Ngoc Linh

Keywords: Multi-machine system, Open loop stability, Shared structures, Permanent Magnet Synchronous Machines, Predictive control

Predictive control of two synchronous machines in parallel supplied by a standard three phase static converter

Résumé

De nos j ours, les sy stèmes e mbarqués so nt d e p lus en pl us no mbreux ce qui i mpacte fortement les systèmes de conversion de l'énergie. Les contraintes associées se t raduisent par u ne r éduction d es masses et d es p ertes af in d 'améliorer l 'efficacité én ergétique d e l a chaine d e c onversion. C'est l e cas d ans l e d omaine de l 'aéronautique où l e c oncept ″d'avion plus électrique″ devient aujourd'hui une réalité. C'est ainsi que la machine synchrone à ai mants p ermanents d evient un a ctionneur d' excellence de pa r s a pui ssance massique importante, son faible coût de maintenance et ses qualités dynamiques.

Lorsque ces machines sont associées pour remplir des fonctions coopératives (surfaces de v ol p ar ex emple) on p eut encore réduire l a m asse em barquée en m utualisant l'électronique de puissance. C'est précisément dans ce cadre que se si tue notre travail, en proposant d es st ructures d 'alimentation réduites, à b ase d'électronique d e p uissance, permettant d'alimenter deux ou plusieurs machines électriques en parallèle et en proposant des lois de commande visant à améliorer le rendement énergétique.

Nous nous sommes intéressés plus particulièrement à la Commande Prédictive de deux Machines S ynchrones a limentées e n pa rallèle par un O nduleur de T ension T riphasé. Ces machines o nt d es car actéristiques i dentiques et doivent s uivre un même pr ofil de vi tesse avec un couple de charge différent et en tout cas indépendant.

L'approche c ommande pr édictive nous c onduit à c onsidérer l 'onduleur de t ension comme un dispositif ayant un nombre fini d'états de commande. Nous devons sélectionner à chaque instant la meilleure solution de commande permettant de minimiser une fonction coût. C ette fonction c oût, r elative à une ou d eux m achines, e st c omposée d' une pa rtie représentant la qualité du couple produit (courant IR q R) et d'une autre partie représentant la qualité de la conversion via les pertes produites (courant IR d R).

Cette ap proche o père n aturellement à f réquence d e d écoupage v ariable. A insi l e document fait é tat d e d ifférentes s olutions étudiées m ontrant le s limites d 'une te lle approche t ant sur l e pl an dyna mique que s ur le pl an de s pe rtes. P our a méliorer c ette solution de base nous développons une approche basée sur l'utilisation de vecteurs virtuels. Ces v ecteurs v irtuels a ugmentent l es p ossibilités d e co mmande et co nduisent à u n fonctionnement à f réquence c onstante a u travers d' une modulation de t ype S VM. L a recherche d 'un ve cteur vi rtuel opt imum e st proposée e t a ppliquée sur un di spositif composé de deux machines de faible puissance. Les différentes propositions sont validées par la voie de la simulation numérique et consolidées par des résultats expérimentaux.

Mots clés:

• Système multi-machines

•

Preface and Acknowledgements

The r esearch s tudy r eported in t his t hesis ha s be en pe rformed i n t he L aboratory o f To ach ieve t hese results, besides my ow n e fforts, i t i s impossible to l ack of t he assistance, very kind support as w ell as the encouragements of many people. I would like to thank all of them for their contribution in my work.

I w ish t o e xpress m y de ep gr atitude t o my s upervisors, P rofessor Mau rice F adel an d

Professor A na L lor, f or t he gui dance, e ncouragement a nd s upport du ring m y gr aduate study. T heir c reative t hinking, kno wledge a nd e xpertise were t he " power s upply" a nd "feedback" of my conducted researches.

I would like to thank President of the University of Toulouse -Marie France Barthet, Professor INPT Maria DAVID for their administrative support, I a m a lso gr ateful t o D r. D amien D idart f or hi s kind a dvices a nd di scussions f or t he experiment in this thesis.

I wo uld l ike t o e xpress my a ppreciation t o t he members of t he t hesis de fense committee, P rofessor S antiago G ómez A rnaltes f rom U niversity o f Madrid, P rofessor Emmanuel Godoy from École Supérieure d'électricité (SUPELEC) for their interest in my research work and the time and effort in considering this thesis.

The USTH program is gratefully acknowledged for its financial support to this research study.

I w ould l ike t o t hank a ll of my c olleagues: C eline, J ulie, E tienne, T rung, H a, T hiet, Thinh … who always support, help and encourage me during the past time.

Toulouse, May 2013 NGUYEN Ngoc Linh

General Introduction

In th e c onventional a ircraft, th e non -propulsive a ircrafts s ystems a re typically distributed by four different secondary power systems: hydraulic, pneumatic, electrical and mechanical. T hese p owers ar e e xtracted f rom t he ai rcraft main en gine b y d ifferent disciplines. For example, mechanical power is obtained from the engine by a driven shaft and distributed to a gearbox to drive lubrication pumps, fuel pumps, hydraulic pumps and electrical generators. Electrical or hydraulic power is distributed throughout the aircraft for driving t he s ubsystems s uch a s f light c ontrol a ctuators, l anding ge ar br akes, ut ility actuators, a vionics, lighting, e tc. Over t he ye ars, t hese s ystems ha ve be come more a nd more c omplicated a nd t he i nteractions be tween different pi eces of e quipment r educe t he efficiency of the whole system.

In or der t o ove rcome t his c omplexity, w ith t he a im t o i mprove efficiency a nd reliability, t he a ircraft manufacturer tr end is toward th e " More E lectric A ircraft" (MEA) concept [START_REF] Mo Re | lectric A ircraft: t he p ast an d t he f uture? I n IEEE Colloquium on E lectrical M achines and Sy stems for t he M ore Electric Aircraf[END_REF] [ROAR07] [START_REF] Abdelhafez A Nd | A Review of M ore-Electric A irecraft[END_REF]. Within the framework of MEA, one of innovations is substituting h ydraulic actuators w ith electromechanical a ctuators (EA). T his re duces weight and decreases maintenance and production costs.

The E A s ystem c ontains f our major c omponents: the e lectrical m otors, t he pow er electronic converters, t he c ontrol dr ives and t he a ctuators. T he m ore us ing of electromechanical actuators (EA) leads to the first demand about an appropriate electrical machine t o meet t he requirements o f ai rcraft t echnology. R elated to t his issue, t he permanent magnet synchronous machine (PMSM) is increasingly being used thanks to its high efficiency throughout the full speed range, high power ratio, and ease of refrigeration, compared t o cl assical wound machines. Mo reover, t he P MSM al so sat isfies t he h igh requirements of the application in term of volumes and weight minimizations [START_REF] Mecrow | Safetyaircraft d esign o f el ectromechanical act uation sy stems i n commercial ai rcraft[END_REF] [CMA+12].

It can be seen that these machines are associated to carry out cooperative functions (for example, in flight control surface or in other mechanical parts such as landing gear, brakes, NGUYEN Ngoc Linh fuel pum ps defrosting s ystem …) . I n or der t o operate su ch t hese systems, each en gine needs a power electronic device which consists of a heavy and bulky cooling component. This el ement w ill increase t he w eight an d s ize o f t he whole sy stem si gnificantly.

Moreover, for the aeronautic applications, which require the high fail-safety, it is necessary to have strictly constraint r edundant systems. These systems ar e essentially t o ensure t he continuous operation of the whole system even when any problem happens. However, they also m ake t he sy stem much h eavier a nd la rger. A s a re sult, i t is in teresting to c onsider whether p ossible t o u se sharing st ructures t o o perate i ndividual sy stems acco rding to t he availability of s equencing w hile r especting t he c ontinuous c ondition of s ystem. It i s precisely in th is c ontext w here our w ork ha s b een de veloped, pow er r educed s tructures based i n p ower el ectronics, ab le t o f eed t wo or m ore el ectric m achines i n p arallel an d providing control laws to improve energy efficiency.

Until r ecently, t ogether w ith t he development of s emiconductor technology, t he introduction of powerful microprocessor and the power electronic devices …, t he systems including m ulti synchronous m achines ( especially P MSM) dr iven by a s ingle i nverter begin to be more interested. Some studies have been carried out in order to test and verify the feasibility of these systems [START_REF] Bidart | Mono inverter dual p arallel P MSM -structure and c ontrol s trategy[END_REF] [APIS08] .

We h ave w orked m ore sp ecifically i n t he Predictive C ontrol of t wo S ynchronous Machines connected in parallel connected to a 2-Level 3-Phase Converter. These machines have i dentical ch aracteristics an d m ust f ollow the sam e sp eed p rofile w ith a l oad t orque which can be different and, in any case, independent.

The frame work of my thesis is presented in four chapters:

• In t he f irst chapter, t he e volution of P MSM i n many a pplication f ields w ill be presented. M ulti-machine sy stems in t he ae ronautic f ield as w ell as the i dea o f using sharing power electronic structure are considered. Typical structures of multimachine sy stem w hich h ave b een developed in t he literature a re p resented. T he features of each structure will be studied. A comparison among these structures will also be synthesized. Finally, the most suitable structure for our application will be selected;

• In the second chapter, before approaching to multi parallel PMSM drive system, a general description about a PMSM drive system is presented first. In this part, the principle operation and stability analysis of a PMSM are considered. Some control strategies t o c ontrol this ki nd of dr ive s ystem w ill be pr esented a nd a nalyzed. studied and e xplained in de tail. T wo a pproaches to p redictive c ontrol will b e described a nd c ompared. S ome s imulations will be done i n or der t o ve rify t he performance of control methods.

• In t he f ourth c hapter, a n e xperimental b ench has be en d eveloped in laboratory LAPLACE with t he a im of ve rifying t he ope ration of s ystem unde r pr oposed control st rategies. E xperimentation r epresents a hypothetical d isplacement o f t wo flaps of an air plane. The studied system converts electrical energy supplied by the EDF network and then distributes it to two mechanical sources. In this mechanism, two PMSMs will be connected in parallel and supplied by a three-phase two-level inverter.

• The conclusions and perspectives will be pointed out in the final.

Introduction

Nowadays, along with the development of electro-technical technology as w ell as the economic i ssues i n i ndustries, there ar e m ore an d m ore sy stems co nsisting o f m ultiple electric m achines. A c lassical s tructure, w hich i s c alled multi-machine m ulti-inverter system ( MMS), h as b een w ell-developed f or t hese s ystems. A ccording t o this s tructure, each machine is supplied by its own inverter. As a result, problem of using a large number of power electronic components need to be considered. In addition, classical structure also requires a c ontroller f or e ach m achine. T herefore, t he r equirement of us ing po werful computers to implement several control modules is unavoidable. Because of these reasons, the p rice, a s w ell as the si ze o f the w hole sy stem, w ill i ncrease c onsiderably. I t i s interesting de veloping sharing s tructures w ith a r eduction of t he nu mber of p ower components and simplified control algorithm.

As s hown i n Figure 1.1, t he MM S can b e r epresented as a m ulti-machine m onoinverter s ystem. A ccording t o t his configuration, a 𝑁-legs i nverter i s u sed t o pow er 𝑛 machines. Each leg of inverter is made up of two electronic switches connected in series. This kind of system is marked as multi-machine system -MS(m,n) [START_REF] Bouscayrol | Structures d 'alimentation et st ratégies d e co mmande pour d es syst èmes m ultimachines a synchrones[END_REF].

Many s tudies ha ve be en c arried out t o r educe t he num ber of pow er e lectronic components by de creasing t he num ber of i nverter's l egs. I t i s a t t he origin of t he i dea o f using the shared structures in multi-machine systems. Such these structures can be found in many applications in the textile industry [START_REF] Charlemagne | Modélisation et commande d'un système de transport de bande textile[END_REF] [START_REF] Belhadj | D irect torque c ontrol w ith a n optimized obs erver f or multi-machine sy stems[END_REF], transportation industry such as the traction drives [START_REF] Kemp | Developments in electric traction[END_REF][PEPDRdF02] [START_REF] Pérez-Pinal | Multi-motor s ynchronization technique applied in traction devices[END_REF][BPDD+06], robotic mobile [START_REF] Bouscayrol | Structures d 'alimentation et st ratégies d e co mmande pour d es syst èmes m ultimachines a synchrones[END_REF] [BPDdF96] and ot her … H owever, t hese s tudies ha ve just onl y a pplied f or i nduction machines. M eanwhile, w ith the s trong de velopment of t he s emi-conductor a nd material industry t oday, t he permanent magnet s ynchronous machines ( PMSM) be come more a nd PMSM with Multi-Machine Systems more popular and appear in many applications. This type of machine shows the superiority in the efficiency, the size and weight, and the flexibility… compared with other machines.

In this chapter, the evolution of PMSM in many application fields will be presented in a first time. After that, a special study of multi-machine system in the aeronautic field is considered. In this part, the possibility and the idea of shared structures are presented and analyzed. Next, the typical structures of multi-machine system which have been developed

in th e l iterature a re p ointed o ut. T he c haracteristics o f e ach structure w ill b e s tudied. A comparison a mong t hese s tructures w ill a lso be s ynthesized. F inally, the m ost s uitable structure for our application will be selected in the conclusion.

Evolution of permanent magnet synchronous machine (PMSM)

The pe rmanent m agnet synchronous machine ( PMSM) dr ive ha s e merged a s a t op competitor f or a f ull r ange of motion c ontrol a pplications. I n s pecialty, t he P MSM i s widely us ed i n m achine t ools, r obotics, a nd a ctuators a nd also be ing considered in highpower applications such as vehicular propulsion and industrial drives. It is also becoming viable for commercial or residential applications. The PMSM is known for having a lot of advantages compared with other types of electric machine:

+ high efficiency: PMSMs use permanent magnets to create a constant rotor magnetic field i nstead o f u sing a rotor circuit. T herefore, t he co pper losses are el iminated.

Moreover, without the rotor circuit, the only heat is produced on the stator, which is much easier to cool than the rotor;

+ high flux density: R ecent advances in high-energy density magnets (NdFeB) have allowed a chievement of ve ry hi gh flux de nsities. T hese magnets pr ovide hi gh torques and allow to build the smaller and lighter machines;

+ lower maintenance, great longevity and reliability: Because there are no mechanical commutators and brushes, the friction is lower and the durability is higher. Besides, without brushes and mechanical commutators, the maintenance cost is reduced and the risk of sparking or corrosive environments is eliminated; + high t orque to i nertia r atio: P MSMs us e pe rmanent m agnets → PMSM w ill h ave low inertia; this makes for a fast response under a given electric torque;

+ high power density (it is represented by the ratio 𝑘𝑊/𝑘𝑔);

+ …
Evolution of permanent magnet synchronous machine (PMSM)

PMSM with Multi-Machine Systems

In P MSMs, p ermanent m agnets ar e m ounted i nside ( interior p ermanent m agnet synchronous m achine -IPMSM) o r o utside (surface mounted pe rmanent m agnet synchronous machine -SPMSM) of the rotor (Figure 1.2). Meanwhile, the stator structure is identical to the induction motors as well as other types of AC motor. The windings thus are constructed to produce a sinusoidal flux density in the air gap of the machine.

Today, t he P MSMs pl ay a n i mportant r ole i n the v ariable speed ap plications. T hese applications have the output capacity ranging from a few hundred 𝑊 (brushless DC micromotor de veloped i n M EMS t echnology [START_REF] Merzaghi | Development of a hybr id MEMS B LDC micromotor[END_REF]) t o m ore t han 1𝐺𝑊 (AC s ynchronous generator [START_REF] Bernot | Alternateurs syn chrones de gr ande pui ssance ( partie 1)[END_REF]). F or s mall pow er machines, t heir r otor us ually consists of several permanent m agnets. T hese p ermanent m agnet sy nchronous m otors are also ca lled a s "brushless motor" because the rotor flux is still created by the magnets, they do not have the brushes collector.

Besides, se veral h igh performance ap plications su ch as r obotics an d aer ospace actuators in w hich th ere is a re quirement f or high to rque p er in ertia ra tio a re a lso th e typical applications of PMSM. In the next part, examples in different power ranges using PMSM will be presented.

Small power applications of PMSM

In t he r ange of s mall pow er ( according t o t he standard N F C 51 -200, the pow er i s under 600W), the brushless motors are mostly used in continuous current mode and called BLDC motors.

In order to control the system with the BLDC motors, an electrical sensor, which are connected to t he m otor, a llows the de tection of t he r otor pos ition a nd e nsures t he These t ypes of motor a re us ually us ed i n t he s ystems w hich a re r equired t o c ontrol exactly the position (hard disk drives, DVD rewrite …). They are also used in the systems that need t o operate a t a f ixed s peed p recisely su ch as t he m echanical t imer i n t he household devices. Other applications which use the small brushless motors are the fans of the micro-computers or t he medical i nstruments such as the drills. All of t hese examples are illustrated in Figure 1. 3.

Medium power applications of PMSM

In t his section, t he studied systems ar e required i n the range of power from 500 W to 100kW a nd us ing or l ikely t o us e t he P MSM. T hese m achines a re mostly pol y-phase (usually three-phase) and always associated with a v oltage inverter controllable in current in order to ensure the autopilot.

One of t he f irst a pplications in t his pow er r ange i s s upported by t he m otorization of various electric v ehicles. I n d eed, cu rrent t rends l ead to t he d evelopment o f el ectric accumulators (batteries Lithium/ion …) and the development of electric motors in order to replace t he internal c ombustion e ngines w hose pe rformance i s ve ry l ow w hile t he f ossil fuel b ecomes ex hausted. Mo reover, b rushless m otors a lso p ossess interesting characteristics w hich ar e su itable for e mbedded sy stems. S everal el ectrical o r h ybrid vehicles such as T oyota-Prius or the Citroen-C0 were equipped with PMSM. In the same tendency, the studies on t wo-wheeled vehicles are also carried out (scooter, electric bikes …).

Some machine t ools an d i ndustrial r obots also use t he b rushless motors because they provide goo d pe rformance ( good low-speed t orque, s trong overload c apacity, r eliability, speed …) and require little maintenance. All the examples described above are illustrated in Figure 1.4 Thanks to many advantages, the PMSM can also be used in various driving mechanism for the specialty systems where the weight and volume play an important role. Therefore, if there are already the applications using the PMSM for some systems in cars (ventilations, air conditioning …), it is also possible to increase their penetration into other areas such as aerospace where the problem "more electric aircraft" is increasingly mentioned.

High power applications of PMSM

Despite the high cost of the magnets, PMSMs are increasingly used or considered for higher pow er a pplications i n w hich t here a re strictly r equirements a bout t he di mension space such as the electric traction drives or marine propulsion systems.

The A GV ( high sp eed electric r ailcar), a p roduct o f A lstom co mpany, al so u ses t he PMSM f or the t raction dr ive a nd e lectrodynamics br aking s ystems of t he t rain. T he successor o f T GV t rain r econciles al so the articulated structure (like T GV) w ith a distributed drive s ystem t hroughout t he train. T his di stribution br ings t o a reduction in consumption of 15% and a division the train into small parts as well as small power units.

The p ower/weight ra tio o f th e tra in i s 22k W/tone a nd c ruising s peed of t he t rain i s 360km/h ( against 320k m/h for t he T GV). T he us ed e ngines ha ve a no minal pow er of 720kW and the power/weight ratio of 1kW/kg (against the maximum value 0.7kW/kg for asynchronous motor) [START_REF]AGV t echnical doc ument i n t he s ite i nternet Alstom T ransport[END_REF].

Similarly, the PMSM can be used for naval propulsion. In this context, the motors used must be compact, lightweight, reliable and resistant to the marine environment (vibrations, humidity, s alinity, te mperature … ). In s ome s pecial c ases a s m ilitary applications, th ey also are required as discrete as possible. Although PMSM that can generate the high power are q uite expensive t o manufacture, t hey st ill appeared o n some su bmarines an d military redundant systems. These systems are essentially to ensure the continuous operation of the whole system even when any problem happens. However, they also make the system much heavier and larger.

Multi-machine system in aeronautic field

On t he ot her ha nd, dur ing e ach s pecific pe riod of t he flight, t here a re j ust a c ertain number of dr ive systems w hich r equire t he p ower el ectronic d evices and ar e u sed w hile + The s econdary pa rt c ontrols t he a erodynamic c onfiguration of t he a ircraft during landing and takeoff processes (lift system, spoilers …). There are three main parts in the secondary actuation system: the flaps and the slats which are used for take off, landing and increasing lift at low speed, the airbrakes which increase the dr ag. T hese c omponents a re no t a ctually required f or f light but they cannot be omitted. Many studies in science and technology have been developed in order to upgrade the flight control system for t he l ast many years [START_REF] Ménio | Transition da ns l es ccommande des vol électrique[END_REF]. At t he be ginning, flight c ontrol systems entirely use mechanical actuators: the pilot controls directly the rudder by taking the effect on the joystick. And then, with the support of the hydro-mechanical technology, the hydraulic actuators allow to operate heavier loads, however, the flight control issue is still always mechanical.

In the 1950s, military aircrafts began to be equipped by electric flight control systems.

This technology, which is also called as "fly by wire", allowed reducing the weight of the system as well as i mproving t he r esponse t ime o f t he electro-hydraulic s ervo-system [START_REF] Schroer | F light c ontrol goe s di gital [ Part T wo, N ASA a t 50[END_REF].

PMSM with Multi-Machine Systems PMSM with Multi-Machine Systems (while respecting the c onditions of s afety).This t rend is today of major i mportance f or aeronautic o f industry. T his i mprovement w ill br ing the a dvantages of e liminating completely the hydraulic power, decreasing the production and maintenance cost as well as the weight of the system. An example of such EMA device is presented in Figure 1.9.

Electromechanical actuators (EMAs)

EMA drives for flight controls are based on direct drive architecture [START_REF] Lemmens | Synchronization and efficiency analysis of a direct-drive multi-motor application[END_REF] built up by an el ectric motor co nnected d irectly t o t he r oller-screw t hat m oves a n act uator ( Figure 1.10). The complete control block diagram for an EMA drive includes the position, speed torque and flux c ontrols. B esides i t a lso i ncludes t he s upervisory a nd c ommunication system.

The more the EMA are used, the electric machines are also becoming more and more important in the general electric aircraft. As mentioned before, thanks to many advantages compared with other types of electric machines such as the high efficiency throughout the full s peed r ange, hi gh power ratio, an d ease o f r efrigeration, t he P MSM i s i ncreasingly being us ed f or a ctuator dr ives. I ndeed, t he f laps or t he slats of bot h w ings must be replaced simultaneously, which means that several PMSMs have to perform the same task at the s ame t ime. T herefore, th e multi-synchronous m achines s ystems ne ed t o be considered.

Shared structures

The objective of the study developed in this chapter is to look for a suitable structure of multi-synchronous machines m ono-inverter s ystem. T his s tructure must meet th e requirements of aeronautic applications about weight and volume especially in the case of imposing two or three redundant electronic systems with PMSM. The developed system is composed o f sev eral identical P MSMs w hich m ove d ifferent l oads a t t he sam e speed.

There are two ways to approach to this problem.

+ The f irst w ay is ju st u sing o nly o ne in verter to o perate s everal P MSMs in parallel. According to this method, the weight of system will be decreased.

+ The second way is reusing a cl assical structure in which an individual inverter can operate just one machine or two machines in parallel.

In t he ne xt pa rt of t his chapter, t he s ynthesis of e xisting s olutions w hich ha ve be en applied for multi-asynchronous machines system is presented. Then a short comparison is carried out in order to find the most suitable structure which can be applied to the multisynchronous machines system, especially in aeronautics applications.

Typical structures of multi-machine system PMSM with Multi-Machine Systems

Typical structures of multi-machine system

Classical structure

In th e c lassical s tructure o f m ulti-machine system, eac h p hase o f a m achine i s connected and controlled by its own inverter's leg. For n three-phase machines, so there are m = 3n inverter's legs. The structure MS(3n, n) is described in the Figure 1.11.

In this case, as well as the cases which are mentioned afterwards, all the inverter's legs are connected to a DC bus that provides the voltage 𝑉 𝐷𝐶 .

In

In t his c onfiguration, t here is no m utual c ontrol be tween di fferent i nverter. All machines operate independently from each others. The synchronism of the system can be obtained easily. However, using one inverter for each machine as well as one drive system needs an individual controller; such system can be expensive and bulky. These problems can be considered as the most negative point of this structure. Shared s tructures w ith 𝑁 < 3𝑛 then ha ve be en de veloped a nd s tudied; e specially f or induction machines. However, according to some specific conditions, it is also possible to apply these structures to the case of synchronous machines. , n�. T he nu mber of t he m achines 𝑛 must be multiple of 𝑦. Some different sharing structures will be presented in the following parts of this section.

Presentation of the shared structures

Sharing common legs structure: n machines with 2n legs

As shown in Figure 1.13, a system which is constructed under this structure consists of a f our-leg i nverter f eeding t wo m achines. The tri -phase vol tage obt ained w ill be r egulated t o f ollow a reference v alue b y a controller. T he c ontrol s ignal i s t hen s ent t o t he i nverter. I t s hould be not ed t hat, t he voltage applied to the inverter need to refer to the point 𝑂. Therefore, the reference voltage must be calculated according to the equation in (1.3) This structure can be used to operate multiple machines which are required to rotate at the same speed ether in the same direction or in opposite.

The most advantage of this structure is that only a four-leg inverter is used in order to run t wo t hree-phase machines (compared with six l egs i n the case of classical structure).

�𝑉 𝐴 1 𝑂 � 𝑟𝑒𝑟 = �𝑉 𝐴 1 𝑁 1 � 𝑟𝑒𝑔 + �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 (𝑉 𝐵𝑂 ) 𝑟𝑒𝑟 = �𝑉 𝐵𝑁 1 � 𝑟𝑒𝑔 + �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 (𝑉 𝐶𝑂 ) 𝑟𝑒𝑟 = �𝑉 𝐶𝑁 1 � 𝑟𝑒𝑔 + �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 �𝑉 𝐴 2 𝑂 � 𝑟𝑒𝑟 = �𝑉 𝐴 2 𝑁 2 � 𝑟𝑒𝑔 + �𝑉 𝑁 2 𝑂 � 𝑟𝑒𝑟 (1.3)
Based on the same principle (share the legs of the inverter), another structure expressed as MS (2n + 1, n) is a lso de veloped [START_REF] Dujic | A General PWM Method fos a ( 2n+1)-Leg I nverter S upplying n T hree-Phase Mac hines[END_REF]. A (2n + 1) leg in verter w ill b e u sed to operate 𝑛 machines. This means that there is just one leg is shared by only one phase of all machines. T his s tructure c an be us ed i n r olling-up applications i n paper i ndustry

[JVD+08].

Sharing the midpoint structure: n machines and 2n legs

According to t his s tructure, t wo phases o f ea ch m achine w ill b e co nnected t o two separated legs of an inverter and the third phase of both machines is shared and connected to the neutral point O of the continuous DC bus which feed the inverter. This configuration is called MS(2n, n) and shown in Figure 1.15.

Using this structure, the two motors can run at different speed. One of t he a dvantages of t his s tructure i s t hat the m otors can ope rate i ndependently with each others, while the number of power electronic components can be reduced.

legs for n machines

In accordance with its name, the mixed structure, which is represented in Figure 1.17, is a combination of two structures which are mentioned above. According to this structure, a three-leg inverter is used to operate two motors. One phase of both motors will be shared and c onnected t o a c ommon l eg o f i nverter ( Figure 1.17, t wo p hases 𝐶 1 and 𝐶 2 ) w hile another phase of both motors (𝐴 1 and 𝐴 2 ) is linked to the neutral point O. , 𝑛�.

As shown in Figure 1.17, the phase A of two machines is connected to the midpoint O ⇒ 𝑉 𝑂𝐴 1 = 𝑉 𝑂𝐴 2 = 0 and the phase C of two machines is linked to the same leg

⇒ 𝑉 𝑂𝐶 1 = 𝑉 𝑂𝐶 2 = 𝑉 𝑂𝐶 .
The diagram of voltage vectors of two machines under the mixed structure is shown in the Figure 1.18.

Moreover, like in the structure with common leg, two possibilities can happen: PMSM with Multi-Machine Systems

+ if 𝑉 𝐵 2 𝑁 2 = 𝑉 𝐵 1 𝑁 1 ⇒ Ω 1 = Ω 2 (Figure 1.18(a)) + if 𝑉 𝐵 2 𝑁 2 = -2𝑉 𝐵 1 𝑁 1 ⇒ Ω 1 = -Ω 2 (
�𝑉 𝐵 1 𝑂 � 𝑟𝑒𝑟 = �𝑉 𝐵 1 𝑁 1 � 𝑟𝑒𝑔 + �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 �𝑉 𝐵 2 𝑂 � 𝑟𝑒𝑟 = �𝑉 𝐵 2 𝑁 2 � 𝑟𝑒𝑔 + �𝑉 𝑁 2 𝑂 � 𝑟𝑒𝑟 (𝑉 𝐶𝑂 ) 𝑟𝑒𝑟 = �𝑉 𝐶𝑁 1 � 𝑟𝑒𝑔 + �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 with �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 = -�𝑉 𝐴 1 𝑁 1 � 𝑟𝑒𝑔 �𝑉 𝑁 2 𝑂 � 𝑟𝑒𝑟 = -�𝑉 𝐴 1 𝑁 1 � 𝑟𝑒𝑔 if 𝐵 2 = 𝐵 1 �𝑉 𝑁 1 𝑂 � 𝑟𝑒𝑟 = -�𝑉 𝐶 1 𝑁 1 � 𝑟𝑒𝑔 if 𝐵 2 ≠ 𝐵 1 (1.5)
This s tructure, l ike t he s tructure s haring t he midpoint, h as t he s ame di sadvantage because of having t o a ccess t o t he neutral poi nt of t he DC bus t o a djust t he pot ential. I n spite of decreasing the number of inverter's legs, this structure just is used in some specific applications.

Parallel structure: 3 legs for n machines

This s tructure be longs to t he t ype M S(3, 𝑛), a t hree-leg i nverter w ill o perate n machines s imultaneously. S ome s tudies c an b e found i n t he l iterature [KFM00] [PE02].

According to these studies, the case of two machines in parallel is focused. As s hown i n According to this structure, all the machines will receive exactly same voltages in both frequency and amplitude. These machines will be set up at the same initialization; the same Contrary to those previous structures, it is impossible to control all the phase voltages of t hese machines. S uitable m ethods ne ed t o be c onsidered a nd de veloped i n or der t o guarantee the stability of such these systems.

Some other structures

Some other structures, which are more complex such as MS� , n�, have also been studied [START_REF] Bouscayrol | Structures d 'alimentation et st ratégies d e co mmande pour d es syst èmes m ultimachines a synchrones[END_REF]. These structures which are based on the same principle of using common l egs or m ixed s tructure w ere us ed i n many a pplications h aving more t han t wo machines.

Another principle has been also developed in the field of multi-machine system which is us ing i nverters or m achines w ith s pecial topology [SH07]. However, s uch t hese particular systems are not easily adaptable in replacing the redundancy systems.

Comparison of these different structures

In previous parts, many different structures developed for multi-machine systems have been presented. To see which structure is more suitable according to the requirements of aeronautic applications in section 1. 3.2.3, a comparison of those structures is provided in T he s tructures us ing c ommon l egs will de crease t he num ber of inverter's legs. This means that the number of power electronic components is also r educed. H owever, us ing c ommon l egs w ill l ead t o t he pr oblem that t he current passing through these legs will be higher. Therefore, there are different electronic devices inside one inverter.

