(différence deux à deux des courants). Ainsi, les signaux obtenus après les transformations des courants sont, table 6.13 :

<table><caption>TABLE 6.13 – Définition des différentes grandeurs dans le plan de la commande.</caption><thead><tr><th>Axe</th><th>Mode</th><th>Nom</th><th>Fonction principale</th></tr></thead><tbody><tr><td>d</td><td>Σ</td><td>I<sub>dΣ</sub></td><td>Réglage du flux dans la machine</td></tr><tr><td>d</td><td>δ</td><td>I<sub>dδ<sub>12</sub></sub></td><td>Équilibrage du flux généré entre le sous-système fractionné 1 et 2</td></tr><tr><td>d</td><td>δ</td><td>I<sub>dδ<sub>23</sub></sub></td><td>Équilibrage du flux généré entre le sous-système fractionné 2 et 3</td></tr><tr><td>q</td><td>Σ</td><td>I<sub>qΣ</sub></td><td>Réglage du couple dans la machine</td></tr><tr><td>q</td><td>δ</td><td>I<sub>qδ<sub>12</sub></sub></td><td>Équilibrage du couple généré ou de la puissance générée entre le sous-système fractionné 1 et 2</td></tr><tr><td>q</td><td>δ</td><td>I<sub>qδ<sub>23</sub></sub></td><td>Équilibrage du couple généré ou de la puissance générée entre le sous-système fractionné 2 et 3</td></tr></tbody></table>

Un essai de démarrage de 0 à la vitesse de 600 tr/min a été réalisé sur le figure 6.45. Tous les courants de référence sont nuls sauf la consigne de couple, $I_{q\Sigma} = 27A$. Les courants $I_{d\delta_{12}} = I_{d\delta_{23}} = 0$ signifient que les courants doivent avoir la même amplitude et $I_{q\delta_{12}} = I_{q\delta_{23}} = 0$ signifie que chaque sous-système fractionné apporte la même puissance à la machine.

<figure><img src="image_68.png" /><figcaption>FIGURE 6.45 – Essai de démarrage du prototype à bobinages fractionnés. a) Consignes de courants dans les plans de commande, b) Courants machine, c) différence des rapports cycliques entre deux bobinages, d) Vitesse machine.</figcaption></figure>

Sur la figure 6.45 a), la mesure du courant dans l'axe $q\Sigma$ et donc l'image du couple atteint sa valeur finale en un temps en adéquation avec la constante de temps électrique liée à l'axe $q\Sigma$. Sur la figure 6.45 b), nous observons l'évolution des différents courants mesurés. Leur amplitude est constante en fonction du temps. Ce dernier fait n'est pas si trivial. En effet, il existe une disparité entre les différentes forces électromotrices, mais surtout entre les différentes inductances des différents bobinages fractionnés selon leur position par rapport à l'entrefer. Cela entraine que les rapports cycliques appliqués aux onduleurs d'alimentation des bobinages fractionnés doivent être différenciés. Cela s'observe parfaitement sur la 6.45 c). Pour finir, sur la 6.45 d) nous observons l'évolution de la vitesse de rotation en fonction du temps.
Sur la figure 6.46, différents essais dynamiques sont réalisés. A $t = 0$, le sous-système fractionné 3 est déconnecté (ou mis à une valeur 0). De $t = 20\,ms$ à $t = 40\,ms$, la consigne de couple est annulée. À partir de $t = 40\,ms$, la consigne de couple est remise à sa valeur antérieure. De $t = 60\,ms$ à $t = 100\,ms$, une consigne négative sur le courant $I_{q\delta_{12}}$ est demandée.

<figure><img src="image_69.png" /><figcaption>FIGURE 6.46 – Essais dynamiques du prototype. a) Consignes de courants dans les plans de commande, b) Les courants dans chacun des sous-systèmes fractionnés.</figcaption></figure>

A $t = 0$, les trois courants dans le sous-système fractionné 3 s'annulent. Pour maintenir le couple constant, les deux autres sous-systèmes augmentent la valeur de leurs courants. De $t = 20\,ms$ à $t = 40\,ms$, la consigne de couple étant nulle, tous les courants s'annulent. De $t = 60\,ms$ à $t = 100\,ms$, la consigne négative sur le courant $I_{q\delta_{12}}$ entraine une dissymétrie sur la répartition des courants entre les sous-systèmes 1 et 2 tout en maintenant le couple constant. Ainsi, dans ce cas, la contribution au couple totale est plus importante pour le sous-système 2. Il est remarquable d'observer qu'une modification des consignes du mode différentiel n'affecte ni le couple, n'entraine pas de modification de la vitesse moteur, ni du bruit acoustique généré.

La campagne d'essais sur le prototype a été concluante sur le bien-fondé du brevet. Qui plus est, ces différents essais ont ouvert la voie à plusieurs nouvelles études notamment sur le power sharing entre les différentes sources sans modification du point de fonctionnement en couple/vitesse de la machine électrique.

## 6.3.2 Projet de recherche associé à la thématique CTAF

La thématique autour des machines à bobinages fractionnés et leur intégration dans une chaine de traction offre un terrain de recherche fertile dans de nombreux domaines. Parmi ce large champ, un aspect est en cohérence avec les différents sujets traités dans ce mémoire. Ainsi, seul cet aspect sera traité dans la suite du mémoire.
### 6.3.2.1 Dimensionnement "système" d'une chaîne de traction à alimentation fractionnée

**Positionnement scientifique** Parler de dimensionnement "système" est très sensible, il est difficile d'en donner une définition universelle. Néanmoins, dans la suite de ce mémoire, le dimensionnement "système" correspondra au dimensionnement global d'une chaîne de traction type *source électrique* / *convertisseur de puissance* / *machine électrique* en utilisant des modèles dégradés de chacune des parties. Par modèle dégradé, nous entendons un modèle fidèle, mais avec une précision diminuée. Ainsi, ce sont plus les interactions entre les composants qui sont mis en avant plutôt que de coller à la physique des composants mis en jeu. Nous entendons donc répondre à plusieurs interrogations :

- En considérant la chaîne de traction complète, existe t'il un nombre optimum de bobinages fractionnés ?

- Comment le degré de complexité du modèle influe sur cet optimum ?

- Comment le cahier des charges influe sur cet optimum ?

**Modèle coté machine électrique** Pour la machine électrique, les paramètres que nous souhaitons faire évoluer sont soit structurels (type de machine, type d'aimants, etc.), géométriques (diamètre externe, entrefer, etc.) soit dimensionnant. Ces paramètres dimensionnant sont définis par :

$N_p$ : Le nombre de phases, combien de phases possède la machine électrique.

$N_s$ : Le nombre de secteurs, en combien de secteurs la machine a été dimensionnée.

$N_f$ : Le nombre de bobines fractionnées, sur une dent en combien de bobines fractionnées a été réparti le bobinage initial.

Au vu du nombre de paramètres à considérer, il est difficile d'envisager une optimisation de l'ensemble directement. Nous proposons donc une optimisation par étapes :

1. Choix du cahier des charges et du volume enveloppe. A cette étape, il est important de définir sur quelle application l'étude sera menée et les critères à considérer (maximisation du rendement, coût, résilience aux défauts, etc.).

2. Choix de la structure à étudier (topologie, aimants).

3. Pré-dimensionnement de la machine électrique isolée. Vis-à-vis des critères définis, nous obtenons une machine électrique *référence* qui servira comme point de départ à l'optimisation système.

4. Obtention d'un modèle d'optimisation système à partir de la machine pré-dimensionnée. Le choix du modèle dégradé est crucial pour la suite de l'optimisation. Le modèle le plus naturel est le modèle de la machine dans le repère de Park étendu décrit dans la partie 6.3.1.6.

5. Optimisation système complète. Une optimisation système est effectuée à partir de la machine référence et pour des petites variations autour de ces paramètres pré-optimaux.

6. Étude de sensibilité de l'optimum. L'étude de sensibilité a pour but de tester la robustesse de l'optimum obtenue vis-à-vis des critères à optimiser, mais également de comparer cette machine à la machine obtenue par pré-dimensionnement.

Le point crucial dans l'obtention d'un modèle pour le dimensionnement système reste la définition de ce modèle par rapport à la machine de référence. Cette définition sous entend beaucoup de choix forts. Par exemple, pour une machine à grand nombre de phases, considère-t-on une alimentation électrique suivant un harmonique ou plusieurs harmoniques ? Pour une machine sectorisée et la machine à bobinages fractionnés, quelle est la stratégie de commande des différents secteurs ou bobinages fractionnés ?

