Olivier Bethoux
email: olivier.bethoux@centralesupelec.fr

Eric Laboure

Remy Ghislain

Eric Berthelot

Efficiency-optimal power partitioning for improved partial load efficiency of electric drives

Keywords: Inverters, motor drives, permanent-magnet (PM) machines, losses minimization, efficiencyoptimal partitioning of the power, Lagrange multipliers method. 1

In this paper, the global power conversion efficiency is improved for a fault-tolerant drive architecture. The fault-tolerant architecture is obtained by combination of a 3-phase open-end winding machine and a 3H-bridge inverter. This combination offers more degrees of freedom for

INTRODUCTION

Efficiency improvement in energy conversion process is becoming a primary development goal in many applications, in connection with rising energy prices and resources conservation. This is, in particular, the case in various areas such as embedded applications [START_REF] Woojoo | Optimizing the Power Delivery Network in a Smartphone Platform[END_REF], communication networks [2], data centers [START_REF] Candan | A Series-Stacked Power Delivery Architecture with Isolated Differential Power Conversion for Data Centers[END_REF], electric grid generation [START_REF] Beik | An Off-Shore Wind Generation Scheme With High Voltage Hybrid Generator, HVDC Interconnections and Transmission[END_REF][START_REF] Thang | Analysis and Design of Grid-Connected Photovoltaic Systems With Multiple-Integrated Converters and a Pseudo-DC-Link Inverter[END_REF][START_REF] Khajesalehi | Modeling and control of quasi Z-source inverters for parallel operation of battery energy storage systems: Application to microgrids[END_REF], or automotive applications [START_REF] Muta | Development of new-generation hybrid system THS II-Drastic improvement of power performance and fuel economy[END_REF]. The following part reviews best practices for enhancing the power converter efficiency for electric drives. Some efficient techniques can be found in the power converters for photovoltaic systems Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 3 mainly. Indeed, photovoltaic inverters achieved in 2015 a rated power efficiency higher than 99%, driven by the market demand. This significantly exceeds the reference value of 95 % of year 2000 [START_REF] Kolar | Extreme efficiency power electronics[END_REF][9][START_REF] Falk | Efficiency and grid compatibility of photovoltaic inverters -state-of-the-art and future trends[END_REF][START_REF] Husev | Three-level three-phase quasi-Z-source neutral-point-clamped inverter with novel modulation technique for photovoltaic application[END_REF]. Similarly, pushed by high standards [START_REF] Whitney | Data center efficiency assessment[END_REF], converter stages of telecom power modules produce less than 1% of power losses at rated power [START_REF] Kasper | Hardware verification of a hyper-efficient (98%) and super-compact (2.2kW/dm3) isolated AC/DC telecom power supply module based on multi-cell converter approach[END_REF]. Indeed, the clear requirement of this application, in terms of compactness and reliability, involves high power density while keeping a low operating temperature. Therefore, minimizing losses of power electronics systems gains increasingly in importance and power conversion efficiency tends to be a key criteria among the performance indices [START_REF] Kolar | What are the Big Challenges in Power Electronics[END_REF].

The rated power operating point is often used to define the performance of a system. However, in case of energy optimization considerations, the typical mission profile has to be considered, leading to assess the losses on different operating points [START_REF] Biela | Mission profile optimized modularization of hybrid vehicle DC/DC converter systems[END_REF]. This is particularly important for achieving a relevant energy efficiency optimization [START_REF] Merkert | Characterization and Scalable Modeling of Power Semiconductors for Optimized Design of Traction Inverters with Si-and SiC-Devices[END_REF]. As an example, PV inverter certification requires not only full-load efficiency but also partial load values, namely at 10%, 20% and 50% of rated power [START_REF] Ho | A Comparative Performance Study of an Interleaved Boost Converter Using Commercial Si and SiC Diodes for PV Applications[END_REF].

Automotive applications also lead to a wide range of operating points [START_REF] Azib | An Innovative Control Strategy of a Single Converter for Hybrid Fuel Cell/Supercapacitor Power Source[END_REF]. In such applications, it is clearly demonstrated that the powertrain losses must be reduced for low power and low speed in order to increase the vehicle's autonomy [START_REF] Gunther | Driving cycle-based design optimization of interior permanent magnet synchronous motor drives for electric vehicle application[END_REF]. For instance, to assess vehicles powertrain efficiency, a realistic average driving cycle such as ARTEMIS is used [START_REF] Hage Hassan | Radial output space mapping for electromechanical systems design[END_REF]. There are various ways of enhancing a power converter's efficiency: these include using new components [START_REF] Khajesalehi | Modeling and control of quasi Z-source inverters for parallel operation of battery energy storage systems: Application to microgrids[END_REF], more complex topologies enabling soft-switching [21], or modular design [START_REF] Ling | High step-up interleaved boost converter with low switch voltage stress[END_REF][START_REF] Ayoubi | Four-phase interleaved DC/DC boost converter interfaces for super-capacitors in electric vehicle application based on advanced sliding mode control design[END_REF], new designs with better integration [START_REF] Van Wyk | Integrating active, passive and EMI-filter functions in power electronics systems: a case study of some technologies[END_REF], system considerations and innovative control procedures, e.g. Differential Power Processing [START_REF] Candan | A Series-Stacked Power Delivery Architecture with Isolated Differential Power Conversion for Data Centers[END_REF]. All these considerations mainly lead to a global approach [START_REF] Kolar | What are the Big Challenges in Power Electronics[END_REF],

Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 4 e.g. of the optimization of the combination of the machine and the associated power electronics converters.

For single-cell converters, it should be noted that high efficiency at partial load leads to a significant increase of the performance of the system at full load [START_REF] Kolar | Extreme efficiency power electronics, 7th International Conference on Integrated Power Electronics Systems (CIPS2012)[END_REF]. Thus, partial load efficiency improvement is very difficult to achieve for classical converter topologies. Indeed, an efficiency of 90% at 10% partial load corresponds to a rated power efficiency of 99% under the assumption of constant losses over the entire output power range.

Experiencing the difficulty in obtaining a flat efficiency curve has led to the alternative solution of splitting a single converter into several smaller converters working in parallel [START_REF] Su | A Novel Phase-Shedding Control Scheme for Improved Light Load Efficiency of Multiphase Interleaved DC-DC Converters[END_REF]. The number of operating cells is defined to optimize the global power efficiency. Indeed, some of them can be disconnected and set off to improve efficiency when working at partial load.

Choosing the number of operating converters with regard to the demanded power ensures that each subsystem works within a small power range around its best efficient operating point. This established control procedure is known as Phase Shedding or efficiency-optimal partitioning [START_REF]LP8754 Multi-Phase Six-Core Step-Down Converter[END_REF].

Toyota uses this strategy to manage a 114 kW boost converter implemented in the hydrogenpowered Mirai fuel cell electric vehicle (EV) [START_REF] Hasuka | Development of Boost Converter for MIRAI[END_REF]. The DC converter regulating the voltage delivered by the fuel cell is made of 4 parallel legs. The associated control determines in realtime the optimal number of cells to be used; this strategy leads to reduce the boost losses by ten per cent at 15 kW operating point [START_REF] Hasuka | Development of Boost Converter for MIRAI[END_REF]. This design also induces significant system improvements [START_REF] Kolar | Extreme efficiency power electronics, 7th International Conference on Integrated Power Electronics Systems (CIPS2012)[END_REF]: among the benefits, one could easily identify input and output current ripple reduction [START_REF] Zumel | Efficiency improvement in multiphase converter by changing dynamically the number of phases[END_REF], a better continuity of service and also a fast transient response if using multiphase interleaved dcdc converters, as in the case of the power management of microprocessors or double-data rate Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

5

(DDR) memory [START_REF] Peterchev | Architecture and IC implementation of a digital VRM controller[END_REF]. With the goal of reducing power loss, the aforementioned system consideration has not yet been exploited in a motor drive.

Therefore the present paper applies a similar point of view to the approach in multiple parallel converters [START_REF] Kolar | Extreme efficiency power electronics[END_REF][START_REF]LP8754 Multi-Phase Six-Core Step-Down Converter[END_REF][START_REF] Hasuka | Development of Boost Converter for MIRAI[END_REF][START_REF] Zumel | Efficiency improvement in multiphase converter by changing dynamically the number of phases[END_REF][START_REF] Peterchev | Architecture and IC implementation of a digital VRM controller[END_REF] to an electric motor drive designed to be fault-tolerant. This drive is the combination of a three-phase open-end-winding permanent magnet synchronous machine (PMSM) and a three-phase inverter made of three full H-bridges. This architecture has been patented for EV powertrain by the VALEO group, a world's leading automotive supplier [START_REF] Sousa | Combined Electric Device for Powering and Charging[END_REF][START_REF] Sousa | Method and Electric Combined Device for Powering and Charging with Compensation Means[END_REF][START_REF] Sousa | Method of exchanging electrical energy between an electrical network conveying a dc or ac electrical quantity and an electrical energy storage unit for hybrid or electric vehicle[END_REF][START_REF] Bouchez | Charge transfer device and associated management method[END_REF].