+ The second criterion is the ratio between the maximum voltage that a machine can r eceive (𝑉 𝑁𝑙 ) 𝑙𝑙𝑥 and t he maximum vol tage which can be ge nerated from the inverter (𝑉 𝑂𝑙 ) 𝑙𝑙𝑥 . Denote this ratio as 𝜂 𝑉 = (𝑉 𝑁𝑖 )𝑙𝑙𝑥 (𝑉 𝑂𝑖 )𝑙𝑙𝑥 . For the classical and parallel s tructure, th is r atio is 𝜂 𝑉 = 1 while w ith o ther structures t his ra tio is always sm aller t han (𝜂 𝑉 < 1). T herefore, f or t he s tructures w hich make t he ratio 𝜂 𝑉 < 1. , the value of the DC bus 𝑉 𝐷𝐶 needs to be regulated.

+ For the structures that can be used for only one machine, it is possible to obtain independent c ontrol be tween di fferent m achines e asily. H owever, f or s haring structures, it is n ecessary to lo ok for s uitable c ontrol structures as well a s control algorithms to ensure the stable operation of each machine.

The parallel structure presents the best advantage of using as less as possible number of inverter's legs. Moreover, it is not necessary to modulate the voltage of the DC bus (𝑉 𝐷𝐶 ) in order to obtain the same machine voltage when compared with the case of conventional structure. It is not too difficult to apply this structure in real applications. The drawback of this structure comes from sharing all inverter's legs. Because being connected to the same legs, all phases of all the machines will not be controlled independently. In fact, when 𝑛 machines are put into parallel, the current passing through each leg will be 𝑛 times higher than that of the classical structure. The problem will come from the design of the electronic devices. However, using these oversized components in the redundancy systems, which are inactive most of time, is is still an acceptable solution.

Therefore, parallel structure is selected as t he most feasible solution for the aeronautic application based in the secondary flight control system. In the case of detecting a failure in an inverter of a machine, it w ill be possible to switch to this structure to guarantee the continuity of the whole system.

Conclusions

The rapid development of semi-conductor and material technology has allowed the use of increasingly the PMSM in industry, replacing the current electric machine types. Along with this, it is the development of embedded systems with the strictly requirements about the p erformance, q uality, w eight, s ize … T his is t he c ase i n aer onautics f ield w here t he concept of "More Electric Aircraft" now becomes a reality.

When these machines are associated to carry out cooperative functions (for example, in flight surfaces), it is a time that a sh aring structure needs to be considered and developed.

This c oncept ha s a lready be en a pplied f or the a pplications w ith induction m otors.

However, f or P MSM, there is still l imit. A s tructure i ncluding t wo or more s ynchronous machines is really interesting.

In this first chapter, several structures of multi-machine systems has been presented and compared. According to the comparison given and analyzed, parallel structure seems to be the m ost s uitable f or the a eronautic applications. It us es t he least number of pow er electronic d evices, do es not r equire a ccessing t o t he D C vol tage bu s and s o i t is qui te simple to implement. However, according to this structure, because all machines obtain the same vol tage f rom o nly one i nverter, it is difficult t o satisfy i ndependently the requirements of all machines, especially in the case of different loads.

In the next chapter, a two-parallel PMSM system will be described. The solutions used to control this system will be introduced and analyzed.

Introduction

As mentioned i n pr evious c hapter, multi-machine sy stems b ecome more an d more popular in variable s peed a pplications. T he pr oblem of u sing m any pow er e lectronic devices w hich i ncreases t he w eight an d si ze o f t hese systems is s olved by the i dea of sharing structures.

In a ddition, t he de velopment of t he s ystem w ith m ultiple machines a lso l eads t o a higher at tention i n sel ecting su itable el ectrical machines. I n t his f ield, t he P MSMs gradually become dominant compared with others such as DC machines, IMs...

In the special field of aeronautic applications where the efficiency as well as the weight and size are the most important things, the parallel structures along with PMSMs seem to be the best choice.

Combining parallel structures with PMSM leads to the idea of multiple Besides, both machines will receive the same voltage (magnitude and frequency):

� |𝑉 𝑠1 | = |𝑉 𝑠2 | = |𝑉 𝑠 | 𝜔 𝑒1 = 𝜔 𝑒2 = 𝜔 𝑠 (2.1)
For this system, the aim is to keep the mechanical speeds of two machines equal and follow the speed reference under different load conditions:

𝜔 𝑙1 = 𝜔 𝑙2 = 𝜔 𝑟𝑒𝑟 𝑠. 𝑡 𝑇 𝑙𝑙𝑙𝑠1 ≠ 𝑇 𝑙𝑙𝑙𝑠2 (2.2)
In t his ch apter, a g eneral d escription ab out a P MSM d rive sy stem w ill b e p resented first. In th is p art, th e p rinciple of o peration a nd s tability analysis o f a P MSM a re considered. In order to operate this drive system, some typical methods will be presented in the next part. Finally, two control strategies for mono-inverter dual-parallel PMSM drive will be described.

Generalities about PMSM drive system

PMSM n ormally i s n ot a s elf-started m otor t ype. T herefore, i n variable sp eed In t his f igure, a P MSM i s su pplied b y a t hree-phase t wo-level vol tage i nverter. A controller will be used to generate the control signals of the inverter in order to control the PMSM.

Principle of operation of PMSM

In t he c ase of a PMSM, t he r otor a cts a s a n i nductor, c ontaining 𝑛 𝑝 pairs of pol es (North a nd South) pe rmanent magnets. M eanwhile, t he s tator of a t hree-phase P MSM normally has a sine distributed three-phase winding. When these windings are excited with a t hree-phase ba lanced supply, a rotating m agnetic f ield is developed. It i s a ssumed t hat this f ield is sinusoidally di stributed in t he a ir ga p. T he mutual i mpact be tween t he s tator and rotor magnetic fields will produce forces to the rotor and will make the rotor rotate.

At steady state, PMSM is a constant speed motor. This speed is calculated by equation

(2.
3):

𝜔 𝑙 = 60 𝑓 𝑠 𝑛 𝑝 (2.3)
where 𝜔 𝑙 is the mechanical speed of the rotor (𝑟𝑝𝑁);

𝑓 𝑠 is the electrical frequency of the voltage supply (𝐻𝑧);

𝑛 𝑝 is the number of pole pairs; As mentioned i n p revious c hapter, there a re t wo t ypes of PMSM de pending on the rotor s tructure. I n S PMSM, magnets a re m ounted on t he s urface of t he r otor. S ince t he permeability of magnets is close to unity and the saliency is really small, the inductances expressed i n t he d -q c oordinates a re a pproximately e qual. I n t his t hesis, j ust non -silent typed P MSMs a re c onsidered. T herefore, the f ollowing relation is always c onfirmed:

𝐿 𝑠 = 𝐿 𝑠 = 𝐿.

Electromechanical description

The space vector form of the stator voltage equation in the stationary reference is given in Equation (2.4):

Mono-Inverter Dual-Parallel PMSM drive system where 𝑅 𝑠 , 𝑢 �⃗ 𝑠 , 𝚤 ⃗ 𝑠 and 𝜓 �⃗ 𝑠 are the resistance of the stator winding, complex space vectors of the t hree pha se s tator voltages, c urrents a nd flux l inkages, all e xpressed i n t he s tationary reference frame fixed to the stator, respectively. They are defined as: ahead of the a-axis, respectively. Next, the direction of permanent magnet flux is chosen as the d-axis, while the q-axis is 90 𝑙 ahead of the d-axis. The angle between the rotor d-axis and the stator a-axis is defined as 𝜃 𝑒 . Note that, as the machine operates at steady state, the d-q reference frame also rotates at a speed of 𝜔 𝑒 = 𝑟𝜃 𝑒 /𝑟𝑡, while the stator a-b-c axes are fixed i n sp ace. In terms of phase variables, the electrical dynamic equation of PMSM can be written as: In Figure 2.5: Where 𝑢 𝑠𝑠 and 𝑢 𝑠𝑠 are the d-q axis voltages, 𝑖 𝑠𝑠 and 𝑖 𝑠𝑠 is the d-q axis stator currents, and 𝜓 𝑝 is the polar flux.

𝑢 �⃗ 𝑠 = 2 3 �𝑢 𝑠𝑙 (𝑡) + 𝑒 𝑗120
𝑢 𝑘 = 𝑅 𝑠
+ the index 𝑘 = (𝑟,
The equations of flux components in d-q reference frame also are obtained:

� 𝜓 𝑠𝑠 = 𝐿𝑖 𝑠𝑠 + 𝜓 𝑝 𝜓 𝑠𝑠 = 𝐿𝑖 𝑠𝑠 (2.16)
The equivalent circuits of the machine in d-q rotational frame are illustrated in Figure 2.6.

Based o n t he g eneral r otating m oment eq uation o f t he el ectrical m achine, t he electromagnetic torque is:

𝑇 𝑒 =

Stability analysis of PMSM

The e quation of t he m achine i n complex do main, w hich i s shown i n (2.21), i s considered.

𝑉 = 𝐸 + 𝑍𝐼 (2.21)
where 𝑍 is the complex stator impedance and follows these equations below: (2.23)

The power developed is calculated by:

𝑃 𝑒 = 3ℛ𝑒�𝐸. 𝐼 * � (2.24)
Let t he ap plied v oltage be t he r eference: 𝑉 = �𝑉�∠0 𝑙 then 𝐸 = �𝐸�∠𝛿 = �𝐸� cos 𝛿 + 𝑗�𝐸� sin 𝛿. Therefore, the power can be written as follow:

𝑃 𝑒 = 3�𝐸��𝑉� �𝑍� 2 (𝑅 𝑠 cos 𝛿 -𝑋 sin 𝛿) - 3�𝐸� 2 𝑅 𝑠 �𝑍� 2 𝑃 𝑒 = 3�𝐸� 𝑍 �𝑉 cos(𝛼 -𝛿) -�𝐸� cos 𝛼� (2.25)
The electromagnetic torque will be expressed as:

𝑇 𝑒𝑙 = 𝑃 𝑒 𝜔 𝑟𝑙 = 3𝑘 𝑒 𝑍 �𝑉 cos(𝛼 -𝛿) -�𝐸� cos 𝛼� (2.26)
where 𝑘 𝑒 is the EMF constant of PMSM, which is determined in (2.7).

Finally, an expression of the torque versus the load angle 𝛿 is represented in Figure 2.7. The stable operation zone of the synchronous machine is characterized by the slope of the curve 𝑇 𝑒 = 𝑓(𝛿), that is the sign of

𝜕𝑇 𝑒 𝜕𝛿
. Therefore, the system is stable if, at constant speed, the load angle is limited in the range as shown in following relation:

This a ngle depends on t he pos ition of t he r otor f lux ( 𝜃 𝑒 ) a nd th e s tator f lux (th e position of 𝑉 �⃗ ).

𝑇 𝑒 = 𝐾 𝑉 𝑠 𝜔 𝑙 sin(𝛿)
(2.28)

- 𝜋 2 < 𝛿 < 𝜋 2 (2.29)

A three-phase two-level voltage inverter

As mentioned, in order to control a PMSM, either in torque or in speed, it is necessary to s upply v oltages w hich ha ve t he va riable a mplitude a nd f requency a t th e te rminals o f machines. T o modulate t hese qua ntities, a vol tage i nverter i s i mplemented. I n t his w ork, the inverter chosen is a three-phase two-level type which is shown in Figure 2.9

As represented in this figure, the inverter has three legs composed of two bidirectional switches. In this study, the use switch is IGBT type with diode for anti-parallel.

In the case which the neutral of machine is not connected, we will have the equation in (2.30) always be verified:

Following this relationship, the phase voltages can be obtained by the equation (2.31)

Control strategies for PMSM drive

The pr inciple of c ontrolling AC machines is well-known a nd s uch s tructure has be en

𝑣 𝑙 + 𝑣 𝑏 + 𝑣 𝑐 = (𝑣 𝐴𝑂 + 𝑣 𝑂𝑁 ) + (𝑣 𝐵𝑂 + 𝑣 𝑂𝑁 ) + (𝑣 𝐶𝑂 + 𝑣 𝑂𝑁 ) = 0 (2.30) � 𝑣 𝑙 𝑣 𝑏 𝑣 𝑐 � = � 𝑣 𝐴𝑂 + 𝑣 𝑂𝑁 𝑣 𝐵𝑂 + 𝑣 𝑂𝑁 𝑣 𝐶𝑂 + 𝑣 𝑂𝑁 � = 1 3 � 2 -1 -1 -1 2 -1 -1 -1 2 � � 𝑣 𝐴𝑂 𝑣 𝐵𝑂 𝑣 𝐶𝑂 � (2.31)

Field oriented control

Field Oriented Control (FOC) i s one of t he most popul ar ve ctor c ontrol methods a nd gives PMSM high performances. This method allows controlling the flux and the torque in AC machine i n s imilar w ay a s f or D C motor. I n t his m ethod, t he motor e quations a re transformed to a coordinate system that rotates in synchronism with PM flux (d-q frame).

In th is f rame, th e stator c urrent v ector is s plit i nto tw o c urrent components, 𝑖 𝑠𝑠 and 𝑖 𝑠𝑠 , which are flux and torque production currents respectively. Therefore, this method allows, separately a nd i ndirectly, t o c ontrol of t he f lux a nd t orque qua ntities by us ing c urrent control l oop w ith s ome kinds of c ontrollers s uch a s PI c ontroller, hys teresis controller… Also a speed controller is needed in the outer loop; the speed control can be achieved by closing the loop by a feedback outside the inner torque control loop. The speed feedback can b e d erived f rom t he sam e r otor an gular po sition s ensor, w hich i s us ed t o obt ain t he rotor position feedback.

In Figure 2.10, a g eneral scheme of t he FOC i s presented. The t hree phase measured stator cu rrents ar e t ransformed i nto d -q f rame. T his c onversion p rovides the f eedback currents 𝑖 𝑠𝑠 and 𝑖 𝑠𝑠 . Also a sensing of the DC-link voltage is required in order to feed the space vector algorithm.

One o f t he easiest s trategies a nd m ost w idely used i n t he industry is constant t orque angle. The torque angle is defined as the angle between the q-axis current 𝑖 𝑠𝑠 and the rotor permanent flux 𝜓 𝑝 . This angle will be maintained at 𝜋 2 � according to this control strategy. It also means that the d-axis current is maintained equal to zero (𝑖 𝑠𝑠 = 0). This means that a linear current controller results in linear control over the torque. This control strategy is not r ecommended f or I PMSM w ith hi gh s aliency r atio �𝐿 𝑠 /𝐿 𝑠 �. H owever, f or S PMSM where 𝐿 𝑠 = 𝐿 𝑠 , t his s trategy i s c ommonly us ed. The main a dvantage of t his c ontrol strategy is that it minimizes the current/torque ratio.

The co ntrol sy stem co nsists o f sep arate co ntrollers f or t he t orque a nd t he c urrent.

Measurement or estimation of the rotor angle is needed in the transformation of the d-and q-axis current components into fixed coordinate system and vice versa.

Direct Torque Control (DTC)

Direct torque control is another strategy for AC motor control which was introduced in Japan by T akahashi and Noguichi [START_REF] Takahashi | A new quick-response and high-efficiency control strategy of an induction motor[END_REF] and also in Germany by Depenbrock [START_REF] Depenbrock | D irect sel f-control ( dsc) of inverter-fed i nduction machine[END_REF].

Their in novative s tudies d epart f rom th e id ea of t ransformation from ab c t hree-phase coordinate to 𝛼𝛽 coordinate.

The pr inciple of t his m ethod r elies on a ba ng-bang c ontrol i nstead of a de coupling control w hich is th e c haracteristic of v ector c ontrol.

T heir technique (b ang-bang c ontrol)

works very well with the on-off operation of inverter with power electronic components.

A general scheme of DTC is presented in Figure 2.11. According to this scheme, two hysteresis controllers a re us ed, one f or t orque error c orrection a nd on e f or f lux l inkage error correction.

The principle of DTC, as its name implies, is to control the electromagnetic torque and flux l inkage di rectly a nd i ndependently by t he use of s ix or e ight vol tage s pace ve ctors Rewriting the equation of electromagnetic torque of a PMSM as follow:

𝑇 𝑒 = 𝑛 𝑝 |𝜓 𝑠 | 𝐿 𝜓 𝑝 sin 𝜎 (2.32)
The torque thus can be controlled by controlling the standard stator flux and the angle 𝜎 which i s f ormed by t his f lux a nd the f lux c reated by t he m agnets. H owever, t his f low also can be approximated by the integration of the stator voltages (Equation (2.33)). + Predicting the future plant behaviors;

+ Optimizing the control signal according to a cost function.

Based on t he pr inciple of pr edictive c ontrol and D TC, a D irect P redictive T orque Control (DPTC) strategy is applied to the PMSM drive system. A general scheme of DPTC for PMSM is shown in Figure 2.15.

Due to the pre-calculation of the system behavior up to the prediction horizon [START_REF] Camacho | Model Predictive Control[END_REF],

predictive control is ge nerally i nevitable of hi gh c omputational e ffort. E specially, i n t he area of electrical drives, a h igh sampling frequency is necessary. Therefore, in DPTC for PMSM drive system, the prediction horizon is limited to one step ahead. Besides, the set of tested control signal also is finite. Normally, this set includes the number of configurations generated f rom a n i nverter. I n Figure 2.15, a g eneral sc heme o f D PTC for P MSM i s illustrated.

The one-step ahead discrete model of drive system will be built.

� 𝑖 𝑠𝑠 (𝑘 + 1) 𝑖 𝑠𝑠 (𝑘 + 1) � = � 1 -𝑇 𝑠 𝑅 𝑠 𝐿 𝑇 𝑠 𝜔 𝑒 (𝑘) -𝑇 𝑠 𝜔 𝑒 (𝑘) 1 -𝑇 𝑠 𝑅 𝑠 𝐿 � � 𝑖 𝑠𝑠 (𝑘) 𝑖 𝑠𝑠 (𝑘) � + � 𝑇 𝑠 𝐿 0 0 𝑇 𝑠 𝐿 � � 𝑢 𝑠𝑠 (𝑘) 𝑢 𝑠𝑠 (𝑘) � + � 0 𝑇 𝑠 𝜓 𝑝 𝐿 𝜔 𝑒 (𝑘) � (2.34)
To complete the model of the whole system, a discrete model of the inverter is used:

� 𝑢 𝐴𝑁 (𝑘) 𝑢 𝐵𝑁 (𝑘) 𝑢 𝐶𝑁 (𝑘) � = 𝑉 𝐷𝐶 3 � 2 -1 -1 -1 2 -1 -1 -1 2 � � 𝑠 𝐴 (𝑘) 𝑠 𝐵 (𝑘) 𝑠 𝐶 (𝑘) � (2.35)
where 𝑠 𝑋 (𝑋 = 𝐴, 𝐵, 𝐶) is the states of inverter's leg. With a standard three-phase two-level inverter, there are eight configurations. Among these configurations, the first state (0,0,0)

and the eight one (1,1,1) give the same output voltage. Therefore, the set of control signal is limited to seven different states of inverter.

From t his m odel, t he f uture be haviors of t he s ystem un der t he f inite s et of c ontrol signal can be predicted. The future errors can be pre-calculated through a cost function.

In D PTC c ontrol, the torque a nd the f lux l inkage w ill b e c onsidered i n t he c ost function:

𝑔 𝑇 = �𝑇 𝑟𝑒𝑟 -𝑇 𝑒𝑙 (𝑘 + 1)� 2 + 𝛾 𝜓 �𝜓 𝑠 𝑟𝑒𝑟 -𝜓 𝑠 (𝑘 + 1)� 2 (2.36)
The factor d enotes t he so-called weighting f actor, w hich i ncreases o r d ecreases t he importance of the torque or flux control. If the same importance is assigned to both control objectives, this factor would correspond to the ratio between the nominal magnitudes of the torque 𝑇 𝑛 and the stator flux �𝜓 �⃗ 𝑠𝑛 � [KRET10]:

𝛾 𝜓 = � 𝑇 𝑛 �𝜓 �⃗ 𝑠𝑛 � � 2 (2.37)
Moreover, with SPMSM whose the inductances in d and q axis are equal 𝐿 𝑠 = 𝐿 𝑠 = 𝐿, the to rque i s d irectly p roportional t o th e q -current 𝑖 𝑠𝑠 . T herefore, a nother c ost f unction including the current components can be considered:

Mono-Inverter Dual-Parallel PMSM drive system

𝑔 𝑙 = �𝑖 𝑠𝑠 𝑟𝑒𝑟 -𝑖 𝑠𝑠 (𝑘 + 1)� 2 + 𝛾 𝑙 ‖𝑖 𝑠𝑠 (𝑘 + 1)‖ 2 (2.38)
There are also several modified cost functions with other criteria such as the power loss of the machine, the power loss in switching states of inverter…

Control structure for Mono-inverter Dual-parallel PMSM system

In pr evious s ection, ge neral d escriptions a bout a P MSM dr ive s ystem ha ve be en presented. Moreover, the st rategies u sed t o control this drive sy stem also ar e i ntroduced.

All these analyses will be very useful for the consideration of mono-inverter dual-parallel PMSM system.

In t his section, t he s tructures w hich a pplied f or t his kind of dr ive s ystem w ill be considered.

Master-slave structure

General description

In this part, a master/slave configuration, which was applied to the mono-inverter dualparallel P MSM d rive s ystem [START_REF] Bidart | Commande c oopérative de s s ystèmes m onoconvertissmono multimachines synchrones[END_REF], i s pr esented. S ome c onsiderations a re t aken i nto account: + Both machines must be operated at the same speed 𝜔 𝑙1 = 𝜔 𝑙2 ; + The loads applied to each machine are different and not mechanically linked;

As s hown in Figure

Consequence due to a load variation

In order to understand and anticipate the operation of two parallel PMSM connected to an inverter, behaviors of master and slave machines will be considered in case of variation of the load of one machine.

Figure 2.17 illustrates the e volution of mono-inverter du al-parallel P MSMs d rive system under master/slave configuration with different loads.

According t o Figure 2.17, t he i mposition of a s peed Ω 𝑙 (point A ) to th e master machine leads to two consequences:

+ First, th e s elf-synchronism of P MSM w ill i mpose a pul se w ith 𝜔 𝑒_𝑙 to vo ltage applied at the terminals of the machines (point B); + Secondly, a load torque 𝑇 𝑙𝑙𝑙𝑠_𝑙 , which depends on the speed, will be applied (point

H)
Two machines are connected in parallel with an inverter, the speed of stator field of the slave machine thus is also imposed:

𝜔 𝑒_𝑠𝑙 = 𝜔 𝑒_𝑙 (point C).
To assure the synchronism of slave machine 𝑆𝑀 𝑠𝑙 , its speed has to be the same with the master m achine: 𝜔 𝑙_𝑙 = 𝜔 𝑙_𝑠𝑙 (point D ). S imilar to th e m aster, a load torque 𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 , which can be different from 𝑇 𝑙𝑙𝑙𝑠_𝑙 , is also applied to the slave machine (point E).

Mono-Inverter Dual-Parallel PMSM drive system At st eady s tate, b oth m achines m ust p rovide r espectively the el ectromagnetic t orque 𝑇 𝑒_𝑙 (point I) and 𝑇 𝑒_𝑠𝑙 (point F). In order to obtain this torque along with a current angle 𝜓 𝑙 = 0, for master machine it is necessary to provide a certain voltage 𝑉 (point J). Under this voltage, a load angle 𝛿 𝑙 is defined (point K).

This voltage (with amplitude 𝑉 and frequency 𝜔 𝑒 ) is also applied to the slave machine.

A load angle 𝛿 𝑠𝑙 is also defined and correlative to a load torque 𝑇 𝑒_𝑠𝑙 (point G).

From t he st eady st ate, the st ability o f t he sl ave m achine i n t he p resence o f a l oad change f rom one machine i s t hen s tudied. A nd t he c hoice of master machine w ill be pointed out.

The choice of master machine

At st eady s tate, b ecause t he t wo m achines a re i dentical a nd s upplied by t he s ame voltage s ource w ith s ame va lue of v oltage 𝑉 𝑠 and f requency 𝑓 𝑠 , t hey w ill ha ve t he s ame torque c haracteristic 𝑇 𝑒 = 𝑓(𝛿). T he c omparison of t wo l oads a pplying t o t wo motors gives the following relationship:

𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 𝑇 𝑙𝑙𝑙𝑠_𝑙 = 𝑇 𝑒_𝑠𝑙 𝑇 𝑒_𝑙 = sin(𝛿 𝑠𝑙 ) sin(𝛿 𝑙 ) (2.39)
Referring to the stability analysis of PMSM, the condition of 𝛿 𝑙 is always guaranteed for the load angle of the master machine because the value of 𝐼 𝑙 is limited. To investigate the stability of the slave motor, two cases comparing the loads of two machines are pointed out:

+ If 𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 ≤ 𝑇 𝑙𝑙𝑙𝑠_𝑙 ⇒ 𝛿 𝑠𝑙 ≤ 𝛿 𝑙 ⇒
The stability is ensured for both machines;

+ If 𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 > 𝑇 𝑙𝑙𝑙𝑠_𝑙 ⇒ 𝛿 𝑠𝑙 > 𝛿 𝑙 ⇒ It i s pos sible f or t he l oad a ngle of t he s lave machine to be out of the stability condition. The stability of the system then is not guaranteed.

To illustrate this, a vector diagram showing the positions of the two machines at steady state is r epresented i n Figure 2. [START_REF]PMSM with Multi-Machine Systems9 T[END_REF]. B oth st udies ar e t hen considered: 𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 > 𝑇 𝑙𝑙𝑙𝑠_𝑙 (Figure 2.19a) and 𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 < 𝑇 𝑙𝑙𝑙𝑠_𝑙 (Figure 2.19b). These figures are shown for a given electrical value of motor angle of master machine 𝜃 𝑒_𝑙 . This value varies over time, and when the steady state is reached, the trend is the same as the angle of slave machine 𝜃 𝑒_𝑠𝑙 .

In t hese d iagrams, co mparisons between t wo l oad a ngle (𝛿 𝑙 , 𝛿 𝑠𝑙 ) are co nsistent w ith the theory described above.

Mono-Inverter Dual-Parallel PMSM drive system For such these systems, therefore, in order to ensure the stability of whole system, it is necessary t o co ntrol the m achine w hich h as t he h ighest m echanical l oad ( the master machine). T he m achine w ith t he hi gher l oad a lso ha s the bigger load a ngle, thus, i t is possible to compare t he l oad an gle o f each m achine ( 𝛿 1 and 𝛿 2 ) to d etermine w hich machine is the master one.

However, w hen t he same v oltages 𝑉 𝑠 ���⃗ are a pplied to th e tw o machines, th e following relationship always is verified:

𝜃 𝑒1 + 𝛿 1 = 𝜃 𝑒2 + 𝛿 2 (2.40)
From t he r elation i n E quation (2.40), a nother r elation c an b e br ought o ut a s f ollows:

𝛿 1 < 𝛿 2 ⇔ 𝜃 𝑒1 > 𝜃 𝑒2 .
As a result, to determine which machine is the master one and to be controlled, it must be compared the electrical position 𝜃 𝑒1 and 𝜃 𝑒2 : the machine with the smaller e lectrical p osition will be c hosen a s the master one . T his s olution, pr oposed i n a patent [FBM+07], i s p resented i n Figure 2.18. T he va lue of 𝜃 𝑒 is a lready us ed f or t he autopilot, no additional sensor is needed.

As s hown i n Figure 2.16, t he p osition a nd c urrents o f t he t wo machines a re continuously r ecorded. A l ogic s ignal "Enable" al lows ch oosing t he master machine i n order t o carry out t he speed a nd c urrent c ontrol. This signal i s c reated by c alculating t he difference (𝜃 𝑒2 -𝜃 𝑒1 ) and pa ssing t his di fference t hrough a hys teresis c omparator. I f "Enable = 1" the master machine is MS1 and MS2 is the slave, on the contrary, if "Enable = 0" MS2 is the master and MS1 is the slave one. The master selector block is presented in According to t he an alysis ab ove, the se lection o f t he m aster m achine h as b een developed. However, this principle is just appropriate to the case of positive speed. In the case where the speed of the motors is negative, the value of load angle 𝛿 is also negative and the machine that is in open loop will be stable as its angle 𝛿 > -𝜋 2

(or (𝛼 -𝜋) in the case of consideration of 𝑅 𝑠 ).

Applying the same principle as above, the slave machine is stable if the absolute value of the load torque applied to the master machine respects the following relations:

+ If�𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 � ≤ �𝑇 𝑙𝑙𝑙𝑠_𝑙 � ⇒ |𝛿 𝑠𝑙 | ≤ |𝛿 𝑙 | ⇒ 𝛿 𝑠𝑙 ≥ 𝛿 𝑙 ⇒ the st ability is g uaranteed for both machines; + If �𝑇 𝑙𝑙𝑙𝑠_𝑠𝑙 � > �𝑇 𝑙𝑙𝑙𝑠_𝑙 � ⇒ |𝛿 𝑠𝑙 | > |𝛿 𝑙 | ⇒ 𝛿 𝑠𝑙 < 𝛿 𝑙 ⇒ it is p ossible f or th e lo ad
angle of the slave machine be out of the stability zone. The stability of the system is not guaranteed.

In t his cas e, t he m aster machine w ill b e t he o ne t hat h as t he sm allest v alue o f l oad angle 𝛿, i.e the highest value of 𝜃 𝑒 . Besides, when selecting the master machine, the value of the speed, represented by the reference speed 𝜔 𝑟𝑒𝑟 is a parameter need to be taken into account.

Conclusions

Based on t he m ethod of c hoosing t he m aster m achine l ike i n p aragraph Error! Reference source not found., the s tability of the whole system is assured although only master machine i s co ntrolled. T he sl ave m achine i n m any co nditions st ill f ollows t he master one and the synchronism of both machines is obtained.

To i mprove t he pe rformance o f t he sy stem u nder m aster/slave st ructure, t he v alue o f the h ysteresis can b e m odulated i n the " master choice b lock". T his issue h as b een al so

considered in [BPDFM09] [Bid11].
Moreover, because just only one machine is controlled in this structure, other control strategies for high performance of PMSM drive system such as DTC, PTC…can be applied for the master machine.

In t he ne xt pa rt of t his c hapter, a nother a pproach w ith m ono-inverter dua l-parallel PMSM will be considered. This consideration is carried out based on the principle of DTC.

Direct torque control (DTC) for mono-inverter dual-parallel PMSM

DTC h as b een w ell-adapted t o el ectric d rive s ystems i n g eneral, as w ell as P MSM drive s ystem i n pa rticular. The configuration of this DTC drive system is set up as shown in Figure 2.20.

General description

In t his pa rt, D TC method w ill be considered with t he m ono-inverter dua l-parallel

PMSM structure [Ndi09] [FNL13]
. All the characteristic of this drive system as mentioned in the previous section are respected.

The principle is preserved as in the case of one machine. However, with one machine, there are just two error components Δ𝑇 and Δ𝜓 𝑠 which a re c onsidered by t wo hys teresis controllers. Just four cases can then happen as shown in Table 2. 1. Therefore, it is easy to find just only one suitable voltage to meet the control demand.

In t he c ase t wo machines ope rating i n pa rallel, the nu mber of e rror c omponent increases to four Δ𝑇 1 , Δ𝑇 2 , Δ𝜓 𝑠1 and Δ𝜓 𝑠2 . Four hysteresis comparators have to be used in order to control the torque and flux linkage of two machines. Now, there are sixteen cases can be synthesized from four states of two machines (Table 2.2). if for the first machine, both the torque and flux need to be increased, meanwhile for the second machine, they need to be decreased.

