M Eric Semail

M Ngac

Ky Nguyen

] Jury

M Mohamed Benbouzid

M Franck Betin

M Johan Gyselinck

Maître Mme Carole Henaux

Anh Kỳ

Chị Xiaoxiao

Thùy Noé

Hai Anh

Anh Long

Chị Long

Anh Thủy

Bảo Châu

Huy

Tiago, Hussein, Youssouf, Jinlin, Lamine Chi

Neural Network BEV Battery Electric Vehicle CBPWM Carrier Based Pulse Width Modulation CCS-MPC Method (I)-2 Method (I)-3 Method (I)-2 Method (I)-3 Method (I)-2 Method (I)-3

Keywords: Multiphase machine, seven-phase machine, non-sinusoidal back-EMF, faulttolerant control, optimal control under constraint, artificial intelligence, adaptive linear neuron, ADALINE. Nomenclature ADALINE ADAptive LInear NEuron AI Artificial Intelligence ANN Artificial

The electrification of transportation has been considered as one of solutions to tackle the shortage of fossil energy sources and air pollution. Electric drives for electrified vehicles, including pure electric and hybrid electric vehicles, need to fulfil some specific requirements from automotive markets such as high efficiency, high volume power and torque densities, lowcost but safe-to-touch, high functional reliability, high torque quality, and flux-weakening control. In this context, multiphase permanent magnet synchronous machine (PMSM) drives have become suitable candidates to meet the above requirements.

The main objective of this doctoral thesis is to propose and refine fault-tolerant control strategies for non-sinusoidal multiphase PMSM drives that require less constraints on their design. In addition, constraints on current and voltage defined by the inverter and the machine are considered to optimize the machine control under the non-sinusoidal condition without exceeding their allowable limits. Therefore, the system sizing is guaranteed, especially in fluxweakening operations. The proposed fault-tolerant control strategies, based on the mathematical model of multiphase drives, enrich the control field of multiphase drives by providing various control options. The selection of one of the proposed control options can be a trade-off between a high quality torque but a low average value and a high average torque but a relatively high ripple. The control and torque performances of the drives can be refined by using artificial intelligence with a simple type of artificial neural networks named ADALINE (ADAptive LInear NEuron). With self-learning ability, fast convergence, and simplicity, ADALINEs can be applied to industrial multiphase drives. All proposed control strategies in this doctoral thesis are validated with an experimental seven-phase PMSM drive. The nonsinusoidal back electromotive force (back-EMF) of the experimental seven-phase PMSM is complex with the presence of multi-harmonics. Experimental results verify the effectiveness of the proposed strategies, and their applicability in a multiphase machine with a complex nonsinusoidal back-EMF.

v Résumé L'électrification des transports a été considérée comme l'une des solutions pour lutter contre la pénurie de sources d'énergie fossile et la pollution de l'air. Les entraînements électriques pour les véhicules électrifiés, y compris les véhicules électriques purs et électriques hybrides, doivent répondre à certaines exigences spécifiques des marchés automobiles, tels qu'un rendement élevé, des densités volumiques élevées de puissance et de couple, un coût faible avec une protection contre les risques électriques, une fiabilité fonctionnelle élevée, une qualité de couple élevée, et un contrôle de défluxage. Dans ce contexte, les entraînements de machines synchrones à aimants permanents (PMSM) polyphasées sont devenus des candidats appropriés pour répondre aux exigences citées ci-dessus.

L'objectif principal de cette thèse de doctorat vise à proposer et affiner des stratégies de commandes tolérantes aux défauts pour les entraînements de machines PMSM polyphasées non-sinusoïdales qui requièrent moins de contraintes lors de leur conception. Par ailleurs, les contraintes de courant et de tension définies par l'onduleur et la machine sont prises en compte pour optimiser en régime non-sinusoïdal le contrôle de la machine sans dépasser les limites admissibles. Cela permet idéalement un dimensionnement au plus juste et cela tout particulièrement dans la zone de défluxage. Les stratégies proposées de commandes tolérantes aux défauts, basées sur le modèle mathématique des entraînements polyphasés, enrichissent le domaine de contrôle des entraînements polyphasés en offrant de diverses options de contrôle. Le choix de l'une des options proposées de commande peut être un compromis entre un couple de haute qualité mais avec une valeur moyenne faible, et un couple moyen élevé mais avec une ondulation relativement élevée. Les performances de contrôle et de couple peuvent être affinées en utilisant l'intelligence artificielle avec un type simple de réseaux de neurones artificiels nommé ADALINE (neurone linéaire adaptatif). Grâce à leur capacité d'auto-apprentissage, à leur convergence rapide, et à leur simplicité, les ADALINE peuvent être appliqués aux entraînements polyphasés industriels. Toutes les stratégies de contrôle proposées dans cette thèse de doctorat sont validées avec un entraînement d'une machine PMSM à sept phases. La force électromotrice non-sinusoïdale de la machine PMSM à sept phases, relevée expérimentalement, est complexe avec la présence de plusieurs harmoniques. Les résultats expérimentaux vérifient l'efficacité des stratégies proposées, et leur applicabilité dans une machine polyphasée avec une force électromotrice non-sinusoïdale complexe.

Mots clés : Machine polyphasée, machine à sept phases, force électromotrice non-sinusoïdale, commande tolérante aux défauts, commande optimale sous contrainte, intelligence artificielle, neurone linéaire adaptatif, ADALINE.

List of tables

Introduction

In 1830s, the idea of using electric machines for the traction of vehicles, known as Electrified Vehicles (EVs), was introduced. However, constraints on production costs, volume, and mass of batteries hindered their commercialization process. In recent decades, fossil energy sources have been sharply declining, and environmental pollution has been becoming a serious global problem. Meanwhile, significant improvements on batteries have been achieved. Therefore, since 1990s, EVs have been slowly emerging and becoming an effective commercial solution to deal with the energy and environmental crises. An EV can be either a combination of an electric drive and a thermal engine in a hybrid electric vehicle (HEV) or a pure electric vehicle.

The pure electric vehicle can be a fuel cell vehicle (FCV) or a battery electric vehicle (BEV). BEVs and FCVs with zero emission have constraints on production costs and technology. Meanwhile, HEVs with low emission become more interesting in automotive industry because the rate of hybridization can be adjusted [START_REF] Chan | The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles[END_REF].

In general, an EV is a sophisticated combination of electrical and automotive engineering.

Regarding electrical engineering, electric drives play an important role in operations of EVs. They generally consist of electric machines, power electronics converters, energy supplies, control systems, and energy management strategies. The electric drive technologies applied to EVs are mainly dependent on the electric machine technologies: DC, AC induction (IM), and AC synchronous. It is noted that DC technology is no longer attractive due to its maintenance requirements and mechanical limitations. Currently, permanent magnet synchronous machines (PMSMs) are more found in HEVs in which spatial constraints are important. Meanwhile, IMs and DC-excited synchronous machines, without high-cost permanent magnets, are more observed in BEVs [2].

There have been several main requirements for electric drives to guarantee high-performance electromechanical conversions of EVs at variable speeds and variable torques. These requirements will be explained in the following paragraphs as follows:

High efficiency

Electric drives for EVs are required to have high efficiency for optimal energy consumption. In fact, the use of electric drives in vehicles increases the system efficiency because the efficiency of electric machines is higher than that of Internal Combustion Engines (ICEs). In addition, electric drives can recover the regenerating braking energy. However, at the present time, the electric energy storage for EVs is a challenge and therefore high-efficiency electric drives can increase the driving range or reduce the need for a huge battery capacity in EVs. Using PMSMs can be a solution [3].

High volume densities

High volume power density (kW/m 3 ) and high volume torque density (Nm/m 3 ), collectively known as high volume densities, are important criteria for electric drives of EVs, especially of HEVs. In the limited space of EVs, these high volume densities can reduce the size of electric drives. The high volume torque density is mainly important at low speed for accelerations and uphill climbs. Meanwhile, the high volume power density is crucial in the constant power range of EVs. These requirements can be satisfied by using PMSMs with higher volume densities than IMs, DC-excited synchronous machines and DC machines with the same torque and power [START_REF] Chan | The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles[END_REF][3][4][START_REF] Zhu | Electrical Machines and Drives for Electric, Hybrid, and Fuel Cell Vehicles[END_REF].

Low-cost but safe-to-touch

To meet the first and second criteria about high efficiency and high volume densities, PMSMs are considered as a high potential candidate. However, the use of permanent magnets results in high-cost machines, making PMSMs less interesting than IMs. For the mass market of EVs, low production costs can ensure an affordable price of EVs, making them become more attractive to customers. Therefore, a compromise between a low cost, high efficiency and high volume densities need to be made.

Besides the low-cost criterion, EVs need to be safe for humans. In this case, a low-voltage standard (< 48 V) can be a solution for the electrical safety margin. Low-voltage drives can avoid the high-voltage electric shock as well as expensive complex requirements for circuit insulations and powerful electronic devices. Therefore, protection costs are decreased. In addition, the battery management can be easier because the number of battery cells in series is reduced, facilitating the degradation detection of battery cells through variations of cell voltages. In recent years, 48 V electric drives have significantly drawn the attention of EV researchers [START_REF] Cavagnino | Experimental Characterization of a Belt-Driven Multiphase Induction Machine for 48-V Automotive Applications: Losses and Temperatures Assessments[END_REF][START_REF] Dörsam | The New Voltage Level 48 V for Vehicle Power Supply[END_REF][START_REF] Pawsey | 48V technology on the Automotive Horizon[END_REF][START_REF] Sattler | 48 V minihybrid-A new solution for the minimal hybridization of vehicles[END_REF][START_REF] Hackmann | 48 V-The way to a high volume electrification[END_REF][START_REF] Lauer | 48 Volt Hybrid with e-drive features -Excellent fuel efficiency and drivability[END_REF][START_REF] Els | 48 Volt Electrification: the next step to achieving 2020 emissions[END_REF]. Nevertheless, for a given power, currents in DC bus and phase windings in the low-voltage electric drives are obviously much higher than conventional drives, leading to a big challenge of electric drives.

High functional reliability

High functional reliability is a prime consideration in electric transportation systems [START_REF] Jahns | Improved Reliability in Solid-State AC Drives by Means of Multiple Independent Phase Drive Units[END_REF][START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF] in which electric drives have ability to tolerate faults in power converters or electric machine windings. Specifically, power converter components including power semiconductors, gate drives, and capacitors have been reported as the most fragile elements in electric drives [START_REF] Yang | An Industry-Based Survey of Reliability in Power Electronic Converters[END_REF]. Faults in power converters are hardly predicted even when several improvements have been proposed in [START_REF] Yang | Condition Monitoring for Device Reliability in Power Electronic Converters: A Review[END_REF]. Therefore, the fault-tolerant capability for electric machines and power converters has become a solution to avoid breakdowns or even damage of EVs, increasing their functional reliability. Besides, voltages and currents of electric machines in EVs are required to respect their limits in all operating modes such as healthy, faulty, accelerating and decelerating. The possibility of operating well with a derated power under a defective condition makes post-fault electric drives more acceptable, avoiding a costly safety margin in the context of a mass market.

High torque quality

An operation with a low-ripple torque is an important criterion in all electric drives, especially in automotive applications, to ensure smooth accurate driving, especially at low speed [START_REF] Zhu | Electrical Machines and Drives for Electric, Hybrid, and Fuel Cell Vehicles[END_REF]. Torque ripples severely deteriorate the EV performances when producing undesirable mechanical vibrations and acoustic noise, decreasing control precision of drivers [START_REF] Zhong | Active torque ripple reduction based on an analytical model of torque[END_REF][START_REF] Azar | Influence of Electric Loading and Magnetic Saturation on Cogging Torque, Back-EMF and Torque Ripple of PM Machines[END_REF]. Therefore, torque ripples need to be reduced even eliminated in electric drives for automotive applications in all operating modes. It means that electric drives can operate in healthy and faulty conditions with constant or low-ripple torques. Therefore, the technologies for electric machines as well as electric drives with corresponding control strategies need to be properly chosen to improve the performance quality of EVs.

Flux-weakening control

EVs need to operate at high speed in many circumstances, for example a drive on highways. Therefore, electric drives for EVs are required to operate in the flux-weakening region, known as the constant power region [START_REF] Chan | The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles[END_REF]. In fact, an EV can reach a high speed if its power is high enough. However, the use of the flux-weakening strategy can avoid the oversizing of electric drives. In flux-weakening control, the tractive torque is reduced when the rotating speed is above the base speed. Therefore, limits of current and voltage are respected, especially at high speed. In classical three-phase drives, the flux-weakening strategy can be analytically presented.

The above requirements for electric drives used in EVs can be summarized as follows:

1) High efficiency for optimal consumption of embedded energy supply.

2) High volume power and torque densities due to limited spaces in EVs.

3) Low-cost for affordable EVs in the mass market but safe for human touches. 4) High functional reliability with fault-tolerant ability and respect for electrical limits. 5) High torque quality for smooth EVs with low vibrations and acoustic noise. 6) Flux-weakening control for high-speed driving without oversized electric drives.

These requirements are considered as challenges that can be contradictory with the classical solutions. Therefore, flexible compromises need to be made. To alleviate these contradictions, the research on new solutions with more degrees of freedom (DoF) is of interest. In this doctoral thesis, electric drives with n phases (n > 3), known as multiphase drives, having more DoF for control and design compared to classical three-phase drives, will be considered. This thesis will be presented in three chapters. Chapter 1 will discuss several distinct properties that open up opportunities for multiphase drives in automotive applications. Indeed, despite some drawbacks, multiphase drives are considered as a suitable candidate for EVs. After modeling a general multiphase drive, recent studies in the control field of multiphase drives will be categorized and analyzed. These analyses combined with the six above requirements for electric drives allow to address objectives of this doctoral thesis.

Chapter 2 will present the modeling and control strategies of a specific multiphase drive in healthy mode and especially under a single-phase open-circuit (OC) fault. Control strategies for two-phase OC faults are briefly presented in Appendix B. These control strategies are based on the mathematical model of the drive in which a seven-phase PMSM is used for illustrations. The approaches in this doctoral thesis can be applied to other electric drives with different numbers of phases. In addition, constraints on current and voltage are always imposed in control schemes to guarantee high functional reliability and flux-weakening operations of the multiphase drives.

In Chapter 3, the performances, either in healthy mode or faulty mode, of the considered multiphase drive with the strategies proposed in Chapter 2 will be improved by using artificial intelligence. As a result, current control and torque quality will be enhanced regardless of uncertainties and imperfections of the multiphase drive. Moreover, the knowledge of harmonic components in signals of the drives allows a fast response of control (fast convergence) and avoids the calculation burden. Therefore, the applicability of the proposed strategies to industrial electric drives is improved.

Chapter 1. Multiphase Drives: Opportunities and State of the Art

Multiphase drives for automotive applications

Multiphase drives: a suitable candidate for EVs

The use of classical machines with only three coupling phases, or rather the use of Voltage Source Inverters (VSIs) with only three legs, is a limitation imposed by the past. It is noted that these electric drives cannot work properly when one phase is not supplied. This historical property induces an important constraint on the safety margin for VSIs and electric machines.

To release this constraint, electric drives of which VSIs have more than three legs will be considered in this doctoral thesis. These electric drives are called multiphase drives. Owing to more legs as well as phases, multiphase drives have more DoF for design and control. In this subsection, multiphase drives will be compared with different existing solutions regarding the requirements for electric drives in the EV mass market presented in the Introduction of this doctoral thesis.

A general multiphase drive with n phases (n > 3) fed by a n-leg VSI and DC-bus voltage in automotive applications is described in Fig. 1.1. The electric drive provides a traction force to an EV (wheel and chassis) through a gearbox. The first study of a multiphase drive, a five-phase IM fed by a VSI, was proposed in 1969 [START_REF] Ward | Preliminary investigation of an invertor-fed 5-phase induction motor[END_REF]. However, at that time, the attention to this proposed five-phase machine was still limited. The interest in multiphase machines for variablespeed electric drives has only been growing significantly in recent decades. It is thanks to evolutions in some specific areas such as power electronics converters and digital signal processors. Another reason for this emergence is that the in-depth knowledge about multiphase drives has been significantly advanced [START_REF] Parsa | Five-phase permanent-magnet motor drives[END_REF][START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF][START_REF] Levi | Multiphase induction motor drives -a technology status review[END_REF][START_REF] Duran | Multiphase Electric Drives: Introduction[END_REF]. Based on the spatial displacement between two adjacent phases, multiphase machines can be divided into symmetrical (with a spatial phase shift angle of 2π/n) and asymmetrical machines (with several sets of phases such as 2 sets of three phases for a six-phase machine) [START_REF] Klingshirn | High Phase Order Induction Motors -Part I-Description and Theoretical Considerations[END_REF][START_REF] Slunjski | Symmetrical/Asymmetrical Winding Reconfiguration in Multiphase Machines[END_REF]. If the rotor construction is considered, there are mainly induction and synchronous multiphase

n Wheel Chassis F traction F environment B v vehicle T em Gearbox A n-1 i A i n I DC … i B i n-1 … V DC … n-phase machine VSI Chapter 1.
Multiphase Drives: Opportunities and State of the Art 6

machines. An induction machine using a squirrel-cage rotor has been interesting due to lowcost materials. Meanwhile, a synchronous multiphase machine can be with PM excitation, with field windings, or of reluctance type [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF][START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines Part I[END_REF][START_REF] Bojoi | Multiphase electrical machines and drives: A viable solution for energy generation and transportation electrification[END_REF]. Like three-phase PMSMs, multiphase PMSMs are interesting due to their advantages such as high efficiency and high volume densities. Therefore, multiphase PMSMs can meet the first two requirements for electric drives in EVs (see Introduction) but using high-cost PMs is one of their drawbacks.

The other requirements can be met by exploiting the following distinct properties of multiphase drives.

1.1.1.A. Low power per phase rating for safe EVs

As analyzed in the third requirement of EVs (see Introduction), a safe-to-touch EV is supplied by a low voltage. Consequently, the current per phase rating becomes higher than that of an EV supplied by a higher voltage with the same power. Therefore, several transistors with a given commercial affordable rating can be used in parallel to make a synthetic transistor with a higher rating. However, the synchronization of these parallel transistors during their lifetime is a challenge since their aging is not the same. As a failure of a single transistor induces higher currents in the remaining healthy transistors, a safety margin becomes needful to ensure a sufficient reliability. Consequently, the oversizing of the transistors is practically necessary. For example, when a phase of a three-phase machine is obtained by putting k windings in parallel, it is possible to supply the machine with k three-phase star windings by k three-leg VSIs [START_REF] Rui | AN-Power stage of 48 V BSG inverter[END_REF]. As a result, a 3k-phase machine is then obtained with the same properties of a threephase machine. The requirement is a perfect synchronization of all three-leg VSIs. On the other hand, circulations of parasitic currents are observed between the windings of different threephase star windings. Moreover, in case of a fault in one winding of one three-phase star winding, the supply of the entire corresponding three-phase winding is removed. This approach is quite simple and used in industrial solutions. A huge power loss in case of a fault of one transistor and the complicated transistor synchronization are its main drawbacks. Therefore, we will not consider this kind of machines in our work.

Eventually, it is concluded that the use of multiphase machines to split the power across a high number of phases and inverter legs is interesting. The current per phase rating of converters and machine windings is lower compared to conventional three-phase drives with the same DC-bus voltage and power [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF][START_REF] Levi | Multiphase induction motor drives -a technology status review[END_REF]. Therefore, it is possible to use two power electronics switches per leg instead of a set of parallel switches, improving the reliability of the electric drives [START_REF] Locment | Comparison of 3-, 5and 7-leg Voltage Source Inverters for low voltage applications[END_REF]. This feature of multiphase drives enables EVs to be supplied by a low voltage such as 48 V with a power greater than 10 kW, the minimum power for hybridization. As a result, safe-to-touch EVs become more feasible with multiphase drives [START_REF] Runde | ISCAD -Design, control and car integration of a 48 volt high performance drive[END_REF].

1.1.1.B. Fault-tolerant ability for high functional reliability

Fault tolerance is one of the major advantages of multiphase machines [21-23, 27, 31] that can meet the fourth requirement of EVs about high functional reliability (see Introduction). There are various faults including Open Circuit (OC) and Short Circuit (SC) faults that can suddenly happen in power converters (switches), phase lines, or stator windings. These faults can Chapter 1.

Multiphase Drives: Opportunities and State of the Art 7 dramatically interrupt or even damage electric drives. The fault tolerance of multiphase drives is due to having more DoF for control than conventional three-phase drives. This feature, as first presented in [START_REF] Jahns | Improved Reliability in Solid-State AC Drives by Means of Multiple Independent Phase Drive Units[END_REF], allows to preserve a persistent operation of electric drives without any additional hardware. Indeed, the reconfiguration in post-fault operations of multiphase drives only defines new current references to obtain constant or low-ripple torques.

1.1.1.C. Low ripple torques for smooth EVs

A three-phase machine is usually designed with distributed stator windings, inducing a sinusoidal back-EMF. As a result, a constant torque is generated when sinusoidal currents are imposed. Then, time-constant d-q currents are simply obtained. The more recent PMSM generation with tooth concentrated stator windings is interesting for EVs due to lower production costs with short end-windings [START_REF] Boglietti | Electrical Machine Topologies: Hottest Topics in the Electrical Machine Research Community[END_REF][START_REF] Refaie | Fractional-Slot Concentrated-Windings Synchronous Permanent Magnet Machines: Opportunities and Challenges[END_REF]. In this case, a sinusoidal back-EMF, required for a constant torque with sinusoidal currents, can be obtained by a special design of rotor.

With a multiphase machine, this design requirement is relaxed as the number of phases is high. Indeed, according to the multi-reference frame approach [START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF][START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF], a n-phase symmetrical machine is characterized by (n+1)/2 (if n is odd) and (n+2)/2 (if n is even) characteristic planes, known as reference frames. One frame is associated with a group of harmonics. A constant torque can be ideally obtained when only a single harmonic of currents and back-EMFs exists in each reference frame (except the zero-sequence frames). Therefore, an increase in the number of phases results in more reference frames, permitting to have more harmonics in back-EMFs. Consequently, in a well-designed multiphase machine, the torque generated by non-sinusoidal back-EMFs can be constant even in a transient operation by imposing constant d-q currents in each reference plane. This property is not available in a three-phase machine. In conclusion, with a multiphase machine, it is possible to obtain a constant torque by imposing constant d-q currents in different reference planes. Meanwhile, a three-phase machine requires the classical constraint on sinusoidal back-EMFs and currents. Therefore, a multiphase machine leads to less constraints on design than a three-phase machine.

As discussed in [START_REF] Locment | Comparison of 3-, 5and 7-leg Voltage Source Inverters for low voltage applications[END_REF][START_REF] Jahns | Pulsating torque minimization techniques for permanent magnet AC motor drives-a review[END_REF][START_REF] Hung | Design of the most efficient excitation for a class of electric motor[END_REF][START_REF] Nguyen | Approches neuromimétiques pour l'identification et la commande des systèmes électriques : application au filtrage actif et aux actionneurs synchrones[END_REF][START_REF] Mekri | Fault-Tolerant Control Performance Comparison of Three-and Five-Phase PMSG for Marine Current Turbine Applications[END_REF][START_REF] Gopakumar | Modified current source inverter fed induction motor drive with reduced torque pulsations[END_REF], an electric drive with a high number of phases generates lower torque ripples than a conventional three-phase drive. Indeed, with a n-phase drive, the lowest harmonic order of torque ripples is 2n if n is odd, and its back-EMF contains only odd harmonics. For example, the lowest harmonic orders of torque ripples in 3-phase, 5-phase, and 7-phase drives are 6, 10, and 14, respectively. When n is even, the lowest harmonic order of torque ripples in symmetrical machines is n while that in asymmetrical machines is 2n. For example, the lowest harmonic order of torque ripples in a 6-phase symmetrical machine is 6, equivalent to a 3-phase machine. The lowest harmonic order of torque ripples in a 6-phase asymmetrical machine is 12, equivalent to a 12-phase symmetrical machine. Therefore, an increase in the number of phases results in higher-frequency torque ripples. This feature is interesting since the mechanical resonance of an electric drive at high frequency is eliminated. As a result, the torque of a multiphase drive can be smoother than that of a three-phase drive.

This feature of multiphase drives meets the fifth requirement of EVs (see Introduction) about high torque quality for smooth driving, especially at low speed.

Chapter 1.

Multiphase Drives: Opportunities and State of the Art 8

1.1.1.D. More possibilities of stator winding configurations

When an inverter is defined with a given maximum current and bus voltage, the connection changes of the machine windings allow to vary the torque-speed characteristic with an approach of flux-weakening. Combinations of the flux-weakening techniques [START_REF] Soong | Field-weakening performance of brushless synchronous AC motor drives[END_REF][START_REF] Slemon | Achieving a constant power speed range for PM drives[END_REF][START_REF] Wai | A new control technique for achieving wide constant power speed operation with an interior PM alternator machine[END_REF][START_REF] Schiferl | Power capability of salient pole permanent magnet synchronous motors in variable speed drive applications[END_REF] and different connections of stator windings can enlarge the speed range. Indeed, the maximum voltages that can be applied to the terminals of a phase winding are different for different configurations. A higher maximum phase voltage (a wider speed range) results in a smaller phase current (a lower torque) and vice versa. pentacle (c), and corresponding torque-speed characteristics (d) [START_REF] Sadeghi | Wide Operational Speed Range of Five-Phase Permanent Magnet Machines by Using Different Stator Winding Configurations[END_REF][START_REF] Nguyen | Different virtual stator winding configurations of open-end winding five-phase PM machines for wide speed range without flux weakening operation[END_REF].

Multiphase drives have more options of stator winding configurations for high speed operations. Generally, with a n-phase machine, there are (n+1)/2 possibilities to connect stator windings. For example, with a 3-phase machine, there are only 2 connecting possibilities: delta and star [START_REF] Swamy | Extended high-speed operation via electronic winding-change method for AC motors[END_REF]. If the number of phases increases to 5, there will be 3 connecting options including star, pentagon, and pentacle as shown in Fig. 1.2a, b, and c, respectively. Among these connecting options, the pentacle gives the widest speed range and lowest torque while the star connection has the shortest speed range and highest torque as presented in Fig. 1.2d [START_REF] Sadeghi | Wide Operational Speed Range of Five-Phase Permanent Magnet Machines by Using Different Stator Winding Configurations[END_REF][START_REF] Nguyen | Different virtual stator winding configurations of open-end winding five-phase PM machines for wide speed range without flux weakening operation[END_REF]. It is understood that the pentacle makes possible to obtain the highest voltage while the maximum current is smallest.

1.1.1.E. Electromagnetic pole changing by imposing harmonics of current

The payload of torque-speed characteristics is specified and characterized by limits that are defined by maximum torque at low speed and maximum power at high speed (above the base speed). To find an optimal solution for a complex payload, a high number of DoF in multiphase drives can applied.

The basic principle of the classical pole changing is that a decrease in the number of pole pairs can extend the speed range. In three-phase drives, the pole changing methods are implemented by switching different winding configurations as discussed in [START_REF] Osama | A new inverter control scheme for induction motor drives requiring wide speed range[END_REF][START_REF] Osama | Modeling and analysis of a wide-speed-range induction motor drive based on electronic pole changing[END_REF][START_REF] Osama | Experimental and finite-element analysis of an electronic pole-change drive[END_REF][START_REF] Giet | Induction motor with pole-changing winding for variable supply frequency[END_REF]. An appropriate connection of stator windings results in a magnetomotive force (MMF) in the air gap corresponding to the desired number of pole pairs. However, these pole changing methods are physically implemented by electronic devices with a high number of switches.

In a multiphase drive, the principle to extend its speed range is inherited from the classical pole changing methods. However, multiphase machines can work on multiple polarities by injecting different harmonic components of current, called electromagnetic pole changing. This pole changing is implemented without any winding reconfigurations, reducing switching losses as presented in [START_REF] Gautam | Variable speed multiphase induction machine using pole phase modulation principle[END_REF][START_REF] Zahr | Five-phase SPM machine with electronic pole changing effect for marine propulsion[END_REF][START_REF] Gong | Design Considerations of Five-Phase Machine With Double p/3p Polarity[END_REF][START_REF] Zahr | Maximum Torque Per Ampere strategy for a biharmonic five-phase synchronous machine[END_REF]. This distinct feature is only possessed by multiphase machines. Indeed, by using the multi-reference frame approach [START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF][START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF], a n-phase machine is decomposed into (n+1)/2 (n is odd) or (n+2)/2 (n is even) fictitious machines with corresponding characterized frames. Each frame is associated with a given group of harmonic components. In healthy mode, these harmonics of current can be independently controlled to obtain a constant torque. Compared to a three-phase machine, a high number of phases in a multiphase machine allows to use more than one harmonic of current to generate constant torques. For example, a five-phase machine, with p pole pairs, is decomposed into 2 fictitious two-phase machines and one zero-sequence machine. The first fictitious machine associated with the first harmonic is equivalent to p pole pairs while the second fictitious machine associated with the third harmonic is equivalent to 3p pole pairs. If only the currents of the first machine with p pole pairs are used to generate torque, the speed range is presented in Fig. 1.3a. When the currents of the first and third machines are imposed, a wider speed range is derived in Fig. 1.3b. At the starting stage (low speed), the first and second machines contribute to the torque generation with 3p pole pairs, whereas the torque is mainly produced by the main machine with p pole pairs at high speed. A more general torque-speed characteristic using the electromagnetic pole changing with decreases in the number of pole pairs from p1 to p4 is shown in Fig. 1.3c.

In conclusion, in an EV using a multiphase drive, its mechanical gearbox can be replaced by an electromagnetic gearbox, not only reducing switching losses and but also increasing the efficiency and reliability of the EV. In addition, a more comfortable driving can be obtained with the electromagnetic gearbox.

Opportunities for multiphase drives in automotive applications

According to the six requirements for electric drives in EVs (see Introduction), the five special properties in section 1.1.1 enable multiphase drives to possibly become a favored choice for automotive applications. Especially, multiphase PMSMs are more interesting than multiphase IMs due to their high efficiency, high volume torque and power densities. The last two properties of multiphase drives can be combined with flux-weakening strategies to extend the speed range. Different from three-phase drives, flux-weakening strategies cannot be analytically expressed in multiphase drives due to too many currents in d-q frames. Therefore, flux-weakening operations in multiphase drives can be implemented by imposing constraints on current and voltage. However, to fully replace their three-phase counterparts being widely used in EVs, multiphase drives, especially multiphase PMSM drives, need to overcome several drawbacks. There are also possible solutions for these drawbacks in the below discussions.

1) One of the disadvantages of multiphase drives is an increase in the calculation burden for design and control. It is the reason why the potential for the multiphase machines has only been progressively exploited over the past twenty years. This problem has been tackled by using more powerful microprocessors in combination with adapted modeling and control. 2) Another inconvenience of multiphase drives is an increase in the number of inverter legs and the number of wires connecting multiphase machines and VSIs. When the number of phases is not a multiple of three, it is difficult to use previous standards developed for three-phase drives. Therefore, special VSIs must be designed for the multiphase drives. In this case, if transistors in parallel are unnecessary, the number of drivers and heatsink components increases and so do production costs. However, the mass production could cut down these costs. In addition, as the current in each leg is lower, the quantity of Silicon and switch losses in multiphase drives are not intrinsically higher compared to three-phase drives. 3) Time constants of fictitious machines in a multiphase machine may not be of the same order, especially a machine with classical distributed windings (an integer number of slots per pole per phase). If the classical Pulse Width Modulation (PWM) is used, the smallest time constant imposes the carrier frequency for PWM. Consequently, the switching frequency is high with corresponding switching losses [START_REF] Kestelyn | Modélisation vectorielle multimachines pour la commande des ensembles convertisseursmachines polyphasés[END_REF]. Using Space Vector PWM (SVPWM) can be a solution to tackle this problem. Indeed, lowamplitude voltage vectors can be used for fictitious machines with small time constants. Specifically, zero-amplitude voltage vectors can be used if no torque is required in the fictitious machine associated with the smallest time constant (or zerosequence machines in the case of an open-winding configuration). Another solution is to choose stator windings which induce equal time constants. It is the case of fractional windings such as tooth concentrated windings. In addition, with new high-frequency wide-bandgap components, this problem could vanish. 4) Another drawback is an occurrence of parasitic currents that do not contribute to the torque generation. It is possible when the control of a n-phase drive (n > 3) is obtained by an adaptation of the classical three-phase drive control. For example, when only the main harmonic of currents is considered for the torque production, only two components of the n-dimensional voltage vector are mainly considered to control two d-q currents associated with the main harmonic. In fact, it is also necessary to take care of (n-2) other components of the voltage vector to avoid the occurrence of parasitic currents. Consequently, the control of a multiphase drive is more complex than a threephase drive. 5) Social inertia is one of existing obstacles preventing multiphase drives from being popularized in industry. The development of these complex electric drives requires long-term human investments. In a society with market profit rules and short-term Chapter 1.

Multiphase Drives: Opportunities and State of the Art 11 shareholding companies, risks from developing a drive with complex knowledge in the field of energy are high. Therefore, a change from industrial three-phase drives to multiphase drives is a challenge.

Recent projects on multiphase drives

In recent years, there have been several projects on multiphase drives for multiple applications such as project "MotorBrain" in Germany and "Intelligent Integrated Energy Converter (CE2I)" in France. The common points of these projects are an attempt to design converter-integrated multiphase machines with a high power density. EVs can be one of high potential applications of these motor drives. In addition, another integrated drive with a seven-phase modular brushless wound rotor synchronous machine is described in [START_REF] Le Luong | Optimal Design of Modular High Performance Brushless Wound Rotor Synchronous Machine for embedded systems[END_REF].

The two projects MotorBrain and CE2I are briefly summarized as follows:

1.1.3.A. MotorBrain project

The MotorBrain project started in 2011 with Infineon and a total of 30 partners from nine European countries. The MotorBrain prototype is a highly integrated electric motor that unifies the most important components of the powertrain for an electric vehicle. The interdisciplinary team represents all development and production sectors relevant to electromobility. The team includes universities, non-university research facilities, semiconductor manufacturers, electric motor builders, automobile component suppliers and automobile manufacturers. The main challenge is to develop a low-cost integrated drive for automotive applications. Without taking fault tolerance into consideration, the constraint on integration leads to the choice of multiphase machines, giving more DoF for design. Therefore, six-phase and nine-phase prototypes have been developed. The first one is a pure reluctance machine while the latter one is composed of ferrite magnets as shown in Fig. 1.4a. The integration of a motor, a gearbox, and an inverter, with the use of newly developed ferrite magnets, allows to build an engine with a reduced weight about 15%, from 90 kg to 77 kg [58] in comparison with the reference non-integrated three-phase drive. It is interesting to note that in this project, the number of phases higher than three does not come from the fault tolerance requirement.

Chapter 1.

Multiphase Drives: Opportunities and State of the Art 12

1.1.3.B. CE2I project

The smart energy converter project CE2I [START_REF]The integrated smart energy converter project CE2I[END_REF] is undertaken by Laboratory of Electrical Engineering and Power Electronics (L2EP) cooperated with other laboratories and industrial partners in the north region of France from 2015 to 2020. The project is to address interdisciplinary issues related to integrated drives. One of the target outcomes is a low-voltage (< 48 V) electromagnetic actuator with integrated VSIs using wide-bandgap components. To maximize the profit of an imposed safe low voltage of 48 V, an open-winding configuration is chosen for the machine. Therefore, 48 V will be the maximum peak voltage imposed on one phase. GaN transistor used in VSIs is a wide-bandgap component fitted for low voltages.

Considering limited available currents for GaN and associated thermal constraints, a multiphase tooth concentrated winding PM machine (Fig. 1.4b) with H-bridge supply is chosen, allowing tolerance for OC and SC faults of power components. Specifically, two axial multi-leg VSIs, one at each axial extremity of the machine, will supply twenty coils of the machine. The final choice of the number of phases from five to twenty will be made according to a compromise between spatial constraints in the VSIs and the machine, and computational constraints for the control and thermal dissipation [START_REF]The integrated smart energy converter project CE2I[END_REF].

Section summary

Section 1.1 has manly delivered several distinct properties, opening up opportunities for multiphase drives in automotive applications regardless of some drawbacks. Before further discussions about the existing control issue, a general model of a multiphase machine will be discussed in the next section. As previously analyzed, compared to multiphase IMs, multiphase PMSMs possess functional advantages such as high efficiency, high volume torque and power densities. Therefore, for the sake of simplicity, this thesis will focus on the modeling and control of multiphase PMSMs.

General model of a multiphase PMSM

The schematic diagram of a n-phase PMSM is shown in Fig. 1.5 with several assumptions: n phases (A, B, C, …, n-1, n) are equally shifted with a spatial phase shift angle δ=2π/n in the stator; the rotor is non-salient; θ is the electrical position; Ω is the rotating speed of the machine; the magnet circuit saturation is not considered in the back-EMF and flux calculations; and iron losses are not considered.

Natural frame model

The voltage and electromagnetic torque of a n-phase PMSM are given by:

  s di v R i + e dt =+ L (1.
          =            

L

Decoupled stator reference frame model

The decoupled stator reference frames are virtual frames obtained by the Clarke (or Concordia) transformation. The machine parameters (back-EMFs, currents, and voltages) in natural frame are converted into the decoupled stator reference frames as follows:

  α xx  = Clarke T (1.
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 nn nn k k k n k n n k k k n k n                  --  --     -- =  --     --   Clarke T
where x is the n-dimensional vector of a parameter in natural frame; xαβ is the n-dimensional vector of the parameter in the decoupled stator reference frames; k is the number of twodimensional stator reference frames (α-β); k=(n- In other words, according to [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF], after the Clarke transformation, the real n-phase machine is mathematically decomposed into (n+1)/2 (if n is odd) or (n+2)/2 (if n is even) fictitious machines as presented in Figs. 1.6 and 1.7, respectively. Specifically, there are k two-phase fictitious machines with k two-dimensional reference frames from (α1-β1) to (αk-βk). In addition, there is only 1 zero-sequence machine with 1 one-dimensional reference frame (z1) if n is odd. When n is even, there are 2 zero-sequence machines with 2 one-dimensional frames (z1, z2). A fictitious machine with its corresponding decoupled reference frame is associated with a given group of harmonics as presented in Table 1.1. where Lj (j ∈ [1, k] ⸦ ℕ), Lz1 and Lz2 are the inductances of fictitious machine j, zero-sequence machine z1 and z2, respectively. If the rotor presents any saliency, all elements of [Lαβ] need to be calculated with the electrical position of the machine.

It is noted that the natural frame model does not allow an easy development of the control system due to the magnetic coupling between phase windings, representing in [L] from (1.1). By using the Clarke transformation, the inductance matrix [L] becomes [Lαβ] in (1.5), allowing the magnetic decoupling between phase windings in the new reference frames.

All row vectors of [TClarke] are orthogonal to each other. An important property of the Clarke matrix regardless of the even or odd number of phases is that the inversion of the Clarke matrix is equal to its transpose:

    1 T - = Clarke Clarke TT (1.6)
The electromagnetic torque of the machine is equal to the sum of torques generated by all fictitious machines as follows:

1

k em j z1 z2 j T T T T =   = + +      (1.7)

Rotor reference frame model

The rotor reference frames are virtual frames in which the sinusoidal parameters of the machine in the decoupled stator reference frames are converted to constant signals by using the Park transformation. The general Park transformation is given by: where xdq is the n-dimensional vector of the machine parameters in the rotor reference frames; k is the number of 2-dimensional rotating frames (d-q); k=(n-1)/2 if n is odd, and k=(n-2)/2 if n is even; xz1 and xz2 are parameters in one-dimensional zero-sequence frames; [TPark] is the nby-n Park transformation matrix with harmonics h1θ to hkθ that are determined from associated harmonics in Table 1

    
di v R i L p L i e dt di v R i L p L i e dt    = + - +    = + + +  (1.9)
where (vdk, vqk), (idk, iqk), and (edk, eqk) are the voltages, currents, and back-EMFs in d-q frame k, respectively; p is the number of pole pairs; (Ldk, Lqk) are inductances in d-q frame k.

In a non-salient machine, the inductance matrix in the rotor reference frames [Ldq], equal to matrix [Lαβ] in (1.5), can be proved as follows: where Lj (j ∈ [1, k] ⸦ ℕ), Lz1 and Lz2 are the inductances of fictitious machine j, zero-sequence machine z1 and z2, respectively. Therefore, we have (Ldk=Lqk= Lk).

     ( )     ( )       ( )         0
In the rotor reference frames, the machine model inherits the magnetic decoupling property from the decoupled stator reference frames. Ideally, the machine parameters such as currents in the rotor reference frames are time-constant for control.

State of the art in the control field of multiphase drives

To derive a general and systematic view of studies on the control of multiphase drives, the state of the art is organized as follows. First, control techniques for multiphase drives in healthy mode are presented. Then, existing studies on fault-tolerant control are analyzed.

Existing control techniques of multiphase drives in healthy mode

As analyzed in [21-23, 26, 31, 60-62], control techniques for multiphase drives can be generally categorized in three main types: Field-Oriented Control (FOC), Direct Torque Control (DTC) and Model-based Predictive Control (MPC). The inner loop for current, torque, and flux control is considered because the outer loop for speed control in the three control techniques is the same.

1.3.1.A. FOC

FOC, early proposed in [START_REF] Haase | Zur Dynamik drehzahlgeregelter Antriebe mit stromrichtergespeisten Asynchron-Kurzschlußläufermaschinen (On dynamic of the speed controlled static AC drive with squirrel-cage induction machine)[END_REF][START_REF] Blaschke | The principle of field orientation as applied to the new transvector closed loop control for rotating machine[END_REF], has become the most popular control technique with the use of orthogonal transformation matrices. To understand the principle of FOC, the FOC-based current control loop of a n-phase PMSM fed by a VSI is described in Fig. In the early 2000s, methodologies to generally study multiphase drives by using space vector approach have been proposed [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF][START_REF] Semail | Outils et méthodologie d'étude des systèmes électriques polyphasés -Généralisation de la méthode des vecteurs d'espace[END_REF]. The geometrical and graphical properties of the space vector are generalized and adapted for multiphase systems with advancements of matrix calculations. Power converters including PWM voltages and current source inverters are characterized by space vectors. The methodologies can be generalized to various inverters with different numbers of legs. Then, the formalism is first applied to three-phase machines before being verified in five-phase induction machines.

