Rania Mezzi

Nadia Yousfi-Steiner

Marie Cécile Péra

Daniel Hissel

Laurent Larger

An Echo State Network for fuel cell lifetime prediction under a dynamic micro-cogeneration load profile

Improving Proton Exchange Membrane Fuel Cell durability is a key that paves the way to its large scale industrial deployment. During the last five years, the prognostics discipline emerged as an interesting field for Proton Exchange Membrane Fuel Cell state of health prediction and lifetime estimation. The information provided by the prognostic module is crucial for optimizing the control strategy to extend the fuel cell lifetime. In this paper, an approach based on Echo State Network for fuel cell prognostics under a variable load is developed. The novelty of this paper is to perform prognostics under a variable load profile without prior knowledge of this latter. Two solutions are developed in this work. The first one consists of evaluating the remaining useful lifetime under a repeated load cycle. The second one is based on using Markov chains to generate estimations of the future load profile, allowing thus to overcome the need of real future load profile prior knowledge. Both pro-posed solutions give accurate prediction results of proton exchange membrane fuel cell remaining useful lifetime, with low uncertainties.

Introduction

The search for new energy sources is a key issue for humanity. Currently, most energy is produced by fossil fuels. According to the International Energy Agency (IEA), the global energy consumption continues to increase, and its impact on the environment is increasingly dangerous. Two-thirds of greenhouse gas emissions and 80% of carbon dioxide (CO 2 ) production result from energy consumption [START_REF]Climate changetopics[END_REF]. This has motivated researchers to move towards clean energies to reduce CO 2 emissions, and thus reduce global warming.

One of the most promising alternative solutions is the use of the hydrogen energy vector, and in particular the use of fuel cells (FC), which are electrochemical converters that produce electricity. Among the available technologies of fuel cells, the Proton Exchange Membrane Fuel Cell (PEMFC) is a device capable of clean electricity production. For this, the used hydrogen must be produced by electrolysis, using renewable energies. PEMFC are intended to replace combustion engines for vehicular applications and subsequently reduce the grenhouse gases and CO 2 emissions. The fuel cell durability is a major focus area, especially with the increasing need for clean and sustainable energy [START_REF] Pei | Main factors affecting the lifetime of Proton Exchange Membrane fuel cells in vehicle applications: a review[END_REF]. The Proton Exchange Membrane Fuel Cell is one of the most popular FC types, especially for transportation applications [START_REF] Kim | The effects of air stoichiometry and air excess ratio on the transient response of a PEMFC under load change conditions[END_REF]. PEMFCs are time corresponds to the remaining useful lifetime (RUL). A schematic presentation of the prognostic is proposed in Fig. 1.

The fuel cell lifetime depends strongly on the environmental factors and operating conditions [START_REF] Mezzi | Operating conditions control for extending proton exchange membrane fuel cell lifetime[END_REF] such as: ambient temperature, vibrations, air impurities and contaminants. Start-stop, load changing and idling induce cycling conditions including pressure, load, relative humidity, potential and stress. These conditions may accelerate the FC degradation [START_REF] Zhao | A review of polymer electrolyte membrane fuel cell durability for vehicular applications: Degradation modes and experimental techniques[END_REF]. Therefore it could be interesting to take these stress factors into account when predicting the FC degradation. Nevertheless, the objective of this work is to propose an efficient prognostics tool while avoiding additional cost (related to adding sensors). According to the authors in [START_REF] Pei | Main factors affecting the lifetime of Proton Exchange Membrane fuel cells in vehicle applications: a review[END_REF], dynamic load changing has the greatest effect on the fuel cell degradation compared to the listed operating and environmental conditions. The underlying hypothesis for the prediction is that the operating conditions' impact is kept the same in the future as in the past. Consequently, only the load changing effect is taken into account for RUL prediction.

Another important issue in prognostics consists in choosing a suitable Health Indicator (HI). Different health indicators were employed in the literature. Guidelines and metrics for defining the PEMFC health indicators and the end of life criteria according to the load profile were presented by Jouin et al. [START_REF] Jouin | Estimating the end-of-life of PEM fuel cells: Guidelines and metrics[END_REF]. More recently, the authors in [START_REF] Liu | Prognostics methods and degradation indexes of proton exchange membrane fuel cells: A review[END_REF] presented a review of the latest prognostics approaches applied to the PEMFC and the innovations of health indicators. Usually a single HI based on voltage [START_REF] Morando | Proton exchange membrane fuel cell ageing forecasting algorithm based on Echo State Network[END_REF], impedance [START_REF] Lee | Development of a method to estimate the lifespan of proton exchange membrane fuel cell using electrochemical impedance spectroscopy[END_REF], or active surface area [START_REF] Zhang | An unscented Kalman filter based approach for the healthmonitoring and prognostics of a polymer electrolyte membrane fuel cell | PHM society[END_REF] is used for prognostics application. The authors in [START_REF] Chen | A novel health indicator for PEMFC state of health estimation and remaining useful life prediction[END_REF] introduced a fusioned health indicator based on the stack voltage, the power and the inner resistance. The authors estimated the FC State Of Health (SOH) by calculating a Mahalanobis distance and a geodesic distance between the FC intitial health state and the health state at time t. After that, the RUL was estimated based on an Unscented Particle Filter. A comparison between the proposed health indicator and the voltage on a prognostic task was realized, and the authors confirmed that the proposed HI and the voltage gave similar results in terms of lifespan.

The objective of the present work is to develop a tool which is able to estimate the lifespan of a PEMFC and to predict its behavior in order to anticipate failures. The components aging leads to the decrease of stack performance. This can be seen through the stack power drop over time. Under a constant current solicitation, degradations induce the voltage decrease with time, which induces a decline in power. Thus, the common point in the PEMFC degradation mechanisms is their effect on the fuel cell voltage [START_REF] Pahon | Long-term tests duration reduction for PEMFC μ-CHP application[END_REF]. Therefore, in the present work, we choose the voltage as a state of health indicator. This choice has been motivated by the availability of this signal measurement (ease of implementation + non-intrusive measure) without additional cost, wich is highly appreciated on real applications. Besides, under a known current solicitation, the voltage evolution reflects the whole fuel cell state of health and not only one or a few components' SOH. In other words, predicting the fuel cell voltage is a costless and efficient way to prevent fuel cell degradation.

In the last decade, the FC prognostics has naturally become a major research topic. Today, this theme is still under development. Different prognostics approaches have indeed been applied to the FC in recent years. The choice of the prognostic approach is important. It depends on: (i) the type of knowledge that one has about the system, more precisely the influence of the degradations on the system behavior and (ii) the access to the data to measure this degradation. A detailed state of the art of the different methods applied to the fuel cell is drawn in the following paragraphs.

Prognostics approaches 2.2.1. Model based approaches

Analytical models are based on deep knowledge of physico-chemical phenomena and generally depend on many parameters. Therefore the use of an analytical approach to model a PEM fuel cell system is not an recognized for their fast start-up, high efficiency, low operating temperature and pollutants-free emissions [START_REF] Oh | Optimal operation of a 1-kW PEMFC-based CHP system for residential applications[END_REF]. However, the average fuel cell lifetime is still below the 2020 department of Energy (DOE) targets [START_REF]DOE technical targets for fuel cell systems and stacks for transportation applications. Energy[END_REF]. To be competitive with the combustion engine, FC lifetime must exceed 5000 h of operation for a light duty vehicule application with less than 10% performance loss, and 40,000 h for stationary ones with less than 20% performance loss.

The aging process is inevitable, but an accurate prognostics tool may provide relevant feedback to the control to mitigate degradations and so help extending fuel cell lifespan. Prognostic allows the prediction of the FC future behavior and subsequently contributes to the reduction of the maintenance costs and improvement of FC reliability.

In this work, two solutions are proposed and discussed for the prediction of the residual life of the PEMFC under a dynamic load profile without prior knowledge of the latter. The first one consists of repeating the known part of the load profile and the second one is based on the use of Markov chains to generate load profiles from the available part of the charge request. The aim of this work is not to realize a long term prediction of the current profile: it doesn't make sense because what the user will decide to do in the future could hardly be predicted. What one could predict is the behavior of the fuel cell with the voltage as an indicator of the performance, with the underlying hypothesis that the system will be used in the future as it was in the past (the operating conditions' impact is kept the same in the future as in the past). Since the current information is missing, we have proposed different possibilities: a repetitive cycle as a deterministic approach or a cycle generated by Markov chain as a statistical approach.

