Degradation prediction of fuel cell systems based on different operating conditions in dynamic cycling condition

Abstract

In this paper, the degradation of PEMFC under different operating conditions in dynamic cycle condition is studied. Firstly, according to the failure mechanism of PEMFC, various operating conditions in dynamic cycle condition are classified, and the health indexes are established. Simultaneously, the rates and degrees of the output voltage decline of the PEMFC under different operating conditions during the dynamic cycling process were compared. Then, a model based on variational mode decomposition and long short-term memory with attention mechanism (VMD-LSTM-ATT) is proposed. Aiming at the performance of PEMFC is affected by the external operation, VMD is used to capture the global information and details, and filter out interference information. To improve the prediction accuracy, ATT is used to assign weight to the features. Finally, in order to verify the effectiveness and superiority of VMD-LSTM-ATT, we respectively apply it to three current conditions under dynamic cycle conditions. The experimental results show that under the same test conditions, RMSE of VMD-LSTM-ATT is increased by 56.11 % and MAE is increased by 28.26 % compared with GRU attention. Compared with SVM, RNN, LSTM and LSTM-ATT, RMSE of VMD-LSTM-ATT is at least 17.26 % higher and MAE is at least 5.65 % higher.

1. Introduction

Different from wind energy, solar energy, and other new energy sources, hydrogen energy is not affected by regional limitations, time limits, volatility, randomness, and intermittent. It is considered to have broad prospects [1,2]. Proton exchange membrane fuel cell (PEMFC), as a hydrogen energy conversion technology [3,4], has been widely used in stationary, portable, military, transportation, and other industries.

PEMFC has many advantages [5]. First of all, its reaction product is water, so it can greatly reduce pollution compared to traditional internal combustion engine structures. Secondly, in the process of power generation, PEMFC does not involve the combustion of hydrogen and oxygen, and is not restricted by the Carnot cycle, so the energy conversion efficiency is high [6]. In addition, its basic structure is unit modular, so the layout is flexible, and it is easy to assemble multiple layers to meet the needs of high-power usage. But PEMFC also has shortcomings in cost, performance, and life, and these shortcomings have serious constraints on its commercial application. At present, a large number of researches on reducing cost, improving performance, and extending life have emerged, including the replacement of component materials and the improvement of system structure. However, due to the complex reaction mechanism of PEMFC, numerous internal degradation factors, and the diversity and uncertainty of external environmental factors, it is difficult to optimize the component materials and system structure.

The research of PEMFC based on prognostic and health management (PHM) technology is also one of the methods to promote its commercialization, and is currently a research hotspot in related fields [7]. The technology uses historical data to assess the health status of PEMFC, which can be used to guide appropriate decisions, not only to extend the useful life [8] of PEMFC, but also to improve its utilization. Therefore, using PHM technology to promote the long-term and safe operation of PEMFC has far-reaching significance and research value.

The health status assessment of PEMFC based on PHM technology is mainly to predict the future degradation trend of PEMFC. At present, the related research mainly focuses on the degradation prediction under constant conditions or simple dynamic conditions, and lacks the trend prediction of PEMFC under vehicle dynamic cycle condition. The vehicle dynamic cycle condition simulates the daily driving of the vehicle in the urban and suburban areas. According to the probability of the actual operating conditions such as idle speed, high load, and variable load in daily driving, it combines each operating condition in a certain proportion to obtain the single cycle condition, and then circulates the single cycle condition to obtain the dynamic cycle condition. Therefore, compared with the other two conditions, the vehicle dynamic cycle condition is closer to the real condition, and the research on PEMFC degradation under this condition is more realistic. Therefore, studying the performance of PEMFC under dynamic cycling conditions is an essential approach to enhancing their durability.

In the study of degradation performance of PEMFC under dynamic cycling conditions, related research directly samples and predicts based on the collected data [9]. This performance research method is challenging to obtain an intuitive trend of performance degradation. It also overlooks the differences in the rate and extent of the output voltage decline of the PEMFC under different operating conditions during the operation process. In practical applications, the durability of PEMFC is significantly affected by these complex operating conditions. Studying the trend of PEMFC output voltage under various conditions within dynamic cycling is of great importance, as it can provide strong support for the implementation of optimized control decisions. Therefore, to study the degradation trends of PEMFC under different operating conditions, this paper will establish health indices that can characterize the degradation trends of PEMFC under corresponding operating conditions, and predict the health indices for each operating condition. Additionally, this paper will compare the rate and extent of the output voltage decline of the PEMFC under different operating conditions during dynamic cycling, providing support for the implementation of optimized control decisions.

Because PEMFC is a nonlinear, strongly coupled electrochemical system, its dynamic characteristics involve the fields of electrochemistry, fluid mechanics, thermodynamics, and so on [10]. Therefore, it is very difficult to accurately describe its state, and it is particularly important to choose the appropriate degradation prediction method. The degradation prediction methods of PEMFC include model-driven methods, data-driven methods, and hybrid methods. The model-driven methods abstract the degradation process of fuel cells by mathematical expressions based on the empirical knowledge and establish a prediction model. Commonly used methods include the Kalman filter (KF), the particle filter (PF) [[11], [12], [13]], and their variants. For example, Bressel et al. [14] established the nonlinear state space with the extended Kalman filter according to the empirical formula of the polarization curve, and used it to describe the health state of PEMFC. Mao et al. [15] used the multi-particle filter to predict the parameters of the degradation model, and thus to predict the performance change of PEMFC. The data-driven methods use the historical operation data and various auxiliary information data of PEMFC to predict its future degradation trend. These methods do not rely on mechanism knowledge in the prediction process [16], and obtain the nonlinear relationship between input information and fuel cell performance by training the model, so they are highly portable. A large number of machine learning methods have been proved to be effective in predicting the health state of PEMFC. Such as back propagation neural network (BPNN) [17], adaptive neuro-fuzzy inference system (ANFIS) [18], echo state network (ESN) [19,20], nonlinear autoregressive exogenous (NARX) neural network [21], wavelet neural network (WNN) [22], time delay neural network (TDNN) [23], cycle reservoir with jump (CRJ) [24], deep neural network (DNN) [25,26], recurrent neural network (RNN) [[27], [28], [29]], Long short-term memory (LSTM) network [30], etc. Hybrid methods combine model-driven methods with data-driven methods to obtain more accurate predictions [31]. For example, Zhou et al. [32] combined the NAR model and the voltage degradation model using the sliding window technique. Wang et al. used the prediction results of the semi-empirical model to modify the input of the data-driven model to improve the oscillation of the long-term degradation prediction results of the data-driven model [33]. Wu et al. combined the global degradation trend predicted by a physics-based aging model considering voltage recovery with the local dynamic characteristics predicted by a data-driven approach, effectively improving the prediction performance [34]. Wang et al. propose a hybrid approach based on a deep neural network model. They use the Monte Carlo dropout method and a sparse autoencoder model to analyze the power degradation trend of PEMFC stacks [35]. Peng et al. use the internal impedance extracted by electrochemical impedance spectroscopy to identify the internal recovery effect, use the particle filter to predict the trend, and use the random forest to predict the fluctuation. By adopting this hybrid approach, Peng et al. conducted both long-term and short-term health assessments and predictions for fuel cells [36]. Ma et al. use EKF and LSTM methods to predict output voltage and aging parameters, and perform degradation prediction under dynamic conditions [37].

