Robust Fuzzy Logic MPPT Using Gradient-Based Optimization for PEMFC Power System

Abstract

In this study, the design of fuzzy logic control (FLC) systems for proton exchange membrane fuel cells (PEMFCs) maximum power point tracking (MPPT) is improved. The improvement is made possible by using a gradient-based optimizer (GBO), which maximizes the FLC systems’ freedom and flexibility while enabling accurate and speedy tracking. During optimization, the parameters of the FLC membership functions are considered choice variables, and the error integral is assigned to be the objective function. The proposed GBO-FLC method’s results are contrasted with those of other computational methods. The results demonstrated that the proposed GBO-FLC beats the other strategies regarding mean, median, variance, and standard deviation. A thorough comparison between the regular FLC and the upgraded FLC was conducted using a variety of scenarios with varied temperatures and water content. The results demonstrate that the suggested FLC-based GBO design provides a dependable MPPT solution in PEMFCs. The advancement of FLC systems through optimizing power generation in fuel cells is made possible by this work, opening the door for more effective and reliable alternative energy sources.

1. Introduction

1.1. Overview

Today, the energy industry and its profound interrelation with climate change have garnered remarkable global interest. Developing clean and sustainable energy sources stands as the most effective way to counteract the harmful impacts of traditional energy sources [1,2]. Notably, studies have demonstrated that the energy cost for wind and solar photovoltaic (PV) systems is comparable to that of conventional generation methods [3,4]. Fossil-fuel-based energy-producing systems incur higher costs in order to mitigate greenhouse gas emissions [5,6]. However, the intermittent nature of renewable energy sources necessitates the utilization of energy storage devices to overcome their fluctuations [7,8]. Therefore, the integration of these devices contributes to an increase in energy production costs. Alternatively, another viable option is the production of green hydrogen, which can be stored and utilized for energy or other purposes [9,10]. Fuel cell (FC) technologies effectively convert the chemical potential of air, oxygen, and hydrogen gases into clean electricity. These FC technologies make use of generation devices in various scales, including small, medium, and large, to achieve efficient electrical power generation [11].

1.2. Literature Review

In the literature, a wide range of FC types are described and categorized based on factors such as the type of fuel [12], operating temperature [13], and electrolyte membrane [14]. Among these FC types, proton exchange membrane FCs (PEMFCs) have emerged as a promising category. PEMFCs offer higher conversion efficiency and reliability [15], making them the preferred solutions in the realm of clean energy supply and storage alternatives. The superior performance of PEMFCs can be attributed to their inherent and operational characteristics, including fast operational speed, low operating temperatures, contamination-free operation, and high conversion efficiency [16,17]. As a result, various industrial and commercial sectors have expressed considerable interest in leveraging PEMFCs within the energy supply sector. Presenting improved control techniques for PEMFCs can lead to extending their applications and their contribution to fully renewable energy societies [18], power to hydrogen [19], and the energy transition process.

However, the output of PEMFCs exhibits nonlinear behavior, with the generated power being influenced by multiple factors, particularly the water content of the membrane and the temperature [20,21]. Achieving maximum power extraction in PEMFCs requires operating at the optimal point. Therefore, the implementation of effective control systems for MPP tracking (MPPT) in PEMFCs is crucial for attaining higher power extraction under varying operating conditions [22]. For PEMFC applications, it is essential to establish a specific operating point for different conditions. One costly approach to achieving MPPT involves acquiring measurements of temperature and membrane water content and utilizing them to precisely define the MPPT point [23]. In addition, alternative approaches that rely on measured electrical quantities from the power converter have demonstrated effective MPPT while reducing sensor costs. Numerous MPPT techniques utilizing electrical measurements have been proposed in the literature for PEMFCs. One well-known approach is the perturb and observe scheme (P&O), which relies on measuring the current and voltage at the terminals of the PEMFCs to track the corresponding MPP [24]. The P&O operates by applying fixed-step perturbations to the operating duty cycle and monitoring the resulting changes in power and voltage to determine the appropriate increment or decrement actions. This method is straightforward and cost-effective to implement. However, the selection of the step size in P&O must compromise a balance between tracking speed and steady-state fluctuations [25].

Various studies have introduced distinct methods specifically designed for PEMFC systems, which suggest employing variable step sizes, including incremental resistance (INR) and incremental conductance (INC) techniques [26]. The incorporation of variable step sizes in MPPT algorithms enables faster MPPT while reducing steady-state fluctuations. Another approach to achieving MPPT involves determining the reference operating terminal voltage for MPPT and utilizing a secondary stage of control systems to track the voltage. Furthermore, recent advancements in metaheuristic optimization algorithms have shown promising results in achieving optimized performance [27,28]. The application of PID control is explored for MPPT in PEMFCs. Various optimization algorithms have been developed in the literature to optimize the PID parameters. These include the salp-swarm optimization, grey-wolf optimizer, sine-cosine optimizer, particle-swarm optimization, and ant-lion optimizer algorithm [29,30]. In addition to traditional methods, there are alternative MPPT solutions that involve using optimization methods to search for the reference voltage for MPP operations or directly controlling the power conversion systems. The water-cycle optimizers algorithm (WCA) cascaded with a PID controller has been presented in [31], while the dynamic cuckoo searching optimizers algorithm (DCSA) has been proposed in [32]. Furthermore, the extremum-seeking approach with a PID controller has been introduced in [33]. It is important to note that some of these methods require additional temperature sensors, which can result in increased costs for PEMFC systems.

