A New Load Frequency Control Technique for Hybrid Maritime Microgrids: Sophisticated Structure of Fractional-Order PIDA Controller

Abstract

This paper proposes an efficient load frequency control (LFC) technique based on a fractional-order proportional–integral–derivative–accelerator with a low-pass filter compensator (FOPIDA-LPF) controller, which can also be accurately referred to as the PIλDND2N2 controller. A trustworthy metaheuristic optimization algorithm, known as the gray wolf optimizer (GWO), is used to fine-tune the suggested PI^λDND²N² controller parameters. Moreover, the proposed PI^λDND²N² controller is designed for the LFC of a self-contained hybrid maritime microgrid system (HMμGS) containing solid oxide fuel cell energy units, a marine biodiesel generator, renewable energy sources (RESs), non-sensitive loads, and sensitive loads. The proposed controller enables the power system to deal with random variations in load and intermittent renewable energy sources. Comparisons with various controllers used in the literature demonstrate the excellence of the proposed PI^λDND²N² controller. Additionally, the proficiency of GWO optimization is checked against other powerful optimization techniques that have been extensively researched: particle swarm optimization and ant lion optimization. Finally, the simulation results performed by the MATLAB software prove the effectiveness and reliability of the suggested PI^λDND²N² controller built on the GWO under several contingencies of different load perturbations and random generation of RESs. The proposed controller can maintain stability within the system, while also greatly decreasing overshooting and minimizing the system’s settling time and rise time.

1. Introduction

The world strives to transition to a low-carbon life where electricity is produced from clean, renewable energy sources (RESs) at reasonable prices to meet daily demands. An increasing number of countries are dedicated to attaining net-zero emissions by the mid-century or shortly after that [1]. Marine energy is the most abundant untapped renewable energy source; the energy of the waves and winds in the oceans and seas meet the world’s needs several times over [2]. Recently, shipboard power systems (SPSs), such as those installed in ships, ferries, warships, vessels, and other marine gear, have considerably grown in size along with their power demands. Lately, due to the rising costs of fossil fuels and progressively tighter emission regulations, modern shipbuilding industries are currently considering how to power all-electric ships (AES). As a result, they should focus on using integrated electric propulsion systems such as steam turbine propulsion, wind propulsion, and solar propulsion, which would allow the ships’ load needs to be met from the same power source. Such systems are also expected to eliminate the need for a separate generation system for these loads and establish a unified electrical platform. The notion of AESs using electric propulsion and integrated power systems (IPSs) has piqued interest in the global shipbuilding industry [3,4].

Recently, there has been growing interest in integrating energy storage systems (ESSs) and RESs into SPSs to reduce sailing costs, thus making such a system increasingly composed of a hybrid marine microgrid system (HM$\mu$GS). In contrast, the frequency instability issue arises due to the intermittent nature of renewable energy sources (RESs) [5]. To solve this issue, numerous researchers have been working on using energy storage systems such as batteries, superconducting magnetic energy storage systems (SMES), and ultracapacitors as backup resources within hybrid power systems. Nevertheless, cost, conservation, and dumping concerns prevent battery energy storage from being widely used. On the contrary, the flow of costly helium liquid is a significant obstacle for SMES. For such cases, the dynamic tuning of controllable loads such as freezers (FRZ) and heat pumps (HP) may be necessary to reduce the volatility produced by RESs’ deployments. Therefore, the marine bio-diesel generator (MBG) and the non-toxic and eco-friendly solid oxide fuel cell (SOFC) can be considered as supplementary sets in these HM$\mu$GSs in the standalone operation mode. Furthermore, to counteract the sporadic and erratic nature of RESs, much emphasis is placed in the literature on the appropriate size of various RESs/ESSs within HM$\mu$GS [6,7]. However, less emphasis has been placed on developing control techniques for frequency regulation and power management in these HM$\mu$GS [5,6,7,8].

On the other hand, after the tremendous development witnessed by DC microgrids, the attention of those in charge of the marine ship industry turned to work on the shift in ship power systems from AC systems to DC microgrids. However, moving completely from the AC networks to the DC microgrids is difficult, so this process needs time. During this time, hybrid systems that depend on AC and DC systems can be implemented [9,10,11].

In electrical grids, frequency is mainly governed by active power, whereas voltage is primarily regulated by reactive power. As a result, the control standpoint of power systems may be split into two main categories. The first focuses on active power control along with frequency, while the second concerns reactive power control along with voltage [12]. Due to its relevance, frequency control in electrical grids has recently received much attention. Frequency control is included in the primary, secondary, and tertiary stages. The main mission of primary frequency control is intercepting the decline of the frequency to avoid the tripping operation of the under/over frequency protection relays [13]. The responsibility of taking the action of the primary controller is assigned directly to the governor. While the main task of secondary frequency control, which is usually named automatic generation control (AGC) or load frequency control (LFC), is frequency regulation, according to an acceptable range. The main job of tertiary frequency control is to dispatch generating units and auxiliary reserves after a severe disturbance. While RESs are becoming more common in electrical power networks, one of the main challenges that this type of resource faces is the uncertainty of the generated active power, resulting in frequency fluctuations. In addition to demand stochasticity, which is already a permanent source of frequency fluctuations, such fluctuations urgently require more resilient and optimal LFC methods to deal with such problems [14].

