International Journal of Hydrogen Energy 141 (2025) 536–549
Available online 20 November 2024
0360-3199/© 2024 Published by Elsevier Ltd on behalf of Hydrogen Energy Publications LLC.
A novel method of long-term aging prediction for proton exchange
membrane fuel cell under the dynamic load cycling condition
Xiaoran Yu a, Yang Yang a,b,*, Yanjun Liu a, Wenchao Zhu c, Changjun Xie a,b,**
a School of Automation, Wuhan University of Technology, Wuhan, 430070, China
b Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan, 430070, China
c State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan, Hubei, 430073, China
A R T I C L E I N F O
Handling Editor: Ibrahim Dincer
Keywords:
Proton exchange membrane fuel cell
Model uncertainty and state estimation
CNN-LSTM
Long-term prediction
Dynamic load cycling condition
A B S T R A C T
Estimating the remaining useful life (RUL) of proton exchange membrane fuel cell (PEMFC) is beneficial for
deploying control strategies. Long-term aging prediction forms the foundation for RUL estimation, but currently
long-term aging prediction and RUL estimation for dynamic load cycling conditions have not been achieved, and
uncertainty during the prediction and estimation processes has not been considered. Therefore, in this study, a
novel hybrid prediction framework based on Bayesian gated recurrent unit, model uncertainty and state esti-
mation (MUSE), and convolutional neural network-long short-term memory (CNN-LSTM) is proposed. First,
MUSE is used to simultaneously quantify the state of health (SOH) and predict aging trend to generate a guiding
sequence Next, based on the sequence and historical experimental data, CNN-LSTM is trained and multi-step
prediction is performed. Subsequently, Bayesian gated recurrent unit is employed to fuse the MUSE-based pre-
diction sequence with the CNN-LSTM-based prediction, and confidence interval is provided. This results in a
novel model and data-driven long-term aging prediction structure. Finally, SOH during the prediction stage is
evaluated by MUSE and RUL is estimated. The proposed method is validated using experimental data under a
dynamic load cycling condition, demonstrating accurate long-term aging prediction and RUL estimation while
considering uncertainty.
1. Introduction
In the context of energy scarcity and the pressing issue of climate
change, the development and utilization of clean energy sources have
assumed paramount importance. Power to hydrogen is a promising so-
lution for storing variable renewable energy [1]. Hydrogen energy in-
volves four aspects: production, transportation, storage, and utilization
[2]. Fuel cell is one kind of technologies of hydrogen utilization.
Hydrogen can be fed to the fuel cells to provide electric power to drive
vehicles [3]. Proton exchange membrane fuel cells (PEMFCs), as envi-
ronmentally friendly energy conversion devices, are employed in
diverse fields such as distributed power generation and automotive
technology [4]. However, short lifespan and high-cost act as limitations
that hinder its commercialization [5]. Estimating the remaining useful
life (RUL) can provide guidance for maintenance strategies, reduce
consumption [6]. This is significant in extending the service life of
PEMFCs [7].
Model-based method is one kind of the methods to estimating RUL
[8]. Mechanistic or empirical models are combined with estimators like
filtering algorithms to assess the state of health (SOH) of PEMFCs, and
prediction for the degradation trend is provided. Extended Kalman filter
(EKF) [9] and unscented Kalman filter (UKF) [10] were proposed for
evaluating SOH of PEMFCs. Ao et al. proposed a frequency domain
Kalman filter to process data in the frequency domain, ensuring pre-
diction accuracy while reducing runtime [11]. In addition to accuracy
and runtime, stability is also a crucial evaluation criterion, so Zhang
et al. combined multi-point estimation with the square root central
difference Kalman filter to create a new algorithm SR-CDKF. This inte-
gration aims to address the challenges posed by observer mobility and
system noise, ensuring a more stable approach to assess SOH [12].
Particle filter is also used [13]. Its variants have been widely used [14,
15]. As for aging model, Ou et al. proposed a semi-empirical model that
considers polarization behavior in different polarization zones to make
degradation prediction of PEMFC systems under vehicle operating
* Corresponding author. School of Automation, No.122, Luoshi Road, HongShan District, Wuhan City, Hubei Prov. China.
** Corresponding author. School of Automation, Wuhan University of Technology, Wuhan, 430070, China.
E-mail addresses: whutyangyang@whut.edu.cn (Y. Yang), jackxie@whut.edu.cn (C. Xie).
Contents lists available at ScienceDirect
International Journal of Hydrogen Energy
journal homepage: www.elsevier.com/locate/he
https://doi.org/10.1016/j.ijhydene.2024.11.194
Received 3 July 2024; Received in revised form 25 October 2024; Accepted 11 November 2024
International Journal of Hydrogen Energy 141 (2025) 536–549
537
conditions [16]. In Ref. [17], an empirical model for lifetime prediction
coupled with electrical load distribution is built. In this model, the
voltage degradation rate is modeled as a linear function of the charac-
teristics of the load profile which is defined by voltage and current.
However, model-based method can only provide aging trends [7]. A
comprehensive aging model requires a complete understanding of the
degradation mechanisms and characterization of the degradation pa-
rameters [18]. Semi-empirical model is usually used.
With the enhancement of machine learning, data-driven method gets
unprecedented development. Data-driven method directly provides
prediction by learning from historical data [9]. Various algorithms have
been applied, such as backpropagation neural network [19], convolu-
tional neural network (CNN) [20], long-short-term memory (LSTM)
neural network [21], echo state network (ESN) [22], gated recurrent
unit [23] and Transformer [24]. Jia et al. developed a novel model that
fuses multi-head self-attention mechanism and bi-directional LSTM
[25]. The preprocessed historical voltage data and PEMFC system
operating parameters are fed to the model and the global information
modeling is achieved. It demonstrates good performance of short-term
prediction. According to Ref. [26], long-term prediction whose predic-
tion horizon (PH) exceeds 168h holds practical significance for PEMFC
operational strategies. However, achieving accurate long-term predic-
tion remains one of the challenges in the field of fuel cell aging pre-
diction. The distinguishing feature of long-term prediction is the
independence from actual data during forecasting. Wang et al. employed
an autoregressive integrated moving average (ARIMA) model to provide
a navigation sequence for LSTM, creating navigation sequence navi-
gated LSTM (NSD-LSTM) to achieving long-term prediction [21]. But it
can be found ARIMA tends to deviate from actual data when faced with
obvious
fluctuations.
