Machine Learning Prediction of Fuel Cell Remaining Life Enhanced by Variational Mode Decomposition and Improved Whale Optimization Algorithm

Abstract

In predicting the remaining lifespan of Proton Exchange Membrane Fuel Cells (PEMFC), it is crucial to accurately capture the multi-scale variations in cell performance. This study employs Variational Mode Decomposition (VMD) to decompose performance data into intrinsic modes, elucidating critical multi-scale dynamics vital for understanding the complex degradation processes in fuel cells. In addition to VMD, this research utilizes an Improved Whale Optimization Algorithm (IWOA) to optimize a Back Propagation (BP) Neural Network. The IWOA focuses on precise adjustments of weights and biases, enabling the BP network to effectively interpret complex nonlinear relationships within the dataset. This optimization enhances the predictive model’s reliability and stability. Extensive experimental evaluations demonstrate that the integration of VMD, and the learning capabilities of the IWOA-optimized BP network significantly improves the model’s accuracy and stability across multiple predictions, thereby increasing the reliability of lifespan predictions for PEMFCs. This methodology offers a robust framework for extending the operational life and efficiency of fuel cells.

1. Introduction

In recent years, the continuous increase in global energy consumption has led to the gradual exposure of issues such as resource depletion and environmental pollution associated with traditional fossil fuels. The energy crisis has become a key factor constraining the sustainable development of society [1]. Against this backdrop, clean and renewable energy technologies have garnered unprecedented attention. Proton Exchange Membrane Fuel Cells (PEMFC) are widely recognized for their high energy conversion efficiency, low emissions [2], and extensive applications, particularly in transportation [3,4]. However, their long-term operational stability and lifespan pose significant challenges [5,6,7], as performance degradation often occurs over time [8]. During operation, PEMFC undergo performance degradation, which manifests in the degradation of the membrane and deactivation of the catalyst [9]. Conducting research on the prediction of PEMFC degradation is of significant importance for reducing maintenance costs and facilitating the commercialization of this technology [10].

PEMFCs are complex nonlinear systems, the specific mechanisms of which are yet to be thoroughly investigated. Current primary degradation prediction methods can be categorized into model-based, data-driven, and hybrid forecasting approaches [11]. Model-based predictive methods rely on a deep understanding of the degradation mechanisms of PEMFC [12]. These methods utilize mathematical models to describe the physicochemical processes underlying the performance decay of the fuel cells. Mlakar et al. [13] developed a semi-empirical model that accounts for the performance degradation of PEMFCs during operation. The model, which integrated degradation effects, utilized data derived from accelerated stress tests. Khan et al. [14] introduced a dynamic semi-empirical aging model for PEMFCs, incorporating time-based elements to predict the degradation of the fuel cells. This model calculated the membrane water content in PEMFCs, enabling the diagnosis of membrane drying and flooding faults. Additionally, the model parameters were optimized using the Butterfly Optimization Algorithm. Semi-empirical models can also be integrated with filtering algorithms. Song et al. [15] proposed an online prediction method for the remaining useful life of power fuel cells, based on an Adaptive Extended Kalman Filter (AEKF). This method accounted for the degradation characteristics of vehicular fuel cells under actual operating conditions. Also, it dynamically updated the weights of operational and environmental factors utilizing AEKF, thereby enhancing prediction accuracy. Model-based predictive methods offer an intuitive understanding of the aging process and allow predictions to be made without extensive experimental data [16]. However, the accuracy of these models may be constrained by oversimplification or the failure to account for all relevant factors [17].

Data-driven approaches, particularly machine learning techniques, predict outcomes by learning patterns from historical data. These methods do not necessitate an in-depth physical understanding of the aging mechanisms but instead directly extract features from the data [18]. Among these techniques, the Back Propagation Neural Network (BPNN) has been extensively applied to the prediction of fuel cell aging. Huang et al. [19] analyzed the voltage characteristics of the start–stop process of PEMFC buses in actual traffic environments and proposed an Organic Grey BP Neural Network Model (OGNNM) to predict the start–stop voltage of PEMFCs. It provided a foundation for performance optimization and lifespan prediction of fuel cell vehicles. Commonly, optimizing BPNN parameters with optimization algorithms enhances model efficacy [20]. Chen et al. [21] proposed a predictive model that integrates a Back Propagation Neural Network (BPNN) with optimization algorithms. The model takes into account the impact of PEMFC current, hydrogen pressure, temperature, and relative humidity on the aging of PEMFCs. The optimization of the model parameters was carried out utilizing the Mind Evolutionary Algorithm (MEA), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA). Similarly, Zhao et al. [22] developed a Time Convolutional Network (TCN) model optimized with GA, which demonstrated strong accuracy, robustness, and adaptability in predicting PEMFC performance degradation. The advantages of data-driven methods include powerful pattern recognition capabilities, the ability to handle large datasets, and adaptability to complex nonlinear relationships [23,24].

