Mr Eric Bideaux

Mr Mr Mathias Bruno Gerard

Mr Matthieu Jeanneret

Mr Desbois-Renaudin

Sari Ali

Thank

Pascal Martin

Anthony Barre

Yanick Meynet

Lucival Rodrigues Da Fonseca

Minha Mãe

Eleonora Maria Nai Da Fonseca

Amiga Cristiane

Fabíola Simões

I believe that water will one day be employed as fuel, that hydrogen and oxygen which constitute it, used singly or together, will furnish an inexhaustible source of heat and light, of an intensity of which coal is not capable

Synthèse du travail

Le modèle énergétique actuel basé sur l'utilisation de sources d'énergie non renouvelables est économiquement et écologiquement discutables et fait l'objet de discussions au cours de ces dernières années. L'augmentation de la consommation d'énergie et donc les émissions de polluants, démontrent les limites de ce modèle énergétique et ne peut donc pas être soutenu à moyen et long terme sans impact majeur sur l'environnement et le développement mondial.

Dans ce contexte, le secteur des transports, qui est responsable d'une importante part de la consommation énergétique mondial 27 % [START_REF] Joni | World energy outlook 2010[END_REF], étudie activement depuis ces 20 dernières années, des solutions propres et durables pour augmenter le mixte énergétique.

La présence de véhicules hybrides, à propulsion électrique et à l'éthanol dans le marché automobile sont des exemples réussis de cette recherche. Néanmoins, ces diérentes solutions sont destinées à des marchés de niche et ne sont adaptées à certains environnement socio-économiques.

L'hydrogène comme vecteur d'énergie pour le transport, et valorisé par des piles à combustible (PAC), est mis en évidence de plus en plus comme une solution prometteuse en raison de ses caractéristiques fonctionnelles, de performance (rendement supérieur à 50 %) et environnementales (aucune émissions de polluants par le véhicule).

Cependant, l'introduction de l'hydrogène dans le secteur des transports à travers des véhicules à pile à combustible est toujours confrontée à diverses contraintes telles que le coût (du système pile à combustible du véhicule et de l'hydrogène) mais surtout des infrastructures nécessaires pour la production, le transport et la distribution de l'hydrogène.

En outre, la technologie liée à la mise en ÷uvre de véhicules propulsé par une pile à combustible n'a pas encore atteint le niveau de maturité que d'autres solutions du secteur des transports. Certaines diérences notables peuvent être observées à travers la durabilité et le coût du système pile à combustible.

Selon le DOE (Department of Energy des USA) [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF], le véhicule à pile à combustible (VPAC) doit être compétitif par rapport aux technologies concurrentes présentes dans les moyens de transports actuels. Par conséquent, des objectifs de commercialisation à xi Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés atteindre dans le processus de l'élaboration du VPAC ont été établis. A titre d'exemples, la PAC devant atteindre un coût de 30 $/kW en 2017 et une durée de vie de 5000 heures (équivalent à 241.000 km). A titre de comparaison, an d'amorcer le marché, les constructeurs visent un prix d'un VPAC similaire à un véhicule hybride diesel.

Pour atteindre ces objectifs et améliorer les performances du véhicule, les constructeurs automobiles adoptent des architectures hybrides contenant système PAC et batterie. La présence de cet élément de stockage (batterie) apporte un degré de liberté dans l'utilisation de la PAC et permet de contrôler certaines conditions défavorables au fonctionnement de la PAC : performances (consommation d'hydrogène) et durabilité.

Cependant, l'hybridation apporte une complexité du contrôle de puissance entre les deux éléments (SPAC et batterie), qui doit être résolu par une stratégie de gestion d'énergie embarquée dans le véhicule. An de contribuer à l'avancement de cette technologie présente dans les véhicules hybrides à pile à combustible (VHPAC), ce document vise à proposer des solutions pour répondre à certaines limitations de cette technologie. Plus précisément, trois thèmes constituent les objectifs de ce travail:

Le dimensionnement optimal des éléments de la chaine énergétique présents dans un véhicule hybride à pile à combustible.

La gestion d'énergie optimale pour les applications en temps réel et intégrant les contraintes dynamiques du système PAC.

La prise en compte de la durabilité de la pile à combustible dans la gestion d'énergie du véhicule.

En raison de ces objectifs, il est tout d'abord nécessaire de présenter une description des élements importants du VHPAC et de l'hybridation, puis d'élaborer des modèles de ces élements du véhicule adaptés à nos objectifs et à l'utilisation des méthodes d'optimisation pour la synthèse de la stratégie de gestion de l'énergie.

Ainsi, le type de VHPAC est mis en oeuvre présenté à travers de ses principales caractéristiques, puis les modèles et les méthodes d'optimisation utilisées dans les tâches de dimensionnement, de gestion d'énergie et de gestion de la durabilité sont présentés. Tandis que pendant la traction, la puissance électrique demandée est fournie par le système PAC et/ou batterie. Cette gestion de la puissance doit être faite dynamiquement pour proter au maximum de l'énergie de freinage et de la meilleure région de rendement du système PAC.

Une caractéristique importante de cette architecture vient du découplage entre la dynamique de puissance imposée par le véhicule et la dynamique du système PAC. Cet aspect permet à la PAC de travailler dans son meilleur domaine de performance tandis que les batteries donnent la composante dynamique de la puissance suivant des contraintes de taux maximales de charge et décharge et des états de charge minimals et maximals.

Concernant les fonctionnalités possibles pour ce type d'architecture, il y existe six diérentes fonctionnalités: µhybrid, mild-hybrid, full-hybrid, plug-in hybrid et range extender. Notre travail s'est concentré sur une fonctionnalité full hybrid du véhicule. Le full-hybrid apporte des modes de fonctionnement comme le start/stop, le power boost et en plus autorise que la puissance de traction soit fournie uniquement par la batterie.

Néanmoins, comme dans cette fonctionnalité la batterie ne peut pas être rechargé par le réseau électrique, il est imposé en mode d'assistance électrique où l'état d'énergie d'une batterie doit être le même (ou proche) au début et à la n d'un cycle de roulage. Anode : H 2 ←→ 2H + + 2e - Cathode : 1 2 O 2 + 2H La réponse en tension de la pile est fonction de la demande de courant et de ses conditions de fonctionnement en termes de ses variables d'état (pressions, température, stoechiométrie et humidité) (2).

U P AC = f (p cath , p an , St O 2 , St H 2 , T P AC , H r ) (2)
Par conséquent, le pilotage de ces variables a un eet important sur la réponse de la PAC, justiant l'intérêt de l'étude des sous systèmes autour de la PAC. L'architecture des sous systèmes retenus dans ce travail est :

xv Sous système hydrogène L'objectif principal de ce sous système est de contrôler et de réguler le débit (stoechiométrie St H 2 ) et la pression (p an ) de l'hydrogène dans la cellule.

Ceci se fait à partir du pilotage entre les deux valves présentes dans le circuit (g. 0.2). La gure (0.3) synthétise le processus de dimensionnement à partir des outils requis à chaque étape.

Sous système air

Modèles pour le dimensionnement de la chaîne de traction Le schéma de la gure (0.3) montre que pour le dimensionnement de la chaîne de traction, nous allons utiliser les spécications caractéristiques de performance stationaire et dynamique, ce qui nécessite des modèles directes du véhicule et de la chaîne de traction.

Dans notre cas, on appelle modèle direct des modèles qui ont à l'entrée la demande A partir de la puissance électrique demandée (P vehicule ) calculée par le modèle inverse, l'optimum du taux d'hybridation est determinée en utilisant un algorithme d'optimisation qui évalue la répartition de puissance optimale entre le système PAC et batterie. Le critère appliqué cherche à trouver la répartition qui minimise la consommation d'hydrogène pour un cycle de conduite.

Comme l'objectif de cette optimisation est dédié aux aspects énergétiques, et donc les modèles du système pile et batterie peuvent être simpliés sont simpliés (les dynamiques étant considérés comme négligeables) tout en conservent l'information de consommation énergétique.

Concernant l'algorithme d'optimisation, la méthode de la programmation dynamique [START_REF] Vinot | Optimal management of electric vehicles with hybrid storage system[END_REF][START_REF] Scordia | Approche systématique de l'optimisation du dimensionnement et de l'élaboration des lois de gestion d'énergie de véhicules hybrides[END_REF][START_REF] Christophe | Introduction à l'Optimisation[END_REF]

                                                                                                                                                                                                          
M inimisation du critère :

J (P * P AC ) = min P * P AC ´tf 0 g (P P AC (t)) .dt P * P AC = argminJ (P P AC (t))

Contraintes :

P batt min (-3C) ≤ P batt ≤ P batt max (10C) P P AC min ≤ P P AC ≤ P P AC max SoE min (40%) ≤ SoE ≤ SoE max (80%)

Condition limite :

SoE (0) = SoE (t f ) (4)
La résolution de ce problème vise à déterminer le prol de la puissance du système PAC P * P AC qui minimise le critère de consommation d'hydrogène g (P P AC (t)) sur un cycle de conduite. Néanmoins, la solution du problème doit prendre en compte certaines contraintes et conditions limites liées aux éléments et aux fonctionnalités du véhicule.

Par exemple, l'état énergétique (SoE -State of Energy) de la batterie doit respecter certaines limites et conditions représentatives du mode d'assistance électrique choisi comme SoE (0) = SoE (t f ).

Malgré le temps de calcul important, la quantité de mémoire nécessaire et le besoin de connaitre à priori le cycle, la programmation dynamique permet un calcul déterministe de l'optimum global, requis par l'approche de dimensionnement. Elle est donc adaptée à ce problème d'optimisation hors ligne. [START_REF] Christophe | Introduction à l'Optimisation[END_REF] est appliqué.

Le PMP fait partie des méthodes variationnelles utilisées pour optimiser un processus. En termes d'implémentation, la minimisation du critère J pour un système avec une dynamique ẋ = f (x (t) , u (t) , t) est faite en utilisant un co-état λ i pour chaque dynamique. Une représentation avec un Hamiltonien [START_REF]Technologie des voitures electrices et hybrides[END_REF]

Les mêmes considérations ne peuvent pas être faites pour le critère d'accélération maximale. Donc, l'équation dynamique complète du véhicule [START_REF]Toyota prius i-tech[END_REF] est résolue de façon numérique.

M dv(t) dt = F traction -F resistance

         F traction = k red .Γ traction max
R wheel (région couple max)

F traction = P traction v(t)
(région puissance max) [START_REF]Toyota prius i-tech[END_REF] Le couple maximal du moteur est obtenu à partir de la spécication liée au démarrage en pente et grâce à la résolution numérique de l'équation dynamique du véhicule et de la chaine de traction [START_REF]Toyota prius i-tech[END_REF].

A la n de cette étape, le dimensionnement de la chaîne de traction obtenu est vérié par rapport aux performances xées par le cahier des charges.

xxiii pile a combustible possède une réponse dynamique dépendant des interactions entre les sous-systèmes. Les diérentes de dynamique entre les sous-systèmes sont les causes principales de mécanismes de dégradations réversibles et irréversibles dans l'opération d'une pile à combustible. Ainsi, pour représenter la dynamique du système PAC dans la gestion d'énergie, nous avons retenu la dynamique la plus contraignante est retenue. Divers auteurs [START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF][START_REF] Gerard | Oxygen starvation analysis during air feeding faults inpemfc[END_REF][START_REF] Talj | Second order sliding mode control of the motocompressor of a pem fuel cell air feeding system, with experimental validadtion[END_REF] ont indiqué que le circuit d'air dans le système PAC peut impacter fortement le fonctionnement en termes de perte de performance et de dégradations irréversibles. La présence d'éléments de dynamique lente comme le compresseur et les non-linéarités du circuit complet rendent complexe la régulation des variables d'état an d'assurer le point de fonctionnement optimal. Par conséquent, dans ce travail, la dynamique du sous système d'air est choisie pour représenter la dynamique principale du système PAC. Cette dynamique peut être représentée par un système d'équations composé par les dynamiques du compresseur, de la cathode et de la vanne de régulation de pression [START_REF] Ahluwalia | Rapid self-start of polymer electrolyte fuel cell stacks from subfreezing temperatures[END_REF]. Les trois variables d'état de ce système sont la vitesse de rotation du moteur du compresseur, la pression à la sortie de la cathode et le déplacement du clapet de la vanne. Dans [START_REF] Ahluwalia | Rapid self-start of polymer electrolyte fuel cell stacks from subfreezing temperatures[END_REF] la pression aval du compresseur p sm et amont de la vanne peut être rapporée à la pression aval de la cathode p cath(out) en tenant compte des pertes de charges à l'intérieur de la cathode et de l'humidicateur. De plus, les débits d'oxygène q O 2 et d'eau q H 2 O dépendent directement du courant fourni par la pile. Par contre le modèle ne prend pas compte l'échange de matière à travers la membrane entre le compartiment cathodique et anodique.

                       dω dt = 1 J α.u mot -CpTatm ωηcp psm patm γ-1 γ -1 q cp (ω, p sm ) -ϕω dp cath(out) dt = RT cath V cath m molar q cp (ω, p sm ) -q O 2 + q H 2 O -q vanne p vanne(in) ẋvanne = 1 τvanne (-x vanne + k vanne u vanne ) (11)
La dynamique est donc non-linéaire, et son intégration dans le problème d'optimisation rend dicile le développement d'une loi de gestion d'énergie optimale pour des applications en temps réel.

Pour surmonter cette diculté, une commande non-linéaire est appliquée à ce circuit avec un double objectif: la stabilisation et la linéarisation des dynamiques. La technique de commande choisie ici est la commande plate [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF][START_REF] Fliess | A lie-backlund approach to equivalence and atness of nonlinear systems[END_REF]. Dans son implémentation, les sorties plates doivent être choisies pour rendre le système plat.

xxv Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés Un système est dit plat si ses commandes et ses états peuvent être représentés en fonction de sorties plates et leurs dérivées successives. Cependant, le choix de ses sorties n'est pas évidente sur un système non-linéaire car il n'existe pas de méthodes systématiques pour trouver les sorties plates. Dans notre cas, les deux sorties plates y 1 et y 2 ont étés choisies respectivement comme les états ω et p cath(out) . A travers de ce choix on arrive facilement à montrer que le troisième état x vanne est aussi dépendant des sorties plates et de leurs dérivées [START_REF] Akihiro | Drive force control of a parallelseries hybrid system[END_REF]. Il en est le même pour les commandes u mot (13) et u vanne [START_REF] André | The artemis european driving cycles for measuring car pollutant emissions. 7th international symposium, highway and urban pollution[END_REF].

x vanne = ẏ2 -f 4 (y 1 , y 2 , q O 2 , q H 2 O ) -f 6 (y 2 ) f 9 (y 2 ) (12)
u mot = W 1 + c 2 1 y d 1 y d 2 +f 1( y d 1 ,y d 2 )(c3) patm γ-1 γ -1 f comp y d 1 , y d 2 + c 4 y d 1 c 1 (13)
u vanne = f x vanne c 7 +        1 c 5 W 2 -ḟcomp + qO 2 -qH 2 O k hu ζ 1 (ζ 1 f x vanne + ζ 2 ) - [k 2 hu ζ 1 (ζ 1 fx vanne+ζ2) 3 +ζ 1( y d 2 -patm)] k 2 hu (ζ 1 fx vanne+ζ2) 4 -4[(ζ 1 fx vanne+ζ2) 2 (y d 2 -patm)]        (14)
Dans ces deux commandes W i sont les nouvelles commandes, y d i sont les trajectoires désirées de sorties et les débit q O 2 et q H 2 O sont considérés comme des perturbations connues. La fonction f x vanne dépend donc des sorties plates, de leaurs dérivées et des perturbations connues.

L'application de ces commandes du système permet de transformer la dynamique en [START_REF] Ahluwalia | Rapid self-start of polymer electrolyte fuel cell stacks from subfreezing temperatures[END_REF] dans un équivalent linéaire constitué d'une chaîne d'intégrateur comme en [START_REF] André | Driving pattern analysis and driving cycles -European Development of Hybrid Technology approaching Zero Emission Mobility (HYZEM)[END_REF]. Des commandes linéaires simples comme un retour d'état peuvent être alors utilisées pour rendre le système stable et robuste. P * batt (t) = λ 1 -(2a 1 (P vehicule (t))+a 2 )

                     Żd 1 = W 1 Żd 2 = Z d 3 Żd 3 = W 2 (15)
2a 1 ∂H ∂W 1 = 0 = λ 2 ∂H ∂W 2 = 0 = λ 3 (16)
A partir de cette formulation, on peut noter que la commande appliquée a simpliée la résolution du problème. Néanmoins, le co-état λ 1 reste indéterminé. De façon à surmonter cette diculté, une approche prédictive est proposée dans ce travail. Considérant que λ 1 est constant et faisant l'hypothèse que la distribution de puissance demandée par le véhicule dans une fenêtre de temps t h du passé est équivalente à la distribution du futur [START_REF] Bernard | Fuel cell hybrid vehicles: Global optimization based on optimal control theory[END_REF][START_REF] Bernard | Fuel ecient power management strategy for fuel cell hybrids powertrains[END_REF][START_REF] Kermani | Gestion energetique des vehicules hybrides: de la simulation a la commande temps reel[END_REF], la valeur de λ 1 peut être calculée par [START_REF] Assanis | An optimization approach to hybrid electric propulsion system design[END_REF].

λ 1 = 2a 1 t+t h
j=t (P batt (j)) -t+t h j=t P vehtot (j)

t h ∆t + a 2 (17)
Avec un changement de variable sur la somme t+t h j=t (P batt (j)) , le SoE apparaît dans le calcul de λ 1 [START_REF] Ates | Energy management of an fc/uc hybrid vehicular power system using combined neural networks-wavelet transform based strategy[END_REF].

λ 1 = 2a 1 ∆SoE desire .εmax ∆t -t-1 i=t-t h P vehicule (i) t h ∆t + a 2 (18)
∆SoE desire indique la diérence entre la valeur actuelle et la valeur de référence du SOE et cette que cette diérence doit être nulle à la n de la fenêtre de temps. An de proter au maximum de la récupération au freinage, la référence du SoE s'adapte en fonction de l'état d'énergie cinétique et potentielle du véhicule. L'idée principale vient du fait que quand le véhicule a une vitesse diérente de zéro, une partie de l'énergie cinétique peut être récupérée dans les phases de freinage. Par conséquent, dans cette situation le véhicule peut optimiser la récupération en augmentant le domaine du SoE à partir de la réduction de la référence SoE ref . Il en est de même concernant l'énergie potentielle (en fonction de l'altitude h). L'équations dans [START_REF] Baptista | Plug-in-hybrid fuel cell vehicle market penetration scenarios[END_REF] et [START_REF] Bento | La transition vers une economie de l'hydrogène: infrastructures et changement technique[END_REF]

SoE ref = SoE 0 -SoE cinetique -SoE potentielle (20)
Où SoE 0 est la valeur de référence à altitude et vitesse nulle xée à 60 %.

Ce développement a permis de créer une architecture de contrôle pour la gestion d'énergie (g. 0.7) qui permet d'intégrer facilement des autres contraintes, comme nous le montrerons dans la suite avec l'intrégration de la durabilité de la PAC dans la commande optimale. [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF] Les dégradations de la PAC reduisent la durée de vie de la PAC sont fortement liées aux conditions de fonctionnement imposées. Chaque mode de fonctionnement contribue à la dégradation de la PAC avec un taux spécique. Par exemple, des démarrages à froid (< 0°C) amènent certains phénomènes de dégradations irréversibles à l'intérieur de la PAC, modiant ses performances et sa réponse en tension. Cela indique que la contribution à la dégradation de chaque mode de fonctionnement est une information importante pour estimer l'état de santé de la PAC (SoH -State of Health). Pei et al. [START_REF] Pei | A quick evaluating method for automotive fuel cell lifetime[END_REF] ont obtenu des valeurs expérimentalement en mesurant les taux de dégradations pour chaque mode de fonctionnement (arrêt-démarrage, veille, puissance max, etc.). Ce type d'analyse expérimentale n'est pas pris en compte dans la démarche de ce travail. En eet la méthode proposée est basée sur les modèles physiques pour extraire les taux de dégradation de la PAC. A partir du travail réalisé au sein du CEA par Robin et al. [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF] montrant qualitativement la dégradation d'une PAC soumise à des conditions automobiles en utilisant des modèles à des échelles diérentes (système et réactions), l'observation de la surface active équivalente de la cathode (S) permet de donner le taux de dégradation de la PAC en fonction du taux de perte en surface active équivalente. La perte de surface active est due à la dissolution du catalyseur et est fonction des conditions locales de xxix Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés focntionnement de la couche active cathodique (pression, température, stoechiométries, humidité et courant) [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF].

A partir de cette étude, il a été possible de dénir le calcul de l'indicateur SoH en [START_REF] Bernard | Fuel cell hybrid vehicles: Global optimization based on optimal control theory[END_REF] et la dynamique de dégradation de la surface active équivalente en fonction de la puissance fournie par le système PAC (6.1) (g. (0.9)).

SoH (%) = 100.

S S tot

; S tot = surf ace totale initiale

⇒ P * batt (t) = λ 1 +(2(a 1 +λ 4 b 1 )(P vehicule (t))+(a 2 +λ 4 b 2 )) 2(a 1 +λ 4 b 1 ) ∂H ∂W 1 = 0 = λ 2 ∂H ∂W 2 = 0 = λ 3 (23)
Ici, le co-état λ 4 est lié à la contrainte sur le SoH, qui est respectéé à l'aide d'un contrôle par retour d'état. Dans ce travail, les méthodes de mésure du SoH ne sont pas développées, mais la valeur du SoH est considérée comme disponible à tout instant. Ainsi, λ 4 est calculé par l'équation [START_REF] Blurnier | Modelisation de moto-compresseurs en vue de la gestion de l air dans les systemes pile a combustible -simulation et validation experimentale[END_REF], où f durabilité est une fonction qui évite le rapprochement du SoH de sa trajectoire limite SOH limite (g. (0.11)). Le dimensionnement optimal des éléments de la chaine énergétique présents dans un véhicule hybride à pile à combustible. During long time petroleum has been considered an abundant raw material and low cost. However, the dramatic spread of transport throughout in the twentieth century has resulted in increased oil consumption. This trend may not be sustainable in the medium and long term. On the one hand, fossil fuels need million of years to replenish stocks: their production is limited by the physical provision of the planet as well as the possibility of extraction according to technical and economic conditions. On the other hand, the burning of fossil fuels releases large quantities of polluting gases that reinforce the problem of climate change (eg carbon dioxide) or increase the pollution of atmospheric air (eg oxide nitrogen, carbon monoxide).

λ 4 = f durabilité (SoH -SoH limite ) = k 1 e k2( SoH(t)) -1 (24)
In terms of energy security, a fossil fuel economy presents weaknesses, such as the instability that has been a feature of the oil industry from its inception [START_REF] Bento | La transition vers une economie de l'hydrogène: infrastructures et changement technique[END_REF]. In fact, the competition between producers to recover the largest possible amount of oil and the arrival of new producers on the market did not allow optimum rate of production, resulting in overproduction crises. A direct eect of this instability in production are the occurrence of sharp increases in the price of petroleum [START_REF] Joni | World energy outlook 2010[END_REF], which are explained by the combined eect of the increase in consumption and supply constraints, and amplied by the action of speculation. Besides, the production becomes more expensive when new elds, where production is more expensive, replace the old ones that arrived at the end-of-life.

As reservoirs of petroleum are increasingly rare and concentrated in some geographic regions, a situation of dependency is created [START_REF] Onu | World population to 2300[END_REF]. Hence, the consumer countries are in a socio-political vulnerable position in relation to the producer countries, possibly causing geopolitical tensions at international level.

Addressing the fossil fuel economy to the application level, in particular to the transport sector, it can be considered the largest consumer of petroleum products. In terms of energy it accounts 27 % of global energy demand, of which almost all is derived from fossil fuels [START_REF] Joni | World energy outlook 2010[END_REF]. So, if nothing is done to counteract this trend, the growth in transport demand would necessarily lead to increased oil imports and emissions. Meanwhile, global demand for automotives should be tripled by 2050 due to the increase in developing countries [START_REF] Gargett | Transport energy futures: long-term oil supply trends and projections[END_REF][START_REF] Bento | La transition vers une economie de l'hydrogène: infrastructures et changement technique[END_REF]. The continuation of this trend may not be sustainable given the existing resources and environmental impacts. Thus, it is essential that the inevitable increase in the global car eet is oset by a sharp decline in consumption and vehicle emissions.

To deal with the energy demanded and reduction of emissions of greenhouse gases, major technological innovations in vehicles are needed (conception, engine, structure, etc.). Low improvements in the actual technology (ICE vehicles) are not sucient to meet the emissions objectives and to deal with the demand of energy created by the increasing number of vehicles [START_REF] Mcdowall | Technology roadmaps for transition management: The case of hydrogen energy[END_REF]. Technological changes are necessary to deal with this context and among the existing alternatives three appear more adapted: electricity, biofuels and hydrogen [START_REF] Bento | La transition vers une economie de l'hydrogène: infrastructures et changement technique[END_REF].

Technologies as the biofuel for the vehicular applications represent a valid option in the task of fossil fuel replacement, as veried in some parts of the world [START_REF] Sales Cardoso | Enviromental and economic assessment of international ethanol trade options for the german trasnport sector[END_REF][START_REF] Bento | La transition vers une economie de l'hydrogène: infrastructures et changement technique[END_REF]. However, since there is an uncertainty if the production of biofuels will be available for passenger cars, given the potential demand from other sectors, such as aviation, marine, and heavy industry, it is vital to develop a range of sustainable mobility technologies over the long term.

Nowadays, the battery electric vehicles become a reality in terms of the replacement of the conventional power train using internal combustion engine (ICE). The electrication of the vehicles is considered the simplest solution to achieve the emissions goals [START_REF] Onu | World population to 2300[END_REF]. Moreover, compared with the conventional power train, it presents an outstanding weaknesses [START_REF] Chan | Powering sustainable mobility: Roadmaps of electric, hybrid and fuel cell vehicles[END_REF]. Major shortcomings of this alternative are attributed to the energy storage: low capacity, high cost, long charging time, small operating temperature range and low cycling stability [START_REF] Chan | The state of the art of electric, hybrid and fuel cell vehicles[END_REF]. These drawbacks limited their wider use for propulsion of passenger vehicles.

Hydrogen is considered as a well adapted solution to the energetic and environmental challenges. As the electricity, the hydrogen is an energetic vector, which oers multiple advantages when compared to primary energy sources, such as the petrol. The hydrogen has a technical potential to compete with the fossil energy solutions in dierent applications (transport, stationnary applications, power grid storage); however, it is in the transport domain that this energy vector can provide better results, especially when it is part of a competitive of hydrogen economy concept.

Hydrogen economy in the transportation context

Dened rst by the electrochemist John Bockris [START_REF] Bento | La transition vers une economie de l'hydrogène: infrastructures et changement technique[END_REF], the hydrogen economy concept has been continuously developed by dierent authors in order to present an possible context of massive hydrogen adoption.

Since its inception, such idea was always related to the sustainable and non-polluting objectives. Therefore, even if the fossil fuels can be used to produce hydrogen, renewable energies have a main role in the hydrogen economy.

The ideal representation of the hydrogen economy is illustrated in g. 1.1. As can be noticed, the hydrogen and electricity are carriers operating between the energy sources and applications. On the other side, this energy transportation produces non-harmful eects in the environment by the consumption and production of oxygen and water.

Dierently from a typical combustion application, there is a presence of sustainable cyclic exchange without pollutant emissions. In this way, the hydrogen participates to the decarbonization process of the energetic system, meaning a reduction of carbon emissions per energy unit.

Useful Energy Applications

Hydrogen Utilization for hydrogen production and distribution, such fuel cell vehicle has been considered one of the best options to reach the assigned decarbonization targets [START_REF] Mcdowall | Technology roadmaps for transition management: The case of hydrogen energy[END_REF][START_REF] Wipke | National fuel cell vehicle learning demonstration fuinal report[END_REF].

Fuel Cell Vehicle: advantages, challenges and motivations

Compared with ICE vehicles, a power train powered by a fuel cell stack oers many advantages related to its energy eciency, clean and fuel exible characteristics. An eciency peak of 60 % can be reached by FC with a conversion of hydrogen in traction power and producing only water [START_REF] Erdinc | Recent trends in pem fuel cell-powered hybrid systems: Investigation of application areas, design architectures and energy management approaches[END_REF][START_REF] Gargeth | Assessing fuel cell vehicle and the role of policy in japan, korea and china[END_REF]. Besides, as aforementioned, the hydrogen used can be produced from a variety of sources including fossil fuels such as natural gas, renewable such as solar and wind power, biomass and nuclear energy [START_REF] Hwang | Sustainability of hydrogen pathways for fuel cell vehicle applications[END_REF].

Despite all the encouraging reasons, the fuel cell vehicle technology still faces some limitations. Out of all the limiting factors, cost, hydrogen storage technology, fuel cell system dynamics and durability are considered by the automotive constructors as the most challenging factors.

These challenges should be overcome to render the hydrogen fuel cell vehicle a competitive option in the market. Cost and hydrogen storage has been subjected to continuous developments and presented signicant progress towards commercialization target values [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF]. Nevertheless, there is a consensus that the slow dynamics and durability require a special attention in order to reach the stated objectives. Hence, these drawbacks have been dealt using dierent approaches such as technological developments in the elements constructions or in their integration procedure.

The hybridization approach applied to the ICE vehicles is another frequent solution proposed. Such hybridization using batteries would permits to compensate the slow dynamics of the fuel cell system, reduce the degradation eects and aggregate the possibility to obtain a reduction in energy consumption. Even though the extra degree of freedom brings advantages to the fuel cell vehicle, the hybrid conguration demands a more elaborated control strategy regarding all these criteria and an optimum sizing of the hybridization rate.

Represented by an energy management strategy, the power split between the fuel cell system and the energy storage elements (batteries) could be designed using several methods and choosing dierent objectives. Although it exist dierent methods proposed, the energy management strategy is considered as an open subject yet.

Believing that it is a coherent alternative and very well adapted to the needs of the transportation sector, this work is developed around the hydrogen energetic alternative through the adoption of the fuel cell vehicles and dealing with their main technical challenges.

Objectives

Being motivated by the aforementioned limitations, this work intends to make a research contribution in order to nd out a solution for each challenge. Specically, this work aims to include the fuel cell system dynamics and durability aspects in an energy management strategy. It requires an approach that deals with phenomena and features presented in reaction, system and vehicle levels. In order to achieve this objective, control strategies allied with models of dierent scales should be developed and applied. Moreover, this thesis considers that the fuel cell vehicle sizing is also related to the mentioned challenging requirements; therefore, the development of a sizing procedure adapted to the needs composes as well the proposed objectives. Figure 1.2 illustrates these objectives and the outline adopted to meet them. The developments are based in three levels associated to the vehicle, fuel cell system and fuel cell, where each one is responsible for a set of specic tasks:

• Vehicle level: sizing and energy management strategy of the vehicle.

• Fuel cell system level: denition of the dynamics and the control laws to regulate the system.

• Fuel cell level: denition of the degradation dynamics analyzing the phenomena at the electrochemical reaction scale.

An information chain allows to transport the information from the vehicle level down to the fuel cell level and feedback. It starts with the assignment of the automobile conditions to the fuel cell system, which results in the denition of the fuel cell dynamics and the local conditions within the fuel cell. These local conditions are imposed to the fuel cell level, returning the degradation rate that combined with the FC system dynamics. They compose the constraints of the energy management strategy. In this manner, the dierent application levels are brought to the vehicle level in order to solve the energy management strategy problem.

The developments and background knowledge present in this process are described through ve main chapters.

Overview of the chapters

Chapter 2: Fuel Cell Electric Vehicles The rst chapter presents the hybrid fuel cell vehicles and its characteristics. Initially, a state of art is introduced to clarify the advances in the technologies, the commercialization requirements and the remaining issues to be overcome. Thereafter, possible architectures are described and discussed aiming to dene the one to be adopted. In sequel, the basic vehicle modeling concepts are presented, comprising the losses and its longitudinal dynamics. This chapter also presents a brief state of art about the battery technologies and the most adopted types of models for this work. The fuel cell and its system are rstly introduced by a description of its operating principles, emphasizing the interactions among the fuel cell and the other elements composing the system. Due to the importance of these interactions, the main characteristics of each subsystem are described briey with a focus on the eects of their state variables in the fuel cell stack behavior.

Chapter 3: Models and Optimization tools The second chapter is related to the description of the tools used in this work. Due to the presence of several elements and dierent application levels (vehicle, system and reactions), a set of dierent models are necessary to study and observe the important phenomena involved in the objectives of this work. Therefore, the rst part of the chapter is devoted to the denition of the modeling tools and the models used. Each model is presented according to its desired application, which means that the link concerning models features and applications is highlighted. The other part of this chapter is about the optimization tools used in the energy management development and the optimum sizing. A state of art about this subject aims to show an overview of the methods used so far. Among the presented methods, some of them are chosen to be used in this work; thereby, they are described by their main characteristics and assigned as the reference tools for the energy management task.

Chapter 4: Fuel Cell Hybrid Electric Sizing The third chapter deals with the sizing of the Fuel Cell Hybrid Electric Vehicle (FCHEV). Intending to obtain an optimal sizing, a systematic sizing procedure is built to dene the dimensions of the main elements in the powertrain and in the hybrid conguration. Initially, a reference vehicle is chosen and it's functional and performance characteristics are assigned. Combining adapted technical specications, optimization approach, population mobility aspects and economical study, the dimension of the vehicle's elements are obtained, providing a commitment between dierent criterions, such as consumption, performance and cost.

Chapter 5: Online energy management strategy In this chapter is proposed an optimal energy management strategy oriented for online applications. Prior to the developing of such strategy, the dynamics of the fuel cell system is discussed in order to prepare its inclusion in the optimization problem. A control strategy is applied to the dened dynamics, leading to a regulation of the state variables and a simplication of the dynamics structure. Afterwards, the online energy management strategy is dened using the optimal theory and concepts such as the predictive control to overcome some typical drawbacks presented in this task. The proposed energy management strategy is validated by simulations using dynamic models and experimental tests using real systems (fuel cell system and battery), electric motor and a motor test bench.

Chapter 6: Fuel cell durability The study of the fuel cell durability, in the context of the FCV energy management, is presented in this chapter. Through a denition of the fuel cell end of life and a state of art about the degradation mechanism and diagnostics methods, the chapter introduces the necessary background to implement the durability in the optimal power split strategy. In the sequence of the chapter, a durability simplied model is presented with the assumptions made and the main characteristics, which renders it adapted to the optimization task. Such model determines the degradation dynamics to be used as a constraint of the optimization problem. The strategy proposed in chapter 4 is adapted to include the durability aspects in the optimization process. Simulation results are also presented.

Summary of the contributions

Motivated by the challenges presented above, the work can yield interesting results for the advancement in the technology of the fuel cell vehicles. Indirectly, models and tools that are created in this work enabled the development of techniques and the understanding of phenomena and characteristics related to dierent parts of the vehicle and the fuel cell system. Meanwhile, the direct contributions resulting from this work concentrate their eorts on the three main topics contained in the established objectives:

• Optimal sizing

• Optimal energy management

• Fuel cell system durability

The chapters dealing with the respective subjects (chapters 4, 5 and 6) describe, in a dierent manner, the contributions that were also presented through publications in scientic journals and conferences.

Regarding the sizing of the fuel cell vehicle, the systematic approach and especially the optimization feature embedded in the process constituted the content of a national communication [START_REF] Da Fonseca | Dimensionnement optimal des elements energetiques d'un vehicule hybride a pile a combustible[END_REF]. Specically, the denition of the optimal hybridization rate between the fuel cell system and battery represented the major contribution of this publication.

The basis in terms of development and knowledge brought by the optimization tools and models were of fundamental importance in this task.

