Renée El

Melhem Bât

Pascal Blaise

M Stéphane

ÉLECTROTECHNIQUE E E A Électronique

M Philippe Delachartre
email: philippe.delachartre@insa-lyon.fr

M Philippe
email: philippe.normand@univ-lyon1.fr

Mme Sylvie

M Hamamache Kheddouci

M Stéphane Benayoun
email: stephane.benayoun@ec-lyon.fr

Stéphanie Cauvin

M Jocelyn Bonjour

M Christian Montes

Mr Adrien Arnaud Prevost

Mr Laurent Duvernois

Mr Tangue Schuller

Mr Simon

Mohammad Arsalan Mian

Mr Gabriel Asif

Ferreira De Carvalho

Dr Cedric Escudero Silva

Dr Xiaoyang Yue

Dr Chenchen Chu

Dr Ting Shi

Zhang Xiaokang

Lyon

Kcl Kirchhoff'

Keywords: control, FV Fundamental voltage, GaN Gallium nitride, HEV Hybrid electric vehicle, IGBT Insulated gate bipolar transistor

AC

Alternating current.

Back-EMF Back electromotive force. CFI Capacitor-based four-leg inverter. CFNI Capacitor-based four-leg neutral-point-connected inverter. DC Direct current. DCSUR DC-source voltage utilization ratio. DOF Degree of freedom. ESS Energy storage system. EV Electric vehicle. FFT Fast fourier transform. FI Fundamental current. FOC Field-oriented

Acknowledgement

Up to now, I have been spending unforgettable three years at Lyon, in France. It is undoubtedly a pretty precious experience in my life. I feel really grateful and fortunate that I could have the opportunity to carry out a scientific research at the Laboratory of AMPERE with my supervisors, Prof. Xuefang Lin-Shi and Dr. Jean-Yves Gauthier, where I began to touch the door of scientific research, learned to built up my confidence and gained comprehensive research training.

First and foremost, I would like to express the best sincere gratitude to my supervisor, Prof. Xuefang Lin-Shi, who has inspired me a lot in the past three years. She is not only a superexcellent supervisor in my research but also an amiable friend in my daily life. I deeply appreciate her patient guidance and constructive advice in numerous difficulties which I have ever encountered. In addition to the academic guidance, she always encourages me to participate in international conferences so that I can keep tracking the research trends and discuss with many excellent researchers, which is indeed significant to my research insight.

My deepest appreciation also goes to my co-supervisor, Dr. Jean-Yves Gauthier, for his self-giving guidance and help in my research. He is an enthusiastic and passionate scholar, who can always answer my questions and his door is invariably open for me. Every time talk with him, I can always obtain some new ideas and thoughts to improve my research. In addition to the academic guidance, Dr. Jean-Yves Gauthier always encourages me to speak French. After the first year, he tried to talk to me in French patiently. Day by day, my french is improved gradually. As a result, I can give a Travaux Pratique (TP) course (80 h)

Notations

x Denote the predicted value of a variable (x).

x Denote the mean of a variable (x) in one electrical period.

Ef f 1 = AC F U N DAM EN T /DC RM S Normalization of the fundamental AC current (related to the fundamental torque) to the input current.

xviii Ef f 2 = DC RM S/AC F U N DAM EN T AL Normalization of the input current to the fundamental AC current (related to the fundamental torque).

x(k) Denote a variable (x) in the (k) th sampling interval.

x(k + 1) Denote a variable (x) in the (k + 1) th sampling interval.

x_c Denote a variable (x) in the conventional drive.

x_f Denote a variable (x) of the MFCS-I in post-fault conditions.

x_m Denote a variable (x) in the basic MFCS.

x_mI Denote a variable (x) in the MFCS-I.

x_n Denote a variable (x) of the MFCS-I in normal conditions.

x * Denote the reference of a variable (x).

Variables physiques

α F Duty-cycle of the fourth leg in the MFCS-II. e dq0 := [e d e q e 0 ] T d-axis, q-axis and 0-axis back-EMFs.

i s_npj and i s_f pj Projections of the current vector in healthy conditions and in post-fault conditions, respectively. i s Current vector.

i ABC := [i A i B i C ] T Phase-currents of a PMSM in A-B-C coordinate.
L ABC Inductance matrix of a PMSM.

L dq0 := diag(L d L q L 0 ) Inductance matrix in d-q-0 coordinate.

u ABC := [u AN u BN u CN ] T Phase-voltages of a PMSM in A-B-C coordinate.
u dq0 := [u d u q u 0 ] T d-axis, q-axis and 0-axis voltages of a PMSM. ε Error index between the predicted and measured currents.

ε th A threshold for the OPF detection.

ζ Etraj Desired damping ratio of the second-order filter.

a L Multiple of L aux to L 0 .

B m

Coefficient of viscous friction.

C bus DC-bus capacitance.

E

Energy stored in the inductor and capacitor in the equivalent boost.

E * Energy reference of E.

E traj Desired energy trajectory in the equivalent boost.

f ele Electrical frequency of the motor.

i bus DC-bus current.

i lo Equivalent load current in the equivalent boost function.

i N Neutral current.

J m Rotational inertia of the motor.

K ω , K ω_I Two feedback gains of the state feedback controller in the speed-loop.

xx

K Φ Torque coefficient.
k pk Proportional gain of a PI controller.

L σ , L Σ and L ∆ Leakage inductance, average inductance and inductance fluctuation.

L AA , L BB and L CC Self-inductance.

L aux

The series inductor of the MFCS-I.

L E

The equivalent inductance in the MFCS-I and MFCS-II.

m Modulation ratio m 0 = i 0_n /i q_n Proportion between i 0_n and i q_n .

M AB = M BA , M AC = M CA and M BC = M CB Mutual inductance.

m i Current overrating ratio.

n Motor mechanical speed (rpm).

P in

Input power of the MFCS-I.

P mec

Mechanical power of a motor.

R

Stator resistance of a PMSM.

r 0
Step-up ratio of the equivalent boost.

R eq

The equivalent load of the equivalent function in the MFCS-I and MFCS-II.

R f lo

Assessment index of a similar NP in the MFCS-I.

T e Electromagnetic torque.

T f Fundamental period.

T ik

Integral time coefficient of a PI controller.

T L Load torque.

T set OPF detection time.

T s Switching period.

u

Abstract

This thesis focuses on a two-level voltage-source-inverter fed three-phase permanent magnet synchronous machine drive which integrates an equivalent direct current (DC)/DC boost function by utilizing the machine's neutral point (NP). Such a drive system is referred to as multi-functional converter system (MFCS), and can be a promising alternative to the two-stage converter-based electric traction system in electric vehicles. En conséquence, on peut conclure qu'il existe deux opportunités créées par l'utilisation du NP:

• Opportunité 1: Utilisation de l'homopolaire pour obtenir une conversion DC/DC à bas prix.

• Opportunité 2: Possibilité de tolérer un défaut en OPF en créant champ tournant.

Chapitre 2: Analyse d'un MFCS de base avec PMSM

Dans ce chapitre, nous étudions de manière exhaustive le MFCS basé sur une machine synchrone à aimants permanents (PMSM). Tout d'abord, le modèle moyen de ce système peut être construit en se basant sur les lois de Kirchhoff et sur l'approche classique de définition des variables correspondant à une moyenne temporelle sur une période de découpage. En 6/178 coordonnées d-q-0, le modèle est : Le modèle moyen du MFCS-I peut s'écrire de la manière suivante :

                                 α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) α h u bus = - R 3 i N - L 0 3 di N dt + u in i bus = C bus du bus dt = α h i N -(α d i d + α q i q ) i lo (R.
α X = u XN * + u in u bus = α h + u XN 1 * u bus (R.
                                       α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) α h u bus = - R 3 i N -( L 0 3 + L aux ) L E di N dt + u in i bus = C bus du bus dt = α h i N -(α d i d + α q i q ) i lo (R.3)
où L E est une inductance équivalente.

Évidemment, on peut conclure que l'ajout de l'inductance en série augmente de fait l'inductance équivalente du boost équivalent. Par conséquent, l'ondulation des courants homopolaires (ou du courant de neutre) peut être réduite, limitant ainsi l'ondulation totale des courants et les pertes fer. Théoriquement, une inductance série d'une valeur de cinq fois l'inductance homopolaire du moteur peut conduire à un NP flottant similaire à celui obtenu avec un variateur conventionnel.

Cependant, dans cette architecture, le rapport de surélévation effectif du MFCS-I reste limité à 2. Afin de l'augmenter, une nouvelle architecture, nommée MFCS-II, est proposée en utilisant un bras de pont auxiliaire, comme le montre la figure R.8.

10/178

+ - Le modèle moyen du MFCS-II peut également être construit : On en déduit une troisième condition de fonctionnement :

u bus + -u in i N i bus o S A S B S C S' A S' B S' C A B C i A i B i C N motor C bus L aux S F S' F (α F ) (α A ) (α B ) (α C )
                                       α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) (α F -α h )u bus = - R 3 i N -( L 0 3 + L aux ) L E di N dt + u in i bus = C bus du bus dt = (α F -α h )i N -(α d i d + α q i q ) i lo (R.

+ -

u bus + - u in i N i bus o S A S B S C S' A S' B S' C A B C i A i B i C N motor C bus L aux OPF Figure R
P in_f = u in i N _f = -3u in i 0_f = P mec_n η = P in_n = -3u in i 0_n ⇓ i 0_f = i 0_n (R.7)
Trajectoires des courants dans le repère d-q-0 en cas de défaut Environmental concerns and their stringent requirements are motivating the modern industry to reduce carbon dioxide emissions [START_REF] Aghabali | 800-V electric vehicle powertrains: Review and analysis of benefits, challenges, and future trends[END_REF]. In the field of automotive industry, vehicle electrification has been emerged as the best solution for green transportation thus is expected to be in the phase of vigorous growth over the next decades [2][3]. At present, the electrification levels of vehicles can be divided into three categories: hybrid electric vehicle (HEV), plug-in hybrid electric vehicle and pure electric vehicle (EV) [4].

                   i d_f = i d_n -2i 0_n cosθ e i q_f = i q_n i 0_f = i
0 -1 1 2 3 -2 -3 0 ¡ 2¡ 3¡ 4¡ (c) θ e i A_ f i B_ f i C_ f 0 0.5 1 1.5 -0.5 -1 -1.5 0 0.5 -0.5 1 1.5 -1 -1.5 (a) (b) i ¢ i β i s_ fpj (p.u.) m 0 =-0.1 m 0 =-0.2 m 0 =-0.3 m 0 =-0.4 m 0 =-0.5 0 -1 1 2 3 -2 -3 i q_ f i d_ f i 0_ f 0 -1 1 -2 2 (p.u.) m 0 =-0.25 a b c d e f i 0_n I II i A_ f i B_ f i C_ f i q_ f i d_ f i 0_ f 0 -1 1 - 2
In terms of electric traction systems, there are two well-known configurations which are adopted to connect the energy storage system (ESS) with the traction drive [5] interleaved DC/DC converters [START_REF] Lin | An electric-vehicle ipmsm drive with interleaved front-end DC/dc converter[END_REF] and dual-active bridge DC/DC converters [START_REF] Tan | Design and performance of a bidirectional isolated DC-dc converter for a battery energy storage system[END_REF]. With the assistance of the DC/DC converter, the electric traction system is able to obtain a sufficient 22/178 DC-bus voltage in the absence of a high-voltage battery bank, and to flexibly manage the power flow between the traction motor and the ESS [START_REF] Schoenen | Advantages of a variable DC-link voltage by using a DC-dc converter in hybrid-electric vehicles[END_REF]. However, the use of the additional DC/DC converter brings some compromises in turn. On the one hand, it increases the cost and volume of the whole powertrain system. On the other hand, the system reliability may be degraded since more power electronic devices are used.

With the use of more power electronic devices in electric traction drives, the fault-tolerance towards electrical failures (open-circuit or closed-circuit faults) is gaining more and more attentions since the security of passengers is quite crucial. In the literature, the existing faulttolerant solutions for three-phase two-level voltage-source inverter (VSI)-fed motor drives always require auxiliary components (a redundant bridge arm or a pair of split DC-bus capacitors) to achieve the fault-tolerant objective. Obviously, these auxiliary components will further increase the cost, volume and weight of electric traction systems.

In the field of alternating current (AC) motor drive system, the MFCS is a special category, which integrates the functionalities of a DC/DC boost converter and a DC/AC inverter upon only one three-phase two-level VSI. Different from the unconnected neutral point (NP) of a conventional motor drive, the NP of MFCSs is connected to a DC power-source. Roughly speaking, the MFCS is a promising alternative to replace the two-stage electric traction system in EVs and HEVs. However, in the literature, it does not get a lot of attentions. Up to now, the mechanism and operating principle of MFCSs are not clearly revealed and explained. There is also no a comprehensive study that presents the modeling, modulation method, control strategy and experimental evaluations for MFCSs. Besides, the possibility of fault-tolerant operations of MFCSs has not been explored and studied.

Objectives

Carrying a strong interest toward the concept of MFCSs, this thesis is going to investigate MFCSs comprehensively and deeply with the following objectives:

• The first objective of this thesis is to deeply study the basic MFCS. It intends to explicitly reveal the mechanism and evaluate the characteristics of the basic MFCS. In other words, we are going to explain the reason why there is an equivalent DC/DC boost converter function existing in a three-phase AC motor drive by merely connecting its NP. Since the NP is connected, the zero-sequence path of the basic MFCS is circulated. Thus, the common modulation methods that do not consider the zero-sequence components are not applicable in the basic MFCS. To address this problem, a specialized modulation method 23/178 is required.

• Although the circulated zero-sequence circuit of the basic MFCS contributes to an equivalent DC/DC boost function, the clamped NP results in bigger current ripples reversely. In addition, the effective step-up ratio of the equivalent DC/DC boost function in the basic MFCS is limited. Therefore, the second objective is to improve the basic MFCS. Moreover, a comprehensive comparison between an improved MFCS and a conventional two-stage drive is necessary to clarify the advantages and disadvantages of the improved MFCS.

• Due to the innately connected NP in MFCSs, the circulated zero-sequence circuit can naturally provides a path for zero-sequence currents, which is necessary to generate a stable rotating magnetic field when the motor encounters an open-phase fault (OPF).

Thereby, it exists a possibility to realize fault-tolerant operations without using auxiliary components when an OPF occurs in MFCSs. This has never been studied in the literature.

Deservedly, the final objective is to explore and achieve the fault-tolerant possibility.

Organization of the thesis

On the purpose of achieving above objectives, this thesis is organized with four chapters, as shown below:

• Chapter 1 reviews the opportunities of the NP of a three-phase star-connected AC motor in the literature. The NP can be connected with a DC power-source to gain an equivalent DC/DC boost function without extra costs. This is the origin of the basic MFCS. Some variants presented in the literature are also introduced. Besides, the NP can also be used to fault-tolerant applications. According to the state of the art of present three-phase fault-tolerant AC drives, it can be connected with a redundant bridge arm or a pair of split DC-bus capacitors to achieve fault-tolerant operations in case of an OPF. Objective speaking, these fault-tolerant solutions always need auxiliary hardwares.

•

27/178

Introduction

A three-phase AC machine has three branches, which can be connected in two ways -Star (Y) or Delta (∆). For the Y-connected one, it has an extra neutral point (NP). In most cases, the NP is not used and hidden inside the motor shell. However, in some special applications, this NP can be picked out and utilized to obtain additional benefits. For example, multi-functional converter system (MFCS) is a typical example that utilizes the NP to gain an equivalent DC/DC boost converter without using additional components in addition to a motor and a VSI. In the field of on-board chargers, the traction system can be reconfigured as a boost converter with power factor correction (PFC) or as a buck converter by connecting the NP of traction motors.

In addition, in the domain of three-phase fault-tolerant AC drives, the NP can also be utilized to provide a path for zero-sequence currents so that a constant torque can be maintained when an open-phase fault (OPF) occurs. This chapter reviews these special applications, which take advantage of three-phase AC motor's NPs.

This chapter is organized as follows:

• In section 1, the concept of the basic MFCS and its variants are presented by reviewing the limited literature. On-board chargers using the NP of traction motors are also illustrated in this section since its fundamental is similar to MFCSs.

• In section 2, the state of the art of three-phase fault-tolerant AC drives is summarized.

Four typical fault-tolerant topologies are illustrated and compared in detail with respect to the cost and post-fault output capability.

Concept of multi-functional converter system (MFCS) 1.Basic MFCS

The idea of the basic MFCS dates back from about twenty years ago [16][17]. It was firstly proposed for electric powertrain systems in HEVs by Toyota and Fuji Electric Co., Ltd., in Japan. In order to reduce the cost and volume of the two-stage configured powertrain system, engineers and researchers from Toyota and Fuji Electric attempted to remove the frond-end DC/DC stage. To retain the same functionalities provided by the DC/DC stage, the NP of traction motors is deliberately connected so that the traction drive can be an intermediary of energy exchange. The basic MFCS consists of an AC motor, a three-phase two-level VSI, two DC-sources and a neutral wiring. But there is no real DC/DC converter. The main features of the basic MFCS can be summarized as follows [START_REF] Nakai | Overview of multi-functional converter systems[END_REF][19]:

28/178

• Endowing a three-phase AC motor drive with the functionality of a DC/DC converter by exclusively connecting the NP of the motor, which is rarely used in a conventional drive.

• Utilizing the zero-sequence inductance of the motor to function as the filtering inductor and reusing the traction inverter as the equivalent chopper.

The schematic of the basic MFCS is shown in figure 1 it is easy to understand that it can be seen as a three-phase DC/DC converter by adjusting the duty cycle between the two zero switching-states. However, it needs to emphasize that the basic MFCS is different from a conventional three-phase DC/DC converter because the three-phase windings of the motor are coupled. Moreover, the back electromotive force (back-EMF) existing in each phase should be considered when the motor speed is non-zero.

However, there is no electromagnetic torque generated if the basic MFCS is only operated with the two zero switching-states. In order to rotate the motor, the six non-zero switching-states should be utilized. Since the non-zero switching-states are involved, the basic MFCS becomes a hybrid system which deals with not only DC/DC conversion but also DC/AC conversion via only one VSI. As we all know, for a conventional three-phase motor drive whose NP is not used, the inverter's switching-states are determined by the targeted voltage-vector and the adopted modulation method under the framework of field-oriented control (FOC). If the conventional control strategy and modulation method are adopted in the basic MFCS without 29/178 modifications, the control behavior of FOC will affect the power flow between the two power sources. Therefore, the key point is to find the independent control degree of freedom (DOFs)

(a) (b) (c) (d) (e) (f) (g) (h)
for the motor function and the equivalent DC/DC converter function.

In [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF], the characteristic equations of the inverter in the basic MFCS are firstly derived.

By considering the 0-axis elements, the available space voltage vectors of the inverter were constructed in a three-dimension (3-D) coordinate instead of the common α-β plane. In this case, the available area of inverter's output voltages is a cube in the 3-D coordinate rather than a hexagon. Then, the voltage applied to the motor under different voltage vectors was further analyzed. The authors indicated that the d-q equations of the basic MFCS are consistent with 30/178

1.1. Concept of multi-functional converter system (MFCS)
those of a conventional drive. However, the connected NP of the basic MFCS leads to a unique 0-axis equation, as

v 0inv - √ 3v l = Ri 0 + L 0 di 0 dt (1.1)
where v 0inv represents the 0-axis potential of the inverter; R is the resistance of the motor; i 0 is the 0-axis current; L 0 is the 0-axis inductance of the motor.

The available motor control area of the basic MFCS was also analyzed in [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF] by comparing the surface of u 0inv = √ 3v l with the voltage cube in the 3-D coordinate. According to geometrical analysis, the authors indicated that the available motor control area of the basic MFCS changes with the voltage ratio r = v h /v l . The effect of the step-up v h is recognized when r < 3. It is not beneficial any more when r > 3. However, in [START_REF] Nakai | Overview of multi-functional converter systems[END_REF], the authors thought that v h = 2v l is the best condition for driving the motor. Both [START_REF] Nakai | Overview of multi-functional converter systems[END_REF] and [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF] did not present a comprehensive analysis and explanation for the effective step-up ratio of the MFCS. Moreover, there were no adequately experimental results presented to verify the effectiveness of their analysis.

In [START_REF] Shetty | Three phase PWM rectifier with integrated battery for automotive applications[END_REF], the authors connected the battery to the NP of a generator powered by an internal combustion engine in HEV applications. The proposed topology uses a single-stage PWM rectifier to process the power from two different sources. The PWM rectifier is used to convert the AC voltage produced by the generator and to function as a DC/DC boost converter for the battery. Essentially, this application is equivalent to the basic MFCS when the traction motor is operated under regenerative braking modes.

The interest of using motor's NPs is not limited to vehicle applications. The authors in [START_REF] Tschanz | A multi-functional converter for a reduced cost, solar powered, water pump[END_REF] applied the similar idea in a solar powered water pump to reduce the cost and complexity of a classic system where a DC/DC boost converter is typically used to step up the solar panel voltage and achieve maximum power point tracking control, and then to supply a brushless-DC motor via a three-phase two-level VSI for driving a water pump. By connecting the solar panel to the NP of the brushless-DC motor, the inverter's bridges act as not only the inverter function but also the chopper of a DC/DC boost converter simultaneously.

Although the connected NP contributes to a free DC/DC boost function in the basic MFCS, it brings some adverse impacts in turn, e.g. the issue of bigger current ripples. [START_REF] Hackner | Optimization of the winding arrangement to increase the leakage inductance of a synchronous machine with multi-functional converter drive[END_REF] and [START_REF] Hackner | Optimization of the winding arrangement to increase the zero-sequence inductance of a synchronous machine with 168/178[END_REF] applied the basic MFCS on an automotive electric-power-steering system on the purpose of cost reductions. The issue of bigger current ripples was investigated in detail by building a high-frequency model. It indicates that the phase-current ripples in the basic MFCS can be divided into two parts, which are related to the mutual inductance and the zero-sequence inductance of the motor. For the first part, it can be reduced significantly with interleaved PWM technique. However, the interleaved PWM may deteriorate the torque performance 31/178 of the motor, which is not discussed by the authors. In terms of the second part, it can be reduced by increasing the zero-sequence inductance with special winding arrangements. Three different winding arrangements were analyzed to increase the zero-sequence inductance of a synchronous machine with a double-layer tooth-coil winding. Experimental results indicate that the phase-current ripple and neutral current ripple can be reduced by half with the proposed winding arrangement. However, the requirement of special winding arrangements can not be fulfilled in a common motor. In this thesis, we will introduce a method to reduce the current ripple by using a series inductor. The detailed sizing method for the series inductor will be given. In addition to the issue of bigger current ripples, another limitation of the basic MFCS is the limited step-up ratio of the equivalent boost function.

Other variants 1.1.2.1 Variant 1

In a conventional two-stage motor drive system, the DC/DC boost converter requires a large boost-up reactor, which is costly and bulky. Although the basic MFCS can realize a reactor free boost converter, its effective step-up ratio is limited. In [START_REF] Itoh | Investigation of a two-stage boost converter using the neutral point of a motor[END_REF] and [START_REF]Investigation of a two-stage boost converter using the neutral point of a motor[END_REF], the authors followed the basic MFCS and proposed a variant. They investigated a two-stage boost converter by using the NP of a motor, as shown in figure 1.3. Different from the basic MFCS, the proposed circuit connects the NP of the motor to a first-stage boost converter. Thanks to the utilization of the motor's leakage inductance, the volume of the boost-up reactor in the first-stage can be reduced compared to a conventional two-stage drive system. Besides, thanks to the first-stage boost, the step-up ratio of the whole system can be higher than the basic MFCS. Regarding the control scheme, the V/F control collaborated with the six-step operation is used to control the motor.

The cascaded PI is used to regulate the DC-bus voltage. In order to compensate the voltage fluctuation caused by the floating NP, a feed-forward control is applied to the current regulator in the first-stage boost. In addition, the loss evaluation was also implemented to investigate the influence of the imposed DC current in the motor. According to the finite element simulation implemented on a 1.5 kW interior permanent magnet motor, the authors indicated that the torque ripple of the motor was increased by 10% and the copper loss was increased by 25%. A trade-off relationship between the converter and motor losses has been demonstrated.

Variant 2

In HEVs, the energy distribution system occupies a big part of volumes and costs. One part of this system is the DC/DC converter that transfers power between the low-and high-voltage buses, as shown in figure 1.4(a). In order to reduce the cost and volume of the DC/DC converter, [START_REF] Plesko | Novel concepts for integrating the electric drive and auxiliary dc-dc converter for hybrid vehicles[END_REF] and [START_REF] Plesko | Novel concepts for integrating the electric drive and auxiliary DC-dc converter for hybrid vehicles[END_REF] proposed a novel concept for integrating the DC/DC converter functionality into the traction drive system, as shown in figure 1.4(b). By using the traction inverter and the machine to function as a primary bridge leg of an isolated full-bridge DC/DC converter, the total costs and weight of the whole system can be reduced. Furthermore, three different switching strategies were proposed in [START_REF] Plesko | Novel modulation concepts for a drive-integrated auxiliary dc-dc converter for hybrid vehicles[END_REF] to improve the efficiency of the total system. Starting from the phase-shift modulation known from the conventional dual active bridge, the switching instants are optimized regarding efficiency to obtain the optimal modulation. The proposed concept was validated on a scaled prototype with a 3 kW machine. The resulting measurements agree well with simulations and the analytical model. 33/178

Variant 3

For a drive system with two motors, the two NPs can be completely used. Therefore, a two-motor type MFCS was also introduced in [START_REF] Nakai | Overview of multi-functional converter systems[END_REF] and [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF]. The circuit configuration of the two-motor type MFCS is shown in figure 1.5, which consists of two traction inverters, two three-phase AC motors, a high-voltage DC-source connected with the bus line and a low-voltage DC-source interfaced with the two NPs. Compared to the basic MFCS, the two-motor type MFCS can utilize the zero-sequence inductance of two motors and the DOFs of two traction inverters. Thus, the neutral current ripple in the two-motor type MFCS can be reduced into 1/4.5 of that in the basic MFCS, and the available motor control area is wider. A control strategy for the two-motor type MFCS with zero-voltage compensations was introduced in [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF].

However, the authors did not present experimental results to verify the effectiveness of the proposed control strategy when the two motors are operated under dynamic conditions.

Integrated on-board charger

The concept of utilizing motor's NPs can also be found in integrated on-board charger applications [START_REF] Sul | An integral battery charger for four-wheel drive electric vehicle[END_REF]- [START_REF] Xiao | An integrated on-board EV charger with safe charging operation for three-phase ipm motor[END_REF]. A non-isolated single-phase (220 V and 50 Hz AC) integrated battery charger was proposed in [START_REF] Pellegrino | An integral battery charger with power factor correction for electric scooter[END_REF] for an electrical scooter. The battery charger is derived from the powertrain system of the scooter, whose circuit configuration is shown in figure 1.6. In the motoring mode, the switch K1 is open. Therefore, the battery is connected with the DC-bus 34/178 to power the motor. In the charging mode, the switch K1 is closed. Then, a single-phase AC source is connected with the NP of the motor via a single-phase rectifier and a LC filter.

The neutral current can be regulated by means of inverter's common-mode voltage while the differential-mode current components are regulated to zero for current balancing. Thus, the AC drive system actually becomes a three-phase DC/DC boost with the PFC capability. To minimize the current ripple, the phase interleaved technique can be utilized. In [START_REF] Xiao | An integrated on-board EV charger with safe charging operation for three-phase ipm motor[END_REF], the authors introduced an integrated propulsion/charging system for EVs, as shown in figure 1.7. In the propulsion mode, only switch S3 is closed. In this case, the battery is connected with the DC-bus of the traction inverter for propulsion requirements. An integrated battery charger has been reported for a four-wheel EV by Sul and Lee [START_REF] Sul | An integral battery charger for four-wheel drive electric vehicle[END_REF] in 1995. The propulsion system consists of four induction motors and four three-phase VSIs 35/178 with a battery connected with the DC-bus, as shown in figure 1.8. In the motoring pattern, the switch K2 is closed on top. So, the battery is connected to the DC-bus capacitor. The four inverters are connected with the DC-bus together. Each NP of the four motors is floating in the motoring pattern. In the charging mode, the transfer switch K2 changes the connection of the battery. The drive system can be reconfigured as a single-phase battery charger. As shown in figure 1.8, the single-phase AC source is connected between two NPs of motors 1 and 2. Meanwhile, the battery is connected between two NPs of motor 3 and 4. From functionality point of view, motors 1, 2 and inverters 1, 2 function as a single-phase boost converter with the PFC capability. Motors 3, 4 and inverters 3, 4 become two buck converters [START_REF] Haghbin | Grid-connected integrated battery chargers in vehicle applications: Review and new solution[END_REF] to charge the battery. Essentially, the fundamental of above integrated on-board chargers is similar to the concept of MFCSs where the drive system is reconfigured as boost or buck converters according to different requirements by utilizing the NP of traction motors. The difference between these on-board chargers and MFCSs is that the traction motors do not need to produce torque in charging patterns. Thus, the control strategy for integrated on-board chargers is simpler.

State of the art of three-phase fault-tolerant motor drives

In addition to the applications in MFCSs and integrated on-board chargers, three-phase AC motor's NPs can also be utilized to tolerate an OPF. In the field of EVs, system reliability and fault tolerance are very critical [START_REF] Forstner | Fault-tolerant torque control of a threephase permanent magnet synchronous motor with inter-turn winding short circuit[END_REF]. As the dominant power take-off, the electrical drive is desired to be equipped with the ability of continuous operations in case of the occurrence of failures. Many different failures can afflict electrical drives, such as machine faults [START_REF] Cheng | Using only the DC current information to detect stator turn faults in automotive claw-pole generators[END_REF][36],

sensor faults [START_REF] Beng | Sensor fault-resilient control of interior permanent-magnet synchronous motor drives[END_REF], and power converter faults [START_REF] Lu | A literature review of IGBT fault diagnostic and protection methods for power inverters[END_REF]. Among the machine and power converter faults, open-circuit and short-circuit failures are the most common failures in motor drives [39][40].

Considering that the phase winding can always be cut off with the power supply by circuit 36/178 breakers or fuses during short-circuit faults, the short-circuit faults can be eventually converted to open-circuit faults. Therefore, the OPF has received more attention in recent years [START_REF] Zhou | High performance three-phase pmsm open-phase fault-tolerant method based on reference frame transformation[END_REF].

Different from multi-phase drives [START_REF] Sui | Open-circuit fault-tolerant control of five-phase pm machine based on reconfiguring maximum round magnetomotive force[END_REF]- [START_REF] Abdel-Khalik | Postfault operation of a nine-phase six-terminal induction machine under single open-line fault[END_REF] with the inherent fault-tolerance, a standard VSI fed three-phase AC motor drive with Y-connection is unable to maintain a stable torque when an OPF occurs. After the appearance of an OPF, one of the motor phases will be disconnected from the power supply. In this case, the currents in the remaining phases have an equal amplitude but opposite sign. Thus, if there is no fault-tolerant solution adopted, the currents in remaining phases can not generate a circular rotating electromagnetic field. In order to improve the reliability and security of three-phase AC drives, numerous fault-tolerant topologies have been proposed over the past decades [START_REF] Campos-Delgado | Fault-tolerant control in variable speed drives: A survey[END_REF]. In general, these topologies always require assistant devices to achieve fault-tolerant operations. [START_REF] Zhou | High performance three-phase pmsm open-phase fault-tolerant method based on reference frame transformation[END_REF], [START_REF] Scarcella | Fault-tolerant capability of deadbeat-direct torque and flux control for three-phase pmsm drives[END_REF]- [START_REF] Hu | Ensuring a reliable operation of two-level IGBT-based power converters: A review of monitoring and fault-tolerant approaches[END_REF] At present, the well-known fault-tolerant topologies of three-phase AC motors can be divided into four categories [START_REF] Welchko | Fault tolerant three-phase AC motor drive topologies: A comparison of features, cost, and limitations[END_REF][66] [1] :

1. CFI: capacitor-based four-leg inverter.

2. SFI: switch-based four-leg inverter.

3. CFNI: capacitor-based four-leg neutral-point-connected inverter.

4. SFNI: switch-based four-leg neutral-point-connected inverter.

Fault-tolerant solutions without using NP

The CFI is an early attempt to endow a standard three-phase VSI fed AC motor drive with the fault-tolerant capability, as shown in figure 1.10. This topology incorporates three triode for alternating current (TRIACs) and two split DC-bus capacitors which form the fourth leg to tolerate an OPF occurring on any of inverter's legs. In healthy operations, all TRIACs are not triggered thus the condition of the motor is consistent with that of a standard VSI fed drive.

When an OPF occurs, the faulty inverter leg is isolated with the DC-bus and the corresponding TRIAC of the faulty phase is triggered. In this case, the terminal of the faulty phase of the motor is connected to the midpoint of the DC-bus. Essentially, the circuit configuration in post-fault conditions is similar to the three-phase four-switch AC drive where the motor is controlled through only two active inverter legs. Although one of motor phase terminals is clamped to the midpoint of the DC-bus, three-phase windings of the motor can still be used to produce the electromagnetic torque. Therefore, the current output capability in the post-fault case is [1] Open-end winding motor drives are not considered in this manuscript because they need two inverters 37/178 identical to that in the healthy case [START_REF] Zaky | A performance investigation of a four-switch threephase inverter-fed im drives at low speeds using fuzzy logic and pi controllers[END_REF]. Objectively speaking, the CFI is a cost-effective faulttolerant topology in which no additional insulated gate bipolar transistors (IGBT) is employed.