Similarly to other three-phase structures, like neutral leg inverter or additional leg inverter [START_REF] Naidu | Fault-tolerant permanent magnet motor drive topologies for automotive X-By-wire systems[END_REF],

the VALEO patented option has the advantage of improved continuity of service due to its fault tolerant capability. In contrast, this solution has proved to be multifunctional without any additional device (i.e. electrical contactor), permitting EV traction [START_REF] Bruyère | A multiphase traction/fast-battery-charger drive for electric or plug-in hybrid vehicles: Solutions for control in traction mode[END_REF][START_REF] Sandulescu | Control Strategies for Open-End Winding Drives Operating in the Flux-Weakening Region[END_REF], battery charging [START_REF] Lacroix | An integrated fast battery charger for electric vehicle[END_REF], power injection back to the grid during peak demand (vehicle to grid / V2G) and post-fault operating mode [START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF][START_REF] Diao | A Differential Algebraic Estimator for Sensorless Permanent-Magnet Synchronous Machine Drive[END_REF]. These are key features for specific applications like the EV emerging market. In traction mode, its key competitive advantage relies on its high degrees of freedom [START_REF] Kolli | Space-Vector PWM Control Synthesis for an H-Bridge Drive in Electric Vehicles[END_REF]. Unlike the combination of a classical three-leg inverter connected with a star-or deltaconnected PMSM, each winding can be supplied independently. It makes the topology intrinsically fault-tolerant. Additionally in normal mode, thanks to this feature, the drive control can be operated using only one or two phases sequentially instead of three phases simultaneously. This option allows the increase the power dynamically in one or two phases and to cancel the power in the other phases. As the drive is running in normal mode, it opens up the possibility for choosing the best windings dynamically in order to minimize losses, while achieving the demanded torque. Thus, on an electric turn, each of the 3 phases converts power at the most appropriate moment.

Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

6

In short, this article develops a method for normal operating optimized with respect to motor drive energy efficiency; the main idea consists in adjusting the number of phases in operation as a function of the required mechanical torque in order to maximize the drive power efficiency at low torque operating points. The paper is organized as follows. Section 2 addresses the optimized motor phase current shape in the general framework of non-sinusoidal PMSM back-electromotive force (back-emf). In section 3, the drive current optimization method is applied in the three cases of one, two or three phases conducting simultaneously; in all these three cases, the phase current is derived analytically. Section 4 presents experimental validations of the innovative strategy.

Finally, section 5 draws conclusions and perspectives of the suggested approach.

DRIVE CURRENT OPTIMIZATION METHOD

In the case of a sinusoidal magnetomotive force in a 3-phase synchronous machine, it is demonstrated that the optimal currents in the machine phases are sinusoidal and in phase with its electromotive forces [42]. These waveforms are, for a given electromagnetic torque, energetically optimal for the machine, as they minimize the RMS current values and hence the windings' Joule losses. In this case, cancellation of low frequency phase-voltage harmonics also enables the mitigation the machine ferromagnetic losses.

V DC

DC bus 3 H-Bridge inverter

Electric machine (PMSM)

i a i b i c v b v a v c
Now considering a more general drive current optimization, the design framework should integrate both losses in the machine and the power inverter for a 3-phase, 2-phase or 1-phase feeding sequence. In the following, any system loss will be taken into account.

A brief recall, underlying causes of losses in the machine-inverter association, allows better understanding of the guiding principle in developing the proposed strategy, based on an efficiency-optimal power partitioning (or fragmentation of power). The general problem of losses minimization that can be stated in terms of a constrained problem can be solved using the Lagrange multipliers method. The problem resolution using this method will be described at the end of the section.

Losses in the inverter and machine combination

Three loss-types with characteristic dependence on the output power P_O are occurring for a power electronic system [START_REF] Kolar | Extreme efficiency power electronics, 7th International Conference on Integrated Power Electronics Systems (CIPS2012)[END_REF]:

. .

The three parameters of this function are linked to various contributions:

1. expresses the power losses, which are independent of output power. This means:

 Power feeding ancillary equipment, particularly the power needed by the IGBT driving cards as well as the cooling system, the sensors and the control card. In the present case, the strategy cannot act on the three last parameters;

 Losses due to the power semiconductors parasitic capacitance that occur at each switching;

 Machine core losses.

2.

describes losses depending linearly on output power and is a consequence of:  Switch losses due to power switches' dynamic resistance.

Regarding the output power, system power efficiency is defined as:

⁄ (2)
As results of (1), (2) yields an output power -efficiency curve starting from zero at a very small output power, then increasing up to a maximum of 1 2

for , ⁄ and finally decreasing for greater powers. The maximal efficiency point depends on the three aforementioned parameters , and .

In the case of several converters that can share the total power, it is possible to adjust these three parameters. Indeed, regarding the same output power, using two converters instead of one has the effect of:

1. Roughly doubling if only IGBT drivers and parasitic capacitances are considered; 2. Leaving unchanged;

3. Halving , in case of an equally partitioned power. In this situation ′ 2 and ′ 2 ⁄ ), the best efficiency point ′ remains unchanged but occurs at a different operating point, namely an output power twice as large as the previous optimal power. Thus, by adjusting the number of converters in operation at the intersection of the efficiency curves, the resulting global power efficiency curve can be much flatter, leading to a wider high-efficiency range.

Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 9 Indeed, for partial load conditions, as quadratic losses are low, it is more suitable to reduce the constant factor ( ) by limiting the number of converters in operation. On the other hand, for high current loads, it is worth using all the available converters in order to lower quadratic losses, which have a significant impact. Fig. 2 provides a schematic overview of such a strategy in the theoretical case of a converter made of two parallel cells. Each cell can be enabled or inhibited depending on the delivered power. Fig. 3 provides a part of an INTERSIL datasheet device, named ISL78225 [START_REF]Intersil, ISL78225 datasheet, 4-Phase Interleaved Boost PWM Controller with Light Load Efficiency Enhancement[END_REF]. It enables optimal control of a 4-phase DC/DC converter: the power efficiency can reach a value higher than 95% over a wide power range 7 ⁄ .

This overview shows that with efficiency optimization goal, sharing the instantaneous power between one, two or three phases is a pertinent degree of freedom to consider. The main idea developed in this paper consists in extending this power partitioning principle to a motor drive. The 3H-bridge topology of the inverter of this drive (Fig. 1) makes it possible to operate on one, two or three phases of the machine. The power partitioning can therefore be obtained by turning on successively, in synchronism with the rotor position,  phase , phase then phase for a one-phase operation  phases , , phases , then phases , for two-phase operation or by feeding, conventionally, the three phases of the machine continuously in case of threephase operation. For each configuration, an optimal phase-current waveform can be calculated to maximize the global efficiency of the system. The waveforms can be analytically defined using the Lagrange multipliers method.

Method for determining optimal current waveform

The aim is to shape the best current waveform in order to cancel torque ripple and achieve the lowest losses in the inverter-machine combination. This dual objective can be translated as a Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 11 constraint of invariance in time of the instantaneous electromagnetic power (3) and the minimization of the sum of the effective currents in the three drive phases (4).

. . . (3) min
, , , (4)
It is therefore a minimization problem (4) under a unique equality constraint (3) which can thus be addressed using a single Lagrange multiplier . From a formal point of view, the three curves specifying the three currents (with to ) over one electric revolution ( ) can be considered as three functions (5) with a 0 ; 2 domain of definition and an image in R.

: 0 ; 2 → ⟼ (5)
The equality constraint is related to a function implicitly dependent on :

, ,

The performance criterion refers directly to a functional defined as:

: Ω → ⟼ , , . (7)
Where is the function implicitly dependent on described as:

, ,

The Lagrange method based on the calculus of variations requires to define a Lagrange function and an extended functional [START_REF] Elsgolc | Calculus of variations[END_REF][START_REF] Naidu | Calculus of Variations and Optimal Control[END_REF][START_REF] Dieulot | Economic supervisory predictive control of a hybrid power generation plant[END_REF]:

, , , , , , , (9)
A necessary condition for finding four consistent curves , , and is to cancel functional derivative of . This lead to satisfy the set of 4 equations:

2. . 0 2. . 0 2. . 0 . . . 0 (10)
Where the extended function is defined as: .

Solving this Euler-Lagrange equation obtains the phase-current shapes. The following section gives the results for the three modes mentioned previously in the introduction of section 2 and evaluates their performances against the conventional three-phase sinusoidal mode.

