Intelligent Extremum Seeking Control of PEM Fuel Cells for Optimal Hydrogen Utilization in Hydrogen Electric Vehicles

Abstract

In terms of their high efficiency and low environmental impact, proton exchange membrane fuel cells (PEMFC) are becoming increasingly essential in the development of hydrogen electric vehicles. Despite these advantages, optimizing hydrogen consumption remains difficult because of the highly nonlinear behavior of PEMFC systems and their sensitivity to variations in operating conditions. This article outlines an intelligent control approach based on extremum seeking control (ESC), based on an artificial neural network (ANN) model, to improve hydrogen utilization in hydrogen electric vehicles. Experimental data on current, voltage, and temperature are collected, preprocessed, and used to train the ANN model of the PEMFC. The ESC algorithm uses this predictive ANN model to adjust the fuel cell current in real time, ensuring voltage stability while reducing hydrogen consumption. The simulation results demonstrate that the ANN-based ESC system provides voltage stability under dynamic load variations and achieves approximately 2.7% hydrogen savings without affecting the experimental current profile, validating the efficacy of the suggested strategy for effective hydrogen management in fuel cell electric vehicles.

1. Introduction

Hydrogen electric vehicle technology is growing rapidly to meet demands for decarbonization, energy transition, and extended range for transportation systems [1]. Among renewable energy sources, proton exchange membrane fuel cells (PEMFCs) are a promising solution due to their high energy density, low emissions, and quick load response [2,3]. However, PEMFC operating conditions considerably impact efficiency and hydrogen consumption management, which is a major technical and economic constraint [4]. Achieving optimal PEMFC performance requires balancing performance, durability, and rational use of hydrogen [5]. Inefficient operation can lead to high fuel consumption, early membrane wear, and unstable voltage output [6].

Accurate identification of the PEMFC voltage model is a fundamental prerequisite for developing reliable predictive frameworks, particularly when employing data-driven approaches such as artificial neural networks (ANNs) [7,8]. Since the voltage response of a PEMFC results from strongly coupled and highly nonlinear electrochemical phenomena, precise estimation of the underlying model parameters is essential to ensure both physical coherence and high prediction. Recent research has emphasized the need for robust and efficient parameter identification strategies capable of capturing the intrinsic dynamics of the fuel cell across a wide range of operating conditions. Enhancing the accuracy and stability of voltage model identification not only improves the quality of the predictive models built upon these parameters but also strengthens their suitability for advanced control and energy management applications [9].

In this context, advanced control and artificial intelligence methods are becoming effective tools for improving PEMFC efficiency. Extremum Seeking Control (ESC) is a particularly useful real-time optimization technique. In this study, it is specifically applied to dynamically adjust the fuel cell current as a key control parameter [10]. The ESC principle is to find a system’s optimal operating point by changing the setpoint until it nears the ideal value [11,12]. One of the major strengths of this approach lies in its robustness to uncertainties and its capacity to adjust to changing operating conditions [13]. In light of these characteristics, ESC is especially well-suited for complicated nonlinear energy systems like PEMFCs, where maintaining an ideal operating point is crucial to increasing efficiency [14,15]. The combination of artificial neural networks (ANN) and ESC controllers offers a promising approach to the intelligent management of hydrogen fuel cells [16]. While ESC enables adaptive online optimization without the need for analytical models [17], ANN provides powerful predictive capabilities that can accurately represent the complex nonlinear dynamics of the system based on experimental data [18]. By integrating these two techniques, it becomes possible to exploit the advantages of each: the neural network’s learning and generalization of electrochemical behavior. This synergy results in a faster, more accurate, and more robust intelligent control in the face of disturbances and load variations [19,20]. The range, efficiency, and overall operating cost of fuel cell electric vehicles are all influenced by hydrogen management. The system could incur unnecessary energy losses, wasteful fuel consumption, and accelerated fuel cell deterioration if this resource is not properly monitored. It is therefore essential to ensure intelligent and efficient use of hydrogen to maintain performance and increase the PEMFC system’s lifespan [21,22]. Therefore, it is crucial to develop a control strategy that can maintain voltage stability while controlling hydrogen consumption. Keeping that in consideration, the ESC–ANN approach is an appropriate option that dynamically changes the cell current according to its real state, thus ensuring rational use of hydrogen and contributing to an overall improvement in the system’s energy efficiency [23,24].