In
Datasets
with
disturbances,
data-driven
methods often capture more nonlinear information while ignoring
aging trends.
Multiple hybrid strategies can also achieve long-term prediction. In
Ref. [27], EKF-LSTM is constructed. In this method, EKF and LSTM
mutually provide observation values to get rid of the reliance of actual
data. EKF-LSTM can achieve long-term prediction for internal parame-
ters and voltage. SOH estimation based on the prediction of data-driven
method can also make prediction [28]. Decomposition is also a solution.
For example, some researchers decomposed voltage data into linear and
nonlinear parts and then appropriated methods were used to process
these parts [29]. In Ref. [13], voltage data is decomposed into trend,
recovery and disturbance parts, and then PF, equivalent circuit model
and random forest algorithm are used to process them, improving pre-
diction accuracy. While the complexity of the hybrid methods is much
higher, they significantly improve the accuracy of prediction compared
to one prediction method [18].
It can be found the methods in Refs. [27,28] can only provide
degradation trends. The methods in Refs. [13,27,28] are tested in static
operation condition and EKF-LSTM in Ref. [27] is tested under the
condition that is made by extracting a certain current from the dynamic
condition. It is not similar to the real world. Due to the periodicity of
data from the dynamic load cycling condition [30], it is difficult to
extract
an
aging
trend
of
voltage.
Therefore,
the
methods
in
Ref. [13.27–29] are ineffective for long-term prediction of the dynamic
load operating condition which has a clear cyclical variation. And there
are few methods to make long-term prediction for dynamic load cycling
conditions. Lv et al. proposed a Transformer based long-term prognosis
to predict relative power-loss rate, but measurement is used in its
structure and RUL estimation is not made [24].
Additionally, There are few long-term aging prediction methods that
consider uncertainty quantification under dynamic load cycling condi-
tions. Uncertainty objectively exists, and existing hybrid methods lack
consideration of the uncertainty [26]. The existing one-sidedness of
aging models makes it difficult to comprehensively reflect the aging
behavior of PEMFCs [31]. The parameters of aging models are affected
by measurement noise, dynamic model drift, and other factors [32].
During establishing aging models through experiments, the state of fuel
cells can also be influenced. Water management state is highly suscep-
tible to its control strategies [33]. Membrane dry faults hinder the
transfer of proton and lead to irreversible membrane degradation [34].
The possibility of fuel shortage and low gas pressure cannot be ignored
[35]. The model uncertainty caused by these factors may affects SOH
estimation. Uncertainty also exists in prediction results [36]. It can lead
to different results from the same neural network trained on the same
data [37]. Quantifying uncertainty is beneficial for evaluating the us-
ability of results [37] and extracting aging information [38].
Attempts to consider uncertainty exist. In Ref. [21], the prediction
results of NSD-LSTM are assumed to follow a standard Gaussian distri-
bution, the mean of multiple results is taken as the final outcome.
Although this method can provide confidence intervals, uncertainty in
the neural network is not explored. Zhu et al. combined Bayesian theory
with gated recurrent unit (GRU) to propose Bayesian GRU. But it is
verified in short-term prediction of constant operating condition [36].
It can be found that the internal parameters characterizing SOH of
fuel cells do not exhibit as drastic changes throughout the entire aging
experiment as voltage does [27]. It is possible to extract aging trends
from these parameters. To achieve long-term prediction and RUL esti-
mation under the dynamic load cycling condition, it is necessary to
consider the internal parameters. For model-based PEMFC aging pre-
diction, the authors have proposed the model uncertainty and state
estimation (MUSE) method [7]. Compared to other model-based
methods, MUSE has shown superior accuracy and robustness. In
data-driven methods, CNN-LSTM has demonstrated good performance
in predicting the lifespan of PEMFCs [39]. CNN is a traditional deep
learning model. It is made up of input layer, convolutional layer, pooling
layer, fully connected layer and output layer [40]. It is a good helper in
feature extraction. LSTM is an improved version of recurrent neural
network. LSTM addresses long-term dependency by adding a gating
mechanism [41]. To make CNN-LSTM more suitable for extracting aging
information from data of the dynamic load cycling condition, one can
incorporate sequences generated by MUSE into experimental data to
achieve guided prediction. So RUL estimation under the dynamic load
cycling condition can be made by estimating SOH based on prediction
results.
For uncertainty quantification, Ref. [36] has shown that Bayesian
GRU has good performance in uncertainty quantification. Since MUSE
and CNN-LSTM give deterministic results from which it is difficult to
analyze uncertainty, Bayesian GRU can be used to fuse the prediction
results of MUSE and CNN-LSTM to explore the mapping relationship
between MUSE-generated guidance sequence, CNN-LSTM prediction
and actual data and to give confidence intervals.
Therefore, a novel Bayesian GRU fused MUSE and CNN-LSTM hybrid
prediction framework for long-term prediction and RUL estimation of
PEMFCs under the dynamic load cycling condition is proposed. The
main innovations of this work include:
1) A new structure of long-term aging prediction for PEMFC is pro-
posed. MUSE is used to capture the aging trends and generate a
sequence which is added to experimental data to guide the prediction
of CNN-LSTM. Compared with other methods, such as CNN-LSTM,
EKF-LSTM, NSD-LSTM and so on, the proposed method shows
higher accuracy of long-term prediction.
2) The uncertainty of long-term aging prediction under the dynamic
load cycling condition is quantified and the confidence interval of
long-term prediction is provided. The capacity of uncertainty quan-
tification of the proposed method is analyzed.
3) SOH during the prediction stage is obtained and RUL estimation for
PEMFC under the dynamic load cycling condition is achieved. The
proposed method demonstrates excellent performance on full life-
cycle RUL estimation.
The rest of the paper is organized as follows. Section 2 presents the
X. Yu et al.
International Journal of Hydrogen Energy 141 (2025) 536–549
538
experimental setup and data. The details of the Bayesian GRU fused
MUSE and CNN-LSTM hybrid method is described in Section 3. The
experiment results, comparisons, and discussion are included in Section
4. Section 5 concludes the results of this work.
2. Experimental setup and data analysis
The data of the dynamic load cycling condition is from Ref. [30].
