model predictive torque control on an electric variable...

Emile De Brabandere

Transmission for Hybrid Electric VehiclesModel Predictive Torque Control on an Electric Variable

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Jan MelkebeekDepartment of Electrical Energy, Systems and Automation

Master of Science in Electromechanical EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: Joachim DruantSupervisors: Prof. dr. ir. Jan Melkebeek, Dr. ir. Frederik De Belie

Permission of usage

The author gives permission to make this master dissertation available for consultation and tocopy parts of this master dissertation for personal use. In the case of any other use, the copyrightterms have to be respected, in particular with regard to the obligation to state expressly thesource when quoting results from this master dissertation.

Toelating tot bruikleen

De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen vande masterproef te kopieren voor persoonlijk gebruik. Elk ander gebruik valt onder de bepalingenvan het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijkte vermelden bij het aanhalen van resultaten uit deze masterproef.

Emile De Brabandere, June 5th, 2015

i

Preface

Motivation

When I had to make a choice about the subject of my master thesis about one year ago, Idetermined three criteria to be fulfilled. Primarily, the subject of the thesis had to be in a fieldof my interest. Secondly, I wanted to be sure to have a good supervisor, and - even more - agood counsellor, to support the progress of my work. And finally, my preferences tended towardsnovel research, rather than expanding an already widely researched subject.

The thesis I chose answers to all of these criteria. Vehicular technology - automotive in particular- has always been a subject of interest to me. The application area of the subject being hybridelectric vehicles (HEVs), the first criterion was met. When I learned that my counsellor wasactive on the same subject in the form of a PhD. research, I knew that my counsellor would beclosely involved in the subject. Additionally, another student would do a thesis on a differentaspect of the subject, such that I knew many people would be involved in the subject, thusensuring a supportive framework for the research of my thesis. The research on the topic isvery active, and its application area (HEVs) is gaining share in the automotive market. Thetechnology studied is relatively new, and is still in its development phase before being appliedin real applications, thus meeting the final criterion.

Acknowledgments

Prof. Dr. Ir. J. Melkebeek, who, as president of the department and director of EELAB, madeit possible for this research to take place. Who, as a teacher on the topic of electrical machines,inspired me to choose for a MSc. in electrical power engineering. And above all, as a consequenceof his courses, provided me with the necessary knowledge to complete this master dissertation.Dr. Ir. F. De Belie, for evaluating the progress of this work with a critical eye, and providingvaluable feedback in order to redirect the points of focus in the research.Ir. J. Druant, my counsellor, for being very supportive of my everyday efforts on this work.Being available at any time for questions and additional explanations, actively helping to solveproblems, and providing valuable input in brainstorm sessions on what to do next. I sincerelyhope that my effort in this work may be of any value to his PhD. research on the topic of electricvariable transmissions.My classmates, Steven, Lucas, Thomas, and Thibaut for chearing up the atmosphere in thelab, and supporting each other on many occasions where one lost his heart due to one of manysetbacks. With little tips and tricks, we have saved each other some efforts.My parents, for giving me the opportunity to have an education, and providing me with every-

iii

thing within their reach in order to facilitate the process of successfully finishing them.My girlfriend, Charlotte, for putting up with my late night working on this very text, especiallywhen that meant postponing her birthday celebration till after the submission deadline of thiswork. And of course, for being there as a mental support, brightening those valuable momentsoff.Last but not least, I wish to acknowledge my late grandfather, who passed away during the real-isation of this work, for inspiring me into studying engineering. Despite his medical background,his ingenuity in everyday life struck me, and very soon made me decide engineering was what Iwanted to do.