As the central objective of this work, the energy management was addressed in several ways and at dierent stages. Initially, a energy management strategy based on the responses from global optimization methods was the subject of a scientic paper [START_REF] Nai Da Fonseca | Hybrid electric system for an hydrogen fuel cell vehicle and its energy management[END_REF],

which was presented at the SAE world conference exhibition. This strategy combined the optimal characteristics of the o-line optimization methods results to a complementary aspect that added an expert content (rule based) to the solution.

Despite the discovery of this strategy, the solution of the energy management problem converged in direction to the development of a method that allowed online applications with low computation time, with a performance close to the global optimization methods and integrating the dynamic response of the fuel cell system. This evolution is evidenced by a set of publications and a patent led.

The scientic articles [START_REF] Nai Da Fonseca | Flatness control strategy for the air subsystem of a hydrogen fuel cell system[END_REF][START_REF] Nai Da Fonseca | Control of pemfc system air group using dierential atness approach: validation by a dynamic fuel cell system model[END_REF] dealt with a necessary stage in the optimization process that is the control of the fuel cell system dynamics. The application of a nonlinear control method leads to a stabilization of the system dynamic behavior (veried by simulation and by tests in real platforms) and simplify its inclusion in the energy management strategy.

Subsequently, the conception of an energy management strategy adapted to be implemented in fuel cell vehicles was performed and tested. Given the innovative character of the solution, its content was protected through the elaboration of a patent that has already been accepted. Such protection created for the patent initially limited the possible communications, but part of its content has already been published [START_REF] Da Fonseca | Energy management strategy for hybrid fuel cell vehicle[END_REF] in an international conference and must produce some future publications and scientic articles.

Finally, the aim to include the information of durability of the fuel cell system in the energy management task generated the synthesis of a new energy management strategy.

Their results constituted the basis for the elaboration of a communication [START_REF] Da Fonseca | Hybrid fuel cell vehicle energy management includ-ing dynamic and durability constraints[END_REF] presented at international conference and still allow the development of another article.

Fuel cell electric vehicles: State of the Art

As electric cars try to forge more than just a niche in the market, the automotive industry is already looking to another form of clean technology that could overtake today's batterypowered vehicles.

Commitments by automobile manufactures and various government agencies to develop hydrogen fuel-cell cars have surged in recent times. Applications in which these vehicles are involved include personal vehicles, eet vehicles (for municipal and commercial use), transit buses, short-haul trucks (such as delivery trucks and drayage trucks for port facilities), and others [START_REF]Hydrogen energy and fuel cells[END_REF][START_REF] Hwang | Sustainability of hydrogen pathways for fuel cell vehicle applications[END_REF][START_REF]The fuel cell industry review[END_REF][START_REF] Gargeth | Assessing fuel cell vehicle and the role of policy in japan, korea and china[END_REF]. In the case of light-duty vehicles auto makers like Ford, Hyundai, Daimler and others had already announced plans to build vehicles and prototypes that run on hydrogen. As an example, Ford, Daimler and the Renault-Nissan

Alliance sign an agreement to accelerate the development of fuel-cell electric-vehicle technology with a goal of launching the world's rst aordable mass-market FCEV as early as 2017.

Ford already presented some years ago (2007) its HySeries Drive concept that associates batteries and fuel cells in a hybrid conguration to be used in its Edge hydrogen version (gure 2.1). This prototype represented a solution where the fuel cell has the function recharge the battery pack (range extender), not eecting the delivery of power directly to the load (power train). The combined autonomy (battery + hydrogen) allows to reach up to 490 km with a speed up to 135 km/h [START_REF] Fcev | [END_REF]. state, some builders are experiencing to pass from the prototyping in direction to a initial production (1000 cars by Daimler/Ford/Nissan) [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF][START_REF] Erdinc | Recent trends in pem fuel cell-powered hybrid systems: Investigation of application areas, design architectures and energy management approaches[END_REF]. The move within the automotive industry to establish plans to mass produce fuel-cell cars appears to be gaining traction due to the positive perspectives in terms of the acceptation of this technology [START_REF] Baptista | Plug-in-hybrid fuel cell vehicle market penetration scenarios[END_REF]. Figure 2.3 shows the trend for the UK market [START_REF]Uk h2 mobility. synopsis of phase 1 results[END_REF]. As the structure of hydrogen refueling stations (HRS) and the fuel cell technology increases in maturity level, the market tends to absorb more of this technology as an available option to the conventional ICE vehicles.

Figure 2.3: FCEV perspective sales at UK market [START_REF]Uk h2 mobility. synopsis of phase 1 results[END_REF] Relying on this trend, some constructors are closer to the production level. Hyundai has announced that it will oer a fuel-cell version of its ix35 sport utility vehicle (gure 2.4) on lease by the end of this year. Announcing a range of 588 km between charges and performance consistent with the conventional model, it plans to make up to 1,000 fuel-cell cars by 2015 and thereafter 10,000 fuel-cell cars per year [START_REF]Performance hyundai[END_REF][START_REF] Hyundai Fcev | [END_REF]. The established starting date of commercialization reects the evolution of the vehicles technology, which means that many advances have already been made since the early versions, but there are remaining challenges to be overcome until the estimated time.

The Department of Energy of the United States (DOE) [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF] lists the main challenges to the adoption of hydrogen mobility as the following categories: Since the context of this work is focused on technological aspects, economical and institutional aspects will not be presented.

Hydrogen cost The cost of hydrogen is dened by the form of production and delivery.

The production of hydrogen can be done in dierent ways; obviously the main objective at long term is to make the production based on the use of renewable or low carbon resources.

Currently, the production source that proves to be more competitive in relation to petrol is through the natural gas reforming method [START_REF]The fuel cell industry review[END_REF][START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF]. Adopting this method, projections of high production indicates a possible cost near to $ 3 per gallon of gasoline equivalent (gge, roughly equal to 1 kg of hydrogen), representing a competitive value in the market.

However, the obtaining of competitive costs using renewable sources without carbon emission is still an open problem. Projections of high productions presents a cost of $4.905.70/kg at the refueling site when using water electrolysis method [START_REF] Hwang | Sustainability of hydrogen pathways for fuel cell vehicle applications[END_REF][START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF].

The cost of production using wind-powered water electrolysis at a centralized plant is $2.703.50/kg, but the costs associated to the delivery are not included.

Concerning the technologies for delivering, the current advances in technology allow to obtain a projection of a value approximately $2 to $3 per gge (delivering hydrogen from a central production facility to a fueling station 60 miles away) [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF]. These values are subject to necessary infrastructures, resulting in investments and maintenance costs. Hydrogen storage, capacity and cost One of the biggest challenges for hydrogen FCEV technology is to store a sucient amount of hydrogen on board the vehicle for an acceptable range in a practical amount of space. There are dierent technologies for hydrogen storage: modest-pressure gas (350 bar), high-pressure gas (700 bar), cryocompressed hydrogen (CcH2), and material-based storage elements [START_REF] Paster | Hydrogen storage technology options for fuel cell vehicles: Well-towheel costs, energy eciencies and greenhouse gas emissions[END_REF].

Since possible leaks by diusion are controlled (that is already achieved with current technology), the procedure to store hydrogen under pressure allows to deal with the inconvenient low volumetric energy density of hydrogen. In this process, the increased pressure leads to a gain in energy density, to the detriment of energy consumption due to compression.

Cryo-compressed hydrogen storage refers to the storage of hydrogen at cryogenic temperatures in a vessel that can be pressurized (nominally to 250-350 atm), in contrast to current cryogenic vessels that store liquid hydrogen at near-ambient pressures. Cryocompressed hydrogen storage can include liquid hydrogen, cold compressed hydrogen, or hydrogen in a two-phase region (saturated liquid and vapor). Even having achieved a technological advancement state consistent with the needs of the process, the manufactured system costs is a barrier to be overcome [START_REF]Tecnical assessment of cryo-compressed hydrogen storage tank systems for automotive applications[END_REF]. State of current technology shows a cost twice that required for the purpose of commercialization, even assuming a large-scale production.

Materials-based approaches using ammonia borane, hydrides, amides, composite ma-terials, metal-organic frameworks, organic molecules, etc. are being explored extensively.

The hydrogen is included in these materials and by dierent processes it is possible to separate it from its carrier. Materials-based approaches are divided in three classes [START_REF] Paster | Hydrogen storage technology options for fuel cell vehicles: Well-towheel costs, energy eciencies and greenhouse gas emissions[END_REF]:

• The compounds of with high specic surface

• Reversible chemical hydrides

• The rechargeable metallic or intermetallic hydrides

The metallic hydride approach is the most advanced one, allowing to store from 10 kg to 70 kg of hydrogen in 1 kg of metallic hydride, corresponding from 0. Among the mentioned storage ways, high-pressure tanks (700 bar) are the most adopted so far. This storage method has been submitted to some improvements beyond the materials level to get the acceptance of the auto makers. By the system optimization, it could be increased the onboard storage capacity of hydrogen, which allowed to some demonstration and prototype vehicles achieve driving ranges as high as 690 km on a single ll, representing more than the DOE goal (500 km). Despite the goal has been already achieved, it must be applicable for dierent vehicle platforms, without compromising space and performance, and at a competitive cost, which leaves open the problem.

Fuel cell cost, durability and performance Since the rst major application in the U.S. spatial program in the sixties, the fuel cell technology had a remarkable evolution.

Specically in the application of vehicles, the use of this power converter element opened the prospects to a transport that is ecient, clean, accessible and adapted to current personal uses.

The state of development achieved by the fuel cells embedded in the vehicles has reached a level of improvement that allows to obtain eciencies of up to 60%, accounting more than twice of the maximum eciency of a current internal combustion engine.

Nevertheless, in order to be ready for commercialization, the eciency presented must be followed with a cost, durability and performance of the vehicle consistent with the constrains of the transport sector. (500,000 units per year) an for 80 kW fuel cell systems [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF] Durability In order to be adapted to the current needs of consumers, a fuel cell vehicle must have a lifetime consistent with current technologies on the market. According to what was established by DOE [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF], a minimum of 5000 hours (equivalent to 150,000 miles) of operation shall be provided before the end of its useful life, dened through a degradation value of 10 % in the eciency.

Even if the criterion of end of life is not clear to dynamic applications, the current status of durability presents only half the stated goal (2500 hours, 75,000 miles). Despite this dierence, there has been a signicant evolution in the direction of the DOE goal.

Due to investment in research of new materials and diagnosis, it could achieve a level of durability that validates the rst phase of the DOE goals (2000 hours). Therefore, in order to overcome the 50% remaining dierence, eorts should be continued in the fuel cell conception and optimal management of system.

Performance Despite the good eciency aforementioned, fuel cells still present some performance limitations when it is submitted to some particular operational conditions presented in the transport domain. Cold start operations, environmental condition that impose temperatures of -40°C Ahluwalia and Wang [START_REF] Ahluwalia | Rapid self-start of polymer electrolyte fuel cell stacks from subfreezing temperatures[END_REF] to 50°C or dynamical responses required by the vehicle are clear examples of situations where the performance of the fuel cell is limited. As an element that is inserted in a system, the coordination of the subsystems should be well conducted to not result in an acceleration of the degradation status, when such situations happen.

Besides, despite the fairly homogeneous behavior in terms of eciency, it is not possible to ensure such good eciency during an automobile application. If subjected to a demand that it imposes a point of low operating eciency (very low power and high power), the advantages compared to a conventional vehicle are reduced. Therefore, a hybrid character is usually inserted into the conception of the fuel cell vehicle. Using an element of energy storage it is possible to select the operating points of the stack to ensure good eciency and at the same time the dynamical demands of an automotive application [START_REF] Chan | The state of the art of electric, hybrid and fuel cell vehicles[END_REF][START_REF] Erdinc | Recent trends in pem fuel cell-powered hybrid systems: Investigation of application areas, design architectures and energy management approaches[END_REF], compensating the slow dynamics of the fuel cell system.

The inclusion of a new element in the energetic chain of the vehicle implies the need for an adoption of hybrid architecture. This topic is the subject of the next section.

Vehicle architectures

The demand for new types of powertrains that are more ecient and reduce the emission of pollutants of vehicles caused an evolution in the technology of electric motors.

This technological evolution has led to obtain models that feature all the dynamic performances of a conventional engine coupled to a much higher eciency, which drove the acceptance of electric vehicles in the market. However, due to autonomy limitations of pure electrical vehicle, solutions hybridizing conventional engine (ICE) and electrical motors were necessary to ll this gap. Although of the real gains in terms of fuel economy and emissions reduction presented by this type of hybridization, the long-term goal for new technologies is to completely eliminate local emissions or reducing them to values near zero [START_REF] Onu | World population to 2300[END_REF][START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF][START_REF]The fuel cell industry review[END_REF], qualifying this type of solution as intermediate. This not denitive character will allow the transition from the conventional form of transportation (based on the use of petroleum) to a transportation that adopts energy coming from renewable sources. Hydrogen fuel cell vehicles make part of the sustainable solutions group. This session will present the most adopted architectures, the denition of the chosen architecture and the classication of hybridization functionalities.

Main architectures

The following topologies represent the most adopted technologies [START_REF] Emadi | Topological overview of hybrid electric and fuel cell vehicular power systems architectures and congurations[END_REF][START_REF] Florescu | Gestion optimise des ux energetiques dans le vehicule electrique[END_REF][START_REF] Montaru | Contribuition a l evaluation du vieillissement des batteries de puissance utilisees dans les vehicules hybrides selon leurs usages[END_REF]:

• Electric vehicle architecture

• Series hybrid architecture

• Parallel hybrid architecture

• Series-parallel hybrid architecture (power split)

Such architectures are presented in gure 2.8. characteristics (electric powertrain and batteries). In addition to the good eciency in the powertrain compared to a conventional vehicle, an electric vehicle has the advantage of its reversibility in terms of the electric motor. This allows energy recovering during braking processes, which will be stored in the battery and can be used later, increasing the energy eciency of the vehicle.

Such energy recovering functionality is desired for other type of vehicles. Therefore, hybrid architectures seek to add this functionality to vehicles through the insertion of energy storage elements and energy conversion elements that are reversibles, such as electric motors and batteries.

Series hybrid architecture In series hybrid architecture only one traction element is commonly used. During the braking phase, the mechanical power for braking is converted to electrical power through the electric motor and then absorbed by the battery.

During the traction, the demanded electrical power can come from the not reversible source and/or from the energy storage element (batteries). This power split should be done dynamically in order to take advantage of the best eciency region of the non reversible source. Thus, this architecture allows a decoupling between the dynamic behavior imposed by the vehicle dynamics and the dynamics of the not reversible source, which is done through the energy storage elements. This decoupling allows the irreversible source to have a stabilized behavior around its best eciency point while the battery is responsible for ensuring the power dynamics of the vehicle. When many energy conversion elements are present in the powertrain, this architecture has the disadvantage of putting in series their eciencies, reducing the overall eciency of the vehicle. In the case of a series hybrid vehicle using the internal combustion engine and batteries, successive conversions between mechanical and electrical power degrade the eciency of the vehicle.

Parallel hybrid architecture In parallel conguration, two motors with dierent features are used and connected directly to the transmission. They can participate together of the vehicle traction, acting directly on the wheels. In order to perform this, a system of transmission and gearbox are necessary for the power transmission. For a same hybrid application using a combustion engine, this architecture presents an overall eciency greater than the series architecture. However, parallel conguration presents major disadvantages, such as the complexity of control and rigid connection between the combustion engine and the wheels.

Series-parallel architecture Also referred as power split, this conguration combines the advantages of series and parallel congurations. Through an eective control, the regime of operation of the combustion engine is regulated in order to make it work at its best eciency point. The power generated is split between traction and recharge of the storage element. Since this architecture has more elements, the system control and command becomes more complex.

Architecture for a fuel cell vehicle

In current applications of fuel cell vehicles the series architecture is usually adopted.

The good performance of this conversion element allows to deal with the inconvenience of the eciency serialization without compromising too much the overall eciency of the vehicle. Allied to this, the simple control necessary for this architecture enables a denition of power split that favors the good performance of the fuel cell.

Regarding the possible degrees of freedom to control the power split, the topology used can contain zero, one or two electric DC/DC converters. Being a system with not reversible characteristics, the fuel cell system generally needs an irreversible interface to perform its conditioning and security. However, the energy storage system should have reversible characteristics and consequently may or may not be interfaced with the electric bus through DC/DC converters. The absence of this element would impose a oating voltage level on the bus. Therefore, depending on the chosen storing element, the bus and the elements connected to it may be subject to voltage uctuations.

Adopting batteries used in the new electric vehicles as the storage element, the voltage uctuations are tolerated by equipment connected to the bus (motors and auxiliary elements of the vehicle). From the point of view of eciency gains and control, the inclusion of another degree of freedom (as a battery interface) do not presents enough advantages to justify its adoption, according to Trischiler et al. [START_REF] Tritschler | Optimisation de l architecture electrique et gestion d energie pour un systeme a pile a combustible embarquee dedie a l application agricole[END_REF].

In this work, series hybrid architecture will be adopted for an electric fuel cell vehicle, where there will be only one DC/DC converter for interface and security of the stack. Figure 2.9 shows the architecture adopted in this work.

Hybrid functionalities

The hybridization has dierent levels of functionality. According to Chan et al. [START_REF] Chan | Electric, hybrid and fuel-cell vehicles: Architectures and modeling[END_REF], they can be classied as a function of the relation between the power of the non reversible source and the energy storage element, which are respectively the fuel cell system and batteries for this application.

1. Micro Hybrid: The fuel cell system is responsible for the propulsion power while batteries have an auxiliary function in startup operations and stop-and-go, preventing the fuel cell to be subject to unfavorable conditions during starting procedure and allowing its stop during standstill situations.

2. Mild Hybrid: Adds the power boost mode to the stop-and-go function. The battery provides power during peak power because of the fuel cell system delay. Besides, in this hybrid level, during braking the vehicle's kinetic energy can be converted into electric used to recharge the battery. However, the battery is not authorized to deliver alone the power to propel the vehicle.

3. Full Hybrid: In this case, beyond the features of the other modes, the battery is able and authorized to provide alone the power for propulsion of the vehicle.

Nevertheless, the battery can not be recharged externally and hydrogen used in the fuel cell should represent the total energy expended by vehicle in its mission. This means that, although it can provide power for propulsion, the battery has function of electric assistance. Therefore, at the end of its mission the state of charge should be near to its initial value. This characteristic congures a mode called charge sustaining. Performing a good strategy for the power split between fuel cell and battery, it is possible to obtain important reductions in consumption.

Plug-in Hybrid:

This mode is present when the battery can be recharged through the electric grid. In the vehicle, the battery assumes the source function and not only reversible storage element. The energy management strategy changes in order to permits the battery to reach the end of the mission with depleted state, dening the charge depleting mode. This hybridization mode enables to adopt a strategy close to an electric vehicle behavior (with a range extender) or more elaborated ones that associates the two sources with more complex objectives.

In order to be able to propose more elaborated energy management strategies, only the full hybrid and plug-in hybrid modes will be tackled.

Elements of the FCEV

This session will be destined to the description of the main elements that compose the selected architecture:

Vehicle

For the analysis of energy consumption purposes, the vehicle may be considered through its longitudinal kinematics and dynamics [START_REF] Trigui | Systemic modelling of hybrid vehicles in order to predict dynamic performance and energy consumption-building the vehlib library of models[END_REF][START_REF] Kermani | Gestion energetique des vehicules hybrides: de la simulation a la commande temps reel[END_REF][START_REF] Florescu | Gestion optimise des ux energetiques dans le vehicule electrique[END_REF]. The movement of the vehicle is obtained as a result of forces acting on it. F traction (t) is the traction force resulted from the torque imposed by the traction train Γ wheel (t) through the wheels. Considering that the losses in slippering are negligible, this transformation is represented by the equation 2.1 (R wheel is the wheel radius). In terms of kinematic variables, equation 2.2 shows the transformation of the angular speed of the wheels ω wheel (t) in linear speed v (t).

F traction (t) = Γ wheel (t) R wheel (2.1) v (t) = ω wheel (t) R wheel (2.2)
Performing the balance between traction force and the movement resistance resultant F resistance , it is obtained the acceleration of the vehicle via the fundamental principle of dynamics (equation 2.3). Where M is the total mass of the vehicle.

dv dt = F traction -F resistance M (2.
3)

The resistive force F resistance is mainly composed of three components (equation 2.4): aerodynamic resistance F aero , rolling resistance F roll and resistance due to the route prole F slope (slope prole).

F resistance = F aero + F roll + F slope (2.4)
Aerodynamic resistance F aero It is the force exerted by the wind according to the vehicle axis of movement. It is dependent on air density ρ air , front surface S, air penetration coecient of the vehicle C x and the sum of the square of the speed of the vehicle v and the relative velocity of the air v air . The equation 2.5 denes F aero .

F aero = 1 2 .ρ air .S.C x . (v + v air ) 2 (2.5)
Rolling resistance F roll It represents the resistant eort due to the contact between the tire and the surface bearings. It is proportional to the mass of the vehicle M , gravitational acceleration g, the rolling resistance coecient µ and to the slope α. Mathematically it is represented by the equation 2.6.

F roll = M.g.µ.cos (α)

(2.6)
Slope prole resistance F slope Represents the resistive eort to carry out a movement on a surface with inclination α (equation 2.7).

F slope = M.g.sin (α)

(2.7)
The mathematical representation of the longitudinal dynamics of a vehicle is given by the equation 3.2.

dv dt = Γ wheel (t) R wheel -1 2 .ρ air .S.C x . (v (t) + v air (t)) 2 + M.g.µ.cos (α) + M.g.sin (α) M (2.8)

Drive train

Composed mainly by electric powertrain (electric machine) and gear reducer, the drive train provides the torque that will be applied to the wheels Γ wheel .

Gear reducer The reducer has the function of adjusting torque and speed of the electric machine to the wheels. Ideally, this element allows a total transmission of power.

However, due to losses such as friction losses, the gear reducer has an eciency dependent on the turning speed.

Assuming a constant eciency η red and a transformation gain k red , the gear reducer is represented by the system of equations 3.3.

         Γ machine (t) = Γ wheel (t) η red k red Ω machine (t) = Ω wheel (t) k red (2.9)
Electrical machine An electrical machine can be regarded as a controllable power converter. This means that the electrical power provided to the machine is transformed into mechanical power that is provided to the vehicle. Being a reversible element, it can act in motor mode (as described above) or in generator mode where mechanical power is converted into electricity, case of braking in vehicles with electric power train.

In general, the electrical machines have good eciency when compared to internal combustion engines, reaching values higher than 90% [START_REF] Mccoy | Energy-ecient electric motor selection handbook[END_REF][START_REF] Gao | Hybrid electric vehicle: Overview and state of the art[END_REF]. The construction technology denes the characteristics of performance and operation. For applications in electric traction, three types of electrical machines are commonly used:

• Direct current machine

• Synchronous machine

• Asynchronous machine

The asynchronous machine, specically with permanent magnet, is the most adopted for applications in vehicles having electric traction. This is due to its good eciency, reliability and performance in terms of maximum engine speed [START_REF] Florescu | Gestion optimise des ux energetiques dans le vehicule electrique[END_REF].

For purposes of energy analysis, an electrical machine can be represented by its mechanical inertia and its static response in terms of torque and speed. Thus, by requiring to the machine a given amount of torque at a given speed, it is necessary an electrical power P elec equivalent to the required mechanical power (torque x speed), adding to it the intrinsic losses to operate this machine.

Battery

The batteries act as an essential element in a fuel cell vehicle. For this particular application there are dierent types of batteries. The choice of the most suitable technology must take into account a range of factors such as [START_REF] Tritschler | Optimisation de l architecture electrique et gestion d energie pour un systeme a pile a combustible embarquee dedie a l application agricole[END_REF]: the automotive applications directed their expectations on this technology. Despite the high price, the automotive constructors adopted it for electric and hybrid vehicles due to its high energy and power density.

1.

Li ion batteries

From the point of view of its construction, a cell of a Li ion battery consists of an electrolyte and two electrodes (gure 2.12): a negative electrode based on graphite (Li x C 6 ) and a positive electrode based on lithium metallic oxide (LiM eO 2 ), where M e can be Co, N i, etc [START_REF] Devie | Caracterisation de l usage des batteries Lithium-Ion dans les vehicules electriques et hybrides. Application a l etude du vieillissement et de la abilite[END_REF].

As the cathode has a higher potential, in discharge mode the electrons comes from the external circuit in its direction such the current ows from the cathode to the anode.

Inside the cell, the cations Li + are transferred between the cathode and the anode through the electrolyte and a separator. Each atom of Lithium is liberated from the anode and releases an electron to circulate through the external circuit in the cathode direction. reach the cathode, binding to an electron. In charge mode, the motion of electrons and cations is done in reverse. Figure 2.12 illustrates this process. Electrochemically, this whole process is represented by oxido-reduction reactions.

Regarding the type of application, a battery may have predominant characteristics that favor higher energy density or higher power density. For electric vehicles, energy batteries are the most suitable because they should ensure greater autonomy of the vehicle. In the case of full hybrid fuel cell vehicles, battery has function of electric assistance by providing power. Then power batteries are the most indicated.

In order to obtain these dierent behaviors, the two types of cells have dierent constructive aspects. Adopting the same mass, a cell intended for power will present a larger electrode surface leading to a lower impedance, while an energy cell possess larger capacity and a higher impedance. The result of these structural dierences lead to a kinetic limitation with lower currents for energy cells than for power ones [START_REF] Brown | Diagnosis of the Lifetime Performance Degradation of Lithium-Ion Batteries: Focus on Power-Assist Hybrid Electric Vehicle and Low-Earth-Orbit Satellite Applications[END_REF].

Whatever the type (power or energy), a battery can be represented mathematically with a higher or lower level of detail. Normally for purposes of energy consumption analysis, a model using equivalent electrical circuits is used. These models can go from simple representations where the battery acts as a perfect voltage source in series with a resistance (g. a 2.13) [START_REF] Ceraolo | New dynamical models of lead-acid batteries[END_REF][START_REF] Kermani | Gestion energetique des vehicules hybrides: de la simulation a la commande temps reel[END_REF], up to more complex representations where the dynamic evolution of states is considered.

(a)

R bat (i bat ) i bat v bat E bat (b) R 1 i bat_input v bat E bat (SoC,T) C 1 R 2 C 2 i bat_output
v bat (t) = E bat (SoC (t) , T (t)) -Z bat (i bat , SoC (t) , T (t)) .i bat (t) (2.10)
In this equation, E bat is the open circuit voltage, SoC is the state of charge, T is the temperature and i bat is the current that can be absorbed or delivered. The impedance Z bat is dependent to the mode which the battery is submitted to. This means that the cell has dierent impedances for charge (i bat < 0) and for discharge (i bat > 0).

The state of charge of the battery (SoC ) is calculated through the integral of the current of the battery i bat and using the relation between the actual electrical charge stored Q(t) and the maximal charge that can be stored in the element Q tot (equation 2.11).

SoC (t) = Q (t) Q tot .100 [%]; Q (t) = ˆibatt (t) dt (2.11)
Due to the inuence of SoC in the durability of the batteries during storage and used conditions [START_REF] Montaru | Contribuition a l evaluation du vieillissement des batteries de puissance utilisees dans les vehicules hybrides selon leurs usages[END_REF][START_REF] Devie | Caracterisation de l usage des batteries Lithium-Ion dans les vehicules electriques et hybrides. Application a l etude du vieillissement et de la abilite[END_REF], its operating region may be limited in order to achieve a compromise between storage capacity and durability. For applications in hybrid vehicles, some studies [START_REF] Brown | Diagnosis of the Lifetime Performance Degradation of Lithium-Ion Batteries: Focus on Power-Assist Hybrid Electric Vehicle and Low-Earth-Orbit Satellite Applications[END_REF][START_REF] Safari | Aging of a comercial graphite/ lifepo4 cell[END_REF] show that the operation between 40 % and 80 % SoC do not lead to life time reduction by degrading eects, but the type of cycle may have an inuence on it.

The maximum capacity is also used in the denition of the maximum current of the battery. The maximum current (for discharge mode or minimum in charge mode) is done using a C rate to normalize relation with the maximum capacity. The C rate is a measure of the rate which the battery is discharged in relation to its maximum capacity. A rate of 1C indicates the current that will discharge the battery in one hour. For a battery 100 Ah, this equates to a discharge current of 100 A and for a 5C rate the current would be 500 A.

For the objectives considered in this work (sizing and energy management), a quasistatic map, obtained experimentally, is used to represent the behavior of the battery.

The voltage response is function of the SoC and current (equation 2.12). Besides, the battery voltage has a dynamic component represented by a rst order transfer function with a time constant τ batt about 0.1 s.

v batt = k τ batt s + 1 f (SoC, I batt ) (2.12)
Normally presented in a real battery system, a BMS (Battery Management System) provides indicators and observation of important measurements that are used by the vehicle control unit. One of the calculated variables is the state of energy of the battery

SoE.

In spite of the similarities with the SoC information, some studies already have shown that the SoE represents a valid and reliable information in terms of autonomy and capacity [START_REF] Devie | Caracterisation de l usage des batteries Lithium-Ion dans les vehicules electriques et hybrides. Application a l etude du vieillissement et de la abilite[END_REF]. Equation 2.13 gives the SoE calculation, where ε max is the maximum energetic capacity of the battery and ε is the actual energy in the battery and P batt the power supplied by the battery.

SoE (%) = 100. ε ε max ; ε = -ˆPbatt dt (2.13)
Finally, the battery pack is a series and parallel association of cell to meet a voltage level (resulting from an association in series) and a level of storage capacity (resulting from the association in parallel).

Fuel cell system

In this part of the chapter will be described the characteristics of a fuel cell system. First, will be presented the available fuel cell technologies, with emphasis on applications in the transportation eld. Then, the system will be described from their groups and elements.

Fuel cell

Although the principle of reverse electrolysis (to obtain electricity by reacting hydrogen and oxygen) was discovered in 1802 by Sir David Henry, only through the technological requirements imposed by space programs on the 60s and 70s that this operating principle gained the status of technological application. Since then, the fuel cell has undergone a signicant evolution up to the current level of development.

Just as an electrochemical battery, the operating principle of a fuel cell is based on an oxido-reduction reaction. The oxidation happens at the negative electrode, while the reduction happens at the positive electrode. In order to minimize the entropy of the chemical reaction and thus reduce the heat built, the chemical reaction must occur in an orderly manner and without direct contact between the reactant gases. An electrolyte is used to achieve this orderly procedure. The electrons pass through the external circuit while only the cations pass through the electrolyte to reach the electrode. The semireactions in 2.14 represent this process.

Anode :

H 2 ←→ 2H + + 2e - Cathode : 1 2 O 2 + 2H + + 2e -←→ H 2 O .............................................................. H 2 + 1 2 O 2 ←→ H 2 O (2.14)
Fuel cell technologies The electrolyte and the operating temperature dene the types of technologies for fuel cells. These technologies should not be considered as competitors. Each type has a favored niche of application to which its advantages and disadvantages ts.

The high temperature fuel cells have the advantage of being able to use directly hydrogenated compounds as fuel; however, it requires the adoption of high temperature materials and needs a signicant time for heating up to the operating temperature.

The low-temperature fuel cells do not have the warming up problem, but the lack of activation energy for the reaction requires the use of catalysts, which are made from precious metals increasing the cost and make them more sensitive to poisoning by pollutants such as carbon monoxide and some sulfur compounds [START_REF] Candusso | Hybridation du groupe electrogene a pile a combustible pour l alimentation d un vehicule electrique[END_REF].

SOFC and PEMFC represent respectively the two types of cells mentioned. They are considered the most promising technologies, showing a level of maturity able to lead them to be implemented in market applications.

For applications in transport domain PEMFC is considered the most suitable due to its operating temperature, compactness, power density and good eciency [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF][START_REF] Baptista | Plug-in-hybrid fuel cell vehicle market penetration scenarios[END_REF].

Consequently, this technology will be emphasized and used in this work.

PEMFC

Each cell in a PEMFC stack comprises a pair of bipolar plates, positive electrode (cathode), negative electrode (anode) and electrolyte (membrane) (gure 2.14). Electrolyte For the PEMFC technology, the electrolyte corresponds to a membrane.

It lies between the two active layers of the electrodes. Its function is to ensure the passage of protons H + from the anode to the cathode. It presents transport phenomena, such as the transport of water that governed by two mechanisms [START_REF] Poirot-Crouvezier | Modelisation dynamiques des phenomenes hydrauliques, thermiques et electriques dans un groupe electrogene a pile a combustible destine a l application automobile[END_REF]:

• Electro-osmose: the formation of water molecules by passing of ions H + from the anode to the cathode.

• Diusion: transport of the water from the cathode to the anode due the presence of a concentration gradient.

These transport phenomena must be taken into consideration in the construction and operation of the fuel cell, since to ensure good proton conductivity it is necessary a good state of membrane hydration.

MEA (Membrane Electrode Assembly) and the stack The association of the electrodes and the membrane dene the core of each cell (MEA). With the assistance of bipolar and terminal plates, the MEA stacking congures the fuel cell stack, which operates as an association in series of cells. Thus, the output voltage of the stack is the sum of each individual cell voltage and the current required to stack is the current that will be passed through each cell. The gas transport (H2 and air) is achemined in parallel in each cell. The gure 6.5 shows a stack for automotive applications.

Figure 2.15: Fuel cell stack for automotive applications [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF] PEMFC electric response The variation of the global reversible energy of the overall reaction is obtained as a function of the Gibbs free energy [START_REF] Candusso | Hybridation du groupe electrogene a pile a combustible pour l alimentation d un vehicule electrique[END_REF][START_REF] Poirot-Crouvezier | Modelisation dynamiques des phenomenes hydrauliques, thermiques et electriques dans un groupe electrogene a pile a combustible destine a l application automobile[END_REF] Where P H 2 and P O 2 are the partial pressures of hydrogen and oxygen respectively, R is the perfect gas law constant and P atm is the atmospheric pressure.

The reversible voltage E rev is equal to 1.23 V for the water production in liquid state, while its value is 1.18 for the water production in steam state. Nevertheless, when the cell delivers a current i f c , the value of its voltage U cell becomes dierent from E rev due to the irreversible losses. The main losses are: activation η act , ohmic η ohm and concentration losses η conc [START_REF] Tritschler | Optimisation de l architecture electrique et gestion d energie pour un systeme a pile a combustible embarquee dedie a l application agricole[END_REF][START_REF] Candusso | Hybridation du groupe electrogene a pile a combustible pour l alimentation d un vehicule electrique[END_REF][START_REF] Poirot-Crouvezier | Modelisation dynamiques des phenomenes hydrauliques, thermiques et electriques dans un groupe electrogene a pile a combustible destine a l application automobile[END_REF]. Equation 2.18 presents the cell voltage U cell as a function of these dierent variables.

U cell = E rev (T, P H 2 , P O 2 ) -η act (i f c , H r , T ) -η ohm (i f c , H r , T ) -η conc (i f c ) (2.
U cell = β 1 + β 2 T F C + β 3 T F C ln p cath -β 5 i F C β 6 + . . . . . . + β 7 T F C ln (p an ) + β 8 i F C e -1268 T F C + β 9 ln (H r ) (2.19)
The voltage response of the complete stack is given by the sum of the individual cells response (U stack = U cell .N cell ).

The amount of oxygen and hydrogen required for reaction, and the amount of water created are directly dependent on the current required to stack. Equations 2.20, 2.21 e 2.22 present these relations.

q O 2 (F C) = N cell i F C 4F (2.20) q H 2 (F C) = N cell i F C 2F (2.21) q H 2 O(F C) = N cell i F C 2F (2.22)
As a result of the reaction, the heat created is obtained from the electric losses that are converted to heat ow Q F C .

Even though the fuel cell has a static response, a dynamic content is imposed to this response by the dynamics of each subsystem. This is achieved through the inuence of subsystems state variables (p cath , T F C , p an , H r and i F C ) on the cell voltage response (equation 3.7).