To guide the dynamic optimization of the PEM Fuel Cell, this study proposes an intelligent control method that combines an ESC algorithm with an Artificial Neural Network (ANN) model. The research process is organized as follows: The development and modeling of the PEMFC are introduced first. This section provides an overview of the principles of PEM hydrogen fuel cell systems, explains the experimental dataset and selected input variables, and illustrates how the ANN model was used to model the PEMFC. Intelligent control design is based on the developed ANN model, which reliably predicts cell voltage based on variations in temperature and current. Next, the intelligent Extremum Seeking Control (ESC) strategy is developed. After recalling the fundamental concept of the ESC algorithm, this section explains the integration of the ANN model within the ESC control loop and highlights the role of the ESC in hydrogen optimization, allowing the system to continuously adjust the current of the fuel cell to the minimum hydrogen consumption point. Then, a presentation of the implemented ESC–ANN algorithm, which combines the predictive capacity of the ANN with the adaptive learning feature of the ESC to achieve real-time optimization of the PEMFC operating point. Finally, the experimental results and analysis section discusses the outcomes obtained under a dynamic current profile, focusing on the evolution of the current command, the cumulative hydrogen consumption, and the efficiency improvement achieved by the proposed control strategy. Overall, the results confirm that the ESC–ANN framework effectively enhances fuel utilization efficiency while maintaining stable voltage behavior, demonstrating its potential for hydrogen electric vehicles’ intelligent energy management.

2. Fuel Cell Modeling and ANN Model Development

2.1. Fundamentals of Hydrogen Hydrogen Fuel Cell PEMs

A proton exchange membrane fuel cell (PEMFC) is an advanced electrochemical energy conversion device that directly converts the chemical energy of hydrogen and oxygen into electrical energy, with heat and water as the only byproducts, as shown in Figure 1 [25]. Unlike conventional combustion-based energy systems, a PEMFC operates without any intermediate thermal processes, enabling it to achieve high energy efficiency and produce no greenhouse gas emissions at the point of use.

The fundamental electrochemical reaction governing the operation of PEMFCs takes the following form:

$$2{\mathrm{H}}_2+{\mathrm{O}}_2\to 2{\mathrm{H}}_2\mathrm{O}+\mathrm{Electrical}\mathrm{Energy}$$

(1)

This overall reaction can be broken down into two half-cell reactions that occur at the anode and cathode:

•   Cathode reaction (oxygen reduction):

$${\displaystyle \frac{1}{2}}{\mathrm{O}}_2+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\to {\mathrm{H}}_2\mathrm{O}+\mathrm{Q}\left(\mathrm{heat}\right)$$

(2)

•   Anode reaction (hydrogen oxidation):

$${\mathrm{H}}_2\to 2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}$$

(3)

A single Proton Exchange Membrane Fuel Cell (PEMFC) typically produces a voltage range between 0.6 and 0.8 V under nominal load conditions. Since this voltage is insufficient for most practical applications, individual cells are electrically connected in series to form a fuel cell stack, thereby increasing the total output voltage and power. The theoretical open-circuit voltage $E_{\mathrm{rev}}$ of a single cell is approximately 1.23 V under standard temperature and pressure conditions. However, the actual output voltage is lower due to several irreversible losses, which arise from various physicochemical mechanisms within the cell. The real operating voltage of the cell can be expressed as [26]:

$$V_{\mathrm{cell}}=E_{\mathrm{rev}}-V_{\mathrm{act}}-V_{\mathrm{ohm}}-V_{\mathrm{conc}}$$

(4)

$$E_{\mathrm{rev}}=1.229+4.3085\times {10}^{-5}T\left[ln\left(P_{{\mathrm{H}}_2}\right)+0.5ln\left(P_{{\mathrm{O}}_2}\right)\right]-0.85\times {10}^{-3}(T-298.15)$$