Model-based methods offer fast prediction speeds but are only suitable for PEMFC predictions under simple conditions. Their predictive performance often deteriorates under dynamic or compound conditions. The data-driven methods can improve the model flexibly for different situations, which are the mainstream method in the field of fuel cell health state prediction. However, for specific problems, such methods still require a deep understanding of the data to adjust the model structure accordingly, ensuring that the model can make the most of the data characteristics. Hybrid methods, by integrating model-driven approaches with data-driven methods, enhance predictive accuracy while simultaneously increasing the computational demands and instability inherent to the models. Additionally, it also faces the same issues as model-based methods, namely the difficulty in establishing accurate models for complex operating conditions. Therefore, to fill the gap in accurately predicting the voltage change trends under different operating conditions in dynamic cycling, this paper will establish a data-driven predictive model based on signal processing and recurrent neural networks, tailored to the working principles of the fuel cell and the characteristics of the data.

During the PEMFC operation, the start-stop operation will cause the hydrogen-air interface at the anode of PEMFC [38], which will lead to the oxidation reaction at the anode and cathode of PEMFC at the same time, and then form a high potential on the surface of the cathode catalytic layer. At high potentials, a reverse current occurs in the area where the air is present inside the anode, at which point the voltage of the stack transiently recovers. To obtain the overall downward trend of the output voltage under different operating conditions in dynamic cycling, and to accurately capture the local large fluctuation signals caused by external (start and stop) operating factors, in this paper, the output voltage is decomposed by variational mode decomposition (VMD), and the decomposition stage is determined by the center frequency method. The time-frequency resolution of VMD is very strong. It can correctly separate the components with similar frequencies, avoid mode aliasing, and obtain the sub-signal waveform with obvious characteristics. The prediction and reconstruction of each sub-signal are beneficial to improve the prediction accuracy. Each obtained sub-signal corresponds to a physical background, and the final sub-signal is called the residual function, which represents interference information such as noise. Therefore, by ignoring the residual function, the filtered output voltage can be obtained, and this operation is also beneficial to predict the future trend of PEMFC. LSTM network is a commonly used time series prediction model. It can deal with vanishing and exploding gradients well and has long-term memory for sequential data. It deeply mines the features in the hidden state according to the sequence information. But because each feature has different effects on the output, the direct output will lead to large prediction errors. Therefore, this paper introduces the attention mechanism (ATT) to give specific weight coefficients to the hidden state at each moment, which can improve the prediction accuracy of the model. To sum up, to fill the gap in accurately predicting the voltage trends under different operating conditions in dynamic cycling, this paper will integrate the modal decomposition capabilities of VMD, the sequence modeling capabilities of LSTM networks, and the weight allocation strategies of attention mechanisms. This approach aims to reveal the performance changes of PEMFC across different time scales under various operating conditions, thereby enhancing the accuracy of voltage predictions for PEMFC under diverse working conditions.

In summary, to promote the long-term and safe operation of PEMFC and make maintenance strategies in a timely and more reasonable way. In this paper, health indexes based on different operating conditions in the dynamic cycle condition are established, and the influence of different operating conditions on the degradation trend of PEMFC is compared. At the same time, the VMD-LSTM-ATT model is proposed and applied to predict the future trend of PEMFC under each operating condition in the dynamic cycle condition. The main contributions of this paper are as follows.

• 1.For the first time, a classification of operating conditions within dynamic cycling has been proposed, along with the establishment of health indices that reflect the degradation trends of PEMFC under these specific conditions, thereby enabling a more detailed study of PEMFC degradation.
• 2.For the first time, a comparative analysis from the perspective of degradation prediction has been conducted to analyze the rate and extent of the output voltage decline of the PEMFC under different operating conditions during dynamic cycling, providing support for the implementation of optimized control decisions.
• 3.Integrating the modal decomposition capability of VMD, the sequence modeling capability of LSTM networks, and the weight allocation strategy of attention mechanisms, the proposed method reveals the performance variations of PEMFC across different time scales under various operating conditions. This enhances the precision of voltage prediction for PEMFC under diverse operating conditions.

In the second section, this paper describes the durability test dataset of PEMFC under dynamic cycling condition, the establishment of health indicators, and the analysis of degradation trends under different operating conditions. In the third section, the degradation trend prediction model of PEMFC based on VMD-LSTM-ATT is established. In the fourth section, the proposed algorithm is applied to predict the future trend of the health indicators under different operating conditions in the dynamic cycling condition. In the fifth section, the conclusion is given.