Comme il a été mentionné dans la partie 6.3.1.6, pour l'obtention d'un modèle pour la machine à bobinages fractionnés, nous allons utiliser un découplage des équations du système en deux étapes. Premièrement, nous allons appliquer une transformation de Park "classique" sur chacun des trois sous-systèmes, blocs $abc/dq$ ou $dq/abc$ sur la figure 6.47.

Nous nous retrouvons alors avec trois sous-systèmes dans le repère de Park, figure 6.48. A ces trois sous-systèmes sont appliqués une seconde transformation pour garder les modes communs, $Σ$ et différentielles, $δ$. Ces blocs de transformations sont notés $dq/Σδ$ ou $Σδ/dq$.
<figure><img src="image_70.png" /><figcaption>FIGURE 6.47 – Schéma de principe de la commande de la machine à commutation du flux double excitation. Transformations dans le plan de commande.</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>FIGURE 6.48 – Principe du découplage.</figcaption></figure>

Le choix de ces deux transformations successives n'est pas anodin. En effet, l'objectif initial est d'arriver à des sous-systèmes découplés après transformations malgré le fort couplage magnétique des bobinages fractionnés. Ces deux transformations permettent de s'en rapprocher fortement, mais il reste de couplage non négligeable dans le mode Σ. C'est la raison pour laquelle il est nécessaire d'ajouter des termes de découplage dans ce mode, bloc decoupling.

Ce modèle de la machine permet de la considérer comme une série de paramètres : inductances et mutuelles, force électromotrice, etc. et ainsi de permettre le couplage entre le dimensionnement et les autres éléments de la chaîne de traction comme l'onduleur et la source électrique. Ainsi, les équations de la machine à pôles lisses dans le mode commun Σ s'écrivent :

$$ egin{cases} v_{d\Sigma} = R \cdot i_{d\Sigma} + L_{\Sigma\Sigma} \cdot \frac{di_{d\Sigma}}{dt} - \Omega_e \cdot L_{\Sigma\Sigma} \cdot i_{q\Sigma} \\ v_{q\Sigma} = R \cdot i_{q\Sigma} + L_{\Sigma\Sigma} \cdot \frac{di_{q\Sigma}}{dt} + \Omega_e \cdot L_{\Sigma\Sigma} \cdot i_{d\Sigma} + \frac{d\Phi_{\Sigma}}{dt} \end{cases} \quad (6.3) $$

Et dans le mode différentiel<fnref n="12" /> :

$$ egin{cases} (v_{d\delta}) = [R] \cdot (i_{d\delta}) + [L_{\delta\delta}] \cdot \frac{di_{d\delta}}{dt} + [L_{\delta\Sigma}] \cdot \frac{di_{d\Sigma}}{dt} - \Omega_e \cdot ([L_{\delta\delta}] \cdot (i_{q\delta}) + [L_{\delta\Sigma}] \cdot i_{q\Sigma}) \\ (v_{q\delta}) = [R] \cdot (i_{q\delta}) + [L_{\delta\delta}] \cdot \frac{di_{q\delta}}{dt} + [L_{\delta\Sigma}] \cdot \frac{di_{q\Sigma}}{dt} + \Omega_e \cdot ([L_{\delta\delta}] \cdot (i_{d\delta}) + [L_{\delta\Sigma}] \cdot i_{d\Sigma}) + \left(\frac{d\Phi_{\delta}}{dt}\right) \end{cases} \quad (6.4) $$

Avec $R$ la résistance des différents bobinages, $L_{ij}$ l'inductance ou la matrice inductance entre les modes $i$ et $j$, $\Phi_i$, le flux propre projeté dans le mode $i$ et $\Omega_e$ la vitesse électrique telle que $\Omega_e = p\Omega$ avec $\Omega$ la vitesse de rotation.

Avec ces équations, le couple s'exprime par :

$$ \Gamma = \langle p \cdot \frac{d\Phi_{\Sigma}}{d\theta_e} \cdot i_{q\Sigma} + p \cdot \left(\frac{d\Phi_{\delta}}{d\theta_e}\right) \cdot (i_{q\delta})^t - p \cdot ([L_{\delta\Sigma}] \cdot i_{q\Sigma} \cdot (i_{d\delta})^t - [L_{\delta\delta}] \cdot i_{d\Sigma} \cdot (i_{q\delta})^t) \rangle \quad (6.5) $$

<footnote n="12">12. (x) désigne un vecteur et $[A]$ une matrice</footnote>
Avec $p$ le nombre de paires de pôles et $\theta_e$ l'angle électrique. $< x >$ désigne la valeur moyenne de $x$. En considérant des mutuelles inductances entre le mode commun et différentiel négligeables devant les inductances propres et un flux propre porté par le mode différentiel faible devant celui du mode commun (cela ne se vérifierait pas dans le cas d'une dissymétrie de la répartition du flux des aimants entre les bobinages fractionnés), le couple s'écrit finalement :

$$ \Gamma \approx \langle p \cdot \frac{d\Phi_{\Sigma}}{d\theta_e} \cdot i_{q\Sigma} \rangle \qquad (6.6) $$

À partir des équations <fnref n="13" /> 6.3, 6.4 et 6.6, il est possible de remonter à la majorité des critères importants pour la machine : rendement, point de fonctionnement, etc. Le couplage entre le pré-dimensionnement et l'optimisation système se fait via ces équations. L'idée sous-jacente est que l'optimum du point de vue système est dans le voisinage des optimums trouvés lors du pré-dimensionnement. Ainsi, les différents paramètres sont obtenus par des petites variations autour des paramètres issus du pré-dimensionnement :

- Les matrices de transformation qui dépendent de la structure étudiée et notamment du nombre de bobinages fractionnés.

- La matrice des inductances et mutuelles dont sont issues les matrices après transformations $L_{ij}$.

- La résistance de chacun des bobinages fractionnés.

- Le flux propre de la machine qui permet d'obtenir les flux propres du mode commun et du mode différentiel.

<figure><img src="image_72.png" /></figure>

Les points durs à développer :

- Obtention du modèle pour l'optimisation système à partir de la machine de référence

- Extension du modèle pour des petites variations des paramètres dimensionnants.

- Introduction de la saturation, notamment sur le suivi des variations des valeurs d'inductances.

### Modèle coté convertisseur de puissance

Comme pour la machine électrique, côté convertisseur de puissance, il existe de nombreux niveaux de modèles pour le dimensionnement. Nous pouvons aller du dimensionnement de la puce de silicum (ou autres), aux fils de bonding, au dimensionnement de la structure en elle-même (multi niveaux, à résonance, etc.), jusqu'au modèle le plus simple possible qui permet d'obtenir le prix du convertisseur en fonction des calibres en courant et en tension des éléments.

Le niveau de granularité dans ce projet ne peut pas être au niveau de la puce silicium, mais bien au niveau de certains éléments de la structure. La structure référence de base reste l'onduleur triphasé avec un filtrage capacitif sur le bus continu, figure 6.49.

<figure><img src="image_73.png" /><figcaption>FIGURE 6.49 – Structure de base pour l'électronique de puissance. Le signe $x$ désigne un bobinage fractionné parmi tous.</figcaption></figure>

<footnote n="13">13. et des équations modélisant la charge et les pertes dans la machine.</footnote>
À partir de cette configuration de base, nous allons modéliser l'influence du nombre de bobinages fractionnés sur le calibre en tension des interrupteurs (puisque le calibre en courant reste constant pour garder les ampères-tours constants). Plus le nombre de bobinages fractionnés est important plus le calibre en tension peut-être réduit en respectant une vitesse de base pour la machine électrique constante. Ainsi, à la faveur de cette diminution de bus continu, nous pourrons modéliser l'influence d'un changement de technologie des interrupteurs de puissance; passage de transistor IGBT à des transistors MOSFET silicium puis à transistor à base de SIC ou de GaN [265, 266, 267, 268, 269]. De plus, l'augmentation du fractionnement diminue les inductances vues par l'onduleur obligeant une augmentation de la fréquence de découpage pour maintenir une ondulation de courant acceptable. L'augmentation de la fréquence de découpage nécessitera une analyse globale de la compatibilité électronMagnétique (CEM) [270, 271, 272], et pourra déboucher sur l'obligation de placer un filtre CEM pour la protection en $\frac{dV}{dt}$ des bobinages fractionnés.

Pour la commande de l'onduleur de puissance, il est envisagé l'utilisation d'une MLI vectorielle tout à fait classique dans ce genre de systèmes [273, 274], figure 6.50.

<figure><img src="image_74.png" /><figcaption>FIGURE 6.50 – Structure de base pour l'électronique de puissance.</figcaption></figure>

Cette structure de commande n'est pas influencée par les paramètres de dimensionnement ce qui permet de la garder fixe, quelle que soit la configuration étudiée.