However, the voltage output capability has been decreased by a half due to the loss of voltage vectors [START_REF] Hu | A fault-tolerant induction generator system based on instantaneous torque control (itc)[END_REF]. It needs to note that the issue of DC-bus midpoint voltage fluctuations and the lack of zero vectors result in the increase of torque ripples and deteriorate the performance of the fault-tolerant control [START_REF] Zhu | Comprehensive analysis and reduction of torque ripples in three-phase four-switch inverter-fed pmsm drives using space vector pulse-width modulation[END_REF]. For the first defect, [START_REF] Zhu | Comprehensive analysis and reduction of torque ripples in three-phase four-switch inverter-fed pmsm drives using space vector pulse-width modulation[END_REF] proposed a p-z coordinate to compensate the voltage fluctuation. Essentially, it is equivalent to synthesize the voltage vector by considering the voltage fluctuation since the midpoint voltage is measurable through voltage sensors. The use of p-z coordinate can simplify the intermediate calculations, e.g., sector judging can be avoided, the computation burden can be reduced. To address the second problem, the zero vector can be synthesized by two inverse vectors. It is pointed out that using the nearest three primary vectors can minimize the torque ripple caused by modulations [START_REF] Hu | Fault diagnosis and tolerance with low torque ripple for open-switch fault of im drives[END_REF]. In the SFI, a redundant inverter leg constructed by two auxiliary IGBTs and three TRIACs are employed to recover the original circuit structure when an OPF occurs, as shown in figure 1.10.

A B + - u in o' + - u bus N C S A S B S C S' A S' B S' C Motor TRIACs split dc-bus
Essentially, the situation of the motor remains unchanged before and after the fault. Therefore, the rated output capability can be completely remained in the SFI. However, compared to the CFI, the SFI suffers from higher hardware costs. midpoint of the DC-bus [START_REF] Liu | A strategy for improving reliability of field-oriented controlled induction motor drives[END_REF] and an auxiliary inverter leg [START_REF] Bolognani | Experimental fault-tolerant control of a pmsm drive[END_REF] via a TRIAC. In order to maintain a constant flux trajectory and preserve the torque output, the phase-current in remaining phases needs to be increased in magnitude by a factor of √ 3 [START_REF] Kiselev | Finite-control-set MPC for open-phase fault-tolerant control of pm synchronous motor drives[END_REF]. Meanwhile, the phase angle should be shifted 30 • away from the faulted phase [START_REF] Guo | Adaptive torque ripple suppression methods of three-phase pmsm during single-phase open-circuit fault-tolerant operation[END_REF]. As a result, the torque output capability of the CFNI and SFNI is reduced by a factor of 1/ √ 3 due to the current limits. The increase in phase-currents is in the form of zero-sequence currents flowing through the NP, and is necessary to guarantee a circular flux trajectory [START_REF] Mendes | Fault-tolerant operating strategies applied to three-phase induction-motor drives[END_REF]. The amplitude of neutral currents in post-fault conditions is increased by a factor of √ 3 compared to that in remaining phases [START_REF] Meinguet | Control strategies and reconfiguration of four-leg inverter pmsm drives in case of single-phase open-circuit faults[END_REF]. The NP voltage of the motor in the CFNI is clamped to half of the DC-bus voltage while it can switch between 0 and the whole DC-bus voltage in the SFNI due to the auxiliary leg. This implies that the CFNI can only apply ±1/2 of the DC-bus voltage across each active phase while the SFNI is able to utilize the whole DC-bus voltage. Therefore, the reachable space vector voltage is decreased from 0.58 to 0.5 per unit (pu) in the CFNI, while it can be maintained invariable in the SFNI [START_REF] Tousizadeh | Performance comparison of fault-tolerant three-phase induction motor drives considering current and voltage limits[END_REF]. In addition, the CFNI also suffers from the issue of DC-bus midpoint fluctuations.

Fault-tolerant solutions using NP

The SFNI requires higher hardware costs to achieve the fault-tolerance.

A B + - u in + - u bus C S A S B S C S' A S' B S' C Motor split dc-bus TRIAC N o'

Comparison

In an effort to quantify the post-fault output capability and the overall cost associated with the four typical fault-tolerant topologies, two evaluation criteria are defined as:

• PPRF = maximum kVA rating under fault rated kVA rating in health

• SOCF = weighted kVA rating of all silicon devices rated kVA rating of six basic IGBTs

The PPRF defines a normalization of the maximum available apparent power magnitude (kVA) in post-fault operations to the nominal apparent power magnitude (kVA) under healthy conditions. The SOCF defines a normalization of the weighted power magnitude (kVA) sustained by all the silicon devices under post-fault operations to the rated apparent power magnitude (kVA) sustained by six basic IGBTs in healthy conditions. Based on above definitions, the bigger PPRF means the drive system can output larger power in fault-tolerant conditions. The bigger SOCF means the drive system requires more costs to achieve the fault-tolerance.

To implement this calculation, each silicon device is considered to withstand a DC-bus voltage of 1 pu and convey a phase-current root mean square (RMS) of 1 pu. Therefore, a healthy drive system has a SOCF of 1. To simplify the comparison, the cost of a TRIAC is assumed to be the same as an IGBT [START_REF] Welchko | Fault tolerant three-phase AC motor drive topologies: A comparison of features, cost, and limitations[END_REF]. Based on these hypotheses, the characteristics of each existing fault-tolerant topology can be evaluated respectively. For example, the CFI has a PPRF of 0.5 because the reachable voltage vector is degraded by a factor of 0.5 while the available current is consistent. Regarding its SOCF, it can be calculated by ( 3 A comprehensive comparison of these fault-tolerant topologies is summarized in table 1.1. From table 1.1, we can see that both CFI and SFI do not require an accessible NP. However, they can only tolerate an OPF which occurs in the inverter. The SFI has a higher PPRF than the CFI while it also requires a higher SOCF (1.83 to 1.5). The CFNI and SFNI require an available NP so that they can tolerate an OPF which can occur in the motor or the inverter.

The SFNI has a higher PPRF (0.58 to 0.50) than the CFNI while it requires a higher SOCF (1.87 to 1.29). In general, all these existing fault-tolerant topologies require assistant devices (at least one TRIAC) to achieve the fault-tolerant operation. These assistant devices increase not only the cost of the whole drive system but also complicate the control circuit because additional control signals need to be designed for auxiliary TRIACs and IGBTs. This will indirectly increase the cost and affect the reliability of the drive system.

Conclusion

The NP of three-phase Y-connection AC motors can be utilized to obtain extra benefits.

In the basic MFCS and its variants, AC motor's NPs are directly or indirectly connected with the DC-source to achieve a DC/DC converter function without or with less passive and active components. In some integrated on-board chargers, the traction drive can be configured as a boost converter with PFC or a buck converter to charge the battery by connecting its NP.

41/178

Essentially, the connected NP leads to the circulation of zero-sequence currents in the traction drive. Thus, the leakage inductance of traction motors become an available resource to construct a DC/DC converter. However, only few researchers focus on the concept of MFCSs and there is no much literature that can be found concerning MFCSs. Objectively speaking, the mechanism of MFCSs is not well explained by a clear analytic model. There is no a closed-loop control strategy presented for MFCSs in the literature. The advantages and disadvantages of the basic MFCS are not comprehensively elaborated while it is compared to a conventional two-stage drive.

From another point of view, the connected NP in AC drives also contributes to the fault-tolerance towards OPFs. In the CFNI and SFNI, the NP is connected to the midpoint of split capacitors and a fourth leg respectively. As a result, the motor is able to output a stable torque even if one of motor phases is disconnected from the power supply. However, the use of additional hardwares is inevitable in the CFNI and SFNI. To be specific, the CFNI requires 1

TRIAC and a pair of split capacitors. The SFNI requires 1 TRIAC and 2 extra IGBTs.

Considering the innately connected NP in MFCSs, it exists a possibility that the MFCS can also achieve the fault-tolerance without using auxiliary components compared to the CFNI and SFNI. Moreover, the modulation method for the basic MFCS is not normalized.

In this chapter, we intend to investigate a PMSM-based basic MFCS comprehensively. It is organized as follows:

• Starting from the very beginning, Section 1 presents the mathematical model of the PMSM from A-B-C to d-q-0 coordinates by considering the zero-sequence components, which are usually neglected in a conventional drive.

• In Section 2, the average model of the PMSM-based basic MFCS is built based on time-averaging equivalent modeling approach and Kirchhoff's laws. To facilitate the understanding of the mechanism of the basic MFCS, three different equivalent circuits are illustrated.

• In Section 3, the classic SVPWM technique is discussed if it is used in the basic MFCS.

The theoretical analysis indicates that it can not provide an independent control DOF to the equivalent boost function of the basic MFCS. To address this issue, a customized zero-sequence voltage injected PWM (ZSVIPWM) is proposed. Then, the characteristics of the PMSM-based basic MFCS is analyzed with respect to its DC-source voltage utilization ratio (DCSUR) and the effective step-up ratio of the equivalent boost function.

• In Section 4, elementary experiments are carried out on a 52.5 W PMSM platform to verify the phenomenon of the integrated boost function. An experimental comparison between the basic MFCS with a 15 V DC-source and a conventional drive with a 30 V DC-source is given.

PMSM model with zero-sequence components

The two-phase equivalent model built in direct-quadrature (d-q) plane is typically used to analyze a conventional PMSM drive and to design the control strategy. Due to the Y-connection winding layout, there is no zero-sequence current circulated in motor windings when the NP is not connected. Thus, the zero-sequence model of PMSMs is usually neglected by researchers.

44/178

As mentioned before, the connected NP in a MFCS will provide a circulated path for the zero-sequence current. Therefore, the zero-sequence model of a PMSM should be considered in a MFCS.

Throughout the whole derivation, the following assumptions are given:

• Stator windings are symmetric in three phases;

• Stator electric potentials are sinusoidally distributed along air-gap as far as the mutual effects with the rotor are concerned;

• Eddy current and hysteresis effects are neglected;

• Resistances are independent of temperature and frequency;

• Magnetic saturation effects are negligible.

Electrical dynamics in A-B-C coordinate

The schematic diagram of a three-phase PMSM with one pole is depicted in figure 2.1.

Modeling of the PMSM in d-q-0 coordinate frame starts with the basic model in A-B-C coordinate frame. By specifying the positive directions, the flux linkage equations are given as

S N i A u A A X A B Y C Z i B i C u B u C B C ¡ f ¢ e £ e d-axis
Φ ABC =      L AA M AB M AC M BA L BB M BC M CA L CB L CC      L ABC i ABC + Φ f      cosθ e cos(θ e -2π/3) cos(θ e + 2π/3)      (2.1a) with                              L AA = L σ + L Σ + L ∆ cos 2θ e L BB = L σ + L Σ + L ∆ cos 2 (θ e -2π/3) L CC = L σ + L Σ + L ∆ cos 2 (θ e + 2π/3) M AB = M BA = -L Σ /2 + L ∆ cos 2 (θ e + 2π/3) M BC = M CB = -L Σ /2 + L ∆ cos 2θ e M AC = M CA = -L Σ /2 + L ∆ cos 2 (θ e -2π/3) (2.1b)
where

• Φ ABC := [Φ A Φ B Φ C ] T denotes the stator flux linkages in A-B-C coordinate frame; • i ABC := [i A i B i C ] T denotes the phase-currents in A-B-C coordinate frame;
• L ABC is an inductance matrix, including self-inductances (L AA , L BB and L CC ) and mutual inductances (M AB = M BA , M AC = M CA and M BC = M CB );

• Φ f denotes the permanent magnet flux linkage;

• θ e denotes the electrical position of the rotor;

• L σ denotes the leakage inductance;

• L Σ denotes the average inductance;

• L ∆ denotes the inductance fluctuation (for a surface-mounted PMSM, L ∆ = 0; for an interior PMSM, L ∆ < 0).

According to Faraday's law of electromagnetic induction, the stator voltage equations are 46/178 derived by

u ABC =      R 0 0 0 R 0 0 0 R      R i ABC + dΦ ABC dt =Ri ABC + L ABC di ABC dt + dL ABC dt i ABC -Φ f ω e      sinθ e sin(θ e -2π/3) sin(θ e + 2π/3)      = Ri ABC + L ABC di ABC dt -2L ∆ ω e      sin2θ e sin2(θ e + 2π/3) sin2(θ e -2π/3) sin2(θ e + 2π/3) sin2(θ e -2π/3) sin2θ e sin2(θ e -2π/3) sin2θ e sin2(θ e + 2π/3)      i ABC -Φ f ω e      sinθ e sin(θ e -2π/3) sin(θ e + 2π/3)      (2.2)
where

• u ABC := [u AN u BN u CN ] T denotes the phase-voltages in A-B-C coordinate frame;
• R is a resistance matrix, including the stator resistance (R);

Obviously, in A-B-C coordinate, the mathematical model of the PMSM contains inductance terms which vary with the rotor position θ e which in turn varies with time. This induces complexity in analyzing machine problems. By applying the Park transformation (power invariant) to (2.1a), the flux linkage equations 47/178 in d-q-0 coordinate frame can be expressed by

Electrical dynamics in d-q-0 coordinate

Φ dq0 = T C Φ ABC = T C      L ABC i ABC + Φ f      cosθ e cos(θ e -2π/3) cos(θ e + 2π/3)           = T C L ABC T -1 C i dq0 + T C Φ f      cosθ e cos(θ e -2π/3) cos(θ e + 2π/3      =      L σ + 3 2 L Σ + 3 2 L ∆ 0 0 0 L σ + 3 2 L Σ -3 2 L ∆ 0 0 0 L σ      L dq0 i dq0 + Φ m      1 0 0      (2.3)
where

• L dq0 := diag(L d L q L 0 ) is the inductance matrix in d-q-0 coordinate frame, including
the d-axis inductance (L d ), q-axis inductance (L q ) and 0-axis inductance (L 0 ). We can thereby conclude that the 0-axis inductance (L 0 ) is equal to the leakage inductance (L σ );

• i dq0 := [i d i q i 0 ] denotes the d-axis, q-axis and 0-axis currents;

• Φ m = 3 2 Φ f is a constant coefficient (If the Park transformation (amplitude invariant) is adopted, Φ m = Φ f );
According to inverse Park transformation (power invariant), the voltage equations of (2.2) can be represented by

T -1 C u dq0 = RT -1 C i dq0 + dT -1 C Φ dq0 dt (2.4)
where

• u dq0 := [u d u q u 0 ] T denotes the motor voltages in d-q-0 coordinate frame.
By left-multiplying T C to (2.4), we obtain

u dq0 = T C RT -1 C i dq0 + T C dT -1 C Φ dq0 dt (2.5)
Then, substituting (2.3) into (2.5), the voltage equations of PMSMs in d-q-0 coordinate can be expressed as:

48/178

d-q-0 model of PMSMs obtained by the Park transformation (power invariant):

Voltage equation:

     u d u q u 0      =      R 0 0 0 R 0 0 0 R           i d i q i 0      +      L d 0 0 0 L q 0 0 0 L 0                 di d dt di q dt di 0 dt            +      -ω e L q i q ω e (L d i d + Φ m ) 0      e dq0 (2.6)
where

• u 0 = u AN + u BN + u CN √ 3 ; • i 0 = i A + i B + i C √ 3 ;
• e dq0 := [e d e q e 0 ] T denotes the d-axis, q-axis and 0-axis back-EMFs.

Torque equation:

According to power conversion law, the electromagnetic torque can be derived by • p is the number of pole pairs;

T e = p ω e (e A i A + e B i B + e C i C ) = p ω e e T ABC i ABC = p ω e (T -1 C e dq0 ) T T -1 C i dq0 = p ω e e T dq0 (T -1 C ) T T -1 C i dq0 = p ω e e T dq0 i dq0 = p[Φ m i q + (L d -L q )i d i q ] = p[ √ 3 2 Φ f i q + (L d -L q )i d i q ] (2.
According to (2.6), it can be seen that the 0-axis back-EMF term (e 0 ) is 0. Therefore, the zero-sequence model of a PMSM can be considered as a first-order system.

For a surface-mounted PMSM (L d = L q ), the electromagnetic torque can be simplified by

T e = pΦ m i q (2.8)
If the Park transformation (amplitude invariant) is adopted, corresponding d-q-0 model can be derived in the same way. In this case, the voltage and torque equations are summarized below:

49/178

d-q-0 model of PMSMs obtained by the Park transformation (amplitude invariant):

Voltage equation:

     u d u q u 0      =      R 0 0 0 R 0 0 0 R           i d i q i 0      +      L d 0 0 0 L q 0 0 0 L 0                 di d dt di q dt di 0 dt            +      -ω e L q i q ω e (L d i d + Φ f ) 0      (2.9)
where

• u 0 = u AN + u BN + u CN 3 ; • i 0 = i A + i B + i C 3 ;
Torque equation:

T e = 3 2 p[Φ f i q + (L d -L q )i d i q ]
(2.10)

Mechanical dynamics

According to Newton's second law, the motion equation of the motor can be formulated by

J m dω m dt = T e -B m ω m -T L (2.11)
where

• J m denotes the rotational inertia;

• ω m denotes the mechanical angular velocity of the rotor;

• B m denotes the coefficient of viscous friction;

• T L denotes the load torque;

Analytical model of a PMSM-based basic MFCS

The circuit configuration of a conventional three-phase PMSM fed by a three-phase twolevel VSI is shown in figure 2.2. The NP (N) of the motor is untapped. Therefore, there is no zero-sequence current generated inside the motor.

The circuit structure of a PMSM-based basic MFCS is illustrated in figure 2 In order to study and analyze the characteristics of the PMSM-based basic MFCS, the system model including the inverter circuit will be built in this section. Throughout the derivation, the following assumptions are given:

+ -

u bus + - u in i N i bus o S A S B S C (¡ A ) (¡ B ) (¡ C ) S' A S' B S' C A B C i A i B i C N motor C bus
• All the switching devices are considered as ideal switches whose voltage drop, dead time and non-linearity are neglected;

• The DC-bus capacitor is regarded as an ideal capacitor whose equivalent series resistance is neglected;

• The DC power-source is considered as an ideal DC source with a constant and stable voltage.

• The switching frequency is considered as constant;

51/178

Average model

Based on aforementioned hypotheses, the switching functions (S X and S X ) of the VSI are defined as

S X =   
1 when the upper switch is ON 0 when the lower switch is ON , (x = A, B, C)

S X = S X =   
1 when the lower switch is ON 0 when the upper switch is ON , (x = A, B, C)

(2.12)

According to time-averaging equivalent modeling approach and Kirchhoff's laws, the average model of the basic MFCS (illustrated in figure 2.3) can be built within one switching period. On the basis of Kirchhoff's voltage Law (KVL), the average voltage equations of the basic MFCS can be derived as

     u AO u BO u CO      u invABC =      α A α B α C      α ABC u bus =      u AN u BN u CN      u ABC +      u in u in u in      (2.13)
According to Kirchhoff's current Law (KCL), the average current equations can be given as

       i N = -(i A + i B + i C ) i bus = C bus du bus dt = -α T ABC i ABC (2.14)
where

• u invABC := [u AO u BO u CO ]
T denotes the terminal voltages of the inverter;

• α ABC := [α A α B α C ]
T denotes the duty-cycles of inverter's legs;

• u bus is the DC-bus voltage;

• u in is the DC-source voltage;

• i N is the neutral current, which can also be seen as the input current of the whole system;

• i bus is the DC-bus current;

• C bus is the DC-bus capacitance;

• All above variables represent the mean values within one switching period; And all the positive directions are defined by the arrows shown in figure 2.3.

52/178

By using the Park transformation (power invariant), (2.13) can be transformed from

A-B-C to d-q-0 coordinates, as      u invd u invq u inv0      u invdq0 =      α d α q α 0      α dq0 u bus =      u d u q u 0      u dq0 +      0 0 √ 3u in      (2.15)
where

• u invdq0 := [u invd u invq u inv0 ]
T denotes the terminal voltages of the inverter converted to d-q-0 coordinate;

• α dq0 := [α d α q α 0 ] T includes the values of duty-cycles converted into d-q-0 coordinate frame;

In the same way, (2.14) can also be transformed from A-B-C to d-q-0 coordinates, as

       i N = -(i A + i B + i C ) = - √ 3i 0 i bus = C bus du bus dt = -α T ABC i ABC = -(T -1 p α dq0 ) T T -1 p i dq0 = -α T dq0 i dq0 (2.16)
By substituting (2.6) into (2.16) and by combining with (2.14), the average model of the basic MFCS in d-q-0 coordinate can be rewritten as:

d-q-0 average model obtained by the Park transformation (power invariant):

[a]                                  u invd = α d u bus = u d = Ri d + L d di d dt -ω e L q i q u invq = α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) u inv0 = α 0 u bus = Ri 0 + L 0 di 0 dt u 0 + √ 3u in i bus = C bus du bus dt = -(α d i d + α q i q + α 0 i 0 ) (2.17a)

53/178

By replacing i 0 with -i N / √ 3, it further yields

                                 α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) α h u bus = - R 3 i N - L 0 3 di N dt + u in i bus = C bus du bus dt = α h i N -(α d i d + α q i q ) i lo (2.17b)
where

• α h = α 0 √ 3 = α A + α B + α C 3
denotes the average duty-cycle of the inverter.

• i lo is seen as an equivalent load current.

[a] The model obtained by the Park transformation (power invariant) will be used in the following analysis and the construction of equivalent circuits.

In (2.17b), the first two equations are similar to the well-known d-q model of a PMSM, which are usually used to design the classic FOC. However, the last two equations are unique in the basic MFCS due to the circulated zero-sequence path. The circulated zero-sequence current will enable the power flow between the DC-bus capacitor and the DC-source. Therefore, the last two equations can not be ignored in the basic MFCS.

In steady-state, the motor power is assumed to be constant. According to power conservation law, the input current (i.e. the neutral current i N ) of the basic MFCS is constant. Thus, i 0 is also constant. In this case, the derivative term (L 0 di 0 dt ) in (2.17a) and the derivative term (2.17b) are equal to 0. By further neglecting the voltage drop across the resistance R, we obtain

( L 0 3 di N dt ) in
u bus ≈ u in α 0 / √ 3 = u in α h = 3u in α A + α B + α C (2.18)
According to (2.18), it can be observed that the DC-bus voltage u bus is able to reach higher value than the DC-source voltage u in as long as α h is smaller than 1. This phenomenon is similar to a DC/DC boost converter.

If the Park transformation (amplitude invariant) is used, (2.17a) and (2.17b) become 54/178 d-q-0 average model obtained by the Park transformation (amplitude invariant): [a] voltage equation:

                                 u invd = α d u bus = u d = Ri d + L d di d dt -ω e L q i q u invq = α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) u inv0 = α 0 u bus = Ri 0 + L 0 di 0 dt u 0 +u in i bus = C bus du bus dt = - 3 2 (α d i d + α q i q + 2α 0 i 0 ) (2.19a) Replacing i 0 by -i N /3, it further yields                                  α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) α h u bus = - R 3 i N - L 0 3 di N dt + u in i bus = C bus du bus dt = α h i N - 3 2 (α d i d + α q i q ) i lo (2.19b)
where

• α h = α 0 = α A + α B + α C 3
denotes the average duty-cycle.

• i lo is seen as an equivalent load current.

[a] The model obtained by the Park transformation (amplitude invariant) will be used in the following control design.

Equivalent circuits

Although (2.18) reveals the existence of an equivalent DC/DC boost function which is integrated in the basic MFCS, how does the boost interact with the motor is still indistinct.

Therefore, this should be studied in detail. As we all know, using equivalent circuit is an effective means to simplify the analysis and understanding of a complex circuit system. In this subsection, different equivalent circuits will be presented to facilitate the mechanism of the basic MFCS. To construct the equivalent circuit, the d-q-0 model obtained by using the Park transformation (power invariant) is used.

55/178

d-q-0 equivalent circuit

Actually, (2.17a) presents the electrical relationship of the basic MFCS in d-q-0 coordinate.

Based on that, we can know that the d-q voltages (u invd and u invq ) of the inverter are still equal to the d-q voltages (u d and u q ) of the motor. This indicates that the connection of motor's NP does not affect the relationship between the inverter and the motor in d-q plane. In other words, the conventional FOC can still be used to control the torque and flux of the motor through d-axis and q-axis currents.

However, the connected NP does induce some changes on the 0-axis. The zero-sequence item of (2.17a) implies that the connected NP enables the zero-sequence circuit of the drive system by providing a zero-sequence path for the zero-sequence current to flow. The zerosequence voltage (u inv0 ) of the inverter is different from the zero-sequence voltage (u 0 ) of the motor. Based on (2.17a), we can draw the corresponding d-q-0 equivalent circuit of the PMSMbased basic MFCS, as displayed in figure 2.4. It consists of a typical d-q circuit and a unique zero-sequence circuit. The d-axis and q-axis are coupled through i d and i q . The zero-sequence circuit is coupled with the d-q circuit to u bus and i bus (see the fourth equation of (2.17a)).

+ -

+ - + - R R R L d L q -¡ e L q i q ¡ e (L d i d +¢ m ) L 0 £3u in i d + - + - + - u 0 d-axis equivalent circuit q-axis equivalent circuit zero-sequence circuit + - i q u invq =¤ q u bus =u q i 0 ¤ 0 u bus d-q coupling u invd =¤ d u bus =u d i bus ¤ 0 i 0 ¤ q i q ¤ d i d N i bus =-α T dq0 i dq0

Visualized equivalent circuit

Although the d-q-0 equivalent circuit reveals the link between the d-q circuit and the zerosequence circuit, the physical significance of the equivalent boost function is still not intuitive.

Normally, a typical DC/DC boost converter consists of a DC-source, a filtering inductor, a 56/178 chopper, a capacitor and a load. Thus, what components of the basic MFCS do construct the equivalent boost function is still confusing in the d-q-0 equivalent circuit.

To further reveal the equivalent boost function, (2.17b) is referenced to establish a more visualized and comprehensible equivalent circuit since the neutral current i N is more intuitional than i 0 . The visualized equivalent circuit is shown in figure 2.5. It is well-known that the invention of FOC makes it easier to control an AC motor by decoupling its electromagnetic torque and flux linkage. In d-q plane, an AC motor can be seen as an equivalent DC motor.

Therefore, we can use an equivalent DC motor drive to represent the motor function of the basic MFCS. To embody the physical significance of α d and α q , we use two virtual H-bridges named as 'H1' and 'H2' to drive the equivalent DC motor hypothetically. The two H-bridges are constructed by eight virtual switches S v1 , S v1 , S v2 , S v2 , S v3 , S v3 , S v4 and S v4 . To be specific, α d represents the difference between the duty-cycles of S v3 and S v4 ; α q represents the difference between the duty-cycles of S v2 and S v5 . Based on the above hypotheses, it is obvious that the physical significance of the first two equations of (2.17b) is well visualized in the dual H-bridge circuit. Essentially, it is similar to a DC motor drive. In order to embody the physical significance of the last two equations of (2.17b), we introduce a virtual chopper 'C1', which is constructed by two virtual switches S v1 and S v1 .

Because the physical limitation of inverter's duty cycles (α A , α B and α C ) is between 0 and 1, the reachable range of α h = (α A + α B + α C )/3 is also (0-1). Therefore, α h can be perfectly represented by the virtual chopper in the average model, where α h denotes the duty-cycle of S v1 .

In this case, the third equation of (

i d + α q i q ).

Generalized equivalent circuit

To further elaborate the power conversion process, it is reasonable to separate all the zero-sequence components from the motoring function by cutting off the NP hypothetically. Once the NP is disconnected, the basic MFCS can be divided into two parts, one is the zero-sequence part and the other is the conventional part where the NP is not connected.

Based on the above hypotheses, generalized equivalent circuits are presented in figure 2.6.

These equivalent circuits can represent a basic MFCS based on any kind of three-phase AC machines, such as induction machines and synchronous reluctance machines. During motoring patterns, the zero-sequence part functions as a DC/DC boost converter, to raise the DC-bus voltage. The conventional part acts as a common motor drive whose NP is not used. The system power flows from the zero-sequence part to the conventional part, i.e. the DC-source is discharged. During regenerative braking patterns, the motor becomes a generator and the conventional part functions as an active rectifier. In this case, the zero-sequence part functions as a DC/DC buck converter, and the system power flows from the conventional part to the zero-sequence part, i.e. the DC-source is charged. Therefore, from the functionality point of view, the generalized equivalent circuits further reveal that the basic MFCS system is similar to a two-stage converter. However, compared to a real two-stage converter, it employs fewer switching devices and passive components by taking advantage of its zero-sequence circuit.

58/178

Modulation method and steady-state characteristic

The proposed average model and equivalent circuits indicate that the zero-sequence circuit of the basic MFCS can be used as a boost. The average duty cycle (α h ) of the inverter can be used as the control DOF for the equivalent boost function. However, how to implement the control DOF (α h ) into the modulation method is a complex issue.

Discussion of SVPWM used in the basic MFCS

As we all know, a VSI-fed three-phase AC motor drive usually uses PWM techniques to realize the conversion from DC to AC [START_REF] Wong | Fundamental study of 3 dimensional pulse width modulation[END_REF]. Normally, SPWM and SVPWM are two typical and commonly used modulation methods. In particular, SVPWM is chosen preferentially due to its high voltage utilization and low harmonics ratio [START_REF] Abassi | Performance analysis of foc and dtc for pmsm drives using svpwm technique[END_REF]. Compared to SPWM, the modulation ratio m of SVPWM can be improved by 15% in the absence of over-modulation [START_REF] Xiao | Strategy and implementation of harmonicreduced synchronized svpwm for high-power traction machine drives[END_REF]. In the classic SVPWM technique, the reference voltage vector is synthesized in α-β plane divided into six sectors. In each switching period, adjacent vectors and zero vectors are utilized to synthesize the reference voltage.

From the carrier-based modulation technique point of view, the classic SVPWM can be approximatively treated as a particular form of SPWM where the modulating signal is superposed with a third harmonic [81][82]. As shown in figure 2.7, by superposing the third harmonic, the amplitude of fundamental voltages is improved and the linear modulation range can be extended. In fact, the superposed third harmonic is a typical type of zero-sequence voltages. This zero-sequence voltage does not generate zero-sequence currents in a conventional three-phase drive system due to the Y-connection winding layout. In other words, the DOF of zero-sequence voltages is well utilized in the classic SVPWM to the purpose of improving the voltage output capability of a three-phase two-level VSI. However, if the classic SVPWM is used in the basic MFCS, the superposed third harmonic voltage will induce third harmonic currents caused by the connected NP thus further resulting in voltage oscillations on the DC-bus.

We calculate the value of α h [1] when the classic SVPWM is applied. Figure 2.8 presents the evolution of α h with respect to the electrical position and the modulation ratio m. It can be seen that for a given m, the shape of α h is similar to a triangular wave in one electrical period. The mean value of α h is 0.5 and its frequency is three times that of the fundamental wave. The amplitude of α h increases with the increase of m. Because the modulation ratio m is related to the operating condition of the motor, α h is directly determined by the control of the motor. In the case of m = 1.15, the peak and valley of α h can reach around 0.66 and 0.34 [1] The calculation is detailed in Appendix B.

59/178

Zero-sequence voltage injected PWM

Since the classic SVPWM is inapplicable in the basic MFCS based on the above analysis, we propose a carrier-based modulation method, which is called zero-sequence voltage injected PWM (ZSVIPWM). In order to control the equivalent boost function, a certain amount of 60/178 zero-sequence voltage is required and should be independent of the control of the motor. The requisite zero-sequence voltage (u ZS ) is defined as

u ZS = (u AN + u BN + u CN ) 3 = u 0 √ 3 (2.20)
According to (2.17a), u ZS can be represented by

u ZS = u 0 √ 3 = α h u bus -u in (2.21)
In steady-state, the derivative term of (2.17b

) L 0 3 di N
dt is 0. By combining with (2.21), we obtain that u ZS is equal to -Ri N 3 . In motoring conditions, the DC-source should provide power to the motor. Therefore, the input current (i.e. neutral current) i N is positive. Based on this, we know that the required zero-sequence voltage u ZS is negative. On the contrary, in regenerative braking conditions, u ZS is positive. In the following, we consider the motoring condition for analysis.

By considering the zero-sequence voltage u ZS , the actual phase-voltage reference (u XN * )

in the basic MFCS is not a standard sinusoidal signal. It should be the sum of the fundamental voltage reference (u * XN 1 ) and the required zero-sequence voltage (u ZS ). Then, by combining with (2.21), we obtain

u XN * = u XN 1 * + u ZS = u XN 1 * + α h u bus -u in , X ∈ {A, B, C} (2.22)
Due to the clamped NP in the basic MFCS, the carrier signal of ZSVIPWM is asymmetrical.

To be specific, the positive and negative amplitudes of the carrier signal are respectively equal to u bus -u in and -u in , as shown in figure 2.9. Moreover, the actual modulating signal (u XN * )

shows a downward offset due to the required zero-sequence voltage (u ZS ) compared to the fundamental voltage reference (u * XN 1 ). In this case, by comparing the modulating signal u XN * with the carrier signal, the actual PWM signals can be generated.

In digital control system, the control input is the duty cycle (α X , X ∈ {A, B, C}). If the regular-sampling method (see figure 2.10) is adopted, the final duty-cycle α X used for digital control can be derived by Consequently, (2.23) is the final algorithm of ZSVIPWM where α h is an independent control DOF for the equivalent DC/DC boost function and the fundamental voltage reference u * XN 1 is the control DOF for the motor. Especially, when α h = 0.5, (2.23) actually corresponds to the algorithm of the classic SPWM by using the regular sampling method. However, if the classic SPWM is adopted, u bus can be only regulated to double u in in an open-loop way since no independent control DOF can be provided by the classic SPWM. This issue is similar to the classic SVPWM. By comparison, we can say that the proposed ZSVIPWM skillfully decouple the control DOFs of the equivalent boost function and the motor function. This makes it possible to actively regulate the power flow between the motor and the DC-source or to control the DC-bus voltage.