Some recent studies on FOC-based control of multiphase drives can be summarized as follows:

1) Five phase and dual three-phase induction machines with sinusoidal MMFs have been studied in [START_REF] Jones | A Synchronous Current Control Scheme for Multiphase Induction Motor Drives[END_REF][START_REF] Che | Current Control Methods for an Asymmetrical Six-Phase Induction Motor Drive[END_REF][START_REF] Yepes | Current Harmonics Compensation Based on Multiresonant Control in Synchronous Frames for Symmetrical n-Phase Machines[END_REF]. Due to more DoF than three-phase counterparts, non-sinusoidal MMFs in multiphase induction machines can be properly controlled in [START_REF] Abdelkhalik | Eleven-phase induction machine: steady-state analysis and performance evaluation with harmonic injection[END_REF][START_REF] Abdel-Khalik | Optimum Flux Distribution With Harmonic Injection for a Multiphase Induction Machine Using Genetic Algorithms[END_REF][START_REF] Zheng | Dual-Plane Vector Control of a Five-Phase Induction Machine for an Improved Flux Pattern[END_REF][START_REF] Abdel-Khalik | Improved Flux Pattern With Third Harmonic Injection for Multiphase Induction Machines[END_REF][START_REF] Abdel-Khalik | Effect of Current Harmonic Injection on Constant Rotor Volume Multiphase Induction Machine Stators: A Comparative Study[END_REF][START_REF] Mengoni | High-Torque-Density Control of Multiphase Induction Motor Drives Operating Over a Wide Speed Range[END_REF]. The injection of current harmonics allows to improve torque quality. 2) Studies based on FOC for three-phase, five-phase, dual three-phase and seven-phase PMSMs with different stator winding topologies have been proposed in [START_REF] Locment | Comparison of 3-, 5and 7-leg Voltage Source Inverters for low voltage applications[END_REF][START_REF] Nguyen | Different virtual stator winding configurations of open-end winding five-phase PM machines for wide speed range without flux weakening operation[END_REF][START_REF] Zahr | Five-phase SPM machine with electronic pole changing effect for marine propulsion[END_REF][START_REF] Gong | Design Considerations of Five-Phase Machine With Double p/3p Polarity[END_REF][START_REF] Zahr | Maximum Torque Per Ampere strategy for a biharmonic five-phase synchronous machine[END_REF][START_REF] Lu | Flux-weakening strategies for a five-phase PM synchronous machine[END_REF][START_REF] Sandulescu | FPGA implementation of a general Space Vector approach on a 6-leg voltage source inverter[END_REF][START_REF] Lu | Computation of optimal current references for flux-weakening of multi-phase synchronous machines[END_REF][START_REF] Sandulescu | Control Strategies for Open-End Winding Drives Operating in the Flux-Weakening Region[END_REF][START_REF] Bruyere | Modeling and control of a seven-phase claw-pole integrated starter alternator for micro-hybrid automotive applications[END_REF][START_REF] Bruyere | Modélisation et commande d'un alterno-demarreur hetaphase pour application automobile micro-hybride[END_REF][START_REF] Sandulescu | Modélisation et commande d'un système à trois phases indépendantes à double fonctionnalité : Traction électrique et chargeur forte puissance pour application automobile[END_REF][START_REF] Scuiller | Maximum Reachable Torque, Power and Speed for Five-Phase SPM Machine With Low Armature Reaction[END_REF][START_REF] Zahr | Machine pentaphasée a double polarité pour électrification du domaine des transports par effet boite de vitesse électromagnétique[END_REF][START_REF] Hu | Current Control for Dual Three-Phase Permanent Magnet Synchronous Motors Accounting for Current Unbalance and Harmonics[END_REF][START_REF] Nguyen | Adaline Neural Networks-based sensorless control of five-phase PMSM drives[END_REF][START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF][START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Fall | Contribution à l'étude de machines électriques polyphasées à aimants permanents en vue d'une meilleure gestion de la commande en modes dégradés pour des applications liées aux energies renouvelables marines[END_REF][START_REF] Moraes | New Electrical Inversed-Series Connection for Even-Phase Symmetrical PMSMs[END_REF][START_REF] Moraes | Homopolar Current's Copper Losses Analysis for Different Modulations in Open-End Winding Five-Phase Drives[END_REF]. In these studies, most of these machines have non-sinusoidal back-EMF waveforms, enabling to increase the torque density. Especially, a bi-harmonic fivephase PMSM with a dominant third harmonic of back-EMF is introduced in [START_REF] Zahr | Maximum Torque Per Ampere strategy for a biharmonic five-phase synchronous machine[END_REF][START_REF] Zahr | Machine pentaphasée a double polarité pour électrification du domaine des transports par effet boite de vitesse électromagnétique[END_REF]. This special machine can easily exploit corresponding current harmonics to have an electromagnetic gearbox, extending the speed range without using physical electronics switches. Meanwhile, studies [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF] calculate current references to obtain maximum torque-speed characteristic by considering limits of peak values of phase currents and voltages for a sinusoidal five-phase PMSM. An alternative to FOC is DTC that was introduced in [START_REF] Takahashi | A New Quick-Response and High-Efficiency Control Strategy of an Induction Motor[END_REF][START_REF] Depenbrock | Direct self-control (DSC) of inverter-fed induction machine[END_REF][START_REF] Buja | Direct torque control of PWM inverter-fed AC motors -a survey[END_REF]. In general, DTC is based on the modeling to estimate the magnetic flux and torque. The control of the magnetic flux and torque is implemented in decoupled stator reference frames without the inner current control loop. DTC techniques are categorized by the way to define the stator voltages. The stator voltages can be either obtained by an optimal Switching Table (ST-DTC) or by a constant switching frequency (PWM-DTC). The inner loop of a n-phase PMSM drive using ST-DTC is described in Fig. 1.9. The ST-DTC is to compare the estimated and reference values of the stator flux (𝜙 𝑒𝑠𝑡 and 𝜙 𝑟𝑒𝑓 ) as well as of the electromagnetic torque (𝑇 𝑒𝑚_𝑒𝑠𝑡 and 𝑇 𝑒𝑚_𝑟𝑒𝑓 ). Then, stator voltage vectors are selected from a look-up table to define switching signals of VSI (𝑠 1,𝑛 ). With ST-DTC, hysteresis controllers are usually applied to force the controlled variables to rapidly track their reference values without PWM. This feature makes ST-DTC simple with a fast torque response. However, ST-DTC leads to variable switching frequencies and parasitic high-frequency components in currents, resulting in higher-ripple torques than FOC [START_REF] Duran | Multiphase Electric Drives: Introduction[END_REF]. In addition, the number of voltage space vectors is exponentially proportional to the number of phases, making the size of the look-up table dramatically increase [START_REF] Liu | A review of drive techniques for multiphase machines[END_REF]. Study [START_REF] Karampuri | Extension of the DTC Technique to Multiphase Induction Motor Drives Using Any Odd Number of Phases[END_REF] proposes a general ST-DTC method for the odd number of phases more than three. However, the ST-DTC technique has not been extended yet to any phase number higher than six.

1) Some studies on ST-DTC for five-phase and asymmetrical six-phase induction machines are introduced in [START_REF] Zheng | A Novel Direct Torque Control Scheme for a Sensorless Five-Phase Induction Motor Drive[END_REF][START_REF] Gao | Low speed control improvements for classic Direct Torque Control of a 2-level 5-phase inverter-fed induction machine[END_REF][START_REF] Taheri | Efficiency Improvement in DTC of Six-Phase Induction Machine by Adaptive Gradient Descent of Flux[END_REF]. Specifically, the stator voltages in the secondary plane are reduced to minimize the stator current components that do not generate the torque. 2) Meanwhile, studies on ST-DTC for five-phase PMSM drives using the multi-machine multi-converter system concept have been proposed in [START_REF] Kestelyn | Direct torque control of multi-phase permanent magnet synchronous motor drive: application to a five-phase[END_REF][START_REF] Parsa | Sensorless Direct Torque Control of Five-Phase Interior Permanent-Magnet Motor Drives[END_REF].

Alternatively, PWM-DTC technique imposes constant switching frequencies and requires a PWM as described in Fig. 1.10. Specifically, DTC algorithm in PWM-DTC is used to generate voltage references before PWM defines inverter switching states. This algorithm is based on the deadbeat solution in which reference values of the stator flux and electromagnetic torque are obtained in just one sampling time [START_REF] Habetler | Direct torque control of induction machines using space vector modulation[END_REF]. This method has several advantages compared to ST-DTC with variable switching frequencies such as lower torque ripples and smaller high- frequency components in currents. However, because of controlling in decoupled stator reference frames, PWM-DTC requires a high switching frequency to guarantee a good performance. In addition, as ST-DTC, the calculation burden is a drawback of PWM-DTC. An application of PWM-DTC can be found in [START_REF] Bojoi | Direct torque control for dual three-phase induction motor drives[END_REF] for a dual three-phase induction motor drive.

1.3.1.C. MPC

Although first developed in the 1970s, MPC has recently become one of the most promising and widely used techniques [START_REF] Duran | Multiphase Electric Drives: Introduction[END_REF][START_REF] Cortes | Predictive Control in Power Electronics and Drives[END_REF]. This approach can be considered as an improvement of DTC. According to the optimization and control actions, MPC techniques can be divided into two types, including Continuous Control Set MPC (CCS-MPC) and Finite Control Set MPC (FCS-MPC). CCS-MPC applies an average model of the system with continuous reference signals and a fixed frequency. Meanwhile, FCS-MPC uses the finite number of switching states in the inverter, and it has variable switching frequencies. Indeed, the FCS-MPC scheme for a n-phase PMSM drive in Fig. 1.11 shows that it is based on the accuracy of the system model to predict future behaviors of the system variables (𝑖 𝛼𝛽 𝑘+1 ). The anticipation with a minimization of a cost function (min(J)) allows to define the optimal VSI switching states (𝑠 1,𝑛 ). It is noted that FCS-MPC can easily add extra control objectives such as copper losses or constraints on current and voltage. In general, MPC has a faster response than FOC and a more flexible control structure than DTC. However, it still possesses high computational costs due to a high number of iterations and high switching frequencies. Importantly, MPC requires accurate knowledge of the system for the model prediction [START_REF] Duran | Multiphase Electric Drives: Introduction[END_REF]. Multiphase Drives: Opportunities and State of the Art 20 1) General MPC schemes for power electronics and devices are categorized in [START_REF] Cortes | Predictive Control in Power Electronics and Drives[END_REF]. MPC-based control structures are described in [START_REF] Arahal | Multi-phase current control using finite-state model-predictive control[END_REF][START_REF] Barrero | A Proof of Concept Study of Predictive Current Control for VSI-Driven Asymmetrical Dual Three-Phase AC Machines[END_REF] for asymmetrical six-phase induction machines, and in [START_REF] Duran | Reduction of Common-Mode Voltage in Five-Phase Induction Motor Drives Using Predictive Control Techniques[END_REF][START_REF] Lim | FCS-MPC-Based Current Control of a Five-Phase Induction Motor and its Comparison with PI-PWM Control[END_REF][START_REF] Martín | Sensitivity of predictive controllers to parameter variation in five-phase induction motor drives[END_REF] for five-phase induction machines. 2) Studies for five-phase PMSMs are conducted in [START_REF] Kestelyn | Investigation on model predictive control of a five-phase permanent magnet synchronous machine under voltage and current limits[END_REF][START_REF] Bermudez | Model predictive optimal control considering current and voltage limitations: Real-time validation using OPAL-RT technologies and five-phase permanent magnet synchronous machines[END_REF], taking into account limits of current and voltage.

Existing control strategies for post-fault operations

1.3.2.A. Possible faults in multiphase drives

To analyze existing fault-tolerant control strategies, possible electric faults may happen in multiphase drives should be presented. In Fig. 1.12, different types of faults in a multiphase drive include SC or OC in inverter switches, phase windings or connecting lines between the machine and the inverter. Among them, OC faults have been more reported than SC faults in electric drives [START_REF] Duran | Recent Advances in the Design, Modeling, and Control of Multiphase Machines Part II[END_REF][START_REF] Cao | Overview of Electric Motor Technologies Used for More Electric Aircraft (MEA)[END_REF] since a SC fault normally leads to an OC fault with the inner protection of the corresponding switches [START_REF] Wintrich | Application Manual Power Semiconductors[END_REF]. Therefore, this present doctoral thesis will mainly focus on OC faults in multiphase drives. When a fault happens in an electric drive, detections, diagnoses, and fault isolations of the fault need to take place, avoiding possible damages of the electric drive. There have been many studies on fault diagnoses, detections and isolations such as [START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF][START_REF] Trabelsi | Real-Time Switches Fault Diagnosis Based on Typical Operating Characteristics of Five-Phase Permanent-Magnetic Synchronous Machines[END_REF][START_REF] Trabelsi | Experimental Investigation of Inverter Open-Circuit Fault Diagnosis for Biharmonic Five-Phase Permanent Magnet Drive[END_REF][START_REF] Duran | A Simple, Fast, and Robust Open-Phase Fault Detection Technique for Six-Phase Induction Motor Drives[END_REF][START_REF] Salehifar | Fault Detection and Fault Tolerant Operation of a Five Phase PM Motor Drive Using Adaptive Model Identification Approach[END_REF][START_REF] Meinguet | Fault-tolerant permanent-magnet synchronous machine drives: Fault detection and isolation, control reconfiguration and design considerations[END_REF][START_REF] Meinguet | An on-line method for stator fault detection in multi-phase PMSM drives[END_REF][START_REF] Nguyen | Fault-Tolerant Operation of an Open-End Winding Five-Phase PMSM Drive With Short-Circuit Inverter Fault[END_REF] by exploiting special characteristics of multiphase drives.

In healthy mode, multiphase drives are properly controlled with smooth torques by the control techniques as presented in the previous subsection. Their phase currents have the same waveform with rated RMS values. In post-fault operations, the number of active phases is lower, and the remaining phases are no longer symmetrical. If the pre-fault current references are not changed, there will have uncontrollable currents and pulsating torques [START_REF] Sandulescu | Control Strategies for Open-End Winding Drives Operating in the Flux-Weakening Region[END_REF][START_REF] Nguyen | Fault-Tolerant Operation of an Open-End Winding Five-Phase PMSM Drive With Short-Circuit Inverter Fault[END_REF][START_REF] Kianinezhad | Modeling and Control of Six-Phase Symmetrical Induction Machine Under Fault Condition Due to Open Phases[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. Specifically, there are two main consequences of the fault occurrence in a multiphase drive as follows:

n B T em A n-1 i A i n I DC … i B i n-1 … V DC … … SC switch OC line OC phase Inter-turn SC SC phase OC switch … VSI n-phase machine Chapter 1.
Multiphase Drives: Opportunities and State of the Art 21 1) Limits of peak or Root Mean Square (RMS) currents may not be respected. In other words, current waveforms of remaining healthy phases of the drive are deteriorated in post-fault operations. The peak current limit is based on the instantaneous peak currents during a short-time operation, and it is related to VSI components. Meanwhile, RMS current limit is mainly defined from the thermal limits of the stator windings. 2) Smooth electromagnetic torques cannot be guaranteed. In general, the uncontrolled healthy phase currents create uncircular rotating MMFs per reference frame, generating torque ripples. These torque ripples can cause noises and vibrations, reducing the operating quality of the drive.

1.3.2.B. Categorization based on criteria of new current references for fault-tolerant operations

From the consequences in the previous subsection, new current references need to be defined when one or more phases are open-circuited, called the reconfiguration or fault-tolerant control. New current references are required to generate smooth torques and comply some of the several additional criteria as follows:

1) Constraints on current limit: the remaining healthy phase currents need to respect their RMS or peak value limits [START_REF] Guzman | Novel control techniques in multiphase drives: Direct control methods (DTC and MPC) under limit situations[END_REF][START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Dwari | Fault-Tolerant Control of Five-Phase Permanent-Magnet Motors With Trapezoidal Back EMF[END_REF][START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF]. 2) Constraints on voltage limit: the peak phase voltages are required to be within their limits to avoid inverter saturation. In other words, flux-weakening operations can be guaranteed in faulty modes [START_REF] Guzman | Novel control techniques in multiphase drives: Direct control methods (DTC and MPC) under limit situations[END_REF][START_REF] Mengoni | High-Torque-Density Control of Multiphase Induction Motor Drives Operating Over a Wide Speed Range[END_REF][START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF]. 3) Copper loss optimization: the remaining healthy phase currents are designed to obtain either minimum total copper losses [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF][START_REF] Dwari | An Optimal Control Technique for Multiphase PM Machines Under Open-Circuit Faults[END_REF][START_REF] Mohammadpour | A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections[END_REF][START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF][START_REF] Hyung-Min | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF][START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF][START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF][START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF][START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF][START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF][START_REF] Xiong | Fault-tolerant FOC for five-phase SPMSM with non-sinusoidal back EMF[END_REF][START_REF] Bermudez | An Experimental Assessment of Open-Phase Fault-Tolerant Virtual-Vector-Based Direct Torque Control in Five-Phase Induction Motor Drives[END_REF] or equal copper losses in the healthy phases [124, 126-135, 137, 138].

The above criteria can be combined with each other in a fault-tolerant strategy. For example, the first and second criteria can guarantee high functional reliability in low-speed and fluxweakening operations for post-fault multiphase drives.

1.3.2.C. Categorization based on types of MMFs for fault-tolerant operations

The principle of a fault-tolerant strategy is based on fault types and characteristics of multiphase drives such as the number of phases and machine design. One of the first attempts [START_REF] Jahns | Improved Reliability in Solid-State AC Drives by Means of Multiple Independent Phase Drive Units[END_REF] shows that the number of stator phases and the type of excitation waveforms (sinusoidal or nonsinusoidal) affect degrading performances. Accordingly, fault-tolerant control studies can be classified into two groups according to types of MMFs for IMs and PMSMs as follows:

1) Sinusoidal multiphase machines: Fault-tolerant strategies for sinusoidal multiphase machines have been found in [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Kianinezhad | Modeling and Control of Six-Phase Symmetrical Induction Machine Under Fault Condition Due to Open Phases[END_REF][START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF][START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF][START_REF] Hyung-Min | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF][START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF][START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF][START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF][START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF][START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF][START_REF] Fnaiech | Fuzzy Logic and Sliding-Mode Controls Applied to Six-Phase Induction Machine With Open Phases[END_REF][START_REF] Sun | Enhanced Optimal Torque Control of Fault-Tolerant PM Machine Under Flux-Weakening Operation[END_REF][START_REF] Sui | Open-Circuit Fault-Tolerant Control of Five-Phase PM Machine Based on Reconfiguring Maximum Round Magnetomotive Force[END_REF]. These machines have more DoF for control because high-order harmonics of back-EMF do not exist, allowing the presence of high-order harmonics of current for optimization purposes.

Chapter 1.

Multiphase Drives: Opportunities and State of the Art

22

Fault-tolerant control strategies for sinusoidal machines cannot be simply applied to non-sinusoidal machines due to torque ripples. 2) Non-sinusoidal multiphase machines: Studies [61, 120-122, 124, 125, 134-136, 142-149] have been proposed fault-tolerant strategies for non-sinusoidal multiphase machines. These strategies are often more complicated than those of sinusoidal machines. However, these non-sinusoidal multiphase machines have several advantages such as easy-to-manufacture, lower production costs, higher torque densities, lower copper losses (if concentrated windings with short end-windings are chosen), and the ability for electromagnetic pole changing (see section 1.1.1.E).

1.3.2.D. Categorization based on modeling of multiphase drives for fault-tolerant operations

The modeling of a multiphase drive in post-fault conditions can be derived by the vector space decomposition [START_REF] Zhao | Modeling and control of a multi-phase induction machine with structural unbalance[END_REF], or the multi-reference frame theory with generalized space vectorial formalism [START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF][START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF][START_REF] Semail | Outils et méthodologie d'étude des systèmes électriques polyphasés -Généralisation de la méthode des vecteurs d'espace[END_REF]. As discussed in the general modeling (see section 1.2), using Clarke transformation [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] allows to decompose a n-phase machine into several decoupled fictitious machines (reference frames). There are (n+1)/2 reference frames for the odd number of phases and (n+2)/2 reference frames for the even number of phases. In these reference frames, currents and voltages are independently controlled in healthy mode. However, in faulty modes, the decoupling property no longer exists. Therefore, there have been two options for the modeling as follows:

1) Clarke transformation of healthy mode [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] is preserved but current references in rotor reference frames need to be redefined [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF]. The control structure is unchanged in faulty modes, making these strategies simpler. New d-q current references of fictitious machines that create most of the torque are designed to be constant. Consequently, other d-q currents in other fictitious machines are timevariant. 2) Reduced-order transformation matrices [126-129, 132-135, 149] with dimensions equivalent to the number of healthy phases are applied. Indeed, when a n-phase drive loses one phase, the machine in this post-fault condition can be considered as an asymmetric (n-1) phase machine. Hence, the post-fault drive can be mathematically re-modeled with (n-1) phases in which new (n-1)-dimensional Clarke and Park matrices are defined with the aim of preserving MMFs under the OC fault. In new decoupled reference frames, new d-q current references with optimization options are determined to obtain constant torques.

1.3.2.E. Categorization based on control techniques for fault-tolerant operations

A popular control technique for a post-fault multiphase drive is FOC in which the speed control loop is regulated by a proportional or PI controller to generate torque references [START_REF] Duran | Recent Advances in the Design, Modeling, and Control of Multiphase Machines Part II[END_REF]. Meanwhile, the inner loop controller for current can be one of the following options:

Chapter 1.
Multiphase Drives: Opportunities and State of the Art 23 1) PI: Studies [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF] apply the classical FOC scheme with PI controllers because main d-q current references, generating most of the torque, are designed to be constant. 2) MPC and DTC: With a faster response than FOC, the use of MPC has been found in [START_REF] Bermudez | Model predictive optimal control considering current and voltage limitations: Real-time validation using OPAL-RT technologies and five-phase permanent magnet synchronous machines[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Guzman | IGBT-Gating Failure Effect on a Fault-Tolerant Predictive Current-Controlled Five-Phase Induction Motor Drive[END_REF][START_REF] Guzman | Speed Control of Five-Phase Induction Motors With Integrated Open-Phase Fault Operation Using Model-Based Predictive Current Control Techniques[END_REF] while some studies [START_REF] Bermudez | An Experimental Assessment of Open-Phase Fault-Tolerant Virtual-Vector-Based Direct Torque Control in Five-Phase Induction Motor Drives[END_REF][START_REF] Bermudez | Open-Phase Fault-Tolerant Direct Torque Control Technique for Five-Phase Induction Motor Drives[END_REF] have been conducted with DTC.

The MPC scheme is also flexible to apply constraints on current and voltage. However, MPC and DTC possess some drawbacks as described in the control techniques (see section 1.3.1). 3) Hysteresis control: To deal with time-variant current references in natural frame, studies [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF][START_REF] Dwari | Fault-Tolerant Control of Five-Phase Permanent-Magnet Motors With Trapezoidal Back EMF[END_REF][START_REF] Mohammadpour | A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections[END_REF][START_REF] Parsa | Fault-Tolerant Interior-Permanent-Magnet Machines for Hybrid Electric Vehicle Applications[END_REF][START_REF] Mohammadpour | Global Fault-Tolerant Control Technique for Multiphase Permanent-Magnet Machines[END_REF] apply hysteresis controllers for current control. However, noises, losses and variable switching frequencies can be considered as several drawbacks of the control. 4) Dual PI: Studies [START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF][START_REF] Xiong | Fault-tolerant FOC for five-phase SPMSM with non-sinusoidal back EMF[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF][START_REF] Xiong | A Fault-Tolerant FOC Strategy for Five-Phase SPMSM With Minimum Torque Ripples in the Full Torque Operation Range Under Double-Phase Open-Circuit Fault[END_REF][START_REF] Abdel-Khalik | Effect of Stator Winding Connection on Performance of Five-Phase Induction Machines[END_REF] use dual PI or proportional integral resonant (PIR) controllers for time-variant current references. Demands for multiple parameter adjustments and the knowledge of frequencies are some drawbacks of this approach. 5) Robust control: A FOC-based control scheme with a robust controller for speed or current such as a fuzzy logic (FL) controller and a sliding mode controller has been found in [START_REF] Fnaiech | Fuzzy Logic and Sliding-Mode Controls Applied to Six-Phase Induction Machine With Open Phases[END_REF][START_REF] Liu | Enhancing Fault-Tolerant Ability of a Nine-Phase Induction Motor Drive System Using Fuzzy Logic Current Controllers[END_REF][START_REF] Betin | Shaft Positioning for Six-Phase Induction Machines With Open Phases Using Variable Structure Control[END_REF]. This scheme can enhance the fault-tolerant ability of the drive in various types of faults (OC faults in inverter switches or lines between inverters and machines).

Objectives of the doctoral thesis

This doctoral thesis is dedicated to enriching control methods of multiphase drives with the aim of highlighting advantages of multiphase drives in automotive applications. Multiphase drives can meet the six requirements for electric drives in EVs (see Introduction) only when these drives are properly controlled under a variety of operating conditions. For example, an EV driven by multiphase drives with two driving modes is described in Fig. 1.13. The torque reference can be generated by either the accelerator pedal (controlled by the driver) or the speed controller (an autonomous operation, for example). Economical mode generally enables EVs to work with high efficiency in the low-torque region of the torque-speed characteristic. Meanwhile, high-performance mode allows EVs to operate with maximum torques at each speed for accelerations. A switch between these two driving modes is made when the required torque results in an excess of electrical limits (currents and voltages). This doctoral thesis will focus on high-performance mode under healthy or faulty conditions. The electromagnetic torque will be maximized under limits of currents and voltages.

Chapter 1.

Multiphase Drives: Opportunities and State of the Art Therefore, according to the above analyses and state of the art in the control field of multiphase drives (see section 1.3), the main objective of this doctoral thesis is to propose control strategies for non-sinusoidal multiphase PMSM drives under OC faults (see section 1.

3.2.A).

Importantly, constraints on current and voltage will be always considered in all operating modes. Another objective of this doctoral thesis is to enhance performances, including the control and torque quality, under impacts of uncertainties and imperfections of the electric drives by using artificial intelligence.

Several main points of these objectives of this doctoral thesis can be justified and more explained as follows:

1) Regarding the type of electric machine, non-sinusoidal multiphase PMSMs are the controlled object in this doctoral thesis. As analyzed in section 1.1.2, besides the general properties of the multiphase concept, multiphase PMSMs are chosen for this doctoral thesis due to their high efficiency and high volume densities. These advantages meet the first two requirements of EVs. In addition, non-sinusoidal back-EMFs are considered due to several advantages such as easy-to-manufacture, low production costs, high torque densities, and the ability for the electromagnetic pole changing (see section 1.1.1.E). The control of non-sinusoidal machines is more challenging and interesting due to high-order back-EMF harmonics. 2) Regarding the control technique in this doctoral thesis, proposed control strategies in all operating modes will be based on FOC technique and PI controllers as healthy mode. Besides the recent techniques MPC and DTC, the classical FOC technique with PI controllers is still interesting, especially for industry due to its quality, simplicity, and robustness (less dependent on the machine model). In addition, this classical FOC has been widely used in industry. Therefore, the proposed control strategies in this thesis are derived from the mathematical model using the multi-reference frame theory [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF]. Currents in rotating reference frames are expected to either be constant or slowly vary, especially currents generating most of the torque.

PI

Current controllers & Multiphase drives in EVs

Economical mode

High-performance mode (This thesis)

Accelerator pedal (Torque control) Chapter 1.

Multiphase Drives: Opportunities and State of the Art 253) Regarding constraints on current and voltage, these constraints guarantee high functional reliability and flux-weakening operations of the drives. From the state of the art (see section 1.3), few studies considering fault-tolerant control and nonsinusoidal back-EMFs under constraints on current and voltage have been conducted, except [START_REF] Guzman | Novel control techniques in multiphase drives: Direct control methods (DTC and MPC) under limit situations[END_REF] with MPC and DTC techniques for 5-phase machines. 4) Regarding enhancements of control and torque performances under impacts of uncertainties and imperfections of electric drives by artificial intelligence, ADALINE (ADAptive LInear NEuron), a simple type of artificial neural networks, is chosen. One of the uncertainties can be "dead-time" voltages (the inverter nonlinearity) while the presence of multi-harmonics in back-EMFs causes an imperfection in machine design. ADALINEs are chosen in this doctoral thesis thanks to their self-learning ability and easy implementation. Moreover, the knowledge of harmonic components in signals of the drives allows a fast response of control (fast convergence) and avoids the calculation burden, increasing their applicability to industrial electric drives. Indeed, this knowledge is easily obtained and generally proportional to the number of phases. ADALINEs just need to find the right amplitudes of these harmonics. These advantages make ADALINEs more favorable compared to other approaches such as FL, PIR, and so on. FL control requires experience of experts while PIR control requires multiple parameter adjustments and the knowledge of frequencies. By using ADALINEs, quality of torque and current control will be significantly improved in either healthy mode or faulty mode.

Conclusions

According to the requirements presented in the Introduction of this doctoral thesis, section 1.1

of Chapter 1 has presented several distinct advantages and some possible drawbacks of multiphase drives. Thanks to these important advantages of multiphase drives, EVs driven by multiphase drives, especially PMSM drives, have high potential to be commercialized in near future. However, there have been some drawbacks of multiphase drives regarding the calculation burden, high electronics drive costs for a high number of VSI legs, the complexity of modeling and control, and social inertia. These drawbacks can be overcome by advances in microprocessors, mass productions, enhancements in power electronics technologies as well as long-term investments of industrial companies and so on. In addition, the modeling of a general multiphase PMSM has been briefly described in section 1.2. This modeling has facilitated the analyses of recent studies on control strategies of multiphase drives in healthy and faulty modes in section 1.3. High-performance driving mode of EVs can be feasible if the objectives of this doctoral thesis presented in section 1.4 are achieved. Then, justifications and explanations of the objectives have been delivered to clarify the purposes of this work.

Chapter 2. Modeling and Control of Multiphase Drives

This chapter is dedicated to presenting the modeling and control of multiphase drives under heathy and faulty modes. The healthy-mode issues are discussed in section 2.1 in which the modeling and a control scheme under constraints on current and voltage are introduced. In this scheme, FOC technique and PI controllers are used. Current references are calculated by an optimal strategy that maximizes torques under the considered constraints. Section 2.2 describes critical problems of post-fault operations without reconfigurations through analytical expressions and experimental results. Especially, section 2.3 proposes three main fault-tolerant control methods by exploiting mathematical models of multiphase drives. The control scheme is almost the same as healthy mode with on FOC technique and PI controllers. The critical problems in post-fault operations including ripple torques and electrical limit excesses are solved. Therefore, torque-speed characteristics with flux-weakening operations of post-fault multiphase drives will be verified by numerical and experimental results. Conclusions of the chapter are presented in section 2.4.

Modeling and control of a multiphase drive in healthy mode

This section presents the modeling and control of a multiphase machine in healthy mode. To effectively illustrate characteristics of the multiphase drive, a seven-phase PMSM drive is considered as a case study. Compared to a five-phase machine, the seven-phase machine possesses more DoF for control.

The case study: a seven-phase PMSM

The schematic diagram of a seven-phase PMSM is described in Fig. 2.1 with rotating speed Ω, electrical position θ, and electromagnetic torque Tem. To model the machine, several assumptions are considered as follows:

1) Seven phases of the machine are equally shifted with the spatial angular displacement δ equal to 2π/7.

A C B G θ δ N S D E F Ω T em Chapter 2.
Modeling and Control of Multiphase Drives 282) The machine has a non-salient rotor; hence, the inductances in fictitious machines can be easily calculated from Clarke transformation matrix as in (1.5).

3) The saturation of the magnetic circuits is not considered in the calculations of the back-EMFs and the fluxes.

To simplify the control scheme, machine parameters are converted from natural frame into the decoupled α-β and d-q frames for the seven-phase machine as follows:

         TT d1 q1 d9 q9 d3 q3 z α1 β1 α2 β2 α3 β3 z T A B C D E F G x x x x x x x x x x x x x x x x x x x x x = = Park Park Clarke T TT (2.1) with   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
                                   = Clarke T ( ) n 18 1 2 1 2 1 2 1 2 1 2 1 2 1 2                          cos( ) sin( ) 0 0 0 0 0 sin( ) cos( ) 0 0 0 0 0 0 0 cos(9 ) sin(9 ) 0 0 0 0 0 sin(9 ) cos(9 ) 0 0 0 0 0 0 0 cos(3 ) sin(3 ) 0 0 0 0 0 sin(3 ) cos(3 ) 0 0 0 0 0 0 0 1         -    = -    -    Park T
where x is an arbitrary parameter of the machine such as current, back-EMF and voltage; [TClarke] is the 7 by 7 Clarke transformation matrix; δ =2π/7 is the spatial angular displacement; θ is the electrical position; [TPark] is the 7 by 7 Park transformation matrix in which the 1 st (θ), 9 th (9θ), and 3 rd (3θ) harmonic components are considered. The selection of these harmonic components in the Park matrix depends on main harmonics existing in the back-EMFs of the machine. In this study, the 5 th harmonic is much smaller than the 9 th one; hence, 9θ is used instead of 5θ.

In the new decoupled reference frames, the real machine is decomposed into 4 fictitious machines including 3 two-phase fictitious machines (FM1, FM2, FM3) and 1 zero-sequence machine (ZM) as described in Fig. 2.2. Each fictitious machine with a corresponding reference frame is associated with a given group of harmonics as shown in Table 2.1. Specially, the d-q reference frame for the second fictitious machine is denoted by (d9-q9) because the ninth harmonic is considered in the back-EMFs.

Chapter 2. The electromagnetic torque Tem of the real machine is equal to the sum of torques generated by all its fictitious machines as expressed in (2.2).

Modeling and Control of Multiphase Drives

em 1 2 3 z T T T T T = + + + (2.2)
Energetic Macroscopic Representation for modeling and control

2.1.2.A. The representation of the electric drive model

Energetic Macroscopic Representation (EMR) is a functional description originally developed in the control team of L2EP laboratory in 2000s to analyze an energetic system [158,[START_REF] Bouscayrol | Graphic formalisms for the control of multi-physical energetic systems[END_REF]. In other words, EMR is a graphical tool using block diagrams to easily organize the model and facilitate the control of an energetic system. In addition, the number of DoF are pointed out for the control. This point is particularly important for multiphase machines whose number of DoF is high in comparison with three-phase machines.

In fact, there have been existing other graphical approaches based on the facilitation of visual human sense. The idea is to choose a graphical approach which is adapted at first for the deduction of control and then for energetic system. Two of other tools for system model representations are Bond Graph (1959) and Causal Ordering Graph (COG 1996). Bond Graph is based on the derivative causality while the integral causality is used in COG. As COG, EMR is based on the integral causality. Consequently, the control scheme is deduced from the model representation by the principle of inversion in which the control is considered as a functional inversion of the model. As Bond Graph, EMR can point out power flows. Therefore, it is thus well adapted to the graphical representation of an energetic system as electrical drives. From the above reasons, EMR is chosen to represent multiphase drives in this doctoral thesis. Thanks to EMR, the modeling and control of a complex system such as a multiphase drive become more synthetic.

More descriptions about EMR elements are presented in Appendix A of this doctoral thesis.

To understand the representation of a system model by EMR, the case study with the considered seven-phase drive is described in Fig. 2.3. Two adjacent elements of EMR are connected by two arrows with two opposite directions, representing the action and reaction between them. This connection is based on causality principle (integral). EMF helps to highlight controlled variables of the system such as a duty cycle (conversion factor) for an inverter. In addition, EMF can determine either the power flow or the control path. Four main elements of EMR can be generally described as follows:

1) Energy sources (oval green pictograms) represent the environment of the studied system, delivering or receiving energy. In Fig. 2.3, the electric source (ES) is a DCbus voltage VDC to feed an inverter while the mechanical source (MS) represents the mechanical load of the drive system. 2) Accumulation elements (rectangle orange pictograms with diagonal lines) represent the energy storage because the transfer functions of these elements include integral operators. In Fig. 2.3, the accumulation elements present in the fictitious machines and in the rotor shaft. 3) Conversion elements (square orange pictograms for mono-physical conversions or circle orange pictograms for multi-physical conversions) convert energy without any energy accumulations. In Fig. 2.3, the mono-physical conversion element is the inverter while the multi-physical conversion elements are the electromechanical elements in the fictitious machines. 4) Coupling elements (overlapped orange pictograms) are for energy distributions. In Fig.

2.3, the first coupling element is the Clarke transformation (or combined with Park transformation). The second coupling element is to calculate the electromagnetic torque from four fictitious machines.

Specifically, mathematical descriptions for all elements in Fig. 2.3 are expressed in (2.3)-(2.7).

Inverter (conversion element):

DC T DC v m V I m i =   =  (2.3) with   T A B C D E F G v v v v v v v v =   T A B C D E F G i i i i i i i i = MS ES DC bus Inverter Coupling 1 Coupling 2 Fictitious machines Shaft Mechanical Load FM2 FM3 ZM FM1 (2.3) (2.4) (2.5) (2.6) (2.7)
Chapter 2.

Modeling and Control of Multiphase Drives

31   T A B C D E F G m m m m m m m m =
where v and i are the 7-dimensional phase voltage and current vectors; m is the conversion factor vector; VDC and IDC are the voltage and current of the DC bus, respectively.

Coupling 1 (coupling element):

       11 dq dq vv ii -- =    =   Park Clarke Clarke Park TT TT (2.4) with   T d1 q1 d9 q9 d3 q3 z dq v v v v v v v v =   T d1 q1 dq1 v v v = ;   T d9 q9 dq2 v v v = ;   T d3 q3 dq3 v v v =   T d1 q1 d9 q9 d3 q3 z dq i i i i i i i i =   T d1 q1 dq1 i i i = ;   T d9 q9 dq2 i i i = ;   T d3 q3 dq3 i i i =
where vdq and idq are the 7-dimensional voltage and current vectors in d-q frames, respectively; (vdq1, v dq2, v dq3) and (idq1, idq2, idq3) are the 2-dimensional voltage and current vectors of fictitious machines FM1, FM2 and FM3 in d-q frames, respectively; vz and iz are the voltage and current of zero-sequence ZM, respectively.

Fictitious machines (accumulation and conversion elements):

( ) where edq is the back-EMF vector in d-q frames; Lj and Lz are the inductances of the fictitious machines in d-q frames calculated as in (1.5); Rs is the stator winding resistance of one phase; s is the Laplace operator; Tj and Tz are the torques of the fictitious machines; Ω is the rotating speed of the machine.

( )   0 with 1,3 1
1 dqj dqj dqj js T dqj dqj j z z z zs zz z j i v e L s R ie T Ω i v e L s R ie T Ω   =-  +   =     =-  +   =   (2.
Coupling 2 (coupling element):

23 em 1 z T T T T T = + + + (2.6) Chapter 2.

Modeling and Control of Multiphase Drives

32

where the electromagnetic torque of the machine Tem is equal to the sum of torques of the fictitious machines (T1, T2, T3, Tz).

Shaft (accumulation element):

( ) ( )

1 1 m em load mm f Ω T T J f s =- + (2.7)
where Tload is the load torque applied to the machine; fm is the friction coefficient of the rotorload bearings; Jm is the moment of inertia of the electric drive and mechanical load.

2.1.2.B. The representation of the electric drive control

The control scheme will be designed according to the model representation with the principle of inversion. Indeed, the control elements are determined based on the inversion of their corresponding models with three inversion rules as follows:

1) Conversion elements are directly inverted from their corresponding models.

2) Accumulation elements require closed-loop controls with controllers. The pictograms of these controllers are light blue parallelograms with oblique lines. 3) Inversions of coupling elements may require criterion inputs which lead to a strategy of the energy management (in blue parallelograms).

The control structure of the studied seven-phase drive with speed and current control is described in Fig. 2.4 where all control elements are presented in light blue parallelograms. More descriptions about EMR elements are presented in Appendix A of this doctoral thesis.

MS ES

DC bus

Inverter Coupling 1 Coupling 2 Fictitious machines Shaft Mechanical Load

Strategy ZM FM1 FM2 FM3 (2.3) (2.4) (2.5) (2.6) (2.7)
Chapter 2.

Modeling and Control of Multiphase Drives

33

The close-loop control of speed and currents can be implemented with PI or other advanced controllers. From the speed reference Ωref, torque reference Tem_ref is calculated by the speed controller.

When no electrical constraints on voltage and current are considered, current references can be generally calculated as in (2.8) by using the vectorial approach, also called Maximum Torque Per Ampere (MTPA), as introduced in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. Then, current references for each fictitious machine can be determined as follows:

2 em_ref ref T Ω ie e = (2.8)    dq_ref ref ii = Park Clarke TT (2.9) with   _ T d1_ref q1_ref d9_ref q9_ref d3_ref q3_ref z_ref dq ref i i i i i i i i =   T d1_ref q1_ref dq1_ref i i i = ;   T d9_ref q9_ref dq2_ref i i i = ;   T d3_ref q3_ref dq3_ref i i i =
where e is the norm of the back-EMF vector e in natural frame.

In other words, the total torque reference Tem_ref is distributed to fictitious machines (T1_ref, T2_ref, T3_ref, Tz_ref) according to the MTPA strategy with repartition coefficient vector kR as follows:

    _ T 1_ref 2_ref 3_ref z_ref R em ref T r1 r2 r3 rz R T T T T k T k k k k k  =   =   (2.10) 7 2 1 r1 2 2 2 2 1 9 3 E k E E E + E = ++ (2.11) 9 7 2 r2 2222 1 9 3 E k EEEE = +++ (2.12) 3 7 2 r3 2222 1 9 3 E k EEEE = +++ (2.13) 7 7 2 rz 2 2 2 2 1 9 3 E k E E E + E = ++ (2.14)
where Eh is the amplitude of h th harmonic of the back-EMFs; the 1 st harmonic of the back-EMFs is considered in the first fictitious machine; the 9 th harmonic is considered in the second machine instead of the 5 th harmonic; the 3 rd harmonic is in the third machine while the 7 th harmonic is in the zero-sequence machine.

Then, current references for all fictitious machines in d-q frames are given by:

Chapter 2.
Modeling and Control of Multiphase Drives

34   0 2 with 1, 3 j_ref dqj_ref dqj dqj z_ref z_ref z j T Ω ie e T Ω i e   =     =   (2.15)
where j e is the norm of the back-EMF vector ej of the fictitious machine j.

In variable torque/speed applications such as automotive, the determination of the electromagnetic torque reference depends on constraints (limits of currents and voltages, for example) and optimizations (high efficiency, high torque quality, and minimum copper losses, for example). Therefore, in this study, the torque generation with the current control loop at each rotating speed is considered. In a multiphase machine, as the torque is the sum of torques generated by fictitious machines, a strategy of the torque repartition that considers the above constraints and optimizations must be defined by kR, as described in Fig. 2.5.