Description of a fuel cell system

Equation 2.18 indicates that to obtain a good response of a fuel cell stack, a good conditioning of the system state variables, such as current, pressures, temperatures and hygrometry is required. For this reason, a system that regulates these variables close to an operating point is needed in order to garanty the best response of the fuel cell stack. Figure 2.17 illustrates all the components required in a possible PEMFC system for vehicular applications. • Hydrogen subsystem

• Air/ oxygen subsystem

• Cooling subsystem • Electric subsystem
Hydrogen subsystem The main objective of this circuit is to control and regulate the ow and pressure of hydrogen within the fuel cell. This is accomplished through a coordinated control between the two valves present in the circuit (gure 2.17). The inlet valve allows the passage of the hydrogen ow from the tank in direction to the anode chamber of the cell. Normally this valve controls the pressure at the anode side.

Thus, when the pressure in the anode is lower than the desired value the valve opens proportionally to this pressure dierence. This ensures the pressure regulation and the amount of hydrogen inside the anode chamber. However, the presences of transport phenomena through the membrane cause the passage of water and nitrogen from the cathode side to the anode, thereby reducing the fraction of hydrogen in the anode and the cell performance. In order to overcome this problem, a purge procedure is necessary.

This procedure is performed by opening the outlet valve for a short time. The dierence in pressure between the anode and the environment causes a strong ow that allows to dispel water and nitrogen accumulated at anode. Beside, there is a depressurization at anode, which causes the lling of the anode chamber with hydrogen by opening the inlet valve. Equation (2.23) can be used to represent the anode chamber pressure (p an )

behavior, when it is considered as a unique volume. Where q purge is the outlet ow of the purge valve, q H 2 (input) is the hydrogen inlet ow.

dp an dt = RT an V an q H 2 (input) -q H 2 (F C) -q purge (2.23)
The circuit conguration illustrated in gure 2.17 represents an operating mode called dead-end. To operate with higher stoichiometries (between 1.2 and 1.5), an hydrogen recirculation is applied from the anode outlet to its inlet. This recirculation mode is accomplished through the use of a pump (active system) or through an ejector system (passive system) [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF].

Even if the recirculation mode presents a better ow of hydrogen, in this work the dead-end conguration has been adopted.

Despite the improvement obtained in terms of the stack performance, the depressurization and especially the amount of hydrogen eliminated from the anode during the purge procedure require the denition of a strategy that adapts well the demands of the stack. Nowadays, a periodic purge strategy is the most used. The frequency and duration of the outlet valve opening are dependent on the current required by the stack. Concerning the cathode pressure (p cath ), it can be calculated using the ideal gas law (2.24) as in the anode case, considering it as an unique volume without pressure drop and mass transfer across the membrane (O2, N2 or H2 crossover). In (2.24), q cp is the air ow delivered to the stack, q valve is the gas ow through the control pressure valve towards to the environment, R is the ideal gas constant and T cath is the cathode temperature.

dp cath dt = RT cath V cath q cp -q O 2 (F C) + q H 2 O(F C) -q valve (2.24)
Typically the oxygen is delivered in the form of air rather than through reservoirs, as in the case of hydrogen. This requires an air compressor which has eects on the temperature and humidication of the supplied air. Thus, this circuit has also to ensure the conditioning in temperature and humidity of the air delivered to the stack.

Basically, it consists of three elements (gure 2.18): air compressor, humidier and control pressure valve.

Air compressor For powers above 1 kW, it is necessary to use air compressors to ensure the ow into the stack. Typically for this type of application, it is required a capacity to deliver large air ow with a relatively low compression ratio. Among the compressors adapted for FC applications, the following models are the most used [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF][START_REF] Blurnier | Modelisation de moto-compresseurs en vue de la gestion de l air dans les systemes pile a combustible -simulation et validation experimentale[END_REF]:

• Scroll compressors • Claw compressors • Turbochargers (centrifugal compressors)
To choose the type of compressor, it should take into account the fact that the presence of this element leads to an increase of energy consumption and a reduction of the dynamic response of the stack. According to dierent authors [START_REF] Vahidi | Current management in a hybrid fuel cell power system: A model-predictive control approach[END_REF][START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF], the turbochargers are best adapted for fuel cell in vehicle applications. They have a good dynamic response, low power consumption and compactness, but they have a drawback in terms of control. Compared to other technologies, the centrifugal compressor has a highly nonlinear dynamic response.

In our work, a compressor of this type is an association of an electric motor and a compression stage. In the compression stage, the outlet air ow q cp is calculated in 2.25 by an experimental static map and highlights its strong dependence on the rotational speed of the motor ω and the compression ratio (Π compression = psm patm ). ω is function of the applied control u mot and the resistive torque C r (eq. 2.27), which is dependent on ω, q cp and Π compression . Equations 2.26 [START_REF] Vahidi | Current management in a hybrid fuel cell power system: A model-predictive control approach[END_REF] represents the dynamics of the motor-compressor system.

q cp = f comp (ω, Π compression ) (2.25) dω dt = 1 J (α.u mot -C r -ϕω) (2.26) C r = C p T atm ωη cp (Π compression ) γ-1 γ -1 q cp (2.27)
Where J is the motor inertia, α is the torque constant amplier and ϕ is the viscous friction coecient, C p is the air specic heat capacity, T amb is the ambient temperature, η cp is the compressor eciency, p sm is the outlet pressure, p atm the atmospheric pressure and γ is the air specic heat ratio (γ = 1.4).

As pointed out by Venturi et al. [START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF], a small change in pressure leads to a large change in ow for this type of compressor. Therefore, a control scheme that couples rotation speed and pressure must be used in order to make a good ow regulation.

Humidier In order to ensure a humidity level that is consistent to the operating conditions of the stack, it is necessary to have the presence of a humidier in the air circuit to prevent the drying of the fuel cell membrane.

Humidiers can be either passive or active [START_REF] Glises | New design of a pem fuel cell air automatic climate control unit[END_REF]. In the case of this study, the passive option is chosen because it meets the needs of the system and does not consume energy. This type of humidier is also known as the membrane humidier. It has two chambers separated by a membrane. Inside one passing inlet dry air and humidied oulet air into the other. Transport of water and heat is performed by the membrane.

In order to take advantage of the water inside the cathode (created by the reaction), the other edge of the humidier is connected to cathode outlet, so there is a transport of the humidity from the outlet reacted gas to the inlet air (gure 2.18).

Considering as a constant control volume where a heat ow and mass balance is done, this component has its heat transfer behavior given by the Fourier equation in (2.28), where µ is the overall heat transfer coecient, S is the contact surface and ∆T mem is the dierence of temperature across the membrane.

dQ dt = µ.S.∆T mem (2.28)
The vapor mass transfer in the humidier is driven by a concentration gradient of the humidity at the boundary of the membrane. Linked to the dierent transport phenomena through the membrane, the vapor mass transfer denes a dynamic pressure behavior inside the humidier, modeled by (2.29). V hu is the volume of each chamber of the humidier, p hu is the pressure in the chamber, R is the ideal gas constant, T hu is the temperature in the humidier, m molar is the molar mass of the gas (air). The ows variables q i represents the input, and output ows and the transport phenomena.

V hu dp hu dt = RT hu m molar i q i (2.29)
The volume of the humidier chamber, denes the pressure dynamic of the humidier.

Moreover, the ows and the geometry of the component create a pressure drop between the compressor outlet and the cathode inlet, as well as between the cathode outlet and the pressure valve inlet.

Control pressure valve The valve on the outlet of the humidier (gure 2.18) is responsible of the pressure control inside the cathode by regulating the output ow.

Typically, this element has a dynamic behavior where the output ow q valve is dependent on the pressure dierence between its extremities p valve and the aperture surface of the orice valve S valve .

This relationship can be represented by a non-linear equation such as 2.30, where the air ow dynamics in the valve is neglected. The parameters ζ i are dened by the constructor data and p valve is the pressure dierence across the valve. In this equation, the uid is considered very slightly compressible due to the small operational value of ∆p valve . This mass ow response q valve is linked to S valve by the opening and closing of the mechanical part of the valve, where x valve is the displacement of the valve operating mechanism.

In terms of dynamic response, the transfer function between the mechanic displacement of the valve x valve and the control input of the valve u valve can be represented by a rst order in (2.31). This dynamic is assumed to be identical on both directions (opening and closing) with a time constant τ valve and a gain k valve .

q valve = (ζ 1 x valve + ζ 2 ) ( p valve ) (2.30) ẋvalve = 1 τ valve (-x valve + k valve u valve ) (2.31)
Allied to this dynamic behavior, characteristics of discontinuities such as death zone, hysteresis, etc. may be present, which complicates the pressure regulation. Cooling subsystem Since the fuel cell reaction is an exothermic reaction, it is necessary to evacuate the heat produced. The temperature of the fuel cell stack should be regulated within the acceptable region of operation and as close as possible of its optimal value (table 2.2). Temperature regulation allows the performance of the fuel cell stack to be improved and avoids its degradation. Cooling systems are used for this purpose. For stacks with signicant power (case of automotive applications), the cooling is made by heat exchange using passing uid. A cooling circuit passes through the cell and through a heat exchanger (radiator), in which heat is exchanged in a natural way or by forced convection using fans. The liquid circulates in the circuit with the help of a pump (gure 2.17).

The cooling circuit can be composed by a radiator, a thermovalve and a pump (g.

(2.19)). In the cooling process, the thermal capacities are used to calculate the temperature at dierent places of the circuit. The most important thermal capacities are:

• Fuel cell stack heat capacity

• Water inside the fuel cell heat capacity

• Radiator heat capacity

• Water inside the radiator heat capacity

The temperature of the fuel cell stack and the radiator are calculated by the equation 2.32. For each element i (stack or radiator), T i is the temperature, m i is the mass, C p i is the heat capacity and Q j are the heat ows involved.

dT i dt = 1 m i C p i Q j (2.32)
At the outlet of the two mentioned elements is calculated the temperature of the cooling liquid T out , which depends on the liquid ow q water , the temperature dierences, the convection coecient h i and the thermal exchange surface S.

dT out dt = 1 m water C pwater [q water (T in -T out ) + h i S (T i -T in )] (2.33)
In order to control the temperature of the stack, the thermovalve acts passively to pass the liquid either through a by pass or through the radiator. If more heat exchange is necessary to reduce the temperature of the stack, a control is applied on the fan that is placed on the radiator.

Electric subsystem The electric subsystem performs the interface between the electric bus and the fuel cell stack. This interface transports the power from the stack to the electric bus, converting the voltage and current levels. A static DC/DC converter is used as interface. As the fuel cell does not act as reversible element, the DC/DC converter must be irreversible to protect the stack in case of a reverse current.

In the case of series hybrid applications, the converter has an important function in controlling the power output of the fuel cell stack system. Specically for the architecture used in this work (with just one converter), the energy management is controlled by the DC/DC converter.

The converters can be step-up (boost), step-down (buck) or both (buck-boost). They are chosen according to the nominal voltage of the stack and the desired bus voltage and the DC bus. When it exceeds the voltage of the electric bus, a buck converter should be used. In case of a stack nominal voltage below the bus voltage, the converter must be of boost type. Finally, a buck-boost converter is adopted when the operating voltage of the stack can be higher or lower than the bus voltage.

For our application, a buck DC/DC converter is used to represent this subsystem.

Equation 2.34 is used, such that the current i F C is imposed to the fuel cell stack while it replies with the voltage U F C . Where L is the inductance and

U F C is the fuel cell stack voltage L di F C dt = U F C -α buck U bus (2.34)
A control strategy is implemented in order to regulate the output current. The voltages of the fuel cell and the electric bus U bus are considered as measurements in order to calculate the value of the duty cycle α buck .

Conclusion of the chapter

The aim of this chapter was to present briey the features of hybrid fuel cell vehicles in order to identify the issues involved. Among them there are the objectives of this thesis.

In the rst part was presented a state of art about the fuel cell vehicles. This state of art has show the positive trend to the advancement of this technology and the challenges that remain for its full consolidation as a viable transportation alternative.

The architectures available today were presented and analyzed in order to extract their main features. We could then dene the hybrid series architecture as the most suited to fuel cell vehicle applications. In this architecture, the non reversible source is the fuel cell stack and the battery is the reversible energy storage element. This combination of elements allows the elaboration of energy management strategies that optimize energy consumption, compensate the dynamic constraints of the fuel cell stack system and act against the cell degradation.

According to our specic application, a description of the main characteristics of each element of the architecture was made. This allows to understand the general operation principle of the elements and how they act on the vehicle in terms of dynamic and energy response. The technologies involved show that there are operational conditions that should be targeted. Specically for the fuel cell system, it can be seen that it is a complex system which requires the control of state variables to obtain a proper functioning of the cell.

In order to obtain the optimal sizing of the vehicle and the optimal energy management including the dynamics and durability aspects, it is important the prior knowledge about the main characteristics of the vehicle components. Therefore, in the next chapter modeling for sizing and optimization will be done based on the information presented in here. It will present the mathematical models developed in accordance with the technologies presented and the most indicated optimization tools for the problem involving the chosen architecture.

Introduction

Even if the fuel cell vehicle is considered as an important option to compose the set of transport solutions, the technology involved has not achieved yet a desired maturity level, requiring more eorts in research and developments.

Having the objectives to propose new methodologies to overcome some remaining technical limitations, the scope of this work is essentially focused essentially on two key topics:

1. Optimal sizing methodology 2. Optimal energy management strategy, integrating the aspects of dynamics and durability of the fuel cell system.

These subjects demand studies that combine multiple disciplines as mechanics, electrics, chemistry, automatic control, etc. Furthermore, to reach all the goals, it is also required a basic structure adapted and dened in introduction to each of these topics. Therefore, this chapter is oriented to present the tools that will be necessary to reach the objectives.

Such tools are:

• Models (forward/backward, simulation, control)

• Optimization tools for the energy management This chapter presents two main parts that are related to the previously established objectives. However, important concepts about modeling and optimization are introduced before the description of the required structure. Besides, the initial part of the chapter includes also a state of the art about the energy management for hybrid applications and a presentation of the modeling tool that is used in these works.

In the sequence, the structure needed to perform the optimal sizing task is described through the dierent models adopted and the optimization tool selected to compose the sizing process. An outline in accordance to the steps of the sizing procedure establishes the order to present the models and the optimization method.

The nal part of the chapter deals with the description of the models and optimization tools that are used to the design of an energy management strategy (EMS). The constraints brought by the inclusion of the fuel cell dynamics in the EMS require an optimization method capable to perform this inclusion. Moreover, to obtain the fuel cell system dynamics, it is necessary a model that has the ability to represent it. In this part of the chapter, the FC system dynamic model and the chosen optimization method are presented.

Modeling and energy management strategy for FCHEV

In this section, the modeling and EMS for hybrid architectures, in particular for the FCHEV, are approached by the presentation of important concepts, state of the art and the tools related to their developments.

Modeling tools and concepts

The models of the FCHEV for this work are constructed using the modeling environment VEHLIB [START_REF] Trigui | Systemic modelling of hybrid vehicles in order to predict dynamic performance and energy consumption-building the vehlib library of models[END_REF] developed by IFSTTAR in Matlab-Simulink (c) . The VEHLIB is a library that contains models of vehicles and elements that compose them. The elements of the library were built using quasi-static models based on the mapping of components originated from experimental results. This allows to obtain a reasonable calculation time, keeping the coherence in terms of the energy consumption. Route, HYZEM Autoroute and Mountain cycle [START_REF] André | The artemis european driving cycles for measuring car pollutant emissions. 7th international symposium, highway and urban pollution[END_REF][START_REF] André | Driving pattern analysis and driving cycles -European Development of Hybrid Technology approaching Zero Emission Mobility (HYZEM)[END_REF] are the most adopted to analyze the behavior of the models. Figure 3.2 illustrates the extra-urban drive cycle, while the other cycles are present in the appendix 1.

Using VEHLIB, causal models can be created with a formalism based on the Bond-Graph theory [START_REF] Bideaux | Design of a hybrid vehicle powertrain using an inverse methodology[END_REF][START_REF] Sueur | Bond-graph approach for structural analysis of {MIMO} linear systems[END_REF].

The models may have two dierent representations that can be called behavior models and design models, depending on the objectives and the type of problems to solve [START_REF] Mester | Conception Optimale Systémique des Composants des Chaînes de Traction Electrique[END_REF].

Such type of problems are known as forward and backward, and they are identied using the principle of causality. This cause and eect relationship indicates that the method to solve a forward problem, which means calculation from the source (cause) to the eect, should be dierent from the method to solve a backward problem (from the eect to the cause). Therefore, the behavior and design models, called as well as forward and backward models, are mainly dierentiated by the sense and type of calculations used to solve the model's equations [START_REF] Dupriez-Robin | Dimensionnement d'une propulsion hybride de volier, basé sur la modélisation par les ux de puissance[END_REF]:

• Forward models: The starting point is the setpoint (cause) of the closed loop system, and the aim of the calculation is to calculate the system response (eect) to follow this setpoint. In the case of the vehicle, the eect is the speed of the vehicle and the setpoint is the driver speed requirements. This modeling is the most accurate to the reality and can be adapted to the control of real time systems without too much implementation eorts.

• Backward models: The starting point is the eect of the forward model. Considering the vehicle, this starting point is the speed. From this speed, it is possible to calculate the other variables up to the traction power necessary to meet the assigned speed. This type of modeling does not represent the reality as the forward one, but it allows to simplify the decision about the command to be applied. After the overview about the modeling aspects, concepts and tools, the sequence of this section introduces a general study of the energy management strategy.

Energy management strategy: concepts and state of the art

As already mentioned, a full hybrid architecture, such as the adopted in this work, oers an extra degree of freedom that can be used to optimize the power sharing between battery and fuel cell system. Nevertheless, the energy management involved in this power split should tackle with the following operating modes:

• Fuel cell system traction mode: traction power delivered by the fuel cell system

• Battery traction mode: traction power delivered by the battery

• Hybrid traction mode: traction power delivered by the fuel cell system and battery at the same time

• Energy recovery mode: recovery power during braking procedures is stored in the batteries

The aim of a good energy management strategy is to appropriate all these modes during the mission and to nd the battery and fuel cell system power proles that represent the minimum of hydrogen consumption to accomplish the mission. Such minimization is not only used to design the FCHEV energy management strategy, but also can be applied in the scope of sizing procedure, dening the most adapted dimensions of FC system and battery in terms of hydrogen consumption.

The EMS can be performed using dierent methods, as presented in the sequence.

State of the art: Energy management

For a better understanding of the energy management strategies for hybrid vehicles with multiple on-board power sources, numerous studies were performed by various authors can be found in the literature. The mainly adopted approaches are grouped according to their characteristics in terms of methodology and implementation.

Intelligent based strategies There exist many studies using articial intelligencebased energy management strategy such as fuzzy logic and neural networks for use in PEMFC powered systems. Among them, fuzzy logic based controllers (FLC) have a predominant position due to its independence of a full mathematical plant model and training procedure. Fuzzy logic incorporates an alternative way of thinking, which allows modeling complex systems using higher level of abstraction originating from our knowledge and experience. However, this knowledge represents a drawback due to the dependence of a certain level of expertise embedded in the strategy. This means that some strong assumptions can be made according to a denition of expertise, which is dependent on the designer's knowledge about the problem.

Many researchers focused on fuzzy logic approach and applied it to hybrid systems with PEMFC for dierent applications. Among them, Gao et al. [START_REF] Gao | Energy management strategy based on fuzzy logic for a fuel cell hybrid bus[END_REF] have proposed a management algorithm for the power split in a hydrogen powered hybrid heavy vehicle (bus) composed of PEMFC, as the main power source, and a battery/UC (ultra capacitor) combination as the reversible converter/storage unit. Caux et al. [START_REF] Caux | On-line fuzzy energy management for hybrid fuel cell systems[END_REF], regulated the energy management in PEMFC/Battery powered hybrid light vehicular systems using fuzzy logic. As a contribution to the mentioned studies, Erdinc et al. [START_REF] Erdinc | A wavelet-fuzzy logic based energy management strategy for a fuel cell/battery/ultra-capacitor hybrid vehicular power system[END_REF][START_REF] Erdinc | Modeling and analysis of an fc/uc hybrid vehicular power system using a wavelet-fuzzy logic based load sharing and control algorithm[END_REF], Li and Liu [START_REF] Li | Optimal fuzzy power control and management of fuel cell/battery hybrid vehicles[END_REF] and Eren et al. [START_REF] Eren | A fuzzy logic based supervisory controller for an fc/uc hybrid vehicular power system[END_REF] inserted the control of internal dynamics of PEMFC system in the fuzzy logic based energy management approach in order to better provide a safe and ecient operation of PEMFC. Eren et al. controlled both the power distribution in the hybrid structure and the oxygen stoechiometry of PEMFC as the outputs of the FLC.

Another articial intelligence method used for the control of hybrids system is the neural network approach. Neural network models are computer programs that are designed to emulate human information processing capabilities. The ability of neural network system to spontaneously learn from examples and to provide adequate and fast responses to new information, which are not previously stored in memory, has generated increasing acceptance for this technology in various engineering elds. The drawbacks of this approach are about the need of training process using a large database obtained experimentally and the lack of stability proof.

Some studies using the neural network approach for energy management can be found in the literature. Ates et al. [START_REF] Ates | Energy management of an fc/uc hybrid vehicular power system using combined neural networks-wavelet transform based strategy[END_REF] provided a neural network supervisory controller using Adaptive Linear Neural Network (ADALINE) that is combined with a multi-level Haar wavelet transform to dene the power delivered by the fuel cell system in a FCHEV.

Askarzadeh et al. [START_REF] Askarzadeh | Articial neural network training using a new ecient optimization algorithm[END_REF] and Prokhorov [START_REF] Dv Prokhorov | Toyota prius hev neurocontrol and diagnostics[END_REF] also used the same neural network type, but combined with fuzzy logic in order to be adapted to dierent drivers.

To summarize, these methods represent a black box approach, which simplies the modeling requirements, especially in the case of complex systems, however, they can not guarantee the global optimization either the stability of the solution. Besides, they are strongly dependent to the designer knowledge and the database used.

Frequency decoupling strategies In vehicular FC systems, transient changes in load demand may often occur. This can cause a dynamic stress into the fuel cell membrane due to pressure oscillation and possible oxygen starvation, reducing the lifetime of the fuel cell [START_REF] Wu | A review of pem fuel cell durability: Degradation mechanisms and mitigation strategies[END_REF]. Therefore, frequency decoupling techniques have been applied to prevent the fuel cell system from load changes at high frequencies.

The most basic approach is using a transfer function for a delay with a suitable time constant, such as a rst order low pass lter. Liu et al. [START_REF] Liu | High frequence decoupling strategy for the pem fuel cell hybrid system[END_REF] applied this concept to PEMFC hybrid system. These lters decouple the main signal with a pre-dened cut-o frequency, preventing the FC from transient variations. However, the loss of important edge information is a signicant phenomenon in conventional ltering techniques.

In order to deal with this drawback, wavelet transform based decoupling strategy for PEMFC has been applied by some authors such as Uznoglu and Alam [START_REF] Uzunoglu | Modeling and analysis of an fc/uc hybrid vehicular power system using a novel wavelet based load sharing algorithm[END_REF], Zhang et al. [START_REF] Zhang | Wavelet-transform-based power management of hybrid vehicles with multiple on board energy sources including fuel cell, battery and ultracapacitor[END_REF] and Ates et al [START_REF] Ates | Energy management of an fc/uc hybrid vehicular power system using combined neural networks-wavelet transform based strategy[END_REF]. The advantage of this type of analysis is that the wavelet transform decomposes a signal into a series of short duration waves or local basis functions (wavelets) on the time axis which allows the analysis of the local phenomena in signal consisting of many transients.

This kind of strategy allows a real-time application, but it presents a diculty to deal with the durability, dynamic aspects and the consumption at the same time.

Optimization based strategy In these types of energy management approaches, the optimal reference power signals for the on board power sources are calculated by minimization of a cost function. This cost function generally represents the fuel consumption or emissions especially for vehicular applications. If the optimization is performed over a xed drive cycle, a global optimum solution can be found. Many approaches such as optimal control theory, linear programming, dynamic or stochastic programming and non-linear model predictive are used for solving the global optimization problem. In fact, the global optimal solution is non-causal because it nds the minimization of the cost function based on the knowledge of the future and past power demands. Therefore, for on-line application these approaches are not indicated. Nevertheless, they might be a basis of designing rules for online implementation or comparison for evaluating the quality of other control strategies. Moreover, an on-line optimization-based control strategy can be found by using the denition of an instantaneous cost function. Such cost function should include an equivalent fuel consumption to guarantee the self-sustainability of the electrical path. Even if the solution of this problem is not globally optimal, it can be used for online application.

In terms of global optimization, the Dynamic Programming (DP) and the Pontryaguin's Maximum Principle (PMP) [START_REF] Christophe | Introduction à l'Optimisation[END_REF] are often used in vehicle energy management and optimal sizing. Scordia et al. [START_REF] Scordia | Approche systématique de l'optimisation du dimensionnement et de l'élaboration des lois de gestion d'énergie de véhicules hybrides[END_REF] adopted the dynamic programming coupled with neural networks to design a power split strategy for hybrid vehicles. Vinot et al. [START_REF] Vinot | Optimal management of electric vehicles with hybrid storage system[END_REF] studied the two approaches for a hybrid electric vehicle battery/UC. As done by Vinot et al., Delprat et al. [START_REF] Delprat | Control of a parallel hybrid powertrain: optimal control[END_REF] used the PMP and an iterative method to design a energy management strategy for conventional hybrid vehicles (ICE/batteries).

For on-line applications, some authors have based their energy management controllers on these optimization methods (DP and PMP). Rodatz et al. [START_REF] Rodatz | Optimal power management of an experimental fuel cell/supercapacitor-powered hybrid vehicle[END_REF] presented an equivalent fuel consumption minimization strategy (ECMS) in order to minimize the amount of hydrogen in a PEMFC/UC hybrid vehicle. A similar approach is used by Xu et al. [START_REF] Xu | Optimal vehicle control strategy of a fuel cell/ battery hybrid city bus[END_REF] for a fuel cell/ battery hybrid bus and by Trischler [START_REF] Tritschler | Optimisation de l architecture electrique et gestion d energie pour un systeme a pile a combustible embarquee dedie a l application agricole[END_REF] for a PEMFC/battery tractor. Bernard et al. [START_REF] Bernard | Fuel ecient power management strategy for fuel cell hybrids powertrains[END_REF] have proposed to complete the previous controls with a predictive approach in the ECMS for a PEMFC/battery vehicle.

Each of the abovementioned approaches provided an important contribution to the energy management issue of PEMFC powered hybrid systems, ensuring the stability of the response. Despite the advances obtained with these methods, it is still necessary to develop a more computationally ecient solution that approaches the local optimization to the global optimization, but taking into account the operating constraints.

An optimization based strategy is adopted to perform the energy management strategy in this work. The knowledge of the dynamics and the consumptions aspects allow to represent the energy management issue in a optimization problem, that can be solved by the DP, PMP, and methods based on these approaches and theories.

Optimization problem

The objective of the energy management is to dene the power split that minimize the cost function g (P F C (t)) (hydrogen consumption according to the power supplied by the fuel cell system), while respecting the constraints and boundary conditions. This can be summarized in an optimization problem dened in (5.29). The trajectory of P * F C is the control trajectory input that minimizes the function J in a drive cycle. Moreover, P * F C must be well adapted to the dynamics of the system (fuel cell system and battery).

                                                                                                  
Cost M inimization :

J (P * F C ) = min P * F C ´tf 0 g (P F C (t)) .dt P * F C = argminJ (P F C (t))
Constraints :

P batt min (-3C) ≤ P batt ≤ P batt max (10C) P F C min ≤ P F C ≤ P F C max SoE min (40%) ≤ SoE ≤ SoE max (80%)
Boundary condition :

SoE (0) = SoE (t f ) (3.1)

Conclusion about the modeling and EMS concepts

The concepts and study peformed in this section will be used as a knowledge basis to describe the elements adopted to accomplish the objectives. Forward and backward models are be adopted for dierent applications. Concerning the energy management strategy, optimization based methods were selected to solve the optimization problem in order to nd the power split that minimizes the hydrogen consumption.

In the sequence, the optimal sizing objective has its models and optimization method described according to the stages that compose it. Thereafter, is the turn to describe the elements (models and optimization methods) related to the objective about the conception of an online energy management strategy.

Models and optimization methods for the optimal sizing

The sizing of a hybrid vehicle can be performed using dierent methodologies, technical specications and sizing criteria (cost, consumption, etc.). According to the choice made, it would be necessary a set of tools that will be involved in each element's sizing stage and validation. Besides, the optimization included in the process aggregates optimal algorithms used for the sizing purpose. Victor Mester [START_REF] Mester | Conception Optimale Systémique des Composants des Chaînes de Traction Electrique[END_REF] structured the optimal sizing process by the requirements, models and tools used at each step. Figure (3.5) summarizes this idea for our particular case.

In the drive train sizing, depending on the type of the technical specication, a static or a dynamic requirement is imposed to the sizing procedure, which demands either stationary or dynamic models of the drive train elements.

In the case of the fuel cell system and battery, the optimization requirement (reduction of hydrogen consumption) leads to the adoption of backward models to discover the electric power that must be shared between these elements. In order to solve the optimization problem, the algorithm used needs the information about the energy con-sumption during the drive cycle. Therefore, the models of these elements should be simple, but must keep the energy content information.

Both the drive train dimension and the FC system and battery sizes are validated using the technical specications and the drive cycles. To perform such validation, a complete model of the vehicle is used. As the objective of this task is to analyze the behavior of the vehicle and its elements, a forward model is adopted.

In the sequel of this section, the models and optimization algorithms chosen for these sizing steps are described, initiating by the drive train.

Drive train sizing

As illustrated in (3.5), the technical specications imposed to the vehicle are responsible for the sizing of drive train elements. They are mainly related to the performances of the vehicle such as maximal acceleration and speed or even the ability to take-o with a high slope. The models and the calculations used to dene the sizing of the elements are depending on the type of these performance requirements. This means that a stationary performance index, like maximum speed, only requires an analytical solution of the model, while dynamic performance index, such as maximum acceleration, needs a solution of the dynamic equations related.

The elements present in this part of the vehicle include the vehicle itself (dynamic behavior), the gear reducer and electrical motor. The models of these elements used for the sizing procedure are:

Vehicle model:

dv dt = Γ 2x wheel (t) R wheel -1 2 .ρ air .S.C x . (v (t) + v air (t)) 2 + M.g.µ.cos (α slope ) + M.g.sin (α slope ) M (3.2)
By this equation, the force balance in the vehicle allows to calculate its speed and the acceleration Through the traction torque applied.

Gear reducer: Electric motor: The model of the electric motor and its power electronics are represented by an association of a torque Γ motor vs speed Ω motor static map and an inertia component. Through the static map is possible to calculate the losses in the motor and to include them in the electric power required to provide the demanded mechanical power.

         Γ machine (t) = Γ 2x wheel (t) η red k red Ω motor (t) = Ω 2x wheel (t) k red (3.
In terms of sizing procedure, the maximum power and maximum torque presented in the motor static map are linked to several performance indexes such as maximum acceleration and maximum speed.

Drive train model structure: Allying these three models, it is possible to achieve the drive train sizing objectives. Figure (3.6) illustrates the drive train structure that will be used in its respective sizing procedure.

Being dened the models involved in the sizing of the drive train elements, the sequence deals with the fuel cell system and battery dimensions, which requires the denition of an optimization method and models adapted to it.

Fuel cell system and battery sizing

As pointed out in g. (3.5), the dimensioning of FC system and battery are obtained by the application of an optimization method that minimizes the hydrogen consumption. This method should solve the optimization problem stated in (5.29), ensuring the global minimization of the criterium. However, the implementation of this type of optimization method requires the knowledge of the drive cycle and models of the vehicle elements (vehicle, drive train, FC system and battery) that are dedicated for this task.

The global optimization method and the models used in this part of the sizing process are described in the sequence.

Global optimization method: Dynamic Programming

Among the existing global optimization methods, the Dynamic Programming (DP) was chosen as optimization tool due to the fact that it presents all the characteristics necessary to solve the optimization problem.

Dynamic programming is a method that uses the graph theory to transport a system governed by a state equation described in discrete time x (t + 1) = f (x (t) , u (t) , t) from a state A to state B regarding some constraints [START_REF] Scordia | Approche systématique de l'optimisation du dimensionnement et de l'élaboration des lois de gestion d'énergie de véhicules hybrides[END_REF]. Therefore, at each time t, it is possible to choose u (t) among all eligible values in order to move the system from initial state to nal state. The main idea is to choose a sequence of commands u (t) to minimize the cost criterion in (3.4).

J (u * ) = min u(0),...,u(t f -1)

t f t=0 g (x (t) , u (t) , t) (3.4)
By knowing the initial state, the dynamics of the system, all possible commands and mission, the set of possible states are determined. With all possible commands and states, it is possible to nd the optimal control sequence u * = u 1 , u 2 ...u t , ..., u t f that minimize the criterion. The Hamilton-Jacobi-Bellman equation is applied to nd this sequence. Belleman optimality principle is the basis of the equation of Hamilton-Jacobi-Belleman. It states that A series of decision is optimal if and only if, whatever the state and instant considered, the following decisions constitute an optimal decision series for the sub problem which has this state and this instant has initial consitions [START_REF] Christophe | Introduction à l'Optimisation[END_REF].

Specically for the DP method applied on the fuel cell hybrid electric vehicle, the FC dynamics is neglected in order to simplify the optimization process. This leads to build a 2D graph to solve the optimization problem (gure 3.7). The two dimensions in the graph are the time and the energy of battery (ε), which is replaced, for practical applications, by the SoE information.

Therefore, the graph implementation depends on the calculation of the feasible area.

The construction of the graph begins with the denition of the SoE grid, which is made using the SoE bounds and discretization ( SoE). Thereafter, the edges are built using the possible range of the control input P F C and the battery power P batt . They have a discretization given by P F C and P batt . The operating range of the battery is reduced • Calculate the hydrogen cost for all the feasible edges.

• For each edge of the graph, store the minimal hydrogen consumption (best ecient)

to go from the starting point of the cycle until the current edge, and store the best path to go from the preceding point to the current one.

• At the end of the evaluation process, recalculate the best path fo the entire graph:

The sequence of P batt that achieves minimum hydrogen consumption is obtained, and thus the control trajectory P * F C in known.

As can be noticed, the DP method is not well adapted for online application due to its non causal characteristic. Moreover, it presents a computation time and a memory allocation that can increase fast according to the drive cycle, battery size and precision of the graph. However, the mentioned drawbacks do not represent impediments for design and sizing applications. They can be overcome by the utilization of powerful calculation platforms, with enough memory and applying typical drive cycles. Besides, at a given discretization, the DP provides global optimization responses that can be used as as benchmark to evaluate other optimization methods.

In order to be developed and implemented, the DP requires models to represent the hydrogen consumption of the FC system, the battery behavior and the electric power related to the drive cycle.

In the following, the models used in this optimization process are presented.

Models for the optimization process

Drive train model for the dynamic programming optimization A backward modeling approach is used to obtain the electrical power prole demanded by the vehicle to follow a speed prole. Therefore, it must be eectuate an inversion of the drive train model. Consequently a speed prole is imposed as an input to the vehicle model that replies with the traction torque to be applied on the wheels. Equation (3.5) shows this inversion, presenting the speed as the input and the torque as the output.

Γ 2x wheel (t) = R wheel M. dv dt + 1 2 .ρ air .S.C x . (v (t) + v air (t)) 2 • • • • • • + M.g.µ.cos (α slope ) + M.g.sin (α slope )]} (3.5)
This torque is transformed by the gear reducer (3.3) to represent the torque that the electric motor must deliver.

The angular speed on the wheels are obtained by (3.6) and is transported to the motor level (Ω motor ) using the gear reducer equation (3.3).

Ω wheel (t) = R wheel .v (t)

(3.6)
For this optimization purpose the inertia of the motor is ignored, so Ω motor and Γ motor dene together the mechanical power necessary to the traction. The losses in the electric motor (obtained from its static map) are added to this power, resulting in the electrical power necessary to follow the cycle. This inversion process is illustrated in g. (3.8).