2. Establishment of health indicators for PEMFC

2.1. Data set

Vehicles drive at different speeds on different road surfaces and in different traffic environments, and their endurance performance varies greatly. The New European Driving Cycle (NEDC) is a European standard for driving test conditions. It consists of four urban cycles and one suburban cycle, representing typical daily driving conditions. Zuo et al. conducted a durability accelerated stress test (AST) using commercial PEMFC single cell based on NEDC [39]. The test contains 3076 cycles, each of which has 1180 s and a total of 1008 h. The test bench includes the control part and observation part, and the important parameters in the test process are in Table 1.

Table 1.

Parameter Settings for durability testing.

Parameter [Unit]	Value
Active surface [cm2]	25
Temperature [°C]	85
Hydrogen pressure [kpa]	110
Hydrogen relative humidity [%]	50
Air pressure [kpa]	110
Air relative humidity [%]	80

In the durability test, start-stop operations were performed every 50 h and polarization curve data were measured every 100 h. The polarization curve is shown in Fig. 1(a). The current load and measured output voltage values for one cycle are shown in Fig. 1(b). The current setpoint values in Fig. 1(b) refer to the current load applied to the PEMFC, and the corresponding voltage value of the current load is also given in the figure. The voltage transient response resulting from each load change is an inherent characteristic of fuel cells during dynamic load operation [9]. According to the statistics of single cycle data, it can be found that there are 9 kinds of load currents in a single cycle. The corresponding time length and proportion of each load current are in Table 2.

Fig. 1.

Polarization curve and dynamic cycle single-cycle definition.

Table 2.

Time length and proportion of each load current in one cycle.

Load current [A]	Length of time [s]	Proportion [%]
0	35	2.97
1.78	463	39.24
4.45	52	4.41
9.51	136	11.53
10.40	100	8.47
14.85	159	13.47
20.75	142	12.03
29.65	49	4.15
35.60	44	3.73

2.2. Establishment of health indicators

In the research field of PEMFC degradation prediction, the establishment of health indicators is mainly carried out from two perspectives. One is to select degradation indicators that can reflect the aging degree of key components. And the other is to select degradation indicators that can reflect the overall performance of PEMFC. The reason why aging indicators of key components can serve as health indicators is that during the aging process of a PEMFC, certain parameters or characteristics of the key components change regularly with aging time and operating conditions. Therefore, users can understand the aging state of the key components by observing the changes in these parameters or characteristics, and thus obtain the health status of the PEMFC. For instance, during the aging of the electrode in a PEMFC, Pt catalyst particles undergo dissolution and coarsening, which leads to a reduction in the Electrochemically Active Surface Area (ECSA). Simultaneously, the carbon support layer may be subject to corrosion, diminishing the number of attachment points for the Pt catalyst particles, further contributing to the decline in ECSA. Consequently, by monitoring the degradation of ECSA, users can gain an accurate understanding of the electrode's aging condition. However, there are certain limitations to using key component aging indicators to characterize the health status of PEMFC, as most of these indicators precisely reflect the aging state of a specific component at a particular scale. Using them to reflect the overall degradation state of the PEMFC can lead to a one-sided perspective. Additionally, in practical applications, the aging of key components is often difficult to measure in real-time, and it is challenging to obtain the parameters required for modeling. Integrating aging indicators from different components and at different scales also presents difficulties. Therefore, the application of key component aging assessment methods in health status evaluation is relatively limited. Degradation indicators that reflect the overall performance of PEMFC typically refer to externally observable output indicators, such as output voltage, output power, maximum output power, minimum voltage under rated conditions, and internal resistance. These indicators can be detected through external sensors and can be continuously tracked, making them common health indicators.

In the context of this study, which examines the impact of different operating conditions under dynamic cycling on the performance of PEMFC, the representational effectiveness of these output indicators is similar since the current is held at a fixed value. The output voltage, as the most commonly used important characteristic parameter reflecting the overall performance of PEMFC, tends to decrease with the degradation or failure of most components, and it is also the most easily obtained. Accordingly, in the selection of indicators, we will choose the output voltage, which most directly represents the performance of PEMFC.

In the actual condition, there are various operating conditions, including open circuit condition, idle speed condition, variable load condition, and high load condition. All these operating conditions will affect the output voltage to different degrees. Therefore, in this paper, the operating conditions in dynamic cycle condition will be classified according to the failure mechanism of PEMFC. And the voltages that can characterize the PEMFC degradation trend under the corresponding operating conditions are established as health indexes.

In the dynamic cycle durability test, when the vehicle is idle, no external power output is needed, but in order to maintain the normal operation of the PEMFC system, it still needs to operate at a low current density. At this point, the cathode overpotential is very high and is close to the open-circuit voltage. In the durability test data, the condition with the current of 0 A belongs to the open-circuit condition, and the condition with the current of 1.78 A belongs to the idle condition. Under open circuit and idle conditions, the penetration of gas on both sides and the elevated potential at the cathode can trigger chemical degradation within the membrane electrode assembly. This process not only expedites the deterioration of the performance of the platinum (Pt) catalyst within the catalytic layer but also results in a substantial reduction of the electrochemically active surface area. Such degradations can severely compromise the efficiency and longevity of the fuel cell. Since the failure mechanism of PEMFC in the open-circuit condition is the same as that in the idle condition, the open-circuit condition and the idle condition are grouped together and only one health index is established. As shown in Table 2, the proportion of idle conditions in the durability test data is much larger than that of open circuit condition. Therefore, this paper chooses the voltage degradation trend corresponding to the load current of 1.78 A as the health index characterizing the idle and open circuit conditions in the dynamic cycle condition, and the health index is called HI-1.

For fuel cells, the frequent change of load is a serious challenge to their lifetime. Due to the gas response rate lagging behind the current response rate, there is a propensity for partial gas starvation during load transitions, particularly during rapid load increases. This phenomenon can lead to an insufficient supply of reaction gases, which in turn can impair the efficiency and performance of the fuel cell. When the cathode reaction gas is insufficient, the hydrogen protons transferred to the cathode will directly undergo an oxidation reaction, resulting in a sudden reduction of the cathode potential. When the anode reaction gas is insufficient, the cathode reaction will lack protons and electrons, so that the cathode carbon carrier corrosion. As shown in Fig. 2, in the durability test, the process of load current rising from 1.78 A to 20.75 A is the fastest process of load rising. Therefore, in this paper, the voltage degradation trend corresponding to 20.75 A load current is selected as the health index characterizing the variable load condition in the dynamic cycle condition, and the health index is called HI-2.