The present paper is organized as follows. Section 2 is dedicated to introducing the fuel cell prognostics and how to estimate the Remaining Useful Lifetime. Then a state of the art of the different prognostics approaches applied to the PEMFC is presented. A description of the chosen prognostics tool, its training algorithm and the optimization algorithm used to set its parameters is given in Section 3. An explanation of the developed methdology to predict the fuel cell lifetime under a variable load is then presented. In Section 4, we present the prognostics results under a micro CHP load profile. Experimental data generated in the framework of the European project SAPPHIRE are used to validate the proposed approach. The generated data corresponds to the stack degradation voltage measurement under a micro CHP load profile.

Research context

Fuel cell prognostics

Prognostics allows the prediction of the future system behavior. It is an important concept in decision making, aiming at alleviating the premature and/or accelerated aging of a system.

Prognostics enables the estimation of the remaining time before the system performance reaches the end of life threshold. This estimated capable of managing variable mission profiles was developed [START_REF] Jouin | Degradations analysis and aging modeling for health assessment and prognostics of PEMFC[END_REF].

The authors in [START_REF] Kimotho | PEM fuel cell prognostics using particle filter with model parameter adaptation[END_REF] also used PFs for the FC prognostics, which enabled them to win the challenge of IEEE 2014 PHM [START_REF]PHM Data Challenge[END_REF]. The same voltage evolution models, proposed in [START_REF] Jouin | Contribution au pronostic d'une pile à combustible de type PEMFC : approche par filtrage particulaire[END_REF], were tested in addition to two others (a polynomial model and a logarithmic one). However, the particularity of this work consisted in taking into account the recovery of fuel cell performance after a characterization (IV curve and EIS measurements) by introducing a rejuvenation factor, thus making it possible to more effectively adapt the parameters of the aging model to follow the degradation evolution. Incorporating rejuvenation into prediction improved its results and the majority of the calculated errors weree less than 5%. However, this method requires a priori knowledge of the characterizations periodicity.

A boundgraph was presented in the work of Jha et al. [START_REF] Jha | Particle filter based hybrid prognostics of proton exchange membrane fuel cell in bond graph framework[END_REF] to model the FC degradation. The obtained model as well as the experimental data were the same as those used in the work of Bressel et al. [START_REF] Bressel | Extended Kalman filter for prognostic of proton exchange membrane fuel cell[END_REF]. The same health indicator α (t) was also chosen and its estimate was obtained in the form of a probability density using PFs. The method was validated by simulation on data from two degradation tests (constant load and variable load: cogeneration). The obtained results were compared to the work of Bressel et al. [START_REF] Bressel | Extended Kalman filter for prognostic of proton exchange membrane fuel cell[END_REF]. The authors confirmed that the obtained results using the particle filters are better in terms of precision, however they were more expensive in computation time. Besides, it is important to emphasize the importance of initializing the PF parameters to obtain good prediction results and the need of data to choose the initial parameters. The choice of the initial parameters, in particular the number of particles, their initial distribution as well as the parameters of the model, are essential to ensure the convergence of the filter towards the real state. To evaluate the effect of the initialization of the parameters on the performance of the particle filter, two particle filters were compared in the work of Baraldi et al. [START_REF] Baraldi | Model-based and data-driven prognostics under different available information[END_REF]. The exact value of the parameters was used for the first while, for the second, only a distribution was given. The second test corresponded to the introduction of uncertainty on the model. The authors proved that the results are better in the first case, with more accurate predictions.

The developed approach in [START_REF] Jha | Particle filter based hybrid prognostics of proton exchange membrane fuel cell in bond graph framework[END_REF] was incorporated in a multiagent system to estimate the RUL of a multistack PEMFC system [START_REF] Liu | Prognostics of a multistack PEMFC system with multiagent modeling[END_REF]. Several PFs were used in the multi agent system. Each agent communicated with the others to improve the prognostics accuracy. The authors affirmed that the multiagent approach gave better prognostics results than a single PF.

Several papers in the literature also modeled the FC equivalent circuit using the Electrochemical Impedance Spectroscopy (EIS) characterization. The authors in [START_REF] Lechartier | Proton exchange membrane fuel cell behavioral model suitable for prognostics[END_REF] proposed a combination of a static model and a dynamic model. The static model is based on a Butler-Volmer law model using the physical parameters of the fuel cell. The dynamic model is based on an equivalent electrical circuit. This combination of models makes it possible to integrate the dynamic evolution of aging for the purpose of prognosis.

The work presented by the authors in [START_REF] Lee | Development of a method to estimate the lifespan of proton exchange membrane fuel cell using electrochemical impedance spectroscopy[END_REF] consists of estimating the remaining useful lifetime of a high temperature single cell by monitoring the time constant of the cathode (product of charge transfer resistance and double layer capacity). To do this, the first step was to estimate the charge transfer resistance and the double layer electrical capacity of the cathode from the EIS performed every 168 h at different current levels. These parameters reflected the degradation state of the cathode active surface area, and is also indicated in the work of Laffly et al. [START_REF] Laffly | Polymer electrolyte membrane fuel cell modelling and parameters estimation for ageing consideration[END_REF]. Thereafter, the temporal evolution of these parameters as well as the time constant of the cathode were modeled. The equivalent circuit used for the identification of the parameters is presented in Fig. 2.

The inverse of the cathode time constant was calculated for several currents. Given the absence of data for other FCs, the authors used the data obtained from different current levels as data from other FCs. The end of life of the PEMFC was reached when the inverse of the time constant of the cathode reached 0. Then the remaining life L res was given by equation ( 1): easy task, as it is a multi-physics and multi-scale system with coupled phenomena. Moreover, all the degradation phenomena that occur in the PEMFC are not fully known yet, which makes the design of a complete physical model of FC degradation particularly difficult. Indeed, many degradation phenomena happen at micro and nano scales. This makes it complicated to take into account the different levels and scales of degradation, especially as many of the parameters used to model degradations may not always be available [START_REF] Pecht | A prognostics and health management roadmap for information and electronics-rich systems[END_REF]. The PEMFC degradation mechanisms are also strongly correlated [START_REF] Wang | A review of polymer electrolyte membrane fuel cells: technology, applications, and needs on fundamental research[END_REF]. Therefore reproducing them via a model is very complicated. The lack of information and measurements imposes the neglect of some phenomena. Therefore, the available physical models in the litterature generally describe only some specific degradations. For instance, the authors in [START_REF] Jahnke | Invited) Physical modeling of performance, membrane and catalyst degradation in PEMFC[END_REF] consider that the membrane and the electrodes are the critical components. Other components degradation are not taken into account. This difficulty in setting up a physical model comes from: (i) either the lack of physical models regarding degradation phenomena, (ii) or when the model to be set up is too complex and / or too computationally expensive. If complete and detailed degradation models would exist, they could be useful for system design and/or understanding phenomena. Nonetheless, their application for online prognostic would be limited by the complexity of their implementation on a real system. Therefore, semi-empirical, empirical and equivalent circuit models were widely used in the literature. The model parameters are generally identified using filter approaches.

Particle filters (PF) and Kalman filters (KF) have been successfully used for PEMFC aging models parameters estimation. Bressel et al. [START_REF] Bressel | Extended Kalman filter for prognostic of proton exchange membrane fuel cell[END_REF] used an extended Kalman filter to predict the FC state of health and its rate of degradation. The developed approach was applied to two different FCs. The first one was subjected to a constant load profile, and the second one was operated under a dynamic load profile, representative of a micro CHP application [START_REF] Bressel | Remaining useful life prediction and uncertainty quantification of proton exchange membrane fuel cell under variable load[END_REF]. For this, an empirical degradation model was proposed, and the model parameters were determined from experimental data. The authors consider that the parameters most influenced by the degradation of the PEMFC are mainly the limit current and the ohmic resistance estimated using an optimization algorithm. The variation of these two parameters was expressed as a function of a single parameter α (t), which is the health indicator. The authors emphasize the dependence of the prognosis results on the initial configuration of the algorithm (the initial state parameters). The end-oflife criterion was defined according to the authors as follows: the FC endof-life was estimated when the predicted state of health reaches a defined threshold α max = 10% for a constant load profile and 75% for a dynamic load profile.