Emile De Brabandere, June 5th, 2015

iv

Model Predictive Torque Control onan Electric Variable Transmission forHybrid Electric Vehicles

by

Emile De Brabandere

Master’s dissertation submitted in order to obtain the academic degree of Master of Science inElectromechanical Engineering

Academic year 2014-2015

Supervisors: Prof. dr. ir. J. Melkebeek, Dr. ir. F. De BelieCounsellor: Ir. J. DruantFaculty of Engineering and ArchitectureGhent University

Department of Electrical Energy, Systems, and AutomationChairman: Prof. dr. ir. J. Melkebeek

Problem statement

Hybrid electric vehicles (HEVs) are considered as the best temporary solution to bridge the gapbetween the petrol fueled car and the full electric car. To achieve a high efficiency, the power tothe wheels of a HEV is split into a part coming from the internal combustion engine (ICE) anda part exchanged with the battery using a pair of electric motor/generators. This power splitbetween the ICE and the electric motors/generators is nowadays performed using a planetarygear system. The aim of an electric variable transmission (EVT) is to replace the two electricmotors and the gear system of a conventional HEV into one electromagnetic power split device.The aim of this master thesis is to control the torque and the flux linked to the different rotorsof an EVT using a model based predictive control (MBPC) approach. This method has theadvantage of very fast dynamics and flexibility to control many variables at once. The controlwill be tested in a hardware-in-the-loop simulation where the EVT is emulated using an FPGA(field programmable gate array).

Chapter 2 elaborates on the functionalities and applications of the EVT, presents the differentimplementation forms of the EVT, and compares them. In chapter 3 the control algorithm is

v

explained and tis suitability for application to the EVT evaluated. Chapter 4 presents the math-ematical derivation of a machine model for the EVT, which is required both for the controllerand emulating the EVT. In chapters 5 and 6 the applied control strategy is evaluated with Mat-lab/Simulink simulations. Different aspects of the control method are studied, and additionalfeatures are presented. Finally, chapter 7 demonstrates the feasibility to realise MBPC on anEVT using an FPGA.

Keywords: model based predictive control, electric variable transmission, hybrid electric vehi-cles, FPGA, predictive torque control

vi

Model Predictive Torque Control on anElectric Variable Transmission for Hybrid

Electric VehiclesEmile De Brabandere

Supervisors: ir. Joachim Druant, dr. ir. Frederik De Belie, and prof. dr. ir. Jan Melkebeek

Abstract—In this article a method for controlling aninduction machine (IM) based electric variable trans-mission (EVT) is presented. The proposed algorithm isa finite-set model based predictive (FS-MBPC) controlalgorithm, which will be used to control the torques onthe interrotor and rotor, and the flux level. The flexibilityof this control algorithm allows for including otheraspects in the control as well. Through simulations inMatlab/Simulink, the effectiveness of the control methodand the proposed extensions is evaluated. Finally, thereal time applicability of the control method is verifiedby implementing the control method on FPGA. As noreal EVT is available, the machine has to be emulated inthe FPGA as well such that hardware-in-the-loop (HIL)simulations are obtained.

Index Terms—model based predictive control, electricvariable transmission, hybrid electric vehicles, FPGA,predictive torque control

I. INTRODUCTION

Whereas full electric vehicles are struggling withdriving range, hybrid electric vehicles (HEVs) areregarded as the intermediate solution to bridge thegap between fossil fueled and full electric vehicles. InHEVs the power delivered to the wheels of the vehiclehas two sources: a fossil fuel tank combined with aninternal combustion engine (ICE), and a battery com-bined with an electric motor. To join these power flows,current HEVs (such as the Toyota Prius) use a plan-etary gear set. In recent years, an induction machinebased electromagnetic converter called EVT has beenpresented [1][2] to take over the functionality of themechanical planetary gear set and combine numerouscomponents of the THS into a single machine. Eversince this EVT has been presented, other researchershave presented alternative system configurations. Theproposed systems can be divided in two main groups,depending on whether or not permanent magnets (PMs)are used in its construction. The more recent researchtends towards PM based EVTs as they exhibit highertorque densities. This article considers the originalEVT, which is an IM based machine. Fig. 1 shows

the cross section of the considered configuration aspresented in [3].

