Anne Jamet
email: anne.jamet@inserm.fr

Julien Guglielmini

Bryan Brancotte

Mathieu Coureuil

Daniel Euphrasie

Julie Meyer

Johanna Roux

Jean-Philippe Barnier

Emmanuelle Bille

Agnès Ferroni

Jean-François Magny

Christine Bôle-Feysot

Alain Charbit

Xavier Nassif

Sylvain Brisse
email: sylvain.brisse@pasteur.fr

Affiliation

High-Resolution Typing of Staphylococcus epidermidis Based on Core Genome Multilocus Sequence Typing To Investigate the Hospital Spread of Multidrug-Resistant Clones

Keywords: Staphylococcus epidermidis, cgMLST, typing

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Staphylococcus epidermidis, a member of coagulase-negative staphylococci, is an abundant and ubiquitous resident of the human skin and mucosal microbiota [START_REF] Oh | Biogeography and individuality shape function in the human skin metagenome[END_REF]. The most frequently isolated microorganisms in intensive care units (ICU)-acquired bloodstream infections are coagulase-negative staphylococci, among which S. epidermidis is prominent [START_REF] Wisplinghoff | Nosocomial bloodstream infections in US hospitals: analysis of 24,179 cases from a prospective nationwide surveillance study[END_REF]. S. epidermidis infections are almost exclusively associated with health care and occur primarily in patients with implanted medical devices, such as intravascular devices or orthopaedic and cardiac implants. In addition, neonates and immunocompromised patients represent a population at particular risk for nosocomial infections [START_REF] Becker | Coagulase-negative staphylococci[END_REF]. The ability of S. epidermidis to form biofilms contributes to the chronicity and difficulties to treat these infections [START_REF] Becker | Coagulase-negative staphylococci[END_REF][START_REF] Harris | Biofilm Morphotypes and Population Structure among Staphylococcus epidermidis from Commensal and Clinical Samples[END_REF].

The proportion of MRSE (methicillin resistant S. epidermidis) among clinical isolates is greater than 70% [START_REF] Mendes | Molecular epidemiology of Staphylococcus epidermidis clinical isolates from U.S. hospitals[END_REF], leading to the probabilistic prescription of vancomycin in patients with suspected S. epidermidis infections. Resistance to linezolid is rare but emerging [START_REF] Mendes | Molecular epidemiology of Staphylococcus epidermidis clinical isolates from U.S. hospitals[END_REF][START_REF] Tewhey | Mechanisms of linezolid resistance among coagulase-negative staphylococci determined by whole-genome sequencing[END_REF]. The emergence and global spread of multidrug-resistant S. epidermidis clones causing invasive diseases has been largely documented [START_REF] Lee | Global spread of three multidrug-resistant lineages of Staphylococcus epidermidis[END_REF][START_REF] Li | Clonal Emergence of Invasive Multidrug-Resistant Staphylococcus epidermidis Deconvoluted via a Combination of Whole-Genome Sequencing and Microbiome Analyses[END_REF]. In addition, the transmission of S. epidermidis strains among patients and between patients and medical staff, mostly in neonatal intensive care units, is possible [START_REF] Becker | Coagulase-negative staphylococci[END_REF][START_REF] Roach | A Year of Infection in the Intensive Care Unit: Prospective Whole Genome Sequencing of Bacterial Clinical Isolates Reveals Cryptic Transmissions and Novel Microbiota[END_REF][START_REF] Soeorg | Molecular epidemiology of Staphylococcus epidermidis in neonatal intensive care units[END_REF][START_REF] Milisavljevic | Genetic relatedness of Staphylococcus epidermidis from infected infants and staff in the neonatal intensive care unit[END_REF]. However, the significance of S. epidermidis transmission is likely to be underestimated, and the identification of subtypes that could represent higher risks for transmission or pathogenesis, remains difficult using available subtyping methods.

The classification of S. epidermidis isolates has so far largely relied on Multilocus Sequence Typing (MLST) based on the sequencing of 7 conserved loci [START_REF] Thomas | Improved multilocus sequence typing scheme for Staphylococcus epidermidis[END_REF]. This typing method allows the assignment of a Sequence Type (ST) to each unique allelic profile. STs can be grouped into clonal complexes (CCs) with eBURST algorithm [START_REF] Feil | eBURST: inferring patterns of evolutionary descent among clusters of related bacterial genotypes from multilocus sequence typing data[END_REF]. While an extensive strain-level diversity has been observed between healthy individuals (1), only a few healthcare associated clones are found in hospital environments worldwide [START_REF] Harris | Biofilm Morphotypes and Population Structure among Staphylococcus epidermidis from Commensal and Clinical Samples[END_REF][START_REF] Conlan | Staphylococcus epidermidis pan-genome sequence analysis reveals diversity of skin commensal and hospital infection-associated isolates[END_REF]. Indeed, more than 50% of the clinical strains and less than 10% of the community strains belong to a clonal complex encompassing ST2 and ST5 [START_REF] Harris | Biofilm Morphotypes and Population Structure among Staphylococcus epidermidis from Commensal and Clinical Samples[END_REF][START_REF] Mendes | Molecular epidemiology of Staphylococcus epidermidis clinical isolates from U.S. hospitals[END_REF][START_REF] Conlan | Staphylococcus epidermidis pan-genome sequence analysis reveals diversity of skin commensal and hospital infection-associated isolates[END_REF]. The global dissemination of these two genotypes suggests an adaptation to the hospital environment. ST2 strains are almost all resistant to methicillin [START_REF] Argudin | Biofilm formation of ica operon-positive Staphylococcus epidermidis from different sources[END_REF][START_REF] Lee | Functional analysis of the first complete genome sequence of a multidrug resistant sequence type 2 Staphylococcus epidermidis[END_REF]. Similar to the closely related S. aureus species, the mecA gene confers resistance to methicillin and is found within the SCCmec (staphylococcal cassette chromosome mec) genomic island [START_REF] Meric | Ecological Overlap and Horizontal Gene Transfer in Staphylococcus aureus and Staphylococcus epidermidis[END_REF][START_REF] Gill | Insights on evolution of virulence and resistance from the complete genome analysis of an early methicillin-resistant Staphylococcus aureus strain and a biofilm-producing methicillin-resistant Staphylococcus epidermidis strain[END_REF].