Greenlight Innovation G20 test bench is used. A single PEMFC is tested.
Original durability data and polarization curve data are directly ob-
tained. The full load current is 35.6A, and the experiment runs for
approximately 1008h, including 3076 dynamic cycling loads. Each cycle
consists of 35 current loads. 16-dimensional data including current,
voltage, and power is stored. Polarization curve tests are conducted at
100-h intervals. The primary control parameters for the experiment are
shown in Table 1. The durability experimental data are depicted in
Fig. 1.
In Fig. 1, it is evident that periodicity of voltage is obvious, making it
challenging to extract the information of aging trend, disturbance and
recovery. However, the polarization curve exhibits a downward trend
over time, which allows for the extraction of SOH using aging models
and polarization curves.
3. Methodology
In this section, the overall framework of the proposed method is
introduced, focusing on key components such as aging model estab-
lishment, MUSE, CNN-LSTM, and the Bayesian GRU.
As Fig. 2 shows, first, MUSE is integrated with an aging model and it
generates a guiding sequence. Next, the sequence and historical data are
input to CNN-LSTM to fit the voltage and make long-term prediction.
Then Bayesian GRU is used to explore the mapping relationship between
the guided sequence, CNN-LSTM voltage fitting, and experimental data.
This step allows to fuse prediction results from both model-based and
data-driven methods while quantifying uncertainty. This approach
provides long-term prediction with confidence intervals under the dy-
namic load cycling condition. Finally, based on the fused prediction
result, MUSE is used again to quantify SOH during the prediction stage,
enabling to estimate RUL.
3.1. Establish of the aging model
In a PEMFC, hydrogen gas is delivered to the anode, where it in-
teracts with a catalyst and splits into protons and electrons. The protons
pass through the exchange membrane to the cathode, while the elec-
trons flow through an external circuit, generating electric energy. At the
cathode, oxygen molecules react with the protons to form water,
releasing heat [40]. The output voltage of a fuel cell can be depicted as
[27]:
Vst = ncell × (Ecell −Eact −Eohm −Econ)
(1)
where ncell is the number of fuel cells. Ecell is the reversible voltage (V).
Eact, Eohm and Econ represent the voltage drop of the activation loss,
ohmic loss and concentration loss. These voltage drops can be calculated
as follows [41]:
Eact = jRact + RT
αa+c
ln
(j + jloss
jex
)
(2)
Eohm = jRohm
(3)
Table 1
Control parameters.
Items
Values
Active surface (cm2)
25
Hydrogen inlet-pressure (kPa)
110
Air inlet-pressure (kPa)
110
Operating temperature (◦C)
85
Hydrogen relative humidity (%)
50
Air relative humidity (%)
50
Fig. 1. Experimental data of the durability test (a) Voltage and current in one dynamic load cycle (b) Overall voltage (c) Polarization curve.
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International Journal of Hydrogen Energy 141 (2025) 536–549
539
Econ = jRcon + RT
βa+c
ln
(jmax −j
jmax
)
(4)
where j is the current density of PEMFC, jloss represents the crossover
current density, jex represents exchange current density and jmax repre-
sents the limiting current density. R is gas constant (8.3145J/mol/K), F
is Faraday constant (96485C/mol). αa + c is the transfer coefficient, βa + c
is a parameter related to the number of electrons transferred in the re-
action, ‘a+c’ means that reactions on both cathode and anode are
considered. T is temperature. Ract is the activation resistance. Rcon is the
concentration resistance. Rohm is the ohmic resistance. Ract, Rcon and
Rohm can be combined as the equivalent resistance (Requ)), reflecting the
overall resistance of the PEMFC, as [41]:
Requ = Ract + Rohm + Rcon
(5)
Then, combined with Eqs. (2)–(5), Eq. (1) can be changed into
Vst = Ecell −jRequ −
RT
αa+cF ln
(j + jloss
jex
)
−
RT
βa+cF ln
(jmax −j
jmax
)
(6)
Except for j, R, T, F, the other parameters should be identified to get
their evolution in the entire durability test. Among these parameters, jex
exhibits a decreasing trend and Requ exhibits an increasing trend, so they
can be utilized to express SOH of the PEMFC. Between 300h and 700h,
the change in Requ is twice the change in jex, as shown in Fig. 3 (a). For
the sake of clarity, only the fitting result at t = 0h is shown in Fig. 3(b).
The results of parameter identification of some other parameters are
shown in Fig. 4.
Fig. 2. The structure of the proposed method.
Fig. 3. Results of polarization curve fitting
(a) Identified value of jex and Requ (b) Polarization curve fitting at t = 0h
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International Journal of Hydrogen Energy 141 (2025) 536–549
540
The overall trend of parameters characterizing SOH of fuel cells can
be considered as a linear trend [6] so jex and Requ can be expressed as
functions of time. The variable g can be introduced to reset the trends of
jex and Requ, which can be considered as SOH, as follows:
{ jex = jex0(1 −g(t))
Requ = Requ0(1 + 2g(t))
(7)
where jex0 is the initial value of jex and Requ0 is the initial value of Requ.