Hybrid models, which combine the advantages of mechanistic models and data-driven approaches, provide more accurate predictions in certain application scenarios. Tian et al. [25] introduced a method that integrated a voltage recovery model with multi-kernel relevance vector machines. To further enhance prediction accuracy, Bayesian optimization algorithms were employed to optimize the weight coefficients of the kernel functions. This approach has been verified to achieve a prediction accuracy of up to 95.35%. Similarly, Wang et al. [26] combined a semi-empirical model of voltage loss with a machine learning method based on a sliding window approach. This integration utilized the predictions from the semi-empirical model to correct inputs to the machine learning method, effectively suppressing fluctuations in long-term forecasting. However, the complexity of hybrid models may increase the risk of overfitting, particularly when data availability is limited. In contrast, focusing solely on data-driven methods might be more advantageous for the rapid development and deployment of effective predictive models.

In response to the aforementioned challenges, this paper presents a VMD-IWOA-BP predicting method, with the following main innovations:

(1)

PEMFC stack voltage data are analyzed using Variational Mode Decomposition (VMD) to extract multi-scale features, enhancing the prediction of degradation by capturing subtle aging changes more effectively than traditional methods.

(2)

This study employs the Improved Whale Optimization Algorithm (IWOA) combined with a Back Propagation (BP) Neural Network to optimize network parameters, improving learning efficiency and fitting complex nonlinear relationships.

(3)

A comprehensive VMD-IWOA-BP prediction model has been developed, integrating multi-scale data analysis, parameter optimization, and neural network prediction. This approach enhances the model’s accuracy, reliability, and adaptability across different operating conditions and aging stages.

By employing this integrated approach, this paper offers a novel technical avenue for the prediction of PEMFC degradation, with the potential to achieve improved forecasting outcomes in practical applications. The structure of this paper is as follows: Section 2 initially presents the experimental data utilized, followed by the proposal of a PEMFC degradation prediction method based on VMD-IWOA-BP. Section 3 provides the experimental predictions of the presented method, with conclusions drawn in Section 4.

2. Predictive Model Construction

2.1. The Working Principle of PEMFC

PEMFCs efficiently convert chemical energy into electrical energy utilizing electrochemistry [27]. As shown in Figure 1, hydrogen is supplied to the anode and oxygen to the cathode. At the anode, hydrogen is split into protons and electrons. The protons pass through the Proton Exchange Membrane (PEM) to the cathode, while the electrons flow through an external circuit, generating electricity. At the cathode, protons, electrons, and oxygen react to produce water, the only emission, making the process environmentally friendly and highly efficient [28,29].

In this study, the RUL of the PEMFC is indirectly predicted by focusing on voltage degradation over time. The model is designed to forecast voltage decay utilizing historical operational data, with voltage decline serving as a key indicator of fuel cell health. As voltage decreases below a defined threshold, it signals the end of the fuel cell’s RUL, making it a reliable proxy for estimating the remaining operational lifespan.

2.2. Basic Principles

2.2.1. Variational Mode Decomposition (VMD)

VMD [30] is an adaptive signal processing technique utilized for decomposing complex signals into intrinsic oscillatory modes that span various time scales. The essence of VMD lies in the use of a complete set of sinusoidal basis functions, allowing the signal to be represented as a linear combination of these sinusoidal waves. This process enables the decomposition of the signal across multiple scales, facilitating multi-scale signal analysis.

VMD decomposes the voltage signal $U\left(t\right)$ into $K$ sub-signals ${\left\{u_k\right\}}_{k=1}^K$, with each sub-signal represented as an intrinsic mode function (IMF).

$$u_k=A_k\left(t\right)\mathrm{c}\mathrm{o}\mathrm{s}\left({\varphi }_k\right(t\left)\right)$$

(1)

Each IMF is characterized by an amplitude $A_k\left(t\right)$, time variable $t$, and phase ${\varphi }_k\left(t\right)$. VMD’s decomposition principle is grounded in the minimization of the following variational problem:

$$\left\{\begin{array}{c}\underset{\left\{u_k\right\},\left\{w_k\right\}}{\min }\{\sum _{k=1}^K{‖{\partial }_k\left(\delta \left(t\right)+{\displaystyle \frac{j}{{\omega }_k}}\right)u_k\left(t\right)e^{-j{\omega }_{k}t}‖}_2^2\}\\ s.t.{\sum }_{i=1}^K{u}_k\left(t\right)=f\left(t\right)\end{array}\right.$$

(2)

