ (s)
2	PKO-TDn(1+PIDn)	−0.1279	0.0307	8.25	−0.1213	0.04311	7.72	−0.0354	0.006844	8.32
3	PKO-TDn(1+PIDn)	−0.1402	0.0642	8.632	−0.171	0.04321	8.75	−0.03802	0.008682	9.44

, the comparison between scenarios 2 and 3 confirms that the GRC exerts a strong effect on LFC performance, even after the GDB and a 0.1 s communication time delay are incorporated into both scenarios to provide a more demanding and realistic test of the proposed method.

With the GRC relaxed to ±0.05, the regulating units are permitted larger ramp rates; so, the system absorbs the disturbance more readily and returns to steady state with smaller deviations and shorter recovery intervals. Constraining the rate to ±0.025 limits how quickly generation can follow the load change, which degrades the dynamic behavior across all three monitored signals.

The undershoots of $∆F_1$, $∆F_2$, and $∆P_{tie}$ grow by roughly 9.6%, 41%, and 7.4%, respectively, while their settling times lengthen by about 4.6%, 13.3%, and 13.5%. The overshoot is also affected, rising sharply by close to 109% in $∆F_1$ and by about 26.9% in $∆P_{tie}$, whereas the $∆F_2$ overshoot stays essentially unchanged. Taken together, these results show that the more permissive ±0.05 limit allows for faster corrective action and yields markedly better damping and overall stability than the tighter ±0.025 constraint. In addition, the results demonstrate that the proposed method decisively overcomes the destabilizing influence of all three nonlinearity sources acting together, GRC, GDB, and CTD, and consistently returns the system to steady state, confirming its effectiveness and reliability under demanding and realistic conditions.

5.4. Case 2—Scenario 1: Fixed Load Increase Under Uniform RES Profiles

In this scenario, a +0.1 pu $\Delta P_D$ is applied in area 1 at t = 10 s. Area 1 is supplied by PV with a uniform profile, and area 2 is supplied by WT with a uniform profile. The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded for system performance assessment and for analyzing the impact of RES penetration in the PS. Figure 18 illustrates the uniform solar and wind power injected into the PS. As shown in Figure 19 and

Table 14. ITAE and parameters of controllers with optimizations in case 2 all scenarios.