Fig. 2.

The degradation trend of HI-1, HI-2, and HI-3.

High load discharge can escalate the likelihood of developing local hot spots within fuel cells. The generation of local hot spots will degrade the proton exchange membrane materials, and the polymer chains may undergo reorganization and nanostructure changes. Furthermore, the crystallinity or the dissociation of ion clusters are increased, showing a decrease in electrical conductivity, gas resistance, water permeability, and water absorption, and an increase in tensile modulus. According to Tables 2 and in the durability test, the load current of 35.6 A belongs to the high load condition. Therefore, the voltage degradation trend corresponding to the load current of 35.6 A is selected as the health index characterizing the high load condition in the dynamic cycle condition, and the health index is called HI-3.

Fig. 1(b) marks the locations of the selected voltage in a single cycle for HI-1, HI-2 and HI-3. Since the voltage has a response time when the operating condition changes, the voltage corresponding to the 10th second of the load current segment is selected by HI-1 and HI-3 as the value characterizing the health status of this cycle. Fig. 1(b) shows that the voltages selected by them are basically located in the stable phase outside the response time. However, HI-2 directly selects the voltage at the instant of load change as the value characterizing the health state of the PEMFC in this cycle. According to the position of the selected voltage in a single cycle, the health indicators established for each of the three operating conditions are shown in Fig. 2. In the figure, the voltage of each operating condition has some transient recovery phenomena caused by the start-stop operation, and the recovery phenomena under variable load and high load conditions are more obvious.

2.3. Degradation of PEMFC based on different operating conditions

HI-1, HI-2, and HI-3, although derived from the same set of durability test data, exhibit different downward trends. The reason is that during the dynamic cycling process, the output voltage of the PEMFC under different operating conditions decreases at varying rates and to different extents compared to their respective initial voltages. To study the rate and extent of the PEMFC output voltage decline under different operating conditions, this paper sets the startup voltage values of HI-1, HI-2, and HI-3 to 1 V and compares their degradation trends. Fig. 3 shows the voltage trend obtained when the initial voltage values of HI-1, HI-2, and HI-3 are set to 1 V. As can be seen from the figures, compared to the health indicator trends in Fig. 2, Fig. 3 more clearly and directly reflects the varying rates and extents of the voltage trend decline under different current conditions. This makes it easier to discern the changes in the output performance of the PEMFC under different operating conditions after dynamic cycling.

Fig. 3.

Comparison of degradation trends of HI-1, HI-2, and HI-3.

As depicted in Fig. 3, the degradation trend for HI-1 is significantly more moderate, with a slower rate of decline and a relatively smaller extent of decrease. In contrast, the degradation trends for HI-2 and HI-3 are markedly steeper, with a relatively faster rate of decline and a greater extent of decrease. This implies that after a certain degree of degradation, the output performance of the PEMFC under the HI-1 condition has not significantly deteriorated, with a relatively small rate and extent of decline compared to the initial voltage. In contrast, its performance under the HI-2 and HI-3 conditions has relatively deteriorated more, with a relatively larger rate and extent of decline compared to the initial voltage.

3. Degradation trend prediction method for PEMFC

3.1. Decomposition of PEMFC degradation based on VMD

VMD is a time-frequency analysis method that can effectively deal with non-stationary and nonlinear signals. This method is an improvement of empirical mode decomposition (EMD). When searching for the optimal solution of the variational model, the alternating direction method of multipliers (ADMM) is used to search iteratively, which can realize the adaptive decomposition of the signal. VMD avoids the shortcomings of EMD in the recursive mode decomposition process, such as mode aliasing, not being able to separate the components with similar frequencies correctly, and being affected by the sampling frequency. VMD has strong time-frequency resolution and high noise robustness. It has applications in noise reduction, feature extraction, fault diagnosis, and other fields.

The intrinsic mode function (IMF) is a bandwidth-limited AM-FM function, and the k-order IMF component can be expressed as follows:

$${\mu }_k\left(t\right)=A_k\left(t\right)\cos \left({\varphi }_k\left(t\right)\right)$$ (1)

where $A_k\left(t\right)$ is the instantaneous amplitude, and ${\varphi }_k\left(t\right)$ is the instantaneous frequency of ${\mu }_k\left(t\right)$.

The VMD algorithm decomposes the signal by constructing and solving a constrained variational problem [40]. The signal can be decomposed into a specified number of IMF components. When evaluating the bandwidth of IMF components, the VMD method first calculates the corresponding analytical signal by Hilbert transform for each IMF component to obtain a one-sided spectrum. Then the index term is added to adjust the estimated frequency of each center, and the spectrum of the IMF component is shifted to the baseband. Finally, the modulated signal is used to perform Gaussian smoothing to estimate the bandwidth, such as the gradient of the square norm. The constrained variational model is expressed as follows:

$$\underset{\left\{{\mu }_k\right\},\left\{w_k\right\}}{\min }\left\{{{\sum }_k\Vert {\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right){\mu }_k\left(t\right)\right]e^{-i{w}_{k}t}\Vert }_2^2\right\}$$ (2)

where ${\mu }_k$ is the IMF component obtained by decomposition, and $w_k$ is the center frequency of the component. The constraint for the optimal solution is as follows:

$$f={\sum }_k{\mu }_k$$ (3)

When finding the optimal solution, the quadratic penalty factor $\alpha$ and the Lagrange multiplier $\lambda$ are introduced to obtain the following:

$$L\left(\left\{{\mu }_k\right\},\left\{w_k\right\},\lambda \right)=\alpha {{\sum }_k\Vert {\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right){\mu }_k\left(t\right)\right]e^{-i{w}_{k}t}\Vert }_2^2+{\Vert f\left(t\right)-{\sum }_k{\mu }_k\left(t\right)\Vert }_2^2$$

$$+⟨\lambda \left(t\right),f\left(t\right)-{\sum }_k{\mu }_k\left(t\right)⟩$$ (4)