Fig. 1: Induction machine based EVT [3]

Fig. 2 shows how the EVT fits in the the drivetrainof a HEV. The ICE is mounted to the primary shaft(which is connected to the inner rotor, further calledrotor), and the load - the wheels of the vehicle - areconnected to the secondary shaft (connected to theouter rotor, further called interrotor). The controlleris used to determine the switching actions of theconverters supplying the windings of the rotor and thestator. In this article, abstraction is made of all thecomponents surrounding the EVT: the ICE, battery,converterss, and load have all been modeled as idealcomponents. The converters have been chosen to betwo level voltage source inverters without modulation.

II. MODEL BASED PREDICTIVE CONTROL

The additional degree of freedom (a second rotat-ing part) compared to conventional electric machines

LOAD!

EVT!

ICE!

PEC 1!

PEC 3!

+!–!

BATT.!

REF.!

MEAS.!

CO

NT

RO

LLER!

Fig. 2: Overview of the considered system

makes the control on the machine more challenging.MBPC has been shown to be very suitable for multipleinput-multiple output systems as the EVT. MBPC usesa model of the system in order to predict the result ofcontrol actions on the system’s future behaviour. Asthe name suggests, a model of the system is requiredfor the controller.

A. Finite-Set Model Based Predictive Control

The discrete nature of the converters (64 possibleswitching states for both converters combined) allowsto implement a specific type of MBPC, called finiteset MBPC (FS-MBPC). The control method consistsof three steps:estimation: based on measurements at the currentupdate instant k an estimation is made for the state atthe next update instant k+1. The estimation is based onthe optimal control action, determined in the previousexecution of the algorithm.prediction: starting from the estimated values at k+1,the output at update-instant k+2 is calculated for everypossible switching state.optimisation: for each switching state a cost functionis evaluated. The state with minimum cost is appliedat time instant k+1. All other calculations are clearedand the algorithm is iterated.

The variables to be controlled are the torque on boththe primary and the secondary axis and the flux level.The cost function can be defined by the user and caninclude many different aspects:

• assigning a greater cost to greater deviations byusing exponential functions (e.g. squared error)

• letting one control variable dominate the costfunction if its deviation crosses a certain threshold(thus ensuring recovery of the considered controlvariable)

• in order to control a variable within a smalltolerance band without allowing any outliers, onecan choose to assign an infinite cost to any controlaction leading to predictions outside this interval

• in order to limit a control variable to a givenmaximum, an infinite cost can be assigned to anycontrol action leading to predictions above thismaximum. Analogously, a minimum for a controlvariable can be implemented

• an integrating term can be included in the costfunction to penalise deviation of the average valuew.r.t. the reference

• etc . . .

Some of the above listed characteristics can betranslated into a specific shape of the cost functionas shown in fig. 3.

xpi(tk+1) – x*(tk+1)!0!

tolerance!band!

zero cost band!

Cost!

uppe

r lim

itat

ion!

Fig. 3: Different cost functions

III. EVT MODELING

A number assumptions, borrowed from [4], allowto express the machine equations in a simple analyticform:

• a three phase symmetrical machine• the windings are sinusoidally distributed (applica-

ble to both stator and rotor windings)• there is no skin- or proximity effect• there is no saturation• there is no influence of the slots on the field

Additionally, the squirrel-cage interrotor is replaced byan equivalent [5] wound rotor, such that the machine

equations become (the equations are expressed in a qd-reference frame, rotating at an arbitrary speed ωb):

V1q = R1I1q + pΨ1q − ωbΨ1d

V1d = R1I1d + pΨ1d + ωbΨ1q

(1a)

V2q = 0 = R2I2q + pΨ2q − (ωb − ω2)Ψ2d

V2d = 0 = R2I2d + pΨ2d + (ωb − ω2)Ψ2q

(1b)

V3q = R3I3q + pΨ3q − (ωb − ω3)Ψ3d

V3d = R3I3d + pΨ3d + (ωb − ω3)Ψ3q

(1c)

In which p represents the Laplace operator. The fluxesare defined as:

Ψ1

Ψ2

Ψ3

=

L11 L12 L13

L21 L22 L23

L31 L32 L33

I1I2I3

(2)

The differential equations are linerised in order toimplement the model in FPGA. A first order Eulerapproximation is used with a time step equal to thesample time of the controller (50µs).