(5)

where

$E_{\mathrm{rev}}$ denotes the reversible voltage achieved in an open-circuit thermodynamic balance,

$P_{{\mathrm{H}}_2}$ and $P_{{\mathrm{O}}_2}$ represent the partial pressures of oxygen and hydrogen.

$$V_{\mathrm{act}}=-\left[{\beta }_1+{\beta }_{2}T+{\beta }_{3}Tln\left(C_{O_2}\right)+{\beta }_{4}Tln\left(I\right)\right]$$

(6)

where

$V_{\mathrm{act}}$: represents the activation voltage drop due to the slow reaction on the active surfaces of the cathode and anode,

${\mathrm{C}}_{{\mathrm{O}}_2}$ is the dissolved oxygen concentration at the cathode, and it can be calculated by Henry’s Law:

$$C_{O_2}={\displaystyle \frac{P_{O_2}}{5.08\times {10}^6}}exp\left({\displaystyle \frac{498}{T}}\right)$$

(7)

$$V_{\mathrm{ohm}}=I(R_m+R_c)$$

(8)

where

$V_{\mathrm{ohm}}$: stands for ohmic losses caused by the ionic resistance of the membrane and electrical resistances in the electrodes and interconnections.

$$V_{\mathrm{con}}=-Bln\left(1-{\displaystyle \frac{D}{D_{max}}}\right)$$

(9)

where $V_{\mathrm{conc}}$: corresponds to concentration losses that occur due to mass transport limitations at high current densities.

$D_{max}$ stands for the limiting current density,

D is the operating current density,

B is the semi-empirical factor associated with the PEMFC.

The voltage–power relationship of a Proton Exchange Membrane Fuel Cell (PEMFC) provides essential insight into its energy conversion efficiency and overall operational performance. It is typically obtained by multiplying the cell voltage $V_{cell}$ by the corresponding current $I_{cell}$ at each operating point, giving the instantaneous output power [27]:

$$P_{\mathrm{cell}}=V_{\mathrm{cell}}\times I_{\mathrm{cell}}$$

(10)

Figure 2 illustrates the relationship between the fuel cell voltage, power output, and current. As the current increases, the voltage decreases due to activation, ohmic, and concentration losses, while the power output initially rises, reaches a practical maximum, and then declines. The operating region near the maximum power point corresponds to optimal fuel efficiency and stable operation. Beyond this region, excessive current may cause unstable chemical reactions and reduced system performance [28].

2.2. Experimental Data and Input Variables

The data used in this study come from a Nexa 1200 W (Heliocentris, Academia GmbH, Berlin, Germany) proton exchange membrane fuel cell experimental system. This system comprises a complete module integrating the air management and hydrogen supply subsystems. It is equipped with instruments to measure in real time the main physical parameters required to characterize the cell’s operation.

The experimental bench presented in Figure 3 was designed to evaluate the performance of hydrogen fuel cells under electrical loads in controlled experimental conditions at the ISA laboratory of ENSA, Ibn Tofail University, Kenitra, Morocco. The variables measured include the current I, the output voltage V, the operating temperature T, and the hydrogen pressure $P_{H_2}$. These quantities were recorded at a high sampling frequency to capture the dynamic phenomena of the fuel cell during rapid load variations. The tests consisted of applying a variable current profile that reproduced the actual conditions of a hydrogen electric vehicle. This profile made it possible to observe the dynamic response of the voltage and collect a set of data representative of the electrochemical behavior of the system. The dataset contains 35,776 valid samples, divided into 70% for training, 15% for validation, and 15% for testing. Before training the model, preprocessing is applied, including outlier removal and variable normalization. The input variables selected for modeling are therefore current I (operating range: 1.69 A to 49.32 A) and temperature T (operating range: 18.07 °C to 56.55 °C), while cell voltage V is the output variable. The hydrogen pressure $P_{H_2}$ was not used as an input variable in the ANN model itself. Instead, it was monitored and utilized for comparison purposes to evaluate the hydrogen consumption before and after the integration of the ESC-ANN control strategy. These parameters were chosen for their direct influence on the electrochemical performance of the cell [29]. The setup includes:

