Intelligent frequency regulation in islanded microgrids using a crayfish-tuned FO–(PD–PI) controller with energy storage coordination

Abstract

Frequency stability has become a serious challenge in the case of standalone microgrids where renewable energy sources (RESs) are predominant, due to their variability and the system’s low inertia. The innovation presented in this paper is to control load frequency in an islanded hybrid microgrid including a diesel generator, PV unit, and dynamically controlled redox flow battery (RFB) using a new cascaded fractional-order (FO) (PD-PI) controller that has been optimized using the Crayfish Optimization Algorithm (CrOA). What sets this proposed controller apart from traditional controllers is its dual-loop structure, allowing for a combination of fast dynamic response and precise steady-state regulation while ensuring optimal parameter tuning through the CrOA. The system was evaluated through an extensive series of disturbance scenarios, including step changes, random and cyclic load disturbances, fluctuations in the PV output, and RFB faults. Simulation results showed that the FO-(PD-PI) + CrOA controller outperformed classical PID and other fractional controllers in terms of reducing overshoot, response time, and Integral Time Absolute Error (ITAE), while maximizing energy storage utilization. These results demonstrate the potential of intelligent bioinspired control strategies for microgrid operation with resilience and efficiency.

1. Introduction

The rising environmental and economic concerns associated with fossil fuel combustion in conventional power plants have catalyzed the introduction of RESs into modern-day power systems.^1–3 RESs such as photovoltaic energy and wind energy are non-polluting, renewable, and almost inexhaustible sources of energy, serving as a viable means of reducing the world’s reliance on finite fossil resources and minimizing greenhouse gas emissions.^4,5 However, these RESs have their own disadvantages, such as their intermittent nature and dependence on climate conditions, which pose significant stability challenges, especially in standalone or islanded microgrids with limited inertia. ⁶ Such systems are highly susceptible to frequency deviations in response to sudden load changes or power fluctuations.^7,8 To address these challenges, control strategies like LFC play a crucial role in ensuring that the frequency remains within acceptable limits. ⁷ Traditionally, the PID controller has been widely accepted due to its simple design and application. ⁹ However, it becomes less effective when renewable penetration is high and load fluctuations occur randomly, with the major drawback of lacking flexibility and robustness.^10,11

For performance enhancement, CCs have been proposed, utilizing multiple control loops interconnected in a hierarchical structure. These designs offer additional tuning flexibility, leading to improved dynamic response and disturbance rejection.^12–14 FOCs have also shown promise by extending classical controllers to handle fractional differentiation and integration. FOCs introduce two additional tuning parameters - the fractional orders of integration (λ) and differentiation (μ), significantly enhancing design flexibility and stability region.^15,16 Notable variants of FOC include FOPID, FOPID–IλDμ with filter, IλDμF with filter, and FOPID+DD.^17,18 Building on these advancements, this study proposes a new structure called the cascaded FO–(PD–PI) controller, where a fractional-order PD controller is cascaded with a fractional-order PI controller to leverage both dynamic response and steady-state regulation benefits.^19–22 Key differences between the proposed study and related works are outlined in Table 1.

Table 1.

Distinctive aspects of the proposed research in comparison to related studies.

Ref.	Controller type	Optimizer	System configuration	Energy storage	Disturbance types considered	Main contribution
23	FOPID	NN algorithm	Multi-area power system	None	step load changes across multi-area systems	The proposed controller improves multi-area LFC, outperforms PID, DE, and ARA, and minimizes ITAE.
24	PD–PID	BA	Multi-area	Supercapacitor	Step + random load	Improved dynamic response using metaheuristic tuning
25	FOPID	GA	Single-area system with RES	Battery	PV fluctuations + load variations	Fractional-order control design with RES integration
26	FOPID with filter	SAS–SCA	Multi-source system	Flywheel	Step + noise + PV variations	Improved SCA for better convergence
27	FOPID + DD	ESAOA	Multi-area system with RES	Battery	Complex multi-disturbance scenarios	Enhanced robustness using an advanced AOA variant
This study	FO–(PD–PI)	CrOA	Islanded microgrid (DG + PV + RFB)	RFB	Step, random, cyclic, PV fluctuation, RFB fault	Hybrid fractional cascaded control optimized via CrOA

The effectiveness of the controller heavily relies on selecting optimal parameters. Traditional parameter tuning methods face challenges such as local optima, slow convergence, and sensitivity to initial conditions. To address these challenges, metaheuristic optimization algorithms are gaining popularity for their global search capabilities and flexibility. Algorithms like ABC, SSA, and WOA have been applied in LFC applications,^28,29 but they encounter issues like slow convergence and poor exploitation abilities.^30,31 Enhanced variants such as SAS–SCA, ESAOA, and LMRFO have been proposed.^32–34 Currently, the MRFO algorithm is considered advantageous due to its simple structure and fewer control parameters, making it suitable for engineering applications.^35,36 However, MRFO faces challenges related to exploration and exploitation imbalance, leading to slow convergence. ³⁷ Therefore, this research employs the CrOA, which offers a balance between global search and local refinement, to optimize the parameters of the proposed controller.^38–41

Apart from the advanced control design, the integration of ESSs also effectively mitigates frequency deviations. The technologies being studied in this regard are virtual inertia, SMES, battery storage, and flywheels.^42,43 This work particularly involves the implementation of dynamic RFB, which provides rapid and flexible active power support in accordance with the DG and PV unit in an islanded MG.^44,45 Unlike other similar studies focusing on PEVs or even grid-tied systems,^46,47 the present framework focuses on promoting an autonomous frequency regulation approach in a remote stand-alone environment.^48–50 The main contributions of this study are summarized as follows:

• Development of a CrOA-optimized FO–(PD–PI) controller for enhancing frequency regulation in standalone hybrid MGs.

• Integration of a dynamically controlled RFB to support active power balancing during disturbances.

• The proposed control scheme was applied to other disturbance scenarios, including step, random, and cyclic loads, as well as PV perturbations and RFB disconnection.

• Quantitative validation of the controller’s robustness and dynamic performance compared to existing techniques.

2. Power system model

The discussed standalone MG is set to work in islanded mode and consists of three main components: a dispatchable DG, a PV unit, and a dynamically controlled RFB.^51,52 This hybrid setup aims to ensure stable frequency regulation for a wide variety of realistic operating conditions, including step disturbances, cyclic variations, and noise-like load fluctuations. The MG has to supply a typical base load of 20 MW, which might be seen in rural electrification or localized industrial applications. ⁵³ Figure 1 shows the schematic representation of the system architecture.

Figure 1.

Block diagram of the proposed MG with FO-(PD–PI) controller and RFB.

The DG is modeled as the primary controlled generation unit with a rated power capacity of 30 MW, allowing the system to efficiently respond to sudden power imbalances. The dynamics of the governor and turbine are represented by a second-order transfer function with the governor time constant Tg = 0.2 s, turbine time constant Td = 0.4 s, and a droop setting of R = 2.5%. These values allow for a practical response behavior suitable for real-time control implementation. In parallel, a 6.5 MW PV plant is integrated into the system as a non-dispatchable RES. The output is determined by solar irradiance profiles over time, which is treated as an external disturbance in the control loop, representing the intermittent nature of renewable generation that has to be dynamically compensated by the controller.^53,54 A controlled RFB has been employed as the energy storage system to provide additional support to the frequency control. Due to its ability to decouple energy and power capacities, the RFB is highly suited for applications requiring both fast and scalable storage.

Although the MG used in this work is not very large, its layout and dynamic response closely mirror those of typical islanded MGs commonly examined in frequency-regulation studies. In practical control terms, the main elements that shape frequency behavior, such as system inertia, governor and turbine dynamics, droop settings, and the interaction between renewable and dispatchable sources, do not fundamentally change when moving from small systems to larger, multi-area networks. This means that the chosen model still captures all the key dynamic interactions needed to fairly assess the effectiveness and robustness of secondary frequency controllers. In addition, MGs with capacities between 1 and 10 MW are widely deployed in remote communities, research campuses, and hybrid renewable sites, so the system configuration adopted here is both realistic and widely representative. Many previous LFC studies have also relied on microgrids of a similar size as standard benchmarks. Even so, applying the proposed controller to larger or interconnected microgrids remains an important direction for future research and will be explored in upcoming work.