The high-resolution whole-genome sequencing (WGS) approach has been used to study S. epidermidis linezolid-resistant isolates epidemiology [START_REF] Tewhey | Mechanisms of linezolid resistance among coagulase-negative staphylococci determined by whole-genome sequencing[END_REF][START_REF] Dortet | Long-lasting successful dissemination of resistance to oxazolidinones in MDR Staphylococcus epidermidis clinical isolates in a tertiary care hospital in France[END_REF]. A major insight from WGSbased epidemiological studies is the description of long-lasting endemic persistence of S. epidermidis clones across wards of hospitals [START_REF] Chong | Molecular Epidemiology of a Vancomycin-Intermediate Heteroresistant Staphylococcus epidermidis Outbreak in a Neonatal Intensive Care Unit[END_REF][START_REF] Rodriguez-Lucas | Long-term endemic situation caused by a linezolid-and meticillin-resistant clone of Staphylococcus epidermidis in a tertiary hospital[END_REF].

The population structure of S. epidermidis is divided into three main phylogenetic lineages [START_REF] Meric | Ecological Overlap and Horizontal Gene Transfer in Staphylococcus aureus and Staphylococcus epidermidis[END_REF][START_REF] Meric | Disease-associated genotypes of the commensal skin bacterium Staphylococcus epidermidis[END_REF], which encompass six previously defined clonal groups [START_REF] Thomas | Differing lifestyles of Staphylococcus epidermidis as revealed through Bayesian clustering of multilocus sequence types[END_REF][START_REF] Tolo | Do Staphylococcus epidermidis Genetic Clusters Predict Isolation Sources?[END_REF]. These subsets of S. epidermidis show distinct associations with clinical infections and carriage. Besides, they have distinctive genetic and phenotypic features including association with biofilm synthesis genes, SSCmec and antimicrobial resistance. However, both asymptomatic carriage and invasive isolates are found in all major phylogenetic groups [START_REF] Meric | Disease-associated genotypes of the commensal skin bacterium Staphylococcus epidermidis[END_REF], and the epidemiological relationships of carriage and invasive isolates need to be deciphered at microevolutionary timescales.

Genotyping of bacterial isolates is best performed with WGS for molecular epidemiological studies [START_REF] Salipante | Application of whole-genome sequencing for bacterial strain typing in molecular epidemiology[END_REF][START_REF] Guglielmini | Genus-wide Leptospira core genome multilocus sequence typing for strain taxonomy and global surveillance[END_REF][START_REF] Ruppitsch | Defining and Evaluating a Core Genome Multilocus Sequence Typing Scheme for Whole-Genome Sequence-Based Typing of Listeria monocytogenes[END_REF][START_REF] Cody | Core Genome Multilocus Sequence Typing Scheme for Stable, Comparative Analyses of Campylobacter jejuni and C. coli Human Disease Isolates[END_REF]. Core-genome MLST (cgMLST) provides the high resolution of genome-wide methods while maintaining the benefits of standardization, reproducibility and portability of classical 7-gene MLST [START_REF] Cody | Core Genome Multilocus Sequence Typing Scheme for Stable, Comparative Analyses of Campylobacter jejuni and C. coli Human Disease Isolates[END_REF][START_REF] Maiden | MLST revisited: the gene-by-gene approach to bacterial genomics[END_REF]. While infections with multidrug resistant S. epidermidis strains are an increasingly important issue in hospitals worldwide, an effective tool for international strain comparison is still lacking.

The main aims of this study were i) To define a cgMLST scheme for S. epidermidis genotyping; ii) To provide a publicly accessible international database for S. epidermidis genomic typing; and (iii) To investigate the genetic relatedness of isolates recovered from infants hospitalized in several wards of a French hospital, including a Neonatal Intensive Care Unit (NICU). We also explored the population structure of S. epidermidis and the association of previously reported marker genes and lineages.

Methods

Routine procedures for bacterial isolation from clinical samples

The 25 isolates sequenced here were recovered from one healthy adult and from cultures performed routinely on clinical samples in the clinical microbiology laboratory at Necker-Enfants Malades Hospital (Paris, France). The strains were grown on Columbia agar supplemented with 5% horse blood and incubated 24-48 h at 37°C (INCUCELL incubator, MMM group, Germany). Coagulase-negative staphylococci isolates were identified at the species level by MALDI-TOF MS (Bruker Daltonics, Bremen, Germany).

Core genome MLST (cgMLST) scheme definition

For the determination of the cgMLST scheme, 5 closed S. epidermidis genomes and 299 draft genome assemblies available at NCBI GenBank (on the 22th of December 2015) were downloaded. We also downloaded 83 draft genome assemblies deposited by Meric et al. [START_REF] Meric | Ecological Overlap and Horizontal Gene Transfer in Staphylococcus aureus and Staphylococcus epidermidis[END_REF] in the Dryad repository. All used assemblies are publicly available from the Institut Pasteur instance of the BIGSdb database at https://bigsdb.pasteur.fr/epidermidis/.

Low quality assemblies were excluded based on the following criteria: assembled genome size >150 % of the reference genome; organism identified as non S. epidermidis as determined using Kraken v0.10.5 [START_REF] Wood | Kraken: ultrafast metagenomic sequence classification using exact alignments[END_REF] and Average Nucleotide Identity < 95% compared to ATCC 12228 strain reference (GCF_000007645) [START_REF] Haubold | andi: fast and accurate estimation of evolutionary distances between closely related genomes[END_REF]. This resulted in the selection of 371 genomes for core genes definition. Supplementary table S1 provides a detailed description of the strain collection used for the establishment of the cgMLST scheme (e.g., date of isolation, geographic origin and source).

From this set we inferred the species core genome using the CoreGeneBuilder pipeline default settings [START_REF] Guglielmini | Genus-wide Leptospira core genome multilocus sequence typing for strain taxonomy and global surveillance[END_REF] and S. epidermidis ATCC 12228 strain (GCF_000007645) as a reference. The pipeline's first step relies on the eCAMBer software [START_REF] Wozniak | eCAMBer: efficient support for large-scale comparative analysis of multiple bacterial strains[END_REF], which consists of a de novo annotation of the genomes (except the reference) using Prodigal [START_REF] Hyatt | Prodigal: prokaryotic gene recognition and translation initiation site identification[END_REF] and the harmonization of the positions of the stop and start codons. In the next step, the core genome is inferred with a bidirectional best hits (BBH) approach as previously described by Touchon et al. [START_REF] Touchon | Organised Genome Dynamics in the Escherichia coli Species Results in Highly Diverse Adaptive Paths[END_REF]. We used CoreGeneBuilder default settings. A gene was considered as part of the core genome if it was found in at least 95% of the genomes under study. This resulted in an initial core genome containing 1623 genes. We then filtered out some genes based on the following criteria. (i)

First, we removed 14 potential paralogous loci. The rationale behind this is that the presence of paralogs inside a typing scheme can lead to ambiguities, as a candidate gene might be attributable to two different core gene loci. To detect those potential paralogs, we compared each allele of each locus against all the alleles of all the other loci using the software BLAT [START_REF] Kent | BLAT---The BLAST-Like Alignment Tool[END_REF]. If a single hit was found between two different loci (more than 70% protein identity between two alleles), we removed both loci. (ii) Second, we removed 840 loci whose length varies among alleles, which is useful in reducing ambiguities during the genotyping process.