Each sub-signal $u_k$ is modulated around a specific central frequency ${\omega }_k$, involving the Dirac function $\delta \left(t\right)$, imaginary unit $j$, and partial derivative ${\partial }_k$. To ensure effective decomposition, VMD requires the sum of all modes to equate the original voltage signal $U\left(t\right)$:

$${\sum }_{k=1}^K{u}_k\left(t\right)=U\left(t\right)$$

(3)

The method addresses the variational model by introducing a Lagrange multiplier $\lambda \left(t\right)$ and a regularization parameter $\alpha$, ensuring the decomposition aligns with the original signal dynamics:

$$\begin{array}{c}L\left(u_k,{\omega }_k,\lambda \right)=\alpha {{\sum }_{k=1}^K‖{\partial }_k\left(\delta \left(t\right)+{\displaystyle \frac{j}{{\omega }_k}}\right)u_k\left(t\right)e^{-j{\omega }_{k}t}‖}_2^2\\ +{‖U\left(t\right)-{\sum }_{k=1}^K{u}_k\left(t\right)‖}_2^2+〈\lambda \left(t\right),U\left(t\right)-{\sum }_{k=1}^K{u}_k\left(t\right)〉\end{array}$$

(4)

Compared to traditional Empirical Mode Decomposition (EMD), VMD exhibits greater robustness to noise and disturbances. The adaptive mechanism of VMD ensures that each mode has a concentrated and finite bandwidth. This adaptability renders VMD highly flexible and accurate in processing nonlinear and non-stationary signals.

2.2.2. Improved Whale Optimization Algorithm (IWOA)

The Whale Optimization Algorithm (WOA) is a metaheuristic algorithm inspired by the social behavior of humpback whales, as introduced by Mirjalili et al. [31]. Similar to other heuristic algorithms, the essence of WOA primarily lies in the update of the whale positions or solutions. WOA is mathematically modeled on the unique bubble-net feeding behavior of humpback whales:

• Encircling prey

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

(5)

$$\stackrel{⃑}{X}\left(t+1\right)={\stackrel{⃑}{X}}^{\ast }\left(t\right)-A·\stackrel{⃑}{D}$$

(6)

In the WOA, the current best solution position, denoted as ${\stackrel{⃑}{X}}^{\ast }\left(t\right)$, is updated with each iteration. Here, the target prey of the humpback whale is modeled as the optimal solution to the problem at hand, where $t$ represents the current iteration number. The coefficients $A$ and $C$ are vector coefficients, and $\stackrel{⃑}{D}$ is the displacement vector moving towards the optimal solution. The formulas for calculating $A$ and $C$ are as follows:

$$A=2a{r}_1-a$$

(7)

$$C=2r_2$$

(8)

In the iterative process, the coefficient $a$ linearly decreases from 2 to 0. The variables $r_1$ and $r_2$ are random numbers generated within the interval $[0,1]$.

2.

Bubble-net attacking method (exploitation phase)

$$\stackrel{⃑}{X}\left(t+1\right)=\left\{\begin{array}{c}{\stackrel{⃑}{X}}^{\ast }\left(t\right)-A·\stackrel{⃑}{D},p<0.5\\ {\stackrel{⃑}{X}}^{\ast }\left(t\right)+\stackrel{⃑}{D}·e^{bl}·\cos \left(2\pi l\right),p\ge 0.5\end{array}\right.$$

(9)

In the model, $p$ is a random number within the range $[0,1]$, $l$ is a random number within the range $[-1,1]$, and $b$ is a constant defining the logarithmic spiral shape. The mathematical modeling encompasses two types of humpback whale hunting behaviors: the shrinking encircling mechanism and the spiral updating mechanism. Since these behaviors coexist in actual predation, the parameter $p$ is utilized to randomly select the updating mechanism.

3.

Bubble-net attacking method (exploitation phase)

When $A\ge 1$, it indicates that the humpback whale is outside the encircling ring, and thus, a random search update is employed.

$$\stackrel{⃑}{D}=\left|C·{\stackrel{⃑}{X}}_{rand}\left(t\right)-\stackrel{⃑}{X}(t)\right|$$

(10)

$$\stackrel{⃑}{X}\left(t+1\right)={\stackrel{⃑}{X}}_{rand}\left(t\right)-A·\stackrel{⃑}{D}$$

(11)

In the model, ${\stackrel{⃑}{X}}_{rand}\left(t\right)$ is the position of a randomly selected whale. Additionally, when $A<1$, the spiral encircling mechanism is employed.