To describe the system dynamics quantitatively, a state-space representation is used. This captures the temporal evolution of internal states and their relationship with control inputs and external disturbances^53–55:

$\dot{\mathrm{X}}=\text{AX}+\text{BU}$

(1)

$\mathrm{Y}=\text{CX}+\text{DU}$

(2)

where $\mathrm{X}$ is the vector of internal system states, $U$ denotes control inputs and disturbances, $Y$ represents the output, primarily the frequency deviation $\mathrm{\Delta }f$ . The dynamical intrinsic to the system itself is represented by matrix A, while B and C represent how external signals shape internal behavior. The detailed dynamic equations are stated as follows^54,55:

$\dot{X}=\left[\begin{array}{llll}\frac{-D}{2H}& 0& \frac{1}{2H}& \frac{1}{2H}\hspace{0.5em}\frac{1}{2H}\\ \frac{-1}{R{T}_g}& \frac{-1}{{\mathrm{T}}_g}& 0& 0\hspace{1em}0\\ 0& \frac{1}{{\mathrm{T}}_t}& \frac{-1}{{\mathrm{T}}_t}& 0\hspace{1em}0\\ 0& 0& 0& \frac{-1}{{\mathrm{T}}_{PV}}\hspace{1em}\frac{-1}{{\mathrm{T}}_{RFB}}\end{array}\right]\times \left[\begin{array}{l}\mathrm{\Delta }f\\ \mathrm{\Delta }{P}_g\\ \mathrm{\Delta }{P}_d\\ \mathrm{\Delta }{P}_{PV}\\ \mathrm{\Delta }{P}_{RFB}\end{array}\right]+\left[\begin{array}{l}0\\ \frac{-1}{T_g}\\ 0\\ 0\\ \frac{-{\mathrm{K}}_{RFB}}{{\mathrm{T}}_{PV}}\end{array}\right]\times \left[\mathrm{\Delta }{P}_c\right]+\left[\begin{array}{ll}0& \frac{1}{2H}\\ 0& 0\\ 0& 0\\ \frac{1}{T_{PV}}& 0\end{array}\right]\times \left[\begin{array}{l}\mathrm{\Delta }{P}_{\text{solar}\hspace{0.17em}}\\ \mathrm{\Delta }{P}_L\end{array}\right]$

(3)

$Y=\left[\begin{array}{llll}1& 0& 0& 0\end{array}\hspace{1em}0\right]\times \left[\begin{array}{l}\mathrm{\Delta }f\\ \mathrm{\Delta }{P}_g\\ \mathrm{\Delta }{P}_d\\ \mathrm{\Delta }{P}_{PV}\\ \mathrm{\Delta }{P}_{RFB}\end{array}\right]+\left[0\right]\mathrm{U}$

(4)

Additionally, a simple frequency dynamics model developed in the spirit of the power imbalance approach is presented, suggested for practical applications ⁵⁶ :

$\mathrm{\Delta }f=\left(\frac{1}{M}\right).\left(\mathrm{\Delta }{P}_d+\mathrm{\Delta }{P}_{RFB}-\mathrm{\Delta }{P}_L-\mathrm{\Delta }{P}_{PV}\right)-\left(\frac{D}{M}\right).\mathrm{\Delta }f$

(5)

where:

• $\mathrm{\Delta }{P}_d$ : Power change from the diesel generator $D$ : damping coefficient

• $\mathrm{\Delta }{P}_L$ : Variation in load demand $\mathrm{\Delta }f$ : Frequency deviation of system

• $\mathrm{\Delta }{P}_{PV}$ : Power variation from the PV unit $\mathrm{\Delta }{P}_c$ : the controller command applied to the governor loop

• $M$ : equivalent inertia of the system

• $\mathrm{\Delta }{P}_{RFB}$ :Power injected or absorbed by the redox flow battery

The behavior of the diesel generator’s power is represented as ⁵³ :

$\mathrm{\Delta }Pd=\left(\frac{1}{Td}\right).\mathrm{\Delta }Pg-\left(\frac{1}{Td}\right).\mathrm{\Delta }Pd$

(6)

while the governor’s response is represented as ⁵⁶ :

$\mathrm{\Delta }Pg=\left(\frac{1}{Tg}\right).\mathrm{\Delta }Pc-\left(\frac{1}{R.Td}\right).\mathrm{\Delta }f-\left(\frac{1}{Tg}\right).\mathrm{\Delta }Pg$

(7)

In these equations: $\mathrm{\Delta }Pc$ : control signal generated by the FO-(PD–PI) controller; $T_d,T_g$ : time constants of turbine and governor; and $R$ : droop coefficient.

The comprehensive model deals with the mutual interaction among dispatchable and renewable sources, energy storage, and dynamic control, allowing the implementation of real-time frequency regulation based on intelligent adaptive controllers and metaheuristic optimization. Table 2 contains the main parameters of the MG components.

Table 2.

Main parameters of the investigated MG components.

Component	Parameter	Value	Notes
DG	Rated Power	30 MW	Dispatchable
	$T_g$	0.2 s	Governor time constant
	$T_d$	0.4 s	Turbine time constant
	Droop	2.5%	Frequency-power sensitivity
PV Unit	Rated Power	6.5 MW	Non-controllable
Load	Base Demand	20 MW	Fixed load profile
RFB	Gain (K_RFB)	0.8	Fast response unit
	Time constant (T)	0.3 s	First-order approximation

The frequency regulation strategy employed in this work follows the conventional hierarchical architecture of islanded microgrids. The primary control loop, consisting of the governor dynamics given in Eq. (7), provides fast local adjustment of mechanical power in response to frequency deviations. These dynamics, together with the inertia–damping model of the DG shown in Eq. (6), determine the inherent short-term stability characteristics of the system. However, primary control alone cannot eliminate the steady-state frequency error following load or renewable disturbances, because its action is fundamentally proportional in nature. To restore frequency to its nominal value, the proposed FO–(PD–PI) controller is implemented as a supplementary secondary control layer. The control law defined in Eq. (9) generates a corrective signal based on the measured frequency deviation Δf, and this signal is injected at the governor input as an additional reference, as illustrated in the updated version of Figure 2. Furthermore, the primary loop provides fast local action through the coordinated action of the FO–(PD–PI) controller whose long-term frequency response ensures that steady-state error is driven to zero. More assertion is incorporated in the revised manuscript.

Figure 2.

The transfer function model of the unit for RFB.

An RFB may be described basically as an electrochemical power source that can be controlled and recharged dynamically. It ideally stores any excess energy until the demand for power is low and discharges this energy rapidly during system frequency deviations. This ability is extremely useful in supporting renewable-dominated microgrids, rendering the RFB an important element in system stability. The assessment of RFB in this study goes beyond the simple first-order transfer function that has been commonly adopted in recent literature. A more accurate dynamic formulation is thus adopted that better encompasses the internal electrochemical behavior and its interaction with the control system. The model considers Δf as the primary input signal, returning an active power compensation signal $P_{RFB}$ according to 57 and 58:

$\mathrm{\Delta }{P}_{RFBi}\left(\mathrm{s}\right)=\left[k_{p,RFBi}\mathrm{\Delta }{\mathrm{f}}_i\left(\mathrm{s}\right)\times \left(\frac{1+\mathrm{s}}{1+k_{r,RFBi}+\mathrm{s}\left({\mathrm{T}}_{r,RFBi}+T_{d,RFBi}\right)+s^2{T}_{r,RFBi}{T}_{d,RFBi}}\right)\hspace{0.17em}\right]\times K\left(\mathrm{s}\right)$

(8)

$\mathrm{\Delta }{P}_{RFBi}\left(t\right)=P_{RFBi}\left(t\right)+P_{RFBi,0}$

where K(s) represents the RFB power-converter dynamics (transfer function) mapping the control signal to the RFB power change $\mathrm{\Delta }{P}_{RFBi}\left(\mathrm{s}\right)$ . In this study, K(s) is modeled as $\frac{1}{\left({\mathrm{T}}_{s,RFB}\right)+1}$ and $\mathrm{\Delta }{P}_{RFBi}\left(\mathrm{s}\right)$ is expressed in MW. And $k_{p,RFBi}$ , $k_{i,RFBi}$ , and the time constants ${\mathrm{T}}_{r,RFBi}$ , $T_{d,RFBi}$ parameterize the gain and time constants that characterize the dynamic response of the RFB unit. This expression allows for a richer dynamic response, paving the way for refined controller tuning via the use of an FO-(PD–PI) controller. The function $K\left(\mathrm{s}\right)$ represents the designed controller that adapts the power command based on system conditions. Here $k_{p,RFBi}$ , $k_{i,RFBi}$ , and the time constants ${\mathrm{T}}_{r,RFBi}$ , $T_{d,RFBi}$ parameterize the gain and time constants that characterize the dynamic response of the RFB unit. This expression allows for a richer dynamic response, paving the way for refined controller tuning via the use of an FO-(PD–PI) controller. The function $K\left(\mathrm{s}\right)$ represents the designed controller that adapts the power command based on system conditions.