Ainsi, nous allons nous intéresser :

- Aux différents calibres en tension et courant des éléments de l'onduleur et de filtrage.

- Nous évaluerons également la modification possible des technologies des composants.

- A l'influence sur la CEM de l'ensemble (modification de la fréquence de découpage, de la tension d'alimentation du bus continu, etc.)

<figure><img src="image_75.png" /></figure>

Les points durs à développer :

- Modèle de l'électronique de puissance à éléments discrets.

- Discontinuité du modèle pour les modifications de technologie des composants.

**Modèle coté source d'alimentation électrique** De même que précédemment, le modèle coté source peut aller du dimensionnement des membranes d'échanges dans une pile à combustible, aux différentes réactions dans une batterie lithium jusqu'à un modèle de source de tension parfaite ou simplement du coût de revient [275, 276, 277, 278]. Pour coller avec le niveau de modélisation du projet, nous considérons des sources de tensions parfaites avec des éléments parasites (R, L et C) en série et parallèle [279, 280, 281, 282]. Les paramètres de dimensionnement principaux seront la technologie choisie et les différents calibres en courant et tension de la source.

**Attendus de l'optimisation "système" par rapport à une optimisation en ilots** Pour poser le cadre, une optimisation en ilots correspond au cas où l'on optimise tous les éléments de manière indépendante, donc sans interactions entre les différentes parties. Contrairement à l'optimisation "système" dont toutes les parties seront dimensionnées conjointement. Le schéma de cette optimisation est donné sur la figure 6.51.
<figure><img src="image_76.png" /><figcaption>FIGURE 6.51 – Schéma de principe de l'optimisation "système". Les flèches rouges représentent les données qui s'échangent entre les modèles.</figcaption></figure>

L'optimisation en ilots correspond à l'étape de pré-dimensionnement de l'algorithme proposé. Cette optimisation peut être sommaire (maximisation du couple de la machine) jusqu'à très pointues (prise en compte avancée des pertes, des vibrations, etc.). Les différents éléments sont alors optimaux selon les critères d'optimisation choisis. Cependant, les interactions des différentes parties ne permettent pas de conclure quant à l'optimalité de l'ensemble. Nous pouvons citer comme exemple, le dimensionnement en puissance réactive de la machine électrique qui modifie les calibres du convertisseur de puissance, ou encore, le dimensionnement des éléments de filtrage du bus continu qui va influer sur la qualité de couple de la machine électrique.

Parmi les avantages attendus de l'optimisation, nous pouvons attendre :

- Amélioration des performances de la chaîne de traction.

- Appréhender les limites du fractionnement en prenant en compte l'environnement de la machine électrique.

**Difficultés et Verrous scientifiques** La principale difficulté se situe au niveau de l'écriture mathématique des échanges entre les modèles de la machine électrique, de l'électronique de puissance et de la source électrique. En effet, une variation du nombre de bobines fractionnées modifie la structure de l'électronique de puissance et des sources. Cela implique que l'algorithme d'optimisation sera capable de gérer des variables discrètes et en même temps des variables continues. De plus, le grand nombre de degrés de liberté va entrainer la formulation d'une fonction coût avec un ou plusieurs minimums locaux et un ou plusieurs minimums globaux. Plusieurs combinaisons peuvent amener à un résultat identique.

Pour traiter ce type de problème, il est courant d'utiliser des algorithmes non déterministes comme les algorithmes par essaim particulaire, génétique, recuit simulé, etc. [283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295]. Ces algorithmes que l'on décrit souvent comme algorithmes exploratoires permettent de couvrir efficacement la zone des paramètres même si nous ne pouvons pas justifier que le minimum global est atteint à la fin de l'optimisation.
<figure><img src="image_77.png" /></figure>

Les points durs à développer :

- Formaliser les échanges entre les différents blocs de l'optimisation.

- Caractère non unique du minimum global avec la présence de nombreux minimum locaux.

- Paramétrage de l'algorithme d'optimisation choisi.

- Cohabitation de variables continues et discrètes.

### 6.3.2.2 Dernière publication significative liée au projet

L'article suivant a été publié dans la revue IEEE transactions on Industrial Electronics en 2022 et porte sur les enjeux du fractionnement et la comparaison avec d'autres structures complémentaires.
# Control of a Segmented Three-Phase Synchronous Motor with Highly Coupled Sub-Windings

Antoine Cizeron, Javier Ojeda, Member, IEEE,
Eric Monmasson, Senior Member, IEEE, and Olivier Béthoux

**Abstract**—This article addresses the control issue of a structure enabling the segmentation of three-phase electric drives. The proposed segmentation process offers an additional degree of freedom in terms of voltage rating for power electronics but results in a three-phase system with highly coupled sub-windings. Thus, for such a segmented motor, the true challenge lies in controlling the machine global electromagnetic torque while balancing the power among the different inverters. To tackle this issue, the dq standard synchronous frame is extended to common and differential modes. This approach enables to design a suitable control structure based on an analytical model and physical considerations. The developed model is calibrated with realistic numerical values obtained from a finite element analysis. Finally, an experimental setup allows developing a proof of concept which shows the interest of the proposed control method compared to a standard multiple dq-frame method. The key finding is that the above-mentioned control objective of the segmented synchronous motor can be achieved without additional inductors and despite the severe constraint imposed by such highly coupled segmentation.

**Index Terms**—AC motor drives, segmented motor drives, highly coupled segmentation.

## I. INTRODUCTION

The current trend in transport electrification is to supply the electric motor drives with a DC bus voltage of several hundred Volt. It induces electrical risks to persons and requires complex insulation monitoring [1]. Lowering the electric drives voltages down to 48V standard drastically mitigates electrical hazards, which is of particular benefit to transportation applications. It also significantly reduces the insulation system stress [2] and enables using silicon MOSFETs or wide band-gap technologies, both having low switching and conduction losses [3], [4]. For a given three-phase motor and a given current rating, it is possible to reduce the voltage rating by using several independently supplied winding sets [5]. These structures, often referred to as segmented drives, are

also relevant approaches in ensuring continuity of service and enhancing drive energy efficiency since each phase is made of several sub-windings [6]. Improvements in connector technology [7], [8] as well as the attractive benefit of integrating power converters close to the motor end windings [3], [9], [10] highly mitigate the drawback of modular structures, namely the increased number of terminals. On the basis of these considerations, a recent patent [11] suggests the innovative Highly Coupled Segmentation (HCS) of motor drive structures. It opens up the possibility of dramatically lowering the drive voltage rating. However, it also induces high magnetic coupling between each winding sets [12] and may challenge the related torque and flux control structure. The key point of this article is to demonstrate that this issue can be addressed by an appropriate controller.

Among the numerous ways to obtain multiple three-phase winding sets, two categories can be distinguished [13] and are compared in Table I. The first category concerns the multiphase motor having $k \times 3$ phases (MK3P) which can be split in $k$ three-phase winding sets with different phase angles [14]–[17]. Most of the MK3P drives require an individual design of a specific multiphase motor and a multiphase control frame. The main expected benefit is a reduced torque ripple [10], [18]. However, the open-circuit fault of a winding set influences the torque harmonics and requires a dedicated fault-tolerant structure [17]. The second category corresponds to the segmented drives, which are obtained through the reconfiguration of the windings of an existing three-phase stator in several independant winding sets [3], [19]–[21]. The aim of the segmentation approach is to enable an effective drive resilience, an efficient drive power sharing and a reduced power supply voltage. In this context, the only requested constraint is to keep the torque ripples similar to the one produced by the initial three phase motor (without segmentation of its stator windings) driven by its conventional inverter. This category is declined into two segmentation processes in Table I.

Multi-Sector Segmentation (MSS) layout is the standard approach in segmenting the windings of a three-phase motor. Each elementary three-phase winding set is housed in a distinct stator sector [6], [21], [25]–[27]. As any other segmentation process, the MSS approach induces no change in the motor low harmonic torque ripple distribution [21] while enabling the parallelization of low voltage DC sources [3]. However, an additional radial force control is required in the MSS structure

Manuscript received October 26, 2021; revised February 8, 2022; accepted June 6th, 2022.

A. Cizeron and O. Béthoux are with the Laboratoire de Génie Electronique et Electronique de Paris, 91192, Gif-sur-Yvette, France. (e-mail: antoine.cizeron@geeps.centralesupelec.fr).