According to the DTC, each requirement pair of torque and flux will be correlative to a voltage vector (Table 2.1). However, the choice of this voltage vector also depends on the position of the stator flux vector in space vector. Under different load conditions, the stator flux ve ctors of t wo machines c an be l ocated i n di fferent sectors. T his pr oblem i s a lso illustrated in Figure 2.21.

In this figure ( Therefore, i t i s n ecessary t o bu ild a ne w opt imal s witching t able t o adapt to bo th machines in parallel configuration. It will be discussed in the next part. The pr inciple us ed t o determine t he v oltage vector w ith th e a im o f meeting th e requirements of both machines is pointed out as follows:

+ In t he c ase of III (Table 2.2), t he r equirements of t wo m achines a re absolutely different. Therefore, the null vector is chosen; + In t he seco nd case ( II ), t wo machines ha ve one r equirements i n c ommon, e ither the torque or the flux evolution. In this case, the suitable voltage of each machine will n ot b e of d ifferent sign. L et's s upposed th at th e s uitable v ector f or th e fi rst machine i s noted as 𝑉 1 and f or t he s econd m achine i s 𝑉 2 . I f t here is an av erage vector located in the middle of 𝑉 1 and 𝑉 2 , this vector will be chosen (Figure 2.22).

+ If the two vectors are in adjacent sectors and the requirements for the torque of both machines are the same, but requirements for the flux will be different. The optimal voltage will be chosen depending on the position of flux vector of each machine (as shown in Figure 2.23.

+ The f inal c ase is the easiest case. B ecause t he t wo m achines h ave t he sam e requirements f or t he torque a nd the f lux, t he same vol tage ve ctor w ill be the optimal for both of them. According to these analyses, an optimal table is built and represented in Table 2. 3. + Although l ocating in t wo a djacent s ectors, in s ome c ase, r equirements of bot h machines can be the same for torque and flux.

𝚫𝝍
+ Meanwhile, in the critical case shown in Table 2.2, the requirements for torque and flux of bot h m achines a re c ompletely oppos ed to e ach ot her. T he vo ltage ve ctor will be chosen according to the average rule (Figure 2.22). It means that an average vector located in the middle of two vectors will be chosen.

+ Another sp ecial case c an be ha ppened, t he t wo s uitable v oltage v ectors f or t wo machines are also two adjacent vectors. The vector will be determined based on the position of the two flux vectors (as shown in Figure 2.23).

As a result, an optimal voltage vector is defined for each case. They are summarized in The test is carried out under these following conditions. Three set points of speed will be considered: These set-points are plotted in Figure 2.24.

0 0 0 0 5 V 4 V 5 V 6 V 6 V 1 V 0 0 0 1 5 V 0/ 7 V 0/ 7 V 6 V 0/ 7 V 1 V 0 0 1 0 0/ 7 V 4 V 5 V 0/ 7 V 6 V 0/ 7 V 0 0 1 1 3 V 2 V 3 V 4 V 4 V 5 V 0 1 0 0 6 V 5 V 6 V 1 V 1 V 2 V 0 1 0 1 4 V 6 V 1 V 5 V 2 V 6 V 0 1 1 0 2 V 4 V 5 V 3 V 6 V 4 V 0 1 1 1 3 V 2 V 3 V 4 V 4 V 5 V 1 0 0 0 6 V 5 V 6 V 1 V 1 V 2 V 1 0 0 1 5 V 1 V 2 V 6 V 4 V 1 V 1 0 1 0 1 V 3 V 4 V 2 V 5 V 3 V 1 0 1 1 3 V 2 V 3 V 4 V 4 V 5 V 1 1 0 0 6 V 5 V 6 V 1 V 1 V 2 V 1 1 0 1 0/ 7 V 1 V 2 V 0/ 7 V 3 V 0/ 7 V 1 1 1 0 2 V 0/ 7 V 0/ 7 V 3 V 0/ 7 V 4 V 1 1 1 1 2 V 1 V 2 V 3 V 3 V 4 V Table 2.4:
0 0 0 0 1 V 2 V 2 V 3 V 4 V 3 V 0 0 0 1 0/ 7 V 2 V 0/ 7 V 3 V 4 V 0/ 7 V 0 0 1 0 1 V 0/ 7 V 2 V 0/ 7 V 0/ 7 V 3 V 0 0 1 1 5 V 6 V 6 V 1 V 2 V 1 V 0 1 0 0 2 V 3 V 3 V 4 V 5 V 4 V 0 1 0 1 3 V 1 V 4 V 2 V 3 V 5 V 0 1 1 0 1 V 5 V 2 V 6 V 1 V 3 V 0 1 1 1 5 V 6 V 6 V 1 V 2 V 1 V 1 0 0 0 2 V 3 V 3 V 4 V 5 V 4 V 1 0 0 1 4 V 2 V 5 V 3 V 4 V 6 V 1 0 1 0 6 V 4 V 1 V 5 V 6 V 2 V 1 0 1 1 5 V 6 V 6 V 1 V 2 V 1 V 1 1 0 0 2 V 3 V 3 V 4 V 5 V 4 V 1 1 0 1 4 V 0/ 7 V 5 V 0/ 7 V 0/ 7 V 6 V 1 1 1 0 0/ 7 V 5 V 0/ 7 V 6 V 1 V 0/ 7 V 1 1 1 1 4 V 5 V 5 V 6 V 1 V 6 V Table 2.5:
+ From 𝑡 = 0(𝑠) → 0.
𝑅 𝑠 2.06(Ω)

𝐿 𝑠𝑠 = 𝐿 𝑠𝑠 = 𝐿 9.15 × 10 -3 (𝐻) 𝜓 𝑝 0.29(𝑊𝑏)
Numbers of pole pairs �𝑛 𝑝 � 3

Inertia (𝐽) 7.2 × 10 -4 (𝑘𝑔. 𝑁 2 )
Table 2.6: Parameters of machine using in simulation The varies of load of two machine are represented in Figure 2.25.

All the simulation results are shown in Figure 2.26

From the simulation results, it can be seen that the speeds of two machines can follow the s et po int ( Figure 2.26a). D ynamic r esponse of t he s ystem i s ve ry f ast ( ~0.02s).

However, the responses at standstill are not good. There is a ripple in speed. System is also stable with the load variations. The steady state is reestablished in a short time (~0.03s).

The electromagnetic torqueses of two machines are shown in

Conclusions

In this chapter, the mono-inverter dual-parallel PMSM drive system has been described and analyzed. The main components of this system are presented and considered in detail. However, in practical application, this table cannot satisfy absolutely the requirements of both machines under different load conditions. The system still operates in stable partly because of the self-synchronism ability of synchronous machine. In order to build a good optimal lookup table of voltage vector, it needs further studies.

The use of DTC method for two machines also leads to the idea of considering another method t o c ontrol t he mono-inverter dua l-parallel P MSM drive s ystem. T his m ethod i s based on t he pr inciple of pr edictive c ontrol. I n s tead of u sing hys teresis c omparators t o find t he r equirements f or t wo m achines, a n op timization issue w ill be solved i n or der t o satisfy all the requirements. The analysis and testing of this method will be presented in the next chapter.

Introduction

As a nalyzed i n pr evious c hapter, i n the pr ogress of l ooking for a c ontrol s trategy f or mono-inverter dua l-parallel P MSM drive s ystem, two s olutions w hich are m aster-slave structure with PI controllers and Direct Torque Control (DTC), have been pointed out.

In m aster-slave st ructure w ith PI c ontroller, operation of w hole s ystem i s f airly w ell.

The stability can be assured according to the choice of master machine. However, the slave machine will not be regulated in this case.

DTC i s a nother s olution f or s uch ki nd of dr ive s ystem. Purpose of t his s olution i s considering the i nformation ( electromagnetic t orque a nd f lux l inkage) o f bot h machines.

Based on this principle, a su itable voltage vector will be chosen and defined in a l ook-up For this kind of drive system, we just change the cost function as well as the optimization algorithm in order to get the different performance of the system.

However, it is difficult for either DTC or PTC to satisfy absolutely the requirements of each machine. They just try to harmonize in the best way the demands of both machines.

Because ea ch v oltage vector f ound i n the optimization p rocess might be not m eet exactly the demands of each machine, the two PMSMs will sometimes operate in voltage supply m ode. T herefore, the r esponses of the PMSMs u nder voltage s upply will b e considered first in this chapter. After that, a general description about the PTC for monoinverter dual-parallel PMSM drive system will be presented. Each component of this drive system i s s tudied a nd e xplained in detail. T wo a pproaches to p redictive c ontrol will b e

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM described a nd c ompared. S ome simulations w ill be done i n or der t o ve rify t he performances of control methods.

PMSM under voltage supply mode

As mentioned above, a PMSM in voltage supply mode will be considered. In this case, amplitude (𝑉 𝑠 ) and frequency (𝜔 𝑠 ) of voltage supply and load torque (𝑇 𝑙𝑙𝑙𝑠 ) will be the known parameters. At each value of (𝑉 𝑠 , 𝜔 𝑠 , 𝑇 𝑙𝑙𝑙𝑠 ), we will obtain a sp ecific steady state of a PMSM. In the next paragraph, the calculation of variables of a PMSM at steady state will be presented.

Steady state calculation

Steady s tates of P MSM w ill be c alculated by c onsidering the c ontinuous model of PMSM (section 2.2.2.2). It is noted that, at steady state all the differential components will equal to zero. As a result, all equations will be rewritten as in (3.1).

At steady state, we have 𝜔 𝑒 = 𝜔 𝑠 .

From the third and the forth equation in (3.1), q-current of PMSM at steady state will be obtained:

𝐼 𝑠𝑠 = 𝑓 0 � 𝜔 𝑒 𝑛 𝑝 � + 𝑇 𝑙𝑙𝑙𝑠 𝑛 𝑝 � 3 2 𝜓 𝑝 (3.2)
From the first and the second equation in (3.1) and using the relationship between the d-q voltages and 𝑉 𝑠 and the load angle 𝛿: � 𝑢 𝑠𝑠 = -𝑉 𝑠 sin 𝛿 𝑢 𝑠𝑠 = 𝑉 𝑠 cos 𝛿 , we have: Load angle (rad)

⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ 𝑢 𝑠𝑠 = 𝑅 𝑠 𝑖 𝑠𝑠 -𝜔 𝑒 𝐿𝑖 𝑠𝑠 𝑢 𝑠𝑠 = 𝑅 𝑠 𝑖 𝑠𝑠 + 𝜔 𝑒 𝐿𝑖 𝑠 + 𝜔 𝑒 � 3 2 𝜓 𝑝 𝑇 𝑒𝑙 = 𝑛 𝑝 � 3

Small deviation model

In the previous part, the calculation about steady state of a PMSM has been established. we can obtain all the roots of this polynomial. In general, a four-order polynomial will give us four roots. At a steady state at which the real part of all roots is negative, it w ill be a stable state. Otherwise, if any root has a positive real part, this point will be unstable.

In Figure 3.4, an example of a map of r oots of PMSM i s r epresented. These r oots in and 𝑇 𝑙𝑙𝑙𝑠 = 0𝑁𝑁). We also have the following values:

⎣ ⎢ ⎢ ⎢ ⎡ 𝛿𝚤 𝑠 δ𝚤 𝑠 δ𝜔 𝑒 δ𝛿 ̇⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ - 𝑅 𝑠 𝐿 𝑠 𝜔 𝑒 𝑖 𝑠0 - 𝑉 𝑠 𝐿 𝑠 cos 𝛿 0 -𝜔 𝑒 - 𝑅 𝑠 𝐿 𝑠 -�𝑖 𝑠0 + � 3 2 𝜓 𝑝 𝐿 𝑠 � - 𝑉 𝑠 𝐿 𝑠 sin 𝛿 0 0 𝑝 2 � 3 2 𝜓 𝑝 𝐽 - 𝑟 0 𝐽 0 0 0 -1 0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ������������������������������� 𝐴 � 𝛿𝑖 𝑠 𝛿𝑖 𝑠 𝛿𝜔 𝑒 𝛿𝛿 � + ⎣ ⎢ ⎢ ⎡ 0 0 - 𝑝 𝐽 0 ⎦ ⎥ ⎥ ⎤ 𝛿𝑇 𝑙𝑙𝑙𝑠 (3.8)
Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM An input parameters 𝜔 𝑒 are varied and the roots of 𝑃(𝑠) will evolve as shown in this map. According to this map, a limit value of 𝜔 𝑒 is determined: 𝜔 𝑙𝑙𝑙𝑙𝑙 ≈ 380(𝑟𝑟𝑟. 𝑠 -1 ).

From t he analysis above, all t hree parameters 𝑉 𝑠 , 𝜔 𝑒 and 𝑇 𝑙𝑙𝑙𝑠 will vary as in section (3.2.1). At each steady state we will have a correlative small deviation model as well as a polynomial characteristic 𝑃(𝑠). Solving 𝑃(𝑠) and considering the roots of 𝑃(𝑠) give us a limit value of stable operation point of PMSM. The limit surface is shown in Figure 3.5. The structure of this kind of drive system is represented in Figure 3.6. All conditions of a mono-inverter dual-parallel PMSM are preserved as described in Section 2.1.

⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ 𝐼 𝑠𝑠0 = 0 𝐼 𝑠𝑠0 = 𝑓 0 � 𝜔𝑒 𝑛 𝑝 � 𝑛 𝑝 � 3 2 𝜓 𝑝 𝑉 𝑠 = ���𝜔 𝑒 𝐿 𝑠 𝐼 𝑠𝑠0 � 2 + �𝑅 𝑠 𝐼 𝑠𝑠0 + 𝜔 𝑒 � 3
Inside this structure, two control loops are used: The two control loops will be considered in detail in the following part of this chapter.

Firstly, the outer loop with R-S-T discrete controller will be presented.

R-S-T discrete controller

Introduction

In the context of computer control, which is now widespread in industry, it is necessary to de velop an e fficient de sign a nd i mplementation m ethodology for di gital c ontroller.

Based o n t hese asp ects, t he t wo-degrees-of-freedom R -S 𝑧 -𝑑 𝐵�𝑧 -1 � 𝐴(𝑧 -1 ) , the equation of the R-S-T controller is given by: 𝑆(𝑧 -1 )𝑢(𝑡) + 𝑅(𝑧 -1 )𝑦(𝑡) = 𝑇(𝑧 -1 )𝑦 * (𝑡 + 𝑟 + 1) (3.10) where 𝑢(𝑡) and 𝑦(𝑡) are the input and output of the plant and 𝑦 * (𝑡 + 𝑟 + 1) is the desired tracking t rajectory, w hich i s e ither generated by a t racking r eference model (𝐵 𝑙 /𝐴 𝑙 ) or stored in the computer memory.

The polynomial characteristic of closed loop control is then written as follows:

𝑃(𝑧 -1 ) = 𝐴(𝑧 -1 )𝑆(𝑧 -1 ) + 𝑧 -𝑠 𝐵(𝑧 -1 )𝑅(𝑧 -1 ) (3.11)
Solving this equation allows the determination of 𝑆(𝑧 -1 ) and 𝑅(𝑧 -1 ). In order to do that, a pole p lacement st rategy i s u sually used [START_REF] Iftikhar | A Control Strategy to Stabilize PWM DC-DC Converters wwith Input Filter Using State-Feedback and Pole-Placement[END_REF]. T his t echnique c an be us ed f or pl ant models of any order, with or without time delay and featuring stable or unstable zeros.

In t he f ollowing pa ragraph, a R -S-T d igital c ontroller w ith a nti w ind-up f or speed/torque control in the case of PMSM will be presented.

R-S-T anti-windup Speed/Torque controller

From the electromechanical description of a PMSM, the relationship between the speed and torque in Laplace domain is built as follows:

𝐺(𝑠) = 𝜔 𝑙 (𝑠) 𝑇 𝑒𝑙 (𝑠) = 1 𝐽𝑠 + 𝑓 0 (3.12)
Performing Z-Transform, a PMSM model in Z-domain will be obtained:

𝐺(𝑧 -1 ) = 𝑏 1 𝑧 -1 𝑟 0 + 𝑟 1 𝑧 -1 = 1 𝑓 �1 -𝑒 -𝑇𝑠 𝜏 𝑚 � � 𝑧 -1 1 -𝑒 -𝑇 𝑠 𝜏 𝑚 � 𝑧 -1 (3.13)
where

𝜏 𝑙 = 𝐽 𝑟 0
is the mechanical time constant.

Based o n t he co efficients 𝑟 0 , 𝑟 1 and 𝑏 1 obtained f rom t he model of P MSM i n Zdomain, along with the support of pole placement technique, the parameters of RST speed controller will be calculated as follow:

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 𝑠 0 = 1 𝑠 1 = 1 𝑟 0 = 𝑝 1 -𝑟 1 + 1 𝑏 1 𝑟 1 = 𝑝 2 + 𝑟 1 𝑏 1 (3.

Predictive Torque Control

Predictive In the following part, a discrete model of considered system will be constructed step by step. where:

𝐴 𝑠 𝑠𝑠 (𝑘) = � 1 -𝑇 𝑠 𝑅 𝑠 𝐿 𝑇 𝑠 𝜔 𝑒 (𝑘) -𝑇 𝑠 𝜔 𝑟𝑒 (𝑘) 1 -𝑇 𝑠 𝑅 𝑠 𝐿 � ; 𝐵 𝑠 𝑠𝑠 (𝑘) = � 𝑇 𝑠 𝐿 0 0 𝑇 𝑠 𝐿 � 𝑃 𝑠 𝑠𝑠 (𝑘) = � 0 𝑇 𝑠 � 3 2
𝜓 𝑝 𝐿 𝜔 𝑒 (𝑘) �

(3.17)

𝑟𝑖 𝑠 𝑠𝑠 𝑟𝑡 = 𝐴 𝑠𝑠 𝑖 𝑠 𝑠𝑠 + 𝐵 𝑠𝑠 𝑢 𝑠 𝑠𝑠 + 𝑃 𝑠𝑠 𝐴 𝑠𝑠 = � - 𝑅 𝑠 𝐿 𝜔 𝑒 -𝜔 𝑒 - 𝑅 𝑠 𝐿 � ; 𝐵 𝑠𝑠 = � 1 𝐿 0 0 1 𝐿 � ; 𝑃 𝑠𝑠 = � 0 -� 3
⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑠𝑠1 𝑘+1 𝑖 𝑠𝑠1 𝑘+1 𝑖 𝑠𝑠2 𝑘+1 𝑖 𝑠𝑠2 𝑘+1 ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡1 -𝑇 𝑠 𝑅 𝑠 𝐿 𝑇 𝑠 𝜔 𝑒1 𝑘 0 0 -𝑇 𝑠 𝜔 𝑒1 𝑘 1 -𝑇 𝑠 𝑅 𝑠 𝐿 0 0 0 0 1 -𝑇 𝑠 𝑅 𝑠 𝐿 𝑇 𝑠 𝜔 𝑒2 𝑘 0 0 -𝑇 𝑠 𝜔 𝑒2 𝑘 1 -𝑇 𝑠 𝑅 𝑠 𝐿 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ������������������������������� 𝔽(𝑘) ⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑠𝑠1 𝑘 𝑖 𝑠𝑠1 𝑘 𝑖 𝑠𝑠2 𝑘 𝑖 𝑠𝑠2 𝑘 ⎦ ⎥ ⎥ ⎥ ⎤ + ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑇 𝑠 𝐿 0 0 0 0 𝑇 𝑠 𝐿 0 0 0 0 𝑇 𝑠 𝐿 0 0 0 0 𝑇 𝑠 𝐿 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ����������� 𝔾(𝑘) ⎣ ⎢ ⎢ ⎢ ⎡ 𝑢 𝑠𝑠1 𝑘 𝑢 𝑠𝑠1 𝑘 𝑢 𝑠𝑠2 𝑘 𝑢 𝑠𝑠2 𝑘 ⎦ ⎥ ⎥ ⎥ ⎤ + ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 0
𝑇 𝑠 � 3 2 𝜓 𝑝 𝐿 𝜔 𝑒1 𝑘 0 𝑇 𝑠 � 3 2 𝜓 𝑝 𝐿 𝜔 𝑒2 𝑘 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ��������� ℍ(𝑘) (3.18) ⎣ ⎢ ⎢ ⎢ ⎡ 𝑢 𝑠𝑠1 (𝑘) 𝑢 𝑠𝑠1 (𝑘) 𝑢 𝑠𝑠2 (𝑘) 𝑢 𝑠𝑠2 (𝑘) ⎦ ⎥ ⎥ ⎥ ⎤ = � 2 3 ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ cos 𝜃 𝑒1 (𝑘) cos �𝜃 𝑒1 (𝑘) - 2𝜋 3 � cos �𝜃 𝑒1 (𝑘) - 4𝜋 3 � -sin 𝜃 𝑒1 (𝑘) -sin �𝜃 𝑒1 (𝑘) - 2𝜋 3 � -sin �𝜃 𝑒1 (𝑘) - 4𝜋 3 � cos 𝜃 𝑒2 (𝑘) cos �𝜃 𝑒2 (𝑘) - 2𝜋 3 � cos �𝜃 𝑒2 (𝑘) - 4𝜋 3 � -sin 𝜃 𝑒2 (𝑘) -sin �𝜃 𝑒2 (𝑘) - 2𝜋 3 � -sin �𝜃 𝑒2 (𝑘) - 4𝜋 3 � ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ � 𝑉 𝐴𝑁 (𝑘) 𝑉 𝐵𝑁 (𝑘) 𝑉 𝐶𝑁 (𝑘) � (3.19)
Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

� 𝑉 𝐴𝑁 (𝑘) 𝑉 𝐵𝑁 (𝑘) 𝑉 𝐶𝑁 (𝑘) � = 𝑉 𝐷𝐶 3 � 2 -1 -1 -1 2 -1 -1 -1 2 � ������������� 𝔻 � 𝑠 𝐴 (𝑘) 𝑠 𝐵 (𝑘) 𝑠 𝑐 (𝑘) � (3.20)
where 𝑠 𝑋 (𝑋 = 𝐴, 𝐵, 𝑜𝑟 𝐶) is represented for two states of the inverter's legs.

𝑠 𝑥 = � 0 ⇔ 𝑉 𝑋𝑂 = 0 1 ⇔ 𝑉 𝑋𝑂 = 𝑉 𝐷𝐶 (3.21)
With a three-phase t wo-level voltage inverter, there are two switching states for each leg, so there are eight possible switching states represented by eight different real vectors

[𝑠 𝐴 𝑠 𝐵 𝑠 𝐶 ] 𝑙 𝑇 (Table 3.2). Configurations 0 a nd 7 lead both to zero voltages, therefore there is usually only one configuration is really used.

𝒔 𝑨 0 1 1 0 0 0 1 1
𝒔 𝑩 0 0 1 1 1 0 0 1 𝒔 𝑪 0 0 0 0 1 1 1 1

B. Cost functions with secondary terms

Some systems have a primary goal or a more important control objective that must be achieved i n or der to p rovide an a dequate system be havior, a nd a dditional s econdary constraints or r equirements t hat s hould a lso be a ccomplished t o i mprove s ystem performance, ef ficiency o r p ower q uality. I n t hese case s, the co st f unction p resents a primary and secondary terms, where the importance of the secondary term can vary within a w ide r ange, de pending on t he a pplication a nd i ts s pecific ne eds

C. Cost functions with equivalent important terms

Unlike t he previous c ase, t here ar e sy stems i n w hich se veral v ariables n eed to b e controlled s imultaneously w ith the s ame p riority. In t hese cases, the cost f unction ca n include several terms with equivalent importance, and it is the job of the weighting factors to compensate the difference in nature of the variables. Such in the case of torque and flux control [START_REF] N Guyen | P redictive T orque Control -A solution f or m ono i nverter -dual pa rallel P MSM s ystem[END_REF], w here bot h va riables ne ed t o be c ontrolled a ccurately i n or der t o ha ve proper system performance.

Application Cost function

Torque and flux control In t he c ase of pa rallel PMSM, i n order to control t he s peed of bot h m achines, it is necessary to control the torque applied to each machine. This is because of the relationship between t he t orque an d t he sp eed p resented i n t he m echanical eq uation o f P MSM.

However, PMSM which is considered in this work, is a non-silent type (𝐿 𝑠𝑠 = 𝐿 𝑠𝑠 = 𝐿).

The el ectromechanical t orque 𝑇 𝑒 is proportional di rectly t o t he q-axis c omponent of t he current: 𝑇 𝑒 = 𝑛 𝑝 �3/2𝜓 𝑝 𝑖 𝑠𝑠 . T herefore, instead of c onsidering t he t orque, t he q -axis current of two machines will be added to the cost function.

However, i f onl y the q -axis currents of t wo m achines a re c onsidered, t he d -axis currents will be free. The power losses in term of (𝑅 𝑠 𝑖 𝑠𝑠 2 ) might be large. This is the reason why the d-axis currents will be also added to the cost function.

Finally, the cost function chosen for this system will be built as follows:

𝑔 = �𝑖 𝑠𝑠1 𝑟𝑒𝑟 -𝑖 𝑠𝑠1 (𝑘 + 1)� 2 + �𝑖 𝑠𝑠2 𝑟𝑒𝑟 -𝑖 𝑠𝑠2 (𝑘 + 1)� 2 +�𝑖 𝑠𝑠1 (𝑘 + 1)� 2 + �𝑖 𝑠𝑠2 (𝑘 + 1)� 2 (3.24)
This c ost f unction w ill be c onsidered t hrough out t his work. Along with t his c ost function, it is also necessary to choose a suitable solving algorithm in order to optimize it.

Several a lgorithms h ave b een st udied an d sy nthesized i n [START_REF] Linder | Model-Based Predictive C ontrol of Electrical D rives[END_REF]. However, i n t his work, the considered predictive control is along with a finite control set. It means that only a limit number of discrete control signals is tested with model of the system. The question here is how to choose the most suitable control signals into this set.

In the next paragraph, predictive control strategies will be presented.

Predictive control strategies

The t echniques f or o ptimum co ntrol o f e lectrical m achine drive ar e b ased o n t he principle that an optimum voltage space vector determined will be applied to the machine in the next switching cycle. As described before, commonly a two-level inverter fed from a DC voltage bus is used as voltage source for the machine. Hence, only a finite number of possible voltage space vectors 𝑢 𝑠 can really exist.

An optimum control which considered the fact that a two-level inverter just can realize only eight naturally output states is also called as Direct Predictive Control. It allow direct inverter control, the control algorithm directly determines the optimum inverter switching state for the next sampling cycle. There is no modulation technique and so the switching frequency is variable. This strategy is shown in Figure 3.10a.

Direct Predictive Torque Control (DPTC)

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM Another a pproach is u sing a m odulation t echnique w hich d iscretizes t he r eference space v oltage v ector g enerated f rom t he co ntroller ( Figure 3.10b) be fore a pplying t o t he inverter. In this approach, along with the real vectors from a two-level inverter, a group of virtual vectors will be created and considered thanks to special strategies.

There is another technique in which the on-off time of switches of inverter is calculated We w ill c onsider f irstly th e " Direct P redictive T orque C ontrol (DPTC)" in th e next section.

Direct Predictive Torque Control (DPTC)

As mentioned in previous section, one simple method called Direct Predictive Control Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM configuration found in this process will be applied directly to the inverter. No modulation technique needs to be used. So, it also belongs to variable switching frequency methods.

The scheme of this control structure is shown in Figure 3.11.

Principle

The principle of DPTC is followed these steps:

+ At 𝑘 instant time, the system states including the positions �𝜃 𝑙1 (𝑘), 𝜃 𝑙2 (𝑘)�, the A diagram of the control algorithm is shown in Figure 3.12.

Simulation results for DPTC

In order to verify the performances of the system under direct predictive torque control strategy, a s imulation ha s be en bui lt i n the M atlab/Simulink e nvironment. PMSM presented in Table 2.6 continues to be used.

The test is carried out under these following conditions. Three set points of speed will be considered:

+ From 𝑡 = 0(𝑠) → 0.4(𝑠): the reference speed is set at 𝜔 𝑟𝑒𝑟1 = 75(𝑟𝑟𝑟/𝑠);

+ From 𝑡 = 0.4(𝑠) → 0.8(𝑠): the motor is reversed and run at 𝜔 𝑟𝑒𝑟2 = -75(𝑟𝑟𝑟/𝑠); In this simulation, two sampling times are used. For the speed control loop, due to the slower mechanical response time, the sampling time is chosen equal to 𝑇 𝑠𝑐 = 1 × 10 -3 (𝑠).

Meanwhile, f or t he pr edictive c ontroller, in o rder to impose the dyna mic r esponse of system as well as r educe t he r ipple o f sy stem's st ates ( torque, cu rrents … ), a faster sampling time is used: 𝑇 𝑝𝑐 = 5 × 10 -5 (𝑠).

All the simulation results (speed, torque, and current responses) will be presented and analyzed in the next paragraphs.

According to the Figure 3.15, the speed responses of two machines (a blue for the first machine and a green for the second) are presented and compared with the reference speed which is drawn in red. It can be seen that the speeds of two machines follow the reference value. D ynamic r esponse of sy stem i s q uite f ast. T he t ransient t ime w hen t here is a st ep change in set point is very small (≈ 0.05𝑠).

Moreover, the system is also stable. At several points of time 𝑡 = 0.2(𝑠), 0.6(𝑠) and 1(𝑠), the load applied to the second machine is changed between two values: 1(𝑁𝑁) and 4(𝑁𝑁) while t he l oad ap plied t o t he f irst machine i s co nstant. T his ch ange makes t he system be out of the steady state, however, the steady state immediately is re-established in ≈ 0.03(𝑠). The speed response at standstill is not good. T here is a ripple in both motors.

This p roblem is in evitable w ith DPTC b ecause it is a f eature o f v ariable s witching frequency method where low speed operation is not encouraged.

The t orque responses o f t wo machines ar e shown i n Figure 3.16. At steady state, the torque of each machine corresponds to the load applied in order to guaranty the following relation: However, t he r ipple i n t he t orque i s important. T he pe ak t o pe ak va lue of r ipple i s approximate 2.8(𝑁𝑁). T his r ipple can b e reduced i f t he sam pling t ime us ed in DP TC strategy is smaller. This issue will be discussed in the next section.

The representation of machine currents in d-q axis is expressed in Figure 3.17. The qcurrent is directly proportional to the electromagnetic torque of each machine. Therefore, the machine with heavier load will have bigger q-current. Meanwhile, the d-current of the machine with heavier load is always smaller. Optimal process lets this term come to minimum also respected the following relation:

As r esults, when optimal process makes t he l eft si de of (3.26) come to t he minimum value, i t a lso d oes n early t he sam e t hing w ith the r ight si de. W e can see t hat i t w ill b e happened when 𝑖 𝑠𝑠1 = -𝑖 𝑠𝑠2 . ) will m ake t he sy stem goes further from the steady state and so the ripple is large. In the following consideration, two sampling frequencies will be studied: It i s easy to r ealize t hat w ith sm aller sam pling t ime t he r ipple i s d ecreased considerably. Theoretically, the amplitude of this ripple can be calculated by the following equation: In Figure 3.18(a), the peak to peak value of ripple is around 0.5(𝑁𝑁) (compared with 2.8(𝑁𝑁) in Figure 3.18(b)). This result can be also found with the relation in (3.27).