Control scheme with an optimal control strategy under constraints on RMS current and peak voltage in healthy mode

Developed from the control scheme in Fig. 2.5, an optimal strategy in which the electromagnetic torque is maximized under constraints on current and voltage is considered to determine current references. In fact, the constraint on peak phase currents has been considered in [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF] for five-phase sinusoidal machines. The peak current limit can be defined from rated parameters of VSI components. However, in this doctoral thesis, the concern about the RMS current limit is preferred for a long-term operation of the drive for good efficiency without any Chapter 2.

Modeling and Control of Multiphase Drives

35 overheated machine windings. The thermal issue is more important when the drive is covered and placed in a limited space of EVs. By using this optimal control strategy, flux-weakening operations are also guaranteed. The details of the optimal strategy are described as follows:

( ) ( )     () with em j lim j RMS_lim Maximize T v t V j = A, B, C, D, E, F, G RMS i t I        (2.16)
where vj(t) and ij(t) are the instantaneous voltage and current of phase j; Vlim is the limit of peak phase voltages; IRMS_lim is the limit of RMS phase currents.

Determinations of Vlim and IRMS_lim can be explained as follows:

1) Conventionally, in a wye-connected stator winding machine, the limit of the peak phase voltages Vlim is roughly equal to a half of the DC-bus voltage in the PWM technique. In an open-end winding configuration (H-bridge), Vlim can be equal to the DC-bus voltage. The voltage limit for the offline optimization, denoted by Vlim_opt, can be chosen to be smaller than the value of Vlim due to the imperfection of the experimental systems. 2) RMS current limit IRMS_lim is associated with the maximum thermal limit of stator windings and permanent magnets. Thus, IRMS_lim can be defined from the rated RMS current. If the RMS current limit is respected in all phases, the winding insulation will be guaranteed for a long-term operation of the drive.

The control scheme with the optimal strategy in (2.16) to maximize torques under constraints on RMS current and peak voltage for healthy mode is described in Fig. 2.6. The details of the optimal strategy are explained as follows:

1) A current calculation method (MTPA, for example) providing the way to define current references is combined with an offline optimization.

2) The offline optimization with fmincon function is used to determine proper current references that generate maximal torques under constraints on current and voltage as described in (2.16). The fmincon function is a nonlinear programming solver in MATLAB to find the minimum or maximum value of a constrained nonlinear multivariable function [160]. 3) After calculations in the offline optimization, optimal d-q current references idq_opt are stored in a look-up table with their corresponding rotating speed Ω. These stored current references will be selected for current control according to the rotating speed of the drive.

Chapter 2.

Modeling and Control of Multiphase Drives

Control performances of a seven-phase PMSM drive in healthy mode

2.1.4.A. Assumptions and experimental setup descriptions

To control in healthy mode, there are several assumptions for the considered seven-phase PMSM drive as follows:

1) The machine windings are wye-connected; hence, the current control of the zerosequence machine is unnecessary because its current is always zero. 2) The control scheme in Fig. 2.6 is used with the aim of maximizing the electromagnetic torque under constraints on RMS current and peak voltage. PI controllers are used for the current control loop. MTPA strategy combined with the offline optimization is applied to calculate current references.

3) The saturation of magnetic circuits is not considered in calculations of back-EMFs and fluxes.

The control scheme in Fig. 2.6 is verified by an experimental electric drive as shown in Fig. The experimental seven-phase PMSM is introduced in [START_REF] Locment | Conception et modélisation d'une machine synchrone à 7 phases à aimants permanents et flux axial : commande vectorielle en modes normal et dégradé[END_REF][START_REF] Locment | Design and study of a multiphase axial-flux machine[END_REF]. This machine has an axial flux with double rotors. Different configurations of the double rotors result in several back-EMF waveforms. For example, when the two rotors have different numbers of poles and spatially shifted an angle of 7 degrees, the back-EMF waveform and its harmonic spectrum are presented in Fig. 2.8. Besides the 1 st harmonic of the back-EMFs, the 9 th and 3 rd harmonics account for the highest proportions (12.5 and 32.3% of the 1 st harmonic, respectively). Each of these harmonics present in one fictitious machine. Several minor harmonics in fictitious machines can be described as follows:

1) In FM1, besides the 1 st harmonic, the 13 th harmonic is equal to 5.02% of the 1 st harmonic. 2) In FM2, besides the 9 th harmonic, the 19 th harmonic is equal to 1.98% of the 1 st harmonic.

2.1.4.B. Optimal calculations under constraints on RMS current and peak voltage

In the control scheme in Fig. 2.6, the offline optimization based on (2.16) considers only the main harmonics of the back-EMFs including 1 st , 3 rd , and 9 th . Therefore, the calculated (optimal) torque in healthy mode is constant when the 1 st , 3 rd , and 9 th harmonics of current are imposed.

The constraints on RMS current and peak voltage can be specified based on parameters of the considered drive (see Table 2.2) as follows:

1) The rated RMS phase current is 5.1 A; hence, the RMS current limit IRMS_lim is selected to be 5.1 A.

2) The VSI is fed by a DC-bus voltage VDC of 200 V; hence, the limit of phase voltage references Vlim is 100 V, equal to a half of VDC. However, high-order harmonics of the considered experimental back-EMFs (11 th , 13 th , and 19 th , for example) may cause additional values to phase voltage references. Therefore, to respect the peak voltage limit Vlim in experiments, the peak voltage limit for the offline optimization Vlim_opt is selected to be 75 V based on simulation assessments.

The optimal torque-speed characteristic is shown in Fig. 2.9a with a speed range from 0 to 73 rad/s and the base speed of 38 rad/s. The torque before the base speed is 33.3 Nm and reduces after the base speed to respect the considered constraints.

In healthy mode, phase currents as well as phase voltages are identical in all phases. Therefore, the RMS currents of all phases at a speed are equal to the highest RMS current among all phases, denoted by IRMS. Thanks to the offline optimization, current references are determined so that IRMS is equal to IRMS_lim (5.1 A) at every speed as shown in Fig. 2.9b. In addition, the peak voltages of all phases at a speed are equal to the highest peak voltage among all phases, denoted

2.1.4.C. Verification with experimental results for healthy mode

2.1.4.C.1. Current control performance:

To obtain the torque-speed characteristic as the optimal result in Fig. 2.9a, six d-q currents obtained from seven measured phase currents need to properly track their constant references at every speed. These current references are calculated from the offline optimization in Fig. However, in Figs. 2.11-2.12, measured currents in both (d9-q9) and (d3-q3) frames, generating minor torques, are time-variant with a main frequency of 14θ even though their references are constant. These current harmonics in (d9-q9) and (d3-q3) frames are generated by unwanted harmonics (11 th and 19 th , for example) of the experimental back-EMF and by the inverter nonlinearity. This issue will be more explained in sections 3.2.1 and 3.2.2 of Chapter 3 of this doctoral thesis.

2.1.4.C.2. Torque performance:

In healthy mode, the total torque is mainly generated by three dominant harmonic components of back-EMFs (1 st , 3 rd , and 9 th ) and corresponding current harmonics. If the back-EMF values are ideally constant in d-q frames, the constant torque will be generated by constant d-q currents. It means that the back-EMFs should contain only one harmonic per reference frame for the best performance [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF]. However, the considered experimental back-EMFs contain more than one harmonic per frame (see Fig. 2.8), causing torque ripples in healthy mode. In this doctoral thesis, a torque ripple ∆T is given by:

( ) ( ) 100% em em ave max T min T ΔT T - = (2.17)
where max(Tem) and min(Tem) are maximum and minimum values of the instantaneous electromagnetic torque Tem, respectively; Tave is the average torque.

In addition, in this doctoral thesis, experimental torques are estimated values calculated from measured currents and estimated back-EMFs, denoted by Tem_exp. Meanwhile, optimal torques are calculated values obtained from the offline optimization as shown in Fig. 2.9a of section 2.1.4.B, denoted by Tem_opt.

The total torque and the fictitious machine torques at 20 rad/s in healthy mode are shown in Fig. 2.13a. The optimal torque (Tem_opt) is constant at 33.3 Nm while the experimental total torque (Tem_exp) has an average value of 33.5 Nm and a ripple of 12% at a main frequency of Chapter 2.

Modeling and Control of Multiphase Drives 41 14θ. These ripples are caused by the interactions between current harmonics and unwanted back-EMF harmonics. For example, the 1 st harmonic of currents interacts with the 13 th harmonic of back-EMFs, and the 3 rd harmonic of currents interacts with the 11 th harmonic of back-EMFs. A ripple-free torque can be possible by applying the MTPA strategy as described in (2.8). However, all d-q current references for control are no longer constant, possibly making PI controllers less effective at high speed. A viable solution with an ADALINE will be presented in section 3.2.3 of Chapter 3 of this doctoral thesis. In Fig. 2.13a, the torques in first, second and third fictitious machines (T1_exp, T2_exp, T3_exp) are proportional to their amplitudes in the back-EMF harmonic spectrum. Therefore, FM1 generates the highest torque (T1_exp) then FM3 (T3_exp).

2.1.4.C.3. Constraints on RMS current and peak voltage:

In Fig. 2.14, all measured phase currents always respect their constraint even in the fluxweakening region (38 to 73 rad/s). Indeed, the RMS current of all phases (IRMS) that is directly calculated from the measured phase currents in real time is stably controlled at 5.1 A, equal to IRMS_lim. It means that the local thermal limit of each phase winding can be guaranteed in a longterm speed-variable operation. Fig. Similarly, Fig. 2.15 shows that experimental voltage references, known as output signals of current controllers, respect their limit (Vlim=100 V) at high speed. It is noted that the voltage limit (Vlim_opt=75 V) used for the offline optimization is suitable. Fig. 2.15 shows a change in the voltage reference waveform before and after the base speed (20 and 55 rad/s).

2.1.4.C.4. Comparative summary:

The general comparisons between optimal and experimental results at 20 rad/s are described in Table 2.3. It is noted that the experimental results are in good accordance with the optimal values in terms of the RMS current and total copper loss. The optimal results are obtained from numerical calculations of the offline optimization (see section 2.1.4.B). In addition, the experimental average torque is like the optimal average torque. However, the experimental instantaneous torque has a ripple of 12% while the optimal instantaneous torque is timeconstant, as previously discussed in the torque performance section. 2.3, the experimental peak voltage reference is higher than its optimal value due to extra voltages generated by high-order harmonics in the experimental back-EMFs. This point can verify the necessity of using the peak voltage limit Vlim_opt=75 V for the offline optimization instead of Vlim=100 V.

Control of a multiphase drive in an OC fault without reconfigurations An OC fault in a seven-phase PMSM drive

In general, an electric drive under OC faults is modeled by considering more constraints on the stator windings of its electric machine [START_REF] Meinguet | Fault-tolerant permanent-magnet synchronous machine drives: Fault detection and isolation, control reconfiguration and design considerations[END_REF]. For example, Fig. 2.16 describes three situations causing an OC fault in phase A of a seven-phase PMSM drive. The first two situations are when either the line cable or the phase-A winding is opened. The last situation is when two switches (𝑠 1 and 𝑠̅ 1 ) in the inverter leg of phase A are simultaneously opened 1 .

Fig. 2.16. An OC fault happens in phase A of a seven-phase PMSM drive.

When the OC fault happens in phase A, the current of phase A must be zero. The fault reduces the number of DoF of the drive for control. The number of DoF means the number of independent variables that can be used for control. In this work, these variables are phase currents or currents in decoupled reference frames. The decrease in the number of DoF can be explained either in natural frame or in α-β frames (or d-q frames) as follows:

1) In natural frame, the number of DoF is 7 for a seven-phase machine when there are no constraints on stator windings. It means that 7 current references of the seven phases can be independently imposed. If the stator windings are wye-connected, the sum of the seven phase currents must be zero. Therefore, the number of DoF is reduced to 6 in this case. When phase A is open-circuited, iA must be zero, making the number of DoF become 5. 2) In α-β frames, the number of DoF can be explained according to (2.1) for currents.

Without any constraints on stator windings, the number of DoF is 7. It means that 7 currents in α-β frames can be independently imposed. When the stator windings are wye-connected, the zero-sequence current ih must be zero. Therefore, only 6 α-β currents can be independently imposed, reducing the number of DoF to 6. If an OC fault happens in phase A, iA must be zero; hence, after applying the inversion property of Clarke matrix in (1.6), the current transformation in (2.1) becomes:

1 SC faults of switches normally lead to OC faults with the inner protection of the switches [START_REF] Wintrich | Application Manual Power Semiconductors[END_REF] as discussed in section 1.3.2.A of Chapter 1.

D B A C i A i D i B i C V DC T em i F i E i G E F G I DC OC switch OC switch OC line OC phase 7-phase PMSM VSI Chapter 2.
Modeling and Control of Multiphase Drives Therefore, the currents in α-β frames must respect (2.19). It means that another constraint on α-β currents has been imposed on the drive. As a result, the number of DoF decreases from 6 to 5 as previously explained in natural frame.

45 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
B C D E F G i i i i i i                                 =      ( ) ( ) ( ) ( ) ( ) ( ) ( )
In this faulty condition, if current references for the post-fault operation are kept as in healthy mode, the current control will be no longer guaranteed due to the coupling between currents in fictitious machines.

Control performances in an OC fault without reconfigurations

It is assumed that an OC fault happens in phase A as discussed in the previous subsection. To see impacts of the OC fault, the experimental drive in section 2.1.4.A is used. The OC fault of phase A is safely created by sending an interrupting signal from a control interface to the inverter of the drive. This signal helps open simultaneously two IGBTs of the VSI leg of phase A, disconnecting phase A from the power source.

2.2.2.A. Current control performance

As analyzed in section 2.2.1, the current control cannot be guaranteed due to the coupling between currents in d-q frames. Current responses in three d-q frames cannot properly track their references at 20 rad/s as presented in Fig. 2.17. Therefore, fault detections and reconfigurations are necessary. Modeling and Control of Multiphase Drives 46

2.2.2.B. Torque performance

The distortion of current control reduces the torque quality of the drive. As described in Fig. 2.18, although the average torques are similar, the experimental OC torque (Tem_est_OC_No) without reconfigurations has a ripple of 50%, much higher than that of the healthy torque (Tem_est_HM) with only 12%.

2.2.2.C. Constraints on RMS current

The post-fault current waveforms of the remaining healthy phases are deteriorated and no longer identical as shown in Fig. 2.19, leading to unequal RMS currents in the remaining phases as shown in Table 2.4. The highest RMS current appears in phase B (IRMS_B), increasing about 1.55 times from 5.1 A (Fig. 2.19a) to 7.9 A (Fig. 2.19b). In other words, the RMS current limit IRMS_lim is not respected in the post-fault operation when new current references are not imposed.

T em_exp_HM T em_exp_OC_No

Chapter 2.

Modeling and Control of Multiphase Drives 47

2.2.2.D. Comparative summary

The comparative summary of healthy mode and a faulty mode with an OC fault in phase A without reconfigurations is described in Table 2.5. Per unit (pu) based on parameters of healthy mode is applied to evaluate the increase in corresponding parameters under the post-fault operation compared to healthy mode. It is noted that per unit is not used for the torque ripple value because the torque ripple in healthy mode, as a base, can be zero. As previously analyzed, the fault occurrence dramatically increases all important parameters such as the torque ripple (12 to 50%), the highest RMS current (1.55 pu), the highest peak voltage reference (1.74 pu), and the total copper loss (1.27 pu). Therefore, a reconfiguration of the control for the post-fault drive, called fault-tolerant control, is necessary.

Fault-tolerant control for a multiphase drive Introduction to proposed fault-tolerant control methods

From the analyses in section 2.2, new current references for post-fault operations need to be defined to avoid the oversizing of the drive. Before proposing fault-tolerant control methods, several assumptions of the considered seven-phase machine are described as follows:

1) Phase windings of the machine are symmetrically distributed in the stator. Constraints on stator winding configurations (such as a wye connection in healthy mode) are removed to have general analyses of fault-tolerant control. 2) In the offline optimization for fault-tolerant control, only the 1 st and 3 rd harmonics of the back-EMFs are considered to facilitate the offline optimization with theoretical smooth torques. Indeed, the number of DoF for control in the post-fault condition is reduced from 7 to 6 (no constraint on stator winding configurations), or from 6 to 5 (wye-connected stator windings) (see section 2.2.1). Therefore, only 4 d-q currents in the first and third fictitious machines that generate most of the torque are usually imposed as constants. Meanwhile, d-q currents in the second fictitious machines are consequently time-variant. As a result, the harmonics of back-EMFs in the second fictitious machine (9 th ) should not be considered in the offline optimization to theoretically obtain smooth torques.

3) The saturation of magnetic circuits is not considered in calculations of back-EMFs and fluxes.

Owing to the equal spatial displacement of phases, phase A is assumed to be open-circuited without loss of generality. Therefore, the current of phase A is always equal to zero (iA=0).

Because there are no constraints on stator windings, the number of DoF becomes 6, compared to 7 in healthy mode. In other words, only 6 currents can be independently imposed either in natural frame or decoupled α-β and d-q frames when phase A is open-circuited. Similarly, when two phases are open-circuited, there will be 5 currents that can be independently imposed. Faulttolerant control strategies for two-phase OC faults are presented in Appendix B. This chapter will focus on an OC fault in one phase (phase A).

Under the considered fault, the transformation of phase currents from natural frame into the decoupled frames is similar to (2.1), (2.4), and (2.9) as follows:

     0 d1 α1 q1 1 B α2 d9 C β2 q9 D E α3 d3 F β3 q3 G z z i i i i i i i i i i i i i i i i i i i i             = =              Park Park Clarke T T T (2.20)
In this doctoral thesis, new current references in single-phase OC faults are derived by exploiting the transformation equation in (2.20). The general scheme of three methods (I), (II),

Chapter 2.
Modeling and Control of Multiphase Drives 49 and (III) is described in Fig. 2.20 where α-β and d-q frames are equivalent. The principles of these methods are explained as follows: 1) Method (I): New current references are determined from decoupled α-β or d-q frames to obtain smooth post-fault torques. As a result, currents in natural frame are derived from the determined currents in the decoupled frames. Method (I) is similar to the study in [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF] where the current references are calculated from the decoupled α-β frames. However, in this doctoral thesis, method (I) with different possible options is systematically described. 2) Method (II): New current references are determined by finding new transformation matrices with a 6-by-6 dimension to obtain smooth post-fault torques. Two current design options will be proposed to provide either control robustness or similar distributions of copper losses in the remaining healthy phases. Method (II) is developed based on the principle in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] where five-phase non-sinusoidal machines are considered. 3) Method (III): New current references are directly determined from natural frame to obtain the waveform uniformity of healthy phase currents, resulting in higher average torques compared to methods (I) and (II) under current limits. Another purpose of method (III) is to distribute equally copper losses to the remaining heathy phases. However, torque ripples with method (III) are inevitable. Method (III) is developed from a solution in [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF] where sinusoidal machines are considered with a various number of phases. However, the approach of method (III) in this doctoral thesis is different from the existing study with analytical expressions and more proposed solutions.

Especially, different from [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF][START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], in this doctoral thesis, new current references with methods (I), (II) and (III) are required to maximize electromagnetic torques and respect constraints on current and voltage as described in (2.16). Therefore, a control scheme with the optimal strategy under constraints on RMS current and peak voltage for faulty modes in Fig. 2.21 is applied. In fact, the scheme in Fig. 2.21 for faulty mode is developed from the control scheme in Fig. 2.6 for heathy mode by adding fault information (types and positions of faults), and by using method (I), (II) or (III) to calculate current references. Then, new current

  0 α1 1 B α2 C β2 D E α3 F β3 G z i i i i i i i i i i i i i              =                   Clarke T Method (I) Method (II) Method (III)

Natural frame α-β frames

Chapter 2.

Modeling and Control of Multiphase Drives 50 references stored in a look-up table can be selected according the fault information and rotating speed. However, the fault detection issue is not within the scope of this doctoral thesis.

Method (I): new current references determined from decoupled reference frames

2.3.2.A. Principle of method (I)

When phase A is open-circuited, six of seven currents in decoupled frames (iα1, iβ1, iα2, iβ2, iα3, iβ3, iz) or (id1, iq1, id9, iq9, id3, iq3, iz) can be independently determined, and the last current is consequently derived. Thanks to the relationship between phase currents and decoupled frames in (2.20), currents of the remaining healthy phases can be derived from currents in the decoupled frames.

In the considered non-sinusoidal seven-phase machine, it is assumed that the 1 st and 3 rd harmonics of the back-EMFs account for the highest proportions, presenting in the first and third fictitious machines, respectively. Other harmonics, 9 th for example, exist in small proportions. Therefore, in method (I), the determination of currents in decoupled frames should be based on the following principles:

1) The 1 st and 3 rd harmonic currents (iα1, iβ1) and (iα3, iβ3), creating most of the torque, are chosen to be sinusoidal in α-β frames. Equivalently, these corresponding currents in d-q frames (id1, iq1) and (id3, iq3) are constant, facilitating the control with conventional PI controllers at high speed. Therefore, 4 currents (iα1, iβ1, iα3, iβ3), equivalent to (id1, Chapter 2.

Modeling and Control of Multiphase Drives 51 iq1, id3, iq3), in the first and third machines have been determined. In other words, the first and third harmonic sinusoidal MMFs are guaranteed in each fictitious machine. 2) Among three currents (iα2, iβ2, iz), equivalent to (id9, iq9, iz), only 2 of the three currents can be independently imposed. The last current is consequently derived from previously determined currents.

From (2.20), the relationship between currents (iα1, iβ1, iα3, iβ3) and (id1, iq1, id3, iq3) is given by: According to the first principle, desired currents (id1, iq1, id3, iq3) are constant, enabling desired currents (iα1, iβ1, iα3, iβ3) to be sinusoidal. Indeed, the desired currents (iα1, iβ1, iα3, iβ3) and (id1, iq1, id3, iq3) are described in Fig. 2.22a and Fig. 2.22b, respectively. Consequently, desired sinusoidal currents (iα1, iβ1) and (iα3, iβ3) create two circles as presented in Fig. 2.23. These circular relationships are to describe the first and third harmonic sinusoidal MMFs, creating constant torques.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos sin 0 0 sin cos 0 0 0 0 cos 3 sin 3 0 0 sin 3 cos 3 d1 α1 q1 1 α3 d3 β3 q3 i i i i i i i i      - -         =           (2.
Finally, there are 3 remaining currents (iα2, iβ2, iz) or (id9, iq9, iz) to be defined. When the 9 th harmonic is considered in the second fictitious machine, the relationship between (iα2, iβ2, iz) and (id9, iq9, iz) can be expressed by:

( ) ( ) ( ) ( )
2 cos 9 sin 9 0 sin 9 cos 9 0 0 0 1

d9 α2 q9 z z i i i i i i       =-          (2.22) Chapter 2.

Modeling and Control of Multiphase Drives

52

According to the second principle, the number of currents that can be independently imposed is 6, leading to only 2 remaining currents to be imposed. Therefore, only two of three currents (iα2, iβ2, iz) can be independently imposed, and the last current will be consequently derived. Therefore, in method (I), three options are proposed to define currents (iα2, iβ2, iz) as described in Table 2.6. These proposed options can be specified as follows:

1) Method (I)-1: Currents in the second machine (iα2, iβ2) are imposed to be zero, aiming at reducing copper losses. Consequently, the zero-sequence current (iz) is different from zero. It means that this option is feasible in two cases [START_REF] Yanagihara | Fault-tolerant Control for 7-phase Non-sinusoidal Permanent Magnet Machines with One Opened Phase[END_REF]. The first case is when there is a neutral line connecting the neutral point of the stator windings and the power source. The second one is when the stator windings are configured in an openend topology. 2) Method (I)-2: β-axis current in the second machine (iβ2) and zero-sequence current (iz) are imposed to be zero, enabling to reduce copper losses [START_REF] Yanagihara | Fault-tolerant Control for 7-phase Non-sinusoidal Permanent Magnet Machines with One Opened Phase[END_REF]. As a result, α-axis current in the second machine (iα2) is different from zero. This option can be applied without adding the neutral line connection or changing the stator winding topology.

The study in [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF] applies the vectorial approach to obtain the current references (iα2, iβ2) with the aim of reducing copper losses. The current references in [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF] and method (I)-2 of this doctoral thesis are similar when phase A is open-circuited. When other phases are opened, study [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF] uses non-zero current references (iα2, iβ2). In addition, the study in [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF] does not consider any constraints on current and voltage as well as flux-weakening operations. 3) Method (I)-3: Only zero-sequence current (iz) is imposed to be zero. The currents (iα2, iβ2) in the second machine are determined according to a simple dual three phase system (iB+iD+iF=0 and iC+iE+iG=0) [START_REF] Vu | Torque optimisation of seven-phase BLDC machines in normal and degraded modes with constraints on current and voltage[END_REF][START_REF] Vu | Control strategies for non-sinusoidal multiphase PMSM drives in faulty modes under constraints on copper losses and peak phase voltage[END_REF]. Like method (I)-2, this option can be applied without adding the neutral line connection or changing the stator winding topology.

It is worth noting that an option with currents (iα2= iz=0) is an impossible solution due to the coupling in the post-fault condition developled from (2.18)-(2.19) as follows:

( )

21 7 2 0 α1 αz α3 2 ii ii    = + + +      (2.23)
where iα1 and iα3 are previously imposed as sinusoidal currents; hence, iα2 and iz cannot be zero at the same time.

Chapter 2.

Modeling and Control of Multiphase Drives

53

The details of methods (I)-1, (I)-2, and (I)-3 will be presented in the following subsections.

2.3.2.B. Method (I)-1

With (iα2=iβ2=0), the current transformation in (2.20) becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1
0 B C D E F G i i i i i i                                 =      ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5
i i i i i                                   (2.24)
where the spatial angular displacement δ is 2π/7. Therefore, calculations of zero-sequence current iz and phase currents (iB, iC, iD, iE, iF, iG) from four main currents (iα1, iβ1, iα3, iβ3) are given by:

  ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( )
 

i i i i i i i i ii i i i i i i i i i i i i i i i i i i i                     = - + = + + + + = + + + + = + + + + = + + + + = ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( )   2
1 α3 β3 α1 1 α3 β3 z z G i i i i i i i i i i                             + + + +     = + + + +  (2.25)
Therefore, all new current references for the OC fault in phase A have been determined. It is noted that current iz is time-variant, making the sum of all phase current references different from zero. Therefore, method (I)-1 can be applicable only when there is a neutral line connecting the neutral point of the wye-connected stator windings and the power source, or only when the stator windings are configured in an open-end topology. In addition, currents in the second fictitious machines (id9, iq9) for control are always zero according to (2.22). Therefore, method (I)-1 have 6 constant d-q currents for control.

2.3.2.C. Method (I)-2

With (iβ2=iz=0), the current transformation in (2.20) becomes:

Chapter 2.
Modeling and Control of Multiphase Drives

54 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0
0 B C D E F G i i i i i i                                 =      ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5
i i i i i                                   (2.26)
Therefore, calculations of current iα2 and phase currents (iB, iC, iD, iE, iF, iG) from four main currents (iα1, iβ1, iα3, iβ3) are described as follows: ii

  ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) 2
i i i i i i i i i i i i i i i i i i i i i i i i i                        = - + = + + + + = + + + + = + + + + = + + + + ( )   ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( )
 

i i i i i i i i i i i i                                = + + + +    = + + + +   (2.27)
In this option, the wye-connected winding topology considered in heathy mode can be used in post-fault operations without additional hardware or changing the stator winding topology. Current iα2 depends on currents in the first and third fictitious machines (iα1, iα3), resulting in time-variant currents (id9, iq9) according to (2.22). Therefore, method (I)-2 have 5 constant d-q currents for control.

2.3.2.D. Method (I)-3

With (iB+iD+iF = iC+iE+iG =0) and (iz=0), the current transformation in (2.20) becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1
0 B C D E F G i i i i i i                                 =      ( ) ( ) ( ) ( ) ( ) ( ) ( )
F C E i i i i i i i i i i i i                                    + + = + + =  (2.28)
Therefore, currents (iα2, iβ2) and phase currents (iB, iC, iD, iE, iF, iG) are calculated from four main currents (iα1, iβ1, iα3, iβ3) as follows:

Chapter 2.
Modeling and Control of Multiphase Drives

55   ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( )   ( ) ( ) (
α1 α3 α2 β α1 1 α3 β3 2 ii ii i i i i                       = - +   - + + - - + + + +      -       - + - + - + + + +      =    ++          (2.29) ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) 2
α1 1 α3 β3 α1 1 α3 β3 α1 1 α3 β3 α1 α2 β2 α2 β2 α 1 2 β2 α E 2 B C D i i i i i i i i i i i i ii i i i i i ii i i i i                        = + + + + + = + + + + + = + + + + + = + + ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( ) ( )   2
α3 β3 α1 1 α3 β3 F α1 1 α3 β3 β2 α2 β2 α2 β2 G ii i i i i i i i i i i i i ii i                              + + +    = + + + + +     = + + + + +  (2.30)
In this option, like method (I)-2, the wye-connected winding topology can be used in post-fault operations without additional hardware or changing the stator winding topology. Currents (iα2, iβ2) are functions of currents in the first and third fictitious machines, resulting in time-variant currents (id9, iq9) for control according to (2.22). Therefore, method (I)-3 have 5 constant d-q currents for control.

2.3.2.E. Comparative analyses of calculated results by the proposed options of method (I)

2.3.2.E.1. Preserving the torque as in healthy mode:

The post-fault torques, more precisely the average torques, are kept the same as in healthy mode when main current references (iα1, iβ1, iα3, iβ3) and (id1, iq1, id3, iq3) in the three options of method (I) are preserved as in Figs 2.22 and 2.23. Specifically, d-q currents are imposed with (id1=0 A, iq1=12.7 A, id3=0 A, iq3=4.1 A). Parameters of the experimental seven-phase drive described in Table 2.2 are used for calculations. If the first and third harmonics of the back-EMFs are considered, the torques with the three proposed options of method (I) are constant and equal to their average value of 33.3 Nm.

For clear comparative analyses of the three options of method (I), only currents (iα2, iβ2, iz), (id9, id9, iz), and phase currents (iB, iC, iD, iE, iF, iG) at 20 rad/s are presented in Figs 2.24-2.26. It is noted that current waveforms of the remaining healthy phases in each option of method (I) are no longer identical as in healthy mode. The zero-sequence current iz in method (I)-1 is timevariant with a high amplitude (21.8 A) compared to the phase currents. This current generates torque ripples if the experimental back-EMFs contain the harmonics in the zero-sequence machine (7 th , 21 st , and so on). If method (I)-1 is chosen, it means that a constraint on machine design (no harmonics associated with the zero-sequence machine in back-EMFs, for example) is imposed. Meanwhile, currents (id9, iq9) with methods (I)-2 and (I)-3 are time-variant, creating torque ripples if the back-EMFs contain the harmonics in the second machine (9 th , for example). Calculated RMS values in all phase currents and the zero-sequence current of the three options are described in Table 2.7. It is noted that the RMS currents in the remaining healthy phases in each option of method (I) are different and higher than their rated value (5.1 A). In the three options, the highest RMS currents are often in phases D and E. In addition, method (I)-1 leads to the RMS zero-sequence current of 13.4 A, possibly causing the oversizing of the drive. Finally, comparisons between the three options of method (I) and healthy mode for the same torque at 20 rad/s are described in Table 2.8. Highest RMS current among all phases IRMS, highest peak voltage among all phases Vpeak, and the total copper loss Ploss are considered. It is noted that when the torque is preserved, IRMS and Ploss in fault-tolerant operations can be doubled, dramatically exceeding its limit. Voltage Vpeak is significantly higher than healthy mode (1.26 pu). Method (I)-2 provides the best results among the three options over all considered criteria. Indeed, method (I)-2 results in the lowest IRMS (1.55 pu), the lowest Vpeak (1.03 pu) and the lowest Ploss (1.47 pu). Meanwhile, method (I)-3 leads to the highest values of the considered parameters, especially the highest IRMS (1.96 pu) and the highest Ploss (2.1 pu).

2.3.2.E.2. Optimal calculations under constraints on RMS current and peak voltage:

When the control scheme in Fig. 2.21 with the optimal control strategy in (2. [START_REF] Yang | Condition Monitoring for Device Reliability in Power Electronic Converters: A Review[END_REF]) is considered, the torques by the three options of method (I) will be reduced to respect constraints on current and voltage. Therefore, flux-weakening operations can be guaranteed. These constraints in faulty modes are the same as in healthy mode (in section 2.1.4) with IRMS_lim=5.1 A and Vlim_opt=75 V to respect the experimental voltage limit Vlim=100 V. As previously assumed, only the first and third harmonic components of the back-EMFs are considered in the offline optimization.

In Figs. 2.27-2.29, main currents (id1, iq1, id3, iq3) or (iα1, iβ1, iα3, iβ3) at 20 rad/s are re-determined by fmincon to respect the considered constraints. It is worth noting that currents (id1, iq1, id3, iq3) in the three options of method (I) are no longer the same. These currents are also different from the currents preserving the heathy torque. The other currents (id9, iq9, iz) or (iα2, iβ2, iz) are consequently calculated from the main currents. Consequently, the current waveforms of the remaining healthy phases in Figs. 2.27c-2.29c are changed compared to the previous case in Figs 2.24c-2.26c.

Calculated RMS values of all phase currents and the zero-sequence current under the constraints with method (I) at 20 rad/s are described in Table 2.9. In the three options, the highest RMS current appearing in phases D and E is equal to IRMS_lim (5.1A). In addition, method (I)-1 results in the RMS zero-sequence current of 7.2 A. Finally, comparisons between the post-fault operations using the three options of method (I) and healthy mode is described in Table 2.10. When IRMS_lim is respected, method (I)-2 generates the highest Tem at 21.7 Nm (0.65 pu). Meanwhile, the torques with method (I)-1 and method (I)-3 are lower at 17.7 Nm (0.53 pu) and 19.1 Nm (0.57 pu), respectively. Additionally, compared to healthy mode and methods (I)-1 and (I)-2, the calculated Vpeak with method (I)-3 is lowest (0.85 pu), leading to the highest base speed. Meanwhile, the calculated Vpeak in healthy mode is lowest, resulting in the lowest base speed. Total copper losses Ploss in fault-tolerant operations are lower than that of healthy mode (0.55 to 0.64 pu) due to the current constraint. Indeed, in each method, the highest RMS current IRMS appearing in phases D and E is equal to IRMS_lim=5.1 A (1 pu) while RMS currents of the other phases are less than 5.1 A, resulting in lower Ploss than healthy mode. Especially, method (I)-2 having similar RMS currents (see Table 2.9) results in the highest Ploss among the three options.

When the operating speed varies, the optimal torque-speed characteristics, all RMS phase currents and all peak phase voltages by the three options of method (I) in terms of speed are expressed in Fig. 2.30. It is noted that method (I)-2 has the highest fault-tolerant torque-speed characteristic. In addition, all options of method (I) respect the constraints on RMS current and peak voltage, allowing flux-weakening operations.

2.3.2.F. Verification with experimental results for method (I)

According to Fig. 2.30, Tables 2.10 and 2.11, method (I)-1, feasible in a system with a neutral line or in an open-end winding topology, generates the lowest torque and the shortest speed range among the three options of method (I). It is possible to physically change the existing wye-connected winding machine in the test bench (see Fig. 2.7) to a neutral line system an open-end winding system to facilitate method (I)-1. However, the high zero-sequence current with method (I)-1 (iz in Fig. 2.27) will interact with the 7 th and 21 st harmonics of back-EMFs (9.4 and 3.2%, respectively) in the experimental machine to generate high torque ripples with low frequencies. In addition, the torques and phase currents will contain switching frequency components [START_REF] Moraes | Homopolar Current's Copper Losses Analysis for Different Modulations in Open-End Winding Five-Phase Drives[END_REF]. From the above reasons, in the scope of this doctoral thesis, method (I)-1 is not verified with the experimental test bench.

Only methods (I)-2 and (I)-3 with the control scheme in Fig. 2.21 are validated in the experimental test bench (see section 2.1.4.A). Although the torques with methods (I)-2 and (I)-3 are sensitive to the 9 th harmonic (12.5%) of the experimental back-EMFs, these methods can be easily implemented in the test bench without any hardware modifications.

2.3.2.F.1. Current control performance:

To experimentally obtain the optimal torque-speed characteristics (see Fig. 2.30a), six d-q currents need to properly track their references (see Figs 2.28-2.29). At a speed, methods (I)-2 and (I)-3 have time-constant currents (id1, iq1, id3, iq3) and time-variant currents (id9, iq9).

Currents in (d1-q1), creating most of the torque, are well controlled in speed ranges as described in Fig. 2.31. As in healthy mode (see section 2.1.4), currents in (d3-q3) and (d9-q9) frames have current harmonics caused by the unwanted back-EMF harmonics and the inverter nonlinearity. These types of disturbances can be eliminated by ADALINEs that will be presented in Chapter 3. Therefore, the control of currents in (d3-q3) in speed ranges is described in Fig. 2.32 with current harmonics. Similarly, the time-variant currents (id9, iq9) are controlled at 20 rad/s with errors as shown in Fig. 2.33. However, these currents create a small proportion of the torque.

2.3.2.F.2. Torque performance:

An operation with three operating states at 20 rad/s as shown in Fig. 2.34 is investigated, including a healthy operation, an OC fault without reconfigurations, and a fault-tolerant operation with one of two methods (I)-2 and (I)-3. The first two operating states have been deeply investigated in sections 2.1.4 and 2.2. It is noted that the average torques in the third state with methods (I)-2 and methods (I)-3 are in good accordance with the optimal (calculated) values in Table 2.10. To respect the constraints on RMS current, the average torque is reduced from 33.5 Nm in healthy mode to 21.6 Nm (0.64 pu) with method (I)-2, and to 19 Nm (0.57 pu) with method (I)-3 in the post-fault operation. However, the experimental torques have ripples

Chapter 2.
Modeling and Control of Multiphase Drives 62 of 31% with method (I)-2, and 32% with method (I)-3. The main reason is that the considered experimental back-EMFs have a significant proportion of the 9 th harmonic (12.5%). This harmonic interacts with the time-variant currents (id9, iq9) in methods (I)-2 and (I)-3 as described in Fig. 2.33, creating ripple torques. These torque ripples can be eliminated by an ADALINE that will be presented in Chapter 3. However, these torque ripples are much lower than the post-fault operation without reconfigurations with 50%. When the operating speed varies, the experimental torque-speed characteristics with methods (I)-2 and (I)-3 in comparison with healthy mode (Tem_opt_HM and Tem_exp_HM) are described in Fig. 2.35. By using these methods, the experimental OC fault torques (Tem_exp_OC) are in good accordance with the optimal torques (Tem_opt_OC) (see Fig. 2.30). The experimental base and maximum speeds are the same as calculated results in Table 2.11.

2.3.2.F.3. Constraints on RMS current and peak voltage:

Table 2.12 shows that experimental RMS currents in all phases at 20 rad/s are in good accordance with optimal values. The experimental highest RMS currents appear in phases D and E in the both options with (4.5 and 5.1 A) for method (I)-2, and (4.5 and 5 A) for method (I)-3.

Method (I)-2 Heathy mode

Without reconfigurations Method (I)-3

T em_exp_HM T em_opt_HM T em_opt_OC T em_exp_OC

Method (I)-2

Method (I)-3

T em_exp_HM T em_opt_HM T em_opt_OC T em_exp_OC

Chapter 2.

Modeling and Control of Multiphase Drives 63 When the operating speed varies, the constraints on RMS current and voltage are still respected in the experimental results with methods (I)-2 and (I)-3, including the flux-weakening regions. Measured phase currents by the two methods in terms of speed are described in Fig. 2.36 and Fig. 2.37. The waveforms of measured phase currents at 20 and 55 rad/s are also plotted. The waveforms of the remaining healthy phase currents are changed at high speed, but the highest RMS current is always less than or equal to its limit (IRMS_lim=5.1 A).

2.3.2.F.4. Comparative summary:

A comparative summary of experimental performances in a healthy operation, an OC fault without reconfigurations, and fault-tolerant operations with methods (I)-2 and (I)-3 is presented in Table 2.13. The highest RMS currents IRMS, torques Tem, highest peak voltages Vpeak, and total copper losses Ploss in the two methods are compared. The performances of current control, torque generations, and constraints have been analyzed in the previous subsections. In general, the calculated and experimental results in Table 2.10 and Table 2.13 are in good accordance.

However, the experimental voltage references in Table 2.13 are higher than the calculated values in Table 2.10. In addition, the experimental Vpeak with healthy mode is no longer highest as shown in calculated results, related to the lowest base speed. The reason is that additional voltages caused by high-order harmonics of the experimental back-EMFs are added to the voltage references.

The experimental total copper losses Ploss with method (I)-2 (0.58 pu) and method (I)-3 (0.49 pu) are much lower than healthy mode (1 pu) and the OC fault without reconfigurations (1.27 pu). These copper losses are also slightly lower than calculated values due to the existing errors in current control. Method (II): new current references determined from reduced-order transformation matrices

2.3.3.A. Principle of method (II)

The principle of methods using reduced-order transformation matrices is to design new current references for post-fault operations by modifying classical transformation matrices, especially matrix Clarke. An overview on this type of methods has been presented in [START_REF] Vu | An Overview of Methods using Reduced-Ordered Transformation Matrices for Fault-Tolerant Control of 5-phase Machines with an Open Phase[END_REF]. Studies [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF][START_REF] Hyung-Min | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF][START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF] consider sinusoidal five-phase machines while methods in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF][START_REF] Xiong | Fault-tolerant FOC for five-phase SPMSM with non-sinusoidal back EMF[END_REF][START_REF] Xiong | A Fault-Tolerant FOC Strategy for Five-Phase SPMSM With Minimum Torque Ripples in the Full Torque Operation Range Under Double-Phase Open-Circuit Fault[END_REF] are applied to non-sinusoidal five-phase machines. In general, for n-phase machine, when one phase is opened, this type of methods is to determine new current references by using new (n-1)-by-(n-1) transformation matrices. In other words, the post-fault n-phase machine can be considered as an asymmetrical (n-1)-phase machine with (n-1)-by-(n-1) transformation matrices. In this doctoral thesis, the case study with a 7-phase non-sinusoidal machine is considered. In addition, the constraints on RMS current and peak voltage are considered; hence, flux-weakening operations at high speed are guaranteed.

Like method (I), method (II) can generate constant post-fault torques when the back-EMFs consist of the first and third harmonics. Moreover, one option of method (II) can guarantee constant torques even when the back-EMFs consist of harmonic components in the second fictitious machines (9 th , for example). This feature can be considered as an advantage of method (II) over method (I).

To find new transformations as well as new current references for a 7-phase non-sinusoidal machine, two following steps are taken as follows:

1) When only new 6-by-6 Clarke matrix for the fundamental currents are determined, the generated sinusoidal currents can generate constant torques in a sinusoidal back-EMF machine. However, in a non-sinusoidal back-EMF machine, there are inevitable ripple torques due to the interactions between the fundamental currents and the higher-order harmonics of the back-EMFs (3 rd , for example).