J. Ojeda and E. Monmasson are with the Systèmes et Applications des Technologies de l'Information et de l'Energie laboratory, 91192, Gif-sur-Yvette, France.
<table><caption>TABLE I: Comparison of electric machines based on multiple three-phase winding sets.</caption><thead><tr><th rowspan="2">Objectives</th><th rowspan="2">Subdivision types</th><th rowspan="2">MK3P</th><th colspan="2">Segmentation processes</th></tr><tr><th>MSS</th><th>HCS</th></tr></thead><tbody><tr><td>Power-sharing capability</td><td></td><td>✓ [14]</td><td>✓ [6]</td><td>✓ [11]</td></tr><tr><td>Degraded mode operation (open-circuit fault of a subdivision)</td><td></td><td>x<br>(Requires harmonic control [17])</td><td>x<br>(Requires radial force control [22])</td><td>✓</td></tr><tr><td>Torque harmonics reduction</td><td></td><td>✓ [23]</td><td>x</td><td>x</td></tr><tr><td>Maximum segmentation level</td><td></td><td>Number of three-phase winding sets</td><td>Number of poles per phase</td><td>Number of turns per phase [11]</td></tr><tr><td>Interleaved PWM implementation</td><td></td><td>✓ [10]</td><td>✓ [21]</td><td>x<br>(Requires external coupler [24])</td></tr><tr><td>Feed forward compensation</td><td></td><td>Matrix with MDQF [14]</td><td>Scalar with MDQF [21]</td><td>Matrix with MDQF,<br>Scalar with the proposed CDM method</td></tr></tbody></table>

in order to take advantage of its power sharing capability [6] or to implement a fault-tolerant scheme [22]. Moreover, the MSS approach is limited by the number of poles per phase $N_p$. Consequently, it cannot adapt all electrical machine to a low voltage standard specification.

A recent international patent suggests an alternative segmentation option enabling to extend the drive modularity level up to the theoretical limit of the winding turn [11]. As such, this innovative architecture, referred to as the HCS, allows a full degree of freedom in the sizing of current and voltage ratings [28], [29]. It also facilitates the integration of the power converters close to the motor windings, thus limiting electromagnetic compatibility issues and boosting inverters heat power transfer. Since the number of turns per phase $N_t$ is usually higher than the number of poles per phase $N_p$, the HCS is significantly more flexible than the MSS and enables using low voltage power supply for almost any electric drive. To enlighten their differences in terms of magnetic configuration and control strategy, the MSS and HCS layouts are both implemented on an existing three-phase Hybrid Excitation Flux Switching Machine (HEFSM). The choice of the hybrid motor topology is only due to laboratory availability. The HCS approach can easily be adapted to any kind of synchronous motor since it has already been used to achieve an on-board 3-phase charger in an electrical vehicle [13], [29] and to mitigate the Pulse Width Modulation (PWM) noise [24].

Concerning the control of a segmented motor, the Multiple *dq*-Frame (MDQF) method independently treats each elementary winding set. It can easily be used to control the MSS structure since its magnetic couplings are low between different sectors [21]. However, contrary to the MSS structure, the HCS structure presents large mutual inductances between each winding set. They are namely in the order of their self inductances [29], which is specific regarding the other segmented drives [19]–[21] and mars the effectiveness of the feed forward compensation, even in a healthy mode (Table I). As a matter of fact, the traditional MDQF control structure requires to combine the HCS structure with additional external coupled inductors [24]. Then, for reasons of cost, performance and space, the present article focuses on the feasibility of controlling the HCS structure without any additional coupled inductors and proposes an adapted Common and Differential

Mode (CDM) control scheme, based on physical considerations, and easy to design and tune.

The main contributions of the paper are:

1) an in-depth analysis of the magnetic specificity of the proposed HCS structure.

2) a comparison between MSS and HCS structures based on inductance measurements.

3) the analysis, design and tuning of a *dq*-based CDM controller applied to a segmented three-phase synchronous motor with HCS structure.

4) an experimental validation as a proof of concept of the proposed CDM controller and its comparison with the MDQF control, both implemented on the HCS structure.

5) an experimental demonstration of the promising capability of the CDM controller in terms of power-sharing between inverters, even in case of faulty operation.

The rest of this paper is organized as follows. Section II introduces the HCS configuration in comparison with the well-known MSS approach. Using Finite Element Analysis (FEA), it assesses the coupling value of the 3-phase motor sub-windings for both HCS and MSS approaches. It hence permits to highlight the need to set up a specific control law based on a dedicated transform applied in the rotating *dq*-frame. In section III, the *dq*-based suggested components are investigated in details and then the suggested control architecture is described. Section IV reports the experimental results obtained using a laboratory setup based on a 1 kW HCS drive. Finally, section V provides conclusions and perspectives.

## II. COMPARISON OF MS AND HC APPROACHES

### A. Three-phase motor segmentation

Based on an already existing conventional three-phase electric drive (Fig. 1), segmenting the initial windings in multiple electrically independent winding sets provides an attractive mean to lower the electric drive rated voltage. An elementary winding set is made of three sub-windings belonging to different phases and is powered by its dedicated inverter. The elementary winding set and its associated inverter constitute an elementary drive. The number of elementary drives is called the scaling factor, defined as $N_{sc}$. In this context, the electric motor is driven by several independent inverters as illustrated in Fig. 2 for the specific case of $N_{sc} = 3$. The segmented
<figure><img src="image_78.png" /><figcaption>Fig. 1: Conventional drive of a three phase motor.</figcaption></figure>

<figure><img src="image_79.png" /><figcaption>Fig. 2: The reduced voltage power supply of a segmented motor in the case of $N_{sc} = 3$ elementary drives.</figcaption></figure>

<figure><img src="image_80.png" /><figcaption>Fig. 3: Three poles pairs, three-phase motor with sub-windings corresponding to a) the MSS layout, b) the proposed HCS layout and c) their corresponding electrical angles.</figcaption></figure>

power supply has to produce in the air-gap the same global magnetomotive force (mmf) than the original power supply (Fig. 1). Thereby, based on Ampere law, each elementary drive conducts the initial motor rated current while the Faraday law shows that their rated voltage is $N_{sc}$ times smaller than in the initial drive.

In this paper, the control issue of the studied HCS approach is compared with the one of the standard MSS method. Fig. 3a and 3b show the spatial stator layouts of the MSS and the HCS structures, respectively. Obviously the innovative HCS layout is more flexible than the MSS segmentation one and could lead to an ultimate voltage reduction when each turn has its own inverter. As the electrical phase shift between sub-windings of a single phase is zero for both cases, the $N_{sc}$ electric angles of each structure is set to an identical control value. An ordinary PWM technique is implemented [30]. Moreover,

a common PWM carrier is used for the $N_{sc}$ elementary drives [12]. The key control difference between the two approaches lies in the coupling factor between sub-windings of the same phase, represented by double dotted lines in Fig. 2. It is low in the MSS structure [21] and high in the HCS structure [12].

## B. Machine modelling in the rotating dq-frame

To design the control of the different elementary winding sets, a low frequency model has to be defined. It is hence based on the phenomena involved at the fundamental frequency. Magnetic saturation, saliency and iron losses are not taken into account. The considered segmentation process of $N_{sc} = 3$ elementary drives (Fig. 3) results in a total of $3N_{sc} = 9$ sub-windings in both configurations. Each sub-winding has its own voltage equation based on Faraday's law. The related $3N_{sc}$ equations lead to the following dynamical model

$$V - V_N = [R] \cdot I + [L] \cdot \frac{dI}{dt} + E \quad (1)$$

where $[R]$ is a diagonal matrix containing the resistive terms of each sub-winding. As the number of turns, and consequently the conductor length, is the same for each elementary sub-winding, it is assumed that this term presents insignificant discrepancies between all the sub-windings. Thus, the resistance term is considered as a scalar $R$ in the different frame. $V$ are the voltages applied at the middle point of the inverter legs while $V_N$ are the voltages of the related neutral points. $I$ is the sub-winding current vector and $E$ is the back-emf term.