Numerous control approaches were proposed to address the LFC issue of power systems with RESs’ penetration by applying distinctive controllers such as proportional–integral (PI) [15], proportional–integral–derivative (PID) [16], proportional–integral–derivative–acceleration (PIDA) [17], fractional-order PID (FOPID) [18], intelligent fractional-order integral [19], ultralocal model control incorporated with LFC [20], non-integer order PID (NOPID) [21], model predictive control [22], non-integer order model predictive control [23], etc. Recently, various innovative control techniques were published in the literature, such as a modified PID controller structure based on the grasshopper optimization algorithm [24], adaptive sliding mode control [25], a fuzzy logic controller [26], and sampled-data control [27]. Although the aforementioned control techniques gave a good response in the frequency regulation of power systems, their designs did not consider the low-inertia issue of modern power systems resulting from the high penetration of RESs. Moreover, most of these studies did not consider the intermittent nature of RESs for short-term frequency stability. The marine microgrid can be categorized as a low-inertia microgrid containing intermittent RESs [5]. Recently, the authors of [28] proposed a linear matrix inequality (LMI) controller approach to handle the LFC issue in maritime microgrids. Likewise, the authors of [29] suggested a fractional-order fuzzy proportional–derivative-one plus integral controller to handle the LFC issues in marine microgrids. Moreover, the HMµG studied in [29] does not consider RESs and non-critical loads. The above-mentioned LMI controller and fractional-order controller have a highly complex structure that results in a long-elapsed time, which would not be practical for actual implementation. Likewise, the authors of [30] investigated a sine–cosine wavelet technique for the frequency regulation of a mobile MµG. Moreover, the above-mentioned control strategy is intricately designed. Additionally, these studies do not consider the power-sharing among RESs and non-critical loads such as heaters and FRZ. Recently, the authors of [5] proposed a two-stage controller, PI-one plus proportional–derivative PI − (1 + PD) controller, to enhance the frequency stability of the MµG. The proposed two-stage controller was designed using the grasshopper optimization algorithm (GOA), which outperformed the genetic algorithm (GA), particle swarm optimization (PSO), firefly algorithm (FA), and cultural algorithm (CA). Motivated by all of the above, this work presents a new, less complicated multi-stage P${\mathrm{I}}^{\mathsf{\lambda }}$DND²N² controller for the LFC of the MµGs, which performs better at transient error reduction in system disturbances.

Aside from controller design, optimal controller parameter modification is a crucial stage for improving system dynamics. As a result, various heuristic algorithmic techniques, such as the genetic algorithm (GA) [15], particle swarm optimization (PSO) [16], the butterfly algorithmic technique (BOA) [18], the mine blast technique (MBA) [31], the cuckoo search technique (CS), the firefly technique (FA) [32], and the flower pollination technique (FPA) [33], are employed to adjust the controllers’ parameters [8,34]. This work uses a reliable metaheuristic optimization algorithm, called the gray wolf optimizer (GWO) [35], to optimize the proposed PI^λDND²N² controller.

The following points serve to highlight the main contribution of this work.

• This study proposes a new PI^λDND²N² controller for the LFC of a self-contained HMμGS, where the proposed controller structure is based on the combination of a PIDND²N² controller and fractional-order integration.
• The proposed PI^λDND²N² controller is optimally designed using a reliable metaheuristic optimization algorithm, the GWO.
• The proficiency of the GWO algorithm is checked against other powerful optimization techniques that were extensively researched: ant lion optimization (ALO) and PSO.
• For assessing the performance of the suggested controller, the utilized PI^λDND²N² controller’s performance is compared to the performances of other controllers used in the literature under load/RESs fluctuations.
• Finally, the proposed controller allows the power system to mitigate the random load variations and intermittent fluctuations that occur in renewable energy power. As a result, the system stability increases, which significantly reduces overshooting and minimizes the settling time and rise time of the system.

The remaining sections of the article are listed as follows: In Section 2, the considered system, including all its components, is described and modeled. The suggested controller structure and the objective function used for the optimization problem are detailed in Section 3. Section 4 provides an overview of the optimization algorithm used to fine-tune the gains of the proposed controllers. Section 5 provides the simulation results performed by the MATLAB program and their comments. Finally, in Section 6, the paper’s findings are summarized.

2. System Modeling

The suggested HM$\mu$GS, shown in Figure 1, is schematically described in this section. Furthermore, the studied hybrid system model includes AWPG, WDG, MBG, SOFC, and thermostatically controlled HP and FRZ components. The following subsection discusses the modeling of the intended HM$\mu$GS.

Table 1. The studied HM$\mu$G parameters.

Parameters	Value	Parameters	Value
$K_{VR}$	1.000	$K_{SOFC}$	1.000
$K_{CE}$	1.000	$T_{WDG}$	0.200
$T_{VR}$	0.080	$K_{HP}$	1.000
$T_{CE}$	0.400	$T_{HP}$	0.100
$K_{WDG}$	1.000	$K_{FRZ}$	1.000
$T_{WDG}$	5.000	$T_{FRZ}$	0.265
$K_{AWPG}$	1.000	M	0.120
$T_{WDG}$	0.300	D	1.000

presents the values of the studied HM$\mu$GS parameters.