2) The third harmonic currents are injected to eliminate the ripple torques by determining new 6-by-6 Clarke matrices for the third harmonic currents. Therefore, the nonsinusoidal currents interact with the non-sinusoidal back-EMFs to generate constant torques. There are two current design options that are considered as two options of method (II).

Chapter 2.

Modeling and Control of Multiphase Drives 66

2.3.3.A.1. The use of fundamental currents:

When phase A is open-circuited, its phase current becomes zero. Therefore, the first column of the classical Clarke transformation matrix in (2.1), associated with phase A, should be removed.

The transformation with the classical Clarke matrix in the post-fault condition for seven-phase machines can be written as follows:

α1 B 1 C α2 D β2 E α3 F β3 G z i i i i i i i i i i i i i         =           fault Clarke T (2.31)   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
                                   = fault Clarke T ( ) 18 1 2 1 2 1 2 1 2 1 2 1 2                  (2.32)
In [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝒇𝒂𝒖𝒍𝒕 ], the first, third and fifth row vectors are no longer orthogonal to each other; hence, three fictitious machines associated with three reference frames which are decoupled in healthy mode are coupled in faulty mode. Consequently, currents cannot be properly controlled without any reconfigurations as analyzed in section 2.2.

According to the principle proposed in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] for non-sinusoidal five-phase machines, new transformation matrices for non-sinusoidal seven-phase machines can be proposed according to the preservation of the fundamental MMF as in healthy mode. In other words, the post-fault seven-phase machine can be considered as an asymmetrical 6-phase machine with 6-by-6 transformation matrices. The relationship between new reference currents in natural frame (iB1, iC1, iD1, iE1, iF1, iG1) and currents in new d-q frames (id11, iq11, id91, iq91, ix1, iz1) is described as follows: 2) sin( 3) sin( 4) sin( 5) sin(6 ) 2 cos(2 ) cos( 4) cos( 6) cos( 8) cos( 10) cos( 12) sin( 2) sin( 4) sin( 6) sin( 8) sin( 10) sin(12 ) 7 sin(3 ) 6) sin( 9) sin( 12) sin( 15) sin(18

        11 T T B1 C1 D1 E1 F1 G1 d11 q11 d91 q91 x1 z1 i i i i i i i i i i i i -- = Clarke Park TT 11 (2.33)   ( ) ( ) ( ) ( ) ( ) ( ) cos( ) 1 cos(2 ) 1 cos(3 ) 1 cos(4 ) 1 cos(5 ) 1 cos(6 ) 1 sin( ) sin(
                         - - - - - - = Clarke T 1 sin(
) 1/2 1/2 1/2 1/2 1/2 1/2                    (2.34)
  cos( ) sin( ) 0 0 0 0 sin( ) cos( ) 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

    -     =         Park T 1 (2.

Modeling and Control of Multiphase Drives

67

The index "1" in all elements in (2.33) means that the fundamental currents are considered. As previously discussed, the post-fault 7-phase machine can be considered as an asymmetrical 6phase machine. Therefore, (d11-q11) and (d91-q91) are 2 reference frames associated with 2 twophase fictitious machines while x1 and z1 are 2 reference frames associated with 2 zero-sequence machines, respectively. If the stator windings are wye-connected, the zero-sequence current iz1 is always zero.

New Clarke matrix [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆

𝟏

] is determined from [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝒇𝒂𝒖𝒍𝒕 ] in (2.32) by: 1) Removing its fifth row because it is not orthogonal to the first and third row vectors, and frames (α3-β3) or (d3-q3) in healthy mode become x1 in faulty mode. 2) Adding coefficient -1 to its first row to respect the orthogonal property between rows.

After the above modifications, the remaining row vectors of the new matrix [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 ] in (2.34) are orthogonal to each other, allowing to control independently currents in new d-q frames.

To prove the robustness of method (II), the non-sinusoidal back-EMFs are assumed to consist of the 1 st , 3 rd , and 9 th harmonics. The torque has ripples due to the interactions between the fundamental currents and these considered harmonics of the back-EMFs. Indeed, from (2.33)-(2.35), the machine torque Tem1 with its average torque Tave1 and harmonics can be generally described as follows:

       24 T TT em1 ave1 n3 q11 1 2 d91 q91 x1 n nf T = e i T + E i f ( θ)+ f ( θ) + e i i i = M 1 (2.
  - - -   -   = -     -   --   M 1 (2.37)
where en and i are the 7-dimensional vectors of the speed-normalized back-EMFs and currents in natural frame; En3 is the amplitude of the 3 rd harmonic of en; f1 and f2 are trigonometric functions of 2θ and 4θ, respectively; enf is a 6-dimensional vector of back-EMFs derived from en by removing the back-EMF of phase A; [M1] is a constant 6-by-3 matrix described in (2.37) and obtained from the 3 rd to 5 th columns of

[𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 ] -𝟏 .
The generated torque is constant if a sinusoidal back-EMF machine is considered. In (2.36), currents (id91, iq91, ix1) can be used to eliminate the impact of the 1 st , 3 rd and 9 th harmonics of enf on torque ripples by simply imposing currents (id91, iq91, ix1) to be zero. However, torques with frequencies of 2θ and 4θ in Tem1 cannot be eliminated because of the presence of the 3 rd harmonic of back-EMFs with amplitude En3. Therefore, the 3 rd harmonic currents need to be injected to compensate the pulsating torques 2θ and 4θ.

2.3.3.A.2. The injection of the third harmonic currents:

The 3 rd harmonic components of current in natural frame (iB3, iC3, iD3, iE3, iF3, iG3) are calculated from new d-q frame currents (ix3, id93, iq93, id33, iq33, iz3) as follows:

Chapter 2.

Modeling and Control of Multiphase Drives

68         11 T T B3 C3 D3 E3 F3 G3 x3 d93 q93 d33 q33 z3 i i i i i i i i i i i i -- = Clarke Park
TT 33 (2.38) 2) sin( 3) sin( 4) sin( 5) sin( 6) cos( 2) cos( 4) cos [START_REF] Cavagnino | Experimental Characterization of a Belt-Driven Multiphase Induction Machine for 48-V Automotive Applications: Losses and Temperatures Assessments[END_REF] cos( 8) cos( 10) cos(12 ) 2 sin(2 ) sin( 4) sin( 6) sin( 8) sin( 10) sin( 12) cos(3 ) 1 cos(6 ) 1 cos(9 ) 1 cos(12 ) 1 cos(15 ) 1 cos(18 ) 1 7 sin

  ( ) ( ) ( ) ( ) ( ) ( ) sin( ) sin(
                        = - - - - - - Clarke T 3
(3 ) sin( 6) sin( 9) sin( 12) sin( 15) 1) Eliminating its first row and because it is not orthogonal to the third and fifth row vectors, and frames (α1-β1) or (d1-q1) in healthy mode become x3 in faulty mode. 2) Adding coefficient -1 to its fifth row due to the requirement of orthogonality. As a result, the orthogonal property of the remaining row vectors in [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑 ] enables to independently control currents in the new d-q frames. From (2.33)-(2.40), the machine torque Tm13 including its average torque Tave13 and harmonics can be expressed by:

sin(18 ) 1/2 1/2 1/2 1/2 1/2 1/2              (2.39)   1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 cos(3 ) sin(3 ) 0 0 0 0 sin(3 ) cos(3 ) 0 0 0 0 0 0 1         =     -     Park T 3 (2.
         24 m13 ave13 n3 q11 n1 q33 1 2 TT T nf d91 q91 x1 x3 d93 q93 T = T + E i E i f ( θ)+ f ( θ) +e i i i i i i +     +    
-  --  -  = -   --  - - -  M 3 (2.42)
where En1 is the amplitude of the 1 st harmonic of speed-normalized back-EMFs en; [M3] is a constant 6-by-3 matrix in (2.42) obtained from the 1 st to 3 rd columns of [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑 ] -𝟏 . In (2.41), it is noted that to eliminate torque ripples f1(2θ) and f2(4θ), the 3 rd harmonic space current iq33 needs to be calculated as follows:

n3 q33 q11 n1 E ii E =- (2.43) Chapter 2.
Modeling and Control of Multiphase Drives 69 Therefore, the 1 st and 3 rd harmonic currents (id11, iq11, id33, iq33) must be determined either to preserve the healthy torque or to respect constraints on RMS current and peak voltage.

To reduce the impact of harmonics in enf on torque Tm13, the other currents (id91, iq91, ix1) and (ix3, id93, iq93) can be defined according to two current design options. Therefore, method (II) is proposed with two options as follows:

1) Robust Control Approach (Method (II)-RCA): guarantees ripple-free torques even when the existing back-EMF harmonics in the second fictitious machines, for example, the 9 th harmonic component.

2) Similar Copper Losses (Method (II)-SCL): obtain similar copper losses in healthy

phases, avoiding the overheated phase windings.

The details of the two options of method (II) are described in the following subsections.

2.3.3.B. Method (II)-RCA (Robust Control Approach)

In this option of method (II), 6 currents (id91, iq91, ix1) and (ix3, id93, iq93) are imposed to be zero. Therefore, besides the 1 st and 3 rd harmonics, if the back-EMFs contain other harmonics such as 9 th in the second fictitious machine, the generated torque Tm13 is still constant and equal to Tave13 according to (2.41). Thus, this option is called Robust Control Approach. As a result, the 1 st harmonics of the 6 healthy phase currents in natural frame are calculated from (2.33) as follows:

2.3.3.C. Method (II)-SCL (Similar Copper Losses)

In this option, the main idea is to distribute similar copper losses to all healthy phases; hence, the local thermal limit of each machine phase can be respected easily. Due to the back-EMF

𝑖 1 𝑖 1 𝑖 1 𝑖 1 𝑖 1 𝑖 1 2 Chapter 2.
Modeling and Control of Multiphase Drives 70 harmonic property, the 3 rd harmonic currents account for a small proportion compared to the 1 st harmonic currents. Therefore, RMS currents as well as copper losses mainly depend on the 1 st harmonic currents. Then, the 1 st harmonics of the 6 healthy phase currents in natural frame can be determined according to [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF] and illustrated in Fig. 2 ii

 = +  (2.62)
Then, the total torque Tm13 with Tave13 and its harmonics can be expressed by:

      TT T nf_9 m13 ave13 d91 q91 x1 x3 d93 q93 T = T +e i i i i i i     +     MM 13 (2.63)
where enf_9 is a 6-dimensional vector of the 9 th harmonic of back-EMFs derived from enf by removing 1 st and 3 rd harmonics.

The torque ripples caused by the interaction between the 9 th harmonic of back-EMFs enf_9 and 6 variable currents (id91, iq91, ix1) and (ix3, id93, iq93) make method (II)-SCL less robust than method (II)-RCA. However, for the optimal calculation in the control scheme (see Fig. 2.21), only the first and third harmonics of the back-EMFs are considered. Therefore, in this case, method (II)-SCL can generate time-constant torques.

2.3.3.D. Summary of current design options in method (II)

The summary of the two options of method (II) is described in Table 2.14. As previously discussed, 4 constant currents (id11, iq11, id33, iq33) while the other 6 currents can be zero or timevariant subject to different objectives. After determining d-q currents (id11, iq11, id91, iq91, ix1, iz1) and (ix3, id93, iq93, id33, iq33, iz3) with either method (II)-RCA or method (II)-SCL, the total phase current reference of a remaining healthy phase is equal to the sum of its 1 st and 3 rd harmonic currents as shown in Fig. 2.42. If the pre-fault control scheme as described in Fig. 2.6 (or Fig. 2.21) with six current controllers is reused, six current references (id1, iq1, id9, iq9, id3, iq3) need to be calculated. In this doctoral Chapter 2.

Modeling and Control of Multiphase Drives 72 thesis, the new total phase current references are transformed from natural frame into d-q frames by classical matrices, resulting in time-variant d-q current references for control. Time-variant d-q current references are also presented in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] when reduced-order matrices for the 1 st harmonic currents are reused. The reason for the use of these matrices is that a general transformation for the total currents, including the 1 st and 3 rd harmonics, has not been found.

2.3.3.E. Comparative analyses of calculated results with methods (II)-RCA and (II)-SCL

2.3.3.E.1. Preserving the same torque as in healthy mode:

With the considered drive (see Fig. 2.7), the average torque with the rated RMS current of 5.1 A is 33.3 Nm in healthy mode as discussed in section 2.1.4. According to Fig. 2.42, to preserve the pre-fault torque, currents (id91, iq91, ix1) and (ix3, id93, iq93), the remaining healthy phase currents, and d-q current references for control (id1, iq1, id9, iq9, id3, iq3) with methods (II)-RCA and (II)-SCL are shown in Fig. 2.43 and Fig. 2.44. Notably, constant currents (id11=0 A, iq11=15.7 A, id33=0 A, iq33= -5 A) are not plotted in Fig. 2.43a and Fig. 2.44a for the sake of clarity. Six currents (id91, iq91, ix1) and (ix3, id93, iq93) are zero with method (II)-RCA and timevariant with method (II)-SCL. The waveforms of the remaining healthy phase currents with method (II)-RCA are very different to each other (see Fig. 2.43b) while these waveforms with method (II)-SCL are alike (see Fig. 2.44b). The highest peak phase current with method (II)-RCA (18.7 A) is higher than that of method (II)-SCL (14.3 A).

(a) (b) (c) Fig. 2.43. Desired d-q currents (a), phase currents (b), d-q current references for control (c), with method (II)-RCA for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s with (id11=0 A, iq11=15.7 A, id33=0 A, iq33= -5 A).

(a) (b) (c) Fig. 2.44. Desired d-q currents (a), phase currents (b), d-q current references for control (c), with method (II)-SCL for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s with (id11=0 A, iq11=15.7 A, id33=0 A, iq33= -5 A).

(II)-RCA (II)-RCA (II)-RCA (II)-SCL (II)-SCL (II)-SCL

Chapter 2.

Modeling and Control of Multiphase Drives

73

In Fig. 2.43c, with method (II)-RCA, currents in (id9, iq9) are zero while the other currents are time-variant. This point once again proves the robustness of method (II)-RCA when its torque is unaffected by the back-EMF harmonics in the second machine (9 th , for example). Meanwhile, method (II)-SCL has six time-variant d-q currents (see Fig. 2.44c). Therefore, the torque with method (II)-SCL is sensitive to the back-EMF harmonics in the second machine like methods (I)-2 and (I)-3.

Calculated RMS currents in all phases by the two options of method (II) are described in Table 2.15. It is noted that every calculated RMS current is higher than that of healthy mode (5.1 A).

The RMS currents with method (II)-RCA vary from 5.3 to 10.6 A with the highest value in phase B and G. Meanwhile, these values with method (II)-SCL are similar from 7.6 to 7.9 A with the highest value in phases C and F. It is concluded that method (II)-RCA creates higher RMS currents than method (II)-SCL for the same torque. When only the first and third harmonics of the back-EMFs are considered, comparisons between the two methods and healthy mode are described in Table 2. [START_REF] Yang | Condition Monitoring for Device Reliability in Power Electronic Converters: A Review[END_REF]. It is noted that the torques by the two methods are constant in this case. In addition, method (II)-RCA generates a higher peak voltage (1.28 pu) and a higher total copper loss (2.25 pu) than method (II)-SCL does (0.99 pu and 1.95 pu, respectively). When the back-EMFs have additional harmonics in the second machine, for instance, the 9 th harmonic with 12.5%, the torque with method (II)-RCA is still constant while method (II)-SCL generates a torque ripple of 17.6% as described in Fig. 2.45. Modeling and Control of Multiphase Drives 74

2.3.3.E.2. Optimal calculations under constraints on RMS current and peak voltage:

If the control scheme in Fig. 2.21 with an optimal strategy in (2.16) is considered, the torques obtained by the two options of method (II) will be lower than the pre-fault torque (33.3 Nm). All currents are re-determined to respect the considered constraints. According to the considered drive, the RMS current limit is 5.1 A and the voltage limit for the offline optimization is 75 V. Different from method (I), currents iq11 and iq33 have a constraint as described in (2.43) to eliminate ripple torques. Therefore, only iq11 will be tuned by fmincon, then iq33 and the other currents will be derived from iq11. As a result, the current waveforms in of all considered currents under constraints are the same as in Fig. 2.43 and Fig. 2.44 but with lower amplitudes generated by the lower d-q currents (id11=0 A, iq11=7.6 A, id33=0 A, iq33= -2.4 A).

RMS currents in all phases under the constraints with method (II) at 20 rad/s are described in Table 2.17. In the two options of method (II), the highest RMS currents are reduced to the RMS current limit of 5.1 A. Finally, comparisons between the post-fault operations using the two options of method (II) and healthy mode at 20 rad/s are described in Table 2.18. When the RMS current limit of 5.1 A is respected, method (II)-SCL generates the higher torque at 21.6 Nm (0.65 pu), like method (I)-2. Meanwhile, the torque with method (II)-RCA is lower with 16 Nm (0.48 pu), less than all options of method (I). Additionally, the highest peak voltages Vpeak by the two options of method (II) are less than that of healthy mode (0.77 and 0.78 pu), making their base speeds higher than that of healthy mode. Total copper losses in fault-tolerant operations are lower than that of healthy mode (0.52 pu for method (II)-RCA and 0.82 pu for method (II)-SCL) due to the current constraint. When the operating speed varies, the optimal torque-speed characteristics, all RMS phase currents and all peak phase voltages by the two options of method (II) in terms of speed are described in Fig. 2.46. It is noted that method (II)-SCL results in a higher fault-tolerant torquespeed characteristic than method (II)-RCA does. In addition, all options of method (II) respect the constraints on RMS current and peak voltage even in the flux-weakening region. Notably, in the flux-weakening region with method (II), all RMS currents are reduced because only iq11 Chapter 2.

Modeling and Control of Multiphase Drives

75 is tuned by fmincon, and the other currents are derived from iq11. If id11 and id33 are varied to be nonzero, the high torque ripples will appear. Table 2.19 describes the calculated base and maximum speeds of the healthy and faulty modes with the two proposed options of method (II). Method (II)-RCL has a base speed of 44 rad/s (1.16 pu) while method (II)-SCL has a base speed of 46 rad/s (1.21 pu). These base speeds are higher than that of healthy mode due to lower voltages Vpeak (see Table 2.18). The post-fault operations with method (II) have the same maximum speed that is lower than that of healthy mode. In addition, the maximum speed with method (II) (0.67 pu) is shorter than with method (I) (0.88 pu) (see Table 2.11).

2.3.3.F. Verification with experimental results for method (II)

Methods (II)-RCA and (II)-SCL with the control scheme in Fig. 2.21 are validated in the experimental test bench (see section 2.1.4.A). As methods (I)-2 and (I)-3, methods (II)-RCA and (II)-SCL can be easily implemented in the test bench without any hardware modifications.

2.3.3.F.1. Current control performance:

To experimentally obtain the optimal torque-speed characteristics (see Fig. 2.46a), six d-q currents (id1, iq1, id9, iq9, id3, iq3) need to properly track their references with waveforms as described in Fig. 2.43c and Fig. 2.44c. At a speed, method (II)-RCL has 4 time-variant current references (id1, iq1, id3, iq3) and 2 time-constant current references (id9, iq9). Meanwhile, 6 current references with method (II)-SCL are time-variant. The current control performances at 20 rad/s are described in Figs 2.47-2.49. Because most current references are time-variant, the current control quality will be reduced at high speed when conventional PI controllers are applied. This phenomenon will be clearly observed in the torque-speed performance. The use of PIR controllers, as in [START_REF] Xiong | Fault-tolerant FOC for five-phase SPMSM with non-sinusoidal back EMF[END_REF][START_REF] Chuan | Mitigating the Torque Ripple in Electric Traction using Proportional Integral Resonant Controller[END_REF], can be a solution to deal with time-variant references at high speed

2.3.3.F.2. Torque performance:

An operation with three operating states at 20 rad/s is shown in Fig. 2 When the operating speed varies, the experimental torque-speed characteristics with methods (II)-RCA and (II)-SCL in comparison with healthy mode (Tem_opt_HM and Tem_exp_HM) are described in Fig. 2.51. By using these methods, the experimental post-fault torques (Tem_exp_OC) are in good accordance with the optimal torques (Tem_opt_OC) from Fig. 2.46a. However, the torque ripples increase in both options of method (II) when the rotating speed increases. The reason is that most current references are time-variant, the current control quality is reduced at high speed when conventional PI controllers are applied. The experimental base and maximum speeds are the same as calculated results in Table 2.19.

Method (II)-RCA Heathy mode

Without reconfigurations Method (II)-SCL

T em_exp_HM T em_opt_HM T em_opt_OC T em_exp_OC

Method (II)-RCA Method (II)-SCL

T em_exp_HM T em_opt_HM T em_opt_OC T em_exp_OC

Chapter 2.

Modeling and Control of Multiphase Drives 78

2.3.3.F.3. Constraints on RMS current and peak voltage:

Table 2.20 shows that experimental RMS currents in all phases at 20 rad/s are in good accordance with optimal values. The experimental highest RMS currents in both options of method (II) are within their limit 5.1 A. Notably, RMS phase currents with method (II)-SCL are similar in phases. When the operating speed varies, the constraints on RMS current and voltage are still respected in the experimental results, including the flux-weakening regions. Measured phase currents obtained by the two methods in terms of speed are described in Figs. 2.52-2.53. The waveforms of measured phase currents at 20 and 47 rad/s are also plotted. The amplitudes of the remaining healthy phase currents are changed at high speed; hence, the highest RMS current is reduced at speeds over the base speed to respect the constraint on peak voltage.

2.3.3.F.4. Comparative summary:

A comparative summary of experimental performances in a healthy operation, an OC fault without reconfigurations, and fault-tolerant operations with methods (II)-RCA and (II)-SCL is presented in Table 2.21. The highest RMS currents, torques, highest peak voltages, and total copper losses in the two methods are compared. The performances of current control, torque generations, and constraints have been analyzed in the previous subsections. In general, the calculated and experimental results in Table 2.18 and Table 2.21 are in good accordance. However, the experimental voltage references are higher than the calculated values in Table 2.18 due to the additional voltages caused by high-order harmonics of the experimental back-EMFs.

Method (III): new current references determined from natural frame 2.3.4.A. Principle of method (III)

It is noted that new current references of the remaining healthy phases with unidentical waveforms in methods (I) and (II) can theoretically generate constant torques in a nonsinusoidal machine (with the 1 st and 3 rd harmonics of back-EMFs). However, under the constraints on RMS current, the highest post-fault average torque of the considered drive is equal to 65% of the healthy torque. This torque is relatively low in case only one of seven phases is open-circuited. Accordingly, method (III) is proposed with the aim of improving the average torque by using identical waveforms of the remaining healthy phase currents when one

Method (II)-SCL

Chapter 2.

Modeling and Control of Multiphase Drives

81

phase is open-circuited [START_REF] Vu | Torque optimisation of seven-phase BLDC machines in normal and degraded modes with constraints on current and voltage[END_REF][START_REF] Vu | Control strategies for non-sinusoidal multiphase PMSM drives in faulty modes under constraints on copper losses and peak phase voltage[END_REF][START_REF] Zhang | Torque Optimization of a Seven-Phase Bi-harmonic PMSM in Healthy and Degraded Mode[END_REF][START_REF] Vu | Sensitivity of Torque Control for Seven-phase BLDC Machine with One Opened Phase under Constraints on Voltage and Current[END_REF][START_REF] Zhao | Fault Tolerant 7-phase Hybrid Excitation Permanent Magnet Machine[END_REF]. The waveform uniformity of these phase currents allows to: 1) Distribute perfectly equal copper losses among healthy phases, avoiding overheated machine windings. 2) Maximize currents to produce more average torques when the constraint on current is considered. Indeed, under the constraint on RMS current, every remaining healthy phase current can reach their current limit, resulting in the maximum average torque.

In method (III) of this doctoral thesis, analytical torque expressions in natural frame will be used to determine parameters of new phase currents when one phase is open-circuited.

Sinusoidal and non-sinusoidal waveforms of currents and back-EMFs are analyzed.

2.3.4.B. Sinusoidal phase currents for sinusoidal back-EMFs

In case of an OC fault in phase A, the speed-normalized sinusoidal back-EMFs of the remaining healthy phases can be described as follows:

               =-      =-      =-        =-      =-      =-     (2.64)
where En1 is the amplitude of the speed-normalized sinusoidal back-EMFs; θ is the electrical position. According to [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], the fundamental magnetomotive force in faulty modes needs to be preserved to generate the same constant torque as in healthy mode. In this doctoral thesis, current references in natural frame are generally determined to have the same amplitude with different initial phase angles as follows:

( ) ( ) ( ) sin sin sin B m1 B C m1 C D m1 D EB FC GD iI φ iI φ iI φ ii ii ii    =+   =+   =+   =-   =-   =-  (2.65)
where Im1 is the amplitude of all healthy phase currents; φB, φC, and φD are initial phase angles of currents in phase B, C

B D B D ave m1 n1 C B D B D 2 m1 n1 C T I E T I E                  -+        = - - -                -+        = - + + - +              
where Tave is the average torque; T2θ is the ripple torque at a frequency of 2θ. To eliminate the ripple torque T2θ in (2.66), the initial phase angles of the phase currents in (2.65) are simply determined based on trigonometric calculations as follows:

2 2 2 1 2 arccos 2 2 7 2 3
with and

BD C1 BD 2 12 φφ φk φφ k kk    +  =+    -   + = -=  +       (2.67)
As a result, the electromagnetic torque of the machine is perfectly constant. According to (2.67), the initial phase angles and their corresponding torques are calculated with the two following possibilities:

( ) 2
 +  =   -  =    = = -   (2.68) ( ) 2 20 2 21 0.45 sin BD C BD em ave m1 n1 C φφ φ φφ T T I E φ  +  =   -  =-    ==   (2.69)
In (2.68) and (2.69), the torque values are proportional to the amplitudes of the fundamental components of currents and back-EMFs. In addition, the torques Tem also depend on the sine function of φC.

It is noted that the torque in (2.68) is much higher than (2.69). Thus, the maximal torque and the corresponding initial phase angles are given by: Chapter 2.

Modeling and Control of Multiphase Drives

   =-    =-     =-   ==  (2.70)
These obtained results are similar to the study in [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF] for seven-phase induction machines.

2.3.4.C. Sinusoidal phase currents for non-sinusoidal back-EMFs

The back-EMFs with the first and third harmonic components in the remaining healthy phases are given by:

       
                      = - + - +                  = - + - +                  = - + - +                  = - + - +            =    
                            - + - +                     = - + - +              (2.71)
where ke is the amplitude ratio of the third harmonic to the first harmonic of the back-EMFs; φe is the phase shift angle between the third and first harmonic components of the back-EMFs. If the sinusoidal currents in (2.65) T I E

T I E k                            -+       = - - -             -+       - + + - +             = -+       + - - + + -             ( )   ( )  
                              +       -+        = - + + + - + +              
where Tave is the average torque; T2θ and T4θ are the ripple torques at frequencies of 2θ and 4θ, respectively. Compared to (2.66), the torque in (2.72) has additional components associated with ke including an extra term of T2θ and the ripple torque T4θ.

. An attempt to eliminate all ripple torques:

To eliminate T4θ in (2.72), the initial phase angles of the currents need to respect:

+  =+    -   + = -=  +       (2.73)
After eliminating T4θ, the remaining torques including Tave and T2θ with corresponding initial phase angles of the currents can be calculated as follows:

( ) ( ) ( )   2 4
T I E T I E k         +  =   -  =-   =-   = - + + -+  (2.74) or ( ) ( ) ( )   2
T I E T I E k         +  =   -  =-   =-   = + - -+  (2.
T I E k         =-    =-     =-   =   = - +   = +  (2.76)
It is noted that the amplitudes of the torque ripples depend on the amplitude (En1ke) and the phase shift angle (φe) of the third harmonic in the back-EMFs.

2.3.4.D. Non-sinusoidal phase currents for non-sinusoidal back-EMFs

The injection of the third harmonic component aims at improving the average torque with reasonable existing ripples. The phase currents considering the additional third harmonic components are expressed by:

( ) ( )     ( ) ( )     ( ) ( )     sin sin 3 sin sin 3 sin sin 3 B m1 B i B i C m1 C i C i D m1 D i D i EB FC GD iI φ k φ iI φ k φ iI φ k φ ii ii ii          = + + + +   = + + + +   = + + + +   =-   =-   =-  (2.77)
where ki is the amplitude ratio of the third harmonic to the first harmonic of the currents; φi is the phase shift angle between the third and first harmonic components of the currents. As a result, the electromagnetic torque consists of a constant torque, ripple torques with frequencies of 2θ, 4θ, and 6θ as follows:

em ave 2 4 6 T T T T T    = + + + (2.
T I E +k k T I E                       -+        - - -               =  -+            - - + - - + -                       -  -+   = ( )   ( )   ( )  
i i C i k +k                             +      + - +            -+         + - + - + + -+                  -+           - - + + - + +                      ( )   ( )  
T I E k                                  -+        - + + + - + +               =  -+            - + + + - + +                       = ( )   96 sin 2cos 3 sin 6 3 sin 6 3 7 2 7 2 B D B D i i e C i e k                                     -+            - + + + + - + + +                     
Compared to (2.72), Tave, T2θ and T4θ in (2.78) have extra terms associated with ki. In addition, the torque also contains a ripple with frequency 6θ associated with both ki and ke.

2.3.4.D.1. An attempt to eliminate all ripple torques:

To eliminate T6θ in (2.78), the initial phase angles of the currents need to respect:

   +  =+    -      + = -=  +            (2.79)
After eliminating T6θ, the remaining torques including Tave, T2θ and T4θ with corresponding initial phase angles of the currents are expressed by:

( ) ( )   ( ) ( ) ( )   ( ) ( )   2
T I E k k T I E k k T I E k k                     +  =   -   =-      = - - + -   = - + + -+ - + +   = - + + - + +     (2.
T I E k k T I E k k T I E k k                     +  =   -   =-      = + + -   = - + - -+ - + +   = + + + + +     (2.81)
After nullifying T6θ, to continue eliminating T4θ in (2.80), the initial phase angles of the currents need to respect:

  ( ) ( ) ( ) ( )   2
T I E k k T I E k k               +  =   -   =-       = - = -   =-   = - -   = - + + -+ + + +  (2.82)
Meanwhile, to eliminate T4θ in (2.81), the initial phase angles of the currents need to respect:

  ( ) ( ) ( ) ( )   2
T I E k k T I E k k               +  =   -   =-       = - = -   =-   =+   = - + - -+ + + +  (2.83) Remarks:
1) From (2.82) and (2.83), it is concluded that all ripple torques cannot be eliminated.

2) Compared to (2.70) and (2.76), the average torques in (2.82) and (2.83) are lower.

3) From the above remarks, the current parameters in (2.82) and (2.83) will not be used in method (III).

2.3.4.D.2. An attempt to increase the average torque:

From the above remarks, if the initial current angles in (2.70)

T I E k + k T I E k k T I E k k                    =-    =-    =-    =-   = - + +   = + - +   = + +      (2.84) Remarks:
1) It is noted that the average torque in (2.84) can be higher than (2.70) and (2.76), especially when φi is equal to φe. Accordingly, non-sinusoidal currents in (2.77) with parameters in (2.84) can meet the general requirement of method (III) to generate a high average torque.

2) The amplitudes of the torque harmonics depend on not only the amplitude (En1ke) and the phase shift angle (φe) of the third harmonic in the back-EMFs but also the amplitude (Im1ki) and the phase shift angle (φi) of the third harmonic of the phase currents. 3) All torques including Tave and torque harmonics are proportional to the current amplitude Im1. Therefore, the total torque ripple ∆T calculated in (2.17) is unaffected by Im1. 4) Phase angle φi equal to φe enables to not only maximize the average torque Tave but also reduce torque ripples T2θ and T4θ. Indeed, in this case, two terms have the same cosine functions with frequency 2θ in T2θ (and 4θ in T4θ) but opposite signs.

To simply illustrate the impact of current parameters in (2.84) on torque generations, parameters of the machine drive in section 2.1.4.A with ke=0.32 and φe=0.9 are considered.

Fig. 2.56a shows the impact of φi on the torque generations when ki is supposed to be ke. It is noted that when φi is equal φe, the average torque Tave is highest and the torque ripple ∆T is lowest. In addition, when φi varies, the torque ripple rapidly increases.

Meanwhile, the impact of ki on the torque generation is described in Fig. 2.56b when φi is supposed to be φe. The value of ki equal to ke gives the best values of the torque ripple and average torque, and the torque ripple rapidly increases when ki exceeds ke. In conclusion, the remaining healthy phase currents with waveforms like the back-EMF waveform, such as in Fig. 2.57, can generate the highest average torque and lowest torque ripple in the case of the uniformity of current waveforms. Notably, this approach cannot guarantee a smooth torque with a machine having a high proportion of the third harmonic in its back-EMFs. Indeed, Fig. 2.58 shows that the torque ripple and average torque are proportional to ke. However, with an increase in ke, the improvement of the average torque is not significant while the torque ripple dramatically rises.

Chapter 2.

Modeling and Control of Multiphase Drives 90

2.3.4.E. Summary of current design options in method (III)

The possible average torque in (2.84) is highest among the above considerations, followed by the torque in (2.76). It is noted that (2.76) is a special case of (2.84) when sinusoidal currents are considered (ki=0). Therefore, method (III) can be divided into two design options denoted by method (III)-1 and method (III)-2. These options are described in Table 2.22 with phase current expressions in (2.77).

2.3.4.F. Comparative analyses of calculated results with methods (III)-1 and (III)-2

2.3.4.F.1. Preserving the same torque as in healthy mode:

In the considered drive, the average torque, with the rated RMS current of 5.1 A, is 33.3 Nm in healthy mode as discussed in section 2.1.4.

The remaining healthy phase currents and d-q current references, determined with methods (III)-1 and (III)-2, are shown in Fig. 2.59 and Fig. 2.60. In each option of method (III), the waveforms of the remaining healthy phase currents are the same as shown in Fig. 2.59a and Fig. 2.60a. Method (III)-1 has two constant currents (id1, iq1) and four time-variant currents (id9, iq9, id3, iq3) for control. Meanwhile, all currents for control with method (III)-2 are time-variant. However, it is noted that the main d-q currents (id1, iq1), generating most of the torque, are constant in method (III)-1 and slightly fluctuate in method (III)-2. Calculated RMS currents in all phases by the two options of method (III) are described in Table 2. [START_REF] Duran | Multiphase Electric Drives: Introduction[END_REF]. It is noted that all calculated RMS currents in each option of method (III) are the same and slightly higher than healthy mode (5.1 A). When only the first and third harmonics of the back-EMFs are considered, comparisons between the two options of method (III) and healthy mode are described in Table 2.24. The torques generated with method (III) are not constant as in methods (I) and (II). Specifically, method (III)-1 generates a higher torque ripple (24%) than method (III)-2 does (19%). Compared to methods (I) and (II) (see Table 2.8 and Table 2.16), method (III) has lower RMS currents IRMS and total copper losses Ploss for the same torque.

2.3.4.F.2. Optimal calculations under constraints on RMS current and peak voltage:

Similar to methods (I) and (II), if the control scheme in Fig. 2.21 with an optimal strategy in (2.16) is considered, the torques obtained by the two options of method (III) become lower than the pre-fault torque (33.3 Nm). All currents are re-determined to respect constraints on RMS current and peak voltage. According to the considered drive, the RMS current limit is 5.1 A and the voltage limit for the offline optimization is 75 V. Different from method (I), only Im1 in method (III) will be tuned by fmincon to respect the current constraint, including fluxweakening region. The reason is that the torque ripple is unaffected by Im1 but it is sensitive to the other parameters of phase currents (ki and φi) as described in Fig. 2.56. After considering the RMS current constraint, the RMS currents of all phases with method (III) at 20 rad/s are equal to the RMS current limit of 5.1 A.

(III)-2 (III)-2

Chapter 2.

Modeling and Control of Multiphase Drives 92

Comparisons between the post-fault operations using the two options of method (III) and healthy mode are described in Table 2.25. When the RMS current limit of 5.1 A is respected, method (III)-2 generates the higher torque at 26.2 Nm (0.79 pu) while the torque with method (III)-1 is slightly lower with 25.9 Nm (0.78 pu). It is noted that method (III) generates higher average torques than both methods (I) and (II) (only 0.65 pu). Additionally, the highest peak voltages Vpeak by the two options of method (III) (0.84 and 0.97 pu) are less than that of healthy mode, making their base speeds higher than that of healthy mode. Total copper losses in faulttolerant operations with both options of method (III) are 0.86 pu, higher than those of method (I) (0.55 to 0.64 pu) and method (II) (0.52 to 0.82 pu) (see Table 2.10 and Table 2.18). When the operating speed varies, the optimal torque-speed characteristics, all RMS phase currents and all peak phase voltages by the two options of method (III) in terms of speed are described in Fig. 2.61. It is noted that method (III)-2 has a higher fault-tolerant torque-speed characteristic as shown in Fig. 2.61a. In Fig. 2.61b, the two options of method (III) respect the constraints on RMS current when RMS currents of all remaining healthy phases are equal to 5.1 A at low speed. Notably, in the flux-weakening region with method (III), all RMS currents are reduced because only im1 is tuned by fmincon, and the other parameters of phase currents are fixed to avoid any increases in the torque ripples. The maximum peak voltages are within their limit along the speed range as described in Fig. 2.61c. Table 2.26 describes the calculated base and maximum speeds of the healthy and faulty modes with the two proposed options of method (III). Like methods (I) and (II), the post-fault operation with method (III) has shorter speed ranges than the healthy operation. The two proposed options of method (III) have similar post-fault maximum speeds (0.66 and 0.67 pu). These maximum speeds are like those of method (II) (0.67 pu) but lower than those of method (I) (0.81 to 0.88

Method (III)-1 Method (III)-2 Methods (III)-1-2 Chapter 2.
Modeling and Control of Multiphase Drives 93 pu). The base speed with method (III)-2 is 38 rad/s (1 pu) while that of method (III)-1 is 45 rad/s (1.18 pu).

2.3.4.G. Verification with experimental results for method (III)

Methods (III)-1 and (III)-2 with the control scheme in Fig. 2.21 are validated in the experimental test bench (see section 2.1.4.A). As methods (I)-2, (I)-3, and (II), method (III) can be easily implemented in the test bench without any hardware modifications.

2.3.4.G.1. Current control performance:

To experimentally obtain the optimal torque-speed characteristics (see Fig. 2.61), six d-q currents (id1, iq1, id9, iq9, id3, iq3) need to properly track their references as presented in Fig. 2.59b and Fig. 2.60b. At a speed, method (III)-1 has 2 time-constant current references (id1, iq1) and 4 time-variant current references (id3, iq3, id9, iq9). Meanwhile, 6 current references in method (III)-2 are time-variant with small fluctuations in main currents (id1, iq1).

The current control performances at 20 rad/s are described in Figs. 2.62-2.64. Because most current references are time-variant, the control quality may be reduced at high speed when conventional PI controllers are applied. This phenomenon will be clearly observed in the torque-speed performance.

2.3.4.G.2. Torque performance:

(a) (b) Fig. 2.65. (Experimental result) Torque in three operating states including healthy mode, an OC fault in phase A without reconfigurations, and when method (III)-1 (a) and method (III)-2 (b) are applied.

To see the effectiveness of methods (III)-1 and (III)-2, an operation with three operating states at 20 rad/s as shown in Fig. 2.65 is investigated, including a healthy operation, a post-fault operation without reconfigurations, and a fault-tolerant operation with one of two methods (III)-1 and (III)-2. The first two operating states have been deeply investigated in sections 2.1.4 and 2.2. It is noted that the average torques in the third state with methods (III)-1 and (III)-2 are in good accordance with the optimal (calculated) values in Fig. 2.61a. To respect the constraints

(III)-1 (III)-2 (III)-1 (III)-2

Method (III)-1 Heathy mode

Without reconfigurations Method (III)-2

Chapter 2.

Modeling and Control of Multiphase Drives 95 on RMS current, the average torque is reduced from 33.5 Nm in healthy mode to 25.6 Nm (0.76 pu) with method (III)-1 and 26.1 Nm (0.78 pu) with method (III)-2 in the post-fault operation. However, the experimental torques have ripples of 33% with method (III)-1, and 31% with method (III)-2. Especially, the differences between the maximum and minimum torques are about 8 Nm, relatively high compared to the previous methods.

When the operating speed varies, the experimental torque-speed characteristics with methods (III)-1 and (III)-2 in comparison with healthy mode (Tem_opt_HM and Tem_exp_HM) are described in Fig. 2.66. By using these methods, the experimental post-fault torques (Tem_exp_OC) are in good accordance with the optimal torques (Tem_opt_OC) from Fig. 2.61. However, the torque ripple increases in method (III)-2 when the rotating speed increases. The reason is that all current references in method (III)-2, especially (id1, iq1), are time-variant; hence, the current control quality will be reduced at high speed with conventional PI controllers. Meanwhile, method (III)-1 does not have this phenomenon because its current references (id1, iq1) are time-constant. The high fluctuations of torques in method (III) will be solved in Chapter 3 by using ADALINEs.

The experimental base and maximum speeds in Fig. 2.66 are the same as calculated results in Table 2.26.

(a) (b) Fig. 2.66. (Experimental result) Post-fault torque-speed characteristics under an OC fault in phase A with method (III)-1 (a), and with method (III)-2 (b).

2.3.4.G.3. Constraints on RMS current and peak voltage:

Table 2.27 shows that experimental RMS currents in the remaining healthy phases at 20 rad/s are similar and around the RMS current limit (5.1 A). The experimental highest RMS currents in both options of method (III) are 5.3 A due to the imperfection of current control. Similarly, phase voltage references with the two options of method (III) in terms of speed are presented in Fig. 2.69 and Fig. 2.70. Their waveforms in terms of time at 20 and 47 rad/s are also plotted. It is noted that the peak values of phase voltage references after the base speeds are maximal but relatively under their limit (Vlim=100 V).

Method (III)-1

I RMS =5.3 A Method (III)-2 I RMS =5.3 A Chapter 2.

Modeling and Control of Multiphase Drives

2.3.4.G.4. Comparative summary:

A comparative summary of experimental performances in a healthy operation, an OC fault without reconfigurations, and fault-tolerant operations with methods (III)-1 and (III)-2 is presented in Table 2.28. Especially, the highest RMS currents, torques, highest peak voltages, and total copper losses are compared. In general, the calculated and experimental results in Table 2.25 and Table 2.28 are in good accordance. However, the experimental voltage references are higher than the calculated values due to the additional voltages caused by highorder harmonics of the experimental back-EMFs.