The equations for the torque on the secondary andprimary axis respectively follow from Lorentz’s law:

T2 = 3

2Np(I2qΨ2d − I2dΨ2q)

T3 = 32Np(I3qΨ3d − I3dΨ3q)

(3)

The estimation step in the control algorithm requiresthe knowledge of the control variables at update instantk in order to estimate its future state at update instantk + 1. Stator voltage and currents can be measuredat its terminals, which is also true for the rotor. Fluxmeasurements however, are avoided, such that the fluxvalues are estimated based on measurement of statorand rotor quantities.

IV. CONTROL DESIGN

First, the proposed control method is evaluatedthrough simulations in Matlab/Simulink. The generalform of the cost function that will be evaluated is thefollowing:

CostSi= CostΨ1

(Ψ∗1(tk+1),Ψ1,pi(tk+1), toleranceΨ1

)

+W2 ·(T2,pi(tk+1)− T ∗

2 (tk+1)

T ∗2 (tk+1)

)2

+W3 ·(T3,pi(tk+1)− T ∗

3 (tk+1)

T ∗3 (tk+1)

)2

(4)

One of the assumptions for modeling the EVT de-manded that the saturation state of the machine belinear. In order for this assumption to be valid, theflux level must be limited, such that no magnetisationoccurs. The stator flux level for which the maximalmagnetic flux density in the machine amounts to 1.6T

(a rule of thumb to determine the limit for unsaturatedstate) is determined to be 0.82Wb. In order for theflux to be controlled below this level, the followingcost function is presented:

Ψ1,pi(tk+1) – Ψ1*(tk+1)!0!

Cost!∞

Fig. 4: Implemented flux cost function

A. Flux control

In [6] a FOC method for the EVT is presented. Inconventional FOC, the flux is kept at its maximal valueunless field weakening has to be applied. This meansthat the magnetisation current is always (except at highspeed) maintained at its maximal value. Moreover, inFOC field weakening must be applied to limit the back-emf when the speed of the machine increases. As theEVT has two rotating parts, and also two electricalterminals (both on stator and rotor), two constraintson the back-emf can be identified:

E1 = ωΨ1 = (s2ω + ω2)Ψ1 (5)

E3 = s3ωΨ3 = (s2ω + ω2 − ω3)Ψ3 (6)

The back-emf of the stator is proportional to theabsolute speed of the interrotor, whereas the back-emf of the rotor is proportional to the relative speedbetween the rotor and the interrotor. Increasing eitherω2 or ω2 − ω3 should thus result in field weakening.

In the proposed control method however, the fluxlevel is not controlled at a reference value, but onlycontrolled not to exceed a certain level that wouldcompromise the validity of the model. This meansthat the controller only controls the torques, and theflux follows from the control actions chosen to controlthem. In contrast to FOC, the controller never demandsan impossible flux level to be reached: the resultingflux values automatically satisfy the limitations on theback-emfs. This can be considered as an automaticform of field weakening.

Additionally, as an extension to the presented controlalgorithm, a second optimisation step is added in whichthe control action resulting in the lowest joule losses

is selected from a subset of switching states thatensure good torque control. To determine whether thetorque control is good or not, all control actions witha corresponding cost below an a priori determinedthreshold are forwarded to the second optimisationstep.

B. Parameter optimisation

In order to assess the quality of the control, thecumulative squared error (CSE) is defined:

CSEx =

∫ [x∗(t)− xm(t)

]2dt (7)

The CSE value of a control variable x shows the corre-spondence of the control variable to its reference value.This specific measure was chosen for two reasons:

1) the expression for the error is positive semidef-inite, such that each error either increases themeasure or keeps it the same;

2) by squaring the deviation, large errors affect thequality measure more than small ones.