Advanced neural network (NN) controllers have gained significant traction in MPPT applications, with the introduction of the radial-basis function NN (RBFN) in [34], the proposal of the artificial NN with incremental conductance (INC) method in [35], and the development of the adaptive MPPT in [36]. However, these methods require complex training data and expert design. In contrast, fuzzy logic-based controllers (FLCs) have been widely utilized in MPPT [27]. FLC provides a variable-step-size MPPT control approach, resulting in rapid tracking with minimal fluctuations in the steady state. Furthermore, the utilization of FLC in MPPT applications eliminates the requirement for PID controllers and their complex design calculations. Additionally, recent optimization algorithms can be employed to enhance the design of FLC and consequently achieve improved MPPT performance. Performance comparisons of Sugeno-based and Mamdani-based designs of FLC have been covered by Luta et al. [37]. Additionally, Luta et al. [38] presented a comparison of Mamdani FLC-based MPPT with PSO-based control. The results have confirmed the improved performance of applying FLC in MPPT applications. Moreover, enhanced performance can be achieved by employing appropriate design algorithms. Priyadarshi et al. [39] proposed the firefly optimization algorithm with asymmetrically designed inputs/outputs membership functions (MF), showcasing its effectiveness in improving performance. Several other optimization algorithms have been proposed for FLC design, including elitist invasive weeds optimization (EIWO) [40], modified hybrid shuffled frogs leaping optimizers algorithm (MSFLA) [41], differential evaluations optimizations algorithm (DEOA), and the hybrid BAT optimizer algorithm [42].

1.3. Paper Contribution

However, there are still various challenging tasks that need to be addressed to enhance the performance of MPPT techniques for PEMFCs. These challenges include improving tracking speeds, optimizing design parameters, mitigating oscillations during steady-state operations, reducing the cost of sensors used, and simplifying implementation complexity. Another issue is the random behavior of optimization algorithms in single-run design methods. Statistical analysis is important to compare the different optimization algorithms fairly. Moreover, MPPT performance is highly affected when using the designed control method, especially when using the FLC. In which the freedom of their design can be maximized and better benefitted with improved design processes. In light of these challenges, this study presents an improved design and MPPT controller for PEMFCs using FLC systems. The chief contributions of the current study can be outlined:

• An enhanced design approach is suggested for maximizing the power point tracking (MPPT) control of PEMFCs through the utilization of FLC.
• The proposed method uses adaptive variable duty cycle MPPT control to ensure rapid MPPT during transient conditions and to minimize fluctuations during steady-state operations.
• An optimized approach is introduced for determining the parameters of FLC systems in order to maximize their flexibility and effectiveness using the gradient-based optimizer (GBO) algorithm.
• The GBO algorithm provides the optimized FLC parameters to accurately and swiftly track the desired outcomes. Thence, the overall performance is significantly enhanced, resulting in improved accuracy and speed.
• Performance comparisons are presented between the GBO algorithm and several other featured algorithms from the literature.
• Furthermore, statistical tests are employed in this study to guarantee a fair and objective comparison of the various metaheuristic algorithms used.

The rest of the sections of this manuscript are structured as follows: Section 2 introduces the modeling of PEMFC and discusses its output characteristics. Section 3 presents the operating principle of FLC MPPT and describes the design optimization process. Section 4 outlines the proposed optimization method using the GBO for FLC in MPPT. The obtained results, along with discussions and performance comparisons, are displayed in Section 5. In conclusion, Section 6 delivers the main findings.

2. Model of PEMFC

PEMFCs are devices that facilitate converting chemical energy into electricity, as represented by Equation (1):

$$2H_2+O_2 \to H_{2}O+electricity+heat$$

(1)

Figure 1 illustrates the polarization curve of a PEMFC, which consists of three distinct regions [27]. The first region is the activation loss region, where the chemical response delay and electron excitation take place. The second region corresponds to concentration loss, which is attributed to losses caused by mass transfer. The third region represents the ohmic loss resulting from the material characteristics of PEMFC. The electrical characteristics of PEMFC can be modeled using Equation (2):

$$V_{fc}=E_{Nernest}-V_a-V_o-V_c$$

(2)