OPTIMAL CURRENT WAVEFORM FOR THE 3 MODES

Based on the consideration defined above, the phase-current shapes can be calculated for the three suggested modes. To demonstrate the generality of the proposed method and the possibility to take into account any imperfections of the machine, the harmonics of the electromotive force will be considered in the following calculations. These harmonics are also taken into account when determining the reference behavior, namely, when the machine is fed by sinusoidal currents.

3.1.Reference classic modes

The motor back-electromotive force is a periodic function of electric angle and is not necessarily sinusoidal. For each phase, this voltage (with to corresponding respectively to 0

to 2) can be described by:

Ω, Ω Ω Ω √2 • 2 3 . 2 3 (11)
Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

13

The Lagrange method based on the calculus of variations requires to define a Lagrange function

where Ω is the motor shaft angular rotor speed and Ω its rated value.

Experimental back-electromotive forces of the test bench permanent magnet synchronous machine is given in Fig. 4. Indeed, in this figure, back-electromotive forces are actually not purely sinusoidal. In the basic case of sinusoidal currents, the average converted power is only driven by the backelectromotive force's fundamental component. Moreover if a null zero sequence condition is added for the current calculation, a given torque leads to the following phase current:

. 2 3 ( 12
)
Where

T Ω 3 Ω ⁄

This classic mode is used as a benchmark for the other three modes.

Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 14

Mode 1: phases functioning alternately

This mode will be associated with small power demand, and hence, aims at condensing electric power on a single phase alternately. The current in the single phase used is minimal if the current is injected into the phase when the absolute value of the back electromotive force is the highest.

Therefore this implies that one electric revolution of the machine is regularly divided into six angular sectors (Fig. 5). The optimal waveform can thus be computed on a given sector, and currents in the other sectors can be derived from this one.

Studying the first sector for example ( ∈ 2 ⁄ 6 ⁄ ; 2 ⁄ 6 ⁄ ) the optimization is written as: (

/ 13) min , min . 3 ⁄
Obviously, the solution is the unique solution of [START_REF] Kasper | Hardware verification of a hyper-efficient (98%) and super-compact (2.2kW/dm3) isolated AC/DC telecom power supply module based on multi-cell converter approach[END_REF]. Generalizing this result, phase k current is given by:

Ω ⁄ (15)
Fig. 6 illustrates for -phase the first mode current waveform in comparison with the classic mode one. Obviously, each phase reference current shows abrupt variations at the beginning and at the end of the phase conduction. This pattern inherently limits the first mode to low speed.

Indeed, as the current is switched on while the related back-emf is high, the current rising slope is all the more limited than the rotor angular speed his high. Conversely, at the end of the phase conduction sector, the current falling slope is increased by the electromotive force and therefore, it increases with the machine rotation speed.

Mode 2: 1 phase inhibited alternately

As a consequence, for higher load, it is suitable to use two phases at the same time instead of one.

Hence, the new strategy involves alternately selecting two phases and thus turning-off one phase cyclically. The losses minimization leads to choose the phase having the lowest back

This involves solving the following set of equations:

2. . 0 2. . 0 0 (18)
The solution is defined as follows:

Ω ⁄ ; (19)
Despite appearances and for a given electromagnetic torque, the current shape is independent of the rotation speed, since the back electromotive force is proportional to speed. Fig. 7 shows the current of a-phase as well as the shape of the conventional mode reference current expressed in [START_REF] Whitney | Data center efficiency assessment[END_REF]. As in Mode 1, each phase reference current exhibits four abrupt variations per electric period. In contrast, the magnitude of the current step is lower, namely 2.91 times smaller.

Moreover, the second mode has degree of freedom meaning that it can take into account extra constraints, such as current slope limitations. This latter issue is discussed in section IV dealing with the proposed modes' implementation. This specific feature makes it possible to use this mode with limited slew rate at higher speed.

Fig. 7: Second mode optimal current waveform compared to the classic one.

3.4.Mode 3: phases functioning continuously

In the case of a high load operation, it is necessary to use all phases simultaneously. It is rare for back electromotive forces to be purely sinusoidal, hence the optimal currents should also contain some harmonics. Solving [START_REF] Falk | Efficiency and grid compatibility of photovoltaic inverters -state-of-the-art and future trends[END_REF] provides precisely the optimal waveform solution. In this mode, optimal 3-phase currents are defined as:

Ω ⁄ (20)
Fig. 8 illustrates the third mode optimal current shape. Note that, with the PMSM under test, it is very close to the classic one. If the zero torque ripple constraint expressed by (3) yields to infinite solutions of current waveforms, on the basis of the minimum RMS value (4) it must be concluded that the optimal waveform is not the classic sinusoidal one. Because of the presence of back-emf harmonic components, it should also be noted that the classic mode does not strictly satisfy the zero torque ripple constraint.

Optimization overview

The open-end winding drive architecture enables the drive of each motor phase independently.

Thus, it not only allows post-default operation but it also permits to optimize drive current waveform. The present section shows how to conveniently exploit the mentioned degree of freedom in normal mode with the purpose of minimizing losses. The previous section described three different modes of operation, and for each one, gave the best analytical current shape. The range of application of each mode must now be defined. This can be done by using key indicators associated to each mode. These indicators are no-load losses, amplitude and RMS value of the phase currents. The values of these indicators calculated relatively to those of the conventional operation mode for the same electromagnetic torque are given in Tab. 1.

1. Clearly, Mode 1 is more suited to low-load. In this mode, no-load losses are three times lower than for conventional mode. However, the very large magnitude and RMS value of the phase currents penalize this mode of operation by greatly increasing the Joule losses at high load. This is therefore not optimal to achieve high torque values with Mode 1.

2. It is particularly interesting to note that the second mode RMS current is very similar to that of the classic mode. This means that this mode will be the best choice for a wide range of electromagnetic torques as no-load losses are, in this case, only two-thirds of the classic mode.

3. Last but not least, Mode 3, corresponding to a conventional mode of operation but with optimized current waveforms, does not lead to significant differences in the RMS current.

This mode does not provide a significant gain in performance and is therefore not of great interest.

After this brief review, the following section intends to give a proof of concept related to these results.

Accepted

INNOVATIVE APPROACH PROOF OF CONCEPT

An experimental test bench has been developed to validate the suggested strategy. The goal is twofold. First, it has to be shown experimentally that a simple current control scheme can track the various proposed current references. Secondly, trials should demonstrate that the innovative proposed strategy improves electromechanical conversion efficiency.

Experimental Test Bench

The experimental setup (Fig. 9) is built around the following devices:

 A permanent

21

(Nrated = 3500 r.min -1 ) in case it is fed with high voltage (in the order of 1 kV). To achieve this, a boost power converter ensuring the connection of the battery with the inverter permits to meet DC bus high voltage requirement [START_REF] Sousa | Combined Electric Device for Powering and Charging[END_REF]. This additional degree of freedom is not considered in this study. The present work only focuses on the optimization of the inverter degree of freedom. Consequently, the battery is directly connected to the inverter DC bus and a 300 V level leads to a base speed of 1230 r.min -1 , which represents 0.35 pu;

 A power inverter (ARCEL) configured in 3 full H-bridges using 6 IGBT modules (1200V / 150 A @ 25°C). Each H-bridge output separately supplies a specific motor phase; FPGA board. This system enables to achieve analogue-to-digital conversions, the PWM vector control and current controllers. Programs to be run on this system are written by the user in a Matlab-Simulink language. The host PC also provides real time facilities to monitor the processes through Control-desk software.



24

A critical point of the proposed method is the capability of the current regulators to track the suggested reference currents. Therefore the following subsection describes this issue.

4.2.Implemented current regulators for each mode

First mode current controller

In the first mode, the motor is a single input (the unique phase voltage , with , , ) single output (the related phase current ) system subject to a measured disturbance (the related electromotive force emf ). In a 60° angle sector, the current dynamic can be described by the following classical differential equation: (21) with L the phase self-inductance, R the phase resistance. Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

25

To tackle the current tracking issue, the combination of the back-emf compensation and an integral and proportional controller (IP), as defined in [START_REF] Lin | On-line gain-tuning IP controller using real-coded genetic algorithm[END_REF], is adopted. As the phase voltage is supplied by a power inverter, the IP gains are tuned to meet the conditions for validity of the above average model. This leads to adopt a closed-loop bandwidth twenty times smaller than the switching frequency, namely 1 .

In the sectors where the phase has to be inhibited, all the related IGBT switches are off, while the integral value is imposed to zero permitting a good starting transient behavior when the phase is switch on again. When the phase is switched off, the related diodes ensured the continuity of current and also permits the most efficient and rapid extinction of the current in the phase.

Figure 12 summarizes the implementation of the first mode current controller.