Although widely utilized for its simplicity in structure, minimal parameter configuration, and ease of understanding and implementation, WOA faces certain limitations in practical applications. These include instability in the quality of the initial solutions and a tendency to become trapped in local optima. To address these issues, the following enhancement strategies are proposed in this paper:

• Improvement in initial population generation: The incorporation of chaos mapping techniques is introduced for generating the initial population. Among various chaos mappings, the Tent map is selected for its remarkable chaotic properties and ergodicity. The mathematical expression of the Tent map is defined as follows:

$$x_{i+1}=\left\{\begin{array}{c}2x_i,0\le x_i<0.5\\ 2\left(1-x_i\right),0.5\le x_i<1\end{array}\right.$$

(12)

This mapping generates a chaotic and highly random sequence of values within the interval [0, 1]. By repeatedly applying the Tent map, the initial population $\stackrel{⃑}{X}\left(0\right)$ for the WOA is generated.

$$\stackrel{⃑}{X}\left(0\right)={\left\{x_i\right\}}_{i=1}^N$$

(13)

The population size $N$ facilitates the Tent map’s ability to exhibit strong chaotic behavior for most initial values, offering an effective mechanism to help the algorithm escape local minima and explore the search space more comprehensively. Additionally, the Tent map evenly covers the entire available state space, thereby enhancing the diversity of the initial population in the WOA.

2.

Enhancement of global search capability: Simulated Annealing (SA) is employed to accept suboptimal solutions with a certain probability, preventing the algorithm from prematurely converging to local minima. Integrating the principles of SA into the WOA enhances its global search capabilities by allowing the acceptance of inferior solutions under controlled probabilities.

$${\stackrel{⃑}{X}}^{\prime }(t+1)=\stackrel{⃑}{X}\left(t+1\right)+\in ·∆$$

(14)

A random perturbation is applied to generate a new position, where $\in$ represents a small random number and $∆$ indicates the direction of the random vector. The difference in the objective function, $∆m(t+1)$, is calculated to determine the change in solution quality:

$$∆m(t+1)=m\left({\stackrel{⃑}{X}}^{\prime }(t+1)\right)-m\left(\stackrel{⃑}{X}\left(t+1\right)\right)$$

(15)

In the optimization process, the objective function values for the solutions, $m\left({\stackrel{⃑}{X}}^{\prime }(t+1)\right)$ and $m\left(\stackrel{⃑}{X}\left(t+1\right)\right)$, are evaluated. If $∆m(t+1)<0$, it indicates that the new solution ${\stackrel{⃑}{X}}^{\prime }(t+1)$ is superior to the original solution $\stackrel{⃑}{X}\left(t+1\right)$, thus ${\stackrel{⃑}{X}}^{\prime }(t+1)$ is accepted as the new current solution. Conversely, if the change in the objective function value is not favorable, a suboptimal solution may still be accepted based on the Metropolis criterion, with an acceptance probability $P_{t+1}$.

$$P_{t+1}=e^{{\displaystyle \frac{∆m(t+1)}{T_{t+1}}}}$$

(16)

In this context, $T$ represents the temperature in SA, which decreases with each iteration, progressively reducing the probability of accepting inferior solutions:

$$T_{t+1}=T_t·\alpha$$

(17)

This reduction follows an exponential decay function, where $\alpha$, a constant, controls the rate of temperature decline to prevent the algorithm from cooling too rapidly and potentially settling into a local optimum (see Algorithm 1).

Algorithm 1 Improved Whale Optimization Algorithm

1: procedure IWOA 2:  Initialize the parameters $a$, $A$, $C$, $l$, $p$ 3:  Generate initial population $\stackrel{⃑}{X}\left(0\right)$ utilizing Tent chaos map with Equations (12) and (13) 4:  Calculate the fitness of each search agent 5:  ${\stackrel{⃑}{X}}^{\ast }$: the best search agent 6:  $t$: 0 7:  while (t < MaxIterations) do 8:    foreach search agent ⃗ Ps do 9:        Update $a$, $A$, $C$, $l$, $p$ 10:      if ($p$ < 0.5) then 11:        if ($\left|A\right|<1$) then   /* Shrinking encircling mechanism */ 12:          Update the position of the current agent with Equations (5) and (6) 13:        else /* Search randomly for prey */ 14:          Select a random search agent ${\stackrel{⃑}{X}}_{rand}\left(t\right)$ 15:          Update the position of the current agent with Equations (10) and (11) 16:        end if 17:      else /* Spiral updating mechanism */ 18:        $\stackrel{⃑}{X}\left(t+1\right)=\stackrel{⃑}{D}·e^{bl}·\cos \left(2\pi l\right)+{\stackrel{⃑}{X}}^{\ast }\left(t\right)$ 19:      end if 20:    end for 21:    Generate a new position with Equation (14) 22:    Calculate the fitness of each search agent 23:    Calculate the difference $∆m(t+1)$ with Equation (15) 24:    if ($∆m(t+1)$ < 0) then 25:      Update the best search agent 26:    else 27:      Acceptance of inferior solutions under controlled probabilities with Equations (15)–(17) 28:    $t=t+1$ 29:  end while 30:  return ${\stackrel{⃑}{X}}^{\ast }(\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{I}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s})$ 31: end procedure