Scenario	Algorithm Controller	ITAE	Area	Parameters
				${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
1	PKO-TDn(1+PIDn)	21.9239	Area 1	0.993	0.570	0.422	534.094	1	0.979	1	1
			Area 2	0.988	0.166	0.928	1	1	0.948	0.766	900.686
	GWO-TDn(1+PIDn)	23.6983	Area 1	1	0.610	0.359	183.023	1	0.993	0.488	845.174
			Area 2	1	0.385	0.742	2.117	0.741	0.990	0.049	642.868
	PSO-TDn(1+PIDn)	26.7529	Area 1	0.992	0.484	0.946	240.867	0.700	0.846	0.911	454.232
			Area 2	0.994	0.149	1	757.879	0.736	0	0.338	643.089
	PKO-PID	37.0438	Area 1	-	0.938	-	-	0.945	1	-	-
			Area 2	-	0.305	-	-	0.729	1	-	-
	GWO-PID	38.2655	Area 1	-	0.627	-	-	0.386	1	-	-
			Area 2	-	0.533	-	-	0.316	0.997	-	-
	PSO-PID	40.7485	Area 1	-	0.260	-	-	0.551	0.829	-	-
			Area 2	-	0.275	-	-	0.055	0.995	-	-
2	PKO-TDn(1+PIDn)	11.5592	Area 1	1	0.479	0.972	1.011	0.982	0.006	0.999	318.155
			Area 2	1	0.976	0.445	695.554	0.974	0.966	0.145	783.744
	GWO-TDn(1+PIDn)	16.1475	Area 1	0.889	0.194	0.319	409.593	0.434	0.947	0.713	270.974
			Area 2	1	0.927	0.630	253.113	0.976	0.457	0.49	456.294
	PSO-TDn(1+PIDn)	15.2312	Area 1	1	0.186	0.587	833.547	1	0.518	0.757	862.131
			Area 2	0.988	0.749	0.545	502.182	0.807	0.805	0.107	531.774
	PKO-PID	20.5254	Area 1	-	0.989	-	-	0.999	1	-	-
			Area 2	-	1	-	-	0.999	1	-	-
	GWO-PID	21.4565	Area 1	-	0.363	-	-	0.402	1	-	-
			Area 2	-	0.741	-	-	0.939	1	-	-
	PSO-PID	23.2232	Area 1	-	0.665	-	-	0.377	1	-	-
			Area 2	-	0.684	-	-	0.639	1	-	-
3	PKO-TDn(1+PIDn)	28.2608	Area 1	1	0.385	0.683	1	0.935	0.474	0.738	1000
			Area 2	1	0.825	0.611	591.675	1	1	0.384	484.001
	GWO-TDn(1+PIDn)	30.9221	Area 1	1	0.755	0.384	162.241	0.237	0.915	0.183	524.869
			Area 2	0.971	0.323	0.803	321.535	0.970	0.665	0.773	592.184
	PSO-TDn(1+PIDn)	33.7697	Area 1	1	0.046	0.348	171.614	0.050	0.908	0.422	492.020
			Area 2	0.990	0.288	0.849	374.178	0.400	0.9	0.896	1
	PKO-PID	39.7584	Area 1	-	0.277	-	-	0.910	1	-	-
			Area 2	-	0.696	-	-	0.985	1	-	-
	GWO-PID	44.4464	Area 1	-	0.315	-	-	0.296	1	-	-
			Area 2	-	0.332	-	-	−0.164	1	-	-
	PSO-PID	46.0625	Area 1	-	0.315	-	-	0.204	0.995	-	-
			Area 2	-	0.467	-	-	0.726	0.828	-	-
4	PKO-TDn(1+PIDn)	16.5372	Area 1	0.961	0.849	0.349	823.583	1	0.997	0.537	331.455
			Area 2	0.997	0.999	0.468	834.849	0.964	0.941	0.996	423.283
	GWO-TDn(1+PIDn)	19.8685	Area 1	1	0.234	0.604	408.723	0.956	0.505	0.398	910.161
			Area 2	0.966	0.530	0.067	105.333	0.994	0.754	0.152	274.736
	PSO-TDn(1+PIDn)	20.2288	Area 1	1	0.798	0.585	660.437	0.867	0.404	0.331	112.168
			Area 2	0.999	0.9	0.462	592.284	0.459	0.9	0.687	182.425
	PKO-PID	27.748	Area 1	-	0.999	-	-	1	1	-	-
			Area 2	-	1	-	-	1	1	-	-
	GWO-PID	28.927	Area 1	-	0.386	-	-	0.534	1	-	-
			Area 2	-	0.869	-	-	0.928	0.957	-	-
	PSO-PID	29.1459	Area 1	-	0.553	-	-	0.618	1	-	-
			Area 2	-	0.748	-	-	0.8115	1	-	-

and

Table 15. Comparative performance analysis in case 2 all scenarios.