The structure of the inductance matrix $[L]$ could be simplified by using a proper ordering. Within the matrix expression, the sub-windings are ordered by their elementary drive number (Fig. 2) and then by their phase one. Hence, the vector $X$ in the natural frame includes the following $3N_{sc}$ components

$$X = (x_{a1} \ x_{a2} \ x_{a3} \ x_{b1} \ x_{b2} \ x_{b3} \ x_{c1} \ x_{c2} \ x_{c3})^T. \quad (2)$$

Based on this ordering method, the resulting inductance matrix $[L]$ is arranged as a conventional three-phase structure. Equation (3) shows the equivalent three-phase inductance matrix for the considered segmented winding layouts (Fig. 3).

$$[L] = \begin{pmatrix} [\mathcal{L}_a] & [\mathcal{M}_{ab}] & [\mathcal{M}_{ac}] \\ [\mathcal{M}_{ba}] & [\mathcal{L}_b] & [\mathcal{M}_{bc}] \\ [\mathcal{M}_{ca}] & [\mathcal{M}_{ba}] & [\mathcal{L}_c] \end{pmatrix} \quad (3)$$

where $[\mathcal{L}_a]$ is a $(N_{sc} \times N_{sc})$ matrix containing the self-inductances of each sub-winding of phase $a$ and the mutual inductances representing the magnetic coupling among them. It is assumed that all phases are symmetric, so the self-inductance matrices are equal to $[\mathcal{L}_s]$ and the mutual inductance matrices are equal to $[\mathcal{M}_s]$. In this way, the inductance matrix becomes

$$[L] = \begin{pmatrix} [\mathcal{L}_s] & [\mathcal{M}_s] & [\mathcal{M}_s] \\ [\mathcal{M}_s] & [\mathcal{L}_s] & [\mathcal{M}_s] \\ [\mathcal{M}_s] & [\mathcal{M}_s] & [\mathcal{L}_s] \end{pmatrix}. \quad (4)$$

The dq transformation is applied to each of the $N_{sc}$ electrically independent three-phase elementary drives. $X$ becomes the vector $X_{dq}$ in the standard dq-frame whose length is $2N_{sc}$
$$ \mathbf{X}_{dq} = (x_{d1} \ x_{d2} \ x_{d3} \ x_{q1} \ x_{q2} \ x_{q3})^T = \begin{pmatrix} \mathbf{X}_{d} \\ \mathbf{X}_{q} \end{pmatrix}. \quad (5) $$

Similarly, in the multiple dq-frame, the inductance matrix is simplified under non-saliency hypothesis as the following ($2N_{sc} \times 2N_{sc}$) matrix

$$ [\mathbf{L}_{dq}] = \begin{pmatrix} [\mathcal{L}_{dq}] & [0] \\ [0] & [\mathcal{L}_{dq}] \end{pmatrix}, \quad (6) $$

where $[\mathcal{L}_{dq}]$ is a symmetric ($N_{sc} \times N_{sc}$) matrix containing the self inductances $L_{dq,i}$ of the three-phase elementary drive indexed i and the mutual inductances $M_{dq,ij}$ between the elementary drives i and j

$$ [\mathcal{L}_{dq}] = [\mathcal{L}_{d}] = [\mathcal{L}_{q}] = \begin{pmatrix} L_{dq,1} & M_{dq,12} & M_{dq,13} \\ M_{dq,21} & L_{dq,2} & M_{dq,23} \\ M_{dq,31} & M_{dq,32} & L_{dq,3} \end{pmatrix}. \quad (7) $$

These terms have to be assessed in the MSS and HCS layouts in order to show the limit of the MDQF control method applied to the proposed HCS structure.

### C. Control comparison of the MSS and HCS structures

In order to compare the two segmentation processes described in Fig. 3, an already existing three-phase HEFSM has been rewound with $N_{sc} = 3$ coils per phase per sector. Each coil has dedicated terminals enabling the implementation of either the MSS or HCS configurations (Fig. 3). Both structures are powered similarly by $N_{sc}$ independent inverters (Fig. 2).

As the hybrid excitation of the motor is not exploited in this article, the excitation current remains constant at its rated value for all experiments. Moreover, the saliency of this motor is under 5 %. Hence, it is considered as a simple non-salient three-phase permanent magnet synchronous motor.

Inductance measurements are performed on each sub-winding of each phase for both configurations. As an example, $[\mathcal{L}_a]$ and $[\mathcal{M}_{ab}]$ are shown in Fig. 4 and 5 for the MSS and HCS segmentation approaches, respectively. The inductance values corresponding to different sectors present discrepancies even though their conductors are arranged in a similar way (Fig. 4a). This is caused by disparities in the magnetic circuit. The rotor and the stator may have a small concentricity defect, technically inducing small discrepancies in the air-gap thickness from one sector to another. As far as the HCS windings are concerned, such discrepancies have no impact on their inductance values since they have a symmetrical configuration around the motor.

The coupling factor between i and j sub-windings is defined as

$$ k_{ij} = M_{ij} / \sqrt{L_i L_j}. \quad (8) $$

Obviously, the MSS structure has low negative mutual inductance values between the sub-windings belonging to different sectors. As shown in Fig. 4a, the resulting coupling factor between two coils of the same phase is lower than 10 %. Conversely, the HCS structure shows high positive mutual inductances between sub-windings of the same phase, namely with a coupling factor higher than 90 % (Fig. 5a). Note that, within one elementary drive, the coupling factor of two sub-windings (e.g. $a_1$ and $b_1$) is close to 25 % for both the MSS and the HCS structures (Fig. 4b and 5b).

<figure><img src="image_81.png" /><figcaption>Fig. 4: Inductance values of the prototype with the MSS layout (a) $[\mathcal{L}_s]$ and (b) $[\mathcal{M}_s]$ inductance sub-matrices ($\mu H$).</figcaption></figure>

<figure><img src="image_82.png" /><figcaption>Fig. 5: Inductance values of the prototype with the HCS layout (a) $[\mathcal{L}_s]$ and (b) $[\mathcal{M}_s]$ inductance sub-matrices ($\mu H$).</figcaption></figure>

<figure><img src="image_83.png" /><figcaption>Fig. 6: Comparison of $[\mathcal{L}_{dq}]$ in (a) the MSS and (b) the HCS layouts ($\mu H$). Results are equivalent following d and q axes.</figcaption></figure>

Based on the previous inductance measurements, the resulting $[\mathcal{L}_{dq}]$ matrix is computed and depicted in Fig. 6a and 6b for the MSS and HCS structures, respectively. Regarding the MSS approach, the mutual inductances values in the multiple dq-frame are negligible compared with the self inductances ones (Fig. 6a). Conversely, for the HCS approach, the mutual inductances values of the $[\mathcal{L}_{dq}]$ matrix are in the same order of magnitude than the self inductances ones (Fig. 6b). Obviously, these properties impact the control structure.

Fig. 7 illustrates the general control scheme of the segmented HEFSM and clarifies the chosen notations. As the current is accurately measured to be efficiently tracked, equation (9) shows that the control law should integrate feedforward terms to improve the controller dynamic performance:

$$ \mathbf{V}_{dq} = R \mathbf{I}_{dq} + [\mathbf{L}_{dq}] \frac{d}{dt} \begin{pmatrix} \mathbf{I}_{d} \\ \mathbf{I}_{q} \end{pmatrix} + \mathbf{E}_{dq} + \Omega_{e} [\mathbf{L}_{dq}] \begin{pmatrix} -\mathbf{I}_{q} \\ \mathbf{I}_{d} \end{pmatrix}. \quad (9) $$

The last term is particularly useful since it permits to fully decouple the d and q axes. However, this compensation term is only effective if the non-diagonal terms (i.e. the mutual inductances) are precisely known. As a matter of fact, the poor MSS layout coupling makes possible to choose the standard MDQF control architecture (Fig. 8) where the
<figure><img src="image_84.png" /><figcaption>Fig. 7: General control structure of the HEFSM with $N_{sc} = 3$ elementary drives.</figcaption></figure>

<figure><img src="image_85.png" /><figcaption>Fig. 8: Current controller implementing the MDQF method with the same PI regulator on each component.</figcaption></figure>

<figure><img src="image_86.png" /><figcaption>Fig. 9: Current controller implementing the proposed CDM method with specific PI regulator for $\Sigma$ and $\delta$ components.</figcaption></figure>

Decoupling block refers to this feedforward term. However, the HCS layout strong positive coupling makes the MDQF control law ineffective regarding the decoupling purpose. Consequently, the suggested approach consists in finding a new frame enabling a good decoupling of the related components. Besides, the adopted transform should not rely on the machine parameters knowledge. Based on these key ideas, the Common and Differential Modes (CDM) control law is proposed and its scheme is given in Fig. 9. The following section gives an in-depth description of the CDM control architecture guidelines and assesses its relevancy based on additional FEA.

### III. DQ-BASED CDM CONTROL ARCHITECTURE

#### A. Additional dq-based transform

As for any 3-phase motor, the standard synchronous dq-frame remains the right option to impose the desired torque and magnetic flux references. However, the HCS structure is

characterized by high magnetic coupling between the elementary winding sets. Consequently determining relevant dq-based variables is a key issue to comprehensively control the HCS structure. In the perfect scenario, the latter would

1. have a direct link with the torque/flux as well as the internal power transmission variables.

2. ensure a proper decoupling permitting to switch a $2N_{sc}$ MIMO system into $2N_{sc}$ SISO systems.