The activation polarization is denoted as $V_a$, the ohmic polarization as $V_o$, and the concentration loss as $V_c$. The thermodynamic potential is represented as $E_{Nernest}$ [40]. This can be deduced from Equation (3):

$$\begin{gathered} E_{Nernest}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2F$}\right.[\Delta G-\Delta S\left(T-T_{ref}\right)+RT\left(\ln \left(P_{H_2}\right)+\frac{\ln \left(P_{O_2}\right)}{2}\right)] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=1.229-0.85\times {10}^{-3}\left(T-298.15\right)+4.3085\times {10}^{-5}\times T \left(\ln \left(P_{H_2}\right)+\frac{\ln \left(P_{O_2}\right)}{2}\right) \end{gathered}$$

(3)

The Faraday constant is denoted as F, and the Gibbs free energy is represented by ΔG. The entropy is denoted as ΔS, and the gas constant is R. The temperature of the FC is symbolized by T, while the reference temperature is denoted as $T_{ref}$. The partial pressure of hydrogen and oxygen are represented as $P_{H_2}$ and $P_{O_2}$, respectively. These quantities can be described by Equations (4) and (5):

$$P_{H_2}=\frac{R_{ha}\times P_{H_{2}O}}{2}[\frac{1}{\frac{R_{ha}\times P_{H_{2}O}}{P_a}\times e^{\frac{1.635\left(\frac{i}{A}\right)}{T^{1.334}}}}-1]$$

(4)

$$P_{O_2}=R_{hc}\times P_{H_{2}O}[\frac{1}{\frac{R_{hc}\times P_{H_{2}O}}{P_c}\times e^{\frac{1.635\left(\frac{i}{A}\right)}{T^{1.334}}}}]$$

(5)

$$\ln \left(P_{H_2}\right)=2.95\times {10}^{-2}{T}_c-9.18\times {10}^{-5}{T}_c^2+1.44\times {10}^{-7}{T}_c^3-2.18$$

(6)

The anode and cathode humidities are captured as $R_{ha}$ and $R_{hc}$, respectively. The pressures at the inlet for the anode and cathode are represented as $P_a$ and $P_c$, respectively [27]. The electrode area is denoted as A, and the current from the cell is denoted as i. The water vapor saturation pressure is represented as $P_{H_{2}O}$, and $T_c$ is calculated as $T-T_{ref}$. The activation polarization ($V_a)$ can be determined using Equation (7).

$$V_a=-\left({\xi }_1+{\xi }_{2}T+{\xi }_{3}T\ln \left(C{O}_2\right)+{\xi }_{4}T\ln \left(i\right)\right)$$

(7)

The semi-empirical constants are denoted as ${\xi }_1,{\xi }_2,{\xi }_3, \mathrm{and} {\xi }_4$. The oxygen concentration at the cathodic electrode is represented as $C{O}_2$ and can be calculated using Equation (8).

$$C{O}_2=\frac{P_{O_2}}{5.08\times {10}^6}x{e}^{\left(\frac{498}{T}\right)}$$

(8)

The mass concentration loss is represented by Equation (9).

$$V_c=-\beta \ln \left(1-\frac{i}{i_{lim}}\right)$$

(9)

The diffusion parameter is represented as $\beta$, and the limiting current for the fuel cell is presented as $i_{lim}.$ The ohmic loss ($V_o)$ can be presented by Equation (10)

$$V_o=i\left(R_m+R_c\right)$$

(10)

Resistance within the connectors is denoted as $R_c$, and ohmic membrane resistance is represented as $R_m$ and is determined using Equation (11).

$$R_m={\rho }_m\left(\frac{l}{A_m}\right)$$

(11)

The membrane thickness is denoted as $l$, and the surface area is represented as $A_m$. The membrane resistivity, denoted as ${\rho }_m$, is determined using Equation (12). The membrane material water content is represented as $\lambda$.

$${\rho }_m=\frac{181.6 [1+0.03\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.+0.062\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$303$}\right.{(\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.)}^{2.5}]}{\left[\lambda -0.634-3\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.\right]e^{4.18\frac{T-303}{T}}}$$

(12)

3. Parameter Identification Process

The power conversion system of PEMFCs based on a boost topology DC/DC power converter is depicted in Figure 2. The boost topology allows for stepping up the output terminal voltage of the PEMFC by utilizing a single inductor, capacitor, semiconductor switch, and diode. The FLC MPPT utilizes the measured output terminal voltage and current of the PEMFC to track the MPP under various operating conditions. This is achieved by adjusting the duty cycle input to the power semiconductor switch, thereby controlling the system. Therefore, by employing this topology with PEMFC, it can simultaneously track MPPT and step up the low terminal voltage of the PEMFC. The relationship between the input and output voltages in the boost topology is dependent on the operating duty cycle and may be mathematically formulated using the following:

$$V_o=\frac{1}{1-D}\times V_{in}$$

(13)

where V_o represents the converter output voltage and V_in represents the input voltage coming from the PEMFC side. The duty cycle of the converter, as denoted by D, is adjusted and controlled using the FLC MPPT controller. The value of D is typically varied based on the defined MPP of the PEMFC, and for each operating condition, an optimum D needs to be searched for to ensure MPPT. Therefore, different operating conditions, such as variations in membrane water content and temperature, require different values of D, which are dynamically adjusted through FLC MPPT control.