The prediction of the RUL of a PEMFC under a cyclic load profile, using a Kalman filter, was proposed in Zhang et al. [START_REF] Zhang | An unscented Kalman filter based approach for the healthmonitoring and prognostics of a polymer electrolyte membrane fuel cell | PHM society[END_REF]. A first order simplified physical model of catalyst aging was developed. This model established a link between the operating conditions (temperature) and the degradation of the electrochemical surface area. The evolution of the active surface was considered as an indicator of degradation. The FC end of life corresponded to a percentage of loss of the active surface (~50% of the active surface at t = 0). The obtained simulation results of the RUL estimation were satisfactory, however the active surface reduction is not the only indicator of the FC degradation and monitoring only this single parameter can lead to an error in prognostics.

Jouin et al. [START_REF] Jouin | Degradations analysis and aging modeling for health assessment and prognostics of PEMFC[END_REF] used a PF for PEMFC prognosis purposes. Three empirical models of voltage evolution were compared: a linear model, a log-linear model and an exponential model. The model parameters are estimated using the PF. Part of the data was used for filter training, its parameters were chosen by performing a sensitivity analysis and the initializations of the particles followed a uniform distribution. This method was applied to 2 stacks of 5 cells subjected to a constant load profile. The authors stated that the results were better using the loglinear model. However, the uncertainties and the dispersion of the predictions were not negligible [START_REF] Jouin | Prognostics of PEM fuel cell in a particle filtering framework[END_REF]. In addition, the model did not take into account variations in load and operating conditions. Thus, a fourth physical aging model describing the degradations of the PEMFC and

L res = L av p k (t) -t (1)
With L av being the average value of the lifetimes estimated at different currents and p k , the ratio between the time constant of the PEMFC tested and the time constant calculated from the model developed in [START_REF] Lee | Development of a method to estimate the lifespan of proton exchange membrane fuel cell using electrochemical impedance spectroscopy[END_REF]. The proposed method has not been validated experimentally.

Other approaches, such as data driven and hybrid methods, are generally used when it is not possible to set up a complete physical model of the system.

Data driven approaches: Machine learning and statistical models

Data driven approaches have been successsfully applied to estimate the PEMFC state of health since it can be performed without a deep understanding of the aging process. Furthermore, data driven approaches are able to deal with systems having several interactions and features acting on their performance. Indeed, data based approaches can model the parameters' correlations and the operating conditions effects. Different data based prognostics approaches exist in the literature such as Adaptive Neuro-Fuzzy Inference System (ANFIS) [START_REF] Vural | Performance prediction of a proton exchange membrane fuel cell using the ANFIS model[END_REF], Summation Wavelet-Extreme Learning Machine (SW-ELM) [START_REF] Javed | Data-driven prognostics of proton exchange membrane fuel cell stack with constraint based summation-wavelet extreme learning machine[END_REF], Echo State Network (ESN) [START_REF] Morando | Proton exchange membrane fuel cell ageing forecasting algorithm based on Echo State Network[END_REF], ELM, etc. These approaches require the availability of a large datasets.

Statistical and black box models were widely applied to PEMFC prognostics. The authors proposed in [START_REF] Ibrahim | Wavelet-based approach for online fuel cell remaining useful lifetime prediction[END_REF] an autoregressive integrated moving average (ARIMA) model and a polynomial model to predict the PEMFC power. The data used for prediction came from two different fuel cells ran under two different load profiles (a constant load profile and a variable one) for 1000 h. The data was processed using a discrete wavelet transform (DWT) in order to decrease the observations number (168 h of operation were represented by 21 h). A comparison of the polynomial and ARIMA model was done with and without DWT, and the authors confirmed that the best results were obtained using a combined ARIMA-DWT. Then the RUL was estimated using this model and the failure threshold was set at a loss of 5.5% of the initial power. The authors highlighted the size decrease of the required database for the prognostics but they did not comment about a potential information loss when compressing data.

Onanena et al. [START_REF] Onanena | Fuel cells static and dynamic characterizations as tools for the estimation of their ageing time[END_REF] proposed a pattern-recognition-based approach to estimate the remaining useful lifetime of the fuel cell. Dynamic characterizations (Electrochemical impedance spectroscopy) were used first, then both static (polarization curves) and dynamic characterizations were used to estimate parameters of the stack, giving an idea about the evolution of the FC state of health. Within the 29 estimated parameters, 24 were selected to predict the FC remaining useful lifetime using linear regression. This method was validated on 2 different PEMFCs to test the reproducibility.

Silva et al. [START_REF] Silva | Proton exchange membrane fuel cell degradation prediction based on Adaptive Neuro-Fuzzy Inference Systems[END_REF], proposed a method based on the use of an Adaptive Neuro Fuzzy Inference System, to predict the variation of the FC voltage. The ANFIS was already used by Fennie et al. [START_REF] Fennie | Fuzzy logic-based state -of-health determination of PEM fuel cells[END_REF] for FC diagnosis as well as the prediction of polarization curves in the work of Vural et al., [START_REF] Vural | Performance prediction of a proton exchange membrane fuel cell using the ANFIS model[END_REF]. The voltage was split into two components: normal operation and external perturbations. Only the normal operation was used to train the ANFIS. The performance of this approach strongly depends on the quantity of data as well as its quality. The authors used data from two fuel cells (1000 h long term test under a constant current) to validate the proposed method.

A Grey Neural Network Model (GNNM) optimized using a PSO algorithm was proposed by Chen et al. in [START_REF] Chen | Degradation prediction of proton exchange membrane fuel cell based on grey neural network model and particle swarm optimization[END_REF]. The advantage of this work is that different operating conditions such as the relative humidity, the temperature and the hydrogen pressure were taken into account when performing the FC prognostics. The optimized PSO-GNNM was iteratively trained with newly measured data using the moving window method. The influence of the window size variation on the PEMFC degradation prediction performance under a static load current was investigated. The method was applied on three datasets, carried out under a constant load profile. The proposed method performance was also compared to ANFIS. The authors confirm that the method they have proposed is more accurate when predicting the voltage, with a training base lower than the used one to train the ANFIS model.

A PEMFC performance prediction model using a supervised learning technique, the relevance vector machine (RVM), was proposed in Wu et al. [START_REF] Wu | A modified relevance vector machine for PEM fuel cell stack aging prediction[END_REF]. Experimental aging data was used to construct an RVM based on the voltage signals of a PEMFC. This model was then confronted with degradation data from two experimental tests of a 1.2 kW fuel cell. The results obtained were then compared to a conventional support vector machine (SVM). RVM and SVM are data-based approaches both based on the concept of kernel functions. SVM is used mainly for classification and nonlinear regression problems. Although few differences between these two models were observed, better predictions were made using a modified RVM model compared to conventional SVM, especially in the case of limited available data for prediction.

The SW-ELM algorithm was proposed by the authors in [START_REF] Javed | Data-driven prognostics of proton exchange membrane fuel cell stack with constraint based summation-wavelet extreme learning machine[END_REF]. ELM is a popular learning algorithm for Single hidden Layer Feedforward Neural Networks (SLFN). SW-ELM is an improved version of ELM. It is the combination of neural networks and wavelet theory. This algorithm also used the PEMFC voltage drop as a health indicator assuming that the aging process is irreversible. The algorithm was validated using the data from the PHM 2014 challenge [38]. Satisfactory performance was observed. This approach was confronted with two other prognosis algorithms, namely the Extreme Learning Machine (ELM) and the Leaky-Echo State Network (Leaky-ESN). The prognosis algorithms were run around the half-life of two PEMFCs with a lifetime of 1150 and 1750 h under a constant load profile. However, according to the authors, the predictions of the SW-ELM algorithm were closer to the real value than the predictions of ELM and Leaky-ESN. The authors also used the prognostic algorithm developed to predict the stack voltage under a variable load profile (cogeneration) [START_REF] Javed | PEM fuel cell prognostics under variable load: A data-driven ensemble with new incremental learning[END_REF]. The prediction results were encouraging, however the load profile was an input of the prognosis algorithm (so it is supposed to be known in advance).

More recently, a data driven method based on the use of deep learning particularly Grid Long Short-Term Memory (G-LSTM) Recurrent Neutral Network (RNN) for predicting the fuel cell stack voltage was developed in [START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF]. A comparison between RVM and G-LSTM on a prognostics task was performed on different data sets of different sizes and load profiles. The authors noticed the influence of the dataset and the voltage recovery phases effect on the accuracy. The authors affirmed that the accuracy decreased when using LSTM, compared to G-LSTM especially at the ripple parts of the voltage. However G-LSTM was able to predict the voltage trend accurately despite the voltage ripples. The effect of changing the sliding windows sizes was introduced, and the authors confirmed that small sliding windows gave smaller voltage prediction errors.