(iii) We also removed 10 loci for which alleles contained ambiguous characters. (iv) Last, we removed 582 loci with variation within the start and stop codons; the rationale is that allele calling relies on BLASTn, which may not produce full-length alignments when the start or stop codons differ, resulting in incomplete alleles. Note that these sets of filtered loci can overlap. After applying these four steps, 699 core genes were retained and constitute our cgMLST scheme.

cgMLST allele definitions and tagging of publicly available genomes using BIGSdb

The BIGSdb server is designed to host separately a database of allele sequence definitions and a database of genomic assemblies with attached source information [START_REF] Jolley | BIGSdb: Scalable analysis of bacterial genome variation at the population level[END_REF]. Newly imported assemblies were scanned using BIGSdb to designate alleles of each locus. A locus was tagged when its sequence had > 90% nucleotide sequence identity over at least 90% of the alignment length with a reference allele. In case of exact match with an allele found in the database, the locus was designated by the allele number. In case of partial match, if the locus corresponded to a protein-coding sequence (beginning with a start codon and ending with a stop codon), the allele was added to the allele database. Then, a subsequent scan allowed tagging isolates with novel alleles. A locus was considered missing when there was no match with alleles in the database.

To evaluate the conservation of the chosen genes in an independent sample of the S. epidermidis population, we analyzed 289 additional genome assemblies from NCBI GenBank and from Harris et al. (4) that were not yet available at the time of scheme definition. The validation set is described in Supplementary table S1.

All assemblies, allele sequences designations and profiles are available at http://bigsdb.pasteur.fr/epidermidis/.

Assessment of the population structure of S. epidermidis

The population structure of S. epidermidis was assessed based on 703 available genomes. To obtain the alignment required for the phylogenetic analysis, we extracted, for each genome assembly, the allelic sequences of the 699 loci. For each locus the alleles sequences were translated to proteins, aligned using MAFFT [START_REF] Katoh | MAFFT multiple sequence alignment software version 7: improvements in performance and usability[END_REF] and the G-INS-i algorithm, and backtranslated to nucleotide sequences, thus resulting in codon alignments. We finally concatenated the 699 alignments into one supermatrix, containing 703 sequences (isolates) and 539 151 sites (the nucleotide positions of missing alleles were replaced by gaps in the concatenate of alignments of individual loci). A phylogenetic tree was obtained with the maximum likelihood (ML) criterion after purging recombination using Gubbins v2.2.0 [START_REF] Croucher | Rapid phylogenetic analysis of large samples of recombinant bacterial whole genome sequences using Gubbins[END_REF] with default parameters resulting in an alignment of 31,385 sites. RAxML v8.2.8 (39) was used for the phylogenetic inference under the GTRCAT model (final tree optimized under the GTRGAMMA model). The tree was drawn using iTOL [START_REF] Letunic | Interactive Tree Of Life (iTOL) v4: recent updates and new developments[END_REF].

The same methodology was applied to compare clusters obtained based on cgMLST allelic profiles with those observed by phylogenetic analysis of the concatenated allele sequences of the 699 core loci.

Patients and isolates

To evaluate the ability of cgMLST to identify potential transmission events within hospital settings, we selected 25 isolates including 24 clinical isolates from patients hospitalized in several wards between November 2014 and August 2016.

The 25 isolates were recovered from 21 hospitalized individuals (16 hospitalized neonates and 5 hospitalized children) and one non-hospitalized healthy adult. All neonates have stayed in a NICU, 3 children have stayed in a gastroenterology ward and 2 children have stayed in an orthopedic ward. One isolate was sequenced per patient except for 3 patients who had 2 isolates sequenced (patients 3, 7 and 21). Information on patients and sources of isolates is summarized in Table 1.

The study of S. epidermidis clinical isolates collected from the Necker-Enfants Malades Hospital was approved by the ethics review board of our institution and registered with clinicaltrials.gov Identifier#NCT03374371. All experiments were performed in accordance with the guidelines and regulations described by the Declaration of Helsinki and the low Huriet-Serusclat on human research ethics and informed consent was obtained for all participating subjects.

Whole-genome sequencing and assembly

DNA extractions were performed on pure culture using DNeasy Blood and Tissue Kit (Qiagen) following manufacturer's instructions with addition of lysostaphin (Sigma) to the Gram-positive lysis buffer (final concentration 20 μg/mL).

Whole-genome shotgun libraries sequenced on an Illumina MiSeq sequencing platform (2 x 150 bp paired-end reads) were prepared with the Nextera XT Kit (Illumina). Whole-genome shotgun libraries sequenced on an Illumina HiSeq-2500 sequencing platform (2 x 130 bp paired-end reads) were prepared with the Ovation Ultralow V2 kit (NuGEN).

Reads were trimmed using Trimmomatic v0.36 [START_REF] Bolger | Trimmomatic: a flexible trimmer for Illumina sequence data[END_REF] and assembled using SPAdes v3.9.0 [START_REF] Bankevich | SPAdes: a new genome assembly algorithm and its applications to single-cell sequencing[END_REF].

De novo assemblies quality was assessed with QUAST [START_REF] Gurevich | QUAST: quality assessment tool for genome assemblies[END_REF]. In silico MLST was performed on the de novo assemblies with MLST v2.6 (https://github.com/tseemann/mlst). De novo assemblies were imported and scanned in BIGSdb.