2.2.3. Back Propagation (BP) Neural Network

The Back Propagation (BP) algorithm is a commonly utilized supervised learning technique that optimizes neural network weights through gradient descent. Errors are minimized by adjusting weights and biases based on the gradients of the loss function relative to the network parameters. As shown in Figure 2, a basic neural network is often referred to as a multi-layer perceptron (MLP) or artificial neural network (ANN). It consists of three layers. The input layer receives the data. The hidden layer processes the data using weighted sums and nonlinear activation functions. Finally, the output layer generates the prediction or decision.

The BP algorithm adjusts the connection weights between layers utilizing stochastic gradient descent (SGD), allowing the network to learn the mapping between input and output data. Through training with historical data, the network can predict the aging state of new input data, providing valuable insights for fuel cell maintenance and lifespan prediction. This method captures complex nonlinear patterns in fuel cell degradation, enhancing both accuracy and reliability in aging predictions.

2.3. VMD-IWOA-BP Predictive Method

This paper presents a VMD-IWOA-BP-based prediction methodology for the remaining useful life (RUL) of PEMFC. This comprehensive approach integrates signal processing, algorithm optimization, and predictive analysis, significantly enhancing the accuracy and robustness of prediction outcomes. The detailed implementation process is illustrated in Figure 3.

As depicted in the green box of Figure 3, the input data, after being processed by VMD, is decomposed into several sets of Intrinsic Mode Functions (IMF), denoted as ${IMF}_K$. These IMFs not only capture the multi-scale characteristics of the original data but also provide abundant information for the subsequent prediction model. Subsequently, the IMFs are partitioned into training and testing sets according to a predetermined ratio, providing data support for the training and validation of the BP neural network model. During the model training process, as indicated by the red box in the figure, the IWOA is introduced to optimize the weights and thresholds of the BP neural network. In this context, the Mean Squared Error (MSE) is employed as the fitness function $m(\stackrel{⃑}{X}\left(t\right)$, serving as a measure to evaluate the efficacy of the solution vector $\stackrel{⃑}{X}\left(t\right)$ within the optimization process:

$$\left\{\begin{array}{c}m(\stackrel{⃑}{X}\left(t\right)={\displaystyle \frac{1}{n_t}}{\sum }_{i=1}^{n_t}{\left(U_{p_t-i}-U_{m_t-i}\right)}^2\\ \stackrel{⃑}{X}\left(t\right)=[W,b]\end{array}\right.$$

(18)

In this study, the weights $W$ and biases $b$ of the BP neural network are integrated into the population $\stackrel{⃑}{X}\left(t\right)$ of the WOA. Also, $n_t$ represents the number of the predicted output voltage in the train set. $U_{p_t-i}$ denotes the predicted output voltage of train set, and $U_{m_t-i}$ is the actual output voltage from the train set. The optimized BP model is trained on historical data to predict future output voltage components. Finally, the predicted voltage components are superimposed to obtain a comprehensive prediction result. This prediction process not only enhances the model’s ability to capture the degradation trends in PEMFC performance but also improves the accuracy and reliability of the prediction outcomes.

3. Results and Discussion

To validate the predictive accuracy of the proposed VMD-IWOA-BP method, publicly accessible datasets were selected as the basis for experimentation. The experimental process was conducted on the Matlab R2022b software platform, leveraging its robust computational and visualization capabilities to complete the simulation training. The methodological design of this study aims to provide a scientific and precise analytical tool for PEMFC remaining useful life prediction through rigorous experimental validation.

3.1. Data Sources and Evaluation Metrics

The dataset utilized in this study comes from the IEEE PHM 2014 Data Challenge, focusing on PEMFC durability analysis. As shown in Figure 4, the experiments involved two PEMFC stacks, each with five cells. The first stack, FC1, was tested under static conditions at a constant 70A current, while the second, FC2, underwent dynamic tests with high-frequency triangular waveform current variations around 70A. Weekly experiments, including polarization curve measurements and Electrochemical Impedance Spectroscopy (EIS), were conducted to monitor PEMFC performance changes, providing essential data for model construction and validation.