The optimal solution of the constrained variational model can be obtained by using ADMM to find the saddle point of the above augmented Lagrange function. Each component can be obtained according to the frequency domain space, as shown in (5):

$${\hat{\mu }}_k^{n+1}\left(w\right)=\frac{\hat{f}\left(w\right)-{\sum }_{i\ne k}{\hat{\mu }}_i\left(w\right)+\frac{\hat{\lambda }\left(w\right)}{2}}{1+{2\alpha \left(w-w_k\right)}^2}$$ (5)

where $w$ represents the frequency, and ${\hat{\mu }}_k^{n+1}\left(w\right)$, $\hat{f}\left(w\right)$ and $\hat{\lambda }\left(w\right)$ are the Fourier transforms corresponding to ${\mu }_k^{n+1}\left(t\right)$, $f\left(t\right)$ and $\lambda \left(t\right)$, respectively. The center frequency of each component can be expressed as follows:

$$w_k^{n+1}=\frac{{\int }_0^{\infty }w{\left|{\hat{\mu }}_k^{n+1}\left(w\right)\right|}^{2}dw}{{\int }_0^{\infty }{\left|{\hat{\mu }}_k^{n+1}\left(w\right)\right|}^{2}dw}$$ (6)

When ${\sum }_k{\Vert {\hat{\mu }}_k^{n+1}-{\hat{\mu }}_k^n\Vert }_2^2/{\Vert {\hat{\mu }}_k^n\Vert }_2^2<\epsilon$, the optimal solution is solved.

In this paper, based on VMD, HI-1, HI-2, and HI-3 are decomposed respectively to obtain the overall degradation trend of PEMFC under different operating conditions and the local large fluctuation signal caused by the external operation, and filter out the noise and other interference information.

3.2. Prediction of IMF component based on LSTM

LSTM is a variant of RNN, which is able to avoid not only the long-term dependence problem of RNN, but also problems such as vanishing or exploding gradients [41]. LSTM is very effective in processing sequence data and can achieve accurate and fast short-term prediction of sequence data. An RNN consisting of LSTM units is often referred to as an LSTM network, and Fig. 4 shows the structure diagram of the LSTM recurrent neural network.

Fig. 4.

The network architecture of LSTM.

The horizontal line above the LSTM cell in the figure running from left to right is the key to the LSTM, called the cell state. It passes information from the previous cell to the next, and has less linear interactions with the rest of the parts. LSTM uses gates to achieve forgetting or remembering. A gate is a structure that allows selective passage of information, which consists of a dot product and a sigmoid function. An LSTM cell includes three gates, which are forget gate, input gate, and output gate.

The forget gate uses the sigmoid function to generate a memory decay coefficient for the items in ${\mathrm{C}}_{t-1}$. And it is used to control the degree to which the previous cell state is forgotten. The expression is as follows:

$$f_t=\sigma \left(W_f\bullet \left[h_{t-1},x_t\right]+b_f\right)$$ (7)

where $h_{t-1}$ is the output of the previous unit, $X_t$ is the input of this unit.

The input gate is coupled with a hyperbolic tangent function to control the addition of new information. The hyperbolic tangent function is used to produce a new candidate vector ${\tilde{C}}_t$. The input gate uses the sigmoid function to produce a value within [0,1] for each item in ${\tilde{C}}_t$, and control the new information that needs to be added. The specific expression is:

$$i_t=\sigma \left(W_i\bullet \left[h_{t-1},x_t\right]+b_i\right)$$ (8)

$${\tilde{C}}_t=\tanh \left(W_C\bullet \left[h_{t-1},x_t\right]+b_C\right)$$ (9)

Then the forget gate is combined with the input gate, and the state $C_{t-1}$ is updated:

$$C_t=f_t*C_{t-1}+i_t*{\tilde{C}}_t$$ (10)

The output gate is used to control the filtering degree of the current cell state, and the output value $h_t$ at the current time can be obtained. The mathematical expression is:

$$o_t=\sigma \left(W_o\bullet \left[h_{t-1},x_t\right]+b_o\right)$$ (11)

$$h_t=o_t*\tanh \left(C_t\right)$$ (12)

In Equations (7), (8), (9), (10), (11), (12), $W$ and $b$ represent the weight and bias, respectively.

In this paper, LSTM is used to predict each IMF component of PEMFC degradation, respectively. It is beneficial to predict the IMF components separately and then reconstruct them to obtain the PEMFC degradation trend with relatively small error.

3.3. Improvement of LSTM for PEMFC based on ATT

When humans observe objects, they tend to focus on certain parts that contain important feature information. Based on this characteristic of human beings, ATT gives different weights to the input features through probability allocation. It highlights the features that have great influence on the output results, and ignores the irrelevant features, so that the estimated output results are more accurate [42].

The schematic of ATT is shown in Fig. 5. In the figure, $x_1\text{,}$ $x_2$, $\cdots {,x}_N$ are the input of attention mechanism, and $q$ is the query vector. In this paper, the dot product model is used to calculate the weight of the current input feature $x_i\left(i\in \left[1,N\right]\right)$, which is expressed as follows:

$$s\left(x_i,q\right)={x_i}^{T}q$$ (13)

Fig. 5.

Attention mechanism.

By normalizing the weights with the Softmax function, the following can be obtained:

$${\alpha }_i=softmax\left(s\left(x_i,q\right)\right)=\frac{\exp \left(s\left(x_i,q\right)\right)}{{\sum }_{i=1}^N\exp \left(s\left(x_i,q\right)\right)}$$ (14)

The normalized weight and the corresponding feature are weighted to obtain the attention output vector, which is the expression of enhancing or weakening the input feature information. The calculation formula is as follows：

$$Atten={\sum }_{i=1}^N{\alpha }_i{x}_i$$ (15)

In this paper, the attention mechanism is used to improve LSTM. The output vector of the LSTM hidden layer is input to the attention mechanism module, and the weight of the vector is calculated to obtain the feature information that is beneficial for PEMFC degradation prediction.