Scenario	Algorithm Controller	${∆\mathit{F}}_1$		${∆\mathit{F}}_2$		${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
		US	OS	US	OS	US	OS
1	PKO-TDn(1+PIDn)	−0.01389	0.02059	−0.03062	0.03483	−0.01109	0.00501
	GWO-TDn(1+PIDn)	−0.02489	0.01935	−0.03315	0.03497	−0.01249	0.00505
	PSO-TDn(1+PIDn)	−0.02552	0.04314	−0.03599	0.04380	−0.01366	0.01083
	PKO-PID	−0.03502	0.03192	−0.05194	0.04090	−0.01971	0.00815
	GWO-PID	−0.04403	0.03782	−0.06523	0.04545	−0.02507	0.01020
	PSO-PID	−0.03777	0.04530	−0.07184	0.05265	−0.02625	0.01036
2	PKO-TDn(1+PIDn)	−0.1022	0.01419	−0.04452	0.01174	−0.01823	0.001620
	GWO-TDn(1+PIDn)	−0.08123	0.01749	−0.04865	0.01463	−0.02014	0.002504
	PSO-TDn(1+PIDn)	−0.08184	0.01733	−0.04980	0.01272	−0.01947	0.002220
	PKO-PID	−0.07855	0.01997	−0.04304	0.01620	−0.01742	0.003231
	GWO-PID	−0.12850	0.02589	−0.07816	0.02021	−0.02901	0.003435
	PSO-PID	−0.09980	0.02528	−0.06280	0.01945	−0.02357	0.004048
3	PKO-TDn(1+PIDn)	−0.03117	0.02655	−0.01370	0.01841	−0.00731	0.00643
	GWO-TDn(1+PIDn)	−0.01770	0.02095	−0.01282	0.02613	−0.00966	0.00698
	PSO-TDn(1+PIDn)	−0.03923	0.03146	−0.02341	0.03013	−0.00926	0.00907
	PKO-PID	−0.03945	0.03302	−0.02201	0.03238	−0.01151	0.00908
	GWO-PID	−0.04011	0.04728	−0.02660	0.06134	−0.01784	0.01121
	PSO-PID	−0.04016	0.03868	−0.02586	0.03816	−0.01313	0.01222
4	PKO-TDn(1+PIDn)	−0.01392	0.01128	−0.01264	0.00793	−0.00272	0.00292
	GWO-TDn(1+PIDn)	−0.02829	0.01708	−0.02073	0.01209	−0.00524	0.00431
	PSO-TDn(1+PIDn)	−0.01734	0.01391	−0.01576	0.00909	−0.00341	0.00358
	PKO-PID	−0.02575	0.01686	−0.01965	0.01485	−0.00412	0.00429
	GWO-PID	−0.03980	0.02400	−0.02841	0.01581	−0.00686	0.00602
	PSO-PID	−0.03460	0.02047	−0.02596	0.01697	−0.00563	0.00532

, the PKO-TDn(1+PIDn) controller provides the strongest overall damping performance. The ITAE reduction from 23.6983 (GWO) and 26.7529 (PSO) to 21.9239 corresponds to 7% and 18% improvement for the multi-stage controller, respectively, and 41% and 46% improvement vs. the PID family. It achieves much smaller undershoot in both $∆F_1$ and $∆F_2$, improving by 44–46% compared with the GWO-TDn(1+PIDn) and PSO-TDn(1+PIDn) methods and by 41–68% relative to the entire PID-group. Overshoot behavior also improves in most cases; PKO-TDn(1+PIDn) reduces overshoot in $∆F_2$ and $∆P_{tie}$ by 20–54% compared with GWO and PSO with the same controller and 23–62% compared with PID-based approaches. A single exception occurs in $∆F_1$ overshoot, where it is slightly higher than the GWO-TDn(1+PIDn) value by 6%, but still noticeably better than the PSO-TDn(1+PIDn) and PID results. Overall, PKO-TDn(1+PIDn) delivers the most balanced and effective transient performance among all compared configurations.

5.5. Case 2—Scenario 2: Fixed Load Increase Under Random RES Profiles

This scenario uses the same disturbance (a +0.1 pu $\Delta P_D$ in area 1 at t = 10 s) but replaces the uniform RES profiles with random PV and WT profiles in areas 1 and 2, respectively. The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded for system performance assessment under random RES variability. Figure 20 illustrates the random solar and wind power injected into the PS.

Based on Figure 21 and Table 14 and Table 15, the results confirm that the PKO-TDn(1+PIDn) controller delivers the best overall transient performance. For $∆F_1$, it reduces undershoot by 20.3% compared with GWO-TDn(1+PIDn) and by 15.4% compared with PSO-TDn(1+PIDn) while improving over the PID-based controllers by 20–28%. Its overshoot is also smaller, decreasing by 16.4% relative to GWO-TDn(1+PIDn), by 18.8% relative to PSO-TDn(1+PIDn), and by 43–77% compared with the PID group. For $∆F_2$, PKO-TDn(1+PIDn) reduces undershoot by 42.8% vs. GWO-TDn(1+PIDn) and by 9% vs. PSO-TDn(1+PIDn), with an additional 12–43% improvement over the PID-based schemes.