The Echo State Network has good performances in terms of computational complexity and convergence rate. A comparison between several ANN (feed-forward, ANFIS and cascade neural network) in a voltage prediction task was performed by the authors in a previous work [START_REF] Mezzi | Contrôle tolérant au vieillissement dans des systèmes pile à combustible PEMFC[END_REF]. ESN showed better results than the others in terms of accuracy and errors' dispersion. In [START_REF] Morando | Contribution au Pronostic de Durée de Vie d'une Pile à Combustible à Membrane Echangeuse de Protons Approche par réseaux de neurones à réservoir (Reservoir Computing[END_REF] the authors proposed an optimized echo state Machine (LSSVM). The RPF improved the standard PF algorithm by diminishing the degeneration phenomenon and loss of diversity among the particles. The prognostics results were more accurate than PF and RPF performed alone, since LSSVM is able to capture the non linearities of voltage degradation. RPF estimated the parameters of the log-linear model, chosen by the authors as the state model of voltage drop, and received the volatge estimated by LSSVM as a new system observation. Consequently RPF predicted the system state until the voltage reached the failure threshold. Similar work was proposed by the authors in [START_REF] Cheng | A prognostic framework for PEMFC based on least squares support vector regression-particle filter[END_REF] based on the use of particle filters and Least Squares Support Vector Regression instead of LSSVM.

More recently, the authors in [START_REF] Xie | Prognostic for fuel cell based on particle filter and recurrent neural network fusion structure[END_REF] associated a PF and a Long-Short Term Recurent Neural Network (LSTM) to perform prognostics of PEMFC. The FC degradation was described by a linear model. Its parameters were identified using a PF in the training phase and a LSTM to update these parameters during the prediction phase. A comparison between PF and the fusion between PF and LSTM on RUL estimation and short term degradation prediction was performed. The results showed that the fusion method is more accurate and more robust to external disturbances than the PF alone for RUL estimation. Compared to PF, LSTM, and Nonlinear Autoregressive Neural Network (NARNN), the fusion method showed its superiority in short term degradation prediction with an accurate degradation prediction regardless of time increase.

A hybrid method based on the use of a data-driven approach particularly ANFIS to predict the future degradation trend was developped in [START_REF] Liu | Remaining useful life estimation for proton exchange membrane fuel cells using a hybrid method[END_REF]. ANFIS parameters were optimized using the Particle Swarm Optimization (PSO) algorithm. Complete degradation data were obtained from ANFIS, and the RUL was estimated by using a semiempirical degradation model of PEMFC developped in [START_REF] Bressel | Extended Kalman filter for prognostic of proton exchange membrane fuel cell[END_REF] and an Adaptive Unscented Kalman Filter (AUKF). The authors in [START_REF] Liu | Remaining useful life estimation for proton exchange membrane fuel cells using a hybrid method[END_REF] confirmed that the use of an AUKF algorithm solved the problem of the traditional UKF initial parameters setting.

The developed tools described above have been applied in majority to prognostics under constant load. The prognostics under dynamic load was less tackled in the literature. Only few papers present prognostics tools under micro-CHP application. In this use case, a variable load profile is used as an input for the prognostics tool. Therefore, in the existing papers in the literature, the authors used the real load profile as an input of the prognostic tool. This a priori knowledge of the real load profile is a strong assumption, and remains a bottleneck of prognostics development in actual applications. The work presented in this paper is focused on the prognostics under a dynamic load profile, with generation of the future load profile: first, by repeating the known part of the load profile and then using the Markov chains to make the estimation.

In order to reach this goal, we propose a prognostics tool based on the use of Neural Networks. Data-driven approaches are suitable for complex systems since they intelligently estimate the system future states from the available data using articifial intelligence tools. A comparison between different machine learning tools: ESN, ANFIS, Feed Forward Neural Networks and Cascade-forward neural network on a prognostic task under constant load was realized in [START_REF] Mezzi | Contrôle tolérant au vieillissement dans des systèmes pile à combustible PEMFC[END_REF]. The obtained results by the ESNs are very encouraging and promising. The authors confirmed that the ESN is the most accurate algorithm for direct prediction compared to the mentioned tools. Therefore, the ESN is chosen for our application due to its fast training procedure and its accuracy when predicting time series. This prognostic tool already has been used in [START_REF] Morando | Proton exchange membrane fuel cell ageing forecasting algorithm based on Echo State Network[END_REF] for Fuel Cells RUL prediction under constant and variable loads. The results of short-term, long-term prediction and RUL estimation using ESN were encouraging. Nevertheless, the authors used the real load profile, supposed to be unknown at the moment they start the prognostics, as an input of the prognostic tool when performing prognostics under a variable load, while in reality, the load profile can not always be known in advance. The authors in [START_REF] Jouin | Prognostics of PEM fuel cells under a combined heat and power profile Õ[END_REF] also proposed a tool for FC prognostics under a dynamic load using the real load profile as an input. The use of the real load profile as an input of the prognostic tool is network, using a Genetic Algorithm (GA), to predict the FC voltage degradation under a constant and dynamic load. A discrete wavelet transform was used to process the voltage signal. Then, an algorithm based on the analysis of the signal to be processed using the Hurst coefficient analysis was introduced [START_REF] Morando | Proton exchange membrane fuel cell ageing forecasting algorithm based on Echo State Network[END_REF]. A mathematical relationship between the Hurst coefficient and the spectral radius (the parameter having the greatest influence on the ESN performance according to the authors) was established. The authors also proposed to split the voltage signal into two components: the trend and the dynamical part. They estimated them separately, to afterwards built the voltage signal again since the voltage contains different dynamics: the degradation information is flooded among the other dynamics. Recently, multiple inputs and multiple outputs echo state network (MIMO-ESN) was proposed in [START_REF] Vichard | Degradation prediction of PEM fuel cell based on artificial intelligence[END_REF]. The objective of the authors in [START_REF] Vichard | Degradation prediction of PEM fuel cell based on artificial intelligence[END_REF] was to develop a degradation model in order to reproduce and predict the performance evolution of the FC according to the operating conditions, in particular the ambient temperature. The authors in [START_REF] Hua | Remaining useful life prediction of PEMFC systems based on the multi-input echo state network[END_REF] used the stack voltage, current, temperature and the pressures of the reactants to predict the RUL and the stack voltage as a health indicator. These parameters have a great effect on the stack voltage, which makes their use as inputs of the ESN interesting. The authors confirmed that MIMO-ESN had a better performance than single input ESN. The authors in [START_REF] Vichard | Degradation prediction of PEM fuel cell based on artificial intelligence[END_REF] and [START_REF] Hua | Remaining useful life prediction of PEMFC systems based on the multi-input echo state network[END_REF] used the real experimental measurements of the parameters for the prediction phase. Nevertheless, in real case prognostics, the measurements of these parameters can't be available in advance for the prediction phase. Consequently, the effect of the parameters variation on the stack voltage can't be evaluated. Among all the operating parameters cited above, the stack current is the most relevant one.

Hybrid approaches

Hybrid approaches are based on the use of the two previously presented methods, in order to take advantage of each of their strengths [START_REF] Lei | Machinery health prognostics: A systematic review from data acquisition to RUL prediction[END_REF]. Indeed, the choice between a data-based or model-based method is made according to: the degree of available physical understanding of the system, the possibility of describing physical phenomena with equations, an affordable number of parameters to be evaluated, and the quantity and quality of available data. In practice, model-based approaches may require the use of experimental data to adjust parameters of the developed model. Vice versa, data-based approaches may require some physical knowledge of the system. The hybrid approach accuracy must be better than the two methods (data and model based) when they are used individually [START_REF] Sutharssan | A review on prognostics and health monitoring of proton exchange membrane fuel cell[END_REF]. Knocking down the locks linked to the hybridization of both data and model-based methodologies is not systematic [START_REF] Bressel | Remaining useful life prediction and uncertainty quantification of proton exchange membrane fuel cell under variable load[END_REF]. It is still necessary to have a physical understanding of the system to: (i) model the degradation phenomena in a physical way, and (ii) know how to recognize where the lack of knowledge exists in order to replace the physical model by a Black Box model. In addition, to set the part of the model performed by a black box, it is still necessary to have an experimental dataset. Ultimately, the realization of a hybrid prognosis model therefore requires physical knowledge of the system, experimental data, but also advanced knowledge in Prognostics and Health Management (PHM) in order to combine the two previous methods intelligently.