Method (III)-1

Method (III)-2

Chapter 2.

Modeling and Control of Multiphase Drives 98

Comparative analyses of fault-tolerant control methods (I), (II) and (III)

In this chapter, the seven fault-tolerant methods have been presented and classified into three main control methods (I), (II), and (III). Each method has its own advantages and disadvantages.

2.3.5.A. Comparisons in terms of control facilitation

Regarding the current control, if FOC technique and classical PI controllers are applied, timeconstant current references can guarantee the control quality, resulting in smooth torques. From Table 2.29, all d-q currents in healthy mode are constant and the zero-sequence current is zero. In fault-tolerant methods, method (I)-1 has the highest number of constant d-q currents (6 currents) while methods (I)-2 and (I)-3 have only 4 constant d-q currents for control. However, method (I)-1 is the only option that requires hardware modifications with a high zerosequence current. This current not only results in high-frequency components in torques and phase currents but also high torque ripples related to back-EMF harmonics associated with the zero-sequence machine (7 th and its multiples). Methods (I)-2 and (I)-3 deal with time-variant currents in the second fictitious machine (sensitive to the 9 th harmonic of back-EMFs). Method (III)-1 has 2 constant d-q currents in the first (main) fictitious machine, making its torque ripple less dependent on the rotating speed. Method (II)-RCA also has 2 constant d-q currents in the second fictitious machine, leading to its robustness (unaffected by back-EMF harmonics associated with the second fictitious machine). The other methods including (II)-SCL and (III)-2 have 6 time-variant d-q currents for control. Therefore, in their torque-speed characteristics, significant increases of torque ripples can be witnessed when the rotating speed increases. However, this problem can be solved in Chapter 3 by using ADALINEs.

2.3.5.B. Comparisons in terms of torque, copper loss, and speed range

The optimal torque-speed characteristics in healthy and under an OC fault in phase A with the seven proposed control options are plotted in Fig. 2.71. These characteristics also represent the Chapter 2.

Modeling and Control of Multiphase Drives 99 comparison between average torques generated with the proposed methods. It is noted that a combination of these proposed methods can improve the operating performances of the drive under the faulty condition. In this case, the highest post-fault average torque for the starting period and a longer speed range are obtained while RMS phase currents and peak phase voltages are within their limits. Table 2.30 compares healthy mode, an OC fault without reconfigurations, and post-fault operations using the seven proposed fault-tolerant methods in this chapter. Torque performances, total copper losses, base speeds, and maximum speeds are considered in these comparisons. Due to a high torque ripple (46% or 50%) and a high total copper loss (1.4 pu or 1.27 pu) under the OC fault without reconfigurations, the use of fault-tolerant control strategies for the electric drive is necessary. It is noted that calculated results for an OC fault without reconfigurations are obtained from MATLAB Simulink while results with proposed methods are obtained from the offline optimization. In general, the optimal (calculated) and experimental results, as summarized in Table 2.30, are in good accordance. However, the experimental machine contains unwanted harmonics of the back-EMFs, resulting in higher torque ripples than optimal results in all operating modes, including healthy mode. In healthy mode as well as in a post-fault operation with each proposed For the torque performance, the highest average torque is generated with method (III)-2 at 0.79 pu in the offline optimization (opt=0.79 pu), and at 0.78 pu in experiments (exp=0.78 pu) thanks to the uniformity of remaining phase current waveforms. However, its torque fluctuation is relatively high compared to the other methods (opt=19% and exp=31%). Method (II)-RCA shows its robustness when its torque is unaffected by the harmonics of the second machine in the experimental machine. Therefore, its experimental torque ripple is lowest with 19%. However, under the constraint on RMS current, its average torque is lowest (opt=exp=0.48 pu). Method (I)-2 has the highest average torque among the three options of method (I) (opt=0.65 pu and exp=0.64 pu).

The total copper loss dramatically increases to 1.27 pu in the OC fault without reconfigurations. This value reduces to the lowest value in method (II)-RCA (opt=0.52 pu and exp=0.48 pu). Meanwhile, the highest total copper loss is generated in method (III) (opt=0.86 pu and exp=0.83 pu). Among the three options of method (I), method (I)-2 generates the highest total copper loss (opt=0.64 pu and exp=0.58 pu).

Regarding the base and maximum speeds, it is noted that post-fault speed ranges are always shorter than that of healthy mode. Among the proposed methods, method (I) results in largest speed ranges while methods (II) and (III) have shorter speed ranges. Method (I)-2 enables the drive to reach the highest speed (0.88 pu) while the shortest speed range is created with method (III)-2 (0.66 pu). Basically, these speed range reductions are caused by putting more constraints on currents in either d-q or natural frames to respect limits of currents and voltages or to obtain smooth torques. For example, in method (III), at high speed, the amplitude of phase currents needs to be immediately reduced instead of alternating current phase angles to guarantee smooth torques. Therefore, in this case, torques decline rapidly when the rotating speeds exceed the base speeds. Chapter 2.

2.3.5.C. Comparisons of three remarkable methods using radar charts

Modeling and Control of Multiphase Drives

To compare methods (I), (II), and (III) more clearly, three remarkable options are selected from the seven proposed options to plot radar charts in Fig. 2.72. In addition, healthy mode and the OC fault without reconfigurations are considered. Torque performances (Tave, ∆T), total copper losses (Ploss), base speeds (Ωbase), maximum speeds (Ωmax), and difficulty are considered in these comparisons. The difficulty is a heuristic assessment according to hardship in calculations and implementations of the considered methods. Method (III)-RCA is the most difficult one because of its calculation burden for all cases of faults.

It is noted that experimental results in all operating modes are in good accordance with calculated results. Only experimental torque ripples are significantly higher than calculated results. Healthy mode is considered as the ideal operating mode with the best performances.

Meanwhile the OC fault without reconfigurations generates the worst parameters such as the highest torque ripple (opt=46% and exp=50%) and the highest total copper loss (opt=1.4 pu and exp=1.27 pu). Therefore, fault-tolerant control methods are required for post-fault operations. In these radar charts, method (I)-2 is chosen because it has the highest maximum speed (0.88 pu). Meanwhile, method (II)-RCA is selected because of its robustness with the lowest experimental torque ripple (exp=19%). Method (III)-2 is added to the radar charts because it generates the highest average torque (opt=0.79 pu).

2.3.5.D. Comparisons with recent studies

Compared to existing studies, for example studies in [87, 120, 126-129, 133-135, 137, 149], the proposed methods in this doctoral thesis can be applied to non-sinusoidal machines with different choices in terms of torque performance (average torque and torque ripple). Especially, the proposed methods in this doctoral thesis take into account the thermal limit (RMS current) and flux-weakening operations by imposing constraints on RMS current and peak voltage.

Besides the methods with several constant d-q currents, two proposed methods in this doctoral thesis using 6 time-variant current references for control (methods (II)-SCL and (III)-2) can be compared with the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. Indeed, the existing approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] can generate ripple-free torques regardless of one or two open-circuited phases or the type of back-EMFs. However, d-q current references usually fluctuate with very high amplitudes. Therefore, hysteresis controllers are applied in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] to directly control currents in natural frame, resulting in variable switching frequencies, increases in switching losses, high-frequency currents and electromagnetic compatibility (EMC) problems. In addition, if a current limit is considered, unidentical waveforms of healthy phase currents in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] will generate low average torques. Meanwhile, methods (II)-SCL and (III)-2 in this doctoral thesis have low fluctuations of currents in frame (d1-q1) which generates most of the torque. Therefore, the pre-fault control scheme with conventional PI controllers can be used in faulty mode. Moreover, the uniformity of phase currents in these methods results in higher average torques if a current constraint is considered.

Conclusions

This chapter has presented the modeling and control the multiphase PMSM drive with nonsinusoidal back-EMFs under heathy and faulty modes. Constraints on RMS current and peak Chapter 2.

Modeling and Control of Multiphase Drives 102 voltage have been always considered. A case study with a seven-phase non-sinusoidal machine and a single-phase OC fault has been chosen to illustrate theories in this chapter. In case of twophase OC faults, fault-tolerant control strategies and experimental results are presented in Appendix B of this doctoral thesis.

Specifically, the modeling and control of the drive in healthy mode have been introduced in section 2.1. In section 2.2, an investigation into the multiphase drive performances in a postfault operation has shown serious deteriorations with a high-ripple torque, a significant increase in RMS currents as well as in the total copper loss. The reason is that the coupling issue prevents the post-fault drive from continuing to use the current references of healthy mode. Therefore, a reconfiguration with fault-tolerant control strategies is required to guarantee a smooth torque and a safe post-fault operation for the drive as well as for EVs.

In section 2.3, three main proposed fault-tolerant control strategies with seven options have been introduced by exploiting the mathematical model of the multiphase drive. Specifically, new current references have been determined in decoupled frames (methods (I)-1-2-3), in natural frame (methods (III)-1-2), or by using new transformation matrices (methods (II)-RCA-SCL). Methods (I) and (II) have theoretically obtained constant torques when the first and third harmonics of back-EMFs are considered. However, their average torques have been relatively low if the RMS phase currents are limited. Therefore, method (III) has been proposed to obtain higher average torques by using identical phase currents. Under the current constraint, these currents with an identical waveform have been effectively exploited to produce more torques. However, torque ripples have been inevitable in method (III). Therefore, there is theoretically a compromise between a high average torque with a high ripple and a smooth torque with a low average value. Therefore, a combination of more than one strategy according to the payload of EVs can take advantage of each method.

The quality of the drive control will be improved by applying ADALINEs in Chapter 3 of this doctoral thesis. Indeed, negative effects of unwanted back-EMF harmonics and other uncertainties in the drive on currents and torques will be eliminated.

Chapter 3. Enhancements of Multiphase Drive Performances with Adaptive Linear Neurons

This chapter is to present improvements of the quality of current control and torque discussed in Chapter 2, including healthy and faulty operations. Specifically, section 3.1 will present basic knowledge about ADALINE (ADAptive LInear NEuron). This choice has been justified in section 1.4 of Chapter 1. Then, section 3.2 will describe how unwanted back-EMF harmonics and the nonlinearity of the inverter deteriorate the quality of current control and torques in healthy mode. These problems will be tackled by using six ADALINEs for current harmonics and one ADALINE for torque ripples with appropriate harmonic injections. Determinations of harmonic components existing in currents and torques allow a fast response of control (fast convergence) and reduce the calculation burden. The improvements of current control and torque performances under a post-fault operation will be described in section 3.3. Indeed, method (III) in Chapter 2, producing the highest average torque with high ripples, will be improved by a single ADALINE like the one for torque ripple eliminations in healthy mode. When harmonic components of the post-fault torque are identified, the ADALINE can directly eliminate torque ripples without the calculation burden. Another application of ADALINEs is to improve current control quality in faulty modes, indirectly enhancing the torque quality. Specifically, time-variant d-q current references with method (II) in Chapter 2 have deteriorated the current tracking performance at high speed, resulting in more torque ripples. A new control scheme with one ADALINE will replace time-variant current references by timeconstant references for control. This chapter will be concluded in section 3.4.

Introduction to adaptive linear neurons Artificial neural networks and adaptive linear neurons

Artificial Neural Network (ANN), as a part of Artificial Intelligence (AI), was introduced in 1940s when the modeling of biological neurons was conducted [START_REF] Haykin | Neural networks: A comprehensive foundation[END_REF]. The main idea of ANNs is to implant humanoid neurons in a machine, enabling the machine to learn from the past data and intelligently react to unprecedented problems during its operation. Thanks to the technological evolution in the domain of numerical calculations, applications of ANNs became more popular in 1990s. ADALINE is an early single-layer artificial neuron that was developed in 1960 [START_REF] Widrow | Adaptive Inverse Control[END_REF][START_REF] Widrow | Adaptive neural networks and their applications[END_REF]. Its outputs are a linear combination of its inputs. A general structure of an ADALINE is described in Fig. 3.1. An ADALINE includes m inputs represented by a mdimensional vector xin, and m corresponding weights with a m-dimensional vector w while its output y is equal to the weighted sum of the inputs. Therefore, the output can be expressed as a dot product of the input vector and the weight vector as follows:

1 ()

m in k ink k= y w x w x t ==  (3.1)
where

  ( ) ( ) ( ) T in in1 in2 inm x x t x t x t = and   T 1 2 m
w w w w = . The Least Mean Square (LMS) or Widrow-Hoff learning rule can be used to update the weights. This learning rule aims at minimizing the squared error between the reference output yref and the output of the ADALINE y as given by: ( ) ( )

2 y ref E y j y j  =- (3.2)
In fact, this rule is the stochastic gradient descent for linear regression as expressed by:

( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) 2 2 2 ref y in ref ref ref in y j y j E wx y j y j y j y j y j y j x w w w  -          = - = - - = - -          (3.3)
where j is the order of iteration. Therefore, the updating rule with learning rate η at the (j+1) th iteration becomes:

( ) ( ) ( ) ( ) ( )

1 ref in
w j w j y j y j x j   + = + - (3.4) According to [START_REF] Nguyen | Approches neuromimétiques pour l'identification et la commande des systèmes électriques : application au filtrage actif et aux actionneurs synchrones[END_REF], the LMS algorithm can be applied with unknown inputs. The initial values of the weights can be chosen or set to zero. After several iterations, the weights will converge to their optimal values when the output y reaches its reference value yref.

To guarantee the stability of an applied system, the learning rate with LMS algorithm should be a value between 0 and 1 [START_REF] Widrow | Adaptive neural networks and their applications[END_REF]. The learning rate depends on parameters of the applied system such as: values of the inputs, the output error, and the sampling time in calculations. Specifically, if the values of the inputs are relatively big compared to the maximum value of the learning rate, the learning rate should be divided by the norm of the inputs to guarantee the stability. When the output error is relatively big compared to the output reference value, the learning rate must be high to quickly reduce the output error. If the output error becomes small, the learning rate must be decreased or even equal to zero to guarantee the stability. In addition, the learning rate is proportional to the sampling time in calculations. With a small value of the sampling time, the learning rate is required to be small. The selection of the learning rate has not been specified for all applications. Therefore, the learning rate must be chosen according to analyses of the above parameters of the applied system. Fig. 3.2 illustrates how the learning rate affects the learning process of the weights. If the learning rate is set too low as in Fig. 3.2a, the training process of weights will be very slow, fast responses of the applied system cannot be guaranteed. However, if the learning rate is set too high as in Fig. 3.2b, the divergence can happen, leading to instability of the applied system. The relatively suitable one is presented in Fig. 3.2c.

Possible applications of ADALINEs in the electric drive

ADALINEs have been applied to various domains such as telecommunication [START_REF] Widrow | Adaptive Inverse Control[END_REF], electrical engineering [START_REF] Nguyen | Approches neuromimétiques pour l'identification et la commande des systèmes électriques : application au filtrage actif et aux actionneurs synchrones[END_REF][START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF][START_REF] Wang | Current Harmonics Suppression Strategy for PMSM with Non-Sinusoidal Back-EMF Based on Adaptive Linear Neuron Method[END_REF][START_REF] Sediki | ADALINE approach for induction motor mechanical parameters identification[END_REF]. This doctoral is to apply ADALINEs in several cases with the aim of refining the considered electric drive as discussed in Chapter 2. Indeed, their applications categorized into healthy and faulty modes are presented in next sections. Besides several advantages such as self-learning and simplicity, ADALINEs have some drawbacks such as: the risk of instability due to an inappropriate learning rate η; and the calculation burden when the number of inputs and weights increases. Therefore, the structures and parameters of ADALINEs need to be properly designed based on properties of specific applications, minimizing these drawbacks.

Control quality in healthy mode

ADALINEs are applied to tackle the existing problems in healthy mode presented in Chapter 2 for a seven-phase machine. Specifically, torque ripples in healthy mode are caused by unwanted harmonics of currents and the back-EMFs of the considered machine as follows:

1) Current harmonics exist in d-q frames although the current references for control are time-constant, especially in (d3-q3) and (d9-q9) frames with a main frequency of 14θ as discussed in Chapter 2. This problem can happen in all electric drives with imperfect back-EMF machines and the inverter nonlinearity. In this case, PI controllers cannot properly filter these current harmonics; hence, PI controllers combined with ADALINEs can be a suitable solution. 2) However, the above combination of PI controllers and ADALINEs for current harmonic eliminations cannot nullify the existing torque ripples in Chapter 2. These torque ripples are also caused by the unwanted harmonic components in the imperfect back-EMFs of the considered machine. In a seven-phase machine, these torque ripples have a main frequency of 14θ. Therefore, these torque ripples need to be eliminated by another ADALINE. The origins of the current harmonics in d-q frames and torque ripples will be analyzed to effectively define the structure of the ADALINEs, avoiding the calculation burden. In general, these eliminations using ADALINEs can be applied to other electric machines with a different number of phases and various back-EMF waveforms.

Impacts of unwanted back-EMF harmonics and the inverter nonlinearity

3.2.1.A. Impacts of unwanted back-EMF harmonics

According to the multi-reference frame theory [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF], control of a wye-connected electrical machine is ideal when only one harmonic is associated with each d-q frame. For example, the back-EMFs of a seven-phase machine should have only 3 harmonics distributed among 3 fictitious machines (reference frames). In this case, a constant torque can be generated by constant d-q currents. Due to wye-connected stator windings with a nullified zero-sequence current, impacts of the time-variant zero-sequence back-EMFs on phase currents and torques are automatically eliminated. However, the back-EMFs may contain more than one back-EMF harmonic in each fictitious machine, called unwanted back-EMF harmonics. For example, more than one harmonic associated with each reference frame are generally described in Table 3.1 for a seven-phase machine. If the back-EMFs contain only the 1 st , 3 rd , and 9 th harmonics, the values of the back-EMFs in (d1-q1), (d3-q3), and (d9-q9) frames are constant.

Table 3.1. Fictitious machines, d-q reference frames, and several associated odd harmonics in natural frame of a seven-phase machine.

Fictitious machine Reference frame Associated harmonic The first machine (FM1)

d1-q1 1, 13, 15 The second machine (FM2) d9-q9 5, 9, 19 The third machine (FM3) d3-q3 3, 11, 17

Without loss of generality, to see impacts of unwanted back-EMF harmonics on current control, it is assumed that there are two associated harmonics per reference frame as follows:

1) The 1 st and 13 th harmonics associated with (d1-q1).

2) The 9 th and 19 th harmonics associated with (d9-q9) 2 .

3) The 3 rd and 11 th harmonics associated with (d3-q3).

Therefore, the unwanted back-EMFs are the 11 th , 13 th , and 19 th harmonics. In this case, the back-EMF of a phase in natural frame is generally given by: (

E j E j φ E j + φ E j + φ E j + φ E j + φ e                    -- + -- + + -- +               =           -- + -- + --                 (3.5)
where ej is the back-EMF of phase j (from 1 to 7, representing phases A to G, respectively); (E1, E3, E9, E11, E13, E19) and (0, φ3, φ9, φ11, φ13, φ19) are the amplitudes and phase shift angles of the 1 st , 3 rd , 9 th , 11 th , 13 th , and 19 th harmonics of the back-EMF, respectively.

By applying the classical Clarke and Park transformation matrices in section 2.1 of Chapter 2, the back-EMFs in d-q reference frames become:

(

       =+   = + +   =+    = + +    =+   = + +   (3.6)
In (3.6), it is worth noting that the back-EMFs in d-q reference frames are constant if there are only the 1 st harmonic (E1), the 9 th harmonic (E9), and the 3 rd harmonic (E3). However, the presence of the unwanted back-EMF harmonics results in harmonics in d-q frames. The back-EMFs have frequencies: 14θ in (d1-q1), 28θ in (d9-q9), and 14θ in (d3-q3). Especially, these back-EMF harmonics generate corresponding current harmonics in d-q frames. The current harmonic amplitudes in d-q frames depend on the harmonic distribution in the back-EMFs, and especially on the rotating speed.

3.2.1.B. Impacts of the inverter nonlinearity

Besides the current harmonics caused by the unwanted back-EMF harmonics, the nonlinearity of the inverters also creates extra current harmonics in d-q frames. The dead time, the time interval in which both of switches of one inverter leg are off, can mainly cause the nonlinearity of inverters. To represent this nonlinearity, according to [START_REF] Grandi | Analysis of dead-time effects in multi-phase voltage source inverters[END_REF], a "dead-time" voltage in a phase of a 7-phase VSI can be generally expressed by:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
j j j v V j j j jj                     -- + -- + -- +                 = - -- + -- + -- +                 -- + --       ( ) 12 sin 19 ( 1) ... 19 7 j                  + -- +           (3.7)
with

dead dead DC PWM T VV T =
where vj_dead is the "dead-time" voltage of phase j; Vdead is a constant voltage; Tdead is the inverter dead time; TPWM is the switching period of the inverter; VDC is the DC-bus voltage. Indeed, using the Fourier analysis, the "dead-time" voltage is expressed by an average signal composed of odd harmonics in (3.7). The harmonic amplitudes are inversely proportional to their orders. Therefore, without loss of generality, the considered harmonics can be up to 19θ. In addition, harmonics with frequencies equal to multiples of the number of phases (7 phases in this doctoral thesis) are null.

The "dead-time" voltages of all phases are transformed into d-q reference frames as follows:

( )   ( ) ( )    ( )   ( )   ( ) ( ) ( )  ( ) ( )   ( )   ( ) ( ) 7
v V v V v V v V v V v V             = - + = - + + = - +     = - + +     = - + =-     ( )   1 3 1 11 1 17 cos 14 d                ++   (3.8)
In (3.8), it is noted that the "dead-time" voltages in rotating frames (d1-q1) and (d3-q3) have a frequency of 14θ while these voltages in (d9-q9) have frequencies of 14θ and 28θ. There are no "dead-time" voltage harmonics in the zero-sequence frame because there are no frequencies equal to multiples of the number of phases as shown in (3.7). Therefore, current harmonics in d-q reference frames are also caused by the inverter nonlinearity with the "dead-time" voltages. The harmonic amplitudes do not depend on the rotating speed but Vdead that is related to Tdead, TPWM, and VDC.

3.2.1.C. Summary of the impacts of unwanted back-EMF harmonics and the inverter nonlinearity

From the previous subsections, the current harmonics in rotating reference frames caused by the unwanted back-EMF harmonics in (3.6) and the "dead-time" voltages in (3.8) are summarized in Table 3.2. Based on the previous analyses, in a specific drive, the amplitudes of the total current harmonics mostly depend on the rotating speed, harmonic spectrum, and Vdead but not the current reference values. Fig. 3.3. The control scheme of current ix under the impacts of the unwanted back-EMF harmonic ex and the inverter nonlinearity with "dead-time" voltage vx_dead without any compensations (x can be d1, q1, d9, q9, d3, or q3).

There are six d-q currents to be controlled as described in Fig. 2.6 in section 2.1 of Chapter 2 for a wye-connected winding topology. Current ix (x can be d1, q1, d9, q9, d3, q3) is controlled in the scheme as described in Fig. 3.3 where a conventional PI controller is applied. Transfer function (1/(Lxs+Rs)) represents the fictitious machine model with their inductance and resistance in the corresponding d-q reference frame. This control scheme under the impacts of the unwanted back-EMF harmonic ex and the "dead-time" voltage 𝑣 𝑥_𝑑𝑒𝑎𝑑 without any PI compensations. In Fig. 3.3, parameters in red represent unchangeable internal properties of the drive.

Eliminations of current harmonics in rotating frames

3.2.2.A. The conventional control scheme with the back-EMF compensation

Theoretically, the current harmonics in d-q frames, generated by the unwanted back-EMF harmonics, can be eliminated by imposing corresponding estimated back-EMF harmonics as calculated in (3.6), called the back-EMF compensation. The conventional control scheme using the back-EMF compensation (ex_com) for a current (id1, iq1, id9, iq9, id3, or iq3) is described in Fig. 3.4.

Fig. 3.4. The conventional control scheme of current ix under the impacts of the unwanted back-EMF harmonic ex and the "dead-time" voltage vx_dead with the back-EMF compensation ex_com (x can be d1, q1, d9, q9, d3, or q3).

However, the accuracy of the back-EMF compensation depends on the parasitic time constant of the drive, resulting in phase angle differences between the estimated back-EMFs and the experimental back-EMFs. In addition, the estimated back-EMFs based on the no-load operation are different from the experimental back-EMFs under loads. Therefore, the back-EMF compensation may increase, instead of decreasing, the amplitudes of current harmonics. In addition, current harmonics in d-q reference frames are also caused by the inverter nonlinearity with the "dead-time" voltages. Obviously, these current harmonics cannot be eliminated by the back-EMF compensation.

3.2.2.B. The proposed control scheme to eliminate current harmonics in rotating frames

As described in Table 3.2, the current harmonics have frequencies of 14θ and 28θ in d-q reference frames. Their amplitudes depend on the back-EMF harmonic distribution, rotating speed, inverter dead time, switching period, and DC-bus voltage. Due to the complexity of the real drive, these values need to be automatically learned in real time to correctly eliminate the current harmonics. With the knowledge of the existing harmonics, the use of ADALINEs is suitable because a fast response of control is guaranteed, and the calculation burden is avoided.

Studies in [START_REF] Nguyen | Approches neuromimétiques pour l'identification et la commande des systèmes électriques : application au filtrage actif et aux actionneurs synchrones[END_REF][START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] have proved the effectiveness of ANNs as well as ADALINEs thanks to their self-learning, simplicity, and fast convergence. Recently, an ADALINE has been used to eliminate parasitic current harmonics caused by unwanted back-EMFs at high speed in healthy operation for a 3-phase machine as proposed in [START_REF] Wang | Current Harmonics Suppression Strategy for PMSM with Non-Sinusoidal Back-EMF Based on Adaptive Linear Neuron Method[END_REF].

Therefore, based on the control structure in Fig. 2.6 in section 2.1 of Chapter 2, a new current control scheme is a combination of a PI controller and an ADALINE as proposed in Fig. 3.5.

Current ix (x can be d1, q1, d9, q9, d3, or q3) is controlled by a PI controller with an adaptive compensating voltage (𝑣 𝑥_𝑐𝑜𝑚 ) generated by an ADALINE. By using the compensating voltage (𝑣 𝑥_𝑐𝑜𝑚 ), all current harmonics in d-q frames generated by either the unwanted back-EMF PI harmonics or the "dead-time" voltages (inverter nonlinearity) can be eliminated. This compensation is called the ADALINE compensation. Fig. 3.5. The proposed control scheme of current ix with current harmonic eliminations by using the ADALINE compensation (𝑣 𝑥_𝑐𝑜𝑚 ) (x can be d1, q1, d9, q9, d3, or q3).

The inputs of the ADALINE are based on current harmonics as described in Table 3.2. The output of the ADALINE is the compensating voltage vx_com that is generally calculated from harmonics 14θ and 28θ as follows:

_ cos( 14) sin( 14) cos( 28) sin( 28)

x com 1_x 2_x 3_x 4_x v w w w w         = + + +     (3.9)
where (w1_x, w2_x) are weights associated with harmonic 14θ; (w3_x, w4_x) are weights associated with harmonic 28θ.

To online update these weights, the error vx_com_error that is the difference between the output vx_com and its desired value vx_com_ref needs to be found. Because vx_com is designed to compensate the unwanted back-EMF harmonics and "dead-time" voltages, the desired output vx_com_ref is the sum of the back-EMF (ex) and "dead-time" voltages (vx_dead). Therefore, the error vx_com_error can be calculated as follows:

( )

_ _ _ _ x com error x com ref x_com x x_dead x_com v v v e + v v = - = - (3.10)
As previously discussed, the back-EMF (ex) and "dead-time" voltages (vx_dead) cannot be measured during the operation of the experimental drive. Therefore, vx_com_ref is difficult to be determined, leading to the difficulty in determining the error vx_com_error for the updating laws. However, it is noted that the compensating voltage error vx_com_error is proportional to the current error ierror between ix_mea and ix_ref. Indeed, when vx_com_error is small, the proper compensating voltage is injected to reduce the impacts of the unwanted back-EMF harmonics and the inverter nonlinearity. Therefore, ierror becomes small or even zero if the PI controller is well designed.

In addition, current ix_mea can be measured from the drive; hence, the updating laws of harmonic weights can be expressed in Fig. 3.5 with current error ierror. Weights of a harmonic are estimated by a learning process with learning rate η, current error ierror, and corresponding inputs (harmonics).

It is noted that there are six ADALINEs combined with six PI controllers for six d-q currents (id1, iq1, id9, iq9, id3, and iq3) in a wye-connected winding seven-phase machine. The presence of harmonics (14θ and 28θ) in voltage 𝑣 𝑥_𝑐𝑜𝑚 in one ADALINE depends on the reference frame of the controlled current according to ADALINE structure has only 2 weights (w1_x, w2_x) for harmonic 14θ. Meanwhile, if x is d9 or q9, 4 weights (w1_x, w2_x, w3_x, w4_x) are used for 2 harmonics 14θ and 28θ. Therefore, the number of weights is optimized to avoid the calculation burden.

The proposed control scheme, eliminating current harmonics caused by unwanted back-EMF harmonics and the inverter nonlinearity, can be generally applied to different electric machines with different back-EMF characteristics and a different number of phases. In a specific case, the harmonic components of currents in d-q frames need to be determined to avoid the calculation burden. Generally, the proposed control scheme can improve the current control quality even when current references in d-q frames are time-variant.

3.2.2.C. Verification with experimental results

The considered experimental drive (see section 2.1.4.A of Chapter 2) is applied to verify effectiveness in eliminating current harmonics in rotating frames of the proposed control scheme in Fig. 3.5 over the back-EMF compensation in Fig. 3.4.

3.2.2.C.1. The back-EMF compensation:

The conventional current control scheme in Fig. 3.3 is applied to the experimental drive (see section 2.1.4.A). It is noted that the back-EMF compensation results in high amplitudes of current harmonics in all d-q frames as described in Fig. 3.6. This current control performance is worse than the case without any compensations in section 2.1.4 of Chapter 2. As previously discussed, the reason of these distortions is that the compensating back-EMFs are not the same as experimental back-EMFs in terms of phase shift angles even amplitudes. Indeed, the experimental drive contains the parasitic time delay, uncertainties of back-EMFs, inverter "dead-time" impacts, and delays caused by calculations. In conclusion, a self-learning compensation instead of the back-EMF compensation needs to be implemented in the real-time experimental drive. Therefore, the ADALINE compensation is a suitable solution.

3.2.2.C.2. The ADALINE compensation:

The effectiveness of the ADALINE compensation in current harmonic eliminations is shown in Fig. 2.9. The current control performances are significantly improved, especially in (d9-q9) and (d3-q3) frames. Fig. 3.8a and Fig. 3.9a show the convergence of harmonic weights within Chapter 3. Enhancements of Multiphase Drive Performances with Adaptive Linear Neurons 112 20 s when the learning rate 𝜂 is 0.0001. According to the LMS rule, 𝜂 is between 0 and 1. In this doctoral thesis, the value of 𝜂 has been determined through simulations before applying to the experimental drive. It is noted that this value is proportional to the calculating step (sampling time) of the simulating and experimental systems. In addition, with more powerful processors of the computer and dSPACE, the sampling time is smaller; hence, the response time is shorter. Current control performances without and with the ADALINE compensation in frames (d9-q9) are described in Fig. 3.8b and Fig. 3.8c, respectively. Similarly, Fig. 3.9b and Fig. 3.9c are for currents in (d3-q3) frame. The current of phase A without and with the ADALINE compensation is shown in Fig. 3.10a. The current waveforms in the two cases are slightly different but the Meanwhile the variations of current references (torque references) at 20 rad/s do not affect the current control quality as described in Fig. 3.12. As previously discussed, the reason is that the current harmonics in rotating frames significantly depend on the rotating speed but not on the d-q current references. To see demonstration videos, please click on this link: https://youtu.be/jJcokkMIusI or scan the nearest QR code in this page.

3.2.2.D. Torque performance after eliminating current harmonics in rotating frames

When the current harmonics in d-q frames are previously eliminated by the ADALINE compensation, all d-q currents are well controlled. In other words, phase currents only contain the imposed harmonics, for example 1 st , 3 rd , and 9 th in the considered drive. However, the torque ripple still exists at 11% as shown in Fig. 3.13, like the torque ripple of 12% without current harmonic eliminations (see section 2.1.4 of Chapter 2). The reason is that the remaining current harmonics (1 st , 9 th , and 3 rd ) interact with unwanted back-EMF harmonics (13 th , 19 th , and 11 th ) to generate torque ripples with frequencies of 14θ and 28θ as described in Table 3.3. For example, the torque in the first fictitious machine T1_exp has a frequency of 14θ because the 1 st harmonic component of currents interacts with the 13 th harmonic of back-EMFs. Similarly, torques in the second and third fictitious machines (T2_exp and T3_exp) have frequencies of 28θ and 14θ, respectively. Finally, the experimental torque of the considered machine Tem_exp that is the sum of the three torques generated by the three fictitious machines has frequencies of 14θ and 28θ. Eventually, the current harmonic eliminations can improve current control quality but cannot guarantee a smooth torque. In the next subsection, the above torque ripples will be nullified by another single ADALINE.

T 2_exp T em_exp T 3_exp T 1_exp

Direct eliminations of torque ripples in healthy mode

3.2.3.A. The proposed control scheme to directly eliminate torque ripples in healthy mode

The principle of the proposed control scheme is to add compensating currents to the existing constant current references of the offline optimization. These compensating currents are directly determined from harmonic components of the electromagnetic torque. In other words, an additional torque, called compensating torque Tem_com, needs to be added. This torque is expected to be equal to the difference between the constant torque reference Tem_ref and the experimental torque Tem_exp with ripples in Fig. 3.13. The torque reference Tem_ref is a constant torque in the optimal torque-speed characteristic, as described in Fig. 2.9a of Chapter 2, to respect the constraints on current and voltage. Therefore, the desired value of the compensating torque is described as follows:

_ _ _ em com ref em ref em_exp T T T =- (3.11)
To determine the compensating torque, the control structure with only one ADALINE is proposed in Fig. 3.14. The inputs are harmonic components presenting in the experimental torque in Table 3.3. The output of the ADALINE is the compensating torque Tem_com that is the weighted sum of the inputs as follows: _ cos [START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF] sin( 14) cos( 28) sin( 28)

em com 1_tor 2_tor 3_tor 4_tor T w w w w         = + + +     (3.12)
where (w1_tor, w2_tor) and (w3_tor, w4_tor) are weights of two harmonics 14θ and 28θ, respectively.

According to the ADALINE principle, the error Tem_com_error, the difference between the desired compensating torque Tem_com_ref and the output of the ADALINE Tem_com, needs to be determined for weight updating laws. Meanwhile, as previously mentioned, the desired compensating torque is calculated as in (3.11). Therefore, the error can be calculated as follows: 14) sin( 14) cos( 28) sin( 28) From (3.13), the error of the compensating torque for updating laws can be described in (3.14) where Tem_exp_withcom is the experimental torque with the compensating torque. The details of the updating laws are shown in Fig. 3.14. Therefore, the weights of harmonic components are updated by the learning rate η, torque error Terror, and corresponding harmonics.

( ) ( ) _ _ _ _ _ _ _ __ em com
Thereafter, as introduced in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], compensating currents in natural frame icom, calculated from compensating torque Tem_com and back-EMFs e, are expressed by: Then, these currents are transformed into d-q frames idq_com and added to constant current references idq_ref generated by the offline optimization in Fig. 2.6 of Chapter 2. Eventually, the total current references 𝑖 𝑑𝑞_𝑟𝑒𝑓 * are used to create the electromagnetic torque Tem_exp_withcom in the seven-phase drive. The total current references in d-q frames are no longer constant due to the compensating currents.

It is worth noting that the approach in Fig. 3.14 only aims at learning only pulsating components of the torque. Therefore, d-axis current references that do not generate torques are almost zero, facilitating current control with conventional PI controllers. Meanwhile, the approach in [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] considers the entire torque in the learning process, resulting in all time-variant current references, similar to [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. For the sake of simplicity, in the next subsection, study [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] will be used to compared with the proposed structure in Fig. 3.14 of this doctoral thesis. This proposed structure in Fig. 3.14 to eliminate torque ripples caused by unwanted back-EMF harmonics can be generally applied to different electric machines with different back-EMF characteristics and a different number of phases. In those cases, the harmonic components of the electromagnetic torque need to be determined to reduce the number of weights, avoiding the calculation burden. Theoretically, these harmonics are multiples of the number of phases.

3.2.3.B. Comparisons with the vectorial approach by real-time simulation results

With the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], a constant torque can always be obtained regardless of operating modes by using current references as generally calculated in (2.8)-(2.9) in Chapter 2. The performances of the vectorial approach are compared with the proposed structure using an ADALINE in Fig. 3.14. The comparisons are based on real-time simulations by Simulink in MATLAB and the considered drive from section 2.1.4 of Chapter 2 with the unwanted back-EMF harmonics (11 th , 13 th , and 19 th ). The learning rate η in all ADALINEs is chosen to be 0.0001. The average torque is expected to be unchanged regardless of control strategies. It is noted that, the case using only ADALINEs for current harmonic eliminations in rotating frames has been previously discussed in section 3.2.2.D. These eliminations of current harmonics cannot eliminate torque ripples. Therefore, for the sake of simplicity, an operation of the drive consists of 4 stages as follows:

1) Stage 1: Use original constant d-q current references idq_ref from the offline optimization in Fig. 2.6 of Chapter 2 with inevitable torque ripples. 2) Stage 2: Apply the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF] to define new current references (timevariant), as calculated in (2.8)-(2.9) in Chapter 2, to generate a theoretically constant torque. 3) Stage 3: Apply the proposed structure in Fig. 3.14 using the ADALINE for torque ripple eliminations. 4) Stage 4: Combine the ADALINE for torque ripple eliminations in Fig. 3.14 and the six ADALINEs for current harmonic eliminations 3 in Fig. 3.5. At a speed of 20 rad/s, as described in Fig. 3.15a, torques in stages 3 and 4 are almost constant, much better than stage 1 with a ripple of 11%, and slightly better than stage 2 with a ripple of 1.3%. Therefore, the effectiveness of the single ADALINE for torque ripple eliminations (stage 3) and its combination (stage 4) with the six ADALINEs for current harmonic eliminations is not clear at low speed. It is worth noting that current references in stages 2, 3, and 4 are not constant. The current references in stage 2, generated by the vectorial approach in [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF], are time-variant with high amplitudes. Meanwhile, these currents of the proposed structure with ADALINEs in stages 3 and 4 have lower amplitudes, especially the d-axis currents. This important characteristic affects current control quality at higher speeds.

Indeed, at a speed of 60 rad/s, as shown in Fig. 3.15b, the torque in stage 4 is still constant while stage 2 with the vectorial approach has a ripple of 5.4%. The torque in stage 3 is no longer constant with a ripple of about 1%. It is noted that the ADALINE for torque ripple eliminations combined with the six ADALINEs for current harmonic eliminations in stage 4 results in the highest control and torque quality. Thanks to the six ADALINEs for current harmonic eliminations as presented in the previous subsection, the currents are better controlled.

Eventually, from now on in this chapter, the torque ripple elimination in Fig. 3.14 is always combined with the current harmonic elimination in Fig. 3.5, generally called the torque ripple elimination. However, for the sake of simplicity, only the learning process to eliminate torque ripples will be discussed in the next subsections.

3.2.3.C. Verification with experimental results

If the optimal torque speed-characteristic for the torque ripple elimination is the same as the original characteristic as described in Fig. 2.9a of Chapter 2, the constraint on RMS current is mostly respected but the peak phase voltage references will exceed its limit 100 V. Therefore, the offline optimization in Fig. 2.6 of Chapter 2 needs to be recalculated with a smaller value of Vlim_opt instead of 75 V. In Fig. 3.16a, according to simulation assessments, a new optimal torque-speed characteristic for the torque ripple elimination is calculated with Vlim_opt=55 V, called the reduced characteristic. The base speed is reduced from 38 to 25 rad/s. Calculated torque values at speeds over the new base speed are smaller than those in the original characteristic, leading to a smaller speed range. The experimental torque-speed characteristics without and with the torque ripple elimination are described in Fig. 3.16b and Fig. 3.16c. It is noted that the torque ripple at 20 rad/s, before the base speed, is significantly reduced from about 12 to 2.8% after the torque ripple elimination. In the flux-weakening regions, the ripple of the original characteristic at 55 rad/s is 9% while the torque ripple of the new characteristic at 35 rad/s is 4.7%. In the flux-weakening The measured phase currents are described in Fig. 3.17a with a slight increase in the RMS current from 5.1 to 5.3 A. The reason is that the compensating currents with harmonics are imposed to eliminate torque ripples. Meanwhile, the phase voltage references almost respect the peak voltage limit of 100 V as described in Fig. 3.17b. Indeed, as previously discussed, the voltage limit for the offline optimization Vlim_opt is reduced from 75 V to 55 V to respect the voltage constraint in the experimental drive. The ADALINE learning process to eliminate torque ripples at 20 rad/s are described in Fig. 3.18a. With a learning rate η of 0.001, harmonic weights (w1_tor, w2_tor, w3_tor, w4_tor) converge within 7 s. An increase in the learning rate η can make the convergence of the harmonic weights faster, but the instability of the control system may happen. With more powerful processors of the computer and dSPACE, the convergence time could be reduced. After the learning process, the torque ripples are significantly reduced from 12 to 2.8%. More clearly, torque and current control performances in one period from the 40 th second after the learning process are shown in Fig. 3.18b. The time-variant d-q currents are properly controlled, especially currents (id1, iq1), generating most of the torque. Like the simulation results in Fig. 3.15, the d-axis currents (id1, id9, id3) are almost zero, facilitating current control with conventional PI controllers. The dynamic performances of the proposed control structure to eliminate torque ripples are validated when the operating speed suddenly changes from 20 to 35 rad/s, over the base speed, and return to 20 rad/s. According to the torque-speed characteristic in Fig. 3.16c, this speed variation also leads to changes of the torque as well as current references to respect the constraint on peak phase voltage references. Fig. 3.19 shows adaptations of the torque, torque harmonic weights, and current control performances in response to the changes of the rotating speed and current references. An increase in the rotating speed from 20 to 35 rad/s results in a decrease in the average torque from 33.5 to 28.7 Nm. In addition, the corresponding torque ripple increases from 2.8 to 4.7%. However, this increase in the torque ripple is mainly due to the decrease in the average torque but not due to the ADALINE performance. To see demonstration videos, please click on this link: https://youtu.be/ylb0cc0_n7U or scan the nearest QR code in this page.

I RMS =5.3 A (

Control quality in faulty mode

Among the fault-tolerant control methods proposed in Chapter 2, method (III)-2 results in the highest average torque but high torque ripples (see section 2.3.4.G of Chapter 2). Meanwhile, method (II)-RCA with robustness, unaffected by the back-EMF harmonics in the second fictitious machine (5 th and 9 th , for example), suffers from the time-variant d-q current references. Consequently, at high speed, the current control quality with conventional PI controllers is reduced, leading to higher torque ripples (see section 2.3.3.F of Chapter 2).