The key idea is to exhibit the main physical dynamics of the segmented machine, namely the rapid modes linked to the leakage inductances and the slow mode related to the self and mutual inductances. Additionally, to maintain a proper decoupling between torque and flux variables, the suggested additional transform, named $[T_{\Sigma\delta}]$ treats independently the $d$ and $q$ axes. It can then be written as

$$[T_{\Sigma\delta}] = \begin{pmatrix} [T_d] & [0] \\ [0] & [T_q] \end{pmatrix} = \begin{pmatrix} [T] & [0] \\ [0] & [T] \end{pmatrix} \quad (10)$$

where $[T]$ is the $(N_{sc} \times N_{sc})$ matrix applied equally to the $d$ and $q$ axes and defined as

$$[T] = \begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{pmatrix}. \quad (11)$$

Transformation (10) is applied to $X_{dq}$ (5), thus leading to $X_{\Sigma\delta}$, the $2N_{sc}$-component vector in the suggested $\Sigma\delta$-frame. It is structured as follows:

$$X_{\Sigma\delta} = [T_{\Sigma\delta}] \cdot X_{dq} = (x_{d\Sigma} \quad X_{d\delta} \quad x_{q\Sigma} \quad X_{q\delta})^T. \quad (12)$$

The first component $x_{d\Sigma}$ is named common mode variable of the $d$-axis. The remaining two components, $x_{d\delta12}$ and $x_{d\delta23}$, included in $X_{d\delta}$, are the differential mode variables of the $d$-axis and reflect the difference between two elementary drives. Similar reasoning applies on the $q$-axis.

A FEA assessment justifies the relevancy of this choice. Indeed, the basis for this approval is typically that the HCS sub-windings are located around the same stator tooth and hence embrace almost the same flux lines. The FEA study is carried out using Ansys Maxwell 2D software and it is applied on the HCS winding configuration (Fig. 3b). First, the test sets a non-zero common mode current for the $N_{sc} = 3$ sub-windings of phase $a$, setting the 2 differential mode currents to zero. Fig. 10a shows the resulting magnetic field lines. They cross radially the rotor-stator air-gap and hence impact the interaction of magnetic flux between both mechanical
parts and leads to an electromagnetic torque. Moreover, as by design, the air-gap thickness is very small, the common mode inductance value is large. It results that the common mode dynamic is low. Second, the two other numerical experiments enforce a non-zero differential mode current. Fig. 10b and 10c exhibits the corresponding magnetic field lines which obviously remain located in the stator. They solely cross the stator slots, thicker than the air-gap. As a result, the differential mode inductance value is much lower than the common mode one. It proves that the dynamics of the common and differential modes are highly different and should be accounted for separately. Besides, based on the FEA, it is also clear that the differential currents have no impact on either the air-gap magnetic flux or the machine torque. They mainly modify the power sharing between elementary drives.

## B. Impact on the inductance matrix

The common and differential modes transformation [$\mathcal{T}$] is applied to the inductance matrix obtained in (7). It derives:

$$[\mathcal{L}_{\Sigma\delta}] = [\mathcal{T}] \cdot [\mathcal{L}_{dq}] \cdot [\mathcal{T}]^{-1} = \begin{pmatrix} L_{\Sigma} & \mathbf{L}_{\Sigma\delta} \\ \mathbf{L}_{\delta\Sigma} & [\mathcal{L}_{\delta}] \end{pmatrix}. \quad (13)$$

The inductance matrix in the dq-based $\Sigma\delta$ frame is computed either using the FEA or the previous inductance measurements performed on the HCS prototype (Fig. 11). Although the 2D assumption slightly overestimates the various coupling impacts, both results are clearly consistent. They confirm the previous FEA assessment based on physical considerations. They lead to the conclusion that:

1) The differential modes has no impact on the air-gap magnetic flux and the electromagnetic torque ($d\Sigma$ and $q\Sigma$ modes, respectively). Accordingly the [$\mathcal{L}_{\Sigma\delta}$] inductance matrix could be simplified as:

$$[\mathcal{L}_{\Sigma\delta}] = \begin{pmatrix} L_{\Sigma} & [0] \\ \mathbf{L}_{\delta\Sigma} & [\mathcal{L}_{\delta}] \end{pmatrix}. \quad (14)$$

2) The common mode dynamic is imposed by the $L_{\Sigma}$ value which is very large in comparison with the other terms of the related matrix.

3) Each differential mode has small inductances compared to $L_{\Sigma}$ (hundred times smaller), leading to a relative high dynamic with regard to the common mode one.

4) The common mode slightly impacts the differential ones. Moreover, since the former dynamics is very slow in comparison to the latter one, the differential mode controller easily reject this perturbation.

## C. HCS dedicated controller

Combining (9), (12) and (14), the common mode components are governed by

$$\left\{ \begin{aligned} v_{d\Sigma} &= R \cdot i_{d\Sigma} + L_{\Sigma} \cdot \frac{di_{d\Sigma}}{dt} + e_{d\Sigma} - \Omega_e \cdot L_{\Sigma} \cdot i_{q\Sigma} \\ v_{q\Sigma} &= R \cdot i_{q\Sigma} + L_{\Sigma} \cdot \frac{di_{q\Sigma}}{dt} + e_{q\Sigma} + \Omega_e \cdot L_{\Sigma} \cdot i_{d\Sigma} \end{aligned} \right. \quad (15)$$

where $e_{d\Sigma}$ and $e_{q\Sigma}$ represent the back emf due to permanent magnets and excitation coils along the $\Sigma$-axis.

To achieve an effective transient decoupling of $d\Sigma$ and $q\Sigma$ components, it is crucial to combine the feedback action with a specific feedforward term. Fig. 9 shows this additional term

<figure><img src="image_87.png" /><figcaption>Fig. 10: Flux lines without excitation around three sub-windings of phase a in the HCS structure (Fig. 3b) for the common and differential components defined by (11).</figcaption></figure>

<figure><img src="image_88.png" /><figcaption>Fig. 11: Comparison of [$\mathcal{L}_{\Sigma\delta}$] ($\mu H$) obtained through a) FEA and b) the inductance measurements of the HCS approach shown in Fig. 6b. Results are equivalent following $d$ and $q$ axes.</figcaption></figure>

<figure><img src="image_89.png" /><figcaption>Fig. 12: Comparison of back electromotive force measurements and FEA results. ($\dot{E} = 12.5$ V, $f_e = 200$ Hz).</figcaption></figure>

<table><caption>TABLE II: HCS back-emf $\Sigma\delta$ components ($f_e=200Hz$)</caption><thead><tr><th></th><th>Experimental</th><th>FEA</th></tr></thead><tbody><tr><td>$e_{q\Sigma}$</td><td>37.5 V</td><td>37.5 V</td></tr><tr><td>$e_{q\delta12}$</td><td>35 mV</td><td>23 mV</td></tr><tr><td>$e_{q\delta23}$</td><td>15 mV</td><td>10 mV</td></tr></tbody></table>

as a Decoupling block permitting to compensate the last terms of (15). Apart from measuring the rotor speed and the sub-winding currents, the compensation term only requires the knowledge of $L_{\Sigma}$ value. Moreover, the two differential modes are independently controlled (Fig. 9). Indeed, the measured inductance matrix is practically diagonal (Fig. 11b) while the common component current evolves slowly and is hence easily rejected by the feedback term.

Furthermore, the 2D FEA model also enables to compute the back-emf which is then compared to the no-load voltages measured on the prototype in its HCS configuration (Fig. 3b). These estimated variables are actually consistent with the real voltages measured in the HCS structure (Fig. 12). Moreover, Table II shows that the two differential mode components of $E_{\delta}$ are thousand times lower than the common mode one.
<table><caption>TABLE III: PI Controllers tuning
(10 dB gain margin and 60 ° phase margin criteria)</caption><thead><tr><th></th><th>r (Ω)</th><th>l (μH)</th><th>τ = l/r (ms)</th><th>K<sub>p</sub> (10<sup>-3</sup>)</th><th>T<sub>i</sub> (ms)</th></tr></thead><tbody><tr><td>MDQF,Σ</td><td>0.2</td><td>760</td><td>3.8</td><td rowspan="2">2.1</td><td rowspan="2">1.24</td></tr><tr><td>MDQF,δ</td><td>0.2</td><td>10</td><td>0.05</td></tr><tr><td>CDM,Σ</td><td>0.2</td><td>760</td><td>3.8</td><td>100</td><td>4.0</td></tr><tr><td>CDM,δ</td><td>0.2</td><td>10</td><td>0.05</td><td>1.4</td><td>0.05</td></tr></tbody></table>

<figure><img src="image_90.png" /><figcaption>Fig. 14: View of the triple three-phase inverter (left) and the HEFSM motor with HCS layout (right).</figcaption></figure>

<figure><img src="image_91.png" /><figcaption>Fig. 13: Open-loop transfer function Nichols plot for the MDQF and CDM tuning parameters described in Table III.</figcaption></figure>

Hence, feeding each elementary drive with an equal current set point yields to a similar electromagnetic torque contribution.