The authors in [START_REF] Zhou | Degradation prediction of PEM fuel cell using a moving window based hybrid prognostic approach[END_REF] proposed a hybrid prognosis approach. It consists of merging a model-based prognostic tool (an empirical model of FC voltage degradation and a particle filter framework), a data based approach (non-linear autoregressive neural network NARNN) and the sliding window techniques. The authors of this article highlighted the importance of finding a compromise between good accuracy and a relatively long prediction time, especially for data-based methods (the prediction error increases with the prediction horizon). The proposed approach was validated on 3 different stacks operating under different aging profiles. The results were also compared to the results of an integrated autoregressive and moving average model (ARIMA) [START_REF] Ibrahim | Wavelet-based approach for online fuel cell remaining useful lifetime prediction[END_REF].

The authors in [START_REF] Cheng | A hybrid remaining useful life prognostic method for proton exchange membrane fuel cell[END_REF] proposed a prognostics tool based on the fusion of a Regularized Particle Filter (RPF) and a Least Square Support Vector

x (n + 1) = f (w in u(n + 1) + wx(n) + w back y(n)) (2) x(n + 1) = (1 -A).x(n) + A.x (n + 1) (3)
The ESN has K inputs u(n), N internal units x(n) and L outputs y(n). w in is a NxK input weights matrix, w is a NxN internal units matrix, w back is a NxL output feedback optional weight matrix and f(.) is an activation function of internal units, typically tanh, or sigmoid functions and A is the leaking rate. The output equation is:

y(n + 1) = f out (w out .(u(n + 1), x(n + 1)) ) ( 4
)
f out is the output activation function, which is usually linear and w out ∈ R L×(k+N) is the output weights matrix.

Training w out is like solving the linear equation system below:

w out .X = Y target (5)
With X = [u x] and f out = Identity. The goal is to minimize the quadratic error E(Y target , w out .X). Several solutions to this problem exist, the one chosen here is presented by equation [START_REF] Mezzi | Operating conditions control for extending proton exchange membrane fuel cell lifetime[END_REF].

w out = Y target X T ( XX T + βI ) -1 (6)
With I being the identity matrix and β a regularization variable. Other alternatives exist in the literature such as weighted regression, Cholesky decomposition, etc. However, the solution adopted in our work is the one described by [START_REF] Mezzi | Operating conditions control for extending proton exchange membrane fuel cell lifetime[END_REF], because of its stability and its low sensitivity to noise [START_REF] Lukoševičius | Reservoir computing approaches to recurrent neural network training[END_REF].

The weight matrices w in , w and w back are created randomly. Usually they are initialized from a uniform distribution with mean 0 and variance 1 [START_REF] Jaeger | The 'echo state' approach to analysing and training recurrent neural networkswith an Erratum note[END_REF]. Only the output weights matrix is optimized. w in and w back are generally dense. They can be scaled and the input can be shifted too (by adding a constant value to u(n)). The scaling of w in and shifting of the input depends on how much processing unit nonlinearity is required by the task. w is usually sparse, the percentage of connected weights is defined by the connectivity (a parameter fixed by the user), and is rescaled by dividing it firstly by the largest absolute eigen value of w, called the spectral radius ρ(w). Then, the resulting matrix is multiplied by the desired spectral radius, typically smaller than 1 [START_REF] Jaeger | A tutorial on training recurrent neural networks, covering BPPT, RTRL, EKF and the "echo state network" approach[END_REF], in order to respect the Echo State Property (ESP). The Echo State Property states that the effect of a previous input u(n) and a previous state of the reservoir x(n) on a future state x(n + k) should disappear gradually with time. The ESP can be obtained even when the spectral radius is greater than 1 for non-zero inputs (including bias inputs to neurons). This property is violated when the spectral radius is greater than 1, for reservoirs having tanh as an activation function and zero input.

Reservoir parameters optimization

The choice of reservoir parameters such as the spectral radius, the connectivity, the neurons number and the leaking rate is essential to obtain good performances. It is important to point out that the parameters' values actually depend on task requirements. The choice of the spectral radius for example depends on the amount of memory and nonlinearity that the task requires. The closer the spectral radius is to 1, the longer the reservoir memory is. Larger ρ(w) drives the reservoir internal states into more nonlinear regions of tanh (saturation zone). Hence scaling w in and w has the same effect on the amount of nonlinearity of the ESN. The leaking rate A can tune the dynamics of the reservoir. A good choice of A is a key for adjusting the reservoir dynamics to match the input flow timescale [START_REF] Jaeger | Optimization and applications of echo state networks with leaky-integrator neurons[END_REF].

According to Jaeger [START_REF] Lukoševičius | A practical guide to applying echo state networks[END_REF], to obtain a "rich" set of dynamics, the reservoir should be big (an important number of neurons), w should be sparse (low connectivity: less than 20% of possible connections between the matrix weights) and randomly connected. Sparse connectivity provides a relative decoupling of subnetworks. This encourages the development of individual dynamics.

Ferreira et al [START_REF] Ferreira | An approach to reservoir computing design and training[END_REF] confirmed that the reservoir parameters' settings have a great effect on its performance. Thus a good tuning of reservoir parameters is essential. The use of optimization algorithms is recommended. In their work [START_REF] Ferreira | An approach to reservoir computing design and training[END_REF], the authors chose genetic algorithm for parameters' tuning. Morando et al. [START_REF] Morando | ANOVA method applied to proton exchange membrane fuel cell ageing forecasting using an echo state network[END_REF] used the ANOVA (ANalysis Of VAriance) method to determine the parameters that have the greatest impact on the model output. The studied parameters are: the reservoir size, the spectral radius, the connectivity, and the leaking rate. The result of this study showed that the most influential parameter is the spectral radius ρ(w), followed by the number of neurons and then the connectivity c. Therefore, to be effective, a setting priority for parameter design can be achieved by optimizing only the parameters that have a large influence on the results. The designated parameters are then optimized using GA [START_REF] Morando | Contribution au Pronostic de Durée de Vie d'une Pile à Combustible à Membrane Echangeuse de Protons Approche par réseaux de neurones à réservoir (Reservoir Computing[END_REF]. In the literature, many optimization tools were applied to optimize the reservoir global parameters, such as stochastic gradient descent approaches used by Jaeger et al. [START_REF] Jaeger | Optimization and applications of echo state networks with leaky-integrator neurons[END_REF], the authors reported that this method suffers from poor stability properties. The evolutionary algorithms were widely used to optimize the reservoir global parameters: Ishii et al. [START_REF] Ishii | Optimization of parameters of echo state network and its application to underwater robot[END_REF] was among the first to use the genetic algorithm for ESN parametes' tuning for the "Twin Burger" underwater robotic application. Zhong et al. [START_REF] Zhong | Genetic algorithm optimized double-reservoir echo state network for multi-regime time series prediction[END_REF] used genetic algorithm too, applied to a double reservoir ESN for multi-regime time series prediction. Amaya et al. [START_REF] Amaya | Prognostic of RUL based on Echo State Network Optimized by Artificial Bee Colony[END_REF] proposed the Bee Colony algorithm and an hypothesis that states that the future load profile is known in advance, which is not always the case in reality. In this work, the authors propose solutions to overcome the need to prior knowledge of the load profile. The novelty of the proposed paper is the RUL prediction of PEMFC following a combined heat and power generation (µCHP) profile, without a priori knowledge of the latter. A presentation of the Echo State Network algorithm is given in the next section.

PEMFC prognostics under a variable load profile

Mathematical formulation of Echo State Networks

An echo state network is composed of an input layer, a randomly connected dynamic reservoir and an output layer. The basic concept is that the reservoir allows a nonlinear dynamic transformation of the input signal which offers to the readout the possibility to extract the desired output using a simple linear regression. The particularity of this tool is that only the output layer neurons are trained in order to transform the ESN output into the desired output signal. Hence the ESN training is fast compared to other artificial neural networks. The ESN architecture is depicted in Fig. 3.

The reservoir update is done according to equations ( 2) and (3).

1. Random generation of N initial candidates, while respecting the limits of the search space. 2. Then, after the evaluation of the candidates cost function, the center of mass is defined as being the most suitable individual giving the most satisfactory solution.

Calculation of new candidates around the center of mass. A random

number is generated, between -1 and + 1. This can be formalized in equation:

x new = x c + l.r k ( 7
)
where:

• x c is the center of mass,

• r is a random number, • l is the upper limit of the search space if r > 0, and lower limit if r < 0,

• k is the current iteration.

Each new individual x new belongs to an interval around the center of mass. This interval is reduced according to the current iteration.

Return to step 2 until the stop criterion is reached.