2.1. Archimedes Wave Power Generation (AWPG)

For the proposed HM$\mu$GS, AWPG is based on using a permanent magnet synchronous generator (PMSG). As such, the active speed and force of the studied AWPG can be expressed as [36]

$$v_{FG}=\frac{dx}{dt}$$

(1)

$$F_{WV}=m_{ft}\frac{d{v}_{FG}}{dt}+{\beta }_G{v}_{FG}+{\beta }_{\omega }{v}_{FG}+K_{C}x$$

(2)

The sinusoidal wave strength can be expressed as [5]

$${\mathit{F}}_{\mathit{W}\mathit{V}}=\mathit{F}\mathbf{sin}\left({\mathit{\omega }}_{\mathit{W}\mathit{V}}.\mathit{t}\right)+\frac{\mathit{F}}{2}\mathbf{sin}\left({2\mathit{\omega }}_{\mathit{W}\mathit{V}}.\mathit{t}\right)+\frac{\mathit{F}}{3}\mathbf{sin}\left({3\mathit{\omega }}_{\mathit{W}\mathit{V}}.\mathit{t}\right)$$

(3)

The AWPG linearized transfer function can be written as

$$G_{AWPG}\left(s\right)=\frac{K_{AWPG}}{{1+T}_{AWPG}.s}$$

(4)

2.2. Wind-Driven Generation (WDG)

Wind power output depends on the wind velocity and the wind turbine (WT)’s inherent identification due to the unpredictable nature of variable speed induction generators. As a result, the producible mechanical power of WDG [37] could be represented as

$$P_{WDG}=\frac{\pi \rho r^2}{2}{v}_{wind}^3{C}_P(\lambda ,\beta )$$

(5)

For the considered wind station, the rate of change in WDG can be expressed as [38]

$$P_{WDG}=\left\{\begin{array}{c}0,\hspace{1em}\hspace{1em}{V}_{WD}<V_{cut-in}\parallel V_{WD}>V_{cut-out}\\ P_{rated},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{V}_{rated}\le V_{WD}\le V_{cut-out}\\ \left(\begin{array}{c}0.001312V_{WD}^6-0.04603V_{WD}^5+0.3314V_{WD}^4\\ +3.687V_{WD}^3-51.1V_{WD}^2+2.33V_{WD}+366\end{array}\right),else\end{array}\right.$$

(6)

$$∆P_{WDG}=\left\{\begin{array}{c}0,\hspace{1em}\hspace{1em}{V}_{WD}<V_{cut-in}\parallel V_{WD}>V_{cut-out}\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{V}_{rated}\le V_{WD}\le V_{cut-out}\\ \left(\begin{array}{c}[0.007872V_{WD}^5-0.23015V_{WD}^4+1.3256V_{WD}^3\\ +11.061V_{WD}^2-102.2V_{WD}+2.33].∆V_{WD}\end{array}\right),else\end{array}\right.$$

(7)

The WDG transfer function can be written as

$$G_{WDG}\left(s\right)=\frac{K_{WDG}}{{1+T}_{WDG}.s}$$

(8)

2.3. Marine Bio-Diesel Generator (MBG)

The power generation from a modern MBG set is equivalent to that of a regular diesel generator. Furthermore, the inherently non-toxic, environmentally benign, and good air quality properties of the proposed materials are motivating the considerations for their usage as proposed in this study. MBG’s linearized model is defined as follows (see Figure 2) [39]:

$$∆u_{MBG}=U\left(s\right)-(R^{-1}\times ∆F)$$

(9)

$$∆P_{VR}=∆u_{MBG}-(R^{-1}\times ∆F)$$

(10)

$$∆P_{GCE}=∆P_{VR}\times \left(\frac{K_{VR}}{1+T_{VR}s}\right)$$

(11)

$$∆P_{MBG}=∆P_{GCE}\times \left(\frac{K_{CE}}{1+T_{CE}s}\right)$$

(12)

2.4. Solid Oxide Electrolyte Fuel Cell (SOFC)

Generally, a SOFC is an energy storage device that produces electricity by converting oxygen (O2) and concentrated organic hydrogen (H2) energy using a porcelain-based electrolyte and two electrodes. Due to its higher efficiency (i.e., 80%), temperature, and reduced polluting gas emissions, it is gaining popularity among other fuel cell units. The linearized SOFC model is expressed as [5]

$$G_{SOFC}\left(s\right)=\frac{K_{SOFC}}{{1+T}_{SOFC}.s}$$

(13)

2.5. Non-Sensitive Loads (Heat Pump (HP) and Freezer (FRZ))

Due to their better-regulating capability, HP and FRZ are controllable loads that could be used for load frequency regulation and, thus, enhance the HM$\mu$GS stability. The HP and FRZ transfer functions can be expressed as [5]

$$G_{HP}\left(s\right)=\frac{K_{HP}}{{1+T}_{HP}.s}$$

(14)

$$G_{FRZ}\left(s\right)=\frac{K_{FRZ}}{{1+T}_{FRZ}.s}$$

(15)

2.6. Dynamic Model of the Proposed HM$\mu$GS

According to the suggested HM$\mu$GS, as described in Figure 1b, the change in power difference (ΔP_D) between the net generated power (ΔP_G) and the sensitive load power demand (ΔP_SN) could be expressed as

$$∆P_G=\left(∆P_{AWPG}+∆P_{WDG}+∆P_{MBG}\pm ∆P_{SOFC}-∆P_{HP}-∆P_{FRZ}=∆P_{SN}\to 0\right)$$

(16)

$$∆P_D=∆P_G-∆P_{SN}$$

(17)

As a result, the model of HM$\mu$GS frequency response can be stated as

$${\mathit{G}}_{\mathit{H}\mathit{M}\mu \mathit{G}\mathit{S}}\left(\mathit{s}\right)=(\frac{∆\mathit{F}}{∆{\mathit{P}}_{\mathit{D}}})=\frac{{\mathit{K}}_{\mathit{H}\mathit{M}\mu \mathit{G}\mathit{S}}}{\mathit{D}+\mathit{M}\mathit{s}}$$