The control parameters (the weight factors W2, andW3, and the tolerances on both torques T2 and T3) havebeen optimised to result in the best possible controlquality CSET2+CSET3. As the cost is dimensionless,only relative values of the weight factors matter. Theoptimal set of parameters turned out to be: W3 = 1,W2 = 1.2589, tolerance on T2 = 1%, tolerance onT3 = 0.5%. The stator flux is allowed to exceed thelimit value by 2.5%.

C. Simulation results

Fig. 5 shows the simulated resulting dynamic con-trol of the torque. The control method is shown toeffectively control the torque. Moreover, it followsstep changes in the reference without any transientbehaviour whatsoever.

0 0.2 0.4 0.6 0.8 1−40

−20

0

20

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Fig. 5: Response of torque control to dynamic reference

The operating points encountered in the followingsimulations have been verified not to reach the limits ofback-emf, such that strictly speaking, field weakeningmust not be applied. For the sake of obtaining figuresshowing the effect clearly, the cost function parametershave been chosen differently for the following twosimulations: W2 = W3 = 1, tolerance on T2 = 10%and tolerance on T3 = 2.5%. Fig. 6 shows the effect of

an increase in the relative speed between both rotors.Although not strictly necessary, the control of the flux

0 0.2 0.4 0.6 0.8 1−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b](b) Flux lapse

0 0.2 0.4 0.6 0.8 150

100

150

200

250

time [s]

Mec

hani

cal s

peed

s [r

ad/s

]

Ω2

Ω3

(c) Speed lapse

Fig. 6: Effect of step increase in Ω3

responds by lowering its level in order to stay furtherremoved from the limit of the back-emf imposed by eq.6. Fig. 7 shows the effect of increasing the speed ofthe interrotor. Again, field weakening is not necessarilyapplied. Nevertheless, the controller reacts to the speedchange by lowering the flux level, such that it remainsfurther removed from the limit imposed on back-emfby eq. 5.

D. Experimental verification on FPGA

To demonstrate the real time feasibility of the con-trol method, the controller is programmed on FPGA.Additionally, the model of the EVT is included inthe FPGA to be emulated. A more simple controlstrategy has been applied in FPGA for convenience ofprogramming. Here, the flux remains controlled, andthe cost function takes the form:

CostSi= W1 · |(Ψ1,pi(tk+1)−Ψ∗

1(tk+1)|+W2 · |(T2,pi(tk+1)− T ∗

2 (tk+1)|+W3 · |(T3,pi(tk+1)− T ∗

3 (tk+1)|(8)

The cost function parameters are set as follows: W1 =100, W2 = W3 = 1, tolerance on torques T2 andT3 = 10%, and tolerance on flux Ψ1 = 2.5%. Theresulting control on torque and flux is shown in fig. 8,demonstrating the effectiveness of the control methodand its feasibility to be executed in real time.

0 0.2 0.4 0.6 0.8 1−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux lapse

0 0.2 0.4 0.6 0.8 160

80

100

120

140

160

time [s]

Mec

hani

cal s

peed

s [r

ad/s

]

Ω2

Ω3

(c) Speed lapse

Fig. 7: Effect of step increase in Ω2 - increasing absolutespeed

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux control

Fig. 8: Response of control on torque step (up)

V. CONCLUSION

In this article, the feasibility of applying MBPCto the EVT has been studied. The control method isadapted in two ways: first the explicit control of theflux level is omitted, next a second optimisation step isadded to the algorithm in order to reduce joule losses.The effectiveness of the proposed control method isverified using simulations in Matlab/Simulink. Omit-ting the explicit control of the flux level has as abenefit that the resulting flux level will automaticallyapply field weakening if necessary. The real timeapplicability has been proven through hardware-in-the-loop simulations carried out on FPGA.