Second mode current controller

In the second mode, the motor is a multivariable system with two inputs and two outputs, the two conducting phase voltages and currents respectively. In a 60° angle sector, the current dynamic is governed by the following differential equation: [START_REF] Ling | High step-up interleaved boost converter with low switch voltage stress[END_REF] with M the mutual inductance between two conducting phases, and , being one of the following couple: , , , , , .

The proposed control strategy is similarly based on IP regulator combined with emf compensation. To do so, the linear dynamic model has to be decoupled using a specific transform. Based on linear algebra, and more specifically on matrix diagonalization and eigen decomposition, the following matrix transform is suggested:

1 √2 1 1 1 1 (23)
It leads to the decoupled model in the new frame , :

0 0 (24)
Using these virtual values, the current control design is fully similar to the first mode one.

Figure 13 shows an overview of the second mode current controller.

Third mode current controller

In the third mode, the motor is a multivariable system with three inputs and three outputs described by the following 3-equations system:

This current control issue has already been studied in many research works using classic Park transform [42]. The key feature of this power architecture is that the zero sequence mode has also to be controlled, which is challenging because the zero sequence inductance L0 is small leading to fast natural dynamic and high current ripples due to inverter switching [42,[START_REF]Intersil, ISL78225 datasheet, 4-Phase Interleaved Boost PWM Controller with Light Load Efficiency Enhancement[END_REF][START_REF] Lin | On-line gain-tuning IP controller using real-coded genetic algorithm[END_REF]. Figure 14 depicts the third mode current controller. Several tests are conducted and analyzed using this experimental setup and the current regulators explained above.

i b i c en. / inh. 1 2 3 1 2 3 T ref References i ,ref i ,ref θ i  current controller i  current controller θ i a Sector selector T -1 -emf Current selector θ  d/dt i  i  i k2 i k1 e  e 

Self-control modes behavior

The first goal of the experiments is to test the ability of the implemented control scheme to track the current reference efficiently in the three suggested modes. Trials presented in this section are performed under the same rotation speed, i.e. 150 r min and 600 r min . At the first speed ( ), Fig. 15 shows the 3-phase currents of each mode, which produce the same average electromagnetic torque (i.e. 0.706 ). It also shows the instantaneous motor torque. The oscilloscope is triggered when the electric angle reaches π value. In each situation, it is clear that the real-time control implementation enables to track precisely the related reference.

29

To produce the same torque, peak and RMS currents are decreasing from first to third mode.

These results are consistent with the theoretical study (see Tab. 3).

However, it may also be noticed that in Modes 1 and 2, the current shape presents slight differences during a small duration at the switching time between two different sectors:

First, switching from a sector to another requires steep current edges, which are difficult to track with a limited DC bus voltage. In the first two modes, this problem results in the small switching time required to reach the current reference when the current starts from zero. As the energy optimization has led to select a phase conduction regarding its back-emf, it is important to notice that the back-emf value contributes to improve the current decay. Conversely, the emf value reduces the ability of the inverter to enforce a high current rising slope. It is for this latter reason that the first two modes are intrinsically limited to low speed. Indeed, the back-emf magnitude increases with speed linearly. As a consequence, the real currents show an asymmetrical waveform different to the symmetrical references. This mismatch increases with motor speed.

Secondly, magnetic couplings inside the motor may lead briefly to unwanted current behavior. In Mode 1, this is noticeable during the current impulse occurring in the inhibited phase while the current in the two other phases switches. Regarding the second mode, this is also visible in the slight overcurrent occurring in the turn-on phase during a sector switch.

The discrepancy between current references and real currents explains the torque ripple which occurs at each sector switching. This phenomenon becomes more pronounced as speed increases as illustrated by Fig. 16. With regard to Mode 1, as the current waveform solution is unique, it does not make it possible to consider suboptimal solutions at high speed. Consequently Mode 1 is limited to the low speed range. Conversely, Mode 2 presents a degree of freedom which was first Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 30 used for minimizing the current RMS value by focusing solely on achieving a constant torque.

However, the chosen constraint can be modified leading to a different solution.

That is the reason why the next subsection suggests an enhancement of Mode 2 at high speed while subsection 4.5 analyses losses measurement with the aim of assessing the real profits of using such an approach, despite its small imperfections.

4.4.

Mode 2 with limited slew rate (Mode 2,lsr)

As noted in the description of previous experiments, the starting current edge is intrinsically all the more difficult to achieve than the motor speed increases. Nonetheless, using 1 phase inhibited alternately strategy allows to meet the constraint of a constant instantaneous torque with unlimited possibilities. Consequently it is possible to consider extra constraints like the limited current slope due to finite DC bus voltage. Specifically, while the speed increases, the actual current waveform becomes too asymmetrical and it is better to modify the constraint in order to guarantee a real zero torque ripple. This new mode is called Mode 2,lsr for Mode 2 with limited slew rate. To achieve this, it is then mandatory to start each phase conduction from a zero current and to constraint the current rate in a given range around zero, while still enforcing continuous constant torque. The optimization problem is now expressed as follows:

⁄ /

/6 0 and 2 ⁄ /6

⁄

Note that setting the first initial current to zero imposes the second initial current to zero using [START_REF] Su | A Novel Phase-Shedding Control Scheme for Improved Light Load Efficiency of Multiphase Interleaved DC-DC Converters[END_REF]; this latter is obviously the final condition of the first current.

This optimization problem can be easily solved numerically by discretizing the electric angle . It was done for various slew rate range. In case of 1.5 3 ⁄ ⁄ , Fig. 17 shows the experimental value of the motor torque and the related 3-phase currents at and 0.706 . The symmetrical current waveform and consequently the absence of torque ripple are reflected in Fig. 17. It also shows that the current has three behaviors on a half conduction sector: it is first limited in slew rate, then follows a curve similar to Mode 2 waveform and finally has a restricted slope. As a consequence, Mode 2,lsr is a good option for increasing the speed range of the studied power partitioning strategy of motor drives.

To illustrate the good dynamic behavior of the suggested modes, it is chosen to show the torque tracking while connecting the torque reference to a square-wave generator having a frequency of 20 Hz, a minimum value of 0.24 pu and a maximum value of 0.71 pu. It may be observed that for low torques, total losses are greater when more phases operate. Thus, under these operating conditions, the mode that appears best for good performance in term of power losses is the Mode 1. This confirms the theoretical results. However, the use of Mode 1 involves more quadratic losses in comparison with the other two modes of operation. Therefore overall losses in Mode 2 become lower than in Mode 1 above a threshold value of 0.65 pu.

Above this middle-value, it is better to select Mode 2,slr. Indeed Mode 2,lsr presents similar losses to those of Mode 2, except in high current RMS values. Mode 2,lsr also produces a slightly smaller torque at the rated current level. These features are not surprising, given the fact that the number of phases conducting simultaneously is identical and the current waveforms are roughly similar. The partial discrepancy leads to a slightly higher RMS value for Mode 2,slr than for Mode 2. Conversely it also enables to use Mode 2,slr at higher speed which is indeed a definite advantage to this latter suggested mode. In addition, because of the rated current (10 A), the Surprisingly, losses in Mode 3 are higher than expected. Indeed, for high-load, where quadratic losses are dominant, Mode 2 and Mode 3 were expected to lead to very similar losses. However, in Mode 3, the three phases of the machine are fed simultaneously, currents in the machine are therefore defined by the response of two equivalent fictitious machines, namely the main machine and the zero-sequence machine. As the zero-sequence inductance is low (roughly a tenth of the phase self-inductance), the machine is, in Mode 3, subject to higher currents at the switching frequency, due to zero-sequence voltages generated by the inverter, and therefore this mode leads to much higher losses than in the two other modes. However, it may be noted that Mode 3 is mandatory to implement high torques.

CONCLUSION AND PERSPECTIVES

This paper studies the combination of a 3-phase open-end winding synchronous machine and a 3H-bridge inverter. This motor drive architecture, covered by several patents owned by VALEO group, provides degrees of freedom. One of its principal characteristics is its capability to be fault tolerant.

The original feature in the innovative approach is based on the optimization of the drive normal operation mode. It makes better use of the architecture degrees of freedom. The idea consists in adjusting the number of phases in use to the demanded torque. By so doing, the power efficiency at low torque operating points has been increased significantly. For instance, at half rated torque (21.25 Nm) the global losses have been diminished by one-third (Fig. 19).

Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

38

The optimization method is based on the control-oriented model of the synchronous machine. It relies on a general approach using the Euler-Lagrange method. The result is an analytical solution and is easy to adjust to various synchronous motors because it is based on a generic formulation.

An experimental setup enables to validate the theoretical considerations. First, it allows to prove that the drive control scheme is able to track the innovative optimal current references in each of the three proposed modes. Overall losses measurements show the benefit of Mode 1 for low torques and Mode 2 for middle torques. Mode 3 is very close to basic sinusoidal mode and allows to produce high torques.

