Zhixue Zheng
email: zhixue.zheng2@femto-st.fr

Marie-Cécile Péra

Daniel Hissel

Laurent Larger

Nadia Yousfi Steiner

Samir Jemei

Fault Diagnosis of PEMFC Systems In The Model Space Using Reservoir Computing

Keywords: PEMFC system, fault diagnosis, reservoir computing, model space, dynamical conditions I

Artificial neuron network provides a promising solution for fault diagnosis of fuel cell systems. A recently proposed novel framework of recurrent neuron network named Reservoir Computing is focused with only its output weights to be trained, which is rather advantageous for online adaption in real-time applications. In a previous work, its simplicity and efficiency has been demonstrated. This paper focus on a novel attempt of performing fault diagnosis directly in the reservoir computing based model space (current-voltage model) instead of the original data space (voltage signal). No additional feature extraction procedure is needed and abnormal health states could be detected directly in the model space (in the form of evolution of output weights).

INTRODUCTION

Fault diagnosis plays a critical role in reinforcing the durability and reliability of polymer electrolyte membrane fuel cell (PEMFC) systems, which are regarded as two main bottlenecks for the commercialization phase. According to whether an analytical PEMFC model exists, existed fault diagnosis methods can be categorized into two general kinds: model-based and data-driven methods [START_REF] Petrone | A review on model-based diagnosis methodologies for PEMFCs[END_REF], [START_REF] Zheng | A review on non-model based diagnosis methodologies for PEM fuel cell stacks and systems[END_REF]. Among the data-driven methods, artificial neural network (ANN), especially the recurrent neural network (RNN) type, provides a promising solution as it demonstrates a strong capability in the modeling or pattern recognition tasks of nonlinear dynamical systems, while no deep knowledge about the underlying physical processes is required [START_REF] Jemei | A new modeling approach of embedded fuel-cell power generators based on artificial neural network[END_REF]- [START_REF] Kamal | Fault detection and isolation for PEM fuel cell stack with independent RBF model[END_REF]. Theoretically, it can approximate arbitrary dynamical systems with arbitrary precision, also called "the universal approximation property", under mild and general assumptions.

Concerning the practical applications especially for realtime fault diagnosis of PEMFC systems, the online adaption capability of the method is highly desirable [START_REF] Li | Diagnosis for PEMFC systems: a data-driven approach with the capabilities of online adaptation and novel fault detection[END_REF]. Traditional RNN methods generally suffer from a high computational complexity, a slow convergence rate and the existence of bifurcations (resulting in local optima or even non-converging of the training process). To overcome their limitations, a fundamentally novel framework for RNN design and training was proposed independently by Prof. Jaeger in 2001 under the name of Echo State Network (ESN) [START_REF] Herbert | The "echo state" approach to analysing and training recurrent neural networks-with an erratum note[END_REF] and by Prof. Maass under the name of Liquide State Machine (LSM) [START_REF] Maass | Real-time computing without stable states: A new framework for neural computation based on perturbations[END_REF]. Together with the later appearing Back-Propagation De-Correlation (BPDC) learning rule, they are collectively referred to as Reservoir Computing (RC).

RC differs from traditional RNN in that a large number of neurons are used (tens to thousands vs. tens of neurons in traditional RNN) and in that only the output weights need to be trained (usually in a linear way, while all the connection weights are trained in the traditional RNN) [START_REF] Herbert | Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication[END_REF]. Therefore, it can greatly facilitate the training phase and further the online adaptation process in the real-time applications. Excellent performances of RC are reported in the literature in various practical applications, including the speed recognition task (word error rate 0% vs. 0.6% of previous methods) [START_REF] Verstraeten | Reservoir-based techniques for speech recognition[END_REF], the non-linear wireless channel equalization task (improved by two orders of magnitude) [START_REF] Jaeger | Optimization and applications of echo state networks with leaky-integrator neurons[END_REF], the prediction of the chaotic dynamics (improved by three orders of magnitude) [START_REF] Jaeger | Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication[END_REF].