The overshoot improvement in $∆F_2$ follows the same pattern, with reductions of 5–16% over the TDn(1+PIDn)-based ones and 44–78% over the PID-based ones. $∆P_{tie}$ also shows clear gains; undershoot decreases by 12.1% compared with GWO-TDn(1+PIDn), by 23.1% compared with PSO-TDn(1+PIDn), and by 21–30% relative to the PID group, while overshoot is reduced by 26–35% vs. the TDn(1+PIDn)-based methods and 53–76% vs. PID. Overall, PKO-TDn(1+PIDn) provides the strongest damping and the smallest transient deviations in scenario 2.

5.6. Case 2—Scenario 3: Random Load Change Under Uniform RES Profiles

In this scenario, the load in area 1 is subjected to a randomly varying disturbance $\Delta P_D$. The PV and WT power profiles follow the same uniform pattern used in scenario 1. The corresponding system responses, $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded to evaluate system behavior under random loading conditions and uniform RES profiles. Figure 22 shows the random load change subjected to the PS.

Based on Figure 18 and Figure 23, and Table 14 and Table 15, it is evident that the TDn(1+PIDn) controller clearly outperforms all PID-based methods. The GWO-TDn(1+PIDn) controller achieves the smallest undershoot in $∆F_1$ and $∆F_2$, reducing the $∆F_1$ undershoot by 43% and the $∆F_2$ undershoot by 6% compared with the next-best controller. The PKO-TDn(1+PIDn) method provides the lowest overshoot in both areas, lowering the $∆F_2$ overshoot by 30% relative to GWO-TDn(1+PIDn). For $∆P_{tie}$, PKO-TDn(1+PIDn) again gives the best result, reducing undershoot by 21% and overshoot by 8%. Overall, the TDn(1+PIDn) family provides substantial percentage improvements in damping and stability, confirming its superiority under the most severe conditions.

5.7. Case 2—Scenario 4: Random Load Change Under Random RES Profiles

In this scenario, a random load disturbance $\Delta P_D$ is applied in area 1 at t = 10 s, following the same loading pattern as in scenario 3. The PV and WT power profiles follow the same random pattern used in scenario 2. The system responses $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded to evaluate system behavior under random loading conditions and random RES variability.

Based on Figure 20, Figure 22 and Figure 24 and Table 14 and Table 15, it is evident that the PKO-TDn(1+PIDn) delivers the best performance among all methods. For $∆F_1$, it reduces undershoot by 19.7% and overshoot by 34.0% relative to the next-best TDn(1+PIDn) controller. In $∆F_2$, it again shows the smallest deviations, improving undershoot by 19.8% and overshoot by 34.4%. For $∆P_{tie}$, PKO-TDn(1+PIDn) reduces undershoot by 22.7% and overshoot by 32.3% compared with the next-best TDn(1+PIDn) design. Overall, the PKO-TDn(1+PIDn) method provides consistent and significant improvements of 20–34% across all transient indices.

5.8. Case 3—Scenario 1: Fixed Load Increase Under Uniform RES Profiles with ESSs

This scenario uses the same disturbance (a +0.1 pu $\Delta P_D$ in area 1 at t = 10 s), and the PV and WT injections follow uniform profiles, matching the loading and RES conditions previously considered in case 2 scenario 1.

ESSs are incorporated into the system to enhance frequency support during the disturbance. Area 1 is supported by VRFB and SMES, while area 2 is supported by VRFB and HAFC. The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$, are recorded for system performance assessment and for studying the impact of ESS integration under uniform RES profiles, and the results are compared with the RES-only case. According to Figure 25 and

Table 16. Comparison of ITAE and parameters of the proposed method in scenario 1 for case 2 and case 3.