To verify the predictive accuracy of the proposed method, Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) are adopted as evaluation metrics for prediction performance. The corresponding formulas are as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(U_{p-i}-U_{m-i}\right)}^2}$$

(19)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|U_{p-i}-U_{m-i}\right|$$

(20)

Here, $n$ represents the number of data points in the test set, $U_{p-i}$ denotes the predicted output voltage of test set, and $U_{m-i}$ is the actual output voltage from the test set. RMSE measures the standard deviation of the differences between predicted and actual values, reflecting the magnitude of prediction errors and imposing a heavier penalty on larger errors. Consequently, it serves as a more stringent performance evaluation metric. MAE, on the other hand, represents the average of the absolute values of prediction errors, which is unaffected by the direction of the errors, providing an unbiased measure of error. Together, these two indicators comprehensively assess the predictive accuracy of the model. Generally, smaller values of RMSE and MAE indicate better prediction performance of the model.

3.2. Data Processing and Decomposition

In conducting data analysis and model prediction, ensuring the quality of raw data is paramount. Given that the original dataset contains noticeable noise and outliers, these factors can adversely impact the accuracy of model predictions. To enhance data usability and precision, this study adopts various data preprocessing techniques, including the moving average algorithm, to optimize the dataset.

As depicted in Figure 5, the preprocessed data, when compared to the raw data, exhibits a clearer and smoother curve while preserving the original degradation trends. This processing not only improves data readability but also provides a more robust foundation for subsequent model construction and prediction. Notably, Figure 5 reveals that FC1 exhibits a relatively gradual decay trend, whereas FC2 demonstrates a steeper downward trend accompanied by more extreme values that deviate from the overall trend to some extent. These observations underscore the pivotal role of data preprocessing in uncovering inherent data patterns and enhancing the performance of predictive models.

The VMD method is applied to the preprocessed data to extract its IMFs. During the VMD process, the modal number $K$ is set to 8, decomposing the data into eight sets of IMFs. These IMFs not only capture the multi-scale characteristics of the data but also provide abundant information for model training and validation. To further optimize model performance, these IMFs are meticulously partitioned into training and testing sets. Subsequently, to effectively mitigate the interference of outliers and enhance the model’s generalization ability, a normalization process is performed on the IMFs. Normalization serves as a crucial step, unifying the numerical scales across different IMFs, which not only improve computational efficiency but also optimize algorithm performance. Additionally, normalization contribute to bolstering the model’s generalization capabilities, mitigating biases that might arise from improper data initialization during model training.

Taking FC1 as an example, Figure 6 presents the normalized voltage IMF components of the test set. As shown in Figure 6, the normalized IMF components retain their original trends while demonstrating a more consistent numerical distribution. This results in a more reliable data foundation for subsequent model training and prediction, improving the overall accuracy and stability of the model’s performance.

As observed in Figure 6, each component exhibits distinct characteristics and behaviors. Specifically, $IM{F}_1$, acting as the trend component, demonstrates relatively stable fluctuations, primarily reflecting the overall trend in voltage variation in fuel cell FC1. This indicates that $IM{F}_1$ captures the long-term changes and primary trends within the data, providing crucial information for understanding the macro-level variations in fuel cell performance.

In contrast, $IM{F}_2$ through $IM{F}_8$ exhibits significantly larger fluctuations, primarily reflecting short-term variations and high-frequency changes within the data. These high-frequency components may be associated with instantaneous load changes, environmental factors, or microscopic reactions within the fuel cell during operation. Their variations may be more intricate and difficult to predict, yet they offer abundant detailed information for a profound understanding of the dynamic characteristics of fuel cell performance.

A meticulous analysis of these IMF components enables a more comprehensive grasp of the multi-scale features of fuel cell performance variations. This multi-scale approach facilitates the development of more accurate prediction models for battery performance degradation and provides a scientific basis for fuel cell health management and lifetime prediction.

3.3. Degradation Prediction under Different Predictive Methods

The structure of the BP network utilized in this study is shown in

Table 1. Parameters of BP network.

Parameters	Value
Learning rate	0.01
Activation	sigmoid
Number of input layers	5
Number of hidden layers	10
Number of output layers	1

. The network consists of five input layers, ten hidden neurons, and one output layer. The hidden layer utilizes a sigmoid activation function to capture nonlinear relationships. A learning rate of 0.01 ensures steady learning and stable convergence, balancing model performance and preventing overfitting.

Table 2. Comparison of different prediction models.

