text:
Arthur T Ételin
email: arthur.tetelin@onera.fr

C L Ément Le Touze
email: clement.letouze@onera.fr

Philippe Villedieu
email: philippe.villedieu@onera.fr

VECTORIAL LIMITATION FOR MULTISLOPE MUSCL SCHEMES

Keywords: Cell-Centered Finite-Volume Methods, Unstructured Meshes, Multislope MUSCL Schemes, Vectorial Slope Limitation, Frame Invariance

In finite volume schemes with MUSCL interpolation of scalar variables at the faces of control volumes, a slope limiting function is used in order to prevent non-physical oscillations of the solution. More particularly, these functions are designed to ensure a certain monotonicity criterion at each face of the control volume, criterion which then ensures a stability property of the scheme. For vectorial variables, these slope limiting functions are generally applied componentwise, but this may result in a frame-dependance, as well as a loss of accuracy due to false detection of extrema. In this paper, a new vectorial interpolation method is introduced, which is frame-invariant, second-order accurate and stable in a sense that will be defined.

INTRODUCTION

CEDRE software is a numerical code developed by the french institute ONERA to solve multi-physics problems in energetics [1]. Like other CFD codes, it solves hyperbolic systems of conservation equations, which, by focusing only on the convective part, read:

∂Q ∂t + ∇ • f (Q, λ) = 0, (1)
where Q represents the vector of conserved variables, λ is the velocity vector field, and f the physical flux. In CEDRE, the spatial discretization relies on a finite volume cell-centered framework on general unstructured meshes [2], which allows us to write the semi-discretized formulation of equation ( 1) on a multi-dimensional domain:

∂Q i ∂t = - 1 |K i | j∈V(i) |S ij |Φ Q n ij , Q n ji , n ij , λ ij , (2)
where K i denotes a cell of the mesh, S ij the face between cells K i and K j . Notation |.| denote either the length, area or volume measure, depending on the dimension of the object. V(i) is the face neighborhood of the cell K i , Φ is the numerical flux and n ij is a face normal vector pointing from cell K i to cell K j . Lastly, Q n ij represents a set of interpolated values at the face center M ij of face S ij . With a first order scheme, these interpolated values are equal to the numerical variables Q i which represents the mean value of Q over the cell K i , located at the cell centroid B i . To reach a higher order of spatial accuracy, CEDRE code relies on a MUSCL approach, on which we will be focusing in this article.

As it is well known by Godunov's theorem [START_REF] Godunov | Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics[END_REF], no linear reconstruction can be both highorder accurate and ensure the monotony of the scheme, that is why Van-Leer introduced the MUSCL approach [START_REF] Van Leer | Towards the ultimate conservative difference scheme. I. The quest of monotonicity[END_REF][START_REF] Van Leer | Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme[END_REF][START_REF] Van Leer | Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection[END_REF][START_REF] Van Leer | Towards the ultimate conservative difference scheme. III. Upstream-centered finite-difference schemes for ideal compressible flow[END_REF][START_REF] Van Leer | Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method[END_REF]. This method consists in evaluating the gradient of the variable, which is then limited by a limitation function [START_REF] Kemm | A comparative study of TVD-limiters-well-known limiters and an introduction of new ones[END_REF][START_REF] Waterson | Design principles for bounded higher-order convection schemes -a unified approach[END_REF] in order to ensure the scheme stability. For a one dimensional scheme, this stability is reached through a TVD property as introduced by Harten [START_REF] Harten | High resolution schemes for hyperbolic conservation laws[END_REF], but in higher dimension, it is known that the TVD property is incompatible with high order accuracy [START_REF] Goodman | On the Accuracy of Stable Schemes for 2D Scalar Conservation Laws[END_REF]. Since then, many other have studied this issue [START_REF] Colella | Multidimensional upwind methods for hyperbolic conservation laws[END_REF][START_REF] Spekreijse | Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws[END_REF][START_REF] Jameson | Analysis and Design of Numerical Schemes for Gas Dynamics, 1: Artificial Diffusion, Upwind Biaising, Limiters and their Effect on Accuracy and Multigrid Convergence[END_REF] in order to develop a MUSCL method on multi-dimensional meshes. These multi-dimensional MUSCL methods can be grouped in two categories: monoslope methods, in which a single limited gradient is computed for the entire cell [START_REF] Barth | Finite Volume Methods: Foundation and Analysis[END_REF], and multislope methods, in which a limited directional gradient is computed for each face [START_REF] Clauzon | Analyse de schémas d'ordre élevé pour les écoulements compressibles[END_REF]. The method that we will study here is of multislope type, and has been introduced by Le Touze [START_REF] Le Touze | Multislope MUSCL method for general unstructured meshes[END_REF].

As far as the authors know, while the limitation framework has already been well studied for scalar variables, such as temperature or pressure, limitation methods of vectorial variables such as velocity have received far less attention than their scalar counterpart. On a software such as CEDRE, vectorial limitation was obtained until now by limiting vectors componentwise, but this kind of procedure turns out to be frame-dependant, and a loss of accuracy can also be observed. For the vectorial limitation problem, one can divide solutions found in the literature in three major methods. The first one has been introduced by Luttwak and Falcovitz for a monoslope framework [START_REF] Luttwak | Slope limiting for vectors: A novel vector limiting algorithm[END_REF][START_REF] Luttwak | VIP (Vector Image Polygon) multi-dimensional slope limiters for scalar variables[END_REF][START_REF] Luttwak | On the Extension of Monotonicity to Multi-Dimensional Flows[END_REF]. As the stability of a scalar variable is usually defined through a maximum principle, the authors defined vectorial variable stability by using the convex hull of vectors from cell neighborhood, and called it VIP (Vector Image Polygon / Polyhedron). A VIP method consists in the computation of a vector at the time step n + 1, followed by its projection on the VIP set if it doesn't already lie into it. With this process, a vector will satisfy a maximum principle componentwise, whatever basis we choose, which is the natural extension of the scalar maximum principle to vector variables. The second method has been developped by Maire et al. [START_REF] Maire | Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme[END_REF][START_REF] Maire | A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids[END_REF]. It consists in computing a local basis for vectors to be reconstructed. Then vectors are projected onto this basis, and the vector reconstruction is computed componentwise in this basis. The last method has been developped by Zeng and Scovazzi [START_REF] Zeng | A frame-invariant vector limiter for flux corrected nodal remap in arbitrary Lagrangian-Eulerian flow computations[END_REF]. It consists in determining an axis on which the vector field will be projected in order to get scalar variables. Then, a simple scalar limiter is computed, in order to get the general vectorial limiter.

The VIP method has the advantage to present the most accurate extension of the scalar maximum principle to the vectorial case. However, its algorithmic cost seems to be important and even prohibitive on large-scale simulations, as we have to compute intersection of vectors with a convex hull for each reconstructed variables. On the other hand, projection methods seems to have a lower computation cost, but the choice of the axis or basis of projection remains arbitrary, and the vectorial aspect of the variable is dropped by these methods. For all these reasons, we propose a new vectorial limitation method in the context of multislope MUSCL methods. In section 2, we will quickly detail the multislope scalar MUSCL scheme, and we will develop a new way of limiting vectors variables in section 3. Nevertheless, we will only focus here on final results. Intermediate demonstrations and numerical tests will be presented in a forthcoming paper.

SCALAR MULTISLOPE MUSCL SCHEME 2.1 General reconstruction

U H + ij = d k=1 β + ij k U + ij k . Similarly, one can write U H - ij = d k=1 β - ij k U - ij k .
Hence, along the axis B i M ij , a scalar monodimensional framework has been created. Thus, we can define the forward and backward slopes, as well as their ratio:

p + ij = U H + ij -U i B i H + ij , p - ij = U i -U H - ij B i H - ij , r ij = p - ij p + ij . (3)
This gives us the reconstructed scalar value at M ij :

U ij = U i + B i M ij ϕ ij (r ij )p + ij , (4)
where ϕ denotes the limiting function, ensuring a L ∞ stability property under a CFL condition with a second order accuracy where the solution is smooth. One of the drawback of this method is that the ratio r ij can't be used directly in a vectorial framework, as slopes will also be vectors.

One way to use it is to define complex slopes in the complex plane spanned by the slopes vectors. But actually, we will follow another approach that simplifies the stability proof, even in the scalar case, and thereafter makes the vectorial extension more natural. To do so, we recast the reconstruction (4) so that the slope ratio r ij disappears.

Limitation function

We will focus here on the limitation function. We consider here a special case of equation ( 1), that is the scalar advection equation, as it is easier to study its behaviour and its stability. This equation reads:

∂u ∂t + ∇ • (λu) = 0, ( 5
)
where u is the advected scalar. In the scalar case, the limitation function is shaped in order to achieve a second order accuracy on smooth area, and to limit the solution around discontinuities.

Particularly, the limitation part of the limiter is designed so that the discretized scalar advection scheme with an explicit euler scheme as time-discretization, namely

U n+1 i = U i - ∆t |K i | j∈V(i) |S ij |Φ(U ij , U ji , n ij , λ ij ), (6)
achieve a maximum principle under a CFL condition, where U denotes here the discrete scalar solution of equation [START_REF] Van Leer | Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme[END_REF]. From [START_REF] Le Touze | Multislope MUSCL method for general unstructured meshes[END_REF], the conditions on the shape of the limiting function read as follows:

0 ≤ ϕ(r ij ) ≤ min(η + ij , r ij η - ij ), (7)
where η + ij and η - ij are geometrical parameters:

η + ij = B i H + ij B i M ij , and η - ij = B i H - ij B i M ij .
The secondorder accuracy is reached with another inequality:

min(1, r) ≤ ϕ(r) ≤ max(1, r). (8)
This inequality means that the reconstructed scalar should be a convex combination of forward and backward slopes, which can be written as a κ-scheme [START_REF] Waterson | Design principles for bounded higher-order convection schemes -a unified approach[END_REF][START_REF] Van Leer | Upwind-difference methods for aerodynamic problems governed by the Euler equations[END_REF]

U ij = U i + B i M ij 1 + κ 2 p + ij + 1 -κ 2 p - ij , (9)
where κ is a real parameter between -1 and 1. Taking now into account the monotonicity constraints, it means that κ can no longer be a constant but must depend on the specific slopes for the current face, which leads to write:

U ij = U i + B i M ij 1 + κ ij 2 p + ij + 1 -κ ij 2 p - ij , (10)
where

κ ij = κ(p + ij , p - ij ) = κ(r ij )
is the local value accounting for both second-order accuracy and monotonicity constraints (see Figure 2). This formulation of the reconstructed scalar is far more convenient as the former one (4) for a vectorial framework as it doesn't use directly the slope ratio r ij . This is why we will use it to define our vectorial method in section 3. To get a limited reconstruction, κ ij will therefore be defined as a function, whose maximum and minimum admissible values depend on the limitation inequalities [START_REF] Van Leer | Towards the ultimate conservative difference scheme. III. Upstream-centered finite-difference schemes for ideal compressible flow[END_REF]. One can easily rewrite them as an inequality on the reconstructed variable:

min(U i , U H + ij ) ≤ U ij ≤ max(U i , U H + ij ) min(U i , U H - ij ) ≤ U f ij ≤ max(U i , U H - ij ) ( 11
)
where we have defined

U f ij = 2U i -U ij .
If the scheme were monodimensional along the axis B i M ij , and if the variable U had been supposed linear on the cell (M ij , M f ij ) (in blue in Figure 1), then U f ij would have been the real reconstructed value at M f ij . For an unstructured mesh, the point M f ij doesn't play any role and is not explicitely defined but one can still define the hypothetical value of the discrete solution at this point, that's why we call it the fictitious reconstruction. One can see from [START_REF] Harten | High resolution schemes for hyperbolic conservation laws[END_REF] that even if this value doesn't need to be computed in practice, we still need to define it in order to establish the scheme stability property as inequalities on both U ij and U f ij are required.