REFERENCES

[1] M. J. Hoeijmakers, “Electromechanical converter,” Sept. 122003. WO Patent 2003075437 A1.

[2] M. J. Hoeijmakers, “Electromechanical converter,” Jan. 16 2007.US Patent 7164219.

[3] M. J. Hoeimakers and J. A. Ferreira, “The electric variable trans-mission,” IEEE Transactions on Industry Applications, vol. 42,no. 4, pp. 1092–1100, July 2006.

[4] J. Melkebeek, Elektrische aandrijftechniek. Gent: Cursusnota’s,Universiteit Gent, EESA, EELAB, 2012.

[5] J. Melkebeek and L. Vandevelde, Bouw en berekening vanelektrische machines. Gent: Cursusnota’s, Universiteit Gent,EESA, EELAB, 2007.

[6] J. Druant, “Field oriented control for an induction machine basedelectrical variable transmission,” unpublished.

Contents

Permission of usage i

Preface iii

Overview v

Extended abstract xii

List of abbreviations & symbols xv

1 Introduction 1

2 Electric variable transmission 3

2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 System overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Current technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Emerging technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Induction machine based EVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Conceptualisation of the EVT . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.3 Variants on the induction machine based EVT . . . . . . . . . . . . . . . 17

2.4 Permanent magnet based EVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Magnetic-geared EVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.3 Double layer permanent magnet EVT . . . . . . . . . . . . . . . . . . . . 19

2.4.4 Double-stator permanent magnet brushless machine EVT system . . . . . 20

2.4.5 Single layer permanent magnet EVT . . . . . . . . . . . . . . . . . . . . . 22

2.5 Switched reluctance based EVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 EVT comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 System control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7.1 Powertrain control level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7.2 EVT control level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Model based predictive control 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xiii

3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.2 Control variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.3 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Field Programmable Gate Array (FPGA) . . . . . . . . . . . . . . . . . . . . . . 38

3.6.1 Advantages and drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.2 Useful features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Modelling 434.1 The need for modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.2 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.3 Machine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.4 Machine parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.5 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.6 Linearisation: Euler approximation . . . . . . . . . . . . . . . . . . . . . . 504.2.7 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.8 Dynamics feedback loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.9 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.10 Flux estimation - torque calculation . . . . . . . . . . . . . . . . . . . . . 52

4.3 Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.4 Homopolar components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 Internal combustion engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Electrical energy storage device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Control design 595.1 Simulink model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1.1 Field oriented control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Cost function analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Control quality assessment . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.2 Qualitative influence of parameters . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Omitting flux control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Efficiency optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Control extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4.2 Qualitative influence of parameters . . . . . . . . . . . . . . . . . . . . . . 82

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Response analysis 896.1 Torque step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Speed step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xiv

7 Experimental validation 957.1 FPGA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.2.1 Optimisation step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.2.2 Experimental verification of the control method . . . . . . . . . . . . . . . 977.2.3 Torque step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Conclusion 101Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

xv

List of abbreviations & symbols

Abbreviations

2L-VSI two level voltage source inverterCSE cumulative squared errorCVT continuously variable transmissionsEVT/E-CVT electric variable transmission/electric-CVTFOC field oriented controlFPGA field programmable gate array(FS)-MBPC (finite-set) model based predictive controlHEV hybrid electric vehicleHIL hardware-in-the-loopICE internal combustion engineMIMO multiple input - multiple outputPEC power electronic converterTHS Toyota Hybrid System

Symbols

subscripts1 referring to stator quantities2 referring to interrotor quantities3 referring to rotor quantitiesa, b, c referring to phase quantitiesαβ indicating Clarke componentsα1β1 Clarke reference frame attached to statorα3β3 Clarke reference frame attached to rotorqd indicating Park componentsquantitiesT torque [Nm]Ψ magnetic flux [Wb]Ω mechanical speed [rad/s]ω electrical speed [rad/s]J mass moment of inertia [kgm2]