ESS Contribution	Algorithm Controller	ITAE	Area	Parameters
				${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
Without ESSs	PKO-TDn(1+PIDn)	21.9239	Area 1	0.993	0.570	0.422	534.094	1	0.979	1	1
			Area 2	0.988	0.166	0.928	1	1	0.948	0.766	900.686
With ESSs	PKO-TDn(1+PIDn)	20.9967	Area 1	0.999	0.942	0.453	1.014	0.999	0.999	0.311	319.381
			Area 2	0.998	0	0.812	166.909	0.998	0.991	0.849	1

and

Table 17. Comparative performance analysis of scenario 1 in case 2 and case 3.

ESS Contribution	Algorithm Controller	${∆\mathit{F}}_1$		${∆\mathit{F}}_2$		${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
		US	OS	US	OS	US	OS
Without ESSs	PKO-TDn(1+PIDn)	−0.01389	0.02059	−0.03062	0.03483	−0.01109	0.00501
With ESSs	PKO-TDn(1+PIDn)	−0.00783	0.01283	−0.02138	0.01752	−0.01206	0.00508

, it is shown that with ESS support, the PKO-TDn(1+PIDn) controller shows major improvements in scenario 1. $∆F_1$ improves by 44% in undershoot and 50% in overshoot, and $∆F_2$ improves by 30% in undershoot and 50% in overshoot.

$∆P_{tie}$ undershoots improve moderately by 14%, while $∆P_{tie}$ overshoot increases slightly by 1.4%. The ITAE value also decreases, confirming faster settling and better damping. Overall, ESSs significantly enhance frequency stability by lowering oscillations by 30–50%. They also support better tie-line behavior, helping the system recover faster and reducing power swings between the areas.

5.9. Case 3—Scenario 2: Fixed Load Increase Under Random RES Profiles with ESSs

This scenario uses the same disturbance (a +0.1 pu $\Delta P_D$ in area 1 at t = 10 s), and the PV and WT injections follow random profiles, matching the loading and RES conditions previously considered in case 2 scenario 2. ESSs are added to the system to enhance frequency stability and support RESs during the disturbance. Similar to scenario 1, area 1 incorporates VRFB and SMES, and area 2 incorporates VRFB and HAFC. The resulting responses ($∆F_1$, $∆F_2$, $∆P_{tie}$) are compared with the RES-only case to study the ESS impact.

It is noticeable from Figure 26 and

Table 18. Comparison of ITAE and parameters of the proposed method in scenario 2 for case 2 and case 3.

ESS Contribution	Algorithm Controller	ITAE	Area	Parameters
				${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
Without ESSs	PKO-TDn(1+PIDn)	11.5592	Area 1	1	0.479	0.972	1.011	0.982	0.006	0.999	318.155
			Area 2	1	0.976	0.445	695.554	0.974	0.966	0.145	783.744
With ESSs	PKO-TDn(1+PIDn)	9.3698	Area 1	1	0.490	0.989	1.097	0.994	0	0.994	562.589
			Area 2	0.999	0.910	0.551	1	0.899	0.999	0.932	685.462

and

Table 19. Comparative performance analysis of scenario 2 in case 2 and case 3.

ESS Contribution	Algorithm Controller	${∆\mathit{F}}_1$		${∆\mathit{F}}_2$		${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
		US	OS	US	OS	US	OS
Without ESSs	PKO-TDn(1+PIDn)	−0.1022	0.01419	−0.04452	0.01174	−0.01823	0.001620
With ESSs	PKO-TDn(1+PIDn)	−0.04709	0.009927	−0.02229	0.007583	−0.01416	0.002362

that, in scenario 2, adding ESSs helps the system recover more smoothly from disturbances.

Frequency deviations in both areas drop sharply, with undershoot reduced by about 50% and overshoot lowered by 30–35%. The tie-line response also becomes more stable; undershoots improve by 22%, and overshoot decreases dramatically by 85%. Overall, ESSs make the system more stable, less oscillatory, and faster in returning to normal conditions.

5.10. Case 3—Scenario 3: Random Load Change Under Uniform RES Profiles with ESSs

This scenario uses the same random load disturbance, and the PV and WT injections follow uniform profiles, matching the loading and RES conditions previously considered in case 2 scenario 3. Both areas use the same ESS arrangement. The system responses ($∆F_1$, $∆F_2$, $∆P_{tie}$) are compared with the RES-only case to study the ESS impact in this scenario.