Relation between the ripple and the sampling time of DPTC

+
Δ𝑖 = 𝑉 𝐷𝐶 × 𝑇 𝑠 𝐿 (3.27) 𝑖 𝑠𝑠1 2 + 𝑖 𝑠𝑠2 2 ≥ (𝑖 𝑠𝑠1 + 𝑖 𝑠𝑠2 ) 2 2 (3.26) a) 𝑇 𝑝𝑐 = 1 × 10 -5 (𝑠) b) 𝑇 𝑝𝑐 = 5 × 10 -5 (𝑠)
Reducing the sampling time can improve the performance of the system. However, it will be a ffected a l ot b ecause of t he c omputational c ost. T oday, many s trong c omputers with high speed processor allow setting up t he sampling time at 10𝜇𝑠. Nevertheless, with the problem of price, it is not an optimal solution.

Simulation results showed that it is possible to control with DPTC strategy

Optimization problem

Optimization p rocess w ill let th e v alue o f c ost f unction c ome to a n optimal v alue. To consider this issue, the optimization problem will be announced as follows:

We will consider the case when optimization process lets cost function 𝑔 come to zero value "fmincon" procedure will search through the whole space vector under the constraints as pointed out in equation (3.28). All voltage vectors which satisfy these conditions will be tested w ith the co st f unction 𝑔. T he optimal vol tage f ound b y " fmincon" pr ocedure i s represented i n 𝛼𝛽-frame: 𝑉 𝛼 * , 𝑉 𝛽 * . T hese vol tages w ill be transformed i nto 𝑟𝑏𝑐-frame an d synthesized by a m odulation t echnique be fore be ing applied to m achines. A d etailed description of "fmincon" procedure is presented in Appendix A.1

The machines in Table 2.6 will be used to consider the operation of "fmincon" under Matlab/Simulink environment. Simulation results are shown in Figure 3.20.

It can be seen that optimization with "fmincon" function give us nearly the same results about dynamic responses. The time response for speed, current is fast (~0.05s). Moreover, the ripple of speed at standstill is improved considerably (Figure 3.20a).

In Figure 3.20b, i t i s clearly that the t hree ph ase c urrent of bot h m achine i s s mooth.

The ripple is also improved.

Using " fmincon" f unction, t he a mplitude an d p hase an gle o f v oltage v ector ar e searched t hrough out the w hole c onverter c ontrol s pace. Therefore, the r ipples i n d -q currents are quite small (≈ 0.1𝐴) (Figure 3.20c). However, it is clearly that the d-current of both machines are different from zero. It means that cost function 𝑔 does not come to zero.

Moreover, the ripple is improved but their average values are not much different compared with DPTC. It means that the more exactly reference voltage in "fmincon" procedure just reduces the amplitude of ripple; it does not create a new steady state.

Conclusions

The simulation results show that DPTC can be used to control this drive system. The principle o f th is solution is s imple to unde rstand and ha s e asy vi sualization. If us ing a small enough sampling time (𝑇 𝑝𝑐 ≈ 1 × 10 -5 𝑠), a good performance of the whole system can be obtained.

However, i t w ill b e d ifficult t o reduce t he s ampling t ime t o su ch v alue as sm all as

𝑇 𝑝𝑐 ~1 × 10 -5 (𝑠). T his pr oblem de pends much on t he ha rdware. I n s uch s mall s ampling time, some ha rdware limitations, d ue t o t he hi gh c omputational c ost, can a ppear. The computational time thus can affect to the operation of the whole system.

As mentioned above, the limitation of sampling time along with the use of only eight real voltage vectors from a two-level inverter leads to a problem of strong in. These vectors have large amplitude of issue is also verified with "fmincon" function where the amplitude and the phase angle of the reference voltage can be placed anywhere in the whole control region of the converter.

Based on theoretical analyses, t he idea of us ing more vol tage ve ctors, which ar e also called v irtual v ectors, with v ariable amplitude a nd s maller a ngle s tep i s pr oposed. A strategy with such a group of virtual voltage vectors will be studied in the next section. As results, a P redictive T orque C ontrol -"Split an d S eek" ( PTC-SS) st rategy i s o btained [NFL13].

Predictive Torque Control -"Split and Seek" (PTC-SS)

The advantages of us ing not onl y t he di screte real s tates o f an i nverter b ut al so t he virtual v oltage v ectors placed a ny where i n t he w hole c ontrol region of a n i nverter is Because all virtual voltage vectors are considered in the (𝛼𝛽) plane, the model of this system needs to be described in 𝛼𝛽 frame too.

Predictive model in (𝜶𝜷) frame:

Firstly, voltages in the (𝑟𝑞) rotor flux reference frame must be rewritten in (𝛼𝛽) plane.

⎣ ⎢ ⎢ ⎢ ⎡ 𝑢 𝑠𝑠1 (𝑘) 𝑢 𝑠𝑠1 (𝑘) 𝑢 𝑠𝑠2 (𝑘) 𝑢 𝑠𝑠2 (𝑘) ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎡ cos 𝜃 𝑟𝑒1 (𝑘) sin 𝜃 𝑟𝑒1 (𝑘) -sin 𝜃 𝑟𝑒1 (𝑘) cos 𝜃 𝑟𝑒1 (𝑘) cos 𝜃 𝑟𝑒2 (𝑘) sin 𝜃 𝑟𝑒2 (𝑘) -sin 𝜃 𝑟𝑒2 (𝑘) cos 𝜃 𝑟𝑒2 (𝑘)⎦ ⎥ ⎥ ⎤ ������������������� 𝕄(𝑘) � 𝑉 𝛼 (𝑘) 𝑉 𝛽 (𝑘) � (3.29)
The vol tage i n (𝛼𝛽) frame can b e cal culated f rom t hree-phased vol tage by us ing Clark's transformation.

� 𝑉 𝛼 (𝑘) 𝑉 𝛽 (𝑘) � = 𝑉 𝐷𝐶 � 2 3 ⎣ ⎢ ⎢ ⎡1 - 1 2 - 1 2 0 √3 2 - √3 2 ⎦ ⎥ ⎥ ⎤ � 𝑠 𝐴 (𝑘) 𝑠 𝐵 (𝑘) 𝑠 𝐶 (𝑘) � (3.30)
The model of system will be obtained as shown in (3.31)

⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑠𝑠1 𝑘+1 𝑖 𝑠𝑠1 𝑘+1 𝑖 𝑠𝑠2 𝑘+1 𝑖 𝑠𝑠2 𝑘+1 ⎦ ⎥ ⎥ ⎥ ⎤ = 𝔽(𝑘) ⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑠𝑠1 𝑘 𝑖 𝑠𝑠1 𝑘 𝑖 𝑠𝑠2 𝑘 𝑖 𝑠𝑠2 𝑘 ⎦ ⎥ ⎥ ⎥ ⎤ + 𝔾(𝑘)𝕄(𝑘) � 𝑉 𝛼 (𝑘) 𝑉 𝛽 (𝑘) � + ℍ(𝑘) (3.31)
The a im of t his a pproach, w ith t he s upport of SVM t echnique, i s t o f ind t he m ore exactly voltage vector at each control period. Thank to this, the performance of the system can be improved. The ripple in the states such as torque, currents will be reduced.

However, how to define these virtual vectors effectively is really a problem. In the next part of t his section, a method which i s cal led "S plit an d Seek" w ill be considered a nd described in order to solve this problem [NFL13].

"Split and Seek" approach

As mentioned above, this approach uses virtual voltage vectors to be added to the finite To ove rcome t his pr oblem, a ne w s olution i s proposed. I nstead of c onsidering all t he virtual vectors in each control cycle, the searching process will be divided into two steps:

+ The first step will look for the angle, around one chosen "real" vector;

+ The second step will look for the magnitude of the vector.

Angle seeking

This idea is based on the fact DPTC strategy with only eight real vectors can operate the system. Only one problem with DPTC is that such a real vector with large magnitude can be repeated for several sampling periods or jumps directly to another one with an angle step of Here, f rom the ba sic ve ctor f ound i n t he pr evious s earch, a gr oup of vi rtual vol tage vector defined i n two a djacent s ectors (l eft a nd rig ht s ide o f th e b asic v ector) w ill b e considered. I f we d efine t he d ifference o f t he angle be tween t wo consecutive vectors Δ𝛼 = 10 𝑙 , we will obtain a group of ten voltage vector (as shown in Figure 3.23). These vectors h ave t he sam e magnitude, equal t o t he 𝑉 𝑙𝑙𝑥 . We w ill c ontinue e valuating t hese vectors through the c ost function and w e w ill c hoose t he one w ith minimum value of 𝑔.

This vector will give us the best value of angle.

With t his process, t he number of vi rtual ve ctors ne ed t o be considered is sixteen (six basic vectors and ten virtual around the basic one).

This is the end of step 2. Through this step, we can eliminate the problem in which a reference voltage vector can jump a large angle (𝜋/3) between two sampling period. Now, we continue considering a step 3 with magnitude seeking.

Magnitude seeking

After positioning the best voltage vector by angle seeking process, we continue looking for t he be st magnitude of t his ve ctor i n or der to reduce t he o vershoot w hich c an happen when a maximum voltage vector is applied to the machine.

Therefore, the voltage vector found in the angle seeking process will be passed through the opt imal magnitude s earching algorithm. I n this pr ocess, a gr oup of vol tage ve ctors which a re r epresented by t he opt imal a ngle, will be va ried i n t heir magnitude.

Simulation results for PTC-SS

In order to verify the ability as well as the improvement of this solution compared with the D PTC, the sam e t est i s c arried o ut w ith t he PTC-SS. T he machine i n Table 2.6

continues to be used. The testing conditions are repeated, the reference speed and loads of two machines are represented as in Section 3.4.2.

In t his simulation, t he two s ampling t imes a pplied f or s peed c ontrol a nd pr edictive control ar e k ept as t he sam e as t he case o f DPTC. A ll simulation results ar e sh own i n Besides, t he sp eed r esponse o f t wo machines a t st andstill is al so improved. T here i s n o ripple compared with DPTC (Figure 3.25a). In addition, through Figure 3.25(b) and (c), i t is easy to realize that the ripple in torque and current is reduced considerably. The torque ripple is approximate 0.

3(𝑁𝑁).

A comparison about the spectrum and the THD of one phase current of two machines in bot h pr edictive s olutions will be c arried o ut i n t he f ollowing pa rts to e valuate t he performance of proposed algorithm.

A comparison with DPTC a) About the losses:

According t o t he e lectromagnetic torque e quation of P MSM, t he t orque i s di rectly proportional t o the q -current. I t m eans that t he d -current doe s not play a ny r ole in producing t he t orque a nd s o i s not t ransformed t o mechanical e nergy. I t w ill c ause t he losses in the resistance of winding of PMSM.

The total losses of two operating machines are calculated by the following equation:

𝑊 𝑙𝑙𝑠𝑠 = 𝑅 𝑠 ��𝑖 𝑠𝑠1 2 + 𝑖 𝑠𝑠2 2 �Δ𝑡 𝑙 𝑠𝑖𝑚 𝑙=0 (3.33)
The value of 𝑊 𝑙𝑙𝑠𝑠 is shown in Table 3. 4. Losses are decreased considerable thank to the new solution. At three range of speed, the losses of PTC-SS are just a h alf of this in DPTC. I t c an be r ecognized that t he s maller r ipple i n t he c urrent br ings to t his improvement.

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

In the next point, the spectrum and THD factor of phase current will be studied.

b) Spectrum and the THD of one-phase current:

The simulation will be carried out with both methods under several reference speeds:

𝜔 𝑟𝑒𝑟 = � 𝜔 1 = 18 × 𝜋 (𝑟𝑟𝑟. 𝑠 -1 ) 𝜔 2 = 25 × 𝜋 (𝑟𝑟𝑟. 𝑠 -1 ) 𝜔 3 = 32 × 𝜋 (𝑟𝑟𝑟. 𝑠 -1 ) (3.34)
with the different load conditions: 𝑇 𝑙𝑙𝑙𝑠1 = 2.5(𝑁𝑁) and 𝑇 𝑙𝑙𝑙𝑠2 = 5(𝑁𝑁).

The f undamental f requencies w hich ar e co rrelative w ith each r eference sp eed ar e calculated by:

𝑓 𝑟𝑢𝑛𝑠 = 𝑛 𝑝 × 𝜔 𝑟𝑒𝑟 2 × 𝜋 = � 𝑓 1 = 27 (𝐻𝑧) 𝑓 2 = 37.5 (𝐻𝑧) 𝑓 3 = 48 (𝐻𝑧) (3.35)
The sampling time for predictive controller is set at 𝑇 𝑝𝑐 = 5 × 10 -5 (𝑠).

In The THD values corresponding to each considered case ar e also summarized in Table 3.5. This value defines the level of harmonic content in alternating signal. The smaller this value is the better the quality is. For current harmonics, the equation of THD is:

where 𝐼 1 represents f or t he f undamental c omponent, 𝐼 ℎ is s tand f or t he hi gher or der harmonic components.

It can be seen that in DPTC, the THD is very high (>20%). Meanwhile, PTC-SS gives us t he sm all T HD v alues ( <1%). I t means that t he qua lity of s ystem under P TC-SS approach is improved. The ripple is reduced respectively. Through t his s ection, a n ew approach to the pr edictive control na med a s PTC-SS is presented a nd t ested with t he m ono-inverter dua l-parallel P MSM dr ive s ystem. T he simulation r esults ha ve s hown t hat PTC-SS applied in th is work has p ositive ef fect. T he searching algorithm doe s not increase m uch t he c omputation e ffort but gi ves t he much better results. The ripple of system's variables is reduced considerably. Consequently, the losses represented in term (𝑅 𝑠 𝑖 𝑠𝑠 2 ) are also reduced.

Conclusions

In t his c hapter, t he pr edictive t orque c ontrol f or mono-inverter dua l-parallel P MSM drive system is presented. Two approaches have been studied and applied to this system.

After the theoretical analyses, simulations are carried out to verify the performances of two approaches.

The si mulation r esults have i nitially sh own t hat p redictive t orque co ntrol can b e applied to this drive system. Just using an inverter to operate two PMSMs in parallel, the two methods are simple in design and give the good performances.

Under the simulation environment, results are correct and satisfactory. The system can operate stable in many ranges of speed as well as under the load change. However, with a real system, there are many issues which need to be considered. The inverter is not ideal any more. The computation time also has an effect to the performance of the system. The error in m easurement pr ocess a nd so on. Therefore, it is very in teresting to v erify th e algorithm with a real system. All the equipments are installed and organized in the following way:

• Motor coupling system: composed of two identical PMSMs with a resolver already installed. Resolver coils are available from the motor through a 12-pin connector in the motor housing.

• Measurement system:

-Hall effect sensors type LEM LA 25-S for current measurements;

-Resolver-to-digital converter that measures the position of the rotor;

-A multi-meter displays the DC voltage bus.

• Power supply system:

-An auto-transformer to control the electrical energy from EDF net work;

-A voltage source that provides DC voltage bus for the inverter;

-A three leg switch-mode inverter that uses IGBT switches.

• Control system: This experimental bench has been developed in laboratory LAPLACE with the aim of creating a system r epresenting t he hypothetical displacement of t wo flaps of an air plane [START_REF] Bidart | Commande c oopérative de s s ystèmes m onoconvertissmono multimachines synchrones[END_REF]. In mechanical part, the parallel structure of two PMSMs will be set up. Detailed descriptions of e ach pa rt of t he w hole s ystem will be di scussed i n t he ne xt pa rts of t his chapter.

Motor coupling system

This system contains two permanent magnet synchronous machines, which are denoted The experimental platform Besides, f or t he be st r epresentation of motion of t he f laps, t he a xes c hosen ha ve a maximum 1 meter length and the speed of movement of the plates is limited to 1𝑁. 𝑠 -1 . At the terminals of axis1 and axis2, there are latches. If a plate passes through the limit of the axis, the latches will send a signal to control the card to stop the system immediately. The used s ensors a re i nductive t yped whose pr inciple of operation an d ch aracteristics ar e described in Appendix B.5. 3 The system also has two resolvers which are mounted on SM1 and SM2. They are used to m easure the a bsolute pos ition of t he r otors. In Figure 4.2, t he gr ey cable i s us ed t o transfer the information of the position, and the orange is the power cable. The experimental platform

Power supply system

The power supply system is represented in Figure 4. 4. It includes the EDF network, a three-phase v ariable auto-transformer an d a p ower co nverter w hich i s co mposed o f a rectifier, a filter and a three-leg IGBT inverter.

Power supply system

The power supplied by EDF network is three-phase, 400𝑉/50𝐻𝑧. In order to adjust the value of the DC voltage (𝑉 𝐷𝐶 ) imposed by the continuous DC bus, and to avoid a suddenly change of t he vol tage on t he c apacitors dur ing t he s tartup, a t hree-phase v ariable au totransformer (400/450/20 AUTC of Automelec) is used. In this way, the voltage applied to the terminals of the rectifier can be adjusted. In addition, thanks to its inductive nature, the current returning to EDF network will be thus smoother.

The characteristics of the auto-transformer are given in Appendix 0. To co mpensate an y o ffset v alue i n t he measurement p rocess ( due t o sensors o r C AN bus), a procedure for initializing the values of current to zero is carried out. Thus, while the electric po wer i s not ye t applied t o the s ystem, a n a verage va lue of t he current measurement is calculated through a large number of sampling periods. The four measured average values are then stored and the obtained offset values are then subtracted from the measured current.

In addition, as will be explained in the following paragraph, in the step of initialization, the absolute values of position of the two machines are measured.

Mechanical position measurement

As d escribed i n Appendix B. and 𝜃 2𝑛𝑢𝑙 , read through the absolute position encoders.

In the next part, these positions of two motors will be used to calculate their speeds.

Speed calculation

The rotational speeds of the two machines SM1 and SM2, respectively denoted as 𝜔 𝑙1

and 𝜔 𝑙2 , can be calculated from the position 𝜃 𝑙1 and 𝜃 𝑙2 . and us ing t he ba ndpass frequency 𝜔 0 = 500(𝑟𝑟𝑟/𝑠), we obtain:

𝜔 𝑙 = 𝑟𝜃 𝑟𝑡 � 𝑠𝑙𝑔𝑙𝑙𝑙𝑙 ≈ Δ𝜃 𝑡 𝑘+1 -𝑡 𝑘 = Δ𝜃 𝑇 𝑠 (4.2)
𝐿(𝑧) = 𝑇 𝑠 𝜔 0 (𝑧 + 1) 𝑇 𝑒 𝜔 0 (𝑧 + 1) + 2(𝑧 -1) = 250(𝑧 + 1) 250(𝑧 + 1) + (𝑧 -1) (4.4)
The experimental platform An e xample of t he s peed o btained with a nd w ithout this filter at s teady s tate 𝜔 𝑟𝑒𝑟 = 50𝑟𝑟𝑟. 𝑠 -1 is presented in Figure 4.6.

dSPACE system

dSPACE system is a tool used to build real-time control systems. It is used to connect different ki nd of s ignals ( analogue or di gital) t hat m easurement de vices obt ain t o a computer, as well as send signals send signals from the computer to the device in order to control them. dSPACE system is an interface between the physical world (motors, inverter, measure sensors … ) and t he co mputer, f rom where t he l ab t ests i s co ntrolled an d t he r esults are displayed. dS PACE s ystem model used i n t his t hesis i s C P1103 s ystem w ith di fferent

MATLAB/Simulink and Control-desk

The dSPACE programming has been carried out using MATLAB/Simulink. The model is e qual to the one us ed i n c hapter 3 f or t he simulation w ith t he e xception of s ome modifications:

-A measurement block is added for receiving the DC voltage, currents and positions.

In this block, the initialization procedure is also done;

The experimental platform -The model of the inverter and two PMSM is removed. They are replaced by the real inverter and motor coupling system;

-The S VM bl ock i s c hanged f or t he SVM t hree-phase generation i ncluded in t he RTI library in Simulink;

-The controller blocks remain in the system.

The sy stem has b een o rganized i n t wo main p arts: measurements a nd c ontroller. The measurement bl ock ne eds t o be s ynchronized by a P WM interrupt b lock. T he i nterrupt signal w ill be ge nerated a t t he m iddle of a modulation pe riod. W ith a S VM s ymmetric, middle point of a modulation period will be zero voltage vector. There is no state switch in any I GBT aw w ell as t he st ate o f system i s t he most st able. T herefore, the measurement will give the most exactly values.

This process is illustrated in Figure 4.7.

After a ll the pr ogramming i s c omplied a nd do wnloaded t o dS PACE c ard, a C ontrol-Desk i nterface s oftware pr ogram pr ovides al t he ne cessary f unctions f or c ontrol, da ta acquisition and modification of the parameters of Simulink blocks in real time.

Digital control implementation

The used Simulink structure comprises several tasks: + Performing the measurement of variables (currents, positions …) and calculates the speeds of the two machines;

+ Regulating these parameters in order to achieves the autopilot of the two machines. The experimental platform

In t his section, t he c ommand s tructure us ed f or t he c ontrol of t he va riables of bot h PMSMs and the parameters selected for the numerical control are presented.

Speed controller

The speed controller as presented in previous chapter is RST type. In this thesis, antiwindup function is also added to the controller. The parameters of this type of controller are calculated base on the model of object in Z-domain.

During initialization process, the unit delay block always takes a saturation value from a saturation block. Therefore, it is necessary to reset the state of this block at each starting time to zero (Figure 4.8).

The parameters of this controller are summarized in Table 4.1

Predictive torque controller

In this work, PTC-SS algorithm has been verified in the real system. The controller will update t he measured v ariables ( currents, pos itions …) t o a pply t o t he di screte m odel of mono-inverter dua l-parallel P MSMs, t o c alculate t he e volution of s ystem i n t he ne xt sampling period. Besides, the two references from the two RST speed controllers are also used to build the cost function.

In addition, to meet the requirement of fast response, the sampling time used for this controller is s et u p a t 𝑇 𝑙𝑐 = 10 -4 𝑠. It is about tw enty times f aster c ompared w ith th e electrical time constant of the motor:

𝜏 𝑒 = 𝐿 𝑠 𝑅 𝑠 ≈ 1.8 × 10 -3 (𝑠) (4.5
)

𝑟 0 + 𝑟 1 1 𝑠 0 𝑠 1 . 𝑧 -1 𝑟 0 + 𝑟 1 . 𝑧 -1

Reset the state of unit delay

The experimental platform

The core of t his controller i s an S-function. This function will c arry out t he " Split & Seek" s trategy t o opt imize the c ost f unction to f ind th e suitable v oltage v ector f or th e operation of two machines. According to this S-function, six basic vectors (𝑉 1…6 ) placed at 𝜋/3 intervals in space vector will be considered first. Next, a group of ten virtual vectors placed at 𝜋/18 intervals around the chosen basic vector will be created and tested.

In this experiment, the DC voltage is set up at value 𝑉 𝐷𝐶 = 50𝑉. The maximum voltage which can be modulated by SVM technique is:

𝑉 𝑙𝑙𝑥 = 𝑉 𝐷𝐶 √3 = 50 √3 ≈ 29(𝑉) (4.6)
Therefore, t he a mplitude o f te sted v oltage v ectors w ill b e d iscrete with th e P TC-SS algorithm, and set to a 5𝑉 step from 0 to 𝑉 𝑙𝑙𝑥 .

Experiment's results

Test with one machine

In a f irst ti me, P TC-SS wi ll b e v erified w ith o ne P MSM. T he u sed machine h ere i s SM1,since it is possible to vary its load.

Diagram of control drive is shown in Figure 4.9.

Parameters Value

Sampling time 𝑇 𝑠𝑐 10 -3 (𝑠)

Damper factor 𝜉 0.95

Natural frequency 𝜔 𝑛 120(𝑟𝑟𝑟. 𝑠 -1 ) The experimental platform

𝑠 0 1 𝑠 1 1 𝑟 0 0.04 𝑟 1 -0.0376
The value of torque is updated by user through a human-machine interface to dSPACE.

Control signal then will be sent to a variable speed drive which controls the torque of SM3.

Speed response of motor is presented in Figure 4.11. This result shows that the motor follows the set-point and also satisfies the change of set-point. When motor goes to the end of the first half of road, it is reversed. Because of design of the system, the motor operates like starting. We can see the same phenomenon happened at two moments: the starting and at 𝑡 = 5(𝑠).

There are some peaks in speed of motor during the test. It is because of the change in load applied to the machine. Besides, it can be seen that under variation of load, system is still stable. Steady state is reestablished in a short time (~0.2s). The experimental platform

In Figure 4.12, current of PMSM in q-axis 𝑖 𝑠𝑠 and its reference value 𝑖 𝑠𝑠 𝑟𝑒𝑟 are plotted.

With considered PMSM, this current is proportional directly with electromagnetic torque.

It can be seen that under the changes of set-point as well as the variation of load, the value of 𝑖 𝑠𝑠 follows its reference accurately.

In Figure 4.13, the d-current 𝑖 𝑠𝑠 of PMSM is shown. Its value oscillates around zero.

The amplitude of ripple is around 0.4A.

It is clear that the predictive controller operates quite well with PTC-SS algorithm for only one machine. The cost function considers two criteria 𝑖 𝑠𝑠 and 𝑖 𝑠𝑠 as follows:

As results, both components are well controlled. Next, the test thus will be continued with two machines in parallel.

Test with two machines in parallel

In this test, information of both machines is considered in only one cost function:

Position and phase-currents of two PMSMs are updated to PTC-SS control block every sampling t ime. These va riables will be us ed t o predict the evolution of bot h machines i n the next period. Diagram of control drive is shown

𝑔 = �𝑖 𝑠𝑠1 𝑟𝑒𝑟 -𝑖 𝑠𝑠1 (𝑘 + 1)� 2 + �𝑖 𝑠𝑠2 𝑟𝑒𝑟 -𝑖 𝑠𝑠2 (𝑘 + 1)� 2 + �𝑖 𝑠𝑠1 (𝑘 + 1)� 2 +�𝑖 𝑠𝑠2 (𝑘 + 1)� 2 (4.8)
The experimental platform From Figure 4.14, t he l oad a pplied t o S M2 i s 𝑇 𝑟2 while t he l oad a pplied t o S M1 i s

𝑇 𝑟1 + 𝑇 𝑟3 .
In th is test, to rque f rom S M3 is s et e qual to zero (𝑇 𝑒𝑥𝑙 ≈ 0𝑁𝑁). T herefore, loads between two machines are always slightly different. The test is carried out under the same set-point 𝜔 𝑟𝑒𝑟 = 20(𝑟𝑟𝑟. 𝑠 -1 ). Two machines will be reversed at 𝑡 = 5𝑠.

The sp eed r esponse i s p resented i n Figure 4.15. A ccording to t his, t he speed of bot h

PMSMs follows the reference. Response during transient time is not really good. There are some ove rshoots dur ing t his pe riod of t ime. Although t here i s also the os cillation, t he steady state is still obtained.

In Figure 4.16, q -current o f b oth m achines an d i ts r eference ar e sh own. T here is a difference b etween t he real v alue f rom t he motor an d r eference v alue. T heoretically, qcurrent determines the torque of the machine and so the speed indirectly. In this case, the real value is not equal to reference but the speed of two machines still comes to the set- Results presented in Figure 4.16 and Figure 4.17 show the limitations of the PTC-SS method. S ince bot h m achines a re c onsidered in t he c ost f unction t hat m eans t he opt imal voltage v ector f ound by t he a lgorithm s hould s atisfy t he requirements of bot h machines. This is, in general case, difficult, since requirements can be different due to the different load t orque applied t o the m achines. T his i s s howed by t he f act t hat the c ost f unction is minimized but not equal to zero.

The optimal vector found by t he algorithm will be then a compromise of requirements of both machines. This will lead to d-currents different from zero (reference value) and qcurrent w ith va lues c orresponding to t he va lues w hich w ill a llow p roviding the c orrect torque.

In fact, when considering the information of both machines, it is sure that the amount of c omputation w ill be i ncreased c onsiderable c ompared w ith t he c ase of onl y one machine. T his i s a r eally p roblem with r eal time a pplication. T herefore, P TC-SS will b e tested with master/slave structure.

Test with two machines in parallel under master/slave configuration

In this test, master/slave structure is reused but instead of using classical PI controller, predictive controller with "Split and Seek" algorithm is applied. Diagram of control drive is shown in Figure 4.18. According to this structure, there is always just one machine controlled at a moment.

Therefore, the amount of computation will be the same with the case of one machine and will be smaller than that in the case of both machines. The experimental platform

In t his test, t he l oad c ondition is the s ame a s i n pr evious t est. T herefore, t he f irst machine S M1 al ways has h eavier load. I t w ill b e t hen ch osen as m aster m achine an d i ts requirements will be considered in the cost function.

In Figure 4.19, speed response of two PMSMs and reference value are plotted.

According t o Figure 4.19, t he speed r esponse of bot h m achines i s better w ith no overshoot during the transient time (<5% set-point). However, the oscillation still exists at steady state. It might be a result of using discrete control signal in control algorithm.

The dq -currents o f b oth m achines ar e r epresented in Figure 4.20 and Figure 4.21.

Because of being the master one and controlled, the responses of SM2 is quite good. We can see t hat q -current o f S M1 al ways f ollows t he r eference; meanwhile, i ts d -current is kept close to zero.

However, unl ike the S M1, t he slave m achine (SM2) d oes n ot f ollow t he references.

The d-current value is different from zero.

Conclusions

In t his c hapter, p redictive c ontrol with " Split and S eek" a lgorithm ha s be en ve rified with a real experimental bench under three tests:

• Control one PMSM;

• Control two PMSMs in parallel;

• Control two PMSMs in parallel under master/slave configuration.

From the obtained results, some conclusions can be pointed out:

• The oscillation and ripple always exist because of the controller just considered the limit number of discrete control signal. However, in all test, steady state of whole system is obtained.

• In t he dr ive s ystem w ith one P MSM, P TC-SS gi ve us t he i nteresting results. T he system responses very fast (response time ≈ 0.05𝑠) under the changes of exterior condition s uch a s s et po int o r l oads. I t is also stable w ith the va ry of load. T he steady state will be reestablished after ≈ 0.05𝑠;

• For a t est with both machines (no master/slave configuration), the system also can obtain desired steady state. However, the performance is just acceptable. In fact, a large amount of c omputation a ffected t o t he qua lity of s ystem. B esides, it i s a lso difficult to guaranty th e s tability o f th e s ystem if th e lo ads o f tw o machines a re much different.

• Considering P TC-SS w ith m aster/slave st ructure, t here ar e so me ad vantages. T he amount of computation is reduced considerably. Besides, the stability of system can be assured based on the choice of master machine.

In conclusion, the initial results show that predictive control strategy can be developed and adapts to this drive system.

Conclusions and Perspectives

The principle objective of this thesis is applying predictive control strategy to a drive system w hich c onsists of t wo pe rmanent m agnet s ynchronous machines ( PMSM) connecting in parallel with a three-phase two-level inverter.

Related to t he co ntrol of multi P MSM sy stem, in C hapter 1, t he e volution of P MSM with m ulti-machine sy stems, esp ecially i n a eronautic f ields, w as p resented. Several structures of multi-machine systems has been introduced and compared. Among them, the parallel structure composed of one inverter and several PMSMs showed many advantages.

This structure thus is selected to be studied in this thesis.

In c hapter 2 , t he m athematic model o f P MSM, stability analyses a s w ell as s tudies about t he r esponse of P MSM unde r vol tage s upplying m ode a re c onsidered i n de tailed.

Typical control methods for this kind of drive system are also introduced. Moreover, some works in th e li terature with a m ono-inverter dual-parallel PMSM w ere i ntroduced. A control structure which uses Master/Slave configuration gave us a vey good performance for the master machine. Besides, thanks to the master-choice technique, the stability of the slave machine is also guaranteed.