A future work perspective is to better understand the impact of the various PWM vector modulations on the drive overall losses and design an energy management strategy. Additionally, future research should focus on the PMSM local behavior while operating in Modes 1 or 2 at high current rms value. The issue is that it has to be ascertained whether there are no local core saturation and possibly refine the PMSM design accordingly. At this stage of the analysis, only large-scale phenomena can be observed. Looking at losses, torque response and voltage waveforms show no core saturation. This further study will specify this issue and identify remedial action for developing a final product.

6.

Fig. 1 : 3 -

 13 Fig. 1: 3-phase open-end winding architecture.

FigFig. 3 :

 3 Fig. 2: 2-Module Power Converter efficiency regarding the strategy used.

Fig. 4 :

 4 Fig. 4: 3 back electromotive forces at constant speed (1500 r.min -1 ).

Fig. 5 :Fig. 6 :

 56 Fig. 5: 6 sectors related to first mode.

  Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 16 electromotive force absolute value. Hence, for each specific position, the electric power is concentrated on the two phases with the highest electromagnetic potential. This also leads the division of an electric revolution into six sectors of the same size ( 3 ⁄ ). They are 6 ⁄ out of phase with the six first mode sectors. Considering the first sector (

Fig. 8 :

 8 Fig. 8: Third mode optimal current waveform compared to the classic one.

  Sensors, namely 3 Hall effect sensors (LEM) for each phase and a 2500-pulse position encoder (IVO Industries) on the motor shaft;  An AGILENT voltage source set to 300 V with an 11 A rated current. It powers the inverter DC bus;  A 5.5-kW induction machine (IM) is mechanically coupled to the PMSM under test and is controlled using OMRON 2G2 drive in the speed control mode. Thus it provides loading onto the PMSM;  An active load (AMREL) connected to the OMRON drive dissipates the power supplied by the IM;  A host PC connected to a DSPACE real-time control system mainly composed of a DS1006 processor board, a DS3001 incremental encoder interface board and a DS5203

Fig. 9 :Fig. 10 :

 910 Fig. 9 : Experimental setup view.

Fig. 11 depicts

 11 Fig.11depicts both power architecture and its related control structure. The structure and the parameters tuning of the current controllers are based on the motor's dynamics. However, the current behavior depends on the mode used. Therefore the description of the current regulators is outlined in the 3 modes considered.

Fig. 11 :

 11 Fig. 11 : Control scheme of the PMSM drive.

Fig. 12 :

 12 Fig. 12 : Current controller of the first mode.

Fig. 13 :

 13 Fig. 13 : Current controller of the second mode.

integrator reset 3 PWMFig. 14 :

 314 Fig. 14 : Current controller of the third mode.

  Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

Fig. 15 : 3 -Fig. 16 : 3 -

 153163 Fig. 15 : 3-phase current and related torque in the 3 suggested modes at .

  Fig 18 shows expected behavior of currents and torque. Specifically, the torque response time is in the range of a millisecond, as expected.

Fig. 17 : 3 -Fig. 18 : 3 - 36 Fig. 19 :

 1731833619 Fig. 17 : 3-phase current and related torque in Mode 2,lsr in steady state at

  Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. 37 maximum torque value achieved in Mode 1 is limited to 0.85 pu, whereas Mode 2 enables 0.97 pu and Mode 3 reaches 1.03 pu. Losses in Mode 3 change almost linearly with torque increments. To discern the intersection of the curves corresponding to Mode 2 and Mode 3, it is necessary for the current in the phases to exceed the rated value.

  article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

	Tab.1: Key indicators for each proposed modes
	Indicator	Mode 1	Mode 2	Mode 3
	,
	,	1.662	1.263	1.059
	,
	,	1.268	1.031	1.002
		1	2	1
		3	3
				20

, and , and are the relative magnitude current, effective current and no-load losses respectively. Tab. 1: Key indicators for each proposed mode.

  Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

	magnet synchronous machine (PMSM) designed by LEROY SOMER. Its
	key features are summarized in Tab 2. The motor has concentrated windings (stator) and

interior-magnets (rotor) as shown in Fig.

10

. It can reach high speed value

  Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014. Accepted article for publication in a future issue of ELSEVIER Electric Power Systems Research -Citation information: 10.1016/j.epsr.2016.09.014.

		TAB.2: PMSM CHARACTERISTICS
	Symbol	Parameter	Value
	R	Stator phase resistance	1.72 
	L d	d-axis self-inductance	14 mH
	L q	q-axis self-inductance	12.5 mH
	L 0	0-axis self-inductance	1.3 mH
	p	number of motor pole pairs	4
	M	amplitude of the induced flux	0.494 Wb
	KSM,1	1 st harmonic back-e.m.f. constant	1.417 V.s
	KSM,3	3 rd harmonic back-e.m.f. constant	0.0354 V.s
	KSM,5	5 st harmonic back-e.m.f. constant	0.0354 V.s
	T rated	Rated torque	42.5 Nm
	Nrated	Rated speed	3500 r.min -1
	Vrated	Rated voltage	970 V
	Irated	Rated current	10 A
		Tab. 2 : PMSM characteristics.
			23


Olivier Béthoux
email: olivier.bethoux@centralesupelec.fr

Eric Labouré
email: eric.laboure@centralesupelec.fr

Eric Berthelot
email: eric.berthelot@centralesupelec.fr

Ghislain Remy
email: ghislain.remy@u-psud.fr

Real-Time Optimal Control of a 3-phase PMSM in 2-Phase Degraded Mode 1 

Keywords: fault-tolerant control, optimization, losses minimization, permanent-magnet synchronous motor (PMSM) drive, phase loss, remedial strategy, real-time. I

This paper aims to optimize real-time control for the degraded mode of a fault-tolerant power architecture, but not the fault detection and isolation procedure itself. Such power architecture is dedicated to electric vehicles in which it performs the three following essential functions: traction, battery charging and electric grid assistance. For safety reason, in degraded mode, the power control is limited to the traction mode. Thus, for a given torque the proposed innovative strategy uses a novel current/voltage transform which leads to an efficient real-time control of the torque. The key idea is to drive the current without a priori restrictions on its waveform, while minimizing the Joule losses, i.e. the effective value of the current. It has been validated on a laboratory test bench. The studied system is based on a 3phase open-end-winding synchronous machine powered by a 3 full H-bridges inverter. The last section of the paper analyses the comparison between the classic sinusoidal current waveforms and the proposed ones while operating on the two remaining motor phases. It results in a 14% increase of the torque produced by the permanent magnet machine under test, and a 14% decrease of the global system losses in traction mode. As a result, the new control strategy enhances traction performance in degraded mode and increases electric vehicle autonomy in postfailure condition.

renewable energy sources in a smart electric grid [START_REF] Rigas | Managing Electric Vehicles in the Smart Grid Using Artificial Intelligence: A Survey[END_REF], [START_REF] Santiago | Electrical Motor Drivelines in Commercial All-Electric Vehicles: A Review[END_REF]. Several drives [START_REF] Gao | Integrated Energy Management of Plug-in Electric Vehicles in Power Grid With Renewables[END_REF] and several architectures [START_REF] Sakr | A review of on-board integrated chargers for electric vehicles[END_REF] meet the above criteria. Since 2011, innovative solutions (fig. 1) have been patented for EV power system by the VALEO group [START_REF] Sousa | Combined Electric Device for Powering and Charging[END_REF]- [START_REF] Bouchez | Charge transfer device and associated management method[END_REF]. Precisely the patented power architecture enables to share the three main power functions without using any electric contactor. It is based on a three-phase open-end-winding permanent magnet synchronous machine (PMSM) fed with a power inverter made of three full H-bridges. The DC bus is powered by a high voltage battery using an interleaved buckboost converter.