Therefore, the aging model is established as follows:
Ecell = Erev −jRequ0(1 + 2g(t)) −
RT
αa+cF ln
(
j + jloss
jex0(1 −g(t))
)
−
RT
βa+cF ln
(jmax −j
jmax
)
(8)
3.2. MUSE based guidance sequence generation
3.2.1. MUSE
MUSE is a unique method that quantifies the uncertainty by quan-
tifying the form of the role of model uncertainty on the system state
transfer. A parameter δ and a matrix d are introduced into the process
equation. δ represents the amplitude of the influence and d represents
the mode of action of the uncertainty. Thus, the state space model is as
follows:
{
xk+1 = (A + diδ)xk + wk
yk = h(xk, uk) + vk
(9)
where xk represents the system state, xk = [gk, gk′]T. A represents the
state transition matrix, A = [1, Ts; 0, 1]T. yk is voltage. h( ) represents
observation equation. wk and vk represent process noise and measure-
ment noise, respectively. The elements of d is either 1 or 0. δ is initialized
to 1. 15 possible positions are set, i.e. d1 = [1, 0; 0, 0], d2 = [0, 1; 0, 0],
…, d14 = [0, 1; 1, 1], d15 = [1, 1; 1, 1], d∈{di, i = 1, 2, …, 15}. Concise
steps of MUSE are as below [7]:
(1) Prediction
̂δi,k|k−1 = ̂δi,k−1|k−1
(10)
̂xi,k|k−1 =
(
A + ̂δi,k−1|k−1di
)̂xi,k−1|k−1
(11)
Pδ
i,k|k−1 = Pδ
i,k|k−1
(12)
Pδx
i,k|k−1 =
[
Pδ
i,k−1|k−1
Pδx
i,k−1|k−1
][ dîxi,k−1|k−1
A + ̂δi,k−1|k−1di
]
(14)
(2) Update
Newton iteration method is used. ζi,k|k = [δ, xk|k]T. The boundary
condition is set as | ζiter+1 i,k|k-ζiter i,k|k |< 0.01
ζiter+1
i,k|k
= ζiter
i,k|k −
[
∇T
ζ ei
(
ζiter
i,k|k
)
⋅∇ζei
(
ζiter
i,k|k
)
+ Si
(
ζiter
i,k|k
)]−1
∇T
ζ ei⋅ei
(
ζiter
i,k|k
)
(15)
(3) d estimation
For di, when k is given, its possibility is calculated by
μi,k = λi,kμi,k−1
∑
15
i=1
λi,kμi,k−1
(16)
Fig. 4. Results of identification of some other parameters
(a) Identification value of jloss and β (b) Identification value of jmax and Erev.
Fig. 5. Flowchart of MUSE
Px
i,k|k−1 =
[
dîxi,k−1|k−1
A + ̂δi,k−1|k−1di
]
⎡
⎣
Pδ
i,k−1|k−1
Pδx
i,k−1|k−1
(
Pδ
i,k−1|k−1
)T
Px
i,k−1|k−1
⎤
⎦
[ dîxi,k−1|k−1
A + ̂δi,k−1|k−1di
]
+ Qk
(13)
X. Yu et al.
International Journal of Hydrogen Energy 141 (2025) 536–549
541
λi,k = (2π)−0.5⃒⃒Γi,k
⃒⃒× exp
[
−0.5
(
Yk −h
(
uk, ̂xi,k|k−1
))TΓi,k
−1(
Yk −h
(
uk, ̂xi,k|k−1
))]
(17)
Γi,k = ∇xg|ˆxi,k|k−1Px
i,k|k−1∇T
xg|ˆxi,k|k−1 + Rk
(18)
(4) Final estimation
̂ζk|k =
∑
15
i=1
μi,k̂ζi,k|k
(19)
In the historical stage, xk can be gotten by MUSE. For the prediction
stage, xk is gotten by the equation below:
xk+1 = Axk
(20)
xk in the both historical and prediction parts are plugged into the
observation equation to get the guidance sequence. The flow chart of
MUSE is shown in Fig. 5.
3.2.2. The principle of CNN-LSTM prediction guided by MUSE
In CNN-LSTM, data is input to the CNN, where the convolutional
layers are used for feature extraction. The pooling layers explore hidden
information. Next, the extracted feature sequence is input to the LSTM
layer for modeling. CNN acts as the front-end feature extractor, while
LSTM captures time dependency. This model effectively enhances the
performance of time series prediction [33].
The structure of prediction used in this work is different from NSD-
LSTM in Ref. [21]. In the training stage, a set of slices is constructed:
{(tm+1,Vgui,m+1), (t1,V1), (t2,V2), …, (tm, Vm), (tm+1,Vm+1)}, {(tm+2, Vgui,
m+2), (t2, V2), (t3, V3), …, (tm+1, Vm+1), (tm+2, Vm+2)}, …, {(ts, Vgui,s),
(ts-m-1, Vs-m-1), (ts-m, Vs-m), …, (ts-1, Vs-1), (ts, Vs)}. This set is used for
training CNN-LSTM. m+2 is the width of slice.
In the prediction stage, at ts+1, {(ts+1, Vgui,s+1) (ts-m, Vs-m),(ts-m+1, Vs-
m+1), …,(ts-1, Vs-1),(ts, Vs)} is input to CNN-LSTM to obtain (ts+1, Vpre,
s+1). At ts+2, {(ts+2, Vgui,s+2), (ts-m+1, Vs-m+1), (ts-m+2, Vs-m+2), …, (ts, Vs),
(ts+1, Vpre,s+1)}is input to CNN-LSTM to obtain (ts+3, Vpre,s+3). Repeat
this process to obtain the predicted values (Vpre,s+1, Vpre,s+2, …, Vpre,s +
PH). The process is illustrated in Fig. 6. Vgui,s+1 is the (s+1)th element in
guidance sequence.
3.3. Bayesian GRU based fusion
In traditional deep learning, model parameters are usually gotten
through maximum likelihood estimation, and model weights and biases
are restricted to point estimates [42]. In a Bayesian context, it works
with probability distribution. This approach helps Bayesian networks
capture the uncertainty associated with network weights. The Bayesian
GRU follows a similar structure to GRU but represents its weights as
distributions. During training, Bayesian GRU leverages variational
inference to obtain posterior distributions for model parameters. In this
study, guiding sequence and CNN-LSTM prediction results are regarded
as inputs and experimental data is regarded as the target. Bayesian GRU
is trained to explore the mapping relationship between the guiding
sequence, CNN-LSTM prediction, and experimental data. This can give a
long-term prediction with confidence interval.
3.3.1. GRU
Compared to LSTM, GRU has a simpler structure and fewer weight
parameters, which reduces the risk of overfitting [43]. A GRU consists of
an update gate zt and a reset gate rt. The reset gate directly affects the
hidden state. The update equation for GRU is as follows [43]:
rt = σ(Wrx ⋅Xt + Wrh ⋅ht−1 + br)
(26)
zt = σ(Wzx ⋅Xt + Wzh ⋅ht−1 + bz)
(27)
̃ht = tanh(W˜hh ⋅rt ⊙ht−1 + Whx ⋅xt + b˜h)
(28)
ht = (1 −zt) ⊙ht−1 + zt ⊙̃ht
(29)
where Xt is the input of current step, ht is the output of current step, σ is
the activation function, br is the deviation vector. The matrix W with
different subscripts belongs to the weight matrix.