Another approach is applying Direct Torque Control strategy to this system. With this method, a n ew opt imal l ook up t able i s bui lt in or der to m eet t he r equirements of bot h machines, even in the case of different load.

Model P redictive C ontrol ( MPC) theory has been us ed in m any f ield of s cience.

Recently, this c ontrol m ethod a lso a pplied more a nd m ore in va riable s peed a pplications where power electronic devices are used a lot. It gradually becomes competitive to the well known c ontrols as: F ield O riented C ontrol (FOC), D irect T orque C ontrol ( DTC).

Therefore, in Chapter 3, Predictive Control strategy is paid attention to apply to the monoinverter dual-parallel PMSM drive system.

NGUYEN Ngoc Linh

In t his t hesis, a uthor h as c oncentrated on F inite S et M odel P redictive C ontrol (FS-MPC), w hich m eets v ery w ell d iscrete n ature o f p ower co nverters. In v iew o f sw itching frequency, the FS-MPC methods are considered in two approaches:

• Variable S witching F requency (VSF) -Section 3.4:Direct P redictive T orque Control (DPTC);

• Constant S witching F requency (CSF) -Section 3.5: P redictive T orque C ontrol -"Split and Seek" (PTC-SS).

Both approaches are described in detail in Chapter 3 of this thesis. Several theoretical analyses and simulation results have been carried out in order to verify the effect of each approach. I n ge neral, they gi ve t he s atisfactory r esults w hich pr ove the c apability of controlling our ki nd of dr ive s ystem with pr edictive c ontrol. T aking into consideration advantages of constant switching frequency for converter operation, approach with the use of Virtual Voltage Vector in PTC-SS has shown some advantages compare with DPTC:

• lower line current THD factor;

• requiring lower sampling frequency (as DPTC with the same performance); Therefore, t his ap proach i s ch osen to t est w ith a r eal d rive system i n Chapter 4. T he first experiment r esults a lso s how that t he dr ive s ystem c an ope rate with t his m ethod.

However, t hese r esults are not r eally a s good a s i n s imulation. I t i s r easonable b ecause there are many conditions which are idealized in simulation such as: no computation time, all r esponses a re i mmediately, no error in c omputation, no noi se … T herefore, i t i s necessary to have more studies to improve the performance of the system.

In the author's opinion, although the obtained results are not as well as desired, PTC-SS is still a new approach for predictive control strategy.

• First, it is necessary to find solutions in order to optimize the research of the best voltage v ector a s w ell as im proving th e calculation time. It i s possible using a dynamic s tep s ize i n a ngle a nd m agnitude i nstead of us ing t he f ixed s teps a s i n Section 3.5. It can be carried out based on the following principle: when the system is far from the steady state, the step needs to be large in order to cover all the space vector. When the system is close to the steady state, the step will be small to narrow the r ange of c onsideration. A s r esults, t he vol tage ve ctor w ill ge t t he m ore exact value;

• Moreover, with t wo m achines, w hen c onsidering t he s tates of bot h m achines, t he computational efforts show the bad effect to performance of the system. When the number o f m achines i s i ncreased more, it really becomes a ser ious problem.

Besides, t he i ssues o f i ncluding the cr iteria o f each m achine w ill m ake t he co st NGUYEN Ngoc Linh function become complex and bulky. Sometimes, a criterion may be not suitable or does not play any role in control problem. Therefore, it is necessary to extend this work to control more than two machines. NGUYEN Ngoc Linh

Résumé en français

Système multi-machines dans le domaine aéronautique

Dans le cadre de l'avion plus électrique (ou More Electric Aircraft -MEA), la tendance est à l 'utilisation de machines de plus en plus électriques, que ce so it pour les actionneurs de c ommande de vol ou pour l es a utres équipements tels q ue l e t rain d 'atterrissage, l es freins, les pompes à essen ce, les systèmes de dégivrage, les systèmes de climatisation, ou encore les systèmes de verrouillage. Comme représenté sur la Figure 2, la MSAP peut être utilisée dans de nombreux systèmes de l'avion. Les systèmes de contrôle de vol sont présentés dans Figure 3, ils se composent de deux parties d'actionnement:

• La partie primaire est l'élément essentiel de chaque vol., Il contrôle l'évolution de l'aéronef selon trois axes: le roulis (par les ailerons sur les bords de fuite des ailes), le tangage (par les gouvernes de profondeur sur le bord de fuite de l'empennage), le lacet (par la gouverne de direction).

• La pa rtie s econdaire r eprésente l a c onfiguration a érodynamique de l 'avion l ors de cas peuvent être synthétisés à partir de quatre états de deux machines (Table 2).

Selon l e D TC, c haque pa ire c ouple-flux doit ê tre corrélée à u n vecteur tension.

Toutefois, le choix de ce vecteur dépend aussi de la position spatiale du vecteur flux. Sous Un schéma de l'algorithme de commande est représenté dans la Figure 11.

La recherche de l'amplitude

Après a voir défini un v ecteur d e t ension ap proprié p ar le processus d e r echerche de l'angle, nous continuons à moduler l'amplitude de ce vecteur afin de minimiser encore, si possible, la fonction coût.

Par conséquent, le vecteur de tension élu par le processus de recherche de l'angle sera passé par la recherche d'une amplitude optimale. Dans ce processus, un groupe de vecteurs

Résultats expérimentaux

L'algorithme P TC-SS est c hoisi pour l'appliquer à u n sy stème r éel selon t rois configurations: avec une machine; avec deux machines en parallèle; avec deux machines en parallèle en config maitre/esclave.

Test avec une machine

Le schéma du mécanisme de commande est représenté sur la Figure 17. La réponse en vitesse de la machine est représentée sur la Figure 18 Dans ce test, la fonction de coût est définie comme suit:

𝑔 = �𝑖 𝑠𝑠 𝑟𝑒𝑟 -𝑖 𝑠𝑠 (𝑘 + 1)� 2 + (𝑖 𝑠𝑠 ) 2
Par conséquent, le courant d-q de la machine est considéré sur la Figure 19.

Test avec deux machines en parallèle

Dans ce t est, l es i nformations d es d eux machines so nt c onsidérées d ans u ne seu le fonction de coût:

La position et l es c ourants de pha se de de ux M SAPs s ont mis à j our dans l e bloc de contrôle P TC-SS c haque t emps d 'échantillonnage. C es va riables ser ont u tilisées p our prédire l'évolution des deux machines dans la prochaine période. Le schéma du mécanisme de commande est présenté sur la Figure 20.

� 𝑉 𝛼 𝑉 𝛽 � = � 2 3 ⎣ ⎢ ⎢ ⎡1 - 1 2 - 1 2 0 √3 2 - √3 2 ⎦ ⎥ ⎥ ⎤ � 𝑉 𝐴 𝑉 𝐵 𝑉 𝐶 � (A.3)
Consequently, the magnitude and angle will be obtained: From the , the switching time duration can be calculated as follows:

|𝑉 𝑟𝑒𝑟 | = �𝑉 𝛼 2 + 𝑉 𝛽 2 𝛼 = tan -1 � 𝑉 𝛽 𝑉 𝛼 � (A.4)
"FMINCON" and SVM technique

𝑡 1 = √3𝑇 𝑝𝑢𝑙𝑠𝑒 �𝑉 𝑟𝑒𝑟 � 𝑉 𝐷𝐶 �sin � 𝜋 3 -𝛼 + 𝑛 -1 3 𝜋�� 𝑡 2 = √3𝑇 𝑝𝑢𝑙𝑠𝑒 �𝑉 𝑟𝑒𝑟 � 𝑉 𝐷𝐶 �sin �𝛼 - 𝑛 -1 3 𝜋�� 𝑡 0/7 = 𝑇 𝑝𝑢𝑙𝑠𝑒 -𝑡 1 -𝑡 2 (A.5)
where 𝑛 = 1 … 6 which is represented to sector 1 to 6, 0 ≤ 𝛼 ≤ 60 𝑙 . 1

𝑠 𝐴 = 𝑡 1 + 𝑡 2 + 𝑡 0 2 � 𝑠 𝐵 = 𝑡 2 + 𝑡 0 2 � 𝑠 𝐶 = 𝑡 0 2 � 2 𝑠 𝐴 = 𝑡 1 + 𝑡 0 2 � 𝑠 𝐵 = 𝑡 1 + 𝑡 2 + 𝑡 0 2 � 𝑠 𝐶 = 𝑡 0 2 � 3 𝑠 𝐴 = 𝑡 0 2 � 𝑠 𝐵 = 𝑡 1 + 𝑡 2 + 𝑡 0 2 � 𝑠 𝐶 = 𝑡 2 + 𝑡 0 2 � 4 𝑠 𝐴 = 𝑡 0 2 � 𝑠 𝐵 = 𝑡 1 + 𝑡 0 2 � 𝑠 𝐶 = 𝑡 1 + 𝑡 2 + 𝑡 0 2 � 5 𝑠 𝐴 = 𝑡 2 + 𝑡 0 2 � 𝑠 𝐵 = 𝑡 0 2 � 𝑠 𝐶 = 𝑡 1 + 𝑡 2 + 𝑡 0 2 � 6 𝑠 𝐴 = 𝑡 1 + 𝑡 2 + 𝑡 0 2 � 𝑠 𝐵 = 𝑡 0 2 � 𝑠 𝐶 = 𝑡 1 + 𝑡 0 2 �
The SEMIKRON device

B.1 The auto-transformer

The a uto-transformer us ed i s A TC 400/ 450/20 w hich is a pr oduct o f A utomelec society. This is a variable three-phase autotransformer, whose purpose is to:

-Adjust the value of the DC voltage 𝑉 𝐷𝐶 imposed on the DC bus;

-Avoid a sudden change in voltage on the capacitors during startup.

In a ddition, t hanks t o i ts i nductive na ture, the auto-transformer s moothes t he cu rrent which is returned to the EDF network.

The characteristics of this device are given in Table B.1

A th ree-phase vol tage, whose a mplitude i s c ontrolled by a c ontrol kno b, c an t hen be supplied to the rectifier device of SEMIKRON.

B.2 The SEMIKRON device B.2.1 The rectifier

The rectifier is a three-phase diode rectifier type. In order to eliminate all harmonics, this re ctified v oltage is f iltered i mmediately b efore p utting into th e in verter by tw o capacitors 1𝜇𝐹. The inverter is powered by a DC voltage ranging from 0 to 400V. The PMSM

B.2.2 The inverter

Technical data of the experimental

In a ddition, t he pow er m odules t hat c ontain t he I GBTs a re pr otected a gainst s urges through r esistance 𝑅 𝑔 . S ettling s peed of th e c urrent is th en li mited, w hich s ignificantly reduces surges induced by parasitic inductances due to the cables.

The D C vol tage bu s is then transformed i nto a s ystem of three-phase vol tage 𝑉 𝑙,𝑏,𝑐 which is applied to both permanent magnet synchronous machines.

B.3 The PMSM

The t wo P MSM us ed, de noted a s S M1 a nd S M2 i n t his t hesis, is a model of F esto manufacturer (motor MTR-AC-70-3S-AB).

These models a re c omposed of a s ynchronous m otor, a br ake a nd a ho lding pos ition sensor encoder type resolver.

B.3.1 The motor

It i s a pe rmanent m agnet s ynchronous m achine c omprising a ro tor a nd a s tator. The characteristics of this motor is given by the manufacturer and shown in Table B.2

B.3.2 The holding brake

On machines "MTR-AC-70-3S-AB", a failsafe brake holding current spring is built.

To unlock the brake, a DC voltage 𝑈 𝑏𝑟 whose value is given in Table B.3 must be applied to the terminals of the electromagnet. This voltage, which is identical for the two motors, is created by a power supply and applied to the two brakes of the desired motor rotation.

B.4 Mechanical properties B.4.1 Load machine

Load m achine S M3 i s the "MT R-AC-100-3S-AB" motor t ype f rom Festo c ompany.

This is a permanent magnet synchronous machine which is chosen from the series models like M S1 a nd M S2. B esides, a br ake a nd a hol ding pos ition s ensor r esolver a re t hus included in the model. The main features of the machine are given in Table B. 4.

This machine is controlled by current (and thus torque) to produce an extra torque 𝑇 𝑒𝑥𝑙 to SM1. The current control of SM3 is carried out by an industrial variable speed drive.

B.4.2 The SEC-AC-305 (Smart Electromotor Controller)

The us ed c ontroller, shown i n All technical specifications of this device is summary in • Guides on these DGEs provides superior load and moment capacity;

Technical data of the experimental

• Use when medium to high loads need to be moved;

• Wide range of options for third-party motor mounting kits

• Comprehensive range of mounting accessories for multi-axis combinations.

The c omponents of t his a xis a re s hown i n oscillates, the sensor is considered "active" and delivers a DC voltage (𝑓 𝑠𝑐 = 1). When the metal plate enters this field, it is disturbed and then reduced. The attenuation of this field then di sables t he s ensor. N o vol tage pr oduces ( 𝑓 𝑠𝑐 = 0). F our di gital i nput a re t hen reserved for t hese sensors. A l ogic ci rcuit allows the power delivery until all sensors are activated to the high state.

B.6 The Human-Machine Interface

As explained in the thesis, the programming of the DS1103 card is carried out through the use of RTI library compatible with Matlab/SIMULINK.

After compiling the program to this card, ControlDesk interface software will be use to build a human-machine interface (HMI). The HMI will provide all the necessary functions for c ontrolling, da ta a cquisition a nd modification of t he pa rameters of Simulink bl ock i n real time. The acquired data can then be saved as ".mat" data file and processed by Matlab.

A screenshot of HMI is presented in

Résumé en français

  Stabilité en boucle ouverte • Mutualisation • Machine synchrone à aimants permanents • Commande prédictive • Commande directe de couple v DEDICATION This dissertation is dedicated to my late father, NGUYEN Xuan Ngoc, and to my mother, TRUONG Thi Quynh Mai, who always had confidence in me and offered me encouragement and support in all my endeavors It is also dedicated to my darling wife, THUONG Phuong Nga, and my lovely unborn child for their care, love, understanding, and patience

  Plasma and Conversion of Energy (LAPLACE), ENSEEIHT, INP Toulouse. The work has been c arried out i n t he period f rom A pril 2010 t o A pril 2 013 unde r t he s upervision of Professor INPT Maurice FADEL and Associate Professor INPT Ana LLOR.

Finally, t wo

  c ontrol s tructures f or m ono-inverter dua l-parallel P MSM w ill b e introduced; NGUYEN Ngoc Linh • The third chapter is the main part of this thesis. It will focus on Predictive Control of two Synchronous Machines connected in parallel connected to a 2-Level 3-Phase Converter. A general d escription about t he P TC f or mono-inverter du al-parallel PMSM d rive sy stem w ill b e p resented. E ach c omponent o f t his drive system i s

Figure 1 . 1 :

 11 Figure 1.1: Multi-Machine System MS(m,n)

Figure 1 . 2 :

 12 Figure 1.2: Two main types of PMSM

Figure 1 . 3 :

 13 Figure 1.3: Examples of application of brushless motors in small power range

Figure 1 . 4 :

 14 Figure 1.4: Examples of applications of brushless motors in medium power range

Figure 1 . 5 :

 15 Figure 1.5: Examples of applications of brushless motors in high power range

Figure 1 .

 1 Figure 1.8 shows an example of secondary flight control of an airbus A300. The two lift devices (slats and flaps) allow their wings to maintain the lift at low speed.

Figure 1 . 6 Figure 1 . 7 :

 1617 Figure 1.6 The possibility of using PMSM in aeronautics[START_REF] Allano | Petits m oteurs él ectrique. T echniques de l 'ingénieur[END_REF]

Figure 1 . 8 :

 18 Figure 1.8: Wing of an airbus A300

Figure 1 . 9 :

 19 Figure 1.9: An advanced Motor Device EMA

(1. 1 )Figure 1 .

 11 Figure 1.10: Direct drive architecture for EMA [ROAR07]

Figure 1 .

 1 Figure 1.11: The classical structure multi-machine multi-inverter MS(3n,n)

Figure 1 .

 1 Figure 1.12: Diagram of voltage vectors under structure MS(3n,n)

In s hared s

  tructures, se veral m achines can h ave so me co mmon p hases. T hese p hases can be connected to the same legs of an inverter or a common point (the neutral point of a DC b us). T his i s b ecause o f t he f act t hat o ne inverter leg can g enerate an alternative voltage from the DC voltage, so a multi-phase (𝑥 legs) inverter can fed 𝑦 machines. These structures are marked as MS(x, y) [Bou95]. However, a multi-machine system may include several multi-phase inverters which are independent f rom e ach ot her. T herefore, a sy stem MS (m, n) will ha ve n y multi-phase inverters an d can b e ex pressed as MS�x n y

Figure 1 .

 1 Figure 1.14 shows t he voltage ve ctors of t hree pha ses of two m achines unde r t wo cases: rotating in the same direction and rotating in the opposite direction. + 𝑉 𝐴 2 𝑁 2 = 𝑉 𝐴 1 𝑁 1 ⇒ Ω 1 = Ω 2 (Figure 1.14(a)) + 𝑉 𝐴 2 𝑁 1 = -2𝑉 𝐴 1 𝑁 1 ⇒ Ω 1 = -Ω 2 (Figure 1.14(b))

Figure 1 . 2 ,Figure 1 .

 121 Figure 1.13: The structure using a common leg MS � 4𝑛 2 , 𝑛�

Figure 1 .

 1 16 represents the di agram of t hree-phase v oltage v ectors o f t wo motors. E ach m otor c an o perate independently [Bou95][FOM07]. PMSM with Multi-Machine Systems Because the third phase of each machine is fixed to the neutral point 𝑂, for any motor, the re lation 𝑉 𝑂𝐴 𝑖 = 0 is a lways tru e. A t th e e quilibrium o f a ll m achines, the re lation 𝑉 𝑂𝑁 𝑖 = -𝑉 𝐴 𝑖 𝑁 𝑖 is also assured.As in the case of using the common legs of the inverter, the reference voltage applied to th e inverter's le gs w ill d iffer f rom th e v oltage c alculated by t he regulator. F or t he 𝑖 𝑙ℎ machine, the reference voltage will be obtained from the relation in(1.4).

Figure 1 .Figure 1 .

 11 Figure 1.15: The structure sharing the midpoint MS(2𝑛, 𝑛)

Figure 1 .Figure 1

 11 Figure 1.17: Mixed structure MS� 3n 2 ,n�

Figure 1 .

 1 Figure 1.19, each l eg of t he i nverter is sh ared b y al l t he m achines. T herefore, t he dimension of the power electronic components of the inverter must be in accordance to the number of m achines i n p arallel. B esides, these m achines m ust b e as similar as p ossible about electrical parameters.

(a) Case 1 : 1 Figure 1 .

 111 Figure 1.18: Diagram of voltage vectors under mixed structure: MS� 3n 2 ,n�

Figure 1 . 19 :

 119 Figure 1.19: Parallel structure MS(3,n)

Figure 2 . 1 :

 21 Figure 2.1: Mono-inverter dual-parallel PMSM drive

  applications, a P MSM is u sually s upplied b y the v ariable f requency sources su ch as voltage s ource i nverter (VSI), c urrent s ource inverter ( CSI)… G eneral structure of s uch drive system is presented in Figure2.2.

Figure 2 . 2 :

 22 Figure 2.2: Structure of a PMSM drive system

Figure 2 . 3

 23 Figure 2.3 is an example of a three-phase motor with 𝑛 𝑝 pole pairs (𝑛 𝑝 = 2). It consists of 3𝑛 𝑝 pairs of windings which are spaced 2𝜋/�3 × 2𝑛 𝑝 �(𝑟𝑟𝑟).

Figure 2 . 3 :

 23 Figure 2.3: Stator windings of a three-phase PMSM 2 pole pairs

Figure 2 . 4

 24 illustrates a co nceptual cr oss-sectional v iew o f a t hree-phase, two-pole PMSM. Equation i n (2.4) is a general s tator vol tage equation and i t is true f or a ll k inds of rotating field electrical machines including the PMSM.

Figure 2 . 4 :

 24 Figure 2.4: Two-pole three-phase PMSM

Figure 2 . 6 :

 26 Figure 2.6: Equivalent circuit of a PMSM in d-q frame

Figure 2 . 7 :

 27 Figure 2.7: Evolution of T em =f(δ) with the consideration of 𝑅 𝑠

Figure 2 . 8 :

 28 Figure 2.8: Evolution of T em =f(δ) without consideration of 𝑅 𝑠

  developed in many studies[Leo01][Vas90]. In this section, the basic structure with fieldoriented c ontrol f or P MSM d rives will b e p resented f irst[START_REF] Ittrich | Vector C ontrol of T hree-Phase AC Machines[END_REF]. S ome ot her s tructures such a s D irect T orque Control (DTC)[START_REF] Vas | Sensorless Vector and Direct Torque Control[END_REF], P redictive Torque C ontrol (P TC)… w ill also be considered after that.

Figure 2 . 9 :

 29 Figure 2.9: Diagram of a three-phase two-level inverter

Figure 2 .

 2 Figure 2.10: General scheme of FOC for a PMSM

Figure 2 .

 2 Figure 2.11: General scheme of DTC for a PMSM

Figure 2 .

 2 Figure 2.12: The basic voltage vectors and six sectors

  𝜓 𝑠 = �(𝑉 -𝑅𝐼)𝑟𝑡 ≈ � 𝑉𝑟𝑡 (2.33) a) Vectors increasing torque b) Vectors decreasing torque

Figure 2 .

 2 Figure 2.14: The evolution of torque under DTC

Figure 2 .

 2 Figure 2.15: General scheme of DPTC for PMSM

Figure 2 .

 2 Figure 2.16: Parallel structure under master/slave configuration

Figure 2 .

 2 Figure 2.17: Evolution of system under master/slave configuration with different loads

Figure 2

 2 Figure 2.18.

Figure 2 .Figure 2 . 19 :

 2219 Figure 2.18: Master choice block

Figure 2 .

 2 Figure 2.20: General scheme of DTC for dual-parallel PMSM drive system

Figure 2 .

 2 21a), the flux vectors of two machines are in the same sector.The s ignals from a ll h ysteresis comparators require the reduction o f b oth th e to rque a nd flux of t wo m achines. T he s uitable vol tage ve ctor in t his case f or b oth m achines i s t he same. However, the problem will happen when the demands, which depends on the output of the hysteresis regulators of each machine, are different (Figure2.21b) or when two flux vectors are located in different sectors(Figure 2.21c).

  to a pply to t he dua l-parallel P MSM sy stem, w ill b e c reated. T he two cases w ill b e considered: a) Two stator flux vectors are in the same sector

  Figure 2.21: Different situation for selecting a suitable voltage vector

Figure 2 .

 2 Figure 2.22: Average rule

  4(𝑠): the reference speed is set at 𝜔 𝑟𝑒𝑟1 = 25𝜋(𝑟𝑟𝑟/𝑠); + From 𝑡 = 0.4(𝑠) → 0.8(𝑠): the motor is reversed and run at 𝜔 𝑟𝑒𝑟2 = -25𝜋(𝑟𝑟𝑟/ 𝑠); + From 𝑡 = 0.8(𝑠) → 1.2(𝑠): the motor is kept standstill.

Figure 2 .

 2 Figure 2.24: The evolution of reference value

Figure 2 .

 2 26b. At steady state, t heir values are correlative t o t he v alue o f t he t otal load ap plied 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑙𝑙𝑙𝑠 + 𝑓 0 𝜔 𝑙 . In Figure 2.26c, the flux linkage of both machines in (𝛼𝛽)-frame is presented. Those fluxes e volve in a c ircular trajectory a nd a re limited b y a h ysteresis b and (a mplitude ≈ 0.02𝑊𝑏).

Figure 2 .

 2 Figure 2.25: The variation of load of each machine in simulation

Figure 2 .

 2 Figure 2.26: Responses of system under DTC

Figure 3

 3 Figure 3.1: dq-current at steady state of a PMSM

Figure 3 . 2 :

 32 Figure 3.2: Load angle at steady state

  a nd o mitting t he 2 -order di fferential c omponent, a s mall de viation model of PMSM can be obtained as shown in (3.8).In (3.8), a c haracteristic m atrix 𝐴 is defined. F rom t his m atrix, w e c an b uild a f ourorder polynomial characteristic 𝑃(𝑠). The root of polynomial characteristic 𝑃(𝑠) will give us the information about the stability of a PMSM. By using a built-in function in Matlab,

Figure 3 . 4

 34 Figure 3.4 are calculated from a sp ecific steady state of PMSM (the state with 𝐼 𝑠𝑠0 = 0𝐴

Figure 3 . 4 :

 34 Figure 3.4: Map of roots of P(s)

Figure 3 . 5 :

 35 Figure 3.5: Example of limitation of stable zone of a PMSM

  -T c ontroller is b ecoming a standard and plays a crucial role [Lan98] [OG05] [LZ06]. The canonical structure of R-S-T controller was represented in Figure 3.7.This structure has two degrees of freedom: the digital filters 𝑅 and 𝑆 are designed in order to ach ieve t he d esired regulation pe rformances a nd t he di gital f ilter 𝑇 is d esigned afterwards in order to achieve the desired tracking performance. This structure thus allows obtaining d ifferent l evels of pe rformance i n t racking and r egulation. T he c ase of a controller operating on the regulation error corresponds to 𝑇 = 𝑅. Digital PID can also be represented in this form leading to particular choices of 𝑅, 𝑆 and 𝑇.

Figure 3 .

 3 Figure 3.7: The R-S-T digital controller [Lan98]

  14) Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM where 𝑝 1 , 𝑝 2 are the desired poles: � 𝑝 1 = -2𝑒 -𝜉𝜔 𝑛 𝑇 𝑠 cos �𝜔 𝑛 𝑇 𝑠 �1 -𝜉 2 � 𝑝 2 = 𝑒 -2𝜉𝜔 𝑛 𝑇 𝑠 (3.15) With the setting 𝑠 0 = 1 and 𝑠 1 = 1, the RST controller is digital equivalent form of PI controller. The structure of RST Anti-Windup controller is shown in Figure 3.8. In (3.15), 𝜉 and 𝜔 𝑛 are d amper f actor an d n atural f requency. T hese parameters w ill b e ch osen according to the desired dynamic response of the system.

  Torque Control strategy has been studied and applied in many applications with both induction machines[START_REF] Correa | P redictive T orque C ontrol f or Inverter-Fed I nduction M achines[END_REF] [RKE+10]… and permanent magnet synchronous machines[START_REF] Geyer | M odel P redictive Direct T orque C ontrol of pe rmanent m agnet s yschronous motors[END_REF] [MD11]… A t orque m agnitude c ontrol along with a c omplement criterion w ill ev aluate a co st f unction f or each sw itching state av ailable i n a t wo-level inverter. T he v oltage v ector w ith th e lo west va lue of t he c ost f unction i s s elected t o be applied in the next sampling interval.Based on the principle of Model Predictive Control (MPC)[START_REF] Bordons | Model P redictive Control[END_REF], a model which precalculates t he f uture b ehavior o f t he sy stem i s o ne o f t he cr ucial p oints o f a P redictive Control scheme. Generally, most systems to be controlled are continuous-time systems, so the description of the system should be made in time or Laplace domain. In contrast to the controlled systems, t he c ontrollers, w hich u sually are realized w ith the h elp o f digital computers, are often discrete-time controllers. For the controller design, it is in most cases sufficient to replace it by a continuous-time model, parameterize it and afterwards realize it as a d iscrete-time digital controller. Nevertheless, it is more advantageous to consider the discrete character of the controller already in the first approach of the controller design and hence t o u se a d iscrete-time sy stem d escription. T his al so i mplies t hat t he c ontrolled

Figure 3 .

 3 Figure 3.8: RST anti-windup Speed/Torque controller for a PMSM

3. 3 . 3 . 1 . 3 Figure 3 . 9 :

 331339 Figure 3.9: A standard three-phase two-level inverter

  directly b ased o n t he er ror b etween p redicted an d r eference v alues. This e rror will b e predefined based on desired performance of the system [PW05] From a bove a nalyses, two a pproaches t o pr edictive t orque c ontrol ( PTC) w ill be considered and verified. The first one based on t he principle of Direct Predictive Control and can be named as Direct Predictive Torque Control (DPTC). The second one is along with modulation technique.

Figure 3 .

 3 Figure 3.10: Two strategies for predictive control [LKKS10]

  phase currents �𝑖 𝑙𝑏𝑐1 (𝑘), 𝑖 𝑙𝑏𝑐2 (𝑘)� are updated into the controller; + Along w ith th e s et o f e ight re al s tates o f in verter {(𝑠 𝐴 , 𝑠 𝐵 , 𝑠 𝐶 )} 1…8 , a ll of t hese quantities a re considered i n the di screte m odel o f d rive sy stem ( as p resented i n (3.22)) in order to p redict a ll th e p ossible system tra nsitions: 𝑋 𝑙 (𝑘 + 1 ) = 𝑓{𝑥(𝑘), (𝑠 𝐴 , 𝑠 𝐵 , 𝑠 𝐶 ) 𝑙 }, f or 𝑖 = 1 … 8. This pr ediction function i s directly de rived from the discrete model and system parameters; + To determine which of the control actions must be selected, the cost function built in (3.24) will be evaluated with all the predictions above and the desired reference values 𝑥 * (𝑘 + 1). T he r eferences h ere ar e t he d esired torques f or b oth m achines generated f rom t he t wo R ST sp eed co ntroller. The f uture reference v alue n eeded 𝑥 * (𝑘 + 1) can b e ass umed eq ual t o t he ac tual v alue 𝑥 * (𝑘) because t he sa mpling time 𝑇 𝑠 is small enough and it can be considered constant over 𝑇 𝑠 ; + The e valuation of t he cost f unction for the e ight pr edictions will le ad to e ight different co st v alues. The c ontrol a ction l eading t o the m inimum cost value

Figure 3 .

 3 Figure 3.11: Direct predictive torque control (DPTC) scheme

Figure 3 .

 3 Figure 3.12: Flow chart of direct predictive control algorithm

Figure 3 .

 3 Figure 3.13: The evolution of reference value

Figure 3 .

 3 Figure 3.15: Speed of two machines

Figure 3 .

 3 Figure 3.17: Currents in d-q frame for two machines

Figure 3 .

 3 Figure 3.16: Electromagnetic torque of two machines

Figure 3 .

 3 18a: 𝑇 𝑝𝑐 = 1 × 10 -5 (𝑠); + Figure 3.18b: 𝑇 𝑝𝑐 = 5 × 10 -5 (𝑠) Results are shown in Figure 3.18.

Figure 3 .

 3 Figure 3.18: Ripple in torque under different sampling time for DPTC

  With D PTC, f or e ach control c ycle, o nly eight re al p ossibilities f rom a tw o-level inverter ar e co nsidered. O ptimization p rocess thus c an co me t o a l ocal m inimum. T o consider t his pr oblem t heoretically, opt imization pr ocess will be c arried out by using "FMINCON" function in optimization toolbox of Matlab program.

Figure 3 .

 3 Figure 3.19: "FMINCON" procedure

2𝑉 𝐷𝐶 3 and 3 .