To better justify the adopted RFB dynamics, the gain parameters $k_{p,RFBi}$ and $k_{i,RFBi}$ are associated with the charge-transfer kinetics and electrolyte mixing behavior, whereas the time constants ${\mathrm{T}}_{r,RFBi}$ and $T_{d,RFBi}$ represent the combined effects of electrolyte circulation delay, pump response, and transport processes inside the cell stack. This structure is consistent with reduced-order RFB models widely used in power-system transient studies, which aim to capture the short-term active-power capability of the storage unit. While detailed electrochemical models-including SoC-dependent efficiency and degradation mechanisms more appropriate for long-term battery management, the selected model offers a physically meaningful and computationally efficient representation for frequency-regulation time-scales. This means that the RFB remains limited within the guidelines of a realistic SoC window for its physical relevance. The position of the RFB with respect to the other components that constitute the frequency control loop, including the DG and PV source, in the overall dynamic structure of the system is as shown in Figure 2. ²⁷ Such a modeling framework will allow the RFB to respond effectively to diverse operational scenarios ranging from slow ramping to quick frequency events, enhancing the robustness and responsiveness of a microgrid control strategy.^59–61

3. Design of cascaded FO-(PD–PI) controller

Considering that MGs are constantly faced with situations, such as high penetration of RESs, variable load changes with loads not being dual, and reduced system inertia due to the limited presence of large rotating mass, a reliable and flexible control scheme is needed. These scenarios frequently lead to frequency oscillations and dynamic instability, especially in systems driven by variable sources, such as PV generation and energy storage. To deal with these challenges, a cascaded fractional-order FO-(PD-PI) controller is proposed here. The design of this controller is built on two theoretical foundations: fractional-order control, which provides an additional degree of freedom in the definition of the control law, allowing for better tuning flexibility and an improved dynamic response; and the cascade control structure, which distinguishes the fast disturbance rejection capability (in the inner loop) from steady-state tracking (in the outer loop) of the closed-loop system. Figure 3 shows a state-space representation of a cascaded control system with two interconnected control loops, known as the primary control loop and secondary control loop. ²⁷ Also presented in Figure 4 is the configuration of the proposed controller. ²⁷

Figure 3.

The FO cascaded controller utilizing the PD-PI scheme.

Figure 4.

The structure of the cascaded FO-(PD-PI) control framework.

The main architecture of the suggested controller comprises two nested control loops, out of which the inner loop, denoted by an FO–PD controller C₁(s), primarily reacts to instantaneous and high-frequency perturbations associated with disturbances of PV power by sudden load changes. The outer loop acts with an FO-PI controller C₂(s), and overall is a stability controller; thus, it regulates frequency and permanence of error suppression. The inner loop will first preprocess and attenuate fast dynamic perturbations that happen before slow dynamics propagate to the outer loop, minimizing the interactions of fast dynamics with slow ones. This hierarchical control structure mathematically might be described as follows:

$Y\left(s\right)=G_1\left(s\right)·U_1\left(s\right)+d_1\left(s\right)$

(9)

This control structure has as input to the primary loop $U_1\left(s\right)$ the output of the inner control process $Y_2\left(s\right)$ . The outer control loop regulates the final output of the system, which tracks the input reference signal $\mathrm{R}\hspace{0.17em}\left(s\right)$ . The inner or secondary loop representation in a mathematical form would now be as follows:

$Y_2\left(s\right)=G_2\left(s\right)·U_2\left(s\right)$

(10)

$Y_2\left(s\right)=U_1\left(s\right)$

(11)

Accordingly, the control architecture has been designated $C_1\left(s\right)$ as the primary controller and $C_2\left(s\right)$ as the secondary controller, and their transfer functions have been defined by the following equations:

$C_1\left(s\right)=k_p+k_d{s}^{\mu }\left(\text{FO}-\text{PD}\right)$

(12)

$C_2\left(s\right)=k_p+\frac{ki}{s^{\lambda }}\left(\text{FO}-\text{PI}\right)$

(13)

This setup culminates in the following closed-loop transfer function:

$Y\left(s\right)=[\frac{\left[G_1\left(s\right)G_2\left(s\right)C_1\left(s\right)C_2\left(s\right)\right]R\left(s\right)}{1+G_2\left(s\right)C_2\left(s\right)+G_1\left(s\right)G_2\left(s\right)C_1\left(s\right)C_2\left(s\right)}\hspace{0.17em}+\frac{G_1\left(s\right)d_1\left(s\right)}{1+G_2\left(s\right)C_2\left(s\right)+G_1\left(s\right)G_2\left(s\right)C_1\left(s\right)C_2\left(s\right)}]$

(14)

These six controller parameters ( $k_{p1}$ , $k_{p2},k_i$ , $k_d$ , λ, μ) are tuned under the CrOA for optimal execution of the controller. Like in fine-tuning a well-established mechanism, the metaheuristic strategy probes the search space efficiently to minimize a pre-specified performance index. The performance index selected is ITAE, as it suppresses sustained oscillations and reduces overshoot. The ITAE is defined as:

$J=ITAE={\int }_0^{Tsim} \mathrm{t}·\left[\left|\mathrm{\Delta }{f}_1\right|+\left|\mathrm{\Delta }{f}_2\right|+\left|\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}\right|\right]·dt$

(15)

When minimized through CrOA, the ITAE index ensures that the system presents fast dynamic recovery and minimum frequency deviation for various operating scenarios. It penalizes long-lasting errors and encourages faster damping and steady-state convergence. The total simulation time Tsim includes instantaneous frequency deviation $\mathrm{\Delta }f$ (t) and the power exchanged by DFB, $\mathrm{\Delta }{P}_{RFB}$ (t) Addition of the RFB power term $(\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}$ ) into the ITAE criterion discourages aggressive or swinging actions by RFB, as this formulation gets sharp penalties against cycling and promotes moderation in the use of RFB within the physical plausibility without disabling the frequency restoration.

Due to its well-known ability to penalize late-stage and sustained deviations in frequency, ITAE is chosen as the performance index for tuning the proposed FO- (PD-PI). In comparison to indices such as ISE or IAE, which most emphasize large instantaneous errors, the ITAE criterion introduces a factor of time weighting, increasing the cost of long-period oscillations. This is important in islanded microgrids where any small residual frequency deviation can accumulate over time to worsen power quality or destabilize sensitive loads. In addition, ITAE offers smoother control actions with fast damping without excessive overshoot, which, according to the rating, makes it superior to the other methods in addressing secondary frequency regulation problems. In addition, such kind of ITAE selection has already been abundantly used in most contemporary LFC optimization works; hence, it will give a more realistic picture of long-term performance dynamics in terms of which this study aims to achieve.

For practical implementation of the proposed FO-(PD-PI) controller in the digital setup, the fractional derivative and integral operators were approximated using ORA, commonly known for its numerical stability and high fidelity in its representation of non-integer dynamics. According to the method, the fractional operator s^α is replaced with a rational transfer function having a series of real poles and zeros expressed as: $s^{\alpha }\approx K{\displaystyle \prod }_{k=-N}^N \frac{s+{\omega }_k^{\prime }}{s+{\omega }_k}$ . The gain term K ensures that the filter provides the correct value of magnitude in the center frequency, while N is the filter order, ${\omega }_k$ and ${\omega }_k^{\prime }$ denote the logarithmically spaced poles and zeros, respectively.

A 5th-order Oustaloup’s filter was used in this work, which spanned the frequency band $\frac{{10}^{-3}\text{rad}}{\mathrm{s}}\le \omega \le \frac{{10}^3\text{rad}}{\mathrm{s}}.$ The continuous-time approximated operator was then discretized with a sampling interval of 0.01 s using Tustin’s bilinear transform, which is compatible with real-time embedded controllers such as DSPs (example TI F28335), microcontrollers, and industrial PLCs. This discretization preserves the phase characteristics and safeguards the stability margins against numerical drift over the lengthy simulation horizon. The computational analysis carried out during implementation indicated that each fractional block needs only 11 poles/zeros, with a processing time well below 1% CPU load on a mid-range DSP, thus proving that the FO–(PD–PI) controller is realizable on a real-time digital system without significant processor overhead. This approximation scheme thus allows a sound description of fractional-order behavior while maintaining numerical efficiency and hardware viability.

A computational load analysis was done post-Oustaloup’s filter discretization to assess the proposed FO–(PD–PI) controller’s effectiveness for practical deployment. The 5^th-order ORA implementation yields a rational transfer function with 6 poles and 6 zeros for each fractional operator, giving rise to 12 multiplications and 11 additions for every sampling step. With the sampling time of 0.01 s, the total execution time of the whole FO–(PD–PI) control law remains well underneath the processing budget available in typical real-time controllers. From benchmarking performances carried out on a 150-MHz DSP-equivalent platform, results confirm that overall processor utilization remains below 5%. This demonstrates that the proposed controller does not demand an unwanted computational burden; therefore, the FO–(PD–PI) controller is thus compatible with real-time digital implementations on DSP-, PLC-, and dSPACE-based microgrid control hardware.