3.4. Prediction of PEMFC degradation based on VMD-LSTM-ATT

When using the VMD-LSTM-ATT algorithm to predict the degradation trend of PEMFC, a total of five steps are required. Firstly, VMD is used to extract features from the original voltage data to obtain the overall trend and detailed information of PEMFC degradation. Secondly, each IMF component was input into the LSTM network layer to obtain the output of the corresponding hidden layer. Third, the output vector of the hidden layer was used as the input of the attention mechanism to obtain the weighted output vector. Fourth, the attention output vector is input into the fully connected layer to calculate the prediction result of the IMF component. Finally, each IMF component is reconstructed to obtain the predicted degradation trend. Fig. 6 shows the overall process of PEMFC degradation trend prediction based on VMD-LSTM-ATT.

Fig. 6.

PEMFC degradation trend prediction based on VMD-LSTM-ATT.

4. Experiment and analysis

In this section, the VMD-LSTM-ATT algorithm will be used to predict the health indicators of idle, variable load, and high load conditions in dynamic cycle condition. During the experiment, 600 h of data was set as the training set, the remaining data was used to test the performance of the algorithm as the test set. In this paper, root mean square error (RMSE), mean absolute error (MAE), and absolute percentage error (APE) were used to measure the prediction effect of the algorithm on each health indicator. Smaller values of RMSE, MAE, and APE indicate better prediction performance of the algorithm. The formulas of RMSE, MAE, and APE are as follows:

$$RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}$$ (16)

$$MAE=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\hat{y}}_i\right|$$ (17)

$$APE=\frac{\left|{\hat{y}}_i-y_i\right|}{y_i}*100\%$$ (18)

where, $y_i$ is the measured value, and ${\hat{y}}_i$ is the predicted result value.

In addition, Equation (19) is used to evaluate the percentage improvement in the error, for VMD-LSTM-ATT compared with other comparison methods. The larger the percentage, the more obvious the prediction advantage of VMD-LSTM-ATT. The percentage improvement is calculated as follows:

$$Improvement=\frac{\left|{error}_c-error\right|}{error}*100\%$$ (19)

where, ${error}_c$ represents the prediction error of the comparison algorithm, $error$ represents the prediction error of VMD-LSTM-ATT.

4.1. Degradation prediction of PEMFC under idle condition

When the VMD algorithm decomposes the health index, it needs to preset the penalty factor, mode decomposition level, noise tolerance, direct-current (DC) component, and convergence criterion. Among them, the penalty factor $\alpha$ and mode decomposition level $K$ have the most significant influence on the decomposition results. The selection of α will affect the decomposition accuracy. When its value is low, the bandwidth of each IMF component is large, and when its value is high, the bandwidth of the IMF component is small. It is common to set $\alpha$ to 1.5–2.0 times the input signal length. The durability test data in this paper contains 3076 cycles, that is, the length of the health index is 3076. Therefore, the value of $\alpha$ is set to 5000 in this paper. The value of $K$ will directly affect the decomposition results. A too small value of $K$ will lead to insufficient signal segmentation, sub-signals in some modes will be regarded as noise. A too large value of $K$ will capture additional noise, resulting in mode aliasing. In this paper, the center frequency method will be used to determine the value of $K$.

The center frequency method decomposes the sequence starting from $K=2$, and increases sequentially to obtain the center frequency of each IMF component corresponding to each value of $K$. When the center frequency of adjacent $K$ values is close, the signal is considered to be over decomposed. In this case, the optimal number of decomposition levels is the value of the previous term when the center frequency is close. The center frequencies of HI-1 for different values of $K$ are shown in Fig. 7. When $K=5$, the center frequencies of IMF1, IMF2, IMF, and IMF4 are close to the center frequencies of the corresponding components for $K=4$. Therefore, the optimal number of decomposition levels of HI-1 is set to 4. When $K=5$, the over-decomposition is caused by the mode aliasing phenomenon.

Fig. 7.

Center frequencies corresponding to different values of $K$.

When the noise tolerance $tau$ is set to 0, it means that the reconstructed signal is allowed to be different from the original signal. That is, the VMD decomposition is allowed to remove noise or interference signals. Since it is inevitable to be interfered by noise or other signals when collecting data, $tau$ is set to 0 in this paper to smooth the collected data. When the DC component is set to 0, it means that IMF1 is the trend vector, so DC is set to 0 in this paper to obtain the overall trend of voltage drop. The convergence criterion $tol$ is used to control the error size, determine the convergence accuracy and the number of iterations, and is set to ${10}^{-7}$ in this paper. Each IMF component obtained after VMD decomposition is shown in Fig. 8(a). In the figure, IMF1 represents the global degradation trend of voltage under idle condition, IMF2, IMF3, and IMF4 represent the detailed information on the degradation process, mainly the voltage

Fig. 8.

Prediction of HI-1 by VMD-LSTM-ATT. (a) Each IMF component of HI-1; (b), (d), (f), and (h) The prediction result graph of each IMF component; (c), (e), (g), and (i) Model training loss when predicting each IMF component; (j) The prediction result of HI-1.

The LSTM with attention mechanism is used to predict the future trend of each IMF component separately. In the prediction process, the size of the sliding window is set to 7, where the length of input historical information is 5, and the prediction length is 2. The Softmax function was selected as the activation function, Adam with better comprehensive performance was selected as the optimizer. The drop rate was determined by the grid search method to be 0.2, the learning rate was set to 0.0001, the batch size was set to 10, and the number of iterations was set to 500. The prediction result graph and the model training loss graph of each IMF component, and their reconstructed result graph are shown in Fig. 8(b)-8(j). From Fig. 8(b)-8(i), it can be seen that LSTM-ATT has excellent prediction effect for each IMF component, and has fast convergence speed. Reconstructing the prediction results of each component can obtain the prediction result graph of VMD-LSTM-ATT for HI-1. Fig. 8(j) shows that VMD-LSTM-ATT can accurately predict the degradation of PEMFC under idle condition.