Despite its simplicity and efficiency, it is still a novel concept in the fuel cell (FC) fault diagnosis domain. An initial attempt of applying RC for PEMFC system diagnosis was made in [START_REF] Zheng | Brain-inspired computational paradigm dedicated to fault diagnosis of PEM fuel cell stack[END_REF]. Four fault types were targeted, including CO poisoning, low air flow rate, defective cooling and natural degradation. An excellent classification rate of the 99.9% in the training phase and 93.4% in the testing phase was obtained respectively. Meanwhile, the influence of a set if RC key parameters was studied. Nevertheless, only the static operating conditions were studied. This work was further developed in [START_REF] Morando | Fuel Cells Fault Diagnosis under Dynamic Load Profile Using Reservoir Computing[END_REF], [START_REF] Morando | Reservoir Computing optimisation for PEM fuel cell fault diagnostic[END_REF] under PEMFC dynamic profiles and with an automatic optimization procedure separately. These previous works considered the stack voltage as a reliable health indicator of the FC system and took it as the only RC input and performed pattern recognition in a supervised way (each fault corresponds to a certain voltage pattern).

In this paper, an innovative framework of performing PEMFC fault diagnosis utilizing RC is developed, inspired by the idea of "learning in the model space" proposed in [START_REF] Chen | Learning in the model space for cognitive fault diagnosis[END_REF]. It differs from the previous works in that it performed fault diagnosis directly in the RC based model space instead of the original data space. A brief introduction of RC basics is made in the following section. The experimental work performed by the authors is introduced in section 3. Section 4 provides more details on the RC based fault diagnosis methodology. The conclusion and perspectives is given in the final section.

II. RESERVOIR COMPUTING BASICS

A. Principal of Rerservoir Computing (RC)

RC differs from the traditional RNN training methods in two basic aspects. First, the internal connections are assumed to be fixed and are not updated during training. Second, only the output weights need to be determined in the training process and they can easily be calculated by linearly regression techniques [START_REF] Lukoševičius | Reservoir computing approaches to recurrent neural network training[END_REF]. The training process is thus strongly simplified, and for online adapting algorithm, this is rather advantageous. A schematic illustration is shown in Fig. 1. Generally, RC consists of three distinct parts: an input layer, a reservoir and an output layer. The input layer is connected to the reservoir via a randomly generated input weight matrix (W in ), which remains unchanged during training. The reservoir contains many randomly interconnected nodes (via W) which are also left untrained [START_REF] Lukoševičius | Reservoir computing approaches to recurrent neural network training[END_REF]. When excited by the input signals, the reservoir exhibits complex transient dynamics, which are further read out by the output layer via an output weight matrix (W out ). As shown in Fig. 1 (B), the update equations of a typical structure are

x (n) = tanh(W in [1; u(n)]+Wx(n-1)) x (n) = (1-α)x(n-1)+αx (n) (1)
Where u(n) is the input vector, x(n) is a vector of reservoir neuron activations and x ̃ (n) is its update at time step n. and W in and W are respectively the input matrix and recurrent matrix, α is the leakage rate.

The linear readout layer is defined as

y(n) = W out [1;u(n);x(n)] (2)
Where y(n) is the output vector, W out is the output weight matrix and it can be calculated by minimizing the Mean Square Error (MSE) between y(n) and y target (n). The most universal and stable solution to calculate W out is the ridge regression, also known as regression with Tikhonov regularization [START_REF] Lukoševičius | A practical guide to applying echo state networks[END_REF]:

W out = Y target X T (XX T + βI) -1 (3)
Where β is the regularization coefficient and I is the identity matrix.

B. RC model based fault diagnosis

The fundamental idea of applying RC for fuel cell fault diagnosis is to represent directly the original data (e.

III. PEMFC SYSEM DESCRIPTION

A. 1 kW experimental test bench

A three-month experimental characterization of the ZSW 5cell fuel cell stack (BZ 100_13) was performed within the French FCLAB laboratory (FR CNRS 3539), with its active surface 96 cm 2 , taking into consideration the influence of the operating current Istk and seven operating parameters, including: the air/H2 stoichiometry (FSC/FSA), the air pressure (Pair), the stack temperature (Tstk), the anodic/cathodic relative humidity (RHH2/RHair) and the cooling water flow rate (Fwater).