3.3.2. Variational inference
The goal of variational inference is to create a variational distribu-
tion q0(ω) that closely approximates the true posterior p(ω|D). Typically,
q0(ω) is defined by a series of simpler and tractable distributions. To
achieve this estimation, the difference between the two probability
distributions is computed using the Kullback-Leibler (KL) divergence,
defined as follows [44]:
KL[q(x)‖p(x)] = Еq(x)
[
log q(x)
p(x)
]
=
∫
q(x)log q(x)
p(x) dx
(30)
After that, the minimization of KL divergence between q0(ω) and p
(ω|D) can be calculated as follows:
KL[q0(ω)‖p(ω|D)] = Еq0(ω)
[
log q0(ω)
p(ω|D)
]
= KL
[
[q0(ω)‖p(ω)] −Еq0(ω)[log p(D|ω)] + log p(D)
(31)
where KL[q0(ω)||p(ω|D)] is used to represent the similarity between
q0(ω) and p(ω). Eq0(ω)[log p(D|ω)] is the loss of probability, representing
the degree of model fitting to the data. The minimum of KL[q0(ω)||p(ω|
D)] can be optimized by evidence lower bound (ELBO):
L ELBO(D, θ) = = Еq0(ω)[log p(D|ω)] −KL[q0(ω)‖p(ω)]
= Еq0(ω)[log p(D|ω)] −Еq0(ω)[log q0(ω)] + Еq0(ω)[log p(ω)]
(32)
To reduce computational complexity, ELBO is estimated using un-
biased Monte Carlo sample from the variational distribution. The
expression then becomes:
L ELBO(D, θ) ≈
∑
K
k=1
log p
(
D
⃒⃒ω(k))
−log qθ
( ω(k))
+ log p
( ω(k))
.
(33)
where K is the number of samples from q0(ω). θ is the optimized
Fig. 6. Guided long-term prediction.
X. Yu et al.
International Journal of Hydrogen Energy 141 (2025) 536–549
542
parameter in backpropagation.
4. Result and discussion
The proposed method is compared with MUSE, CNN-LSTM, EKF-
LSTM, NSD-LSTM and adaptive EKF- nonlinear autoregressive exoge-
nous neural network (AEKF-NARX). Among these methods, MUSE was
previously proposed by the authors, and CNN-LSTM was built by open-
source code. EKF-LSTM, NSD-LSTM and AEKF-NARX are obtained by
compiling and debugging the code according to the algorithm structure
in the relevant papers [27,28,45]. The performance of long-term pre-
diction of these methods is discussed. Three metrics are used: root mean
square error (RMSE), mean absolute percentage error (MAPE) and per-
centage error of RUL estimation (%ErFT) [4]. Smaller absolute value of
these metrics indicates higher prediction precision. Accuracy (Acc) is
used to evaluate the overall performance of RUL estimation over the
entire lifecycle [28]. Higher Acc means better overall estimation
performance.
RMSE =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
∑n
i=1
(
ypred,i −yact,i
)2
n
√
√
√
√
(34)
MAPE = 1
n
∑n
i
⃒⃒⃒⃒⃒
ypred,i −yact,i
yact,i
⃒⃒⃒⃒⃒
(35)
%ErFT,i = RULest,i −RULact,i
RULact,i
× 100
(36)
Acc = 1 −1
n
∑n
i
⃒⃒%ErRUL,i
⃒⃒
(37)
where ypred,i is the prediction value, yact,i means the actual value, RULest,i
means the estimated RUL and RULact,i denotes the actual RUL. n is the
total amount of data. For this study, the fuel cell is considered to have a
constant aging rate, so its actual RUL is linear [28].
To validate the uncertainty quantification capability of the proposed
method, these metrics are used: Gaussian negative log-likelihood (G-
NLL), prediction interval coverage probability (PICP), and prediction
interval average width (PINAW). G-NLL characterizes the accuracy of
the prediction by measuring the cumulative error between the predic-
tion distribution and the distribution of actual data. PINAW is the
average width of the confidence interval. Small G-NLL indicates accu-
rate prediction. PICP is used to assess how well the confidence interval
covers the actual data.
Fig. 7. Datasets (a) Dataset 1, voltage (b) Dataset 1, current density (c) Dataset 2, voltage (d) Dataset 2, current density (e) Dataset 3, voltage (f) Dataset 3, cur-
rent density.
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543
G-NLL =
∑
n
i=1
log σ2
i + log 2 π +
(
yact,i −ypred,i
)2/
σ2
i
2
(38)
PICP = m
n × 100
(39)
PINAW = 1
n
∑
n
i=1
(
yupper
pred,i −ylower
pred,i
)
(40)
where μi and σi represent the mean and variance of prediction, respec-
tively. m represents denotes the number of true values within the con-
fidence interval. yupper pred, iandylower pred, irepresent represent the
upper and lower bounds of the confidence interval, respectively.
4.1. Data preprocessing
The quantity of original data is too huge to deploy algorithm, so it is
necessary to take samples. Interval is set as 1h and a new Dataset is
constructed, called Dataset 1. Moreover, load current exhibits an
apparent hierarchical pattern. There are 9 current levels: 0A, 1.78A,
4.45A, 9.51A, 10.4A, 14.85A, 20.75A, 29.65A and 35.6A [41]. A
pseudo-steady condition can be obtained by extracting the data at a
current level [21]. To comprehensively validate the performance of the
proposed method, two levels with the largest proportion are chosen:
1.78A and 14.85A. They are called Dataset 2 and Dataset 3. Dataset 1
contains the original features. It is a full current dataset. Dataset 2 and
Dataset3 seem to show decreasing trends, belonging to constant current
Fig. 8. (a) Long-term prediction of Dataset 1 (a) PH = 300h (b) Partial enlargement, PH = 300h (c) PH = 200h (d) Partial enlargement, PH = 200h (e) PH = 100h (f)
Partial enlargement, PH = 100h
Table 2
Long-term prediction metrics of different methods in Dataset 1.