(18)

3. Proposed Control Methodology and Objective Function

Over the past few decades, research in control engineering has increasingly been conducted since the introduction of fractional-order (FO) control by I. Podlubny [40]. Using the fractional operators in the controller allows for the general differential and integral notation of any real number. There are two definitions to describe the FO concept. The first is Riemann–Liouville to define the order derivative of a function f(t), as demonstrated in (19) [41].

$${\mathit{D}}_{\mathit{l}\mathit{b},\mathit{u}\mathit{b}}^{\mathit{q}}\mathit{f}(\mathit{t})=\frac{\mathbf{1}}{\mathit{\Gamma }(\mathit{n}-\mathit{q})}{\left(\frac{\mathit{d}}{\mathit{d}\mathit{t}}\right)}^{\mathit{n}}\left({\int }_{\mathit{l}\mathit{b}}^{\mathit{u}\mathit{b}}\frac{\mathit{f}\left(\mathit{\tau }\right)}{{(\mathit{t}-\mathit{\tau })}^{\mathit{q}-\mathit{n}+1}}\mathit{d}\mathit{\tau }\right)$$

(19)

where $\mathit{\Gamma }\left(\mathit{z}\right)={\int }_0^{\mathit{\infty }}{\mathit{t}}^{\mathit{z}-1}{\mathit{e}}^{-\mathit{t}}\mathit{d}\mathit{t},\mathsf{𝕽}\left(\mathit{z}\right)>0$ is the Gamma function, the parameter q is bounded as n − 1 < q < n, q is the order of the FO calculus, and lb and ub are the lower and upper bands, respectively.

Then, taking the Laplace transformation for the fractional Riemann–Liouville derivative yields the following [19]:

$$\mathsf{𝓛}({\mathit{D}}_0^{\mathit{q}}\mathit{f}(\mathit{t}))={\mathit{s}}^{\mathit{q}}\mathit{F}(\mathit{s})-{\sum }_{\mathit{k}=0}^{\mathit{n}-1}{\mathit{s}}^{\mathit{k}}({\mathit{D}}_0^{\mathit{q}-\mathit{k}-1}\mathit{f}(\mathit{t})){|}_{\mathit{t}=0}$$

(20)

The second definition associated with the basic FO concept is Caputo’s definition, by which the time-domain representation for q order of a function f(t) can be expressed as

$${\mathit{D}}_{\mathit{l}\mathit{b},\mathit{u}\mathit{b}}^{\mathit{q}}\mathit{f}(\mathit{t})=\left\{\begin{array}{cc}\frac{1}{\mathit{\Gamma }(\mathit{n}-\mathit{q})}\left({\int }_{\mathit{l}\mathit{b}}^{\mathit{u}\mathit{b}}\frac{{\mathit{f}}^{\mathit{n}}\left(\mathit{\tau }\right)}{{(\mathit{t}-\mathit{\tau })}^{1-\mathit{n}+\mathit{q}}}\mathit{d}\mathit{\tau }\right)& \mathit{n}-\mathbf{1}<\mathit{q}<\mathit{n}\\ {(\frac{\mathit{d}}{\mathit{d}\mathit{t}})}^{\mathit{n}}\mathit{f}(\mathit{t})& \mathit{q}=\mathit{n}\end{array}\right.$$

(21)

Again, taking the Laplace transformation for Equation (21) yields Equation (22), which indicates the integral order has an initial condition and denotes the physical meaning.

$$\mathsf{𝓛}({\mathit{D}}_0^{\mathit{q}}\mathit{f}(\mathit{t}))={\mathit{s}}^{\mathit{q}}\mathit{F}(\mathit{s})-{\sum }_{\mathit{k}=0}^{\mathit{n}-1}{\mathit{s}}^{\mathit{q}-\mathit{k}-1}{\mathit{f}}^{\mathit{k}}(0)$$

(22)

where s is the Laplace operator.

The recursive approximation is usually applied to implement the fractional-order definition [42]. The Laplace transformation of the qth derivative is

$${\mathit{s}}^{\mathit{q}}\approx \mathit{K}{\prod }_{\mathit{k}=-\mathit{N}}^{\mathit{N}}\frac{\mathit{s}+{\mathit{\omega }}_{\mathit{k}}^{\prime }}{\mathit{s}+{\mathit{\omega }}_{\mathit{k}}}$$

(23)

where

$$\begin{array}{l}\mathit{K}={\mathit{\omega }}_{\mathit{h}}^{\mathit{q}}\\ {\mathit{\omega }}_{\mathit{k}}^{\prime }={\mathit{\omega }}_{\mathit{b}}{\left(\frac{{\mathit{\omega }}_{\mathit{h}}}{{\mathit{\omega }}_{\mathit{b}}}\right)}^{\frac{\mathit{k}+\mathit{N}+(1-\mathit{q})/2}{2\mathit{N}+1}}\\ {\mathit{\omega }}_{\mathit{k}}={\mathit{\omega }}_{\mathit{b}}{\left(\frac{{\mathit{\omega }}_{\mathit{h}}}{{\mathit{\omega }}_{\mathit{b}}}\right)}^{\frac{\mathit{k}+\mathit{N}+(1+\mathit{q})/2}{2\mathit{N}+1}}\end{array}$$

and N is the approximation order of an effective frequency range [${\mathit{\omega }}_{\mathit{b}},{\mathit{\omega }}_{\mathit{h}}$].