Feature	This Paper	DNM [33]	SDN [34]	FD3 Framework [35]
Problem domain	Fuel cell aging, RUL prediction	General prediction and classification	Greenhouse time series, seasonal prediction	Carbon emissions prediction
Model	BP	DNM	SDN	DNM
Decomposition	VMD	None	STD	CEEMD
Optimization algorithm	IWOA	BBO\PSO\GA \ACO\ES\PBIL	None	None

compares various prediction models and their applications. The presented VMD-IWOA-BP model is specifically designed for the complex task of fuel cell aging prediction, which requires capturing long-term degradation and nonlinear dynamics. In contrast, models like the Dendritic Neuron Model (DNM) [33] are designed for general prediction tasks. Single Dendrite Neuron (SDN) [34], which utilizes Seasonal-Trend Decomposition (STD), is more suitable for handling seasonal data. The FD3 framework [35], utilizing Complete Ensemble Empirical Mode Decomposition (CEEMD), is applied to carbon emissions prediction. These models are more suited for simpler, short-term prediction tasks. While SDN and FD3 utilize decomposition techniques, they focus on short-term trends and lack global optimization, limiting their ability to handle the complexity of fuel cell degradation. By combining VMD for multi-scale feature extraction with IWOA for global optimization, the VMD-IWOA-BP model offers superior accuracy and adaptability for long-term fuel cell prediction.

The WOA is selected for this study due to its superior performance, as shown by the experimental results. As illustrated in Figure 7, WOA consistently demonstrates faster convergence and achieves lower fitness values when compared to other optimization methods such as Genetic Algorithm (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO). These findings indicate that WOA is particularly effective in identifying optimal solutions early in the iteration process, resulting in a quicker convergence.

Figure 7 reveals that the initial fitness value of the IWOA is lower compared to WOA. This is attributed to the application of chaotic mapping in IWOA, which ensures a more uniform distribution of the initial population. By enhancing the diversity of the population, chaotic mapping allows the algorithm to start closer to optimal solutions, thereby accelerating the convergence process from the very beginning.

The results in

Table 3. RMSE of different optimization algorithms.

Dataset	VMD-GA-BP	VMD-DE-BP	VMD-PSO-BP	VMD-WOA-BP	VMD-IWOA-BP
FC1	0.0112	0.0120	0.0110	0.0102	0.0021
FC2	0.0066	0.0081	0.0038	0.0030	0.0024

demonstrate that under the same experimental settings, WOA outperforms other traditional optimization algorithms in terms of prediction accuracy. Therefore, WOA is selected as the preferred optimization algorithm for this task due to its superior performance.

Simulations are conducted utilizing BP, VMD-BP, VMD-WOA-BP, and VMD-IWOA-BP models, with the same dataset trained on 60% of the data as shown in Figure 8. The results from Figure 8 reveal that the predictive errors of the VMD-IWOA-BP model are significantly lower than those of the other methods, demonstrating its superior predictive capabilities. In the comparison depicted in Figure 8, the conventional BP network exhibits relatively large prediction errors due to the lack of effective data preprocessing and parameter optimization mechanisms. Although the VMD-BP model utilizes VMD technology to decompose the data, enhancing its representation capability, it still employs traditional methods for parameter optimization, which do not fully exploit its potential.

Table 4. Prediction performance metrics of FC1.

Method	RMSE	MAE	Time (s)
BP	0.0237	0.0139	0.9062
VMD-BP	0.0136	0.0106	8.7393
VMD-WOA-BP	0.0102	0.0073	48.9033
VMD-IWOA-BP	0.0021	0.0009	56.9953

and

Table 5. Prediction performance metrics of FC2.

Method	RMSE	MAE	Time (s)
BP	0.0125	0.0082	0.9354
VMD-BP	0.0065	0.0034	7.8048
VMD-WOA-BP	0.0030	0.0015	43.8735
VMD-IWOA-BP	0.0024	0.0009	56.9243

demonstrate the superior performance of the VMD-IWOA-BP model compared to other models, as evidenced across two distinct datasets. This discrepancy highlights the model’s robust adaptability and versatility. Particularly notable is the model’s performance on the FC1 dataset, where there was a significant reduction in prediction errors, with RMSE decreasing by 91% and MAE by 93%. Such improvements represent a substantial leap in predictive accuracy.

On the FC2 dataset, the model also demonstrated superior predictive capabilities, with reductions in RMSE and MAE by 81% and 91%, respectively. This further confirms the robust performance of the VMD-IWOA-BP model in processing various types of fuel cell data.

In this study, both IWOA and WOA are set with the same maximum number of iterations (50). However, the function evaluations differ due to the introduction of the Simulated Annealing (SA) mechanism in IWOA. This mechanism allows for additional evaluations between the optimal and suboptimal solutions, enhancing the global search capability by preventing the algorithm from becoming trapped in local optimum. The analysis shows that while the VMD-IWOA-BP model offers the best accuracy, it comes at a higher computational cost, with runtimes increasing from under 1 s (BP) to approximately 57 s. However, the significant reduction in RMSE (from 0.0237 to 0.0021 for FC1) justifies the increased computational time in applications where precise predictions are critical. The time increase from VMD-WOA-BP (48 s) to VMD-IWOA-BP (57 s) is moderate. This results in a substantial boost in accuracy, making the added computational cost worthwhile when balancing accuracy with processing time. Thus, for high-accuracy demands, the improved model’s efficiency remains reasonable.