VECTORIAL MULTISLOPE MUSCL SCHEME

Stability formula

The vectorial MUSCL scheme derives naturally from the stability formula of the vectorial version of the fully discretised scheme (6):

V n+1 i = V n i - ∆t |K i | j∈V(i) |S ij |Φ(V ij , V ij ), (12)
which is the discrete formulation of the vectorial advection equation:

∂v ∂t + (λ • ∇)v = 0, ( 13
)
where v is the exact vectorial solution of ( 13), and V represents its vectorial discrete solution.

In order to get the stability formula, one has to make some assumption on the numerical flux function Φ. We will firstly assume that it is consistent:

∀V ∈ R d , Φ ij (V, V) = (λ • n ij )V ij . (14)
Secondly, we assume that there exists two scalar parameters A ij and B ij such that

A ij (V ij -V i ) = Φ ij (V ij , V i ) -Φ ij (V i , V i ), B ij (V ji -V i ) = Φ ij (V ij , V ji ) -Φ ij (V ij , V i ). ( 15
)
One can notice that if the numerical flux is the classical upwind numerical flux:

Φ ij (U, V) = max(λ • n ij , 0)U + min(λ • n ij , 0)V, ∀U, V ∈ R d , (16)
then these scalar parameters A ij and B ij exist and are equal to:

A ij = max(λ • n ij , 0), B ij = min(λ • n ij , 0). ( 17
)
If we introduce the fictitious vectorial reconstruction V f ij = 2V i -V ij , and two other parameters:

ν + ij = ∆t |S ij | |K i | A ij , ν - ij = -∆t |S ij | |K i | B ij , (18)
we get:

V n+1 i = V i   1 - j∈V(i) ν - ij + ν + ij   + j∈V(i) ν - ij V ji + j∈V(i) ν + ij V f ij . (19)
In order to get a convex combination, we have to assume that the numerical flux is monotonous:

A ij ≥ 0, B ij ≤ 0, (20)
which means that the coefficients ν + ij and ν - ij are both positive. From this point, one can define the following CFL condtition 1 -

j∈V(i) ν - ij + ν + ij ≥ 0, (21)
which ensures that the formula ( 19) is a convex combination of V i , V ji and V f ij . In order to get the vectorial scheme stability, we still have to choose the way we will limit the reconstruction V ij and its fictitious counterpart V f ij in the same time. 3. We compute the intersection of the two previous intervals, I ∩ I f . If this intersection is empty, then it means that we are on a kind of vectorial extremum. As in the scalar case, a vectorial extremum means that the method degenerates at first order, and V ij = V i .

4. If I ∩ I f is not empty, we choose by any kind of way a value κ included in I ∩ I f . One can for instance choose the closest value to a unique κ 0 previously chosen, or we can choose the value which maximizes or minimizes the norm V i -V ij on the interval I ∩ I f . One can also choose for example the mean value of I ∩ I f . Different choices of κ = f (I ∩ I f ) imply different kinds of limitation functions, and eventually different interpolation schemes.

CONCLUSION

In this paper, we have presented a new way of limiting vectors in the framework of multislope MUSCL schemes for finite volume methods. This limiting method consists in requiring that the reconstruction vector and its fictitious counterpart lie in a truncated circular sector, which ensures a control over the norm and the direction of V n+1 i . Nevertheless, we didn't presented detailed calculus, and extension of the angular bound to three-dimensional vectors has not been presented here. Moreover, results for this method are only theoretical so far, as no numerical tests have been presented in this paper. All these aspects will be addressed in a forthcoming paper.

Figure 1 :

 1 Figure 1: Multislope MUSCL method.

Figure 2 :

 2 Figure 2: Combination of the second-order zone in green and the monotonicity constraints in red to get the stability zone for the limiter, and equivalence in terms of κ values.

Figure 4 :

 4 Figure 4: Bounds on V n+1 i . In grey is represented the convex hull of vectors from a given neighborhood. Red dashed line represents the norm maximum principle. Purple dashed line represents the angular bound. Left figure represents a case where vectors from the neighborhood point roughly in the same direction. Here, an angular bound seems to be required. Right figure represents a case where cell K i is a convergence or divergence point, and therefore no angular bound holds for V n+1 i .

Monotonicity criterion

From the formula [START_REF] Luttwak | Slope limiting for vectors: A novel vector limiting algorithm[END_REF], one can observe that with the right choice of limitation for the reconstruction V ij , we can get almost any kind of control over V n+1 i . In the scalar case, a monotonicity criterion is applied on the reconstructed variable in order that U n+1 i achieve a maximum principle. On the vectorial case, the control we want over V n+1 i should ideally be the vectorial extension of this maximum principle. This vectorial extension corresponds in fact to the VIP area from Luttwak and Falcovitz [START_REF] Luttwak | Slope limiting for vectors: A novel vector limiting algorithm[END_REF][START_REF] Luttwak | VIP (Vector Image Polygon) multi-dimensional slope limiters for scalar variables[END_REF][START_REF] Luttwak | On the Extension of Monotonicity to Multi-Dimensional Flows[END_REF], that is the convex hull of vectors from a given cell neighborhood. Hence, if we write our reconstruction under a κ-scheme form:

we only have to find an interval I = [κ -, κ + ] such that for any value of κ ij within I, we have that both V ij and its associated fictitious V f ij = 2V i -V ij lie into the convex hull of the neighborhood vectors. Nevertheless, this is the best method only from a theoretical point of view. Indeed, as already explained in the introduction, the algorithmic cost of both the convex hull construction and the "lying in the Convex Hull" test may be prohibitive in practice. For this reason, we introduce here an alternative method to the convex hull. This alternative limitation takes the form of a truncated circular sector area around each reconstructed vector V ij and its associated fictitious reconstruction V f ij (see Figure 3). This method ensures some kind of control over the norm and the direction of V n+1 i as illustrated in Figure 4. To summarize, this area is constructed as follows:

1. V ij and its fictitious counterpart have both to satisfy an upper bound on their norm:

This condition ensures that the vector norm V n+1 i is bounded. It is theoretically possible to build a stable numerical scheme only with this condition in the sense that the norm won't tend toward infinity. But this stability area is much bigger than the well designed convex hull previously mentionned. Especially, if all vectors point roughly to the same direction, we can assume that V n+1 i should also point to this direction. With only the norm condition, such a control on the vector direction can not be guaranteed. That's why we introduce the next condition.

2. V ij has to lie in an angular sector defined by V i and V H + ij on the one hand, and its fictitious counterpart V f ij has to lie on the angular sector defined by V i and V H - ij on the other hand. This condition ensures that V ij and its associated fictitious V f ij won't oscillate, which implies an angular stability for V n+1 i . We have to notice that this condition can occur only for two dimensional vectors. Indeed, in a three dimensional framework, V ij won't automatically lie in the same plane defined V i and V H + ij . A similar problematic occurs for the fictitious reconstruction. It means that the angular sector has to be redefined for vector of dimension 3, which won't be done here.

Arthur Tételin, Clément Le Touze and Philippe Villedieu 3. Both previous conditions are enough to ensure a vectorial stability. But if all vectors are colinear, the current limitation won't tend toward the scalar case limitation, as in the scalar case, a lower bound is also required on the reconstruction U ij as shown in formula [START_REF] Harten | High resolution schemes for hyperbolic conservation laws[END_REF]. If we impose a lower bound on the vectors norm:

the scheme won't be satisfying because it will prevent to use an admissible reconstruction for smooth vector fields. This is why we decided not to impose a minimum constraint on the norm, but to truncate the angular sector defined by the two previous bounds. For the angular sector defined for V ij , we draw the line going from min(

A similar line is drawn for the angular sector associated with the fictitious reconstruction V f ij . Green lines represent the set of all second-order accurate reconstructions. If V ij and V f ij lie both on the green line and within the red area, then the reconstruction is admissible. Here, V f ij lies outside the truncated circular sector. The reconstruction V ij is thus not accepted.

Hence, the reconstruction and limitation process is as follows:

1. We compute the two limit values κ -and κ + such that V ij lies on the boundary of its truncated circular sector. It means that for any value κ lying inside the interval I = [κ -, κ + ], the reconstructed vector V ij assiociated with κ will lie inside its stability area.

2. We compute the two limit values κ f -and κ f + such that V f ij lies on the boundary of its truncated circular sector. It means that for any value κ lying inside the interval I f = [κ f -, κ f + ], the fictitious reconstructed vector V f ij associated with κ will lie inside its stability area.


L E Diplôme D'état De Docteur

E N Médecine

Par Mme

Mathilde Rausch

Thèse Soutenue Publiquement

L A Faculté

M Le Pr

Sébastien Schmerber

M Le Dr

Ihab Atallah

Jean-Eric Berger

Mme Le Dr Amandine

Briault M Le Pr

Bruno Degano

M Le De

Gil Frey

LE RETRAIT DES PROTHÈSES TRACHÉALES MÉTALLIQUES NON COUVERTES EN BRONCHOSCOPIE RIGIDE

AVERTISSEMENT

Ce document est le fruit d'un long travail approuvé par le jury de soutenance.

La propriété intellectuelle du document reste entièrement celle du ou des auteurs. Les utilisateurs doivent respecter le droit d'auteur selon la législation en vigueur, et sont soumis aux règles habituelles du bon usage, comme pour les publications sur papier : respect des travaux originaux, citation, interdiction du pillage intellectuel, etc.