In MPPT control schemes, the terminal voltage and current of PEMFCs are measured to calculate the slope of the output power with respect to voltage (ΔPFC/ΔVFC), which serves as an effective criterion for determining the location of the MPP. The slope is continuously evaluated to track changes in the operational circumstances of the PEMFC. The controller regulates the duty cycle to track the MPP, aiming for a zero slope at the MPP.

The proposed controller in this study utilizes only the measured PEMFC output voltage (VFC) and current (IFC). The FLC is designed with two inputs based on the estimated error (E), derived from the calculations of ΔPFC/ΔVFC for MPPT control, and the change in error (ΔE) in each sampling period. These two inputs are mathematically modeled for any sampling period (k) as follows [38]:

$$E\left(k\right)=\frac{P_{FC}\left(k\right)-P_{FC}\left(k-1\right)}{V_{FC}\left(k\right)-V_{FC}\left(k-1\right)}$$

(14)

$$\Delta E\left(k\right)=E\left(k\right)-E\left(k-1\right)$$

(15)

where E(k) represents the inputted error at instant k, ΔE(k) represents the change in error at instant k, P_FC(k) and P_FC (k − 1) represent the measured output power at sampling instants k and (k − 1), respectively, and VFC(k) and VFC(k − 1) represent the measured terminal voltage at sampling instants k and (k − 1), respectively. The FLC MPPT scheme solely relies on the measured electrical quantities of current and voltage from the PEMFCs. This simplifies the implementation and eliminates the need for expensive sensors, distinguishing it from other existing MPPT controllers. The principal stages of the FLC MPPT scheme are illustrated in Figure 3 and can be defined as follows:

• Fuzzification stage:
  In this stage, the MFs are calculated for the inputs of the FLC. The calculation of these MFs depends on the selected shape and type of MFs for each input. In the proposed method, triangular MFs are chosen and designed for the inputs. The purpose of this stage is to convert the physically measured electrical signals into a suitable form for the FLC method.
• Fuzzy-rule based evaluation stage:
  In this step, a set of fuzzy rules is applied to evaluate the calculations obtained from the previous fuzzification stage.
• Defuzzification stage:
  In this stage, the output of the fuzzy rule-based evaluation is inputted to the defuzzification step. Afterward, it outputs the MF for the converter duty cycle by utilizing the calculated values and increments for the duty cycle.

Figure 3. Main stages of FLC in MPPT applications.

Figure 3. Main stages of FLC in MPPT applications.

The suggested FLC utilizes MFs that are divided into seven levels for both the input and output sides. These levels consist of three positive levels (PLS3, PLS2, and PLS1), three negative levels (NLS3, NLS2, and NLS1), and a single zero level (ZLS0). The shape of the employed MFs for the input and output sides is depicted in Figure 4a.

Table 1. The included fuzzy rules table.

Input MFs		ΔE
		NSL3	NSL2	NSL1	ZSL0	PSL1	PSL2	PSL3
E	NSL3	NSL3	NSL3	NSL3	NSL3	NSL2	NSL1	ZSL0
	NSL2	NSL3	NSL3	NSL3	NSL2	NSL1	ZSL0	PSL1
	NSL1	NSL3	NSL3	NSL2	NSL1	ZSL0	PSL1	PSL2
	ZSL0	NSL3	NSL2	NSL1	ZSL0	PSL1	PSL2	PSL3
	PSL1	NSL2	NSL1	ZSL0	PSL1	PSL2	PSL3	PSL3
	PSL2	NSL1	ZSL0	PSL1	PSL2	PSL3	PSL3	PSL3
	PSL3	ZSL0	PSL1	PSL2	PSL3	PSL3	PSL3	PSL3

presents the fuzzy rules, which comprise a total of 49 rules based on the designed MFs. The output MF consists of seven different levels, allowing for more adaptive control of the step size. This adaptive control enables fast tracking and reduces fluctuations in steady-state.

The classical FLC design, as depicted in Figure 4a, lacks the ability to adjust and optimize the MFs as fixed divisions are used in its design. However, in the proposed FLC design, the shape of the MFs is adjusted, allowing for better flexibility and optimization. This enhanced flexibility enables the optimization of the FLC design, leading to improved performance using the proposed method. Figure 4b illustrates the flexibility in designing and adjusting the various parts of the MFs in both directions. In the proposed method, the optimization process is carried out simultaneously for the input and output MFs, allowing for a better and more optimized design compared to conventional approaches. As a result of the optimization, the distances between the points of the MFs will vary to achieve their optimized values. Figure 4c displays the resulting shape of the MFs in the proposed method. During the optimization process, each MF is designed with seven points, resulting in a total of 21 points that are simultaneously optimized using the proposed gradient-based optimizer (GBO) method. In the proposed design, a symmetrical design is maintained for the MFs around the zero value, ensuring a balanced representation of the input and output variables. Specifically, each MF has 3 adjustable points, resulting in a total of 9 adjustable points in the proposed process. Figure 5 depicts the different stages involved in the proposed FLC optimization process. The authors have implemented the FLC in Matlab, which allows for greater flexibility in optimizing the FLC. In this scheme, the objective function is established using the integral-absolute error (IAE), which corresponds to the MPPT criterion based on ΔPFC/ΔVFC and presented by the following relation.