• A NEXA 1200 PEM fuel cell module along with its monitoring software,
• Three metal hydride canisters from Heliocentris, each with a hydrogen storage capacity of 800 NL,
• A Nexa 1200 DC/DC converter,
• A BK Precision power supply was used to start the fuel cell,
• Hall effect sensors to take Voltage and current measurements,
• Two Metrix AX502 power supplies to power the Hall effect sensors,
• A programmable DC electronic load,
• A MicroLabBox-dSPACE DS1202 device running Control Desk software,
• Computer for data acquisition.

2.3. Modeling PEMFC Using an Artificial Neural Network

In our previous research [30], we developed an artificial neural network (ANN) model designed to accurately predict the voltage of a proton exchange membrane fuel cell (PEMFC) based on real experimental data from a Nexa 1200 W system. The ANN architecture consists of 4 hidden layers with 128, 64, 32, and 16 neurons respectively. The activation function used is the Rectified Linear Unit (ReLU) function. The model is trained using the ADAM optimizer, and the loss function employed is MSE. The Training was performed over 500 epochs with a batch size of 64.

The results showed an excellent correlation between the experimental and predicted values, with a coefficient of determination of 0.907, a root mean square error (RMSE) of 0.207 V, and a mean absolute percentage error (MAPE) of 1.05%, confirming the robustness of the proposed model for static battery modeling as shown in Figure 4.

In this article, this approach is extended and improved by integrating the ANN model into an Extremum Seeking Control (ESC) adaptive control strategy. This integration constitutes a significant development: the model is no longer used only for voltage prediction, but becomes an active component of the control loop. The objective is to enable the system to search for the optimal operating point in real time, ensuring minimum hydrogen consumption while maintaining voltage stability. The ESC–ANN combination enables the predictive accuracy of the neural model to be exploited to guide the online optimization algorithm, without requiring explicit knowledge of the physical model of the stack. This intelligent approach opens the way to adaptive, self-optimizing energy management suited to PEMFC found in hydrogen electric vehicles.

3. Intelligent Extremum Seeking Control of PEMFC

3.1. Principle of Extremum Seeking Control (ESC)

Extremum seeking control (ESC) is a real-time optimization approach that guarantees that a dynamic system functions at its optimal point without the need for an explicit mathematical model of the objective function. In order to push the system toward the extremum (minimum or maximum) of a specified cost, the controller continuously modifies the input signal, observes the corresponding changes in output, and modifies the control variable [31]. The goal of ESC in the case of a proton exchange membrane fuel cell (PEMFC) is to reduce hydrogen consumption while preserving the necessary voltage level, improving overall energy efficiency. The ESC algorithm uses a small perturbation added to the control variable (the fuel cell current in this work) to estimate the gradient of the performance function online. This approach is particularly well-suited to PEMFC systems, which feature highly nonlinear and time-varying dynamics that are difficult to model accurately. The general structure of an ESC loop is composed of [32,33]:

• A perturbation signal $d\left(t\right)=asin\left(\omega t\right)$ injected into the control input,
• A demodulation mechanism that extracts information about the gradient of the performance function,
• An adaptation law that updates the control input based on the estimated gradient.

Mathematically, the ESC principle can be described as follows [34]:

$$u\left(t\right)=\widehat{u}\left(t\right)+asin\left(\omega t\right)$$

(11)

where

$\widehat{u}\left(t\right)$ is the mean control signal,

a is the perturbation amplitude,

$\omega$ is the perturbation frequency.

$J\left(u\right)$ is the cost function to be minimized, depending on the control input u.

The corresponding measured output $y\left(t\right)=J\left(u\right(t\left)\right)$ is passed through a demodulator and a low-pass filter to estimate the gradient ${\displaystyle \frac{dJ}{du}}$. The adaptation law updates $\widehat{u}$ according to [35]:

$$\dot{\widehat{u}}\left(t\right)=-ky\left(t\right)sin\left(\omega t\right)$$

(12)

where k is a positive adaptation gain determining the convergence speed.