Il est mis à disposition de toute personne intéressée par l'intermédiaire de l'archive ouverte DUMAS (Dépôt Universitaire de Mémoires Après Soutenance).

Si vous désirez contacter son ou ses auteurs, nous vous

VPPB, la relecture attentive de ce travail. J'ai hâte de re travailler avec toi, j'espère qu'on arrivera à faire des tâches séparées (pas trop longtemps quand même). Tu vas devenir une bête d'orl et tu le sais. A Judith, merci pour tous tes gâteaux et pour avoir aussi bien repris les reines pour nous représenter. A Émilie, pour ces moments mignons de ton premier semestre. J'ai hâte de voir comment tu as grandi. A Marin, pour ces bons souvenirs à Nice (dont les moments où on s'inquiète vraiment). A Damien et Theotime que j'ai hâte de découvrir l'année prochaine. Aux petits nouveaux qui arrivent (Matthieu et Mael), vous n'imaginez pas la chance que vous avez de débarquer dans cette équipe. Merci aux infirmières du service et de la consultation, merci pour votre aide depuis le début et votre confiance, même quand je demandais « comment ils font d'habitude les autres ? ». A Ayette, la maman des internes, toujours présente dans les meilleurs et les pires moments, merci pour ton soutien, les discussions de qualité, les fromages cachés au fond du frigo, ce fameux weekend sans sommeil. A Martine, merci pour ton aide à aller au bout de cette consultation et tes tiramisus. A Stéphanie et Véronique, pour votre présence et votre motivation sans faille dans le service.

Merci à l'équipe d'ORL de Chambéry : Au Dr Manipoud pour votre confiance et votre rigueur, mes premières méatotomies et parotidectomies, merci de m'accueillir à nouveau. Au Dr Guichard, pour toutes ces histoires, les montages photos et Jacques Brel. Cette fois c'est vrai, Mathilde est revenue. Au Dr Filidoro, pour ta joie de vivre. Aux infirmières du service et de consultation (Sandrine, Nicole, Françoise, Christine, Chapi, Charline, Melissa et toutes les autres) et aux secrétaires (Virginie, Audrey et Mallory) pour leur aide précieuse. A Mehdi, Dr Lagtoubi, merci du fond du coeur, tu as rendu ces quelques mois fabuleux, pour nos épistaxis "arrêt au stand de F1" et les afterwork tapas.

Merci à l'équipe de vasculaire et thoracique de Chambéry qui m'a accueillie au tout début de ce périple : Au Dr Blaise, merci pour ta confiance lors de ces anastomoses, merci pour nos blocs « orteils » en autonomie et ta joie de vivre. Au Dr David, pour ma première amputation dont je me rappellerai toute ma vie. Au Dr De Lambert, pour ces moments robots et l'expression du bébé ataxique qui restera. Au Dr Frassinetti, pour cette motivation chirurgicale sans faille, peu importe la localisation du problème. A Noémie, Antoine et Florent qui ont répondu à toutes mes questions de premier semestre. A Guillaume, Lylia, Aurore et toutes les autres, pour ces gardes et ce pot de qualité.

Merci à l'équipe de CMF-Plastique de Grenoble : J'ai bien failli rester avec vous… Au Pr Giot, aux Dr Coudurier, Deneubourg, Lafontaine, Gelin et Ton Van (aka Thon-Vanessa), merci pour votre accueil chaleureux, les moments conviviaux au café, pour tout ce que vous m'avez appris au bloc et en dehors.

Au Dr Marzloff pour être devenue plus qu'une chef, merci pour mon nouveau sourire, pour ton efficacité chirurgicale. Aux infirmières de la consultation, Christine, Alex, Nath et Malika, pour l'ambiance de fou que vous mettez dans ce 7 ème et votre expertise pansement. A mes co-internes du semestre, Alexandra, Clémence, David, Giovanni, Haizam et Nathan pour avoir rendu ce semestre plus qu'agréable, pour ces moments confinés en chambre de garde. Au trio ADelay, CBeyer et NFereira, une des plus belles rencontres, pour être les plus traquenards (sauf toi NF), et de véritables amis (VF) sur qui je peux compter. J'espère que nos retrouvailles régulières vont se poursuivre.

Enfin, merci à l'équipe d'ORL d'Annecy pour cette année passée à vos côtés. Merci de m'avoir accueillie six mois de plus au pied levé, de m'avoir fait autant progresser chirurgicalement, merci pour les café-débats et la bonne humeur. Au Dr Angel, merci pour ta confiance, ton calme légendaire et ton savoir infini. Merci pour ce service, une fois qu'on met un pied dedans, on n'a plus envie de s'en aller. Au Dr Lagarde, tu n'auras finalement pas la page entière mais le coeur y est ! Merci pour ta bonne humeur et tous ces moments « astuces de chirurgiens » dont j'ai bien pris note. Au Dr Paoletti, pour ta playlist que je connais enfin presque par coeur, pour tous ces bons moments au bloc (et pour cette dernière MLES !). Au Dr Berta, pour ton énergie qui ne s'éteint jamais, ces vendredis au bloc, tes « Tuuu vaaas y aaaarrriver » qui énerve, le vieux et son chien. Merci de m'avoir fait confiance sur mes premiers curages, et sur tout le reste. Au Dr Sandré, par où commencer ? Depuis mon premier semestre j'ai toujours dit que je voulais être une orl comme toi, et je le pense de plus en plus chaque jour. Merci pour tes avis sur les vertiges quand je n'y connaissais rien, pour Mr Rubillon, pour ces belles électriques, ces congrès, l'avion qu'on a raté, notre complicité à Annecy, le sheitan qui sommeille en toi. Et on n'est pas en retard, on a tout le temps dans notre vie, même s'il y en a pas une pour rattraper l'autre. A notre future installation. A Lucille, Sanela et Hugo pour ce semestre à se voler l'ordinateur et plein de bonne humeur. A Pauline et Astrid, parce qu'on se marre bien dans ce bureau, c'est toujours un plaisir de vous voir en arrivant dans cette concertation. (Paupau, pour PLK, le tel rose et la piscine Molitor). A Corinne, pour être effectivement la meilleure cadre de tout l'univers. A Patichoud pour être un soleil dans le brouillard de la consultation, même triplée. A Martine, Adeline, Delphine, Juliette, Dilsah, Claire, Lysiane, Christalle et toutes les autres, pour votre aide et votre sourire.

A Pauline et Alexis, pour m'avoir donné le goût de l'ORL alors que je n'étais qu'une externe.

Merci à ma famille,

A mes parents, merci pour le soutien que vous m'avez toujours apporté, pour mes 13 déménagements, votre éducation et votre amour. A ma mère, pour être toujours disponible à toute heure dans les bons et les mauvais moments, pour être un modèle de bienveillance et d'amour. A mon père, pour m'avoir aiguillée vers la médecine quand je ne savais pas où aller, pour m'avoir donné cet esprit pratique et bricoleur qui me sert tous les jours.

A Thomas, mon humain préféré, même après 18 ans de vie commune. J'ai toujours voulu faire comme toi depuis petite et je continue. Merci d'être une belle source d'inspiration et le meilleur clown. A Zidane ! A Manu et Sophie, merci pour toutes ces vacances partagées, ces weekends, les réunions de famille, les soirs où on a fait trop de bruit (« si j'étais le fils du voisin… »). Merci de votre soutien à distance, on s'est bien éparpillé tous les quatre mais on se retrouve toujours. Bravo pour vos petits choux. A Olivia, Émilie et Luca, pour vos vidéos mignonnes, merci de nous faire rire sans le savoir et d'égayer encore plus les réunions de famille.

A Corinne et la famille Rau-composée, merci pour ces bons moments, les repas où on peut plus marcher ni conduire, et pour toutes les futures salades d'oranges qu'on va partager. A Didier et sa tribu, merci pour la joie de vivre ambiante, les sorties ski et les vacances à Cannes.

A mon grand-père, pour tes blagues, les journées à l'ateliers et les sorties vélos. A ma grand-mère, qui me répétait petite que je finirais soit Miss France soit Docteure. C'est chose faite. Merci pour tout ton amour. A Hodge, 7 ans que tu me réveilles toutes les nuits mais merci pour tous les ronronnements. A Meringue et Agathe, coeur coeur.

A mes amis,

A Flora, pour être là depuis cette visite de classe en grande section. Merci pour cette belle amitié qui perdure, inséparables de Bandol à Marseille, en passant par Val Thorens et la Corse. Merci pour les guerrières qu'on devient, feu les ploucs. A Clément, pour être une leçon de vie à toi tout seul. Pour ce beau voyage à Boston. Aux Bonhommes, vous avez rendu cette deuxième P1 incroyable. A toutes nos réunions malgré nos vies éparpillées. A nos weekends marseillais réussis. A Steve, un pingouin parti trop tôt. A Camille, pour cette amitié parfaite, sans filtre, que j'aime tant. Merci d'être toujours là. A Marie, pour cette D4 en tête à tête qui nous a plus que soudées et pour notre chow-chow. Tu m'inspires tous les jours, sur tous les plans. A Smaranda, pour ces moments russes, Fascination, et ton enthousiasme.

A Eloïse pour être de bonne humeur en toute circonstance, toujours la plus motivée. A Pitou, pour être le plus fou et le plus drôle, toujours infatigable. A Thomas, on aurait pu être de la même famille… pour ce running gag et ton intestin fragile. A Antoine, pour ces fous rires (dont celui au-dessus du puis), et bravo pour tous tes projets. A Jonas, pour cette période d'externat qu'on idéalise tant, les musiques qui vont avec, pour le pacte. A Mathilde, tellement de choses me viennent en tête, mais peut être qu'en premier c'est ta cicatrice frontale à Sziget… Merci pour tous ces moments et ton rire que j'adore. A Nathalie, la nouvelle annécienne, pour les fous rires au sucre, les émerveillements devant Rebecc, The Royals, et tout ce qui va suivre, maintenant que je suis libre. A Clémentine et Laurianne, qu'il me tarde de revoir… vous me manquez.

A Hugo, aka le plus stylé, pour cette amitié qui continue malgré tout, même à distance. A Guillaume, aka le plus gangster, pour ces moments dans le 6 et pour ta réussite musicale. Aux « garçons » du 6, pour tous ces moments entre l'ayers, le sucre, Sziget… A Antonine, pour cette tradition du début du mois qui perdure, pour cette colocation, tous ces vendredis et samedis soir riches en couleurs. A Maud, pour notre belle colocarton et l'envie de frange incessante.