Therefore, this section provides various solutions to improve the torque performance with ADALINEs in faulty mode as follows:

1) An ADALINE is used to directly eliminate torque ripples in method (III)-2. Its structure is like the one in healthy mode, but harmonic components of the torque are redefined to avoid the calculation burden. 2) An ADALINE is applied to method (II)-RCA to improve current control quality, resulting in higher-quality torques. Indeed, the ADALINE is to separate harmonics of measured phase currents in natural frame, enabling current control with time-constant references. Therefore, the torque quality can be improved at high speed.

Direct eliminations of torque ripples in faulty mode with method (III)-2

3.3.1.A. Harmonic components of torque and the proposed control structure

In method (III)-2, new current references are designed in natural frame with identical waveforms, including the first and third harmonic components. As discussed in Chapter 2, when only the first and third harmonic components of the back-EMFs are considered, the torque consists of ripples with frequencies of 2θ, 4θ, and 6θ and the total calculated ripple of 19%.

More generally, when the 9 th , 11 th , and 13 th harmonics are considered in the back-EMFs besides the 1 st and 3 rd harmonics, possible harmonic components of the torque can be up to 22θ as presented in Table 3.4. Therefore, the structure of an ADALINE for method (III)-2 is described in Fig. 3 where the back-EMF of phase A is zero in efault. However, in real-time calculations, only main harmonics of the torque, determined from the back-EMF or torque harmonic spectrum, can be selected instead of all possible harmonics to reduce the number of weights, avoiding the calculation burden. A reduced optimal torque-speed characteristic is defined to respect the constraints on peak voltage as described in Fig.

3.3.1.B. Verification with experimental results

T em_opt T em_exp

With torque ripple elimination (Reduced characteristic) torque of 26.1 Nm. At 30 rad/s, the average torque decreases to 16.3 Nm in the reduced characteristic due to the peak voltage limit. It is worth noting that the torque ripple reduces from 32% to 10% after the ripple elimination.

The measured phase currents are described in Fig. 3.22a with a slight increase in the highest RMS current from 5.1 to 5.4 A. The reason is that the compensating currents with harmonics are imposed to eliminate torque ripples. Meanwhile, the phase voltage references almost respect the peak voltage limit of 100 V as described in Fig. 3.22b. Indeed, as previously discussed, the voltage limit for the offline optimization Vlim_opt is reduced from 75 to 55 V to respect the voltage constraint in the experimental drive. The ADALINE learning process to eliminate torque ripples at 20 rad/s are described in Fig. 3.23. According to the experimental seven-phase machine in this doctoral thesis, the harmonic components for the ADALINE to eliminate torque ripples can be down to 14θ with 7 even harmonics. Therefore, there are 14 harmonic weights that converge within 5 s with the learning rate η of 0.001. An increase in the learning rate η can make the convergence of the harmonic weights faster, but the instability of the control system may happen. With more powerful processors of the computer and dSPACE, the convergence time could be reduced. After the learning process, the torque ripples are significantly reduced from 31 to 5%. More clearly, torque and current control performances in one period from the 10 th and 20 th second, equivalent to without and with the torque ripple elimination, are respectively shown in Fig. 3.24a and Fig. 3.24b. The current control quality in these figures is similar. However, because of generating most of the torque, new current references in (d1-q1) frame in Fig. 3.24b are significantly modified by the ADALINE with additional compensating currents to eliminate torque ripples. The dynamic performances of the proposed control structure to eliminate torque ripples are validated when the operating speed changes from 20 to 30 rad/s, over the base speed, and then returns to 20 rad/s. According to the torque-speed characteristic in Fig. 3.21c, these speed variations also lead to changes in the torque reference as well as current references to respect the constraint on peak voltage. Fig. 3.25 shows adaptations of the torque and harmonic weights in response to the changes of the rotating speed and current references. An increase in the rotating speed from 20 to 30 rad/s results in a decrease in the average torque from 26.1 to 16.3 Nm. In addition, the corresponding torque ripple increases from 5 to 10%. However, this increase in the torque ripple is due to the decrease in the average torque but not due to the ADALINE performance. To see demonstration videos, please click on this link: https://youtu.be/pqxElkrpU0Y or scan the nearest QR code in this page. ] if the first and third harmonic components of the measured phase currents are not separated. The time-variant current references reduce the control quality at high speed as shown in section 2.3.3.F.2. Therefore, a solution to separate harmonics of measured phase currents is necessary. Fig. 3.26. Time-variant d-q current references generated with method (II)-RCA for the pre-fault control scheme.

3.3.2.B. Proposed control structure to use time-constant d-q current references

In [START_REF] Nguyen | Neural networks for phase and symmetrical components estimation in power systems[END_REF], a PLL based on ADALINE has been proposed and it has been concluded that the ADALINE-based PLL gives a better performance compared to the conventional PLL. That is thanks to several advantages of neural networks such as online self-learning, fast convergence, and simple implementation in real time. Therefore, an ADALINE is a good option to separate harmonic components of measured phase currents, allowing to use more constant d-q current references for control.

In this doctoral thesis, a real-time current learning (RTCL) using an ADALINE is proposed. RTCL is to learn and separate measured phase currents into several harmonic components, for example the 1 st and 3 rd harmonics. These separated harmonics are transformed into d-q frames by using [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 ] and [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑 ] to obtain constant d-q currents in method (II). Therefore, the new control structure for method (II) is represented by EMR in Fig. 3.27 in which the offline optimization considering constraints on RMS current and peak voltage is implemented. It is noted that, the new control structure has 10 PI controllers instead of 6 PI controllers in the prefault control scheme in Fig. 2.21 of Chapter 2.

The details of RTCL are described in the block diagram of the new control structure in Fig. 3.28. The real-time current learning RTCL aims at separating harmonics of measured phase currents to estimate useful current harmonics in natural frame. The harmonic separation allows to obtain constant d-q currents used as feedback signals of PI controllers. The most important part of RTCL is a simple ADALINE to estimate the measured current of only a single arbitrary phase among the remaining healthy phases. Then, the other phase currents are derived from their mathematical relationships with the ADALINE-based estimated current by using current design options RCA and SCL (see sections 2.3.3.B and 2.3.3.C of Chapter 2). The minimization of the number of ADALINEs avoids the calculation burden, enabling to easily implement RTCL in real time. For example, if phase B is chosen, the ADALINE-based estimation for the measured current of phase B is presented in Fig. 3.29. The estimated current iB_est is equal to the sum of several main harmonics depending on the back-EMF harmonic distribution as follows: where (w1, w2) are weights for harmonic θ; (w3, w4) are weights for harmonic for 3θ; and (w5, w6) are weights for harmonic for 9θ. In the considered seven-phase machine, the 1 st , 3 rd , and 9 th harmonics are considered with the idea of using one harmonic per two-phase fictitious machine. In Fig. 3.29, estimated current iB_est is compared to measured current iB_mea to obtain current To sum up, the principle to implement RTCL when phase B is chosen to be estimated by the ADALINE is described as follows:

      _ 1 2
V DC I DC  MS T em  T Load ES DC bus Inverter 7-phase PMSM Shaft Mechanical Load RTCL A B C D E F G i i i i i i i i       =           Strategy Optimization Look-up table Fault-tolerant control method (II) B1 C1 D 1 m 1 E1 F1 G 1 i i i i i i i       =         A_ref B_ref C_ref D_ref ref E_ref F_ref G_ref v v v v v v v v         =         B3 C3 D 3 m 3 E3 F3 G 3 i i i i i i i       =         d11 dq11 q11 i i i   =     d33 dq33 q33 i i i   =     d93 dq93 q93 i i i   =     d91 dq91 q91 i i i   =     PI PWM Inverter & 7
1) Estimate the measured current of phase B by using the ADALINE in Fig. 3.29. Due to the back-EMF assumption, the 1 st and 3 rd harmonics of current are considered as useful harmonics to be obtained. Therefore, the 1 st and 3 rd harmonic currents of phase B in natural frame (iB1_est and iB3_est) are estimated. are determined from differences between the measured phase currents and the estimated 1 st harmonic currents as described in RTCL block of Fig. 3.28.

The flowchart for dynamic performances is described in Fig. 3.30 with changes in rotating speed or current references. With fixed current references, the harmonic weights of the ADALINE do not change when the rotating speed varies because current waveforms are unchanged at all speeds. Therefore, the new current control structure using RTCL can be applied to variable-speed applications with a given torque.

When current references (idq_ref means iq11_ref and iq33_ref) (as well as the torque reference) are required to change, the drive needs to return to the pre-fault control scheme in Fig. 2.21 with time-variant current references in Fig. 3.26 as discussed in section 2.3.3 of Chapter 2. Without returning to the pre-fault control scheme, the drive will be unstable because the ADALINE is an inner loop of the current control, providing the false feed-back signals for current controllers.

After switching to the pre-fault control scheme, the learning process of phase-B current is activated with the ADALINE to determine new values of the harmonic weights. After that, the new control scheme can be re-applied to improve the control quality of the drive. In this doctoral thesis, the switch of these two schemes is manually made. This point can be considered as a drawback of the new control scheme.

3.3.2.C. Verification with experimental results

The new control structure to improve current control quality can be applied to methods (II)-RCA and SCL. However, as previously mentioned, only method (II)-RCA will be verified in this chapter, and the verification for method (II)-SCL is similar.

The experimental drive in section 2.1.4 of Chapter 2 continues to be applied. To validate the effectiveness of the proposed control scheme, the speed of the electric drive is limited to 35 rad/s although the maximum speed of the original characteristic presented in Chapter 2 is 49 rad/s. The reason is that the voltage references slightly increase compared to the original ones, reducing the speed range. The experimental machine has been designed for wind turbines with a smaller speed range compared to automotive applications. The following results demonstrate the idea of this work. If the speed could increase to higher values, the proposed strategies would be more effective.

At low speed, there are no differences in the current control and torque performances between the original method (II)-RCA (the pre-fault control structure) and method (II)-RCA with RTCL (the new control structure). However, at 35 rad/s, the new control structure with RTCL generates lower torque ripples compared to the pre-fault control structure as described in Fig. 3.31. Indeed, in the third and final stages of the 4-stage operation, torque ripples reduce from 26% to 15% when the new control structure with RTCL is activated. Phase current waveforms in different stages are presented in Fig. 3.32 in which the constraint on RMS current is always respected in method (II)-RCA. Meanwhile, the peak values of phase currents slightly increase when the new control structure is used. To activate the new control structure in the final operating stage, the learning process of phase B has been previously taken from the 25 th second when the pre-fault control scheme is still active. The harmonic weights converge within 4 s as described in Fig. 3.33a when the learning rate η is 0.001. When the learning process of phase-B current is done as in Fig. 3.33b, the new control scheme can be activated. The real-time current learning RTCL will continue to determine currents of the other healthy phases in real time based on the harmonic separation in Fig. 3.33c. After all measured phase currents are analyzed with the estimated first and third harmonic components, 10 PI controllers for constant d-q currents (id11, iq11, id91, iq91, ix1, id33, iq33, i93, i93, ix3) are used in the new control scheme. Current control performances are described in Fig. 3.34 with the estimated values obtained from RTCL to make comparisons with their reference values. It is noted that currents (iB1, iC1, iD1, iE1, iF1, iG1) and (id11, iq11, id91, iq91, ix1) are directly estimated from the ADALINE and mathematical calculations. Therefore, these values are perfectly constant without high frequencies of the experimental drive. Meanwhile, currents (iB3, iC3, iD3, iE3, iF3, iG3) and (id33, iq33, id93, iq93, ix3) contain high frequencies because these currents are derived from the subtraction of the measured phase currents and the estimated first harmonic components. The proper control performances of the first harmonic components, generating most of the torque, make the torque quality higher compared to the pre-fault control scheme. The dynamic performances of the new control scheme, as previously described in Fig. 3.30, are verified in Fig. 3.35. Variations of the rotating speed in Fig. 3.35a do not have any significant influences on torque quality (a ripple from 14 to 15%). As previously discussed, variations of current references Fig. 3.35b (iq11_ref from 5 to 8 A) require a switch from the new control scheme to the pre-fault control scheme to guarantee the stability of the drive. It is worth noting that the torque quality is improved whenever the new control scheme with RTCL is activated, specifically reductions from 25 to 19% (iq11_ref =5 A) and from 21 to 14% (iq11_ref =8 A). To see demonstration videos, please click on this link: https://youtu.be/pnWGwRDVfxg or scan the nearest QR code in this page.

Conclusions

This chapter has presented enhancements of the quality of current control and torque in a multiphase drive by using a simple type of artificial intelligence, ADALINE. The knowledge of harmonics in the drive has helped ADALINEs reduce the weight updating time and avoid the calculation burden. However, the number of used ADALINEs can be flexible according to the computation power of processors. ADALINEs for the control of parameters in the main fictitious machines that generate most of the torque should be prioritized.

In healthy mode, ADALINEs have been applied to eliminate current harmonics in d-q frames caused by unwanted back-EMF harmonics and inverter nonlinearity. In addition, torque ripples generated by the unwanted back-EMF harmonics have been directly nullified by a single ADALINE. By using these (seven) ADALINEs, every machine can generate a constant torque with high quality current control. However, the speed range should be reduced to respect the constraint on peak voltage.

In faulty mode, torque ripples in the proposed fault-tolerant methods in Chapter 2 have been directly eliminated by the same way as in healthy mode with a shorter speed range. An ADALINE has been applied to eliminate torque ripples in the post-fault operation using method (III)-2 where the highest average torque is obtained. As healthy mode, the most important step of torque ripple eliminations using ADALINEs is to determine harmonic components of the torque, allowing to avoid the calculation burden.

Torque Speed

Old control scheme

Current references change

New control scheme

Old control scheme

New control scheme

A new control scheme using a single ADALINE has improved current control quality of the post-fault drive with method (II), indirectly improving the torque quality. Time-variant current references for control have been replaced by time-constant values. The key part of the new control structure is a real-time current learning (RTCL) that allows to separate harmonic components from measured phase currents, providing feedback signals of constant d-q currents. With a robustness to the back-EMF harmonic components in the second fictitious machine (9 th , for example), method (II)-RCA has been chosen to verify the effectiveness of the new control scheme.

General conclusions and Perspectives

General conclusions

Electrified vehicles have been considered as alternatives of ICE vehicles to cope with the shortage of fossil energy sources and the air pollution. In general, all types of electrified vehicles are driven by electric drives. To ensure high-performance electromechanical conversions, several requirements for the electric drives such as high efficiency, high volume densities, low-cost but safe-to-touch, high functional reliability, high torque quality, and fluxweakening control need to be met. In this context, multiphase PMSM drives are suitable candidates to meet the above requirements from electrified vehicles.

This doctoral thesis has proposed, systemized, and refined fault-tolerant control strategies for non-sinusoidal multiphase PMSM drives. Besides several advantages such as easy-tomanufacture, low production costs, high torques, and electromagnetic pole changing, nonsinusoidal back-EMFs of electric machines in the considered drives make the control more complicated but interesting to study. In addition, in this doctoral thesis, limits of RMS current and peak voltage have been always considered from healthy to faulty operations; hence, fluxweakening operations can be guaranteed. A case study with seven-phase machines and a singlephase OC fault has been used for verifications of the proposed strategies. Two-phase OC faults are briefly described in Appendix B. Most results in this doctoral thesis have been obtained from an experimental test bench.

The doctoral thesis has been organized into three chapters. In Introduction and Chapter 1, the context and state of the art have been discussed to clarify the necessary of this doctoral research.

In Chapter 2, the mathematical modeling has been described to develop control strategies based on FOC for the seven-phase PMSM drive. In healthy mode, the current control and torque performances with the current and voltage constraint consideration have been presented. Although, there are no faults happening in the drive but measured currents in d-q frames and the electromagnetic torque consist of harmonic components. When one phase is open-circuited, a post-fault operation without any reconfigurations leads to very high torque ripples and the breaking of the electrical limits. Therefore, three main fault-tolerant control strategies have been cohesively proposed with the exploitation of the mathematical model of the drive. Method (I) with three options has provided constant d-q currents in the first and third fictitious machines, creating most of the torque. This feature allows the drive to obtain smooth torques if the back-EMF harmonics in the second fictitious machine are small. Method (II) has applied new transformation matrices to generate new current references. Especially, method (II)-SCL can obtain the similar average torque as in method (I)-2. Meanwhile, method (II)-RCA proves its robustness when its torque is unaffected by the harmonics in the second fictitious machine. However, method (II) generates time-variant d-q current references in the first and third fictitious machines, reducing current control quality as well as torque quality at high speed. Methods (I) and (II) can generate a maximum average post-fault torque equal to 65% of the torque in healthy mode. Therefore, method (III) has been proposed with the aim of improving the average torque to 79% under the constraints on current and voltage. Indeed, the uniformity

Multiphysical conversion

Multi-domain conversion of energy which can be with or without tuning input (red line). For example, an electrical machine converts electrical energy to mechanical energy (and vice versa).

Coupling

Mono-domain coupling or distribution of energy between more than two subsystems. For example, the coupling can be the distribution of energy to fictitious machines of a multiphase machine, or the coupling of torques from fictitious machine to create the total electromagnetic torque.

A.2. The control scheme of the system

The control scheme obtained from the model representation in EMR is based on the principle of inversion. The control is considered as a functional inversion of the model.

Element Pictogram Description

Conversion inversion

Direct inversion of an energy conversion element in which the output is the reference of the next control block. Information from the system model (dash line) can be used. For example, this element can be a calculation of current references from a torque reference and back-EMFs in a fictitious machine of a multiphase drive.

Accumulation inversion

Indirect inversion of an accumulation element to control its output. It is a closed-loop control with reference, output feedback (mandatory), and disturbance rejection (optional). For example, it can be the closed-loop control of current or speed.

Coupling inversion

Direct inversion of a coupling element (both mono-domain and multi-domain). A distribution or weighting factor must be introduced to manage the power flow. For example, this element distributes a torque reference to fictitious machines of a multiphase drive according to a repartition rate.

Strategy

Strategy block to impose references, distributions, and/or weighting factors on the control scheme. For example, a strategy to maximize torque under constraints on current and voltage.

Estimation

Model copy of the system implemented in the control program, playing the role of estimation of reference model. For example, the estimation of current harmonic components existing in phase currents of an electric machine. travailler avec des machines à forces électromotrices non-sinusoïdales donc avec moins de contrainte lors de la conception et de fait un coût moindre. 4) Plus de possibilités de configuration d'enroulement de stator grâce à plus de phases (permettent de faire varier la caractéristique couple-vitesse). 5) Changement de pôle électromagnétique avec les harmoniques de courant. Cela permet d'avoir un effet équivalent à une boite de vitesse et donc de disposer d'une plage de vitesse à puissance maximale quasi-constante bien adaptée au cahier des charges d'une automobile moyenne gamme (milieu de segment), élément important pour le coût. En plus, une conduite plus confortable peut être obtenue avec la boîte de vitesses électromagnétique.

Strategy

     0 0 0 0 d1 α1 q1 1 α3 d3 β3 q3 α2 d9 C D E F G β2 q9 i i i i i i i i i i i i i i i i i             = =              Park Park Clarke T T T (B.1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1
En particulier, les machines PMSM polyphasées sont plus intéressantes que les IM polyphasées en raison de leur rendement élevé, leurs densités élevées de couple et de puissance. Cependant, pour remplacer complètement les homologues triphasés utilisés majoritairement dans les véhicules électriques, les entraînements polyphasés, en particulier les entraînements de machines PMSM polyphasées, doivent surmonter quelques handicaps : des ressources en temps de calcul augmentant avec le nombre de phases, une plus grande complexité de conception du contrôle notamment pour un fonctionnement de qualité en mode de défaut, et une méconnaissance des spécificités de ces entrainements électriques. C.2, différents types de défauts dans un entraînement polyphasé incluent le courtcircuit (SC) ou le circuit ouvert (OC) dans les interrupteurs de l'onduleur, les enroulements de phase ou les lignes de connexion entre la machine et l'onduleur. En fonctionnement post-défaut, le nombre de phases actives est plus faible et les phases restantes ne sont plus réparties symétriquement dans la machine. Si les références de courant avant défaut ne sont pas modifiées, il y aura des courants incontrôlables et des couples pulsatoires [START_REF] Sandulescu | Control Strategies for Open-End Winding Drives Operating in the Flux-Weakening Region[END_REF][START_REF] Nguyen | Fault-Tolerant Operation of an Open-End Winding Five-Phase PMSM Drive With Short-Circuit Inverter Fault[END_REF][START_REF] Kianinezhad | Modeling and Control of Six-Phase Symmetrical Induction Machine Under Fault Condition Due to Open Phases[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF]. Les récentes études de contrôle tolérant aux défauts peuvent être classées comme suit : 1) Classement basé sur les critères pour élaborer de nouvelles références pour les opérations tolérantes aux défauts : contraintes sur les limites de courant [START_REF] Guzman | Novel control techniques in multiphase drives: Direct control methods (DTC and MPC) under limit situations[END_REF][START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Dwari | Fault-Tolerant Control of Five-Phase Permanent-Magnet Motors With Trapezoidal Back EMF[END_REF][START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF] et de tension [START_REF] Guzman | Novel control techniques in multiphase drives: Direct control methods (DTC and MPC) under limit situations[END_REF][START_REF] Mengoni | High-Torque-Density Control of Multiphase Induction Motor Drives Operating Over a Wide Speed Range[END_REF][START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF], en optimisant les pertes de cuivre [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF][START_REF] Dwari | An Optimal Control Technique for Multiphase PM Machines Under Open-Circuit Faults[END_REF][START_REF] Mohammadpour | A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections[END_REF][START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF][START_REF] Hyung-Min | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF][START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF][START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF][START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF][START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF][START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF][START_REF] Xiong | Fault-tolerant FOC for five-phase SPMSM with non-sinusoidal back EMF[END_REF][START_REF] Bermudez | An Experimental Assessment of Open-Phase Fault-Tolerant Virtual-Vector-Based Direct Torque Control in Five-Phase Induction Motor Drives[END_REF][START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF]. 2) Classement basé sur la conservation de force magnétomotrices pour les opérations tolérantes aux défauts : machines sinusoïdales [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Kianinezhad | Modeling and Control of Six-Phase Symmetrical Induction Machine Under Fault Condition Due to Open Phases[END_REF][START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF][START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF][START_REF] Hyung-Min | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF][START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF][START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF][START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF][START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF][START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF][START_REF] Fnaiech | Fuzzy Logic and Sliding-Mode Controls Applied to Six-Phase Induction Machine With Open Phases[END_REF][START_REF] Sun | Enhanced Optimal Torque Control of Fault-Tolerant PM Machine Under Flux-Weakening Operation[END_REF][START_REF] Sui | Open-Circuit Fault-Tolerant Control of Five-Phase PM Machine Based on Reconfiguring Maximum Round Magnetomotive Force[END_REF] et machines non-sinusoïdales [61, 120-122, 124, 125, 134-136, 142-149]. 3) Classement basé sur la modélisation des entraînements polyphasés pour des opérations tolérantes aux défauts. La transformation de Clarke en mode sain est préservée mais les références actuelles dans les repères de référence du rotor doivent être redéfinies [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF][START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF]. Une autre option est que des matrices de transformation d'ordre réduit avec des dimensions équivalentes au nombre de phases saines sont appliquées [126-129, 132-135, 149]. 4) Classement basé sur les techniques de contrôle pour les opérations tolérantes aux défauts : correcteurs PI [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF], MPC et DTC [START_REF] Bermudez | Model predictive optimal control considering current and voltage limitations: Real-time validation using OPAL-RT technologies and five-phase permanent magnet synchronous machines[END_REF][START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF][START_REF] Bermudez | An Experimental Assessment of Open-Phase Fault-Tolerant Virtual-Vector-Based Direct Torque Control in Five-Phase Induction Motor Drives[END_REF][START_REF] Bermudez | Open-Phase Fault-Tolerant Direct Torque Control Technique for Five-Phase Induction Motor Drives[END_REF][START_REF] Guzman | IGBT-Gating Failure Effect on a Fault-Tolerant Predictive Current-Controlled Five-Phase Induction Motor Drive[END_REF][START_REF] Guzman | Speed Control of Five-Phase Induction Motors With Integrated Open-Phase Fault Operation Using Model-Based Predictive Current Control Techniques[END_REF], correcteurs d'hystérésis [START_REF] Kestelyn | A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time[END_REF][START_REF] Dwari | Fault-Tolerant Control of Five-Phase Permanent-Magnet Motors With Trapezoidal Back EMF[END_REF][START_REF] Mohammadpour | A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections[END_REF][START_REF] Parsa | Fault-Tolerant Interior-Permanent-Magnet Machines for Hybrid Electric Vehicle Applications[END_REF][START_REF] Mohammadpour | Global Fault-Tolerant Control Technique for Multiphase Permanent-Magnet Machines[END_REF], correcteurs résonants proportionnels [START_REF] Che | Postfault Operation of an Asymmetrical Six-Phase Induction Machine With Single and Two Isolated Neutral Points[END_REF][START_REF] Xiong | Fault-tolerant FOC for five-phase SPMSM with non-sinusoidal back EMF[END_REF][START_REF] Tani | Control of Multiphase Induction Motors With an Odd Number of Phases Under Open-Circuit Phase Faults[END_REF][START_REF] Xiong | A Fault-Tolerant FOC Strategy for Five-Phase SPMSM With Minimum Torque Ripples in the Full Torque Operation Range Under Double-Phase Open-Circuit Fault[END_REF][START_REF] Abdel-Khalik | Effect of Stator Winding Connection on Performance of Five-Phase Induction Machines[END_REF], et contrôle robuste [START_REF] Fnaiech | Fuzzy Logic and Sliding-Mode Controls Applied to Six-Phase Induction Machine With Open Phases[END_REF][START_REF] Liu | Enhancing Fault-Tolerant Ability of a Nine-Phase Induction Motor Drive System Using Fuzzy Logic Current Controllers[END_REF][START_REF] Betin | Shaft Positioning for Six-Phase Induction Machines With Open Phases Using Variable Structure Control[END_REF].

C.3. Etat de l'art des entraînements polyphasés

C.4. Objectifs de thèse

Cette thèse de doctorat est dédiée à l'enrichissement des approches existantes de contrôle pour les entraînements polyphasés, dans le but de mettre en évidence l'intérêt des entraînements polyphasés dans les applications automobiles. Les entraînements polyphasés peuvent répondre de manière significative aux exigences des véhicules électrifiés de la section C. Ainsi, en fonction des analyses ci-dessus et de l'état de l'art dans le domaine de la commande des entraînements polyphasés, l'objectif principal de cette thèse de doctorat est de proposer des stratégies de contrôle des entraînements PMSM polyphasés non-sinusoïdales dans des conditions de circuit ouvert. Surtout, les contraintes de courant et de tension seront toujours prises en compte dans tous les modes de fonctionnement. Un autre objectif de cette thèse est de valoriser les performances, notamment la qualité du couple et du contrôle, sous les impacts d'incertitudes et d'imperfections des entraînements électriques en utilisant l'intelligence artificielle.

Plusieurs points principaux de ces objectifs de cette thèse de doctorat peuvent être justifiés et plus expliqués comme suit :

1) Concernant le type de machine électrique, les PMSM polyphasés non-sinusoïdale sont l'objet contrôlé de cette thèse de doctorat. Outre les propriétés générales du concept polyphasé, les PMSM polyphasés sont choisis pour cette thèse de doctorat en raison de leur rendement élevé et de leurs densités volumiques élevées. Ces avantages répondent aux deux premières exigences des EV. De plus, les forces électromotrices non-sinusoïdales sont pris en compte en raison de plusieurs avantages tels que la facilité de fabrication, les faibles coûts de production, les densités de couple élevées, et la capacité de changement de pôle électromagnétique. Le contrôle des machines non-sinusoïdales est plus difficile et intéressant en raison des harmoniques élevés des forces électromotrices.

Economical mode

High-performance mode (This thesis)

Accelerator pedal (Torque control) références [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF]. Les courants dans les cadres de référence rotatifs devront être constants ou varier lentement, en particulier les courants générant la majeure partie du couple.

3) Concernant les contraintes de courant et de tension, ces contraintes garantissent une grande fiabilité de fonctionnement et des opérations d'affaiblissement du flux des entraînements. D'après l'état de l'art, peu d'études ont été menées sur la commande tolérante aux défauts et les forces électromotrices non-sinusoïdales sous contraintes de courant et de tension, sauf [START_REF] Guzman | Novel control techniques in multiphase drives: Direct control methods (DTC and MPC) under limit situations[END_REF]

C.5. Plan de la thèse

Cette thèse de doctorat se compose de 3 chapitres comme suit : Dans un premier temps, la modélisation et le contrôle de l'entraînement en mode sain sont présentés. Les contraintes sur le courant RMS et la tension crête sont appliquées. Dans un second temps, les qualités de courant et de couple de l'entraînement en fonctionnement postdéfaut sont examinés montrant une forte détérioration sur les ondulations de couple ainsi qu'une augmentation importante de la valeur RMS de courants. Le problème de couplage empêche l'entraînement de continuer à utiliser les références actuelles du mode sain pour le fonctionnement après défaut.

C.
Par conséquent, une reconfiguration avec des stratégies de commandes tolérantes aux défauts est nécessaire pour garantir des performances de couple peu ondulé et des opérations postdéfaut sûres pour l'entraînement ainsi que pour les EV. Trois principales stratégies proposées de commandes tolérantes aux défauts avec sept options sont présentées en exploitant le modèle mathématique de la machine polyphasée (FIG. C.4). Plus précisément, des nouvelles références de courants peuvent être conçues dans :

1) Les repères découplés (α-β ou d-q) avec les méthodes (I)-1-2-3.

2) L'utilisation de nouvelles matrices de transformation avec les méthodes (II)-RCA-SCL. 3) Le repère naturel avec les méthodes (III)-1-2. Les méthodes (I) et (II) peuvent permettre théoriquement d'obtenir des couples constants lorsque le premier et le troisième harmonique de back-EMF de la machine sont considérés. Cependant, leurs couples moyens sont relativement faibles si le courant de phase RMS est limité (0.65 pu). Par conséquent, la méthode (III) est proposée pour obtenir des couples moyens plus élevés (0.79 pu) en utilisant des courants de phase identiques mais déphasés. Sous la contrainte de courant, ces courants de forme d'onde identique peuvent être efficacement exploités pour produire plus de couple. Cependant, les ondulations de couple sont inévitables dans la méthode (III) (19%). Finalement, il y a théoriquement un compromis entre des couples moyens élevés avec des ondulations de couple, et des couples sans ondulations avec des valeurs moyennes faibles. De plus, une combinaison de plusieurs stratégies en fonction de la charge utile des véhicules électriques peut tirer parti de chaque méthode (FIG. C.5). Ce chapitre a présenté les améliorations de la qualité du couple et du contrôle de courant d'un entraînement polyphasé en utilisant un type simple de contrôle issu de l'Intelligence Artificielle nommé ADALINE (neurone linéaire adaptatif). En mode sain, les ADALINE sont appliqués pour éliminer les harmoniques de courant dans les repères d-q causés par des harmoniques indésirables de force électromotrice et la non-linéarité de l'onduleur. De plus, les ondulations de couple générées par les harmoniques indésirables de force électromotrice sont directement annulées par un seul ADALINE. En utilisant ces ADALINE, chaque machine peut générer un couple constant avec un contrôle de courant de haute qualité. Cependant, la plage de vitesse doit être réduite pour respecter la contrainte de tension de crête. En mode dégradé, les ondulations de couple dans les méthodes tolérantes aux défauts proposées au chapitre 2 peuvent être directement éliminées ou réduites (31% à 5% à 20 rad/s dans la FIG.

G z i i i i i i i i i i i i i              =                  
C.6) de la même manière qu'en mode sain avec une plage de vitesse plus courte. Un ADALINE est appliquée pour éliminer les ondulations de couple dans le fonctionnement post-défaut en utilisant la méthode (III)-2 où le couple moyen le plus élevé est obtenu. L'étape la plus importante de l'élimination des ondulations de couple à l'aide des ADALINE est de déterminer les composantes harmoniques du couple, ce qui permet d'éviter un temps de calcul très élevé. Un nouveau schéma de contrôle en utilisant un seul ADALINE améliore la qualité de contrôle du courant de l'entraînement post-défaut avec la méthode (II), améliorant indirectement la qualité du couple (26% à 15% à 35 rad/s dans la FIG. C.7). Les références de courant variant dans le temps pour le contrôle sont alors remplacées par des valeurs constantes dans le temps. L'élément clé de la nouvelle structure de contrôle est un apprentissage du courant en temps réel qui permet de séparer les composants harmoniques des courants de phase mesurés, en fournissant des signaux de retour de courants d-q constants. Avec une robustesse aux composants harmoniques de force électromotrice dans la deuxième machine fictive (le 9 ème harmonique, par exemple), la méthode (II)-RCA est choisie pour vérifier le nouveau schéma de contrôle. Lorsque les références de courant sont fixes, la méthode (II)-RCA avec la nouvelle structure de commande n'est pas affectée par la vitesse de rotation. En d'autres termes, la région de fonctionnement avant la vitesse de base est idéale pour appliquer la nouvelle structure afin d'améliorer la qualité du couple. Cependant, si le couple ou les références de courant doivent changer, le fonctionnement doit revenir à l'ancien schéma de commande dans lequel un nouveau processus d'apprentissage est activé pour estimer le nouveau courant de la phase B. Par la suite, le nouveau schéma de commande peut être utilisé. Cela peut être considéré comme un inconvénient du nouveau schéma de contrôle.

C.6. Conclusions générales et perspectives C.6.1. Conclusions générales

Les véhicules électrifiés ont été considérés comme des alternatives aux véhicules utilisant un moteur à combustion interne (ICE) pour faire face à la pénurie de sources d'énergie fossile et à la pollution de l'air. En général, tous les types de véhicules électrifiés sont entraînés par des moteurs électriques. Pour garantir des conversions électromécaniques de haute performance, plusieurs exigences des entraînements électriques tels qu'un rendement élevé, des densités volumiques élevées, un coût faible mais sûr, une fiabilité fonctionnelle élevée, une qualité de couple élevée, et un contrôle de défluxage doivent être satisfaites. Dans ce contexte, les entraînements de machines PMSM polyphasées sont devenus des candidats appropriés pour répondre aux exigences ci-dessus des véhicules électrifiés.

Cette thèse de doctorat a proposé systématiquement et affiné des stratégies de contrôle tolérantes aux défauts pour les entraînements de machines PMSM polyphasées nonsinusoïdales. Outre plusieurs avantages tels que facile-à-fabriquer, des faibles coûts de production, des densités de couple élevées, et des changements de pôles électromagnétiques, les forces électromotrices non-sinusoïdales des machines électriques dans les entraînements considérés ont rendu le contrôle plus compliqué mais intéressant à étudier. De plus, dans cette thèse de doctorat, les limites du courant RMS et de la tension de crête ont toujours été considérées dans les modes de fonctionnement sain et dégradés. Par conséquent, les opérations de défluxage peuvent être garanties. Une étude d'un cas d'une machine à sept phases et un défaut monophasé en circuit ouvert a été utilisée pour les vérifications des stratégies proposées. Les défauts biphasés en circuit ouvert sont brièvement décrits à l'annexe B. Les résultats présentés dans cette thèse de doctorat ont été obtenus à partir d'un banc d'essai expérimental.

Résumé : Les entraînements électriques pour les véhicules électrifiés doivent répondre à certaines exigences spécifiques des marchés automobiles, tels qu'un rendement élevé, des densités volumiques élevées de puissance et de couple, un coût faible avec une protection contre les risques électriques, une fiabilité fonctionnelle élevée, une qualité de couple élevée, et un contrôle de défluxage. Dans ce contexte, les entraînements de machines synchrones à aimants permanents (PMSM) polyphasées sont devenus des candidats appropriés pour répondre aux exigences citées ci-dessus. L'objectif principal de cette thèse de doctorat vise à proposer et affiner des stratégies de commandes tolérantes aux défauts pour les entraînements de machines PMSM polyphasées non-sinusoïdales qui requièrent moins de contraintes lors de leur conception. Par ailleurs, les contraintes de courant et de tension définies par l'onduleur et la machine sont prises en compte pour optimiser en régime non-sinusoïdal le contrôle de la machine sans dépasser leurs limites admissibles. Cela permet idéalement un dimensionnement au plus juste et cela tout particulièrement dans la zone de défluxage. Les stratégies proposées de commandes tolérantes aux défauts, basées sur le modèle mathématique des entraînements polyphasés, enrichissent le domaine de contrôle des entraînements polyphasés en offrant de diverses options de contrôle. Le choix de l'une des options proposées de commande peut être un compromis entre un couple de haute qualité mais avec une valeur moyenne faible, et un couple moyen élevé mais avec une ondulation relativement élevée. Les performances de contrôle et de couple peuvent être affinées en utilisant l'intelligence artificielle avec un type simple de réseaux de neurones artificiels nommé ADALINE (neurone linéaire adaptatif). Grâce à leur capacité d'auto-apprentissage, à leur convergence rapide, et à leur simplicité, les ADALINE peuvent être appliqués aux entraînements polyphasés industriels. Toutes les stratégies de contrôle proposées dans cette thèse de doctorat sont validées avec un entraînement d'une machine PMSM à sept phases. La force électromotrice non-sinusoïdale de la machine PMSM à sept phases, relevée expérimentalement, est complexe avec la présence de plusieurs harmoniques. Les résultats expérimentaux vérifient l'efficacité des stratégies proposées, et leur applicabilité dans une machine polyphasée avec une force électromotrice non-sinusoïdale complexe.

Mots clés :

Machine polyphasée, machine à sept phases, force électromotrice non-sinusoïdale, commande tolérante aux défauts, commande optimale sous contrainte, intelligence artificielle, neurone linéaire adaptatif, ADALINE.

Duc Tan VU

Fault-tolerant control of non-sinusoidal multiphase permanent magnet synchronous machine drives under constraints on current and voltage for automotive applications Abstract: Electric drives for electrified vehicles need to fulfil some specific requirements from automotive markets such as high efficiency, high volume power and torque densities, low-cost but safeto-touch, high functional reliability, high torque quality, and flux-weakening control. In this context, multiphase permanent magnet synchronous machine (PMSM) drives have become suitable candidates to meet the above requirements. The main objective of this doctoral thesis is to propose and refine faulttolerant control strategies for non-sinusoidal multiphase PMSM drives that require less constraints on their design. In addition, constraints on current and voltage defined by the inverter and the machine are considered to optimize the machine control under the non-sinusoidal condition without exceeding their allowable limits. Therefore, the system sizing is guaranteed, especially in flux-weakening operations. The proposed fault-tolerant control strategies, based on the mathematical model of multiphase drives, enrich the control field of multiphase drives by providing various control options. The selection of one of the proposed control options can be a trade-off between a high quality torque but a low average value and a high average torque but a relatively high ripple. The control and torque performances of the drives can be refined by using artificial intelligence with a simple type of artificial neural networks named ADALINE (ADAptive LInear NEuron). With self-learning ability, fast convergence, and simplicity, ADALINEs can be applied to industrial multiphase drives. All proposed control strategies in this doctoral thesis are validated with an experimental seven-phase PMSM drive. The non-sinusoidal back electromotive force (back-EMF) of the experimental seven-phase PMSM is complex with the presence of multi-harmonics. Experimental results verify the effectiveness of the proposed strategies, and their applicability in a multiphase machine with a complex non-sinusoidal back-EMF.

Keywords: Multiphase machine, seven-phase machine, non-sinusoidal back-EMF, fault-tolerant control, optimal control under constraint, artificial intelligence, adaptive linear neuron, ADALINE.

vii 6 -Fig. 1 . 1 .Fig. 1 . 2 .Fig. 1 . 3 .Fig. 1 . 4 .Fig. 1 . 5 .Fig. 1 . 6 .Fig. 1 . 7 .Fig. 1 . 8 .Fig. 1 . 9 .Fig. 1 . 10 .Fig. 1 . 11 .Fig. 1 . 12 .Fig. 1 . 13 .Fig. 2 . 1 .Fig. 2 . 2 .Fig. 2 . 3 .Fig. 2 . 4 .Fig. 2 . 5 .Fig. 2 . 6 .Fig. 2 . 7 .Fig. 2 . 8 . 39 Fig. 2 . 39 Fig. 2 . 40 Fig. 2 . 40 Fig. 2 . 41 Fig. 2 . 42 Fig. 2 . 42 Fig. 2 . 44 Fig. 2 . 45 Fig. 2 . 46 Fig. 2 . 46 Fig. 2 . 20 . 49 Fig. 2 . 21 . 50 Fig. 2 . 22 . 51 Fig. 2 . 23 . 51 Fig. 2 . 24 . 56 Fig. 2 . 25 .Fig. 2 . 26 . 56 Fig. 2 . 27 .Fig. 2 . 28 .Fig. 2 . 29 . 59 Fig. 2 . 61 Fig. 2 . 61 Fig. 2 . 61 Fig. 2 . 62 Fig. 2 . 62 Fig. 2 . 63 Fig. 2 . 63 Fig. 2 . 38 ( 64 Fig. 2 . 39 ( 64 Fig. 2 .Fig. 2 . 41 . 70 Fig. 2 . 42 . 71 Fig. 2 . 43 . 72 Fig. 2 . 44 . 72 Fig. 2 . 73 Fig. 2 . 75 Fig. 2 . 76 Fig. 2 . 76 Fig. 2 . 76 Fig. 2 . 77 Fig. 2 . 77 Fig. 2 . 78 Fig. 2 . 79 Fig. 2 . 54 ( 79 Fig. 2 . 55 (Fig. 2 . 56 . 89 Fig. 2 . 57 . 89 Fig. 2 . 89 Fig. 2 . 59 . 90 Fig. 2 . 60 . 91 Fig. 2 . 92 Fig. 2 . 93 Fig. 2 . 94 Fig. 2 . 94 Fig. 2 . 94 Fig. 2 . 95 Fig. 2 . 96 Fig. 2 . 96 Fig. 2 . 97 Fig. 2 . 97 Fig. 2 . 99 Fig. 2 .Fig. 3 . 1 . 104 Fig. 3 . 2 . 105 Fig. 3 . 3 . 108 Fig. 3 . 4 . 109 Fig. 3 . 5 . 110 Fig. 3 . 112 Fig. 3 . 112 Fig. 3 . 112 Fig. 3 . 113 Fig. 3 . 113 Fig. 3 . 113 Fig. 3 . 114 Fig. 3 . 14 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 . 27 .Fig. 3 . 30 .Fig. 3 .Fig. 3 .Fig. 3 .Fig. 3 .