α X = δ/2 T s /2 = u XN * + u in u bus = α h + u XN 1 * u bus (2.

DC-source voltage utilization ratio

Thanks to the equivalent DC/DC boost function, u bus can reach a higher value than u in in the basic MFCS. However, the reachable AC voltages for the motor are bounded. In other 62/178 words, the effective step-up ratio of the equivalent DC/DC boost has a limit due to the coupling between α d , α q and α h when saturation occurs. If the boosted DC-bus voltage can not improve the fundamental AC voltages, it makes no sense. It may even result in higher switching losses.

Therefore, it is necessary to study the relationship between the step-up ratio of the equivalent DC/DC boost and the DC-source voltage utilization ratio (DCSUR).

The step-up ratio (r 0 ) of the equivalent boost is defined as

r 0 = u bus u in (2.24a)
To evaluate the reachable AC voltages, the DCSUR (r 1 ) is defined as

r 1 = u XN 1amp u in (2.24b)
where

• u XN 1amp is the amplitude of fundamental AC voltages.

For the sake of analysis, the proportion (r 2 ) between the zero-sequence voltage and the DC-source voltage is defined as

r 2 = |u ZS | u in (2.24c)
As mentioned before, the positive and negative amplitudes of the carrier signal are equal to u bus -u in and -u in , respectively. In order to guarantee u XN * to be within the linear modulation region, the amplitude of u XN * (which is defined as u XN amp ) is limited by the lowest one between u in and u bus -u in . Accordingly, the amplitude of fundamental AC voltages (u XN 1amp ) is also limited. Considering different conditions, there are five cases to be discussed:

1) u bus > 2u in (i.e. r 0 > 2)
In this case, the carrier signal is asymmetrical because u bus -u in is larger than u in . The offset of the carrier signal is (u bus -2u in )/2, which is larger than 0. Meanwhile, the actual modulating signal u XN * shows a downward offset because of the injected zero-sequence voltage u ZS . Based on all above conditions, it is obvious that the lower edge of u XN * will firstly reach the limit of -u in with the rise of the modulation ratio, as shown in figure 2.11(a). As a result, the limit of u XN amp is u in and the limit of u XN 1amp is evaluated as

u in -|u ZS | (i.e. r 1 = 1 -r 2 ).
2) u bus = 2u in (i.e. r 0 = 2)

When u bus = 2u in , the carrier signal is symmetrical because the offset (u bus -2u in )/2 of the carrier signal is 0. The actual modulating signal u XN * still shows a downward

63/178

Chapter 2. Investigation of a PMSM-based basic MFCS offset because of the injected zero-sequence voltage u ZS . In this case, the lower edge of u XN * will also firstly reach the limit of -u in , as shown in figure 2.11(b). As a result, the limit of u XN amp is still u in and the limit of u XN 1amp is still evaluated as u in -|u ZS | (i.e.

0 u bus >2u in (r 0 >2) u XN1 * Carrier signal u XN * -u XN1amp (a) -u XNamp u bus -u in -u in u ZS 0 u bus =2u in (r 0 =2) u XN1 * Carrier signal u XN * -u XN1amp (b) -u XNamp u in -u in u ZS (u bus -2u in )/2>0 (u bus -2u in )/2=0 0 u in <u bus <2u in (1<r 0 <2) u XN1 * Carrier signal u XN * -u XN1amp (c) -u XNamp u bus -u in -u in u ZS u ZS <(u bus -2u in )/2<0 0 u in <u bus <2u in (1<r 0 <2) u XN1 * Carrier signal u XN * -u XN1amp (d) -u XNamp u bus -u in -u in u ZS u ZS =(u bus -2u in )/2<0 u XN1amp 0 u in <u bus <2u in (1<r 0 <2) u XN1 * Carrier signal u XN * (e) -u XNamp u bus -u in -u in u ZS (u bus -2u in )/2<u ZS <0 -u XN1amp
r 1 = 1 -r 2 ).
3) u in < u bus < 2u in (i.e. 1 < r 0 < 2) and u ZS < (u bus -2u in )/2

In this case, the carrier signal is asymmetrical. It shows a upward offset because the offset (u bus -2u in )/2 is smaller than 0. However, (u bus -2u in )/2 is larger than u ZS , i.e. r 0 > 2 -2r 2 . This indicates that the downward offset of u XN * is bigger than the upward 64/178 offset of the carrier signal. In this case, the lower side of u XN * will firstly reach the limit of -u in , as shown in figure 2.11 (c). As a result, the limit of u XN amp is u in and the limit of u XN 1amp is evaluated as u in -|u ZS | (i.e. r 1 = 1 -r 2 ).

4) u in < u bus < 2u in (i.e. 1 < r 0 < 2) and u ZS = (u bus -2u in )/2

In this case, the downward offset of u XN * is equal to the upward offset of the carrier signal, i.e. r 0 > 2 -2r 2 , as shown in figure 2.11(d). This implies that the positive and negative edges of u XN * will reach the limits at the same time. Therefore, the limit of the limit of u XN amp is still u in and the limit of

u XN 1amp is still u in -|u ZS | (i.e. r 1 = 1 -r 2 ). 5) u in < u bus < 2u in (i.e. 1 < r 0 < 2) and u ZS > (u bus -2u in )/2
In this case, the downward offset of u XN * is smaller than the upward offset of the carrier signal, i.e. r 0 < 2 -2r 2 , as shown in figure 2.11(e). This indicates that the positive edge of u XN * will firstly reach the limit (u bus -u in ). As a result, the limit of u XN amp is u bus -u in and the limit of

u XN 1amp is u bus -u in + |u ZS | (i.e. r 1 = r 0 + r 2 -1).
Based on the above analysis, we can summarize that for a given r 2 , r 1 equals to 1 -r 2 in the case of r 0 ≥ (2 -2r 2 ). For the sake of explanations, the relationship between r 0 , r 1 and r 2 are visualized in figure 2.12. It can be seen that r 1 decreases with the decline of r 0 in the case of 1 < r 0 < (2 -2r 2 ). Therefore, the largest DCSUR is 1 -r 2 and the point 'a' (r 0 = 2 -2r 2 )

is the optimal operating point for the whole system. When r 2 reduces to r 2 , r 1 becomes r 1 .

In this case, the largest DCSUR increases to 1 -r 2 and the best operating point moves to 'b'.

Further, when r 2 reduces to 0, r 1 becomes r 1 . In this case, the largest DCSUR is 1 and the best operating point moves to the point '2', i.e. u bus = 2u in .

0 1 1 2 r 0 r 2 2r 2 r 2 r 1 r 2 ' r 1 ' r 1 '' a 1-r 2 b 1-r 2 ' r 2 ' 2r 2 '
Figure 2.12: Analysis of DCSUR with respect to different r 0 and r 2 .

65/178

In steady-state, u ZS is equal to -Ri N 3 . This indicates that |u ZS | is equal to the voltage drop across R/3. Generally speaking, the stator resistance of a PMSM is relatively small. Therefore, r 2 is relatively small. If we neglect r 2 , the largest DCSUR of the basic MFCS approximates to 1 and the maximum effective step-up ratio r 0 of the equivalent boost is 2. As emphasized before, the DCSUR of a conventional drive (without using its NP) is only 0.58 [START_REF] Liang | Analytical modeling of current harmonic components in pmsm drive with voltage-source inverter by svpwm technique[END_REF] by using the classic SVPWM. Once more, this proves that the basic MFCS has higher DCSUR than a conventional drive due to the equivalent boost function (1 against 0.58).

Influence of the clamped NP on current ripples

Although the basic MFCS can obtain a free DC/DC boost function, the clamped NP will induce some adverse problems. To explicitly explain the influence of the clamped NP, instantaneous voltage and current waves will be compared between a conventional drive and the basic MFCS in the following.

For a conventional drive whose NP is not connected (thus is floating), the instantaneous NP voltage (u N O ) can be formulated by

u N O = (u AO + u BO + u CO ) 3 = (S A + S B + S C ) 3 u bus (2.25)
when the motor is balanced in three-phase, the instantaneous phase-voltages (u AN , u BN and u CN ) of the motor can be expressed by

     u AN u BN u CN      = u bus 3      2 -1 -1 -1 2 -1 -1 -1 2           S A S B S C      (2.26)
From (2.26), it can be known that there are totally five voltage-levels available to each motor phase while considering eight available switching states. To be specific (taking u AN as an example), u AN equals to 0 V under S 000 and S 111 ; ± 1 3 u bus V under S 110 , S 101 , S 010 and S 001 ; ± 2 3 u bus V under S 100 and S 011 .

For the basic MFCS where the NP is connected to the DC-source (thus is clamped), the instantaneous phase-voltages can be expressed by

     u AN u BN u CN      = u bus      S A S B S C      -      u in u in u in      (2.27)
Apparently, there are only two voltage-levels (u bus -u in and -u in ) available for each motor phase. When u bus is regulated at double u in , the two voltage-levels are ± 1 2 u bus = ±u in .

66/178

For a given switching frequency, the instantaneous voltage and current waveforms of the basic MFCS and a conventional drive are compared in figure 2.13. The DC-bus voltage u bus is consistent in the two drives. Therefore, the basic MFCS only needs a DC-source with the voltage of u bus /2 while the conventional drive needs a DC-source with the voltage of u bus . For the sake of explanations, the subscript _c denotes a variable in the conventional drive, _m denotes a variable in the basic MFCS. From figure 2.13, it can be seen that u N O_c has four voltage-levels (0, 1 3 u bus , 2 3 u bus and u bus ) while u N O_m is clamped to 1 2 u bus throughout. u AN _c has three voltage-levels (0, 1 3 u bus and 2 3 u bus ) to choose, which switches six times in one switching period (T s ) while u N O_m only has two voltage-levels (± 1 2 u bus ) available and merely switches twice. Therefore, it can be observed that the current ripple ∆i A_m in the basic MFCS is much bigger than ∆i A_c in the conventional drive. Consequently, we can conclude that although the connected NP contributes to a free boost function, it will induce bigger current ripples. The increasing current ripples may result in more iron losses in the motor.

Experimental verification

To construct a PMSM-based basic MFCS, a PMSM with an available NP is necessary.

However, many commercial PMSM products do not provide the neutral terminal. In order to obtain the NP, we disassemble a 52.5 W PMSM and pull out its NP. Experiments will be implemented on this PMSM to verify the effectiveness. In the test-bench (see figure 2.14), hall current and voltage sensors are used to measure the phase currents, neutral current and DC-bus voltage. Control programs are implemented on a dSPACE MicroLabBox rapid prototyping system. An oscilloscope (Tektronix oscilloscope MDO3000) is used to observe the instantaneous voltages and currents. The main parameters of the test-bench are shown in table 2.2.

Zero-sequence inductance measurement

Considering that L 0 is a rarely used parameter in a conventional drive, we will introduce a practical method for identifying the value of L 0 . According to (2.6), the zero-sequence model of the PMSM can be seen as a first-order system. In this case, the inductance parameter L 0 can be identified by testing the dynamic response of the first-order system. To configure the zero-sequence circuit, the terminals of three phases are connected together, as shown in figure 2.15(a). Then, a voltage pulse of 1 V magnitude is applied between the terminals and the NP.

Then, the corresponding current response is captured by the oscilloscope, as illustrated in figure 2.15(b). Based on this, L 0 /3 can be estimated by where ∆t _0.632 is the response time of the first-order system.

N PMSM A (a) (b) 1 V i N =5.9 A i N =0 A ¡t _0.632 =1.7 ms i N_ 0.632 =3.75 A
Therefore, the value of L 0 of this PMSM is equal to 0.86 mH.

Validation of the equivalent DC/DC boost function

Based on the modeling analysis and equivalent circuits, we know that the DC-bus voltage Obviously, the DC-bus voltage can be also boosted to 30 V when α h is set at 0.5.

u

Experimental comparisons in steady-states

Figure 2.17 illustrates the steady-state performance of the basic MFCS and a conventional drive when the motor is operated at the nominal condition (4000 rpm and 125 mN•m) [2] . For the basic MFCS, its DC-bus voltage can reach 30 V even if a 15 V DC-source is used thanks to the equivalent boost function. u bus is boosted at 30 V stably while α h is around 0.47 (considering the voltage drop on motor's resistance R, α h is not exactly 0.5). The motor speed remains stable at 4000 rpm with slight fluctuations of 11 rpm (0.27%). The electromagnetic torque T e is maintained at 125 mN•m stably with 10 mN•m (8%) fluctuations. For the conventional drive, a 30 V DC-source is connected to the DC-bus directly. Therefore, the ripple of u bus is caused by the charge-discharge of the DC-bus capacitor in different switching states. However, for the basic MFCS, u bus has to be actively controlled at 30 V by the equivalent boost function. In [2] The control design for the equivalent boost is detailed in Appendix C, the classic FOC is used to control the motor.

start voltage-levels ± 1 3 u bus (±10 V), 0 V and ± 2 3 u bus (±20 V) thanks to the floating NP, as shown in figure 2.18(c).

71/178

72/178

The Fast Fourier Transform (FFT) analysis of u CN , i B and i C in both drive systems is shown in figure 2.18(b) and (d). Firstly, the fundamental voltage (FV) amplitude and fundamental current (FI) amplitude are respectively 11.94 V and 3.57 A in the basic MFCS while they are respectively 11.59 V and 3.37 A in the conventional drive. The fundamental power factor (cosφ) is about 0.88 in the basic MFCS while it is 0.89 in the conventional drive.

Comparatively speaking, these fundamental characteristics are similar in both drives.

Secondly, for the basic MFCS, the zero-sequence current flowing into the motor undoubtedly produces more losses. To calculate the total active power of the motor, a simple method of accumulating the power generated by all the harmonics (including zero-sequence and fundamental components) is adopted. The result indicates that the total active power of the motor in the basic MFCS increases by 6.02 W compared to that in the conventional drive. Therefore, it can be concluded that the increased motor losses in the basic MFCS is about 11.5% of the nominal power at the nominal condition. The increased copper losses caused by the zero-sequence current are the power consumed across R/3, which can be calculated by i N 2 R 3 = 3.98 W (7.6%). Therefore, the increased iron losses are about 2.04 W (3.9%). Although the motor losses of the basic MFCS increase, they can be mitigated by designing a winding layout with larger L 0 . For example, [START_REF] Hackner | Optimization of the winding arrangement to increase the leakage inductance of a synchronous machine with multi-functional converter drive[END_REF] and [START_REF] Hackner | Optimization of the winding arrangement to increase the zero-sequence inductance of a synchronous machine with 168/178[END_REF] indicated that a motor with the two-layer fractional slot winding can obtain larger leakage inductance than the motor with side-by-side placed winding.

Thirdly, the 20 kHz harmonic voltage (20kHV) amplitude of u CN in the basic MFCS is larger than that of the conventional drive (9.48 V to 0.37 V). Thus, it can be seen that the 20 kHz harmonic current (20kHC) amplitude of i C in the basic MFCS is also larger than that in the conventional drive (0.68 A to 0.003 A). However, the 20kHC in the basic MFCS does not affect the electromagnetic torque because the 20kHCs of i B and i C almost have the same amplitude (0.68 A to 0.64 A) and phase position (58.47 • to 59.70 • ). In other words, the 20kHCs do not generate harmonic torque. Therefore, although the phase-current ripple of the basic MFCS is larger, the torque performance can be maintained.

In general, the main characteristics of the basic MFCS compared to the conventional drive are summarized in table 2.2.

Conclusion

A PMSM-based MFCS is a special three-phase AC drive system where the DC power supply is connected with the NP of the motor. Due to this connection, it is possible to reach a higher DC-bus voltage without using a physical DC/DC boost converter. To investigate 73/178 The infeasibility of SVPWM used in the MFCS is explained by studying the distribution of α h with respect to the modulation ratio. It indicates that if SVPWM is used, α h will be completely determined by the control of the motor. Moreover, α h varies at three times of the fundamental frequency, which will result in the issue of voltage oscillations on the DC-bus. To solve this problem, a ZSVIPWM is proposed and the expression of final duty cycles is given for digital implementations. To guarantee the modulating signal to be within the linear modulation region, the DCSUR of the MFCS is analyzed. The result indicates that it can reach 1 while it is only 0.58 in a conventional drive by using SVPWM. Although the MFCS can harvest a free boost function, the clamped NP induces bigger current ripples. From the switching model point of view, the increasing ripples are caused by the decreasing number of available voltage-levels in each phase within one switching period.

Elementary verification is carried on a 52.5 W PMSM to verify the phenomenon of the equivalent DC/DC boost, which is integrated in the MFCS. Simulation and experimental results show that the DC-bus voltage is able to reach double the DC-source voltage. However, the phase-current ripples in the MFCS are much bigger than that in a conventional drive.

Fortunately, this increase does not affect the torque performance of the motor.

74/178

75/178

Introduction

In Chapter 2, the mathematical model, equivalent circuits and modulation method were presented for the basic MFCS. Experimental validations carried out on a 52.5 W PMSM was provided to demonstrate the effect of the integrated boost function. However, the basic MFCS has two salient drawbacks. The first one is the bigger current ripples caused by the clamped NP, which results in more losses (11.5% for the 52.5 W PMSM). The second defect is that the maximum effective step-up ratio of the equivalent boost is only 2.

To address the above issues, we intend to propose two solutions, which lead to two new variants of the basic MFCS. This Chapter is organized as follows:

• In Section 1, we propose a new variant, which is referred to as MFCS-I for the basic MFCS on the purpose of solving the issue of bigger current ripples caused by the clamped NP.

The main idea is to add a series inductor between the NP and the DC-source so that the NP becomes floating again. Due to the change of the circuit structure, the average model and equivalent circuit are built for the MFCS-I.

• To improve the effective step-up ratio, another variant is proposed in Section 2 by using an auxiliary inverter leg, which is called as MFCS-II. In the same way, the corresponding mathematical model and equivalent circuit are built to investigate and analyze its characteristics.

• From the equivalent circuit point of view, the MFCS-I and MFCS-II are similar. Therefore, in Section 3, a unified closed-loop control strategy is proposed for the MFCS-I and MFCS-II. The classic FOC is employed to control the motor and a differential flatness-based controller is presented to control the equivalent boost function.

• In Section 4, experimental results are given to verify the effectiveness. In particular, the MFCS-I is compared in detail with a two-stage converter-based drive.

Improvement 1: using a series inductor (MFCS-I)

As mentioned before, a salient drawback of the basic MFCS is the bigger current ripple caused by the clamped NP. Such a big current ripple will produce more iron losses in the motor thus reducing the efficiency of the whole drive system. In order to suppress the current ripple, a series inductor (L aux ) is added between the NP and the DC-source, as shown in figure 3.1.

76/178

+ -

u bus + - u in i N i bus o S A S B S C S' A S' B S' C A B C i A i B i C N motor C bus L aux

Average model and equivalent circuit

The same modeling method as that of the basic MFCS can be adopted for the MFCS-I. In A-B-C coordinate, the average voltage and current equations of the MFCS-I are given by

     u AO u BO u CO      u invABC =      α A α B α C      α ABC u bus =      u AN u BN u CN      u ABC +      -L aux di N dt + u in -L aux di N dt + u in -L aux di N dt + u in      (3.1a)        i N = -(i A + i B + i C ) i bus = C bus du bus dt = -α T ABC i ABC (3.1b)
In the same way, the above mathematical model can be transformed into d-q-0 coordinate by using the Park transformation (power invariant), as: d-q-0 average model of the MFCS-I obtained by the Park transformation (power invariant):

                                 u invd = α d u bus = u d = Ri d + L d di d dt -ω e L q i q u invq = α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) u inv0 = α 0 u bus = Ri 0 + L 0 di 0 dt u 0 - √ 3L aux di N dt + √ 3u in i bus = C bus du bus dt = -(α d i d + α q i q + α 0 i 0 ) (3.2a) 77/178 Replacing i 0 by -i N / √ 3, it yields                                        α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) α h u bus = - R 3 i N -( L 0 3 + L aux ) L E di N dt + u in i bus = C bus du bus dt = α h i N -(α d i d + α q i q ) i lo (3.2b)
where

• L E is an equivalent inductance which is the sum of L 0 /3 and L aux ;

According to (3.2b), we can further build the equivalent circuit for the MFCS-I, where R eq representing the motor function is seen as an equivalent load for the equivalent boost function, as shown in figure 3.2. Obviously, it can be concluded that the use of the series inductor actually increases the equivalent inductance of the equivalent boost. A bigger inductor contributes to a smaller input current ripple. Therefore, the current ripple of zero-sequence currents or neutral current can be reduced by using the series inductor.

+ -

u in C bus i N L 0 /3 i bus S v1 O + - u bus + - ¡ h u bus ¡ h i N S' v1 (¡ h ) L aux R eq i lo
                                 u invd = α d u bus = u d = Ri d + L d di d dt -ω e L q i q u invq = α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) u inv0 = α 0 u bus = Ri 0 + L 0 di 0 dt u 0 -L aux di N dt + u in i bus = C bus du bus dt = - 3 2 (α d i d + α q i q + 2α 0 i 0 ) (3.3a) Replacing i 0 by -i N /3, it further yields                                          α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) α h u bus = - R 3 i N -( L 0 3 + L aux ) L E di N dt + u in i bus = C bus du bus dt = α h i N - 3 2 (α d i d + α q i q ) i lo (3.3b)
where

• L E is an equivalent inductance which is made up of L 0 /3 and L aux ;

Steady-state characteristic

According to the equivalent circuit and by neglecting the resistance of the motor and the series inductor, the step-up ratio of the equivalent boost function is still

r 0 = u bus u in ≈ 1 α h (3.4)
According to the ZSVIPWM algorithm presented in (2.23), the actual duty cycles of the inverter can be expressed by

α X = α h + u XN 1 * u bus , X ∈ {A, B, C} (3.5)

79/178

Because the limit of actual duty cycles is between 0 and 1, we obtain

0 ≤ α X = α h + u XN 1 * u bus ≤ 1 ∀X ∈ {A, B, C} (3.6)
By simplifying (3.6), the allowable u * XN 1 can be expressed as

-α h u bus ≤ u XN 1 * ≤ (1 -α h )u bus ∀X ∈ {A, B, C} (3.7)
By substituting (3.4) into (3.7), we obtain

-u in ≤ u XN 1 * ≤ 1 -α h α h u in ∀X ∈ {A, B, C} (3.8)
According to (3.8), it can be seen that u XN 1 * is bounded. The lower boundary is -u in and the upper boundary is u in (1 -α h )/α h . Generally speaking, the ideal u XN 1 * is a standard sinusoidal signal. Thus, the amplitude (u XN 1amp ) of u XN 1 * is restricted by the smaller one between u in and u in (1 -α h )/α h . For the case of 0 < α h ≤ 0.5, u XN 1amp is equal to u in . In the case of 0.5

< α h ≤ 1, u XN 1amp equals to u in (1 -α h )/α h .
Therefore, the DCSUR of the MFCS-I can be evaluated by

r 1 = u XN 1amp u in =        1, 0 < α h ≤ 0.5 1 -α h α h , 0.5 < α h ≤ 1 (3.9)
To evaluate the effective step-up ratio of the MFCS-I, the graphs of r 0 and r 1 with respect to α h are drawn in figure 3.3. In Region 1, the step-up ratio r 0 reduces with the rise of α h while the DCSUR r 1 is always maintained at 1. This indicates that, although u bus can be larger than double u in , the reachable AC voltage for the motor is limited. Therefore, the MFCS-I is unexpected to be operated within Region 1 since the boosted DC-bus voltage is unhelpful in expanding the operating range of the motor. On the contrary, the MFCS-I is desired to be operated in Region 2 where the boosted DC-bus voltage can improve the DCSUR. At the point 'F' (α h = 0.5), the step-up ratio is 2 and the DCSUR is 1.

Consequently, although the MFCS-I can solve the issue of bigger current ripples by adding a series inductor, the DCSUR is not improved compared to the basic MFCS.

Series inductor selection

As discussed in Chapter 2, the bigger current ripples of the basic MFCS is mainly caused by the high-frequency component (20kHC), i.e. the ripple in the neutral current. Therefore, it is obvious that the neutral current ripples can be reduced by increasing the value of the filtering inductor of the equivalent boost converter. However, how to choose an appropriate series inductor is a big issue. For a real DC/DC boost converter, the inductor design is usually based on the desired current and voltage ripples at nominal conditions [START_REF] Jun | Analysis of inductor-current of boost dc-dc converters and its design for intrinsic safety[END_REF][85] with a constant duty cycle. However, in the MFCS-I, the duty cycles of the inverter are not constant when the motor speed is not zero. Therefore, this method is not applicable for the MFCS-I. In the following, we will introduce a simple method to select the series inductor based on the concept of a re-floating NP.

According to the analysis in Section 2.3.4, we know that the clamped NP results in the reduction of the number of phase-voltage levels in one switching period. This directly results in the increase of phase-current ripples. By comparison, the NP of a conventional drive is floating.

As a result, its phase-current ripple is smaller. Therefore, if the MFCS-I has a similar NP with the conventional drive, it can be considered that its phase-current ripple is acceptable. Thus, we take the floating NP of the conventional drive as a criterion, the objective of designing L aux is to recreate a similar NP with that of the conventional drive.

In order to analyze the transient voltage of the NP in the MFCS-I, the switching model is required, which can be built as

     u AO u BO u CO      =      S A S B S C      u bus =      u AN u BN u CN      +      u N O u N O u N O      =      u AN u BN u CN      +      -L aux di N dt + u in -L aux di N dt + u in -L aux di N dt + u in      (3.10)
By adding the three terms of (3.10), the zero-sequence switching model of the MFCS-I is given by

u in -S 0 u bus = R 3 i N + ( L 0 3 + L aux ) di N dt (3.11)

81/178

where S 0 = (S A + S B + S C )/3.

By assuming that u bus is regulated at double u in , then by neglecting the resistance term (R/3), the derivative term ( di N dt ) of (3.11) approximates to di N dt ≈ (0.5 -S 0 )u bus L 0 /3 + L aux (3.12) In this case, the transient voltage of the NP (u N O ) can be formulated by

u N O = (u AO + u BO + u CO ) 3 - (u AN + u BN + u CN ) 3 = (u AO + u BO + u CO ) 3 + R 3 i N + L 0 3 di N dt ≈ (u AO + u BO + u CO ) 3 + L 0 3 di N dt ≈ (S A + S B + S C ) 3 u bus u N O_c + L 0 /3 L 0 /3 + L aux (0.5 -S 0 )u bus u bias (3.13)
According to (3.13), it can be seen that u N O consists of two terms, one is the NP voltage (u N O_c ) of a conventional drive, the other is named as the voltage bias (u bias ). Due to the voltage bias, the NP in the MFCS-I is able to be floating again. Notably, when u bias approximates 0 with a large L aux , the transient voltage of the NP in the MFCS-I is the same as u N O_c .

Furthermore, the actual phase-voltages of the MFCS-I can be formulated by

     u AN u BN u CN      =      u AO u BO u CO      -      u N O u N O u N O      = u bus 3      2 -1 -1 -1 2 -1 -1 -1 2           S A S B S C      u ABC_c -      1 1 1      L 0 /3 L 0 /3 + L aux (0.5 -S 0 )u bus u bias (3.14)
According to (3.14), we can see that the actual phase-voltages also include two terms, one is the phase-voltage (u ABC_c ) of the conventional drive, and the other is the voltage bias (u bias ). In the same way, if u bias approximates 0, the actual phase-voltages of the MFCS-I are the same as u ABC_c . The detailed comparisons of voltage and current curves of the MFCS-I and a conventional drive are shown in figure 3.4, where the switching states S 000 , S 100 , S 110 and S 111 are used during the switching period. For the sake of analysis, the subscript _mI is given to denote a variable in the MFCS-I.

L 0 /3 L 0 /3 + L aux u bus 2 (3.15)
Under switching states S 100 and S 110 , the voltage difference is

L 0 /3 L 0 /3 + L aux u bus 6 (3.16)

83/178

If we consider a similar NP with the conventional drive as an evaluation criteria to choose the series inductor, we hope that the bigger voltage difference presented in (3.15) is small enough.

For the sake of analysis, an assessment index (R f lo ) is defined to judge a similar NP, as

R f lo = L 0 /3 L 0 /3 + L aux 1 2 = 0.5 1 + 3a L (3.17)
where

• a L = L aux L 0
represents the rate of the auxiliary L aux compared to L 0 .

The evolution of R f lo with respect to a L is illustrated in figure 3.5. It can be seen that R f lo can be effectively decreased when a L is increased from 0 to 5. When a L = 5, R f lo is 0.03.

This implies that if L aux is chosen as 5L 0 , the voltage difference of phase-voltages between the MFCS-I and a conventional drive is only 0.03u bus . Roughly speaking, it is small enough and can be ignored. In this case, the NP can be considered as a similar NP with that of the conventional drive. If a L is larger than 5, the improved effect is not significant. In turn, too bigger auxiliary will increase the cost and volume of the whole system. In practice, the selection of a L is a trade-off between the current ripples and the cost.

Improvement 2: using an auxiliary leg (MFCS-II)

Although the MFCS-I can suppress the current ripples by employing a series inductor, the effective step-up ratio of the equivalent boost function is still 2 due to the limited control DOF provided by the traction inverter. In order to improve the effective step-up ratio, another improvement named as MFCS-II is proposed in this section, as shown in figure 3.6. Compared to the MFCS-I, an auxiliary inverter leg is employed. The negative pole of the DC-source source is connected with the NP of the motor. The positive pole of the DC-source is connected to an auxiliary leg (fourth leg) through the series inductor.

+ -

u bus + -u in i N i bus o S A S B S C S' A S' B S' C A B C i A i B i C N motor C bus L aux S F S' F (α F ) (α A ) (α B ) (α C )

Average model and equivalent circuit

For the fourth leg, the duty cycle of the upper switch is defined as α F . By using the same modeling method, the average voltage and current relationships of the MFCS-II are given by

     u AO u BO u CO      u invABC =      α A α B α C      α ABC u bus =      u AN u BN u CN      u ABC +      -u in + L aux di N dt + α F u bus -u in + L aux di N dt + α F u bus -u in + L aux di N dt + α F u bus      (3.18a)        i N = i A + i B + i C i bus = C bus du bus dt = -α T ABC i ABC + α F i N (3.18b)
In the same way, the above mathematical model can be transformed into d-q-0 coordinate by using the Park transformation (power invariant), as: 85/178 d-q-0 average model of the MFCS-II obtained by the Park transformation (power invariant): voltage equation:

                                 u invd = α d u bus = u d = Ri d + L d di d dt -ω e L q i q u invq = α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) u inv0 = α 0 u bus = Ri 0 + L 0 di 0 dt u 0 - √ 3u in + √ 3L aux di N dt + √ 3α F u bus i bus = C bus du bus dt = -(α d i d + α q i q + α 0 i 0 ) + α F i N (3.19a) Replacing i 0 by i N / √ 3, it further yields                                        α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) (α F -α h )u bus = - R 3 i N -( L 0 3 + L aux ) L E di N dt + u in i bus = C bus du bus dt = (α F -α h )i N -(α d i d + α q i q ) i lo (3.
                                 u invd = α d u bus = u d = Ri d + L d di d dt -ω e L q i q u invq = α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) u inv0 = α 0 u bus = Ri 0 + L 0 di 0 dt u 0 -u in + L aux di N dt + α F u bus i bus = C bus du bus dt = - 3 2 (α d i d + α q i q + 2α 0 i 0 ) + α F i N (3.20a)
Replacing i 0 by i N /3, it further yields 86/178

                                         α d u bus = u d = Ri d + L d di d dt -ω e L q i q α q u bus = u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) (α F -α h )u bus = - R 3 i N -( L 0 3 + L aux ) L E di N dt + u in i bus = C bus du bus dt = (α F -α h )i N - 3 2 (α d i d + α q i q ) i lo (3.20b)
According to (3.19b), the equivalent circuit of the MFCS-II is shown in figure 3.7. By comparison, we can find that the equivalent duty cycle of the equivalent boost in the MFCS-II becomes (α F -α h ). In other words, the MFCS-II has two control DOFs to regulate the DC-bus voltage. One is the mean duty cycle (α h ) of the traction inverter and the other is the duty cycle α F of the fourth leg.

+ -

u in C bus i N L 0 /3 i bus S v1 O + - u bus + - ¡ h i N S' v1 (¡ F -¡ h ) L aux R eq (¡ F -¡ h )u bus i lo

Steady-state characteristic

According to the equivalent circuit shown in figure 3.7 and by neglecting the resistance of the motor and of the series inductor, the step-up ratio (r 0 ) of the equivalent boost can be formulated by

r 0 = u bus u in ≈ 1 α F -α h (3.21)
In the inverter side, the average terminal voltage of inverter's legs is between 0 and u bus when the duty cycle is respectively set as 0 and 1. For the fourth leg, its terminal voltage can be expressed as α F u bus . Then, the average voltage of the NP u N O can be expressed as (-u in + α F u bus ) by neglecting the voltage drop across the series inductor. Therefore, the average 87/178 phase-voltage of the motor can be expressed by

0 -u N O ≤ u XN ≤ u bus -u N O , X ∈ (A, B, C) ⇓ -(-u in + α F u bus ) ≤ u XN ≤ u bus -(-u in + α F u bus ) (3.22)
If we neglect the superimposed zero-sequence voltage, u XN ≈ u XN 1 . Then, by substituting (3.21) into (3.22), we obtain

- α h α F -α h u in ≤ u XN 1 ≤ 1 -α h α F -α h u in (3.23)
Generally speaking, the fundamental phase-voltage of the motor is desired to be sinusoidal and symmetric. To this end, a simple way is to set α h at 0.5 [1] so that u XN 1 can be symmetric.