 33 are separated with each other by an angle step of 𝜋 This Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

  mentioned i n s ome doc uments [VLF+09] [RVAB06]. M oreover, this s trategy w ill b e associated with the SVM technique to generate the virtual vectors. It will lead to a constant switching frequency operating mode. As shown in Figure 3.21, the virtual voltage vectors can be located in any position in the in verter c ontrol region. E ach virtual v ector is c onsidered in th e (𝛼𝛽) plane a nd represented by a pair of values: the angle (in comparison with 𝛼-axis) and the magnitude. In this figure, the real vectors are represented by solid line vectors while the virtual vectors are illustrated by the dash lines. These real ones divide space vector into six sectors. The control region of a standard inverter under space vector modulation technique is limited by a circle with the radius equals to 𝑉 𝐷𝐶 /√3.

Figure 3 .

 3 Figure 3.21: Control region of a three-phase two-level inverterincluding the virtual states

  control s et (with o nly eight re al v ectors) d efined in D PTC. H owever, if a ll the virtual vectors a re considered i n each co ntrol cy cle, t he co mputational ef fort w ill b e i ncreased considerably. L et's consider a n inverter w hich i s s upplied by a D C v oltage of 𝑉 𝐷𝐶 = 540𝑉. Under SVM technique, the maximum voltage which can be generated in the linear region is: 𝑉 𝑙𝑙𝑥 = 𝑉 𝐷𝐶 √3 = 540 √3 ≈ 312(𝑉) (3.32) Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM If we co nsider, f or i nstance, a cas e w here the d ifference b etween an y t wo v irtual vectors i s Δ𝛼 = 10 𝑙 and Δ𝑉 = 10𝑉, th ere w ill b e tage ve ctors which need t o be evaluated at each sampling time. It is too large compared with the eight real configurations of the inverter.

𝜋 3 .

 3 Therefore, the e ntire v oltage s pace will b e f irstly d ivided into s ix sectors by si x r eal active voltage vectors (as shown in Figure3.21). These six basic vectors will compose the first considered control set. This control set will be evaluated into the model in (3.31) to calculate th e prediction. A value of t he c ost f unction w ill be obt ained corresponding t o each vector. T he ve ctor w hich gi ves us t he m inimum va lue of t he c ost f unction w ill b e chosen as the starting point for the next algorithm step (step 2 in Figure3.24).

Figure 3 .

 3 Figure 3.22: PTC -"Split and Seek" approach

  Figure 3.25: Responses of system under PTC-SS

Figure 3 .

 3 Figure 3.25. I n ge neral, t he dyna mic r esponses of t he two m achines a re s till good . T he small transient time and the stability of system under the change of loads as well as under the d ifference o f l oads b etween t wo machines ar e m aintained. Response t ime i s f ast (~0.05s). T he sy stem i s st able u nder t he ch ange o f set p oint a s w ell as o f l oad. W hen having a suddenly change of load, steady state is reestablished after a sh ort time (~0.05s).

Figure 3 .Table 3 . 4 : 1 𝑘=0;

 3341 26, t he A -phase cu rrent o f each machine under is i llustrated. It can b e realized that under DPTC approach(Figure 3.26a), the phase current ripples a lot with an amplitude is up to ~1(A). Meanwhile, in PTC-SS, the ripple almost disappears. The current is so smooth(Figure 3.26b) Normally, i t c annot identify t he f requency c omponents of a s ignal bur ied i n a no isy time domain (A-phase current in Figure3.26b is an example). Therefore, to check this, the Discrete Fourier Transform (DFT) could be computed to see if a spectral peak is present or not. For this purpose, Matlab has the FFT analysis tool, which performs the computation DFT. We will use this tool to get the spectrum and total harmonic distortion (THD) of these phase currents. Speed range (𝑟𝑟𝑟. 𝑠 -1 ) DPTC PTC-SS -18 × 𝜋 → 18 × 𝜋 3.0887 (J) 1.4634 (J) -25 × 𝜋 → 25 × 𝜋 2.7210 (J) 1.0362 (J) -32 × 𝜋 → 32 × 𝜋 2.7425 (J) 0.8630 (J) The total losses of two machines 𝑋(𝑁𝜔 𝑠 ) = � 𝑥 𝑘 𝑒 -𝑗𝑙𝜔 𝑠 𝑘 𝑁with Mono-Inverter Dual-Parallel PMSM The frequency spectrums of A-phase current of two machines are represented in Figure 3.27 and Figure 3.28. In both figure, there are always the highest peaks which represent for the fundamental frequencies in (3.35). These are fundamental components of signal and are not the noise. However, in the case of DPTC, we can see many smaller peaks. They show a low harmonic content in the current. It also means that the noise buried in these currents is big.On the contrary, frequency spectrums of A-phase current of two machines under PTC-SS j ust h ave o ne cl ear sp ectral p eak. T he noi se unde rlying s inusoidal c urrent s ignal i s almost absent.

Figure 3 Figure 3 .

 33 Figure 3.26: A-phase current of two machines

  his s ystem is c omposed of f our major pa rts: 1) M otor c oupling s ystem, 2) Power E lectronics D rive B oard, 3) dS APCE b ased D S1103 R &D c ontroller c ard a nd CP1103 I/O board and 4) MATLAB/Simulink and Control desk software.

Figure 4 . 1 :-

 41 Figure 4.1: Presentation of the experimental bench

  as S M1 an d S M2. T hey h ave i dentical ch aracteristics and a re co nnected i n p arallel t o a voltage inverter. Their technical specifications are shown in Appendix 0.According to Figure4.2, each of these machines pulls a plate through a ballscrew axis (axis1 and axis2) w hich a llows ob taining t he very s mooth m otion a s w ell as a good positioning accuracy. I n a ddition, t o c reate a di fferent l oad between t wo m otors, a t hird machine w hich i s d enoted SM3 o r "load m achine" is lin ked w ith th e firs t m achine S M1 through a n a luminum pl ate. T orque ge nerated f rom S M3 i s c ontrolled by a s mart electromotor controller (Appendix B.4.2) and take effect on SM1. In consideration of the axis1 and axis3, they are perfectly parallel thanks to an embedded link [Bor81], so a rigid connection is created between these two axes.In fact, the aim of this experiment is to move the two plates horizontally along axis1 and axis2. To do s o, it is necessary to transform a rotational movement of the shaft of the motor i nto t ranslational movement. T herefore, b allscrew ax es ar e u sed i n t his cas e. The conversion is thus made possible by rooling balls, which are arranged in precision grooves between a scr ewed sh aft an d n ut. O wing t o t hese ch aracteristics, ballscrews f or al l applications are essential, with which high speeds, accuracies and reliability are requested over a long service life. The structure of a b allscrew axis is shown in Figure4.3. Besides, all technical specifications of this axis are summary in Appendix B.4.3

Figure 4 Figure 4 . 3 :

 443 Figure 4.2: Motor coupling system

Figure 4 2

 42 Figure 4.4: Power supply system

  to measure th e s tator c urrents 𝑖 𝑙1 , 𝑖 𝑏1 , 𝑖 𝑙2 and 𝑖 𝑏2 , H all ef fect sensors type LEM LA-25-P, whose datasheet is given in Appendix B.5.2, are connected to the analog inputs of t he c ard. T he ne utral po ints of motors a re not l inked; t he c urrents 𝑖 𝑐1 and 𝑖 𝑐2 which circulate in the third phase are obtained by the equation in (4.1) 𝑖 𝑐 = -(𝑖 𝑙 + 𝑖 𝑏 ) (4.1)

Figure 4 . 5 .

 45 scheme in Figure 4.5. The measures of absolute positions 𝜃 𝑛𝑢𝑙 which are obtained by the resolvers are coded by 12 bi ts. H owever, t he c ard D S1103 do es not a llow l inking a ll 20 -bit d igital Input/Output. It is noted that the I/O is needed to activate the read of the AD2S1200 and also to capture the signals CLCK. Therefore, the two position 𝜃 𝑙1 and 𝜃 𝑙2 can not be read simultaneously.

Figure 4 . 5 :

 45 Figure 4.5: Reading of θ m1 and θ m2 : Hardware part

Figure 4 . 6 :

 46 Figure 4.6: Example of filtering speed

  analogue I/O and digital I/O. Controllers have to be programmed and then, by means of the analogue and digital I/O, command signals are sent to the system and measured values are received.

  s imulation ha rdware D S1103 i s ba sed on pow er P C microprocessor supplemented by T exas I nstruments T MS320F240 D SP p rocessor. T he T MS320F240 slave DSP is loaded with firmware that allows access to the different peripheral devices of the slave DSP. Functions needed for motor control (SPI, PWM or SVM) are already predefined in slave DSP so there was no need to compile custom code. Programs f or m aster p ower P C can b e co mpiled d irectly f rom t he C -code us ing R TI library containing the functions provided by the DS1103 processor or Simulink Real-Time Workshop. Besides the PWM/SVM module and SPI interface provided by slave processor, the encoder interface and A/D modules which are controlled by master power PC are used for closed loop motor control to measure the rotor position and stator currents. Detailed technical specifications of dSPACE DS1103 are described in Appendix B.5.1 [dSP03]

Figure 4 .

 4 Figure 4.7: Interupt signal

Figure 4 .

 4 Figure 4.8: RST controller block with anti-windup

Table 4 . 1 :

 41 Parameters of speed controller The experimental platform In this test, the profile of load torque generated from SM3 is set up as follows: a. From 𝑡 = 0(𝑠) → 𝑡 = 1.5(𝑠) and 𝑡 = 9.0(𝑠) → 𝑡 = 10(𝑠): lo ad to rque is s et a t zero. It means that total load applied to SM1 is 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑟1 + 𝑇 𝑟3 ; b. From 𝑡 = 1.5(𝑠) → 𝑡 = 3.0(𝑠): load is set at 𝑇 𝑒𝑥𝑙 = 0.1𝑁𝑁. So, the total load will be decreased: 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑟1 + 𝑇 𝑟3 -𝑇 𝑒𝑥𝑙 ; c. From 𝑡 = 3.0(𝑠) → 𝑡 = 5.0(𝑠): l oad i s set at 𝑇 𝑒𝑥𝑙 = -0.1𝑁𝑁. S o, t he t otal l oad will be 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑟1 + 𝑇 𝑟3 + |𝑇 𝑒𝑥𝑙 |; d. From 𝑡 = 5.0(𝑠) → 𝑡 = 6.5(𝑠): load is set at 𝑇 𝑒𝑥𝑙 = -0.1𝑁𝑁, however, the motor is re versed. So, th e to tal lo ad w ill b e 𝑇 𝑙𝑙𝑙𝑙𝑙 = -(𝑇 𝑟1 +𝑇 𝑟3 -|𝑇 𝑒𝑥𝑙 |). T he a bsolute value of load is the same with (b); e. From 𝑡 = 6.5(𝑠) → 𝑡 = 9.0(𝑠): load is set at 𝑇 𝑒𝑥𝑙 = 0.1𝑁𝑁, however, the motor is reversed. So, the total load will have absolute value as in (c); f. From 𝑡 = 9.0(𝑠) → 10.0(𝑠): load torque is equal to zero. The total load is the same as (a) This profile is illustrated in Figure 4.10.

Figure 4 . 9 :

 49 Figure 4.9: PTC-SS applied to one PMSM

Figure 4 .

 4 Figure 4.11: Speed response of PMSM

  Figure 4.14.

Figure 4 . 2 + 2 ( 4

 4224 Figure 4.13: Current in d-axis

Figure 4 .

 4 Figure 4.14: PTC-SS with two PMSMs

Figure 4

 4 Figure 4.15: Speed response of two PMSMs

Figure 4 .Figure 4 .

 44 Figure 4.16: Current in q-axis of both machines

Figure 4 .

 4 Figure 4.18: PTC-SS with two PMSMs under master/slave

Figure 4 . 19 :

 419 Figure 4.19: Speed response of two PMSMs under master/slave configuration

Introduction•••••••

  Aujourd'hui, l es systèmes e mbarqués s ont de pl us e n pl us n ombreux et ils impactent fortement l es sy stèmes d e co nversion d 'énergie. L es co ntraintes associées concernent la réduction de masse a insi que l a di minution de s pe rtes a fin d' améliorer l 'efficacité énergétique de la chaîne de conversion. C'est bien sûr le cas dans le domaine aéronautique où le concept de «l'avion plus électrique» devient aujourd'hui une réalité. En conséquence, la machine synchrone à aimant permanent devient un a ctionneur par excellence grâce à sa forte densité de puissance, son faible coût d'entretien et ses qualités dynamiques. Lorsque ces machines sont associées pour mener à bien des fonctions coopératives (par exemple les surfaces de vol), la masse embarquée peut encore être réduite par le partage de l'électronique d e p uissance. C 'est p récisément d ans ce contexte o ù n otre travail a é té élaboré, avec l 'idée de développer des st ructures d'électronique d e p uissance capables d'alimenter d eux o u p lusieurs m achines é lectriques en p arallèle et dotées de lois de contrôle adaptées afin d'améliorer l'efficacité énergétique. Nous a vons t ravaillé pl us s pécifiquement s ur l a c ommande pr édictive de de ux machines synchrones connectées en parallèle reliées à un seul convertisseur triphasé. Ces machines o nt d es ca ractéristiques i dentiques et d oivent su ivre l e m ême p rofil d e vitesse avec un couple de charge qui peut être différent et indépendant dans tous les cas. Le cadre de travail de ces travaux est présenté en quatre chapitres: Le p remier ch apitre présente les u sages d e l a m achine sy nchrone à aimant permanent (MSAP) dans de nombreux domaines d'application. Les systèmes multimachines d ans l e d omaine aér onautique a insi q ue l 'idée d e l a st ructure d'électronique de puissance partagée sont ainsi considérés. Les structures typiques du s ystème multi-machine so nt a lors p résentées. L es car actéristiques d e ch aque structure sont ensuite étudiées. Le d euxième ch apitre co nsacré à l a d escription g énérale d 'un sy stème d'entraînement d e MS AP et n otamment l e s ystème d 'entraînement d es MS AP e n NGUYEN Ngoc Linh parallèle. Dans cette partie, l'analyse de la stabilité et le principe de commande de la MSAP sont ainsi considérés. Le t roisième ch apitre est l a p artie p rincipale d e cet te t hèse. L a c ommande prédictive du couple des deux machines synchrones connectées en parallèle, reliées à un s eul convertisseur triphasé deux niveaux est ainsi étudiée. Chaque composant de ce sy stème d 'entraînement est ét udié e t ex pliqué en d étail. D eux ap proches d e commande prédictive seront décrites et comparées. Des simulations permettront de vérifier les performances des méthodes de contrôle. Le quatrième chapitre présente le b anc ex périmental q ui a ét é d éveloppé a u laboratoire LAPLACE avec l 'objectif de vé rifier l e f onctionnement d u s ystème avec l es stratégies d e co ntrôle p roposées. L e b anc r eprésente u n d éplacement hypothétique de de ux volets d' un a vion. L e système é tudié c onvertit l 'énergie électrique f ournie p ar l e r éseau E DF p our al imenter d eux ch arges mécaniques. Dans ce m écanisme, l es d eux MS APs so nt connectées en parallèle su r u n m ême onduleur. Machine Synchrone à Aimant Permanent (MSAP) La machine sy nchrone à ai mants p ermanents ( MSAP) a ém ergé co mme une s olution compétitive dans une large gamme d 'applications de contrôle d u mouvement. E n particulier, l a MS AP est l argement u tilisée d ans l es m achines-outils, l a r obotique et l es actionneurs industriels et el le e st é galement en visagée d ans d es ap plications d e h aute puissance comme la propulsion des véhicules et des entraînements industriels. Elle devient également v iable p our d es ap plications commerciales o u r ésidentielles. L a MSAP est connue pour a voir b eaucoup d' avantages pa r r apport à d' autres t ypes de m achines électriques: Une grande efficacité: les MSAPs utilisent des aimants permanents pour créer un champ magnétique au rotor constant au lieu d'utiliser un c ircuit. Par conséquent, il n'y a p as d e p ertes cu ivre. E n o utre, san s le c ircuit d u r otor, l a seule chaleur es t produite sur le stator, ce qui est beaucoup plus facile à refroidir. Une haute densité de flux: Les progrès récents dans les aimants de haute densité énergétique ( NdFeB) o nt p ermis l a réalisation d e t rès f ortes d ensités d e f lux. C es aimants permettent un couple élevé, le développement de machines plus petites et plus légères, une grande longévité et fiabilité. Parce qu'il n'y a pas de commutateurs mécaniques ni de contacts glissants le frottement est plus faible et la durée de vie est p lus longue. D 'ailleurs, san s c es él éments, l e co ût d 'entretien e st r éduit e t l e NGUYEN Ngoc Linh risque lié a ux c ourants d' extra-coupure est él iminé ce q ui e st p articulièrement appréciable dans certaines conditions d'utilisation. Un rapport entre couple et inertie élevé: En utilisant des aimants permanents, la MSAP au ra u ne f aible i nertie, ce q ui assu re u ne r éponse r apide so us u n co uple électrique d onné u ne d ensité d e p uissance él evée ( il es t r eprésenté par le rapport kW / kg qui traduit la puissance massique) • ... Dans l es M SAPs, d es aimants p ermanents so nt m ontés à l 'intérieur du r otor (machine synchrone à ai mants p ermanents en terres -IMSAP) ou s itués e n pé riphérie du r otor (machine synchrone à aimants montés en surface-SMSAP) (Figure 1). Dans tous les cas, la st ructure du st ator e st i dentique aux machines asy nchrones comme d'autres t ypes d e moteurs à courant alternatif. Les bobinages sont alors construits de manière à produire une densité de flux sinusoïdale dans l'entrefer de la machine. Aujourd'hui, l es MSAPs j ouent un r ôle i mportant d ans l es a pplications à vi tesse variable. C es ap plications o nt l a capacité d e p roduire q uelques W atts (micro-moteur brushless D C dé veloppé en t echnologie M EMS) à pl us d'un GW ( générateur s ynchrone AC). P our l es p etites machines él ectriques, le r otor co mporte g énéralement p lusieurs aimants p ermanents. C es m oteurs sy nchrones à ai mants p ermanents so nt ég alement appelés «moteur sans balais» parce que le flux du rotor est toujours créé par les aimants, ils n'ont pas de collecteur et de balais. a) Machine synchrone à aimants montés en surface (SPMSM) b) Machine synchrone à aimants enterrés (IPMSM)

Figure 1 :

 1 Figure 1: Deux principaux types de MSAP

  Afin de faire fonctionner de tels ces systèmes, chaque moteur a besoin d'un dispositif d'électronique de puissance qui se compose d'une composante de refroidissement lourde et encombrante. Cet élément va augmenter le poids et la taille de l'ensemble du système de manière significative. De plus, les applications aéronautiques exigent des systèmes à haute sécurité i ntégrée, i l f aut donc avoir des systèmes r edondants. Ces sy stèmes servent essentiellement à assurer le fonctionnement continu de l'ensemble du système, même si un problème se produit. Toutefois, ils alourdissent et augmentent le volume du système. D'ailleurs, pendant chaque phase de vol, un certain nombre de systèmes d'entraînement nécessitent des dispositifs d'électronique de puissance. Ils sont alors utilisés tandis que les autres sont inactifs. Par exemple, les systèmes d'ouverture des portes ne sont utilisés que lorsque l'avion est à l'arrêt. Les systèmes du train d'atterrissage, des volets des ailes ou des onduleurs d e p oussée so nt ég alement u tilisés seu lement p our q uelques minutes su r l'ensemble du vol. Il est intéressant d'examiner l'utilisation de structures de partage pour exploiter d es s ystèmes i ndividuels e n f onction d e l a d isponibilité de s équençage, tout e n respectant la continuité de service.

  l'atterrissage et d u d écollage d es p rocessus ( système d e v olets, sp oilers . ..). I l y a deux parties principales dans le système d'actionnement secondaire: les volets 'slat' et 'flap' qui sont utilisés pour le décollage, l'atterrissage et pour l'augmentation de la portance à basse vitesse ; les spoilers qui augmentent la traînée. Ces composants ne sont pas réellement nécessaires pour le vol, mais sont indispensables. NGUYEN Ngoc Linh

Figure 2 :Figure 3 :

 23 Figure 2: Utilisation des MSAP en aéronautique

Figure 4 : 2 , 2 , 2 ,Figure 5 : 2 ,

 422252 Figure 4: Structure classique multimachine/multionduleur SM(3n,n)

Mono-onduleur

  dual-parallele MSAP Selon la Figure 6, un système d'entraînement en comprenant deux MSAPs connectées en p arallèle et al imentées p ar un on duleur triphasé à d eux niveaux es t présenté. D ans ce genre d e sy stème d 'entraînement, les machines doi vent être i dentiques. Cela s ignifie que tous les paramètres des deux machines sont les mêmes et les deux machines recevront la même tension (en amplitude et en fréquence): � |𝑉 𝑠1 | = |𝑉 𝑠2 | = |𝑉 𝑠 | 𝜔 𝑟𝑒1 = 𝜔 𝑟𝑒2 = 𝜔 𝑠 Pour ce t ype de système, l'objectif est de maintenir les vitesses des deux machines qui sont ég ales, d e f açon à su ivre l a v itesse d e r éférence d ans l es d ifférentes co nditions d e charge: 𝜔 𝑟𝑒1 = 𝜔 𝑟𝑒2 = 𝜔 𝑟𝑒𝑟 𝑠. 𝑡 𝑇 𝑙𝑙𝑙𝑠1 ≠ 𝑇 𝑙𝑙𝑙𝑠2 La structure maître/esclave Une co nfiguration m aître/esclave a ét é appliquée au sy stème d 'entraînement monoonduleur bi-machine MSAP. Dans de tels systèmes, notons que : + Les deux machines sont identiques

Figure 6 :

 6 Figure 6: Mono-onduleur dual-parallèle MSAP

Figure 7 :

 7 Figure 7: La structure parallèle avec maître/esclave

Figure 8 :Figure 9 :

 89 Figure 8: Commande directe de couple

Figure 10 :

 10 Figure 10: Commande Prédictive Directe de Couple (DPTC)

Figure 11 :

 11 Figure 11: Schéma de l'algorithme de commande

Figure 12 :

 12 Figure 12: Schéma de la méthode PTC -"Split and Seek"

Figure 14 :oo•o

 14 Figure 14: Deux vecteurs adjacents et les vecteurs virtuels

Figure•

  Figure 16: Présentation du banc expérimental

  Dans ce t est, le profil de couple de charge générée par SM3 est mis en place comme suit: g. De 𝑡 = 0(𝑠) → 𝑡 = 1.5(𝑠) et 𝑡 = 9.0(𝑠) → 𝑡 = 10(𝑠): le couple de charge est fixé à zéro. Cela signifie que la charge totale appliquée à SM1 est 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑟1 + 𝑇 𝑟3 ; h. De 𝑡 = 1.5(𝑠) → 𝑡 = 3.0(𝑠): la charge est fixée à 𝑇 𝑒𝑥𝑙 = 0.1𝑁𝑁. Ainsi, la charge totale sera réduite: 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑟1 + 𝑇 𝑟3 -𝑇 𝑒𝑥𝑙 ; i. De 𝑡 = 3.0(𝑠) → 𝑡 = 5.0(𝑠): la charge est fixé à 𝑇 𝑒𝑥𝑙 = -0.1𝑁𝑁. Ainsi, la charge totale sera 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑇 𝑟1 + 𝑇 𝑟3 + |𝑇 𝑒𝑥𝑙 |;

Figure 17 :

 17 Figure 17: PTC-SS appliqué à une MSAP

Figure 18 : 2 + 2 +Figure 19 :

 182219 Figure 18: Réponse de vitesse de MSAP

Figure 21 :

 21 Figure 21: Réponse en vitesse de deux machines

Figure 23 :

 23 Figure 23: Courants sur l'axe q des deux machines

Figure 25 :

 25 Figure 25: Réponse en vitesse des deux MSAPs en configuration maître/esclave

Figure A. 2 :A. 2 . 2

 222 Figure A.2: Voltage space vector and its components in (𝛼𝛽)

Figure A. 3 : 1 A. 2 . 3

 3123 Figure A.3: Reference vector as a combination of adjacent vectors in sector 1

Figure B. 2 ,

 2 i s t he "S EC-AC-305" de veloped by F esto company. It has been designed for use in control cabinets for supplying AC servo motors and for controlling their torque (current), speed and position.

B. 4 . 3

 43 Spindle axes with recirculating ball bearing guideThree axes used in this work are standard spindle axes with recirculating ball bearing guide (DGE-40-1000-SP-KG-KF-GK-SH) made by Festo company. This type of axis has a stroke-lengths from 100 to 2000 mm. It has some basic characteristics:

Figure B. 3 [ 1 ] 3 ]

 313 and t he b rief descriptions ar e point out in Table B.6 Type Brief description Spindle axis DGE-SP-KF Electromechanical axis with spindle and recirculating ball bearing guide [2] Emergency buffer with retainer A For avoiding damage at the end stop in the event of malfuntion [3] Shock absorber kit C For avoiding damage at the end stop in the event of malfuntion [Shock absorber kit E For avoiding damage at the end stop in the event of malfuntion [4] Switching lug L For sensing the slide position [5] Sensor bracket T Adapter for mounting the inductive proximity sensors on the axis [6] Inductive proximity sensor O/P/R/W For use as a signal generator and safety monitoring [7] Slot nut for slide X For mounting loads and attachments on the slide [8] Centring pins/sleeves Z For centring loads and attachments on the slide [9] Parallel kit For parallel motor attachment

Figure B. 3 :

 3 Figure B.3: A brief description of a ballscrew axis

•B.5. 2

 2 32-bit parallel I/O • Organized in four 8-bit groups • Each 8-bit group can be set to input or output Technical data of the experimental The architecture of the card DS1103 with the link to the input/output of other devices is shown in Figure B.4. Current sensors Instantaneous measurements of stator current are carried out by Hall sensor type LEM LA25-P. During ope ration, t hese s ensors pr ovide a n vol tage c orrelative w ith t he c urrent according to the rate 1V/A. The bandwidth of this sensor type is 150 k Hz. The outputs of the s ensors are c onnected t o a nalog i nput of t he c ontrol board by the s hielded coaxial cables. CAN bus accept a v oltage between ±10V and a reduction ratio of ten. The digital signal is created under 12-bit and changing between -1 and +1.The parameters of this sensor are presented below.

Figure B. 4 :B.5. 3

 43 Figure B.4: Architecture of the card DS1103

  Figure B.5.

FigureFigure 1 . 1 :

 11 Figure B.5: Human-machine interface

Figure 1 . 2 :

 12 Figure 1.2: Two main types of PMSM..................................................................................

Figure 1 . 3 :

 13 Figure 1.3: Examples of application of brushless motors in small power range ...................

Figure 1 . 4 :

 14 Figure 1.4: Examples of applications of brushless motors in medium power range ...........

Figure 1 . 5 :

 15 Figure 1.5: Examples of applications of brushless motors in high power range .................

Figure 1 . 6

 16 Figure 1.6 The possibility of using PMSM in aeronautics [All95] .....................................

Figure 1 . 7 :

 17 Figure 1.7: Flight control actuation systems [Nfo06] ..........................................................

Figure 1 . 8 :

 18 Figure 1.8: Wing of an airbus A300....................................................................................

Figure 1 . 9 :

 19 Figure 1.9: An advanced Motor Device EMA.....................................................................

Figure 1 .

 1 Figure 1.10: Direct drive architecture for EMA [ROAR07] ...............................................

Figure 1 .

 1 Figure 1.11: The classical structure multi-machine multi-inverter MS(3n,n).....................

Figure 1 .

 1 Figure 1.12: Diagram of voltage vectors under structure MS(3n,n)....................................

Figure 1 . 2 ,

 12 Figure 1.13: The structure using a common leg MS � 4𝑛 2 ,𝑛� ...............................................

Figure 1 . 2 ,

 12 Figure 1.14: Diagram of voltage vectors under structure MS� 4𝑛 2 ,𝑛� .................................

Figure 1 .

 1 Figure 1.15: The structure sharing the midpoint MS(2𝑛, 𝑛)...............................................

Figure 1 .

 1 Figure 1.16: Diagram of voltage vectors under the structure MS(2𝑛, 𝑛) ............................

Figure 1 . 2 ,

 12 Figure 1.17: Mixed structure MS� 3𝑛 2 ,𝑛� .............................................................................

Figure 1 . 2 ,

 12 Figure 1.18: Diagram of voltage vectors under mixed structure: MS� 3𝑛 2 ,𝑛� .....................

Figure 1 . 19 :

 119 Figure 1.19: Parallel structure MS(3, 𝑛) .............................................................................

Figure 2 . 1 :

 21 Figure 2.1: Mono-inverter dual-parallel PMSM drive.........................................................

Figure 2 . 2 :

 22 Figure 2.2: Structure of a PMSM drive system...................................................................

Figure 2 . 3 :

 23 Figure 2.3: Stator windings of a three-phase PMSM 2 pole pairs.......................................

Figure 2 . 4 :

 24 Figure 2.4: Two-pole three-phase PMSM...........................................................................

Figure 2 . 5 :

 25 Figure 2.5: Electrical model of one phase of PMSM..........................................................

Figure 2 . 6 :

 26 Figure 2.6: Equivalent circuit of a PMSM in d-q frame......................................................

Figure 2 . 7 :

 27 Figure 2.7: Evolution of 𝑇 𝑒 = 𝑓(𝛿) with the consideration of 𝑅 𝑠.......................................

Figure 2 . 8 :

 28 Figure 2.8: Evolution of 𝑇 𝑒 = 𝑓(𝛿) without consideration of 𝑅 𝑠........................................

Figure 2 . 9 :

 29 Figure 2.9: Diagram of a three-phase two-level inverter.....................................................

Figure 2 .

 2 Figure 2.10: General scheme of FOC for a PMSM.............................................................

Figure 2 .

 2 Figure 2.11: General scheme of DTC for a PMSM.............................................................

Figure 2 .

 2 Figure 2.12: The basic voltage vectors and six sectors........................................................

Figure 2 .

 2 Figure 2.13: Circular trajectory of stator flux linkage in DTC............................................

Figure 2 .

 2 Figure 2.14: The evolution of torque under DTC................................................................

Figure 2 .

 2 Figure 2.15: General scheme of DPTC for PMSM..............................................................

Figure 2 .

 2 Figure 2.16: Parallel structure under master/slave configuration........................................

Figure 2 .

 2 Figure 2.17: Evolution of system under master/slave configuration with different loads ..

Figure 2 .

 2 Figure 2.18: Voltage vector diagram of two PMSM under different loads.........................

Figure 2 . 19 :

 219 Figure 2.19: Master choice block.........................................................................................

Figure 2 .

 2 Figure 2.20: General scheme of DTC for dual-parallel PMSM drive system.....................

Figure 2 .

 2 Figure 2.21: Different situation for selecting a suitable voltage vector...............................

Figure 2 .

 2 Figure 2.22: Average rule ....................................................................................................

Figure 2 .

 2 Figure 2.23: Two vectors are adjacent.................................................................................

Figure 2 .

 2 Figure 2.24: The evolution of reference value.....................................................................

Figure 2 .

 2 Figure 2.25: The variation of load of each machine in simulation......................................

Figure 2 .

 2 Figure 2.26: Responses of system under DTC.....................................................................

Figure 3 . 1 :

 31 Figure 3.1: dq-current at steady state of a PMSM...............................................................

Figure 3 . 2 :

 32 Figure 3.2: Load angle at steady state..................................................................................

Figure 3 . 3 :

 33 Figure 3.3: Limit in operation of a PMSM..........................................................................

Figure 3 . 4 :

 34 Figure 3.4: Map of roots of Ps.............................................................................................

Figure 3 . 5 :

 35 Figure 3.5: Example of limitation of stable zone of a PMSM.............................................

Figure 3 .