Aux belles rencontres de Grenoble : A Léa et Charles pour ce nouvel an, la dameuse et tes verres à vins, à Alexandre, à la Panoplie, à Louis, pour tous ces bons moments, au ski ou au bar. A Micka, merci d'être là. Merci pour tous ces bons moments au Hasard, ouvert et fermé. Si j'en ai marre de la chirurgie un jour, je viens vous aider à ranger les chaises. A Laura et Claire, je vous remets ici parce que vous méritez d'être partout. Notre trio m'aura permis de venir à bout de cette ****** de thèse.

A Maxime, pour ces quelques mois annéciens ensoleillés, des bières Ikea jusqu'à aujourd'hui, pour toutes les cases qu'on n'a pas encore en commun. Merci de m'avoir soutenue dans ce déni de thèse, pour la limite insta, le skate, le crawl unilatéral et l'esprit sportif que tu tentes de m'inculquer. Merci de toujours me faire sourire, même quand il pleut. were initially widely used for benign stenosis, because of their ease of placement, immediate airway patency restoration, low risk of migration and their great adjustment for irregular and malacic airway [START_REF] Cuisnier | Surgical and/or endoscopic treatment of acquired tracheal stenosis in adult patients[END_REF][START_REF] Ducic | Use of endoscopically placed expandable nitinol tracheal stents in the treatment of tracheal stenosis[END_REF][START_REF] Husain | Long-term follow-up of ultraflex metallic stents in benign and malignant central airway obstruction[END_REF] . Post tracheostomy, post intubation stenosis and tracheobronchomalacia are the most common benign etiologies leading to airway stenting [START_REF] Murgu | Central Airway Obstruction: Benign Strictures, Tracheobronchomalacia, and Malignancy-related Obstruction[END_REF] . Most SEUMS is made from nitinol, an alloy of nickel and titanium, which provide shape memory properties. They are considered to incorporate into the tracheal mucosa rapidly in 3 to 6 weeks and are described as a permanent option of stenting and less likely to migrate [START_REF] Ducic | Use of endoscopically placed expandable nitinol tracheal stents in the treatment of tracheal stenosis[END_REF] . Also, the epithelialization within the stent maintains mucociliary clearance, which improves tolerance.

TABLE DES MATIÈRES

The use of SEUMS stent is not recommended in the setting of benign airway stenosis due to late complications such as granulation tissue formation resulting in stenosis and recurrent respiratory infections, migration, stents' fracture, fistulae formation and mucous plugging [START_REF] Saad | Self-expandable metallic airway stents and flexible bronchoscopy: long-term outcomes analysis[END_REF] . The Food and Drug Administration (FDA) published a public health notification in 2005 about SEUMS and does not recommend its use in benign disease unless there is no other alternative [START_REF]FDA public health notification: complications from metallic tracheal stents in patients with benign airway disorders[END_REF] . When stenting is the only option for these patients, silicone or covered stents are the best option to treat benign airway stenosis. Despite this, some patients with benign airway disorders underwent SEUMS placement and presented important stent-related complications. In some cases, SEUMS removal was mandatory and procedures were very risky due to several issues and complications [START_REF] Noppen | Removal of covered self-expandable metallic airway stents in benign disorders: indications, technique, and outcomes[END_REF][START_REF] Lunn | Endoscopic removal of metallic airway stents[END_REF][START_REF] Alazemi | Outcomes, health-care resources use, and costs of endoscopic removal of metallic airway stents[END_REF] .

Few studies described various rigid bronchoscopy removal techniques but mostly for covered stents [START_REF] Noppen | Removal of covered self-expandable metallic airway stents in benign disorders: indications, technique, and outcomes[END_REF][START_REF] Lunn | Endoscopic removal of metallic airway stents[END_REF] . The usual technique is to clamp the upper part of the stent with a forceps and make a rotation movement, without dissection. This approach can lead to mucosal damage and stent fracture in the setting of SEUMS.

In this study, we analyze retrospectively SEUMS removal using rigid bronchoscopy in patients who underwent SEUMS placement for benign disorders. We also describe a novel technique to safely remove these stents as well as the indications, outcomes and complications.

Operative technique

All patients were informed about surgical risks, including life-threatening complications and gave informed consent for surgery. No stent was removed in an emergency and all procedures were scheduled leaving enough time to patients to decide whether they want to be operated on or not, as they play an important role in deciding their own care, in the setting of high-risk procedures.

SEUMS removal is done under general anesthesia. The patient is ventilated by highfrequency jet ventilation connected to the ventilation port of a rigid bronchoscope, with a fraction of inspired oxygen of 100% and a frequency of 120 breaths per minute.

We used size 12 rigid bronchoscope (EFER-DUMON®, Marseille, France) for tracheal stent removal and size 8.5 rigid bronchoscope (KARL STORZ®, Tuttlingen, Germany) for bronchial stent removal. A HOPKINS® telescope 0° (10320A, 5.5mm X 50cm) inserted in the bronchoscope and connected to a camera and a video endoscopy tower is used throughout the whole procedure to get an endoscopic view of the surgical field. A metallic suction tube of 45 cm length and 4 mm diameter (10380AM, KARL STORZ®, Tuttlingen, Germany) is used to aspirate secretions and blood during stent removal.

In the first four procedures, a grasping forceps (BAF52 CR, MEDISURGE©, Marseille, France) is used to grasp the circular extraction loop or if impossible, one of the metallic loops of the proximal edge of the stent. Then removal was performed using the usual technique described in the literature by making twisting movements, which breaks the threads to remove the stent in a piecemeal fashion or as "rolling spaghetti on a fork" fashion (Figure 1) [START_REF] Nashef | Expanding wire stents in benign tracheobronchial disease: indications and complications[END_REF] .

In all other procedures of SEUMS removal, we used a new operative technique (Figure 2 and Video 1). In this technique, the tip of the rigid bronchoscope was placed next to the proximal end of the stent. Next, the stent was carefully dissected from the airway wall around the entire circumference with bronchoscopic hook scissors (BAC00, MEDISURGE©, Marseille, France). The bronchoscope was advanced between the trachea and the stent throughout the dissection. Once the stent was completely dissected from the tracheobronchial wall, we removed it in one piece with a gentle traction using the grasping forceps. After removal, we injected steroids (Depo-Medrol 40mg/ml, PFIZER®, US) in the surgical bed to lower the risk of granulation tissue formation. Injection is performed using a syringe filled with the solution and

RESULTS

Patients' characteristics

Patients in this series (total (n=7), female (n=4) and male (n=3)) underwent SEUMS

Indications for stent removal

Given the risks of the removal, indication was discussed with pulmonologists and patients. The decision was made on several arguments and symptoms, taking in consideration the benefit-risk balance of the procedure.

Patients presented multiple stent-related complications shown in table 1.

All patients presented recurrent infections and required repeated antibiotic treatments.

One patient had a lung abscess. Excessive granulation tissue formation was observed in all patients. Four patients presented significant granuloma formation with airway obstruction and two patients needed fiber CO2 laser debulking before stent removal. In two patients, segments of the stent showed fractures with protrusion into the tracheobronchial airway. Significant stenosis was observed in one of these two patients. There was no case of stent migration in our series.

Eight patients complained about chronic severe cough. Five patients presented chronic dyspnea and needed oxygen therapy.

Table 1: Indications for stent removal

Indications for SEUMS removal n

Recurrent infection 9

Recurrent granulation tissue formation 9

Obstructive granulation tissue 4

Stent fracture 2

Migration 0

Outcomes and complications

Four stents were removed using the usual technique of piecemeal fashion or "rolling spaghetti on a fork" fashion and 5 stents were removed using our new dissection technique.

Stent location was tracheal in 1 case, left main bronchus in 2 cases, bronchus intermedius in 3 cases and tracheobronchial with Y-shape stent in 3 cases.

Operative time ranged from 2,2 to 9,9 hours (mean time 4,8 hours). Six stents were removed without any peroperative complications. Superficial mucosal tear was not considered as a complication in our study, as SEUMS were deeply incorporated in the tracheobronchial wall in all patients.

Peroperative complications occurred in three procedures. In one patient, we left few residual metallic threads. This patient had a history of stent fracture before removal.

One patient got a deep mucosal tear with cartilage exposure but without airway breach or subsequent consequences. One difficult removal resulted in a mediastinal breach secondary to a deep tracheal tear with subsequent pneumothorax and massive subcutaneous emphysema. In this case, we used fibrin glue to seal the tear and we placed a Y-shaped silicone stent to reinforce sealing of the tracheal tear. A chest tube was also placed. The patient needed prolonged intubation and ventilation and developed intensive care-unit acquired weakness. After 37 days of intubation, we performed a temporary tracheostomy to extubate the patient and we removed the silicone stent in the same setting.

The longest operative time in our series was in an 81 years-old female patient and the surgery lasted for almost 10 hours. This patient had a history of tracheobronchomalacia and had a Y-shaped SEUMS for 6 years. The dissection was tedious, especially for the bronchial parts of the stent. There were no significant peroperative complications in this case but immediately after stent removal, the patient showed severe tracheobronchomalacia with oxygen desaturation. A Y-shaped metallic covered stent was deployed. She was extubated the next day in the intensive care unit but showed signs of acute respiratory distress 8 days after surgery and was reintubated with difficult ventilation parameters. Unfortunately, this patient died from acute respiratory failure. Complications are detailed in table 2.

Table 2: Per and postoperative complications of SEUMS removal

Complications n

Cartilage exposure 1

Pneumothorax from mediastinal breach 1

Retained stent pieces 1

Postoperative death 1 Consequently, we observed two life-threatening complications in nine procedures, leading to longer hospital stay or death. One of them was postoperative and directly related to the patient's poor respiratory status. The median time of postoperative intubation in the intensive care unit was 0 day (range 1 to 37 days).

Table 3 shows the comparison between the two techniques used for SEUMS removal in our series. The mean operative time was nearly similar in the two techniques (4,7 hours for the piecemeal technique and 4,8 hours for the dissection technique). We observed two peroperative complications with the "piecemeal" technique (one case of mediastinal breach with pneumothorax and massive subcutaneous emphysema and a case of retained stent piece). Subsequently, the mean postoperative intubation period was longer in the "piecemeal" technique (9,5 days) compared to the dissection technique (2,5 days) that presented one minor complication in the form of cartilaginous exposure that occurred during dissection. Katlenburg-Lindau, Germany). This technique was described in one of our previous publications and its aim is to stiffen the posterior tracheal wall through fibrosis [START_REF] Castellanos | Laser tracheobronchoplasty: a novel technique for the treatment of symptomatic tracheobronchomalacia[END_REF] . A summary of all cases of SEUMS removal in our series is shown in table 4.

Table 4: Descriptions of stents characteristics, removal details, complications and outcomes

Ultraflex stent (Boston Scientific Corporation, Natick, MA) Leufen stent (Leufen Medical GmbH, Berlin, Germany)

DISCUSSION

Management of central airway obstruction in benign pathology is quite challenging.