3.2. Inputs and Outputs of the ESC-PEMFC System

Over time, the mean control value $\widehat{u}\left(t\right)$ converges toward the optimal input $u^{*}$ that minimizes the cost function $J\left(u\right)$. The steady-state oscillations around this optimum remain small if the perturbation amplitude a and frequency $\omega$ are properly chosen. In the case of PEMFC applications, the objective function can be formulated as a combination of the voltage tracking error and the hydrogen consumption rate:

$$J=\alpha {(V_{\mathrm{ref}}-V_{\mathrm{pred}})}^2+\beta {\dot{m}}_{{\mathrm{H}}_2}$$

(13)

where

$V_{\mathrm{ref}}$ is the reference voltage,

$V_{\mathrm{pred}}$ is the predicted cell voltage obtained from the ANN model,

${\dot{m}}_{{\mathrm{H}}_2}$ is the instantaneous hydrogen mass flow rate,

$\alpha$ and $\beta$ are weighting coefficients that balance the trade-off between voltage stability and hydrogen economy.

By continuously adapting the current command $I_{\mathrm{FC}}$ to minimize J, the ESC ensures efficient operation of the fuel cell while avoiding hydrogen over-consumption. This data-driven and model-free nature of ESC makes it especially appealing for real-time control of nonlinear energy systems, where precise analytical models are hard to obtain or subject to uncertainty.

Figure 5 illustrates the operating principle of the proposed Extremum Seeking Control (ESC) loop integrated with the PEMFC. The controller is composed of three main stages: modulation, demodulation, and parameter update. In the modulation stage, a small sinusoidal perturbation $bsin\left(\omega t\right)$ is added to the mean current $\overline{I}\left(t\right)$ to explore the neighborhood of the optimal operating point. The demodulation stage extracts the gradient information of the cost function with respect to the input current. A low-pass filter removes high-frequency components to obtain a smooth estimation of the gradient $\xi \left(t\right)$. Finally, in the parameter update stage, the gradient signal is scaled by the adaptation gain $-k$ and integrated to update the mean control signal $\overline{I}\left(t\right)$. This allows the ESC to iteratively adjust the current toward the optimal value that minimizes hydrogen consumption while maintaining voltage stability. Through this closed-loop mechanism, the controller continuously drives the PEMFC toward its maximum efficiency point without requiring an explicit analytical model of the fuel cell’s nonlinear characteristics [36,37].

3.3. Integration of the ANN Model into the ESC Loop

In this work, the proposed Extremum Seeking Control (ESC) strategy is combined with an Artificial Neural Network (ANN) model of the PEM fuel cell for real-time optimisation of hydrogen. The ANN model provides accurate predictions of the fuel cell voltage $V_{\mathrm{pred}}$ based on measurable operating variables such as the current $I_{\mathrm{FC}}$, temperature $T_{\mathrm{FC}}$ [38].

$$V_{\mathrm{pred}}=f_{\mathrm{ANN}}(I_{\mathrm{FC}},T_{\mathrm{FC}})$$

(14)

This model replaces the complex electrochemical equations of the PEMFC within the ESC loop, significantly reducing computational cost while maintaining high prediction accuracy. At each control step, the ESC algorithm perturbs the input current $I_{\mathrm{FC}}$ with a small sinusoidal signal and evaluates the cost function J using the ANN-predicted voltage. The cost function is then demodulated and filtered to estimate its gradient with respect to the current. The updated control input is computed as:

$$I_{\mathrm{FC}}\left(t\right)=\widehat{I}\left(t\right)+asin\left(\omega t\right),\dot{\widehat{I}}\left(t\right)=-k\xi \left(t\right)$$

(15)

where $\xi \left(t\right)$ represents the estimated gradient of the cost function calculated as:

By embedding the ANN model into the ESC loop, the control strategy can continuously predict the PEMFC voltage under varying operating conditions and adjust the current accordingly to minimize hydrogen consumption while maintaining voltage stability [39].