 3 Figure 3.6: PTC for Mono-Inverter Dual-parallel PMSM drive system..............................

Figure 3 .

 3 Figure 3.7: The R-S-T digital controller [Lan98] ................................................................

Figure 3 .

 3 Figure 3.8: RST anti-windup Speed/Torque controller for a PMSM..................................

Figure 3 . 9 :

 39 Figure 3.9: A standard three-phase two-level inverter.........................................................

Figure 3 .

 3 Figure 3.10: Two strategies for predictive control [LKKS10] .............................................

Figure 3 .

 3 Figure 3.11: Direct predictive torque control (DPTC) scheme............................................

Figure 3 .

 3 Figure 3.12: Flow chart of direct predictive control algorithm............................................

Figure 3 .

 3 Figure 3.13: The evolution of reference value.....................................................................

Figure 3 .

 3 Figure 3.14: The variation of load of each machine in simulation......................................

Figure 3 .

 3 Figure 3.15: Speed of two machines....................................................................................

Figure 3 .

 3 Figure 3.16: Electromagnetic torque of two machines........................................................

Figure 3 .

 3 Figure 3.17: Currents in d-q frame for two machines ..........................................................

Figure 3 .

 3 Figure 3.18: Ripple in torque under different sampling time for DPTC..............................

Figure 3 .

 3 Figure 3.19: "FMINCON" procedure ..................................................................................

Figure 3 .

 3 Figure 3.20: Simulation results after optimization process.................................................

Figure 3 .Figure 3 .

 33 Figure 3.21: Control region of a three-phase two-level inverter including the virtual states95 Figure 3.22: PTC -"Split and Seek"approach ...................................................................

Figure 3 .

 3 Figure 3.23: Two adjacent sectors with virtual voltages.....................................................

Figure 3 .

 3 Figure 3.24: Diagram of the new solution...........................................................................

Figure 3 .

 3 Figure 3.25: Responses of system under PTC-SS.............................................................

Figure 3 .

 3 Figure 3.26: A-phase current of two machines..................................................................

Figure 3 .

 3 Figure 3.27: Spectrum of A-phase current of machine 1...................................................

Figure 3 .

 3 Figure 3.28: Spectrum of A-phase current of machine 2...................................................

Figure 4 . 1 :

 41 Figure 4.1: Presentation of the experimental bench..........................................................

Figure 4 . 2 :

 42 Figure 4.2: Motor coupling system....................................................................................

Figure 4 . 3 :

 43 Figure 4.3: Structure of a ballscrew axis...........................................................................

Figure 4 . 4 :

 44 Figure 4.4: Power supply system.......................................................................................

Figure 4 . 5 :

 45 Figure 4.5: Reading of 𝜃 𝑙1 and 𝜃 𝑙2 : Hardware part ........................................................

Figure 4 . 6 :

 46 Figure 4.6: Example of filtering speed..............................................................................

Figure 4 . 7 :

 47 Figure 4.7: Interupt signal ..................................................................................................

Figure 4 . 8 :

 48 Figure 4.8: RST controller block with anti-windup...........................................................

Figure 4 . 9 :

 49 Figure 4.9: PTC-SS applied to one PMSM........................................................................

Figure 4 .

 4 Figure 4.10: Variation of load during the test....................................................................

Figure 4 .

 4 Figure 4.11: Speed response of PMSM.............................................................................

Figure 4 .

 4 Figure 4.12: Current in q-axis ............................................................................................

Figure 4 .

 4 Figure 4.13: Current in d-axis............................................................................................

Figure 4 .

 4 Figure 4.14: PTC-SS with two PMSMs.............................................................................

Figure 4 .

 4 Figure 4.15: Speed response of two PMSMs.....................................................................

Figure 4 .

 4 Figure 4.16: Current in q-axis of both machines ...............................................................

Figure 4 .

 4 Figure 4.17: Current in d-axis of both machines...............................................................

Figure 4 .

 4 Figure 4.18: PTC-SS with two PMSMs under master/slave .............................................

Figure 4 . 19 :

 419 Figure 4.19: Speed response of two PMSMs under master/slave configuration ...............

Figure 4 .

 4 Figure 4.20: Current in q-axis of both machines...............................................................

Figure 4 .

 4 Figure 4.21: Current in d-axis of both machines...............................................................

9 T Figure 1 :

 1 Deux principaux types de MSAP9T .......................................................................

9 T Figure 2 :

 2 Utilisation des MSAP en aéronautique9T ..............................................................

9 T Figure 3 :

 3 Système de contrôle des surfaces de vol9T ............................................................

1.3.1 The idea of sharing structures Within

t he f ramework o f t he Mo re Electric A ircraft ( MEA), t here i s a trend o f u sing the m ore an d m ore el ectric m achines

[START_REF] Rosero | M oving t owards a more e lectric a ircralf[END_REF]

[AF09], whether f or t he f light control actuators (see section

1.3.2)

, or for other mechanical parts such as the landing gear, brakes, fuel pumps, defrosting systems, air conditioning systems, startup of motors, door locks …

In o rder t o operate su ch t hese sy stems, each e ngine n eeds a p ower el ectronic d evice which c onsists of a he avy a nd bul ky c ooling component. T his e lement w ill i ncrease t he weight a nd s ize of t he w hole system s ignificantly. Moreover, for t he a eronautic applications, w hich re quire th e h igh f ail-safety, it i s n ecessary t o h ave st rictly co nstraint

  t he ve ctor r epresentation of t he c omplex vol tage, 𝑂 is t he middle poi nt of t he D C bus, (𝐴 𝑙 , 𝐵 𝑙 and 𝐶 𝑙 ) represent t he phases of t he machine 𝑖 and 𝑁 𝑙 is the neutral point. In

	general, t he vo ltage re sulting f rom th e c ontrol o f th e machines is th ree-phased a nd
	balanced. T his relation i s t rue f or al l m achines an d i s r epresented by t he f ollowing
	equation:

𝑉 𝐴 𝑖 𝑁 𝑖 + 𝑉 𝐵 𝑖 𝑁 𝑖 + 𝑉 𝐶 𝑖 𝑁 𝑖 = 0

  𝑉 𝐴 𝑖 𝑁 𝑖 = 𝑉 𝐴 𝑖 𝑂 𝑉 𝐵 𝑖 𝑁 𝑖 = 𝑉 𝐵 𝑖 𝑂 𝑉 𝐶 𝑖 𝑁 𝑖 = 𝑉 𝐶 𝑖 𝑂

  Typical structures of multi-machine system PMSM with Multi-Machine SystemsIn or der to ge t t he w ell adapted voltage t o apply to e ach l eg of t he i nverter, t he potential be tween t he ne utral poi nt of i nverter 𝑂 and t he ne utral poi nt of e ach machine 𝑁 1 and 𝑁 2 need to be considered: �𝑉 𝑁 1 𝑂 �

𝑟𝑒𝑟

and �𝑉 𝑁 2 𝑂 � 𝑟𝑒𝑟 .

  However, w ith t his s tructure, a n a daptation m odule ne eds to be us ed t o m odulate t he midpoint 𝑂 of a c ontinuous D C bus . T hanks t o t his, t he two vol tages �𝑉 𝑁 1 𝑂 �

	𝑟𝑒𝑟	and
	�𝑉 𝑁 2 𝑂 �

𝑟𝑒𝑟

can be regulated properly. Besides, it should be paid attention that each shared leg of the inverter in this case has to withstand the higher current compared with the case of non -shared l eg. It is r easonable because there ar e two c urrents w ho w ill p ass t hrough such these legs.

Table

Chapter 2 Mono-Inverter Dual-Parallel PMSM drive system

					Mono-Inverter Dual-Parallel PMSM drive system
	Structure's name	MS(m,n)	Number of legs per machine	Relation (𝑉 𝑁𝑘 ) max (𝑉 𝑂𝑘 ) max	Speed	Volume	Notes Advantages Disadvantages
	Classical	MS(3𝑛, 𝑛)	3	1	independent	large	Independent machines	Non-sharing
	Common legs	MS�	4𝑛 2	, 𝑛�	2	2 3	Ω 1 = Ω 2 or Ω 1 = -Ω 2	average	Easy to implement	Over current in common legs
	Midpoint	MS(2𝑛, 𝑛)	2	1 √3	independent	average	Can apply for 1 machine	Access to neutral point
	Mixed	MS�	3𝑛 2	, 𝑛�	1.5	1 √3	Ω 1 = Ω 2 or Ω 1 = -Ω 2	small	Inherit from the previous	Inherit from the previous
	Parallel	MS(3, 𝑛)	3 𝑛	1	Ω 1 = Ω 2	small	Easy to implement	Over current in all legs
					Table 1.1: Summary of different structures

Table of Content

 of

2.2.2 Modeling of PMSMs

	In a r otating electrical machine, there is always a rotating component and a st ationary
	one. B ecause of t his r otation, t he s elf a nd mutual i nductances, w hich a re responsible f or
	various voltages, become time varying. These time varying inductances sometimes cause
	difficulties for the analysis. Besides, rotor flux in a PMSM is constant and provided by the
	permanent m agnets. T hank t o t his, r otor f lux i s a lways pol arized an d has a cl early an d
	consistently oriented shaft. If using a position sensor (such as resolver, encoder with zero
	pulse), t he position of t he r otor f lux c an be s pecified pr ecisely a nd f or t his r eason, t he
	problem of flux oriented always can be ensured.
	Therefore, in t his s ection, a m odel of a t hree-phase P MSM i n d -q r eference f rame i s
	derived in o rder to c larify t he c oncept of t he a xis t ransformation ( Park) a nd t he r elation
	between three-phase quantities and their equivalent two-phase quantities.
	Throughout the de rivation of t he two-phase ( d-q) mathematical model of P MSM, t he
	following assumptions are pointed out [Kri01]
	+ Stator windings produce sinusoidal EMF distribution. Space harmonics in the air-
	gap are neglected;
	+ Air-gap r eluctance h as a co nstant co mponent as w ell as a si nusoidal v arying
	component;
	+ Balanced three-phase sinusoidal supply voltage is considered;
	+ Although magnetic saturation is considered, eddy current and hysteresis effect are
	neglected;
	+ Presence of damper windings is not considered because PMSMs used today rarely
	have that kind of configuration.

  𝑜 𝑢 𝑠𝑏 (𝑡) + 𝑒 𝑗240 𝑜 𝑢 𝑠𝑐 (𝑡)�

	𝚤 ⃗ 𝑠 =	2 3	�𝑖 𝑠𝑙 (𝑡) + 𝑒 𝑗120 𝑜 𝑖 𝑠𝑏 (𝑡) + 𝑒 𝑗240 𝑜 𝑖 𝑠𝑐 (𝑡)�	(2.5)
	𝜓 �⃗ 𝑠 =	2 3	�𝜓 𝑠𝑙 (𝑡) + 𝑒 𝑗120 𝑜 𝜓 𝑠𝑏 (𝑡) + 𝑒 𝑗240 𝑜 𝜓 𝑠𝑐 (𝑡)�
	The r esultant vol tage, c urrent a nd f lux l inkage space ve ctor s hown i n (2.5) for the s tator
	are calculated by m ultiplying instantaneous phase values by the stator winding orientation
	in w hich the s tator re ference a xis for th e a-phase i s c hosen t o the direction of maximum
	emf. Reference axes for b-and c-stator frame are chosen 120 𝑙 and 240 𝑙 (electrical angle)

  . 𝑖 𝑘 + 𝐿.

	𝑢 �⃗ 𝑠 = 𝑅 𝑠 . 𝚤 ⃗ 𝑠 +	𝑟𝜓 �⃗ 𝑠 𝑟𝑡	(2.4)
	𝑟𝑖 𝑘 𝑟𝑡	+ 𝑒 𝑘	(2.6)

2.2.2.2 Continuous model of PMSM in d-q reference frame

  𝜃 𝑒 represents the electrical angle. This value depends on the rotor position 𝜃 𝑙 and the number of pole pairs 𝑛 𝑝 : 𝜃 𝑒 = 𝑛 𝑝 𝜃 𝑙 . 𝑢 𝑠𝑟 + 𝑗𝑢 𝑠𝑞 and 𝚤 ⃗ 𝑠 𝑠𝑠 = 𝑖 𝑠𝑟 + 𝑗𝑖 𝑠𝑞 . T he e quations (2.15) are ob tained by separating the real and imaginary parts from the above equation:

	Based on the electrical dynamic equation described in (2.6), output mechanical torque The i ndex "𝑟𝑞" in th e rig ht-upper r epresents t o t he f lux r eference f rame ( field o riented
	generated 𝑇 𝑒 in a PMSM is related to the mechanical speed 𝜔 𝑙 as given by Equation (2.8) frame).
		𝑇 𝑒 = is pol ar-flux v ector. B ecause t he real ax is o f t he coordinate s ystem is 𝑒 𝑙 𝑖 𝑙 + 𝑒 𝑏 𝑖 𝑏 + 𝑒 𝑐 𝑖 𝑐 𝜔 𝑙 (2.8) directly orientated to the axis of the pole flux, quadrature component (along the q-axis) of In (2.12), 𝜓 � �⃗ 𝑠𝑠 𝑝
	When the machine is connected to a mechanical load (such as a screw shaft for example), 𝜓 � �⃗ 𝑝 𝑠𝑠 is zero. Accordingly, the pole flux vector just has only the real direct component 𝜓 𝑝 .
	the ex ternal t orques co ming from t hese l oads can b e d enoted as 𝑇 𝑙𝑙𝑙𝑠 . T he m echanical From that, it follows:
	model of a PMSM based on t he fundamental principle of dynamics and is represented in
	(2.9).	𝜓 �⃗ 𝑝 𝑠𝑠 = 𝜓 𝑝𝑠 + 𝑗𝜓 𝑝𝑠 = 𝜓 𝑝 with 𝜓 𝑝𝑠 = 0		(2.13)
		a) Equivalent circuit � 𝜔 𝑙 = 𝐽 Substituting t he E quation (2.13) and E quation (2.12) into E quation (2.11), we wi ll b) Vector diagram 𝑟𝜃 𝑙 𝑟𝑡 (2.9) 𝑟𝜔 𝑙 𝑟𝑡 = 𝑇 𝑒𝑙 -𝑇 𝑙𝑙𝑙𝑠 -𝑓 0 𝜔 𝑙 obtain:
	Figure 2.5: Electrical model of one phase of PMSM + 𝑓 0 : the coefficient of viscous friction of the machine and load; With: 𝑢 �⃗ 𝑠 𝑠𝑠 = 𝑅 𝑠 𝚤 ⃗ 𝑠 𝑠𝑠 + 𝑟�𝐿𝚤 ⃗ 𝑠 𝑠𝑠 � 𝑟𝑡 + 𝑗𝜔 𝑒 (𝐿𝚤 ⃗ 𝑠 𝑠𝑠 + 𝜓 𝑝 )		(2.14)
	𝑏, 𝑐) corresponds to the phase of a PMSM; + 𝑢 𝑘 (the R MS value i s V) is t he stator vol tage a pplied be tween one pha se a nd t he neutral point; + 𝑒 𝑘 (the RMS value is E) is the induced electromagnetic force; + 𝑖 𝑘 (the RMS value is I) is the current in the phase 𝑘; 𝐿 is the equivalent cyclic RMS. It is synthesized by the inductance of the phase winding 𝐿 𝑠 + 𝐽: the total inertia of the machine; + 𝑇 𝑙𝑙𝑙𝑠 : the load torque + 𝑇 𝑒𝑙 : the electromagnetic torque generated by the machine. Calling 𝑇 𝑙𝑙𝑙𝑙𝑙 as the total load applied to the motor. Because the component 𝑠𝜔 𝑚 𝑠𝑙 = 0 at the steady state, the relation in (2.9) will be rewritten as shown in Equation (2.10). Let 𝑢 �⃗ 𝑠 𝑠𝑠 = ⎩ ⎪ ⎨ ⎪ ⎧𝑢 𝑠𝑠 = 𝑅 𝑠 𝑖 𝑠𝑠 + 𝑟(𝐿𝑖 𝑠𝑠 ) -𝜔 𝑒 𝐿𝑖 𝑠𝑠 𝑟𝑡 𝑢 𝑠𝑠 = 𝑅 𝑠 𝑖 𝑠𝑠 + 𝑟�𝐿𝑖 𝑠𝑠 � 𝑟𝑡 + 𝜔 𝑒 𝐿𝑖 𝑠𝑠 + 𝜔 𝑒 � 2 𝜓 𝑝 3 (2.15)
		and the mutual inductances 𝐿 𝑙 ; 𝑇 𝑒𝑙 = 𝑇 𝑙𝑙𝑙𝑙𝑙 = 𝑓 0 𝜔 𝑙 + 𝑇 𝑙𝑙𝑙𝑠	(2.10)
		+ 𝑅 𝑠 is the resistance of a phase of the stator. Normally, because the impedance of the
		inductance which de pends on t he r otating s peed (𝑍 𝐿 = 𝐿𝜔 𝑒 ) is m uch bigger t han
	the resistance: 𝐿𝜔 𝑒 ≫ 𝑅, the resistive drop can be neglected; In or der to get t he m odel of P MSM, ge neral stator vol tage e quation in (2.4) will b e
	transformed t o t he d -q reference f rame ( flux r eference f rame -Figure 2.4). This f rame rotates at the sam e sp eed w ith t he f lux v ector ( 𝜔 𝑒 ). T his r elation i s s hown i n E quation + In addition, three other angles are also shown on the vector diagram (Figure 2.5b): (2.11).
		𝜓 = (𝐸, 𝐼) � , 𝜑 = (𝑉; 𝐼) � and 𝛿 = (𝑉; 𝐸) � . For a maximum excitation flux Φ 𝑀 created by t he rotor in the gap, the effective value 𝑢 �⃗ 𝑠 𝑠𝑠 = 𝑅 𝑠 𝚤 ⃗ 𝑠 𝑠𝑠 + 𝑠𝑠 𝑟𝜓 �⃗ 𝑠 𝑟𝑡 + 𝑗𝜔 𝑒 𝜓 �⃗ 𝑠 𝑠𝑠 (2.11)
	of the electromagnetic force is proportional to speed and is presented in equation (2.7). The relationship between the stator and rotor flux can be described as follows:
		𝐸 = 𝑠𝑠 = 𝐿𝚤 ⃗ 𝑠 𝜓 �⃗ 𝑠 𝑠𝑠 + 𝜓 �⃗ 𝑝 𝑛 𝑝 𝛷 𝑀 √2 𝑠𝑠 𝜔 𝑙 = 𝑘 𝑒 𝜔 𝑙	(2.12)	(2.7)
		Mono-Inverter Dual-Parallel PMSM drive system Mono-Inverter Dual-Parallel PMSM drive system

  𝑛 𝑝 �𝜓 𝑠𝑠 𝑖 𝑠𝑠 -𝜓 𝑠𝑠 𝑖 𝑠𝑠 � = 𝑛 𝑝 �3/2𝜓 𝑝 𝑖 𝑠𝑠

	Equations (2.15) can be rewritten as follow:
	� 𝑟𝑖 𝑠𝑠 𝑟𝑡 𝑟𝑖 𝑠𝑠 𝑟𝑡	= -= -𝜔 𝑒 𝑖 𝑠𝑠 -𝑅 𝑠 𝐿 𝑖 𝑠𝑠 + 𝜔 𝑒 𝑖 𝑠𝑠 + 𝑅 𝑠 𝐿 𝑖 𝑠𝑠 +	1 𝐿 1 𝐿 𝑢 𝑠𝑠 -𝜔 𝑒 �3/2 𝑢 𝑠𝑠	𝐿 𝜓 𝑝	(2.18)
	With t he e quations i n (2.17) and (2.18), a c ontinuous state m odel w hich de scribes
	completely a PMSM in d-q coordinate is obtained. Equations (2.18) can be used to create
	the current state space model of a PMSM as follow:
			𝑠𝑠 𝑟𝚤 ⃗ 𝑠 𝑟𝑡	= 𝐴 𝑠𝑠 𝚤 ⃗ 𝑠 𝑠𝑠 + 𝐵 𝑠𝑠 𝑢 �⃗ 𝑠 𝑠𝑠 + 𝑃 𝑠𝑠
	𝐴 𝑠𝑠 = �	--𝜔 𝑒 -𝑅 𝑠 𝐿 𝜔 𝑒 𝑅 𝑠 𝐿	� ; 𝐵 𝑠𝑠 = �	1 0 𝐿		1 𝐿 0	� ; 𝑃 𝑠𝑠 = �	-𝜔 𝑒 � 0	2 3	𝐿 𝜓 𝑝	�	(2.19)
	From the mechanical equations of PMSM, we also have:
	⎩ ⎪ ⎨ ⎪ ⎧ 𝑟𝜔 𝑒 𝑟𝑡 𝑟𝑡 = 𝜔 𝑠 -𝜔 𝑒 = 𝑛 𝑝 2 �3/2𝜓 𝑝 𝑖 𝑠𝑠 𝐽 𝑟𝛿	-	𝑓 0 𝐽	𝜔 𝑒 -	𝑛 𝑝 𝐽	𝑇 𝑙𝑙𝑙𝑠	(2.20)
												(2.17)
					Mono-Inverter Dual-Parallel PMSM drive system

Table 2 .

 2

1: Optimal switching table for DTC control 2.3.3 Direct Predictive Torque Control (DPTC)

The f irst id eas a bout P redictive C ontrol a pplied to p ower c onverters s tarted in t he 1980's [HS83][KS83]. The pr inciple of t his s trategy i s ba sed on t he c alculation o f t he future s ystem be haviors a nd t he us e of a c ost function t o e valuate t hese be haviors. T he execution of the predictive algorithm can be divided in three main steps: + Updating t he st ates o f t he sy stem and est imating t he v ariables t hat c annot b e measured;

  2.16, t his i s a b lock d iagram d escribing t he master/slave configuration. A block named "Master choice" is added to the system in order to select the master machine. A ccording t o t his c onfiguration, a t e ach moment o f t ime, onl y t he machine considered as master is controlled. The other one, the slave machine, will operate in open loop under the voltage supplying mode.Like this, the value of the voltage supplied by the inverter is entirely due to the control of the master machine. The slave machine does not have any influence on this value.

Practically, according t o th is structure, it is n ot d ifferent c ompared w ith a normal PMSM d rive sy stem. T herefore, t he st ability o f t he master m achine w hich i s al ways controlled will be assured. However, it is possible that the slave machine, which is in open loop, c ould be uns table. T he pr oblem o f c hoosing t he a dequate m achine t o s et as t he master w hile r especting t he st ability co ndition o f t he co mplete sy stem n eeds t o b e considered.

Table 2 . 2 :

 22 Table of the evolution of torque and flux of two PMSMs in parallel

Mono-Inverter Dual-Parallel PMSM drive system

2.4.2.2 Optimal switching table for two machines

In this part, an optimal switching table, which allows selecting a suitable voltage vector

  𝒔𝟏 𝚫𝝍 𝒔𝟐 𝚫𝑻 𝒆𝟏 𝚫𝑻 𝒆𝟐 /𝑉 6 𝑉 6 /𝑉 1 𝑉 1 /𝑉 2 𝑉 2 /𝑉 3 𝑉 3 /𝑉 4 𝑉 4 /𝑉 5 /𝑉 2 𝑉 4 /𝑉 3 𝑉 5 /𝑉 4 𝑉 6 /𝑉 5 𝑉 1 /𝑉 6 𝑉 2 /𝑉 1 /𝑉 5 𝑉 1 /𝑉 6 𝑉 2 /𝑉 1 𝑉 3 /𝑉 2 𝑉 4 /𝑉 3 𝑉 5 /𝑉 4 /𝑉 3 𝑉 3 /𝑉 4 𝑉 4 /𝑉 5 𝑉 5 /𝑉 6 𝑉 6 /𝑉 1 𝑉 1 /𝑉 2

					𝑺 𝟏	𝑺 𝟐	𝑺 𝟑	𝑺 𝟒	𝑺 𝟓	𝑺 𝟔
	0	0	0	0	𝑉 5	𝑉 6	𝑉 1	𝑉 2	𝑉 3	𝑉 4
	0	0	0	1	𝑉 4	𝑉 5	𝑉 6	𝑉 1	𝑉 2	𝑉 3
	0	0	1	0	𝑉 4	𝑉 5	𝑉 6	𝑉 1	𝑉 2	𝑉 3
	0	0	1	1	𝑉 3	𝑉 4	𝑉 5	𝑉 6	𝑉 1	𝑉 2
	0 𝑉 5 0 1 0 0 1 0 1 𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7
	0	1	1	0	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7
	0 𝑉 3 1 1 1 1 0 0 0 𝑉 6 1 0 0 1 𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7
	1	0	1	0	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7	𝑉 0/7
	1 𝑉 2 1 0 1 1 1 0 0 𝑉 6	𝑉 1	𝑉 2	𝑉 3	𝑉 4	𝑉 5
	1	1	0	1	𝑉 1	𝑉 2	𝑉 3	𝑉 4	𝑉 5	𝑉 6
	1	1	1	0	𝑉 1	𝑉 2	𝑉 3	𝑉 4	𝑉 5	𝑉 6
	1	1	1	1	𝑉 2	𝑉 3	𝑉 4	𝑉 5	𝑉 6	𝑉 1

Table 2 .

 2 In t his case, w ith 6 b asic sect ors, t here are t welve co mbination sectors. According to these sectors and requirements about the voltage vector of each machine, another table will be built. Some general conditions also are inferred:

3: Table of optimal voltage vectors for two PMSMs Mono-Inverter Dual-Parallel PMSM drive system b) Two stator flux vectors are in two adjacent sectors

Table 2 .

 2 𝝍 𝒔𝟏 𝚫𝝍 𝒔𝟐 𝚫𝑻 𝒆𝟏 𝚫𝑻 𝒆𝟐 𝑺 𝟏𝟐 𝑺 𝟏𝟔 𝑺 𝟐𝟏 𝑺 𝟐𝟑 𝑺 𝟑𝟐 𝑺 𝟑𝟒

4 and Table 2.5 Mono-Inverter Dual-Parallel PMSM drive system

  𝚫𝝍 𝒔𝟏 𝚫𝝍 𝒔𝟐 𝚫𝑻 𝒆𝟏 𝚫𝑻 𝒆𝟐 𝑺 𝟒𝟑 𝑺 𝟒𝟓 𝑺 𝟓𝟒 𝑺 𝟓𝟔 𝑺 𝟔𝟏 𝑺 𝟔𝟓

Table of optimal voltage vectors for two PMSMs (𝑆 12 → 𝑆 34 ) Mono-Inverter Dual-Parallel PMSM drive system

4.2.3 Simulation tests

  In o rder to verify th e o ptimal ta ble o f D TC f or m ono-inverter dua l-parallel P MSM drive s ystem, a s imulation i s c arried out unde r t he M atlab/Simulink e nvironment. T he parameters of PMSM using in this test are summarized in Table2.6.

Table of optimal voltage vectors for two PMSMs (𝑆 43 → 𝑆 65 ) Mono-Inverter Dual-Parallel PMSM drive system 2.

  The f irst c omponent i s t he P MSM. T hrough t his pa rt, a m odel a s w ell a s t he pr oblems concerned to PMSM such as stability, control is studied.

	The s tructure m aster/slave a pplying t o t his ki nd of s ystem h as b een p resented. T he
	principle to de termine t he m aster machine i s proposed a nd pr oved ba sed on t he studies
	about the stability of PMSM in open loop.
	Another m ethod a lso ha s be en de veloped i n t his c hapter i s based on t he pr inciple of
	direct to rque control. D TC i s a n e ffective m ethod f or hi gh pe rformance m otion c ontrol

thanks to the fast torque response. It also can be applied for a mono-inverter dual-parallel PMSM drive system. DTC for two machines leads to the issue of building another optimal switching table. This table will allow controlling the two PMSMs at the same time under different condition of loads. Simulation results show that with an optimal lookup table we can find a solution for the control of mono-inverter dual-parallel PMSM.

table .

 .

Nevertheless, how t o c hoose only one ve ctor s uitable f or bot h m achines w ill be a difficult question and sometimes there is no solution.

Based on t he principle of DTC, the idea of Predictive Torque Control (PTC) has been proposed. In D TC, a n estimation model i s us ed t o e stimate t he c urrent t orque a nd f lux linkage of each machine. Mean while, P TC u ses a d iscrete model o f t he w hole sy stem t o predict i ts e volution i n t he future. Requirements f or al l the variables w hich need to be controlled such as torque, flux and currents … are considered in a cost function. This cost function will help us choosing the most suitable state of an inverter to apply to the system.

  𝑠 𝑉 𝑠 cos 𝛿 + 𝜔 𝑒 𝐿𝑉 𝑠 sin 𝛿 = (𝑅 𝑠 2 + (𝜔 𝑒 𝐿) 2 )𝐼 𝑠𝑠

						+ 𝜔 𝑒 𝑅 𝑠 �	2 3	𝜓 𝑝	(3.3)
	From this equation, the load angle 𝛿 will be calculated by the following equation:
			𝛿 = 𝛼 -arccos	⎝ ⎛	(𝑅 𝑠 2 + (𝜔 𝑒 𝐿) 2 )𝐼 𝑠𝑠 + 𝜔 𝑒 𝑅 𝑠 � 3 2 𝑉 𝑠 �𝑅 𝑠 2 + (𝜔 𝑒 𝐿) 2	𝜓 𝑝	⎠ ⎞	(3.4)
	with 𝛼 = atan �	𝜔 𝑒 𝐿 𝑅 𝑠	�. I n (3.4), t he a ppearance of " arcos" f unction c an b e l ead t o doub le
	value of load angle. Therefore, the condition of stability of a PMSM, which is presented in
			�𝑅 𝑠 2 + (𝜔 𝑒 𝐿 𝑠 ) 2 𝑖 𝑠0 +	𝜔 𝑒 𝑅 𝑠 � 3 2 �𝑅 𝑠 2 + (𝜔 𝑒 𝐿 𝑠 ) 2 𝜓 𝑟	< 𝑉 𝑠
							(3.1)
							2	𝜓 𝑝 𝑖 𝑠𝑠
						𝑇 𝑒𝑙 -𝑇 𝑙𝑙𝑙𝑠 =	𝑓 0 𝜔 𝑒 𝑛 𝑝
			𝐼 𝑠𝑠 =	-𝑉 𝑠 sin 𝛿 + 𝜔 𝑒 𝐿𝐼 𝑠𝑠 𝑅 𝑠	(3.6)
	To illu strate th ese th eoretical e quations, th e c alculations a bove w ill b e a pplied to a
	PMSM. Parameters of this PMSM are summarized in Table 2.6
			𝑅 𝑠				2.06(Ω)
			𝐿 𝑠𝑠 = 𝐿 𝑠𝑠 = 𝐿	9.15 × 10 -3 (𝐻)
			𝜓 𝑝				0.29(𝑊𝑏)
			Numbers of pole pairs �𝑛 𝑝 �	3
			Inertia (𝐽)				7.2 × 10

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM 𝑅 section 2.2.3, will be used to eliminate an unsuitable value: 𝛼 -𝜋 < 𝛿 < 𝛼.