The electric power obtained from this inversion is shared between the FC system and battery by the DP. Therefore, the characteristics of these elements must to be integrated in the DP method. According to (5.29), it is required a fuel cell system hydrogen consumption model to represent the criterion to be minimized, while the battery model Fuel cell system simplied model for the dynamic programming optimization

The fuel cell voltage response is given by (3.7), where the state variables T F C , p an , p cath and H r are assumed constant on their best operation points.

U cell = β 1 + β 2 T F C + β 3 T F C ln p cath -β 5 i F C β 6 + . . . . . . + β 7 T F C ln (p an ) + β 8 i F C e -1268 T F C + β 9 ln (H r ) (3.7)
The fuel cell stack power is given by (3.8) and presents a parabolic prole when it is ploted in function of i F C [82] (g. (3.9)). For the optimization purpose, the current is limited to avoid the falling part of the power prole.

P F C stack = N cell .U cell .i F C (3.8)
A DC/DC converter, used to link the fuel cell stack to the electric bus, is represented by a constant eciency of 90 % (eq. (3.9)) This allows to determine the power to be supplied by the fuel cell stack.

P F C bus = η DC/DC .P F C stack (3.9)

As the fuel cell system is composed by auxiliaries, an energy consumption model was created to represent these elements, even if they are not directly powered by the fuel cell stack (they are linked to the electric bus). Some of them have a continuous operation, such as the pump in the cooling subsystem, and these elements are represented by a power consumption that depends only on the size of the stack (N cell ). Other elements present behavior that follows the fuel cell stack operation. The air compressor is an example and its electric consumption is related to the current delivered by fuel cell stack (air supplied proportional to the FC current). The auxiliary power consumption modeling is dened by the equation (3.10).

P F C auxiliary = N cell (k aux1 i F C + k aux2 ) (3.10)
Where k aux1 is the gain to transform the current demanded to the stack in the power consumed by the compressor to make the cathode conditioning and k 2 is the power due to the other elements with constant operation. This notion of the stack size in the auxiliary consumption is useful in the optimal sizing procedure, which means that a bigger stack will need a more powerful conditioning and consequently they will consume more energy.

To integrate this consumption model in the optimization algorithm, it is necessary to transform the electric power of the stack and fuel cell system auxiliaries in hydrogen consumption rate (g H 2 /s). Disregarding the losses in hydrogen due to the purge procedure, the hydrogen consumption rate is directly depend on i F C , as presented in (3.11).

q H 2 (F C) = N cell i F C 2F (3.11)
In order to calculate q H 2 (F C) in terms of the power delivered by the stack (P F C stack ) and consumed by the auxiliaries (P F C auxiliary ), a sequence of calculation and interpolation is needed. Figure (3.10) shows this sequence.

In the Fuel Cell Stack block, the current and voltage are obtained from the power input by interpolation. This can be eectuated due to the limitation of the power prole in (3.9), permitting to consider only the bijective part dened by the current range between zero and i F C limit. . where the voltage depends on the SoE and i batt (3.12). Dierently from the model presented in the last chapter, the dynamics related to the voltage response was excluded to simplify the calculations, while the SoE is still calculated by (3.13).

Battery pack simplied model for dynamic programming optimization

U batt = f (SoE, i batt ) (3.12) SoE = 100 ε ε max ; ε = ˆPbatt dt (3.13)
Optimization process permits to dene the optimal dimensions of the fuel cell system and battery, which completes the sizing of the vehicle elements. However, a validation step has to be done to ensure that the obtained dimensioning is still capable to accomplish the drive cycles and to meet the performance index. To verify this, a vehicle behavioral model is used.

Validation of the sizing

The tools used in this part of the sizing process are the models of each part of the vehicle.

As the objective of this step is to verify if the response of the obtained dimension is compatible to meet the specication requirements and to perform the drive cycles, a dynamical forward model of the vehicle is indicated.

Such model is built using the equations of all the elements in the vehicle presented in the chapter 2, including the models of the fuel cell subsystems.

Conclusion about the tools related to the optimal sizing objective

The sizing of a hybrid fuel cell vehicle requires the development of several tools to be used in dierent steps that compose this procedure. The set of tools (presented in this section) is composed by models with dierent approaches (forward and backward) and a optimization tool (dynamic programming) to dene, through the energy management, the size of fuel cell system and battery that minimize the hydrogen consumption.

In the next section the energy management objective is again approached in terms of structure necessary to execute it, but with a dierent scope that will demand other models and optimization tools.

Models and optimization methods for the design of online energy management strategy

The second objective of this work is to develop an optimal energy management strategy capable to be integrated in an FCHEV. To embed such strategy in a real time system as the FCHEV, some important requirements should be regarded in the conception of the optimization algorithm. Dierently from the optimization that is indicated to the optimal sizing procedure, the drawbacks about computation eorts, memory used and knowledge in advance of the entire drive cycle must be overcome to be available for embedded applications. Therefore, this work proposes to develop an optimization algorithm adapted to online embedded applications (without the mentioned drawbacks) that includes the dynamic of the fuel cell system.

Normally, the inclusion of such dynamic is not adapted to global optimization methods, such as the dynamic programming used in the sizing approach. Thus, it requires the adoption of another optimization method and a representation of the fuel cell system dynamic. A special attention should be given to the FC system dynamic in order to simplify its inclusion in the optimization algorithm. Our work proposes a control law for this purpose and to provide a gain in performance and stabilization to the fuel cell system. Nevertheless, to perform the conception and validation of the the control law, it is necessary the development of a dynamic model of the fuel cell system.

Consequently, this section presents the optimization method to be used in the develop of an online energy management strategy and that can easily include extra dynamics constraints such as the fuel cell system dynamic. Besides, the models used to perfom the optimization procedure and to design the control of the fuel cell system dynamic are also described in this section. The structure in (3.11) presents the outline of the development stages and the tools necessary to reach the objective.

In this structure, the rst element to be developed is the

Dynamic fuel cell system model for the validation of a control law

In the chapter 2 it was shown that the fuel cell stack response is highly dependent on its subsystems state variables. Due to this reason, the fuel cell system model used to validate the control law has its subsystems modeled using dynamic equations that enable to observe the behavior of the main state variables. In our work, this model represents a 5 kW (90 cells) real fuel cell system that is integrated in a prototype called PROSPAC (appendix 2).

Adopting the same 2D base model as used by Gerard in [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF], the constructed fuel cell system model is able to describe some phenomena and properties such as mass transport, electrochemical reaction, heat transport, proton conduction and mass conservation by the adoption of physical laws. The parameters of the used functions are tted and veried using experimental data. Local conditions (membrane hydration, current density distribution, temperature distribution, liquid water distribution, etc.) can be also observed in the model.

In the following, the fuel cell system has its element's models presented by their equations and hypotheses made.

Air supply subsystem

Cathode model: To calculate the outlet cathode pressure, its capacity is considered as an unique volume. The state variable (p cath outlet ) is dependent on the average temperature in the cathode and the gas ow balance (eq. 3.14).

dp cath outlet dt = RT cath V cath q cp -q O 2 (F C) + q H 2 O(F C) -q valve (3.14)
The cathode presents as well a pressure drop between the inlet and outlet. It is dependent on the ow that the crosses the cathode q cath , which is considered laminar.

Such relation is presented in (3.15).

∆p cath = p cath inlet -p cath outlet = k cath .q 2 cath (3.15)
Where the parameter k cath is obtained from experimental data.

Compressor model: The compressor model represents the centrifugal air compressor present in PROSPAC system. This element is composed by a brushless dc motor and a compressor stage. Concerning the dc motor, it was considered that the electrical dynamic is much faster than the mechanical one, which enables to neglect it. Equations (3.17) and

(3.18) represent the motor speed and resistive torque imposed to it Vahidi et al. [START_REF] Vahidi | Current management in a hybrid fuel cell power system: A model-predictive control approach[END_REF],

while (3.16) is an experimental static map that denes the air ow delivered as in the real compressor.

q cp = f comp (ω, Π compression ) (3.16) dω dt = 1 J (α.u mot -C r -ϕω) (3.17) C r = C p T atm ωη cp (Π compression ) γ-1 γ -1 q cp (3.18)
Humidier: As in the case of the cathode model, the humidier is considered as a single volume with an iso-temperature characteristic. This allows to calculate the pressure at its outlet by (3.20)

q valve = (ζ 1 x valve + ζ 2 ) ( p valve ) (3.22) ẋvalve = 1 τ valve (-x valve + k valve u valve ) (3.23)

Hydrogen subsystem

This subsystem is mainly represented by the pressure dynamic equation (3.24). The same assumptions of the cathode pressure calculation were made here. The anode also presents a pressure drop between the inlet and outlet (3.25).

dp an outlet dt = RT an V an q H 2 (input) -q H 2 (F C) -q purge (3.24)
∆p an = p an inlet -p an outlet = k an .q 2 an (3.25)

The model integrates the purge strategy designed for the PROSPAC prototype, with constant purge frequence and opening time of the purge valve.

Cooling subsystem

The temperatures in the thermal capacities are calculated by (3.26), considering that the capacities are unique volumes and the parameters do not change according to the temperature and other state variables. The heat transfer phenomena adopted are the convection and conduction.

dT i dt = 1 m i C p i Q j (3.26)
Temperature of the cooling liquid at the outlet of each thermal capacity is calculated by (3.27). In this computation, the water (cooling liquid) ow is assumed constant due to the constant pump cadence. It is also assumed that there are no losses of cooling liquid in the circuit and, as in the previous equation, there are no changes of the parameters values according to the temperature and other state variables.

dT out dt = 1 m water C pwater [q water (T in -T out ) + h i S (T i -T in )] (3
L di F C dt = U F C -α buck U bus (3.28)

Fuel cell

In order to better observe the dynamic phenomena inside the fuel cell, a dierent approach is made in the fuel cell model structure. Instead of considering the cell as a single element (voltage response for the entire cell), a spatial discretization is applied on it. The meshed cell is constituted of the bipolar plate channels and the MEA (Membrane Electrode Assembly) as presented in gure (3.12).

Reactant channels are meshed such that the behavior of each mesh element is represented by a capacity (C ) that performs the gas ow balance in the mesh. The interactions between the mesh elements are represented by resistive elements (R) to indicate the losses in the interfaces of the meshes (pressure drop).

In the MEA, the membrane is modeled using a R-C-R association in order to deal with multiple types of variables involved in the fuel cell reactions. The multiport C elements use the balance of the ow variables (current, gas and heat ow) to calculate the eort ones (voltage, pressures and temperature). R elements perform the losses in the interfaces between the membrane and the two chambers (cathode and anode).

Another eect dealt in the membrane is the water transport, which is resulted from the combination of two mass transfer processes: electro-osmosis and diusion [START_REF] Poirot-Crouvezier | Modelisation dynamiques des phenomenes hydrauliques, thermiques et electriques dans un groupe electrogene a pile a combustible destine a l application automobile[END_REF]. The electrochemical response of the mesh elements is given by the same equation of the cell model used for the sizing procedures (3.7), but with dierent local conditions.

0 Y X Z Channel Channel Cathode Membrane Anode M E A Air H 2 X Y MESH 1 MESH 2 MESH 3 MEA Y Z R R R R C C C R R R C C C R R R C C C R R R Cathode Membrane Anode Channel R R C R C Channel R C R C

Fuel cell system dynamic model and PROSPAC prototype in the control context

The adoption of the dynamic model aids in the validation of the control law and also in its conception. The synthesis of the control strategy will need a denition of a model oriented to the control. Such model is derived from the presented fuel cell system model with simplication hypotheses. Thus, the control strategy based on a simplied model is validated in the complex model to verify its robustness. Afterwards, the control law is veried in the real system, testing the conditions that were not inserted in the validation model, such as unknown non linearities of the elements.

When the control law is completely validated, the resulted dynamics is ready to compose the set of requirements to develop the online energy management strategy.

Models and optimization algorithm for the conception of the online energy management strategy

As illustrated in g. (3.11), models of vehicle's drive train, battery and fuel cell system (hydrogen consumption model) are also required to develop the energy management strategy. Therefore, they are presented in this part followed by the description of the basis optimization method.

Battery model

In order to perform fast calculations, the battery model is a simple one with emphasis on the dynamics of the state of energy SoE:

SoE = 100 ε ε max ; ε = ˆPbatt dt (3.29)

Fuel cell system consumption model

The approach used to obtain this model was dierent from the one in the sizing procedure.

For this energy management strategy, the calculation of the hydrogen consumption should be simplied in order to be adapted to embedded applications, which leads to choose a simple analytical model that relates the hydrogen consumption to the power delivered by the fuel cell system. This fuel cell system representation also requires to consider that all the state variables (pressures, temperatures, etc.) are controlled and regulated on their optimal values.

Dierent authors [START_REF] Bernard | Fuel ecient power management strategy for fuel cell hybrids powertrains[END_REF][START_REF] Thritschler | Energy management strategies for an embedded fuel cell system on agricultural vehicles[END_REF] have shown that the equivalent hydrogen consumption rate can be represented by a quadratic function g(P F C ) (3.30), where P F C is the power delivered by the fuel cell system. This work uses the same quadratic approximation to represent the fuel cell system consumption.

g(P F C ) = a 1 P 2 F C + a 2 P F C + a 3 (3.30)
The gure 3.13 shows an example of this approximation compared with the response of a model that considers the dynamics of each subsystem (Gerard [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF]). The hydrogen consumption rate represents the amount of hydrogen used in the fuel cell reaction and an equivalent consumption resulting from the conversion of the auxiliaries electric energy in hydrogen.

Optimization method

The optimization method, used to construct an algorithm compatible to online applications, must be able to solve in real time the optimization problem (5.29), with a solution that approximates the global optimization response. It is also desired that the method could solve an optimization problem that includes the dynamic of the fuel cell system without too much computational eorts. This requirement leads to the adoption of another global optimization method instead of the dynamic programming. In this work the online energy management strategy is derived from the Pontryaguins' Maximum Principle (PMP). The characteristics of this method presented in the following aids to justify the choice according to the desired objective.

Pontryaguins Maximum Principle The PMP is part of the variational methods used to optimize a process. Variational methods are applicable when it is possible to assess the variation of the criterion according to the change in control. The basic idea is to characterize the optimal control u * by the fact that a control u = u * + u must give a higher criterion as J = J * + J > J * . Expressing J in terms of u, it can be expected to nd characterizations of u * .

In terms of implementation, the minimization of the criterion J for a system with a dynamics given by (3.31) is made by the calculation of the co-states λ i for each dynamic constraint. The equation (3.32) represents this integration.

ẋ = f (x (t) , u (t) , t) , x (t 0 ) = x 0 (3.31) min x(t), u(t) J (u) = t f 0 {g (x (t) , u (t) , t) + λ [ ẋ -f (x (t) , u (t) , t)]} dt (3.32)
The solution using this new criterion is obtained by calculating the Pontryaguins function, called as well as the Hamiltonien [START_REF] Christophe | Introduction à l'Optimisation[END_REF] (equation 3.33). The optimal control u * at time t is obtained by the optimization condition H u (x, u, t) = 0.

H (x, u, λ, t) = λ.f (x, u, t) -g (x, u, t) (3.33)
When g is convex and f is linear, H is concave. This optimality condition corresponds to a maximum of H. In the case of the consumption criterion, the sign of the Hamiltonian constraint is therefore reversed to obtain Pontryagin minimum.

However, the optimal solution do not depends only on H u (x, u, t) = 0. A set of three conditions must be satised (3.34), which does take into account the dynamics of the system in the optimization process.

H u (x, u, t) = 0 ; ẋ = H λ , λ = -H x (3.34)
In the case of the optimization for the FCHEVs, the cost function g(x, u, t) is a simplied quadratic function (3.30) that represents the overall hydrogen equivalent consumption of the fuel cell system. Concerning the dynamics, dierent authors considers only the SoE dynamics (3.29). This consideration allows to obtain a convex Hamiltonien and a possible global optimization. Equation 5.31 represents the Hamiltonien obtained under these assumptions using the power balance (P vehtot = P F C + P batt ) to change the variables in the cost function. In this function, the power demanded by the vehicle is considered as a known information, which leads to adopt P batt as the new control variable.

H (ε, P batt , λ) = g (P vehtot -P batt ) -λ (P batt ) = = a 1 P 2 batt (t) -(2a 1 (P vehtot (t)) + a 2 ) .P batt (t) + • • • • • • a 1 (P vehtot (t)) 2 + a 2 (P vehtot (t)) + a 3 -λ (P batt ) (3.35)
Appling the three conditions on (5.31), an optimal prole of P * batt is obtained. P * batt depends on the known variable P vehtot and the co-state λ (6.7). The condition λ = -H ε shows that the co-state has a constant prole during the entire cycle (λ = cte), but it does not indicate its value. In general, λ is related to the knowledge of the drive cycle in order to obtain the boundary condition (SoE (0) = SoE (t f )). Therefore, assuming that the information of the entire drive cycle is not available in real time, the presence of λ in P * batt calculation represents a limitation for on line applications.

∂H ∂P

batt = 0 = 2aP * batt (t) -(2a (P vehtot (t)) + b) -λ ⇒ ⇒ P * batt (t) = λ + (2a (P vehtot (t)) + b) 2a (3.36)
Despite the mentioned limitation, this approach provides an mathematical solution that is useful to the development of an optimization method well adapted for on line applications and that takes into account other dynamics involved in the fuel cell system.

Conclusion of the chapter

The technology of the fuel cell vehicles still presents limitations in several levels, which avoids its insertion in the market. Considering this context, our work proposes to deal with technical limitations that are dened in terms of the following objectives:

• Optimal sizing of the elements that compose the fuel cell vehicle

• Optimal energy management strategy, including the fuel cell system dynamics and its durability

This particular chapter presented and dened the necessary elements to accomplish the steps that lead to reach the two established objectives. These elements were represented by the models of the vehicle's components and optimization tools. As the objectives require the adoption of dierent model levels and optimization methods, these tools were described according to their purposes:

Optimal Sizing In order to develop a sizing procedure, the steps that compose it require the adoption of drive train, fuel cell system and battery models adapted to the assigned goals: design or validation. In the design part, the drive train elements and the energetic elements (fuel cell system and battery) are sized using dierent approaches. The rst ones (drive train elements) are based on the technical specications, while the others (ful cell system and battery) are based on the optimization of the energy management (hydrogen consumption). Therefore, the models used for the two task diers in terms of assumptions made and modeling approaches, as presented in the chapter. Besides, the sizing of the fuel cell system and battery is performed using an approach that applies the global optimization method called Dynamic Programming in order to minimize a chosen criterion (hydrogen consumption). In the validation part, the dimension obtained must be veried according to the technical specications, which requires a model of the vehicle capable to represent its dynamic and static responses.

Development of Energy Management Strategy

Concerning the synthesis of the online optimal energy management strategy (EMS), its development must consider the inclusion of the fuel cell system dynamics in the optimization problem. Thus, a dynamic model of FC system is presented in this chapter in order to aid in the denition of this new constraint. This model will be used for the validation of a control law, which is responsible to improve the performance of the fuel cell system and to simplify the inclusion of its dynamics in the optimization algorithm. The conception of the energy management strategy also requires the adoption of other models that indicate the power required to accomplish the mission (vehicle and powertrain models) and the energy aspects of the battery (State of Energy) and fuel cell system (hydrogen consumption).

The last element necessary to the development of the EMS is the method used to perform the optimization. Maximum Pontryaguin's Principle was dened as the basis of the EMS that will be developed in the chapter 5. According to the characteristics described in this chapter, this principle shows to be adapted to include the new dynamics and to propose an analytical resolution for the optimization problem, which are important advantage compared with the dynamic programming approach.

In the next chapters the presented objectives will be developed and the elements described in this chapter will be useful during the processes.

Introduction

In the context of the vehicles, the last years shown that the recent technologies introduced new functionalities for the vehicle. Acting in dierent application elds, they provide improvements of the economy, performance, comfort and even in new concepts such as connectivity. In order to reach this development level, it was necessary a breakthrough in several processes involved in the design of the new type of vehicles. Topics as hybridization rate, zero emission drive mode and eco-driving were or are still part of design studies and discussions.

Being an element of the design process, the vehicle sizing has required in recent years a special attention from the automobile constructors. In the FCHEV particular case, the development context indicates that the sizing process can contribute to permit this technology to reach an equivalent maturity level as the hybrid vehicles present in the market so far. A sizing procedure specic for the FCHEV application can improve the hydrogen consumption and even its durability. However, the new functionalities introduce extra requests to the sizing and behave as open issues. The chapter 2 presented the most representative functionalities introduced by the hybrids architectures as:

• Micro hybrid

• Mild hybrid • Full hybrid • Plug-in hybrid
• Range extender Nowadays, the three last topics are the most discussed by the community interested in the fuel cell vehicle development. Therefore, the main objective of the chapter is to present the development of a sizing methodology for the hybrid fuel cell vehicle, considering its features and contributions. Initially, the issue of sizing for a hybrid architecture is studied by the approaches already used, highlighting their requirements and characteristics. An analysis of the sizing characteristics for the three most studied functionalities (full hybrid, plug-in hybrid and range extender) is also performed in this initial part of the chapter, followed by the denition of the type of vehicle adopted in this work (reference vehicle)

and the technical specications (performance requirements) used in the sizing procedure of the elements

In the sequence, the optimal sizing approach is developed with a description of all its steps and results. Combining performance requirements, optimization and other criteria, the method presented in this chapter propose a systematic approach to obtain the optimal dimensions of the main elements in the vehicle.

4.2 FCHEV Sizing methods: state of art, concepts and basic requirements

State of art

The sizing methodology oriented for hybrid vehicles is a subject that has been tackled by several authors. Moore and Lovins [START_REF] Moore | Vehicle design strategies to meet and exceed pngv goals[END_REF] studied the new technology of vehicles with an special attention to the hybrid ones. In their research, the choice about the size of the elements in the vehicle's powertrain must meet a set of requirements obtained from technical specications that are adapted for the type of vehicle. Among these specications there are criteria about performance, drivability and degraded modes.

Other authors also justied the importance of technical specications in the sizing process, but they included extra conditions and methods related to the optimization potential presented in the hybridization.

• Assanis et al. [START_REF] Assanis | An optimization approach to hybrid electric propulsion system design[END_REF] adopted a method that performs an exhaustive research of the powertrain best dimension in terms of consumption. Such method tests the several combinations of powertrain elements and chooses the one that presents the minimal consumption. The approach presents drawbacks associated to the calculation eort necessary and especially to the local minimization obtained.

• den Tillaart [START_REF] Van Den Tillaart | Matador: Management tool for the assessment of driveline technologies and research, chapter subtask 2.4: Test methods. soc correction methods for hevs[END_REF] proposed to work with all the hybrid architectures using a dierent optimization method to each architecture, which complicates the evaluation of the results because the methods can not be directly compared.

• Scordia presented in [START_REF] Scordia | Approche systématique de l'optimisation du dimensionnement et de l'élaboration des lois de gestion d'énergie de véhicules hybrides[END_REF] his proposition of sizing method for hybrid electric/ ICE vehicles. Applying this method for dierent architectures (series, parallel and power-split), he chose a global optimization method to perform the sizing of the battery, electric and ICE. The dynamic programming method was used to optimize the consumption of the vehicle either in charge sustaining or in charge depleting mode, dening as well the dimensions of the elements.

Regarding the applications in the fuel cell vehicle domain:

• Yu et al. [START_REF] Yu | An innovative optimal power allocation strategy for fuel cell, battery and supercapacitor hybrid electric vehicle[END_REF] performed the sizing of a vehicle having fuel cell system, batteries and ultracapacitors with the function to deliver power to the traction elements. The authors represented the dimensioning problem as a minimization of a cost function composed by several criteria, such as energy consumption and battery lifetime.

This combination is made by subjective weighting parameters.

• Wu et al. [START_REF] Wu | Component sizing optimization of plug-in hybrid electric vehicles[END_REF] applied an approach that implements an iterative and evolutionary algorithm to obtain the optimal sizing for a fuel cell/battery vehicle. The algorithm applied is called parallel chaos optimization algorithm and it behaves as a metaheuristic approach, which can not guarantee the convergence of the solution nor the global minimization.

Dierently from the mentioned FCHEV sizing approaches, our work proposes to ally technical specications to global optimization methods in order to compose a systematic sizing process. However, these two components should be adapted to the available hybridization functionalities. Considering this, in the sequence is presented an overview of the requirements and methods for the full hybrid, plug-in hybrid and range extender functionalities.

Sizing requirements related to the hybrid functionalities

Full hybrid In this functionality the batteries have the power assistance function, which means that the total energy at the end of a mission should come from the hydrogen.

Therefore, the charge sustaining mode is implemented in the optimization algorithm as a constraint (SoE initial = SoE f inal ). Such power assistance function of the battery also aects the denition of the technical specications by the inclusion of a specic degraded mode. Considering a situation where the battery can not be used (defect or weak energy level), some of its functions like power boost are no more available, letting the fuel cell system alone in the traction function and possibly limiting the performances or the drivability. Such situation can be observed in a drive condition with slope. In order to tackle this operation mode, it is normally included a requirement where this type of drive condition can be performed using only one power supply element (fuel cell system in our case). For instance, Toyota [START_REF] Akihiro | Drive force control of a parallelseries hybrid system[END_REF] advises 105 km/h with 5 % slope as the degraded mode requirement.

Plug-in hybrid In this functionality the vehicle uses two types of energy (hydrogen and electricity) due to the fact that the embedded batteries can be externally recharged and depleted (until a certain limit) during the vehicle operation. Such characteristic hinders the optimization process because the cost function to be minimized by the optimization algorithm must associate the hydrogen and the electric energy of the battery by the translation to a common quantity. The monetary cost or the pollutant emissions (released during the hydrogen and electricity production) would be the best choices, but the related numbers are not easily available due to the uncertainty in cost and emissions brought by the mix of existing production process. Despite this inconvenient, the two power sources are able to meet the performance and drivability requirements, not needing to integrate degraded mode in the technical specications. However, a battery autonomy range (distance traveled using battery to power the vehicle) replaces the aforementioned constraint.

Range extender As mentioned in chapter 2, this functionality denes the battery as the main element in the electric power delivery function. Thus, this element has a size (energy capacity) compatible to full electric vehicle applications. In this context, the fuel cell system only works to recharge the battery, giving an extra autonomy range to the vehicle. The optimization involved can be derived from the plug-in hybrid with an extra constraint that avoids the traction directly using the power delivered by the fuel cell system. Normally, the extra drive range provided by the fuel cell system is present in the technical specication for vehicles with this functionality, which is used to dene the power and hydrogen capacity of the fuel cell system.

Among these functionalities, our work is intended to size a full hybrid FCHEV using optimization methods and technical specications for a given platform (type of vehicle).

In the sequence, the reference platform (vehicle) is presented followed by the description of the specications that will be used.

Reference Vehicle

In the design process, the vehicle platform can be chosen by selecting one of the two choices:

• Adoption of a reference vehicle commercially available The next section develops a set of technical specications that represents a combination of the reference vehicle performances index and the new requirements that includes the technologies involved.

•

Technical Specications

The set of the technical specications, to which the new types of vehicles are submitted, are normally based on statements and specications made by institutions or groups specialized in the subject. In the case of vehicles having new powertrain and new types of energy embedded, the Partnership for a New Generation Vehicles (PNGV) indicates a set of technical specications and performance requirements [START_REF] Moore | Vehicle design strategies to meet and exceed pngv goals[END_REF].

The PNGV document is very often adopted for sizing works, as in the case of Wu et al. in [START_REF] Wu | Component sizing optimization of plug-in hybrid electric vehicles[END_REF]. These specications are mainly used to dene the criteria to be applied in the sizing of the powertrain. However, it exists an index related to the fuel cell system that characterizes a fuel cell traction mode (all the traction power is due to the FCS).

In the next section, it is shown how the data of the table 4.4 are used in the proposed sizing methodology.

Sizing Methodology

The objective of the methodology described in this section is to accomplish, using a systematic approach, the sizing of the main elements in the powertrain and in the hybrid system of the vehicle. The denitions of the element's dimensions are related to the minimum and maximum limits of each component (power, torque, energy, etc). The elements and their main variables that are submitted to this sizing process are: 4.4). Nevertheless, for the fuel cell system and the battery pack the sizing methodology is not restricted to the adoption of the technical specications.

The methodology is summarized with the structure presented in gure 4.2. Each step of the process is described in the sequence. Electric Motor A synchronous motor technology is adopted in this work. This technology presented an outstanding performance in hybrid applications, such as in the Toyota Prius [START_REF]Toyota prius i-tech[END_REF][START_REF]Technologie des voitures electrices et hybrides[END_REF][START_REF] Kano | Design of slipring-less winding excited synchronous motor for hybrid electric vehicle[END_REF]. Moreover, this type of motor presents a good relationship between the maximum power and the mass of the motor (kW/kg). In general, this technology indicates a value about 2.7 kW/kg [START_REF] Kano | Design of slipring-less winding excited synchronous motor for hybrid electric vehicle[END_REF] for the range of the selected application (light-duty vehicle).

In VEHLIB a quasi-static model represents the characteristics of the motor. A consumption analyses can be performed by using a map that represents the losses in the motor related to torque and rotational speed (gure 4.3). The two variables have constant proles on their respective regions in the map.

0
The maximum torque can be obtained up to a base rotational speed ω base , while the maximum power can be obtained in the range of ω base and the maximum rotational speed ω max . In this work, the maximum rotational speed is established as 5900 rpm in order to comply with the technology of the base model.

Torque (N.m) Speed (rpm) wbase

Iso-power P max Iso-power P min wmax

M dv(t) dt = F traction -F resistance ; F traction = P traction v(t) M dv(t) dt = P traction v(t) -1 2 .ρ air .S.C x . (v (t) + v air (t)) 2 + M.g.µ.cos (α) + M.g.sin (α) (4.1)
Considering that the maximum speed of the vehicle v max is accomplished with a maximum traction power P traction max in steady state ( dv dt = 0) and ignoring the air speed (v air = 0), the equation 4.1 becomes the desired relation between the maximum traction power P traction max , the maximum speed v max and the slope angle α (equation 4.2).

P traction max = 1 2 .ρ air .S.C x . (v max ) 3 + v max .M.g.µ.cos (α) + v max .M.g.sin (α) (4.2)

Through this equation, it is possible to nd the maximum traction power value for dierent maximum speed values.

Imposing the two speed requirements (vehicle maximum speed and vehicle maximum speed in fuel cell mode) present in the FCHEV technical specication (table 4.4), the traction power necessary by the vehicle to reach each maximum speed can be calculated analytically.

Figure 4.5 illustrates the obtained power/speed relation. As can be seen, the adoption of a electric motor allows to obtain a maximum speed of 162 km/h with 42 kW of embedded traction power. Concerning the rst information about the fuel cell system dimension, the gure 4.5 shows that a fuel cell system with 30 kW of maximum power can reach the speed requirements established (110 km/h @ 5 % slope). This power value represents a minimal limit for the rated power of the fuel cell system. M dv(t) dt = F traction -1 2 .ρ air .S.C x . (v (t) + v air (t)) 2 + M.g.µ.cos (α) + M.g.sin (α)

         F traction = k red .Γ traction max
R wheel (max torque region)

F traction = P traction v(t)
(max power region) As the maximum traction torque (Γ traction max ) and the gear reducer gain (k red ) must be used in this step to solve the dynamic equation in 4.3. The initial value of Γ traction max was chosen from the electric motor map (g. 4.4), while the reduction gear gain is obtained by a calculation using the rotational speed ratio between the primary and secondary of the reduction gear.

Transforming the longitudinal maximal speed of the vehicle v max in maximal rotational speed at the secondary Ω wheel(max) (equation 4.4) and considering the maximal rotational speed at primary Ω motor(max) as 5900 rpm, the reduction ratio k red can be calculated by the equation 4. [START_REF]Technologie des voitures electrices et hybrides[END_REF].

Ω wheel(max) = v max .R wheel (4.4) Ω motor(max) = k red .Ω wheel(max) (4.5)
The transmission ratio value obtained from this calculation is 3.42.

Afterwards, the several congurations of the electric motor, according to the dierent power characteristics, are obtained by a homothetic transformation from the initial motor torque vs speed static map (4.4) that presents 27 kW of max power.

To perform the homothetic transformations it was considered the same technology for all the dierent power results. Other assumptions made were based in the work of Desbois-Renaudin et al. [START_REF] Desbois-Renaudin | Vehicules hybrides thermiqueselectriques. etude parametrique de l'architecture hybride parallele. rapport nal d'etude[END_REF]:

• The eciencies are kept for the same technology, no matter the considered power of the motor;

• The motor inertia changes proportionally to its power Such hypothesis leads to dene a homothetic rate k homothetic related to the maximum power desired to the electric motor and the maximum power of the reference motor (4.6) that is applied to the torque, losses and inertia of the motor. However, the maximum rotational speed is maintained at the same value (5900 rpm). k homothetic = P max motor P max ref erence

(4.6)
Applying this approach permited to built motor static maps to represent powers from 27 kW up to 50 kW. Thereafter, these static maps were used in the procedure illustrated in (4.6), which generated the results of gure (4.7). The analysis of the results shows that, out of all the acceleration criteria, the rst criterion (0 =⇒ 100 km/h) is the most constrained one. The constraint that species an acceleration time equal or lower than 13 s is achieved only for traction power values higher or equal to 43 kW . Considering that the transmission presents an eciency of 97 %, the minimal traction power delivered by the motor to meet this acceleration requirement [START_REF] Naganuma | Control system for sensing the dierential pressure between air and hydrogen in a pefc[END_REF] Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés is about 44 kW . As this value is in accordance with the maximum speed requirement (P max motor 42 kW ), it is retain as the maximum traction power of the vehicle.

Next step refers to the maximum torque, which is sized using the requirement of gradeability.

Maximum Torque Step

This part of the process is about the determination of the maximum traction torque and the validation of the mechanical transmission reduction gain.

The reduction ratio obtained in the previous step is veried veried by submitting the vehicle to the gradeability requirement (table 4.4). This is performed by simulation of the gradeability index using the vehicle dynamic model (4.3) and the parameters related to the elements of the powertrain and vehicle.

The results of the simulation indicate that the value of the reduction gain enables the achievement of the gradeability requirement, applying a maximum traction torque Γ motor(max) of 260 N m @ 350 rpm at electric motor level.

Drive train validation step

After the denition of the motor (maximum/ minimum torque and power) and the transmission ration, it is necessary to verify if the drive train proposed is capable to meet the the performance requirements contained in the technical specications.

Using the longitudinal dynamic model of the vehicle, the reduction gear model and the obtained electric motor representation (static map + inertia), the performance index are imposed to the vehicle and the results show that the sizing results permits to accomplish the performance requirements.

Conclusions about the drive train sizing

Throught the obtained results, the elements of the powertrain (motor + mechanic power transmission) are dimensioned according to its macroscopic energetic characteristics. Besides, a lower limit for the fuel cell system maximal power is known, which transforms the maximal speed in fuel cell mode to a FCS power constraint. Analyzing the summary of results in (4.5) it can be seen that the electric motor presents a reasonable power level while the fuel cell system presents maximum power that gets close to this level. The fuel cell system traction mode is responsible for this result, which does not means the best commitment between cost and performance of the vehicle.

The remaining elements in the sizing process are the components of the hybrid system (FCS and battery). They are sized using an approach based on the optimization of the energy consumption, mobility analysis and economic aspects. The next step presents the optimization part.

Optimization step

The hybridization present in the chosen architecture requires a more complex methodology in the process of the FCS and battery sizing, such that the nal conguration reduces the energy consumption.

Dierent types of solutions were already adopted for this problem. Using intelligent methods, such as articial neural networks, Jain et al. [START_REF] Jain | Optimal powertrain component sizing of a fuel cell plug-in hybrid electric vehicle using multi-objective genetic algorithm[END_REF] proposed a multi-objective approach for the components sizing in hybrid vehicles.

A sizing approach based on drive cycles and energy management strategy using fuzzy logic was proposed by Ravey et al. [START_REF] Ravey | Energy-sourcesizing methodology for hybrid fuel cell vehicles based on statistical description of driving cycles[END_REF]. This has as objective to nd a conguration that minimizes the hydrogen consumption in a fuel cell hybrid truck. The result indicates a better downsizing possibility of the fuel cell system.