Many non-surgical techniques have been used such as laser photocoagulation, balloon dilatation and stenting of the airway [START_REF] Ernst | Central airway obstruction[END_REF][START_REF] Galluccio | Interventional endoscopy in the management of benign tracheal stenoses: definitive treatment at long-term follow-up[END_REF] . Nowadays, tracheal stents are commonly used for the treatment of airway stenosis but indications are still debating [START_REF] Dutau | Airway Stenting Revisited: 30 Years, the Age of Reason?[END_REF] . Silicone stents are preferred for patients with benign strictures since FDA and authors warned about the use of SEUMS [START_REF]FDA public health notification: complications from metallic tracheal stents in patients with benign airway disorders[END_REF] . Despite this, some patients with benign airway stenosis underwent SEUMS placement. This was justified by the fact that SEUMS allows epithelialization within the stent lumen, therefore they are less likely to migrate, which could be convenient for patients with long life expectancy. On the other hand, this renders them very tedious and dangerous to remove.

In our series, we used a new technique to dissect embedded uncovered metallic stents from the tracheobronchial wall, while limiting the risk of airway tears. To the best of our knowledge, this is the first description of this technique. Other authors described some removal procedures using the bronchoscope to dissect metallic stents from the tracheobronchial wall, but all stents in these series were covered metallic stents [START_REF] Noppen | Removal of covered self-expandable metallic airway stents in benign disorders: indications, technique, and outcomes[END_REF][START_REF] Chawla | Removal of self expandable metallic airway stent: A rare case report[END_REF] . No wonder that, as described, when they advance the bronchoscope around the stents, they can dissect and remove them easily. One of these studies described removal of covered metallic stents in patients with a history of 1 month to 5 years of stent placement [START_REF] Noppen | Removal of covered self-expandable metallic airway stents in benign disorders: indications, technique, and outcomes[END_REF] . We observe in these series that when the duration of stenting increases, patients show high risk of complications like tracheobronchial tears and peroperative stents' fractures with inability to remove the whole stent.

Few authors described SEUMS removal with rigid bronchoscopy which consist of clamping the upper part of the stent, to separate the stent from the tracheal wall, using optical forceps or Jackson dilators, then alligator forceps to rotate and extract the stent in a piecemeal fashion or "rolling spaghetti on a fork" fashion. One author used this technique in 12 uncovered metallic stents in a series of 30 stents grouping covered and uncovered metallic stents [START_REF] Lunn | Endoscopic removal of metallic airway stents[END_REF] . The mean duration of stenting was 163 days. The main complications for uncovered stents were retained stent pieces (n = 6) and reobstruction requiring silicone stent placement (n = 11). They observed that 91% of the complications occurred in patients with uncovered stents. They also found that complications were more likely to occur the longer a stent has been in place. In our series we focused only on SEUMS removal excluding any case with covered metallic stents because, according to our experience, covered stents are easy to remove and present a low rate of removal-related complications. On the other hand, peroperative complication rate of SEUMS removal in our series is lower than that described in the literature. In fact, we got only one life threatening peroperative complication in the form of tracheal tear with mediastinal breach, pneumothorax and subcutaneous emphysema in which we used the "piecemeal" fashion removal technique. No major peroperative complications occurred in cases where we used the dissection technique despite a long mean stenting time of 7,4 years in our series.

Several other methods of uncovered metallic stent removal have been developed. One author reported for one patient the use of an oversized silicone stent inside the SEUMS to apply pressure on the granulation tissue [START_REF] Khemasuwan | Complex metallic stent removal: decade after deployment[END_REF] . It maintains airway patency and leads to the necrosis of the granuloma, making it easier to remove the stent ten week later. This method is called stent-in-stent technique. Other authors described a combined endoscopic and open approach to remove SEUMS, under spontaneous ventilation [START_REF] Rampey | Combined endoscopic and open approach to the removal of expandable metallic tracheal stents[END_REF] .

They used a rigid bronchoscope with large alligator forceps to pull the stent then performed tracheotomy to remove remaining parts of the stent. They placed a silicone T-tube in a second-stage procedure a few weeks later. This technique has the disadvantage of requiring at least two general anesthesia and a skin incision.

Chung et al removed 18 uncovered stents in a series of 36 stents, using biopsy forceps through the working channel of flexible bronchoscopy [START_REF] Chung | Remove airway ultraflex stents by flexible bronchoscope[END_REF] . Eight uncovered stents were removed in a single procedure and 10 in multiple procedures (2 to 7). The duration of stenting was from 1 day to 3 years. He described 4 cases of retained pieces, 3 cases of mucosal tear with bleeding and 3 patients requiring new silicone stent placement due to reobstruction by granulation tissue. In comparison, our technique allows SEUMS removal in one intervention, in one piece and without skin opening even in patients with a history of long stenting time.

Recently, several authors have described a removal technique under fluoroscopy [START_REF] Xie | The strategy of removing tracheal stents by using an interventional technique under fluoroscopy[END_REF][START_REF] Bi | Fluoroscopic removal of self-expandable metallic airway stent in patients with airway stenosis[END_REF][START_REF] Zhao | Y-shaped airway self-expanding covered metallic stent removal via the interventional technique[END_REF] .

Xie et al reported the removal of 45 uncovered stents using internal stripping, direct removal or stent-in-stent technique, with a low complication rate in a cohort of patients with an average stenting duration of 6 months. Again, these results are difficult to compare with ours as shorter duration of stenting results in less granulation tissue formation, easier stent removal and less complications. In addition, although SEUMS removal with flexible bronchoscopy or fluoroscopy is attractive because it avoids general anesthesia for patients with major comorbidities, it is not possible in our population which has significant granulation tissue after years of stenting. Our technique is suitable for long-term uncovered stent placement with important granulation tissue. Moreover, rigid bronchoscopy allows direct airway access in case of massive bleeding or tracheal collapse and seems safer.

The main surgical complications of stent removal are tracheal wall dehiscence, bleeding, stent fracture, cartilage exposure, pneumothorax and mediastinal breach, which can lead to death. Lunn et al reported one case of pneumothorax, probably due to high pressure of jet-ventilation. One patient underwent thoracotomy and bronchotomy for stent removal in Alazemi's series. He also reported two deaths during hospitalization after removal "for reasons that were not clearly attributable to their removal procedure". In our series, complications occurred in two patients, but only one patient presented life-threatening peroperative complications in the form of mediastinal breach with pneumothorax. This is in accordance with previous reports for uncovered metallic stents removal by rigid bronchoscopy [START_REF] Lunn | Endoscopic removal of metallic airway stents[END_REF][START_REF] Alazemi | Outcomes, health-care resources use, and costs of endoscopic removal of metallic airway stents[END_REF] .

In our series, two patients needed silicone placement after SEUMS removal due to peroperative complication (one case) and important tracheobronchomalacia (one case).

Five patients underwent laser sessions for residual tracheobronchomalacia several months after stent removal, to avoid new stenting. We do not consider these last incidences as removal complications as it is directly related to the initial disease of the patient. Some authors even interpreted that if patients don't need additional surgery or new silicone stent placement after stent removal, it should question the first indication of stenting [START_REF] Lunn | Endoscopic removal of metallic airway stents[END_REF] . In all the series previously discussed, around 50% patients required silicone stents' placement following metallic stents' removal. We only placed silicone stents in two cases at the end of our procedures. In the rest of cases, we preferred to tolerate moderate tracheobronchomalacia and perform laser tracheobronchoplasty afterwards.

Ideal length of stenting time is the remaining question. Some authors now advise to remove SEUMS three months after deployment [START_REF] Bi | Fluoroscopic removal of self-expandable metallic airway stent in patients with airway stenosis[END_REF] . The use of iterative follow-up bronchoscopy or CT to identify and treat early complications is recommended [START_REF] Dialani | MDCT detection of airway stent complications: comparison with bronchoscopy[END_REF][START_REF] Lee | Airway stent complications: the role of followup bronchoscopy as a surveillance method[END_REF] .

Beyond the duration of stenting, primary indication and type of stent must be well discussed, and all alternative treatment should be considered to avoid long term stentrelated complications.

CONCLUSION

Although considered permanent, SEUMS can be removed by rigid bronchoscopy, even if the stent has been in situ for years. This is a very risky procedure and must be performed by a trained operator. Patients should be aware of possible fatal complications. We described a new technique using rigid bronchoscopy and bronchoscopic hook scissors for SEUMS removal through dissection that can help to decrease the risk related to this type of surgery.

  ABREVIATIONS .......................................................................................................................... INTRODUCTION ........................................................................................................................ ENDOSCOPIC REMOVAL OF UNCOVERED METALLIC AIRWAY STENTS .................................... INTRODUCTION ........................................................................................................................ MATERIAL AND METHODS .......................................................................................................

Figure 1 :

 1 Figure 1: "Piecemeal fashion" operative technique. .............................................................................................

Figure 2 :

 2 Figure 2: SEUMS removal operative technique. .....................................................................................................

Table 1 :