In this case, α F becomes the only control DOF for the equivalent boost. Therefore, r 0 and r 1 of the MFCS-II can be evaluated by

         r 0 = u bus u in = 1 α F -0.5 r 1 = u XN 1amp u in = 0.5 α F -0.5
(3.24) [1] α h can also be variable under different operating conditions. In this case, a control allocation strategy between α h and α F is necessary.

88/178

The curves of r 0 and r 1 with respect to α F are shown in figure 3.8. In Region 1, both r 0 and r 1 are positive while they are negative in Region 2. If u in is positive, the MFCS-II is desired to be operated in Region 1 to obtain a positive DC-bus voltage. Thus, the available range of α F is (0.5-1). When α F is 1, r 0 is 2 and r 1 is equal to 1. This special case actually corresponds to the MFCS-I. With the decrease of α F , both r 0 and r 1 increase. When α F is close to 0.5, both r 0 and r 1 can reach high values. This indicates that the MFCS-II is able to achieve a highly effective step-up ratio when the duty cycle of the fourth leg is near 0.5. If u in is negative (connected reversely), the MFCS-II is desired to be operated in Region 2. In addition, if u in is a single-phase AC power source, the MFCS-II can be operated in both Region 1 and Region 2.

Universal closed-loop control strategy

According to the aforementioned analysis, we know that the MFCSs (including the MFCS-I and MFCS-II) are hybrid power conversion systems, which conduct not only DC/AC but also DC/DC conversions simultaneously thanks to the motor function and the equivalent DC/DC boost function. Meanwhile, we know that both the MFCS-I and the MFCS-II have the similar equivalent circuit. The only difference between them is the control DOF of their boost functions.

In the MFCS-I, α h is the control DOF for the equivalent boost. In the MFCS-II, α h is set at 0. For the equivalent boost, it is a typical nonlinear system. Therefore, we adopt a differential flatness-based nonlinear controller. In the following, the detailed design for all controllers will be detailed.

Design of FOC

The classic FOC is a cascaded control framework for AC machines where the current-loop is the inner-loop and the speed-loop is the outer-loop.

PI control for d-q current-loops

For the inner-loop, two PI controllers are utilized to regulate the d, q-axis currents. The detailed design process is presented as follows:

89/178

According to the modeling analysis in chapter 2, the d-q voltage equation of the PMSM can be expressed by

         u d = Ri d + L d di d dt -ω e L q i q u q = Ri q + L q di q dt + ω e (L d i d + Φ m ) (3.25)
By using the state feedback decoupling approach, new control variables v d and v q are given as

     u d = v d -ω e L q i q u q = v q + ω e (L d i d + Φ m ) (3.26)
After the decoupling, the d-axis and q-axis current-loops are simplified as

         L d di d dt = v d -Ri d L q di q dt = v q -Ri q (3.27)
Then, by applying Laplace transformation to (3.27), both d-axis and q-axis current-loops can be expressed by a first-order system plant G k (s), as

G k (s) = 1 L k s + R (3.28)
where k can be replaced by d or q.

In this case, a linear control technique, such as the PI controller can be used. The transfer function of a PI controller P I k (s) is given by

P I k (s) = k pk 1 + 1 T ik s (3.29)
where k pk is the proportional gain, T ik is the integral time coefficient.

When the PI controller is applied, the block diagram of the closed-loop system is shown in figure 3.9 where the inverter non-linearity and the PWM delay effect are neglected. According to pole-zero cancellation method, it is easy to calculate the control gains k pk and T ik .

State feedback control with an integral action for the speed-loop

Assuming that the current-loops have been correctly synthesized with respect to frequency separation principle, the closed-loop current dynamics is faster than the desired mechanical dynamics. Thus, the speed-loop can be designed without considering the current-loops. Considering a motor without a load, the mechanical dynamic can be expressed as

ωm = - B m J m ω m + K Φ J m i * q (3.30)
where K Φ = 1.5pΦ m , i * q is the reference of q-axis current.

The control objective is to ensure that the actual speed ω m is equal to the speed reference ω * m . To guarantee a zero steady-state error, an integral action is necessary, as εω = ω * m -ω m . By adding the integral action, (3.30) can be extended as

  ωm εω   =   -Bm Jm 0 -1 0     ω m ε ω   +   K Φ Jm 0   i * q +   0 1   ω * m (3.31)
For such a system, a control law is given as

i * q = -K ω ω m -K ω_I ε ω (3.32)
where K ω and K ω_I are two feedback gains.

By substituting (3.32) into (3.30), the closed-loop system can be expressed as

  ωm εω   =   -Bm+K Φ Kω Jm -K Φ K ω_I Jm -1 0   Aext   ω m ε ω   +   0 1   ω * m (3.33)
where A ext is the extended matrix.

The closed-loop dynamic depends on the eigenvalues of A ext , one has eig

    -Bm+K Φ Kω Jm -K Φ K ω_I Jm -1 0     = det   sI -   -Bm+K Φ Kω Jm -K Φ K ω_I Jm -1 0     (3.34)
It leads to a characteristic equation, as

P (s) = s 2 + B m + K Φ K ω J m s - K Φ K ω_I J m (3.35)
Obviously, (3.35) represents a second-order system. Therefore, K ω and K ω_I can be calculated by the method of pole assignments.

91/178

Differential flatness-based control for the equivalent boost

Boost converter is a typical nonlinear system in power electronics fields. Usually, the small-signal model is used to design the controller by linearizing the large-signal model at a specific operating point. However, in the MFCS-I or MFCS-II, the motor functions as an equivalent load of the equivalent boost converter. It has a wide operating range, which may change from the null power to the nominal power. Therefore, the linearized small-signal model is inapplicable. To control such a system, a nonlinear controller is required. For example, the sliding mode control [START_REF] Wai | Design of voltage tracking control for DC-dc boost converter via total sliding-mode technique[END_REF][87], passivity-based control [START_REF] Son | Complementary PID controller to passivity-based nonlinear control of boost converters with inductor resistance[END_REF][89] and differential flatness-based control [START_REF] Thounthong | Control of a three-level boost converter based on a differential flatness approach for fuel cell vehicle applications[END_REF][91] can be adopted. In this research, we choose the differential flatness-based control due to its simplicity.

Brief introduction to differential flatness theory

The differential flatness theory was proposed by Fliess et al. using the formalism of differential algebra [START_REF] Fliess | Flatness and defect of non-linear systems: Introductory theory and examples[END_REF]. For a nonlinear system, it can be expressed with the following form:

ẋ(t) = f (x(t), u(t)) (3.36)
where

x(t) ∈ R n , u(t) ∈ R m and (m ≤ n) ∈ N.
This system is said to be differential flat if one can find a variable, called as the flat output.

An output y is flat if the following smooth functions exist [START_REF] Rigatos | Nonlinear control and filtering using differential flatness approaches: applications to electromechanical systems[END_REF] h :

R n × (R m ) p+1 → R m η : (R m ) p → R n ϑ : (R m ) p+1 → R m such that y = h(x, u, u, ..., u (p) ) x = η(y, ẏ, ..., y (p-1) ) u = ϑ(y, ẏ, ..., y (p) ) (3.37)
where p is the number of derivatives.

This implies that the state vector x of a differential flat system and its input vector u can be represented by a function of the flat output and its finite derivatives. If the output vector y can be proved to be flat, the design of trajectory tracking control for each desired output y traj become straightforward [START_REF] Thounthong | Nonlinear differential flatness-based speed/torque control with state-observers of permanent magnet synchronous motor drives[END_REF]. A linear feedback control law can be adopted to ensure that the tracking errors e = (y -y traj ) asymptotically vanish, as

Parameterization of the differential flatness-based controller

By neglecting the resistance of the motor and the series inductor, the mathematical model of the equivalent boost function of the MFCS-I and MFCS-II can be built as

         iN = - u bus L E α E + u in L E ubus = i N C bus α E - u bus R eq C bus (3.39)
In (3.39), although we do not know the value of the equivalent load R eq , it can be calculated by u bus /i lo , where i lo is equal to 3 2 (α d i d + α q i q ) as shown in (3.3b) and (3.20b). Assuming that control dynamics of i d and i q are much faster than the dynamic of u bus , thus, the average model of the equivalent boost can be rewritten by

         iN = - u bus L E α E + u in L E ubus = i N C bus α E - i lo C bus (3.40)
For the equivalent boost of the MFCS-I, the control variable is u := α E = α h . For the equivalent boost in the MFCS-II, the control variable is u := α E = α F -0.5. The state variables of (3.40) are i N and u bus . The flat output candidate is chosen as the energy E which is stored in the DC-bus capacitor and the equivalent inductor L E [START_REF] Gil-Antonio | Trajectory tracking control for a boost converter based on the differential flatness property[END_REF]:

y := E = 1 2 (L E i 2 N + C bus u 2 bus ) (3.41)

93/178

With this candidate, we can find that: E,Ė)

                               i N = u in C bus Ė i 2 lo + u 2 in C 2 bus Ė2 i 4 lo -(2L E + 2C bus u 2 in i 2 lo )( Ė2 C bus 2i 2 lo -E) (L E + C bus u 2 in i 2 lo ) = η 1 (E, Ė) u bus = u in η 1 (E, Ė) - Ė i lo = η 2 (
α E = u 2 in L E + u in η 1 (E, Ė) + i 2 lo C bus -i lo η 2 (E, Ė) - Ë η 2 (E, Ė)u in L E + η 1 (E, Ė)i lo C bus = ϑ(E, Ė, Ë) (3.42)
Obviously, all the states and input can be expressed by the energy E and its first-order and second-order derivatives. Thus, the equivalent boost function can be considered as a flat system with y = E as the flat output.

Then, to track a predefined energy trajectory E traj , a feedback control law is given as follows, where an integral action is also added to guarantee a zero steady-state error. The error will asymptotically converge to zero if the control parameters K dE , K pE and K iE are chosen such that all roots of the characteristic polynomial of the closed-loop system are located in the left half plane. The associated characteristic polynomial can be written as

Ë = Ëtraj -K dE ( Ė -Ėtraj ) -K pE (E -E traj ) -K iE t 0 (E(τ ) -E traj (τ ))dτ (3.
p(s) = s 3 + K dE s 2 + K pE s + K iE ⇓ p(s) = (s + a1)(s 2 + 2ζ E ω E s + ω 2 E ) (3.45)
where ω E and ζ E are respectively the desired damping ratio and natural frequency; a1 > 0,

ω E > 0 and ζ E > 0.
Therefore, K dE , K pE and K iE can be determined by

K dE = 2ζ E ω E + a1, K pE = 2a1ζ E ω E + ω 2 E , K iE = a1ω 2 E (3.46)
Finally, the control law is synthesized according to the inverse dynamic expression presented in (3.42) (α E ).

94/178

Trajectory planing

Trajectory planning is an important step in the realization of a differential flatness-based controller. Since all the states and input are written as functions of the chosen flat output y, the flat output trajectory y traj determines the trajectories of states and input. So, a well-known waveform can be considered so that all the transient behaviors can be predicted where physical constraints can be taken into account in the trajectory planning. To limit the derivative terms in (3.43), a low-pass second-order filter can be used to produce a desired trajectory for the flat output E traj (energy). The transfer function of the second-order filter is given by

E traj E * = ω 2 Etraj s 2 + 2ζ Etraj ω Etraj s + ω 2 Etraj (3.47)
where ζ Etraj and ω Etraj are respectively the desired damping ratio and natural frequency of the second-order filter, which determine the dynamic of the desired energy trajectory; E * is the predefined energy reference, as

E * = 1 2 (L E i 2 N + C bus (u * bus ) 2 ) (3.48)
Note that E * is related to the predefined reference voltage.

Control framework

Finally, the block diagram of the universal closed-loop control strategy for the MFCS-I and MFCS-II is shown in figure 3.11. The speed controller generates the q-axis current reference i * q under motoring conditions. In regenerative braking conditions, i * q is directly given by the desired braking force. The d-axis current reference i * d is set as 0 for a surface-mounted PMSM. Then, the two current references (i * d and i * q ) are given into the current-loop PI controllers, which output the d-axis voltage reference u * d and q-axis voltage reference u * q . By using inverse Park transformation, the fundamental voltage reference u * AN 1 , u * BN 1 and u * CN 1 are obtained.

The differential flatness-based controller is adopted in the zero-sequence control-loop. The energy reference E * is calculated by the voltage reference u * bus and the measured neutral current i N . Then, the desired energy trajectory E traj is obtained by the trajectory planning process presented in (3.47). By using the error feedback control presented in (3.43), the second-order derivative of the energy is obtained. Substituting the second-order derivative of the energy into the inverse dynamic presented in (3.42), the general control variable α E in obtained. For the MFCS-I, α E is directly used to generated the real duty cycles α A , α B and α C based on the ZSVIPWM algorithm presented in (2.23). For the MFCS-II, α h is set as 0.5 in the ZSVIPWM algorithm. Essentially, the classic SPWM is used on the inverter side. Therefore, α E is used to generate the duty cycle of the fourth leg in the MFCS-II, as α F = α E + 0.5.

95/178

Experimental results

The effectiveness of the MFCS-I and MFCS-II is verified on a 1.2 kW PMSM drive platform, as shown in figure 3.12. The PMSM is customized by a chinese company with an available NP. The traction inverter is constructed with an intelligent power module (ON Semiconductor STK581U3C2D-E). A 1.5 kW PMSM is used as the load motor, which is driven by an industrial inverter (LEROY SOMER). The load motor can be operated in either torque-controlled or speed-controlled modes. All the control algorithms are implemented on a dSPACE DS1202 MicrolabBox rapid prototyping system. The main parameters of the 1.2 kW test-bench are presented in table 3.1.

Current ripples analysis of the MFCS-I

To verify the effect of current ripples suppression in the MFCS-I by using a series inductor, standstill and motoring conditions. The basic MFCS is also compared.

In standstill condition

Thanks to the equivalent boost function, the DC-bus voltage of the basic MFCS and MFCS-I can be boosted to double the DC-source voltage, i.e. 360 V. In standstill conditions, the motor speed is null. Therefore, the actual duty cycles of the inverter in the three phases are perfectly consistent. In this case, the transient currents of three phases are also consistent.

Therefore, we only illustrate the current wave of phase-A in the following analysis. In steady states, the experimental results are shown in figure 3.13, which are related to the basic MFCS, the MFCS-I with a 1.3 mH series inductor and the MFCS-I with a 13 mH series inductor.

97/178

For the basic MFCS (see figure 3.13(a)), it can be observed that the neutral voltage (u N O ) is exactly equal to the DC-source voltage (180 V) because the NP is clamped. The phase-voltage (u AN ) has two voltage-levels (180 V and -180 V). As analyzed before, in a conventional drive, the phase voltage is 0 V under the two zero voltage vectors if the motor speed is null. Therefore, the phase-current ripple is 0 in a conventional drive if the motor speed is 0. However, since u AN is not 0 V in the basic MFCS, this voltage applied on each phase will produce current. As a result, we can clearly see the current ripples of i A and i N in figure 3.13(a), whose values are 5 A and 15 A respectively. Obviously, such large ripples are adverse to the motor and will result in big iron losses. Therefore, we can conclude that the zero-sequence inductance (L 0 = L σ ) or leakage inductance of the motor is insufficient to suppress the current ripple in the equivalent boost function.

By adopting a 1.3 mH series inductor, the value of a L can be calculated as 1.3/2.4 = 0.54.

In other words, the inductance of the series inductor is equal to 0.54L 0 . In this case, the value

(a) (b) (c) u AN u NO i A i N u AN u NO i A i N u AN u NO i A i N [500 V/div] [500 V/div]

98/178

of R f lo is further evaluated as 0.19. This means that the voltage bias (see (3.15)) is 0.19u bus ≈ 68 V under the two zero voltage vectors. From figure 3.13(b), we can see that the u AN still has two voltage-levels. However, compared to that in figure 3.13(a), the voltage amplitude is reduced to around 68 V. This verifies the correctness of the theoretical analysis. Because u AN is still not 0 V, there is still a current ripple in i A . However, the ripple of i A is decreased to 2 A compared to that in the basic MFCS. Roughly speaking, the 1.3 mH inductor is still insufficient to suppress the current ripple into an acceptable range. By using a 13 mH series inductor, the value of a L is calculated as 13/2.4 = 5.4. In other words, the inductance of the series inductor is equal to 5.4L 0 . In this case, the value of R f lo is 0.03. This means that the voltage bias (see (3.15)) is 0.03u bus ≈ 11 V under the two zero voltage vectors. From figure 3.13(c), it is obvious that the voltage amplitude of u AN is close to 0. This implies that the NP is almost similar to that of a conventional drive. As a result, the ripples of i A and i N are much smaller and acceptable in practice.

In motoring condition

When the motor is operated at a 1000 rpm speed and a 2 N•m load, the phase-A voltages, neutral voltages, phase-A currents and neutral currents of the basic MFCS, the MFCS-I with a 1.3 mH series inductor and the MFCS-I with a 13 mH series inductor are further illustrated in figure 3.14. For the basic MFCS, u AN only has two voltage-levels (around 180 and -180 V) due to the clamped NP, as shown in figure 3.14(a) and (b). Apparently, both i A and i N have big ripples. Meanwhile, i A is severely distorted whose shape is not a perfect sinusoidal wave. Therefore, we can conclude that the basic MFCS is unpractical in practice.

By using a 1.3 mH series inductor, u AN is not limited between 180 and -180 V. Thanks to the series inductor, u AN and u N O are able to have four voltage levels within one switching period. This shows good agreements with the theoretical analysis. However, the ripple of i A and i N are still relatively large. Meanwhile, the shape of i A is still distorted. By using a 13 mH series inductor, the ripples of i A and i N are smaller, as shown in figure 3.14(e). Meanwhile, the shape of i A is closer to a sinusoidal wave. Therefore, it can be concluded that, the basic MFCS concept can not be directly used in practice because the zero-sequence inductance of the motor is relatively small which is insufficient to act as a suitable filtering inductor in the equivalent boost function. The MFCS-I structure can suppress the current ripple into an acceptable range by employing a series inductor whose inductance is five times the zero-sequence inductance of the motor. For the MFCS-I, its DC-bus voltage is equal to the source-voltage (180 V) before starting its equivalent boost function. Once the control instruction of the equivalent boost is executed, u bus is boosted to 360 V rapidly. At the beginning, the neutral current (i N ) is increased to 6.2 A to charge the DC-bus capacitor. This phenomenon demonstrates the existence of the equivalent boost function even if there is no additional chopper used in the MFCS-I.

99/178

u AN u NO i A i N u AN u NO i A i N u AN u NO i A i N Zoom in (a) (b) (c) (d) (e) (f)
A B o + - u bus C S A S B S C S' A S' B S' C + - u in i A i B i C i N i bus A B o + - u bus N C S A S B S C S' A S' B S' C PMSM C bus i A i B i C i bus + -
In terms of the two-stage converter-based drive, we can observe the similar starting performance by using the same controller. In the initial stage of the starting process, the inductor current i Laux is increased to 6.5 A. This value is larger than that of the MFCS-I because the zero-sequence inductance of the motor and L aux jointly construct the filtering inductance of the equivalent boost in the MFCS-I.

Starting performance of the motor function

Since the DC-bus voltage is maintained at 360 V in both drives, the motor is ready to start. Figure 3.17 shows a starting process of the motor in which a speed step (from 0 to 1000 rpm) is given at 2.5 s. In the MFCS-I, it can be seen that the actual speed can reach the desired value within 0.5 s. Then, it is maintained at 1000 rpm stably. Although its DC-bus voltage shows a 3 V dip when the motor accelerates instantly, it can return to the stable state quickly due to the control for the equivalent boost function. Figure 3.17 zoomed-in views of phase-currents and neutral current. Interestingly, the phase-currents in the MFCS-I are no longer the standard sinusoidal waveforms. They present an offset which is caused by the zero-sequence current (or i N ). Figure 3.17(b) and (d) depict the similar starting process of the two-stage converter-based drive. By comparison, it can be concluded the starting performance of both drives is similar. It needs to point out that the phase-currents in the two-stage converter-based drive are symmetrical because its NP is not used and there is no zero-sequence current flowing into the motor.

Steady-state performance in motoring conditions

To study the influence on phase-current quality caused by the connected NP in the MFCS-I, the steady-state performance of both drives is compared where the motor is operated at the nominal condition (3000 rpm with 4 N•m load). In steady states, the phase-currents, neutral current i N and the inductor current i Laux are displayed in figure 3.18. The corresponding FFT analysis of phase-currents is further presented below. From figure 3.18(a), it can be seen 103/178 that the phase-current offset is around -2.45 A in the MFCS-I, which is equal to -1/3 of the neutral current (the DC RMS of i N =7.35 A). In the two-stage converter-based drive, there is no phase-current offset due to the unconnected NP, as shown in figure 3.18(b).

In terms of the phase-current FFT analysis, the proportion of zero-sequence component is 38% in the MFCS-I while it is almost null in the two-stage converter-based drive. This is reasonable because the DC/DC conversion process is achieved through the motor in the MFCS-I while the DC/DC conversion process is independent to the motor in the two-stage converter-based drive. The second-order, third-order and fourth-order harmonics in the MFCS-I are bigger than those in the two-stage converter-based drive. However, it needs to point out that the even and third-order harmonics do not affect the torque performance due to the symmetrical structure in three phases. Interestingly, the fifth-order harmonic is smaller (0.1% to 0.8%) in the MFCS-I. The phase-current THD is 6.10% in the MFCS-I while it is 5.36% in the two-stage Therefore, it is reasonable to conclude that the connection of motor's NP in the MFCS-I does not affect the dynamic performance of the motor. In addition, the DC-bus voltage in both drives can be well maintained during the dynamic test. Although it joggles slightly (around 2 V) when the motor's power changes instantly, it can recover to 360 V quickly. This verifies the feasibility of the proposed control strategy in which the motor function and the equivalent DC/DC boost function can be controlled independently.

Regenerative braking performance

To emulate a regenerative braking scenario, the load motor is operated in the speedcontrolled mode, which drags the tested PMSM to decelerate from the nominal speed to zero 105/178 In the initial stage, both i N and i Laux are negative, which are equal to -5.75 A and -5.64 A respectively. This implies that both the MFCS-I and the two-stage converter-based drive can recycle the braking energy to charger the DC-source. With the decrease of the motor's speed, the i N and i Laux decrease. When the motor is completely stopped, i N and i Laux become positive (around 0.5 A) again, in order to maintain the DC-bus voltage. Therefore, it can be concluded that the MFCS-I can realize the similar regenerative braking performance even if it does not use a real directional DC/DC converter.

In steady states (3000 rpm and -4 N•m torque), the phase-currents (see figure 3.20(c))

of the MFCS-I show a positive offset, which is still equal to -1/3 of the input current (the DC RMS of i N is 5.75 A). In terms of the phase-current FFT analysis (see figure 3.20(e)), the proportion of the zero-sequence component is 32%. The second-order, third-order and fourth-order harmonics are still larger while the fifth-order harmonic is smaller (0.3% to 0.6%).

By comparing with the two-stage converter-based drive, the phase-current THD in the MFCS-I 106/178 is degraded by 8.5% (6.51% to 6.00%). To evaluate the charging efficiency, defining the index Consequently, by combining with above experimental results, the characteristics of the MFCS-I compared to the two-stage converter-based drive are summarized in Table 3.3. It can be seen that the MFCS-I only requires 6 switching devices while the two-stage converter-based drive needs 8 switching devices. We can say that the cost of the MFCS-I is decreased by 25%.

Regarding the current THD of the MFCS-I, it is degraded by (0.51% to 0.74%) in the motoring pattern and the regenerative braking pattern, respectively. In terms of the system efficiency of the MFCS-I, it is improved by (3.1% to 3.4%) in the motoring pattern and the regenerative braking pattern, respectively.

Performance of the MFCS-II

Compared to the MFCS-I, the MFCS-II is able to have an equivalent boost function, which has a higher effective step-up ratio thanks to the new DOF provided by the fourth leg according to the aforementioned analysis. To verify the higher effective step-up ratio in the MFCS-II, a 40 V DC-source is used. Experimental validations are still carried out on the 1.2 kW PMSM test-bench, which is configured in the form of the MFCS-II. We set α h = 0.5 throughout.

Therefore, the equivalent duty cycle of the equivalent boost function is α E = α F -0.5.

Relationship between u bus and α F

According to the theoretical analysis, the DC-bus voltage u bus approximates (u in /α E ) by neglecting the resistance of the motor in steady states. In open-loop modes, the relationship between u bus and α F can be easily tested. To this end, we decrease the value of α F from 1 to 0.6 with a 0.05 interval gradually. Then, the value of u bus is recorded, as shown in figure 3.21.

At first, α F is equal to 1. In this case, the equivalent duty cycle α E of the equivalent boost 108/178 function is equal to 0.5. Therefore, the theoretical u bus will be 80 V. From figure 3.21, we can see that u bus is equal to 80 V in the initial stage. Actually, the MFCS-II in this special case is similar to the MFCS-I because α F = 1 means the DOF of the fourth leg is not utilized. Then, with the decrease of α F , u bus increases from 80 to 381 V. Theoretically, u bus can reach 400 V when α F = 0.6. The difference between the theoretical analysis and the experimental result can be explained by the effect of dead time and the voltage drop across the switching devices and the motor's resistance.

u bus [100 V/div] ¡ F [0.

Steady-state performance

The steady-state performance of the MFCS-II is presented in figure 3.22, where the equivalent boost is controlled in a closed-loop way. In figure 3.22(a), the motor is operated at 1000 rpm. In figure 3.22(b), the motor is operated at 2000 rpm.

First, it can be observed that the DC-bus voltage u bus can be regulated at 360 V stably even if the DC-source voltage u in is only 40 V. In this case, the step-up ratio of the equivalent boost function is 9. With the 40 V DC-source, the motor can not only be operated in the low-speed condition (1000 rpm) but also the high-speed condition (2000 rpm). Obviously, the effective step-up ratio of the MFCS-II is larger than that of the MFCS-I.

However, it needs to note that, if the DC-source voltage is too small, the neutral current (input current in the MFCS-II) will be large. This means that the zero-sequence current in the motor may be larger than the amplitude of the fundamental current. For example, we can see that i A in figure 3.22(a) shows a big upward offset which is caused by the zero-sequence current. The offset is equal to 1/3 of i N . In figure 3.22(b), i A is almost completely positive.

This phenomenon happens because the zero-sequence current of the motor is bigger than the 109/178

Conclusion

To suppress the big current ripple of the basic MFCS, an improvement named as MFCS-I is proposed by using a series inductor. Based on the concept of a re-floating NP, the value of the series inductor can be easily selected. Theoretically, a series inductor with five times the zero-sequence inductance of the motor can create a similar NP with the conventional drive. To comprehensively evaluate the MFCS-I, a two-stage converter-based drive is compared, where 111/178 the same inductor L aux is used in its boost stage. Experimental results show that, without using a physical chopper, the MFCS-I is also able to boost its DC-bus voltage to double the DC-bus voltage. The starting performance of the equivalent boost is similar with the real boost in the two-stage converter-based drive. Moreover, the starting performance of the motor in the MFCS-I is also similar with that in the two-stage converter-based drive. The steady-state and dynamic performance is further compared in detail. According to the experimental results, the phase-current THD of the MFCS-I is degraded by (8.5% -13.8%) compared to that in the two-stage converter-based drive. But total efficiency of the MFCS-I is improved by (3.1% -3.4%) thanks to the reduced number of switching devices. In addition, the similar dynamic performance of the two drives also reveals that the connection of motor's NP in the MFCS-I does not affect the acceleration and deceleration ability of the motor.

To improve the effective step-up ratio, another improvement named as MFCS-II is proposed by using an auxiliary leg. The theoretical analysis indicates that both the step-up ratio of the equivalent boost and the DCSUR increase with the decrease of α F . Different from a real single-phase DC/DC boost converter which can not achieve an ultra high step-up ratio due to the limit of its duty cycle, the MFCS-II can reach an ultra high step-up ratio when α F approaches 0.5. According to the experimental validations, it can be concluded that the DC-source voltage can reach 360 V even if a 40 V DC-source is supplied. Meanwhile, the motor can still be operated at low-speed (200 rpm) and high-speed (2000 rpm) conditions. However, it needs to note that the phase-currents may be always positive if the motor power is too high.

In conclusion, the MFCS-I is more suitable for the application where the desired DC-bus voltage is less than double the DC-source voltage. If one needs an ultra high step-up ratio, the MFCS-II can be considered.

112/178

Introduction

Open-phase fault (OPF) is one of the most prevalent failures in VSI fed motor drives, which eventually results in the outage of motor windings. The fault may be caused by internal failures of stator windings, open-circuit and short-circuit failures of power electronic switches, and mechanical failures of terminal connectors. If it is not treated with remedial measures, the motor will suffer from big electromagnetic torque ripples hence induce serious mechanical vibrations. Long time operations with the fault will further lead to a secondary damage even irreparable breakdown.

As one of the key parts of EVs, the electrical drive is desired to be equipped with the ability of continuous operations in case of the appearance of an OPF. Especially, the uninterrupted torque output ability should be insured on uphill and downhill conditions in order to guarantee the personal safety of passengers. According to the literature review presented in Chapter 1, there are four typical fault-tolerant topologies (CFI, SFI, CFNI and SFNI). In terms of the two (CFNI and SFNI) who use the NP of the motor, they can tolerate an OPF which occurs no matter on the inverter side or on the machine side. Thanks to the circulation of the zero-sequence path, the motor can still generate a circular rotating magnetic field with only two active phases. However, the CFNI and SFNI always require some assistant hardwares (a pair of split capacitors, two auxiliary IGBT and a TRIAC) to achieve the fault-tolerance. These assistant devices will not only increase the cost of the whole system but also complicate the control circuit because additional control signals need to be added to the main control board.

This will indirectly affect the reliability of the drive system.

Thanks to the inherently connected NP in the MFCS-I, the path needed by the zerosequence current exists throughout. Therefore, the MFCS-I is supposed to possess the faulttolerance towards an OPF without using any auxiliary device. This chapter intends to explore the fault-tolerant potential of the MFCS-I, which is organized as follows

• In Section 1, the fault-tolerant potential of the MFCS-I is discussed. To guarantee a constant torque output and a stable average DC-bus voltage in post-fault conditions, three constraints are found. Based on the three constraints, a new current vector trajectory is proposed. The power conversion process and the post-fault output capability of the MFCS-I are analyzed.

• In Section 2, the corresponding fault-tolerant control strategy is proposed. Since the d-axis and 0-axis current references are not constant in post-fault operations, two current tracking controllers are developed.

115/178

• In Section 3 and 4, the effectiveness of the proposed fault-tolerant scheme is verified on the 52.5 W and the 1.2 kW PMSMs respectively.

Fault-tolerant potential

In the CFNI and SFNI, the path required by the zero-sequence current under post-fault conditions is provided by the fourth leg. This leg can be constructed by either a pair of DC-bus split capacitors or an auxiliary leg. To switch the circuit configuration from the pre-fault to post-fault condition, a TRIAC is also required. In the MFCS-I, the path required by the zero-sequence current exists, thanks to the inherently connected NP. Therefore, it is reasonable to expect that the MFCS-I has the fault-tolerant potential without using any extra hardware.

To realize this possibility, the key point is how to maintain a constant torque output as well as a stable DC-bus voltage under post-fault conditions.

Derivation of post-fault current trajectories

The circuit configuration of the MFCS-I under the post-fault condition (phase-A is open) is shown in figure 4.1. Since the phase-A is disconnected from the power supply, it can not be utilized to generate the rotating magnetic field. In other words, only phase-B and C are available under the post-fault condition. In the following, we are going to find three constraints, which are necessary to realize the fault-tolerant potential for the MFCS-I. For the sake of analysis, the subscript (_f ) denotes a variable which is in post-fault conditions. The subscript (_n) denotes a variable which is in healthy conditions. 116/178

+ -

u bus + - u in i N i bus o S A S B S C S' A S' B S' C A B C i A i B i C N motor C bus L aux OPF

Constraint 1: torque conservation

Generally speaking, the primary objective of fault-tolerant operations in motor drives is to preserve the stable torque output under post-fault conditions. Thus, for a surface-mounted PMSM, the post-fault q-axis current i q_f is desired to be consistent with that (i q_n ) in normal condition. Therefore, it yields the first constraint, as

i q_f = i q_n (4.1)

Constraint 2: post-fault d-q-0 current relationship

According to Clarke transformation (shown in appendix A), we obtain

     i α_f i β_f i 0_f      = 2 3           1 - 1 2 - 1 2 0 √ 3 2 - √ 3 2 1 2 1 2 1 2                i A_f i B_f i C_f      (4.2)
where i α is the α-axis current; i β is the β-axis current.

Once the OPF occurs, i A_f will be 0 A. By considering this condition, we obtain

i 0_f = -i α_f (4.3)
According to the rotation transformation (shown in appendix A) from α-β to d-q coordinates, we further obtain

     i d_f = i α_f cosθ e + i β_f sinθ e i q_f = -i α_f sinθ e + i β_f cosθ e ( 4.4)
Substituting (4.3) into (4.4) and eliminating the term of i β_f , the d-q-0 current relationship in the post-fault condition is found to be

i 0_f = i q_f sinθ e -i d_f cosθ e ( 4.5)
Therefore, (4.5) is the second constraint, which indicates that the value of i 0_f is related to i d_f , i q_f and the position θ e . In healthy conditions, i d_n = 0. i q_n and i 0_n are constant, which are determined by the load and speed of the motor, i.e. the operating power of the motor.