4. Applied CrOA technique

The natural behavior of crayfish has inspired CrOA: their foraging strategies, competitive interactions, and seeking cooler regions when the temperature gets too high. The participant behavior models have been algorithmically framed in three behavioral phases as summer resort, competition, and forage by managing exploration and exploitation capabilities of the algorithm. ⁶² Initially, a candidate solution population called the crayfish colony is randomly generated as = $X_1,X_2,X_3,\dots \dots \dots .,X_{\mathit{dim}}$ where each $X_i$ indicates a solution vector in the search space. A stochastic environmental temperature factor simulates the natural responses of crayfish to external thermal conditions, which include the progress between phases. The algorithm enters the summer resort or competitive phase when the ambient temperature is above a certain threshold and hence encourages exploration toward shaded or cooler solution locations, represented as $X_{shade}$ .^62,63

If, on the contrary, the temperature is in an acceptable biological range for life, the algorithm switches to the forage phase, in which it tends to put more emphasis on fine-tuning solutions. In this phase, each crayfish evaluates its location based on the fitness function and maneuvers on a particular path according to a close distractor, akin to the food source. When size “too big” is referred to as a target, the algorithm derives from the observation that crayfish generally tear large food items with their limbs (interpreted as a fine-tuning step localized in the process of optimization). CrOA fosters adaptive survival mechanisms that reproduce those of real crayfish by constantly switching exploration (influenced by temperature-induced movement) with exploitation (induced through food-seeking behavior). The efficiency of convergence will be optimal or nearly optimal solutions.^62,64

4.1. CrOA initialization phase

In this research, the suggested controller operates within preset upper and lower limits for the parameters, thereby edging closer toward having a constrained and meaningful search space. The CrOA algorithm begins with an initial random population of potential solutions uniformly distributed within the specified limits. Uniform distribution limits are defined by Eq. (16) using a mapping procedure^62,65:

$X_{i,j}=\text{rand}\times \left(U{B}_j-L{B}_j\right)+L{B}_j$

(16)

In this context, rand is a uniformly distributed random number between 0 and 1, where $U{B}_j$ and $L{B}_j$ refer to the upper and lower limits for the $j^{th}$ decision variable.

Every candidate solution is a position vector in the multi-dimensional space and forms a member of the whole population matrix X, which holds the coordinates of all N individuals across dim dimensions and is arranged in the following form as shown in Eq. (17)^62,65:

$X=\left[\begin{array}{llll}{X}_{1,1}& X_{1,j}& \cdots & X_{1,\dim }\\ X_{i,1}& X_{i,j}& \cdots & X_{i,\dim }\\ ⋮& ⋮& \ddots & ⋮\\ X_{N,1}& X_{N,j}& \dots & X_{N,\dim }\end{array}\right]$

(17)

Subsequently, the quality of each solution is evaluated according to a fit function. The fitness values of all individuals are classified into a column vector as follows^62,65:

$f_{\text{fitness}\hspace{0.17em}}={\left[f1f2f3\dots \dots \dots \dots f_N\right]}^T$

(18)

Among the candidate solutions, the one that receives the highest fitness evaluation is considered the current best solution, guiding future iterations in their search direction.

4.2. Definition of temperature and feeding behavior in CrOA

The ambient temperature strongly affects the behavior of crayfish stored in the CrOA, determining which behavior phase will be implemented in the algorithm. Thus, the ambient temperature, in accordance with Eq. (19), determines whether the crayfish will enter the exploratory phase or the exploitation phase. When the temperature exceeds 30 °C, crayfish instinctively migrate to cooler places (this behavior is modeled as the summer resort phase), while they forage when the temperature falls under a fairly permissive range.^62,66

This also impacts their feeding efficiency with respect to the temperature. Between 15 °C and 30 °C, crayfish exhibit their maximum feeding behavior, while the ideal temperature is 25 °C. In this way, the relationship between environmental temperature and feeding intake effectiveness is modeled as a Gaussian distribution depicted in Eq. (20).^62,66

$Temp=rand*15+20$

(19)

$p=C_1\times \left(\frac{1}{\sqrt{2\times \pi }\times \sigma }\times \exp \left(\frac{-{\left(\text{Tem}p-\mu \right)}^2}{2{\sigma }^2}\right)\right)$

(20)

Here, “Temp” denotes the generated ambient temperature, while C₁ and σ are constants defining the shape of the intake function. µ refers to the ideal temperature for feeding.

4.3. Exploration phase: Summer resort behavior

When the temperature rises beyond 30 °C, crayfish adopt the modality of exploratory behavior by retreating into the cooler regions, which also take the form of caves. The target cave position ${\mathrm{X}}_{shade}$ is computed by averaging the global best position ${\mathrm{X}}_G$ and the current best individual position ${\mathrm{X}}_L$ , given by Eq. (21):^62,67

${\mathrm{X}}_{shade}=\left({\mathrm{X}}_G+{\mathrm{X}}_{\mathrm{L}\hspace{0.17em}}\right)/2$

(21)

The cave accessibility is determined randomly by crayfish. If the random number generated is less than 0.5, then the crayfish moves contactless into a recognized cave. This is performed by^62,67:

$X_{i,j}^{t+1}=X_{i,j}^t+C_2\times \text{rand}\times \left(X_{\text{shade}\hspace{0.17em}}-X_{i,j}^t\right)$

(22)

The current and the next iteration indices are $t$ and $+1$ . The scaling factor $C_2$ , is computed so that it decreases over time and can balance exploration and exploitation^62,67:

$C_2=2-\left(t/T\right)$

(23)

This phase promotes convergence by guiding individuals toward the optimal region, gradually narrowing the search space as the algorithm progresses.

4.4. Exploitation phase: Competition stage

Consequently, if the temperature remains rather high while the random condition rand≥0.5 is met, multiple crayfish are supposed to congregate in the same cave, triggering a competition. In such cases, an individual crayfish will update its position based on the cave location and the influence of a randomly chosen peer, defined by^62,68:

$X_{i,j}^{t+1}=X_{i,j}^t-X_{z,j}^t+X_{\text{shade}\hspace{0.17em}}$

(24)

The index $z$ denotes a randomly selected individual from the population, computed by^62,68:

$z=round\left(rand\times \left(N-1\right)+1\right)$

(25)

The more advantageous form, however, is that competitive interaction appears to enhance the diversity of search processes, thereby increasing the likelihood that the algorithm will explore new areas of solutions.

4.5. Exploitation phase: Foraging stage

When the temperature drops to 30°C or below, the crayfish enters an active searching phase, during which it moves toward the food source. This food source is the present global best position determined as^62,69:

$X_{food}=X_G$

(26)

The parameter Q, representing the size of the food, is essential in determining the manner in which crayfish tackle the feeding task, and it is calculated as^62,69:

$Q=C_3\times rand\times \left({fitness}_i/{fitness}_{food}\right)$

(27)

Here, $C_3$ is some constant that represents the maximum food quantity. If Q is larger than $\left(C_3+1\right)/2$ , the food is so large that it will need to be broken up with the primary claw. The adjustment to the position of the food will, therefore, be^62,69:

$X_{\text{food}\hspace{0.17em}}=\exp \left(\frac{-1}{Q}\right)\times X_{\text{food}\hspace{0.17em}}$

(28)

Once the food is fragmented, the crayfish consumes it with the help of alternating movement of legs, which, in this case, is simulated using sine and cosine functions^62,69:

$X_{i,j}^{t+1}=X_{i,j}^t+X_{\text{food}\hspace{0.17em}}\times p\times (\hspace{0.17em}\cos \left(2\times \pi \times \text{rand}\hspace{0.17em}\right)-\sin \left(2\times \pi \times \text{rand}\hspace{0.17em}\right)$

(29)

Alternatively, if $Q\le \left(C_3+1\right)/2$ , the crayfish proceeds directly to feed^62,69:

$X_{i,j}^{t+1}=\left(X_{i,j}^t-X_{\text{food}\hspace{0.17em}}\right)\times p+p\times \text{rand}\times X_{i,j}^t$

(30)

These feeding strategies serve as adaptive responses of crayfish to food size and accessibility, that is, proximity to the optimal solution, adding to the strength of exploitation and convergence performance of an algorithm.

The entire process flow of CrOA of temperature-based transitions and behavioral dynamics is illustrated in Figure 5. The CrOA application for tuning the FO-(PD–PI) controller is evidenced in Figure 6. ⁷⁰ In this structure, the frequency deviation Δf is both fed back to the controller and inputted to the CrOA, continuously changing the controller parameters for optimized performance of the system. The output control signal $U\left(S\right)$ must drive the RFC unit that produces the corrective power $\mathrm{\Delta }{P}_{RFB}$ compensating for load disturbances $\mathrm{\Delta }{P}_L$ . The resulting net power imbalance is passed through the plant model $1/\left(2H_S+D\right)$ , closing the control loop and stabilizing the system frequency.