 61112131415161718191101111121132122232425262728392392402402412422422442452462462204922150222512235122456225226562272282295926126126126226226326323864239642241702427124372244722732752762762762772772782792547925525689257892892599026091292293294294294295296296297297299231104321053310834109351103112311231123113311331133114314333333333333273303333 Fig. 1.1. A n-phase machine fed by a n-leg VSI in an EV. ........................................................ Fig. 1.2. Different possibilities of stator winding configurations for five-phase machines: star (a), pentagon (b), pentacle (c), and corresponding torque-speed characteristics (d) [45, 46]. .............................................................................................................................................. Fig. 1.3. Electromagnetic pole changing in multiphase drives for speed range extensions: p pole pairs (a), a combination of p and 3p pole pairs (b), and a more general case (c). .............. Fig. 1.4. The highly compact electric motor prototype without using rare earth metals in MotorBrain project [58] (a), and the model of an integrated machine in CE2I project [59] (b). ............................................................................................................................................ Fig. 1.5. The schematic diagram of a n-phase multiphase PMSM........................................... Fig. 1.6. Equivalent fictitious machines of a n-phase machine when n is odd. ....................... Fig. 1.7. Equivalent fictitious machines of a n-phase machine when n is even. ...................... Fig. 1.8. The inner control loop of a n-phase PMSM drive based on FOC technique. ............ Fig. 1.9. The inner control loop of a n-phase PMSM drive based on ST-DTC technique. ...... Fig. 1.10. The inner control loop of a n-phase PMSM drive based on PWM-DTC technique. .................................................................................................................................................. Fig. 1.11. The inner control loop of a n-phase PMSM drive based on FCS-MPC technique. .Fig. 1.12. Different types of faults in a n-phase drive. ............................................................. Fig. 1.13. Multiphase drives in EVs with two driving modes. ................................................. Fig. 2.1. Schematic diagram of a seven-phase PMSM. ............................................................ Fig. 2.2. Decomposition of a seven-phase machine into four fictitious machines. .................. Fig. 2.3. Representation of the model of a seven-phase PMSM drive using EMR. ................. Fig. 2.4. The general control scheme of a seven-phase PMSM drive with speed and current control represented by EMR. ................................................................................................... Fig. 2.5. The control scheme of a seven-phase PMSM drive with current control represented by EMR. ...................................................................................................................................Fig. 2.6. The control scheme of a seven-phase PMSM drive in healthy mode with the optimal control strategy under constraints on RMS current and peak voltage, represented by EMR. ........................................................................................................................................Fig. 2.7. Experimental test bench of the seven-phase PMSM drive. ....................................... Fig. 2.8. The measured speed-normalized back-EMF and harmonic spectrum of the experimental seven-phase PMSM. ...........................................................................................

Fig. 1 .

 1 Fig. 1.1. A n-phase machine fed by a n-leg VSI in an EV.

  Fig. 1.2. Different possibilities of stator winding configurations for five-phase machines: star (a), pentagon (b),

  Fig. 1.3. Electromagnetic pole changing in multiphase drives for speed range extensions: p pole pairs (a), a combination of p and 3p pole pairs (b), and a more general case (c).

4 .

 4 The highly compact electric motor prototype without using rare earth metals in MotorBrain project [58] (a), and the model of an integrated machine in CE2I project[START_REF]The integrated smart energy converter project CE2I[END_REF] (b).

Fig. 1 . 5 .

 15 Fig. 1.5. The schematic diagram of a n-phase multiphase PMSM.

  Opportunities and State of the Art 13

Fig. 1 . 6 .

 16 Fig. 1.6. Equivalent fictitious machines of a n-phase machine when n is odd.

Fig. 1 . 7 .

 17 Fig. 1.7. Equivalent fictitious machines of a n-phase machine when n is even.

  Opportunities and State of the Art 15

Fig. 1 . 8 .

 18 Fig. 1.8. The inner control loop of a n-phase PMSM drive based on FOC technique.

Fig. 1 . 9 .

 19 Fig. 1.9. The inner control loop of a n-phase PMSM drive based on ST-DTC technique.

Fig. 1 .

 1 Fig. 1.10. The inner control loop of a n-phase PMSM drive based on PWM-DTC technique.

Fig. 1 . 11 .

 111 Fig. 1.11. The inner control loop of a n-phase PMSM drive based on FCS-MPC technique.

Fig. 1 . 12 .

 112 Fig. 1.12. Different types of faults in a n-phase drive.

Fig. 1 . 13 .

 113 Fig. 1.13. Multiphase drives in EVs with two driving modes.

Fig. 2 . 1 .

 21 Fig. 2.1. Schematic diagram of a seven-phase PMSM.

Fig. 2 . 2 .

 22 Fig. 2.2. Decomposition of a seven-phase machine into four fictitious machines.Table 2.1. Four fictitious machines with corresponding reference frames and associated harmonics of a sevenphase machine (only odd harmonics).

  The third machine (FM3) d3-q3 3, 11, 17,…, 7j ± 3 Zero-sequence machine (ZM) z 7, 21,…, 7j with j ∈ ℕ0

30 Fig. 2 . 3 .

 3023 Fig. 2.3. Representation of the model of a seven-phase PMSM drive using EMR.

Fig. 2 . 4 .

 24 Fig. 2.4. The general control scheme of a seven-phase PMSM drive with speed and current control represented by EMR.

Fig. 2 . 5 .

 25 Fig. 2.5. The control scheme of a seven-phase PMSM drive with current control represented by EMR.

Fig. 2 . 6 .

 26 Fig. 2.6. The control scheme of a seven-phase PMSM drive in healthy mode with the optimal control strategy under constraints on RMS current and peak voltage, represented by EMR.

  Fig. 2.8. The measured speed-normalized back-EMF and harmonic spectrum of the experimental seven-phase PMSM.

  1.26% 1.98% 3.2% 1% 0.29% 1.7% 12.5% 10.3%Chapter 2. Modeling and Control of Multiphase Drives 39 by Vpeak. The value of Vpeak is less than or equal to voltage limit Vlim_opt even at high speed as shown in Fig. 2.9c. Therefore, a flux-weakening operation of the drive can be performed. (a) (b) (c) Fig. 2.9. (Calculated result) Optimal torque-speed characteristic (a), all RMS phase currents (b), and all peak phase voltages (c) in terms of speed under constraints on RMS current and peak voltage in healthy mode.

  2.6 as discussed in section 2.1.4.B. Currents in (d1-q1) frame that create most of the torque are well controlled as described in Fig. 2.10. (a) (b) Fig. 2.10. (Experimental result) Currents in (d1-q1) frame in terms of time at 20 rad/s with id1_ref=0 A and iq1_ref=12.7 A (a), and in terms of speed (b) in healthy mode.

  Fig. 2.11. (Experimental results) Currents in (d9-q9) frame in terms of time at 20 rad/s with id9_ref=0 A and iq9_ref=1.6 A (a), and in terms of speed (b) and in healthy mode. (a) (b) Fig. 2.12. (Experimental result) Currents in (d3-q3) frame in terms of time at 20 rad/s with id3_ref=0 A and iq3_ref=4.1 A (a), and in terms of speed (b) in healthy mode.

  Fig. 2.13. (Experimental result) Torques in terms of time at 20 rad/s (a), and torque-speed characteristics (b) in healthy mode.

Fig. 2 .

 2 Fig.2.13b presents the torque-speed characteristic in a speed range from 0 to 73 rad/s with the base speed at 38 rad/s. The experimental torque Tem_exp is in good accordance with the optimal value Tem_opt as obtained in section 2.1.4.B and equal to the sum of fictitious machine torques.

Fig. 2 .

 2 Fig. 2.15. (Experimental result) Phase voltage references in terms of speed, and voltage reference waveforms at 20 and 55 rad/s in healthy mode.

  Fig. 2.17. (Experimental result) Measured currents in (d1-q1) frame (a), (d9-q9) frame (b), and (d3-q3) frame (c) when phase A is open-circuited without any reconfigurations at 20 rad/s.

Fig. 2 .

 2 Fig. 2.18. (Experimental result) Torques in healthy mode and when phase A is open-circuited without any reconfigurations at 20 rad/s.

  Fig. 2.19. (Experimental result) Measured phase currents in healthy mode (a), and when phase A is opencircuited without any reconfigurations (b), at 20 rad/s.

Fig. 2 .

 2 Fig. 2.20. General scheme of the proposed fault-tolerant control methods when phase A is open-circuited.

Fig. 2 .

 2 Fig. 2.21. The control scheme of a seven-phase PMSM drive for faulty modes with an optimal control strategy under constraints on current and voltage, represented by EMR.

  Fig. 2.22. Desired currents (iα1, iβ1, iα3, iβ3) (a), and (id1, iq1, id3, iq3) (b), at 20 rad/s with (id1=0 A, iq1=12.7 A, id3=0 A, iq3=4.1 A).(a) (b) Fig. 2.23. Circles created by desired currents (iα1, iβ1) (a), and (iα3, iβ3) (b).

Chapter 2 .

 2 Modeling and Control of Multiphase Drives56For control, the time-invariant property of the reference signals can facilitate the current control with conventional PI controllers. Method (I)-1 has 6 constant currents (id1, iq1, id9, iq9, id3, iq3), only iz is time-variant. Methods (I)-2 and (I)-3 have 5 constant currents (id1, iq1, id3, iq3, iz) and 2 time-variant currents (id9, iq9).

25 . 26 .

 2526 Fig.2.24. Desired currents (iα2, iβ2, iz) (a), currents (id9, iq9, iz) (b), and phase currents (c), at 20 rad/s with method (I)-1 for an OC fault in phase A to preserve the healthy-mode torque (id1=0 A, iq1=12.7 A, id3=0 A, iq3=4.1 A). Desired currents (iα2, iβ2, iz) (a), currents (id9, iq9, iz) (b), and phase currents (c), at 20 rad/s with method (I)-2 for an OC fault in phase A to preserve the healthy-mode torque (id1=0 A, iq1=12.7 A, id3=0 A, iq3=4.1 A). Desired currents (iα2, iβ2, iz) (a), currents (id9, iq9, iz) (b), and phase currents (c), at 20 rad/s with method (I)-3 for an OC fault in phase A to preserve the healthy-mode torque (id1=0 A, iq1=12.7 A, id3=0 A, iq3=4.1 A).

27 . 28 . 29 .

 272829 Desired currents (iα2, iβ2, iz) (a), currents (id9, iq9, iz) (b), and phase currents (c), at 20 rad/s with method (I)-1 for an OC fault in phase A under constraints on RMS current and peak voltage (id1=0 A, iq1=6.4 A, id3=0 A, iq3=3.2 A). Desired currents (iα2, iβ2, iz) (a), currents (id9, iq9, iz) (b), and phase currents (c), at 20 rad/s with method (I)-2 for an OC fault in phase A under constraints on RMS current and peak voltage (id1=0 A, iq1=7.9 A, id3=0 A, iq3=3.8 A). Desired currents (iα2, iβ2, iz) (a), currents (id9, iq9, iz) (b), and phase currents (c), at 20 rad/s with method (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage (id1=0 A, iq1=7.9 A, id3=0 A, iq3=0.5 A).

Fig. 2 .

 2 30. (Calculated result) Optimal post-fault torque-speed characteristics (a), all RMS phase currents (b) and all peak phase voltages (c), in terms of speed under constraints on RMS current and peak voltage with methods (I)-1, (I)-2, and (I)-3 when phase A is open-circuited.

Fig. 2 .

 2 31. (Experimental result) Currents in (d1-q1) frame in terms of speed with method (I)-2 (a), and with method (I)-3 (b), when phase A is open-circuited. (a) (b) Fig. 2.32. (Experimental result) Currents in (d3-q3) frame in terms of speed with method (I)-2 (a), and with method (I)-3 (b), when phase A is open-circuited. (a) (b) Fig. 2.33. (Experimental results) Currents in (d9-q9) frame in terms of time at 20 rad/s with method (I)-2 (a), and with method (I)-3 (b), when phase A is open-circuited.

Fig. 2 .

 2 34. (Experimental result) Torque in three operating states including healthy mode, an OC fault in phase A without reconfigurations, and when method (I)-2 (a) and method (I)-3 (b) are applied.

Fig. 2 .

 2 35. (Experimental result) Post-fault torque-speed characteristics under an OC fault in phase A with method (I)-2 (a), and with method (I)-3 (b).

Fig. 2 .

 2 Fig. 2.36. (Experimental result) Phase currents in terms of speed, and current waveforms at 20 and 55 rad/s with method (I)-2 under an OC fault in phase A.

Fig. 2 . 64 Fig. 2 . 38 (

 264238 Fig. 2.37. (Experimental result) Phase currents in terms of speed, and current waveforms at 20 and 55 rad/s with method (I)-3 under an OC fault in phase A.Similarly, phase voltage references with the two methods in terms of speed are presented in Fig.2.38 and Fig.2.39. Their waveforms in terms of time at 20 and 55 rad/s are also plotted. It is noted that the peak values of phase voltage references after the base speeds are relatively under their limit (Vlim=100 V).

Fig. 2 . 39 (

 239 Fig. 2.39 (Experimental result) Phase voltage references in terms of speed, and current waveforms at 20 and 55 rad/s with method (I)-3 under an OC fault in phase A.

Fig. 2 . 40 .

 240 Fig. 2.40. The 1 st harmonic components of the 6 remaining healthy phase currents determined with method (II)-RCA when phase A is open-circuited.

Fig. 2 . 42 .

 242 Fig. 2.42. Calculations of time-variant d-q current references by using new reduced-order and classical transformation matrices for the pre-fault control scheme when phase A is open-circuited.

Fig. 2 .

 2 Fig. 2.45. (Calculated result) Torques under an OC fault in phase A generated with method (II)-RCA and method (II)-SCL at 20 rad/s when the 1 st , 3 rd and 9 th harmonics of the back-EMFs are considered.

Fig. 2 .

 2 46. (Calculated result) Optimal torque-speed characteristics (a), all RMS phase currents (b), and all peak phase voltages (c), in terms of speed under constraints on RMS current and peak voltage with methods (II)-RCA and (II)-SCL when phase A is open-circuited.

Chapter 2 .Fig. 2 .Fig. 2 .

 222 Modeling and Control of Multiphase Drives 76 but several drawbacks have been discussed in section 1.4 of Chapter 1. In this doctoral thesis, this problem can be solved by using ADALINEs in Chapter 3. 47. (Experimental result) Currents in (d1-q1) frame at 20 rad/s with method (II)-RCA (a), and with method (II)-SCL (b), when phase A is open-circuited. (a) (b) Fig. 2.48. (Experimental result) Currents in (d3-q3) frame at 20 rad/s with method (II)-RCA (a), and with method (II)-SCL (b), when phase A is open-circuited. 49. (Experimental results) Currents in (d9-q9) frame at 20 rad/s with method (II)-RCA (a), and with method (II)-SCL (b), when phase A is open-circuited.

Fig. 2 .

 2 50. (Experimental result) Torque in three operating states at 20 rad/s including healthy mode, an OC fault in phase A without reconfigurations, and when method (II)-RCA (a) and method (II)-SCL (b) are applied.

Fig. 2 .

 2 51. (Experimental result) Post-fault torque-speed characteristics under an OC fault in phase A with method (II)-RCA (a), and with method (II)-SCL (b).

Fig. 2 . 79 Fig. 2 .

 2792 Fig. 2.52. (Experimental result) Phase currents in terms of speed, and current waveforms at 20 and 47 rad/s with method (II)-RCA under an OC fault in phase A.

Fig. 2 . 54 ( 80 Fig. 2 .

 254802 Fig. 2.54 (Experimental result) Phase voltage references in terms of speed, and current waveforms at 20 and 47 rad/s with method (II)-RCA under an OC fault in phase A.

56 .

 56 Impacts of φi (a) and ki (b) on the average torque and the ripple torque when new phase current references are determined in (2.77) and (2.84) for an OC fault in phase A.

Fig. 2 . 57 .

 257 Fig. 2.57. The current reference of a remaining phase determined in (2.77) and (2.84), and the considered experimental speed-normalized back-EMF at 20 rad/s for an OC fault in phase A.

Fig. 2 .

 2 Fig. 2.58. Impacts of ke on the average torque and the ripple torque when new phase current references are determined in (2.77) and (2.84) for an OC fault in phase A.

59 . 60 .

 5960 Desired phase currents (a), and d-q currents (b), with method (III)-1 for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s (Im1=9.3 A). Desired phase currents (a), and d-q currents (b), with method (III)-2 for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s (Im1=8.7 A).

Fig. 2 .

 2 61. (Calculated result) Optimal torque-speed characteristic (a), all RMS phase currents (b), all peak phase voltages (c), in terms of speed under constraints on RMS current and peak voltage with methods (III)-1 and (III)-2 when phase A is open-circuited.

Fig. 2 .Fig. 2 .

 22 62. (Experimental result) Currents in (d1-q1) frame at 20 rad/s with method (III)-1 (a), and with method (III)-2 (b), when phase A is open-circuited. 63. (Experimental result) Currents in (d3-q3) frame at 20 rad/s with method (III)-1 (a), and with method (III)-2 (b), when phase A is open-circuited. (a) (b) Fig. 2.64. (Experimental results) Currents in (d9-q9) frame at 20 rad/s with method (III)-1 (a), and with method (III)-2 (b), when phase A is open-circuited.

Fig. 2 .

 2 Fig. 2.67. (Experimental result) Phase currents in terms of speed, and current waveforms at 20 and 47 rad/s with method (III)-1 under an OC fault in phase A.

Fig. 2 .

 2 Fig. 2.68. (Experimental result) Phase currents in terms of speed, and current waveforms at 20 and 47 rad/s with method (III)-2 under an OC fault in phase A.

Fig. 2 .Fig. 2 .

 22 Fig. 2.69. (Experimental result) Phase voltage references in terms of speed, and current waveforms at 20 and 47 rad/s with method (III)-1 under an OC fault in phase A.

Fig. 2 .

 2 Fig. 2.71. (Calculated result) Torque-speed characteristics in healthy mode and under an OC fault in phase A applying the seven proposed control methods.

  base speed and the maximum speed obtained in the offline optimization and experiments are the same.

  Fig. 2.72. (Experimental result) Comparisons using radar charts: optimal calculated results (a), experimental results (b), with Tave, ∆T, and Ploss at 20 rad/s.

Fig. 3 .

 3 Fig. 3.1. A general structure of an ADALINE.

2 .

 2 The effect of learning rate η on the convergence of weights: η is too low (a), η is too high (b), η is suitable (c).

  Fig. 3.6. (Experimental result) Currents in (d1-q1) frame (a), (d9-q9) frame (b), and (d3-q3) frame (c) with the back-EMF compensation for current harmonics in rotating frames at 20 rad/s in healthy mode.

  Fig. 3.7. (Experimental result) Currents in (d1-q1) (a), currents in (d9-q9) (b), currents in (d3-q3) (c) without and with the ADALINE compensation for current harmonics in rotating frames at 20 rad/s in healthy mode. Fig. 3.8. (Experimental result) Harmonic weights for currents in (d9-q9) (a), currents in (d9-q9) frame without (b) and with (c) the ADALINE compensation for current harmonics in rotating frames at 20 rad/s in healthy mode. 9. (Experimental result) Harmonic weights for currents in (d3-q3) (a), currents in (d3-q3) frame without (b), and with (c) the ADALINE compensation for current harmonics in rotating frames at 20 rad/s in healthy mode.

3 .

 3 Enhancements of Multiphase Drive Performances with Adaptive Linear Neurons 113 constraint on RMS current is always respected (IRMS_A_com ≤ 5.1 A). The current harmonics in natural frame generated by the unwanted back-EMFs are mostly eliminated as shown in Fig. 3.10c compared to Fig. 3.10b, especially the 11 th harmonic (5.8 to 0.9%). Relatively, only the 1 st , 3 rd , and 9 th harmonics exist in the phase current. Fig. 3.10. (Experimental result) Phase-A current without (iA_no_com) and with the ADALINE compensation for current harmonics in rotating frames (iA_com) (a), harmonic spectrums of phase-A current without (b) and with (c) the ADALINE compensation at 20 rad/s in healthy mode. The dynamic performances of the proposed current control structure with ADALINEs are verified with variable speeds and variable current references. The changes of the rotating speed (from 20 to 10 and 30 rad/s) slightly reduce the current control quality as shown in Fig. 3.11. Fig. 3.11. (Experimental result) Currents with variable speeds in (d1-q1) frame (a), in (d9-q9) frame (b), and in (d3-q3) frame (c) with the ADALINE compensation in healthy mode. Fig. 3.12. (Experimental result) Currents with variable current references in (d1-q1) frame (a), in (d9-q9) frame (b), and in (d3-q3) frame (c) with the ADALINE compensation in healthy mode at 20 rad/s.

Fig. 3 .

 3 Fig. 3.13. (Experimental result) Torques in one period at 20 rad/s in healthy mode after the eliminations of current harmonics in rotating frames.

Fig. 3 .

 3 Fig.3.14. The proposed structure using an ADALINE to directly eliminate torque ripples in healthy mode.

Fig. 3 .

 3 [START_REF] Yang | An Industry-Based Survey of Reliability in Power Electronic Converters[END_REF]. (Real-time simulation result) Torque and current control performances at 20 rad/s (a), and at 60 rad/s (b) in healthy mode with 4 operating stages (η=0.0001 for torque eliminations in stages 3 and 4).

Fig. 3 .

 3 [START_REF] Yang | Condition Monitoring for Device Reliability in Power Electronic Converters: A Review[END_REF]. (Experimental result) The optimal calculated torque-speed characteristics (a), experimental torquespeed characteristics without (b) and with (c) the torque ripple elimination in healthy mode.

Chapter 3 .

 3 Enhancements of Multiphase Drive Performances with Adaptive Linear Neurons 119 region of the reduced characteristic, torque ripples are slightly higher than the low-speed region because ADALINEs respond to fast coincident variations in speed and current references.

  17. (Experimental result) Phase currents (a), and phase voltage references (b) in terms of speed and time with the torque ripple elimination in healthy mode.

  a) (b) Fig. 3.18. (Experimental result) Torque, torque harmonic weights, and current control performances with the ADALINE learning process (η=0.001) (a), and in one period from the 40 th second at 20 rad/s (b) in healthy mode.

Fig. 3 . 3 .

 33 Fig. 3.19. (Experimental result) Torque, torque harmonic weights (η=0.001), and current control performances in response to the rotating speed from 20 to 35 rad/s, then to 20 rad/s, including current reference variations in healthy mode.

Fig. 3 .

 3 21. (Experimental result)The optimal calculated torque-speed characteristics (a), the experimental torquespeed characteristics without (b) and with (c) the torque ripple elimination when phase A is open-circuited.

Chapter 3 .

 3 Fig. 3.23. (Experimental result) Torque, torque harmonic weights with the learning process at 20 rad/s to eliminate torque ripples when phase A is open-circuited with method (III)-2 and η=0.001.

Fig. 3 .

 3 24. (Experimental result) Torque and current control performances in one period from the 10 th second (without the torque ripple elimination) (a), and from the 20 th second (with the torque ripple elimination and η=0.001) (b) when phase A is open-circuited with method (III)-2.

Fig. 3 .

 3 Fig. 3.25. (Experimental result) Torque and harmonic weights (η=0.001) when the rotating speed changes from 20 to 30 rad/s and then returns to 20 rad/s under an OC fault in phase A.Current control improvements in faulty mode with method (II)-RCA

Fig. 3 .

 3 Fig. 3.27. The new control structure using real-time current learning RTCL for an OC fault with method (II) represented by EMR.

Fig. 3 . 28 .

 328 Fig. 3.28. The detailed structure of RTCL in the new control structure represented by a block diagram.

Chapter 3 .Fig. 3 . 29 .

 3329 Fig. 3.29. The structure of the ADALINE used in RTCL.

Fig. 3 . 30 .

 330 Fig. 3.30. The flowchart of the ADALINE learning process in response to changes of the rotating speed and current references.

Fig. 3 .

 3 33. (Experimental result) The learning process of phase-B current with harmonic weights (η=0.001) (a), phase current convergence (b), separated harmonics (c) by RTCL at 35 rad/s.

Fig. 3 .

 3 34. (Experimental result) Control performances with the new control structure using RTCL at 35 rad/s: (id11, iq11) (a), (id91, iq91) (b), ix1 (c), (id33, iq33) (d), (id93, iq93) (e), ix3 (f).

Fig. 3 .

 3 35. (Experimental result) Dynamic performances with method (II)-RCA in the new control structure using RTCL: variable rotating speed (a), variable current references at 26 rad/s (b).

(B. 4 )

 4 FIG. B.1. (Calculated result) Torque-speed characteristics under constraints on RMS current and peak voltage when two phases are open-circuited.

1 )

 1 FIG. C.1. Une machine à n phases alimentée par un onduleur de source de tension à n bras dans un EV. Selon les six principales exigences relatives aux entraînements électriques dans les EV de la section C.1, les six propriétés spéciales suivantes permettent aux entraînements polyphasés ( FIG. C.1) d'être un choix privilégié dans les applications automobiles :

2 )

 2 Concernant la technique de contrôle dans cette thèse de doctorat, les stratégies de contrôle proposées dans tous les modes de fonctionnement seront basées sur la technique FOC et les contrôleurs PI comme mode sain. Outre les techniques récentes MPC et DTC, la technique FOC classique avec contrôleurs PI reste intéressante, notamment pour l'industrie en raison de sa qualité, sa simplicité, et sa robustesse (moins dépendante du modèle de machine). De plus, ce FOC classique a été largement utilisé dans l'industrie. Par conséquent, les stratégies de contrôle proposées dans cette thèse sont dérivées du modèle mathématique utilisant la théorie des cadres multi-

FIG. C. 4 .

 4 FIG. C.4. Schéma général des méthodes de commande tolérante aux défauts lorsque la phase A est ouverte.

FIG. C. 5 .C. 5 . 3 .

 553 FIG. C.5. (Résultat numérique) Caractéristiques couple-vitesse en mode sain et en mode dégradé de circuit ouvert en phase A, en appliquant les sept méthodes de commande proposées.La qualité de la commande d'entraînement peut être améliorée en appliquant un type simple d'intelligence artificielle nommé ADALINE dans le prochain chapitre de cette thèse de doctorat. En effet, les effets négatifs des harmoniques indésirables de back-EMF sur les couples et les courants peuvent être éliminés.

  .6. (Résultat expérimental) Les caractéristiques couple-vitesse calculées optimales (a), les caractéristiques couple-vitesse expérimentales sans (b) et avec (c) l'élimination d'ondulation de couple lorsque la phase A est ouverte.

FIG. C. 7 .

 7 FIG. C.7. (Résultat expérimental) Couple en mode sain, en mode dégradé sans des reconfigurations, en mode dégradé avec la méthode (II)-RCA dans l'ancienne structure de contrôle, et en mode dégradé avec la méthode (II)-RCA dans la nouvelle structure de contrôle en utilisant RTCL à 35 rad/s.

  en mode dégradé avec la méthode (II)-RCA dans la nouvelle structure de contrôle en utilisant RTCL à 35 rad/s. ................................................................................... 160

	List of tables
	xxv

1. Une machine à n phases alimentée par un onduleur de source de tension à n bras dans un EV. ............................................................................................................................ FIG. C.2. Différents types de défauts dans un entraînement à n phases. ............................... FIG. C.3. Les entraînements polyphasés dans les véhicules électriques avec deux modes de conduite. ................................................................................................................................. FIG. C.4. Schéma général des méthodes de commande tolérante aux défauts lorsque la phase A est ouverte. .......................................................................................................................... FIG. C.5. (Résultat numérique) Caractéristiques couple-vitesse en mode sain et en mode dégradé de circuit ouvert en phase A, en appliquant les sept méthodes de commande proposées. ............................................................................................................................... FIG. C.6. (Résultat expérimental) Les caractéristiques couple-vitesse calculées optimales (a), les caractéristiques couple-vitesse expérimentales sans (b) et avec (c) l'élimination d'ondulation de couple lorsque la phase A est ouverte........................................................... FIG. C.7. (Résultat expérimental) Couple en mode sain, en mode dégradé sans des reconfigurations, en mode dégradé avec la méthode (II)-RCA dans l'ancienne structure de List of figures xxiv contrôle, et

Table 1 .

 1 1. Fictitious machines, reference frames, and associated harmonics of a n-phase machine. ................................................................................................................................... 14

Table 2 .

 2 1. Four fictitious machines with corresponding reference frames and associated harmonics of a seven-phase machine (only odd harmonics). ................................................... 29

Table 2 .

 2 2. Electrical parameters of the experimental seven-phase PMSM drive. ................... 37

Table 2 .

 2 3. Comparisons between optimal and experimental results in healthy mode under constraints on RMS current and peak voltage at 20 rad/s. ....................................................... 43

Table 2 .

 2 4. Experimental RMS currents in all phases when phase A is open-circuited without any reconfigurations at 20 rad/s. .............................................................................................. 46

Table 2 .

 2

5. Comparisons between experimental results in healthy mode and when phase

A is open-circuited without any reconfigurations at 20 rad/s. ......................................................... 47 Table 2.6. Description of the three options in method (I). ....................................................... 52 Table 2.7. Calculated RMS values of all phase currents and the zero-sequence current with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s. ................................................................................................................................ 57 Table 2.8. Comparisons between calculated results with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s. ................................ 57 Table 2.9. Calculated RMS values of all phase currents and the zero-sequence current with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s. .................................................................................................... 58 Table 2.10. Comparisons between calculated results with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s. ........ 59 Table 2.11. Comparisons between the calculated base and maximum speeds with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage. ..................................................................................................................................... 60 Table 2.12. Experimental RMS currents in all phases with methods (I)-2 and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s. ............ 63

Table 2 .

 2 [START_REF] Jahns | Improved Reliability in Solid-State AC Drives by Means of Multiple Independent Phase Drive Units[END_REF]. Comparisons between experimental results with methods (I)-2 and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s. ............[START_REF] Semail | Outils et méthodologie d'étude des systèmes électriques polyphasés -Généralisation de la méthode des vecteurs d'espace[END_REF]

Table 2 .

 2 14. Description of the two options in method (II). ...................................................... 71

Table 2 .

 2 [START_REF] Yang | An Industry-Based Survey of Reliability in Power Electronic Converters[END_REF]. Calculated RMS currents in all phases with methods (II)-RCA and (II)-SCL for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s. .....................[START_REF] Abdel-Khalik | Effect of Current Harmonic Injection on Constant Rotor Volume Multiphase Induction Machine Stators: A Comparative Study[END_REF]

Table 2 .

 2

	List of tables

[START_REF] Yang | Condition Monitoring for Device Reliability in Power Electronic Converters: A Review[END_REF]

. Comparisons between calculated results with methods (II)-RCA and (II)-SCL for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s. ...............

[START_REF] Abdel-Khalik | Effect of Current Harmonic Injection on Constant Rotor Volume Multiphase Induction Machine Stators: A Comparative Study[END_REF]

Table 2 .

 2 17. Calculated RMS currents in all phases with methods (II)-RCA and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s when phase A is opencircuited. ...................................................................................................................................

Table 2 .

 2 18. Comparisons between calculated results with methods (II)-RCA and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s when phase A is opencircuited. ...................................................................................................................................

Table 2 .

 2 19. Comparisons between the calculated base and maximum speeds with methods (II)-RCA and (II)-SCL for an OC fault in phase A under constraints on RMS current and peak voltage. .............................................................................................................................

Table 2 .

 2 20. Experimental RMS currents in all phases with methods (II)-RCA and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s. .............................................

Table 2 .

 2

21. Comparisons between experimental results with methods (II)-RCL and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s. .............................................

Table 2.22. Description of the two options in method (III). .................................................... Table 2.23. Calculated RMS currents in all phases with methods (III)-1 and (III)-2 for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s. .......................... Table 2.24. Comparisons between calculated results with methods (III)-1 and (III)-2 for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s. .......................... Table 2.

25. Comparisons between calculated results with methods (III)-1 and (III)-2 under constraints on RMS current and peak voltage at 20 rad/s. ....................................................... Table 2.26. Comparisons between the calculated base and maximum speeds with methods (III)-1 and (III)-2 for an OC fault in phase A under constraints on RMS current and peak voltage. ..................................................................................................................................... Table 2.27. Experimental RMS currents in all phases with methods (III)-1 and (III)-2 under constraints on RMS current and peak voltage at 20 rad/s when phase A is opened................. Table 2.28. Comparisons between experimental results with methods (III)-1 and (III)-2 under constraints on RMS current and peak voltage at 20 rad/s. ............................................. Table 2.29. Current references in d-q frames for control generated in healthy mode and in a post-fault operation with methods (I), (II), and (III). ...............................................................

Table 2 .

 2 30. Comparisons between healthy mode, an OC fault in phase A without reconfigurations, and post-fault operations with methods (I), (II), and (III) under constraints on RMS current and peak voltage at 20 rad/s. .........................................................................

Table 3 .

 3

	List of tables

1. Fictitious machines, d-q reference frames, and several associated odd harmonics in natural frame of a seven-phase machine. ........................................................................... Table

3

.2. Current harmonics in d-q frames caused by unwanted back-EMF harmonics and the inverter nonlinearity with "dead-time" voltages in a seven-phase machine. ...................

Table 3 .

 3 3. Torque ripples generated by unwanted back-EMF harmonics in fictitious machines of a seven-phase machine. ......................................................................................

Table 3 .

 3 4. Possible harmonic components of the torque generated with method (III)-2. ...... TABLE A.1. Several EMR elements in the model of an energetic system. .......................... TABLE A.2. Several EMR elements in the control scheme of an energetic system. ............

Torque Speed star pentagon pentacle

	Chapter 1.	Multiphase Drives: Opportunities and State of the Art
		9

  [START_REF] Chan | The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles[END_REF]

	em T	T ei ΩΩ em P ==	(1.2)
	where v, i, and e are the n-dimensional vectors of phase voltages, phase currents and back-
	EMFs, respectively; Rs is the resistance of the stator winding of one phase; [L] is the n-by-n
	stator inductance matrix; L is the self-inductance of one phase; Mj is the mutual inductance
	between two phases shifted an angle of (2πj/n) with j ∈ [1, (n-1)/2] ⸦ ℕ if n is odd, and j ∈ [1,
	n/2] ⸦ ℕ if n is even; Tem and Pem are the electromagnetic torque and power of the machine,
	respectively.

  Clarke transformation matrix; xz2 in xαβ and the last row of [TClarke] only exist if n is even; coefficient √2/𝑛 in [TClarke] is to preserve the power in the new reference frames.

	Chapter 1.	Multiphase Drives: Opportunities and State of the Art
		1)/2 if n is odd, and k=(n-2)/2 if n is even; xz1
	and xz2 are parameters in one-dimensional zero-sequence frames (z1, z2); [TClarke] is the n-by-n
		14

Table 1 .

 1 1. Fictitious machines, reference frames, and associated harmonics of a n-phase machine.

						Fictitious machine	Reference frame	Associated harmonic
						1 st fictitious machine	α1-β1	nj ± 1
						2 nd fictitious machine	α2-β2	nj ± 2
						k th fictitious machine	αk-βk	nj ± k
						1 st zero-sequence machine	z1	nj
						2 nd zero-sequence machine (if n is even)	z2	n(j ± 1/2)
						with j ∈ ℕ0, k=(n-1)/2 if n is odd, k=(n-1)/2 if n is even.
	The voltages in the decoupled stator frames are given by:
	v	αβ	=	s R i	αβ	+	  αβ di dt αβ L	+	e	αβ		(1.4)
	where vαβ, iαβ and eαβ are the n-dimensional vectors of voltages, currents and back-EMFs in the
	decoupled stator frames, respectively; for a non-salient machine, the inductance matrix [Lαβ] in
	the decoupled stator frames is diagonal and expressed as follows:

  .1. It is noted that xz2 in xdq and xαβ, and the last row of [TPark] only exist in case of the even number of phases.

	Chapter 1.			Multiphase Drives: Opportunities and State of the Art
			dk
	dk	s dk	dk	qk qk	dk
			qk
	qk	s qk	qk	dk dk	qk
	The voltages in a d-q frame k can be expressed as follows:
					16

  1.8. As previously mentioned, matrices [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] and [𝑻 𝑷𝒂𝒓𝒌 ] are applied to decompose the n-phase machine into

	Chapter 1.	Multiphase Drives: Opportunities and State of the Art
	(n+1)/2 or (n+2)/2 fictitious machines (decoupled reference frames). Therefore, the control of
	flux and torque can be decoupled by independently regulating constant currents in rotor
	reference frames. Reference voltages 𝑣 𝑑𝑞_𝑟𝑒𝑓 , mostly calculated by proportional integral (PI)
	controllers and estimated back-EMF compensations 𝑒 𝑑𝑞_𝑒𝑠𝑡 (optional), are converted into
	natural frame 𝑣 𝑟𝑒𝑓 . These voltages are reference values to determine the duty cycle for the
	generation of switching signals of VSI (𝑠 1,𝑛 ). Specifically, 𝑣 𝑟𝑒𝑓 is compared with carrier
	signals, called Carrier Based PWM (CBPWM), or it is used to define different adjacent space
	vectors applied in different time periods, called Space Vector PWM (SVPWM).
		17

Stator voltage look-up table Hysteresis controllers

	Chapter 1.		Multiphase Drives: Opportunities and State of the Art
		ST-DTC	VSI
			…	n-phase
	Torque and flux estimators	ʃ	Sector identification	machine
		p
			19

  2.7. Electrical parameters of the drive are described in Table2.2. The experimental seven-phase PMSM is mechanically connected to a load drive that is an industrial three-phase synchronous machine. The load drive is controlled to tune the speed of the seven-phase machine. A voltage source inverter (VSI) with seven legs of Insulated Gate Bipolar Transistor (IGBT) is used to

	Chapter 2.
	DC bus	Inverter Coupling 1	Fictitious machines	Coupling 2	Shaft Mechanical Load
		(2.3)	(2.4)	(2.5)	(2.6)	(2.7)
				ZM
				FM3
	ES			FM2		MS
				FM1
					Current
					calculation
					method
					Optimization
					Look-up table
					Strategy
		Parameter			Unit	Value
		Stator resistance Rs		Ω	1.4
		Self-inductance L		mH	14.7
		Mutual inductance M1		mH	3.5
		Mutual inductance M2		mH	-0.9
		Mutual inductance M3		mH	-6.1
		1 st harmonic of the speed-normalized back-EMF	V/rad/s	1.265
		3 rd harmonic of back-EMF over 1 st harmonic	%	32.3
		9 th harmonic of back-EMF over 1 st harmonic	%	12.5
		Number of pole pairs p			3
		Rated RMS current of seven-phase PMSM	A	5.1
		DC-bus voltage VDC		V	200
		Maximum calculation time		μs	350
		PWM frequency			kHz	10
		The friction coefficient

supply the seven-phase machine. A dSPACE 1005 board with I/O interface is used to transfer PWM signals

(10 kHz)

to the IGBT driver of the inverter and collect measured data of speed and currents.

Modeling and Control of Multiphase Drives

37

Fig. 2.7. Experimental test bench of the seven-phase PMSM drive.

Table 2.2. Electrical parameters of the experimental seven-phase PMSM drive. of the rotor-load bearings fm Nm/(rad/s) 0.1 The moment of inertia of the electric drive and mechanical load Jm kg.m 2 0.02

  2.14 also shows a variation of the measured current waveform before and after the base speed of 38 rad/s. Current waveforms at two investigated speeds (20 and 55 rad/s) are totally different but have the same RMS value.

	Chapter 2.		Modeling and Control of Multiphase Drives
	T em_opt				T em_exp
	Fig. 2.14. (Experimental result) Phase currents in terms of speed, and current waveforms at 20 and 55 rad/s in
	T em_exp	T 1_exp	healthy mode.	T 1_exp	T em_opt
		T 3_exp		T 3_exp
		T 2_exp		T 2_exp
			42

Table 2 .

 2 3. Comparisons between optimal and experimental results in healthy mode under constraints on RMS current and peak voltage at 20 rad/s.

	Type of results	RMS current IRMS (A)	Torque Tem Tave (Nm) ∆T (%)	Peak voltage Vpeak (V)	Total copper loss Ploss (W)
	Optimal (numerical)	5.1	33.3	0	45.9	255
	Experimental	5.1	33.5	12	60	255.2

Modeling and Control of

Multiphase Drives 44

Table 2 .

 2 4. Experimental RMS currents in all phases when phaseA is open-circuited without any reconfigurations at 20 rad/s.

	Operating mode	A	B	C	RMS current (A) D	E	F	G
	Phase A is opened without reconfigurations	0.1	7.9	5	5.7	6.8	5.7	5.8

Table 2 .

 2 5. Comparisons between experimental results in healthy mode and when phaseA is open-circuited without any reconfigurations at 20 rad/s.

	Operating mode	Highest RMS current IRMS		Torque Tem		Highest peak voltage Vpeak	Total copper loss Ploss
		(A)	(pu)	Tave (Nm)	Tave (pu)	∆T (%)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.5	1	12	60	1	255.2	1
	OC fault without reconfigurations	7.9	1.55	33.6	1	50	104.1	1.74	323.7	1.27

pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Cavagnino | Experimental Characterization of a Belt-Driven Multiphase Induction Machine for 48-V Automotive Applications: Losses and Temperatures Assessments[END_REF]. Description of the three options in method (I).

	Option	Current design description iα2 iβ2 iz	Objective
	Method (I)-1	0	0	≠0	Reduce copper losses
	Method (I)-2	≠0	0	0	Reduce copper losses
	Method (I)-3	≠0	≠0	0	A simple dual three phase system (iB+iD+iF=0) and (iC+iE+iG=0)

  )

	cos	cos 2	cos 3	cos 5	cos 6	cos 10	sin	sin 3	sin 5
	cos 2	cos 3	cos 6	cos 9	cos 10	cos 15		sin 3	sin 9	sin 15
				sin 2	sin 6	sin 10

Table 2 .

 2 [START_REF] Dörsam | The New Voltage Level 48 V for Vehicle Power Supply[END_REF]. Calculated RMS values of all phase currents and the zero-sequence current with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s.

	Method	A	B	C	RMS current (A) D E	F	G	iz
	Method (I)-1	0	5.2	7.6	9.7	9.7	7.6	5.2	13.4
	Method (I)-2	0	5.7	6.3	7.9	7.9	6.3	5.7	0
	Method (I)-3	0	6.5	6.9	10	10	6.9	6.5	0

Table 2 .

 2 [START_REF] Pawsey | 48V technology on the Automotive Horizon[END_REF]. Comparisons between calculated results with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A to preserve the healthy-mode torque at 20 rad/s.