The contribution presented by Morando et al. [START_REF] Ohyagi | Durability of a PEMFC Pt-Co cathode catalyst layer during voltage cycling tests under supersaturated humidity conditions[END_REF] is directly related to the Big Crunch phase. More specifically, they proposed a modification that relates to the dispersion of individuals around the center of mass.

In the original algorithm, the search space division function was represented by a function inversely proportional to the number of iterations. This function greatly reduced the search space from the second iteration because indeed, 50% of the search space was deleted between the first and the second iteration. This rapid reduction of the search space, and thus this cancellation of potential interesting solutions, can lead to the convergence towards a local and not towards a global optimum. Thus, it may be advantageous to divide the search space slower at the beginning of the algorithm and then speed up the search space division, in order to find global optimum more efficiently. This was done by replacing the original linear function with a delayed exponential function. The two divisional functions of the search space are shown in Fig. 4.

As it can be seen, the division of the search space is slower at first with the red curve. This makes it possible to allocate more iterations to finding the neighborhood of the global optimum. Then, an acceleration of the search space division occurs. This acceleration makes it possible to quickly find the global optimum of the function.

Methodology

As we have shown in Section 2.2, various tools and strategies exist in the literature for PEMFC prognostics under a constant load. However PEMFC prognostics under a variable load is less addressed in the litterature. Under a dynamic charge, long-term load profile variation cannot be predicted. For this reason many existing works assumed that the future power demand was known [START_REF] Jouin | Prognostics of PEM fuel cells under a combined heat and power profile Õ[END_REF]. This assumption is very strong, and can't be verified in a real case application. The objective of the authors in [START_REF] Jouin | Prognostics of PEM fuel cells under a combined heat and power profile Õ[END_REF] is to prove that their prognostics tool is accurate even under a variable load. For that, the real load profile is used as an input of the prediction tool. Nevertheless, in real case applications, the future load profile is not available. Therefore the prognostics tool will not be able to predict the future state accurately because of the load profile unavailability. The aim of this paper is not only to develop an accurate prognostic tool but also to propose a solution to compensate for load profile unavailability. The objective is not to realize a long term prediction of the current profile, because this is impossible. As we have said before, we cannot predict what a user will decide to do in the future. What one could predict is the FC behavior (the voltage) knowing the current. Since the current information is missing, we have proposed two possibilities for load profile generation: a repetitive cycle or many cycles generated by Markov chains. The prognostics tool uses the available voltage data at time t for training the Echo State Network, and the available load profile data for cycle repetition or Markov Chains to generate load profiles. The generated load profiles are used then as inputs of the Echo State Network. The latter predicts the voltage based on the generated load profile and the known part of the voltage and current. Discussions and results' analyse of the RUL prediction under a variable load are presented in Section 4.

Cycle repetition

Cycle repetition consists in repeating the known part of the load Fig. 4. The two division functions of search space according to the generation.

Rigamonti et al. [START_REF] Rigamonti | Echo state network for the remaining useful life prediction of a turbofan engine[END_REF] chose Differential Evolution. The obtained results using the evolutionary algorithms were satisfactory.

In the litterature, the reservoir parameters' setting was one of the main focuses. The ESN results seem to be dependent on the parameters' choice. Many algorithms were proposed for ESN parameters' optimization. So to have a good reservoir, one should choose an appropriate optimization algorithm. It is worthwhile to point out that even the optimization algorithm needs to be tuned in order to give good results. A comparison between different optimization algorithms for a prognostics task was performed in [START_REF] Mezzi | Contrôle tolérant au vieillissement dans des systèmes pile à combustible PEMFC[END_REF]. The chosen optimization algorithms for the comparison are: Genetic Algorithm, Particle Swarm Optimization, Simulated Annealing (SA) and Big Bang-Big Crunch (BB-BC). The results show that those algorithms gave good accuracy results. However BB-BC prensented the best compromise in terms of accuracy and results dispersion. Moreover, BB-BC is easily implementable and only a few parameters of this algorithm need to be chosen by the user. Consequently we choose the BB-BC algorithm for the reservoir parameters' setting. The optimized parameters in this paper are: the reservoir size N, the spectral radius ρ, the regularization variable β, and the leaking rate A. The connectivity is fixed to 10% in this study.

Big Bang Big Crunch

The Big Bang-Big Crunch algorithm, originally introduced by Erol and Eksin [START_REF] Erol | A new optimization method: Big Bang-Big Crunch[END_REF], is an optimization algorithm from the field of artificial intelligence. This algorithm is inspired from the Big Bang BB and Big Crunch BC theory. The Big Bang theory is relatively well known, that of the Big Crunch is much less known. It is based on Einstein's theory of general relativity, and states that the expansion phase of the universe, initially due to the Big Bang, will end. The universe will begin to collapse on itself to form the largest black hole in existence. The aging of the universe can be divided into two distinct phases: the Big Bang phase and the Big Crunch phase. The BB-BC optimization algorithm is therefore based on the representation of these two phases, and is summarized below.

t=1 (real -estimation) 2 n √ √ √ √ √ (8) NRMSE = RMSE real max -real min (9)
Regarding the RUL prediction performance indicators, the relative accuracy (RA) described by equation [START_REF] Morando | Proton exchange membrane fuel cell ageing forecasting algorithm based on Echo State Network[END_REF] assesses the accuracy of an algorithm at a specific time instance. The closer the RA is to 1, the better the accuracy. The Alpha-Lambda metric [START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF], defined by equation [START_REF] Lee | Development of a method to estimate the lifespan of proton exchange membrane fuel cell using electrochemical impedance spectroscopy[END_REF], allows to check if the prediction accuracy at a given time is within desired α bounds.

RA λ = 1 - ⃒ ⃒ ⃒RUL(tλ) -R ÛL(t λ ) ⃒ ⃒ ⃒ RUL(t λ ) (10)
α -λ Performance = { 1if (1 -α).RUL(t λ ) ≤ R ÛL(t λ ) ≤ (1 + α).RUL(t λ ) 0otherwise (11)
Where t λ = P + λ(EOL -P) is the instant when the prediction is performed, RUL(t λ ), R ÛL(t λ ) and α is a percentage value of the acceptance confidence bounds. Difference between the estimated and the real RUL is calculated to study the accuracy. For the dispersion analysis, simulations are performed several times and the errors results are analyzed. The minimum (min), maximum (max), median and mode metrics values are presented in boxplots.

Results

Experimental database: Micro combined heat and power generation mission profile

The experimental data comes from the European project SAPPHIRE "System Automation of PEMFCs with Prognostics and Health management for Improved Reliability and Economy" [START_REF]Projet SAPPHIRE: system automation of PEMFCs with prognostics and health management for improved reliability and economy[END_REF][START_REF] Lechartier | Contribution au pronostic de pile à combustible PEMFC basé sur modèle semi-analytique[END_REF]. The objective of the study consists in observing the PEMFC behaviour on a long term test, particularly under a micro-cogeneration µCHP load profile. The test includes the realization of static and dynamic characterizations: polarization curves and Electrochemical Impedance Spectroscopy to evaluate the PEMFC degradation. A five cells PEMFC with a 100 cm 2 surface area is used to carry out these tests and the test operating conditions are described in Table 1.

Since the studied µCHP load profile realized in the framework of the SAPPHIRE project presents a repetitive cycle, only a part of this load profile is shown in Fig. 5 for visibility reasons. The voltage evolution under this load profile and a zoom of the latter are given in Fig. 6.

Important performance recovery was observed after a sudden shutdown of the test bench after 639 h of continuous test. Therefore, for the next sections, we decided to use the experimental data corresponding to the continuous test, i.e the data set measured during the first 639 h. The measured voltage contains sharp peaks, outliers and slight fluctuations. It is composed of 2,690,154 data points, thus the data length may result in a high computation time. Therefore the stack voltage is first profile, assuming that the PEMFC will experiment exactly the same events cyclically. The load profiles used in the research field to simulate a micro CHP profile, for example, are cyclical. The prognostics under a cyclic load profile is presented separately from Markov chains because the probability for markov chains to generate a cyclic load profile is very low. This can be explained by the fact that the probability of switching from one state to another depends only on the current state. Knowing the current state, the definition of the next state is based on the probabilities which are defined from a preliminary study of the existing time series and the use of a random variable. An example is detailed in Section 4.3.1.