Based on Figure 27 and

Table 20. Comparison of ITAE and parameters of the proposed method in scenario 3 for case 2 and case 3.

ESS Contribution	Algorithm Controller	ITAE	Area	Parameters
				${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
Without ESSs	PKO-TDn(1+PIDn)	28.2608	Area 1	1	0.385	0.683	1	0.935	0.474	0.738	1000
			Area 2	1	0.825	0.611	591.675	1	1	0.384	484.001
With ESSs	PKO-TDn(1+PIDn)	22.5031	Area 1	1	1	0.476	1	1	0.988	0.234	586.983
			Area 2	0.965	0.234	0.766	62.507	1	1	1	1000

and

Table 21. Comparative performance analysis of scenario 3 in case 2 and case 3.

ESS Contribution	Algorithm Controller	${∆\mathit{F}}_1$		${∆\mathit{F}}_2$		${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
		US	OS	US	OS	US	OS
Without ESSs	PKO-TDn(1+PIDn)	−0.03117	0.02655	−0.01370	0.01841	−0.00731	0.00643
With ESSs	PKO-TDn(1+PIDn)	−0.01276	0.01206	−0.00849	0.01288	−0.00712	0.005499

, it is evident that with ESSs, the system performance in scenario 3 improves notably. Frequency deviations in area 1 show the strongest gains, with 59–61% reductions in undershoot and overshoot. Area 2 also improves, with 30–35% lower deviation magnitudes. Tie-line oscillations decrease as well, with 3% better undershoot and 15% better overshoot. Overall, ESSs make the system more stable, less oscillatory, and quicker to settle.

5.11. Case 3—Scenario 4: Random Load Change Under Random RES Profiles with ESSs

This scenario uses the same random load disturbance, and the same PV and WT random profiles, matching the loading and RES conditions previously considered in case 2 scenario 4. Both areas use the same ESS arrangement.

The resulting dynamic responses, including $∆F_1$, $∆F_2$, and $∆P_{tie}$ are recorded for system performance assessment and for studying the impact of ESS integration under random loading conditions and random RES profiles, and the results are compared with the RES-only case for the same conditions and system parameters.

Based on Figure 28 and

Table 22. Comparison of ITAE and parameters of the proposed method in scenario 4 for case 2 and case 3.

ESS Contribution	Algorithm Controller	ITAE	Area	Parameters
				${\mathit{K}}_{\mathit{T}}$	${\mathit{K}}_{\mathit{D}}$	$\frac{1}{\mathit{n}}$	N	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}\mathit{D}}$	NN
Without ESSs	PKO-TDn(1+PIDn)	16.5372	Area 1	0.961	0.849	0.349	823.583	1	0.997	0.537	331.455
			Area 2	0.997	0.999	0.468	834.849	0.964	0.941	0.996	423.283
With ESSs	PKO-TDn(1+PIDn)	12.3843	Area 1	0.989	0.975	0.634	1.004	0.995	0.999	0.649	986.024
			Area 2	0.995	0.991	0.293	44.356	0.969	1	0.401	93.692

and

Table 23. Comparative performance analysis of scenario 4 in case 2 and case 3.

ESS Contribution	Algorithm Controller	${∆\mathit{F}}_1$		${∆\mathit{F}}_2$		${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
		US	OS	US	OS	US	OS
Without ESSs	PKO-TDn(1+PIDn)	−0.01392	0.01128	−0.01264	0.00793	−0.00272	0.00292
With ESSs	PKO-TDn(1+PIDn)	−0.01355	0.007986	−0.008129	0.005345	−0.00266	0.00266

, it is evident that, in scenario 4, adding ESSs leads to consistent performance improvements across the system. $∆F_1$ undershoot improves by 3% and overshoot by 29%, while $∆F_2$ shows stronger gains, with undershoot reduced by 36% and overshoot by 33%. $∆P_{tie}$ also benefits, with 2% improvement in undershoot and 9% in overshoot. Importantly, the overall ITAE drops from 16.5372 to 12.3843, representing a 25% reduction, confirming that ESSs enhance damping, speed up recovery, and reduce the cumulative error over time in system performance.