	Method	Torque Tem =Tave	Highest RMS current IRMS	Highest peak voltage Vpeak	Total copper loss Ploss
		(Nm)	(pu)	(A)	(pu)	(V)	(pu)	(W)	(pu)
	Healthy mode	33.3	1	5.1	1	45.9	1	255	1
	Method (I)-1	33.3	1	9.7	1.90	51.4	1.12	500.5	1.96
	Method (I)-2	33.3	1	7.9	1.55	47.1	1.03	375.5	1.47
	Method (I)-3	33.3	1	10	1.96	57.9	1.26	536	2.1
	pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Sattler | 48 V minihybrid-A new solution for the minimal hybridization of vehicles[END_REF]. Calculated RMS values of all phase currents and the zero-sequence current with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s.

	Method	A	B	C	RMS current (A) D E	F	G	iz
	Method (I)-1	0	3.2	3.9	5.1	5.1	3.9	3.2	7.2
	Method (I)-2	0	3.6	4.3	5.1	5.1	4.3	3.6	0
	Method (I)-3	0	2.9	4	5.1	5.1	4	2.9	0

Table 2 .

 2 [START_REF] Hackmann | 48 V-The way to a high volume electrification[END_REF]. Comparisons between calculated results with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s.

	Method	Highest RMS current IRMS	Torque Tem=Tave	Highest peak voltage Vpeak	Total copper loss Ploss
		(A)	(pu)	(Nm)	(pu)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.3	1	45.9	1	255	1
	Method (I)-1	5.1	1	17.7	0.53	42.8	0.93	144.4	0.57
	Method (I)-2	5.1	1	21.7	0.65	42.3	0.92	162.6	0.64
	Method (I)-3	5.1	1	19.1	0.57	39	0.85	140	0.55
	pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Lauer | 48 Volt Hybrid with e-drive features -Excellent fuel efficiency and drivability[END_REF] describes the calculated base and maximum speeds of the healthy and faulty modes with the three options of method (I). The lowest base speed can be observed in healthy mode at 38 rad/s. The highest base speed is 46 rad/s (1.21 pu) with method (I)-3 while the base speeds Method (I)-2 results in the highest post-fault maximum speed (0.88 pu) while method (I)-1 has the lowest maximum speed (0.81 pu).

	Methods (I)-1-2-3	Methods	(I)-1-2-3

with methods (I)-1 and (I)-2 are the same (1.08 pu). The lowest (healthy mode) and highest (method (I)-3) base speeds are consequences of the highest (healthy mode) and lowest (method (I)-3) values of Vpeak in Table

2

.10. The healthy mode has the highest maximum speed of 73

Table 2 .

 2 [START_REF] Lauer | 48 Volt Hybrid with e-drive features -Excellent fuel efficiency and drivability[END_REF]. Comparisons between the calculated base and maximum speeds with methods (I)-1, (I)-2, and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage.

	Method	(rad/s)	Base speed Ωbase	(pu)	Maximum speed Ωmax (rad/s)	(pu)
	Healthy mode	38		1	73		1
	Method (I)-1	41		1.08	59	0.81
	Method (I)-2	41		1.08	64	0.88
	Method (I)-3	46		1.21	61	0.84
	pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Els | 48 Volt Electrification: the next step to achieving 2020 emissions[END_REF]. Experimental RMS currents in all phases with methods (I)-2 and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s.

								RMS current (A)
	Method	A		B		C		D		E		F		G
		opt	exp	opt	exp	opt	exp	opt	exp	opt	exp	opt	exp	opt	exp
	Method (I)-2	0	0.1	3.6	3.8	4.3	3.9	5.1	4.5	5.1	5.1	4.3	4	3.6	3.8
	Method (I)-3	0	0.1	2.9	3.3	4	3.5	5.1	4.5	5.1	5	4	3.6	2.9	3.2
	opt: optimal calculated results; exp: experimental results.

Table 2 .

 2 [START_REF] Jahns | Improved Reliability in Solid-State AC Drives by Means of Multiple Independent Phase Drive Units[END_REF]. Comparisons between experimental results with methods (I)-2 and (I)-3 for an OC fault in phase A under constraints on RMS current and peak voltage at 20 rad/s. unit where the base values are parameters of healthy mode.

	Method	Highest RMS current IRMS		Torque Tem		Highest peak voltage Vpeak	Total copper loss Ploss
		(A)	(pu)	Tave (Nm)	Tave (pu)	∆T (%)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.5	1	12	60	1	255.2	1
	OC fault without reconfigurations	7.9	1.55	33.6	1	50	104.1	1.74	323.7	1.27
	Method (I)-2	5.1	1	21.6	0.64	31	68.2	1.14	149.1	0.58
	Method (I)-3	5	0.98	19	0.57	32	59	0.98	124	0.49
	pu: per

  [START_REF] Jahns | Pulsating torque minimization techniques for permanent magnet AC motor drives-a review[END_REF]

		0.363 0.521 0.232
		0.815 0.232 0.418
	 	0.452 0.418 0.521 0.452 0.418 0.521
		0.815 0.232 0.418
		0.363 0.521 0.232

  ) are 2 reference frames associated with 2 two-phase fictitious machines while x3 and z3 are 2 reference frames associated with 2 zero-sequence machines. The zero-sequence current iz3 is always zero due to the wye connection of the stator windings.

				40)
	where [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑	] and [𝑻 𝑷𝒂𝒓𝒌 𝟑	] are new 6-by-6 Clarke and Park matrices as presented in (2.39) and
	(2.40), respectively.
	Similarly, (d33-q33) and (d93-q93As [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 ], 6-by-6 matrix [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑	] is defined from matrix [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝒇𝒂𝒖𝒍𝒕 ] in (2.32) by:

MM

		(2.41)
		13
		0.418 0.452 0.521
		0.521 0.363 0.232
	 	0.232 0.815 0.418 0.232 0.815 0.418
		0.521 0.363 0.232
		0.418 0.452 0.521

  .[START_REF] Soong | Field-weakening performance of brushless synchronous AC motor drives[END_REF]. It is noted that their amplitudes are designed to be identical. Expressions for these currents are presented as follows:

	Chapter 2.		Modeling and Control of Multiphase Drives
	with
	C3			0.6591 sin(3 q33	90 )
	0.6591 sin( q11 ii B1  =	+	21.4 ) 	(2.50)
	0.6591 sin( q11 ii C1  =	90 ) + 	(2.51)
	0.6591 sin( q11 ii D1  =	+	158.6 ) 	(2.52)
	0.6591 sin( 158.6 ) q11 ii E1  = -	(2.53)
	0.6591 sin( q11 ii F1  =	90 ) -	(2.54)
	0.6591 sin( q11 ii G1  =	-	21.4 ) 	(2.55)
							𝑖 1	𝑖 1
							𝑖 1
							𝑖 1
							𝑖 1
							𝑖 1
	From (2.33), to obtain the currents as in (2.50)-(2.55), currents (id91, iq91, ix1) need to be
	calculated as follows:
	d91 i	= -	sin(6 / 7) sin(5 / 7)  	( i d11	) sin( )  cos( ) q11 i -	(2.56)
	q91 i		=		8 / 7 sin(6 / 7) 	sin( ) 8 / 7 sin(3 / 7) cos( ) q11 ii d11   +  C1 i   - cos(3 / 7) cos(2 / 7) C1 i     -    	(2.57)
	x1 i	=	8 / 7 sin(2 / 7) 	sin( ) 8 / 7 sin(3 / 7) cos( ) q11 ii d11   +  C1 i   - cos( / 7) cos(2 / 7) C1 i     -    	(2.58)
	However, from (2.41), as currents (id91, iq91, ix1) are time-variant, these currents interact with the
	1 st , 3 rd and 9 th harmonics of back-EMFs in enf to generate torque ripples. By using (2.43),
	currents (ix3, id93, iq93), eliminating the torque ripples caused by the 1 st and 3 rd harmonics of
	back-EMFs, can be expressed by:
	d93 i		= -	sin(6 / 7) sin(5 / 7)  	( i d33	) sin(3 )  cos(3 ) q33 i -	(2.59)
	q93 i		=		8 / 7 sin(6 / 7) 	sin(3 ) 8 / 7 sin(3 / 7) cos(3 ) q33 ii d33   +  C3 i   - cos(3 / 7) cos(2 / 7) C3 i     -    	(2.60)
	x3 i		=	8 / 7 sin(2 / 7) 	sin(3 ) 8 / 7 sin(3 / 7) cos(3 ) q33 ii d33   +  C3 i   - cos( / 7) cos(2 / 7) C3 i     -    	(2.61)
							71

Fig. 2.41. The 1 st harmonic components of the 6 remaining healthy phase currents determined with method (II)-SCL when phase A is open-circuited.

Table 2 .

 2 [START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF]. Description of the two options in method (II).

	Option	id91	iq91	Current design description ix1 ix3	id93	iq93	Objective
	Method (II)-RCA	0	0		0	0		0	0	Torque is unaffected by the back-EMF harmonics of the second fictitious machine
	Method (II)-SCL	time-variant	time-variant	time-variant	time-variant	time-variant	time-variant	Similar copper losses are distributed in the remaining healthy phases

Table 2 .

 2 [START_REF] Yang | An Industry-Based Survey of Reliability in Power Electronic Converters[END_REF]. Calculated RMS currents in all phases with methods (II)-RCA and (II)-SCL for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s.

	Method	A	B	C	RMS current (A) D	E	F	G
	Method (II)-RCA	0	10.6	8	5.3	5.3	8	10.6
	Method (II)-SCL	0	7.6	7.9	7.6	7.6	7.9	7.6

Table 2 .

 2

	Method	Torque Tem=Tave	Highest RMS current IRMS	Highest peak voltage Vpeak	Total copper loss Ploss
		(Nm)	(pu)	(A)	(pu)	(V)	(pu)	(W)	(pu)
	Healthy mode	33.3	1	5.1	1	45.9	1	255	1
	Method (II)-RCA	33.3	1	10.6	2.08	58.9	1.28	572.5	2.25
	Method (II)-SCL	33.3	1	7.9	1.55	45.5	0.99	496.4	1.95
	pu: per unit where the base values are parameters of healthy mode.

[START_REF] Yang | Condition Monitoring for Device Reliability in Power Electronic Converters: A Review[END_REF]

. Comparisons between calculated results with methods (II)-RCA and (II)-SCL for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s.

Table 2 .

 2 [START_REF] Zhong | Active torque ripple reduction based on an analytical model of torque[END_REF]. Calculated RMS currents in all phases with methods (II)-RCA and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s when phase A is open-circuited.

	Method	A	B	C	RMS current (A) D	E	F	G
	Method (II)-RCA	0	5.1	3.8	2.5	2.5	3.8	5.1
	Method (II)-SCL	0	4.9	5.1	4.9	4.9	5.1	4.9

Table 2 .

 2

	Method	Highest RMS current IRMS	Torque Tem=Tave	Highest peak voltage Vpeak	Total copper loss Ploss
		(A)	(pu)	(Nm)	(pu)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.3	1	45.9	1	255	1
	Method (II)-RCA	5.1	1	16	0.48	35.5	0.77	132.1	0.52
	Method (II)-SCL	5.1	1	21.6	0.65	36	0.78	208.6	0.82
	pu: per unit where the base values are parameters of healthy mode.

[START_REF] Azar | Influence of Electric Loading and Magnetic Saturation on Cogging Torque, Back-EMF and Torque Ripple of PM Machines[END_REF]

. Comparisons between calculated results with methods (II)-RCA and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s when phase A is open-circuited.

Table 2 .

 2

	Method	(rad/s)	Base speed Ωbase	(pu)	Maximum speed Ωmax (rad/s)	(pu)
	Healthy mode	38		1	73	1
	Method (II)-RCA	44		1.16	49	0.67
	Method (II)-SCL	46		1.21	49	0.67
	pu: per unit where the base values are parameters of healthy mode.

[START_REF] Ward | Preliminary investigation of an invertor-fed 5-phase induction motor[END_REF]

. Comparisons between the calculated base and maximum speeds with methods (II)-RCA and (II)-SCL for an OC fault in phase A under constraints on RMS current and peak voltage.

  SCL in the post-fault operation. However, the experimental torques have ripples of 19% with method (II)-RCA and 24% with method (II)-SCL. The experimental back-EMFs with a significant proportion of the 9 th harmonic (12.5%) interact with the time-variant currents (id9, iq9) in methods (II)-SCL. Meanwhile, method (II)-RCA is unaffected by the 9 th harmonic, verifying the robustness of method (II)-RCA.

	Chapter 2.	Modeling and Control of Multiphase Drives
	healthy mode to 16 Nm (0.48 pu) with method (II)-RCA, and 21.6 Nm (0.64 pu) with method
	(II)-
	(II)-RCA	(II)-SCL
	(II)-RCA	(II)-SCL
	(II)-RCA	(II)-SCL
		.50, including a healthy
	operation, an OC fault without reconfigurations, and a fault-tolerant operation with one of two
	77

methods (II)-RCA and (II)-SCL. The first two operating states have been deeply investigated in sections 2.1.4 and 2.2. It is noted that the average torques in the third state with methods (II)-RCA and (II)-SCL are in good accordance with the optimal (calculated) values in Table

2

.18. To respect the constraints on RMS current, the average torque is reduced from 33.5 Nm in

Table 2 .

 2 [START_REF] Parsa | Five-phase permanent-magnet motor drives[END_REF]. Experimental RMS currents in all phases with methods (II)-RCA and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s.

								RMS current (A)
	Method	A		B		C		D		E		F		G
		opt	exp	opt	exp	opt	exp	opt	exp	opt	exp	opt	exp	opt	exp
	Method (II)-RCA	0	0.1	5.1	4.9	3.8	3.3	2.5	2.9	2.5	2.4	3.8	3.8	5.1	5
	Method (II)-SCL	0	0.1	4.9	5	5.1	4.8	4.9	4.8	4.9	4.8	5.1	4.9	4.9	5.1

opt: optimal calculated results; exp: experimental results.

Table 2 .

 2

[START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF]

. Comparisons between experimental results with methods (II)-RCL and (II)-SCL under constraints on RMS current and peak voltage at 20 rad/s.

Method Highest

RMS current IRMS Torque Tem Highest peak voltage Vpeak Total copper loss Ploss

		(A)	(pu)	Tave (Nm)	Tave (pu)	∆T (%)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.5	1	12	60	1	255.2	1
	OC fault without reconfigurations	7.9	1.55	33.6	1	50	104.1	1.74	323.7	1.27
	Method (II)-RCA	5	0.98	16	0.48	19	64.4	1.07	123.4	0.48
	Method (II)-SCL	5.1	1	21.6	0.64	24	59.9	1	203.3	0.8

pu: per unit where the base values are parameters of healthy mode.

  and D, respectively. Then, analytical torque expressions in natural

	Chapter 2.				Modeling and Control of Multiphase Drives
	frame can be used to determine parameters of the current references. The electromagnetic
	torque is analytically expressed by:
	em T	ave T =+	2 T 				(2.66)
	with
			sin	 32 2 cos 7 2 7	sin	2	sin	( ) 
			sin	 32 2 cos 7 2 7	sin 2	2	( sin 2	) 
					82

  are imposed, the electromagnetic torque is expressed by:

	em T	=	ave T	+	2 T	4 T  +						(2.72)
	with
			ave		m1 n1	sin	 32 2 cos B D 7 2 7	sin	B	2	D	sin	( )  C
							sin	 32 2 cos B D 7 2 7	sin 2		B	2	D	( sin 2	C	) 
			2			m1 n1	e	sin	96 2 cos B D 7 2 7	sin 2	B	2	D	e	sin 2	C

  With arbitrary values of ke and φC, the ripple torques T2θ in (2.74)-(2.75) cannot be nullified. It means that the attempt to eliminate all ripple torques is unfeasible due to the waveform uniformity of the remaining phase currents in (2.65). 2) In only two special values of ke, ripple torques T2θ in (2.74)-(2.75) can be nullified to obtain constant torques. Specifically, ke must be 3.74 in (2.74) and 0.33 in (2.75). At the same time, φC must be 0.5φe. However, if the first and third harmonic back-EMFs are in phase (φe=0), the average torques Tave in both cases become zero due to (φC=0.5φe=0). 3) In (2.74) and (2.75), the average torques are proportional to the amplitudes of the fundamental components of currents (Im1) and back-EMFs (En1). In addition, the torque values also depend on the sine function of φC. Accordingly, if the initial current angles in (2.70) are applied in (2.72), the average torque Tave is significantly increased 10% compared to (2.75). In this case, only one part of T2θ is eliminated, and the remaining torques become:

	Chapter 2.			Modeling and Control of Multiphase Drives
	2.3.4.C.2. An attempt to increase the average torque:
	C φ	2
	B φ	5 42
	D φ	37 42
	ave T	2.84	m1 n1 I E
	2 T	0.9	m1 n1 e I E k	( cos 2	e	)
	4	0.36		m1 n1 e	( cos 4	e	)
							75)
	Remarks:
		1) The maximum average torque in (2.75)
		is higher than (2.74) but less than in (2.70).
		4) From the above analyses, current parameters in (2.74)-(2.75) will not be used in
		method (III).
							85

Table 2 .

 2 [START_REF] Levi | Multiphase induction motor drives -a technology status review[END_REF]. Description of the two options in method (III).

	Option	φB	Current design description φC φD ki	φi	Objective
	Method (III)-1	-	5 42 	-	2 	-	37  42	0	0	Enhance the average torque by using only sinusoidal currents 2.84 ave m1 n1 T I E =
	Method (III)-2	-	5 42 	-	2 	-	37 42 	ke	φe	Enhance the average torque by using non-sinusoidal currents with the waveform of back-EMFs   2 2.84 1.76 ave m1 n1 e T I E + k =

Table 2 .

 2 [START_REF] Duran | Multiphase Electric Drives: Introduction[END_REF]. Calculated RMS currents in all phases with methods (III)-1 and (III)-2 for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s.

	Method	A	B	C	RMS current (A) D	E	F	G
	Method (III)-1	0	6.6	6.6	6.6	6.6	6.6	6.6
	Method (III)-2	0	6.5	6.5	6.5	6.5	6.5	6.5

Table 2 .

 2

	Method

24.

Comparisons between calculated results with methods (III)-1 and (III)-2 for an OC fault in phase A when the healthy-mode torque is preserved at 20 rad/s.

Torque Tem Highest RMS current IRMS Highest peak voltage Vpeak Total copper loss

									Ploss
		Tave (Nm)	(pu)	∆T (%)	(A)	(pu)	(V)	(pu)	(W)	(pu)
	Healthy mode	33.3	1	0	5.1	1	45.9	1	255	1
	Method (III)-1	33.3	1	24	6.6	1.29	40	0.87	363.1	1.42
	Method (III)-2	33.3	1	19	6.5	1.27	37.6	0.82	352.3	1.38
	pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Slunjski | Symmetrical/Asymmetrical Winding Reconfiguration in Multiphase Machines[END_REF]. Comparisons between calculated results with methods (III)-1 and (III)-2 under constraints on RMS current and peak voltage at 20 rad/s.

	Method	Highest RMS current IRMS		Torque Tem		Highest peak voltage Vpeak	Total copper loss Ploss
		(A)	(pu)	Tave (Nm)	(pu)	∆T (%)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.3	1	0	45.9	1	255	1
	Method (III)-1	5.1	1	25.9	0.78	24	38.6	0.84	218.1	0.86
	Method (III)-2	5.1	1	26.2	0.79	19	44.3	0.97	218.3	0.86
	pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines Part I[END_REF]. Comparisons between the calculated base and maximum speeds with methods (III)-1 and (III)-2 for an OC fault in phase A under constraints on RMS current and peak voltage.

	Method	(rad/s)	Base speed Ωbase	(pu)	Maximum speed Ωmax (rad/s)	(pu)
	Healthy mode	38		1	73	1
	Method (III)-1	45		1.18	49	0.67
	Method (III)-2	38		1	48	0.66
	pu: per unit where the base values are parameters of healthy mode.

Table 2 .

 2 [START_REF] Bojoi | Multiphase electrical machines and drives: A viable solution for energy generation and transportation electrification[END_REF]. Experimental RMS currents in all phases with methods (III)-1 and (III)-2 under constraints on RMS current and peak voltage at 20 rad/s when phase A is opened.When the operating speed varies, the constraints on RMS current and voltage are still respected in the experimental results, guaranteeing flux-weakening operations. Measured phase currents by the two methods in terms of speed are described in Figs. 2.67-2.68. The waveforms of measured phase currents at 20 and 47 rad/s are also plotted. The waveforms of the remaining are changed at high speed and the highest RMS currents reduce at speeds over the base speeds to respect the constraint on peak voltage.

	Method	A	B	C	RMS current (A) D	E	F	G
	Method (III)-1	0.1	5.3	4.8	4.9	5	4.9	5.3
	Method (III)-2	0.1	5.3	4.8	4.9	5	4.8	5.1

Table 2 .

 2 [START_REF] Rui | AN-Power stage of 48 V BSG inverter[END_REF]. Comparisons between experimental results with methods (III)-1 and (III)-2 under constraints on RMS current and peak voltage at 20 rad/s. unit where the base values are parameters of healthy mode.

	Method	Highest RMS current IRMS		Torque Tem		Highest peak voltage Vpeak	Total copper loss Ploss
		(A)	(pu)	Tave (Nm)	Tave (pu)	∆T (%)	(V)	(pu)	(W)	(pu)
	Healthy mode	5.1	1	33.5	1	12	60	1	255.2	1
	OC fault without reconfigurations	7.9	1.55	33.6	1	50	104.1	1.74	323.7	1.27
	Method (III)-1	5.3	1.04	25.6	0.76	33	56.5	0.94	212.5	0.83
	Method (III)-2	5.3	1.04	26.1	0.78	31	62.4	1.04	208.6	0.82
	pu: per

Table 2 .

 2 [START_REF] Locment | Comparison of 3-, 5and 7-leg Voltage Source Inverters for low voltage applications[END_REF]. Current references in d-q frames for control generated in healthy mode and in a post-fault operation with methods (I), (II), and (III).

	Method	id1_ref	iq1_ref	id9_ref	iq9_ref	id3_ref	iq3_ref	iz_ref
	Healthy mode	constant	constant	constant	constant	constant	constant	0
	Method (I)-1	constant	constant	constant	constant	constant	constant	variant
	Method (I)-2	constant	constant	variant	variant	constant	constant	0
	Method (I)-3	constant	constant	variant	variant	constant	constant	0
	Method (II)-RCA	variant	variant	0	0	variant	variant	0
	Method (II)-SCL	variant	variant	variant	variant	variant	variant	0
	Method (III)-1	constant	constant	variant	variant	variant	variant	0
	Method (III)-2	variant	variant	variant	variant	variant	variant	0
	constant: time-constant; variant: time-variant.

Table 2 .

 2 [START_REF] Runde | ISCAD -Design, control and car integration of a 48 volt high performance drive[END_REF]. Comparisons between healthy mode, an OC fault in phase A without reconfigurations, and postfault operations with methods (I), (II), and (III) under constraints on RMS current and peak voltage at 20 rad/s. : per unit where the base values are parameters of healthy mode. * Ωbase and Ωmax in offline optimizations and experiments are the same.

	Method	opt	Tave	exp		opt	∆T	exp	opt	Ploss	exp	Ωbase *	Ωmax *
		(Nm)	(pu)	(Nm)	(pu)	(%)		(%)	(W)	(pu)	(W)		(pu) (rad/s) (pu) (rad/s) (pu)
	Healthy mode	33.3	1		33.5	1	0		12 255	1	255.2	1	38	1	73	1
	OC fault without reconfigurations	33	0.99 33.6	1	46		50 366.5 1.4 323.7 1.27	36	0.95	36	0.49
	Method (I)-1	17.7 0.53	-		-	0		-144.4 0.57	-		-	41	1.08	59	0.81
	Method (I)-2	21.7 0.65 21.6 0.64	0		31 162.6 0.64 149.1 0.58	41	1.08	64	0.88
	Method (I)-3	19.1 0.57	19	0.57	0		32 140 0.55 124 0.49	46	1.21	61	0.84
	Method (II)-RCA	16	0.48	16	0.48	0		19 132.1 0.52 123.4 0.48	44	1.16	49	0.67
	Method (II)-SCL	21.6 0.65 21.6 0.64	0		24 208.6 0.82 203.3 0.8	46	1.21	49	0.67
	Method (III)-1	25.9 0.78 25.6 0.76 24		33 218.1 0.86 212.5 0.83	45	1.18	49	0.67
	Method (III)-2	26.2 0.79 26.1 0.78 19		31 218.3 0.86 208.6 0.82	38	1	48	0.66
	opt: optimal calculated results; exp: experimental results.

pu

Table 3 .

 3 2. Current harmonics in d-q frames caused by unwanted back-EMF harmonics and the inverter nonlinearity with "dead-time" voltages in a seven-phase machine.

	Frame	Current harmonics by unwanted back-EMF harmonics	Current harmonics by "dead-time" voltages
	d1-q1	14θ	14θ
	d9-q9	28θ	14θ, 28θ
	d3-q3	14θ	14θ

Table 3

 3 .2. Specifically, if x is d1, q1, d3 or q3, the

		PI combined with ADALINE to
	w 1_x	eliminate current harmonics in i x	PI
	w 2_x
												-+	+	+ +	-+
	w 3_x	w	1_x	Updating laws cos(14 ) error   = + 1_x w i	Transfer function
	w 4_x	w	2_x	=	w	2_x	+		i	error	sin(14 ) 
		w	3_x	=	w	3_x	+		i	error	cos(28 ) 
		w	4_x	=	w	4_x	+		i	error	sin(28 ) 

Table 3 .

 3 3. Torque ripples generated by unwanted back-EMF harmonics in fictitious machines of a seven-phase machine.

	Fictitious machine	Current harmonics Unwanted back-EMF harmonics	Torque ripples
	The first machine (FM1) with torque T1_exp	1	13	14θ
	The second machine (FM2) with torque T2_exp	9	19	28θ
	The third machine (FM3) with torque T3_exp	3	11	14θ

Table 3 .

 3 4. Possible harmonic components of the torque generated with method (III)-2.

	Back-EMF harmonic
		1 st	3 rd	9 th	11 th	13 th	19 th
	Current harmonic
	1 st	2θ	2θ, 4θ	8θ, 10θ	10θ, 12θ	12θ, 14θ	18θ, 20θ
	3 rd	2θ, 4θ	6θ	6θ, 12θ	10θ, 12θ	10θ, 16θ	16θ, 22θ

base_new Ω base T em_opt T em_exp

  3.21a. Like healthy mode, the base speed reduces from 38 to 25 rad/s. The experimental characteristics without and with the torque ripple elimination are shown in Fig.3.21b and Fig.3.21c, respectively. At 20 rad/s, before the base speed, the torque ripple is significantly reduced from 31 to 5% after the torque ripple elimination with the same average

	ADALINE to eliminate				Updating laws
	torque ripples with method (III)-2	1_tor 2_tor w w	= =	1_tor 2_tor w w	error error T  T  + +	cos(2 ) sin(2 )  
	w 1_tor	3_tor w 4_tor w 21_tor w	3_tor w 4_tor w 21_tor + + w = = =	  +	cos(4 )  sin(4 )  cos(22 ) error error T error T T  	+	-	7-phase drive (in Fig. 2.21)
	w 2_tor	22_tor w	=	22_tor w	+		error T	sin(22 ) 
	w 3_tor
	w 4_tor										+ +
	w 21_tor w 22_tor									   	e	 fault	2	   	e	fault	Strategy (in Fig. 2.21)
	Ω
									Without torque ripple elimination
										(Original characteristic)

  2) Then, iB1_est is used to calculate the 1 st harmonic currents of the other phases (iC1_est, Finally, the 3 rd harmonic currents in natural frame (iC3_est, iD3_est, iE3_est, iF3_est, iG3_est)

iD1_est, iE1_est, iF1_est, iG1_est) by exploiting their mathematical relationships as described in current design options RCA and SCL in sections 2.3.3.B and 2.3.3.C of Chapter 2. 3)

  Fig. 3.31. (Experimental result) Torque in healthy mode, an OC fault without any reconfigurations, with method (II)-RCA in the pre-fault control structure, and with method (II)-RCA in the new control structure using RTCL at 35 rad/s.

			Start
			The pre-fault control
			scheme
			Activate ADALINE to
			learn i B
			The proposed control
			scheme
	no	Ω changes?	yes	no	i dq_ref changes?	yes

Fig.

3

.32. (Experimental result) Phase currents in healthy mode, an OC fault without any reconfigurations, with method (II)-RCA in the pre-fault control structure, and with method (II)-RCA in the new control structure using RTCL at 35 rad/s.

  TABLE A.2 gives details of several EMR elements in the control scheme of the system. More information and the EMR library can be found in the official website of EMR [158].

TABLE A .

 A 2. Several EMR elements in the control scheme of an energetic system.

Techniques de contrôle des entraînements polyphasés en mode sain

  Comme analysé dans[21-23, 26, 31, 60-62], les techniques de contrôle pour les entraînements polyphasés peuvent être généralement classées en trois types principales : contrôle vectoriel orienté (FOC), contrôle direct du couple (DTC), et contrôle prédictif basé sur un modèle (MPC). Seules boucles internes pour le contrôle du courant, du couple et du flux sont analysées car la boucle externe pour le contrôle de la vitesse dans les trois techniques de contrôle est la même.

C.3.1.

C.

3.2. Stratégies de contrôle pour des opérations avec défauts

FIG. C.2. Différents types de défauts dans un entraînement à n phases.

  1 uniquement lorsque ces entrainements sont correctement contrôlés dans des diverses conditions de fonctionnement. Par exemple, un véhicule électrique entraîné par des entraînements polyphasés avec deux modes de conduite est décrit dans la FIG. C.3. La référence de couple peut être générée soit à partir de la pédale d'accélérateur (contrôlée par le conducteur), soit du régulateur de vitesse (une opération autonome, par exemple). Le mode économique permet généralement aux véhicules électriques de fonctionner avec un rendement élevé dans la région à faible couple de la caractéristique couple-vitesse. Pendant ce temps, le mode haute performance permet aux véhicules électriques de fonctionner avec des couples maximaux à chaque vitesse pour les accélérations. Un basculement entre ces deux modes de conduite se produit lorsque le couple requis entraîne un dépassement des limites électriques (courants et tensions). Cette thèse de doctorat portera sur le mode haute performance dans des conditions saines ou dégradés. Le couple électromagnétique sera maximisé dans les limites des courants et des tensions. Les entraînements polyphasés dans les véhicules électriques avec deux modes de conduite.

	Appendices
	FIG. C.3.
	156

  avec les techniques MPC et DTC pour les machines 5 phasées. 4) Concernant les améliorations des performances de contrôle et de couple sous impacts d'incertitudes et d'imperfections des entraînements électriques par l'intelligence artificielle, ADALINE (ADAptive LInear NEuron), type simple de réseaux de neurones artificiels, est retenu. L'une des incertitudes peut être les tensions «deadtime» (la non-linéarité de l'onduleur) tandis que la présence de multi-harmoniques dans les forces électromotrices provoque une imperfection dans la conception de la machine. Les ADALINE sont choisis dans cette thèse de doctorat grâce à leur capacité d'auto-apprentissage et leur mise en oeuvre facile. Par ailleurs, la connaissance des composants harmoniques dans les signaux que doivent compenser les ADALINE permet leur convergence rapide et d'éviter la charge de calcul, augmentant l'applicabilité aux entraînements électriques industriels. En effet, cette connaissance est facilement obtenue et généralement proportionnelle au nombre de phases. Les ADALINES ont juste besoin de trouver les bonnes amplitudes de ces harmoniques. Ces avantages rendent les ADALINE plus favorables par rapport à d'autres approches telles que FL, PIR, etc. Le contrôle FL nécessite l'expérience d'experts tandis que le contrôle PIR nécessite de multiples ajustements de paramètres et nécessite de la connaissance de la fréquence. En utilisant des ADALINE, la qualité du contrôle du courant et la qualité du couple seront considérablement améliorées, soit en mode sain, soit en mode dégradé.

5.1. Chapitre 1-Les entraînements polyphasés : opportunités et état de l'art Section C.1

  a abordé les principales exigences des entraînements électriques pour les véhicules électrifiés, motivant cette thèse de doctorat. Grâce aux propriétés spéciales des entraînements polyphasés présentés dans le premier chapitre, les EV pilotés par des entraînements polyphasés, en particulier les entraînements de machines PMSM, ont un fort potentiel de commercialisation dans un proche avenir. Cependant, les entraînements polyphasés présentent certains inconvénients en ce qui concerne le temps de calcul élevé, le coût plus élevé de l'électronique de commande en raison du nombre élevé de bras de VSI à piloter, la complexité de la modélisation et du contrôle, et l'inertie sociale. Ces inconvénients peuvent être surmontés par les progrès des microprocesseurs, les productions de masse, les améliorations des technologies d'électronique de puissance ainsi que les investissements à long terme des entreprises industrielles, etc.De plus, la modélisation générale d'une machine PMSM polyphasée a été brièvement décrite pour donner un aperçu du modèle mathématique d'une machine polyphasée. Cette modélisation est présentée pour faciliter les analyses des études récentes sur le domaine de contrôle des entraînements polyphasés pour les modes sains et dégradés. A partir de ces analyses, les objectifs de cette thèse de doctorat sont clarifiés. Plus précisément, la technique de FOC pour machine non-sinusoïdale est appliquée dans cette étude en raison de sa qualité, sa simplicité et sa popularité. Le contrôle s'opère en mode sain et en modes dégradés sous contraintes de courant et de tension. Enfin, pour tenir compte de l'incertitude sur les modèles, l'utilisation de structure d'auto-apprentissage de type ADALINE est effectuée afin d'améliorer la qualité de contrôle du couple dans tous les modes de fonctionnement.

C.5.

2. Chapitre 2-Modélisation et contrôle des entraînements polyphasés

  Ce chapitre est dédié à la modélisation et au contrôle des entraînements de machines PMSM polyphasées avec des forces électromotrices non-sinusoïdales en mode sain et en modes dégradés. Les contraintes sur le courant RMS et la tension de crête ont été prises en compte dans les modes sains et dégradés. Une étude d'un cas de machine non-sinusoïdale à sept phases est choisie pour illustrer les théories de cette thèse de doctorat.

Without loss of generality, the selection of the 9 th harmonic comes from the back-EMF harmonic spectrum of the considered machine (see section 2.1.4.A of Chapter 2). If the 5 th harmonic of the back-EMF is considered, analyses and solutions are similar.

The use of the six ADALINEs for current harmonic eliminations in Fig.3.5 aims at eliminating the impacts of the unwanted back-EMF harmonics and the inverter nonlinearity with "dead-time" voltages, improving current control quality even when current references are not constant.

Method (II)-RCA

w

laboratory. I also would like to thank all professors, colleagues, and staffs of L2EP as well as those at ENSAM, including laboratories LISPEN and LMFL.

I would like to express my gratitude to the Ministry of Education and Training, a representative of the Vietnamese government, for the four-year scholarship. I also would like to thank Campus France, a representative of the French government, together with ENSAM and CE2I project for their valuable support.

Methods (II)-RCA-

of current waveforms enables to maximize currents in all remaining phases within their limit. This current maximization allows to generate more torques, especially when the phase current and back-EMF waveforms are similar as described in method (III)-2. However, the fixed waveforms of phase currents in method (III) lead to inevitable torque ripples.

Chapter 3 has tackled the existing problems presented in Chapter 2 by using a simple type of artificial intelligence, ADALINE. ADALINEs have eliminated current harmonics in d-q frames as well as torque ripples in healthy mode. By the same way, the post-fault torque ripples in method (III) have been nullified. One of the drawbacks of the torque ripple elimination is a reduction in the speed range as well as the base speed due to more harmonics injected into phase voltage references. Another application of ADALINEs is to separate harmonic components of measured phase currents in method (II) by using a real-time current learning. This harmonic separation provides time-constant feedback signals for current control in a new control scheme. Therefore, controllers such as PI of time-constant d-q currents can have good quality even at high speed. Variations of the rotating speed do not affect the control quality of the new control scheme. However, the new control scheme needs to deal with instability when current references vary. Therefore, the switch of the pre-fault and new control schemes has been made before any variations of current references.

Finally, this doctoral thesis contributes to enriching the control field of multiphase PMSM drives for automotive applications, especially in faulty conditions. A PMSM with multiharmonics in its back-EMFs can generate a smooth torque regardless of the operating modes without exceeding current and voltage limits, avoiding the oversizing of the drive.

Publications associated with this doctoral thesis are listed as follows:

1) Journal papers:

Perspectives

Among the presented studies in the three chapters of this doctoral thesis, there are several issues that have not been solved yet. These issues could be studied in the future as follows:

1) It would be interesting to research on a general approach to find reduced-order matrices, as in method (II) in Chapter 2, corresponding to each opened phase. 2) An automatic solution to replace the manual switch between the new and old control schemes in method (II), as discussed in section 3.3.2.B of Chapter 3, in response to the variation of current references should be found.

3) The proposed control strategies could be more interesting when they are applied to multiphase machines with a higher maximum speed. The high-speed machines are suitable for automotive applications instead of a low-speed machine for wind turbines that has been considered in this doctoral thesis. 4) Fault-tolerant control with SC faults in transistors or machine windings (turns) could be interesting to study.

Appendices Appendix A. Energetic Macroscopic Representation

Energetic Macroscopic Representation (EMR) is a graphical formalism for modeling and control of energetic systems [START_REF] Bouscayrol | Graphic formalisms for the control of multi-physical energetic systems[END_REF]. Specifically, EMR is a tool to organize the model of an energetic system, then the control scheme for this system is easily developed.

A.1. Representation of the system model

The system model is represented in EMR according to the two principles:

1) Principle of causality: inputs and outputs of an energetic system must follow the integral causality in which the outputs are functional integrals of the inputs. The outputs are delayed with respect to the inputs. For example, if a system contains a capacitor, the capacitor current must be the input while the capacitor voltage must be the output to respect the principle of causality. The reason is that the capacitor voltage is an integral of the capacitor current. A model which imposes a voltage on the capacitor violates the principle of causality. 2) Principle of interaction: subsystems of an energetic system interact with each other via pairs of action and reaction variables in which the product of these variables is the instantaneous power exchanged between these subsystems. For example, when a battery is connected to a resistor, they interact with each other by a pair of variables current and voltage. The battery imposes a voltage on the resistor as an action while the resistor enforces a current to the battery as a reaction.

Several EMR elements of the system model are described in

Element Pictogram Description

Source

Terminal of the system which supplies or dissipates energy. Name of the source can be written in the label such as energy source (ES) and mechanical source (MS).

Accumulation

Accumulation of energy which introduces delay to the system and represents dynamics of the system. For example, it can be an inductor which cause current dynamic delay, or a shaft that connects the rotor of a motor to a mechanical load.

Monophysical conversion

Mono-domain conversion of energy which can be with or without tuning input (red line). For example, an inverter with modulation functions converts electrical DC energy to electrical AC energy.

Label

Appendix B. Two-phase OC faults

When two phases of a seven-phase machine are open-circuited and there are no constraints on the winding configuration, the number of DoF for control reduces from 7 in heathy mode to 5 under these faulty conditions. If the stator windings are wye-connected, the number of DoF is equal to 4. In other words, only 4 currents can be independently imposed on the machine. Like method (I) in Chapter 2 for a single-phase OC fault, currents (id1, iq1) and (id3, iq3) are imposed as constant values to facilitate conventional PI controllers. Therefore, in case of two-phase OC faults, the 4 imposed currents are determined from the optimal strategy considering constraints on RMS current and peak voltage as in Fig. 2.21 of Chapter 2. Two open-circuited phases can be adjacent (phases (A-B), for example) and non-adjacent (phases (A-C) and (A-D), for example). Current references for two-phase OC faults in phases (A-B) can be determined as follows:

Appendix C. Résumé étendu en Français

C.1. Exigences dans les applications automobiles

Dans les années 1830, l'idée d'utiliser des machines électriques pour la traction des véhicules automobiles, également appelées véhicules électrifiés (EV), a été introduite. Cependant, des contraintes sur le coût, le volume et la masse des batteries ont entravé leur processus de commercialisation. Au cours des dernières décennies, les sources d'énergie fossile ont fortement diminué et la pollution de l'environnement est devenue un problème mondial très sérieux. Pendant ce temps, des améliorations significatives sur les batteries ont été réalisées. Par conséquent, depuis les années 1990, les véhicules électriques émergent lentement et deviennent une solution commerciale efficace pour faire face aux crises énergétiques et environnementales. Un EV peut être soit une combinaison d'une propulsion électrique et d'un moteur thermique dans un véhicule électrique hybride (HEV), soit un véhicule purement électrique. Le véhicule électrique pur est un véhicule à pile à combustible (FCV en anglais) ou un véhicule électrique à batterie (BEV). Alors que les BEV et les FCV à émission nulle ont des contraintes de coût et de technologie, les HEV à faibles émissions deviennent plus intéressants dans l'industrie automobile en raison du taux d'hybridation qu'il est possible d'ajuster [START_REF] Chan | The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles[END_REF].

En général, un véhicule électrique est une combinaison sophistiquée d'ingénierie électrique et d'ingénierie automobile. Les entraînements électriques jouent un rôle important dans les performances des EV. Ils se composent de machines électriques, de convertisseurs d'électronique de puissance, de systèmes d'alimentation en énergie, de contrôle et de gestion de l'énergie. Les technologies des entraînements électriques appliquées dans les véhicules électriques dépendent principalement des technologies utilisées dans les machines électriques : DC, AC induction (IM) et AC synchrone. On constate que la technologie DC n'est plus attractive en raison de ses contraintes de maintenance et de ses limitations mécaniques. Actuellement, les machines synchrones à aimants permanents (PMSM en anglais) sont plus présentes dans les HEV où les contraintes spatiales sont importantes. Pendant ce temps, des IM et des machines synchrones à excitation séparée à rotor bobiné, sans aimants permanents coûteux, sont observés dans les BEV [2].

Plusieurs exigences principales des entraînements électriques dans les véhicules électriques sont indiquées comme suit : 1) Le rendement élevé pour une utilisation optimale de l'alimentation en énergie intégrée.

Les entraînements électriques à haut rendement peuvent augmenter l'autonomie ou réduire le besoin en capacité de batterie dans les EV. 2) Les densités volumiques élevées de puissance et de couple en raison des espaces limités dans les EV, en particulier les HEV. 3) Le faible coût pour obtenir des véhicules électriques abordables sur le marché de masse et sans danger pour les humains (l'alimentation par une tension faible de 48 V, par exemple, requiert moins de dispositifs de protection qu'une tension de bus de 500 V). 4) La fiabilité fonctionnelle élevée avec la capacité de tolérance aux défauts pour le transport (des défauts dans les convertisseurs de puissance ou les enroulements de