3.1. Controller Structure

The conventional PID controller has only three parameters to be tuned. The fractional-order PID controller has five parameters, meaning two extra parameters of the integral and fractional orders, λ and µ. Compared to a conventional PID [43], these parameters can enhance the controller’s transient time, stability, and steady-state errors. In addition, they increase the degree of freedom of the controller. The complete transfer function of the fractional-order PID controller (FOPID) is given as

$${\mathit{G}}_{\mathit{c}}(\mathit{s})={\mathit{K}}_{\mathit{P}}+{\mathit{K}}_{\mathit{I}}{\left(\frac{1}{\mathit{s}}\right)}^{\mathit{\lambda }}+{\mathit{K}}_{\mathit{D}}{\mathit{s}}^{\mu }, \lambda > 0 \mathrm{and} \mu > 0$$

(24)

Here, the complete transfer function of the PI^λDND²N² controller is given in (25). Its structure is depicted in Figure 3. It merges two controller forms as PIDND²N², which allows system stability and amplification to reduce overshoots, minimizing system settling time and rise time [44]. The second controller is the fractional integrator to reduce the rise time and steady-state error in the system, where λ has a range of [0, 2].

$$G_c(s)=K_P+K_I{\left(\frac{1}{s}\right)}^{\lambda }+K_{D_1}s\frac{N_1}{s+N_1}+K_{D_2}{s}^2\frac{N_2^2}{{\left(s+N_2\right)}^2}$$

(25)

3.2. Formulation of the Objective Function (J)

In this study, the proposed control system of the investigated HM$\mu$GS aims to achieve three main tasks:

• Enabling the RESs to produce the maximum possible power by utilizing their converters/interfacing devices;
• Controlling the extracted power from dispatchable units (SOFC and MBG);
• Controlling the input power of thermostatic non-sensitive loads (HP and FRZ).

As such, the provided control system is equipped with modern optimization approaches to instantaneously tune its parameters, ensuring active power balancing for keeping the reliability and stability of the HM$\mu$GS. The formulation of the objective function (J) is critical to achieving the optimal dynamics of the system. This research selects the objective function of the integral time absolute error (ITAE), as shown in (26).

$$Minimize {\mathit{J}}_{\mathit{I}\mathit{T}\mathit{A}\mathit{E}}={\int }_0^{{\mathit{t}}_{\mathit{s}\mathit{i}\mathit{m}}}\mathit{t}\left|∆\mathit{F}\left(\mathit{t}\right)\right|.\mathit{d}\mathit{t}$$

(26)

Subject to

$$\left\{\begin{array}{c}{\mathit{K}}_{\mathit{P}\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{K}}_{\mathit{P}\mathit{i}}\le {\mathit{K}}_{\mathit{P}\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\\ {\mathit{K}}_{\mathit{l}\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{K}}_{\mathit{I}\mathit{i}}\le {\mathit{K}}_{\mathit{l}\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\\ {\mathit{K}}_{\mathit{D}1-\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{K}}_{\mathit{D}1-\mathit{i}}\le {\mathit{K}}_{\mathit{D}1-\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\\ {\mathit{K}}_{\mathit{D}2-\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{K}}_{\mathit{D}2-\mathit{i}}\le {\mathit{K}}_{\mathit{D}2-\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\\ {\mathit{N}}_{1-\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{N}}_{1-\mathit{i}}\le {\mathit{N}}_{1-\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\\ {\mathit{N}}_{2-\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{N}}_{2-\mathit{i}}\le {\mathit{N}}_{2-\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\\ {\mathit{\lambda }}_{\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}\le {\mathit{\lambda }}_{\mathit{i}}\le {\mathit{\lambda }}_{\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}\end{array}\right.$$

(27)

where i = 1, 2. The proportion (${\mathit{K}}_{\mathit{P}}$), integral (${\mathit{K}}_{\mathit{I}}$), and differential (${\mathit{K}}_{\mathit{D}}$) gains were tuned in the range of [0, 10]. While the fractional-order operator of the integral controller (λ) and the coefficient of the derivative filter (N) were tuned in the range of [0, 2] and [50, 1000], respectively.

4. Overview of the Gray Wolf Optimization (GWO) Algorithm

Realistically, computational difficulties may arise in discovering globally optimum solutions from an enormous solution space. Consequently, heuristic optimization algorithms were developed to discover feasible solutions within a vast solution space efficiently. Moreover, recent research in meta-heuristic optimization methods, such as that discussed in Section 1, introduced effective techniques for handling complex, real-time, and nonlinear problems.

The valuable feature of the GWO method over other well-known meta-heuristic algorithms is that its function does not require specific input parameters. Furthermore, it is simple and devoid of computational complexity. Its advantages also include the ease with which the GWO algorithm can be translated into programming languages and the ease with which the GWO algorithm can be understood. The authors attempted to develop and fine-tune the suggested load frequency controllers on the proposed HM$\mu$GS based on GWO optimization.