Beyond the dynamic contribution quantified above, the three ESSs differ markedly once cost, service life, energy and power density and siting are taken into account, and these differences shape where each is most sensibly deployed. The SMES offers very high power density and a near-instantaneous response, which suits short, high-power support, though at a high capital cost per unit of stored energy and with the standing burden of cryogenic cooling. The VRFB combines high energy density with a long cycle life and a comparatively efficient round trip, which suits the smoothing of sustained renewable swings, though the electrolyte tank volume needed for large-scale deployment remains a siting consideration. The HAFC sits between these roles, offering long-duration balancing through the hydrogen path, but with a round-trip efficiency of only about 20–40% and a durability that is sensitive to frequent load cycling. Read together, these characteristics suggest a complementary rather than competing role for the three technologies in area 1 (VRFB and SMES) and area 2 (VRFB and HAFC), consistent with the combined improvement reported for the ESS-assisted scenarios above. The previous simulations quantify the combined effect of the paired devices in each area rather than isolating each device’s individual contribution.

5.12. Case 4: Validation by OPAL-RT Simulator

The OPAL-RT simulator is employed to experimentally verify the real-time behavior of the proposed PKO-TDn(1+PIDn) method. Instead of relying solely on offline simulations, MATLAB/Simulink models are deployed to the OPAL-RT simulator to enable hardware-in-the-loop (HIL) execution, ensuring that the controller is assessed under realistic computational conditions in which numerical delays, communication latencies, and execution imperfections naturally occur that are not captured in conventional offline simulations [77,78]. The HIL configuration is composed of several coordinated components that work together to create a realistic real-time testing environment. At the center of this setup is a high-performance host PC with an Intel Core i7-10700 CPU processor, which serves as the command and compilation station that compiles the MATLAB/Simulink models using RT-LAB and an OP4512 real-time simulator that executes the power–system dynamics. When transferring the model from offline simulation to the real-time simulator, the model was discretized using a fixed-step ode4 (Runge–Kutta) solver at a step size of 10 µs, selected to meet the real-time deadline of the OP4512 while preserving the dynamics of interest.

The model is split into a Master subsystem (SM), which runs the plant and controllers in real time under a hard 10 μs frame, and a console subsystem (SC) on the host, which passively monitors $∆F_1$, $∆F_2$, and $∆P_{tie}$ via the OpComm blocks that handle real-time data exchange over the high-speed link between the target and the host, ensuring the captured signals genuinely reflected what the target computed, without any alteration. The only change needed for real-time deployment was replacing the offline variable-step solver with a fixed-step solver at a step size, chosen to meet the timing deadline while still capturing the relevant transient dynamics; controller gains and system parameters are left unchanged. The resulting real-time traces closely matched the offline results in undershoot, overshoot, and settling time, with no execution overruns, and stable closed-loop behavior throughout.

Both units are linked through a high-speed network to enable continuous data exchange and synchronized operation. This setup provides a reliable platform for evaluating the real-time performance and robustness of the proposed control strategy.

This arrangement ensures that the controller is tested under realistic conditions, including computation delays, task-scheduling constraints, and numerical discretization effects, thereby allowing for a more reliable assessment of its real-time operability and robustness. Figure 29 illustrates the OPAL-RT experimental setup.

The OPAL-RT simulation and MATLAB simulation results are compared for case 1 scenario 1 in Figure 30, case 2 scenario 1 in Figure 31, and case 3 scenario 3 in Figure 32. From the figures, the MATLAB results are approximately similar to the OPAL-RT simulation results, demonstrating that the proposed approach maintains its effectiveness when executed on real-time hardware. The consistency between the offline MATLAB simulation and the OPAL-RT real-time implementation was quantitatively evaluated using the maximum absolute error, root mean square error (RMSE), and normalized root mean square error (NRMSE), as presented in

Table 24. Quantitative comparison between MATLAB and OPAL-RT simulation results.