The influence of Istk, under both the static condition and the dynamic excitations was studied. Three forms of excitations were performed, i.e. sinusoidal, square and sawtooth, each form with 2 magnitudes respectively, i.e. 2A and 5A, and 5 exciting frequencies, i.e. 1000 Hz, 100 Hz, 30 Hz, 10 Hz, 5 Hz,

- (A) (B)
1 Hz, 0.1 Hz. Sampling frequencies are 50 kHz, 5 kHz, 2 kHz, 500 Hz, 250 Hz, 50 Hz and 5 Hz respectively. Corresponding Labview control surface is shown in Fig. 3 left.

Table 1 shows the nominal operating parameters of the studied PEMFC system. In the initial stage of this work, the influence of air stoichiometry (FSC) is firstly studied to verify the efficiency of the proposed methodology.

Control & monitoring interface

Excitation signal generation and collection interface

IV. FAULT DIAGNOSIS METHODOLOGIE USING RC

A. RC model

The fuel cell stack is operated under constant DC current 40A with an additional and sinusoidal perturbation of low amplitude (typically 5% of the DC component). The voltage response of the stack is measured. Different operating conditions are applied, mimicking a fault in the air supply system (Fig. 4). As a matter of fact, the air stoichiometry factors are varied from high values to low values (from 4 to 1.5, with 2.8 as nominal value).

A one-input and one-output small RC model is trained to model the voltage response of the stack under different FSCs. The RC critical parameters are initially chosen as: number of internal nodes = 20, leakage rate α = 0.3, regularization parameter β = 1× 10 -8 , input matrix W in and internal recurrent matrix W are generated randomly in the range of [-0.5, 0.5]. It should be noted that these parameters can be further optimized to adapt a certain task. And global optimization methods, such as genetic algorithm, and big bang crunch algorithm can be selected to perform automatic optimization [START_REF] Morando | Reservoir Computing optimisation for PEM fuel cell fault diagnostic[END_REF][START_REF] Lukoševičius | A practical guide to applying echo state networks[END_REF]. 5 shows the trained RC output and Vstk under the nominal condition FSC = 2.8. It could be found that the RC output matches well the real stack voltage, and a Mean Square Error (MSE) is obtained less than 0.003%. Meanwhile, the corresponding output weight matrix W out is calculated and plotted with its 22 elements (i.e. 1+1+20 as indicated in Eq (2)) in Fig. 6. Essentially, W out could be regarded as a group of internal FC system parameters which reflect the nonlinear relationship between the Istk and Vstk under the dynamic excitations. This input-output relationship could certainly be established in other simpler nonlinear functions such as polynomial functions. RC model is selected herein considering its demonstrated capacity in dealing with temporal signals in dynamic systems. And this characteristic will be further explored in the following work for dealing with more complex multi-fault diagnosis under dynamic operating conditions. The output weight matrix is obtained during the training phase and the distribution of its elements is given in Fig. 8. This distribution can be related to the air stoichiometry factors and used for diagnostic. To demonstrate more clearly the influence of the air stoichiometry factor, the 6 th , 9 th , and 20 th elements of the output matrix are selected for further observation, as shown in Fig. 9.

In Fig. 9, it can be observed that the lowest and highest air stoichiometries (i.e. 1.5 and 4, corresponding to number 1 and 8 in the horizontal axis) deviate the most from the nominal condition (corresponding to number 6). The feasibility of utilizing the evolution of output weight matrix elements in the model-space is thus demonstrated. This is rather advantageous for real-time applications where the online adaption capability is highly required, as only linear readouts are needed to be recalculated in the proposed method.

In the following work, supervised algorithms such as fuzzy logic, support vector machine can be further applied for fault classification [START_REF] Li | Online implementation of SVM based fault diagnosis strategy for PEMFC systems[END_REF]. In this paper, an initial attempt is made to perform PEMFC system fault diagnosis in the RC based model space. The RC model establishes a relationship between stack current and stack voltage under various dynamic excitations (sinusoidal type). An excellent performance for the modeling task is obtained (less than 0.003%). In the next step, abnormal health behavior could be detected in the distribution of the output weight distribution. This initial work demonstrates the feasibility of the idea of learning in the model space based on RC method.