 1 Indications for stent removal ........................................................................................................................ Table 2: Per and postoperative complications of SEUMS removal ................................................................ Table 3: Comparison of the two SEUMS removal techniques........................................................................... Table 4: Descriptions of stents characteristics, removal details, complications and outcomes ....... DISCUSSION .............................................................................................................................. CONCLUSION ............................................................................................................................ REFERENCES ............................................................................................................................. INTRODUCTION Les prothèses trachéo-bronchiques ont été développées pour maintenir l'ouverture des voies aériennes supérieures, surtout dans le cadre d'obstructions d'origine maligne, par envahissement endo-bronchique ou compression extrinsèque. Elles sont depuis plusieurs décennies utilisées également pour les obstructions d'origine bénigne, comme la trachéomalacie, les sténoses post intubation ou post trachéotomie et les sténoses des anastomoses des patients greffés pulmonaires. Il existe différents types de prothèses trachéo-bronchiques. La première était en silicone et a été développée par Montgomery en 1965. Elle nécessite une trachéotomie pour être maintenue dans la trachée. Par la suite, des prothèses métalliques autoexpansibles non couvertes, initialement utilisées dans le traitement des sténoses vasculaires ou biliaires, ont été placées dans les voies aériennes supérieures. D'autres modèles couverts par une membrane de polyuréthane ou en silicone sont secondairement apparus, notamment pour limiter la croissance tumorale à travers les mailles de la prothèse. La plupart des prothèses non couvertes sont en nitinol, un alliage de nickel et de titane, et présentent un certain nombre d'avantages. Tout d'abord, elles vont s'intégrer à la muqueuse ; une épithélialisation se fait par-dessus la prothèse ce qui augmente la clairance muco-ciliaire et permet théoriquement une meilleure tolérance. Le nitinol apporte une mémoire de forme qui permet une bonne adaptation aux trachées irrégulières, notamment dans le cadre des trachéomalacies, et diminue également le risque de migration ultérieure. De plus, elles sont auto-expansibles et peuvent donc être placées facilement sous anesthésie locale, par bronchoscopie souple ou rigide, sans franchissement de la sténose par le bronchoscope. Tous ces avantages rendent ces prothèses attractives dans le cas des obstructions bénignes et ont donc été largement utilisées. Les effets sont bénéfiques à court terme mais il existe à long terme un certain nombre de complications. L'intégration et l'épithélialisation de la prothèse rendent le retrait très difficile et dangereux, elles sont donc souvent considérées comme permanentes. Elles peuvent également être à l'origine de fistule avec les organes avoisinants. De plus, un tissu de granulation peut se former à l'intérieur de la prothèse et devenir sténosant, nécessitant des désobstructions répétées. C'est également une source de surinfection bactérienne fréquente. Depuis 2005, il est officiellement non recommandé d'utiliser ces prothèses métalliques non couvertes dans les sténoses d'origine bénigne. Il existe un nombre difficilement évaluable de patients porteur de ce type de prothèse depuis des années et présentant des complications répétées. Pour limiter le nombre de traitements et d'hospitalisations ainsi qu'améliorer la qualité de vie, le retrait de ces prothèses peut être envisagé. Il s'agit d'une intervention risquée, chez des patients aux lourds comorbidités. Nous étudions sur une série de patients le retrait des prothèses trachéales métalliques non couvertes en bronchoscopie rigide, sous anesthésie générale, et discutons des complications per et post opératoires. Nous présentons également une nouvelle technique opératoire, développée dans le service d'ORL et de Chirurgie Cervico-Faciale du CHU de Grenoble-Alpes et pratiquée sur une partie de notre cohorte. Résumé Objectif : Nous étudions le retrait des prothèses métalliques non couvertes en bronchoscopie rigide sur une série de patients et présentons une nouvelle technique de retrait. Matériel et méthodes : Les données de sept patients ayant bénéficié du retrait de ces prothèses entre mai 2017 et mai 2022 ont été analysées rétrospectivement. Toutes les prothèses retirées ont été initialement placées pour une obstruction trachéale ou bronchique d'origine bénigne. Le retrait est réalisé en bronchoscopie rigide sous anesthésie générale en technique classique pour certains patients et selon une nouvelle technique pour les autres. Dans celle-ci, le bronchoscope est inséré progressivement entre la prothèse et la paroi trachéobronchique, tout en disséquant la prothèse de la paroi à l'aide de micro-ciseaux à lames crochet. Résultats : Neuf prothèses métalliques non couvertes ont été retirées chez les sept patients avec succès en bronchoscopie rigide. Cinq prothèses ont pu être retirées à l'aide de notre nouvelle technique. L'ablation des autres prothèses a été réalisée selon la technique couramment utilisée, à l'aide d'une pince crocodile, sans dissection. Les prothèses retirées ont été implantées en moyenne pendant 7 ans (5-12 ans). Les principales indications de retrait étaient les infections récurrentes, la formation de tissu de granulation, entraînant dans certains cas une obstruction trachéo-bronchique, et la fracture de la prothèse. Trois interventions ont présenté des complications peropératoires : persistance de débris prothétiques, déchirure de la muqueuse avec exposition cartilagineuse et brèche médiastinale avec pneumothorax. Deux de ces complications ont été observées lors de la réalisation de la technique usuelle sans dissection. Un patient, à l'état pulmonaire précaire, est décédé d'une défaillance respiratoire 8 jours après le retrait, sans qu'il ne soit observé de complication peropératoire. Conclusion : Bien que considérées comme permanentes, les prothèses trachéales métalliques non couvertes peuvent être retirées en bronchoscopie rigide après plusieurs années in situ. Il s'agit d'une procédure risquée et réservée aux opérateurs entraînés en bronchoscopie. Les patients doivent être prévenus du risque de complications vitales en per et post-opératoire. Mots clés : Prothèse trachéale métallique, non couverte, retrait, bronchoscopie rigide, risque vitale, microdissection INTRODUCTION Airway stenting has become a new appropriate treatment for benign and malignant tracheobronchial strictures in the absence of possible surgical option 1 . The gold standard treatment for airway obstruction remains tracheal resection with anastomosis but this technique does not apply for long segmental stenosis and patients may not support long and open surgery 2 . Self-expandable uncovered metallic stents (SEUMS)

  connected to a disposable flexible 23G and 4 mm injection needle (AF-D1816PN2304, ALTON Medical Instruments, Shanghai, China) introduced in the working channel of a single-use flexible bronchoscope (AMBU® aScopeTM Broncho, Copenhagen, Danemark). Patients were admitted for airway observation postoperatively for 2 to 4 days. They also get respiratory care including nebulized 3% sodium chloride to help clear secretions along with broad spectrum antibiotics. Some patients were also given nebulized N-acetyl-L-cystine 1gr/5ml b.i.d. (ISTENDO®, Delbert, France).

Figure 1 :

 1 Figure 1: "Piecemeal fashion" operative technique. a) Airway obstruction due to embedded SEUMS. b) Piecemeal fashion or as "rolling spaghetti on a fork" fashion operative technique. c) Pieces of SEUMS after removal. d) Post SEUMS removal view of the tracheal wall.

Figure 2 :

 2 Figure 2: SEUMS removal operative technique. a) Airway obstruction due to embedded SEUMS. b) Dissection of the upper part of the SEUMS, with bronchoscope between the stent and the tracheal wall. Bronchoscopic hook scissors dissect the stent from the tracheal wall. c) SEUMS removed in one piece at the end of the procedure. d) Tracheal wall after the removal.

  placements between 2005 and 2017 in different centers. Indications for stent ), postintubation airway stenosis (1 case) and post tuberculosis stenosis (1 case). Nine SEUMS were removed by rigid bronchoscopy between May 2017 and May 2022. The mean age was 59 years (range 45 to 81 years). The median physical status score (ASA) was 4. The mean duration of stenting was 7 years (5-12 years). Two patients had bilateral bronchial SEUMS and underwent stent removal in separate procedures for each side. Each procedure was analyzed separately.

Table 3 : Comparison of the two SEUMS removal techniques Piecemeal technique Dissection technique

 3

	Operating time (hours)	4,7	4,8
	Peroperative complications (n)	2	1
	Mean postoperative intubation	9,5	2,5
	period (days)
	Two patients required new silicone stent placement after the SEUMS removal due to
	threatening tracheobronchomalacia. Four patients needed additional procedures after
	stent removal to treat residual tracheobronchomalacia. In these patients, we performed
	Holmium laser tracheobronchoplasty (Sphinx junior, LISA laser products GmbH

Remerciements

ABRÉVIATIONS

SEUMS: Self-expandable uncovered metallic stent FDA: Food and Drug Administration ASA: American Society of Anesthesiologists

Endoscopic removal of uncovered metallic airway stents

Mathilde Rausch, MD [START_REF] Dalar | Bronchoscopic Treatment in the Management of Benign Tracheal Stenosis: Choices for Simple and Complex Tracheal Stenosis[END_REF][START_REF] Brichet | Sténoses et complications trachéales postintubation1Post-intubation tracheal stenosis and ruptures[END_REF] , Amandine Briault, MD-Msc 2-3 , Marie-Pierre Aboussouan, MD [START_REF] Ducic | Use of endoscopically placed expandable nitinol tracheal stents in the treatment of tracheal stenosis[END_REF] , Jean-Eric Berger, MD [START_REF] Ducic | Use of endoscopically placed expandable nitinol tracheal stents in the treatment of tracheal stenosis[END_REF] , Ihab Atallah, MD-PhD 1-2 Results: Nine SEUMS were removed from 7 patients. The mean duration of stenting was 7 years (5-12 years). Indications for stent removal included mostly recurrent pulmonary infections, obstructive granulation tissue and stents' fractures. We used the usual described technique in the literature known as the "piecemeal fashion" without dissection in 4 SEUMS removals and our new dissection technique in 5 procedures. We reported one major intraoperative complication using the former technique in the form of tracheal tear with mediastinal breach, while we did not experience any major complications with the later one.

One patient died 8 days after surgery from respiratory failure.

Interpretation: Although considered permanent, SEUMS can be removed by rigid bronchoscopy after long-term stenting. This is a very risky procedure and must be performed by a trained operator. Patients should be aware of possible severe complications.

Key words: self-expandable metallic stents; uncovered metallic stents; airway stent removal, rigid bronchoscopy; dissection Abbreviations: SEUMS: Self-Expandable Uncovered Metallic Stent, FDA: Food and Drug Administration, ASA: American Society of Anesthesiologists

MATERIAL AND METHODS

Patients

We retrospectively reviewed data of patients who underwent SEUMS removal by rigid bronchoscopy between May 2017 and May 2022. We collected information about patient demographics, medical comorbidities, indication for stent placement, type and position of stent, indication for removal, characteristics and complications of the procedures, additional stent insertion or surgery. In addition, we collected the performance status of patients that was measured using the American Society of Anesthesiologists (ASA) physical status classification [START_REF] Owens | ASA physical status classifications: a study of consistency of ratings[END_REF] . All patients underwent chest CT scan and flexible bronchoscopy under local anesthesia before stent removal to evaluate airway patency, position of the stent and the amount of granulation tissue.

Informed consent for data collection in medical records was obtained. All procedures performed in the study were in accordance with the ethical standards of our institution and with the 1964 Helsinki declaration and its later amendments.


Cyril Karamaoun

Benoît Haut

Alain Van Muylem

come

Simulation of airway calibre and inflammation interaction by a new model of airway epithelium

• This new model is able to mimic FE NO decreasing with BC by decreasing diffusion area and lengthening diffusion pathways, hence reducing JawNO.

• FE NO is quite insensitive to BC occurring in central airways.

• BC involving distal airways may, depending on its level, partially or even totally mask an epithelial NO production increase

• This emphasizes the importance of the site of BC in the interpretation of a FE NO change.

Simulation of airway calibre and inflammation interaction by a new model of airway epithelium

Corresponding author : ckaramao@ulb.ac.be 1 TIPs (Transfers, Interfaces and Processes), Université libre de Bruxelles, Brussels, Belgium 2 Chest Department, Erasme University Hospital, Université libre de Bruxelles, Brussels, Belgium

Background

-Experimental evidences show that airway bronchoconstriction (BC) interferes with the ability of exhaled nitric oxide (FE NO ) to reflect inflammatory status (Figure 1).

-Intensity but also site of BC likely influence FE NO changes.