Metrics
Method
PH = 300h
PH = 200h
PH = 100h
RMSE
MUSE
0.1006
0.0266
0.0223
CNN-LSTM
0.0211
0.0151
0.0156
EKF-LSTM
0.0879
0.0911
0.0620
AEKF-NARX
0.0753
0.0643
0.0295
NSD-LSTM
0.1126
0.1159
0.1196
The proposed method
0.0138
0.0103
0.0113
MAPE
MUSE
0.1021
0.0268
0.0247
CNN-LSTM
0.0250
0.0171
0.0180
EKF-LSTM
0.1115
0.1140
0.0808
AEKF-NARX
0.0829
0.0700
0.0276
NSD-LSTM
0.1299
0.1357
0.1402
The proposed method
0.0156
0.0117
0.0133
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dataset, as shown in Fig. 7.
4.2. Comparison of long-term prediction
To validate the long-term prediction performance of the proposed
method, the Datasets are split into historical data and validation data.
The historical data is used for training, while the validation data is used
to assess prediction accuracy. Three different PHs are set: 300h, 200h,
and 100h. Fig. 8 shows the results for different methods in Dataset 1.
According to the 3σ rule, the confidence interval for long-term predic-
tion is chosen to be within three standard deviations (±3σ) of the mean.
This means that there is a 99.7% probability that the predicted results
will fall within this interval.
It can be observed that the prediction of the proposed method is
closest to the actual data. Despite the deviation between the guidance
sequence provided by MUSE and the real data, the Bayesian GRU fusion
measures effectively reduce this deviation. And the actual values
consistently fall within the confidence interval. In contrast, other
methods exhibit significant deviation from the actual data. NSD-LSTM is
unable to track the actual data because it is difficult for ARIMA to
capture the periodicity of data. CNN-LSTM cannot provide prediction
with periodicity. It can be concluded that the prediction structure in the
Fig. 9. Cumulative error of Dataset 1 (a) RMSE (b) MAPE.
Fig. 10. long-term prediction of Dataset 2 and Dataset 3 (a) Dataset 2, PH = 300h (b) Dataset 3, PH = 300h (c) Dataset 2, PH = 200h (d) Dataset 3, PH = 200h (e)
Dataset 2, PH = 100h (f) Dataset 3, PH = 100h
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545
proposed method helps to improve the performance of long-term pre-
diction. EKF-LSTM and AEKF-NARX cannot provide good prediction.
Metrics for these methods in Dataset 1 are summarized in Table 2.
It is evident that the proposed method exhibits significant advan-
tages in accuracy compared to the other methods in Dataset 1. When
comparing RMSE, at PH = 300h, the proposed method reduces the error
by 86.28%, 34.60%, 84.3%, 81.67%, and 87.74% compared to MUSE,
CNN-LSTM, EKF-LSTM, AEKF-NARX and NSD-LSTM, respectively.
When comparing the cumulative errors of these methods shown in
Fig. 9, the proposed method has the smallest value, indicating that
guided information and uncertainty consideration help to improve the
performance of long-term prediction. The proposed method has the best
performance of long-term prediction in Dataset 1. To comprehensively
validate the performance of the proposed method, the comparison in
Dataset 2 and Dataset 3 is shown in Fig. 10. The accumulative errors are
shown in Fig. 11.
In Fig. 10(a), it is evident that the prediction of AEKF-NARX and
CNN-LSTM deviates from the actual data. The curve of EKF-LSTM in
Dataset 2 is difficult to exhibit a declining trend. And the prediction of
NSD-LSTM deviates from the actual data, likely due to significant
discrepancy in the trend extracted by ARIMA, consistent with findings in
Ref. [21]. Its prediction is difficult to consistently exhibit nonlinear
variations in both Dataset 2 and Dataset 3. In Fig. 10(a), (c), (e),
EKF-LSTM provides a result with a declining trend. In Fig. 10(b)–(f), it
can be found that the prediction results of MUSE are located in the
middle of the actual data trend, and the prediction curves given by the
proposed method have a similar trend with MUSE. In Fig. 10(a), it can be
seen that the prediction curve of MUSE deviates from the actual data but
the prediction result of the proposed method is located in the middle of
the actual data and has an aging trend similar to that of the actual data,
which demonstrates that the prediction framework proposed in Section
4.3.2 can effectively consider the aging trend and the nonlinear varia-
tion. In Fig. 10, the prediction of the proposed method reflects a
decreasing trend while preserving certain nonlinear variations. This
characteristic is particularly pronounced in Dataset 3. Because the
voltage and current density of Dataset 2 and Dataset 3 do not show
apparent regularity, the fuel cell can be considered to be operating under
the condition of time-varying loads. The proposed method shows good
performance under these conditions, indicating that it has good uni-
versality under different conditions. And the prediction of the proposed
method closely aligns with the actual data. Metrics for these methods in
Dataset 2 and Dataset 3 are summarized in Table 3.
It can be known in Table 3, when comparing MAPE, compared with
MUSE, CNN-LSTM, EKF-LSTM, AEKF-NARX, and NSD-LSTM, the pro-
posed method reduces cumulative errors by 21.82%, 57.74%, 2.27%,
46.42%, and 29.8%, respectively. Notably, in Dataset 2, it is evident that
Fig. 11. Accumulated errors of Dataset 2 and Dataset 3 (a) RMSE, Dataset 2 (b)
MAPE, Dataset 3 (c) RMSE, Dataset 2 (d) MAPE, Dataset 3.
Table 3
Long-term prediction of Dataset 2 and Dataset 3.
Dataset 2
Dataset 3
Metrics
Method
PH = 300h
PH = 200h
PH = 100h
PH = 300h
PH = 200h
PH = 100h
RMSE
MUSE
0.0112
0.0065
0.0052
0.0210
0.0209
0.0203
CNN-LSTM
0.0113
0.0095
0.0117
0.0319
0.0406
0.0215
EKF-LSTM
0.0065
0.0063
0.0057
0.0223
0.0213
0.0306
AEKF-NARX
0.0148
0.014
0.0088
0.0317
0.0680
0.0203
NSD-LSTM
0.0099
0.0092
0.0060
0.0268
0.0286
0.0224
The proposed method
0.0065
0.0060
0.0064
0.0191
0.0179
0.0172
MAPE
MUSE
0.0109
0.0060
0.0051
0.0268
0.0277
0.0264
CNN-LSTM
0.0115
0.0093
0.0113
0.0427
0.0508
0.0279
EKF-LSTM
0.0065
0.0059
0.0052
0.0275
0.0252
0.0373
AEKF-NARX
0.0145
0.0174
0.0088
0.0400
0.0908
0.0259
NSD-LSTM
0.0099
0.0091
0.0055
0.0344
0.0369
0.0289
The proposed method
0.0064
0.0056
0.0052
0.0227
0.0369
0.0196
Table 4
Metrics of uncertainty quantification.