Florian Duprez [START_REF] Dupriez-Robin | Dimensionnement d'une propulsion hybride de volier, basé sur la modélisation par les ux de puissance[END_REF] developed a sizing methodology for a hybrid system (ICE + battery) in a boat. This adopts the optimizations principles of the Dynamic Programming, which are inserted in a main algorithm. Despite the complexity of the problem, an optimal solution was obtained for the energy management of the boat and it was transported to the sizing of the components.

Associating Dynamic Programming and articial neural networks, Julien Scordia [START_REF] Scordia | Approche systématique de l'optimisation du dimensionnement et de l'élaboration des lois de gestion d'énergie de véhicules hybrides[END_REF] presented an optimal sizing procedure for dierent architectures of hybrid vehicles. The dimensions of the elements were well adapted to the energy management proposed in his work.

The choice of global optimization methods made by dierent authors is justied by the oine characteristic of the sizing process. As a good candidate for the optimization task, the Dynamic Programming method (introduced in the last chapter) is chosen to perform the optimization of the hydrogen consumption.

Optimization process

The entire optimal procedure is developed in the block 4 of the gure 4.2. Dierent drive cycles, previous sizing results and a range of power capacity and energy capacity (respectively to FCS and battery) are the inputs of the process, while the hydrogen consumption for each FCS + battery combination is the output. As the FCS and the battery chosen capacities are linked to the total mass of the vehicle, the increase of the embedded power or energy capacity requires more electric power to perform the drive cycle.

The possible capacities used for the battery are presented in the table 4.6. An assumption was made about the fuel cell system mass evolution. It states that there is a linear evolution of the entire system mass with respect of the number of the cells.

Energy (kW

The equation 4.7 shows this relation, where ϕ 1 and ϕ 2 were obtained by interpolation of data granted by CEA. They represent a ratio mass/power for the entire system (fuel cell + auxiliaries + connections) close to 2.3 kg/kW .

M sysF C = ϕ 1 N cell + ϕ 2 (4.7)
Table 4.7 shows the dierent congurations of the fuel cell system used in the process. The optimization algorithm is capable to calculate the optimal power proles of fuel cell system P * F C and battery P * batt related to a complete drive cycle. These optimal pro- les minimize the hydrogen consumption and ensure the constraints of the optimization problem.

Max. Power (kW )

The equivalent hydrogen consumption of the fuel cell system and the prole of the state of the energy SoE of the battery are calculated imposing their respective optimal power proles on energy consumption models. The SoE prole is particularly important to verify the constraint SoE initial = SoE f inal , indicating that all the energy is delivered from the hydrogen.

At the end of the optimization process, the results lead to dene the optimal size of the FCS and battery.

Optimization conditions and results

As presented in gure 4.8, the optimization process is always related to a certain drive condition. Such drive condition is represented by the drive cycles (speed proles) imposed to the vehicle and can represent either normalized drive proles or real ones based on the measurements and recordings made in real vehicles. In our work, speed proles with real characteristics are chosen in the optimization process. Normally, these types of drive cycles reects the driver behaviors when driving in urban, extra-urban and highway conditions. The three proles, used in this work, are present in the appendix 1. However, these types of drive conditions are not adopted with the same frequence by the drivers.

According to the type of travelled path, transportation objectives and the limitation of the drive laws (maximum speed limitation), some of drive cycles are favored than the others. In the context of this work, the transportation characteristics of duty-light vehicles in France are used to compose the sizing optimization.

Transportation context in France

In 2008 a research about mobility was demanded by the French government [START_REF]Enquete nationale transports et deplacement[END_REF]. The informations present in the data base of this research deal with dierent aspects of the population mobility, including the types of transport, distances, frequencies, etc.

In this part of the process, this set of statistical data is used to analyze and cover the existent gaps present in the sizing results.

Out of all the data, some information are particularly useful for the objectives of this part. They correlate the requirements of the population mobility with the energy range of a fuel cell hybrid vehicle:

• Types of displacements (drive cycles)

• Distance travelled per displacement

Regarding the type of displacement using light-duty vehicles, the research characterizes two types of drive cycles as the most representatives: urban and extra-urban. They were identied according to their average speed and the type of the route (city, between cities, rural, etc.)

Considering the total of data related to the type of the route for a driver in a lightduty vehicle, the results of the mobility research show a proportion of 28 % of urban drive routes and 72 % of extra-urban drive routes, as can be seen in the gure 4.10. They indicate that for extra-urban routes the range from 5 to 10 km is the most frequent while for urban routes the range from 2 to 5 km is the most covered.

2.87

Extra-Urban 14.5 Optimization results Three drive cycles were imposed to the optimization algorithm (DP). These drive cycles are presented in appendix 1 and have urban, extra-urban and highway characteristics.

The power proles of these cycles, obtained from the backward model, are presented in the gures 4.12 to 4.14. In this sizing step, some assumptions are made:

• The eciency of the fuel cell system DC/DC converter is considered constant as 95 %.

• The battery charge and discharge eciency have the the same constant value of 90 %.

• The electric motor has a simetric behavior in terms of torque and power (Γ motor(max) = Γ motor(min) , P motor(max) = P motor(min) ), allowing to apply the minimal power during recovery braking procedures. For this cycle, a hybrid conguration composed by 15 kW of fuel cell system and 3 kW h of battery pack shows be the best option.

As can be noticed, for fuel cell systems with less or more than 15 kW there is an over-consumption eect.

For lower values, there is less consumption of the auxiliaries and a reduced mass of the FC system. However, the optimal power prole P * F C requires to operate at a value closer to its maximum power, where the eciency is lower than its best value. Thus, the reductions in the mass and in the consumption of the auxiliaries do not compensate the losses in the fuel cell stack.

For a fuel cell system with a power capacity higher than the optimal, the increasing on the mass and specially on the auxiliaries' consumption leads to the noticed overconsumption.

Concerning the batteries capacities, two aspects of the pack are important in the denition of the optimal value: embedded mass and capacity to recover the energy during braking procedures.

For packs with less than 3 kW h there is a lower embedded mass, but the recovery capacity (recharge) is not enough to store all the available energy during the braking operations. This braking kinetic energy that is not stored can not be transformed in traction energy later. Such situation leads to use hydrogen when the traction is necessary instead of the possible recovery energy.

In the case of battery packs with a capacity greater than 3 kW h, the embedded mass causes the over-consumption. A conguration with more than 3 kW h will not bring more gain in eciency because the cycle do not provides more energy to be recovered.

Consequently, when the battery capacity increases there is no compensation of the mass increasing by the energy recovering. The results obtained when the extra-urban drive cycle was imposed to the vehicle shown a minimum consumption for a hybridization with 35 kW of fuel cell system and 5 kW h for the battery (gure 4.16). Through the results of the fuel cell system dimensions, it is possible to verify that the economy in hydrogen obtained adopting a 35 kW FCS has almost no dierence from the adoption of 25 kW , leading to choose this last conguration as the best in terms of performance/cost commitment.

The eects explained for the previous case are still valid for this cycle, but the power prole required by the vehicle leads to another optimal combination. The decoupling eect of the hybrid series architecture allows to the fuel cell system to adopt an optimal power behavior that tends to follows the average value of the cycle power prole. Therefore, due to the higher average power of this drive cycle (compared with the urban one), a fuel cell system with higher power capacity would be more adapted to deliver the power with better eciency and reduced hydrogen consumption.

Compared with the last drive cycle, the capacity obtained for the battery pack increased due to the potential in terms of energy recovery available in this cycle. For values greater or equal to 6 kW h the system is able to recover enough energy to compensate the losses with the augmentation of the embedded mass.

Concerning the fuel cell system, a value of 35 kW do not necessarily represent the best value. The increase of the maximum power could lead to a reduction in the consumption.

Nevertheless, it can be seen in (4.17) that the performance gains become weaker with the increasing of the maximum power. Moreover, the adoption of higher values of maximum power would lead to obtain a relation between the maximum power of the fuel cell system and the maximal traction power of the vehicle (P F C /P veh ) close to 90 %, compromising the hybridizing and downsizing ideas proposed in this work. Hence, a value of 35 kW is kept as reference for this cycle.

FCS Power capacity (kW)

Figure 4.17: Sizing study of battery and FC system for highway cycle Table 4.9 summarizes the best results for the three cycles.

Cycle

FC system max power (kW ) Battery energy capacity (kW h)

Urban 15 3
Extra-Urban

5

Highway 35 6

Table 4.9: Results of the optimization process

As the fuel cell system power constraint in the table 4.5 limits the value of the maximum power to values greater or equal to 30 kW , a fuel cell system capable to deliver 35 kW is the only available solution among the results of the optimization process. However, another criterion should be applied to verify if the gain in hydrogen economy compensate the extra cost related to a more powerfull fuel cell system.

Concerning the battery size, the optimization results are not able to dene the its energy capacity. The three dierent battery capacities in table 4.9 must to be submitted to other criterions in order to determine the nal battery sizing. The next step in the sizing process proposes the inclusion of the mobility aspects in order to aid in the denition of the battery pack dimension

Mobility statistical step

In the proposed analysis, the information of a distance travelled by a common driver is linked to the energy capacity of the battery. Considering in this work that the vehicle can operate in a battery traction mode (present in a plug-in hybrid functionality), the distance travelled in this mode would dene the capacity of the battery pack.

In this analysis, the data of travelled distance of urban and extra-urban drive cycles present in French transportation research are used. Nevertheless, the distance value can not be well dened using only the data of the gure 4.11.

In order to identify how a certain distance would satises the two types of drive cycles, the same data of the gure 4.11 are presented again in the gure 4.18 using a cumulated representation. Using this distance as the reference and keeping the SoE limits of the optimization problem, the proposition made here states that 40 % of the total energy capacity of the battery pack (∆SoE = 40 %) is sucient to accomplish a distance of 15 km.

The denition of the amount of energy required for this task is obtained by simulating the drive conditions on the forward vehicle model. Drive condition is represented by a drive cycle that is a series composition of the urban and the extra-urban drive cycles of the gures 4.12 and 4.13 with the same proportion presented in g. (4.10). The nal travelled distance is close to 15 km (14.7 km).

• 5 kW h for extra-urban drive cycles

• 6 kW h for highway drive cycles

The viability to add 0.75 kW h and 1.75 kW h to the reference value is veried in terms of the hydrogen savings (in euros) necessary to compensate the cost of the extra capacities.

In the extra-urban context, the optimal consumption for a 4.25 kW h conguration is about 10.5 g H 2 /km, while the best conguration in terms of hydrogen consumption (5 kW h) requires 10.3 g H 2 /km. Transforming the consumption dierence in monetary values, it is obtained 0.0005 ¿/km. This means that it would be necessary more than 990, 000 km to compensate the cost of an extra 0.75 kW h (496 ¿). Taking into account that a fuel cell hybrid vehicle has a target of life time that represents approximately 241, 000 km (150, 000 miles according to DOE [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF]), the reduction in hydrogen consumption do not justies the adoption of a 5 kW h battery.

For highway drive cycles, the dierence in cost consumption between the reference and the best result of the optimization process (6 kW h) is 0.001 ¿/km. This leads to require a life time that allows to reach approximately 992, 000 km to compensate the cost of extra 1.75 kW h, indicating a scenario as unjustiable as the previous one.

It can be concluded that the distance range criterion, dened by the battery traction mode, is a determinant factor on the denition of the battery's energy capacity due to the high cost of this component.

Fuel cell system sizing In the case of the fuel cell system, the reference value given by the fuel cell drive mode constraint (30 kW ). As the optimization results in tab. (4.9) presents only one value that respect the fuel cell system power constraint (P F C(max) 30 kW ), the hydrogen economy obtained adopting this FCS dimension (35 kW ) is compared with the cost related to the 5 kW increased (compared with the reference).

The highway optimization results presents a value of 18.9 g H 2 /km as the optimal consumption for a 4.25 kW h of battery and 30 kW of fuel cell system conguration, while the consumption obtained for the same battery capacity and 35 kW of FCS is 17.8 g H 2 /km. The monetary dierence indicates an economy of 0.0025 ¿/km by using a 35 kW FCS instead of 30 kW . Considering that the extra cost involved must be compensate by the hydrogen savings, it would be necessary 112, 000 km to achieve this objective. This value represents about 46 % of the target distance (241, 000 km), which means that this scenario (P F C(max) = 35 kW ) is economically feasible.

It can be concluded that the hydrogen savings obtained using a larger fuel cell system can compensate the cost of the extra embedded power, for a highway scenario, before the end of life of the vehicle. Therefore, a fuel cell stack of 35 kW is considered as the chosen dimension for this work.

Validation of the sizing

The dimensions of the fuel cell system and battery impacts the mass of the vehicle, which can aect the vehicle performances. Thus, the mass associated to these elements are integrated in the vehicle dynamic model and the performances in the technincal specications are veried if they are reached. This validation step indicated that the inclusion of the FCS and battery combination dened in the previous step in the vehicle still allows the vehicle to meet the assigned performance indexes, validating the size obtained.

Conclusions of the chapter

In this chapter, the models and optimization tools described previously were used in the conception of a sizing methodology for a fuel cell hybrid electric vehicle. Such methodology started with the denition of the reference vehicle and its characteristics.

The performances index of the reference vehicle provided an initial idea of the FCHEV functional requirements. However, the nal set of the specications and technical requirements was obtained combining the performances index of the reference vehicle and new index that reect the technological contribution brought by this transportation solution.

The total content of the functional requirements allowed to size the main elements of the powertrain (electric motor and mechanical power transmission).

Regarding the elements of the hybrid system (fuel cell system and battery pack), it was applied a specic procedure that uses the concepts of global optimization. This procedure lead to obtain the best combinations of FCS and battery pack in terms of reduction in hydrogen consumption for typical drive cycles: urban, extra-urban and highway. This part of the sizing methodology dened the FCS dimension and presented the better options for the battery pack.

In order to dene the energy capacity of the battery, it was rst used an approach based on the population mobility. In the sequence, it was eectuated an analysis of the economical cost about the choice of the battery capacity and the reduction in the hydrogen consumption. Analyzing the results about the elements in the hybrid system, the FCS presents a ratio P F CS(max) /P motor(max) with a value that is close to the indicated by vehicle constructors such as Daimler (80 %) [START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF]. Regarding the battery, it was obtained an energy capacity higher than the value indicated by the same constructor in its Class B hydrogen model (1.4 kW h). This dierence is due to two factors:

• Presence of a battery traction mode (allowing to drive the vehicle in zero-emission zone) that imposes an energetic condition in the sizing process.

• Lower capacity of the battery to deliver and recover power (10 C and -3 C) compared with the data indicated by the constructor (24 C and -7.2 C) [START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF].

In the next chapter, the sizing obtained is incorporated in the models and in the optimization problem with the objective to develop an optimal energy management strategy for online applications.

Introduction

Despite the presence of hybrid vehicles on the market, the energy management required for this kind of vehicles is still an open problem. Several studies have dealt with this topic proposing solutions that uses dierent methods, but there is no consensus of the method that is the most adapted. Besides, the way to treat with this problem diers from authors to authors, which means that the assumptions made will dierentiate the approaches. For instance, in the FCHEV particular case, the fuel cell system dynamics is often ignored in the solutions using optimizations methods as presented in chapter 3. Nevertheless, this dynamics is still considered as technological drawback that should be overcome, being approached in this work as a main objective such that it would be included in the energy management strategy.

Another concern of the energy management methods is about its applicability for a problem that requires online responses without too much computation time and memory capacity, such as the vehicle applications. For our specic problem, the optimizations methods presented in the chapter 3 have shown the ability to solve the typical optimization problem for a FCHEV, but the non causal characteristics of their implementations avoid the applicability for problems addressed to real time systems, such as the hybridization in the vehicle. Therefore, this chapter's goal is to propose an online optimization algorithm capable to solve an optimization problem as stated in the chapter 3 and that includes the dynamics of the fuel cell system in its constraints. In order to achieve these goals, an outline is used to development process.

The dynamics of the fuel cell system have been presented in the rst part of the chapter. This representation is obtained from the dynamic equations of the main elements that compose the system.

Afterwards a control strategy is applied to the fuel cell system in order to control the state variables and to linearize the dynamics. The controller's performance is veried applying it on the validation dynamic fuel cell system model introduced in chapter 3. Thereafter, the control law is tested in the real fuel cell system that integrates the PROSPAC prototype (appendix B), where the perormance of the proposed control law is compared to other strategies frequently used in the automotive applications.

A simplication performed by application of the control law allows to include this new dynamics as a constraint in the optimization problem, which is presented in the sequence of the chapter.

Based on an algorithm that allies the Pontryaguin's maximum principle with a predictive approach, a local optimization algorithm is designed and tested by comparing their results with a global optimization method.

The algorithm is tested and validated by simulations and experimental campaigns.

During the testing procedure, several types of cycles are used to conrm the validity of the strategy. Dierent speed and altitude proles permit to validate the algorithm robustness to maximize the energy recovery under dierent conditions.

Fuel Cell System Dynamics

Composed by several elements in dierent subsystems, the fuel cell system has its dynamics determined by a complex set of interactions among the stack, the dierent subsystems and the dynamic evolutions of the external environment.

These interactions can associate state variables with dierent dynamic behaviors, which lead to undesired eects. Such undesired eects are often present in the case of multi-physical systems. For the specic case of the fuel cell system, several harmful eects result from the possible dierent dynamics.

In this work, the dynamics of the fuel cell system is represented by the dynamics of the subsystems that are the most constrained, meaning the dynamics that can more often generate some harmful eect for the stack. Therefore it is necessary a rst overview of the most representatives harmful eects for the fuel cell stack.

Out of the numerous harmful eects, two particular ones have the special attention due to their common presence in normal operations and their important degradation results: hydrogen and oxygen starvation.

Thounthong et al. [START_REF] Thounthong | Analysis of a fuel starvation phenomenon of a pem fuel cell[END_REF] shown the eects caused by hydrogen starvation in the stack during an electric load demand. The failure to feed the stack with hydrogen as fast as the electrical demand, is responsible for dierent undesirables eects to the system. Beside the losses in the fuel cell stack performance observed with the time, hydrogen starvation causes irreversible degradation eects in dierent parts of the stack leading to a reduction of the life time [START_REF] Schmittinger | A review of the main parameters inuencing long-term performance and durability of {PEM} fuel cells[END_REF][START_REF] Thounthong | Analysis of a fuel starvation phenomenon of a pem fuel cell[END_REF][START_REF] Wu | A review of pem fuel cell durability: Degradation mechanisms and mitigation strategies[END_REF].

On the cathode side, the lack of oxygen during electrical demands imposes the fuel cell stack to operate in a condition that also aects the performance and the characteristics of the stack. This oxygen starvation was discussed by dierent authors. Gerard et al. [START_REF] Gerard | Oxygen starvation analysis during air feeding faults inpemfc[END_REF] presented the eects of the oxygen starvation on the dierent characteristics of the fuel cell. Schimittinger et al. [START_REF] Schmittinger | A review of the main parameters inuencing long-term performance and durability of {PEM} fuel cells[END_REF] highlighted the importance of this phenomenon among the types of harmful phenomena that can happen during an operation of the fuel cell system.

Even though hydrogen starvation can produces highly degradation eects, oxygen starvation is considered the phenomenon that denes the dynamics of the fuel cell system due to the control eort necessary to avoid it.

As the oxygen starvation results from a failure in the interaction between the electric and air supply subsystem, more precisely, between the current demand and the oxygen ow in the fuel cell active layer sites.

Compared to the components of the electric subsystem (DC/DC converter), the elements present in the air supply subsystem have a slower dynamics (air compressor valve, etc.). Consequently, the dynamics of the air supply subsystem is considered to represent the fuel cell system dynamics.

Being also considered by other authors [START_REF] Blurnier | Modelisation de moto-compresseurs en vue de la gestion de l air dans les systemes pile a combustible -simulation et validation experimentale[END_REF][START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF][START_REF] Tawee Pukrushpan | Modeling and control for pem fuel cell stack system[END_REF] as the most representative dynamic of the system, the air subsystem dynamics has a non linear characteristics. Equations 3.17, 3.14 and 3.23 of the chapter 3 show that the dynamics is represented by the non-linear equations of the compressor, cathode capacity and pressure control valve. In these dynamic equations, the control of the compressor and the valve should regulate the air stoichiometry (air ow) and the cathode pressure in order to assure a good reaction.

The application of control strategies would allows a regulated and stable behavior of the output variables to be obtained. This subject is presented in the following section.

Air circuit control

A control strategy for the air supply subsystem must be implemented such that the stoichiometry (ratio of the input oxygen ow over the reacted oxygen ow) is maintained around a constant desired value, whatever the power demand is. Moreover, the control of the oxygen ow should be coupled to the regulation of the pressure in the cathode, keeping a pressure equilibrium between cathode and anode.

The command of the motor-compressor in the air group is the way to regulate the stoichiometry, but it should deal with an important barrier that is the nonlinear characteristics of the components of the air supply subsystem (compressor, valves, capacity volume, transport phenomena) [START_REF] Vahidi | Current management in a hybrid fuel cell power system: A model-predictive control approach[END_REF][START_REF] Venturi | Air supply system for automotive fuel cell application[END_REF].

Several authors dealt with this subject with dierent approaches. Tekin et al. [START_REF] Tekin | Energy consumption reduction of a pem fuel cell motor-compressor group thaks to ecient control laws[END_REF] adopted the fuzzy logic with particle swarm approach to control the motor compressor;

Grujicic et al. [START_REF] Grujicic | Model-based control strategies in the dynamic interactions of air supply and fuel cell[END_REF] applied static and dynamic feedfoward control techniques for the air ow control. Nonlinear approaches using passivity, sliding mode and prediction control techniques were developed in [START_REF] Talj | Modeling and control of the air supply system of pem fuel cells[END_REF][START_REF] Talj | Second order sliding mode control of the motocompressor of a pem fuel cell air feeding system, with experimental validadtion[END_REF][START_REF] Vahidi | Current management in a hybrid fuel cell power system: A model-predictive control approach[END_REF] respectively. Theses approaches obtained good results in terms of oxygen ow.

However, the works mentioned previously assume that the anode pressure follows the cathode pressure value. This means that the cathode pressure is not regulated around a desired value and it changes according to the control. Therefore, in order to maintain the pressure equilibrium between cathode and anode, a fast electro-valve is required to control the anode pressure and to adapt its setpoint with the real time value of the cathode pressure [START_REF] Al-Durra | Study of nonlinear control schemes for an automotive traction pem fuel cell system[END_REF][START_REF] Naganuma | Control system for sensing the dierential pressure between air and hydrogen in a pefc[END_REF].

In order to tackle this problem and to implement a robust solution for the control of the air supply system, this work proposes a non linear control design that enables stoichiometry and pressure control to be coupled. Hence, the oxygen stoichiometry and the cathode pressure are controlled at the same time, keeping the variables close to their desired values. Moreover, the proposed control solution must preserve the stack performance and its durability during all the operation.

The control design is based on the dierential atness theory [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF]. According to this theory, if at outputs exist, it is possible to obtain the full control of all the system state variables and to impose the system to track desired trajectories as imposed on these at outputs.

Dierential atness strategy is designed using a simplied fuel cell system model and is rstly tested using the dynamic model of the fuel cell system introduced in chapter 3

and after implemented and tested in the real fuel cell system (PROSPAC).

Control Model

A simplied model of the air supply system has been designed based on the respective model used in the hybrid system model (chapter 3). Some assumptions are made to reduce the model complexity and dimension:

• Operation in gas phase only (no multiphase ow).

• The humidier is represented only by its equivalent pressure drop without dynamics and transport phenomena.

• The transport phenomena inside the fuel cell are not taken into account. Therefore, diusion through the membrane is not taken into account (no gas crossover or water transport).

• The temperature inside the fuel cell is homogeneous and perfectly controlled.

• The cathode is represented by a global volume and a pressure drop from the inlet to the outlet (gure 5.1).

q cp = aω 2 + bω        ψ 1 ψ 2 - 1 ψ 2 1 -psm patm -1 γ-1 γ        (5.1)
Finally, the model used for the control synthesis is given by equation 5.2.

                       dω dt = 1 J α.u mot -CpTatm ωηcp psm patm γ-1 γ -1 q cp (ω, p sm ) -ϕω dp cath(out) dt = RT cath V cath m molar (q cp (ω, p sm ) -q O 2 + q H 2 O -q valve ) ẋvalve = 1 τ valve (-x valve + k valve u valve ) (5.2)
Values of the parameters in equations 5.1 and 5.2 are presented in appendix C.

The outputs to be controlled are the cathode outlet pressure p cath(out) and the compressor air ow q cp . However, there is a coupling between these two variables, which leads to implement a non-linear control strategy to decouple the outputs, linearize the system and thereafter apply a classical control strategy for the equivalent linear system.

Proposed Control Strategy

A non linear control strategy called dierential atness [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] is applied to the non linear system presented previously. This control strategy permits to dene the control inputs u directly from the desired output trajectories y and its derivatives.

In comparison with other approaches, the dierential atness strategy permits to compensate the known non-linearities of the system, setting a trajectory for the at outputs and ensuring performance for the system. However, a stabilization feedback loops are required to ensure robustness to the system. The theory that denes this control strategy is briey presented, followed by its implementation.

Dierential atness theory

According to Fliess et al. [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF][START_REF] Fliess | A lie-backlund approach to equivalence and atness of nonlinear systems[END_REF], given a non linear system such as 5.3, if all the states x i can be expressed as functions of the outputs and their derivatives, the system in 5.3 is dened to be dierentially at and admits the at output y dened as in 5.4.

ẋ = f (x, u) ; x n ; u m ; (5.3)                      y = [y 1 , y 2 , y 3 , ..., y m ] T ; y m with y = φ x, u, ..., u (θ) (5.4)
The state variables and the input variables can be written as in 5.5.

         x = f 1 y, ẏ, ..., y (κ) u = f 2 y, ẏ, ..., y (κ+1) (5.5)
Where θ and κ are a nite number of derivatives. The summation of the relative degree of each output must have the same value as the system characteristic index [START_REF] Isidori | Nonlinear control systems[END_REF].

Implementation of the dierential atness control strategy

In order to implement the dierential atness control strategy in 5.2, the states (ω, p cath(out) and q valve ) and the inputs (u mot and u valve ) should be function of the at output y 1 and y 2 . Therefore, choosing the at outputs as ω and p cath(out) , two states are directly the at outputs, while the controls (u mot and u valve ) and the other state (x valve ) are developed in sequence.

Considering the dynamic model of the air supply subsystem (5.2) with a simplied representation performed by the adoption of the functions f i as in (5.6). The compressor air ow q cp is represented function of ω and p cath(out) due to the transformation that can be done from the cathode outlet pressure p cath(out) to the supply manifold pressure p sm by the application of the pressure drops in the cathode and humidier k hu + k F C .

In accordance with the model presented in chapter 3, the contribution of the pressure valve ow q valve in the cathode pressure dynamics is represented by two functions (f 5 and f 6 ) that are dependent on x valve and p cath(out) . The cathode outlet pressure p cath(ou) appears in the functions due the application of the pressure drop in the humidier k hu that transforms the valve inlet pressure to p cath(out) .

                     dω dt = f 1 (u mot ) + f 2 ω, p cath(out) + f 3 (ω) dp cath(out) dt = f 4 ω, p cath(out) , q O 2 , q H 2 O + f 5 x valve , p cath(out) + f 6 p cath(out) ẋvalve = f 7 (x valve ) + f 8 (u valve ) (5.6)
Derivating the rst at output y 1 and changing the state variables ω and p cath(out) by the at outputs y 1 and y 2 in (5.6), the control input u mot is obtained from the rst equation of the model as:

u mot = f -1 1 ( ẏ1 -f 2 (y 1 , y 2 ) -f 3 (y 1 )) (5.7)
Equation (5.7) shows that u mot depends on the at outputs and its derivatives ( ẏ1 , y 1 and y 2 ), dening an initial characteristic index ρ 1 = 1.

Derivating the second at output y 2 and replacing the function f 5 by a more detailed representation(5.8), in accordance with q valve model, the state variable can be represented in (5.9). f 5 (x valve , y 2 ) = x valve .f 9 (y 2 )

(5.8)

x valve = ẏ2 -f 4 (y 1 , y 2 , q O 2 , q H 2 O ) -f 6 (y 2 ) f 9 (y 2 ) (5.9)
As can be noticed, x valve depends on the at outputs, their derivatives, q O 2 and q H 2 O .

These two last elements are not considered as state variables, but as known disturbances which are estimated from the FC reaction according to the desired supplied current.

Therefore, the three state variables conrm the adoption of the assigned at outputs.

However, it is necessary to verify if the control of the pressure valve also meets this requirement.

In order to obtain u valve , the at output y 2 is derivated twice as presented in (5.10), where x valve is replaced by the function f 10 to simplify the representation.

ÿ2 = d dt (f 5 (y 1 , y 2 , q O 2 , q H 2 O )) + ẋvalve f 9 (y 2 ) + • • • • • • + f 10 ( ẏ2 , y 1 , y 2 , q O 2 , q H 2 O ) d dt (f 9 (y 2 )) + d dt (f 6 (y 2 )) (5.10)
Replacing ẋvalve by its equation in (5.6), the control u valve is represented by the equation (5.11).

u valve = f -1 8 ÿ2 -d dt (f5(y1,y2,qO 2 ,q H 2 O ))-f10( ẏ2 ,y 1 ,y 2 ,q O 2 ,q H 2 O ) d dt (f 9 (y 2 )) f 9 (y 2 ) • • • • • • + d dt (f 6 (y 2 )) f 9 (y 2 ) -f 7 (f 10 ( ẏ2 , y 1 , y 2 , q O 2 , q H 2 O )) (5.11)
u valve equation also shows its depence only on the at outputs, their derivatives, the known disturbances and their derivatives, indicating that the chosen outputs (ω and p cath(out) ) are suitable to become the at outputs. Consequently, according to [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF][START_REF] Fliess | A lie-backlund approach to equivalence and atness of nonlinear systems[END_REF], the system is dierentially at, admits the at outputs y 1 and y 2 and permits a stable inversion to dene the control inputs. Besides, the characteristc index ρ 2 of the cathode pressure dynamics equals 2 leads to obtain the characteristic index of the system equals 3 (ρ system = ρ 1 + ρ 2 = 3), indicating that the dynamics resulted from the control strategy do not present a residual part.

After the conrmation about the existence of the at outputs, a trajectory generation should be done for them. The output p cath(out) has as desired trajectory a stationary prole (p d cath(out) = cte), where the value is the same at the anode pressure, assuring a balance pressure between cathode and anode.

The denition of the motor speed desired trajectory ω d begins with the assignment of the desired air ow in the fuel cell stack (5.12). This is directly dependent to the desired current trajectory i d F C (eq. 5.12), which is a known variable controlled in the electric subsystem (DC/DC converter). This air ow is supplied by the compressor (q

d air = q d cp )
and is function of two variables: desired supply manifold pressure p d sm and compressor speed ω d (eq. 5.13). In equation 5.13, p d sm is obtained by the equation 5.14 that takes into account the pressure drop at the fuel cell stack and the humidier (k F C and k hu ). Therefore, in order to obtain ω d , the quadratic equation 5.15, obtained from 5.12, 5.13 and 5.14, must be solved. According to the equation 5.15, the motor speed desired prole

ω d is function of i d F C and p d cath(out) . q d air = St O 2 χ O 2 N cell i d F C

4F

(5.12)

q d cp = a ω d 2 + bω d        ψ 1 ψ 2 - 1 ψ 2 1 -p d sm patm -1 γ-1 γ        (5.13) p d sm = p d cath(out) + q d air 2 (k hu + k F C ) (5.14) a ω d 2 + bω d                    ψ 1 ψ 2 - 1 ψ 2       1-     p d cath(out) + St O 2 χ O 2 N cell i d F C 4F 2 (k hu +k F C ) p atm -1     γ-1 γ                          • • • • • • - St O 2 χ O 2 N cell i d F C 4F = 0 (5.15)
Concerning the valve behavior, the trajectory of its pressure dierence ∆p valve , used to calculate the mass ow q valve , is given from equation 5. [START_REF] Askarzadeh | Articial neural network training using a new ecient optimization algorithm[END_REF].

∆p valve = p d cath(out) -k hu q 2 valve -p atm (5.16)
Therefore, applying this equation in the valve mass ow equation (3.22 in chapter 3), the new value of q valve is given by the resolution of the quadratic equation 5.17. The solution is given by equation 5.18 ensures that for all x valve 0 and p d cath(out) p atm there is a q valve 0. The presence of the displacement in the valve control law integrates the content of its mechanical characteristics that approximates to the physic behavior.

q 2 valve -

(ζ 1 x valve + ζ 2 ) 2 1 + k hu (ζ 1 x valve + ζ 2 ) 2 p d cath(out) -p atm = 0
(5.17)

q valve = (ζ 1 x valve + ζ 2 ) p d cath(out) -patm 1+k hu (ζ 1 x valve +ζ 2 ) 2
(5.18)

The system 5.19 presents the new dynamic of the air group, where the constants are replaced by c i (to simplify the notation), f comp calculates the air desired ow supplied by the compressor (q * cp ) and f valve calculates the desired air ow passing across the valve

(q * valve ).                          dy 1 dt = c 1 .u mot -c 2 1 y d 1 y d 2 +fcomp(y d 1 ,y d 2 )(c3) patm γ-1 γ -1 f comp y d 1 , y d 2 -c 4 y d 1 dy 2 dt = c 5 f comp y d 1 , y d 2 -q O 2 + q H 2 O -f valve x valve , y d 2 dx valve dt = 1 c 6 (-x valve + c 7 u valve ) (5.19)
The rst control variable u mot is claculated in 5.20, where W 1 is the new input that replaces the term dy 1 dt .

u mot = W 1 + c 2 1 y d 1 y d 2 +f 1( y d 1 ,y d 2 )(c3) patm γ-1 γ -1 f comp y d 1 , y d 2 + c 4 y d 1 c 1 (5.20)
As mentioned before, the second control variable is calculated in two steps. The rst derivative of the at output y 2 allows to calculate q valve (equation 5.21) and using this result in 5.17 the value of x valve is obtained as in 5.22.

q valve = f comp y d 1 , y d 2 -q O 2 + q H 2 O - 1 c 5 dy d 2 dt = f valve x valve , y d 2 (5.21) x valve = 1 ζ 1   q valve y d 1 , y d 2 , ẏd 2 , q O 2 , q H 2 O y d 2 -k hu q valve y d 1 , y d 2 , ẏd 2 , q O 2 , q H 2 O -p atm -ζ 2   (5.22)
As can be noticed, it is necessary to compute the rst derivative of the desired output trajectory y d 2 to nd the value of x valve . In our case the value of the rst derivative is zero because the desired trajectory for the outlet cathode pressure is a constant value.

Finally, the second control input u valve is obtained from the second derivative of the at output y 2 (5.23). In 5.23, the rst derivative of q valve is calculated using the equation 5.18. It presents the terms of x valve and ẋvalve that are respectively replaced by 5.22 (calculated previously) and the dynamic equation of the valve.

Therefore, the control variable u valve is presented in 5.24, where x valve is replaced by a function

f x valve y d 1 , y d 2 , ẏd 2 , q O 2 , q H 2 O to simplify the presentation and W 2 is a new input that re- places the term d 2 y 2 dt 2 . d 2 y d 2 dt 2 = c 5 ḟcomp y d 1 , y d 2 -qO 2 + qH 2 O -qvalve (5.23)
u valve = f x valve c 7 +        1 c 5 W 2 -ḟcomp + qO 2 -qH 2 O k hu ζ 1 (ζ 1 f x valve + ζ 2 ) - [k 2 hu ζ 1 (ζ 1 f x valve +ζ 2 ) 3 +ζ 1( y d 2 -patm)] k 2 hu (ζ 1 f x valve +ζ 2 ) 4 -4[(ζ 1 f x valve +ζ 2 ) 2 (y d 2 -patm)]        (5.24)
In the sequence, the eects of these control laws in the system dynamics are presented.