-So far, no theoretical studies have been conducted about the influence of BC on the total flux of NO in the conducting airway compartment (JawNO).

Aim

-To present a new model of epithelium NO production and transport coupled with NO transport in the airways.

-To theoretically assess how the site of airway BC influences JawNO and, hence, FE NO .

Cyril Karamaoun 1 , Benoît Haut 1 , Alain Van Muylem 2

Were simulated :

1) JawNO and FE NO as a function of the furthest constricted generation, without production change 2) FE NO with a 30% increase of epithelial NO production and :

Simulations Results

1)

2)

Conclusions

Figure 1 Late phase of DPT challenge -NO production in the airway epithelium and transport from the epithelium to the lumen (JawNO) -Airway narrowing by smooth muscle shortening (Figure 3) -Dimensionless calculations for numerical stability and physical comparisons

• NO axial transport equations in the airways are then solved based on diffusion/convection equations in Weibel's morphometric data. • NO production is assumed as constant by volume of epithelium, i.e. 4.87 10 -12 molNO s -1 cm -3 (Figure 4)

(

  Figure 2

  Figure4


Anthony Roy
email: anthony.roy1@univ-nantes.fr

Ayoub Igourzal
email: ayoub.igourzal@univ-nantes.fr

Jean-Christophe Olivier
email: jean-christophe.olivier@univ-nantes.fr

François Auger
email: francois.auger@univ-nantes.fr

Experimental validation of an optimized power allocation strategy for multi-stack fuel cell

Keywords: Fuel cell, hydrogen, multi-stack, power allocation, Hardware-In-the-Loop

Hydrogen is more and more considered for the decarbonization of energy uses, especially for transport applications. To reach a power range of several hundred kilowatts usually met in high power applications, an association of fuel cells in multi-stack is necessary. An adapted control must therefore be designed to share the requested power among the fuel cells. This paper presents an optimized power allocation strategy based on the minimization of the consumption of the stacks. The polarization curves are considered to take the fuel cell characteristics and its state of health into account. The proposed control strategy is experimentally validated thanks to a computation in real-time on a test bench, emulating four stacks of fuel cells with different states of health. The results show the effectiveness of the developed strategy, allowing the consumption to be reduced by 1.2 % compared to a strategy based on a fair distribution. Moreover, the reconfigurability of the power allocation is validated by considering the unavailability of some of the stacks.

I. INTRODUCTION

In recent years, hydrogen has been increasingly considered for the decarbonization of energy needs. For example, hydrogen can be an interesting energy vector for the industry and transport sectors [START_REF] Roy | A combined optimization of the sizing and the energy management of an industrial multi-energy microgrid: Application to a harbour area[END_REF], and it could be associated with renewable energy sources in a multi-source system. Hydrogen can be used as an energy source for different means of transport such as cars, buses or vessels, thanks to the use of a hydrogen fuel cell coupled or not to other storage solutions [START_REF] Ma | Fuel cell-battery hybrid systems for mobility and offgrid applications: A review[END_REF]. Among the different technologies of fuel cells existing on the market, the Proton Exchange Membrane (PEM) fuel cell is currently the most developed one, as the advantages are numerous, such as the fast start-up speed, the low operating temperature and the high power density [START_REF] Fan | Recent development of hydrogen and fuel cell technologies: A review[END_REF]. Thus, this technology is more and more considered in transport applications, for example for maritime mobility [START_REF] Van Biert | A review of fuel cell systems for maritime applications[END_REF]. However, these transport applications often involve high power requirements, as the load power can reach several hundreds of kilowatts [START_REF] Van Biert | A review of fuel cell systems for maritime applications[END_REF]. Thus, multi-stack architectures have been investigated over the last years, by considering an assembly of fuel cell in series and/or parallel so as to reach desired voltage and current requirements and also to bring modularity and fault tolerance [START_REF] Marx | A review of multi-stack and modular fuel cell systems: Interests, application areas and on-going research activities[END_REF]. Moreover, it allows the replacement of only faulty modules instead of the whole system. However, the association of fuel cells in series and/or parallel requires an appropriate control and energy management system to ensure the reliability, the durability and the cost-effectiveness of the solution. The load power must be allocated among the modules, which can present mismatches due to different state of health (ageing) or operating conditions (temperature, relative humidity...). Several authors have recently investigated the control of multi-stack fuel cell by considering different criteria to define how the modules must operate. The control can be done according to the ageing and the durability, by considering the degradation of stack performance [START_REF] Wang | A Power Allocation Method for Multistack PEMFC System Considering Fuel Cell Performance Consistency[END_REF]- [START_REF] Bahrami | Fuel cell management system: An approach to increase its durability[END_REF]. In [START_REF] Meng | A Distributed Performance Consensus Control Strategy of Multistack PEMFC Generation System for Hydrogen EMU Trains[END_REF], the authors proposed a control strategy based on consensus allowing the modules to reach the same lifetime. Moreover, the operating decisions can be defined in order to minimize the hydrogen consumption [START_REF] Bouisalmane | Hydrogen consumption minimization with optimal power allocation of multi-stack fuel cell system using particle swarm optimization[END_REF], [START_REF] Khalatbarisoltani | Power Allocation Strategy Based on Decentralized Convex Optimization in Modular Fuel Cell Systems for Vehicular Applications[END_REF]. In [START_REF] Somaiah | Distributed Maximum Power Extraction From Fuel Cell Stack Arrays Using Dedicated Power Converters in Series and Parallel Configuration[END_REF], the management of fuel cells is done according to a maximum power point tracking algorithm. Moreover, economic criteria can be considered. In [START_REF] Li | Cost Minimization Strategy for Fuel Cell Hybrid Electric Vehicles Considering Power Sources Degradation[END_REF], the authors proposed an economic optimization taking into account the operating cost of fuel cells and the hydrogen cost. It can be highlighted that the control of multi-stack fuel cell can be done by coupling these technical and economic criteria, as in [START_REF] Li | Incrementoriented online power distribution strategy for multi-stack proton exchange membrane fuel cell systems aimed at collaborative performance enhancement[END_REF]. These control methods can be designed as rule-based oriented strategies or as optimization problems [START_REF] Fernandez | An Adaptive State Machine Based Energy Management Strategy for a Multi-Stack Fuel Cell Hybrid Electric Vehicle[END_REF]. Nevertheless, the literature survey shows that the studies experimentally validating their control strategies as in [START_REF] Jian | Hardware-in-the-loop real-time validation of fuel cell electric vehicle power system based on multi-stack fuel cell construction[END_REF] are more scarce, particularly for those considering multi-criteria strategies, as the design of multi-stacks is recent. Thus, we propose in this article a multi-criteria power allocation strategy for a multi-stack fuel cell association. The operating decisions are defined thanks to an on-line optimization problem, consisting in the minimization of the multi-stack consumption while considering the ageing of the modules. A Power Hardware-in-the-Loop (PHiL) experimental validation is carried out to show the effectiveness of the proposed control, considering an emulation of a set of fuel cell stacks. A comparison with a classical non-optimized strategy based on the equalization of the power allocation is also proposed. This paper is organized as follows: the modelling of the multi-stack fuel cell is presented in Section II. Then, a power allocation method based on an optimization problem is proposed in Section III. The PHiL test bench is described in Section IV and some experimental results are proposed. Section V concludes the paper by giving some perspectives.

II. MULTI-STACK FUEL CELL DESCRIPTION

The topology of the system considered in this article is presented in Fig. 1. The studied multi-stack is made of four stacks of hydrogen fuel cells, but an extension to a larger number of stacks could be easily considered. Each is connected to a DC/DC boost converter allowing the voltage to be increased. These power converters supply a DC load connected to a DC bus voltage, on which a capacitor is also connected. The behaviour of each fuel cell stack is assessed according to the static polarization curve of the fuel cells, following the detailed model presented in [START_REF] Robin | Proton exchange membrane fuel cell model for aging predictions: Simulated equivalent active surface area loss and comparisons with durability tests[END_REF]. The operating voltage of the stack [V] is defined from the current [A], the state of health and several fuel cell parameters such as the operating temperature, the chemical and geometrical characteristics, etc. The state of health represents the degradation of the fuel cell voltage compared to the initial voltage, with = 1 at the beginning of the lifetime [START_REF] Song | Degradation adaptive energy management strategy using fuel cell state-of-health for fuel economy improvement of hybrid electric vehicle[END_REF]. The power of the fuel cell stack [W] is defined as

 =  
An example of fuel cell stack polarization curves is given in Fig. 2. for two values (1 and 0.8), considering for each stack a connection in series of 50 cells with the same . These curves show that for a given power value, the most degraded fuel cell stack presents a higher current. As the hydrogen consumption depends directly on the fuel cell current [START_REF] Li | Cost Minimization Strategy for Fuel Cell Hybrid Electric Vehicles Considering Power Sources Degradation[END_REF], this ageing means that more hydrogen will be consumed for a same output power.

The system is controlled according to the command diagram presented in Fig. 3. The outer loop of the control allows the DC bus voltage to be regulated to the reference value * thanks to a PI controller, by defining the reference value of the DC load current ( * ). This value is then used as input data of the power allocation algorithm, which determines the reference current of each fuel cell stack ( * to * ) according to an optimization problem, as described in the next section of this article. These references are used as input data of the control loop of DC/DC converter input current. PI controllers are implemented in these inner loops for the regulation of the currents, so as to define the pulse width modulated (PWM) signals controlling the transistors of the power converters.

III. POWER ALLOCATION STRATEGY

A power allocation strategy is necessary so as to spread the supply of the load over the fuel cell stacks. Assuming that all the fuel cells behave identically, the simplest strategy would be to equally distribute the requested power between the stacks. However, this strategy may not be optimal in terms of efficiency, as the fuel cell stacks may not be identical, especially because of their ageing. As shown in Fig. 2, a degradation of fuel cell characteristics leads to a higher current, thus to a higher hydrogen consumption. Thus, an optimized power allocation strategy taking the degradation of fuel cells into account has to be designed to maximize the efficiency and to avoid hydrogen overconsumption. Moreover, it must allow the reconfigurability of the system so as to ensure the load supply whatever the situation, including in the event of a stack fault. The power allocation strategy proposed in this article is made of several steps as shown in the flowchart of Fig. 4, resulting in an optimized distribution of the current over the fuel cell stacks, thus an optimized distribution of the power. The main steps of the method presented in the flowchart of Fig. 4 are described hereafter.

Step 1: the optimized current values ( to ) are initialized considering an equally distribution of the demanded current * .