Metrics
Dataset
PH = 300h
PH = 200h
PH = 100h
​
1
94.0555
61.8399
30.8184
G-NLL
2
219.8467
146.6081
73.2850
​
3
139.8390
90.6945
45.2205
​
1
100
100
100
PICP (%)
2
59.33
69.50
63
​
3
67
74
63
PINAW
1
0.1351
0.1205
0.1219
2
0.0120
0.0114
0.0109
3
0.0368
0.0402
0.0344
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the curve of EKF-LSTM first decreases and then increases, failing to
reflect the degradation of PEMFCs. However, due to its prediction falling
within the range of actual data, its precision remains high. Considering
both accuracy metrics and prediction results, the performance of EKF-
LSTM falls short of the proposed method. In Dataset 3, the precision
advantage of the proposed method is similarly significant, with MAPE
cumulative errors reduced by 2.1%, 34.76%, 12%, 49.46%, and 20.96%
compared to MUSE, CNN-LSTM, EKF-LSTM, AEKF-NARX, and NSD-
LSTM, respectively (see Fig. 11). It can be concluded that the pro-
posed method can achieve excellent long-term prediction in pseudo-
steady conditions. It can not only provide aging trends but also reflect
nonlinear variations.
To analyze the uncertainty quantification capability of the proposed
method, G-NLL, PICP and PINAW were calculated as shown in Table 4.
It can be known in Table 4 that in one Dataset, with PH decreasing,
G-NLL decreases. This suggests that the uncertainty in the prediction is
gradually decreasing as more training data are available. When
comparing the same PH value, the G-NLL on Dataset 1 is significantly
lower than that on the other Datasets. From Figs. 8 and 10, it can also be
seen that the prediction results of the proposed method is close to the
actual data in Dataset 1, resulting in a small G-NLL with a small (yact,i-
ypred,i)2, while for Dataset 2, the prediction result mainly reflect the
aging trend, and do not reach the coincidence of prediction results with
the real data in the corresponding time. For Dataset 3, at PH = 200h and
PH = 100h, the prediction results have similar cyclic variations as the
actual data, resulting in smaller G-NLL and larger PICP than that of
Dataset 2. The actual data in Dataset 1 and Dataset 2 have more pro-
nounced cyclic variations than that of Dataset 2, and it can be summa-
rized that the proposed method has a better performance in quantifying
uncertainty in cyclically varying datasets. Uncertainty quantification
performance. For Dataset 1, PICP always reaches 100% and PINAW is
always biggest, indicating that the proposed method has the best un-
certainty quantification performance in Dataset 1. Comparing Dataset 2
and Dataset 3, it can be found that higher PICP and larger PINAW occur
simultaneously. For pseudo-steady-state conditions with strong fluctu-
ations, wider confidence intervals can cover more real data.
4.3. RUL estimation
RUL estimation requires the prior knowledge of failure threshold.
Failure threshold is determined and then RUL is estimated.
4.3.1. Verification of MUSE and determination of failure threshold
MUSE is used to quantify SOH and uncertainty, and the results of
uncertainty quantification are shown in Figs. 12 and 13.
It can be observed that δ exhibits a stable trend, indicating that the
uncertainty of the aging model has a consistent impact on state transi-
tion. Additionally, it can be found that in Fig. 13, the values of δ in these
conditions are different. In Fig. 13, the probability of d2 rapidly ap-
proaches 1. According to the process equation in Eq. (9), it indicates that
uncertainty has an impact on state transition and it causes a tiny drift in
SOH estimation. This indicates that the method can quantify uncertainty
quickly. SOH estimated by MUSE is shown in Fig. 14.
Selecting different data will result in different failure thresholds. It
can be found in Fig. 14 that estimated voltage of MUSE in Dataset 1
achieves a high level of fit with the experimental data. For the Dataset 2
and Dataset 3, the estimated voltage constantly follows the experimental
data, located in the middle of the experimental data. The good perfor-
mance of voltage fitting means excellent estimation of SOH. And the
curve of SOH reflects an obvious trend which is useful for determination
of failure threshold. The effectiveness of MUSE in estimating SOH and
model uncertainty is validated.
The curves of SOH in Fig. 14 are regarded as the reference curve. Its
maximum SOHmax is regarded as the failure threshold, and then the time
of end of life (tEoL) can be determined [11]. For Dataset 1, SOHmax is
0.6283, tEoL is 745h. For Dataset 2, SOHmax is 0.8010, tEoL is 647h. For
Dataset 3, SOHmax is 0.7473, tEoL is 698h.
4.3.2. SOH prediction and RUL estimation
RUL estimation requires information about SOH prediction. Among
Fig. 12. Estimated δ (a) Dataset 1 (b) Dataset 2 (c) Dataset 3.
Fig. 13. Estimated d (a) Dataset 1 (b) Partial enlargement, Dataset 1 (c) Dataset
2 (d) Partial enlargement, Dataset 2 (e) Dataset 3 (f) Partial enlargement,
Dataset 3.
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International Journal of Hydrogen Energy 141 (2025) 536–549
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all the compared methods, only MUSE and EKF-LSTM can obtain SOH.
Therefore, in Fig. 15, Table 5, and Table 6, the proposed method is
compared with MUSE and EKF-LSTM. SOH prediction when PH = 300h
is shown in Fig. 15. %ErFT is listed in Table 4.
It can be observed in Fig. 15 that SOH prediction of the proposed
method is close to the SOH reference. And the estimated RUL is closest to
300h. Predicted SOH has a good fitting with the SOH reference. It is
necessary to explain that EKF-LSTM cannot provide effective SOH pre-
diction because it is difficult for EKF to estimate SOH accurately. It can
be seen in Fig. 15 that the curve of EKF-LSTM at PH = 300h deviates
from the reference so its %ErFT is not listed in Table 5.