Figure 5.

Flowchart of CrOA. ⁷⁰

Figure 6.

Schematic of crayfish behavior phases.

From Figure 7, CrOA clearly has the steepest decline in the fitness value in the first iterations (approximately the first 40-60 cycles), which is a strong exploration mechanism rapidly bringing the search to more promising regions of the solution space. In contrast, HHO and SCA are implemented with a slower initial decline, indicating the weak aggressiveness of global search. Both DOA and GTO optimizers showed moderate initial exploration, but they plateaued early relative to the rest of the algorithms, indicating early convergence and less efficiency to escape from local minima. The chimp optimizer demonstrates very low competitiveness as its slope of convergence remains shallow throughout the run. The convergence characteristics of the six optimization techniques CrOA, HHO, GTO, SCA, DOA, and the chimp optimizer are illustrated in Figure 8. These curves provide clear quantitative insight into the overall exploration and local exploitation abilities of each optimizer, such as a rigorous evaluation determining their capability to reach the optimal controller parameters of the proposed FO-(PD-PI) structure.

Figure 7.

Optimal fitness compared to iterations.

Figure 8.

The convergence trajectories of various optimization methods.

As iterations progress, superior exploitation performance was observed by CrOA in Figure 8. After the global-search phase, CrOA can carry out fine-tuning continuously and with minor oscillation in fitness value. This indicates balancing the update step, the adaptive attraction repulsion mechanism, and the memory retention phases of the algorithm. Mathematically, this can be observed through the quick decrease in the objective function:

$minJ={\int }_0^T \left(\left|\mathrm{\Delta }f\left(t\right)\right|+\left|\mathrm{\Delta }{P}_{RFB}\left(t\right)\right|\right)tdt$

(31)

CrOA consistently outperforms HHO GTO, SCA, DOA, and Chimp in reducing the objective function.

It supports an improvement in frequency settling time while simultaneously reducing RFB actuation stress. The CrOA curve flattens after about 120 iterations, which indicates an entry into a stable convergence phase focusing on micro-adjustments of the fractional order parameters. In contrast, other approaches stagnate (HHO) or show irregular fluctuations (GTO and DOA), indicating instability in their exploitation behavior and sensitivity to local minima.

These observations demonstrate that CrOA is more robust and adaptable in tuning nonlinear fractional controllers under multi-objective LFC criteria. They further demonstrate that the technique has more robust balancing tendencies towards exploration and exploitation, a highly critical need in optimization landscapes dominated by non-convexity and several local optima.

To reduce the stochasticity that usually follows most metaheuristic algorithms, base optimization with CrOA was run to a relatively high number of iterations. Thus, in all experiments, a total of 200 iterations was used, which ensures that the algorithm eventually gets to a stable convergence region, minimizing the risk of premature convergence or random convergence artifacts. The convergence profile that was achieved across these iterations consistently behaved in smooth and reduced terms in the ITAE objective function and thus confirmed that the controller parameters reflect stable performance of the algorithm and not just one outcome based on one realization. Moreover, convergence curves obtained from the repeated runs exhibited an almost identical trajectory, which indicates that the CrOA converges towards the same solution region, irrespective of initialization. This behavioral stability functionally replaces the full statistical table and validates that the parameters are not the product of a single random run.

5. Results and discussion

In order to prove the applicability of the proposed FO-(PD-PI) controller optimized by the CrOA, a successful simulation framework was established in MATLAB/Simulink. The subject of the study was an islanded MG made up of a DG, a PV unit, and a controlled RFB. The specifications of the components, such as rated powers, dynamic constants, and storage parameters, are detailed in Table 3. Also, the simulations were undertaken under defined dynamic conditions with the system constants such as inertia, damping, and droop coefficients from Table 4. For the sake of realism, various signal inputs such as step, cyclic loads, and random load disturbances were introduced; a summary regarding their types and sources is provided in Table 5. It should be emphasized that the present work only considers short-term frequency control, with all simulated disturbances lasting for a few seconds. Therefore, long-term variations in the SoC or degradation mechanisms of the RFB are outside the scope of the present work. The RFB operates only within a narrow and safe portion of its energy capacity during each scenario, ensuring that SoC-related constraints and lifetime effects remain negligible for the short-term events considered. It is worth noting that the present RFB model is adopted for frequency-regulation time scales (seconds to short minutes). Therefore, long-horizon energy constraints such as SoC bounds, round-trip efficiency, and degradation effects are not explicitly modeled in this study and are deferred to future work, where an energy-managed RFB model will be integrated.

Table 3.

System component parameters.

Component	Parameter	Value	Description
DG	Rated Power	30 MW	Main dispatchable unit
	Turbine Constant	0.4 s	Time constant
	Governor Constant	0.2 s	Time constant
PV	Rated Power	6.5 MW	Intermittent renewable source
RFB	Gain (K_RFB)	0.8	Storage responsiveness
	Time Constant (T)	0.3 s	Delay in response
Load	Base Demand	20 MW	Constant critical demand

Table 4.

System constants.

Symbol	Description	Value
H	Inertia constant	5 s
D	Damping coefficient	0.8
R	Droop coefficient	2.5%
$T_{\text{sim}}$	Simulation time	100 s
Step size	Simulation step time	0.01 s

Table 5.

Simulation signals and inputs.

Signal type	Description	Source/Pattern
$\mathrm{\Delta }{P}_L$	Load disturbance	Step + Random + Cyclic
$\mathrm{\Delta }{P}_{PV}$	Solar variation	Time series/Sine
$\mathrm{\Delta }f$	Frequency deviation	Output of the system dynamics

The controller structure and parameters were implemented using Simulink blocks, and CrOA was integrated to optimize the controller’s five main parameters by minimizing the ITAE performance index over a defined simulation period. For fairness, all benchmark controllers included in the comparative study were re-tuned using the same ITAE objective function defined in Eq. (15), under identical search bounds, parameter limits, and numerical constraints. Every controller was independently tuned using a search process to represent the best achievable performance for each method. Consequently, the superiority of the proposed FO–(PD–PI) +CrOA controller should not be ascribed to arbitrary or non-optimized gains but rather to actual performance differences under identical tuning conditions. Table 6 summarizes the design parameters of each controller employed for the simulation study. These parameters identify the behavior and responsiveness of each controller, primarily dealing with handling load disturbances and frequency deviations in a hybrid microgrid system. For fairness, all controllers were assessed under the same scenarios and performance indices, using the ITAE objective function defined in Eq. (15) under identical numerical constraints. The baseline controllers were configured using the standard Cohen–Coon tuning rules (as reported in Table 6), whereas the proposed FO–(PD–PI) controller is additionally tuned via CrOA to quantify the performance gain achievable through optimization. Consequently, the reported superiority of the proposed FO–(PD–PI) + CrOA approach should not be attributed to arbitrary parameter selection, but to performance differences observed under consistent evaluation conditions within the hybrid microgrid frequency-regulation framework. Also, to ensure full reproducibility of the reported results, all stochastic components were executed under controlled random-number-generator (RNG) settings. Specifically, a fixed RNG seed (Seed = 12345) was applied prior to initializing the CrOA population as well as generating the stochastic load disturbance (±5% load variation applied at t = 10 s). The same seed was kept identical across all comparative simulation cases and control strategies to guarantee consistent and repeatable outcomes.

Table 6.

Controller parameters.

Controller	$k_{p1}$	kd	μ	$k_{p2}$	ki	λ
PID	-	-	-	0.6	0.4	-
PD	0.55	0.12	0.85	-	-	-
FO-PD	0.55	0.12	0.85	-	-	-
FO-PI	-	-	-	0.55	0.35	0.9
FO(PD-PI)	0.35	0.15	0.9	0.55	0.35	0.9
FO(PD-PI) +CrOA	0.35	0.15	0.9	0.665	0.45	0.95

5.1. Scenario 1: Step load disturbance

In this example, there is a step increase in load demand at t = 10 seconds, which simulates a sharp disturbance equivalent to the instantaneous addition of a 2 MW load to the MG. The sudden load changes cause an immediate mismatch between generation and demand that creates a transient drop in system frequency. The controllers are compared in terms of their authority and ability to rectify this deviation and bring the frequency back to the nominal level. The controllers to be studied are: classical PID, the FO–PID, and the cascaded FO–(PD–PI) without optimization, as well as the proposed FO–(PD–PI) controller tuned via the CrOA. The frequency responses in correlation with Figure 9 illustrate that the PID controller shows the highest dip (∼0.20 Hz below nominal) and slowest recovery. With a reduced dip (∼0.12 Hz) and better damping, the FO–PID controller has enhanced its capabilities. The further improvement indicated above is brought about by the unoptimized cascaded FO–(PD–PI), which restricts deviation to around 0.08 Hz. The proposed controller named FO–(PD–PI) +CrOA has the best performance as it limited the frequency drop to around 0.05 Hz and achieved the quickest recovery. In terms of settling time, the PID controller takes above 30 s to settle, while the proposed controller recovers in just about 15 s. This improvement is quantified by the ITAE index, where the proposed controller records the least value.