Weight value

ACKNOWLEDGMENT

The work was performed within the second phase of French ANR BIPHOPROC (Brain Inspired PHOtonic PROCessing) project, with the agreement number ANR-14-OHRI-0002-01. This project is also part of the DEMO 3 within the Labex ACTION project (laboratory of excellence, funded by the ANR). Great thanks go to my college Raffaele Petrone, with whom the experimental part was collaborated. Experimental supports from Fabien Harel and Xavier François are also greatly appreciated.

Fig. 1 .

 1 Fig. 1. Traditionnal RNN and RC paradigm: (A) Traditional gradient-descent based RNN training methods adapt all connection weights, including input-to-RNN, RNN-internal, and RNN-to-output weights. (B) In RC, only the RNNto-output weights (bold arrows) are adapted ([13]).

Fig. 2 .

 2 Fig. 2. Framework of the proposed RC based fault diagnosis methodology

Fig. 3 .

 3 Fig. 3. 1 kW test bench with its control interfaces

Fig. 4 .

 4 Fig. 4. Stack current (normalized by divising 100) and voltage (normalized by divising 5) under different air stoichiometries (nominal condition FSC=2.8) with sinusoidal excitation 0.1 HzFig.5shows the trained RC output and Vstk under the nominal condition FSC = 2.8. It could be found that the RC output matches well the real stack voltage, and a Mean Square Error (MSE) is obtained less than 0.003%. Meanwhile, the corresponding output weight matrix W out is calculated and plotted with its 22 elements (i.e. 1+1+20 as indicated in Eq (2)) in Fig.6. Essentially, W out could be regarded as a group of internal FC system parameters which reflect the nonlinear relationship between the Istk and Vstk under the dynamic excitations. This input-output relationship could certainly be established in other simpler nonlinear functions such as polynomial functions. RC model is selected herein considering its demonstrated capacity in dealing with temporal signals in dynamic systems. And this characteristic will be further explored in the following work for dealing with more complex multi-fault diagnosis under dynamic operating conditions.

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

 567 Fig. 5. Stack voltage (FSC = 2.8) under sinusoidal excitation and RC output

Fig. 8 .Fig. 9 .

 89 Fig. 8. Wout distribution under different FSC configurations


3d-multi-scale-study-on-metalpolymer-nano-composites

Manon Goubet, Constantin Matei, Zineb Saghi, Bernard Viala and Jean-Herve Tortai Introduction: Metal-polymer nanocomposites are a special class of materials that are capable to achieve usually incompatible combinations of physical constants like a giant-permittivity () or a highpermeability at RF (µ) and no conductivity (). The objective of this work is to study the size, shape, separation and integrity of the nanoparticles, and the dispersion quality and volume fraction contained in the polymer. These quantities play a key role on the electrical or magnetic properties of the nanocomposite films. Two-dimension (2D) information is obtained by scanning transmission electron microscopy (STEM) with nanoparticle suspensions directly deposited on carbon grids by spin coating. Three-dimensional (3D) information is achieved by focused ion beam (FIB)scanning electron microscopy (SEM) tomography on cross-sectional samples of the nanocomposite film deposited by spincoating on a silicon wafer.