For comparative purposes, the PI parameters of the standard MDQF technique (Fig. 8) and the proposed CDM method (Fig. 9) have been tuned based on the same specifications. The PI regulators settings should provide 10 dB-60 ° gain-phase margins while maximizing the open-loop gain crossover frequency $f_x$. Assuming that for both cases, the plant is sufficiently well decoupled by the aforementioned feedforward compensation terms (9), the relation between the duty cycle $\alpha$ and the related current $i$ is given by a first order system ($r, l$) associated with a pure delay. The latter includes both the computational delay of the microcontroller equal to one sampling period and the stochastic delay induced by the digital PWM. Denoting $T_s = 50\mu s$ the control sampling time, the global delay to be considered is therefore equal to $1.5 \times T_s$. Consequently, the SISO system transfer function to be regulated is as follows:

$$H(s) = \frac{I(s)}{\alpha(s)} = \frac{V_{DC} \cdot e^{-1.5 \cdot T_s \cdot s}}{r + l \cdot s} \quad (16)$$

with $V_{DC} = 48V$ the DC-link voltage of the segmented power supply, $r$ the winding resistance and $l$ the considered inductance. This latter could be either $L_Σ$ or $L_δ$. In the proposed CDM controller, each of the common ($Σ$) and differential ($δ$) components benefit from a specific PI parameter setting whereas in the MDQF controller, the PI parameter setting acts on both components without distinction. In all cases, the open-loop transfer function in $Σ$ or $δ$ component can be written as:

$$H_{OL}(s) = K_p \cdot \left( 1 + \frac{1}{T_i \cdot s} \right) \cdot H(s) \quad (17)$$

where $K_p$ and $T_i$ are the proportional gain and the integral time constant of the PI. Based on the [$C_{Σδ}$] values listed in Fig. 11b, the CDM parameter sets of the $Σ$ and $δ$ are adjusted as shown in Table III. It results in the same $Σ$ and $δ$ open-loop frequency response as depicted in Fig. 13 (CDM, $Σ$ and CDM, $δ$). The resulting gain crossover frequency is $f_{x,Σδ} = 1$ kHz.

However, the MDQF method imposes, because of the intrinsic structure of such controller (Fig. 8), that the PI parameter setting is unique and therefore, affects both the common ($Σ$) and differential ($δ$) components. It is then necessary to choose the best trade-off which maximizes the bandwidth of the $Σ$ component with a phase margin of at least 60 ° and at the same time do not degrade too much the stability on the $δ$ axis (phase margin> 60 ° and gain margin> 10 dB). Thus, this unique parameter set induces a lower bandwidth for the $Σ$ component ($f_{x,Σ} = 50$ Hz) compared to what was obtained with the CDM method, resulting in a larger settling time. The $δ$ component cross-frequency is also low in order to ensure the 10 dB gain margin ($f_{x,δ} = 75$ Hz). Fig. 13 (MDQF, $Σ$ and MDQF, $δ$) shows the related open-loop frequency responses and the adopted PI parameters are given in Table III.

## IV. EXPERIMENTAL RESULTS

To validate the proposed controller, an experimental setup has been implemented (Fig. 14). The synchronous motor described in section II is configured in the HCS structure and driven by $N_{sc} = 3$ basic 3-phase inverters. The latter, located on the same Printed Circuit Board (PCB) are powered by a common 48 V DC source and their switching frequency is set at $F_s = 20$ kHz. The aforementioned control laws are implemented in a dSpace MicroLabBox and the sampling frequency is synchronized with the switching one. In addition, a DC motor imposes a controlled load torque.

The purpose of the following experiments is to highlight the relevancy of the proposed CDM control compared to the standard MDQF method initially developed to drive MSS motor structures. It also aims at demonstrating the capability to drive the HCS motor structure at any speed and to enforce any power distribution reference even in case of faulty operation.

### A. Control Methods Comparison for the HCS Structure

The first test consists in enforcing torque setpoint steps from its rated value to 0, and vice-versa between 0 ms and 40 ms. Meanwhile, the rotor speed is maintained constant by controlling the load torque. Both differential references are set to zero: $I_{dδ}^* = 0$ and $I_{qδ}^* = 0$. Then, a step response
<figure><img src="image_92.png" /><figcaption>Fig. 15: Simulation results at $N = 600 \text{ rpm}$ with a) the MDQF control and b) the proposed CDM control.</figcaption></figure>

<figure><img src="image_93.png" /><figcaption>Fig. 16: Experimental results at $N = 600 \text{ rpm}$ with a) the MDQF control and b) the proposed CDM control.</figcaption></figure>

occurs on the $q\delta_{12}$ component between 80 ms and 120 ms in order to validate the differential mode controller design. The test has been implemented in simulation (Fig. 15) using MATLAB/Simulink software. Worth to be mentioned here, the control feedforward terms used in this simulation are ideal since they perfectly match their actual counterparts (Fig. 6b and (9)).

First, the simulation results fully validate the predictions based on the open-loop frequency response. The standard MDQF method shows poor performances (Fig. 15a). The $q$-axis common mode current component ($i_{q\Sigma}$) exhibits a significant overshoot and a long settling time ($\approx 20$ ms) which is consistent with its low computed open-loop cross frequency ($\approx 50$ Hz). This overshoot may lead to exceed the acceptable current limit in one of the elementary drives. Similarly, the $\delta$-axis settling time also proves to be large, which is in line with its reduced bandwidth (Fig. 13). Conversely the suggested CDM control structure exhibits good and similar performances regarding either $i_{q\Sigma}$ or $i_{q\delta_{12}}$ (Fig. 15b). It also highly mitigates the overshoot on the $q\Sigma$-axis. Each component features the same 1 ms settling time, which is fully consistent with the computed 1 kHz open-loop cross frequency (Fig. 13).

Second, the same test is then experimentally done. An anti-windup method is implemented on the duty-cycle values and, as it is on the actual system, only approximated parameters are used in the feedforward action (Fig. 16). As can be seen, the experimental waveforms are in close agreement with the simulation results (Fig. 15). In all measurements, the residual ripples on $i_{q\Sigma}$ are due to the non-sinusoidal waveforms of back-emf (Fig. 12). Apart from that, a few other differences and similarities are worth to be mentioned.

<figure><img src="image_94.png" /><figcaption>Fig. 17: Measured a) $\Sigma\delta$ and b) sub-winding currents during $q\Sigma$ and $q\delta_{12}$ steps under abc3 elementary drive deactivation with the standard MDQF control.</figcaption></figure>

Regarding the standard MDQF technique, the measured $d$-axis common mode current component ($i_{d\Sigma}$) is deeply impacted by the $i_{q\Sigma}$ transient (Fig. 16a), which negatively induces a magnetic field transient in the air-gap. Indeed, in practice, the feed forward compensation terms (9) imperfectly compensates the coupling terms, since these terms are based on measurements of mutual inductances that are of course affected by uncertainties. Conversely, CDM approach presents very small discrepancies between simulation and experimental results. Even though the $q$-axis dynamics is slowed by the brief duty-cycle value saturation during transient, it is more than ten times faster than the results obtained with the MDQF method. Moreover, using the CDM method, the $d$-axis common mode component is not impacted by the $q$-axis $\Sigma$ component transient since it remains equal to zero (Fig. 16b). This proves that the $dq$ axes decoupling remains efficient even though affected by measurement uncertainties. Indeed, the additional $\Sigma\delta$ transform used in the CDM control method permits to base the feedforward compensation term on a single measurement, namely $L_\Sigma$. This latter measurement is truly quite easy to perform correctly.

Finally, the proposed CDM method is well adapted to drive a HCS structure. Since this method also improves the differential component control, it opens up the possibility to consider for future works the power-sharing between the different elementary drives.