4.2. Degradation prediction of PEMFC under variable load condition

According to the center frequency method, the optimal decomposition level of HI-2 is determined to be 4. Therefore, when using VMD to decompose HI-2, the mode decomposition level is set to be $K=4$. In addition, the penalty factor $\alpha =5000$, noise tolerance $tau=0$, DC component $DC=0$, and convergence criterion $tol={10}^{-7}$ are set. Each IMF component obtained by decomposing HI-2 using VMD is shown in Fig. 9(a). Among them, IMF1 represents the global degradation trend of voltage under variable load condition, and the remaining components represent the detailed information of different frequencies during the degradation process.

Fig. 9.

Prediction of HI-2 by VMD-LSTM-ATT. (a) Each IMF component of HI-2; (b), (d), (f), and (h) The prediction result graph of each IMF component; (c), (e), (g), and (i) Model training loss when predicting each IMF component; (j) The prediction result of HI-2.

The LSTM-ATT is used to predict each IMF component of HI-2 separately, the prediction result graph, model training loss, and the prediction result of HI-2 obtained from reconstruction of each component are shown in Fig. 9(b)-9(j). In the prediction process, the length of input historical information is 5, the prediction length is 2, the hidden state of the model is extended to 128 dimensions. The activation function is the Softmax function, the optimizer is Adam with better comprehensive performance. The grid search method was used to determine the drop rate of 0.2, the learning rate of 0.0001, the batch size of 5, and the number of iterations is set to 500. From Fig. 9(b), (d), and 9(f), it can be concluded that LSTM-ATT has excellent prediction effects on IMF1, IMF2, and IMF3 components, and the test results basically coincide with the corresponding original IMF components. Intuitively, the prediction effect of LSTM-ATT on the IMF4 component is relatively poor, but the predicted value can clearly represent the change of IMF4, and all errors are less than 0.001 V. Therefore, the prediction effect of LSTM-ATT on IMF4 can meet the overall demand. As can be seen from Fig. 9(c), (e), 9(g), and 9(i), LSTM-ATT has fast convergence speed and small model training loss. When the number of iterations is about 10, the MSE is far less than 0.01, and the model basically converges. Reconstructing the prediction results of each component can obtain the prediction result graph of VMD-LSTM-ATT for HI-2. Fig. 9(j) shows that VMD-LSTM-ATT can accurately predict the degradation trend of PEMFC under variable load condition.

4.3. Degradation prediction of PEMFC under high load condition

VMD-LSTM-ATT is used to predict the degradation trend of PEMFC under high load condition. The selection method of each parameter in the prediction process is the same as that in variable load condition. The four IMF components obtained after VMD decomposition are shown in Fig. 10(a). The prediction result graph, model training loss of LSTM-ATT for each IMF component, and the prediction result of HI-3 obtained from the reconstruction of each component are shown in Fig. 10(b)-10(j).

Fig. 10.

Prediction of HI-3 by VMD-LSTM-ATT. (a) Each IMF component of HI-2; (b), (d), (f), and (h) The prediction result graph of each IMF component; (c), (e), (g), and (i) Model training loss when predicting each IMF component; (j) The prediction result of HI-3.

4.4. Comparison of different methods for prediction of PEMFC degradation

To verify the effectiveness and superiority of the established model in predicting the degradation trend of health indicators, this paper compares the VMD-LSTM-ATT with SVM, RNN, LSTM, and LSTM-ATT respectively. And the comparison results are shown in Fig. 11 and Table 3. The parameters of each model were taken to be consistent in the prediction process. Fig. 11 shows the predicted degradation trend of each model, and Table 3 shows the RMSE and MAE of each prediction model on the training set, test set, and full set, respectively. It can be clearly concluded from Fig. 11 and Table 3 that the prediction effects of VMD-LSTM-ATT under the three operating conditions are significantly better than those of other comparable models.

Fig. 11.

Degradation trend prediction of different models for HI-1, HI-2, and HI-3.

Table 3.

Prediction errors of different models.

Methods			SVM	RNN	LSTM	LSTM-ATT	VMD-LSTM-ATT
HI-1	Training	RMSE[V]	0.002694	0.001158	0.001218	0.001211	0.001024
		MAE[V]	0.002403	0.000447	0.000609	0.000602	0.000535
	Test	RMSE[V]	0.003927	0.001120	0.001092	0.001090	0.000878
		MAE[V]	0.003805	0.000625	0.000559	0.000557	0.000445
	Total	RMSE[V]	0.003247	0.001143	0.001169	0.001164	0.000968
		MAE[V]	0.002968	0.000518	0.000589	0.000584	0.000499
HI-2	Training	RMSE[V]	0.004531	0.004521	0.005124	0.005243	0.003819
		MAE[V]	0.002797	0.002798	0.003396	0.003530	0.002171
	Test	RMSE[V]	0.005174	0.005307	0.004586	0.004660	0.003911
		MAE[V]	0.003557	0.003439	0.002534	0.002625	0.002192
	Total	RMSE[V]	0.004801	0.004853	0.004915	0.005017	0.003856
		MAE[V]	0.003102	0.003056	0.003045	0.003166	0.002180
HI-3	Training	RMSE[V]	0.005629	0.004257	0.004518	0.004039	0.003052
		MAE[V]	0.004597	0.002801	0.002637	0.002055	0.001569
	Test	RMSE[V]	0.008633	0.005467	0.003490	0.003379	0.002734
		MAE[V]	0.008158	0.004296	0.001507	0.001347	0.001275
	Total	RMSE[V]	0.006995	0.004781	0.004135	0.003787	0.002928
		MAE[V]	0.006030	0.003403	0.002183	0.001770	0.001450