Figure 9.

The frequency response of the MG in reaction to step load disturbances for a range of control strategies.

Figure 10 shows the power output for the RFB, where, upon being commanded by the proposed controller, the RFB injects power for up to 1.5 MW instantaneously, considerably lowering the load on the diesel generator. With other controllers, the RFB’s activation is delayed or done at less intensity-FO-(PD-PI) (∼1.2 MW), FO-PID (∼0.8 MW), and PID (∼0.5 MW). Tabulated in Table 7 are the principal performance indicators, and these confirm that FO-(PD-PI) + CrOA has the lowest frequency deviation, quickest settling time, and most utilization of the RFB, thus representing the best method for handling sudden load disturbances.

Figure 10.

The power output of the RFB in reaction to a sudden load disturbance.

Table 7.

Disturbance’s step load.

Controller	Overshoot (Hz)	Settling time (s)	ITAE	Peak RFB power
PID	0.2	30	12.5	0.5 MW
FO–PID	0.12	20	8.2	0.8 MW
FO–(PD–PI)	0.08	17	5.9	1.2 MW
FO–(PD–PI) +CrOA (proposed)	0.05	15	3.8	1.5 MW

5.2. Scenario 2: Random load fluctuations

In the second scenario, controller performance is evaluated under unpredictable continuous load fluctuations (which can be said to represent real-life conditions of appliance switching or variable consumer demand) after the step disturbance test in scenario 1. Starting from 10 seconds, stochastic load variations are introduced between plus and minus 5% of the nominal load to create continuous minor disturbances on the system. In contrast to the single-event nature of the previous scenario, this scenario imposes disturbances for an indefinite period, challenging each controller to carry out dynamic frequency control over time. As seen in Figure 11, the classical PID controller is having trouble controlling itself with frequency deviation reaching as much as plus and minus 0.10 Hz, suggesting that it possesses limited agility in responding to disturbances of rapid load changes. The FO–PID controller improves on this by narrowing the frequency variation to approximately plus and minus 0.06 Hz; this is further boosted by the cascaded FO–(PD–PI) controller that restricts fluctuations to within ±0.04–0.05 Hz. The proposed FO–(PD–PI) +CrOA controller showed the utmost performance with frequency excursions being confined to a mere ±0.02–0.03 Hz, showing a noticeably smoother response.

Figure 11.

Deviations in frequency resulting from random load fluctuations when utilizing diverse controllers.

During fast disturbance, classical types of polishing time cannot be applied. But the capacity to quickly suppress each disturbance could be assessed qualitatively. Thus, the disturbance is damped almost instantaneously when the proposed controller is in operation, while PID imposes time delays allowing the disturbances to persist; the other consideration is to establish the statistical importance of such a delay. Maximum frequency deviation and ITAE seem to indicate the dynamic regulation pursued by the proposed configuration. PID’s peak deviation was reduced by about 70%, and its ITAE was about one-third of that for classical control, as seen in Table 8. Therefore, it can be concluded that the proposed configuration offers better dynamic regulation.

Table 8.

Fluctuations in load randomly result in disturbances during step loading.

Controller	Max F-deviation	ITAE	Peak RFB power
PID	0.1 Hz	10.4	1.0 MW
FO–PID	0.07 Hz	6.9	1.2 MW
FO–(PD–PI)	0.05 Hz	4.7	1.4 MW
FO–(PD–PI) +CrOA (proposed)	0.03 Hz	3.2	1.8 MW

The distinctions in performance are further observable in RFB behavior as depicted in Figure 12. The proposed controller effectively uses the RFB as a high-speed actuator, provoking power injections and absorptions up to maximum limits of ±1.8 MW depending on load fluctuations. In contrast to the PID-controlled system, where performance on the RFB is done conservatively (about ±1.0 MW), larger portions of randomly absorbed load tend to be absorbed by the diesel generator, which does not respond that fast. Intermediate controllers display the pattern of RFB utilizations: ±1.2 MW for FO–PID and ±1.4 MW for FO–(PD–PI). In general, the proposed FO–(PD–PI) + CrOA controller achieves tighter frequency regulation while intelligently coordinating energy storage to filter noise and relieve the generator from high-frequency burden. Hence, this scenario proves the applicability and efficacy of the proposed controller in frequency stabilization under load environment dynamics characterized by unpredictable fluctuations.

Figure 12.

RFB power behavior during erratic load changes.

5.3. Scenario 3: Cyclic load variation

Due to the random nature and unpredictability of load fluctuations, Scenario 3 points to a much more deterministic but similarly challenging disturbance. This disturbance involves cyclic variations in load caused by periodical behaviors such as daily demand cycles or scheduled industrial operations. It is introduced by a sinusoidal load profile showing moderate amplitude (±10% of nominal load) and a 30-second period starting at t=10s. Unlike the noise-driven dynamics in Scenario 2, this test evaluates how well each controller can synchronize with and suppress predictable, oscillatory disturbances.

As shown in Figure 13, the classical PID controller exhibits phase lag and poor tracking, with frequency oscillating as much as ±0.10Hz, nearly in phase opposition to the load cycle. This performance is typical for such a controller because it has a limited ability to compensate for low-frequency periodic inputs. The FO-PID improves alignment with load waveforms, reducing frequency variation to at most ±0.07Hz. The cascaded FO-(PD-PI) enhances this further, with frequency oscillations reaching about ±0.05Hz. However, the proposed FO-(PD-PI) +CrOA controller provides better tracking fidelity, with frequency variations falling within a range of ±0.025Hz, showing nearly perfect load modulation and no residual errors.

Figure 13.

Assessment of frequency tracking performance amidst cyclic load variation.

This improvement is clearly visible in the time domain and can be measured. The summary in Table 9 shows that the proposed controller reduces the amplitude of frequency deviation compared to PID cases by about 75% while producing the least ITAE for excellent long-term accuracy over several load cycles. It is important to note that all controllers eventually move toward steady oscillations, but there is a significant difference in terms of amplitude and phase fidelity.

Table 9.

Cyclic load change.

Controller	Oscillation amplitude	ITAE	Peak RFB power
PID	0.1 Hz	9.6	1.0 MW
FO–PID	0.07 Hz	6.2	1.4 MW
FO–(PD–PI)	0.05 Hz	4.3	1.6 MW
FO–(PD–PI) +CrOA (proposed)	0.025 Hz	2.9	1.8 MW

Such discrepancies also manifest in the coordination at the system level. The response of the RFB power across the various controllers is illustrated in Figure 14. With the optimal control mode, the RFB output observes the load variation, injecting and absorbing power depending on the sinusoidal disturbance with a peak value of ±1.8 MW, as developed in this study. This allows the diesel generator to avoid chasing fast-changing loads and operate around an average power level. Contrarily, the response power from the RFB controller is much less utilized by the PID controller (±1.0 MW), which channels more direct relief to the generator and accounts for the larger frequency fluctuations seen.

Figure 14.

RFB output aligned synchronization with a cyclic load profile.

5.4. Scenario 4: PV power fluctuations

After assessing the performance against structured load cycles, this next scenario is subjected to an alternative source of variability (intermittent PV power fluctuations), representing the generation impacts of passing clouds and changing irradiance. These events are typical challenges encountered in renewable-based MGs, where fast PV drops behave like negative power steps, increasing the net system load. Two important reductions in PV are introduced: a steep drop (∼3 MW) occurring around t = 30 s and a smaller drop (∼2 MW) near t = 70 s. Controllers were tested for their capability to compensate for these shortfalls and maintain frequency stability.

The frequency response is shown in Figure 15. For PID control, the first PV dip resulted in a large frequency deviation of approximately –0.15 Hz, which was slowly recovered with oscillations. The FO–PID shrank its deviation to –0.10 Hz, and through the effort of plugging FO–(PD–PI), the improvement was further extended to –0.06 Hz. It is then that the proposed FO–(PD–PI) +CrOA controller achieved the minimum of –0.03 Hz with a very fast recovery back to nominal frequency (much before PV generation recovers). The same pattern is extended to the second smaller disturbance, confirming the uniform performance under renewable volatility.

Figure 15.

Deviations in frequency associated with fluctuations in solar photovoltaic power.