The general working principle is described as follows:

• The ANN is trained to predict the fuel cell voltage V from two main inputs: the current I and temperature T. This approximation avoids the complexities and uncertainties associated with classical physical models.
• The cost function calculation: ESC aims to minimize a trade-off between hydrogen consumption $m_{H_2}$, computed via Faraday’s law as a function of current I, and the voltage tracking error ${(V_{\mathrm{pred}}-V_{\mathrm{ref}})}^2$. The trade-off is weighted by coefficients $w_{H_2}$ and $w_V$.
• Gradient estimation: To estimate the derivative of the cost with respect to the current command I, ESC evaluates the cost function at $I+\delta I$ and $I-\delta I$. These evaluations are performed via the ANN, which predicts the corresponding voltages $V(I+\delta I,T)$ and $V(I-\delta I,T)$.
• Command update: The estimated gradient is used to adapt the current command to minimize the cost. For clarity, Algorithm 1 presents an illustrative pseudocode of ESC-ANN integration.

Algorithm 1 Pseudocode of ESC-ANN integration

For each time step k

$I_k^{ref}\leftarrow$ reference current at step k

$T_k\leftarrow$ temperature at step k

If $k=1$$I_k\leftarrow I_k^{ref}$

Else $I_k\leftarrow I_{k-1}^{cmd}$

End If

$I_u\leftarrow min(I_k+\delta I,I_{max})$

$I_l\leftarrow max(I_k-\delta I,I_{min})$

$V_u\leftarrow \mathrm{ANN}.\mathrm{predict}(I_u,T_k)$

$V_l\leftarrow \mathrm{ANN}.\mathrm{predict}(I_l,T_k)$

Compute cost: $J_u$ and $J_l$

Estimate gradient: $dJ/dI={\displaystyle \frac{J_u-J_l}{2\delta I}}$

Update command: $I_{new}\leftarrow I_k-\alpha \times {\displaystyle \frac{dJ}{dI}}$

Saturate $I_{new}$ around $I_k^{ref}$ within ±5%

Smooth correction and set $I_k^{cmd}$

End For

The integration of the ANN within the ESC framework offers several key advantages. It eliminates the need for complex, uncertain physical parameter identification, enabling a simpler modeling approach. The ANN provides fast and smooth gradient estimation, which supports real-time optimization and enhances robustness against unmodeled dynamics and ageing effects. Additionally, it significantly reduces computational cost compared to traditional mechanistic fuel cell models and avoids the noisy gradients typically encountered in physical models, resulting in improved stability and reliability of the ESC optimization.

4. Implemented ESC-ANN Algorithm

4.1. ESC Controller Parameters

The proposed ESC algorithm was implemented in MATLAB R2022b using the ANN-based PEMFC model. The control objective is to minimize the instantaneous hydrogen consumption while ensuring voltage stability around a fixed reference value. The algorithm continuously adjusts the fuel cell current $I_{\mathrm{FC}}$ according to the estimated gradient of a composite cost function. The cost function J is defined as a weighted compromise between hydrogen flow and voltage tracking error, so Equation (13) becomes:

$$J=w_{{\mathrm{H}}_2}{\dot{m}}_{{\mathrm{H}}_2}+w_V{(V_{\mathrm{pred}}-V_{\mathrm{ref}})}^2$$

(16)

where $w_{{\mathrm{H}}_2}=1.0$ and $w_V=0.02$ are weighting factors that balance hydrogen economy and voltage regulation, respectively.

The current update is then obtained through a finite-difference approximation of the gradient:

$$\xi \approx {\displaystyle \frac{dJ}{dI}},I_{\mathrm{new}}=I_k-\alpha {\displaystyle \frac{dJ}{dI}},J=w_{{\mathrm{H}}_2}{\dot{m}}_{{\mathrm{H}}_2}+w_V{(V_{\mathrm{pred}}-V_{\mathrm{ref}})}^2$$

(17)

where $\alpha =0.15$ is the adaptation step controlling the convergence rate.

A low-pass filter with a temporal constant ${\tau }_f=10$ is applied to the corrective signal to preserve a smooth and physically realistic control action. To ensure complete consistency with the experimental current profile, the resulting control input $I_{\mathrm{cmd}}\left(k\right)$ is then limited to stay within ±5% of the reference current $I_{\mathrm{ref}}\left(k\right)$.