This solution has proved to be simple, fault-tolerant and energy efficient [START_REF] Dogan | Multistatic Reluctance Network Modeling for the Design of Permanent Magnet Synchronous Machines[END_REF]- [START_REF] Diao | A Differential Algebraic Estimator for Sensorless Permanent-Magnet Synchronous Machine Drive[END_REF]. Indeed, this system has already been analyzed and discussed in different ways: 1) Analytical and numerical computing methods to design the synchronous machine (regarding zero-sequence inductance, electromotive force, etc) [START_REF] Dogan | Multistatic Reluctance Network Modeling for the Design of Permanent Magnet Synchronous Machines[END_REF], 2) Innovative techniques for ensuring good performances in electric grid connection mode [START_REF] Lacroix | An integrated fast battery charger for electric vehicle[END_REF], 3) Improved vector control minimizing current ripple in traction mode [START_REF] Bruyère | A multiphase traction/fast-battery-charger drive for electric or plug-in hybrid vehicles: Solutions for control in traction mode[END_REF], new approaches enabling high speed range optimizing air-gap flux-weakening [START_REF] Sandulescu | Control Strategies for Open-End Winding Drives Operating in the Flux-Weakening Region[END_REF], robust methods to track down any possible failures in the drive [START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF] and simple techniques enabling traction mode without a speed sensor [START_REF] Diao | A Differential Algebraic Estimator for Sensorless Permanent-Magnet Synchronous Machine Drive[END_REF]. In real-time control for the degraded mode of a faulttolerant power architecture, Fault Detection and Isolation (FDI), is a major issue [START_REF] Benbouzid | Advanced Fault-Tolerant Control of Induction-Motor Drives for EV/HEV Traction Applications: From Conventional to Modern and Intelligent Control Techniques[END_REF]. In the studied drive, a fault may occur either in the motor itself or in the voltage PWM inverter [START_REF] Zidani | A Fuzzy-Based Approach for the Diagnosis of Fault Modes in a Voltage-Fed PWM Inverter Induction Motor Drive[END_REF]- [START_REF] Yang | An industry-based survey of reliability in power electronic converters[END_REF].

 The fault has to be detected by the IGBT driver (hardware detection) if an IGBT fails in short-circuit [START_REF] Lu Bin | A Literature Review of IGBT Fault Diagnostic and Protection Methods for Power Inverters[END_REF]- [START_REF] Dulau | A New Gate Driver Integrated Circuit for IGBT Devices with Advanced Protections[END_REF].  The fault can be detected by a FDI algorithm in the case of an open-phase fault or an open-switch fault.

Then, two main approaches can be used: the first FDI technique consists in using normalized dc-components [START_REF] Sleszynski | Open-transistor fault diagnostics in voltage-source inverters by analyzing the load currents[END_REF]- [START_REF] Estima | A new approach for real-time multiple open-circuit fault diagnosis in voltage-source inverters[END_REF], while the second FDI method involves computing the magnitude and the frequency of the 3phase currents [START_REF] Meinguet | A Method for Fault Detection and Isolation Based on the Processing of Multiple Diagnostic Indices: Application to Inverter Faults in AC Drives[END_REF]. This current work does not focus on the FDI procedure but intends to enhance drive performances in post-failure functioning, increasing the maximum torque and extending the vehicle's range. Indeed, the above architecture is fault-tolerant since it is able to operate using two out of three phases. For an Fig. 1. Innovative power architecture for EV suggested by VALEO [START_REF] Sousa | Method of exchanging electrical energy between an electrical network conveying a dc or ac electrical quantity and an electrical energy storage unit for hybrid or electric vehicle[END_REF]. electric vehicle, traction mode is vital and its loss could lead to dramatic consequences. In a fault-tolerant architecture, a phase default is detected (caused by a motor winding or a power switch), the combination of inverter and machine is then reconfigured in order to isolate the faulty phase [START_REF] Kolli | Space-Vector PWM Control Synthesis for an H-Bridge Drive in Electric Vehicles[END_REF]. S. Bolognani, et al. demonstrated in [START_REF] Bolognani | Experimental fault-tolerant control of a PMSM drive[END_REF] that using sinusoidal waveforms enables the drive to operate up to 58% of the rated torque. PMSM are designed and optimized on various criteria (cost, main machine and zero-sequence machine inductances, cooling, …) leading to a final implementation with most frequently non-sinusoidal electromotive forces (emf). In this situation, the question arises whether the choice of sinusoidal currents in the two remaining phases is relevant for minimizing the drive losses and for maximizing the available torque. This question directly impacts 1) The EV range by improving global efficiency;

2) The availability and the safety of the EV in degraded mode. Thus, the purpose of the present article is to suggest an optimal strategy to control the PMSM drive in real-time during post-failure operation. The key points are: 1) efficient torque control using a novel current/voltage transform, while keeping a simple and easy-to-tune control algorithm, 2) optimal current references enabling to minimize global drive losses and to maximize drive torque. The paper is organized as follows. Section 2 explains the post-failure drive control structure, describes the developed experimental test bench and shows the drive behavior while using basic sinusoidal current references. Section 3 presents the innovative strategy. Section 4 assesses on an experimental bench its performances against a competing classic approach. Section 5 draws a conclusion and outlines that the proposed control strategy may extend to other drive applications.

II. THREE-PHASE OPEN-END-WINDING SYNCHRONOUS MACHINE IN TWO-PHASE MODE

The present section addresses the problem of driving a three-phase synchronous motor using only two of the three phases. To tackle this issue, an innovative reference frame is introduced, allowing the design of a simple control scheme based on PI-controllers; additionally a dedicated space vector (SV) pulse-width modulation (PWM) technique permits to minimize inverter switching losses. This control architecture is successfully implemented on a laboratory test bench. At the end of the section, the latter is presented and a test with classic sinusoidal current references is discussed. Synchronous motor drive in degraded mode (third phase disconnected).

A. Electrical model in degraded mode

The synchronous machine operates according to Faraday law. Within the context of the loss of the third phase (Fig. 2), the two remaining stator currents and stator voltages are described by the following 2-equations system: [START_REF] Rigas | Managing Electric Vehicles in the Smart Grid Using Artificial Intelligence: A Survey[END_REF] with L the phase self-inductance, M the mutual inductance between two phases, R the phase resistance and the back electromotive forces (back-emf) induced into the two remaining phases by the rotor movement.

At the shaft rotation speed Ω, currents control enables to generate an instantaneous electromagnetic torque:

Ω ⁄ (2)
To this end, the two machine voltages are produced by the voltage inverter and are computed by a current-controller.

B. Current/voltage transform in degraded mode

Compared to classical Park and Concordia transform for 3phase machines [START_REF] Zidani | Induction motor stator faults diagnosis by a current Concordia pattern-based fuzzy decision system[END_REF], in degraded mode a novel transform is needed using 2 of the 3 phases. Unlike fault-tolerant PMSM drive, mostly based on electrically and magnetically isolated windings [START_REF] Mecrow | Fault-tolerant permanent magnet machine drives[END_REF]- [START_REF] Sneessens | Modelling and torque control of a five phase PMSM synchronous motor using tooth concentrated windings technology[END_REF], the studied machine windings are magnetically well coupled to permit an optimized 3-phase or single phase battery charging mode [START_REF] Lacroix | An integrated fast battery charger for electric vehicle[END_REF]. Therefore, we consider our transform as novel in contrast to the solution adopted in [START_REF] Labbé | Two phase operation for a three phase PMSM using a control model based on a Concordia like transform associated to a classic Park transform[END_REF] as explained below.

To tackle this two-dimensional control problem, the adopted approach consists of diagonalizing the inductance matrix using an appropriate transformation. Matrix eigenvectors are proportional to 1 1 and 1 1 and are associated to two distinct eigenvalues, namely and respectively. Subsequently, the matrix is defined by ;

√ 1 1 1 1 (3)
; e

Indeed, the transform proposed in [START_REF] Labbé | Two phase operation for a three phase PMSM using a control model based on a Concordia like transform associated to a classic Park transform[END_REF] is only validated by simulation, the mutual inductance value M must not be neglected to ensure a pure system decoupling in the fictitious frame; furthermore, the power invariance principle is respected, hence simplifying current references computation from torque demand.

C. Control scheme in degraded mode

In a classic way the back-emf are computed and compensated in the proposed control scheme. Current control can now be achieved using two independent controllers. Each one deals with a linear first order system best suited for classic PI controllers. Fig. 3 depicts the control structure suggested for driving the PM in post-failure mode. It works with fictitious variables based on the measures and the transformation and generates the inverter PWM signals directly in the new Γ Δ frame as explained in the following subsection.

D. SV-PWM in degraded mode

A fictitious machine was defined to achieve a simple and efficient control structure of the degraded mode of the 3-phase PMSM working with two out of the three phases. As the PMSM is specifically powered by a voltage inverter, the second crucial issue is to explore its ability to guarantee a space vector control in the degraded mode.