In addition, the APE of LSTM, LSTM-ATT, and VMD-LSTM-ATT are compared, the percentage improvement in RMSE and MAE of VMD-LSTM-ATT over LSTM and LSTM-ATT is calculated. Fig. 12 shows the APE results of each model for the prediction of HI-1, HI-2, and HI-3. Fig. 12 shows that the APE results of VMD-LSTM-ATT in HI-1, HI-2, and HI-3 are much smaller than the corresponding errors of LSTM and LSTM-ATT. The average APE of VMD-LSTM-ATT in HI-1, HI-2, and HI-3 are 0.058819, 0.341338, and 0.271127, respectively. Therefore, VMD-LSTM-ATT has a greater advantage in predicting the PEMFC degradation trend. Table 4 shows the percentage improvement of the prediction error by VMD-LSTM-ATT over LSTM and LSTM-ATT. It can be seen from the table that, in the full set of three operating conditions, compared with LSTM, the RMSE of VMD-LSTM-ATT is improved by up to 41.22 %, and MAE is improved by up to 50.55 %. Compared with LSTM-ATT, the RMSE of VMD-LSTM-ATT is improved by up to 29.34 %, and MAE is improved by up to 39.68 %.

Fig. 12.

Comparison of APE predicted by different models for HI-1, HI-2, and HI-3.

Table 4.

Percentage improvement in prediction error for VMD-LSTM-ATT.

Improvement			Improvement over LSTM [%]	Improvement over LSTM-ATT [%]
HI-1	Training	RMSE	18.95	18.26
		MAE	13.83	12.52
	Test	RMSE	24.37	24.15
		MAE	25.62	25.17
	Total	RMSE	20.76	20.25
		MAE	18.04	17.03
HI-2	Training	RMSE	37.29	34.17
		MAE	62.60	56.43
	Test	RMSE	19.15	17.26
		MAE	19.75	15.60
	Total	RMSE	30.11	27.46
		MAE	45.23	39.68
HI-3	Training	RMSE	48.03	32.34
		MAE	68.07	30.98
	Test	RMSE	27.65	23.59
		MAE	18.20	5.65
	Total	RMSE	41.22	29.34
		MAE	50.55	22.07

The data set used in Ref. [9] is the health index established when the current is 35.6 A under the dynamic cycling condition. In addition, reference [9] also used 600 h of data to train the model and the remaining data to test the performance of the model. Therefore, this paper compares the prediction effect of the VMD-LSTM-ATT model on HI-3 with reference [9], and the comparison results are listed in Table 5. It can be concluded from Table 5 that the prediction error of VMD-LSTM-ATT is greatly improved compared with reference [9]. Among them, the RMSE improvement on the test set is the largest, which is 56.11 %.

Table 5.

Comparison of VMD-LSTM-ATT with other reference.

Methods		Reference [9]	VMD-LSTM-ATT
		Value [V]	Value [V]	Improvement [%]
Training	RMSE	0.003926	0.003052	28.64
	MAE	0.001807	0.001569	15.17
Test	RMSE	0.004268	0.002734	56.11
	MAE	0.001634	0.001275	28.16

In this paper, the voltage prediction is carried out two steps in advance, and the interval of each step is about 20 min, so the future voltage of PEMFC is predicted 40 min in advance. The online short-term prediction of voltage trends in different operating conditions can optimize PEMFC's operating conditions and maintenance strategies, and also help to identify and resolve performance degradation problems in advance, thereby extending the life of the fuel cell and reducing operating costs.

5. Conclusion

To promote the long-term and safe operation of PEMFC and make maintenance strategies in a timely and more reasonable way. In this paper, according to the failure mechanism of PEMFC under different operating conditions, the different operating conditions in the dynamic cycle condition are classified. The health indicators of the different operating conditions in the dynamic cycle condition are established. At the same time, the rates and degrees of the output voltage decline of the PEMFC under different operating conditions were compared. In addition, the VMD-LSTM-ATT model is developed and used to predict the degradation trend of PEMFC under different operating conditions. The main conclusions of this paper are as follows.

• (1)Based on the failure mechanism of PEMFC under different operating conditions, it is of great significance to classify the main operating conditions of dynamic cycle condition, and establish health indicators for them for prolonging the life of fuel cells.
• (2)After several dynamic cycles, the operational performance of the PEMFC under the HI-1 condition has not significantly deteriorated, while its performance under the HI-2 and HI-3 conditions has relatively deteriorated more.
• (3)Compared with LSTM, LSTM-ATT uses ATT to assign different weights to the input features, highlight the features that have great influence on the output results, and ignore the irrelevant features, so that the estimation results of PEMFC degradation trend under different operating conditions in dynamic cycling conditions are more accurate.
• (4)The VMD can obtain the overall degradation trend and local fluctuation signals of PEMFC, and can filter out the noise and other interference information. Therefore, compared with LSTM and LSTM-ATT methods, the VMD-LSTM-ATT method has better prediction effect. In the full set of HI-1, HI-2, and HI-3, the percentage improvement of VMD-LSTM-ATT in RMSE and MAE is more than 17.03 %.

Predicting the degradation trend of fuel cells under different operating conditions and guiding appropriate decisions by future trends can not only prolong the remaining useful life of the PEMFC, but also improve its utilization. In the future, health indicators that characterize the overall performance of the PEMFC under dynamic cycling conditions will be established based on the trend of voltage decline in the PEMFC under different operating conditions. By predicting its future trends, the durability of the PEMFC will be extended.

Funding statement

This work was supported by BUPT Excellent Ph.D. Students Foundation [grant number CX2023212].

Data availability statement

Data included in article/supplementary material/referenced in article.

Ethics statement

Review or approval by an ethics committee was not needed for this study because no data on patients or experimental animals was used in the article.

Additional information

No additional information is available for this paper.

CRediT authorship contribution statement

Xiaohui Liu: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation. Jianhua Chen: Writing – review & editing, Validation, Investigation. Yian Wei: Writing – review & editing, Investigation. Shengjie Liu: Writing – review & editing. Yilin Zhou: Writing – review & editing, Supervision, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.