To better validate the effectiveness of the presented IWOA, an ablation study was conducted. Based on the VMD-WOA-BP model, two modifications are tested separately: the addition of Tent mapping (WOA-Tent) and the inclusion of Simulated Annealing (WOA-SA). The RMSE results from these variations are presented in

Table 6. RMSE of ablation study.

Setup	FC1	FC2
WOA	0.0102	0.0030
WOA-Tent	0.0055	0.0028
WOA-SA	0.0040	0.0027
WOA-Tent-SA	0.0021	0.0024

.

The results of the ablation study clearly illustrate the impact of incorporating the Tent map and SA into the WOA. The baseline WOA model shows high RMSE values of 0.0102 for FC1 and 0.0030 for FC2, indicating its limitations in accurately capturing the complex degradation patterns. Adding the Tent map (WOA-Tent) leads to a significant reduction in RMSE, particularly for FC1 (0.0055), by improving the diversity of the initial population and enhancing the exploration process. Similarly, the use of Simulated Annealing (WOA-SA) further reduces the RMSE, with FC1 achieving 0.0040, demonstrating that SA enhances global search capabilities and prevents premature convergence.

To more clearly demonstrate the advantages of the method presented in this paper,

Table 7. Performance compared with published methods.

Method	FC1	FC2
AEKF-NARX [36]	0.0095	0.0085
Transformer-LSTM [11]	0.0038	0.0041
VMD-IWOA-BP	0.0021	0.0024

presents the RMSE metrics from various studies, where the training set constitutes 60% of the data. Among the comparative approaches, the AEKF-NARX hybrid method [36] combines the strengths of both model-based and data-driven techniques. By leveraging the physical degradation characteristics of fuel cells and the capability of neural networks to process nonlinear time series data, this approach effectively predicts both overall degradation trends and finer details. The Transformer-LSTM hybrid method [11] integrates a Wiener process to model stochastic degradation trends with a Transformer network for capturing long-term dependencies and patterns in the data. Monte Carlo dropout is employed to quantify prediction uncertainty, offering a confidence interval for the results. This combination enables a more robust and adaptable approach to degradation prediction, capturing both global trends and local fluctuations in performance.

The comparative analysis in Table 7 demonstrates the superior prediction accuracy of the VMD-IWOA-BP model. For FC1, the RMSE decreases from 0.0095 with AEKF-NARX and 0.0038 with Transformer-LSTM to 0.0021 with the proposed model. Similarly, for FC2, the RMSE drops from 0.0085 (AEKF-NARX) and 0.0041 (Transformer-LSTM) to 0.0024. These results highlight the model’s ability to capture both long-term degradation trends and finer-scale fluctuations, outperforming other methods in accuracy.

4. Conclusions

This study employs VMD technology to extract multi-scale features from historical data of PEMFC, providing a detailed revelation of the subtle changes occurring during the PEMFC aging process. Compared to traditional single-scale analysis methods, this technique significantly enriches the data foundation for aging prediction. Additionally, the IWOA is utilized to optimize the parameters of the BP neural network. This combined strategy not only enhances the network’s learning efficiency but also significantly improves the model’s ability to fit complex nonlinear relationships. By integrating the global search capability of IWOA with the learning mechanism of the BP neural network, an effective method for parameter adjustment and network training is provided for the PEMFC aging prediction model. Furthermore, by developing a comprehensive VMD-IWOA-BP prediction model, this research effectively combines multi-scale data analysis, parameter optimization, and neural network prediction. This innovative approach to model construction not only increases the accuracy and reliability of aging predictions but also enhances the model’s adaptability to different operational conditions and aging stages. Experimental results demonstrate that the model achieves RMSE of 0.0021 and 0.0024 under dynamic and static conditions, respectively, showcasing its substantial potential for practical applications.

Although this study demonstrates the efficacy of the VMD-IWOA-BP model, certain limitations need to be addressed. The model’s performance heavily depends on the quality and quantity of the input data, and in scenarios where data are limited, the risk of overfitting may increase. Future research could focus on mitigating this risk by incorporating more advanced regularization techniques and exploring transfer learning to leverage data from related systems.

Additionally, the computational cost of combining VMD with IWOA may pose challenges for real-time applications. Future work should investigate more efficient algorithms or hybrid models to reduce the computational load while maintaining high accuracy. Broader implications of this research include the potential to adapt the model to other fuel cell types or energy storage systems, extending its applicability across various fields.