Besides, i t i s n ecessary to sat isfy t he co ndition of "ar cos" f unction. I t m eans t hat i n order to let this function be meaningful, the relation in

(3.5

) must be respected. (3.5) Condition in (3.5) shows us a limit in operation of a PMSM. Under a specific value of 𝑉 𝑠 and 𝑇 𝑙𝑙𝑙𝑠 , speed of a PMSM cannot exceed a specific value. This limitation is illustrated in Figure 3.3. The d-current then can be calculated as in (3.6): -4 (𝑘𝑔. 𝑁 2 ) Table 3.1: Parameters of PMSM Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

  It can b e se en t hat t o o perate a P MSM at a g iven sp eed u nder a co nstant l oad, sev eral values of voltage can be supplied. However, thinking about if all these voltage values are really suitable or not. Will a PMSM operate stably? As a result, a small deviation model of PMSM will be considered in order to know which one is stable operating point[START_REF] Perera | A sensorless, stable V /f c ontrol m ethod f or pe rmanent-magnet s ynchronous motor drives[END_REF].Considering that a P MSM is operating at a st eady state with specific components: 𝑉 𝑠 , 𝐼 𝑠𝑠0 , 𝐼 𝑠𝑠0 , 𝜔 𝑒 and 𝛿 0 . P roposed t hat there i s a s mall c hange of l oad 𝛿𝑇 𝑙𝑙𝑙𝑠 . T his c hange makes all steady states also change in small quantities as in(3.7): 𝐼 𝑠𝑠0 + 𝛿𝑖 𝑠𝑠 𝑖 𝑠𝑠 = 𝐼 𝑠𝑠0 + 𝛿𝑖 𝑠𝑠 𝑢 𝑠𝑠 = 𝑈 𝑠𝑠0 + 𝛿𝑢 𝑠𝑠 𝑢 𝑠𝑠 = 𝑈 𝑠𝑠0 + 𝛿𝑢 𝑠𝑠 𝑇 𝑒𝑙 = 𝑇 𝑒𝑙0 + 𝛿𝑇 𝑒𝑙 𝜔 𝑒 = 𝜔 𝑒0 + 𝛿𝜔 𝑒 𝛿 = 𝛿 0 + 𝛿𝛿

	⎧	𝑖 𝑠𝑠 =
	⎪ ⎪
	⎨
	⎪ ⎪
	⎩

3.3.3.1 Discrete model of considered system 3

  .3.3.1.1 One-step predictive model of dual-parallel PMSMIn this part, the continuous model of PMSM in d-q reference frame which has already been presented in section 2.2.2.2 will be used to build the discrete model of dual-parallel

	𝑖 𝑠 𝑠𝑠 (𝑘 + 1) = 𝐴 𝑠 𝑠𝑠 (𝑘)𝑖 𝑠 𝑠𝑠 (𝑘) + 𝐵 𝑠 𝑠𝑠 (𝑘)𝑢 𝑠 𝑠𝑠 (𝑘) + 𝑃 𝑠 𝑠𝑠 (𝑘)

PMSM. For a PMSM, we already have:

Considering a lin ear, ti me-variant s ystem, t he v alues o f t he el ements o f t he matrices 𝐴 𝑠𝑠 , 𝐵 𝑠𝑠 and 𝑃 𝑠𝑠 are constant. However, in the case of a rotating electric machine, this is not tru e f or th e ro tating s peed 𝜔 𝑟𝑒 , a s i t c hanges its va lues d uring t he o peration of t he machine. M oreover, the ro tating s peed is a lso multiplicatively l inked with d ifferent s tate variables resulting in nonlinearity.

With a small enough sampling time 𝑇 𝑠 , it can be considered that the rotating speed 𝜔 𝑟𝑒 does not change in one sampling period. Thank to this, a first order approximation will be used to generate the discrete time model of a PMSM as shown in

(3.17)

  As mentioned ab ove, this d rive sy stem i ncludes t wo i dentical m achines. T herefore, using equation f rom (3.17), t he d iscrete cu rrent m odel of dual-parallel P MSM w ill b e obtained. 𝜃 𝑒1 (𝑘) and 𝜃 𝑒2 (𝑘) are the e lectrical p ositions o f d -q r eference f rame o f each machine at t ime 𝑘. I n t his work, t hese pos itions are m easured by r esolvers. T he 𝑉 𝐴𝑁 (𝑘), 𝑉 𝐵𝑁 (𝑘) and 𝑉 𝐶𝑁 (𝑘) voltage will be calculated from the discrete state model of the inverter

	2	𝜓 𝑝 𝐿	𝜔 𝑒	�
	in next section.
	3.3.3.1.2 A three-phase two-level inverter
	A s imple d iagram o f a th ree-phase t wo-level i nverter i s s hown i n Figure 3.9. The

(3.16) Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM The d -q vol tages w hich a re us ed in (3.18) are cal culated from t he p hase-to-neutral voltages from the inverter by us ing the Park's transformation. This relation is presented in (3.19)

In

(3.19)

, vector [𝑉 𝐴𝑁 (𝑘) 𝑉 𝐵𝑁 (𝑘) 𝑉 𝐶𝑁 (𝑘)] 𝑇 , w hich i s represented by the pha se-to-neutral v oltages will be expressed as functions of inverter's leg states (Equation (3.20)):

Table 3 . 2

 32 This m odel al lows p redicting t he b ehaviors af ter a sa mpling p eriod i f a sp ecific configuration of [𝑠 𝐴 (𝑘) 𝑠 𝐵 (𝑘) 𝑠 𝐶 (𝑘)] 𝑇 is used.Besides the model, one vital factor of predictive control is the cost function. The main objective o f it is to reduce t he e rror b etween t he p redicted an d t he r eference value.Moreover, t he c ost f unction also admits an y n ecessary term th at c ould re present a prediction f or system variables, s ystem c onstraint or s ystem r equirement. T his f lexibility allows ach ieving easi er more c ontrol ta rgets to i ncrease system p erformance (efficiency, In t hese c ost f unctions, onl y one , or t he c omponents of one va riable, are c ontrolled. This is the simplest case, and since only one type of variable is controlled, no w eighting factors ar e necessary. T he t ypical co st f unction f or t his case i s t he predictive c urrent control [RPS+07]. In this cost function, all the terms are composed of variables of the same nature (same uni t a nd order of m agnitude). M oreover, s ome a re a de composition of a

		⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑠𝑠1 𝑘+1 𝑖 𝑠𝑠1 𝑘+1 𝑖 𝑠𝑠2 𝑘+1 𝑖 𝑠𝑠2 𝑘+1 ⎦ ⎥ ⎥ ⎥ ⎤	= 𝔽(𝑘)	⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑠𝑠1 𝑘 𝑖 𝑠𝑠1 𝑘 𝑖 𝑠𝑠2 𝑘 𝑖 𝑠𝑠2 𝑘 ⎦ ⎥ ⎥ ⎥ ⎤	+ 𝔾(𝑘) �	2 3	ℙ(𝑘)𝔻 �	𝑠 𝐴 (𝑘) 𝑠 𝑐 (𝑘) 𝑠 𝐵 (𝑘)	� + ℍ(𝑘)	(3.22)
	3.3.3.2	Cost function

: Eight real states of a three-phase two-level inverter Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM power quality …). Since these terms can be of different physical nature (current, voltage, reactive po wer, s witching l osses, torque, f lux, e tc.), t hey can l ead to coupling ef fects between variables, or to overestimate the importance of one term respecting to the others in the c ost f unction, making t heir p resence not w orth, a nd e ven not c ontrollable. T his issue has been commonly dealt with by i ncluding weighting coefficients or weighting factors 𝜆, for each term of the cost function: 𝑔 = 𝜆 𝑥 |𝑥 * -𝑥 𝑝 | + 𝜆 𝑦 |𝑦 * -𝑦 𝑝 | + ⋯ + 𝜆 𝑧 |𝑧 * -𝑧 𝑝 | (3.23)

Depending on t he nature of the different terms involved in the formulation of the cost function, some typical types will be presented in the next section. 3.3.3.2.1 Typical cost functions A. Cost functions without weighting factors single va riable into t wo c omponents. T herefore, no weighting f actors a nd t heir corresponding tuning process is necessary. Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

Table 3 .

 3 . S ome e xamples a re: predictive current control with reduction of the switching frequency to improve efficiency [VCA+07], predictive current control with reduction of common mode voltage to prevent motor damage [VRRE08], and predictive current control with reactive power reduction to improve power quality [MAR05][VARP08]. The corresponding cost functions are listed in 3.

Table 3 . 3

 33

: Cost functions with secondary terms Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM 3.3.3.2.2 Cost function for two parallel PMSMs system

  Normally, t his va lue i s c lose t o z ero. H owever, f rom t he s imulation r esults, i t c an be realized that with DPTC value of cost function 𝑔 is not equal to zero. The torque from the machine is different compared with the reference value from speed controller. It le ads to the problem that values of q-currents 𝑖 𝑠𝑠1 , 𝑖 𝑠𝑠2 are different with their references 𝑖 𝑠𝑠1 𝑟𝑒𝑟 , 𝑖 𝑠𝑠2 𝑟𝑒𝑟 .

It is the same with d-currents whose va lues do not c ome t o z ero. A problem poi nted out here is whether the cost function which is presented in (3.

3.3.2.2

) can come to zero or not.

  𝜔 𝑙1 = 𝜔 𝑙2 and so t he el ectrical speeds are: 𝜔 𝑒1 = 𝜔 𝑒2 ;The basic electrical equations at steady state of PMSM can be rewritten to calculate the magnitude of voltage vector 𝑉 𝑠 :

	+ � -𝑉 𝑠 sin(𝛿 1 ) = -𝜔 𝑒1 𝐿 1 𝑖 𝑠𝑠1 𝑟𝑒𝑟 𝑉 𝑠 cos(𝛿 1 ) = 𝑅 𝑠1 𝑖 𝑠1 𝑟𝑒𝑟 + 𝜔 𝑒2 �3/2𝜓 𝑝1	⇒ 𝑉 𝑠 2 = �𝜔 𝑒1 𝐿 1 𝑖 𝑠𝑠1 𝑟𝑒𝑟 �	2	+ �𝑅 𝑠1 𝑖 𝑠𝑠1 𝑟𝑒𝑟 +
	𝜔 𝑒1 �3/2𝜓 𝑝1 �	2	;
	+ � -𝑉 𝑠 sin(𝛿 2 ) = -𝜔 𝑒2 𝐿 2 𝑖 𝑠𝑠2 𝑟𝑒𝑟 𝑉 𝑠 cos(𝛿 2 ) = 𝑅 𝑠2 𝑖 𝑠𝑠2 𝑟𝑒𝑟 + 𝜔 𝑒2 �3/2𝜓 𝑝2	⇒ 𝑉 𝑠 2 = �𝜔 𝑒2 𝐿 2 𝑖 𝑠𝑠2 𝑟𝑒𝑟 �	2	+ �𝑅 𝑠2 𝑖 𝑠𝑠2 𝑟𝑒𝑟 +
	𝜔 𝑒2 �3/2𝜓 𝑝2 �
						𝑒1 𝐿𝑖 𝑠𝑠1 𝑟𝑒𝑟 �	2	+ �𝑅 𝑠 𝑖 𝑠𝑠1 𝑟𝑒𝑟 +
	𝜔 𝑒1 �3/2𝜓 𝑝 �	2	, a n t he s econd one i s �𝜔 𝑒2 𝐿𝑖 𝑠𝑠2 𝑟𝑒𝑟 �	2	+ �𝑅 𝑠 𝑖 𝑠𝑠2 𝑟𝑒𝑟 + 𝜔 𝑒1 �3/2𝜓 𝑝 �
						min(𝑔)
						𝑠. 𝑡 �	0 ≤ 𝑉 𝛼 2 + 𝑉 𝛽 2 ≤ 𝑉 max 2 𝑉 𝛼𝑙𝑙𝑛 ≤ 𝑉 𝛼 ≤ 𝑉 𝛼𝑙𝑙𝑥 𝑉 𝛽𝑙𝑙𝑛 ≤ 𝑉 𝛽 ≤ 𝑉 𝛽𝑙𝑙𝑥	(3.28)
	values will be generated 𝑖 𝑠𝑠1 𝑟𝑒𝑟 , 𝑖 𝑠𝑠2 𝑟𝑒𝑟 and 𝑖 𝑠𝑠1 𝑟𝑒𝑟 ≠ 𝑖 𝑠𝑠2 𝑟𝑒𝑟 , the two currents 𝑖 𝑠𝑠1 𝑝 , 𝑖 𝑠𝑠2 𝑝 must run to
	these va lues. M oreover, t he s um �𝑖 𝑠𝑠1 𝑝 �	2 + �𝑖 𝑠𝑠2 𝑝 �	2	must comes to zero va lue too ⇔
	�	𝑖 𝑠𝑠1 𝑝 → 0 𝑖 𝑠𝑠2 𝑝 → 0	.
		Now, this state will be represented by following features:
		+ �	𝑖 𝑠𝑠1 = 0; 𝑖 𝑠𝑠2 = 0 𝑖 𝑠𝑠1 = 𝑖 𝑠𝑠1 𝑟𝑒𝑟 ; 𝑖 𝑠𝑠2 = 𝑖 𝑠𝑠2 𝑟𝑒𝑟
		+ because the load of each machine is different → 𝑖 𝑠𝑠1 𝑟𝑒𝑟 ≠ 𝑖 𝑠𝑠2 𝑟𝑒𝑟 ;
				� 𝛿 1 = 𝑐𝑜𝑛𝑠𝑡; 𝛿 2 = 𝑐𝑜𝑛𝑠𝑡 𝛿 1 ≠ 𝛿 2	;

. It m eans t hat, u nder di fferent load c onditions, w hen t he two d ifferent r eference + at st eady s tate, t he l oad an gles o f b oth m achines a re co nstant and di fferent:

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM + mechanical speeds of t wo machines ar e eq ual:

2

In this work, the two considered machines are identical. It means that, the parameters of both machines are the same. Therefore, from two equations above, we will obtain two different va lues of t he vol tage s upply: t he f irst o ne is �𝜔 2 . It is clear that two expressions give us two different values. It is unreasonable because we just have only one voltage supply.

Table 3 .

 3

			Fundamental Value (A)			THD (%)
		DPTC	PTC-SS	DPTC	PTC-SS
		MC1	MC2	MC1	MC2	MC1	MC2	MC1	MC2
	𝜔 1	2.675	8.043	2.635	8.028	21.62	14.37	0.79	0.52
	𝜔 2	2.346	7.988	2.326	7.896	27.56	16.19	1.04	0.61
	𝜔 3	2.227	7.85	2.182	7.851	26.70	15.15	1.22	0.66

5: The fundamental and THD values

  The experimental platform and 𝜃 𝑙2 . T hese c ards w ork m ainly around t he component AD2S1200 developed by Analog Devices[START_REF] Analog D Evice | AD2S1200 12 -Bit R /D C onverter w ith R eference Oscillator[END_REF]. This is a complete 12-bit resolution tracking resolver-to-digital converter, i ntegrating a n on-board programmable s inusoidal os cillator t hat pr ovides s ine wave excitation for resolvers.The sy stem al lows o btaining t he v alues o f 𝜃 𝑙1 and 𝜃 𝑙2 is r epresented t hrough t he scheme in

Table 1 :

 1 Résumé des différentes structures

	Nom de la structure	MS(m,n)	Nombre de bras par machine	Relation (𝑉 𝑁𝑘 ) max (𝑉 𝑂𝑘 ) max	Vitesse	Volume	Avantages	Notes Inconvénients
	Classique MS(3𝑛, 𝑛)	3	1	indépendant	grand	machines indépendantes	non partagée
	Bras commune	MS�	4𝑛 2	, 𝑛�	2	2 3	Ω 1 = Ω 2 or Ω 1 = -Ω 2	moyen	facile à implanter		sur-courant dans le bras commun
	Point milieu	MS(2𝑛, 𝑛)	2	1 √3	indépendant moyen	applicable à une machine	accès au point neutre
	Mix	MS�	3𝑛 2	, 𝑛�	1.5	1 √3	Ω 1 = Ω 2 -Ω 2 or Ω 1 =	petit	Idem		Idem
	Parallèle	MS(3, 𝑛)	3 𝑛	1	Ω 1 = Ω 2	petit	facile à implanter		sur-courants dans tous les bras

  Selon cet te structure, il ex iste t oujours u ne se ule m achine co mmandée, à u n i nstant donné. Par conséquent, le volume de calcul sera l e même que dans le cas d'une machine seule et sera plus faible que dans le cas où on pilote les deux machines. Sur la Figure 25, la réponse en vitesse des deux MSAPs est tracée et comparée avec la valeur de référence. Determine 𝑉 𝛼 , 𝑉 𝛽 , 𝑉 𝑟𝑒𝑟 and the angle 𝛼 between the voltage vector and the 𝛼-axis; + Step 2: Determine time duration 𝑡 1 , 𝑡 2 and 𝑡 0 ; + Step 3: Determine the switching time of each power electronic switch (𝑠 𝐴 , 𝑠 𝐵 , 𝑠 𝐶 ) A.2.1 Determine 𝑽 𝜶 , 𝑽 𝜷 , 𝑽 𝒓𝒆𝒇 and angle 𝜶 Considering t he 𝛼𝛽 coordinate, t he 𝛼-axis is chosen a lign w ith t he 𝑟-axis. T he link

	between two coordinate is shown in Equation
	From t his f igure, t he vol tage ve ctor 𝑉 𝑠 ���⃗ can be represented i n 𝛼𝛽 frame a ccording t o t he
	equation in (A.3):

"FMINCON" and SVM technique + Step 1:

  Technical data of the experimentalB.5.3 Limit switches ........................................................................................ 190 9 T B.69 T 9 T The Human-Machine Interface9 T............................................................................ 191

Table B .

 B 1: Technical parameters of the auto-transformer Technical data of the experimental the inverter are s upplied w ith a D C vol tage of 15𝑉. T he c ontrol s ignals f or d rivers th at provide switching commands use the C-MOS technology(0 -15𝑉). case of lack of power or an error occurring on one of the control signals, operatingsystems are blocked and an error signal is generated. If the error signal is not canceled, the inverter is blocked and the security of the system is assured. The driver is able to detect a short circuit in approximately 4𝜇𝑠, the IGBT can withstand a short circuit current up to 10 times the maximum current value (10 × 50𝐴), with a voltage 1200𝑉 for 10𝜇𝑠.In order to protect the inverter against short circuits when IGBT of the same leg closes or ope ns, the de vice i s integrated t o a nother s ecurity m odule t hat ge nerates a d ead t ime (𝑡 𝑠𝑒𝑙𝑠 ≈ 6𝜇𝑠) for e ach s tatus c hange. T he c ontrol s ignals applied t o t he dr ivers are inhibited du ring this time. T he s witching pe riod i s c hosen 𝑇 𝑠𝑤 = 100𝜇𝑠, t he d ead t ime corresponds to 6% of this period. It has no influence on the mechanical part of the system and few harmonics induced on t he electrical quantities also do not play a role with regard to the study.

	Parameters Primary voltage In Parameters Value Unit 400 V Value	Unit
	Secondary voltage 0 -450 V
	Nominal voltage 325	V
	Capacity	15600 VA
	Rated current	4.3	A
	Intensity	20	A
	Rated Torque	1.4	Nm
	Rated speed	6250	rpm
	Motor constant	0.32 Nm. A -1
	EMF constant	0.27 Vs. rad -1
	Resistance	0.955	Ω
	Inductance	1.65	mH
	Peak current	10	A
	Peak torque	3.1	Nm
	Maximum speed 11640	rpm
	Pairs of poles	4

The v oltage i nverter i s composed o f t hree legs. E ach l eg u ses t wo IGBT sw itches, which are bidirectional in current, controllable. SKHI22 divers which control each leg of

Table B .

 B

2: Technical parameters of PMSM (given by manufacturer)

Table B .

 B 3: Characteristics of holding brakeTechnical data of the experimental On 𝐸 𝑠𝑙𝑙𝑙 , a reference sinusoidal voltage 𝑉 𝑟 with high frequency (10 kHz in the case of study) i s a pplied. A magnetic f ield is c reated a nd a n i nduced voltage a ppears acr oss t he primary winding �𝑉 𝑝 �. Across 𝐸 𝑠1 , 𝐸 𝑠2 , t wo signals of t he s ame frequency a nd a mplitude varying according respectively cos 𝜃 and sin 𝜃 are then induced.These signals denoted 𝑉 𝑐𝑙𝑠 and 𝑉 𝑠𝑙𝑛 are analyzed by an analog device AD2S1200. This component, which also c reates t he s ignal 𝑉 𝑟 , i s located o n a car d ca lled "r esolver c ard".Signals are transferred to the two resolver card by two shielded cables.

	Parameters	Value Unit
	Braked voltage	24	V
	Braked power	11	W
	Holding torque of the brake 1.5	Nm
	The power supplying to both the two motors and brakes is carried out by means of two
	shielded cables.

  𝑉 𝑟𝑙𝑙𝑒𝑠 • 230 𝑉 𝐴𝐶 , 50/60 Hz or 310 𝑉 𝐷𝐶 with 𝑈 𝑍𝐾 𝑙 𝐶 as from 35 𝑙 𝐶 Storage temperature -25 𝑙 𝐶 bis 60 𝑙 𝐶 Connecting cable to motor Max. 50m, screened cable C'<200pF/m Depending on the system additional external Technical data of the experimental mains filtes may be necessary in the power supply cables in order to comply with the EMC guidelines. By experience this applies to motor cable lengths > 5m. 𝑅 𝑙 = 20 𝑘Ω, offset adjust ± 0.1 V • Ain1, Ain1/ ± 10V, 𝑅 𝑙 = 20 𝑘Ω, offset adjust ± 0.1 V

	SEC-AC-305 General logic output	Electrically isolated
	Power supply • Dout0: ready-to-operate	• 24 V, 100 mA via external 24 V supply
	• Dout1: freely configurable • supply • 24 V, 100 mA via external 24 V supply • 𝑉 𝐶𝐶 • 24 𝑉 𝐷𝐶 ± 20% (Power supply of the electronics) • Current consumption 24 V • Approx. 0.35A Rated output • With DC supply 1500 VA • With AC supply 1000 VA Peak power max. 10s 3000VA Intermediate circuit voltage Max. 340 V DC, 5 A Rated current per phase 5 𝐴 𝑒𝑟𝑟 Peak current per phase max. 10s 10 𝐴 𝑟𝑟 Braking chopper Internal resistor (external extension only via additional braking module) 100 Ω • Pulse output 1,3 kVA • Continuous output 25 W • Response threshold 375 • Load tolerance 0.5 : 25 s Ambient temperature 0 𝑙 𝐶 bis 50 𝑙 𝐶 At rated output 1kW; motor current 5𝐴 𝑒𝑟𝑟 under following conditions: output reduction Min. 1500V Nominal value inputs for speed and current • Ain0, Ain0/ ± 10V, Monitor settings 2 analogue outputs with 8-bit resolution at X1 • Output level ± 10V voltage output, short-circuit resistant • Representable variables Freely standardized, e.g nominal and actual current values, rotary angle electric or mechanical, nominal and actual speed value General logic inputs Isolated, 12 …30 V, active high • Dino • Bit 0 \ • Din1 • Bit 1, \ target selection for positioning • Din2 • Bit 2, /16 destinations can be selected • Din3 • Bit 3/ from destinations window • Din4 • Control input for final output stage enable at high • Din5 • Regulator enable at high, quit at low • Din6 • Limit switch input 1 • Din8 • Control signal start positioning • Dout2: freely configurable • 24 V, 100 mA via external 24 V supply • Dout3: freely configurable • 24 V, 100 mA via external 24 V supply • Dout4: holding brake • 24 V, 500 mA via external 24 V supply Monitor settings • Overvoltage in intermediate circuit • Approx. 400 V • Undervoltage in intermediate circuit • Programmable, can be switched off • Overcurrent in intermediate circuit • Short-circuit monitoring • Thermo-protection of motor • Normally-closed contact Displays on device • Ready-to-operate • LED • Error and status messages • Seven-segment display Terminal interface Serial • Level • RS 232, 9600 … 57600 bit/s • Plug connector • 9-pin. D-sub Interference emission EN 50 081 part 2, basic standard industrial area Immunity to interference EN 50 081 part 2, basic standard industrial area 30𝑊/ Voltage endurance for cable • Din7 • Limit switch input 2 Table B.5: Technical parameters of PMSM (given by manufacturer)
	• Din9	• Control signal Sync for coupling in master-
		slave mode
	• Reset (electrically)	Reset button on the front
	Technical data of the experimental Technical data of the experimental

Table B.5 Figure B.2: Variable speed drive SEC-AC-305 (Festo) • Thermo-protection of final output stage • 100P o P C heat sink temperature • Fault in rotary angle sensor • Common error

Hardware part of the control B.5.1 DS1103 PPC Controller Board The

  c ontrol boa rd us ed i n t his w ork i s D S1103 P PC C ontroller B oard de veloped b y dSPACE.All technical specifications of this board is summarized in TableB.7

	EAMM-U [10] Motor 1sampling rate timer (decrementer)	(consisting of: housing, clamping component, clamping sleeve, toothed belt gearwheel, toothed belt) Motors specially matched to the axis, with or without gearing, with or • 32-bit down counter • Reload by software • 30-ns resolution
	EMMS [11] Axial kit 1 timer base counter	without brake For axial motor attachment • 64-bit up counter • 30-ns resolution
	EAMM-A		(consisting of: coupling housing, clamping component, motor flange) • 3 timer interrupts
	[12] Slot cover B/S		• 7 incremental encoder index line interrupts For protecting against the ingress of dirt • 1 UART (universal asynchronous receiver and
	[13] Proximity sensor Interupt controller G/H/I/J/N	transmitter) interrupt For use as a signal or safety check • 1 CAN interrupt • 1 slave DSP interrupt
	[14] Cable with socket V	• 2 slave DSP PWM interrupts For proximity sensors • 1 host interrupt
	[15] Slot nut for mounting slot Y	• 4 external interrupts (user interrupts) For mounting attachments • 16 multiplexed channels equipped with 4 sample
	[16] Central support M [17] Foot mounting F Channel	& hold A/D converters(4 channels belong to one For mounting the axis A/D converter. 4 consecutive samplings are necessary to sample all channels belonging to one A/D converter.) For mounting the axis • 4 parallel channels each equipped with one sample
				& hold A/D converter
		• Note: 8 A/D converter channels (4 multiplexed Table B.6: Brief description of a ballscrew axis and 4 parallel) can be sampled simultaneously
	Resolution B.5 Parameter A/D Converter Input voltage range	Specification • ± 10𝑉 • 16-bit
		PowerPC Type		• PPC 750GX
		CPU clock		• 1 GHz
				• 32 KB level 1 (L1) instruction cache
	Processor	Cache		• 32 KB level 1 (L1) data cache
				• 1 MB level 2 (L2)
		Bus frequency		• 133 MHz
		Temperature sensor	• Reads actual temperature at the PPC
	Memory	Local memory Global memory		32 MB application SDRAM as program memory, cached 96 MB communication SDRAM for data storage and data exchange with host
				• One 32-bit down counter
				• Reload by software
		2 general-purpose timers	• 15-ns resolution • One 32-bit up counter with compare register
				• Reload by software
				• 30-ns resolution
				Technical data of the experimental

Predictive Torque Control -"Split and Seek" (PTC-SS) Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

T Les8 T courants dq 8 T des d eux8 T 8 T machines8 T 8 T sont r eprésentées su r8 T 8 T les 8 T Figure 26 8 T et8 T Figure 27. 8 T Parce que8 T seulement la machine maître est 8 T contrôlée8 T , la réponse de cette machine est bonne. 8 T Nous pouvons voi r que 8 T 8 T le 8 T courant s ur l 'axe q 8 T de8 T 8 T MS18 T 8 T suit t oujours8 T 8 T la r éférence8 T , et 8 T le 8 T courant sur l'axe d est maintenu 8 T proche de zéro.8 T Cependant, c ontrairement à8 T 8 T la8 T 8 T SM18 T , 8 T la m achine e sclave8 T 8 T (8 T SM2) 8 T ne r especte p as les8 T 8 T références.8 T 8 T La valeur8 T 8 T du courant sur l'axe d8 T 8 T est différente de zéro aussi8 T . Conclusions L'objectif principal d e cet te t hèse est l'application d 'une st ratégie de co mmande prédictive à un s ystème d' entraînement qui s e compose de deux m achines s ynchrones à aimants p ermanents ( MSAP) r eliées en p arallèle al imentées p ar u n o nduleur t riphasé d e l'onduleur à deux niveaux. Plusieurs an alyses théoriques et d es r ésultats d e si mulation o nt ét é ef fectués af in d e vérifier l'effet de cette approche. Les premiers résultats de l'expérience ont montré que le système d'entraînement peut fonctionner avec cette méthode. Bien que les résultats obtenus puissent être améliorés, la méthode PTC-SS constitue une nouvelle approche de la stratégie de commande prédictive. Sur la base de cette méthode, il est possible de développer une recherche dynamique afin de trouver le vecteur de tension le plus adéquat tout en cherchant à réduite le temps de calcul. Ceci fera l'objet de recherches futures.

Technical data of the experimental

List of symbols

𝜓 𝑝

Permanent magnet flux (𝑊𝑏)

𝜓 𝑠

Predictive Torque Control with Mono-Inverter Dual-Parallel PMSM

The three known parameters will be chosen as follow:

+ the voltage 𝑉 𝑠 is considered from 0 to 𝑉 𝑙𝑙𝑥 = 𝑉 𝐷𝐶 √3 ⁄ ≈ 312(𝑉);

+ the f requency 𝜔 𝑠 is c onsidered f rom 0 t o no minal va lue 𝜔 𝑛𝑙𝑙𝑙𝑛𝑙𝑙 = 300 × 𝜋 ≈ 942(𝑟𝑟𝑟. 𝑠 -1 );

+ three load condition: � 𝑇 𝑙𝑙𝑙𝑠 = 0(𝑁𝑁) 𝑇 𝑙𝑙𝑙𝑠 = 5(𝑁𝑁) 𝑇 𝑙𝑙𝑙𝑠 = 10(𝑁𝑁)

;

In Figure 3.1, currents in dq-axis of this PMSM are shown. Under the same values of (𝜔 𝑠 , 𝑇 𝑙𝑙𝑙𝑠 ), a lthough the vol tage 𝑉 𝑠 increase, q-current doe s not c hange. The q-current at steady state just depends on frequency 𝜔 𝑠 and load torque 𝑇 𝑙𝑙𝑙𝑠 .Otherwise, the increase of 𝑉 𝑠 makes d-current change. At s mall range of s peed, t he c hange of d -current i s very big.

Under s tandstill c ondition (𝜔 𝑠 = 0 𝑟𝑟𝑟. 𝑠 -1 ) and l oad t orque e quals t o z ero (𝑇 𝑙𝑙𝑙𝑠 = 0𝑁𝑁), the change is from 0 to ≈ 150 𝐴. At higher speed, the change is smaller. Moreover, when the speed increases (with the same 𝑉 𝑠 and 𝑇 𝑙𝑙𝑙𝑠 ), the d-current has a tendency toward negative value (the field weakening region).

On t he ot her ha nd, m eanwhile q -current m uch de pends on t he l oad be cause i t represents for electromagnetic torque, d-current changes a little under three values of load.

In Figure 3.2, l oad angle i s s hown. A s i ts na me, va lue of load a ngle 𝛿 will c hange according to the load torque. At a specific pair of value (𝜔 𝑠 , 𝑉 𝑠 ), load angle correlative to heavier load will bigger.

The limit in operation of this PMSM is illustrated in Figure 3. U Problem:

We need to find the minimum value of the cost function:

The constrain condition:

U Implementation: + Update the current state of the system: 𝜃 𝑙1 (𝑘), 𝜃 𝑙2 (𝑘), 𝑖 𝑙𝑏𝑐1 (𝑘), 𝑖 𝑙𝑏𝑐2 (𝑘);