Cluster analysis of clinical isolates using BIGSdb

To validate the ability of cgMLST to cluster related isolates, we first used sequence data from a recently published study of linezolid resistant isolates [START_REF] Tewhey | Mechanisms of linezolid resistance among coagulase-negative staphylococci determined by whole-genome sequencing[END_REF]. Raw Illumina reads (accession number SRP039360) were downloaded from NCBI Sequence Read Archive (SRA) and processed as newly sequenced genomes. Assemblies were imported and scanned in BIGSdb except isolate UCLA-Sa-LR-01, which was discarded due to poor quality of its de novo assembly. We also tested our cgMLST scheme using the 25 newly sequenced clinical isolates (Table 1). The PHYLOViZ tool (integrated as a plugin of BIGSdb) was used to generate minimum spanning trees based on the cgMLST allelic profiles [START_REF] Nascimento | PHYLOViZ 2.0: providing scalable data integration and visualization for multiple phylogenetic inference methods[END_REF]. Here, clusters were defined as groups of genotypes showing ≤ 21 cgMLST allele mismatches with at least one other member of the group.

Data availability

This Whole Genome Shotgun project has been deposited in GenBank under the Bioproject ID PRJNA608162.

All assemblies, sequences and their allelic designations and cgMLST profiles are available

Results

Definition and evaluation of a cgMLST scheme for S. epidermidis using public genomes

Using 371 publically available genomes, we developed a cgMLST scheme consisting of 699 core-genes, which includes 30% of the 2,348 CDS of S. epidermidis ATCC 12228 reference genome. As shown in Supplementary figure S1, the distribution along the S. epidermidis genome of the 699 cgMLST genes was even.

To evaluate the conservation of gene loci of the newly defined cgMLST scheme, a validation set of 289 additional publically available genome assemblies was analyzed (NCBI genomes, last accessed on Sept. 27 th , 2017). These additional genomes were made public after the definition of our cgMLST scheme and thus represent a non-overlapping strain collection.

These genomes had 97.5% tagged loci on average, showing that the cgMLST scheme loci are highly conserved, and can therefore be expected to be mostly present in future sequenced genomes. Supplementary figure S2 shows the distribution of the number of missing loci among the 699 genes of the scheme.

The cgMLST genotyping system is publicly available through the Institut Pasteur BIGSdb server (https://bigsdb.pasteur.fr/epidermidis/).

Population structure of S. epidermidis genomes

We next used 703 genomes to assess the phylogenetic structure of S. epidermidis based on cgMLST gene sequences (Supplementary Figure 1), which confirmed that the population is divided into 3 main clades (A-C) as previously reported [START_REF] Meric | Ecological Overlap and Horizontal Gene Transfer in Staphylococcus aureus and Staphylococcus epidermidis[END_REF]. Most of the isolates belong to clade A, which encompasses clonal groups 5 and 6 associated with the two major nosocomial clones, ST2 and ST5 respectively [START_REF] Thomas | Differing lifestyles of Staphylococcus epidermidis as revealed through Bayesian clustering of multilocus sequence types[END_REF][START_REF] Tolo | Do Staphylococcus epidermidis Genetic Clusters Predict Isolation Sources?[END_REF].

The supplementary figure S3 shows the concordance with clades A, B and C as defined previously by Méric and colleagues [START_REF] Meric | Ecological Overlap and Horizontal Gene Transfer in Staphylococcus aureus and Staphylococcus epidermidis[END_REF], with the clonal groups defined previously [START_REF] Thomas | Differing lifestyles of Staphylococcus epidermidis as revealed through Bayesian clustering of multilocus sequence types[END_REF][START_REF] Tolo | Do Staphylococcus epidermidis Genetic Clusters Predict Isolation Sources?[END_REF], and with classical 7-gene MLST.

In addition to the core genes, we scanned the assemblies for the presence of accessory genes associated with the SCCmec locus (mecA, ccrA and ccrB) and the ica genes associated with biofilm formation capacity through polymeric N-acetyl-glucosamine (PNAG) synthesis. In addition, we searched for the putative commensal marker fdh encoding a formate dehydrogenase [START_REF] Conlan | Staphylococcus epidermidis pan-genome sequence analysis reveals diversity of skin commensal and hospital infection-associated isolates[END_REF].

The distribution of SCCmec associated genes among infection and commensal isolates

showed a positive association between infection strains and the SCCmec locus (Fisher <1e-3)

(Figure 1). The search of the co-occurrence of the icaAD genes, commonly used as a proxy to evaluate the biofilm formation potential of isolates [START_REF] Cafiso | Presence of the ica operon in clinical isolates of Staphylococcus epidermidis and its role in biofilm production[END_REF], yielded no significant association (Figure 1). In contrast, a positive association between commensal isolates compared to infection isolates was observed for the putative commensal marker gene fdh (Fisher <0.005) (Figure 1).

Application of the S. epidermidis cgMLST scheme to hospital linezolid-resistant isolates

To evaluate the ability of cgMLST to cluster S. epidermidis isolates, we analyzed the relationships among isolates from a previous study of linezolid resistant isolates [START_REF] Tewhey | Mechanisms of linezolid resistance among coagulase-negative staphylococci determined by whole-genome sequencing[END_REF].

All genomes had > 93% of tagged loci (95.5% on average). Cluster analysis based on cgMLST allelic profiles showed 4 clusters and 1 outgroup isolate (Figure 2). The same clusters were found after phylogenetic analysis of the concatenated sequences of core genes (Figure 2). These results were concordant with the ones reported by Tewhey et al. based on the alignments of 2,093 core genes (6). However, there was no description of potential epidemiological links between the isolates of this study, which were compared only because of their common phenotypic linezolid resistance.

Application to S. epidermidis epidemiology in a French hospital

To assess the ability of cgMLST to cluster S. epidermidis isolates with defined epidemiological sources, we next sequenced 24 clinical isolates recovered from several wards of a pediatric French hospital and an isolate from a healthy adult (total of 25 sequenced isolates). The mean number of contigs was 211 (range, 137 to 410) (Table S2). After cgMLST genotyping of the genomic assemblies, the mean percentage of tagged loci was 97.6% (range, 96.1 to 100%) (Table S2). As expected, most isolates recovered from hospitalized patients (20 out of 24) belonged to ST2 (n = 12) and ST5 (n = 8) (Table 1).