Markov chains

The idea of repeating the known part of the load profile presented previously implies the following assumption: the PEMFC is expected to experiment exactly the same periodic events during its entire operation. This hypothesis can be strong for real life applications, even for stationnary applications. Thus, the generation of various load profiles from the available part of the database makes it possible to check our algorithm responses to different solicitations. These different created profiles introduce randomness in the forecasting part, making it possible to evaluate the robustness and accuracy of our algorithm in case of load profile change. Therefore, in order to make our algorithm more realistic, a Markov chain profile generation is proposed.

A Markov chain is a mathematical process for describing state-tostate transitions of a system according to probabilistic rules. The main characteristic of a Markov chain is that, no matter how the process has reached its current state, the possible future states are fixed. This translates onto the following hypothesis: the probability of switching from one state to another depends solely on the current state.

A Markov chain is a stochastic process, but it differs from a classical stochastic process because a Markov chain must be "memoryless". That is, the likelihood that future actions will occur does not depend on the actions that led to the current state. This is called the Markov property, and it is generally said that this corresponds to the assumption that the system order is 1. This is a strong assumption, since the FC behavior at time t depends on the traveled path, especially because of the water management and the progressive degradation (caused by the presence of Carbon monoxide, Nitrogen accumulation, etc). However, in the experimental case, the characterizations, starting and stopping the tests and other phenomena described in [START_REF] Pivac | Rejuvenation techniques for PEM fuel cells[END_REF] cause a rejuvenation of the FC which corresponds to a partial erasure of the memory.

Evaluation metrics

In this paper, two prognostics indicators are used. The first one is the voltage prediction and the second one is the remaining useful lifetime (RUL) estimation. This choice has been motivated by several factors: (i) the availability of voltage signal measurement, (ii) the voltage measurement is non-intrusive, and last but not least (iii) no additional sensors are needed. This makes the algorithm implementation easy, costless and non-intrusive. All these advantages are highly appreciated for real applications, especially in the framework of automotive applications.

To evaluate the effectiveness of the prognostics, several metrics are introduced. The first and the second one are dedicated to evaluate the voltage prediction accuracy. The chosen metrics for the accuracy are: (i) the Root Mean Square Error (RMSE) described by equation ( 8) and (ii) the Normalized Root Mean Square Error (NRMSE) described by equation [START_REF] Liu | Prognostics methods and degradation indexes of proton exchange membrane fuel cells: A review[END_REF].

The RMSE will give an error in Volts, since the predicted variable is the average cell voltage. It makes it possible to give an error with respect to a physical parameter, and it is used to evaluate the variation of the measurements. However, its value can be difficult to interpret.

The use of the NRMSE in this case is recommended because of its interpretability and independence from scale.

preprocessed. A resampling of data has been performed, and the voltage length after resampling is 567577. The resampling is done preserving the useful information contained in the signal. Then the resampled voltage signal is divided according to the current level (0, 0.15, 0.25 or 0.35 A/cm 2 ) to be processed separately. The data preprocessing is realized using a detrend function to extract the noise and fluctuations from the data, then the noise is substracted from each of the voltage signals. Finally the whole voltage signal corresponding to the stationary load profile is rebuilt again. A zoom on the preprocessed voltage signal is prensented in Fig. 7.

As shown in Fig. 7, the preprocessed data is smoothed, and peaks and noise are deleted succefully while keeping the useful information contained in the signal.

Lifetime forecasting with cycle repetition

At first, the voltage is predicted under the real load profile, and the results are shown in Fig. 8. In other words, the real load profile, supposed to be unknown during the prediction phase, serves as an input. The objective of this step is to evaluate the prognostics tool performance under a variable load, before adding an other uncertainty due to the lack of knowledge regarding the real load profile: 50% of the collected data is dedicated to learning, i.e. 319 h. The ESN is trained by the available data at instant t of the prediction time, and the voltage is estimated until it reaches the end of life threshold. For stationary applications, the conventional failure threshold is set to 20% power loss versus the original nominal operating point. However, the power loss of the available experimental data is below 20%. Therefore, the failure threshold for the RUL estimation has been set at a loss of 10% of the intial voltage corresponding to one of the current levels: 0, 0.15, 0.25 or 0.35 A/cm 2 . The voltage prediction is launched 10 times in order to take into account the random process of the ESN matrices generation. The prognostics tool performances for the voltage prediction task are evaluated by calculating the RMSE and NRMSE introduced in Section 3.3.

All the calculations are performed using a computer with a processor Intel® Core ™ i5-6500 CPU @3.2 GHz (4CPUs), and a Random Access Memory of 16384 MB.

The ESN optimized parameters identified using the BB-BC algorithm are presented in Table 2.

The obtained reservoir has a high leaking rate and a low spectral radius. This means that the reservoir capacity to forget the past inputs and reservoir states is important. This result is expected when handling a variable load profile. The reservoir needs to adapt its states rapidly depending on the inputs signals dynamics. In our case, one of the inputs accuracy and the error dispersion improves greatly with the increase of the prediction time and subsequently increasing learning base. Fig. 10 shows great relative accuracy, higher that 0.6. As we have explained in Section 3.3, higher RA (close to 1), reflects good prediction capabilities of the algorithm.

In order to take into account the effect of the ESN reservoir random creation, the average RUL is retained for the future. Therefore, the average RUL is calculated at different times and subsequently different training bases of the available experimental data. The results are described in Fig. 11.

The lower and upper bounds present ±10% of the real lifetime. The continuous line represents the true value of the RUL and the real lifetime is 492 h. The first three estimations of the RUL are lower than the real RUL. This can be explained by the fact that during the first 100 h, degradation rate is important, hence the ESN underestimates the RUL. It is important to underline that all the estimated RUL are almost within the bounds of ±10%.

Lifetime forecasting with Markov chains 4.3.1. Load profile generation

As previously stated, any stochastic process is known to have the Markov property if the probability of going to the next state depends only on the current state and not on the past states.

X(1)→X(2)→X(3)→⋯→X(t)→X(t + 1)→⋯

From the equation above, the Markov property means that the movement from X (t) to X (t + 1) depends only on X (t), the current state, and not on previous states. Let's take the example shown in Fig. 12. In this purely random example, we start from a state 1 at time t. Statistical analyzes of previous data made it possible to establish that, when starting from a state 1, there is:

• P(1→1) = 25% probability of remaining in the same state at the next moment,

• P(1→2) = 35% probability to go to state 2 at the next instant,

• P(1→3) = 40% probability to go to state 3 at the next instant.

Normally, these probabilities are defined from a preliminary study of an existing time series, which will serve as a model. Then, after having defined a first state, a random variable R v (rand under Matlab®) will define the following states and generate a cycle. For example, in our case, the state X (t + 1) is defined as follows:

• if 0.00 < R v < 0.25, the state X (t + 1) will remain the same, • if 0.25 < R v < 0.60, state X (t + 1) will be state 2, • if 0.60 < R v < 1.00, state X (t + 1) will be state 3.

And so on for the following states X (t + 2), X (t + 3), …, X (t + n), thus generating a new profile that is relatively similar to the reference profile. Besides, in order to adapt this methodology to the profile generation for the PEMFC, a first constraint is added. In order to have a coherent profile, it is necessary to ensure that the physical phenomena activated in the model profile also occur in the generated profiles. Let us take the example of the cogeneration voltage presented in Fig. 6 again. This signal can also be represented in terms of probability density, and probability to be at a certain voltage level.

The voltage operating range of the PEMFC is divided into three different parts. In the first one, where the voltage is lower than 0.5 V, the concentration overvoltage is dominant. The second, between 0.5 and 0.8 V, corresponds to the linear part of the polarization curve and finally, the third part, with a potential higher than 0.8 V, corresponds to the part where the activation overvoltage are preponderant. These three zones correspond to different physical phenomena, and the idea is, once a profile is generated, to verify that the probability density on these three zones is also respected. In our study case 99% of the voltage is between 0.5 and 0.8 V meaning that the fuel cell is very often in the linear operating zone. Finally, a last constraint is added: the energy of the generated profile must be, ±5%, the same as that of the reference profile. An example of a Markov chain load profile is shown in Fig. 13. The deviation between the real load profile and the generated profile by Markov chain has been evaluated by calculating the NRMSE. 10 profiles have been generated and the NRMSE has been calculated for each one of them. This error varies between 26% and 27% when 50% of the collected data is dedicated to the training basis. For the same learning basis, repeating the same load profile cyclically gives a NRMSE of 28%. This result is expected since the cycle repetition is a particular case of Markov chain load profile generation. The advantage of using the Markov chain is the possibility to generate multiple load profiles. The generated load profiles integrate the effect of random while remaining faithful to the real load profile.