If both i d_f and i q_f are maintained to be consistent with i d_n and i q_n [1] , obviously, i 0_f will be a non-constant value. According to (4.5), it will be a sinusoidal function of θ e whose period [1] A common fault-tolerant strategy in the CFNI and SFNI.

117/178 is equal to the electrical period. Therefore, the neutral current i N _f under post-fault conditions will also be sinusoidal. In this case, the sinusoidal i N _f indicates that the transient input power

P in_f = u in i N _f = -3u in i 0_f
will also be sinusoidal. For the sinusoidal input power, its average value is null. This implies that the DC-source will be incapable to provide the required power to the motor. Therefore, by setting i d_f = i d_n and i q_f = i q_n , the MFCS-I is impossible to work.

Constraint 3: average power conservation

From the healthy to post-fault conditions, the output power of the motor is desired to be invariant because the torque and speed of the motor are expected to be stable. Therefore, in order to maintain the balance between the average input power and output power, the average value of i 0_f should meet:

P in_f = u in i N _f = -3u in i 0_f = P mec_n η = P in_n = -3u in i 0_n ⇓ i 0_f = i 0_n (4.6)
where the superscript x indicates the mean of a variable (x) within one electrical period; the total efficiency η and source-voltage u in are supposed invariable from the healthy to post-fault conditions.

According to (4.6), we know the third constraint, which deduces that the mean of i 0_f should be consistent with i 0_n .

Fault-tolerant d-q-0 current trajectories

Then, by combining the constraints 1 (see (4.1)), 2 (see (4.5)) and 3 (see (4.6)) simultaneously, theoretical i d_f , i q_f and i 0_f which can fulfill fault-tolerant operations are proposed as follows: From (4.7), it can be observed that the post-fault currents are associated with their previous states (i d_n , i q_n and i 0_n ). To be specific, i d_f consists of a DC component (i d_n )

                   i d_f = i d_n -2i 0_n cosθ e i q_f = i q_n i 0_f = i
and a periodic component (-2i 0_n cosθ e ). i q_f is completely the same as i q_n . i 0_f contains a frequency-doubled component (i 0_n cos2θ e ) in addition to a DC component (i 0_n ) and a periodic 118/178 component (i q_n sinθ e -i d_n cosθ e ). By using the inverse Park transformation (amplitude invariant), the post-fault phase-currents can be expressed as: where i d_n is considered as 0 for a surface-mounted PMSM; m 0 = i 0_n /i q_n is defined as the proportion between i 0_n and i q_n .

                         i A_f = 0 i B_f = -
Furthermore, i B_f and i C_f can be rewritten in the form of Fourier series expansions, as shown below

                                           i B_f = a 0 2 + 2 k=1 [a k cos(kθ e ) + b k sin(kθ e )] i C_f = a 0 2 + 2 k=1 [a k cos(kθ e ) + b k sin(kθ e )] a 0 = 3m 0 i q_n , a 1 = √ 3 2 i q_n , a 1 = - √ 3 2 i q_n , a 2 = a 2 = 3 2 m 0 i q_n , b 1 = b 1 = 3 2 i q_n , b 2 = - √ 3 2 m 0 i q_n , b 2 = √ 3 2 m 0 i q_n (4.9)
Therefore, the RMS of remaining phase-currents under the post-fault condition can be

I RM S_f = a 0 2 2 +   a 2 1 + b 2 1 √ 2   2 +   a 2 2 + b 2 2 √ 2   2 = 15m 2 0 + 6 4 • i q_n (4.10)
If an OPF occurs in phase-B or C, the post-fault currents can be derived in the same way.

For clarity, all the representations of post-fault currents are summarized in table 4.1.

Analysis of post-fault power conversion process

Theoretical curves of d-q-0 currents and phase currents are illustrated in figure 4.2(c) and

(d) where m 0 corresponds to -0.15, -0.25 and -0.45 respectively. It can be observed that i d_f is a cosine function of θ e . i q_f is a constant in order to maintain the stable torque. i 0_f is neither a constant nor a standard sine function. With the decrease of m 0 , the unconventional shape of i 0_f becomes conspicuous. According to (4.7), the unconventional shape is caused by the frequency-doubled term (i 0_n cos2θ e ). Although the shape of i 0_f is special, the mean of i 0_f is equal to i 0_n over one electrical period in order to guarantee the balance between the average input power and output power. Regarding the phase-currents which are shown in figure 4.2(d),

i A_f is exactly 0 A because of the fault. i B_f and i C_f do not present the sinusoidal waveforms with a 60 • phase-shift. With the decrease of m 0 , the unconventionality of i B_f and i C_f is noticeable.

According to the formulas of post-fault currents, the post-fault current vector (i s_f ) can be visualized in α -β -0 coordinate. For comparison, the current vector (i s_n ) which is in healthy conditions is also presented. Figure 4.2(a) displays these vector trajectories when the proportion of i 0_n and i q_n is equal to -0.45 (m 0 = -0.45). Besides, their projections on α-β plane are further presented, which are referred to as i s_f pj and i s_npj , respectively. Under the normal condition, the trajectory of i s_n is a standard circle. Its 0-axis component i 0_n is equal to -0.45i q_n . Moreover, the trajectory of i s_npj is also circular, which is exactly the same as that of the current vector in a standard drive (without zero-sequence current components).

Interestingly, under the fault-tolerant condition, both i s_f and i s_f pj are no longer circular.

They present a heart-shaped trajectory. Concretely, the evolution of i s_f pj with respect to m 0 is shown in figure 4.2(b). It can be observed that the heart-shaped feature becomes more and more distinct with the decrease of m 0 . In other words, if the zero-sequence is bigger, the heart-shaped feature is more apparent.

Normally, the mechanical power (output power) P mec_f of the motor needs to be maintained 120/178 when the fault occurs, i.e. the motor speed and torque should be preserved. However, it is obvious that the zero-sequence current i 0_f is not a constant, as shown in figure 4.2(c). The non-constant i 0_f implies that the input power P in_f of the system is variable in one electrical period. In other words, the mismatch between the transient input power and output power is inevitable. According to energy conservation law, the mismatched energy will be absorbed or 121/178 released by the passive components in the system. Therefore, in the post-fault condition, the DC-bus capacitor will function as an energy buffer to balance the mismatched energy between the transient input power of the drive system and the mechanical output power of the motor. To explicitly elaborate the power conversion process in the post-fault condition, i 0_f is divided into two areas ('I' and 'II' marked in figure 4.2(c)) to discuss. In area 'I', P in_f is positive because of the negative i 0_f (i.e. the positive i N _f ). Thus, the DC-source provides power to the drive system. From the operating point 'a' to 'b', P in_f is lower than the power required by the motor because the absolute value of i 0_f is smaller than that of i 0_n . Thus, the DC-bus capacitor has to fill in the power shortage so as to support the motor holding the constant mechanical power.

As a result, u bus_f will decrease in this period. At the operating point 'b', i 0_f is equal to i 0_n .

In this case, the transient input power is balanced with the transient output power. From the operating point 'b' to 'c', P in_f is larger than the power required by the motor because the absolute value of i 0_f is bigger than that of i 0_n . Thus, the DC-bus capacitor has to absorb the energy which is not consumed by the motor. As a result, u bus_f will increase in this process. At the operating point 'c', u bus_f will reach the peak. From the operating point 'c' to 'd, P in_f is lower than the power required by the motor again. Thus, u bus_f will decrease once more. In turn, P in_f is negative in area 'II' because of the positive i 0_f (i.e. the negative i N _f ). This implies that the DC-source begins to absorb power from the drive system. In this case, the DC-bus capacitor not only powers the motor but also charges the DC-source. As a result, u bus_f will continue decreasing. At the operating point 'f', u bus_f will reach the valley. In steady states, the power conversion process is periodic. The oscillation of u bus_f is related to the integral of i 0_f , i.e. the amplitude and frequency of i 0_f .

In According to the above analysis, the energy (∆E _f ) will be stored in the DC-bus capacitor.

Therefore, we obtain

∆E _f = 1 2 C bus u 2 bus_f max - 1 2 C bus u 2 bus_f min ⇓ 2∆E _f C bus = u 2 bus_f max -u 2 bus_min = (u bus_f max + u bus_f min ) (u bus_f max -u bus_f
m 0 (π -2arcsin M 4m 0 ) + √ 2 8m 0 4m 2 0 + M (4 + M ) (4.14)
where M = 8m 2 0 + 1 -1; f ele = ωe 2π is the electrical frequency of the motor.

According to (4.14), we can know that ∆u bus decreases with the decrease of i q_n and the increase of f ele . In other words, ∆u bus will be smaller at high-speed and in low-load conditions.

Analysis of post-fault output capability

The output capability of a drive system in post-fault conditions is restricted to voltage and current limits. In healthy conditions, the DC-bus voltage of the MFCS-I can be raised to twice the source voltage thanks to the equivalent boost function. The line-to-line voltage of the motor is limited by the leg-to-leg voltage of the inverter, hence ±u bus . Although the DC-bus voltage will oscillate with the neutral current based on the analysis of the post-fault power conversion process, its mean value can still be maintained. The oscillation amplitude is relatively small when a sufficient DC-bus capacitor is employed. Therefore, it can be supposed that the situation of remaining phases of the motor does not change before and after the fault.

In this case, the voltage limit for each motor phase is the same. Actually, this scenario is similar to the CFNI where the motor's NP is connected to the midpoint of the DC-bus.

In terms of the torque output capability in post-fault conditions, it is reasonable to premise that the inverter is designed to withstand the rated phase-current RMS of healthy conditions.

When one of motor phases is disconnected, the torque degradation is inevitable in case of the remaining phase-currents violate the current limit.

By using the inverse Park transformation (amplitude invariant), the phase-currents under healthy conditions can be formulated by:

          
i A_n = i d_n cos(θ e ) -i q_n sin(θ e ) + m 0 i q_n i B_n = i d_n cos(θ e -2π/3) -i q_n sin(θ e -2π/3) + m 0 i q_n i C_n = i d_n cos(θ e + 2π/3) -i q_n sin(θ e + 2π/3) + m 0 i q_n (4.15) 123/178 Furthermore, the phase-current RMS under healthy conditions is thus calculated as:

I RM S_n =   i 2 d_n + i 2 q_n √ 2   2 + (m 0 i q_n ) 2 = 1 2 + m 2 0 • i q_n (4.16)
where i d_n = 0 for a surface-mounted PMSM.

In order to quantify the post-fault torque output capability, the current overrating ratio (m i ) is defined. It can be calculated by combining (4.10) with (4.16), as

m i = I RM S_n I RM S_f = 4m 2 0 + 2 15m 2 0 + 4 = 4 Φ f pω m 2u in η 2 + 2 15 Φ f pω m 2u in η 2 + 4 (4.17)
Theoretically, m i denotes the proportion between the phase-current RMS of healthy conditions and that of post-fault conditions. In other words, m i is related to how much equivalent torque the motor can obtain when the post-fault current RMS reaches the current limit. The evolution of m i with respect to the motor speed and system efficiency is displayed in observed that the range of m i is between 0.518 and 0.707 with the change of the motor speed and the system efficiency. m i increases with the decrease of ω m and increase of η. Theoretically, the motor can output 0.518 to 0.707 of the nominal torque in the post-fault condition.

Post-fault equivalent circuit

When the OPF occurs in phase-A, the circuit structure of the MFCS-I is changed. The leg-A of the inverter is isolated from the motor. Thus, the control DOF of leg-A (i.e. the duty cycle α A ) is useless to the whole drive system under the post-fault condition. Essentially, the motor has been degenerated into a two-phase motor.

If we consider the MFCS-I as a two-phase system in post-fault conditions, the average model and equivalent circuit should be rebuilt. According to the time-averaging equivalent modeling approach and Kirchhoff's law, the mathematical model of the MFCS-I under the OPF can be built as

  u BO u CO   =   α B α C   u bus =   Ri B + L BB di B dt + M BC di C dt + e B Ri C + L CC di C dt + M CB di B dt + e C   +   -L aux di N dt + u in -L aux di N dt + u in   (4.18a)        i B + i C + i N = 0 i bus = C bus du bus dt = -α B i B -α C i C (4.18b)
Then, by adding the two equations of (4.18a) and combining with (4.18b), the zero-sequence voltage equation can be formulated as

α h_f u bus = - R 2 i N -( L BB + M BC 2 + L aux ) di N dt - e A 2 + u in = - R 2 i N -( L σ 2 + L Σ 4 + L aux ) L E_f di N dt - e A 2 + u in (4.19)
where

• α h_f = α B + α C 2
is the mean duty cycle in post-fault conditions, whose range is still (0-1).

• L E_f denotes the equivalent inductance. From figure 4.4(b), it can be seen that the MFCS-I in the post-fault condition can be approximated to a special boost converter where the back-EMF (e A /2) of phase-A is seen as a disturbance term to the input voltage u in . The amplitude of this disturbance is related to the motor speed. According to the analysis of post-fault power conversion process, we know that the DC-bus voltage is unable to be maintained at a constant value since i 0_f or i N _f is not constant. In other words, if we choose to preserve a constant torque output of the motor preferentially, we can not maintain a stable voltage on the DC-bus. Obviously, the stable torque output should be guaranteed in priority in the post-fault operations.

Fault-tolerant control strategy 4.2.1 Uniform control framework

For the CFNI and SFNI, the control framework is usually changed in post-fault conditions due to the reconfiguration of motor's NP. In the CFNI, the NP of the motor is connected to the midpoint of the DC-bus when an OPF occurs. In the SFNI, it is connected with the fourth leg after an OPF. Since the NP of the motor is connected, the relationship between the output voltage of the inverter and the applied voltage on motor phases has changed. Therefore, 126/178 both the CFNI and SFNI require to add some modifications in the modulation, transformation matrices or the whole control architecture, which are complicated. For example, in [START_REF] Bolognani | Experimental fault-tolerant control of a pmsm drive[END_REF], the authors proposed a feed forward compensation strategy for the SFNI to compensate the error between the d-q voltage provided by the inverter and the d-q voltage applied to the motor. This compensation strategy can still guarantee a constant d-q voltage so that the current controller can avoid the tracking of sinusoidal components. Therefore, the PI controller can still be used.

In [START_REF] Zhou | High performance three-phase pmsm open-phase fault-tolerant method based on reference frame transformation[END_REF], the authors do not compensate the voltage error caused by the unbalanced condition.

In turn, they proposed a novel transformation matrix so that the position-dependent sinusoidal current commands are transformed to DC variables. Thus, the PI current controller can still be used. In [START_REF] Gaeta | Modeling and control of three-phase pmsms under open-phase fault[END_REF], the authors proposed an analytical model for PMSMs, which are in post-fault conditions. By using the model, a fault-tolerant FOC for the CFNI and SFNI is achieved.

For MFCS-I, the situation of the NP has not changed from pre-fault to post-fault conditions.

Therefore, the concept of a virtual healthy system can be adopted. In this way, the fault-tolerant control can be fulfilled by simply replacing the current references i * d_n , i * q_n and i * 0_n with the fault-tolerant current references i * d_f , i * q_f and i * 0_f . In the virtual healthy system, the control objective for phase-A is to actively force i A_f to be 0 A even if this goal is naturally achieved.

Since the post-fault current references are associated with those in healthy conditions, it is necessary to obtain the value of i * 0_n . In healthy conditions, the DC-bus voltage is stable. Therefore, the power conservation equation can be derived as:

P in = u in i N = -3u in i 0 = P mec η = ω m T e η (4.20)
According to (4.20), the zero-sequence current reference i * 0_n can be calculated by

i * 0_n = -Φ f pω m 2u in η i * q_n (4.21)
Since the drive system in the post-fault condition can be seen as a virtual healthy system with the control objective of forcing i A_f = 0, a uniform control framework is introduced, as shown in figure 4.5. The d-axis current reference i * d_n is set as 0 A for a surface-mounted PMSM without flux weakening control. The q-axis current reference i * q_n is guided by the speed controller. The zero-sequence current reference i * 0_n can be calculated by (4.21) directly. However, in (4.21), the real-time efficiency η is hard to be known accurately in practice. If we adopt an approximated value of η, we will obtain an inaccurate i * 0_n . The inaccuracy in i * In the healthy mode, i * d_n , i * q_n and i * 0_n are directly fed into the current controller. In the post-fault mode, i * d_f , i * q_f and i * 0_f calculated by (4.7) are used as the references for the current controllers. According to figure 4.2, we know that i * d_f and i * 0_f are non-constant. In order to track this kind of current references, a current controller with high bandwidth, fast response and optimum tracking ability is necessary. To this end, we will develop two controllers in the following.

Deadbeat predictive current tracking controller

Model predictive control (MPC) is a promising controller for motor drives due to its merits of simplicity, fast response and multi-variable control flexibility [START_REF] Kazmierkowski | Current control techniques for three-phase voltage-source PWM converters: A survey[END_REF]. The basic principle of MPC is to predict the behavior of the controlled system using its model in the future. Usually, the controller uses this information to choose the optimal actuator state based on a predefined 128/178 optimization criterion [START_REF] Saeed | Low-complexity deadbeat model predictive current control with duty ratio for five-phase pmsm drives[END_REF]. MPC can be classified into two categories: continuous control set (CCS) MPC [START_REF] Cortes | Predictive control in power electronics and drives[END_REF] and finite control set (FCS) MPC [START_REF] Perez | Predictive control of AC-ac modular multilevel converters[END_REF] [START_REF] Donoso | Finite-set model-predictive control strategies for a 3l-npc inverter operating with fixed switching frequency[END_REF]. FCS-MPC only considers a finite number of switching states thus directly gives the optimal switching state. A salient drawback of the FCS-MPC is that it needs a high sampling frequency to guarantee the accurate tracking performance thus requires a high computational burden on the microprocessor. CCS-MPC requires a modulator in the control system. When the choice of the voltage vector is made in order to return the error to 0 at the end of the sample period, the CCS-MPC is often called as deadbeat controller.

In order to verify the proposed fault-tolerant scheme of the MFCS-I quickly and efficiently, the deadbeat predictive controller is adopted to track the time-varying post-fault current references since there is no control parameters needed to be tuned. Because the sampling frequency is much higher than the fundamental frequency, the DC-bus voltage and motor speed measurable by voltage and speed sensors are supposed to be invariable within one sampling period. By applying Euler forward approximation to the voltage model presented (3.3b), the d-q-0 currents at the (k + 1) th sampling period can be expressed by

     i d (k + 1) i q (k + 1) i 0 (k + 1)      =           1 - RT s L d ω e (k)L q T s L d 0 - ω e (k)L d T s L q 1 - RT s L q 0 0 0 1 - RT s 3L E           A      i d (k) i q (k) i 0 (k)      +           T s u bus (k) L d 0 0 0 T s u bus (k) L q 0 0 0 T s u bus (k) 3L E           B      α d (k) α q (k) α 0 (k)      +         0 - ω e (k)Φ m T s L q - u in T s 3L E         C (4.22)
where A, B and C are the matrices of the discrete equation; T s is the sampling period.

In order to eliminate current error at the (k + 1) th sampling interval, the d-q-0 currents at the (k + 1) th sampling period should follow

           i d (k + 1) = i * d i q (k + 1) = i * q i 0 (k + 1) = i * 0 (4.23)
By replacing (4.23) into (4.22), the required d-q-0 duty cycles at the k th sampling period 129/178 can be calculated by

     α d (k) α q (k) α 0 (k)      = B -1           i * d i * q i * 0      -A      i d (k) i q (k) i 0 (k)      -C      (4.24)
Since the deadbeat current controller is completely constructed by the mathematical model of the PMSM, no tuning is required. However, it is sensitive to parameter variations. The steady errors caused by the parameter variations can usually be mitigated through online parameter identification [START_REF] Zhang | Model predictive current control for pmsm drives with parameter robustness improvement[END_REF].

Differential flatness-based current tracking controller

The property of differential flatness allows an alternate representation of a system where the state and input variables can be directly expressed by the flat output and a finite number of its derivatives [START_REF] Fliess | A lie-backlund approach to equivalence and flatness of nonlinear systems[END_REF]. With the differential flatness, state feedback can be chosen in such a way that the closed-loop dynamic behavior is linear. Therefore, the differential flatness-based control is appropriate for trajectory tracking control and has been used in motor drives [105][106].

A differential flatness-based current controller is also proposed for post-fault current tracking. The DC-bus voltage and motor speed are still supposed to be invariable within one sampling period. The output y = [y 1 , y 2 , y 3 ] T , control variable u = [u 1 , u 2 , u 3 ] T and state variable x = [x 1 , x 2 , x 3 ] T are respectively chosen as:

y =      i d i q i 0      , u =      α d α q α 0      , x =      i d i q i 0      (4.25)
Thus, according to the model presented in (3.3b), u can be expressed by:

                   α d = 1 u bus (Ry 1 + L d ẏ1 -ω e L q y 2 ) = f 1 (y 1 , ẏ1 , y 2 ) α q = 1 u bus (Ry 2 + L q ẏ2 + ω e (L d y 1 + Φ m )) = f 2 (y 2 , ẏ2 , y 1 ) α 0 = 1 u bus (Ry 3 + 3L E ẏ3 + u in ) = f 3 (y 3 , ẏ3 ) (4.26)
It needs to note that, in the post-fault condition, we do not utilize the equivalent DC/DC boost function to regulate the real-time DC-bus voltage. We only control the 0-axis current.

In this case, it is obvious that both u and x can be represented by the output y and its first-order derivative. Therefore, the system can be considered as a flat system. Then, to track 130/178 DC-bus voltage u bus is boosted at 30 V stably thanks to the equivalent boost function. The phase-currents (i A , i B and i C ) are sinusoidal. The torque ripple is around 15 mN•m. At 0.6 s, an OPF occurs in phase-A, which results in i A = 0 A instantly. From 0.6 s to 0.7 s, the fault-tolerant control strategy is not triggered. Therefore, it can be seen that i B and i C are distorted during this period. At the same time, the torque ripples increase to 80 mN•m. In addition to the big torque ripples, the DC-bus voltage (u bus ) can no longer be maintained stable.

It begins to oscillate with a 4 V ripple.

At 0.7 s, the fault-tolerant control strategy is engaged. Thus, it can be observed that the torque performance can be well preserved. In steady states, the torque ripples recover to the same value (16 mN•m) with that in the healthy operation. Meanwhile, we can see that the shapes of i B and i C are similar with the theoretical analysis presented in figure 4.2(d), corresponding to the second case where m 0 = -0.25.

132/178

Concerning the DC-bus voltage, the ripple of u bus is around 8 V in the fault-tolerant condition, which is even larger than that in the faulty case where the fault-tolerant control is not triggered. In fact, this phenomenon can be expected and is reasonable. As we emphasized in the analysis of post-fault power conversion process, the zero-sequence current i 0_f will be non-constant in fault-tolerant conditions. This implies that the transient input power provided by the DC-source is non-constant in fault-tolerant conditions. However, the mean of i 0_f is equal to i 0_n . I.e. the average input power is still constant. In order to balance the mismatch between the transient input power and the output power of the motor, the DC-bus capacitor will become an energy buffer. Therefore, u bus oscillates regularly with the zero-sequence current i 0_f within one electrical period. The oscillation amplitude is determined by the power of the motor. However, the mean of u bus can be still maintained at 30 V.

In order to check the tracking performance of the deadbeat predictive current controller, the current references and actual currents under the fault-tolerant condition are detailed in figure 4.7. Obviously, the shapes of the actual i B and i C are identical with their references. This indicates that the deadbeat predictive current controller shows a remarkable performance in tracking a changeful trajectory. Based on above simulation results, the feasibility of the MFCS-I to tolerate an OPF is validated. In conclusion, it is able to maintain a stable torque and a stable average DC-bus voltage by the proposed current trajectory.

133/178

Experimental results

Since the feasibility of fault-tolerant operations is verified by simulations, experimental implementations are further carried out on the real 52.5 W PMSM based MFCS-I test-bench.

To create an OPF, we use a circuit interrupter which is connected in series with phase-A.

Fault-tolerant transition performance without load

The experimental results of a fault-tolerant transition process is shown in figure 4.8 where the motor is operated at 2000 rpm without load. Only the frictional torque is applied on the shaft. First, under the healthy operation, the motor speed (n) is regulated at 2000 rpm stably.

The torque ripple is about 10 mN•m. The DC-bus voltage (u bus ) is boosted at 30 V. Meanwhile, the d-axis current (i d_n ) is 0 A and the q-axis current (i q_n ) is 0.58 A. The zero-sequence current and neutral current (i 0_n and i N _n ) are respectively -0.15 A and 0.45 A. Roughly speaking, all the electrical and mechanical states are stable in the healthy condition. In this case, the zero-sequence current proportion (m 0 ) can be calculated as -0.26.

134/178

When the OPF occurs, i A decreases to 0 A instantly. At the beginning, the fault-tolerant control strategy is not triggered. Therefore, we can see that the torque ripple increases to 50 mN•m. The ripples of i q also show a big fluctuation in amplitude of 1.6 A. The ripples of u bus increase to 2 V as well. The ripples of i d , i 0 and i N are 1.4, 0.4 and 1.2 A respectively. Obviously, if no treatment means is conducted to the system, the motor will continue producing strenuous vibrations and even be damaged.

Then, we start the fault-tolerant control. Once the fault-tolerant mode is triggered, the ripple of T e is decreased to 12 mN•m quickly, which is almost consistent with that in the healthy condition. The ripple of i q is also similar with that in the healthy condition. Therefore, we can say that the performance of T e and i q is preserved rapidly. For greater details, enlarged views are displayed to the right of each graph. First, regarding i d , its shape is distorted before the action of the fault-tolerant control. Since the fault-tolerant control functions, the shape of i d tends to be cosinoidal, which is in line with the theoretical waveform presented in figure 4.2(c).

Meanwhile, i 0 is also similar to the theoretical curve. The period of i d and i 0 is consistent with that of i B and i C , which is equal to the fundamental period (T f = 7.5 ms). As analyzed before, in order to maintain the constant torque, the DC-bus capacitor has to absorb and release some transient energy within one electrical period. Therefore, we can see that u bus is synchronized with the change of i N . The ripple of u bus increases to 3 V, which is in accord with the theoretical result (around 3.1 V) according to (4.14). However, it needs to be emphasized that the mean of u bus is still 30 V.

In order to verify the tracking performance of the deadbeat predictive current controller,

Fault-tolerant transition performance with load

In order to further verify the fault-tolerant transition performance of the MFCS-I, the motor is operated with a load. When the OPF occurs, i A decreases 0 A rapidly. In the same way, the fault-tolerant control is not engaged at the beginning. Therefore, we can see that the ripple of T e is increased to 150 mN•m dramatically. The ripple of i q becomes 4.2 A. The ripples of i d , i 0 and i N are respectively 4, 1.0 and 2.5 A. The motor shows severe vibrations and its temperature starts to rise.

136/178

Then, we switch on the fault-tolerant control strategy. Thanks to the remedial measure, T e returns to 60 mN•m quickly with the ripple of 13 mN•m. In the enlarged views, i d is still a cosine waveform, which is in accord with the theoretical analysis. Moreover, the period of i d and i 0 is still consistent with the fundamental period. With the increase of motor's power, the peak value of i N increases to 5.6 A. This means that the transient energy which should be absorbed or released by the DC-bus capacitor becomes bigger. As a result, it can be seen that the ripple of u bus is increased to 6 V. However, the mean of u bus is still 30 V due to the DC-bus

Dynamic condition

When the motor is operated in post-fault conditions, it still has dynamic requirements, such as acceleration, deceleration and load variations. Thus, in order to further validate the practicability of the MFCS-I to be against an OPF, dynamic experiments under post-fault conditions are further conducted, as shown in figure 4.12. The setup of the dynamic experiment is as follows: the speed reference n * is set as 1000 rpm before 2 s. It is set to 1500 rpm between 2 and 5 s while it is reset to 1000 rpm after 5 s. A 50 mN•m load is imposed at 10 s and removed at 15 s.

From figure 4.12, we can see that i A is always 0 A because of the fault. Even so, the motor speed can still track the reference signal well. The torque performance can be preserved during the acceleration, deceleration, loading and deloading processes. When the motor accelerates 137/178 instantly, u bus declines to 22 V firstly. Thanks to the control to the mean of u bus , the zerosequence current reference i * 0_n decreases from -0.07 to -0.37 A. I.e. i * N _n increases from 0.21 to 1.11 A rapidly so that the DC-source can output more average power to the drive system. Therefore, it can be seen that the mean of u bus can return to 30 V. Similarly, i * 0_n can also be adjusted cleverly during the load change processes. It changes from -0.07 to -0.28 A then returns to -0.07 A again. This demonstrates that it is feasible to use a PI controller to regulate the mean of u bus which enables an appropriate value for determining the value of i * 0_n .

Implementation on the 1.2 kW PMSM

In this section, the effectiveness of fault-tolerant operations of the MFCS-I will be verified on a 1.2 kW PMSM. The test-bench has already been presented in figure 3.12, whose parameters are already displayed in table 3.1. The DC-source voltage is 180 V and the desired DC-bus 138/178 voltage is 360 V. A circuit breaker which is connected in series with phase-A is used to simulate an OPF. In this experiment, the differential flatness-based current tracking controller is adopted, whose parameters are set as:

ζ 1 = ζ 2 = ζ 3 = 1, ω 1 = 2500 rad/s, ω 2 = ω 3 = 5000 rad/s.
Considering that the long-time operation with the OPF without fault-tolerant control is adverse to the lifetime of the motor, a predictive current-based OPF detection method is used to detect the OPF. In this way, the fault-tolerant control strategy can be put into action as soon as possible after the occurrence of the fault.

Open-phase fault Detection

The conventional OPF detection method in the typical three-phase fault-tolerant drives relies on the magnitude of fault-induced current harmonics, which is only related to d,q-axes currents. Normally, the magnitude is proportional to the load [START_REF] Yang | Online open-phase fault detection for permanent magnet machines with low fault harmonic magnitudes[END_REF]. In the MFCS-I, there are not only d,q-axes currents but also the zero-sequence current in healthy conditions. Therefore, the zero-sequence current can be an additional signal for the OPF detection.

According to (3.3a), the voltage equation of the MFCS-I can be discretized. The sampling interval is T s . Thus, in healthy conditions, i d , i q and i 0 in the (k + 1) th interval can be predicted by using their measurements in the (k) th interval, as:

     îd (k + 1) îq (k + 1) î0 (k + 1)      =          1 - RT s L d ω e (k)L q T s L d 0 - ω e (k)L d T s L q 1 - RT s L q 0 0 0 1 - RT s 3L E               i d (k) i q (k) i 0 (k)      +         T s u bus (k) L d 0 0 0 T s u bus (k) L q 0 0 0 T s u bus (k) 3L E              α d (k) α q (k) α 0 (k)      +        0 - ω e (k)Φ f T s L q - u in T s 3L E        (4.30)
where îd (k + 1), îq (k + 1) and î0 (k + 1) are the predicted values; i d (k), i q (k) and i 0 (k) are the measured values.

If the motor is operated in healthy conditions, the measured currents in the (k+1) th interval will show good agreements with the predicted currents. Even if we consider the time delay and the inaccuracy of the system model, the deviation between the measured and predicted currents is relatively small. However, once the OPF occurs, the predicted currents will be far away from the measured currents due to the unbalanced condition, which is caused by the fault. In this case, the deviation between the measured and predicted currents will be bigger. We can use 139/178 this feature to detect the OPF. To take the d-q-0 currents into account simultaneously, an error index ε is defined as follows:

ε = | îd (k + 1) -i d (k + 1)| + | îq (k + 1) -i q (k + 1)| + | î0 (k + 1) -i 0 (k + 1)| (4.31)
If ε exceeds a threshold ε th , the occurrence rate of OPFs is high. [START_REF] Kontarček | Cost-effective three-phase pmsm drive tolerant to open-phase fault[END_REF] In addition to u bus , the phase-currents (i A , i B and i C ), d-q-0 currents (i d , i q and i 0 ) and their references (i * d , i * q and i * 0 ) and corresponding tracking errors (e d , e q and -3 3 142/178 e 0 ) are drawn. In the healthy condition, i A , i B and i C present normal shapes with a downward offset due to the zero-sequence current. i * d , i * q and i * 0 are 0, 0.74 and -0.34 A respectively. In this case, the value of zero-sequence proportion m 0 is calculated as -0.46. Thus, this situation can be approximated to the third case in figure 4.2(c) and (d) where m 0 is equal to -0.45. The actual i d , i q and i 0 can track the references well. The tracking errors e d , e q and e 0 are less than 0.4 A. Then, the OPF is made artificially by the circuit breaker. At that moment, i A becomes 0 A instantly. However, the fault flag is not triggered immediately. Therefore, the motor is operating in the faulty condition without the fault-tolerant control. In that period, it can be seen that all the currents are distorted and the tracking errors e d , e q and e 0 become big. In around 2 ms, the fault flag is triggered and the fault-tolerant control is put into operations.

In the post-fault operation, all the currents recover to stable states. The shapes of i B and i C show good agreements with the theoretical shapes presented in figure 4.2(d), corresponding to the case of m 0 = -0.45. Meanwhile, the shapes of i * d , i * q and i * 0 are also consistent with the theoretical analysis. The tracking errors return to around 0.4 A again, which is almost similar to that in the healthy condition. Therefore, we can conclude that the OPF detection time T set is about 2 ms. Moreover, we can also conclude that the proposed flatness-based current controller is feasible to track the non-constant current references in the post-fault condition.