$$Objective Function={{\displaystyle \int }}_{t=0}^{T_{sim}}abs \left( \Delta E \right)={{\displaystyle \int }}_{t=0}^{T_{sim}}abs \left( \frac{P_{FC}\left(k\right)-P_{FC}\left(k-1\right)}{V_{FC}\left(k\right)-V_{FC}\left(k-1\right)} \right)$$

(16)

where T_sim stands for total simulation time during optimization run for evaluating desired objective function. The targeted output of the proposed optimization scheme is to obtain optimized points for the MFs. The implementation of the suggested FLC is shown in Figure 6.

4. Brief Overview of Optimization Algorithms Adopted

4.1. Gradient-Based Optimizer

The GBO adopts both gradient-based and population-based approaches in its search process. It utilizes Newton’s method for exploration within the search domain and employs a set of vectors along with two additional operators. The GBO has proven good performance in several engineering problems, such as PV parameter extractions for different models [43], economic load dispatch [44], controller design for automatic voltage regulators [45], solving unit commitment problems, etc. The primary objective of GBO is to minimize the objective function. The optimization process of GBO requires decision variables, constraints, and the objective function. Control parameters such as the transition parameter (α) and the probability rate are considered. The number of iterations and the population size depend on the complexity of the problem. Each member of the population is represented by a vector, denoted by Equation (17). The generation of the vectors initially occurs randomly, as shown in Equation (18).

$$X_{n,d}=\left[X_{n,1}, X_{n,2}, \dots ,X_{n,D}\right], n=1,2,\dots ,N, d=1,2,\dots D$$

(17)

$$X_n=X_{min}+rand\left(0, 1\right)\times \left(X_{max}-X_{min}\right)$$

(18)

The random number in the range [0, 1] is denoted as rand(0, 1), while $X_{max}$ and $X_{min}$ represent the decision variable X. To ensure a quality search and achieve good positions, the vector is controlled using Newton’s gradient method [46]. The Gradient Search Rule (GSR) is determined using the Taylor series, as shown in Equations (19) and (20).

$$f\left(x+\mathsf{\Delta }x\right)=f\left(x\right)+f^{\prime }\left(x_o\right)\mathsf{\Delta }x+\frac{f^{″}\left(x_o\right)\mathsf{\Delta }{x}^2}{2!}+\frac{f^{\left(3\right)}\left(x_o\right)\mathsf{\Delta }{x}^3}{3!}+\cdots$$

(19)

$$f\left(x-\mathsf{\Delta }x\right)=f\left(x\right)-f^{\prime }\left(x_o\right)\mathsf{\Delta }x+\frac{f^{″}\left(x_o\right)\mathsf{\Delta }{x}^2}{2!}-\frac{f^{\left(3\right)}\left(x_o\right)\mathsf{\Delta }{x}^3}{3!}+\cdots$$

(20)

Equation (21) represents the first-order derivative, and the new position is denoted by Equation (22). The GSR can be simplified using Equation (23).

$$f^{\prime }\left(x\right)=\frac{f\left(x+\mathsf{\Delta }x\right)-f\left(x-\mathsf{\Delta }x\right)}{2\mathsf{\Delta }x}$$

(21)

$$x_{n+1}=x_n-\frac{2\mathsf{\Delta }x x f\left(x_n\right)}{f\left(x_n+\mathsf{\Delta }x\right)-f\left(x_n-\mathsf{\Delta }x\right)}$$

(22)

$$GSR=randn x \frac{2\mathsf{\Delta }x x x_n}{\left(x_{worst}-x_{best}+ϵ\right)}$$

(23)

4.2. Osprey Optimization Algorithm

The OAA utilizes population-based techniques to find solutions through iterative steps, harnessing the search capabilities of its population members. Each osprey represents a vector that captures the problem variables as shown in Equation (24). The initial position of the osprey is randomly determined, as illustrated in Equation (25).

$$x={\left[\begin{array}{l}{x}_1\\ \dots \\ x_i\\ \dots \\ x_N\end{array}\right]}_{N\times m}={\left[\begin{array}{l}{x}_{1,1}\dots x_{1,j}\dots x_{1,m}\\ \dots \dots \dots \dots \dots \\ x_{i,1}\dots x_{i,j}\dots x_{i,m}\\ \dots \dots \dots \dots \dots \\ x_{N,1}\dots x_{N,j}\dots x_{N,m}\end{array}\right]}_{N\times m}$$