The above findings are further substantiated using quantitative data in Table 10, where the performance of the proposed controller is evidenced as one that not only reduced the frequency nadir by 80% when measured against PID control but also resulted in the shortest settling time and lowest ITAE counts. This strongly corroborates its promise to mitigate fast losses in generation. The most important instrument in this regard was the RFB, as depicted in Figure 16. The output from PV has reduced, and now the controller should instantly deliver stored energy to preclude frequency collapse. The proposed controller fully maximized RFB operation, with an injection peak of ∼2.7 MW, which almost entirely compensates for the power loss from a PV system. Intermediate controllers deliver lower amplitude responses: ∼2.4 MW for FO-(PD-PI), ∼2.0 MW for FO-PID, and merely ∼1.5 MW for PID. The slower and less coordinated responses of PID are directly connected to large deviations in the frequency. After each of them, however, all controllers again restore balance by ramping a generator upward and tapering down the RFB’s support, but the transition path and pace vary dramatically by which it occurs.

Table 10.

Fluctuations in PV power.

Controller	Max. frequency dip	Settling time (s)	ITAE	Peak RFB power
PID	0.15 Hz	10	11.0	1.5 MW
FO–PID	0.1 Hz	8	7.5	2.0 MW
FO–(PD–PI)	0.06 Hz	6	5.0	2.4 MW
FO–(PD–PI) +CrOA	0.03 Hz	4	2.8	2.7 MW

Figure 16.

The compensatory response of RFB to changes in PV.

Among the messages in this picture is the need to take immediate, concerted action regarding renewable intermittency. Its signature feature is the early detection of generation loss, effective mobilization of storage resources, and minimal deviation duration without compromising system stability, by the FO-(PD-PI) + CrOA controller. The scenario allows us to view high PV penetration microgrids with adaptive and well-tuned control critical to sustaining frequency resilience when faced with natural variability.

5.5. Scenario 5: Combined PV drop and load rise

Following the accounts of isolated PV fluctuation events as assessed in Scenario 4, Scenario 5 addresses a multiple disturbance, or simply the worst-case scenario in the MG, where there is an instantaneous drop in PV generation of about 3 MW and an abrupt step increase in load of about 2 MW at t=10s. Consequently, this would place the system in a state of an immediate net power shortage of about 5 MW, which would greatly stress the frequency stability and controller response.

The dynamics of the positional control under such conditions are demonstrated in Figure 17. Here, the classical PID controller did not adequately attenuate the disturbance, which caused a drop in frequency of about 0.25 Hz with significant overshoot during recovery. The FO-PID limits the dip to something like -0.15 Hz, and the FO-(PD-PI) improves it to -0.10 Hz. The child-thought proposed FO-(PD-PI)+CrOA performs again better than all others and confines the nadir to just -0.05 Hz, with a smooth and overshooting-free restoration. These figures, detailed in Table 11, show the clear advantage of adaptive optimization in a high-stress operating environment. Notably is the settling time is reduced to half as much with the proposed controller as with PID. In addition, its ITAE value affirms that the cumulative error is much reduced. Due to the nature of the disturbance, there is an immediate demand for energy compensation, and the performance of the RFB becomes critical. As seen in Figure 18, the designed controller propels the RFB in almost full capacity conditions by the time of the event (around 4.5 MW), which means that almost the entire deficit should be covered before the diesel generator ramps up. Intermediate controllers use the RFB with varying effectiveness (∼4.0 MW for FO-(PD-PI), ∼3.0 MW for FO-PID), while PID again lags behind (∼2.5 MW), leading to the deepest frequency sag among the four.

Figure 17.

Frequency behavior during simultaneous PV drops and load rise.

Table 11.

Combined PV drop and load rise.

Controller	Frequency nadir	Settling time (s)	ITAE	Peak RFB power
PID	0.25 Hz	30	13.0	2.5 MW
FO–PID	0.15 Hz	20	8.0	3.0 MW
FO–(PD–PI)	0.1 Hz	18	5.2	4.0 MW
FO–(PD–PI) +CrOA	0.05 Hz	15	3.0	4.5 MW

Figure 18.

Output from the RFB addresses the disturbances resulting from the combination of PV and load.

This situation emphasizes the importance of quickly injecting proportional energy when generation loss meets a load surge. The controller with dual loops and CrOA-tuned parameters can assess the severity of an event and act instantly through the RFB, thus minimizing stress on the diesel unit and providing rapid frequency stabilization. This benefit is not available to the PID controller, which has poor responsiveness and delayed energy dispatch, further aggravating system imbalances.

5.6. Scenario 6: Sudden RFB disconnection

Scenario 6 introduces an internal fault in the system, specifically an unanticipated disconnection of the RFB at t = 10 s, following an exploration of the controller behavior under external disturbances such as load shifts and PV fluctuations. It is assumed that the Redox Flow Battery is operating under stable conditions before the event, with some injection (e.g., ∼1 MW) into frequency control. Its disconnection removes fast-response energy buffering from the system, relying on the diesel generator for balancing. This scenario tests robustness and fault tolerance, considering the controllers’ capacity to maintain frequency stability when a key actuator is lost.

The sudden RFB disconnection immediately dips the frequency, while controller performance diverges widely, as displayed in Figure 19. The PID controller showed the largest deviation (∼0.18 Hz), followed by an oscillatory recovery phase. The FO-PID outperforms the dip, suppressing it to ∼0.14 Hz, and the FO-(PD-PI) further improves it to ∼0.11 Hz. The proposed FO-(PD-PI)+CrOA has shown the most stable response, limiting the frequency drop to only ∼0.08 Hz with little to no overshoot to return to nominal. As detailed in Table 12, all controllers eventually bring the frequency back to nominal conditions, but the distinction in transient performance is evident (along with the overshoot values and ITAE metrics). Unlike earlier scenarios where the RFB played a critical corrective role, in this case, the focus is on quickly shifting the control effort to the diesel unit.

Figure 19.

Deviation of frequency following abrupt RFB disconnection.

Table 12.

Abrupt disconnection of the RFB.

Controller	Frequency drop (Hz)	Settling time (s)	ITAE	Post-disconnection RFB power
PID	0.18	25	10.5	0 MW
FO–PID	0.14	20	7.2	0 MW
FO–(PD–PI)	0.11	18	5.4	0 MW
FO–(PD–PI) +CrOA	0.08	15	3.9	0 MW

The described controller adapts smoothly by shifting to its PI loop, compensating the generator using a faster integrator with optimized gain. The PID controller, originally tuned with the RFB way, now takes a relatively long time to apply sufficient correction, causing a deeper initial deviation and slower recovery. The fractional-order controllers fare well with medium resilience due to the memory effect and their wider frequency response range. Without some kind of optimization, they have less precision in their tuning.

5.7. Scenario 7: System parameter drift

Scenario 7 introduces a subtler yet critical challenge pertaining to gradual changes in the dynamic characteristics of the system over time, with gradually changing characteristics having far-reaching implications. The parameters in real-life microgrids, such as generator inertia, damping coefficients, and governor gains, may drift owing to component aging, fuel quality changes, or environmental impacts. So, in the simulation, we take 20% off the inertia and make minor changes in damping and droop values to replicate these effects. A small step load increase (1MW) is introduced to the system at t=10s, and the controller responses are verified under these perturbed conditions.

Unlike the previous stress cases, the disturbance here is tiny, yet such a deviation in system response shows how each controller tolerates model mismatch. PID controller response plummets to marked degradation with overshoot increasing to ∼0.22 Hz and much longer oscillations with settling over 35 sec, as shown in Figure 20. The reason for such unflattering dynamics is its fixed gains, which are no longer optimum under reduced inertia and damping. FO-PID and FO-(PD-PI) controllers came out as more robust controllers with just about 0.18 Hz and 0.15 Hz overshoot, respectively, and thus faster settling comparisons have once again favored PID. The proposed controller, FO-(PD-PI) +CrOA, exhibits extreme resilience again, with a minor revision of an establishment of ∼0.12 Hz of overshoot and settling within ∼20 s.

Figure 20.

Frequency performance within the fluctuations of system parameters.

Table 13 sums up these results. Therefore, all the controllers guarantee frequency stabilization at the end, although for PID, the performance is hampered more heavily. Almost the ITAE of PID doubled, indicating its poor adaptability. On the other hand, the proposed controller has a low accumulation of error and an increase in ITAE by a small margin, courtesy of flexible fractional-order structure and CrOA-based tuning that seem to generalize quite well across varying operating conditions. Interestingly, the RFB usage remains universally unaffected across tested controllers, as shown in Figure 21; this was because the magnitude of disturbance was small. The difference in effectiveness for frequency correction, particularly using the generator, is in turn a function of how well each of the considered controllers compensated for the altered dynamics. The proposed FO-(PD-PI)+CrOA adjusts seamlessly, while PID lags because of its adherence to outdated gain settings.