New Dynamics of the System

A new open loop structure of the system is obtained as the result of the proposed control laws. This consists of an integrator chain as can be seen in (5.25).

                     Żd 1 = W 1 Żd 2 = Z d 3 Żd 3 = W 2 (5.25)
For the new linear system in 5.25 it is possible to apply classic control approaches.

The state space representation in (5.26) allows to apply a state feedback (U = -KX) to assure the stability.

   Żd 1 Żd 2 Żd 3    =    0 0 0 0 0 1 0 0 0       Z d 1 Z d 2 Z d 3    +    1 0 0 0 0 1    W 1 W 2 = Ẋ = AX + BU Y 1 Y 2 = 1 0 0 0 0 1    Z d 1 Z d 2 Z d 3    = Y = CX (5.26)
Assigning the poles positions as (-10, 10i), (-10, -10i) and (-50, 0i), there is a guarantee that the system is stable. The K matrix obtained from this pole positioning is:

K = 50 -10.57 -0.63 -0.72 200.18 20 = K 1 K 2 (5.27)
The nal linear system with its control is given in (5.28). • 1 : calculated air ow that must be delivered to the stack (equation 5.12).

                     Żd 1 = W 1 = y d 1 -50.
• 2 : calculated desired trajectory of the supply manifold pressure (equation 5.14).

• 3 : desired motor speed (inversion of the equation 5.1).

• 4 : control law for the motor-compressor (equation 5.20).

• 5 : control law for the pressure valve (equation5.24).

• 6 : motor-compressor (rst equation of system 5.2).

• 7 : cathode of the fuel cell (second equation of system 5.2).

• 8 : pressure valve (third equation of system 5.2). + -q cp q O2 q H2O q valve p cath(out) • Cathode outlet pressure (reference=1.2 bar).

• Compressor speed prole.

• Pressure valve behavior. These variables were obtained from the hybrid system model when it is submitted to a power prole dened in gure 5.3. 2 ) will compensate the low dynamics of the valve during the fast transients. This avoids important pressure drops on the cathode outlet. Moreover, the gure 5.5 shows that the disturbance created by the periodic purge procedure in the anode is compensated by an action in the valve control in order to reject this perturbation in the cathode pressure prole. The cell voltage follows the demanded power prole, which means that the fuel cell state conditions allow to deliver a rated power without a signicant voltage drop at the cell level. Thus, even when the maximum power was demanded, the cell voltage remains close to 0.7 volts. The compressor control, coupled with the control valve, avoids as well the choked ow regime at the compressor, since the pressure ratio has a value greater than the critical regime (Π critic = 0.52) during all the operation. Nevertheless, this coupled control, linked also to the dynamic operational conditions, could aect the durability of the stack directly or indirectly by its eects on dierent internal variables, such as oxygen molar fraction, cell voltage and membrane humidication, for instance [START_REF] Tang | Experimental investigation of dynamic performance and transient responses of kwclass pem fuel cell stack under various loads changes[END_REF].

In terms of degradation, the result of oxygen molar fraction presents a non-homogeneous distribution inside the stack (gure 5.7), however the minimal level assures the operational conditions of the stack without losses in the performance, as presented by the cell voltage response (gure 5.6). In order to obtain a better distribution, a control law that takes into account the geometry inside the stack should be applied. The control law shows good results in the regulation of the air circuit variables and fuel cell voltage response; however, it is important to observe if there is no side eect on other important state variables, such as the humidity in the fuel cell stack. As a variable dependent on a passive element (humidier), the relative humidity is indirectly subjected to actions on the other state variables. Therefore, even if the control of this variable is out of the scope of this work, the relative humidity in the fuel cell is observed in order to assure that the air subsystem control law does not create an undesired eect on it.

During the operation, the humidity remains acceptable (>50%) at anode and cathode (gure 5.7) and converge towards an equilibrium level at the end of the cycle, which reects the good water contents of the membrane (gure 5.7). This avoids degradation eects due to membrane dryness. One can be observed the presence of small oscillations in the humidity response, which are related to the purge in the anode. The purge procedure reduces the humidity in the anode active zone, creating an instantaneous water transport from the cathode active zone to the anode active zone through the membrane, which explains the phenomenon at the three levels (cathode active zone, membrane and anode active zone).

Results in Real Fuel Cell System

The performance of the control law is tested in the real fuel cell system contained in PROSPAC (appendix B). The reference for the oxygen stoichiometry is 2, for the cathode outlet pressure is 1.2 bar and for the stack temperature measured on the cooling liquid at stack's outlet is 60 °C. The most important variables to be observed are:

• Compressor speed prole.

• Cathode pressure prole.

A current demanded prole in gure 5.8 was imposed to the fuel cell system, in order to verify the response of three control strategies:

• Classical approach using two PID control loops (gure 5.9)

• Dynamic prepositioning control strategy (gure 5.10)

• Flat control strategy (gure 5. The speed and pressure responses in gures 5.11 and 5.12 shows the good tracking performances of the at control strategy. As it can be seen, the at control strategy provides a better response time than the prepositioning strategy and its decoupling property avoids the interference of one variable on the other, as it is observed for the PID strategy.

The performance of the at control strategy has inuence on the fuel cell stack response. Figure 6.5 illustrates the voltage and power response for this control strategy

Conclusion on the air supply subsystem control

The non linear control strategy applied to the complex dynamics of the air subsystem provided good dynamic responses in dierent levels of the system. Compared with classical solutions adopted, the at control strategy oered a stabilization of the dynamics with a controllability and stability obtained from its decoupling property.

Regarding the application for vehicles, the results shown a response time lower than 300 ms to reach the demanding setpoints, which is perfectly adapted to the range of the FCHEV applications that requires a response time from one to a couple of seconds.

Moreover, the control strategy enables to represent the fuel cell system by an equivalent linear model in the energy management process. In the next section, the linearized dynamics given in 5.25 is introduced in the energy management optimization problem and used to the denition of a online optimization method.

                     ε = f 1 (P batt ) = -P batt ω = f 2 (W 1 ) = W 1 ṗcath = f 3 (W 2 ) = W 2
(5. [START_REF] Caux | On-line fuzzy energy management for hybrid fuel cell systems[END_REF] In order to solve this problem with an online approach, it is proposed a new optimization method.

Online predictive energy management method

When the complete driving cycle is unknown, the optimization problem can not be solved at once. Thus, a new approach based on the Pontryaguin's Minimum Principle (PMP) [START_REF] Christophe | Introduction à l'Optimisation[END_REF][START_REF] Bernard | Fuel cell hybrid vehicles: Global optimization based on optimal control theory[END_REF] allied with a predicting horizon approach is proposed in this work. This method deals with the constraints over the cycle, compromising as less as possible the optimization step.

Appling the PMP, the optimization problem in (5.29) can be solved using the Hamiltonien formulation (5.31). In this equation, the power balance was applied, such that the control inputs are dened as P batt , W 1 and W 2 .

H (ε, ω, p cath , P batt , W 1 , W 2 , λ 1 , λ 2 , λ 3 ) = g (P vehtot -P batt ) + λ 1 f 1 (P batt ) + λ 2 f 2 (W 1 ) + λ 3 f 3 (W 2 ) = g (P vehtot -P batt ) -λ 1 (P batt ) + λ 2 W 1 + λ 3 W 2 (5.31)
The optimal solution is obtained applying the conditions in (5.32) [START_REF] Christophe | Introduction à l'Optimisation[END_REF][START_REF] Vinot | Optimal management of electric vehicles with hybrid storage system[END_REF][START_REF] Bernard | Fuel cell hybrid vehicles: Global optimization based on optimal control theory[END_REF].

                     ε = ∂H ∂λ 1 ; ω = ∂H ∂λ 2 ; ṗcath = ∂H ∂λ 3 -λ1 = ∂H ∂ε ; -λ2 = ∂H ∂ω ; -λ3 = ∂H ∂p cath ∂H ∂P batt = 0; ∂H ∂W 1 = 0; ∂H ∂W 2 = 0 (5.32)
The conditions about the derivatives of the co-states λ n show that they are constants ( λn = 0), while the minimization of H according to the control inputs deliver the optimal power prole of the battery P * batt and the values of the co-states λ 2 and λ 3 (6.7):

∂H ∂P batt = 0 = 2a 1 P * batt (t) + (2a 1 (P vehtot (t)) + a 2 ) -λ 1 ⇒ ⇒ P * batt (t) = λ 1 -(2a 1 (P vehtot (t))+a 2 ) 2a 1 ∂H ∂W 1 = 0 = λ 2 ∂H ∂W 2 = 0 = λ 3 (5.33)
An analytic function represents the optimal prole P * batt , which depends on the known variable P vehtot and the constant co-state λ 1 . In general, λ 1 is related to the knowledge of the drive cycle in order to obtain the boundary condition (SoE (0) = SoE (t f )).

Therefore, assuming that the information of the entire drive cycle is not available, the presence of λ 1 in P * batt calculation represents a limitation for online applications. To overcome this limitation, a predictive strategy is adopted to dene the value of λ 1 .

Predictive approach

Isolating λ 1 in the response of the optimization problem (equation 5. P vehtot distribution: Observing equation 5.37, it can be seen that the time ordered prediction to the future power P vehtot is not necessary. Therefore, it is not important to know when a specic power demand will be done. Using this observation, this proposed method makes a simple hypothesis that the distribution of P vehtot during a predicted nite horizon is assumed as the same of the last nite horizon ( t+t h j=t P vehtot (j) = t-1 i=t-t h P vehtot (j)).

Despite the simplicity, this prediction represents a valid and computational ecient prediction approach, according to Bernard et al. in [START_REF] Bernard | Fuel ecient power management strategy for fuel cell hybrids powertrains[END_REF].

For this application, the nite horizon is applied as a sliding window of time. The gains (k p and k i ) of the PI controller are obtained using a pole placement procedure. For this objective, the dynamic system is composed by the battery dynamics (SoE calculation) and the control part (equation 5.34), where λ 1 is replaced by (5.38) Applying the Pade approximation [START_REF] Puri | Stable model reduction by impulse response error minimization using michailov criterion and pade's approximation[END_REF] for the presented time delay, the block diagram of gure 5.14 can be represented by the equation (5.39). The aim of this control procedure is to place the poles in the stable region without vanishing P vehtot component.

SoE = P vehtot εmax t h s 2 +2s-4 t h s 3 +(2+t h kp)s 2 +(2kp+t h k i )s+2k i + SoE ref erence kps+k i s 2 +kps+k i (5.39)
With a prediction horizon of 150 s, the values of k p and k i are respectively 0.087 and 0.0014, giving the poles positions at (-0.0567, 0), (-0.0213, 0), (-0.0213, 0), (-0.004, 0) and (-0.004, 0). This prediction horizon value aects the calculus of λ 1 and the optimization results. In our work a brief parametric study about the duration of the prediction horizon was performed, which indicated that there is a value adapted to each type of cycle. Figure (5.15) shows the results for a extra-urban drive cycle, where the value of 150 s presented the best performance in terms of hydrogen consumption. However, for dierent drive cycles it was found other values of t h , indicating that that this subject still needs a better analysis, such as the identication of the drive cycle to adapt the predictive horizon duration.

λ 1 = 2a 1 [kp(SoEreference-SoE(t))+ki ´(SoE ref erence -SoE(t))dt].εmax ∆t -t-1 i=t-t h P vehtot (i) t h ∆t + a 2 (5.40)
The last variable to be dened in (5.40) is the SoE ref erence . It has a dynamic prole, dierently from several authors that adopt the SoE ref erence xed in the middle of the SoE operational range. This permits to increase the entire system eciency, taking advantage of the kinetic and potential energy.

The main idea is related to the fact that when the vehicle has a non zero positive speed, the amount of kinetic energy can be partially recovered during the braking operations.

Consequently, in this situation the vehicle can optimize the energy recovering increasing the SoE region by the reduction of the SoE ref erence value.

The same concept is used for the potential energy. SoE ref erence reduces when the vehicle passes to a higher altitude, which means that it will have an adapted SoE region to maximize the braking recovering energy, compensating the potential energy barrier to reach the current altitude.

Otherwise, when the vehicle is in a lower altitude level (compared with the reference), the potential energy to get down in the level can be stored in the battery due to the recovering braking. In this case, SoE ref erence has a higher value when the level is lower than the reference. To return to the reference level, it is necessary to overcome the potential energy barrier, which will be done by the previous energy stored in the battery. the inclusion of dierent constraints. Furthermore, due to the analytical characteristic, the nal solution calculates a new optimal power value at each time step using a low computational eort and low amount of memory.

The next section present the simulation and experimental results of the the proposed energy management strategy.

Optimization Results

Results from the optimization applied in the model

The backward vehicle model is used in this tests, where the urban, extra-urban and highway drive cycles of the appendix A were imposed. The altitude proles in the cycles are constant and equal to zero.

Results obtained from the predictive optimization method were compared with the Dynamic Programming results in order to verify how close to the global optimization this method is. Figure 5.17 presents the results for the urban drive cycle. As can be noticed, the power proles of fuel cell system for the two optimization methods are dierent due to the drive cycle prediction made. As expected, this result shows that the on line prediction method gives a sub optimal response when compared with the DP method. Despite this dierence, the hydrogen consumption obtained from the on line prediction method (16.2 g H 2 /km) is close to the value of the DP method (16.1 g H 2 /km). As the fuel cell system has a large power region with good eciency, neither the power nor the consumption are the most appropriate variables to evaluate the optimality level of the response. Therefore, the SoE prole was used to verify how close the sub optimality is to the global optimality.

The similarity of the SoE proles in gure 5.17, shows that the proposed online method has a sub optimality that is close to the DP global optimality, with the advantages of the low time consumption, memory allocation and no need of knowledge of the entire drive cycle (time causality).

Concerning the λ 1 prole, it adapts its value according to the predicted SoE and the power demanded drive cycle (P vehtot ), without important losses in terms of optimization.

The same analysis can be extended for the others drive cycles, as can be seen in terms of hydrogen consumption (table 5.1) and proles of power and SoE (gures 5.18 and 5.19). In order to verify the eect of a drive prole including gradation in the optimization process, a real drive cycle with mountain prole was applied to the methods (appendix A). Figure 5.20 presents the grade prole of this cycle.

For this simulation, the limits of SoE were changed to [20 % 80 %] to allow a better convergence for the DP method.

Experimental tests of the optimization algorithm

In order to verify the applicability of the energy management strategy proposed, a downscale FCHEV was implemented using the PROSPAC prototype, a electric motor test bench and an electric motor.

Experimental structure

The fuel cell system and battery in PROSPAC are responsible to deliver the power demanded by a 8 kW electric motor. The vehicle characteristics and drive conditions are imposed to the motor by the test bench structure. Figure (5.22) illustrates this assembly.

The motor bench controls the motor speed simulating the drive behavior in the vehicle. This control is made by imposing an acceleration setpoint (driver pedal) to the motor controller in order to follow a given speed prole (speed control). Besides, the motor bench also simulates the drive conditions and eorts, which means that in the dynamic equation of the vehicle (2.8), the mass and resistant eorts are simulated by the motor bench using a mechanical charge (electric motor). Therefore, the losses in rolling and aerodynamics are calculated by the motor bench and the related torque is applied against the motor.

In order to adapt the model of the vehicle to the elements present in the test structure, a scale factor of 10 was applied to the mass of the vehicle, its frontal surface S vehicle and air penetration coecient C x (table 5.2). This reduces the maximum traction power of the motor to 4.5 kW . Despite the symmetric assumption made during simulation tests about the features of motor model (P motor(max) = -P motor(min) ), in the experimental test campaign, the minimal power of the motor (regenerative power) was limited to -2 kW to guarantee the maximum recharge of the battery at -1 C, as required in PROSPAC specications. mass of the vehicle (kg) 100 S vehicle .C x 0.07 Table 5.2: Vehicle experimental parameters PROSPAC prototype is responsible to provide the power to the electric motor and auxiliaires. The total power is obtained from the combination of the fuel cell system and battery, where the sharing between them is made by the energy management strategy (EMS) embedded in the PROSPAC controller (DSPACE platform). The EMS controls the power provided by the fuel cell system through the command of the DC/DC converter.

Furthermore, the PROSPAC controller regulates and controls the operation of the fuel cell system elements (compressor, valves, etc).

Developed tests

In the developed tests, the extra-urban drive cycle is imposed by the motor bench to the motor. The objectives established are to validate the functioning of the proposed strategy and to analyze the potential about the optimization of the hydrogen consumption. Three energy management strategies were applied in this test:

• Dynamic Programming: the implementation was made imposing an optimal FCS power prole obtained by simulation to the real system.

• Predictive online energy management strategy: the same algorithm used for simulation was adapted and applied to the real system, with a time prediction of 150 s.

• Rule based strategy: an algorithm created by the PROSPAC designer that aims to keep the SoE controlled close to 50 %. In this method, the informations of SoE, demanded power P vehtot and the state conditions of the fuel cell stack (pressures, ows and temperatures) dene the power delivered by the fuel cell system P F C . The state conditions operate as security constraints to limit the power delivered by the fuel cell, while SoE and P vehtot determine the power split. The main objective of this method is to perform an energy management where the battery is only responsible for the peak power and high frequencies, being the responsibility of the fuel cell system to deliver the main power and to keep the SoE stable around 50 %

(recharge the battery).

Figure (5.23) presents the power prole demanded by the motor and the auxiliaries. While the rule base strategy tracks the demanded power, the predictive online method tends to approach the dynamic programming value. The existing dierence between the DP and predictive is caused by the choice about the duration of the predictive horizon and its initials values, which requires a certain time to converge towards the best solution.

Figure (5.25) shows the consumptions for the three strategies as function of the ∆SoE.

These consumptions were estimated by the current relationship in (2.21), meaning that the losses in the purge procedures were neglected. This indicates that, in contrast with the predictive online approach, the SoE sustaining method (Rule Based) does not takes advantage of the braking recovery energy in the SoE control, spending more hydrogen to accomplish the mission.

Conclusions of the chapter

The main objective of this chapter was focused on the synthesis of an online algorithm for the FCHEV energy management. In order to achieve this goal, dierent subjects were discussed leading to a development chain to compose the nal solution.

As this work proposes a solution of the energy management that takes into account the dynamics of the fuel cell system, the rst part of the chapter analyses the fuel cell system considering the dynamic eects of the interactions between the several subsystem that compose it.

Using the information of the most harmful eects created from the failure of the interactions and the dynamics of the elements, it is dened the air supply subsystem as the one having the dynamics to represent the fuel cell system dynamics.

To assure optimal operating points for the fuel cell, a non linear control law using the dierential atness theory is applied to the non-linear dynamics of the air supply subsystem. Cathode outlet pressure and oxygen stoichiometry are the considered at outputs to be controlled, where the control inputs are the air compressor velocity and the pressure valve controls.

The control law is rstly applied to the detailed fuel cell system model, leading to good results in terms of desired trajectory tracking and behavior of the other internal states of the fuel cell stack. Afterwards, the designed atness control strategy is applied to the real fuel cell system present in PROSPAC prototype. These results are compared with the results of two other classical control methods. Among the three methods, the atness control strategy presented a better performance than the others with a more stable prole of pressure and stoichiometry. Furthermore, this control strategy allowed the air supply subsystem with its control to be taken as equivalent to a very simple linear model.

Integrated in the optimization problem, the new fuel cell system dynamic represented an additional new dynamic constraint that the optimal solution must respect in the process of the optimal power split denition.

In order to propose an online solution for the new optimization problem, a predictive energy management method is developed based on one hand on the theory of the Pontryaguin's Maximum Principle and on the other hand on the predictive control using a sliding horizon window to nd out the power demands of the cycle.

Thanks to the fuel cell control linearization, an analytic solution of the optimal power split can be calculated in real time. Finally, the battery SoE is adjust to maximize the recovery of kinetic and potential energy involved in the braking procedure.

The performance of the algorithm is veried applying dierent drive cycles and comparing the results to a global optimization method. Dynamic Programming method is used as reference, where the hydrogen consumption and the proles of the fuel system power and state of energy of the battery are the variables to be compared. The analysis of the results showed that the proposed algorithm is a sub optimal solution that approaches nearly the global optimization, without requiring the prior knowledge of the drive cycle.

The applicability and the performance potential of the proposed method were veried by experimental tests. Using a real fuel cell and battery system to power a scaled vehicle and drive train, the results of the online predictive strategy were compared with the dynamic programming and a rule based (SoE sustaining) ones. The advantages of the proposed method obtained in the simulation steps were conrmed in the experimental tests, showing that the method can approach the global optimization results.

Introduction

The durability aspect of the fuel cell system is one of the main challenges to upscale fuel cell commercialization. In the FCHEV context, the automobile constructers are facing this challenge as a high-priority matter. They apply development eorts in several areas such as materials and systems. Aiming to contribute to this discussion, this chapter intends to include this issue in the energy management strategy.

In order to achieve this objective, is required to rst study the concepts and characteristics involved in the fuel cell durability issue. Considering the presented background, a simplied degradation model is adopted based on works performed in CEA. The model relates a degradation rate to the power demands and the state conditions imposed to the fuel cell during operation. Additionally, the model supposes to determine the state of health of the fuel cell, which will be important in the optimization process.

As a part of the dynamic representation of the system, this model is integrated as a dynamical constraint in initial the optimization problem and the optimization algorithm presented in the previous chapter is extended to take into account this new constraint.

In order to take into account the durability aspect, the optimization criteria is modied and includes the target time of life of the fuel cell (EoL).

The nal optimization structure aggregates constraints and cost functions that are related with the hydrogen consumption, battery SoE and fuel cell State of Health (SoH ).

Important concepts and requirements about durability in a FCHEV

As a promising technology for energy applications, the hydrogen fuel cell has several advantages compared with conventional solutions (ICE, gas, etc). However, it is not yet a mature technology, which makes it not ready to be competitive with the other options. One particular aspect is about the durability of the fuel cell stack. Despite the fast improvements of the materials and systems involved, the fuel cell stacks are still presenting a lifetime that does not reach the assigned targets.

According to the type of application, the durability targets stated by the US Department of Energy (DOE) [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF] are dierent. For stationnary applications it is demanded 40000 hrs of operating hours. For transport applications, the DOE indicates that the fuel cell stack should assure a time of life of 5000 hrs in order to be adapted to the market in 2015.

In this work, we are interested in the targets for the transport application. Even if the recent announcements of the automobile constructors indicates that the durability goal has been reached and the FCHEV is ready to be inserted in the market (depending on the hydrogen structure), the scientic publications only present values that reach the half of the target (2500 hrs). Therefore, our work deals with the durability of the fuel cell system, and specially the fuel cell stack, as an open and major issue. The durability is a complex subject that must be rstly approached by a denition of the fuel cell end of life (EoL) concept.

The EoL of a fuel cell stack is related to its inability to assure basic operations or/and to guarantee a minimal performance. Thus, the end of life of a fuel cell stack can be dened by the observation of a failure or a certain loss in performance.

Failure The occurrence of any failure normally imposes a condition to the user that impedes the operation of the fuel cell (for example a membrane failure). In the case of a FCHEV it leads to a need of a complex and unexpected maintenance to bring the system back to its normal operational conditions. This represents the most undesired situation and should be prevented in advance to avoid any harmful eect. In order to detect these events, some fault detection methods have already been developed [START_REF] Quan Yang | Fault detection and isolation of pem fuel cell system by analytical redundancy[END_REF][START_REF] Escobet | Model-based fault diagnosis in {PEM} fuel cell systems[END_REF][START_REF] Alberto | On-line fault diagnostic system for proton exchange membrane fuel cells[END_REF].

Through the observation of the measured variables of the fuel cell, a fault can be predicted and prevented in a certain range of probability.

Loss in performance Dierently from the failure case, the reversible or irreversible losses in the fuel cell performance lead to a degraded mode that implies overconsumption (eciency) or reduction of the power capacity. Figure 6.4 [START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF] shows the eects of the loss in performance with the time in the polarization curve for a fuel cell stack composed by 5 cells. As can be noticed, the aging eect causes degradation in the performance of the cell, which means that a old stack needs a higher amount of hydrogen than a new stack, in order to provide the same power. Despite the undesirable behavior of the FCHEV, the loss in performance does not stop the basic operational capability of the system. The loss in performance is adopted in this work to evaluate the durability of the fuel cell. However, it must be dened how to evaluate the performance (performance index) and its minimal acceptable limit at which some specications of the vehicle, including the technical ones are still guaranteed. Some performance indexes are dened to indicate the end of life of a fuel cell stack.

In the FCHEV particular case, Pei et al. [START_REF] Pei | A quick evaluating method for automotive fuel cell lifetime[END_REF] used the information of the cell voltage to analyze the end of life of a stack for a bus application. Their proposition outlines that a fuel cell stack is considered at its EoL when its rated power capacity is decreased leading to reduction of 10 % in the cell voltage (g. 6.2). Such situation represents that the fuel cell stack can not guarantee the traction functions of the vehicle (electric power). Such analysis allows to include the losses of fuel cell eciency, which causes an increase of hydrogen consumption to perform the same task (same power delivered). Furthermore, the loss in voltage is related also to the reduction of the maximum power of the stack and consequently to the ability to accomplish some technical requirements of the vehicle. by the vehicle to the powertrain. Such requirements reach also the fuel cell system, leading it to be subjected to conditions that can cause reversible or irreversible degradation eects.

Vehicle operating modes and their degradation mechanisms

At the time, the operational modes considered as the most important in the FC aging context are:

Cold start A start up from subzero temperatures is a procedure that the fuel cells embedded in vehicles are submitted. However, when the fuel cell is operated at subfreezing temperatures, the water generated at the cathode will tend to form ice that can result in a loss in performance of the fuel cell. • Interfacial delamination between the cathode catalyst layer and membrane,

• Cathode catalyst layer pore collapse and densication

• Pt area loss due to particle ripening and dissolution in PFSA ionomer.

The ice formation in the cathode generates these mechanisms that lead to a reduction in the electrochemical surface area in addition to the membrane resistance increasing [START_REF] Borup | Fy 2004 program report[END_REF].

As a side eect, it is obtained a worse quality reaction (less electrochemical surface) and more losses in the membrane.

of production of water related to the power delivered and the dryness caused by the high temperature. These aspects are linked to the degradation eects associated to the performance of the cells. A high temperature and a possible low humidity can change the membrane resistivity. Besides, the temperature can also impact on others parts of the cell, such as the GDL [START_REF] Borup | Fy 2004 program report[END_REF]. The GDL degradation acts on the water diusion, inuencing related degradation mechanisms and leading to losses in performance.

Even if degradations are unavoidable, the degradation rate can be limited with a specic design of the components and using mitigating strategies. Concerning the mitigating strategies, several authors proposed dierent strategies for each degradation mechanisms, but without considering the drawbacks brought by the vehicular applications. The embedded characteristic imposes strong constraints to implement the proposed strategies.

For example, Borup et al. 25 indicated that to avoid the increase of loss of GDL hydrophobicity it would be better to use sparging nitrogen instead of air during shut-down procedures, but the space and application automobile constraint avoids the inclusion of a nitrogen vessel in the vehicle.

Consequently, other works considered dierent approaches to deal with the limitation of the degradation rate, our work also intends to contribute to this subject. Nevertheless, the control of the degradation needs the knowledge of the degradation state of the fuel cell stack. Thus, a diagnostic method should be used to observe and apply the control strategies to minimize the degradation. In the sequence this subject is presented in a brief state of art. The phenomena related to each degradation mechanism cannot be observed separately, they present strong interactions that are dicult to isolate them. Moreover, as a multi-physic, multi-scale and multi-variable complex system, the fuel cell system remains a real challenge in terms of diagnostics. The denition of the State of Health (SoH ) of constraint the fuel cell is dependent on the knowledge of inner variables which are not available for measurements, especially in mobile applications. Dierent diagnostics and observation methods for this purpose have been proposed in the literature. In the sequence an overview of the methods is presented.

The challenge to rebuild the non observed inner states is faced by the researchers using several approaches that sometimes combine signal processing theory, measurements, dynamic modeling and articial intelligence.

Kim et al. [START_REF] Kim | State-of-health diagnosis based on hamming neural network using output voltage pattern recognition for a {PEM} fuel cell[END_REF] studied the durability of the fuel cell using a pattern recognition approach. Based on the measurement of the fuel cell voltage, the authors applied a behavior model-based method that uses the Hamming neural network [START_REF] Meilijson | A single-iteration threshold hamming network[END_REF]. This is intended to indentify a fuel cell output voltage among the known proles that reect the time of life of a typical fuel cell. The information used to verify the degradation state of the cell is its composed resistance (activation and concentration). The estimated value is compared to the new and aged fuel cell values in order to dene its state of health. Despite the facility of implementation of this approach, it method needs an important amount of good quality data, which sometimes can not be assured.

Using an approach that applies the signal processing theory, Bethoux et al. [START_REF] Bethoux | A new on-line state-of-health monitoring technique dedicated to pem fuel cell[END_REF] proposed to estimate the fuel cell impedance by the measurements of the voltage and current at dierent frequencies. Therefore, an AC signal is added to the main input signal (current), which includes a frequency component to the output signal. Using the Fresnel framework, the amplitude and the phase of the impedance are obtained; This allows the parameters values of the fuel cell equivalent electric model to be calculated.

The state of health of the fuel cell is evaluated according to the equivalent resistance of the cell. To be embedded in a realtime system, the calculations required by this procedure should be performed with a dynamics of the degradation control, which can require an important calculation capacity. Besides, this method needs add a high frequency signal to the main current demand, which is not simple for a nal application, such as a FCHEV.

Kurtz et al. [START_REF] Kurz | An impedance-based predictive control strategy for the state-of-health of {PEM} fuel cell stacks[END_REF] considered the specic problem of the diagnostic of the water management in the fuel cell. Due to the inability to identify the related phenomena from the polarization curve, their approach used the impedance of the fuel cell. Electrochemical Impedance Spectroscopy (EIS) was the tool adopted to measure this variable and allowed to distinguish the membrane conductivity, kinetics and mass transport limitations.

Through the EIS results, the authors can detect in advance possible faults in terms of water management. As can be noticed, this method needs a structure that is dicult to apply for embedded application and is mainly used to fault detection.

Lee et al. [START_REF] Ju Hyung Lee | Development of a method to estimate the lifespan of proton exchange membrane fuel cell using electrochemical impedance spectroscopy[END_REF] dealt with the durability subject by a lifespan approach. They proposed Figure 6.3: Fuel cell equivalent circuit [START_REF] Ju Hyung Lee | Development of a method to estimate the lifespan of proton exchange membrane fuel cell using electrochemical impedance spectroscopy[END_REF] a method that uses the EIS to t the values of the parameters presented in a equivalent model. This model is composed by inductances, resistances and capacities that represent the characteristics of the elements in the cell (gure 6.3). Such approach focus on the eect of performance drop due to the cathode catalyst loss, which means that the changes in the resistance and capacity of the cathode indicate the state of health of the cell. As most of diagnostic methods, this one also needs equipments that limit the implementation for online embedded applications.

As noticed by the presented methods, they are oriented to identify one type of phenomena (humidity, catalyst loss, membrane resistivity), considering that the other ones are not aecting the diagnostic. Another major inconvenient is about the lack in knowledge of the interactions between the input variables (current, voltage, pressures, temperature, humidity, etc) and the degradation index, which is composed by several eects.

This avoids to create a model that could be inverted to discover the eect of each state variable in the degradation result.

The research in this area continues and tends to converge towards methods adapted to the requirements and constraints of the FCHEV applications; however, the diagnostic method is an open problem so far.

This work do not have the objective to propose a method, but the importance of the diagnostic to achieve the stated objective (optimal energy management including durability constraint) leads to consider the SoH as an available information, independently of the applied diagnostic method.

In order to develop an energy management strategy that includes the fuel cell durability, the SoH information is not the only durability information required. The durability aspect is included in our optimization process as a new dynamic constraint as well. Therefore, a model that represents the fuel cell stack degradation dynamics is necessary. In this work, such model (degradation dynamics) is obtained based on the results of the work made by [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF]. Next section presents the approach used to dene the degrada-constraint tion dynamic of the fuel cell stack, followed by the optimization problem including such dynamic as a constraint.

6.4 Fuel cell stack degradation dynamic model and optimization problem for EMS application

Degradation dynamic model

Several models designed for study the degradation of PEMFC are available in the litterature. Out of all, the ones adopting phenomenological approach present the advantage to mainly use physical aspects in the construction of the model. This simplies the understanding of the results, but they normally need a powerful calculation platform to solve the model. For the aimed application, the degradation model should be related to the loss in fuel cell performance and be easily calculated to be implemented in the energy management algorithm. Thus, this calculation requirement avoids the use of complex models.

The approach chosen in this work designs the degradation dynamics of a fuel cell stack according to the results of models simulations. The degradation mechanisms of the fuel cell (loss in active surface, carbon corrosion, etc.) are observed when harmful conditions are imposed to the model (power cycling, high power, etc). Besides, degradation result highlights its inuence on the performance instead of the failure.

Robin et al. [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF] performed a study about the degradation of fuel cell submitted to automotive conditions and requirements, which is well adapted to meet our stated objective. Their approach was based on the use of models developed in CEA (Commissariat à l'Energie Atomique) that delivers either informations about the FC performance and element's degradation levels. The required interactions between the system and the fuel cell are achieved combining two dierent models [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF].

Due to the dierence in the models, an indirect coupling procedure is used (more information can be found in [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF]). This needs some additional assumptions in order to make compatible the two models:

• Ideal response of the auxiliary of the fuel cell system

• Open Circuit Voltage (OCV) is not considered in the possible operation modes

• Start/Stop cycles are not considered • Carbon corrosion is neglected • Membrane degradation is neglected • Otswald ripening is neglected
This set of assumptions leads to adopt the cathode Pt surface degradation as the mechanism to be observed and to be considered as the one representing the aging of the fuel cell. Pt surface has an inuence on the cell performance, where a new cell (high surface) presents a better performance than a aged one (low surface). Figure 6.4 and 6.5 illustrates these relationships in terms of voltage and power response. The temperature optimal value at 343 K indicates that the reactions are favored by a high temperature, but the values higher than the optimal tends to disturbs the water distribution in the fuel cell core and dries the membrane.

It is known that a low current makes the fuel cell to operate close to the OCV triggering undesired degradation eects in the catalyst layer, while high current values is stressing the fuel cell components.

The relative humidity shows a prole where the increasing of its value is followed by an improvement of the lifetime. As the only degradation mechanism taken into account is the cathode Pt surface degradation, the high H r values (100 %) other degradation mechanisms (e.g. carbon corrosion) do not create degradation eects as they usually do.

Concerning the pressure, the optimum values are located between 1.5 and 1.8 bar.

According to the author, a low pressure tends to compromise the reaction kinectics, while a high pressure interfere the distribution of reactant gas ows in the cell, reducing the lifetime.

O 2 stoichiometry prole illustrates an optimum at 1.5, which is justied by the fact that a low stoichiometry is insucient to satisfy the reaction rate, while high stoichiometry increases the cathodic pressure and decreases the relative humidity, reducing the fuel cell lifetime

Dierently from the other variables, H 2 has no inuence on the fuel cell durability (considering only the Pt surface degradation). This result is explained by the fact that the anodic conditions were neglected.

Except for the H 2 stoichiometry, the regulation of the state variables at their best operation values can minimize the degradation rate of the fuel cell stack. This regulation can be obtained by an implementation of a control strategy at a fuel cell system level.

In this work, the degradation dynamics of the fuel cell is obtained considering that the state variables are regulated at their reference values. Nevertheless, as the objective is to dene a optimal power split that considers the durability in the strategy, the current is a control variable and is not regulated at its reference value. The denition of its value is given by the energy management strategy.

Despite the lack of precision, the results give good indications to build the degradation dynamic model. This enables, the relationship to be constructed between the active surface degradation rate and the current delivered is the basis to construct the degradation model. As the control variable in the energy management strategy is the power delivered by the fuel cell system, the surface degradation rate vs. current is transformed constraint in order to implement the power as the control variable. Using the same data of 6.6, it is constructed a relationship between the active surface degradation rate and the power delivered by the fuel cell system. Figure 6.7 shows the degradation dynamics which was dened (equation 6.1).