(24)

$$\begin{array}{l}{x}_{i,j}=l{b}_j+r_{i,j}\times (u{b}_j-l{b}_j)\\ i=1,2,\dots \dots N\\ j=1,2,\dots \dots m\end{array}$$

(25)

4.3. Sine Cosine Algorithm

The SCA starts the optimization procedure by conducting a search using random initial solutions. These solutions are iteratively evaluated using an objective function [47]. It is important to note that there is no guarantee of finding a solution within a limited time during a single iteration, as the initial solutions are determined stochastically to get the best solution [48]. As the number of stochastically deduced solutions and the optimization process increases, the chances of reaching the global optimum also increase [49]. Equations (26) and (27) describe the update process for the positions in the SCA.

$$X_i^{t+1}=X_i^t+r_1\times \sin \left(r_2\right)\times \left|r_3{P}_i^t-X_i^t\right|$$

(26)

$$X_i^{t+1}=X_i^t+r_1\times \cos \left(r_2\right)\times \left|r_3{P}_i^t-X_{ii}^t\right|$$

(27)

The current solution is represented as $X_i^t$, while the three random numbers are $r_1$, $r_2$, $r_3$. The place point positions is represented as $P_i$.

4.4. Salp Swarm Algorithm (SSA)

The SSA, introduced by Mirjalili et al. [50], is a suitable method for feature selection. It has been found to effectively identify optimal subsets of features, maximizing classification accuracy [51,52]. Additionally, the SSA can be applied to tune hyperparameters for neural networks [53]. This algorithm is stimulated using the predatory behavior of salps, which form a chain-like swarm and move directly toward the food source. The population in SSA is divided into two categories: leaders and followers, based on their positions within the chain. The leaders are located at the front, while the followers trail behind. Equation (28) represents the position update of the leader during the search process.

$$x_i^1=\left\{\begin{array}{c}{y}_i+r_1 \left(\left(u{b}_i-l{b}_i\right)r_2+l{b}_i\right) r_3\ge 0\\ y_i+r_1 \left(\left(u{b}_i-l{b}_i\right)r_2+l{b}_i\right) r_3<0\end{array}\right.$$

(28)

The location of the first salp is expressed as $x_i^1$, and the position of the food source is denoted as $y_i$ for the $i\mathrm{th}$ dimension. The minimum limit is $l{b}_i$, and the maximum limit is $u{b}_i$. The random numbers used in the algorithm are $r_1$, $r_2$, and $r_3$.

4.5. Whale Optimization Algorithm

The idea of WOA is extracted from the hunting performance of humpback whales. Humpback whales identify the position of their prey and then swim around it to capture it [54]. The WOA mimics this behavior by considering the best current solution as the prey and modifying the locations of the search agents accordingly. Mathematically, the position update in the WOA is represented by Equations (29) and (30).

$$\overrightarrow{D}=\left|\overrightarrow{C} · \overrightarrow{X^{*}}\left(t\right)-\overrightarrow{X}\left(t\right)\right|$$

(29)

$$\overrightarrow{X}\left(t+1\right)=\left| \overrightarrow{X^{*}}\left(t\right)-\overrightarrow{A}-\overrightarrow{D}\right|$$

(30)

The present iteration is represented denoted by t, and the coefficient vectors are represented by $\overrightarrow{A} \mathrm{and}$ $\overrightarrow{C}$. The location vector for the best solution is denoted as $X^{*}$, while $\overrightarrow{X}$ denotes the location vector. The absolute value is represented by ∣ ∣. Equations (31) and (32) are used to calculate the vectors $\overrightarrow{A}$ and $\overrightarrow{C}$.

$$\overrightarrow{A}=2\overrightarrow{a}·\overrightarrow{r}-\overrightarrow{a}$$

(31)

$$\overrightarrow{A}=2·\overrightarrow{r}$$

(32)

The random vector is denoted as $\overrightarrow{r}$, while the vector $\overrightarrow{a}$ linearly declines from 2 to 0 during the iteration.

5. Results

In this case study, the use of a GBO-FLC optimization method is proposed to improve the performance of a PEMFC. The specifications of the PEMFC, as outlined in

Table 2. Specification of PEMFC.

Parameter	Value	Unit	Parameter	Value	Unit
A	232	cm2	NFC	35	-
l	0.0178	cm	ξ1	0.944	-
Imax	2.00	A/cm2	ξ2	0.00354	-
T	343	K	ξ3	7.8 × 10−8	-
n	2	-	ξ4	−1.96 × 10−4	-
R	8.31447	J (mol K)−1	F	96,484,600	C (kmol)−1
kO2	2.11 × 10−5	kmol (atom S)−1	qH2in	10 × 10−5	(kmol (S)−1)
kH2	4.22 × 10−5	kmol (atom S)−1	qO2in	5 × 10−5	(kmol (S)−1)

, serve as the basis for this optimization process. The PEMFC is connected to a boost converter with specific components, including an inductor of 0.5 mH, a capacitor of 500 μF, and a switching frequency of 20 kHz. The effectiveness of the proposed GBO-FLC is evaluated under two different operating conditions: Case 1, representing normal operation with a constant membrane water content and temperature, and Case 2, involving variable water content and temperature.