According to the current information and the position information, the trajectory of the current vectors in 3-D space are further calculated and drawn, as shown in figure 4.14(b). In the healthy condition, the current trajectory i s_n is a circle whose zero-sequence component is -0.34 A, which is equal to i 0 . In the faulty condition, the current trajectory is mismatched.

In the post-fault condition with the fault-tolerant control, the current trajectory i s_f is like a heart. This verifies the correctness of the theoretical analysis.

Low-speed with load

To further test the fault-tolerant performance when the motor is operated with load, we have recorded the new experimental results, as shown in figure 4. [START_REF] Kinoshita | Electric system for electric vehicle[END_REF]. In this case, the motor speed is 104.7 rad/s and the load is around 2 N•m. From figure 4.16, it can be seen that ε is equal to 0.6 in the healthy condition. u bus is boosted at 360 V and u bus_f ilter is also 360 V due to the equivalent boost function. The ripple of the torque T e is around 0.16 N•m. Then, the OPF occurs, it results in the increase of ε. We can see that ε reaches 6. As a result, the torque ripple is increased to 3.8 N•m. Since ε exceeds 1, the fault flag is triggered, which means that the fault-tolerant control strategy is put into operations. Therefore, the motor torque is preserved instantly. In steady states, the torque ripple recovers to around 0.31 N•m. In post-fault operations, the transient DC-bus voltage can not be maintained constant anymore. Its ripple 143/178 is increased to 25 V, which is bigger than that in the no load condition. This phenomenon is reasonable because the amplitude of the zero-sequence current is bigger in the load condition. In other words, the DC-bus capacitor needs to absorb and release more energy within one electrical period in the post-fault condition. However, the average DC-bus voltage u bus_f ilter can still be remained at 360 V due to the control for the mean of u bus .

Then, to analyze the fault-tolerant performance in detail, zoomed in views are further presented in figure 4.17. In the healthy condition, i A , i B and i C present normal shapes with a downward offset due to the zero-sequence current. i d , i q and i 0 are 0, 3.39 and -0.49 A respectively. In this case, the value of zero-sequence proportion m 0 is calculated as -0.14. This situation can be approximated to the first case in figure 4.2(c) and (d) where m 0 is equal to -0.15. The actual i d , i q and i 0 can track the references well. The tracking errors e d , e q and e 0 are less than 0.6 A. Then, the OPF is made artificially by the circuit breaker. At that moment, i A becomes 0 A instantly. However, the fault flag is not triggered immediately. Therefore, the motor is operating in the faulty condition without the fault-tolerant control. In that period, it can be seen that all the currents are distorted and the tracking errors e d , e q and e 0 become big.

In around 3 ms, the fault-flag is triggered and the fault-tolerant control is put into operations.

In the post-fault operation, all the currents recover to stable states. The shapes of i B and i C show good agreements with the theoretical shapes presented in figure 4.2(d), corresponding to the case of m 0 = -0.15. Meanwhile, the shapes of i * d , i * q and i * 0 are also consistent with the 145/178 theoretical analysis. The tracking errors return to around 0.6 A again, which is almost similar with that in the healthy condition. Therefore, we can conclude that the fault detection time T set is about 3 ms. Moreover, we can also conclude that the proposed flatness-based current controller is feasible to track the non-constant current references in the post-fault condition.

According to the current information and the position information, the trajectory of the current vector in 3-D space are further calculated and drawn, as shown in figure 4.17(b). In the healthy condition, the current trajectory i s_n is a circle whose zero-sequence component is -0.49, which is the same as i 0 . In the faulty condition, the current trajectory is mismatched. In the post-fault condition with the fault-tolerant control, the current trajectory i s_f is like an ellipse. This verifies the correctness of the theoretical analysis.

Evaluation of available speed and torque ranges

To evaluate the reachable speed range in post-fault operations, the motor is operated respectively at a very low speed (20.9 rad/s) and the full speed (314.2 rad/s). The steady-state phase-currents and neutral current are shown in figure 4. [START_REF] Nakai | Overview of multi-functional converter systems[END_REF]. By comparison, the unique shapes of remaining phase-currents and neutral current are more distinct under the full-speed condition. This phenomenon is in accord with the theoretical analysis because the absolute value of m 0 at the full speed is larger than that in the low speed condition. The actual current shapes also show good agreements with the theoretical curves. It needs to note that the waveform of i N is reverse compared to the shape of i 0_f because i N is equal to -3i 0 . In conclusion, we can say that the motor is still able to work within a wide speed range under the post-fault operation.

Especially, it can still reach the nominal speed.

Then, to evaluate the post-fault torque output capability, contrast experiments are carried out on the 1.2 kW PMSM respectively. The motor is operated in the healthy condition and fault-tolerant condition respectively. In the healthy operation, the motor is operated at 157.1 rad/s with 4.0 N•m torque (rated torque). The experimental results are shown in figure 4.19.

First, from figure 4.19(a), it can be seen that the RMS value of phase-currents in the healthy condition is 4.565 A, whose offset is around -1.28 A due to the zero-sequence current. The offset is equal to -1/3 of i N whose value is 3.84 A. From figure 4.19(b), the phase-current THD is 4.97% and the zero-sequence proportion is 24.73%. From figure 4.19(c), it can be observed that the torque is stable at 4.0 N•m with the ripple of 0.4 N•m. Therefore, we know that the phase-current RMS is 4.565 A when the motor is operated with the nominal torque in healthy conditions. We can take this value as the rated current limit.

Afterwards, the motor is operated in the post-fault condition with the same speed but 146/178 be seen that the remaining phase-currents includes 23.79% of zero-sequence components and 23.55% second-order components. Therefore, the THD of remaining phase-currents is around 23.67%, which is much bigger than that in the healthy condition. Actually, this phenomenon is reasonable because the zero-sequence components and second-order components make the possibility of keeping the q-axis current at a constant value in the post-fault condition.

Dynamic Performance

In post-fault operations, the motor may still have the requirements of acceleration, decel- N•m load is imposed and removed, the motor speed shows around 10 rad/s fluctuations for a while, and then it turns to the stable state quickly. In terms of the DC-bus voltage, it can be maintained at 360 V stably with only a 2 V ripple during the whole test.

For the post-fault operation, the experimental results are shown in figure 4.22. First, it can be seen that the amplitude of remaining phase-currents is bigger than that in the healthy operation. This phenomenon is in accord with the theoretical analysis because the motor requires bigger currents to generate the same magnetic field in the post-fault condition when phase-A is not available. Regarding the acceleration and deceleration performance, the motor herein almost shows a consistent dynamic with that in the healthy operation. However, it needs to note that the fluctuation on the DC-bus voltage is bigger. According to the theoretical analysis, we have revealed that the transient input power is not constant in post-fault operations, in order to maintain a constant q-axis current. Therefore, the DC-bus capacitor becomes an energy buffer in the post-fault condition to balance the mismatch between the transient input power and the 150/178 output power. From figure 4.22, we can observe that the fluctuation of the DC-bus voltage is 6

V at the condition of 104.7 rad/s with 0.4 N•m, 4 V at the condition of 209.4 rad/s with 0.4 N•m, 30 V at the condition of 104.7 rad/s with 2.1 N•m. I.e. the fluctuation becomes bigger when the load is higher and speed is lower. Obviously, this phenomenon is in accord with the theoretical analysis. Actually, this issue is similar to the midpoint voltage fluctuation in the CFNI which is also caused by a non-constant neutral current. Normally, the issue can be solved by using a sufficient DC-bus capacitor. Since the DC-bus voltage can be measured in each sampling period which can be taken into account in the modulation, the DC-bus fluctuation does not affect the control performance to the motor. Although the real-time DC-bus voltage is oscillating, its mean value is still stable at 360 V due to the control for the mean of the DC-bus voltage.

Consequently, we can say that the MFCS-I can not only tolerate the OPF without using extra costs but also meet variable speed and variable load operations with only two available phases.

The post-fault dynamic performance of the motor can still be maintained.

Conclusion

Conventional fault-tolerant topologies of three-phase two-level VSI fed AC drives require auxiliary hardwares to achieve the fault-tolerant operation towards an OPF. These auxiliary hardwares definitely increase the cost and complexity of the drive system. The two fault-tolerant solutions (CFNI and SFNI) that utilize the NP are popular in industries because they can not only tolerate an OPF which occurs in the inverter side but also in the motor side. Considering the innately connected motor's NP in the MFCS-I, this chapter explored its fault-tolerant potential for the first time without using additional hardwares. To preserve the stable torque output in fault-tolerant operations, a novel current vector trajectory was derived. In post-fault operations, the current vector trajectory is no longer a standard circle. It presents a heart-shaped form. Moreover, the d-axis and 0-axis currents are not constant anymore. The remaining phasecurrents are also not exactly sinusoidal due to the generation of a second-order component. A simple uniform control framework was proposed for the MFCS-I where the fault-tolerant control can be fulfilled by simply replacing the d-q-0 current references. The real-time DC-bus voltage is not regulated anymore. In turn, its mean value is the control objective in this framework. To track the non-constant current trajectories in fault-tolerant conditions, the predictive deadbeat controller and differential flatness-based controller are developed for the current-loop.

Experimental validations are carried out on the 52.5 W and the 1.2 kW PMSMs to verify the effectiveness of the proposed fault-tolerant scheme, respectively. According to the experimental results, the OPF can be well tolerated once the fault-control program is executed. In post-fault operations, the motor can still be operated in a wide speed range. The post-fault torque output capability is around 55% of the nominal torque in the 1.2 kW PMSM based MFCS-I. Both the predictive deadbeat controller and the differential flatness-based controller show a remarkable current tracking performance even if the current references are not constant.

151/178

Moreover, the motor in post-fault operations can still have a consistent dynamic performance compared to that in healthy operations.

Consequently, the MFCS-I has two advantages: One is the integrated DC/DC boost function without using a real chopper. The other is the cost-effective fault-tolerance towards an OPF without using additional hardwares. Finally, compared to the four typical fault-tolerant topologies, the main characteristics of the MFCS-I are summarized in table 4.2.

152/178

General conclusion Conclusion

The main objective of this dissertation was to develop a low-cost and high-reliability alternative for the typical two-stage converter-based three-phase PMSM drive system in HEV and EV applications. Inspired by the original idea of the basic MFCS which utilizes the NP of a three-phase motor to realize a free DC/DC boost function upon a three-phase two-level VSI, a comprehensive study with respect to the modeling, control, improvement and fault-tolerant capability of MFCSs has been conducted. This dissertation starts with an overview of the applications using the NP to obtain additional benefits.

• In chapter 2, the PMSM model considering zero-sequence components is given. For a PMSM with ideally sinusoidal back-EMFs, its 0-axis model is regarded as a first-order system. Then, the average model expressed with inverter's duty cycles of a PMSM-based basic MFCS is built in A-B-C firstly and further converted into d-q-0 coordinate. It reveals that the d-q model of the basic MFCS is still same as that of a conventional drive while the (the rated DC-bus voltage should be 360 V) is supplied. This indicates that the effective step-up ratio of the equivalent boost of the MFCS-II can reach 9 even higher. However, the issue of a throughout positive phase-current should be noticed.

Key contribution:

Two improved solutions are proposed to ameliorate the basic MFCS.

A universal closed-loop control strategy is proposed for different MFCSs for the first time.

• In chapter 4, the possibility of fault-tolerant operations of the MFCS-I towards an OPF without using any auxiliary hardware is discussed for the first time by considering the innately connected NP. To achieve this possibility, three crucial constraints are found and a novel current vector trajectory is proposed for fault-tolerant operations. In post-fault conditions, the q-axis current is still being held constant to preserve a stable torque.

Interestingly, both d-axis and 0-axis currents are not constant anymore. To be specific, tech-benches to verify the effectiveness, which illustrate that the motor torque can be well preserved by using the proposed fault-tolerant scheme when an OPF occurs. The motor can still be operated within a wide speed range and can still output half the nominal torque under the limit of rated currents.

Key contribution:

The fault-tolerant possibility without using any redundant configuration for three-phase PMSMs is investigated and achieved for the first time based on the concept of MFCSs.

Perspectives

Following the works presented in Chapters 1-4, some perspectives can be envisaged:

• Change the power electronics system: With the development of high switchingfrequency power electronic devices (such as SiC and GaN), the inverter can be operated at a higher switching frequency. Therefore, the current ripple of phase-currents in motor drives can be reduced. In this case, the basic MFCS may be directly used in real cases or the MFCS-I with a smaller series inductor will be sufficient. From another point of view, the MFCS concept can also be extended to multi-level (3-level, 5-level or higher level) inverter systems since the current ripple of multi-level inverter systems is smaller. In addition, it will also be interesting to apply the MFCS concept to some grid-connected drives (AC/AC motor drives), where the NP of the motor can be connected to a single-phase AC source.

• More analysis and considerations in motor design: In MFCSs, the motor is actually operated in unconventional conditions where a DC component is superposed to the fundamental components. Thus, more analysis such as magnetic saturation and thermal dissipation of the motor in MFCSs can be further studied. Moreover, some guidance can be given to design a special machine with a larger L 0 .

155/178

• Considering third-harmonic back-EMF: If a PMSM contains third-harmonic back-EMFs, the zero-sequence back-EMF (e 0 ) in (2.6) is not 0. Therefore, it will generate the third-harmonic torque since the zero-sequence current is not null. To eliminate the third-harmonic torque, new modulation method and control strategy should be developed.

• Variable DC-bus voltage control: Since the MFCSs integrate an equivalent DC/DC boost function, its DC-bus voltage can be consistently controlled to be the minimum value, which can exactly meet the operating demand of the motor (Normally, under low-speed conditions, a low DC-bus voltage is sufficient). In this case, the overall operating efficiency of the drive system may be improved with a variable DC-bus voltage when the motor is constantly operated under acceleration or deceleration conditions. To this end, a variable DC-bus voltage control strategy may be significant for MFCSs. Moreover, for the MFCS-II, the average duty cycle (α h ) of the inverter can also be used as a DOF instead of the constant value (0.5) used in this manuscript. In this case, an allocation strategy between α h and α F may be interesting with respect to the required DC-bus voltage.

• Considering interior PMSM (IPMSM) or synchronous reluctance motor (SynRM) in the fault-tolerant control of the MFCS-I: For an IPMSM or an SynRM, the reluctance torque is not zero due to L d = L q . In this case, the proposed post-fault current references are infeasible to maintain a stable torque (i d_f is a cosine function, which will induce a torque fluctuation when L d = L q ). To expand the fault-tolerant capability to the interior PMSM and the SynRM-based MFCS-I, new fault-tolerant current vector trajectory should be proposed.

156/178

Appendix A

Transformation matrices

The matrices of the Park transformation (power invariant) are shown below: The matrices of Clarke transformation are shown below: V 3 , V 4 , V 5 , V 6 , V 0 and V 7 , respectively. Their action time is respectively T 1 , T 2 , T 3 , T 4 , T 5 , T 6 , T 0 and T 7 . The switching period is T s , the amplitude of basic voltage-vectors is 2 3 u bus , the amplitude of zero-vector is 0. When the referenced vector V ref is located in sector 1 (0 < θ e < π 3 ), there is

T C = 2 3           cos(θ e ) cos(θ e - 2π
T clarke = 2 3           1 - 1 2 - 1 2 0 √ 3 2 - √ 3 2 1 2 1 2 1 2           (A.3a) T -1 clarke =           1 0 1 - 1 2 √ 3 2 1 - 1 2 - √ 3 2 1           (A.
V ref T s = V 1 T 1 + V 2 T 2 (B.1)
According to the law of sines, we obtain According to the small-signal model, the system transfer functions can be derived as

H1

38 v

 38 to students in French at INSA de Lyon. Then, I would like to send my sincere thanks to the jury of my thesis: Prof. Babak NAHID-MOBARAKEH, Prof. Olivier BETHOUX and Prof. Demba DIALLO, for their valuable and professional comments and suggestions which significantly improved the quality of this manuscript. iii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.1 Concept of multi-functional converter system (MFCS) . . . . . . . . . . . . . . . 28 1.1.1 Basic MFCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.1.2 Other variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.1.2.1 Variant 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.1.2.2 Variant 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.1.2.3 Variant 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.1.3 Integrated on-board charger . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.2 State of the art of three-phase fault-tolerant motor drives . . . . . . . . . . . . . 36 1.2.1 Fault-tolerant solutions without using NP . . . . . . . . . . . . . . . . . 37 1.2.2 Fault-tolerant solutions using NP . . . . . . . . . . . . . . . . . . . . . .List of FiguresR.1 Systèmes de traction électrique. (a) Configuration à un étage avec un ESS haute tension. (b) Configuration à deux étages avec un ESS basse tension et un convertisseur intermédiaire DC/DC boost. . . . . . . . . . . . . . . . . . . . . . R.2 Schéma électrique du MFCS de base. . . . . . . . . . . . . . . . . . . . . . . . . R.3 Schéma du CFNI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.4 Schéma du SFNI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.5 Circuit équivalent du MFCS basé sur un PMSM. . . . . . . . . . . . . . . . . . . R.6 Schéma de principe de ZSVIPWM. . . . . . . . . . . . . . . . . . . . . . . . . . R.7 Schéma électrique du MFCS-I (ajout d'une inductance en série L aux ). . . . . . . R.8 Schéma électrique du circuit du MFCS-II avec ajout d'un bras de pont. . . . . . R.9 Rapport de surélévation effectif r 0 et DCSUR r 1 du MFCS-II en fonction du rapport cyclique α F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.10 Stratégie de commande universelle pour le MFCS-I et le MFCS-II. . . . . . . . . R.11 Schéma du circuit du MFCS-I lorsque la phase A est ouverte. . . . . . . . . . . . R.12 Trajectoire théorique des courants à imposer après un défaut de circuit ouvert sur la phase A. (a) Représentation spatiale des courants et projection sur le plan α-β dans des conditions normales et après défaut. (b) Évolution de la projection des courants sur le plan α-β en fonction de m 0 . (c) Chronogrammes des courants dans le repère d-q-0 après défaut pour différents m 0 . (d) Chronogrammes des courants dans le repère a-b-c après défaut pour différents m 0 . . . . . . . . . . . . R.13 Cadre commande uniforme (avec ou sans faute). . . . . . . . . . . . . . . . . . . G.1 Electric traction systems. (a) Single-stage configuration with a high-voltage ESS. (b) Two-stage configuration with a low-voltage ESS and an intermediate DC/DC boost covnerter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Schematic of the basic MFCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Specific circuits of the basic MFCS under different switching-states. (a) 111. (b) 000. (c) 100. (d) 110. (e) 010. (f) 011. (g) 001. (h) 101. . . . . . . . . . . . . . ix 1.3 A two-stage boost converter using the NP of a motor. . . . . . . . . . . . . . . . 32 1.4 Energy distribution system in HEVs. (a) Conventional configuration. (b) Integrated configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5 Two-motor type MFCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.6 Single-phase integrated battery charger for an electric scooter. . . . . . . . . . . 35 1.7 A propulsion/charging system for EVs. . . . . . . . . . . . . . . . . . . . . . . . 35 1.8 Single-phase integrated battery charger for a four-wheel EV. . . . . . . . . . . . 36 1.9 Capacitor-based four-leg inverter (CFI). . . . . . . . . . . . . . . . . . . . . . . 38 1.10 Switch-based four-leg inverter (SFI). . . . . . . . . . . . . . . . . . . . . . . . . 39 1.11 Capacitor-based four-leg neutral-point-connected inverter (CFNI). . . . . . . . . 39 1.12 Switch-based four-leg neutral-point-connected inverter (SFNI). . . . . . . . . . . 40 2.1 Schematic diagram of a three-phase PMSM. . . . . . . . . . . . . . . . . . . . . 45 2.2 Circuit configuration of a conventional PMSM drive. . . . . . . . . . . . . . . . 51 2.3 Circuit configuration of a PMSM-based basic MFCS. . . . . . . . . . . . . . . . 51 2.4 d-q-0 equivalent circuit of the PMSM-based basic MFCS. . . . . . . . . . . . . . 56 2.5 Visualized equivalent circuit of the PMSM-based basic MFCS. . . . . . . . . . . 57 2.6 Generalized equivalent circuits of a basic MFCS. (a) In motoring pattern. (b) In regenerative braking pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.7 Effect of superposing a third harmonic voltage in the carrier based modulation. . 60 2.8 The evolution of α h when the classic SVPWM is used. . . . . . . . . . . . . . . 60 2.9 Schematic diagram of ZSVIPWM. . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.10 Schematic diagram of regular sampling method. . . . . . . . . . . . . . . . . . . 62 2.11 Five specific cases of the basic MFCS. . . . . . . . . . . . . . . . . . . . . . . . . 64 2.12 Analysis of DCSUR with respect to different r 0 and r 2 . . . . . . . . . . . . . . . 65 2.13 Detailed comparisons of voltage and current waveforms between the basic MFCS (blue) and a conventional drive (red). . . . . . . . . . . . . . . . . . . . . . . . . 67 2.14 52.5 W test-bench. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.15 Zero-sequence inductance measurement. (a) Zero-sequence configuration. (b) Step response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.16 Simulation and experiment results of the equivalent DC/DC boost effect. (a) Simulation result. (b) Experimental result. . . . . . . . . . . . . . . . . . . . . . 70 2.17 Steady-state performance (motor speed, torque and DC-bus voltage) at the nominal condition. (a) The basic MFCS with a 15 V DC-source. (b) The conventional drive with a 30 V DC-source. . . . . . . . . . . . . . . . . . . . . . 71 x 2.18 Steady-state performance (neutral current, phase-currents, phase-voltages and FFT analysis) at the nominal condition. (a), (b) The basic MFCS with a 15 V DC-source. (c), (d) The conventional drive with a 30 V DC-source. . . . . . . . 3.1 Circuit configuration of the MFCS-I by using a series inductor (L aux ). . . . . . . 3.2 Equivalent circuit of the MFCS-I. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Analysis of the effective step-up ratio of the MFCS-I. . . . . . . . . . . . . . . . 3.4 Detailed comparison of voltage and current curves between the MFCS-I and a conventional drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Evaluation of R f lo with respect to a L . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Circuit configuration of the MFCS-II. . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Equivalent circuit of the MFCS-II. . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Analysis of the effective step-up ratio of the MFCS-II. . . . . . . . . . . . . . . . 3.9 Block diagram of d-q current-loop after decoupling. . . . . . . . . . . . . . . . . 3.10 Block diagram of the differential flatness-based control. . . . . . . . . . . . . . . 3.11 Universal control strategy for the MFCS-I and the MFCS-II. . . . . . . . . . . . 3.12 Test-bench of a 1.2 kW PMSM drive. . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Phase-A voltage, neutral voltage, phase-A current and neutral current in standstill conditions. (a) The basic MFCS without a series inductor. (b) The MFCS-I with a 1.3 mH series inductor. (c) The MFCS-I with a 13 mH series inductor. . . . . 3.14 Phase-A voltage, neutral voltage, phase-A current and neutral current in motoring conditions (1000 rpm and 2 N•m). (a), (b) The basic MFCS without a series inductor. (c), (d) The MFCS-I with a 1.3 mH series inductor. (e), (f) The MFCS-I with a 13 mH series inductor. . . . . . . . . . . . . . . . . . . . . . . . 3.15 Circuit diagrams of two PMSM drives. (a) MFCS-I with a 13 mH series inductor. (b) two-stage converter-based drive. . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Starting performance of the boost function. (a) DC-bus voltage and neutral curent in the MFCS-I with a 13 mH inductor. (b) DC-bus voltage and inductor current in the two-stage converter-based drive. . . . . . . . . . . . . . . . . . . . 3.17 Starting performance of the motor. (a) Motor speed and DC-bus voltage in the MFCS-I with a 13 mH inductor. (b) Motor speed and DC-bus voltage in the two-stage converter-based drive. (c) Phase-currents and neutral current in the MFCS-I with a 13 mH inductor. (d) Phase-currents and inductor current in the two-stage converter-based drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 3.18 Steady-state performance. (a) Phase-currents and neutral current i N in the MFCS-I with a 13 mH inductor. (b) Phase-currents and inductor current i Laux in the two-stage converter-based drive. (c) FFT analysis of phase-currents in the MFCS-I with a 13 mH inductor. (d) FFT analysis of phase-currents in the two-stage converter-based drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.19 Dynamic performance comparison. (a) The MFCS-I with a 13 mH series inductor. (b) The two-stage converter-based drive. . . . . . . . . . . . . . . . . . . . . . . 106 3.20 Regenerative braking performance. (a) Motor speed and neutral current in the MFCS-I with a 13 mH inductor. (b) Motor speed and inductor current in the two-stage converter-based drive. (c) Zoomed in phase-currents and neutral current in the MFCS-I with a 13 mH inductor. (d) Zoomed in phase-currents and inductor current in the two-stage converter-based drive. (e) FFT analysis of phase-currents in the MFCS-I with a 13 mH inductor. (f) FFT analysis of phase-currents in the two-stage converter-based drive. . . . . . . . . . . . . . . 107 3.21 Relationship between u bus and α F . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.22 Steady-state performance of the MFCS-II: DC-source voltage, phase-A current and neutral current. (a) When the motor is operated at 1000 rpm. (b) When the motor is operated at 2000 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.23 Steady-state performance of the MFCS-II when the motor is operated under the condition of 200 rpm with the full load. (a) Phase-currents and neutral current. (b) FFT analysis of phase currents . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.24 Dynamic performance of the MFCS-II. . . . . . . . . . . . . . . . . . . . . . . . 111 4.1 Circuit configuration of the MFCS-I when phase-A is open. . . . . . . . . . . . . 116 4.2 Theoretical currents in the post-fault condition (phase-A is open). (a) Space current vectors and their projections on α-β plane under normal and fault-tolerant conditions. (b) Trend of i s_f pj with respect to m 0 . (c) d-q-0 currents under fault-tolerant conditions for different m 0 . (d) Phase-currents under fault-tolerant conditions for different m 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3 Current overrating ratio (m i ) with respect to motor speed and system efficiency. 124 4.4 Zero-sequence equivalent circuits. (a) In healthy conditions. (b) In the post-fault condition (phase-A is open). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.5 Uniform control framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.6 Simulation results of a fault-tolerant transition of the 52.5 W PMSM based MFCS-I when the motor is operated under 1000 rpm with a 25 mN•m load. . . . 132 xii 4.7 Comparison of the phase-current references and actual phase-currents under the fault-tolerant mode in the 52.5 W PMSM based MFCS-I. . . . . . . . . . . . . . 4.8 Experimental results of a fault-tolerant transition when the motor is operated under 2000 rpm without load in the 52.5 W PMSM based MFCS-I. (a) Phasecurrents, torque, speed and DC-bus voltage. (b) d-axis current, q-axis current, zero-sequence current and neutral current. . . . . . . . . . . . . . . . . . . . . . 4.9 Experimental comparison of phase-current references and actual phase-currents under the fault-tolerant condition when the motor is operated at 2000 rpm without load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Experimental results of a fault-tolerant transition when the motor is operated under 2000 rpm with a 60 mN•m load in the 52.5 W PMSM based MFCS-I. (a) Phase-currents, torque, speed and DC-bus voltage. (b) d-axis current, q-axis current, zero-sequence current and neutral current. . . . . . . . . . . . . . . . . 4.11 Experimental comparison between phase-current references and actual phasecurrents under the fault-tolerant mode when the motor is operated at 2000 rpm with a 60 mN•m load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Experimental results of the 52.5 W PMSM based MFCS-I under fault-tolerant conditions with dynamic tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 An experimental trial for the choose of ε th . . . . . . . . . . . . . . . . . . . . . . 4.14 Experimental results of a fault-tolerant transition process when the motor is operated under a high-speed (262 rad/s) in no load condition: ε, fault flag, DC-bus voltage u bus , average value of the DC-bus voltage u bus_f ilter and torque T e .141 4.15 Experimental results of a fault-tolerant transition process when the motor is operated under high-speed (262 rad/s) in no load condition. (a) Zoomed-in DC-bus voltage and phase-currents. (b) Current vectors during the transition. . 4.16 Experimental results of a fault-tolerant transition process when the motor is operated under low-speed (104.7 rad/s) with load (2 N•m) condition: ε, fault flag, DC-bus voltage and torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Experimental results of a fault-tolerant transition process when the motor is operated under low-speed (104.7 rad/s) with load (2 N•m) condition. (a) Zoomedin DC-bus voltage and phase-currents. (b) Current vectors during the transition. 4.18 Steady-state phase-currents and neutral currents under post-fault operations. (a) At a very low speed (20.9 rad/s). (b) At the full speed (314.2 rad/s). . . . . . . 4.19 Steady-state performance of the MFCS-I when it is operated in the healthy condition (157.1 rad/s and 4 N•m load). (a) Phase-currents and neutral current. (b) THD of phase-currents. (c) Motor torque. . . . . . . . . . . . . . . . . . . . xiii 4.20 Steady-state performance of the MFCS-I when it is operated in the fault-tolerant condition (157.1 rad/s and 2.2 N•m load). (a) Phase-currents and neutral current. (b) THD of phase-currents. (c) Motor torque. . . . . . . . . . . . . . . . . . . . 148 4.21 Experimental results of a dynamic test when the motor is operated under healthy operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.22 Experimental results of a dynamic test when the motor is operated under postfault operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.1 Explanations for the calculations of inverter's mean duty-cycle by using the classic SVPWM. (a) Voltage vectors. (b) When V ref is located in sector 1. (c) The shape of the 7-segment SVPWM in sector 1. . . . . . . . . . . . . . . . . . . . . 161 C.1 Simplified equivalent circuit of the MFCS. . . . . . . . . . . . . . . . . . . . . . 164 C.2 A simple control strategy for the basic MFCS. . . . . . . . . . . . . . . . . . . . 164 xiv List of Tables R.1 Comparaison des HEV commerciaux en ce qui concerne les caractéristiques de conception et les spécifications publiées. . . . . . . . . . . . . . . . . . . . . . . R.2 Principales caractéristiques du MFCS par rapport au variateur conventionnel. . R.3 Caractéristiques du MFCS-I par rapport aux quatre topologies tolérantes aux fautes typiques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.1 Comparison of commercial HEVs with respect to design features and published specifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Characteristics of two-level three-phase fault-tolerant topologies towards OPFs. . 2.1 Basic parameters of the 52.5 W PMSM-based test-bench. . . . . . . . . . . . . . 2.2 The main characteristics of the basic MFCS compared to the conventional drive. 3.1 The parameters of the 1.2 kW test-bench. . . . . . . . . . . . . . . . . . . . . . 3.2 Parameters of controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Characteristics of the MFCS-I compared to the two-stage converter-based drive. 4.1 d-q-0 currents under post-fault conditions considering different fault locations. . 4.2 Characteristics of the MFCS-I compared to the four typical fault-tolerant topologies.152 B.1 Duty-cycle calculations by using the classic SVPWM. . . . . . . . . . . . . . . . xv LPF low-pass filter. MFCS Multi-functional converter system.MFCS-IA variant of the basic MFCS by using a series inductor.MFCS-IIA variant of the basic MFCS by using an auxiliary inverter leg.

  Average duty-cycle of the traction inverter. α ABC := [α A α B α C ] T Duty-cycles of inverter's legs.α dq0 := [α d α q α 0 ] T Duty-cycles of the inverter converted into d-q-0 coordinate. Φ ABC := [Φ A Φ B Φ C ] T Stator flux linkages of a PMSM in A-B-C coordinate. e ABC := [e A e B e C ] T Back-EMFs of a PMSM.

xixu

  invABC := [u AO u BO u CO ] T Terminal voltages of the inverter. u invdq0 := [u invd u invq u inv0 ] T Terminal voltages of the inverter converted to d-q-0 coordinate. ∆E Energy provided by the power source u in from point b to c. ∆i A Current ripple on phase-A. ∆u bus Voltage ripple on the DC-bus capacitor. εω Integral action of the speed-loop. η Total efficiency. ω Etraj Desired natural frequency of the second-order filter. ω m Mechanical angular velocity of the rotor. Φ f Permanent magnet flux linkage. θ e Electrical position of the rotor.

2

 2 Proportion of zero-sequence voltage to the DC-source voltage.

Figure R. 1 :

 1 Figure R.1: Systèmes de traction électrique. (a) Configuration à un étage avec un ESS haute tension. (b) Configuration à deux étages avec un ESS basse tension et un convertisseur intermédiaire DC/DC boost.

Figure R. 2 :Figure R. 3 :Figure R. 4 :

 234 Figure R.2: Schéma électrique du MFCS de base.

entre 0 et 1 ,

 1 la plage accessible pour α h = (α A + α B + α C )/3 est aussi (0-1). Par conséquent, α h peut être parfaitement représenté par le hacheur virtuel dans le modèle moyen, où α h désigne le rapport cyclique de conduction de S v1 . Dans ce cas, la troisième équation de (R.1) est visualisée par le contour 'K' sur la figure R.5. La quatrième équation de (R.1) est représentée par le noeud 'P'. Par conséquent, nous pouvons affirmer que les deux dernières équations de (R.1) peuvent être représentées par un convertisseur élévateur DC/DC équivalent. L'inductance homopolaire 7/178 L 0 /3 joue le rôle d'inductance dans la fonction boost. Enfin, l'ensemble du système MFCS peut être considéré comme un variateur pour moteur à courant continu à deux étages. Le hacheur équivalent du moteur à courant continu équivalent peut être considéré comme une charge équivalente pour le convertisseur élévateur DC/DC équivalent. Le courant de charge équivalent i lo est égal à (α d i d + α q i q ).