RUL prediction using Markov chains

For the following results, 10 ESNs were launched at every prediction time and the mean value of the RUL is saved. Besides the RUL, min, max, median and mean values of the estimated RMSE are presented in Table 4. The used reservoir parameters are presented in Table 2.

For each prediction time, an average value of 72 s is needed to predict 10 RULs corresponding to a generated load profile. The use of Markov chains has slightly increased the computational time. Nevertheless it remains acceptable given the execution frequency of the prognostics algorithm.

The previous table shows that, for all the learning rates, the mean and the median of the estimated RULs are either equal or very close. That means that the results' repartition is a normal distribution (Gaussian type). That gives information about the results' dispersion of the algorithm: 68% of the results are within the mean +/-the standard deviation. The corresponding standard deviations are all lower than 10% of the real corresponding RUL. Thus the next analysis will mainly be focused on the mean values. For the first learning rate, with only 10% of the data reserved for training, the results are very interesting, with a difference between the mean and the real RUL of only 34 h, which is less than 10% of the real RUL (RUL = 425 h). Less accurate results are obtained for the following learning rate, 20%, with a RUL estimated at 393 h, versus a real one of 356 h. The error is approximately 37 h, which corresponds to 10.4% of relative error, which is slightly higher than the 10% fixed limit boundaries. Then, 30% of learning rate gives an estimated RUL of 339 h vs a real RUL equal to 288 h, meaning a relative error of 17%. However, the standard deviation is low, with only 11.3 h, thus the algorithm is also robust against the randomness of the reservoir for this learning rate. This error increase can be caused by the randomness introduced by the generation of the load cycle using a Markov chain. The 40% of learning rate also gives results with more than 10% of error. The estimated RUL is 249 h vs a real RUL equal to 219 h, meaning 13% of relative error, wich is interesting but slightly higher than the limit boundaries. Finally, the 3 last learning rates (50, 60 and 70%), give very interesting results, close to the real RUL with errors below 10%.

Fig. 14 shows the different average RUL values obtained for the different prediction times (i.e on variable learning basis). The idea of the study is to start from 10% learning base, and gradually increase the training base in order to observe the prediction results of the RUL under different prediction times.

As we have seen previously, the continuous line represents the true value of the RUL and the dashed lines are the real Lifetime ±10%. The average estimated RUL values are presented under different prediction times. As described in Fig. 14, the RUL values are all estimated within the ±10% bounds. The first ones are closer to the limiting bounds, then the errors decrease, and subsequently the accuracy improves. The use of an optimized Echo State Network for PEMFC prognostics under a constant load gives good results with low average errors below 5% for voltage prediction. This method has shown its effectiveness under constant load in a previous work [START_REF] Mezzi | Multi-reservoir echo state network for proton exchange membrane fuel cell remaining useful life prediction[END_REF]. However, the errors are higher when performing prognostics under a variable load. The results obtained from the estimation of the RUL with the cycle repetition are satisfying. These results are expected, since the real load profile is repetitive. The

Conclusion

The paper provides a strategy of prognostics under a dynamic load with respect to international criteria such as the end of life threshold defined by the United States Department of Energy. The used tools for the proposed strategy are known and widely used by the scientific community around the world. The novelty of the developed tool is the combination of Echo State Networks and Markov chains to predict the fuel cell lifetime under a dynamic load without a prior knowledge of the load profile. The developed tool is generic, and can be implemented regardless of the country if the Echo state Network and Markov Chains tools are mastered. The results are satisfying, and do not require great effort for data processing beforehand. Then, issues relating to the prior knowledge of the load profile arise: in a real application, the load profile is not known in advance, that is why two propositions have been made and applied for prognostics under a dynamic load in order to overcome the need for the prior knowledge of the load. A first study replays the part of the load profile already known while a second study focuses on the generation of load profiles based on Markov chain. In this work, both proposed approaches give good results on the proton exchange membrane fuel cell remaining useful lifetime prediction for a stationary application.

Fig. 1 .

 1 Fig. 1. Remaining Useful Lifetime estimation.

Fig. 2 .

 2 Fig. 2. Equivalent circuit of a PEMFC used in EIS curve fitting.

Fig. 3 .

 3 Fig. 3. ESN structure.

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. Current profile of the load.

Fig. 10 .

 10 Fig. 10. Relative Accuracy prediction under different prediction times.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Estimation of the RUL at different prediction times with repetition of the charge cycle. Fig. 12. Example of state succession in a Markov chain.

Fig. 13 .

 13 Fig. 13. Load profile example built by Markov chain.

Fig. 14 .

 14 Fig. 14. Distribution of average RUL values obtained for 10 ESN under a variable learning base.

  Echo State Network (ESN) and Liquid State Machines (LSM) are new approaches for Recurrent Neural Network design and training. They were proposed by Jaeger in 2001 and Maas in 2002 respectively. Both approaches are known as Reservoir Computing methods. Prognostics based on Echo State Network have been widely applied in different fields and the results were convincing.

	3.1. Echo State Network

Table 1

 1 Test characterstics.

	Test Operating conditions	Range	Units
	Absolute inlet pressure anode/cathode Anode/cathode stoichiometries Temperature Relative anodic/cathodic hygrometry Current density	2.02650-1.5199 2.5-3.5 60 60-100% 0, 0.15, 0.25 and 0.35	10 5 Pa -• C -A/cm 2

Table 4

 4 Prediction results of the RUL under a stationary load profile.

	10 20 30 40 50 60 70	425 356 288 219 154 85 12	344 352 319 2288 149 78 8	425 431 359 280 169 88 10	391 394 336 249 160 80 9.55	391 393 339 249 161 80 10	21.2 26.7 11.3 14.7 5.8 3.7 0.9	79 75 74 72 70 69 69

Learning rate [%] Real RUL [h] Min [h] Max [h] Mean [h] Median [h] Stadard deviation [h] Computation time [s]

Acknowledgment

This work has been supported by the EIPHI Graduate school (contract "ANR-17-EURE-0002") and the Region Bourgogne Franche-Comté. The authors are thankful for the use of experimental data realized in the framework of the european project SAPPHIRE.

is the variable load profile, which reaches different current levels in a short time, the reservoir should be able to adjust its internal states depending on the load profile variations.

These average RMSE and NRMSE are presented in Table 3 under different learning basis.

The dynamics of FC degradation phenomena are slow (hours scale [START_REF] Wang | PEM fuel cell diagnostic tools[END_REF]), thus the prognostics algorithm can be run every 10 h in order to collect new experimental data. The new experimental data serves to increase the learning base and consequently improves the accuracy. The prognostics algorithm does not need to be run frequently (every 10 h of FC operation). One interesting solution is to execute the prognostics algorithm during the stopping phase of the vehicle. The process unit in this phase is less busy.

The RMSE and the NRMSE calculated on the voltage prediction are very low. The prediction results are good whatever the amount of data dedicated to the training basis. These results show that the chosen prognostics tool offers a great prediction accuracy even with a short training database.

Here, we will overcome the prior load profile knowledge by repeating the load profile already experienced by the PEMFC cyclically, making thus the following assumption: the PEMFC will experience the same events periodically.

In this case of study, the hypothesis is not strong since the load profile is repetitive. The obtained results will allow us to evaluate the proposed prognostics tool performances, particularly the RUL estimation under different prediction times. The prediction time corresponds to the instant when the prediction begins. That means that all the data available until the prediction time is used for the reservoir training.

A study of the RUL dispersion caused by the input layer and the reservoir matrices random creation is also realized and the results are shown in Fig. 9 and Fig. 10. Then, the RUL estimation is performed at different prediction times and subsequently different training bases: from 5% (32 h) to 70% (448 h) of the available experimental data.

At each prediction time, 10 ESNs with random input and reservoir matrices weights are created. Each of these ESNs allows estimating a RUL. We obtain 10 RULs per prediction time. The RULs distribution are presented by boxplots, which allows the vizualization of the results dispersion. The relative accuracy RA is also calculated for each reservoir. Then its distribution is presented at different prediction times (i.e different learning basis) as shown in Fig. 10.

The lower and upper bounds present ±5% of the real RUL at the prediction time t λ . The crosses represent the true value of the RUL and the real lifetime is 492 h. As a reminder, all the data available until the prediction time t is used for the reservoir learning. At prediction time t, the voltage estimation is performed during a prediction horizon H. The prediction horizon H is complete when the FC voltage reaches the failure threshold (a loss of 10% of the intial voltage). As we can see in Fig. 9, the

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.