GWO emulates the natural hunting mechanism and leadership hierarchy of gray wolves (GWs). GWs adapt to living in sets (5–12 on average). As described in [33], four types of GWs, α (alpha), β (beta), δ (delta), and Ω (omega), are used to simulate the hierarchy of leadership. Furthermore, the prey-hunting process in the GWO algorithm consists of three primary strategies: seeking, encircling, and attacking. For modeling the GWs behavioral patterns, α was deemed the fittest solution, followed by β and δ, and the other remaining solutions are categorized as Ω. In GWO, hunting (optimization) operations are driven by α, β, and δ wolves, with Ω always following these three wolves. Both Equations (28) and (29) are used to model the behavior of GW encircling prey.

$$\overrightarrow{D}=\left|\overrightarrow{C}.\overrightarrow{x_P}\left(t\right)-\overrightarrow{x}(t)\right|$$

(28)

$$\overrightarrow{x}\left(t+1\right)=\overrightarrow{x_P}\left(t\right)-\overrightarrow{A}\overrightarrow{D}$$

(29)

where

t: current iteration; and

$\overrightarrow{{\mathit{x}}_{\mathit{P}}}\left(\mathit{t}\right)$: victim’s current position,

while $\overrightarrow{\mathit{A}}$ and $\overrightarrow{\mathit{C}}$ are the coefficient vectors, which are determined by Equations (30) and (31), respectively.

$$\overrightarrow{A}=2\overrightarrow{a}\overrightarrow{r_1}-\overrightarrow{a}$$

(30)

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

(31)

where $\overrightarrow{{\mathit{r}}_1}$ and $\overrightarrow{{\mathit{r}}_2}$ are two random vectors between [0, 1], while, over each course of the iteration, the $\overrightarrow{\mathit{a}}$ component linearly decreases from 2 to 0.

During the hunting process, which is primarily directed by alphas, the GWs’ positions are updated. Even though alphas are the principal actors in the hunting process, betas and deltas do occasionally participate. To determine the GWs’ optimum positions, the three best solutions (for α, β, and δ) are stored, while the rest of the solutions, including Ω, are competing. Equations (32)–(34) are used to update the positions of the GWs.

$$\overrightarrow{D_{\alpha }}=\left|\overrightarrow{C_1}\overrightarrow{X_{\alpha }}-\overrightarrow{X}\right|,\overrightarrow{D_{\beta }}=\left|\overrightarrow{C_2}\overrightarrow{X_{\beta }}-\overrightarrow{X}\right|, \overrightarrow{D_{\delta }}=\left|\overrightarrow{C_3}\overrightarrow{X_{\delta }}-\overrightarrow{X}\right|$$

(32)

$$\overrightarrow{X_1}=\overrightarrow{X_{\alpha }}-\overrightarrow{A_1}(\overrightarrow{D_{\alpha }}), \overrightarrow{X_2}=\overrightarrow{X_{\beta }}-\overrightarrow{A_2}(\overrightarrow{D_{\beta }}), \overrightarrow{X_3}=\overrightarrow{X_{\delta }}-\overrightarrow{A_3}(\overrightarrow{D_{\delta }})$$

(33)

$$\overrightarrow{X}(t+1)=\frac{\overrightarrow{X_1}+\overrightarrow{X_2}+\overrightarrow{X_3}}{3}$$

(34)

It can be seen that the final position varies within the circle that is entirely identified by the α, β, and δ in the search area. In contrast, other GWs update their position by forecasting the prey position. Two parameters are considered to mathematically express the modeling of the approaching methodology to the prey: $\overrightarrow{\mathit{a}}$ that decreases linearly from 2 to 0 and $\overrightarrow{\mathit{A}}$ that would be in the range of [−$\overrightarrow{\mathit{a}}$, $\overrightarrow{\mathit{a}}$]. When $\overrightarrow{\mathit{A}}$ is between [−1, 1], the search agent’s next position could be any position between the prey and the current position.

So, in GWO, the optimum search depends on the positions of α, β, and δ. They diverge while searching for prey and converge while attacking it. Mathematically, when $\overrightarrow{\mathit{A}}$ $>$ 1 or $\overrightarrow{\mathit{A}}$ $<$ −1, the search agent diverges to prey. Another variable in GWO that aids the exploration mechanism is named $\overrightarrow{\mathit{C}}$; according to Equation (31), the value of $\overrightarrow{\mathit{C}}$ varies between [0, 2], which affects defining the distance of the prey, as in Equation (29). As a result, GWO exhibits more random behavior, enhancing exploration and avoiding local optima. The overall flowchart of the GWO algorithm is presented in Figure 4; more details on GWO are given in [35].

5. Results of the Simulation and Discussion

The performance of the proposed PI^λDND²N² controller based on the GWO algorithm is verified in that it enhances the frequency stability of the considered HM$\mu$GS. The numerical results of the studied system are executed using MATLAB/Simulink^® software to assess the robustness and efficacy of the proposed PI^λDND²N² controller. To carry out the optimization (i.e., the GWO algorithm) process, its code is executed in an m-file and interfaced with the HM$\mu$GS model built by the Simulink model. The simulation program runs on a laptop with an Intel^® Core i5-2.6 GHz processor and 8.0 GB RAM. Moreover, the proficiency of the GWO algorithm used to obtain the optimal parameters of the proposed LFC-based PI^λDND²N² controller is verified by comparing its performance with that of the other optimization algorithms used in the literature (e.g., ALO and PSO algorithms). Moreover, the outperformance of the proposed PI^λDND²N² controller is validated over that of the conventional PID controller, which was designed using different optimization algorithms. Under numerous different operating circumstances, the frequency regulation of the considered HM$\mu$GS is validated using the following scenarios.

Scenario 1: Investigation of the proposed PI^λDND²N² controller using various optimization algorithms.