The GBO controller parameters selected for the study are as follows: a population size of 10, a maximum iteration limit of 30, and 30 independent runs. The performance of the proposed GBO-FLC is compared against other algorithm approaches for both Case 1 and Case 2. In the following subsections, a more detailed explanation of each case is provided, and the comparative analysis of the different algorithms is presented.

During the normal operation of the PEMFC, a constant membrane water content of 12 and a steady temperature of 343 K were maintained. To ensure a fair comparison, the population size and iterations are 10 and 20, respectively. The optimization process focused on determining the values of the nine unknown parameters of the FLC, with the objective of minimizing the integral absolute error.

Table 3. Parameters limits and best values obtained using different optimization algorithms.

	GBO	OOA	SCA	SSA	WOA
MF11	1.14988	0.90819	1.5	1.06597	1.47164
MF12	1.00452	0.7888	0.77638	1.19924	1.1768
MF1U	899.35528	464.1454	880.92117	1078.99711	1965.51729
MF21	0.99945	0.90819	0.9	1.26012	1.47164
MF22	0.75383	0.7888	0.97143	0.92453	1.17965
MF2U	162.30635	177.19702	153.88178	220.35592	219.36654
MF31	1.34852	0.90819	1.29706	0.95742	1.12571
MF32	1.17279	0.7888	1.1812	1.17098	1.1768

provides limits of these unknown parameters, along with their optimal values obtained after 30 runs. Statistical evaluation of the results is presented in

Table 4. Numerical evaluations for used algorithms.

	GBO	OOA	SCA	SSA	WOA
Best	1.85736	1.85716	1.86017	1.85768	1.85751
Worst	1.91456	2.13843	1.92361	2.03591	2.34069
Mean	1.86315	1.90068	1.88947	1.93269	1.92757
StD	0.01152	0.05881	0.01831	0.04934	0.13314
Variance	0.00013	0.00346	0.00034	0.00243	0.01773
Median	1.86015	1.88402	1.89466	1.92177	1.86077

.

Based on the results shown in the table, the mean cost function values range from 1.86315 to 1.93269. The GBO algorithm achieves the minimum mean cost function value of 1.86315, followed by SCA with a value of 1.88947, while SSA obtains the highest mean cost function value of 1.93269. The standard deviation values vary between 0.01152 and 0.13314. The GBO algorithm achieves the minimum standard deviation value of 0.01152, followed by SCA with a value of 0.01831, while WOA obtains the largest standard deviation value of 0.13314. Therefore, for the investigated cases, the GBO algorithm outperforms the OOA, SCA, SSA, and WOA algorithms.

Table 5. Results during 30 runs.

Run	GBO	OOA	SCA	SSA	WOA	Run	GBO	OOA	SCA	SSA	WOA
1	1.91456	1.88635	1.87459	2.02962	2.05613	16	1.85736	1.88941	1.89889	1.93571	2.29235
2	1.85768	1.85768	1.91875	1.92471	1.86077	17	1.86022	1.88598	1.86017	1.91452	1.86077
3	1.86017	1.88506	1.90145	1.96071	1.86015	18	1.8577	1.87743	1.91107	1.91605	2.34069
4	1.85768	1.86022	1.86351	2.03591	1.86015	19	1.86015	1.88464	1.89637	1.88307	1.91447
5	1.86015	1.88457	1.89635	1.9615	1.93018	20	1.8773	1.95246	1.89226	1.91189	1.86077
6	1.86015	1.96856	1.86977	1.87156	1.8577	21	1.87202	1.86022	1.87995	1.86028	1.86077
7	1.86313	1.86015	1.87733	1.91452	1.85768	22	1.85767	1.85716	1.87743	1.94017	1.86015
8	1.85736	1.88464	1.89917	1.86034	1.86077	23	1.8575	1.88399	1.92035	2.03369	1.86015
9	1.86015	1.87718	1.90503	1.9615	1.85751	24	1.86015	1.86062	1.86055	1.90088	1.86115
10	1.85768	1.8577	1.87113	1.91882	1.8577	25	1.86015	1.88403	1.8973	1.93018	1.88307
11	1.86015	1.87948	1.87895	1.85768	2.02266	26	1.885	2.13843	1.89387	1.91452	1.90088
12	1.86077	1.92118	1.89316	1.91189	1.86077	27	1.85736	1.86022	1.90102	1.89228	1.96982
13	1.87203	1.96817	1.90727	2.03553	2.26229	28	1.85751	1.87479	1.89546	1.93018	1.85995
14	1.86028	1.86313	1.92361	1.91505	1.87755	29	1.85736	2.00238	1.86313	1.95836	1.86077
15	1.8575	1.97045	1.86067	1.92974	1.8985	30	1.85767	1.88401	1.89554	1.96983	1.86077

provides detailed data over 30 runs.