Step 2: a linear approximation of the I-V characteristic of each fuel cell stack is computed around the last calculated current setpoint, to make the solving of the linear optimization problem of the next step easier. Thus, the fuel cell stack voltage is approximated by the following linear expression (Thevenin model):

 = - + ,  

with the approximated resistance of the ohmic polarization region [Ω] and

, the open circuit voltage obtained when is zero. It must be noted that in this work, the online identification is not considered but it could be easily implemented through a small perturbation method.

Step 3: the problem is solved in order to calculate the reference current of each fuel cell stack ( to ) allowing the total hydrogen consumption to be minimized, while ensuring the load power is met. Thus, the decision variables of the optimization problem correspond to the current of each fuel cell stack, gathered in a vector such as:

 = [ ]  
The aim of the optimization problem is to minimize the hydrogen consumption, which is related to the value of the fuel cell stack current. Thus, the objective function consists in the minimization of the sum of the fuel cell currents:

 = + + + =  
The optimal solution must ensure that the load power is met, thus the following equality constraint must be verified:

 =  
The voltage of each fuel cell stack is determined thanks to the linear approximation considered in the step 2 of the proposed algorithm, according to the equation (2). Thus, this optimization problem corresponds to a linear optimization problem with quadratic constraint, which can be written according to the following classical matrix formalism:

 = argmin = argmin   s.t.:  0.5 + + = 0 
where is the vector containing the optimal solution and A, B, C and D are the matrices integrating the optimization input data of equations ( 2), ( 4) and ( 5) for each stack:

 = ⎣ ⎢ ⎢ ⎢ ⎡ -2 0 0 0 -2 0 0 0 0 -2 0 0 0 0 -2 ⎦ ⎥ ⎥ ⎥ ⎤    = [ , , , , ] 
 = [1 1 1 1]   = -  
To obtain a solution as fast as possible, this optimization problem is solved iteratively by Newton's method. It can be noted that a reformulation is done in a way to get a Lagrangian function. Moreover, the reconfigurability of the system is integrated in this step by considering in the optimization problem only the available fuel cell stacks (the current of non-available fuel cell stacks is set to zero).

Step 4: if the sum of the absolutes values of the differences between the currents of the two last iterations is lower than a predefined tolerance (example: 10 mA), the algorithm is stopped and the current values of the last iteration are considered to be the new reference currents for the fuel cell stacks ( * to * ). Otherwise, the steps 2 and 3 are computed until the stopping criterion is met.

IV. EXPERIMENTAL VALIDATION

The optimized power allocation strategy proposed in the previous section is experimentally validated on a PHiL test bench. The experimental platform and the characteristics of the fuel cell stacks are described in the following subsections. Several experimental results are also given and discussed.

A. Test bench description

An overview of the test bench used for the experiments is given in Fig. 5

B. Emulated fuel cells characteristics

In order to validate the efficiency of the proposed power allocation algorithm, four different fuel cell stack characteristics have been considered, assuming different values. The voltage versus current and power versus current static curves of these stacks are given in Fig. 6 and7. It has been assumed that the stack 1 presents the best state of health value ( = 1), while the stack 4 is the most degraded one ( = 0.7). The nominal power of the considered fuel cell stack at the beginning of its lifetime ( = 1) is 360 W. So as to illustrate how the choice of these values can influence the results of the power allocation optimization, an example of optimized allocation is shown on the curves of Fig. 6 and7, considering a load power of 880 W under = 100 V. Moreover, the linear I-V characteristic around the optimal solution is plotted for each stack. It can be seen that the higher the value, the higher the current, as a less aged fuel cell can provide more power than a more aged one.

C. Experimental results

The designed control algorithm is validated for several load current values, considering the variable current load profile shown in Fig. 8, for a 20 s period. The obtained output current of each emulated fuel cell stack is plotted in Fig. 9 for two allocation strategies: the first one is without optimization (i.e. an equally distribution of the demanded power over the stacks) and the second one corresponds to the proposed algorithm based on an optimized sharing. The power profile of each fuel cell stack for both strategies is presented in Fig. 10. The obtained results show that for the non-optimized strategy, the power is equally shared between the stacks but the currents present slight differences. This phenomenon is due to the fact that the fuel cells polarization curves are not the same. For a same value of power, the current of the most aged fuel cell will be higher than the less aged fuel cell, as the voltage decreases with the ageing. Thus, this sharing strategy is not optimal in terms of hydrogen consumption. For the optimized strategy, the obtained results show the effectiveness of the proposed control algorithm as the power and currents are shared to reduce the solicitation of the most aged fuel cells (FC stacks 3 and 4) and to request more power from the less aged fuel cells (FC stacks 1 and 2). The sharing obtained for the period between 10 s and 15 s corresponds to the optimized distribution presented in Fig. 6. It can be noted that even if the power allocation algorithm computed every 300 ms, the distribution of the currents is kept optimized according to the values, by conserving the ratios between the currents. Also, these profiles show the ability of the proposed control to maintain the stability of the system during the transients. So as to show the benefits brought by the proposed algorithm, the difference ∆ between the total current of the optimized strategy (Fig. 9 bottom) and the nonoptimized strategy (Fig. 9 top) is plotted in the Fig. 11, with = . This profile shows that the optimized control allows the total current to be decreased, up to -0.5 A for the highest load current value and even -2A during a transient. A comparison of the consumption of the stacks for both strategies is given in Table I, for the period of 20 s presented in Fig. 9. The results show that the total consumption is reduced by 1.2 % when using the optimized strategy rather than the non-optimized strategy. Thus, a similar hydrogen reduction can be expected thanks to the optimized power allocation strategy. Thanks to the proposed control, a longer durability of multi-stack can be expected as the less efficient and the more aged modules are less stressed. Moreover, more savings can be expected by increasing the load power. An analysis of the computation time of the power allocation algorithm during the tests shows that the algorithm takes between 30 and 50 µs to provide results, which means that the power can be allocated efficiently in a short time. Finally, the reconfiguration capability of the multi-stack in case of fault has been experimentally validated by switching off some of the stacks, considering a constant load current * of 6 A during the 30 s of the test. The obtained current and power profiles are given in Fig. 12. The stack 4 has been switched off between 8.6 s and 15.3 s, then the stack 3 between 21.1 s and 27.8 s. These profiles show that the dispatch of the requested power has been updated when the fault occurs while maintaining the sharing rules observed previously according to the state of health of the fuel cells (the less aged fuel cells take most of the power). Thus, the power allocation algorithm ensures the reconfigurability of the multi-stack in case of stack fault.

V. CONCLUSION

In this article, an optimized power allocation strategy has been proposed, so as to minimize the consumption of a multi-stack fuel cell system. The proposed optimization problem allows the operating setpoint of each stack to be defined by taking into account the polarization curves, and therefore some possible mismatches between the fuel cells such as the ageing. This strategy has been validated experimentally by considering different SoH values, leading to a lower solicitation of the most degraded fuel cells thus to a hydrogen consumption. The proposed allocation strategy performs better than a classical strategy based on an equally sharing of the requested power. The developed strategy has been applied to a small number of stacks and to a low voltage due to constraints related to the experimental test bench, but it can be applied to any number of stacks and any voltage level. As an extension of the presented work, several perspectives could be explored. For example, others criterions could be integrated in the power allocation strategy, such as the operating cost, the ageing in terms of operating hours, the number of operating fuel cells, etc. The estimation of the polarization curve parameters during the operation could also be investigated, by an online identification of the characteristics of a real fuel cell. Thus, the SoH could be estimated online during the operation, allowing the power sharing to be updated according to the ageing of the stacks. Finally, the use of real fuel cells instead of emulated ones could be considered.

Fig. 1 .

 1 Fig. 1. Overview of the considered multi-stack fuel cell system

Fig. 2 .Fig. 3 .

 23 Fig. 2. Example of polarization curves for two fuel cell stacks with different SoH values

Fig. 4 .

 4 Fig. 4. Flowchart of the proposed power allocation strategy

  . The four fuel cell stacks are emulated by DC voltage sources (ref. PSI 9200-15 from Electro-Automatik manufacturer), which are controlled by the fuel cell stack models computed in real-time in the Opal-RT target (OP4510). Each voltage source is connected to a DC/DC 1 st order approximation of I-V curve around ( a 4 mH inductance and supplies a DC current load. This load is connected to the DC voltage bus, on which a 880 μF capacitor is connected. The nominal voltage of the DC bus is set to 100 V. The power allocation algorithm, the regulation of the DC voltage and the control of the boost converters are implemented using the MATLAB SIMULINK software and computed in real-time in the Opal-RT target, which is equipped with an Intel Xeon E3 3.5 GHz processor. The model is computed with a 60 μs time-step. As the power allocation algorithm can require more than one time step to provide optimized results, is considered that this algorithm is computed every 300 ms to ensure the optimization convergence and no overrun. Thus, until the optimized results are available, the last values of the current references are used to adapt the theoretical setpoint values based on an equally shared distribution of the demanded power. The monitoring of the experiments and the visualization of signals are done thanks to a host computer connected to the Opal-RT target via the TCP/IP protocol.

Fig. 5 .

 5 Fig. 5. Overview of the PHiL test bench

Fig. 6 .

 6 Fig. 6. I-V characteristics of the four considered fuel cell stacks (markers indicate an example of optimized solution for power allocation)

Fig. 8 .

 8 Fig. 8. Profile of load current iload considered in the experiments

Fig. 9 .Fig. 10 .

 910 Fig. 9. Output current profiles of the emulated fuel cell stacks (top: nonoptimized energy management, bottom: optimized energy management)

Fig. 11 .

 11 Fig. 11. Difference between the total current of the two strategies

Fig. 12 .

 12 Fig. 12. Output current (top) and power (bottom) profiles of emulated fuel cell stacks with occurences of stack fault

TABLE I .

 I COMPARISON OF EXPERIMENTAL RESULTS

	Fuel cell	Consumption without	Consumption with
	stack	optimized allocation [mAh]	optimized allocation [mAh]
	1	24,4	29,8
	2	25,2	26.8
	3	26,4	23.8
	4	26,5	20.9
	Total	102,5	101.3

978-1-6654-8240-0/22/$31.00 ©2022 IEEE 2022 IEEE 31st International Symposium on Industrial Electronics (ISIE) | 978-1-6654-8240-0/22/$31.00 ©2022 IEEE | DOI: 10.1109/ISIE51582.2022.9831524 Authorized licensed use limited to: University of Nantes. Downloaded on November 30,2022 at 09:13:28 UTC from IEEE Xplore. Restrictions apply.

Authorized licensed use limited to: University of Nantes. Downloaded on November 30,2022 at 09:13:28 UTC from IEEE Xplore. Restrictions apply.

ACKNOWLEDGMENT

This research was supported by the project "High power fuel cells for vessels propulsion" granted by the French "Pays de la Loire" region.