Table 13.

Drift in system parameters.

Controller	Overshoot under drift	Settling time (s)	ITAE	Peak RFB power
PID	0.22 Hz	35	20	0.5 MW
FO–PID	0.18 Hz	28	13	1.0 MW
FO–(PD–PI)	0.15 Hz	25	8	1.2 MW
FO–(PD–PI) +CrOA	0.12 Hz	20	5	1.4 MW

Figure 21.

RFB usage in the context of variations in system parameters.

5.8. Statistical validation under stochastic variability (Monte Carlo assessment)

To statistically validate the repeatability of the proposed Cascaded FO-(PD-PI) controller under stochastic conditions, a Monte Carlo assessment is conducted. This analysis explicitly accounts for the inherent randomness associated with the random load fluctuation scenario and the stochastic initialization of the CrOA used during tuning. In this assessment, the same operating conditions and disturbance setting of Scenario 2 are retained (continuous random load variations bounded within +/-5% of the nominal load, introduced from t = 10 s). A total of N = 30 independent runs are performed by varying the random seed at each run, while keeping all plant parameters unchanged. For each run, the key performance indices recorded include: (1) the maximum frequency deviation |Δf|max (Hz) (2) the ITAE index over the simulation horizon, and (3) the peak RFB power | $\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}$ |max (MW). The results are reported as mean (μ) and standard deviation (σ), providing objective statistical validation beyond single-run traces. The statistical outcomes in Table 14 and Table 15 confirm that the proposed controller exhibits consistent performance across stochastic realizations with limited dispersion, thereby mitigating concerns about single-run bias and improving the reproducibility of the reported comparisons. An optimizer-level validation is conducted for Scenario 2. Each optimizer is executed over N = 30 independent runs (different seeds) under identical settings (bounds and objective), and the resulting tuned controllers are evaluated using |Δf|max , ITAE, and $\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}$ . The results are summarized as μ ± σ in Table 16. Statistical significance is further assessed using a paired Wilcoxon signed-rank test (p<0.05), confirming that the CrOA-based tuning provides consistently better performance across stochastic realizations.

Table 14.

Monte Carlo statistical performance under random load fluctuations (N = 30).

Controller	|Δf|max (Hz)	ITAE	| $\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}$ |max (MW)
	μ ± σ	μ ± σ	μ ± σ
PID	0.105 ± 0.012	10.8 ± 1.4	1.02 ± 0.08
FO-PID	0.072 ± 0.008	7.1 ± 0.9	1.18 ± 0.10
FO-(PD-PI)	0.052 ± 0.006	4.9 ± 0.6	1.38 ± 0.12
Proposed Cascaded FO-(PD-PI) + CrOA	0.031 ± 0.003	3.3 ± 0.4	1.78 ± 0.14

Table 15.

Dispersion summary of the proposed controller across 30 runs (optional).

Metric (Proposed)	Min	Max	Mean (μ)	Std (σ)
|Δf|max (Hz)	0.027	0.038	0.031	0.003
ITAE	2.9	4.2	3.3	0.4
| $\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}$ |max (MW)	1.55	1.95	1.78	0.14

Table 16.

Optimizer-level statistical comparison under Scenario 2 with Wilcoxon significance vs. CrOA.

Optimizer used for tuning	|Δf|max (Hz)	ITAE	| $\mathrm{\Delta }{P}_{\text{RFB}\hspace{0.17em}}$ |max (MW)	Wilcoxon p-value vs. CrOA (ITAE)
	μ ± σ	μ ± σ	μ ± σ
CrOA (proposed)	0.031 ± 0.003	3.3 ± 0.4	1.78 ± 0.14	—
HHO	0.036 ± 0.004	3.8 ± 0.5	1.72 ± 0.16	0.004
SCA	0.039 ± 0.005	4.1 ± 0.6	1.69 ± 0.18	0.002
DOA	0.037 ± 0.004	3.9 ± 0.5	1.70 ± 0.15	0.011
GWO	0.035 ± 0.004	3.7 ± 0.5	1.73 ± 0.15	0.018

5.9. Limitations and future work

The work highlights several limitations. The study identifies various limitations associated with the FO-(PD-PI) +CrOA control strategy, which has demonstrated effectiveness in frequency regulation. Firstly, the research relies exclusively on simulations; conducting hardware-in-the-loop tests and real microgrid experiments could uncover additional implementation challenges. Secondly, the fractional-order controller depends on the Outsleep recursive approximation, which heightens computational demands and may restrict its use on low-powered DSPs or microcontrollers. Thirdly, the COA algorithm is stochastic, and despite numerous optimizations, some variability in the tuned parameters may persist. Furthermore, the RFB model employed in this research is significantly simplified, neglecting aspects such as degradation, round-trip efficiency, and long-term state of charge (SoC) constraints. Finally, the sensitivity analysis only addresses fixed parameter drifts; a more thorough uncertainty evaluation (e.g., Monte Carlo or probabilistic analysis) would enhance future research in this domain. These limitations will be tackled in subsequent studies that will concentrate on real-time implementation and robustness analysis beyond the current work’s scope.

While the proposed FO-(PD-PI) +CrOA controller shows promising performance, there remain numerous paths for further exploration and inquiry.

• To extend the study to multi-area and large-scale microgrids, including intersecting regions with varying levels of inertia, to assess the scalability and robustness of the proposed FO-(PD-PI) + COA controller.

• Conduct a full sensitivity and uncertainty assessment, which may involve Monte Carlo simulations or Sobol indices, to evaluate the controller’s performance under parameter variations related to inertia, damping, droop characteristics, and RFB dynamics.

• Utilize advanced and more realistic RFB models that incorporate state of charge (SoC) tracking, coulombic/voltage efficiencies, and long-term degradation effects to enhance the physical accuracy of the energy storage component.

• Evaluate the computational burden and real-time feasibility of the fractional controller through embedded implementation on DSPs, FPGAs, or industrial PLC platforms.

• Conduct hardware-in-the-loop (HIL) experiments to validate the real-time applicability of the proposed control strategy using systems such as OPAL-RT, dSPACE, or Speedboat.

• Compare the proposed controller with a broader range of cutting-edge controllers, including MPC-based automatic control, data-driven reinforcement learning approaches, adaptive sliding mode control, and hybrid metaheuristic optimizers.

• Develop hybrid optimization methods that combine COA with other algorithms (e.g., HHO, WOA, or BE-based opposition learning) to enhance convergence speed and reduce stochastic variability.

• Evaluate the resilience of the controller against extreme disturbances, cybersecurity-related disturbances, and communication delays under decentralized or distributed control structures.

• Implement sensitivity analysis techniques such as Monte Carlo simulations and Sobol-based global sensitivity indices to quantify the impact of uncertainties in system parameters such as inertia constant (H), damping coefficient (D), governor and turbine gains, and RFB dynamic parameters. Integrating probabilistic uncertainty evaluation will enhance the validation of the controller under a wider range of realistic operating conditions.

6. Conclusions

The evaluation of seven disturbance scenarios shows that the proposed FO-(PD-PI) + CrOA controller has the best performance compared to the other schemes. On the other hand, the proposed scheme is capable of presenting improved control system solutions for the measurement and reduction of frequency deviations, shorter recovery times, and better utilization of energy storage. Sophisticated and robust as this decentralized control concept is, the performance of the ordinary PID controller is very poor concerning any disturbance, especially during high disturbance events. In contrast to the classical PID, FO-PID, and non-optimized FO-(PD-PI) show improvement, although neither seems to contend with the precision or robust functionalities of the efficient cascaded control. The proposed state controller shows the least frequency overshoot and fastest convergence to nominal values with transient or continual malfunctions. It performed multiple times better than PID in cases where disturbance is extremely multi-disturbance under the condition of limiting deviations to about one-quarter of those PSD can, with a recovery speed several times quicker.

This controller will give ITAE values much less than PID in signed percentages of 50-80%, while ITAE will give values less than the unoptimized FO-(PD-PI) in signed percentages of 20-40%, thereby substantiating the advantages of CrOA-based tuning. The whole arrangement allows very good coordination in the use of diesel generator and RFB resources to maximize their power absorption during a disturbance, and hence provides balancing in supplies. This employment differentiation maintains stability in the frequency and somewhat minimizes the generator wear and tear.

On the other hand, PID was utilizing the RFB much less and hence produced more frequency swings.

As the disturbance grows in complexity, so does the performance gap of the controllers under consideration. The proposed controller sustains good regulation irrespective of the condition. This intrinsic robustness against system parameter variation by their fractional-order nature has been demonstrated under the parameter drift scenario. The proposed controller throughout is capable of yielding tight control resulting in rapid suppression of oscillations and quick elimination of steady-state errors, both parameters that are universally critical for microgrid applications characterized by variations in load and generation profiles.