At each iteration, the ANN predicts the voltage $V_{\mathrm{pred}}$ corresponding to the updated current command, and the hydrogen flow rate ${\dot{m}}_{{\mathrm{H}}_2}$ is computed according to Faraday’s law:

$${\dot{m}}_{{\mathrm{H}}_2}={\displaystyle \frac{I_{\mathrm{FC}}{M}_{{\mathrm{H}}_2}}{2F}}$$

(18)

where $M_{{\mathrm{H}}_2}=0.002016\mathrm{kg}/\mathrm{mol}$ is the molar mass of hydrogen and $F=96,485.3\mathrm{C}/\mathrm{mol}$ is the Faraday constant.

4.2. Adaptation Parameters of ESC

The adaptation mechanism of the ESC algorithm uses a gradient-descent approach. The key adaptation and physical constraint parameters are summarized below in

Table 1. Adaptation parameters.

Parameter	Description	Value
$\alpha$: Adaptation gain	Step size for control update	0.15
$\Delta I$: Current increment	Finite-difference step for gradient estimation	0.2
$T_s$: Logical sampling time	Time interval for hydrogen accumulation	0.1
$I_{min}$: Minimum current	Lower operational limit of PEMFC	1.72
$I_{max}$: Maximum current	Upper operational limit of PEMFC	49.79

.

The key parameters of the ESC, namely the adaptation step $\alpha$, the gradient calculation increment $\Delta I$, and the correction margin around the real current profile (set at ±5%), were determined through an empirical approach based on trial-and-error combined with robustness validation of the system. The choice of $\alpha =0.15$ provides a compromise between fast convergence and limiting oscillations during the current command update. The increment $\Delta I=0.2$ ensures a reliable gradient estimation by finite differences while avoiding the introduction of noise in the calculation. The “profile-aware” constraint limits the current modification to a narrow band around the real profile, thus enhancing the stability and feasibility of the optimized commands.

4.3. Role of the ESC Algorithm in Hydrogen Optimization

The application of the suggested Extremum Seeking Control (ESC) approach is a crucial component in increasing hydrogen usage for fuel cell electric vehicles. The ESC controller finds and maintains the operating point that minimizes hydrogen consumption while maintaining voltage stability and dynamic performance by continuously modifying the fuel cell current in real time. The ESC mechanism actively investigates the nonlinear interplay between current demand, voltage, and hydrogen flow to identify the ideal operating conditions at each instant, in contrast to conventional control techniques that rely on predefined set points. When included in an energy management framework, this control layer allows the fuel cell to give only the power needed by the traction system, with the auxiliary source handling quick transients. As a result, overall hydrogen consumption is decreased, energy efficiency is increased, and system longevity is increased for hydrogen electric vehicles. Such intelligent control strategies are necessary to achieve economical and sustainable operation in hydrogen mobility. They open the door for energy management systems that can adjust to changing driving circumstances and progressive component deterioration while continuously maximizing the use of hydrogen. For next-generation zero-emission automobiles, the ESC algorithm is a promising step toward the implementation of robust, intelligent, and high-efficiency fuel cell systems [40,41].

5. Experimental Results and Analysis

5.1. Voltage Profile

The voltage predicted by the ANN model during ESC optimization was analyzed to evaluate the prediction performance under dynamic conditions. Figure 6 shows the evolution of the voltage predicted by the ANN under ESC. To quantify the accuracy of the model in real operation, the residual error, calculated as the difference between the voltage predicted by the ANN and the measured voltage, is plotted and presented in Figure 7. This error remains low overall and centered around zero. These results confirm the reliability of the ANN model for dynamic voltage estimation, a key element for successful energy optimization.