This curve reects the current behavior in terms of degradation mechanisms, where a low power tends to submit the fuel cell stack to a OCV condition (undesired in terms of degradation of Pt surface), while a high power stress the fuel cell components. For a fuel cell stack of 90 cells, a minimum of degradation rate is obtained at 5.5 kW and the prole can be approximated by a quadratic function as in the equation 6.

dS dt = f 2 (P F C ) = b 1 P 2 F C + b 2 P F C + b 3 (6.1)
In (6.1), S is the current value of the equivalent active surface area. This equivalent surface area considers that the specic cathode Pt surface is equivalent to the active cell surface. P F C is power delivered and the parameters b i are obtained by a quadratic tting of the result in gure 6.7.

Active surface dynamics in (6.1) not only allows to include it in the optimization problem, but also leads to the denition of the fuel cell State of Health SoH. Then equation 6.2 denes the SoH indicator, where S tot (220 cm 2 ) is the initial value of the surface.

SoH

(%) = 100. S S tot (6.2) constraint                                    ε = f 1 (P batt ) = -P batt ω = f 2 (W 1 ) = W 1 ṗcath = f 3 (W 2 ) = W 2 Ṡ = f 4 (P F C ) = b 1 P 2 F C + b 2 P F C + b 3 (6.4)
The SoH boundary condition in (6.3) requires the denition of its minimal value SoH min , which denes the end of life of the fuel cell stack. This minimal value is related to the performance index presented in section 2. Furthermore, the same boundary condition do not only indicate the limit value but also the time scale when it could happens t EoL (end of life). Such characteristic requires a solution able to deal with dierent time references at the same problem. This means, the presence of an instantaneous reference related to the hydrogen consumption (g (P F C (t))), a cycle duration related to the SoE boundary condition and the lifetime duration associated to the SoH boundary condition.

In the sequence is dened the limits of the new constraints (S and SoH), either at the EoL (SoH value at EoL) and during the fuel cell stack lifetime (SoH limit prole).

SoH limits for the optimization problem

In section 2 the performance limit were dened in terms of end of life of a fuel cell system as a loss of 10 % in voltage at a rated power delivered. Because the variable that represents the degradation is the active surface area, the EoL index should be translated in terms of S.

Previous sections explained that the reduction in the active surface is always followed by a reduction in the performance. Consequently, it is possible to determine the value of active surface area that gives a loss of 10 % in the voltage for a FC system rated power (5 kW ).

The procedure to dene this surface imposes a rated power and dierent values of S for the fuel cell stack in fuel cell system model of the chapter 3. Therefore, the cell voltage value is observed for dierent surface values when the stack delivers the rated power. When the voltage presents a reduction in 10 % compared with the value of a new stack (initial active surface value), the related surface value is chosen to dene the minimal limit. The set of voltage results indicates that for a loss about 20 % of active surface area, the cell presents a voltage value 10 % smaller than the value for a total surface area (220 cm 2 ). Therefore, when the value of the active surface area reaches 80 % of its initial value, we can consider that the fuel cell stack achieved the end of life. Such value of limitation is adopted by Robin et al. [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF] as well. Their choice is justied by the fuel cell stack capacity to deliver enough power to meet the transport requirements and specications.

Using the degradation dynamics (6.1), the lower boundary of the fuel cell SoH and equivalent active surface (176 cm 2 for an initial value of 220 cm 2 ) give the durability condition of the fuel cell system that will be integrated in the optimization problem.

In order to be adapted to the automobile targets, our work considers that the SoH min is reached at the desired lifetime assigned by the DOE [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF], which means that SoH ≥ 80 % f or t ≤ 5000 hrs. However, this condition do not provide information about the limit prole of SoH before the EoL. This allows the power split strategy to degradate the fuel cell stack since the beginning of life and reach the SoH min before the EoL, which can not be sustained due to the existence of a minimum degradation rate, no matter the power value chosen. To avoid this situation, it is proposed the denition of a SoH constraint limit prole that determines a SoH boundary to be avoided by the energy management strategy. SoH limit prole evolutes in the time and reaches its end of life value (80 %) when the fuel cell stack accomplishes a desired lifetime of 5000 hrs. This is a limit for the SoH that separates the allowed area over it from the avoidable area under it. The prole proposed is supposed by default linear, which does not means that it could not have another shape. Figure 6.9 illustrates the proposed prole. In the next section this prole is used by an optimal energy management strategy to solve the optimization problem stated.

Optimal energy management strategy with durability constraints

As introduced in the previous chapter, the optimization problem is solved using the Pontryaguin's Maximum Principle (PMP) [START_REF] Christophe | Introduction à l'Optimisation[END_REF]. Using the Hamiltonian function (6.5) that includes the degradation dynamics, the optimal power split can be calculated applying again the three conditions (6.6).

H (ε, ω, p cath , P batt , W 1 , W 2 , λ 1 , λ 2 , λ 3 , λ 4 ) =

g (P vehtot -P batt ) + λ 1 f 1 (P batt ) + λ 2 f 2 (W 1 ) + λ 3 f 3 (W 2 ) + λ 4 f 4 (P F C ) = = (a 1 + λ 4 b 1 ) (P vehtot -P batt ) 2 + (a 2 + λ 4 b 2 ) (P vehtot -P batt ) + • • • • • • (a 3 + λ 4 b 3 ) -λ 1 P batt + λ 2 W 1 + λ 3 W 2 (6.5)                      ε = ∂H ∂λ 1 ; ω = ∂H ∂λ 2 ; ṗcath = ∂H ∂λ 3 ; Ṡ = ∂H ∂λ 4 -λ1 = ∂H ∂ε ; -λ2 = ∂H ∂ω ; -λ3 = ∂H ∂p cath ; -λ4 = ∂H ∂S ∂H ∂P batt = 0; ∂H ∂W 1 = 0; ∂H ∂W 2 = 0 (6.6)
The λ n derivative conditions indicates that they have constant values ( λn = 0) while the condition about the derivative of the Hamiltonian with respect to the control inputs (P batt , W 1 and W 2 ) present the following results:

∂H ∂P batt = 0 = 2 (a 1 + λ 4 b 1 ) P * batt (t) -(2 (a 1 + λ 4 b 1 ) (P vehtot (t)) + (a 2 + λ 4 b 2 )) -λ 1 ⇒ ⇒ P * batt (t) = λ 1 +(2(a 1 +λ 4 b 1 )(P vehtot (t))+(a 2 +λ 4 b 2 )) 2(a 1 +λ 4 b 1 ) ∂H ∂W 1 = 0 = λ 2 ∂H ∂W 2 = 0 = λ 3 (6.7)
Optimization result in (6.7) leads to an algebraic relation between the power demanded by the vehicle and the co-states λ 1 and λ 4 . As in the previous chapter, P vehtot is a known information and λ 1 is calculated using the predictive approach as previously. Despite the presence of λ 4 in the P * batt equation, it was considered that its inuence in the calculation of λ 1 can be neglected. This assumption is based on the dierence in time scales of λ 1 and λ 4 . While λ 1 is calculated using a prediction of hundreds of seconds, the co-state λ 4 has a dynamics adapted to the hours scale. Therefore, in a prediction of λ 1 the value of λ 4 is considered constant. constraint Concerning the calculation of λ 4 , it has to be performed such that it is possible to partially control the fuel cell system SoH by the power setpoint established through the energy management strategy. Some similar approaches are already applied for a battery SoE control. Thritschler et al. [START_REF] Thritschler | Energy management strategies for an embedded fuel cell system on agricultural vehicles[END_REF], Kessels [START_REF] Theodorus | Energy Management for Automotive Power Nets[END_REF] and [START_REF] Michiel Koot | Energy management strategies for vehicular electric power systems[END_REF] are examples of this type of implementation. They use a SoE or SoC reference value and the measurements of these variables to calculate the co-state value, which is performed by a feedback control strategy, as in gure 6.10. Such control permits to the SoE or SoC to follows their references during the drive cycle, assuring the power assistance mode of the battery. However, we do not want that the SoH has a reference value to be followed. As indicated in the last section, the SoH has a limit prole that evolutes in time (g. 6.9), so it has to avoid this limit during the entire lifetime. Therefore, the control strategy for the calculation of λ 4 must only maximize the positive dierence value between the SoH to its limit value SoH limit . This SoH control works as a repelling force linked to the dierence between SoH (t) and SoH limit (t) (g. 6.11). As the smaller is the dierence the greater is the force, otherwise the force tends to zero.

Somehow, this force is related to the value of λ 4 and its calculation uses an exponential function such as:

λ 4 = f 5 ( SoH (t)) =
k 1 e k 2 ( SoH(t)) -1 ; SoH (t) = SoH (t) -SoH limit (t) (6.8)

Where k 1 and k 2 are constants.

Figure 6.12 shows the behavior of this gain function. As can be observed, as closer to the limit the SoH (t) is, as greater the λ 4 value is. While λ 4 can be neglected when SoH (t) presents a important value (it means the fuel cell degradations are not accelerated). The implementation of the SoH control turns the optimization structure to the one presented in gure 6.13.

The nal solution of the optimization problem presents a commitment between the minimization of the hydrogen consumption and the warranty that the durability constraint is controlled. It is expected that this association modify the best operational point (in terms of hydrogen consumption) when SoH is close to its limit, leading to adopt the a lower degradation rate and increasing the hydrogen consumption.

In the next section, the optimization structure proposed is tested under dierent SoH conditions for a real automotive environment (drive cycle). The optimization solution is applied to the model that represents the real system PROSPAC composed by fuel cell system of 5 kW and a 2 kW h battery . In order to verify the eect of the state of health in the energy management, three initial conditions are imposed: SoH (t) = 1 %; SoH (t) = 5 %; SoH (t) = 10 % ( SoH (t) = SoH (t) -SoH limit (t)).

Extra-urban drive cycle (appendix A) is imposed to the system. Power prole related to this cycle, which is required from the hybrid system, is shown in gure 6.14. When this power prole is imposed to the hybrid system, the energy management strategy replies with three dierent fuel cell system power prole related to each ∆SoH, as can be seen in gure 6.15. The results in terms of active surface losses are presented in table 6.1. Active surface losses are represented as percentage of the total allowed. This total is considered as 20 % of the initial surface area (220 cm 2 ); thereby, a reduction of 44 cm 2 represent the total of losses allowed.

SoH (t 0 ) (%) 100.

surf ace losses(cm 2 ) to 5 %, which is followed by a slight increase in the hydrogen consumption.

Nevertheless, an important relative reduction in terms of surface degradation is obtained when the initial condition is SoH (t 0 ) = 1 %. The reduction in surface degradation of almost 16 % (compared to SoH (t 0 ) = 10 %) is followed by an expected increase in terms of hydrogen consumption (11 %) that is not completely lost in the process, but also converted and stored in the battery pack.

Conclusion of the chapter

The development of energy management strategy for FCHEV is again studied in this chapter, but with a dierent approach. Keeping all the characteristics of the solution given in the previous chapter, the power split has been changed such that the durability aspect of the fuel cell system is included and considered in the optimization process as a constraint.

These durability aspects of the fuel cell system are rstly presented in the chapter by the denition of the durability criteria adopted and how it is related to the performance of the fuel cell. Based on studies made by other authors, the loss in voltage was chosen as the performance index that can indicate the end of life of the fuel cell stack. As this information is directly related to the eciency, the loss of 10 % in the cell voltage for a rated power delivered indicates an eciency level that disqualies it for the vehicle applications.

The dierent operation modes imposed by the automotive application would lead Nevertheless, SoH measurement is considered as an available and reliable information to control the fuel cell degradation.

Then, a fuel cell degradation model is introduced to represent the specic phenomenon of loss in the cathode catalyst activity, which is represented by a reduction in the active surface area of the fuel cell. Using the study developed by Robin et al. [START_REF] Robin | Multiscale coupling between two dynamical models for {PEMFC} aging prediction[END_REF], a degradation rate of the active surface in function of the power delivered by the system is used as the degradation dynamic model of the fuel cell. This dynamic equation allows the SoH to be calculated and two new constraint to the optimization (degradation dynamics and limit of SoH) to be dened. Concerning the last one, its limiting value is related to the fuel cell end-of-life, which is given in terms of cell voltage degradation limit. Then, it required a transformation of the voltage loss into active surface reduction. We show that such translation indicated a 20 % of equivalent active surface reduction has to be integrated as the SoH nal limitation at the end of life.

The integration of the new durability constraints in the optimization problem developed in the previous chapter, but taking into account the new constraints. A new structure is proposed as the solution, where a SoH control is implemented to avoid its oncoming to a low limit prole. Such prole denes an available region during all the fuel cell stack life, which tends to assure a time of life target value.

A reasonable commitment between the consumption and durability is observed as the results of the optimization method for dierent degradation conditions. The algorithm shows a capacity to handle with the two tasks needing a low calculation capacity and memory, which are important characteristics for a vehicle application.

The energy management strategy developed in this chapter show its potential by the simulation results; however, they must be considered as trends and not quantitative results. An experimental approach would be desired to verify these qualitative results, but the equipments and required structure were not part of the scope of this work.

Chapter 7

General Conclusion

Transport sector presents a context that demands from the available transport ways ever more performance and especially less pollutants emissions. The environmental constraints has pushed up the development of new suitable solutions with dierent energetic approaches. According to the characteristics of the region, the vehicular technologies are oriented towards the most available sources of energy. Ethanol and GPL are some examples of the energy options used to replace the petrol for conventional vehicles. Although the acceptation of these new types of energy vectors, several studies presented that is necessary a breakthrough not only in the energy source but also in the powertrain of the vehicles.

Therefore, the adoption of electrical powertrain has been often adopted by automobile constructors either to integrate hybrid vehicles or in the full electric vehicles. In the rst option the hybridization allows to reduce the petrol consumption and pollutants emissions. Meanwhile, the full electric vehicle represent, at rst sight, a perfect solution as it does not uses directly fuel; however, the price of the battery, distance range and recharge time are still drawbacks of this technology.

Despite the recent commercialization of hybrid and full electric vehicle, they represent solutions that compose the set of options available. As a complementary solution, the Fuel Cell Hybrid Electric Vehicles (FCHEV) has been subjected of research and developments of research centers and automobile constructors.

Considered as a promising option, this solution allies the good eciency to the drive range and clean features. However, even if some automakers announced the commercialization of FCHEV for 2015, this technology has not reached a maturity and stability level to be compared with the available options in the market. Even if advances have been obtained, high cost, dicult energy management and low durability are still barriers to the full adoption of the hydrogen fuel cell technologies embedded in the vehicles. In order to contribute to the eorts applied to the development of the FCHEV, this work performed a study of dierent functionalities related to the existing drawbacks. They are grouped in three main subjects:

• FCHEV optimal sizing

• Online optimal energy management strategy

• Consideration of the durability aspects (degradation) in the energy management strategy

In the outline of the development chain, rstly a tool basis was created to achieve the objectives. It is composed by a set of optimization tools and models adapted to the dierent elements of the fuel cell hybrid electric vehicle. Matlab/Simulink environment was adopted to the developments.

Concerning the main objectives of this thesis, the optimal sizing was the rst to be dealt. Such task was intended to obtain the best conguration of the vehicle's elements in order to meet a set of technical specications and minimizing the hydrogen consumption.

Moreover, population mobility and economical aspects were also included in this sizing procedure. Adopting a systematically approach, the sizing procedure was performed in four steps:

• Technical requirement: The vehicle powertrain was sized to meet a set of technical requirements composed by specications based on a typical vehicle capability and also on the technological features imposed by the new elements included in the vehicle. The sizing of the powertrain elements resulted in a conguration that keeps the drivability for a typical user and integrated the new functionalities brought by the new technologies embedded.

• Optimization: The hybridization contained in the architecture provides an extra degree of freedom, which can be used to optimize the hydrogen consumption during a mission. Such reduction is obtained by optimizing the power split between the fuel cell system (FCS) and the battery pack at the time. Even if this optimization process is mostly used to dene the energy management strategy, an optimal sizing of the FCS and battery can be obtained as well for a given mission (drive cycle).

The results of this step indicated the best combinations of FCS max power and battery energy capacity in terms of hydrogen consumption. During this optimization process, several aspects were took into account, such as the mass of the systems, the energy consumption of FCS auxiliaries and the battery pack capacity to deliver and recover power.

• Mobility aspects: The distance range of the vehicle was submitted to the requirements of the population mobility aspects. Based on statistical data about the population displacement, a bound of the battery energy capacity was assigned.

Such capacity is related to the ability of the vehicle to accomplish a most frequent distance range in a battery traction mode.

• Economical aspects: The economical point of view was applied in the last part of the sizing procedure. This economical analysis aimed to verify the interest in the increasing of the hybridization performance (reduction of hydrogen consumption) when comparing to the cost of the new dimension of the system. The results indicate that the cost of the battery pack extra capacity is not compensated by the economy in hydrogen consumption during all vehicle's life, while the savings in hydrogen can justify the extra-cost of a more powerful fuel cell system.

To summarize the optimal sizing work, the proposed method presents a systematic approach that allies dierent and complementary tasks useful to determine the capacities of the main elements of the vehicle power train and hybrid system. Furthermore, the results of the hybridization sizing process of indicate the following remarks:

• According to the type of drive cycle and constraints adopted in the optimization problem, the optimal conguration presents dierent results.

• The increasing in mass and energy consumption of FCS and battery should be compensated by an increase of the hybrid system performance, which is not always the case.

• The battery pack power recovery capacity is more eective than the power delivery capacity in the optimization of the hydrogen consumption.

• The cost of the battery is hardly compensated by the gain in performance of the system, which means that it limits the adoption of the best congurations (in terms of hydrogen consumption).

In the sequence, the subject dealt was the development of an optimal online energy management strategy (EMS). Dierently from the global optimization tool used in the sizing procedure, this EMS is oriented to online embedded applications, such as the vehicle ones. Moreover, the proposed online optimization method includes the dynamics of the fuel cell system, which is normally neglected in the literature.

Concerning this EMS subject, the contribution of this work might be divided in two parts:

• Fuel cell system dynamics: Considered as a limiting factor for vehicle applications, the dynamics of the fuel cell system was included in the constraints of the optimization problem. Therefore, it was necessary to dene and develop the dynamical model that represents the main fuel cell system behavior. As the most constrained subsystem in the FCS and presenting a non-linear dynamics that could be responsible for some harmful eects, the air supply subsystem had its dynamic behavior modeled to represent the fuel cell system dynamics. In order to deal with this dynamics, a non-linear control law using the dierential atness theory was applied to regulate the most important state variables: oxygen ow and cathode pressure.

This control strategy has been validated though simulations (with a dynamic FCS model) and tests on real hybrid FC system. Besides, such control simplied the dynamics to an integrator chain, leading to an easier inclusion of it in the optimization problem.

• Predictive online energy management strategy: Requiring an optimization approach better adapted for the vehicle applications than global optimization methods such as Dynamic Programming or Pontryaguin's Maximum Problem (PMP), the energy management strategy to be embedded in the FCHEV was developed using an optimization method that aggregates the PMP theory and a predictive approach. This predictive component is responsible to ll the gap of the unknown information related to the drive cycle imposed to the vehicle. The prediction adopts a sliding horizon of power demanded and integrates the information of battery's state of energy (SoE), allowing to obtain the power assistance mode (SoE (t initial ) = SoE (t f inal )).

Another improvement provided by this proposed optimization strategy concerns the energy recovery during braking procedures. Adapting the reference of the SoE according to the speed and altitude of the vehicle, the battery is prepared to recover the maximum of the kinetic and potential energy available during braking, leading to an optimization in the hydrogen consumption.

The analysis of the simulation and experimental optimization results showed that even though the algorithm presents a local optimization, it has the ability to approximate the global optimization results. Furthermore, the proposed solution is capable to calculate the power split in an analytical way and demanding a small amount of memory, which represents an overcome of the global solution drawbacks and an important feature to be highlighted.

Durability aspect was the last subject dealt in this work. As made with the fuel cell dynamics, the degradation dynamics was integrated in the optimization problem as a constraint. This was eectuated using a model that indicates the fuel cell degradation rate in function of the power demanded. Such degradation rate is observed though the losses at the fuel cell active surface area and enables the denition of the State-of-Health SoH variable. Combining the consumption criterion, battery SoE constraint and durability constraints, it was created three dierent time scales in the optimization problem that mixes instantaneous conditions to drive cycle durations (SoE constraint) and time of life aspects (fuel cell system durability). Therefore, it was required an adoption of a SoH boundary prole, where this prole changes in time up to reach the fuel cell's end-oflife. To respect this boundary prole, a SoH control is integrated in the solution of the optimization problem, such that it assures an assigned lifetime of the fuel cell.

Even if the results represent only trends, it was noticed that the expected commitment between hydrogen consumption, SoE and SoH constraints was presented in the results.

This indicates that although the lack of accuracy, the results lead to consider the proposed optimization structure adapted to the give consistent responses and to allow the evolution of the constraints complexity.

The results and developments presented in this document opened the perspectives for future research and developments. Out of all the possibilities, three applications elds are closely related to this thesis work:

Drive cycle identication methods: Global optimization methods and specially the proposed online predictive optimization method emphasized the importance of the knowledge about the drive cycle. The dierences between the local optimization results and global optimization ones could be reduced using a better estimation about the type of the drive cycle imposed to the vehicle. Consequently, it can increase the performance of the vehicle even in realtime applications.

Such estimation must represent the main characteristics of the drive conditions and should be provided during the driving operation. This type of research has been developed for several applications using dierent approaches, such as GPS data analysis, statistical analysis and intelligent methods for pattern recognition.

Control and observation of the fuel cell state variables: As already mentioned, the fuel cell response is aected by the conditions imposed to its state variables.

Being a multivariable system, a more elaborated control strategy would be a better option to obtain a regulation of the variables in a integrated way. This means a control strategy that aggregates the dynamics of each subsystem to dene commands to be applied on the actuators. The interest of such proposition is related to obtain a better performance of the fuel cell system (energetic gain) and a better control of the degradation rate. Considering these two topics, our work open the prespectives for developments on the control of the fuel cell state variables, as was performed for the air supply subsystem variables.

As presented in the chapter 6, the other state variables (temperature, humidity, etc) are also important in the durability and performance of the fuel cell system, but their related dynamics require the use of more advanced control laws for stabilization and optimization.

Despite the developments and researches conducted in this area, it is still considered an open research eld. Some of the most diculties to advance in this topic are about the several application domains required in the development:

• Modeling

• Parametric identication

• Observation and reconstruction of state variables (software sensors)

• Control strategies (linear and non-linear)

These requirements indicate the need to perform a multi-domain interaction represented by commom eorts of dierent experts (controls, electrochemical, modeling, system design, etc.).

Fuel cell diagnostics: In order to deal with the fuel cell system durability issue, an understanding of the degradation mechanisms and their measurements are very important. All this diagnostic process can be used to understand the degradation and evaluate the state of health of the fuel cell system. Moreover, either through the fault detection or by the estimation about the dynamic performance evolution, the diagnostic methods can oer reliability to the fuel cell system and aid in the control of the degradation process.

Nevertheless, the characteristics of the related areas demand research levels from the nano scale up to the system level, which involves the adoption of models and measurement systems not completely adapted to be coupled and to be embedded in realtime systems. At the time, numerous eorts are been made to overcome these drawbacks, but research groups still faces gaps about the degradation process informations and diagnostic implementation that could be lled adopting multiscale modeling approaches, adapted measurement systems and identication methods to indicate the dynamical behavior of the fuel cell degradation. The main elements that compose the system and their characteristics are:

Fuel cell system The stack embedded in PROPAC is composed of 90 cells and its voltage can vary from 62V to 100V depending on its operating conditions. The nominal operating point (maximum power of the stack) is at 62V 105A and 6.5kW . Pure hydrogen at 1.2 bar is used in the stack with a dead-end conguration. The pressure and ow control at the anode is made by a servovalve that uses the anode inlet pressure information to control the hydrogen ow towards the stack. A purge procedure is done aiming to eliminate the nitrogen and water in liquid state, possibly present in the anode chamber.

On the cathode side, a compressor ensures the air feeding. The air, initially ltered, is compressed and sent to a humidier, where it is enriched with water vapor before entering the cathode. At the outlet of the cathode, the air depleted in oxygen and rich in water vapor is introduced into the secondary of the humidier. Coordinated with the VI Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés compressor, the valve at the outlet of the humidier allows to eectuate the pressure control.

The cooling circuit is composed by a pump, a radiator and a thermovalve. The cooling liquid (de-ionized water) cross the stack and pass through the thermovalve. This element has the function to direct the liquid ow in function of the liquid temperature. This procedure is made passively and progressively using a simple strategy:

1. When the temperature of the cooling liquid is lower than 60°C, the ow is aspirated by the pump and driven towards the hydrogen heater before to get into stack again.

2. When the temperature of the cooling liquid is higher than 70°C, the ow is driven to a heat exchanger (for a possible heat recover) and afterwards to a ventilated radiator to evacuate the heat. Then, the liquid follows the aforementioned path.

3. When the temperature value is in between 60°C and 70°C, the thermovalve remains in a state where it is partially opened for the two mentioned circuits.

Batteries Two battery packs of 24V are embedded in the PROSPAC system. They have technology LiF eP O 4 and were developed by CEA. In this series hybrid conguration, the batteries aims to guarantee the start-up procedure of the FC system and to perform the hybridization in the power split procedure.

Each battery pack has a capacity of 1kW h, constituting 2kW h in total. The operation voltage value is 48V in discharge mode and 58V for a 100% charged. DC/DC converter A buck converter is used as the interface between the fuel cell stack and the voltage bus (where the batteries are located). The input voltage range is from 60V to 100V , while the output voltage is around 55V . This converter presents a maximum power of 6.5kW in the input at steady state and with a minimum eciency of 90% when its in the zone between 74V 35A to 62V 105A.

Control and supervision

The PROSPAC system is entirely controled in an autonomous and automatic way by a DSPACE fast prototyping device of type microau-toboxII. This device ensures the management of all the auxiliaries of the system during all the functioning phases, which are composed by:

FinalementFigure 0 . 1 :

 01 Figure 0.1: Architecture série pour un véhicule hybride à pile à combustible

Figure 0 . 2 :

 02 Figure 0.2: Système pile à combustible

Figure 0 . 3 :

 03 Figure 0.3: Méthode de dimensionnement du VHPAC

3 )

 3 du conducteur (ou cycle de conduite) et comme sortie la vitesse et/ou accélération du véhicule. Par conséquent, les modèles du véhicule, réducteur et moteur électrique ont été construits à partir de la dynamique associée à chaque élément. La dynamique du véhicule permet d'aboutir à l'équation (3). D'autre part, le réducteur est modélisé par un gain avec un rendement xe et le moteur électrique par une représentation quasi-statique qui combine une cartographie de pertes avec une inertie mécanique. xvii dv dt = F traction -F resistance M (Modèles et algorithmes pour le dimensionnement du système PAC et batterie La dénition du taux d'hybridation (dimensionnement du système PAC et batterie) se base sur un modèle inverse du véhicule et les caractéristiques énergétiques/puissance des systèmes PAC et batterie. Contrairement au modèle directe, le modèle inverse a pour entrée le prol vitesse du véhicule et comme sortie la puissance électrique demandée pour que le véhicule suivre la trajectoire imposée. Le modèle du véhicule et des éléments de la chaîne de traction sont donc simpliés (dynamique simplié) en vue de l'inversion.

Figure 0 . 4 :

 04 Figure 0.4: Structure pour le développement de la gestion d'énergie en ligne

Figure 0 . 5 :

 05 Figure 0.5: Méthodologie de dimensionnement du VHPAC

Figure 0 . 8 :

 08 Figure 0.8: Réponses en tension pour une PAC (5 cellules) à diérents temps de durée d'utilisation[START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF]

( 21 )Figure 0 . 9 :Figure 0 . 10 :

 2109010 Figure 0.9: Taux de dégradation de surface active x Puissance fourni système PAC

Figure 0 . 11 :Figure 0 . 12 :

 011012 Figure 0.11: λ 4 en fonction du SoH (t)

Figure 1 . 1 :

 11 Figure 1.1: Hydrogen economy scheme

Figure 1 . 2 :

 12 Figure 1.2: Approach structure

Figure 2 . 1 :

 21 Figure 2.1: Ford Edge FCEV

Figure 2 . 2 :

 22 Figure 2.2: GM Chevy Volt FCEV

Figure 2 . 4 :

 24 Figure 2.4: Hyundai ix35 FCEV

Figure 2 . 5 :

 25 Figure 2.5: Daimler-Mercedes Benz [2]

•

  Economic and Institutional Safety codes and standards Hydrogen supply and delivery structure Manufacturing cost and supplier base Public awareness and acceptance

Figure 2 .

 2 Figure2.6 shows the status of production and delivery cost for dierent methods.

Figure 2 . 6 :

 26 Figure 2.6: Hydrogen production and delivery cost status[START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF]

  by 2017. Meanwhile, advances in the development of the fuel cell and its system should continue, particularly with regard to the development and adoption of equipment in the system with lower cost of production and maintenance. For the fuel cell the advances should be done by replacing expensive noble materials such as platinum by other materials with same performance and lower cost.

Figure 2 . 7 :

 27 Figure 2.7: Projected transportation fuel cell system cost -projected to high-volume

Figure 2 . 8 :

 28 Figure 2.8: Vehicles architectures

Figure 2 . 9 :

 29 Figure 2.9: Fuel cell vehicle architecture

Figure 2 .Figure 2 .

 22 Figure 2.10: Vehicle dynamics

Figure 2 .

 2 Figure 2.11 illustrates a basic prole of this relationship torque vs speed. The torque signal indicates that the machine operates in motor mode (positive torque), or generator mode (negative torque). The operating limits indicate a region of maximum torque (or minimum in the case of generator) and a region of iso-power. Inside, the dierent zones of eciency are present.

Figure 2 . 11 :

 211 Figure 2.11: Electrical machine static representation

Figure 2 . 12 :

 212 Figure 2.12: Functioning of a Li ion battery cell

Figure 2 . 13 :

 213 Figure 2.13: Electric circuit battery representation-(a) perfect source+ internal resistance, (b) dynamic model

Figure 2 .

 2 Figure 2.14: Cell structure

18 )Figure 2 .

 182 Figure 2.16 shows the characteristic curve voltage vs current density for a cell (polarization curve), where can be seen the operating region of each specic loss.

Figure 2 .

 2 Figure 2.16: Polarization curve

Figure 2 . 17 :

 217 Figure 2.17: Example of a fuel cell system

Figure 2 . 18 :

 218 Figure 2.18: Fuel cell air supply subsystem

  water temperature at FC outlet T in_w_FC : water temperature at FC intlet T in_w_R : water temperature at Radiator intlet T out_w_R : water temperature at Radiator outlet Q R : Heat exchange in the Radiator Q FC : Heat exchange in the FC Q fan : Heat exchange by the fan Q losses : Heat due to the losses in the FC T R : temperature of the Radiator T FC : temperature of the FC

Figure 2 .

 2 Figure 2.19: Cooling subsystem

  Figure (3.1) presents this library.

Figure 3 .

 3 Figure 3.1: VEHLIB

  Figure 3.2: Extra-Urban Cycle HYZROUT

Figure 3 .

 3 Figure 3.3 illustrates these two types of models for the same electric powertrain.

Figure 3 .

 3 Figure 3.4 illustrates a fuel cell hybrid electric forward model created using VEHLIB.

  electric motor function f 2 : gear reduction function f 3 : vehicle model function

Figure 3 . 3 :

 33 Figure 3.3: Forward and backward modeling

Figure 3 . 5 :

 35 Figure 3.5: Structure to perform the optimal sizing procedure

3 )Figure 3 . 6 :

 336 Figure 3.6: Structure of the models for the drive train sizing procedure

Figure 3 . 7 :

 37 Figure 3.7: Dynamic programming diagram

Figure 3 . 9 :

 39 Figure 3.9: Typical FC Stack power vs current

Figure 3 . 11 :

 311 Figure 3.11: Structure for the development of the online energy management strategy

Figure 3 . 12 :

 312 Figure 3.12: Cell meshed scheme

3. 4 . 3 . 1

 431 Drive train model As in the case of the optimization for the sizing procedure, the energy management strategy needs to know the electric power demanded by the vehicle during the cycle. Such information is provided by a backward model of the drive train, which transforms the speed prole (drive cycle) in power prole. The equations, static maps and calculations that represent this backward model are the same as used in the optimization for the sizing procedure (eqs. (3.3), (3.5) and (3.6)).

Figure 3 . 13 :

 313 Figure 3.13: Typical equivalent hydrogen consumption map for a fuel cell system

  Adoption of a new platform oriented for the particular application Even if the second option represent the best option for a new technology, it requires more eort in the development of the elements that compose the platform. This drawback is overcome in the rst option, where it presents the advantage to allow the utilization of the known characteristics associated to the already existing elements, serving as a basis for the denition of the technical specications. This work uses an existing vehicle with light-duty characteristics chosen as the reference. It should be able to meet the demands of ordinary drivers in dierent drive conditions. Thus, the vehicle must have performance characteristics that permits to accomplish either urban or highway paths. Having all the necessary features to perform the requirements about the types of missions, a Renault Clio II 1,5L Dci model (gure 4.1) was selected as the reference platform (vehicle).

Figure 4 . 1 :

 41 Figure 4.1: Reference Vehicle: Renault Clio 2 1,5L Dci

1 .

 1 Electric motor a) Maximum rotational speed b) Maximum/ Minimum torque: same value with dierent signs (+ traction and -braking) c) Maximum/ Minimum power: same value with dierent signs (variables are subjected to one or more aspects of the technical specications (table

Figure 4 . 2 :

 42 Figure 4.2: Sizing methodology structure

Figure 4 . 3 :

 43 Figure 4.3: Electric motor map VEHLIB

Figure 4 . 4 :

 44 Figure 4.4: Ideal electric motor map

Figure 4 . 5 :

 45 Figure 4.5: Traction power x maximum speed for the vehicle

Figure 4 .

 4 Figure 4.6 shows the diagram that represents the procedure used to obtain the acceleration time vs maximum traction power.

Figure 4 . 6 :

 46 Figure 4.6: Diagram for the acceleration index

Figure 4 . 7 :

 47 Figure 4.7: Time x Power for the accelerations requirements

Figure 4

 4 : fuel cell system power profile for a drive cycle and an hybrid configuration FC system consumption model P vehicle : vehicle required power for a drive cycle H 2 * /km H 2 * /km : optimal hydrogen consumption per km for a drive cycle and an hybrid configuration Battery pack model SoE* P* Battery P* Battery : Battery power profile for a drive cycle and an hybrid configuration SoE* : State of energy profile for a drive cycle and an hybrid configuration

Figure 4 . 8 :

 48 Figure 4.8: Optimal sizing structure

Figure 4 . 9 :

 49 Figure 4.9: Battery pack conguration

Figure 4 . 10 :

 410 Figure 4.10: Proportion of each type of route

Figure 4 . 11 :

 411 Figure 4.11: Travelled distance distribution for a single travel

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.12: HYZURB-Urban cycle: power and speed prole

Figure 4 .

 4 Figure 4.16 presents the results for the urban drive cycle.

Figure 4 . 15 :

 415 Figure 4.15: Sizing study of battery and FC system for urban cycle

Figure 4 . 16 :

 416 Figure 4.16: Sizing study of battery and FC system for extra-urban cycle

Figure 4 . 18 :

 418 Figure 4.18: Cumulated travelled distance distribution for a single travel

Figure 5 . 1 :

 51 Figure 5.1: Simplied model of air supply subsystem

y 1 - 2 Żd 2 = Z d 3 Żd 3 = 2 ( 5 Figure 5 . 2

 122332552 Figure 5.2 presents the control structure proposed where the systems i are dened as:

Figure 5 . 2 :

 52 Figure 5.2: Control block structure

  the fuel cell system

Figure 5 . 3 :

 53 Figure 5.3: Electrical power prole demanded to the fuel cell

  Figures in 5.4 present the behavior of the above-mentioned variables when the system is subjected to the power prole of the gure 5.3.