In Dataset 1, the proposed method exhibits significantly low cumu-
lative errors. Compared with MUSE, it reduces %ErFT by 79.63%. This
indicates that only the proposed method can achieve RUL estimation for
Fig. 14. Estimation of voltage and SOH (a) SOH estimation in Dataset 1 (b) Voltage estimation in Dataset 1 (c) SOH estimation in Dataset 2 (d) Voltage estimation in
Dataset 2 (e) SOH estimation in Dataset 3 (f) Voltage estimation in Dataset 3.
Fig. 15. SOH prediction when PH = 300h (a) Dataset 1 (b) Dataset 2 (c) Dataset 3.
X. Yu et al.
International Journal of Hydrogen Energy 141 (2025) 536–549
548
the full current dataset. For Dataset 2, compared with MUSE and EKF-
LSTM, the proposed method reduces RUL estimation cumulative errors
by 77.66%, and 75.98%, respectively. In Dataset 3, at PH = 300h, the
proposed method outperforms MUSE in RUL estimation accuracy with
improvement of 90.91%. The accuracy enhancement highlights the
significant advantage of the proposed method in RUL estimation, mak-
ing it deployable for monitoring PEMFC. The data from the last 400h of
lifetime is used to validate performance of RUL estimation on the full
lifecycle. Acc is computed and satisfactory horizon (SH) is used. SH
means the duration in which the estimated RUL falls within an error
range [26]. Bigger SH means stronger performance to give satisfactory
results. In this validation, the PEMFC is considered to undergo uniform
aging over time, resulting in a linear decrease in its lifespan. Its actual
RUL is linear. This viewpoint is also supported by Ref. [28]. The result of
RUL estimation is shown in Fig. 16.
In Fig. 16, it is evident that RUL estimation for Dataset 2 exhibits
obvious fluctuation. The reason may lie in the presence of multiple
anomalous data points in the experimental data, causing the SOH esti-
mation at those specific data points to deviate from the normal data. For
Dataset 1 and Dataset 3, the curves of RUL estimation are almost entirely
within the boundary. For Dataset 3, it can be observed that the estimated
RUL significantly decreases between 470 hours and 530 hours. By
comparing Fig. 7(e), it is evident that around 500 hours, the voltage of
the fuel cell is quite low, approximately 0.62. This low voltage may lead
to a transient fault at that moment. According to Table 5, the proposed
method demonstrates superior accuracy in full-life-cycle RUL estima-
tion. Compared to MUSE, the proposed method achieves a 73.07% in-
crease in Acc for Dataset 1. Furthermore, even under the constant
current conditions, the proposed method maintains a substantial
advantage of accuracy, indicating its superior full lifecycle RUL esti-
mation accuracy in scenarios with high uncertainty.
5. Conclusions
In order to realize long-term aging prediction and RUL estimation of
PEMFC under the dynamic load cycle condition, a Bayesian GRU fused
MUSE and CNN-LSTM hybrid method is proposed. In this method, MUSE
is used to generate guidance sequences to guide the training and pre-
diction of CNN-LSTM. Bayesian GRU is utilized to fuse the prediction
results of MUSE and CNN-LSTM. The main conclusions are as follows:
1) Under the dynamic load cycling condition, the proposed method
shows excellent performance of long-term aging prediction. The
prediction curve of Dataset 1 is close to the actual curve. For the
pseudo-steady conditions used in this paper, the results of the pro-
posed method not only can show the aging trends but also reflect the
nonlinear variations. For Dataset 1, compared with MUSE, CNN-
LSTM, EKF-LSTM, AEKF-NARX and NSD-LSTM, the proposed
method reduces under the full current dataset the cumulative RMSE
76.32%, 31.36%, 85.31%, 79.07% and 89.83%, respectively.
2) The capacity of uncertainty quantification of the proposed method in
Dataset 1 is best. Bayesian GRU fuses the prediction results of MUSE
and CNN-LSTM, providing confidence intervals for long-term
Table 5
%ErFT of different methods.
Method
PH = 300h
PH = 200h
PH = 100h
Dataset 1
MUSE
−72.67
–66
−64
EKF-LSTM
–
–
–
The proposed method
3.33
19
23
Dataset 2
MUSE
−42
−6.50
88
EKF-LSTM
50
25
52
The proposed method
7
3.50
23
Dataset 3
MUSE
¡22
−65.50
−29
EKF-LSTM
¡25
10.50
99
The proposed method
2
¡56
11
Table 6
Acc and SH for the last 400h
Dataset
Metrics
MUSE
EKF-LSTM
The proposed method
1
Acc
0.3958
−5.1500
0.6850
SH/h
162
–
341
2
Acc
0.4513
−0.4322
0.6374
SH/h
247
–
286
3
Acc
0.7128
−0.4287
0.8125
SH/h
243
–
379
Fig. 16. RUL on the full life cycle (a) Dataset 1 (b) Dataset 2 (c) Dataset 3.
X. Yu et al.
International Journal of Hydrogen Energy 141 (2025) 536–549
549
prediction. It is the first time to quantify the uncertainty of long-term
aging prediction under the dynamic load cycle condition. The veri-
fication of Dataset 1 shows that PICP of the long-term prediction
result is 100%, indicating that the proposed method has strong
ability of uncertainty quantification.
3) The proposed method can realize high-precision estimation of RUL.
The highest Acc obtained in the full lifecycle estimation of RUL is
0.8125. This method can be used to estimate RUL on the full lifecycle
of PEMFC.
CRediT authorship contribution statement
Xiaoran Yu: Writing – review & editing, Writing – original draft,
Validation, Software, Methodology, Formal analysis, Data curation,
Conceptualization. Yang Yang: Writing – review & editing, Validation.
Yanjun Liu: Methodology. Wenchao Zhu: Supervision. Changjun Xie:
Resources, Project administration, Funding acquisition.
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.
Acknowledgements
This work was supported by the National Key Research and Devel-
opment Project of China (2020YFB1506802); and the Key Research and
Development Project of Guangdong Province (2020B0909040004).
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