## B. Faulty Operation

A second test is performed to illustrate the promising abilities of the HCS structure controlled with both the standard MDQF method (Fig. 17) and the proposed CDM method (Fig. 18). For both control methods, one of the three inverters is suddenly turned off at 0 ms while keeping $i_{q\Sigma}^*$ constant to 18 A. This configuration corresponds to an open-winding fault operation. Then, the common mode component setpoint $i_{q\Sigma}^*$ is set to 0 A between 20 ms and 60 ms and the remaining
<figure><img src="image_95.png" /><figcaption>Fig. 18: Measured a) $\Sigma\delta$ and b) sub-winding currents during $q\Sigma$ and $q\delta_{12}$ steps under abc3 elementary drive deactivation with the proposed CDM control.</figcaption></figure>

differential component setpoint $i_{q\delta_{12}}^*$ is set to -6 A between 100 ms and 140 ms. These two step responses evaluate the fault ride through capacity of the structure.

Using the MDQF method, the common component ($\Sigma$) is deeply affected by the open-winding fault and the differential component step response interferes with the common component behavior (Fig. 17). This is mainly due to the poor dynamical performances of the MDQF control, described in Fig. 13. Conversely, using the CDM method, the torque control remains effective even when only two out of three elementary drives are operating (Fig. 18a). Additionally, the power-sharing between the two remaining elementary drives is still operational since the differential component control can still be realized in such degraded mode (Fig. 18b).

## V. CONCLUSIONS AND PERSPECTIVES

The novel stator winding segmentation process studied in this article enables to supply an electric machine with a reduced DC-link voltage without changing its operational range or increasing the rated current of the feeding inverters. With its multiple degrees of freedom and high magnetic couplings, this attractive topology offers important control challenges which the present study addresses. The scope of the current contribution is limited to designing a proof of concept based on an existing three-phase machine configured as a segmented three-phase motor facing high positive magnetic coupling between each elementary drives. The developed approach consists in extending the standard $dq$ transform with a common and differential modes transform. A comprehensive dynamical model is developed using analytical equations and FEA method. It shows that the extended transformation is suitable to easily track the control objectives, i.e. the global machine operating point (the torque and air-gap magnetic field) while ensuring a balanced power distribution between all the elementary drives. An experimental setup has been developed. The experiments show that the aforementioned model

is reliable, enabling to tune properly the control parameters. The stability enhancement provided by the proposed CDM control method compared to the standard MDQF technique has been shown. Moreover, the CDM control architecture can enforce balanced power distribution between elementary drives whatever the electric machine operating point and even under open-fault condition. As a consequence, the elementary drive power allocation ability seems to be an interesting opportunity for future research lines.

## APPENDIX

* HCS: *Highly Coupled Segmentation*

* MSS: *Multi-Sector Segmentation*

* MDQF: *Multiple dq-Frame*

* CDM: *Common and Differential Modes*

* MK3P: *Multiphase with $k \times 3$ Phases*

* FEA: *Finite Element Analysis*

* HEFSM: *Hybrid Excitation Flux Switching Machine*

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<figure><img src="image_96.png" /></figure>

**Antoine Cizeron** received the MSc degree in electrical engineering from Université Paris-Saclay in 2019. Since then, he has been working toward the Ph. D. degree in the Group of electrical engineering - Paris (GeePs, UMR CNRS 8507) and the Systèmes et Applications des Technologies de l'Information et de l'Energie laboratory (SATIE, UMR CNRS 8029). His current research interests include modeling and control of motor drives.

<figure><img src="image_97.png" /></figure>

**Javier Ojeda** (M'06) was born in Buenos Aires, Argentina. He received the PhD. degree in electrical engineering from Ecole Normale Supérieure de Cachan, France, in 2009. Since 2010, he is an assistant professor in the Ecole Normale Supérieure Paris-Saclay, SATIE Laboratory and EEA teaching Department. His current research interests include modeling and control of multiphase motors.

<figure><img src="image_98.png" /></figure>

**Eric Monmasson** (M'96-SM'06) received the Ing. and Ph.D. degrees from the Ecole Nationale Supérieure d'Ingénieurs d'Electrotechnique d'Electronique d'Informatique et d'Hydraulique de Toulouse, Toulouse, France, in 1989 and 1993, respectively. Since 2003, he is a Full Professor at the University of Cergy-Pontoise, Cergy-Pontoise, France. He is also with the Systèmes et Applications des Technologies de l'Information et de l'Energie laboratory (SATIE, UMR CNRS 8029), Cergy-Pontoise.

He is the author or coauthor of 3 books and more than 200 scientific papers. His current research interests include the control of power electronics, electrical motors and generators, and FPGA-based and SoC-based industrial control systems. Dr. Monmasson was the Chair of the technical committee on Electronic Systems-on-Chip of the IEEE Industrial Electronics Society from 2008 to 2011. He is also a member of the steering committee of the European Power Electronics Association and he was the Chair of the number one technical committee of the International Association for Mathematics and Computers in Simulation (2011-2017). He is an Associate Editor of the IEEE Transactions on Industrial Informatics and the Editor-in-Chief of the IEEE Industrial Electronics Magazine.

<figure><img src="image_99.png" /></figure>

**Olivier Béthoux** received the M.Sc. degree from Ecole Centrale de Lille, Villeneuve-d'Ascq, France, in 1990, and the Ph.D. degree from The University of Cergy-Pontoise, Pontoise, France, in 2005. From 1992 to 1997, he was a Research Engineer with the Alstom Research Center, France. From 1999 to 2017 he has been an Assistant Professor first in the Ecole Nationale d'Electronique et de ses Applications (ENSEA) and then in the Paris-Saclay University, France. Since 2017, he is a Full Professor at the Sor-

bonne Université, Paris, France. Since 2006, He has also been with the Group of electrical engineering - Paris (GeePs, UMR CNRS 8507), Paris. He is the author or coauthor of 2 books and more than 150 scientific papers. His main research interests include the analysis, design, control, and fault diagnosis of power electronics and electric machines. He is also deeply involved in advanced control techniques and energy management of fuel cells and photovoltaic systems.
# Chapitre 7

## Conclusion et perspectives

L'écriture d'une habilitation à diriger des recherches est un moment d'introspection sur ce qu'ont été les premières années de recherche sur une thématique et un moment de projection sur ce que seront les prochaines années en termes de thématiques, d'organisation de la recherche, de perspectives d'industrialisation, etc. Qui plus est l'écriture est périlleuse entre d'un côté l'envie d'être pédagogique et donc de décrire finement son travail et de l'autre, être capable de synthèse et de recul sur son travail. C'est sur cette crête périlleuse que doivent s'engager celles et ceux qui souhaitent passer cette étape importante de la vie d'un chercheur.

Dans ce mémoire, j'ai décrit trois thématiques de recherches qui me semblent pertinentes par rapport aux contextes scientifiques actuels. Les trois thématiques rentrent dans le cadre de l'amélioration des contraintes d'usage des machines électriques en s'appuyant sur :

- L'amélioration de la résilience aux défauts des machines électriques par l'augmentation du nombre de phases actives. Les machines à grand nombre de phases tendent à sortir du cadre strict des applications nécessitant une grande résilience aux défauts pour se démocratiser à toutes les applications. Le nombre de phases devient un critère d'optimisation de la structure tout comme le nombre de paires de pôles pour offrir à ces machines de nouveaux degrés de liberté d'alimentation et de commande tout à garantissant une résilience accrue.

- L'amélioration de la détection des défauts par le diagnostic vibratoire couplé à l'optimisation de la machine électrique. La machine électrique doit devenir une structure simple à diagnostiquer pour être capable de détecter et de localiser des défauts, d'anticiper des défauts graves, voire de s'autoréparer. Tout cela ne peut être possible qu'en couplant la méthode de détection des défauts à la structure même de la machine et à son dimensionnement.

- A l'intégration d'une chaîne de traction batterie - électronique de puissance - machine par le développement d'une structure fractionnée. Les chercheurs sont en quête d'une réponse à l'optimisation globale d'une chaîne de traction ; à savoir la tension d'alimentation globale, l'adéquation entre la batterie et la machine électrique et l'intégration physique de la machine avec son alimentation. C'est à tous ces défis que tentent de répondre les structures fractionnées tout en ajoutant de nouveaux degrés de liberté sur le dimensionnement et la commande.

L'écriture de ce mémoire commença en 2019 pour s'achever en 2022. A l'évocation de cette fourchette d'années pour l'écriture d'un mémoire, nous pourrions penser que celle-ci fut un chemin de croix notamment en considérant la crise sanitaire mondiale. Néanmoins, le temps d'écriture a plutôt été un moment agréable entre une introspection scientifique mais également personnelle. Ces temps furent pour moi des temps de réflexion sur ce qui me motive dans la recherche, sur ce qui je souhaite développer et sur les chemins que je souhaite emprunter. Loin de me dissuader d'arrêter la recherche notamment en ces temps difficiles, l'écriture de l'habilitation à diriger des recherches a été pour moi un accélérateur d'envie pour les années à venir.
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