5.2. Current Profile

The evolution of the fuel cell current during the experimental cycle for both the real test profile and the ESC-optimized command is illustrated in Figure 8. The real current profile is derived from experimentally recorded data representing realistic operational conditions of the PEM fuel cell. The current is progressively increased every 5 min, reaching a maximum of 49.79 A, before decreasing in steps. Throughout the test, each step lasts 5 min, enabling the analysis of its electrical behavior under varying loads. The profile was selected to rigorously test the ESC-ANN control strategy’s robustness and adaptability. The ESC-optimized current closely follows the real profile within a ±5% margin, demonstrating effective current correction as designed. The ESC controller introduces small and smooth variations around the original current trajectory. These adaptive modifications balance voltage deviation and hydrogen consumption by minimizing the instantaneous cost function. The overall form and dynamics of the experimental current profile are completely maintained in spite of these changes, demonstrating that the ESC algorithm does not alter the electrical behavior of the system. The fuel cell can maintain steady output performance while operating closer to its local efficiency optimum thanks to this adaptive fine-tuning.

5.3. Convergence Behavior of the ESC–ANN Framework

To clearly demonstrate the convergence of the proposed ESC–ANN approach, the gradient signal $\xi \left(t\right)$ and the instantaneous cost function $J\left(t\right)$ are explicitly presented in Figure 9. The gradient signal represents the sensitivity of the cost function with respect to the control current as a key indicator of the extremum-seeking process. Its convergence toward zero confirms that the ESC algorithm reaches a stable operating point. Additionally, the evolution of the cost function illustrates the progressive reduction of the objective, which combines the minimization of hydrogen consumption and voltage regulation. The corresponding figures provide direct visual evidence of ESC convergence, validating the effectiveness of the proposed control strategy.

5.4. Instantaneous Hydrogen Flow and Cumulative Hydrogen Consumption

Figure 10 shows the instantaneous hydrogen flow of the PEMFC under the action of the ESC controller. Throughout the operating cycle, Both curves show the same general trend over the operating cycle, which reflects the fuel cell’s stepwise current evolution. However, the ESC-optimized profile exhibits slightly moderated hydrogen flow peaks and smoother transitions, especially when the current increases quickly.

This performance shows that the controller successfully reduces the instantaneous hydrogen peaks that usually happen during brief load variations, resulting in more reliable and effective fuel use. Instead of just reducing the flow of hydrogen, ESC improves recovery during current drops, limits overconsumption during high-demand phases, and enhances the fuel cell’s dynamic response.

Figure 11 illustrates the cumulative hydrogen consumption over the full test cycle for both the reference case and the ESC-controlled operation. The curve obtained with ESC shows a systematically gentler slope compared to the reference profile, indicating a continuous reduction in hydrogen usage throughout the cycle.

By the end of the test, the ESC strategy leads to an overall hydrogen saving of about 2.74%, computed according to [42].

$$\mathrm{Saving}={\displaystyle \frac{H_{2,\mathrm{ref}}-H_{2,\mathrm{ESC}}}{H_{2,\mathrm{ref}}}}\times 100$$

(19)

This improvement demonstrates how well the suggested ESC optimization technique reduces hydrogen usage. Even while the percentage of savings may seem modest, in real fuel cell electric vehicles, its cumulative effect becomes substantial over extended periods of operation. The outcomes show the controller’s potential for onboard energy management in hydrogen-powered systems by highlighting its capacity to maintain high energy efficiency and stability.

6. Conclusions

This work presents an intelligent optimization strategy based on extremum seeking control (ESC) applied to a PEMFC system for hydrogen electric vehicles. Using a voltage prediction model built from an artificial neural network (ANN) trained on actual experimental measurements, the controller continuously adapts the fuel cell current to reduce hydrogen consumption. The simulation results clearly demonstrate the effectiveness of the proposed approach, with an overall reduction of approximately 2.7% in hydrogen consumption, achieved without modifying the original current profile or compromising the dynamic response of the system. The integration of the ESC-ANN framework eliminates the need for explicit physical models, confirming the relevance of the ESC approach for the energy optimization of PEMFC systems, particularly in the context of hydrogen mobility, where consumption control and operational stability are essential to the reliability and overall efficiency of powered electric vehicles. Looking forward, future work will focus on expanding the framework to incorporate fuel cell aging and degradation models, aiming to improve long-term performance prediction and control robustness.