After successful isolation, each of the two healthy Hbridges can still generate three voltage levels, namely 0, and . As a result, the inverter is only able to produce nine discrete space vectors. Using the suggested fictitious machine approach, the output voltage vectors are described in Γ Δ frame by using the innovative transform in (3). The available discrete voltages are normalized with respect to bus voltage and listed in Table I while Fig. 4 illustrates them in Γ Δ frame. Both H-bridge inverters can provide an average value of the voltage setpoint

Δ Δ Δ (6)
The set of three voltage vectors is chosen to guarantee that any Δ is positive. The main issue is to select the most relevant one [START_REF] Kolli | Space-Vector PWM Control Synthesis for an H-Bridge Drive in Electric Vehicles[END_REF], [START_REF] Vafakhah | Interleaved Discontinuous Space-Vector PWM for a Multilevel PWM VSI Using a Three-Phase Split-Wound Coupled Inductor[END_REF]. As depicted in Fig. 4, eight sectors are delimited by eight non-zero voltages. From this representation, it clearly appears that reducing voltage differences leads to decreased voltage ripples, which implies using adjacent vectors of the sector under consideration. Reducing switching losses is important [START_REF] Xiong | New Physical Insights on Power MOSFET Switching Losses[END_REF] to improve the inverter reliability because thermal stress (due to losses) is a key factor in the acceleration of the failure mechanism process [START_REF] Coquery | Power Module Lifetime Estimation from Chip Temperature Direct Measurement in an Automotive Traction Inverter[END_REF]. Indeed, the switching losses are mostly an important part of the semiconductor power losses [START_REF] Xiong | New Physical Insights on Power MOSFET Switching Losses[END_REF]. Hence, switching frequency should decrease; leading to generate higher current ripple magnitude and so additional losses [START_REF] Van Der Geest | Influence of PWM switching frequency on the losses in PM machines[END_REF]. To reduce it, as switching frequency cannot be increased, the trade-off has to be done by the switching control strategy. Then, it is crucial to build a sequence guaranteeing an equitable switching distribution between legs. It is also important to obtain a limited number of switching and to avoid over-switching while changing from one sector to another.

In conclusion, the reference vector is located in a unique sector associated to a single set of three voltages (zero voltage and two active vectors). For example in Fig. 4, sector I is delimited by the three discrete voltages , and . The proposed switching sequence associated to this sector is illustrated in Fig. 5. Each single H-bridge needs only one switching, 1 turn-on or 1 turn-off during 2 ⁄ . Finally, the minimization of the switching losses and their equitable distribution requirements are therefore fulfilled.

TABLE I REMAINING STATOR VOLTAGES IN A-B AND -

FRAMES VECTORS V a V b V VD V1 0 0 0 0 V2 0 -1 1 √2 ⁄ 1 √2 ⁄ V3 0 +1 1 √2 ⁄ 1 √2 ⁄ V4 -1 0 1 √2 ⁄ 1 √2 ⁄ V5 -1 -1 √2 0 V6 -1 +1 0 √2 V7 +1 0 1 √2 ⁄ 1 √2 ⁄ V8 +1 -1 0 √2 V9 +1 +1 √2

E. Torque control implementation using classic references

The suggested remedial control structure has been implemented and validated on an experimental test bench specifically designed for the study of the EV power architecture patented by VALEO (Fig. 1). Following a brief presentation of the experimental setup (Fig. 6), validation tests based on classic sinusoidal references are discussed underlying the performance limits. To overcome the limitations due to this basic strategy and enhance remedial strategy performance, the next section proposes a comprehensive optimization.

1) SM under test and experimental setup

The PM motor drive under test consists of an 8-pole 3phase open-end-winding PMSM (LEROY SOMER) supplied by an inverter composed of 3 full H-bridges powered by the same DC bus. The latter is fed by a 3.3 kW DC supply source (AGILENT) adjusted to its 300 V rated voltage. The PMSM is coupled to an induction machine (IM) mechanically. Using a dedicated industrial drive (OMRON), the IM operates in speed control. It allows mimicking the PMSM mechanical load and can work in either regenerative braking or traction modes. The key PMSM features are summarized in Table II. Each of its phases is equipped with a Hall effect current sensor (LEM). An incremental encoder (IVO Industry) coupled with the PMSM shaft enables the precise determination of the rotor position (its resolution is of 2500 points per revolution).

The control of the PMSM -inverter combination is implemented in a DSPACE real-time system. It is composed of a DS 1106 main card and secondary DS 3001 and DS 5203 cards. DS 3001 enables to determine the absolute rotor position. DS 5203 is based on a FPGA Xilinx Virtex® 5 driven with a 100 MHz clock. The suggested 20 kHz SVPWM is implemented in it. It also performs analog-to-digital conversions at the switching frequency rate. This frequency is used to synchronize the DS1106 microprocessor in which current control is implemented.

2) Control structure validation using classic strategy

To validate the proper functioning of the global system with the suggested control part, the first test consists in using the classic sinusoidal waveform strategy. Within this context, outputs calculated by the "current references" function block are based on the torque reference T and the electric angle .

The required fictitious current references , , can therefore be inferred from the suggested transform as described in [START_REF] Sousa | Combined Electric Device for Powering and Charging[END_REF].

V  V  V  V  V  V  V  V  V  V  45°S ector I V  V 1 V 9 V 3 t 1 t 9 t 3 T SW /2 T SW /2 V 1 V 9 V 3 t 9 t 3 t 1
As the ultimate objective is to ensure continuity of service in traction mode, the remedial strategy is applied immediately after a phase default occurs. This default is simulated on phase "c" and it is assumed that the fault is detected and isolated instantaneously and the phase "c" is immediately disconnected by its phase isolator. For this test, the PMSM delivers 20 N.m of torque and is loaded by the IM which maintains a 600 r.min -1 constant rotational speed. The oscilloscope is triggered at a time when the phase fault is simulated (Fig. 7). On the left-hand side of the picture are captured phase and DC bus currents in normal condition, while the post-failure behavior is shown on the right. The remedial control structure fully allows to track the calculated current references: remaining currents become √3 times larger than in normal mode and are with 60° phase shift. However, it is important to note that a significant current ripple appears on the DC current feeding the inverter. The oscillation frequency is twice the electric frequency. This natural phenomenon occurs because of back-emf harmonics which have not been yet considered. It is fully consistent with the simulation analyses presented in [START_REF] Yang | An industry-based survey of reliability in power electronic converters[END_REF] where authors mention an additional torque ripple due to harmonics in emf and saliency effect.

This proves that the classic strategy does not sufficiently address the problem of driving the proper remedial currents. Therefore, section III tackles their optimization and section IV compares, on the developed test bench, the suggested solution with the classic sinusoidal strategy.

III. CURRENT OPTIMIZATION IN DEGRADED MODE

Most classical control strategies of PMSM aim to achieve the required torque while minimizing the Joule losses [START_REF] Baudart | Torque Control Strategy of Polyphase Permanent-Magnet Synchronous Machines With Minimal Controller Reconfiguration Under Open-Circuit Fault of One Phase[END_REF]. Indeed, Joule losses induce motor and inverter thermal stresses which affect their reliability [START_REF] Song | Evaluation Methodology and Control Strategies for Improving Reliability of HEV Power Electronic System[END_REF] and also reduce the global car driving range due to a decline in power efficiency. However these classic approaches usually consider pure sinusoidal back-emf in the PMSM. They also force pure sinusoidal currents in both the normal and degraded modes of operation [START_REF] Labbé | Two phase operation for a three phase PMSM using a control model based on a Concordia like transform associated to a classic Park transform[END_REF], and may improve the drive performance by adding a third-harmonic component [START_REF] Bianchi | Strategies for the fault-tolerant current control of a five-phase permanent-magnet motor[END_REF], [START_REF] Dwari | Fault-tolerant control of five-phase permanent magnet motors with trapezoidal back EMF[END_REF]. Generally speaking, the addressed optimization problem consists in defining the best trajectory between two given points, namely the current waveform for one motor revolution, while respecting a constraint, namely a constant torque. The Euler-Lagrange method has historically been developed for this class of problem [START_REF] Naidu | Calculus of Variations and Optimal Control[END_REF], [START_REF] Ngo | Optimal Control of the Gearshift Command for Hybrid Electric Vehicles[END_REF] and successfully used for current optimization in PMSM [START_REF] Baudart | Torque Control Strategy of Polyphase Permanent-Magnet Synchronous Machines With Minimal Controller Reconfiguration Under Open-Circuit Fault of One Phase[END_REF], [START_REF] Ede | Effect of optimal torque control on rotor loss of fault-tolerant permanent magnet brushless machines[END_REF]. Consequently, the Euler-Lagrange procedure is used in this particular case and applied in its generality. The obtained analytical form will be discussed subsequently.

In the event of any back-emf, this section aims to minimize the Joule losses while achieving a constant electromagnetic torque T in steady state operation. The criterion used is therefore described by the functional , defined by: , Subject to the one simple constraint of constant power defined by Ω:

, , 0

To address this single constraint mathematical optimization, a new single Lagrange-multiplier variable λ(t) is introduced and the minimum of the unconstrained augmented functional , , is sought :

, , , , (10)
A necessary condition for , , optimality is that the partial derivatives of the Lagrange function are zero at any time [START_REF] Naidu | Calculus of Variations and Optimal Control[END_REF], which means that:

2 0 2 0 0 (11)
Solving this set of three equations with the three unknowns , , yields the following solution:

. ⁄ . ⁄ 2 ⁄ (12)
Consequently, it leads to a substantially different strategy than the classic solution which arises from the sinusoidal waveform constraint.