Figure 5 . 4 :

 54 Figure 5.4: Results of the simulation-air system variables

Figure 5 . 5 :

 55 Figure 5.5: Pressure valve control input during the transients and disturbances

Figure 5 . 6 :

 56 Figure 5.6: Cell voltage

Figure 5 . 7 :

 57 Figure 5.7: Fuel cell internal variables

Figure 5 . 8 :Figure 5 . 10 :

 58510 Figure 5.8: Demanded current prole

Figure 5 . 11 :Figure 5 . 12 :

 511512 Figure 5.11: Compressor speed for the control strategies

Figure 5 . 13 :

 513 Figure 5.13: Fuel cell stack electrical response-atness strategy

Figure 5 .

 5 [START_REF] André | The artemis european driving cycles for measuring car pollutant emissions. 7th international symposium, highway and urban pollution[END_REF] presents the control block diagram.

Figure 5 . 14 :

 514 Figure 5.14: Control block diagram

Figure 5 .

 5 Figure 5.15: hydrogen consumption vs prediction horizon

  Allied with a xed reference value (SoE 0 ), the two aforementioned components (SoE kinetic and SoE potential ) dene the SoE ref erence function (equation 5.41).SoE ref erence = SoE 0 -SoE kinetic -SoE potential (5.41)The value of SoE 0 is given by the stationary desired value (SoE (0) = 60 %) and brings the power assistance mode (SoE (0) = SoE (t f )) to the SoE ref erence function . SoE kinetic is obtained from the equation 5.42, where η braking chain is the eciency of the vehicle braking chain, v (t) is the current vehicle speed and v max is maximum speed. SoE kinetic = (SoE max -SoE min ) η braking chain Kinectic Energy(v(t)) Kinectic Energy(vmax) = = (SoE max -SoE min

Figure 5 . 16 :

 516 Figure 5.16: Optimization control block

Figure 5 . 17 :

 517 Figure 5.17: Urban cycle optimization results

Figure 5 .

 5 Figure 5.18: Extra-urban cycle optimization results

Figure 5 . 19 :

 519 Figure 5.19: Highway cycle optimization results

Figure 5 .

 5 Figure 5.21 presents the proles of P F C , SoE and λ 1 . The information about the grade changes the SoE ref erence in the proposed algorithm, which leads to a result very close to what is obtained with the DP method. Furthermore, the dierence in hydrogen

Figure 5 . 21 :

 521 Figure 5.21: Mountain cycle optimization results

Figure 5 . 22 :

 522 Figure 5.22: Experimental test structure

Figure 5 . 24 :

 524 Figure 5.24: Fuel cell system power and battery SoE proles for the EMS strategies

Figure 5 . 23 :

 523 Figure 5.23: Demanded Power prole

Figure 5 . 25 :

 525 Figure 5.25: Hydrogen consumption vs ∆SoE for the EMS strategies

  Regarding the time of life denitions and the performance index that indicate it, the rst part of the chapter establishes the background needed to understand to the phenomena and operation modes related to the degradation mechanisms. This part is composed by a denition of a fuel cell end of life constraint performance, the operation modes and mechanisms present in the degradation process, and a state of art of the methods used in the diagnostics of the fuel cell state of health (SoH ).

Figure 6 . 1 :

 61 Figure 6.1: Polarization curves for a fuel cell stack (5 cells) in dierent life times[START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF]

Figure 6 . 2 :

 62 Figure 6.2: Durability index [95]

  According to Yang et al. 130, this operation aects the durability of the Membrane Electrode Membrane (MEA) by the occurrence of degradation mechanisms. Yang et al. 130 stated that the cold start has found three principal degradation mechanisms:

6. 3 . 2

 32 Diagnostic methods for the fuel cell stack durability All the aforementioned vehicle operating modes are responsible by a degradation contribution in the decay of the fuel cell time of life. Pei et al. 95 determined experimentally each contribution, allowing to observe the evolution of the degradation of the stack in realtime. However, such approach needs an important number of experimental tests in a long duration in a real FCHEV to characterize these durability decay rate. The structure necessary to perform these test campaigns is out of the scope of our work. Therefore, in this work, the State of Health SoH of a fuel cell stack is measured by the observation of variables and intrinsic phenomena, such as the active surface value (catalyst layer), membrane resistivity, etc.

Figure 6 . 4 :constraintFigure 6 . 5 :Figure 6 . 6 :Figure 6 . 6

 64656666 Figure 6.4: Polarization curves for dierent active surface values [102]

Figure ( 6 . 8 )Figure 6 . 8 :

 6868 Figure 6.8: Method used to obtain the limit active surface area value

Figure 6 . 9 :

 69 Figure 6.9: SoH limit proposed prole

Figure 6 . 10 :

 610 Figure 6.10: SoE or SoC control block diagram

Figure 6 . 11 :

 611 Figure 6.11: SoH limit prole

Figure 6 . 12 :

 612 Figure 6.12: λ 4 in function of SoH (t)

Figure 6 . 13 :

 613 Figure 6.13: New optimization structure including SoH control

Figure 6 .

 6 Figure 6.14: Drive Cycle

Figure 6 . 15 :

 615 Figure 6.15: FC system power prole

Figure 6 . 16 :

 616 Figure 6.16: SoE prole

  the fuel cell to reach this end of life level too early. Requirements such as start/stop and load cycling are examples of type of vehicle requirements. However, they represent conditions in the vehicle level that are related to several phenomena within the fuel cell stack. The understanding of the causes and eects of theses phenomena is necessary as a background before including the durability aspect in the optimal energy manage-constraint ment strategy. Therefore, the chapter introduced a brief review of the most important degradation mechanisms and which components of the fuel cell that are aected.As important as the knowledge of the degradation mechanisms is the diagnosis of the fuel cell state of health SoH. Such information allows to follow the performance of the fuel cell and to propose actions that reduce degradation. Several methods used to evaluate the SoH are existing in the literature. They adopt approaches based on dierent identication tools. Examples of identication methods are presented and it can be concluded that despite the evolution in diagnostic of the fuel cell, this subject still needs some developments before being integrated in realtime applications, such as FCHEV.

Figure A. 3 :Figure A. 4 :

 34 Figure A.2: Extra-Urban Cycle HYZROUT

• 9 J 32 . 10 - 6 kg.m 2 ζ 2 - 4 . 5 α 3 .

 9321062453 Stopped mode • Battery mode VII • Fuel cell system start-up • Fuel cell system nominal state • Hybrid management for the entire system • Stop procedure of the fuel cell system • Security stop procedure of the fuel cell system • Urgency stop procedure The communication between the auxiliaries and the sensors is done by analogical signals, digital signals and two CAN bus. VIII Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés 45 N m.V -1 N cell 80 C p 1 kJ.kg -1

  + + 2e -←→ H 2 O .............................................................. H 2 + 1 2 O 2 ←→ H 2 O

[START_REF]Performance renault clio[END_REF]

La structure élémentaire est composée de couches actives (anode et cathode), d'une membrane et de couches de diusion. L'ensemble est nommé Assemblage Membrane Electrode (AME).

  Comme dans le cas du sous système hydrogène, ce sous système contrôle et régule le débit (stoechiométrie St O 2 ) et la pression (p cath ) de l'air dans la cellule. Les éléments qui composent ce circuit sont, les plus souvent, un compresseur d'air, un humidicateur et une vanne de régulation de pression. Le contrôle du débit d'air est eectué par le compresseur, tandis que la vanne de régulation de pression en sortie pilote la pression de la cathode. L'humidicateur est utilisé pour humidier l'air

	Dimensionnement chaîne de traction
	Cahier de charges			Validation
	(vitesse, accel, etc)
	Type
	de
	spécification: Statique (eg. vit max) Dynamique (eg. accel max)	Modèles utilisés: Eléments chaine de traction: résolutions stationaires et	Dimensions Chaine
		dynamiques	Traction	Modèles utilisés: Modèle direct	Dimensionnement
	Dimensionnement Système PAC et Batterie		VHPAC	optimal
	Cycles de
	conduite
	(urban, routier, etc)	Modèles utilisés:
		modèles inverses:
	Objectif:	véhicule Modèle direct:
	Minimization de la	Système PAC
	consommation énergétique	Batterie	Dimensions
	(hydrogène)	Gestion d'énergie : Algorithme	Système PAC et Batterie
		d'optimisation
	Modèles et méthodes d'optimisation utilisés
	Chaque objectif de ce travail nécessite d'outils et des modèles adaptés aux problèmes
	posés. Dans cette section, nous allons maintenant présenter les outils (modèles et algo-
	rithmes d'optimisation) utilisés dans la suite du mémoire.

à l'entrée de la cathode an d'augmenter les performances et durabilité de la pile. Dans cette étude un humidicateur passif par diusion de l'eau de l'air humidié en sortie via l'air sec en entrée est utilisé.

Sous système de refroidissement

Comme la réaction électrochimique de la PAC est exothermique, un circuit de refroidissement est nécessaire pour réguler la température et évacuer le ux de chaleur créé. Ce circuit est normalement composé d'un radiateur, d'une vanne thermostatique et d'une pompe à eau. La régulation de température de la pile est faite à partir de la vanne et d'un ventilateur au niveau du radiateur. Sous système électrique Ce sous système constitue l'interface entre le bus électrique et la PAC. Un convertisseur DC/DC irréversible est utilisé pour cette interface et permet la régulation de courant. Après avoir présenté les principales caractéristiques du VHPAC, la section suivante présente les modèles et les méthodes d'optimisation utilisés dans chaque partie de ce travail. Outils pour le dimensionnement optimal du VHPAC Dans la méthode de dimensionnement du véhicule, les diérents éléments de la chaine de traction et des sources de puissance (système PAC et batterie) sont soumis à des xvi

  En conséquence, la nouvelle dynamique équivalente du système PAC est intégrée au problème d'optimisation et sa résolution est fait premièrement par l'application du Principe du Maximum de Pontryaguine, qui donne un résultat très simple (16):

xxvi Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés

  montrent les contribution des énergies cinétiques et potentielles au calcul du SoE ref , où η chaine f reinage est le rendement moyen de la chaîne de traction pendant les phases génératives (moteur en mode générateur). xxvii SoE cinetique = (SoE max -SoE min ) η chaine f reinage SoE P otentielle = (SoE max -SoE min ) η chaine f reinage Energie P otentielle(h(t)-h ref )

	Energie cinetique(v(t))
	Energie cinetique(vmax)
	(19)
	Energie P otentielle(hmax-h min )

  déterminée selon les exigences de l'application, 30 kW en mode stationnaire par exemple ). Ainsi, les dégradations irréversibles sont observés par le changement de la réponse en tension de la PAC. Donc, pour fournir une même puissance, la densité de courant pour fournir une puissance donnée sera diérente pour une PAC en début de vie et qu'une PAC en n de vie (g. (0.8)[START_REF] Gerard | Etude des interactions pile/systeme en vue de l optimisation d un generateur pile a combustible: Interactions coeur de pile/ compresseur; -Interactions coeur de pile/ convertisseur[END_REF]). Une conséquence est une perte de rendement et donc une consommation d'hydrogène diérent (la PAC dégradée consomme plus d'hydrogène pour fournir la même puissance). A partir de cette analyse, il est possible de dénir la n de vie d'une PAC lorsque sa tension à puissance nominale est 10 % inférieure à celle observée en début de vie[START_REF] Pei | A quick evaluating method for automotive fuel cell lifetime[END_REF].

												Sys. PAC
			Calculateur Véhicule								Contrôle
												Local
		SOE REFERENCE
	SOE CONSTANT	+	+	+	PI	fPI	Lambda1 Calcul	LAMBDA1	OPTIMISATION	PPAC*	Système PAC	PPAC	+	Pvehicule	VEHICULE	vvehicule
	REFERENCE	-	-	-										+
														Pbatt
							Pvehicule (t..Tf)	Pvehicule distribution
		Conversion	Conversion		SOE									Batterie
		Energie	Energie
		cinetique=>	potentielle =>
		SOE	SOE
	Pedal d'accel			Consigne de couple		Pvehicule
							ESTIMATION		Pvehicule_estimé
														Altitude
											vvehicule
			Figure 0.7: Schéma block d'optimisation
	Gestion d'énergie intégrant la durabilité de la PAC

La troisième contribution de ce travail est l'introduction de la durabilité de la pile à combustible dans la gestion d'énergie du véhicule. L'objectif de cette implémentation est de contrôler le taux de dégradation de la PAC pendant les diérents modes opératoires d'un VHPAC, et en conséquence de maximiser sa durabilité pour atteindre l'objectif établi par le DOE (5000 hrs de durée de vie pour le système PAC). Les grandeurs liées à la durabilité doivent tout d'abord être dénies. Parmi ces grandeurs, le critère de n de vie peut être caractérisée en fonction de la performance de la PAC, ainsi on peut dénir un niveau de performance minimal permettant d'assurer les conditions de fonctionnement minimales dans le contexte des applications automobiles.

Pei

et al. 95 ont dénit que le critère d'observation est la tension de la PAC lorsqu'elle fournie sa puissance maximale xxviii Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés

(

  Modeling tools and concepts . . . . . . . . . . . . . . . . . . . . . Drive train sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . Results from the optimization applied in the model . . . . . . . . 5.5.2 Experimental tests of the optimization algorithm . . . . . . . . . 5.5.2.1 Experimental structure . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

		5.5.1
	3.2	Modeling and energy management strategy for FCHEV . . . . . . . . . .
		3.2.1
		3.2.2	Energy management strategy: concepts and state of the art 5.5.2.2 Developed tests . . . . . . . . . . . . . . . . . . . . . . . . .
	Table of Contents 3.3.2 Fuel cell system and battery sizing . . . . . . . . . . . . . . . . . 3.2.2.1 State of the art: Energy management . . . . . . . . . . . 3.2.3 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 5.6 Conclusions of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 6 Optimal energy management strategy including fuel cell durability Conclusion about the modeling and EMS concepts . . . . . . . . . 3.3 Models and optimization methods for the optimal sizing constraint . . . . . . . . . 3.3.1 6.1 6.2 Important concepts and requirements about durability in a FCHEV . . . General Introduction	xxxv
	1 General Introduction 3.3.2.1 Global optimization method: Dynamic Programming . .
	1.1 Context
	7 General Conclusion
	Bibliography
	A Drive Cycles	I
			3.4.3.3 5.3.4 Results in Real Fuel Cell System . . . . . . . . . . . . . . . . . . Fuel cell system consumption model . . . . . . . . . . .
			3.4.3.4 5.3.5 Conclusion on the air supply subsystem control . . . . . . . . . . Optimization method . . . . . . . . . . . . . . . . . . .
	3.5	5.4	New Optimization problem and predictive online energy management strat-

1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Hydrogen economy in the transportation context . . . . . . . . . . . . . . 1.3 Fuel Cell Vehicle: advantages, challenges and motivations . . . . . . . . . 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Overview of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Summary of the contributions . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.2 Models for the optimization process . . . . . . . . . . . . 3.3.3 Validation of the sizing . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Conclusion about the tools related to the optimal sizing objective 3.4 Models and optimization methods for the design of online energy management strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Dynamic fuel cell system model for the validation of a control law 3.4.1.1 Air supply subsystem . . . . . . . . . . . . . . . . . . . 3.4.1.2 Hydrogen subsystem . . . . . . . . . . . . . . . . . . . . 3.4.1.3 Cooling subsystem . . . . . . . . . . . . . . . . . . . . . 3.4.1.4 Electric subsystem . . . . . . . . . . . . . . . . . . . . . 3.4.1.5 Fuel cell . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Fuel cell system dynamic model and PROSPAC prototype in the control context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Models and optimization algorithm for the conception of the online energy management strategy . . . . . . . . . . . . . . . . . . . . . 3.4.3.1 Drive train model . . . . . . . . . . . . . . . . . . . . . . 3.4.3.2 Battery model . . . . . . . . . . . . . . . . . . . . . . . 4 Fuel Cell Hybrid Electric Vehicle Sizing 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 FCHEV Sizing methods: state of art, concepts and basic requirements . . 4.2.1 State of art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fuel Cell System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Air circuit control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Control Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Proposed Control Strategy . . . . . . . . . . . . . . . . . . . . . . 5.3.2.1 Dierential atness theory . . . . . . . . . . . . . . . . . 5.3.2.2 Implementation of the dierential atness control strategy 5.3.2.3 New Dynamics of the System . . . . . . . . . . . . . . . 5.3.2.4 Control Structure . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . egy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Online predictive energy management method . . . . . . . . . . . 5.4.2.1 Predictive approach . . . . . . . . . . . . . . . . . . . . 5.5 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés 6.3 Degradation of FC in a FCHEV: Operation modes, degradation mechanisms and diagnostic methods . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Vehicle operating modes and their degradation mechanisms . . . . 6.3.2 Diagnostic methods for the fuel cell stack durability . . . . . . . 6.4 Fuel cell stack degradation dynamic model and optimization problem for EMS application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Degradation dynamic model . . . . . . . . . . . . . . . . . . . . . 6.4.2 Optimization problem with durability constraint . . . . . . . . . . 6.4.3 SoH limits for the optimization problem . . . . . . . . . . . . . . 6.5 Optimal energy management strategy with durability constraints . . . . . 6.6 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusion of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 2 .

 2

	Energy density
	2. Power density

1: Features of dierent technologies of batteries Historically, technologies Ni/Cd and Ni/MH were the rst to be adopted for hybrid vehicular applications. With the increasing development of technology based on Li ion,

Table 2

 2

	Acronym	Name	Temperature	Electrolyte
	PEMFC	Polymer Electrode Membrane Fuel Cell	50°C -80°C	Polymer
	MT-PEMFC	Medium Temperature PEMFC	90°C -120°C	Polymer
	HT-PEMFC	High Temperature PEMFC	120°C -180°C	Polymer
	DMFC	Direct Methanol Fuel Cell	25°C -50°C	Polymer
	DEFC	Direct Ethanol Fuel Cell	25°C -50°C	Polymer
	SOFC	Solid Oxide Fuel Cell	700°C -1000°C	Ceramic
	MCFC	Molten Carbonate Fuel Cell	650°C	Liquid
	PAFC	Phosforic Acid Fuel Cell	150°C -200°C	Liquid
	AFC	Alkaline Fuel Cell	60°C -120°C	Liquid

.2 presents the existing fuel cell technologies and its main features.

Table 2 .

 2

2: Fuel cell technologies

[START_REF] Tritschler | Optimisation de l architecture electrique et gestion d energie pour un systeme a pile a combustible embarquee dedie a l application agricole[END_REF]

  . Neglecting irreversible losses within the cell, this energy G is given by the equation 2.15. Where n is the number of electrons involved in the reaction, F is Faraday's constant and E max the maximum voltage of the cell in open circuit.

	RT nF	ln	P H 2 P atm	P O 2 P atm	(2.17)

G = -nF E max (2.15)

Through this equation and using the relationship of equation 2.16, one can estimate the thermodynamic potential of equilibrium at a temperature T . As the entropy S is dependent on temperature and pressure, the expression of the reversible voltage E rev is a function of the partial pressures and the temperature. It is obtained through the Nerst equation (equation 2.17)

[START_REF] Candusso | Hybridation du groupe electrogene a pile a combustible pour l alimentation d un vehicule electrique[END_REF]

.

G = H -T S

(2.16)

E rev = E max -

  ∆p hu = p hu inlet -p hu outlet = k hu .q 2Control pressure valve: This element is modeled by the ow and displacement equations in(3.22) and (3.23) respectively. In(3.22) it was considered a laminar resultant ow, while in the displacement equation the inuence of temperature is neglected due to the lack of information and for the sake of simplicity.

		dQ dt	= µ.S.∆T mem		(3.19)
	V hu	dp hu outlet dt	=	RT hu m molar i	q i	(3.20)
						hu	(3.21)

and the heat transfer inside it by

(3.19)

. Besides, the model of this component has the same type of pressure drop as in the cathode (3.21).

  .27)3.4.1.4 Electric subsystemBuck converter model in (3.28) is adopted to integrate the fuel cell system model. As can be noticed, this equation do not takes into account the losses in switching and the changes in the parameters due to the temperature eects. Such informations were not available to integrate in the model, but for the assigned purposes the model is enough detailed.

Table 4 .

 4

	Performance
	M ax.Speed	160 km/h
	time f or :0 → 100 km/h	14.3 s
	400 m start → stop	19.5 s
	1000 m start → stop	36.2 s

1: Technical data of reference vehicle By these technical characteristics, this vehicle is capable to reach the performance levels described in table 4.2 [1].

Table 4 . 2

 42

: Performance of the reference vehicle

In order to adopt the aforedescribed platform as a reference for the sizing of the new powertrain, a numerical model of this vehicle is used. The model integrates the VEHLIB modeling tool and was created and validated using experimental results obtained from tests on a real Renault Clio II 1.5L Dci. As stated in the last chapter, VEHLIB has a focus on the energy consumption analyses, meaning that some inherent dynamics of the vehicle were simplied.

Despite the importance of the performance index in table 4.2, it is necessary to take into account other index adapted to the new technologies involved in the electrical powertrain.

Table 4

 4

	Constraints		Description
	Acceleration time	097 km/h	12.2 s
		6497 km/h	5.3 s
		0137 km/h	23.4 s with hybrid traction
	048.3 km/h	5 s with motor-only traction
	Gradeability	88.5 km/h at 6.5% grade
	for 1200 s with the SoC (State of Charge) range of 20%70%
	Maximum speed		145 km/h
	Table 4.3: PNGV hybrid vehicle performance requirements [126]
	The performance index that compose the technical specications are shown in table
	4.4.
	Constraints		Description
	Acceleration time		0100 km/h	13 s
			80120 km/h	20 s
			3060 km/h	6 s
	Gradeability	0-10 km/h in 2s at 20% grade
		with the SoC (State of Charge) range of 2070 %
	Maximum speed (vehicle)		145 km/h
	Maximum speed (FC mode)		110 km/h at 5% grade
	Table 4.4: FCHEV requirements

.

[START_REF]Daimler fcev performance[END_REF]

shows the main technical references of the document for hybrid vehicles.

As can be noticed in table 4.3, the PNGV only establishes minimal performance limits that a general vehicle must meet. Consequently, a new set of specications is dened for our specic case. This appropriates the informations of the work performed by Desbois-Renaudin et al.

[START_REF] Desbois-Renaudin | Vehicules hybrides thermiqueselectriques. etude parametrique de l'architecture hybride parallele. rapport nal d'etude[END_REF]

in agreement with the data in the tables 4.3 and 4.2.

Such composition aggregates the typical drive conditions presents in Europe roads (with a highlight in France) with the minimal limits imposed by the PNGV and the inherent performances of the reference vehicle.

Table 4

 4

	.5 summarizes

Table 4 .

 4

5: Powertrain parameters

Table 4 .

 4 6: Battery capacity valuesIf the pack should have a nominal voltage of 128 V , where each element of the pack has a nominal voltage of 3.2 V . Therefore, the pack presents a physical conguration with N parallel branches composed by 40 elements in series (gure 4.9).

		h) 1	2	3	4	5	6	7	8	9	10
	N°elements 120 240 400 520 680 800 920 1080 1200 1320
	Mass (kg)	10.3 20.6 34.4 44.8 58.6 68.9 79.3	93	103.4 113.7

Table 4 .

 4 10 15 20 [START_REF] Borup | Fy 2004 program report[END_REF] 30 35 N°Cells 7: Fuel cell system power congurations Dynamic programming method is implemented in the optimization block of the gure 4.8 and is used to solve the optimization problem stated in chapter 3.

		110 164 218 273 328 382
	Mass (kg)	23 34.5 46 57.5 69 80.5

Table 4

 4

.8 indicates the average distances for each type of route.

Table 4 .

 4 8: Average distance for each type of cycleEven if these data do not dene the drive cycles to be used in the optimization process, they can guide the analysis about the sizing coherence of the optimization results and the mobility context in France.

Table 4 .

 4 The data in the table 4.11 summarize the nal sizing of the vehicle. 11: Size of the vehicle elements

		P max (kW )	44
	Electric motor	Γ max (N.m) 260
		Ω max (rpm) 5900
	Mechanical transmission	k red η red (%)	3.42 97
	FC system	P max (kW )	35
	Battery	ε max (kW h) 4.25

  Considering a time horizon t h and a calculation time step t, the sum of λ 1 over t h

	Where the distributions	t+t h j=t (P batt (j)) and t+t h j=t P vehtot (j) are calculated dier-
	ently:
										34) and knowing
	that it has a constant value during the drive cycle, a nite horizon can be applied to
	equation 5.35.
					P * batt (t) =	λ 1 -(2a 1 (P vehtot (t)) + a 2 ) 2a 1	(5.34)
					λ 1 = 2a 1 P * batt + (2a 1 (P vehtot (t)) + a 2 )	(5.35)
	is:
	t+t h j=t	λ 1 =	t h t	λ 1 =	t+t h j=t	(P batt (j)).2a 1 -	t+t h j=t	(2a 1 (P vehtot (j)) + a 2 )	(5.36)
	Hence, λ 1 is calculated in 5.37.
			λ 1 =	2a 1		t+t h j=t (P batt (j)) -t+t h j=t P vehtot (j) t h	+ a 2	(5.37)
								∆t

  Thus, at every t there is a new value of P vehtot distribution to be used in the algorithm. A certain amount of memory is necessary to store the values of P vehtot in the past time window t h , nevertheless it does not represent a strong drawback compared to the application of the Dynamic Programming.P batt distribution: The calculation of P batt distribution is made considering that the λ 1 prole will bring the battery SoE to its reference value (SoE ref erence ) at the end of the predicted horizon t h (SoE (t + t h ) = SoE ref erence ). Consequently, a change of variable can be done such that SoE information appears in the λ 1 calculation. Replacing the term t+t h j=t (P batt (j)) by ε max . SoE desired / t, the calculation of λ 1 is obtained from 5.38.Instead of using the dierence between SoE ref erence and SoE (t) to represent the ∆SoE desired target, a PI (proportional + integrator) controller is applied, which increases the robustness against the presence of noise in the SoE measurement and uncertainties in the battery model.

	λ 1 =	2a 1	∆SoE desired .εmax ∆t	-t-1 i=t-t h P vehtot (i) t h	+ a 2	(5.38)
				∆t

  ) .η braking chain SoE potential is calculated in 5.43, where h (t) is the vehicle current level, h ref is the reference level, h max is the maximum level and h min is the minimum level. SoE P otential = (SoE max -SoE min ) η braking chain P otential Energy(h(t)-h ref ) P otential Energy(hmax-h min ) =

																(5.43)
		= (SoE max -SoE min ) .η braking chain	(h(t)-href) (hmax-h min )
	The gure 5.16 presents the entire solution integrated with the physical part. P * F C is
	the optimal control applied, obtained from P vehtot -P * batt .
												FC Sys
			Vehicle controller								Local
												Control
		SOE REFERENCE
	SOE CONSTANT	+	+	+	PI	fPI	Lambda1 Calculation	LAMBDA1	OPTIMIZATION	PFC*	FC System	PFC	+	Pvehtot	VEHICULE	vvehicle
	REFERENCE	-	-	-										+
														Pbatt
							Pvehtot (t..Tf)	Pvehtot distribution
		Kinetic Energy	Potential Energy		SOE									Battery
		to	to
		SOE	SOE
		conversion	conversion
	Driver			Torque setpoint			Pvehtot
	Pedal						ESTIMATION	Pvehtot_estimated
														Level
											vvehicle
															(5.42)
									(v(t)) 2
									(vmax) 2

table 5 .

 5

	1) justies the use of the online prediction

  1.

		10	x 10 -7
									Simulation data
	Pt Surface Degradation Rate (m²/s)	5 6 7 8 9	y = 2.677e-14*x 2 -2.876e-10*x + 1.18e-06			Quadratic fitting
		1000 4	2000	3000	4000	5000	6000	7000	8000
						Power (W)
		Figure 6.7: Active Surface degradation rate x FCS power delivered

Table 6 .

 6 1: Active surface losses and hydrogen consumption It can be noticed a reduction in the degradation when the SoH (t 0 ) goes from 10

		44 cm 2	(%)	consumption (	g H 2 km	)
	10	0.0502		11.9
	5	0.0495		12.1
	1	0.0431		13.3

  Table C.1: Parameters values for the control model IX Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés

		.K -1	F	96400 C.mol -1
	T atm	300 K	St O 2	1.8
	η cp	0.75	χ O 2	0.21
	k hu	0.0084 bar.m -3 h	k F C	0.008 bar.m -3 h
	m molar	18 g/mol	k valve	9 mm/V

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2013ISAL0110/these.pdf © [R. Naiff da Fonseca], [2013], INSA de Lyon, tous droits réservés

Acknowledgements

Simulation results indicate that the amount of energy necessary to accomplish the mission is 1.7 kW h with 25 % for urban and 75 % for extra-urban drive cycles. Therefore, the total energy capacity of the pack (SoE = 100 %) is 4.25 kW h, which represents that the battery should have at least this capacity value to perform the battery mode functionality.

After the determination of the minimal limit of the battery size, an economical approach nishes the denition of the battery pack and fuel cell system dimension.

Economic feasibility step

The previous steps in the battery and FCS sizing process lead to obtain the optimal results in table 4.9 and the sizing constraints in 4.10.

Sizing variable

Constraint Criterion used to dene the constraint min P F C (max) 30 kW

Fuel cell system drive mode min ε Batt (max)

kW h

Battery drive mode Table 4.10: FCS and Battery sizing constraints This part of the methodology is intended to dene the FCS and battery pack dimansions through the comparison of the reduction in terms of hydrogen consumption with the cost related to the choice of a certain energy capacity of the battery pack and maximal FCS power capacity. In order to do this comparison, the variables units should be transformed to a common base, which is the cost in euros.

The cost data were obtained from the DOE indications [START_REF] Department | The department of energy hydrogen and fuel cells program plan[END_REF] and from the article of Xu et al. [START_REF] Xu | Optimal sizing of plug-in fuel cell electric vehicles using models of vehicle performance and system cost[END_REF], published in 2012 that develops a sizing study of a hybrid fuel cell bus.

Two values are specially important for our study:

• FCS cost: 56 ¿/kW (DOE)

• Battery cost: 496 ¿/kW h The optimization problem stated in the chapter 3 is changed in order to take into account the new equivalent dynamic for the fuel cell system as developed in the last section (5.30).

This requires the proposed solutions to consider the fuel cell system dynamics in 5.25

as a constraint to be respected during the power split denition process.

Cost M inimization :

Constraints :

Boundary condition :

Online energy management strategy

The advantages brought from the adoption of this energy management strategy permits to obtain a exible control structure, which can be adapted to receive new constraints for the optimization problem. Therefore, in the next chapter, the issue of the fuel cell system durability is presented and is included in the optimization problem and in the energy management strategy.

Optimal energy management strategy including fuel cell durability constraint Start/Stop Being subjected to ordinary drive conditions, a vehicle must have the possibility to perform start/stop procedures very often. In this operation, it is demanded a dynamic power prole and consequently a dynamic ow of fuel. This induces a transient condition in which the distribution of the fuel in the cell is not homogeneous and the proportion of H 2 /air drives a reverse current in some parts of the cell [START_REF] Borup | Fy 2004 program report[END_REF]. Moreover, this can cause a carbon corrosion on the supports of the electrodes and the collapse of the Pt catalyst layer [START_REF] Taniguchi | Analysis of electrocatalyst degradation in {PEMFC} caused by cell reversal during fuel starvation[END_REF]. Another problem created by this operation mode is about the dissolution of platinum in the membrane [START_REF] Coulon | Modélisation de la dégradation chimique de membranes dans les pile à combustibles à membrane électrolyte polymère[END_REF]. As a result, starting and stopping the fuel cell can induce considerable losses in its performance due to the reduction of the catalyst layer.

Idling The idling condition present in the vehicle operation requires that the fuel cell stack operates at its high potential and no current. It can induce performance loss caused in part by a loss of electrochemically active surface area [START_REF] Cheng | A review of {PEM} hydrogen fuel cell contamination: Impacts, mechanisms, and mitigation[END_REF]. In this loss of surface area, the platinum dissolution plays a major role, especially of the cathode catalyst, where high potentials are encountered [START_REF] Watanabea | Activity and stability of ordered and disordered co-pt alloys for phosphoric acid fuel cells[END_REF][START_REF] Zhai | The stability of pt/c catalyst in h3po4/pbi {PEMFC} during high temperature life test[END_REF]. The dissolved platinum can be clustered with other platinum particles (forming larger particle) or diuse in inaccessible parts of the MEA.

Furthermore, according to [START_REF] Wu | A review of pem fuel cell durability: Degradation mechanisms and mitigation strategies[END_REF] the OCV (Open Circuit Voltage) condition present in this mode promotes the occurrence of peroxide (H O • ) and hydroperoxide (H OO • ) radicals, which are considered to be responsible for chemical attack on the membrane and catalyst [START_REF] Buchi | Study of radiation-grafted fep-g-polystyrene membranes as polymer electrolytes in fuel cells[END_REF], aecting the performance and damaging the cell. High power load A technical specication of maximum speed requires to the vehicle to maintain a rated traction power which is provided by the powertrain elements. Depending on the power split strategy adopted, it may be required from the fuel cell system to provide its maximum power for a long duration. Such behavior causes an increase of the heat ow created by the fuel cell stack and a humidity that depends on the balance Both equation 6.1 and 6.2 must be integrated in the optimization problem to establish new constraints concerning the fuel cell system durability.

Load changing

Optimization problem with durability constraint

Due to the durability constraints of (6.1), the new optimization problem (6.3) enables a solution of optimal power split to be calculated, that takes into account the durability constraints.

Cost M inimization :

Constraints :

Boundary condition :

Appendix A

Drive Cycles

The drive cycles used in this work are presented in this appendix :

PROSPAC Prototype

This appendix presents a description of real hybrid system (FC system + battery) prototype. It is used for the development of dynamic models, control law design and tests, and energy management.

The PROSPAC system (gure B.1) is a mobile electric generator system. It can delivers a net power of 5kW . Besides, the system should respond to the following overload characteristics:

• 100% (5kW ) in steady state L'hydrogène à travers de l'utilisation des piles à combustible (PAC), est de plus en plus considéré comme une option énergétique possible au secteur des transports grace à ces caractéristiques fonctionnelles. Cependant, la technologie liée à la mise en oeuvre de véhicules alimentés par une pile à combustible n'a pas encore atteint le niveau de maturité requis. Ainsi, ce travail propose de traiter certaines de ces limitations qui existent encore. Plus précisément, trois thèmes représentent les objectifs de ce travail. Le premier thème est abordé à travers l'élaboration d'une méthodologie de dimensionnement adapté à un véhicule hybride à la pile à combustible. Dans une approche systématique, le dimensionnement proposée combine les exigences de performance présents dans les spécications techniques du véhicule, les algorithmes d'optimisation, l'analyse de la mobilité de la population et la viabilité économique de la conception. Le deuxième objectif établi a été développé à l'aide d'une approche d'optimisation de la répartition de puissance entre la batterie et le système PAC. Par l'adoption d'une méthode d'optimisation globale combinée à une stratégie de commande prédictive et l'inclusion de la dynamique du système PAC, un algorithme de gestion d'énergie pour des applications de temps réel a été conÃu. Enn, la durabilité de la pile à combustible a été incluse dans ce travail par l'intégration de sa dynamique de dégradation dans le problème d'optimisation lié à la gestion d'énergie. Cette dynamique représente une contrainte à prendre en considération lors de la répartition de puissance entre le système PAC et la batterie.

Mots-clés : Hydrogen, Fuel Cell Vehicle, Optimal Energy Management, Optimal Sizing, Fuel Cell Durability, Non Linear Control Laboratoire de Recherches : AMPERE, UMR CNRS 5005 Directeur de Thèse : Eric BIDEAUX Président du Jury : XXXXXXXXXXXXX Composition du Jury Alexandre TROFINO, Christophe TURPIN,Rachid OUTBIB, Luc LORON, Denis CANDUSSO, Eric BIDEAUX, Mathias GERARD, Bruno JEANNERET, Loic BOUILLO,Matthieu DESBOIS-RENAUDIN and Ali SARI