Vectors are indicated by underlined symbols and matrices are indicated by boldface symbols.

xvii

Only abbreviations that are used repeatedly are included in this list, all others are declared inthe text. All abbreviations and symbols are explained in the text.

xviii

Chapter 1

Introduction

Since fossil fuel resources are estimated to run out if the present consumption rate is maintained,and the awareness of the link between emissions and climate change is proven, alternative energysources are being studied. Electricity has been around for a long time as an energy carrier,and renewable energy sources (wind and solar energy) are gaining a larger share in the energygeneration. It is believed that electrical energy (generated from renewable energy sources) shouldreplace fossil fuel energy in order for the energy consumption to be sustainable.

Transportation is one of the main consumers of fossil fuels, such that electrifying this area iscurrently a hot topic. Whereas some car manufacturers aim at developing full electric vehicles,others are still aiming at an intermediate solution. The main drawback of full electric vehiclestoday is their limited driving range as a result of limited electrical storage capacity on board.Most full electric vehicles are designed for city usage, such that a long driving range is not apriority. Nevertheless, Tesla Motors is now transforming the market of full electric vehicles,and bringing them (almost) to comparable performance levels as conventional vehicles. Mostcar manufacturers however, are investing in the development of hybrid electric vehicles (HEVs),which are considered the intermediate solution to bridge the gap between fossil fueled vehiclesand full electric vehicles. The first commercially successful HEV was developed by Toyota: thePrius. Until today, the Toyota Hybrid System (THS) as applied in the Prius is still consideredas a benchmarking system for development of new HEVs.

HEVs are being developed by car manufacturers with entirely different objectives, resulting ina range going from economic city vehicles all the way up to two seater hypercars which areannihilating performance of the most powerful fossil fueled vehicles. Different HEV systems areavailable on the market: plugin HEVs have the possibility to charge the battery using externalmeans (much like you would fuel a conventional car), whereas in non-plugin HEVs, the batteriesare only charged by the electric generator in the vehicle. Another distinction can be made onwhether the vehicle can operate in full electric mode or not. Mild hybrids cannot be drivensolely on the electric motor, whereas the full hybrid can.

As two energy sources are available in HEVs (the fossil fuel and the battery), there are twopower flows that deliver power to the wheels of the vehicle. Many HEVs combine both powerflows and deliver them to the wheels (as is the case with the Toyota Prius). The managementof these power flows is done through a power split device, which in the current systems, takesthe form of a planetary gear system. Recently, an electric machine has been presented which

1

CHAPTER 1. INTRODUCTION

can replace the pair of motors/generators and the gear system with a single electromagneticconverter: the electric variable transmission (EVT). in an EVT, power is transmitted solelythrough electromagnetic fields, such that it solves one of the most obvious drawbacks of theplanetary gear system: mechanical contact between different parts. The elimination of thisdrawback results in a number of advantages. Moreover, the number of components in the systemcan be greatly reduced by combining functionalities in a single machine. In [1], an overview isprovided of the different EVTs that have been researched, as well as their possible applications.The EVT is described as a new breed of electric machines, providing dual mechanical portsinstead of only one.

The drivetrain and each of its components must be controlled in order to fulfill the desires of thedriver of the vehicle. Two levels of control can de distinguished: the highest level determines thecomposition of the different power flows, whereas the lower one controls each of the componentsin the drivetrain to realise the operating point demanded by the higher control level. In thistext, the lower control level of the EVT will be studied. The electromagnetic converter is - aswith other electrical machines - controlled through its electrical ports. The output variables ofthe EVT are the torques and speeds that are generated on both mechanical ports. Due to itsnature, an EVT is a more complex machine than another electrical machine. Its multiple input -multiple output (MIMO) character requires a complex control method. Model based predictivecontrol is proven very suitable for MIMO systems, such that this is the control method chosento be applied to the EVT. topologies

2

Chapter 2

Electric variable transmission

2.1 General description

As indicated in the introductory chapter, the EVT finds it application in HEVs. However, beforegoing into details about the topologies and operation, it is instructive to consider the machineindependently of any of these aspects. To this extent, the concept of black box modelling of theEVT is introduced.