Figure R. 5 :

 5 Figure R.5: Circuit équivalent du MFCS basé sur un PMSM.

Figure R. 8 :

 8 Figure R.8: Schéma électrique du circuit du MFCS-II avec ajout d'un bras de pont.

4 )Figure R. 9 : 178 Figure R. 10 :Chapitre 4 :

 49178104 Figure R.9: Rapport de surélévation effectif r 0 et DCSUR r 1 du MFCS-II en fonction du rapport cyclique α F .

Figure R. 12 :

 12 Figure R.12: Trajectoire théorique des courants à imposer après un défaut de circuit ouvert sur la phase A. (a) Représentation spatiale des courants et projection sur le plan α-β dans des conditions normales et après défaut. (b) Évolution de la projection des courants sur le plan α-β en fonction de m 0 . (c) Chronogrammes des courants dans le repère d-q-0 après défaut pour différents m 0 . (d) Chronogrammes des courants dans le repère a-b-c après défaut pour différents m 0 .

  commande est introduit, comme le montre la figure R.13. Dans la boucle externe, la vitesse du moteur et la tension moyenne du bus DC sont les principaux objectifs de contrôle. Dans la boucle interne, les courants d-q-0 doivent suivre les réféences dans (R.8). Afin de suivre les références de courant non constantes, une commande deadbeat et une commande basée sur la platitude différentielle sont proposées.

Figure R. 13 :

 13 Figure R.13: Cadre commande uniforme (avec ou sans faute).

Figure G. 1 :

 1 Figure G.1: Electric traction systems. (a) Single-stage configuration with a high-voltage ESS. (b) Two-stage configuration with a low-voltage ESS and an intermediate DC/DC boost covnerter.

. 1 .Figure 1 . 1 :

 111 Figure 1.1: Schematic of the basic MFCS.

Figure 1 . 2 :

 12 Figure 1.2: Specific circuits of the basic MFCS under different switching-states. (a) 111. (b) 000. (c) 100. (d) 110. (e) 010. (f) 011. (g) 001. (h) 101.

Figure 1 . 3 : 178 1. 1 .

 131781 Figure 1.3: A two-stage boost converter using the NP of a motor.

Figure 1 . 4 :

 14 Figure 1.4: Energy distribution system in HEVs. (a) Conventional configuration. (b) Integrated configuration.

Figure 1 . 5 :

 15 Figure 1.5: Two-motor type MFCS.

Figure 1 . 6 :

 16 Figure 1.6: Single-phase integrated battery charger for an electric scooter.

Figure 1 . 7 :

 17 Figure 1.7: A propulsion/charging system for EVs.

Figure 1 . 8 :

 18 Figure 1.8: Single-phase integrated battery charger for a four-wheel EV.

Figure 1 . 9 :

 19 Figure 1.9: Capacitor-based four-leg inverter (CFI).

From the functionality point 178 1. 2 .Figure 1 . 10 :

 1782110 Figure 1.10: Switch-based four-leg inverter (SFI).

Figure 1 . 11 : 178 Chapter 1 .Figure 1 . 12 :

 1111781112 Figure 1.11: Capacitor-based four-leg neutral-point-connected inverter (CFNI).

+ 6 )

 6 /6 = 1.5 due to the use of three extra TRIACs. The SFI has a PPRF of 1 because the situation of the motor is not changed after the fault. In terms of its SOCF, it can be calculated by (2 + 3 + 6)/6 = 1.83 due to the use of three TRIACs and two auxiliary IGBTs. The CFNI has a PPRF of 0.5 because the attainable voltage vector is reduced from 0.58 to 0.5 pu and the torque output 40/178 capability is decreased by a factor of 1/ √ 3. The SOCF of the CFNI can be calculated by (1 * √ 3 + 6)/6 = 1.29 because the TRIAC connected with the NP has to be designed sufficiently to withstand the neutral current, which is larger than the phase-current by a factor of √ 3. In the same way, the PPRF of the SFNI is evaluated as 0.58 and the SOCF is (3 * √ 3 + 6)/6 = 1.87.

Figure 2 . 1 :

 21 Figure 2.1: Schematic diagram of a three-phase PMSM.

A

  much simpler form leading to a legible model can be obtained by appropriate transformation of stator variables. To meet the condition of power conservation, we use the Park transformation (power invariant) in the following analysis, whose transformation matrices (T C and T -1 C ) are shown in Appendix A. In the control part, the Park transformation (amplitude invariant) is used since it has no square root operation, whose transformation matrices (T P and T -1 P ) are also presented in Appendix A. If the Park transformation (amplitude invariant) is used, the expression of motor's torque equation should be multiplied by a factor of 3/2.

  7) where • e ABC := [e A e B e C ] T denotes the back-EMFs in A-B-C coordinate;

Figure 2 . 3 :

 23 Figure 2.3: Circuit configuration of a PMSM-based basic MFCS.

Figure 2 . 4 :

 24 Figure 2.4: d-q-0 equivalent circuit of the PMSM-based basic MFCS.

Figure 2 . 5 :

 25 Figure 2.5: Visualized equivalent circuit of the PMSM-based basic MFCS.

Figure 2 . 6 :

 26 Figure 2.6: Generalized equivalent circuits of a basic MFCS. (a) In motoring pattern. (b) In regenerative braking pattern.

Chapter 2 .Figure 2 . 7 :

 227 Figure 2.7: Effect of superposing a third harmonic voltage in the carrier based modulation.

Figure 2 . 8 :

 28 Figure 2.8: The evolution of α h when the classic SVPWM is used.

Figure 2 . 9 :Figure 2 . 10 :

 29210 Figure 2.9: Schematic diagram of ZSVIPWM.

Figure 2 . 11 :

 211 Figure 2.11: Five specific cases of the basic MFCS.

Figure 2 . 13 :

 213 Figure 2.13: Detailed comparisons of voltage and current waveforms between the basic MFCS (blue) and a conventional drive (red).

Figure 2 .

 2 Figure 2.14: 52.5 W test-bench.

Figure 2 . 15 :

 215 Figure 2.15: Zero-sequence inductance measurement. (a) Zero-sequence configuration. (b)Step response.

Figure 2 . 16 :Figure 2 . 17 :

 216217 Figure 2.16: Simulation and experiment results of the equivalent DC/DC boost effect. (a) Simulation result. (b) Experimental result.

Figure 2 . 18 :

 218 Figure 2.18: Steady-state performance (neutral current, phase-currents, phase-voltages and FFT analysis) at the nominal condition. (a), (b) The basic MFCS with a 15 V DC-source. (c), (d) The conventional drive with a 30 V DC-source.

1 79 3. 1 . 3 81 3. 2 85 3. 2 . 1 87 3. 3 89 3. 3 . 1 89 3. 3 . 2 92 3. 3 . 3 95 3. 4 96 3. 4 . 1 96 3. 4 . 2

 17913812852187389318932923395496419642 Average model and equivalent circuit . . . . . . . . . . . . . . . . . . . 77 3.1.2 Steady-state characteristic . . . . . . . . . . . . . . . . . . . . . . . . . Series inductor selection . . . . . . . . . . . . . . . . . . . . . . . . . . Improvement 2: using an auxiliary leg (MFCS-II) . . . . . . . . . . Average model and equivalent circuit . . . . . . . . . . . . . . . . . . . 85 3.2.2 Steady-state characteristic . . . . . . . . . . . . . . . . . . . . . . . . . Universal closed-loop control strategy . . . . . . . . . . . . . . . . . Design of FOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential flatness-based control for the equivalent boost . . . . . . . Control framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . Current ripples analysis of the MFCS-I . . . . . . . . . . . . . . . . . Comparison between the MFCS-I (with the 13 mH series inductor) and a real two-stage converter-based drive . . . . . . . . . . . . . . . . 101 3.4.3 Performance of the MFCS-II . . . . . . . . . . . . . . . . . . . . . . . 108 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 3 . 1 :

 31 Figure 3.1: Circuit configuration of the MFCS-I by using a series inductor (L aux ).

Figure 3 . 2 :

 32 Figure 3.2: Equivalent circuit of the MFCS-I.

Figure 3 . 3 :

 33 Figure 3.3: Analysis of the effective step-up ratio of the MFCS-I.

Figure 3 . 5 :

 35 Figure 3.5: Evaluation of R f lo with respect to a L .

Figure 3 . 6 :

 36 Figure 3.6: Circuit configuration of the MFCS-II.

  19b) If the Park transformation (amplitude invariant) is used, (3.19a) and (3.19b) become d-q-0 average model of the MFCS-II obtained by the Park transformation (amplitude invariant):

Figure 3 . 7 :

 37 Figure 3.7: Equivalent circuit of the MFCS-II.

Figure 3 . 8 :

 38 Figure 3.8: Analysis of the effective step-up ratio of the MFCS-II.

5 throughout.

 5 So (α F -0.5) becomes its control DOF for the equivalent boost function. Therefore, we can use a universal closed-loop control strategy to control MFCS-I and MFCS-II. It has two control objectives. The first one is the motor torque or speed and the second one is the equivalent boost. For the motor, the classic FOC controller can still be used since the connected NP does not affect the d-q model. It consists of two current controllers and one speed controller.

Figure 3 . 9 :

 39 Figure 3.9: Block diagram of d-q current-loop after decoupling.

  [y (p) -y (p) traj ] + K p-1 [y (p-1) -y (p-1) traj ] + ... + K 0 [y -y traj ] = 0 (3.38) 92/178 where the set of control gains K p-1 ,...K 0 can be chosen such that the closed-loop characteristic polynomial is Hurwitz. Different from input-output linearization control for nonlinear systems, flatness-based control does not lead to zero dynamics. Consequently, the stability of the whole closed-loop system can be guaranteed. The block diagram of the differential flatness-based control is shown in figure 3.10[95][96].

Figure 3 . 10 :

 310 Figure 3.10: Block diagram of the differential flatness-based control.

43 )

 43 Defining the tracking error as e E = (E -E traj ) and conducting the time derivative to(3.43), it yields ... e E + K dE ëE + K pE ėE + K iE e E = 0 (3.44)

Figure 3 . 11 :

 311 Figure 3.11: Universal control strategy for the MFCS-I and the MFCS-II.

Figure 3 . 12 :

 312 Figure 3.12: Test-bench of a 1.2 kW PMSM drive.

[Figure 3 . 13 :

 313 Figure 3.13: Phase-A voltage, neutral voltage, phase-A current and neutral current in standstill conditions. (a) The basic MFCS without a series inductor. (b) The MFCS-I with a 1.3 mH series inductor. (c) The MFCS-I with a 13 mH series inductor.

Figure 3 . 14 : 178 3. 4 . 2

 31417842 Figure 3.14: Phase-A voltage, neutral voltage, phase-A current and neutral current in motoring conditions (1000 rpm and 2 N•m). (a), (b) The basic MFCS without a series inductor. (c), (d) The MFCS-I with a 1.3 mH series inductor. (e), (f) The MFCS-I with a 13 mH series inductor.

Figure 3 . 15 :Table 3 . 2 :K 60 ζ 3 . 4 . 2 . 1 Figure 3 . 16 :

 31532603421316 Figure 3.15: Circuit diagrams of two PMSM drives. (a) MFCS-I with a 13 mH series inductor. (b) two-stage converter-based drive.

Figure 3 . 17 :

 317 Figure 3.17: Starting performance of the motor. (a) Motor speed and DC-bus voltage in the MFCS-I with a 13 mH inductor. (b) Motor speed and DC-bus voltage in the two-stage converter-based drive. (c) Phase-currents and neutral current in the MFCS-I with a 13 mH inductor. (d) Phase-currents and inductor current in the two-stage converter-based drive.

Figure 3 . 18 :8% 3 . 4 . 2 . 4

 3183424 Figure 3.18: Steady-state performance. (a) Phase-currents and neutral current i N in the MFCS-I with a 13 mH inductor. (b) Phase-currents and inductor current i Laux in the two-stage converter-based drive. (c) FFT analysis of phase-currents in the MFCS-I with a 13 mH inductor. (d) FFT analysis of phase-currents in the two-stage converter-based drive.

Figure 3 . 19 :

 319 Figure 3.19: Dynamic performance comparison. (a) The MFCS-I with a 13 mH series inductor. (b) The two-stage converter-based drive.

Ef f 2 =Figure 3 . 20 :

 2320 Figure 3.20: Regenerative braking performance. (a) Motor speed and neutral current in the MFCS-I with a 13 mH inductor. (b) Motor speed and inductor current in the two-stage converter-based drive. (c) Zoomed in phase-currents and neutral current in the MFCS-I with a 13 mH inductor. (d) Zoomed in phase-currents and inductor current in the two-stage converterbased drive. (e) FFT analysis of phase-currents in the MFCS-I with a 13 mH inductor. (f) FFT analysis of phase-currents in the two-stage converter-based drive.

Figure 3 . 21 :

 321 Figure 3.21: Relationship between u bus and α F .

Figure 3 . 22 :

 322 Figure 3.22: Steady-state performance of the MFCS-II: DC-source voltage, phase-A current and neutral current. (a) When the motor is operated at 1000 rpm. (b) When the motor is operated at 2000 rpm.

Figure 3 .

 3 Figure 3.23 shows the steady-state performance of the MFCS-II when the motor is operated at 200 rpm with the full load (4 N•m). Obviously, this verifies that the motor can still operate at the nominal load condition in the MFCS-II. The FFT analysis of phase-currents is further presented on the right. The THD of phase currents is around 4.87%.

Figure 3 . 23 : 178 3. 4 . 3 . 3

 323178433 Figure 3.23: Steady-state performance of the MFCS-II when the motor is operated under the condition of 200 rpm with the full load. (a) Phase-currents and neutral current. (b) FFT analysis of phase currents

Figure 3 . 24 :

 324 Figure 3.24: Dynamic performance of the MFCS-II.

Figure 4 . 1 :

 41 Figure 4.1: Circuit configuration of the MFCS-I when phase-A is open.

Figure 4 . 2 :

 42 Figure 4.2: Theoretical currents in the post-fault condition (phase-A is open). (a) Space current vectors and their projections on α-β plane under normal and fault-tolerant conditions. (b) Trend of i s_f pj with respect to m 0 . (c) d-q-0 currents under fault-tolerant conditions for different m 0 . (d) Phase-currents under fault-tolerant conditions for different m 0 .

  figure 4.2(c), the energy (∆E _f ) provided by the DC-source (u in ) from point 'b' to 'c' can be calculated by ∆E _f = c b P in_f (t)dt = 3u in c b i 0_f (t)dt (4.11)

figure 4 .Figure 4 . 3 :

 443 figure 4.3 where Φ f , p and u in correspond to the parameters of the 1.2 kW PMSM. It can be

• 2 ¢ 2 LFigure 4 . 4 :

 2244 Figure 4.4: Zero-sequence equivalent circuits. (a) In healthy conditions. (b) In the post-fault condition (phase-A is open).

0_n

  may result in the under-voltage or over-voltage hazard on the DC-bus over a large time scale. Therefore, to maintain a stable DC-bus voltage, a PI controller is used to regulate the mean of the DC-bus voltage. The control parameters of the PI controller can be designed in the way 127/178 of a common DC/DC boost converter. Based on the analysis of post-fault power conversion process, the oscillation period of the DC-bus voltage is equal to the electrical period. Over one electrical period, the mean of the DC-bus voltage is invariable. Therefore, a low-pass filter (LPF) is required to obtain the actual mean of the DC-bus voltage (u bus_f ilter ). The cutoff frequency of the LPF should be lower than the fundamental frequency of the motor. Finally, i * 0_n is determined by the combination of the direct calculation (4.21) and the DC-bus voltage controller.

Figure 4 . 5 :

 45 Figure 4.5: Uniform control framework.

Figure 4 . 6 :

 46 Figure 4.6: Simulation results of a fault-tolerant transition of the 52.5 W PMSM based MFCS-I when the motor is operated under 1000 rpm with a 25 mN•m load.

  i B ,i C [A] i* A ,i* B ,i* C [A] Time [s]

Figure 4 . 7 :

 47 Figure 4.7: Comparison of the phase-current references and actual phase-currents under the fault-tolerant mode in the 52.5 W PMSM based MFCS-I.

Figure 4 . 8 :

 48 Figure 4.8: Experimental results of a fault-tolerant transition when the motor is operated under 2000 rpm without load in the 52.5 W PMSM based MFCS-I. (a) Phase-currents, torque, speed and DC-bus voltage. (b) d-axis current, q-axis current, zero-sequence current and neutral current.

figure 4 .Figure 4 . 9 :

 449 figure 4.9 illustrates the detailed comparison between the theoretical and actual remaining

Figure 4 .

 4 10 depicts the experimental results where the motor speed is 2000 rpm and the load is 60 mN•m. First, under the healthy operation, n is controlled at 2000 rpm steadily. T e is about 60 mN•m, which is almost half the nominal torque. The torque ripple is around 9 mN•m. u bus is still boosted at 30 V stably. i d_n is 0 A and i q_n is about 1.79 A. i 0_n is -0.46 A and i N _n is 1.37 A.

Figure 4 . 10 :

 410 Figure 4.10: Experimental results of a fault-tolerant transition when the motor is operated under 2000 rpm with a 60 mN•m load in the 52.5 W PMSM based MFCS-I. (a) Phase-currents, torque, speed and DC-bus voltage. (b) d-axis current, q-axis current, zero-sequence current and neutral current.

Figure 4 . 11 :

 411 Figure 4.11: Experimental comparison between phase-current references and actual phasecurrents under the fault-tolerant mode when the motor is operated at 2000 rpm with a 60 mN•m load.

Figure 4 . 12 :

 412 Figure 4.12: Experimental results of the 52.5 W PMSM based MFCS-I under fault-tolerant conditions with dynamic tests.

Figure 4 . 13 :

 413 Figure 4.13: An experimental trial for the choose of ε th .

46 FaultFigure 4 . 15 :

 46415 Figure 4.15: Experimental results of a fault-tolerant transition process when the motor is operated under high-speed (262 rad/s) in no load condition. (a) Zoomed-in DC-bus voltage and phase-currents. (b) Current vectors during the transition.

FaultFigure 4 . 16 :FaultFigure 4 . 17 :

 416417 Figure 4.16: Experimental results of a fault-tolerant transition process when the motor is operated under low-speed (104.7 rad/s) with load (2 N•m) condition: ε, fault flag, DC-bus voltage and torque.

Figure 4 . 18 :

 418 Figure 4.18: Steady-state phase-currents and neutral currents under post-fault operations. (a) At a very low speed (20.9 rad/s). (b) At the full speed (314.2 rad/s).

Figure 4 . 19 :

 419 Figure 4.19: Steady-state performance of the MFCS-I when it is operated in the healthy condition (157.1 rad/s and 4 N•m load). (a) Phase-currents and neutral current. (b) THD of phase-currents. (c) Motor torque.

Figure 4 . 20 :

 420 Figure 4.20: Steady-state performance of the MFCS-I when it is operated in the fault-tolerant condition (157.1 rad/s and 2.2 N•m load). (a) Phase-currents and neutral current. (b) THD of phase-currents. (c) Motor torque.

  eration, loading and deloading. To assess the dynamic performance in post-fault operations, contrastive experiments are further carried out on the 1.2 kW PMSM where it is operated in healthy conditions and fault-tolerant conditions respectively. The dynamic setup is set as follows: ω m is 104.7 rad/s before 2 s. It is increased to 209.4 rad/s at 2 s and is reset to 104.7 rad/s at 6 s. A 2 N•m load is applied and removed at 10 s and 15 s respectively.

Figure 4 . 21 :

 421 Figure 4.21: Experimental results of a dynamic test when the motor is operated under healthy operations.

Figure 4 . 22 :

 422 Figure 4.22: Experimental results of a dynamic test when the motor is operated under post-fault operations.

0• In chapter 3 ,

 3 -axis model can be equivalent to a DC/DC boost function due to the connection of the NP. Moreover, three equivalent circuits are developed to facilitate the understanding of the equivalent boost function, which indicates that the average duty cycle of the inverter can be a control DOF for the equivalent boost. To enable the average duty cycle of the inverter (i.e. the control DOF of the equivalent boost), a specialized modulation technique named ZSVIPWM is proposed. As a result, the motor function and the equivalent boost function can be controlled independently. According to the theoretical analysis, the maximum effective step-up ratio of the equivalent boost is 2 and the DCSUR of the basic MFCS is 1. Simulation and experiment are carried out on a 52.5 W test-bench to verify the phenomenon of the equivalent boost and the proposed ZSVIPWM. By comparison, it can be concluded that the basic MFCS supplied by a 15 V DC-source (without a real boost stage) can have the similar steady-state and dynamic performances with a standard drive 153/178 supplied by a 30 V DC-source. However, as a compromise, the current ripple of the basic MFCS is bigger.Key contribution:We explicitly explain the mechanism of the basic MFCS. A specialized modulation technique named ZSVIPWM is proposed so that the control DOFs of the equivalent boost function and the motor function can be synthesized together for digital implementations. two improved solutions are proposed to ameliorate the basic MFCS, which lead to two new variants named MFCS-I and MFCS-II respectively. For the former, a series inductor connected with the NP is employed to reduce the current ripple. A sizing method of the series inductor is given based on a concept of a re-floating NP. Normally, a series inductor with the value of 5 times the zero-sequence inductance of the motor is sufficient. For the latter, an auxiliary inverter leg is used to improve the effective step-up ratio of the equivalent boost. Theoretical analysis shows that both the step-up ratio and DCSUR of the MFCS-II can be increased when the duty cycle of the auxiliary leg is decreased from 1 to 0.5. To control different MFCSs, a universal closed-loop control strategy is proposed, in which the classic FOC can still be used to control the motor function and a differential flatness-based controller is used to regulate the DC-bus voltage.Experimental results carried on a 1.2 kW experimental platform are given to verify the effectiveness of the MFCS-I and MFCS-II. Specifically, the MFCS-I is compared with a conventional two-stage converter-based drive in detail. The results illustrate that the efficiency of the MFCS-I is improved by a factor of 3.1%-3.4% than that of the two-stage converter-based drive due to the reduction of switching losses. In the MFCS-II, the motor can be operated in a wide range (from 200 rpm to 2000 rpm) even if a 40 V DC-source

  154/178 the post-fault d-axis current is a cosine function. The post-fault 0-axis current contains and a constant component and a second-order component in addition to the the sinusoidal component. The remaining phase-currents in post-fault conditions are also not exactly sinusoidal. The post-fault power conversion process is analyzed in detail, which indicates that the DC-bus capacitor becomes an energy buffer to balance the mismatch between the transient input power and output power. A deadbeat predictive current controller and a differential flatness-based current controller are developed to track the non-constant current references. Simulation and experiment are carried on a 52.5 W and a 1.2 kW

  the Park transformation (amplitude invariant) are shown below:

  3b) The matrices of rotation transformation are shown below: e ) sin(θ e ) 0 -sin(θ e ) cos(θ e ) e ) -sin(θ e ) 0 sin(θ e ) cos(θ e six basic voltage-vectors and two zero-vectors (see figure B.1(a)) are defined as V 1 , V 2 ,

Figure B. 1 :Figure C. 1 :Figure C. 2 :

 112 Figure B.1: Explanations for the calculations of inverter's mean duty-cycle by using the classic SVPWM. (a) Voltage vectors. (b) When V ref is located in sector 1. (c) The shape of the 7-segment SVPWM in sector 1.

  Average value of the DC-bus voltage.u bus_f maxPeak value of u bus .Cette thèse porte sur un système d'entraînement de machine synchrone à aimants permanents triphasée alimentée par un onduleur qui intègre un étage de conversion continu/continu, en utilisant le point neutre de la machine. Cette architecture, appelée système de convertisseur

	u bus_f min	valley value of u bus .
	u bus	DC-bus voltage.
	u in	DC-source voltage.
	u N O	Neutral point voltage.
	u XN 1amp	Amplitude of the fundamental phase-voltage.
	u * XN 1	Fundamental voltage reference.
	u XN amp	Amplitude of the actual phase-voltage.
	u * XN	Phase-voltage reference.
	u ZS	Zero-sequence voltage of the motor.

bias Voltage bias in the MFCS-I. xxi u bus_f ilter xxii Résumé multifonctionnel (MFCS), peut être avantageusement utilisée dans des applications de traction automobile dans le cas où une élévation de tension est nécessaire. Premièrement, une analyse approfondie est présentée afin de comprendre et de modéliser le MFCS. Une méthode de modulation spécifique est, de plus, proposée utilisant la particularité que le courant homopolaire n'est pas nul. Deuxièmement, deux nouvelles variantes nommées MFCS-I et MFCS-II ont été conçues afin d'améliorer le MFCS de base. Pour la première, une inductance en série est ajoutée pour diminuer l'ondulation du courant. Pour la seconde, l'ajout d'un bras d'onduleur auxiliaire améliore le rapport d'élévation effectif de la tension. Pour contrôler ces variantes de MFCSs, une stratégie de commande universelle est développée dans laquelle une commande vectorielle peut être prise en compte pour la fonction moteur et un contrôleur basé sur la platitude différentielle est utilisé pour réguler la tension du bus continu. Troisièmement, compte tenu de la connexion intrinsèque du point neutre dans le MFCS-I, son potentiel de tolérance aux fautes contre un défaut de phase ouverte est exploré pour la première fois. Une étude théorique permet de formuler les conditions pour atteindre cette propriété et une nouvelle trajectoire vectorielle de courant est proposée, générant un couple constant et une tension moyenne stable du bus continu après un défaut. Pour finir, les idées et les stratégies de contrôle présentées ci-dessus sont systématiquement validées expérimentalement sur des plateformes de prototypage rapide de lois de commande. xxiii

Table R .

 R Différents types de convertisseurs ont été étudiés dans la littérature, tels que les convertisseurs élévateurs DC/DC à une seule branche[START_REF] Chen | Electrified automotive powertrain architecture using composite DC-dc converters[END_REF], les convertisseurs DC/DC entrelacés[START_REF] Lin | An electric-vehicle ipmsm drive with interleaved front-end DC/dc converter[END_REF] et les convertisseurs DC/DC à double pont actif[START_REF] Tan | Design and performance of a bidirectional isolated DC-dc converter for a battery energy storage system[END_REF]. Avec l'aide de ce convertisseur, le système de traction électrique est capable d'obtenir une tension de bus DC suffisante en l'absence d'un pack batteries haute tension et peut gérer de manière flexible le flux de puissance entre le moteur de traction et l'ESS[START_REF] Schoenen | Advantages of a variable DC-link voltage by using a DC-dc converter in hybrid-electric vehicles[END_REF]. Cependant, l'utilisation du convertisseur DC/DC supplémentaire apporte à son tour quelques inconvénients. D'une part, cela augmente le coût et le volume de l'ensemble du système de conversion et, d'autre part, la fiabilité du système peut être dégradée car plus de dispositifs électroniques de puissance sont utilisés. Le premier objectif de cette thèse sera d'apporter une vision explicite du mécanisme de conversion du MFCS de base. Il s'agira d'expliquer comment la fonctionnalité d'élévation de tension (convertisseur DC/DC équivalent) apparaît dans ce système grâce à la simple connexion du point neutre du moteur à courant alternatif triphasé. Étant donné que le point neutre est connecté, un courant homopolaire circule. Ainsi, les méthodes de modulation conventionnelles ne sont pas applicables dans le MFCS de base. Pour répondre à ce problème, une méthode de modulation spéciale est nécessaire. • Certains inconvénients existent dans le MFCS de base comme le fait que les ondulations de courant sont plus importantes, ce qui augmente les pertes fer ou encore que le rapport d'élévation du convertisseur DC/DC équivalent est limité. Par conséquent, le second objectif sera d'améliorer ce MFCS de base. D'autre part, une comparaison complète avec le système conventionnel à deux étages est nécessaire pour clarifier les avantages et les inconvénients.

	résumées comme suit :
	• Doter un variateur de fréquence triphasé de la fonctionnalité d'un convertisseur DC/DC
	en connectant exclusivement le NP du moteur qui est rarement utilisé dans un variateur
	conventionnel.
	• Utiliser l'inductance homopolaire du moteur en tant qu'inductance de filtrage et réutiliser
	l'onduleur de traction comme hacheur équivalent.
	Comme le montre le schéma du MFCS de base sur la figure R.2, la source DC haute tension
	(v
	Caractéristique de conception	2008 LS 600h Hybrid Camry 2004 Prius	2010 Prius
	Puissance maximale du moteur	165 kW	105 kW	50 kW	60 kW
	Tension du bus DC	288-650 V DC	250-650 V DC 200-500 V DC 200-650 V DC
	Puissance nominale du convertisseur DC/DC	36,5 kW	30 kW	20 kW	27 kW
	Tension de la batterie	288 V	244,8 V	201,6 V	201,6 V

1: Comparaison des HEV commerciaux en ce qui concerne les caractéristiques de conception et les spécifications publiées. Nous pouvons remarquer que la tension du bus DC est généralement 2 à 3 fois plus grande que la tension de la batterie. Ceci est possible grâce à l'utilisation d'un convertisseur 2/178 DC/DC intermédiaire. Le niveau de fiabilité devient un critère de plus en plus important de manière générale et tout particulièrement dans le cadre de la traction électrique afin de garantir un meilleur niveau de service et une plus grande sécurité pour les passagers. Une manière de l'augmenter consiste à utiliser des systèmes qui sont capables de continuer à fonctionner malgré l'apparition d'un défaut électrique (circuit ouvert ou court-circuit). Dans le cadre de solutions conventionnelles d'alimentation des moteurs triphasés à partir d'un onduleur à deux niveau de tension (VSI), il est nécessaire d'ajouter des dispositifs auxiliaires afin d'atteindre cet objectif de tolérance aux fautes. Cependant, ces dispositifs auxiliaires augmentent le coût, le volume et le poids des systèmes de traction électrique. Dans le domaine des variateurs pour moteurs à courant alternatif (AC) triphasé, les systèmes de conversion multifonctionnels (MFCS) forment une catégorie spéciale de convertisseurs qui intègrent les fonctionnalités d'un convertisseur élévateur DC/DC et d'un onduleur DC/AC sur un seul VSI triphasé à deux niveaux. Ceci est rendu possible grâce à la connexion du point neutre du moteur électrique à la source d'alimentation DC, contrairement aux systèmes conventionnels pour lesquels le point neutre n'est pas connecté. Le MFCS est ainsi une alternative prometteuse pour remplacer le système de traction électrique à deux étages dans les EVs et les HEVs. Cependant, dans la littérature scientifique, le MFCS n'a pas reçu une grande attention jusqu'à présent, engendrant certaines lacunes concernant les explications du mécanisme et du principe du MFCS. Il n'y a que peu d'études approfondies à propos de la modélisation, de la méthode de modulation, de la stratégie de commande et de l'évaluation expérimentale. Objectifs Sur ces bases et sur le potentiel fort intérêt applicatif de ce système, cette thèse va étudier le MFCS de manière globale et approfondie avec les objectifs suivants : 3/178 • • Compte tenu de la connexion naturelle du point neutre dans ces systèmes, il existe une possibilité de réaliser des opérations tolérantes aux fautes sans utiliser de composants auxiliaires lorsqu'un défaut en phase ouverte (OPF) se produit dans les MFCS. Le dernier objectif sera donc d'explorer cette piste et de tester cette capacité. Chapitre 1 : Opportunités apportées par le point neutre d'un moteur triphasé Une machine à courant alternatif triphasé a trois enroulements, qui peuvent être connectés de deux manières -étoile (Y) ou triangle (∆). dans le cas de la connexion en Y, le point neutre (NP) devient accessible. Dans la plupart des cas, le NP n'est pas utilisé et caché à l'intérieur de la coque du moteur. Cependant, dans certaines applications spéciales, ce NP peut être sorti et utilisé pour obtenir des avantages supplémentaires. Le MFCS est un exemple qui utilise le NP pour obtenir un convertisseur boost DC/DC équivalent pour un moteur almenté par un VSI classique sans utiliser de composants supplémentaires. L'idée du MFCS de base remonte à environ vingt ans où il a d'abord été proposé pour les systèmes de transmission électrique dans les véhicules électriques hybrides par Toyota et Fuji Electric Co., Ltd., au Japon. Afin de réduire le coût et le volume du système de groupe motopropulseur configuré à deux étages, les ingénieurs et les chercheurs de Toyota et Fuji Electric ont tenté de supprimer l'étage DC/DC. Pour conserver cette fonctionnalité, le NP des moteurs de traction est alors volontairement connecté. Le MFCS de base se compose d'un 4/178 moteur AC avec neutre sorti, d'un VSI triphasé à deux niveaux et de deux sources DC, mais pas de convertisseur DC/DC. Les principales caractéristiques du MFCS de base peuvent être h ) est connectée du côté du bus DC. Et contrairement à une configuration conventionnelle, la source DC basse tension (v l ) est connectée au NP du moteur. Étant donné que le NP est connecté à une source d'alimentation, le MFCS de base est similaire à un convertisseur DC/DC multiphase à trois branches où les trois enroulements du moteur auront le rôle d'inductances.