Case/Scenario	${∆\mathit{F}}_1$			${∆\mathit{F}}_2$			${∆\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$
	Max |Error|	RMSE	NRMSE (%)	Max |Error|	RMSE	NRMSE (%)	Max |Error|	RMSE	NRMSE (%)
C1-S1	0.0053	0.00089	1.4	0.0015	0.00018	1.1	0.00066	0.00008	1.1
C2-S1	0.057	0.0046	13.24	0.0295	0.0039	5.92	0.0094	0.0011	6.98
C3-S3	0.015	0.0021	8.53	0.0082	0.0011	5.35	0.0028	0.00064	5.04

.

The obtained results demonstrate a high level of agreement between both simulators for all investigated scenarios. For the deviations including $∆F_1$, $∆F_2$, and $∆P_{tie}$, the maximum absolute errors remain within acceptable limits, confirming that the discretized real-time model preserves the dynamic behavior of the original offline model.

For the C1–S1 operating condition, the NRMSE values are limited to 1.4%, 1.1%, and 1.1% for $∆F_1$, $∆F_2$, and $∆P_{tie}$, respectively, indicating excellent matching between MATLAB and OPAL-RT results. Under more severe operating scenarios (C2–S1 and C3–S3), the highest observed NRMSE remains below 13.24%, while most signals maintain errors below 7%. These deviations are mainly attributed to the fixed-step discretization and real-time execution constraints of the OPAL-RT simulator.

The small differences between offline and real-time responses confirm the feasibility of implementing the proposed controller in a real-time environment. As the realized controller is a fixed low-order linear structure, its online computational burden is negligible, and no execution overruns or instabilities occurred during operation.

Therefore, the OPAL-RT results validate not only the dynamic performance of the proposed PKO-TDn(1+PIDn) method but also its practical applicability for hardware-in-the-loop testing.

6. Conclusions

In this study, the LFC was investigated for a two-area interconnected PS comprising non-reheat thermal plants. The capability of the proposed control scheme to handle a nonlinear multi-area system with different control parameters for each area was demonstrated by studying the test system while considering nonlinearity sources such as GRC, GDB, and CTD. By integrating a multi-stage TDn(1+PIDn) controller with PKO and ITAE serving as the OF, dynamic challenges such as frequency deviations and tie-line power fluctuations were reduced. The PKO-TDn(1+PIDn) method was compared in detail to a number of other methods, such as GWO-TDn(1+PIDn), PSO-TDn(1+PIDn), PKO-PID, GWO-PID, and PSO-PID. The inclusion of PV in area 1 and WT in area 2 intensifies the fluctuations within the system. System performance was assessed under a range of operating conditions, such as step and random load changes, and uniform and random RES power profiles. ESSs were added, VRFB and SMES in area 1, alongside VRFB and HAFC in area 2, to mitigate RES variability by absorbing surplus energy and rapidly injecting power during deficits, thereby reducing overshoot, undershoot, and settling time across all tested scenarios. The proposed PKO-TDn(1+PIDn) method outperformed the other methods and consistently delivered superior damping characteristics across these diverse operating conditions. The HIL experiments on the OPAL-RT OP4512 simulator validated the proposed controller’s practical applicability, showing strong agreement with offline simulations with no overruns or instability, which confirmed reliable implementation on FPGA-based real-time hardware. In short, the tilt-derivative and filtered-PID stages tuned by PKO accounted for the improved transient indices, while the layered ESS support suppressed the frequency and tie-line deviations introduced by solar and wind intermittency.

Future work may include:

• Extending the approach to larger multi-area and inverter-dominated microgrids, including coordinated grid-forming inverter control, where advanced robust strategies such as sliding-mode control have proven effective [79].
• Modeling the long-term degradation and lifetime economics of the storage units, the HAFC in particular, under frequent LFC service, drawing on evidence of PEMFC health sensitivity to operating temperature and durability-enhancing strategies [80].
• Evaluating different energy storage technologies, such as pumped hydro storage, supercapacitors, flywheels and hybrid systems, within the same control framework.
• Studying the impact of EV integration and FACTS devices on tie-line power control and overall stability.