Nineteen isolates were grouped into 6 clusters (≤ 21 allele differences), whereas there were 6 ungrouped isolates (Figure 3). Five clusters were observed among isolates from the NICU during one year (span time of clusters ranging from 49 to 354 days), whereas another cluster contained isolates from the orthopedic and the gastroenterology wards. The maximum number of differing alleles among isolates of the same cluster was 21, whereas the minimum number of differing alleles between isolates of distinct clusters was 39 (Figure 3). For the 3 pairs of isolates recovered from 3 individual patients, collected between 7 to 38 days apart, we observed a maximum of 7 differences. The timeline shown in Figure 4 illustrates that isolates belonging to the same cluster were sometimes found in patients hospitalized in distinct wards (i.e., cluster IV in patients hospitalized in gastroenterology and orthopedic units) and in patients hospitalized up to 1-year apart (i.e., cluster I for patients 3, 16 and 18).

Discussion

S. epidermidis is a leading cause of bacteremia and device-related infections and can cause severe infections in immunocompromised patients and neonates. S. epidermidis infections are mostly caused by strains belonging to multidrug-resistant nosocomial clones ST2 and ST5 [START_REF] Mendes | Molecular epidemiology of Staphylococcus epidermidis clinical isolates from U.S. hospitals[END_REF][START_REF] Soeorg | Molecular epidemiology of Staphylococcus epidermidis in neonatal intensive care units[END_REF][START_REF] Dortet | Long-lasting successful dissemination of resistance to oxazolidinones in MDR Staphylococcus epidermidis clinical isolates in a tertiary care hospital in France[END_REF][START_REF] Shelburne | Whole-genome sequencing of Staphylococcus epidermidis bloodstream isolates from a prospective clinical trial reveals that complicated bacteraemia is caused by a limited number of closely related sequence types[END_REF]. Our results corroborate the predominance of ST2 and ST5 among hospitalized patients. Although the 7-gene MLST has played an important role to recognize the worldwide emergence and nosocomial endemic persistence of multidrug-resistant clones, there is a need for higher resolution genotyping, allowing strain comparison within endemic clones.

Moreover, standardization is paramount for worldwide tracking of strain dissemination and recognizing subtypes with medically important properties. We therefore explored a core genome MLST (cgMLST) approach, which provides a high resolution and has the potential for a universal typing method both for epidemiological investigations and for population biology studies. Standardized cgMLST schemes are available for many pathogens including Klebsiella pneumoniae, Campylobacter jejuni, Campylobacter coli, Staphylococcus aureus and Listeria monocytogenes [START_REF] Cody | Core Genome Multilocus Sequence Typing Scheme for Stable, Comparative Analyses of Campylobacter jejuni and C. coli Human Disease Isolates[END_REF][START_REF] Bialek-Davenet | Genomic definition of hypervirulent and multidrug-resistant Klebsiella pneumoniae clonal groups[END_REF][START_REF] Moura | Real-Time Whole-Genome Sequencing for Surveillance of Listeria monocytogenes, France[END_REF][START_REF] Leopold | Bacterial wholegenome sequencing revisited: portable, scalable, and standardized analysis for typing and detection of virulence and antibiotic resistance genes[END_REF]. However, a cgMLST scheme has been lacking until now for S. epidermidis.

The potential of the developed S. epidermidis cgMLST scheme to delineate closely related isolates was investigated by comparing genomes obtained from children hospitalized in several wards of the same hospital. Our analysis showed that the cgMLST scheme of 699 genes had an adequate discriminatory power to delineate distinct clusters of very closely related isolates within the same ST. The maximum allelic distance among epidemiologically related isolates (2 isolates recovered from 2 patients hospitalized several months apart in the orthopedic ward) was 21, and we chose this cutoff to define clusters. This relatively high number of differences is likely to reflect the fact that clones are able to persist several years in hospitals where they diversify and are likely transferred to patients on multiple occasions [START_REF] Lee | Global spread of three multidrug-resistant lineages of Staphylococcus epidermidis[END_REF][START_REF] Chong | Molecular Epidemiology of a Vancomycin-Intermediate Heteroresistant Staphylococcus epidermidis Outbreak in a Neonatal Intensive Care Unit[END_REF][START_REF] Rodriguez-Lucas | Long-term endemic situation caused by a linezolid-and meticillin-resistant clone of Staphylococcus epidermidis in a tertiary hospital[END_REF]. For instance, Dortet et al. investigated a long-lasting outbreak spanning 2 years caused by linezolid-resistant S. epidermidis clones and report 22 ST2 isolates with 13 SNPs along their core genome [START_REF] Dortet | Long-lasting successful dissemination of resistance to oxazolidinones in MDR Staphylococcus epidermidis clinical isolates in a tertiary care hospital in France[END_REF]. Our findings also show that several clones can co-exist and persist over months in a NICU. This is concordant with previous observations showing that patients acquire genetically related strains belonging to multidrug-resistant endemic clones that persist in clinical settings [START_REF] Soeorg | Molecular epidemiology of Staphylococcus epidermidis in neonatal intensive care units[END_REF][START_REF] Krediet | Molecular epidemiology of coagulase-negative staphylococci causing sepsis in a neonatal intensive care unit over an 11-year period[END_REF].

In addition, we leveraged the ability of the BIGSdb database tool to analyse the presence of genes of medical interest to compare the distribution of several putative marker genes associated with infection or carriage. We confirmed a positive association between infection strains and the SCCmec locus [START_REF] Meric | Disease-associated genotypes of the commensal skin bacterium Staphylococcus epidermidis[END_REF] and a positive association between commensal isolates and the fdh gene [START_REF] Conlan | Staphylococcus epidermidis pan-genome sequence analysis reveals diversity of skin commensal and hospital infection-associated isolates[END_REF]. The co-occurrence of icaAD genes has been proposed as a marker for biofilm formation ability (45) linked with isolate virulence. However, the presence of the ica locus does not always correlate with biofilm production in vitro [START_REF] Cafiso | Presence of the ica operon in clinical isolates of Staphylococcus epidermidis and its role in biofilm production[END_REF]. For instance in a study of 200 S. epidermidis isolates from hospitalized patients in Germany, among 124 icaADpositive isolates, only 49% were able to produce biofilm in vitro, whereas only 50% of in vitro biofilm-producing isolates were ica-positive [START_REF] Mertens | Genetic determinants and biofilm formation of clinical Staphylococcus epidermidis isolates from blood cultures and indwelling devises[END_REF]. The weak correlation between ica presence and biofilm formation ability is corroborated by Du et al. who reported biofilm production in less than 60% of their icaA-positive isolates [START_REF] Du | Molecular analysis of Staphylococcus epidermidis strains isolated from community and hospital environments in China[END_REF].

In another study, when nosocomial isolates associated with infection or contamination were compared only mecA, but neither icaA nor fdh, was differentially found among infection and contamination isolates [START_REF] Tolo | Do Staphylococcus epidermidis Genetic Clusters Predict Isolation Sources?[END_REF].