  [START_REF] Aghabali | 800-V electric vehicle powertrains: Review and analysis of benefits, challenges, and future trends[END_REF] où α d est le rapport cyclique sur l'axe d; α q est le rapport cyclique sur l'axe q; α h est le rapport cyclique moyen; u bus est la tension du bus DC; u d et u q sont les tensions aux bornes du moteur

sur les axes d et q respectivement; i d et i q sont les courants dans le moteur sur les axes d et q respectivement; i bus est le courant du bus; i N est le courant dans le neutre; L d , L q et L 0 sont respectivement les inductances d-q-0 du moteur; R est la résistance du moteur; u in est la tension de la source DC; C bus est la capacité du condensateur du bus DC; i lo est un courant équivalent. D'après l'équation (R.1), il est possible de construire un circuit équivalent du MFCS, comme le montre la figure R.5. Il est bien connu que l'utilisation de la commande vectorielle facilite la commande d'un moteur à courant alternatif en découplant son couple électromagnétique et sa gestion du flux. Dans le repère d-q, un moteur à courant alternatif peut être considéré comme un moteur à courant continu équivalent. Par conséquent, nous pouvons utiliser un hacheur équivalent pour alimenter un moteur à courant continu équivalent afin de représenter la fonction motrice du MFCS. Pour incarner la signification physique de α d et α q , nous utilisons deux ponts en H virtuels nommés « H1 » et « H2 » pour piloter le moteur à courant continu équivalent. Les deux ponts en H sont construits par huit commutateurs virtuels S v1 , S v1 , S v2 , S v2 , S v3 , S v3 , S v4 et S v4 . Pour être plus précis, α d représente la différence entre les rapports cycliques de conduction de S v3 et S v4 ; α q représente la différence entre les rapports cycliques de conduction de S v2 et S v5 . Ceci permet de représenter physiquement les deux premières équations de (R.1). Afin d'obtenir une signification physique pour les deux dernières équations de (R.1), nous introduisons un hacheur virtuel 'C1' qui est construit par deux commutateurs virtuels S v1 et S v1 . Étant donné que les rapports cycliques de l'onduleur (α A , α B et α C ) sont compris

Table R . 2 :

 R2 Principales caractéristiques du MFCS par rapport au variateur conventionnel.

	u bus -u in	u XN1	*	Carrier signal
	0
		u ZS
				u XN	*
	-u in
	Figure R.6: Schéma de principe de ZSVIPWM.
	Caractéristique	Évaluation
	Performances en régime permanent Similaires
	Ratio d'utilisation de la tension	Plus grand (1 vs 0,58)
	Fonction boost intégrée	OUI vs NON
	Ondulation du courant	Plus grande
	Pertes	Plus importantes (cas d'une PMSM de 52,5 W : 11,5%)
		8/178

2) où u * XN est la référence de tension de phase réelle; u * XN 1 est la référence de tension fondamentale. Enfin, des expériences comparatives ont été réalisées sur un banc d'essai de 52,5 W. Les principales caractéristiques du MFCS par rapport au variateur classique sont résumées dans le tableau R.2. Les performances en régime permanent sont similaires. La fonction d'élévation de la tension (convertisseur DC/DC équivalent) permet bien d'augmenter le ratio d'utilisation de la tension (DCSUR). Cependant, l'ajout de la fonctionnalité boost mène à une ondulation du courant du MFCS beaucoup plus importante ainsi qu'un rendement de conversion moins bon.

Chapitre 3 : Améliorations du MFCS de base et stratégie générique de commande en boucle fermée

  Schéma électrique du MFCS-I (ajout d'une inductance en série L aux ).

	+ u bus	i bus S A	S B		S C	motor
		S' A C bus A	S' B i A	B	S' C C i B	i C	N	L aux	i N	+ -u in
	-
	o
	Figure R.7:
						9/178

Le MFCS de base présente deux inconvénients majeurs. Le premier est la plus grande ondulation de courant causée par le fait que la tension de neutre est fixe, ce qui entraîne plus de pertes de fer. Le deuxième problème est que le rapport d'augmentation effectif maximum du boost équivalent n'est que de 2.

Afin de supprimer l'ondulation du courant, une évolution de cette architecture, nommée MFCS-I, est proposée en ajoutant une inductance en série, comme le montre la figure R.7.

Contrainte 1 : conservation du couple

	.11: Schéma du circuit du MFCS-I lorsque la phase A est ouverte.
	Pour garantir un couple de sortie constant et une tension de bus DC moyenne stable dans
	ces conditions d'après défaut, trois contraintes sont à respecter et sont décrites successivement
	dans les sous-sections suivantes.
	D'une manière générale, l'objectif principal d'une commande tolérante aux fautes pour un
	variateur est de préserver, si possible, le couple généré même dans les conditions de défauts. Ainsi,
	pour un PMSM monté en surface, le courant après défaut i q_f doit être identique au courant
	i q_n avant l'apparition du défaut (conditions normales de fonctionnement). Par conséquent, cela
	donne la première contrainte :
	i q_f = i q_n	(R.5)
	14/178

Contrainte 2 : relation entre les courants d-q-0 après défaut

	Le défaut impose un courant de phase à zéro, cela engendre alors une dépendance entre
	les courants dans le repère d-q-0 qui ne sont plus indépendants. Cette relation est la suivante :
	i 0_f = i q_f sinθ e -i d_f cosθ e	(R.6)

Contrainte 3 : Préservation de la puissance moyenne

	Entre les conditions de fonctionnement saines et les conditions de fonctionnement en
	défaut, la puissance de sortie du moteur devrait être invariante si tout se passe bien car le couple
	et la vitesse du moteur devraient être maintenues. Côté source, il n'est pas forcément toujours
	possible de maintenir une puissance constante, mais il est nécessaire tout de même de garantir
	une puissance moyenne constante sur une période électrique. La valeur moyenne du courant
	i 0_f (représenté par une barre) est directement liée à cette puissance moyenne dans ce système.

s_f pj et i s_npj correspondent

  q_n sinθ e -i d_n cosθ e + 2i 0_n cos 2 θ e = i q_n sinθ e -i d_n cosθ e + i 0_n + i 0_n cos2θ e aux projections de ces trajectoires sur le plan α-β.

	(R.8)
	La figure R.12(a) décrit la trajectoire du courant (appelée i s_f ) dans un exemple de
	fonctionnement où la proportion de i 0_n et i q_n est égale à -0,45 (m 0 = -0, 45). A titre de
	comparaison, la trajectoire en régime de fonctionnement normal (i s_n ) y est aussi représentée.
	Les tracés i Dans les conditions normales, la trajectoire i s_n est un cercle. Sa composante sur l'axe
	0 i 0_n est constante et égale à -0,45i q_n . En revanche, dans les conditions de défaut,

i s_f et i s_f pj ne

  sont plus circulaires : ils présentent une trajectoire en forme de coeur.

	15/178

Table R . 3 :

 R3 Caractéristiques du MFCS-I par rapport aux quatre topologies tolérantes aux fautes typiques.

	fonction boost équivalente puissent être synthétisés ensemble.
	• Afin de réduire l'ondulation de courant et d'améliorer le rapport d'augmentation de
	tension effectif du MFCS de base, deux solutions améliorées ont été proposées et testées
	expérimentalement, qui ont conduit au MFCS-I et au MFCS-II. Ensuite, une stratégie
	de commande universelle en boucle fermée a été proposée où un FOC classique peut
	être utilisé pour contrôler le moteur et un contrôleur basé sur la caractéristique de la
	platitude différentielle a été utilisé pour contrôler le boost équivalent. Une comparaison
	expérimentale est menée entre un entraînement basé sur un convertisseur à deux étages et
	le MFCS-I.
	• La possibilité de tolérance aux fautes du MFCS-I vis-à-vis d'un défaut en phase ouverte
	sans utiliser de composants auxiliaires a été étudiée pour la première fois. Pour réaliser
	cette possibilité, une nouvelle trajectoire vectorielle de courant a été proposée et validée
	Topologie expérimentalement. Fonction boost gratuite Bus DC divisé NP accessible CFI NON OUI NON SFI NON NON NON Perspectives CFNI NON OUI OUI Suite aux travaux présentés ici, quelques perspectives peuvent être envisagées : Composants PPRF SOCF localisation de la faute semi-conducteurs Côté moteur Côté onduleur auxiliaires 3 (TRIAC) 0.50 1.50 √ X 2 (IGBT) 1 1.83 √ X 3 (TRIAC) 1 (TRIAC) 0.50 1.29 √ √
	SFNI •	NON	NON	OUI	2 (IGBT) 1 (TRIAC)	0.58	1.87	√	√
	MFCS-I	OUI	NON	OUI	0	(0.52-0.71) (résultat théorique) 0.55 (résultat expérimental)	1	√	√
	Conclusion et perspectives
	Conclusion
	L'objectif principal de cette thèse a été de développer une alternative à faible coût et à
	haute fiabilité pour le système d'entraînement d'une PMSM typique basé sur un convertisseur
	à deux étages dans les applications HEV et EV. Inspiré par l'idée originale du MFCS de
	base qui utilise le point neutre d'un moteur triphasé pour réaliser une fonction d'élévation de
	tension DC/DC gratuite sur un VSI triphasé à deux niveaux, une étude complète impliquant la
	modélisation, le contrôle, l'amélioration et la capacité de tolérance aux fautes a été menée. Les
	principales contributions de ces travaux peuvent se résumer comme suit :
	• En construisant le modèle moyen et en développant les circuits équivalents, le mécanisme
	du MFCS de base est clairement explicité. Une technique de modulation spécifique nommée
	ZSVIPWM a été proposée afin que les degrés de liberté de contrôle de la machine et de la
					18/178

Éléments concernant le système d'électronique de puissance : Avec

	le développe-
	ment des dispositifs de commutation à large bande interdite (tels que les composants
	à base de carbure de silicium (SiC) ou de nitrure de gallium (GaN)), l'onduleur pour-
	rait fonctionner à une fréquence de découpage plus élevée. Dans ce cas, le problème de
	l'ondulation de courant intrinsèque au principe de fonctionnement du MFCS peut être
	réduit. Par conséquent, la bobine à ajouter en série dans le MFCS-I pourrait être de taille
	très réduite voire même inutile et donc le MFCS de base pourrait être directement utilisé.
	Du point de vue de l'architecture, le concept de MFCS peut également être étendu aux
	systèmes d'onduleurs multi-niveaux (3 niveaux, 5 niveaux ou autres) car l'ondulation de
	courant des systèmes d'onduleurs multi-niveaux est plus faible. De plus, il serait également
	intéressant d'appliquer le concept de MFCS dans certains variateurs connectés au réseau
	(variateurs de moteur AC/AC), où le point neutre du moteur peut être connecté à une
	source AC monophasée.
	• Considérations

autour du moteur (analyse et conception) : Dans

	pour concevoir une machine spéciale avec un plus grand L 0 .
	•
	les MFCS, le
	moteur est utilisé dans des conditions non conventionnelles avec une composante continue
	sur les courants et des ondulations de courants plus importantes. Des analyses plus
	détaillées pourraient être menées concernant la saturation magnétique et la dissipation
	thermique du moteur dans ces conditions. De plus, des conseils pourraient être donnés
	19/178

Force contre-électromotrice du moteur non sinusoïdale : Dans

	le cas où un PMSM
	contiendrait des back-EMFs de troisième harmonique, la back-EMF homopolaire (e 0 ) dans
	(2.6) ne serait pas nulle. Par conséquent, elle générera un couple additionnel de troisième
	harmonique puisque le courant homopolaire du MFCS n'est pas nul. Pour éliminer ce
	couple de troisième harmonique, une nouvelle méthode de modulation et une nouvelle
	stratégie de contrôle devront être développées.
	• Choix de la tension du bus DC : La tension de bus DC a été régulée à une valeur
	constante dans les travaux présentés ici, mais, étant donné que le MFCS intègre une
	fonction boost DC/DC équivalente, cette tension peut être choisie de manière optimale afin
	de répondre au plus juste à la demande en tension du moteur (lors d'un fonctionnement
	à basse vitesse, un niveau de tension faible suffirait). Dans ce cas, l'efficacité globale du
	système d'entraînement pourrait être améliorée. À cette fin, une stratégie de génération
	de trajectoire pour la consigne de la tension de bus DC pourrait être intéressante dans le
	cadre des MFCS.
	• Machines

avec saillance magnétique comme la machine synchrone avec aimants enterrés (IPMSM) ou la machine synchrone à réluctance variable (SynRM) :

	Dans le cadre de la commande tolérante aux fautes du MFCS-I, nous avons considéré
	ici que le couple ne venait que du courant i q . Mais pour une IPMSM ou une SynRM, le
	couple de réluctance n'est pas nul car L d est différent de L q . Dans ce cas, les références
	de courant proposées ne permettent pas de maintenir un couple stable. Pour étendre la
	capacité de tolérance aux fautes à ce type de machines, une nouvelle trajectoire vectorielle
	de courant devrait être proposée.
	20/178

General introduction Context and motivations

	The present work of this thesis focuses on the modeling, analysis and exploration of
	fault-tolerant capabilities for multi-functional converter systems (MFCSs). This thesis is jointly
	funded by China Scholarship Council (CSC) and Laboratoire Ampère, Villeurbanne, France
	(CNRS UMR5005, Université de Lyon, Institut National des Sciences Appliquées de Lyon, Ecole
	Centrale de Lyon, Université Claude Bernard Lyon 1) for 42 months. The work mainly involves
	with the knowledge of two disciplines, electrical engineering and automation. Specifically, it
	is mostly related to the research fields of electrical machines, power electronics and automatic
	controls.

Table G . 1 :

 G1 Comparison of commercial HEVs with respect to design features and published specifications.

	Design

Feature 2008 LS 600h Hybrid Camry 2004 Prius 2010 Prius

	Motor peak power rating	165 kW	105 kW	50 kW	60 kW
	DC-bus voltage	288-650 V DC	250-650 V DC	200-500 V DC 200-650 V DC
	DC/DC converter power rating	36.5 kW	30 kW	20 kW	27 kW
	Battery voltage	288 V	244.8 V	201.6 V	201.6 V
	four HEVs (2008 Lexus LS600h, hybrid Camry, 2004 Prius and 2010 Prius)[11] is listed in table
	G.1. It can be seen that the DC-bus voltage is usually regulated at around 2-3 times the battery
	voltage by using an intermediate DC/DC boost converter. Different types of DC/DC converters
	have been studied in previous researches, such as single-phase DC/DC boost converters[12],

Chapter 1 Opportunities of three-phase motor's neutral point (NP) Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Concept of multi-functional converter system (MFCS) . . . . . . .

	a 52.5 W test-bench to verify the effectiveness.
	• Chapter 3 proposes two improved solutions for the basic MFCS while considering the
	issues of bigger current ripples and the limited effective step-up ratio of the equivalent
	boost function. The first one is referred to as MFCS-I, which uses a series inductor
	connected between the NP and the DC-source to reduce the current ripple. A sizing
	method of the series inductor is proposed based on the concept of a re-floating NP. The
	second solution is referred to as MFCS-II, which employs an auxiliary leg to improve the
	effective step-up ratio. Then, a universal closed-loop control strategy is proposed for both
	MFCS-I and MFCS-II. Experiments are carried out on a 1.2 kW drive platform to verify
	the effectivenss.
	• Chapter 4 explores the possibility of fault-tolerant operations of the MFCS-I towards
	an OPF without using any auxiliary hardware. First, the fault-tolerant potential of the
	MFCS-I is discussed. Three constraints of post-fault currents are found to guarantee a
	constant torque and a stable average DC-bus voltage in post-fault conditions. Second, a
	novel current vector trajectory which can fulfill the three constraints is proposed. Detailed
	analysis concerning the power conversion process and post-fault output capability is also
	presented. Then, two current tracking controllers are introduced to track the variable
	current trajectories. Experiments are carried out on the 52.5 W and 1.2 kW test-benches 1.1.1 Basic MFCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	to verify the effectiveness, respectively. 1.1.2 Other variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	Finally, conclusions and perspectives are given in the end. 1.1.3 Integrated on-board charger . . . . . . . . . . . . . . . . . . . . . . . .
	1
	Chapter 2 comprehensively investigates a permanent magnet synchronous motor (PMSM)-
	based basic MFCS. First, the mathematical model of PMSMs with zero-sequence com-
	ponents is derived. Then, the average model of the PMSM-based basic MFCS is built.
	Three equivalent circuits are proposed to facilitate the understanding of the mechanism of
	the basic MFCS. Second, the infeasibility of conventional pulse-width modulation (PWM)
	techniques used in the basic MFCS is discussed. To address this issue, a specialized
	zero-sequence voltage injected PWM is proposed. Then, the DC-source voltage utilization
	ratio of the basic MFCS is analyzed. Finally, comparative experiments are carried out on
	24/178 25/178

.2 State of the art of three-phase fault-tolerant motor drives . . . . .

	1.2.1 Fault-tolerant solutions without using NP . . . . . . . . . . . . . . . .
	1.2.2 Fault-tolerant solutions using NP . . . . . . . . . . . . . . . . . . . . .
	1.2.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1 . 1 :

 11 Characteristics of two-level three-phase fault-tolerant topologies towards OPFs.

	Topology Name	Split DC Bus	Accessible NP	Auxiliary Silicon Components	Post-Fault Power Rating Factor (PPRF)	Silicon Overrating Cost Factor (SOCF)	Tolerate Fault Occurring in Motor Side Inverter Side
	CFI	YES	NO	3 (TRIAC)	0.50	1.50	X	√
	SFI	NO	NO	2 (IGBT) 3 (TRIAC)	1	1.83	X	√
	CFNI	YES	YES	1 (TRIAC)	0.50	1.29	√	√
	SFNI	NO	YES	2 (IGBT) 1 (TRIAC)	0.58	1.87	√	√

3 Modulation method and steady-state characteristic . . . . . . . . .

  Validation of the equivalent DC/DC boost function . . . . . . . . . . According to the brief introduction of the basic MFCS in Chapter 1, we know that it can obtain a cost-free DC/DC boost function by utilizing the NP of the motor. Therefore, it can be a promising alternative to the two-stage converter-based drive, which employs an intermediate DC/DC boost converter to raise the DC-bus voltage. However, in the literature, the operating principle and mechanism of the basic MFCS are not clearly revealed by an analytic model.

	Introduction
	Chapter 2
	Investigation of a PMSM-based basic
	MFCS
	2.4.1 Zero-sequence inductance measurement . . . . . . . . . . . . . . . . .
	42/178 2.4.2 43/178

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 PMSM model with zero-sequence components . . . . . . . . . . . . 2.1.1 Electrical dynamics in A-B-C coordinate . . . . . . . . . . . . . . . . 2.1.2 Electrical dynamics in d-q-0 coordinate . . . . . . . . . . . . . . . . . 2.1.3 Mechanical dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analytical model of a PMSM-based basic MFCS . . . . . . . . . . 2.2.1 Average model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Equivalent circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.1 Discussion of SVPWM used in the basic MFCS . . . . . . . . . . . . . 2.3.2 Zero-sequence voltage injected PWM . . . . . . . . . . . . . . . . . . . 2.3.3 DC-source voltage utilization ratio . . . . . . . . . . . . . . . . . . . . 2.3.4 Influence of the clamped NP on current ripples . . . . . . . . . . . . . 2.4 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Experimental comparisons in steady-states . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  The zero-sequence inductance L 0 /3 plays the role of the filtering reactor in the boost function. Finally, the basic MFCS system can be seen as a two-stage DC motor drive. The equivalent DC motor drive function as an equivalent load for the equivalent DC/DC boost converter. The equivalent load current (i lo ) is equal to (α d

2.17b

) is visualized by the contour 'K' in figure

2

.5. The fourth equation of (2.17b) is visualized on the node 'P' in figure 2.5. Therefore, it is reasonable to say that the last two equations of (2.17b) can be represented by an equivalent DC/DC boost 57/178 converter.

Table 2 . 1 :

 21 Basic parameters of the 52.5 W PMSM-based test-bench.

	Parameter Value	Unit	Parameter	Value Unit
	R	0.5	Ω	p	4	-
	L d	1.1	mH	L q	1.1	mH
	L 0	0.86	mH	C bus	1000	µF
	Φ f	0.0056	Wb	Nominal speed	4000 rpm/s
	J m	0.0005 Kg • m 2 Nominal power	52.5	W
	B

m 0.0001 N •m/rad Nominal torque 125 mN•m 68/178

  bus is associated with the average duty cycle α h . Therefore, a simple way to test the effect of the equivalent DC/DC boost function is to adjust α h in an open-loop manner. Simulation verification is firstly conducted to verify the equivalent DC/DC boost functionality before conducting experiments. The simulation model is built in MATLAB/SIMULINK environment, in which the PMSM with an available neutral terminal is provided by the Simscape library.Simulation step is set as 1e-6 s and the ode1(Euler) solver is chosen. The switching frequency is set at 20 kHz. The DC-source voltage u in is 15 V. At the beginning, α h is 1. At 0.5 s, α

h is decreased to 0.5. Corresponding simulation results are shown in figure

2

.16(a). It can be seen that u bus is 15 V before 0.5 s. When α h is decreased to 0.5 suddenly, u bus is boosted to 32 V. In steady states, it is equal to 30 V. This verifies the existence of a voltage boosting functionality 69/178 integrated in the basic MFCS. Figure

2

.16(a) shows the experimental results with the same setup.

Table 2 . 2 :

 22 The main characteristics of the basic MFCS compared to the conventional drive.

	Characteristic	Evaluation
	Steady-state performance Similar
	DCSUR	Larger (1 vs 0.58)
	Integrated boost function YES vs NO
	Current ripple	Bigger
	Losses	Bigger (11.5% for the 52.5 W PMSM)
	such a special system, the mathematical model of PMSMs with zero-sequence considerations is
	given firstly. Then, according to time-averaging equivalent modeling approach and Kirchhoff's

law, the average model of the MFCS is built in A-B-C coordinate, and then is converted into d-q-0 coordinate. By building different equivalent circuits, it is easy to understand that the zero-sequence circuit of the MFCS is equal to an equivalent DC/DC boost converter where the average duty cycle (α h ) of the inverter can be the control DOF.

Table 3 . 1 :

 31 The parameters of the 1.2 kW test-bench.

	Parameter	Value	Unit	Parameter	Value Unit
	Stator resistance R	0.5	Ω	Pole pairs p	4	-
	d-axis inductance L d	1.7	mH	d-axis inductance L q	1.7	mH
	0-axis inductance L 0	0.0024	H	DC-bus capacitance C bus	940	µF
	Flux linkage Φ f	0.1053	Wb	Nominal speed	3000 rpm/s
	Mechanical inertial J m 0.0009 Kg • m 2 Nominal power	1.2	kW
	Friction coefficient B m 0.001 N •m/rad Nominal torque	4	N•m
	DC-source voltage u in	180	V	DC-bus voltage u ref bus	360	V

Table 3 . 3 :

 33 Characteristics of the MFCS-I compared to the two-stage converter-based drive.

	PMSM drives	Switching Devices	Current THD at Motoring Pattern	Current THD Braking Pattern at Rengenerative	Ef f 1	Ef f 2
	Two-stage converter	8	5.36%	6.00%	0.848	0.935
	MFCS-I	6	6.10%	6.51%	0.877	0.964
	Comparison	Decreased by 25% Degraded by 13.8% Degraded by 8.5% Improved by 3.4% Improved by 3.1%

  q_n sinθ e -i d_n cosθ e + 2i 0_n cos 2 θ e = i q_n sinθ e -i d_n cosθ e + i 0_n + i 0_n cos2θ e

	(4.7)

  2m 0 i q_n cosθ e cos(θ e -2π/3) -i q_n sin(θ e -2π/3) + i q_n (sinθ e + 2m 0 cos 2 θ e ) i C_f = -2m 0 i q_n cosθ e cos(θ e + 2π/3) -i q_n sin(θ e + 2π/3) + i q_n (sinθ e + 2m 0 cos 2 θ e )

	(4.8)

Table 4 .

 4 1: d-q-0 currents under post-fault conditions considering different fault locations.

	Fault location	i d_f	i q_f	i 0_f
	Phase A	-2i 0_n cosθ e	i q_n	i q_n sinθ e + i 0_n [1 + cos(2θ e )]
	Phase B	-2i 0_n cos(θ e -2π 3 ) i q_n i q_n sin(θ e -2π 3 ) + i 0_n 1 + cos[2(θ e -2π 3 )]
	Phase C	-2i 0_n cos(θ e + 2π 3 ) i q_n i q_n sin(θ e + 2π 3 ) + i 0_n 1 + cos[2(θ e + 2π 3 )]
			119/178

  bus_f max denotes the peak value of u bus ; u bus_f min denotes the valley value of u bus ; ∆u bus If the average value (u bus_f ) of the DC-bus voltage is regulated at the voltage reference (u * bus ), we can consider that (u bus_f max + u bus_f min )/2 ≈ u * bus . By taking this into (4.12), the DC-bus voltage ripple can be calculate by

		∆u bus ≈	∆E _f bus C bus u *	(4.13)
	Then, by substituting (4.11) into (4.13), the DC-bus voltage ripple can be expressed as
	∆u bus ≈	3u in i q_n 2πC bus u * bus f ele
				(4.12)
			min )
			∆u bus
	where u denotes the DC-bus voltage ripple.
		122/178

  introduced an analytic method to estimate the appropriate value of ε th by quantifying the effects caused by inverter's dead time, voltage drop of semiconductor devices, parasitic parameters and sensor noise. However, in practice, it still needs to consider a redundant margin for the theoretically calculated value. In our work, an experimental trial is adopted to set ε th , which is simpler and more effective in case of false alarms. To consider different operating condition and avoid false alarms, we have done a progressive trial where the motor speed and load are increased gradually.

The experimental results are shown in figure

4

.13. It can be observed that the motor speed is increased from 52.3 rad/s to the nominal value (314.2 rad/s). The load is increased from around 0 to the nominal value (4 N•m). From figure

4

.13, we can see that, although the value of ε changes a little when the speed and load change, it is always smaller than 1. Therefore, we choose ε th = 1 as a safe indicator to detect the OPF in the following fault-tolerant experiments.

Table 4 . 2 :

 42 Characteristics of the MFCS-I compared to the four typical fault-tolerant topologies.

	Topology CFI	Free Boost Function NO	Split DC Bus YES	Accessible Motor Neutral NO	Auxiliary Silicon Components 3 (TRIAC)	Post-Fault Power Rating Factor (PPRF) 0.50	Silicon Overrating Cost Factor (SOCF) 1.50	Tolerate Fault Occurring in Motor Side Inverter Side X √
	SFI CFNI	NO NO	NO YES	NO YES	2 (IGBT) 3 (TRIAC) 1 (TRIAC)	1 0.50	1.83 1.29	X √	√ √
	SFNI	NO	NO	YES	2 (IGBT) 1 (TRIAC)	0.58	1.87	√	√
	MFCS-I	YES	NO	YES	null	(0.52-0.71) (theoretical range) 0.55 (experimental case)	1	√	√

  According to H1(s) and H3(s), the PI parameters for the current-loop and voltage-loop of the equivalent boost can be tuned. Multi-functional converter for three-phase motor drives: modeling, analysis and fault-tolerant Control (Convertisseur multifonctionnel pour l'alimentation des moteurs triphasés : modélisation, analyse et commande tolérante aux fautes) Cette thèse porte sur un système d'entraînement de machine synchrone à aimants permanents triphasée alimentée par un onduleur qui intègre un étage de conversion continu/continu, en utilisant le point neutre de la machine. Cette architecture, appelée système de convertisseur multifonctionnel (MFCS), peut être a vantageusement utilisée dans des applications de traction automobile dans le cas où une élévation de tension est nécessaire. Premièrement, une analyse approfondie est présentée afin de comprendre et de modéliser le MFCS. Une méthode de modulation spécifique est, de plus, proposée utilisant la particularité que le courant homopolaire n'est pas nul. Deuxièmement, deux nouvelles variantes nommées MFCS-I et MFCS-II ont été conçues afin d'améliorer le MFCS de base. Pour la première, une inductance en série est ajoutée pour diminuer l'ondulation du courant. Pour la seconde, l'ajout d'un bras d'onduleur auxiliaire améliore le rapport d'élévati on effectif de la tension. Pour contrôler ces variantes de MFCSs, une stratégie de commande universelle est développée dans laquelle une commande vectorielle peut être prise en compte pour la fonction moteur et un contrôleur basé sur la platitude différentielle est utilisé pour réguler la tension du bus continu. Troisièmement, compte tenu de la connexion intrinsèque du point neutre dans le MFCS-I, son potentiel de tolérance aux fautes contre un défaut de phase ouverte est exploré pour la première fois. Une étude théorique permet de formuler les conditions pour atteindre cette propriété et une nouvelle trajectoire vectorielle de courant est proposée, générant un couple constant et une tension moyenne stable du bus continu après un défaut. Pour finir, les idées et les stratégies de contrôle présentées ci -dessus sont systématiquement validées expérimentalement sur des plateformes de prototypage rapide de lois de commande. MOTS-CLÉS : Système de convertisseur multifonctionnel, machine synchrone à aimants permanents triphasée, point neutre, commande tolérante aux fautes.

	(s) = THESE DE L'UNIVERSITE DE LYON OPEREE AU SEIN DE L'INSA LYON ĩN (s) D(s) = u * bus L 0 /3 s + u * bus R eq (L 0 /3)C bus + (1 -D s ) (L 0 /3)C bus i N s s 2 + 1 R eq C bus s + FOLIO ADMINISTRATIF (1 -D s ) 2 (L 0 /3)C bus	(C.4)
	NOM : ZHANG Prénoms : Xiaokang TITRE : NATURE : Doctorat	H2(s) =	ũbus (s) D(s) H3(s) = =	-i N s C bus s 2 + R eq C bus s + (1 -D s ) DATE de SOUTENANCE : 24/03/2022 u * bus (L 0 /3)C bus 1 (1 -D s ) 2 s + (L 0 /3)C bus ũbus (s) ĩN (s) = H2(s) H1(s) Numéro d'ordre : 2022LYSEI021	(C.5) (C.6)
	Ecole doctorale : ÉLECTRONIQUE, ÉLECTROTECHNIQUE et AUTOMATIQUE (EEA)
	Spécialité : Génie Electrique
	RESUME : Laboratoire (s) de recherche : Laboratoire Ampère -UMR CNRS 5005 -INSA Lyon
	Directeur de thèse: LIN-SHI Xuefang
	Président de jury :		DIALLO Demba
	Composition du jury :
	Rapporteurs :		NAHID-MOBARAKEH Babak, BETHOUX Olivier
	Examinateur :		DIALLO Demba
	Directrice de thèse :		LIN-SHI Xuefang
	Co-encadrants de thèse :		GAUTHIER Jean-Yves
				165/178

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2022LYSEI021/these.pdf © [X. Zhang], [2022], INSA Lyon, tous droits réservés

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2022LYSEI021/these.pdf © [X. Zhang], [2022], INSA Lyon, tous droits réservés 1.2. State of the art of three-phase fault-tolerant motor drives

able to put an end to this journey without you.

Finally, I would like to thank my motherland (China) and China Scholarship Council (CSC) for the funding support.

the desired current references so that the tracking errors (e 1 = y 1 -y 1ref , e 2 = y 2 -y 2ref , e 3 = y 3 -y 3ref ) asymptotically vanish, feedback control laws are given as follows:

( ẏ1 -ẏ1ref ) + K d1 (y 1 -y 1ref ) + K d2 (y 1 -y 1ref )dt = 0 ( ẏ2 -ẏ2ref ) + K q1 (y 2 -y 2ref ) + K q2 (y 2 -y 2ref )dt = 0 ( ẏ3 -ẏ3ref ) + K 01 (y 3 -y 3ref ) + K 02 (y 3 -y 3ref )dt = 0 (4.27) where K d1 = 2ζ 1 ω 1 , K d2 = ω 2 1 , K q1 = 2ζ 2 ω 2 , K q2 = ω 2 2 , K 01 = 2ζ 3 ω 3 and K 02 = ω 2 3 can be designed according to the following characteristic polynomials: 2 1 e 1 = 0 ë2 + 2ζ 2 ω 2 ė2 + ω 2 2 e 2 = 0 ë3 + 2ζ 3 ω 3 ė3 + ω 2 3 e 3 = 0 (4. [START_REF] Plesko | Novel modulation concepts for a drive-integrated auxiliary dc-dc converter for hybrid vehicles[END_REF] where

, ω 3 are the desired dominant damping ratios and natural frequencies.

Therefore, we obtain:

Substituting (4.29) into (4.26), it is clear that the closed-loop system is asymptotically stable when K d1 , K d2 > 0, K q1 , K q2 > 0 and K 01 , K 02 > 0. Once the flat output is stabilized, the whole system will be exponentially stable because the state variable is completely expressed in the form of the flat output.

Implementation on the 52.5 W PMSM

Simulations and experiments are firstly implemented on the 52.5 W PMSM to verify the effectiveness of the proposed fault-tolerant scheme. The parameters of the 52.5 W PMSM have been presented in table 2.1. In the following test, the DC-source voltage is 15 V and the DC-bus voltage is regulated at 30 V. The deadbeat predictive current controller is used to track the current references in both healthy and fault-tolerant conditions. From figure 4.14, it can be observed that the value of ε is around 0.7 in the healthy condition.

Simulation results

Thus, the fault flag is not triggered. Meanwhile, u bus is boosted at 360 V stably thanks to the equivalent boost function. u bus_f ilter is also 360 V in the healthy condition. In terms of the torque T e , it is around 0.4 N•m with the ripple of 0.18 N•m. Although there is no load torque applied on the motor, the friction torque is the main load to the motor.

Therefore, the duty-cycles α A , α B and α C can be calculated by

]

]

When V ref is located in the other sectors, the duty-cycles can be calculated in the same way. The results are summarized in tableB.1. According to tableB.1, it is easy to calculate the inverter's mean duty-cycle by α h = (α A + α B + α C )/3 and draw its curve.

Table B.1: Duty-cycle calculations by using the classic SVPWM.

Sector