2D morphological analysis of the nanoparticles during formulation: Two types of TEM samples were taken during the nanocomposite formulation: after dispersion and after surface functionalization. The liquid samples were deposited on TEM grids by spin coating and dried on a hot plate. The observations were realized with a FEI Tecnai Osiris S/TEM operating at 200kV. The 2D distribution of particle size was performed with IPSDK software [START_REF]IPSDK Smart Image Processing[END_REF]. The High-angle annular dark field STEM (HAADF-STEM) images were segmented and the nanoparticles were labelled to measure Feret diameter and circularity (see fig 3). Overlapping nanoparticles were considered as agglomerates during the segmentation step thereby the circularity was used to discriminate single nanoparticles. Nanoparticles' mean Feret diameter allows to confirm the supplier's data, here the mean diameter (28nm) for 28 nanoparticles labelled, which was conform to the diameters given by the supplier (30 to 50nm). 3D morphological analysis of the film using FIB-SEM tomography: The acquisition on the nanocomposite film was performed on a Zeiss Crossbeam 550, with a 4 nm isotropic voxel size and using an InLens secondary electron detector. The collected FIB-SEM volume was about 13 µm x 15 µm x 5 µm, for a two-day acquisition time. After acquisition, the stack was cropped and aligned in Fiji [START_REF] Schindelin | Fiji -an Open Source platform for biological image analysis[END_REF] with MultiStackReg plugin. Previous work on nanocomposite films was reported by Takacs & al. in [START_REF] Takacs | Non-conductive ferromagnets based on core double-shell nanoparticles for radio-electric applications[END_REF], where the segmentation process was made on Avizo software, based on gloabal thresholding and watershed. Nowadays, artificial intelligence allow new methods of image segmentation: machine learning and deep learning [START_REF] Seo | Machine learning techniques for biomedical image segmentation: An overview of technical aspects and introduction to state-of-art applications[END_REF]. In this work, Ilastik [START_REF] Berg | ilastik: interactive machine learning for (bio)image analysis[END_REF] was used to segment the FIB-SEM volume. The pixel classification workflow (Pxl_C) is based on the random forest classifier algorithm, where the user chooses a predefined number of features and successive annotations are provided to train the classifier and assign features to pixels of the image. Once segmented, the nanoparticles were labelled by connected component labelling. In parallel, the segmented nanoparticles were also classified and separated in the Object Classification workflow of Ilastik (Obj_C): two criteria for object detection and segmentation are applied to select objects with the highest probability of the nanoparticle class and preserve their shape. Pixel classification (Pxl_C) and object classification (Obj_C) workflows allow working on subvolume before applying the segmentation on the total volume. Here, training was made on one subvolume (2µmx2µmx200nm) for pixel classification and object classification before batch processing on the total volume (13µmx15µmx5µm).

In order to select the more accurate description of the film properties, a comparative study have been made on Pxl_C and Obj_C segmentation efficiency. 3D rendering of the obtained volumes was made on Tomviz [START_REF] Levin | Tutorial on the Visualization of Volumetric Data Using tomviz[END_REF], see figure 2. IPSDK software was used to extract quantitative data from the subvolume and the total volume of both segmentations. Obj_C seemed less sensitive to noise than Pxl_C. This can be explained by the fact that the Obj_C segmentation works on the probability of belonging of the object to the right category, whereas Pxl_C measured were based on the connected component labelling with no noise reduction step. More nanoparticles are detected in Pxl_C (19 098 in Pxl_C, 12 315 in Obj_C), the mean Feret diameter is also inferior to Obj_C value (24,1nm in Pxl_C, 31,4 in Obj_C). The volume fraction remains nevertheless almost identical (1,18% in Pxl_C, 1,07% in Obj_C), meaning that the segmentation efficiency has a low impact on this parameter. This work is based on a machine learning approach for the segmentation of nanoparticles; another approach is possible with deep learning. The StarDist library allows the segmentation of convex objects in 2D and 3D images [START_REF] Schmidt | Cell Detection with Star-convex Polygons[END_REF], [START_REF] Weigert | Star-convex Polyhedra for 3D Object Detection and Segmentation in Microscopy[END_REF]. Knowing that Co/C nanoparticles are spherical, this approach would give much more accurate results.

Conclusion:

A combination of 2D TEM and 3D FIB-SEM tomography techniques was used to characterize nanoparticles at different steps of the fabrication of a metal-polymer nanocomposite film. TEM samples of nanoparticle suspensions were prepared by direct spin coating on grids leading to a better characterization of 2D morphological parameters. FIB-SEM tomography using multiple film cross-sections benefited of improved post image processing resulting in more accurate 3D spatial information.

Acknowledgment: This work was partly supported by PTA (Upstream Technological Platform), cooperated by CNRS RENATECH and CEA Grenoble, France

Figure 1 :

 1 Figure 1: (a) HAADF-STEM image of Co@C//Pyr-PS a . (b) Segmented image. (c) Classified and labelled nanoparticles.

Figure 2 :

 2 Figure 2 : Top (from left to right): a selected 2D FIB-SEM cross-section through the Co@C//Pyr-PS a volume; segmented image using random forest classifier (Ilastik); labelled image using connected

component labelling (Pxl_C); labelled image using the object classification approach implemented in Ilastik. Bottom: 3D rendering of the selected sub-volume.