The effectiveness of the GWO algorithm used to fine-tune the suggested PI^λDND²N² controllers’ parameters in the considered HM$\mu$GS is confirmed by evaluating its effectiveness against the other optimization algorithms used in the literature (e.g., ALO and PSO algorithms). This scenario is addressed in the following three different operation cases:

Case A: Assessment of system performance in the event of a sudden load change.

In this case, the proposed PI^λDND²N² controllers in the considered HM$\mu$GS are fine-tuned using the proposed GWO algorithm, in addition to the other two optimization algorithms (i.e., ALO and PSO algorithms), considering a 20% sudden load change in the sensitive load demand at t = 5 s. Additionally, this case considers the studied system without RESs (i.e., AWPG and WDG). The optimization algorithms used are set with 100 iterations and 30 search agents. The optimal parameters of the proposed PI^λDND²N² controllers, which are designed using various optimization algorithms (i.e., GWO, ALO, and PSO), are given in

Table 2. The optimal parameters of the proposed PI^λDND²N² controllers using various optimization algorithms.

Parameter		PIλDND2N2 Controller-Based GWO (Proposed)	PIλDND2N2 Controller-Based ALO	PIλDND2N2 Controller-Based PSO
Controller 1	KP1	5.2761	6.9075	10
	KI1	8.6231	5.2305	0.1000
	KD1-1	8.8277	1.9913	0.4634
	KD2-1	0.0505	2.5289	0.1000
	N1-1	519.9601	97.1532	50.0000
	N2-1	103.6735	483.6617	647.7456
	λ1	0.1190	0.1308	1.0033
Controller 2	KP2	9.3471	9.5982	6.6812
	KI2	9.9671	10	10
	KD1-2	2.6624	0.5318	0.1000
	KD2-2	0.0724	0.2243	0.1000
	N1-2	374.9135	350.5638	962.5905
	N2-2	436.3411	413.2159	50.0000
	λ2	1.0000	1.0000	1.0000
ITAE		0.0048	0.1416	0.1372

. Figure 5 shows the convergence curve of the optimization algorithms that are used. The proposed GWO algorithm outperforms the other algorithms in terms of convergence characteristics and objective function. Figure 6 shows the frequency deviation of the considered HM$\mu$GS with the proposed PI^λDND²N² controllers, which were designed using various optimization algorithms under the operating conditions of this case. The results show that the suggested controller based on the GWO algorithm gives superior performance, with less overshoot, less undershoot, and a faster settling time than other optimization algorithms (i.e., ALO and PSO).

Case B: Assessment of system performance in the event of a sudden load change and the fixed generation of RESs.

In this case, the performance of the considered HM$\mu$GS with the proposed PI^λDND²N² controllers, which are designed using various optimization algorithms, is examined under a 0.2 pu sudden load change in the sensitive load demand at t = 0 s onwards, a 0.1 pu of WDG during (0 ˂ t ˂ 100 s) and 0.15 pu at t = 100 s onwards, and a 0.1 pu of AWPG at t = 0 s onwards. Table 2 indicates the optimal parameters of the proposed controllers designed using the various optimization algorithms used in this case. Figure 7 displays the frequency deviation of the investigated grid with the proposed PI^λDND²N² controllers based on various optimization algorithms. Figure 7 makes it clear that the investigated HM$\mu$GS with the suggested PI^λDND²N² controllers based on the GWO algorithm is more efficient and reliable than using other optimization algorithms across all the difficulties encountered by the system.

Case C: Assessment of system performance in the event of a random load change as well as random RESs generation.

Under this case study, a random generation of WDG and AWPG, as shown in Figure 8, and a random load change in the sensitive load demand, as shown in Figure 9, are considered to examine the studied HM$\mu$GS with the proposed PI^λDND²N² controllers, which are designed using various optimization algorithms. The profile of the load change in the sensitive load demand shown in Figure 9 was obtained using the random load model shown in Figure 10, which was extracted from [45]. The optimal gains of the suggested controllers designed using the various optimization algorithms used in this case are given in Table 2. Figure 11 shows the frequency deviation of the analyzed system with the suggested PI^λDND²N² controllers based on various optimization algorithms. The simulation results show that the suggested PI^λDND²N² controllers based on the GWO algorithm could dampen the high-frequency deviations resulting from high load/RESs fluctuations compared to other optimization algorithms. Therefore, the proposed PI^λDND²N² controllers based on the GWO algorithm give superior performance in improving the frequency stability of the considered HM$\mu$GS than using the other optimization algorithms used in the literature. The system performance specifications in terms of maximum undershoot (MUS), maximum overshoot (MOS), and maximum settling time (Ts) for the investigated system are given in

Table 3. System performance specification for Scenario 1, under various optimization algorithms.

Optimal Controller		MUS (Hz)	MOS (Hz)	TS (s)
Case A	PIλDND2N2-GWO (Proposed)	7.135 × 10−3	0.000 × 10−3	1.984
	PIλDND2N2- ALO	9.402 × 10−3	0.000 × 10−3	3.018
	PIλDND2N2-PSO	17.850 × 10−3	4.447 × 10−3	5.050
Case B	PIλDND2N2-GWO (Proposed)	3.556 × 10−3	3.566 × 10−3	1.857
	PIλDND2N2-ALO	4.687 × 10−3	4.729 × 10−3	3.134
	PIλDND2N2-PSO	8.926 × 10−3	8.934 × 10−3	4.743
Case C	PIλDND2N2-GWO (Proposed)	5.690 × 10−3	5.271 × 10−3	1.599
	PIλDND2N2-ALO	7.509× 10−3	6.990 × 10−3	3.067
	PIλDND2N2-PSO	14.160 × 10−2	13.200 × 10−3	4.011

. From these numerical results, it is evident that the suggested PI^λDND²N² controllers based on the GWO algorithm surpass the other employed algorithms, leading to superior results. Therefore, it is most suitable for the frequency regulation of the considered HM$\mu$GS.