Specifically, even though considering pure sinusoidal backemf shape, (13) provides current waveforms completely different from the ones given by [START_REF] Sousa | Method of exchanging electrical energy between an electrical network conveying a dc or ac electrical quantity and an electrical energy storage unit for hybrid or electric vehicle[END_REF] and exhibited in [START_REF] Yang | An industry-based survey of reliability in power electronic converters[END_REF] for instance.

√ , sin 2 cos 2 3 ⁄ ⁄ sin 2 3 ⁄ 2 cos 2 3 ⁄ ⁄ (13)
Indeed the current references in the two healthy phases no longer have sinusoidal shapes and are phase shifted by 3 ⁄ instead of 6 ⁄ . The power output by phase is therefore quite different. Optimizing without preconceived waveforms increases the current level when the absolute value of the phase back-emf is large. Similarly, current and related backemf cross the zero point simultaneously. It induces that around this point the phase power is negligible which implies a maximum power in the other remaining phase. It is for that reason that current and back-emf are in phase and that the phase current peak occurs close to the time when the other back-emf is zero. The current peak value is slightly higher with the calculated waveforms than with pure sinusoidal shapes. Moreover, it is important to notice in this latter case that there is an additional torque ripple with a magnitude of about 13 percent of the mean torque. Now, considering the real back-emf measured on the PMSM under test, we are able to calculate the current reference waveforms minimizing the unconstrained augmented functional [START_REF] Lacroix | An integrated fast battery charger for electric vehicle[END_REF]. Fig. 8 depicts current references in both cases, namely classic sinusoidal waveforms and calculated waveforms for an operating condition defined by 20 and 600 . We can verify in Fig. 8 that the calculated currents effectively lead to a constant electromagnetic torque. These very encouraging results must be confirmed on the developed test bench in order to assess the increase of the maximal torque and also the power efficiency enhancement. To achieve this result, the current control implementation relies on the control scheme described in the previous section (Fig. 3) and using the novel transform. Thus the proposed current reference is controlled in the suggested -frame. The main electrical variables are depicted in the -frame in Fig. 9: the back-emf is required for forward compensation, each phase current is zero at the same time as the corresponding emf and the voltage is the control value driving the SVPWM block. In our control scheme, two PI controllers are used. The time constant and gain of these two PI are calculated in order to guaranty the closed-loop stability and ensure a sufficient gain at the fundamental frequency of the reference currents in order to reduce errors at this frequency.

IV. INNOVATIVE APPROACH PROOF OF CONCEPT

The present section assesses and benchmarks the performance of both the classic strategy and the suggested one using the test bench described in section II.

In order to compare the two approaches, Tab. III gives RMS and peak-to-peak current values in steady state, namely at 20 Nm and 600 r.min -1 . Obviously, constraining the current waveform to be sinusoidal yields a greater RMS current: the RMS value of suggested currents is 6 percent lower than in the classic scenario. By contrast, relaxing the current shape constraint provides a higher peak value, namely 13 percent higher than in the classic scenario. Indeed, as both current and back-emf are zero simultaneously, the second remaining current has to be greater around this point so as to compensate the phase power decrease. Fig. 10 presents the two the two situations under test. In concrete terms, this enables to enhance accordingly the vehicle range in post-failure operation. Similarly, the torque is increase by 14 % while operating at the same RMS current value. In practical terms, since the effective current is limited for thermal concern, this allows to perform a higher torque in remedial strategy, hence enhancing the traction mode function availability. Last but not least, Fig. 12 shows that the overall suggested strategy is relevant to achieve transient objectives of a drive. For this validation, the torque reference consists in a square waveform characterized by a 15 Nm low level and a 25 Nm high level. Fig. 12 shows that the current transient is effective with a response time of about 4 ms. This ensures the immediate response to driver demand, i.e. acceleration or braking.

V. CONCLUSION AND PERSPECTIVES

This paper aims to optimize the post-failure traction mode of a fault-tolerant power architecture dedicated to electric vehicles. The studied electric powertrain is based on a 3-phase PMSM. Considering degraded mode operation on two of the three phases, the study tackles the problem of optimal constant torque generation, and additionally, the drive losses are minimized. The related current waveforms are expressed analytically, which leads to a generic solution easy to transpose to any synchronous machine.

In order to validate the innovative strategy, an experimental test bench has been developed and a control scheme has been designed, implemented and validated. As soon as a default is detected in one phase, the control system switches from d-q reference frame to the suggested post-failure reference frame; it enables a proper functioning in degraded mode. In particular, the post-failure control structure permits 1) A simple PI controller tuning through truly decoupled system and back-emf compensation. 2) Optimized functioning of the inverter through a dedicated SVPWM. 3) A good switching from normal mode to remedial mode. Moreover, a comparative study has been carried out between a classic strategy based on sinusoidal current waveform and the proposed strategy with no preconceived waveform consideration. Regarding the combination of the inverter and the PMSM under test, the major conclusions that can be drawn from this paper are listed as follows.

1) Motor drive losses are reduced by 14%, thereby enhancing the electric vehicle range in degraded mode, which provides the driver with a higher chance to reach a maintenance point. 2) Moreover, the reduction of thermal stress benefits the reliability in post-failure mode, which is a second keypoint. 3) Maximal motor drive torque is increased by 14%, thereby improving the capacity to control the electric vehicle when a failure occurs, which benefits security issues. It has been shown experimentally that optimizing the current control strategy enhances PMSM drive performances in post-failure mode significantly. Additionally, the proposed strategy fits most PMSM drives since the current references are given analytically Possible future work in the area is to generalize this approach to the other types of electric motors used in electric vehicles, namely induction machines and variable reluctance machines.

Fig. 2 .

 2 Fig. 2.Synchronous motor drive in degraded mode (third phase disconnected).

  matrix which 1) enables to diagonalize the real inductance matrix and 2) preserves the power invariance principle since ⋅ . Using the suggested transformation (3) in (1) leads to the following new set of equations which describes the fictitious machine associated to the innovative transform T: currents, voltages and back-emf expressed in the new frame.

Fig. 3 .

 3 Fig. 3. SM drive control structure in degraded mode. generation is a 3-dimensional problem: two dimensions are devoted to the fictitious machine and one is dedicated to time. Therefore, 3 different discrete voltage vectors among the 9 inverter possibilities, are necessary for solving this SV-PWM problem. Noting 1 ⁄ the desired switching period, the 3 associated durations Δ Δ Δ are computed using the following equation: * * 1/2 1 1 1

0 Fig. 4 .Fig 5 .

 045 Fig. 4. 2 H-Bridges inverter switching states and space vectors in - frame.

Fig. 6 .

 6 Fig. 6. Experimental setup view.

Fig. 7 .

 7 Fig. 7. Experimental setup view.

Fig. 8 .

 8 Fig. 8. Optimal current references at 600 and 20 a) Considering pure sinusoidal waveform constraint; b) Considering no waveform constraint.

Fig. 9 .

 9 Fig. 9. PMSM drive fictitious variables in the -frame with optimal currents in degraded mode at 600 and 20 .experimental currents using a trigger moment based on backemf crossing zero; one can see that phase b current reaches its maximal value around this point in time. Nevertheless, the entire power balance of the drive in remedial strategy shown in Fig.11clearly highlights that losses are significantly reduced for a given torque; 14 percentage points represent a significant difference between

Fig. 11 .

 11 Fig. 11. Losses comparison of both remedial strategies 600 .

Fig. 12 .

 12 Fig. 12. Torque transient in post-failure operation mode: 600 . and 15 ↔ 25 .

TABLE II LEROY

 II SOMER LS 132 S SYNCHRONOUS MOTOR PARAMETERS

	Symbol	Parameter	Value
	R	Stator phase resistance	1.72 Ω
	Ld	d-axis self-inductance	14 mH
	Lq	q-axis self-inductance	12.5 mH
	p	number of motor pole pairs	4

M

amplitude of the induced flux 0.494 Wb KSM,1

1 st harmonic back-e.m.f. constant 1.417 V s KSM,3

3 rd harmonic back-e.m.f. constant 0.0354 V s KSM,5

5 th harmonic back-e.m.f. constant 0.0354 V s Irated Rated current 10 A

TABLE III RMS

 III AND PEAK-TO-PEAK CURRENT VALUES @ 20 , 600 .

	Strategy	RMS current	peak-to-peak current
	constraint of sinusoidal waveform	7.70 A	11.53 A
	No waveform constraint	8.15 A	13.01 A

Fig. 10. Steady state currents in post-failure operation mode using the suggested strategy: 600 . and 17 . .

ACKNOWLEDGEMENTS

This work was supported by the French automotive cluster MOVEO in a FUI program, through the SOFRACI project.