ANOVA and Tukey tests were accomplished to verify the superiority of GBO in comparison with other methods. ANOVA measures mean differences between groups and compares the variability between groups and within groups. This test helps clarify whether the differences between group means are statistically significant.

Table 6. ANOVA results.

Source	df	SS	MS	F	Prob
Columns	4	0.098	0.024	4.91	0.001
Error	45	0.723	0.005
Total	49	0.820

presents the ANOVA test results, while Figure 7 illustrates the corresponding ranking. The results confirm that GBO consistently produces the best results, as evidenced by its lowest mean fitness and fluctuations compared to the other algorithms. The statistical analysis further supports the conclusion that GBO outperforms the competing algorithms in terms of optimization performance.

The results of ANOVA are further validated via the Tukey test, as depicted in Figure 8. The mean of the GBO group is observed to be lower compared to the OOA and SCA groups, and significantly lower compared to the SSA and WOA groups. This provides strong evidence that the GBO algorithm exhibits superior performance compared to the other algorithms under consideration.

Figure 9 illustrates the variation of the objective function. As depicted in Figure 9a and Table 5, the IAE values converge to 1.86015, 1.88457, 1.89635, 1.9615, and 1.93018, respectively, for GBO, OOA, SCA, SSA, and WOA. GBO achieves rapid convergence to the optimal solution, while OOA requires more time to reach its best solution. In run number 15, as shown in Figure 9b and Table 5, the IAE values converge to 1.8575, 1.97045, 1.86067, 1.92974, and 1.8985, respectively, for GBO, OOA, SCA, SSA, and WOA. Once again, GBO demonstrates fast convergence to the optimal solution, while OOA requires more time to reach its best solution.

Figure 10 displays the time responses of power, current, voltage, and duty cycle for a normal case study using both classical FLC and FLC-GBO. The dynamic performance of FLC-GBO is noticeably superior to classical FLC.

The PEMFC’s temperature and membrane water content are altered in the second scenario. At first, the water content is 10, and the temperature is set to 40 °C. The water content rises to 14 at 0.1 s while the temperature stays at 40 °C. The temperature then rises to 70 °C while the water content stays at 14 at 0.25 s. In this case, the PEMFC’s power climbs from 5000 W to 6880 W in 0.1 s and then from 6880 W to 9180 W in 0.25 s. Figure 11 displays the PEMFC’s power, voltage, current, and duty cycle responses in detail. In comparison to the standard FLC, the optimized FLC delivers a maximum power of 6880 W with a short tracking time.

6. Conclusions

An improved design methodology and optimization approach have been proposed in this study for FLC-based MPPT in PEMFC applications. The flexible and accurate tracking ability of fuzzy FLC has been effectively optimized using the advanced GBO method. The utilization of the GBO optimization method has enabled the simultaneous optimization of FLC membership functions (MFs) for both the input and output sides. This approach offers enhanced design flexibility, allowing for precise control over the positioning of points within the triangular MFs in the proposed FLC design. The proposed FLC design has been implemented and tested under various operating conditions, including normal operation and scenarios with variable temperature and water content. Extensive comparisons were conducted using statistical analyses to assess the performance of the GBO optimizer in comparison to other recent optimizers, namely OOA, SCA, SSA, and WOA. The results demonstrated that the FLC system utilizing the GBO optimizer exhibited a superior response compared to classical MPPT designs. Moreover, the statistical analysis, including mean cost function values and standard deviation, revealed the superior performance of the GBO optimizer in comparison to the studied optimizers. The obtained objective functions over 30 runs showed that GBO has a minimum mean cost function value of 1.86315, followed by the SCA with a value of 1.88947, while the WOA has a mean cost function of 1.92757 and the SSA has the highest mean cost function value of 1.93269. Meanwhile, the standard deviation value of the proposed GBO is 0.01152, which is only 8.65% of the standard deviation of the WOA method. ANOVA and Tukey tests were conducted to validate the superiority of the GBO algorithm over other algorithms. The ANOVA test results confirmed that GBO consistently produced the best results with the lowest mean fitness and fluctuations. The Tukey test further confirmed the significant difference in performance, showing that the GBO group achieved lower means compared to other groups. These findings validate the effectiveness and competitiveness of the proposed FLC design with GBO optimization in maximizing the power point tracking (MPPT) capability of PEMFCs. The proposed method can also help extend the applications of PEMFCs with better efficiency and faster response. Moreover, it can contribute to the extensive use of green hydrogen, power to hydrogen technology, and, in general, the energy transition process. Future research will include comparisons with more recent optimization techniques, the extension to type-2 FLC systems, and the application of the proposed method for PV systems MPPT control.