In conclusion, we developed a cgMLST scheme for WGS-based typing of S. epidermidis, integrated it in a publically available BIGSdb web application platform and illustrated its application to the molecular epidemiology of S. epidermidis in a French university hospital.

We hope that the novel genotyping tool will contribute to allowing reproducible comparisons among clinical laboratories for the epidemiological surveillance and biological studies of S.

epidermidis clones causing infections. There was a positive association between infection isolates and mecA, ccrA and ccrB loci compared to commensal isolates (** Fisher<0.001) and there was a positive association between commensal isolates and fdh locus compared to infection isolates (* Fisher <0.005). The minimum-spanning tree based on cgMLST allelic profiles is shown on the left side. The tree was generated from allelic profiles of 699 core genes and visualised with PHYLOViZ.

Each node represents an isolate and is coloured according to the 7-loci ST. In silico MLST performed on the de novo assemblies revealed that isolates UCLA-Sa-LR-14 (497) and UCLA-Sa-LR-21 (503) had been erroneously assigned to ST24 and ST186-like respectively in the original publication [START_REF] Tewhey | Mechanisms of linezolid resistance among coagulase-negative staphylococci determined by whole-genome sequencing[END_REF]. We found that in fact both strains belonged to ST186. Black The patient numbers indicated in the coloured boxes and the patient isolate numbers indicated between brackets correspond to those in Table 1. The clusters to which isolates belong correspond to those in Figure 3. The isolate sequence type (ST) is indicated on the right side of the graph.

  /bigsdb.pasteur.fr/epidermidis/.

Figure 1 .

 1 Figure 1. Distribution of gene presence and absence for mecA, ccrAB, ica and fdh loci among 252 infection isolates and 194 commensal isolates from the BIGSdb database. mecA encodes penicillin-binding protein 2 (PBP2); ccrA and ccrB encode cassette chromosome recombinase A and B respectively; icaA and icaD genes are involved in polysaccharide intercellular adhesin (PIA) production; fdh encodes a formate dehydrogenase.

Figure 2 .

 2 Figure 2. Comparison of cluster analysis based on cgMLST allelic profiles with phylogenetic analysis based on core genes alignments for 20 linezolid-resistant isolates.

  numbers next to the nodes indicate the BIGSdb identifiers of each isolate. Red numbers on the connecting lines indicate the number of core gene allele differences between adjacent nodes. Link lengths are loosely related to the number of allelic differences. The tree based on 699 core gene sequences alignments is shown on the right side. The tip labels indicate the BIGSdb identifiers of each isolate. Clusters reported by Tewhey et al. based on the alignments of 2,093 core genes (6) are indicated by circles and letters (A, B, C and D).

Figure 3 .

 3 Figure 3. Comparison of clustering analyses based on cgMLST allelic profiles and on core gene sequences alignment, for 25 S. epidermidis isolates sequenced in this study.The minimum-spanning tree based on cgMLST allelic profiles is shown on the left side. The tree was generated from allelic profiles of 699 core genes. Each node represents an allelic

Figure 4 .

 4 Figure 4. Timeline of the 21 hospitalized patients from which 24 S. epidermidis isolates were sequenced.

Acknowledgements

The authors want to thank Pr C. Bréchot for having stimulated the collaboration between the Necker Hospital and Institut Pasteur in the frame of the Pasteur International biobanking network (PIBnet) programme. We thank Bryan Brancotte for maintenance of the Institut Pasteur web application.

Funding information: Contrat de Recherche Clinique, French Ministry of Health (CRC17040); the sponsor was Assistance Publique des Hôpitaux de Paris (APHP) (Département de la Recherche Clinique et du Dévelopement).

Author statements

Conflicts of interest

The authors declare that there are no conflicts of interest Ethical approval: The study of S. epidermidis clinical isolates collected from the Necker-Enfants Malades Hospital has been approved by the ethics review board of our institution and registered with clinicaltrials.gov Identifier#NCT03374371. All experiments were performed in accordance with the guidelines and regulations described by the Declaration of Helsinki and the low Huriet-Serusclat on human research ethics and informed consent was obtained for all participating subjects.


S Clénet

T Henneron

J Korecki

Sensor placement for field reconstruction in rotating electrical machines

Keywords: Electrical machines, Field reconstruction, Sensor placement, Finite Element Method

A method is proposed to place sensors in an electrical machine in order to be able to reconstruct the magnetic field distribution. This method is based on the Empirical Interpolation Method combined with the Maxvol technique. The results applied on a surface mounted permanent magnet machine at no load show that the field distribution can be accurately reconstructed even when the sensor location is imposed in the airgap of the rotating machine.

I. INTRODUCTION

eing able to reconstruct the magnetic flux density distribution in an electrical machine from several local field measurements would be of great interest enabling to estimate global and local quantities (iron losses, torque…) in real time. With the development of IoT (Internet of Things) combined with the progress in microelectronics, we can expect, in a next future, to equip electrical machines with wireless sensors of magnetic field, temperature… at low cost. The question, which arises, is then: are we able to reconstruct the field distribution in an electrical machine from a limited number of sensors and where shall we locate them?

It seems that this issue of sensor placement has not been addressed in the case of electrical machines especially for internal field reconstruction.

Besides, in numerous research fields, methods have been proposed to find the best position of sensors. Some methods are based on the derivation of the sensor location from Partial Derivative Equations (PDE) model solved analytically [START_REF] Veldman | Sensor and Actuator Placement for Proportional Feedback Control in Advection-Diffusion Equations[END_REF] or numerically like the (Discrete) Empirical Interpolation Method ((D)EIM) [START_REF] Barrault | An "empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations[END_REF][START_REF] Chaturantabut | Nonlinear Model Reduction via Discrete Empirical Interpolation[END_REF]. This last method has been applied in the case of nuclear reactor [START_REF] Argaud | Sensor placement in nuclear reactors based on the generalized empirical interpolation method[END_REF]. Moreover, in the field of computational electromagnetics, the (D)EIM has been already applied with success to reconstruct the distribution of the electromagnetic field in the regions where the behavior law is nonlinear, when constructing a reduced order model from a Finite Element (FE) model [START_REF] Henneron | Model-Order Reduction of Multiple-Input Non-Linear Systems Based on POD and DEI Methods[END_REF] [START_REF] Montier | Structure Preserving Model Reduction of Low-Frequency Electromagnetic Problem Based on POD and DEIM[END_REF]. For all these reasons, the (D)EIM seems to be a good candidate for sensor placement for field reconstruction on the whole electrical machine.