Scenario 2: Comparison of the suggested PI^λDND²N² controller’s performance and the PID controller designed using various optimization algorithms.

To validate the superiority of the suggested PI^λDND²N² controller based on the GWO algorithm, its performance is compared with both the PID controller based on the GWO algorithm and the PID controller based on the GOA [5]. The optimal parameters of the suggested PI^λDND²N² controllers based on the GWO and other compared PID controllers are given in

Table 4. The optimal parameters of the suggested PI^λDND²N² controllers and the compared PID controllers.

Parameter		PIλDND2N2 Controller-Based GWO (Proposed)	PID Controller-Based GWO	PID Controller-Based GOA [5]
Controller 1	KP1	5.2761	2.1759	0.526
	KI1	8.6231	4.2154	10.110
	KD1-1	8.8277	9.9547	0.112
	KD2-1	0.0505	-	-
	N1-1	519.9601	-	-
	N2-1	103.6735	-	-
	λ1	0.1190	-	-
Controller 2	KP2	9.3471	5.4278	5.014
	KI2	9.9671	8.4178	5.558
	KD1-2	2.6624	2.8938	1.615
	KD2-2	0.0724	-	-
	N1-2	374.9135	-	-
	N2-2	436.3411	-	-
	λ2	1.0000	-	-

. This scenario is addressed in the following three different operation cases:

Case A: Assessment of system performance inthe event of a sudden load change.

The performance of the considered HM$\mu$GS with the suggested PI^λDND²N² controllers based on the GWO algorithm is compared with the PID controller based on the GWO algorithm and the PID controller based on the GOA [5] under a 20% sudden load change in the sensitive load demand at t = 5 s. Additionally, this case considers the studied system without RESs (i.e., AWPG and WDG). Figure 12 shows the system frequency deviation using each of the suggested PI^λDND²N² controllers based on the GWO algorithm, the PID controller based on the GWO algorithm, and the PID controller based on the GOA [5]. The results demonstrate that the suggested PI^λDND²N² controller performs better than the other PID controllers. Moreover, the proposed controller exhibits optimal performance by achieving reduced overshoot, reduced undershoot, and a faster settling time than the other used controllers.

Case B: System performance evaluation in the event of a random load change as well as random RESs generation.

To evaluate the effectiveness and robustness of the suggested PI^λDND²N² controllers based on the GWO algorithm, the random generation of WDG and AWPG shown in Figure 8 and the random load change in the sensitive load demand shown in Figure 9 are considered in the studied HM$\mu$GS. Moreover, the optimal parameters of the suggested PI^λDND²N² controllers based on the GWO and the other compared PID controllers used in this case are given in Table 4. Figure 13 displays the frequency deviation of the system studied with the suggested PI^λDND²N² controllers based on the GWO algorithm. As shown in Figure 13, the proposed PI^λDND²N² controller shows a reliable behavior compared to the PID controller based on the GWO algorithm and the PID controller based on the GOA [5], despite the high load/RESs fluctuations. The system performance specifications in terms of the MOS, MUS, and Ts for the investigated system under Scenario 2 are given in

Table 5. System performance specification for Scenario 2.

Optimal Controller		MUS (Hz)	MOS (Hz)	TS (s)
Case A	PIλDND2N2-GWO (Proposed)	7.137 × 10−3	0	2.128
	PID—GWO	8.818 × 10−3	6.583 × 10−4	5.093
	PID-GOA [5]	9.773 × 10−3	0	6.036
Case B	PIλDND2N2-GWO (Proposed)	5.609 × 10−3	5.279 × 10−3	1.716
	PID-GWO	8.753 × 10−3	6.532 × 10−3	3.016
	PID-GOA [5]	9.187 × 10−3	7.114 × 10−3	6.027

. This analysis shows that the suggested PI^λDND²N² controllers based on the GWO algorithm give better results than the PID controller based on the GWO algorithm and the PID controller based on the GOA [5]. Therefore, it is most suitable for the frequency regulation of the considered HM$\mu$GS when considering high load/RESs fluctuations.

6. Conclusions

This paper discussed supporting the frequency response of the low-inertia self-contained hybrid maritime microgrid system containing a marine biodiesel generator and renewable energy sources by proposing a combination of a PIDND²N² controller and a fractional-order integration, referred to as a PI^λDND²N² controller. The associated gains of the suggested PIDND²N² controller were optimally computed using the gray wolf optimization approach. The simulation results validated that the proposed controller provides optimal dynamic stability metrics compared to the conventional controllers based on PID controllers. In addition, the proposed controller allowed the power system to mitigate the random load variations and intermittent fluctuations that occur in renewable energy power. By employing the proposed controller, the system could maintain stability and avoid fluctuations, resulting in a reduction in overshooting by over 19%. Additionally, the settling time and rise time of the system were shortened by more than 71%, leading to more efficient and effective performance. Future work will focus on distributing the inertia property required for the maritime microgrid to compensate for the high penetration of RESs among the distributed energy resources in the power system. Moreover, the robust integral scheme for the nonlinear system can be applied to handle the mismatch uncertainties in the system.