In this paper, a method based on (D)EIM is investigated to place sensors in order to reconstruct the field distribution in a rotating electrical machine. The (D)EIM is coupled with the Maxvol technique to improve its efficiency [START_REF] Goreinov | How to find a good submatrix[END_REF]. The method is tested on a permanent magnet machine.

II. METHOD FOR SENSOR PLACEMENT

A. Description of the problem

Let us consider a FE model of an electrical rotating machine. We denote ne the number of mesh elements, θ the rotor position and B(θ), 2ne×1, the vector containing the radial and tangential components [bi(θ)]1≤i≤2ne of the magnetic flux density on each element. Let us introduce the vector B sens (θ), M×1, of the magnetic flux density measured by M sensors which represent in fact M entries of B(θ). We denote ϕ, M×1, the vector of the indices of these M entries. We have then:

B sens (θ)= [b sens 1(θ),…,b sens M (θ)] t =[bϕ(1)(θ),…,bϕ(M)(θ)] t (1)
The problem of sensor placement consists then in finding the vector of indices ϕ and the 2ne×M matrix R=[ri] 1≤i≤M, such that the error between B(θ) and the reconstructed field

B reconst (θ)=R B sens (θ) (2)
is minimum. Once ϕ and R are determined, the magnetic flux density distribution B reconst (θ) is reconstructed from B sens (θ) gathering M radial or tangential components of the magnetic flux density of B(θ).

The problem of choosing the best vector of indices ϕ, which is equivalent to finding the best sensor locations, is combinatorial, which is computationally unfeasible. In fact, the complexity of this combinatorial optimization problem is proportional to 2 n M which represent the number of the Mcombinations of the 2ne entries of B(θ). For example, let us consider a mesh with ne=30000 elements and a sensor number M=10, the M-combination number of sensors is equal to 1.62 10 38 . As an alternative, we have applied the (D)EIM [START_REF] Veldman | Sensor and Actuator Placement for Proportional Feedback Control in Advection-Diffusion Equations[END_REF][2] combined with the Maxvol algorithm [START_REF] Chaturantabut | Nonlinear Model Reduction via Discrete Empirical Interpolation[END_REF]. It enables to obtain quickly a suboptimal sensor location ϕ as well as the reconstruction matrix R. The complexity of this algorithm is proportional to the number of sensors M. In the following, we present the (D)EIM and then the Maxvol algorithm.

B. (D)EIM

The aim of the (D)EIM is to approximate the 2ne×1 vector B(θ) by a 2ne×1 vector B reconst (θ) in the form [START_REF] Barrault | An "empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations[END_REF][START_REF] Chaturantabut | Nonlinear Model Reduction via Discrete Empirical Interpolation[END_REF]:

θ ≈ θ = θ = ∑ c θ (3)

B

with Ui the i th column of a 2ne×M matrix U and c(θ) is a vector which entries [c1(θ),…,cM(θ)] being linear combinations of M "well-chosen entries" of the vector B(θ). Then, it is possible to "measure" only M entries of B(θ) to compute ci(θ) and thus, by applying (1), to determine an approximation B reconst (θ) of B(θ). In order to determine these M entries, snapshots of B(θ) (i.e. solutions of the FE model) are calculated for N angular positions of the rotor (θ1,…,θN). Then, the approximation B reconst (θ) is sought in a subspace K M spanned by an orthogonal basis U=[U1,…,UM]. This basis can be determined by a Gram-Schmidt Process or a Singular Value Decomposition (SVD) of the snapshot matrix B =[B(θ1), …, B(θn)]. In our case, we have considered the M singular vectors Ui associated to the M most significant singular values of the SVD of the matrix B. Then, by applying a greedy algorithm, the M entries of B(θ) are determined and stored in the M×1 index vector ϕ. Only the entries [bϕ(1)(θ),…,bϕ(M)(θ)] need to be calculated to determine the vector c(θ) and so B reconst (θ) from (2). Let P denote the N×M matrix such that the j th column is the ϕ(j) th column eϕ(j) of the identity matrix INxN. All the entries of P are equal to zero except the M entries pϕ(j),j (1≤j≤M). The vector c(θ) is then given by:

c(θ) = (P t U) -1 P t B(θ) (4)
The M×M matrix (P t U) -1 can be precalculated and the M entries of the vector P t B(θ) are equal to the M entries of B(θ) gathered in the M×1 vector B sens (θ) introduced in (1). Then, combining (3) and ( 4), the expression of the matrix R is given by: R =U(P t U) -1 (5) which enables to reconstruct the field distribution B reconst (θ) by [START_REF] Barrault | An "empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations[END_REF].

A. Maxvol technique

The algorithm Maxvol is an iterative process which aims at finding a M×M submatrix M sub of the N×M matrix M (N>M) with an almost maximum volume (the volume of a square matrix M sub is given by vol(M sub )=|det(M sub )|. The problem of finding the matrix with the maximum volume is combinatorial. The Maxvol technique enables to obtain a suboptimal solution in a reasonable time [9]. The input is the index set ϕ obtained with the (D)EIM describes in the previous section. The output is new index set ϕ from which a new matrix R is recalculated from (5).

III. APPLICATION

A. Machine studied A 5-phase buried permanent magnet synchronous machine is considered. The machine has 7 pole pairs and 20 slots leading to a fractional number of slots per pole pair. The mesh of the ferromagnetic part is given in Fig. 1 and the number of elements is equal to ne=37552.

B. Evaluation of (D)EIM combined with Maxvol technique

In order to evaluate the proposed method of sensor placement for the reconstruction of the magnetic flux distribution, we consider the operating point at no load. The supply currents of the machine are equal to zero, the magnetic field is only created by the permanent magnets. We consider a rotation interval of 360/7°, which corresponds to an electrical period. The field distribution Bj (1≤j≤N) has been calculated for N=90 angular positions uniformly distributed on this interval. We have then constructed the 2ne×N matrix of snapshots B by concatenating the 2ne×1vectors Bj (1≤j≤N). Then, we have applied the (D)EIM and the Maxvol technique (see II-B and II-C) to determine the position of the M sensors ϕ as well the reconstruction matrix R (see [START_REF] Barrault | An "empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations[END_REF]). In order to evaluate the quality of the reconstructed field as a function of the number of sensors M, we define the reconstruction error εM:

ε = log " # ∑ $ % & % '()*+,-,/ $ 0 1 %23 ∑ $ % $ 0 1 %23 4 (6)
With 5 , being the reconstruction field at the j th position using M sensors. We have calculated the error εM when applying the (D)EIM alone and when combining the (D)EIM with the Maxvol technique. We present in Fig. 2 the evolution of the error ε as a function of the sensor number M (1≤j≤N-1). We do not consider the case M=N because, by principle of the (D)EIM, the error ε is equal to zero. We can see that the (D)EIM and (D)EIM/Maxvol method has almost the same behavior with a decrease of the error as a function of the sensor number which tends to zero. We can see also that the Maxvol technique enables to improve slightly the quality of the reconstruction by relocating more conveniently the sensors. It should be noticed that the (D)EIM is very fast, even combined with the Maxvol technique, leading to very short computation time of ϕ and R whatever the number of sensors (less than 1 minute with Scilab® for the proposed example).

C. Location of the sensors

We also study the evolution of the position of the sensors as a function of the number of sensors. In Fig. 3, we indicate the location of the sensors for M=20. We can see that all of them are close to the airgap and that the number of sensors measuring either the radial (Mr=11) or the tangential (Ms=9) components of the magnetic flux density are almost the same. We present also the location, where the error εmax which is equal to the maximum entry of the matrix B-B recons , which corresponds to the next point that would be selected by the (D)EIM. We can see that it is still located in the airgap. Globally, whatever the number of sensors, we note that the sensors are almost always located in the airgap or very close to it either on the rotor or stator side. The field distribution varies not so much in the stator and in the rotor as a function of the position, whereas this variation is much more complex within the airgap or close to it leading to always a higher difference between the snapshot B and the reconstructed field B recons in the airgap. This difference being the criterion for the choice of the sensor position by the (D)EIM/Maxvol, the sensors are always chosen in that region.

D. Modification of the selection process

For further practical applications of the proposed method, the sensors should be located in regions where it is easy to implement them. The airgap seems to be the best fitting region. The sensors can be inserted easily since the region is quite accessible. Moreover, this insertion can be done almost at the end of the process of fabrication before assembling the stator and the rotor together. It leads only to few modifications on the existing manufacturing processes to add the sensors. It has been shown previously that the sensors are located in the airgap or very close to it without imposing any constraint on the position (The sensors should be located inside the machine). If we impose the sensors to be located inside the airgap, we can expect to keep a good quality of reconstruction. In order to evaluate it, we have run the (D)EIM/Maxvol technique by restricting the sensor location to the airgap. In Fig. 4, we present the evolution of the error εM (see [START_REF] Veldman | Sensor and Actuator Placement for Proportional Feedback Control in Advection-Diffusion Equations[END_REF]) as a function of the number of sensors when applying the (D)EIM/Maxvol technique with and without imposing the sensor location in the airgap. We can see an error decreasing according to M in both cases. The comparison, with the original (D)EIM/Maxvol technique, shows that there is a small accuracy loss for a sensor number M lower than 20 but which remains acceptable (the maximum error ratio is equal to 1.2 for M=7). In the following, we will use only the (D)EIM/Maxvol/Airgap technique.

E. Local field reconstruction

We have reconstructed the magnetic flux density for different numbers M of sensors. In order to illustrate our method, we have selected to two points, presented in Fig. 5, one located in the rotor and one in the stator. In Fig. 6 and Fig. 7, we have compared the evolution of the reconstructed components (radial and tangential) for three sensor numbers (M=5,10 and 20) to the components obtained by the FE model, which is our reference. We can see that for M=5 the quality of the reconstructed field is quite poor. We can notice that a good approximation can be obtained with M=10 sensors in the stator (point 2). In Fig. 5, we present the sensor location in the airgap. However, in the rotor, it appears spurious oscillations on the radial components which can be met when applying interpolation method like the (D)EIM. However, we can see that with M=20 sensors, we have a good agreement with the reference. This result can be generalized to the whole machine.

IV. CONCLUSION

In this paper we have tested a method to automatically determine the sensor locations in order to reconstruct the magnetic flux density distribution in a permanent magnet machine as a function of the position. It appears surprisingly that the method places the sensors mainly in the airgap of the machine which will facilitate the insertion of such sensors inside the machine. The chosen machine has a fractional number of slots per pole pair leading to a complex evolution of the magnetic flux density distribution as a function of position of the rotor. Thus, to obtain an accurate reconstruction of the field on the whole machine, the sensor number is quite high. However, we have also seen if the targeting domain of field reconstruction is reduced (for example the stator yoke), this number can be reduced. It appears through the obtained results that it is potentially possible to reconstruct the magnetic flux density distribution in the machine by means of a limited number of sensors.

Fig. 1 .

 1 Fig.1. Geometry and mesh of the ferromagnetic parts of the studied machine

Fig. 2 .

 2 Fig.2. Evolution of the error 6 (see (6)) as a function of the number of sensors for the and DEIM/Maxvol

Fig. 3 .

 3 Fig.3. Locations of the sensors (o : radial component × : tangential component ♦ : error maximum) for M=20.

Fig. 4 .

 4 Fig.4. Comparison of the evolution of the error εM (see (6)) as a function of the number of sensors obtained with the original DEIM/Maxvol technique and the DEIM/Maxvol/Airgap technique where the sensor locations are imposed in the airgap.

Fig. 5 .

 5 Fig.5. Locations of the sensors (o : radial component ♦ : selected points to compare the evolution of the reconstructed field as a function of the number M) for M=10.

Fig. 6 .

 6 Fig.[START_REF] Montier | Structure Preserving Model Reduction of Low-Frequency Electromagnetic Problem Based on POD and DEIM[END_REF]. Radial (top) and tangential (bottom) components of the magnetic flux density reconstructed with different numbers of sensors M compared to the reference computed with the FE model in the rotor (point 1 in Fig.5).

Fig. 7 .

 7 Fig.[START_REF] Goreinov | How to find a good submatrix[END_REF]. Radial (top) and tangential (bottom) components of the magnetic flux density reconstructed with different numbers of sensors M compared to the reference computed with the FE model in the stator (point 2 in Fig.5).

ACKNOWLEDGMENT

This work has been carried out within the framework of CE2I project. CE2I is co-financed by European Union with the financial support of European Regional Development Fund (ERDF), French State and the French Region of Hauts-de-France.