The EVT can be represented as a black box with three ports (see fig. 2.1). Two of these portsare of mechanical nature, and take the form of the axes that allow connection of external devicesto the EVT. The remaining port is of electrical nature, and allows interaction of the EVT withan electrical energy storage device. The mechanical ports both take the form of a shaft enteringthe blackbox, the electrical port is represented by a contact of negative polarity, and one ofpositive polarity.

Figure 2.1: Blackbox representation of the EVT

In its normal operating mode, the primary axis will be connected to the shaft of an internalcombustion engine (the fossil fuel energy source), the secondary shaft is connected to the load(in the case of a HEV, i.e. the driven set of wheels). The electrical port is where the electricalenergy storage device (e.g. a battery) is connected to. This is the setup of the machine that willbe studied in this text.

In its blackbox representation, the EVT may be seen as a power converter: mechanical and

3

CHAPTER 2. ELECTRIC VARIABLE TRANSMISSION

electrical power enter the blackbox, are converted inside it, and exit either as mechanical orelectrical power. Of course, the power balance over the blackbox must hold, but the amount ofpower entering the EVT, may be redistributed in different portions of mechanical and electricalpower. To fulfill its destined functionality, it is important to note that each of the ports (i.e.both mechanical ports and the electrical ports) are bidirectional ports: power can flow in eitherdirection through the ports.

From this discussion the general description of the EVT can be summarized as follows: the EVTmust be able to transform and redistribute incoming power (both electrical and mechanical),into outgoing power (both electrical and mechanical). Hence, the machine will be called anelectromechanical converter.

2.1.1 Functionality

As discussed above, the EVT is an electromechanical converter, which allows mechanical andelectrical power to flow bidirectionally through two mechanical and one electrical port. Depend-ing on the direction of power flow through each of the ports, different functionalities can beidentified.

The objective of the EVT is to replace and improve a number of parts present in conventionalvehicles and HEVs now available on the market. It is thus mandatory that the functionality ofthe replaced parts are taken over by the EVT. Below, a number of functionalities is highlighted,in order to understand the different operating modes of the EVT. For simplicity, each operatingmode is discussed separately, but it is important to keep in mind that many of these modesoccur at the same time in real applications.

Starter motor

In conventional vehicles, the ICE is connected through gears with the starter motor. This motorin turn is connected to the vehicles battery. In order to be able to fire the engine, its shaft mustbe rotated by an external force. The starter motor extracts energy from the vehicle’s batteryin order to bring the shaft of the ICE into movement. This combined with injection of fossilfuel and ignition (either by pressure and glow plugs in diesel engines, or by spark ignition inpetrol engines), fires the ICE. Obviously, the starter motor only works during short periods, butits presence is essential for firing an ICE. Due to its rare operation, the starter motor is oftenselected for low cost, and thus has very low efficiency. Moreover, although the device is usedonly for start-up, its weight has to be carried by the vehicle wherever it goes.

As the primary shaft of the EVT is connected to the shaft of an ICE, and power should be ableto flow bidirectionally, it appears that the EVT can be used as a starter motor for the ICE. Inthis operation mode, no power is transferred through the secondary shaft, electrical power flowsfrom the electrical energy source towards the EVT, and mechanical power flows from the EVTtowards the shaft of the ICE (see fig. 2.2).

4


Pm

Pe

Figure 2.2: Starter motor mode of the EVT

Generator

Once the ICE is running, it generates mechanical power and drives ancillary services of thevehicle. The ICE provides energy to all of the systems either directly, through a belt or gearconnection, or indirectly, via a generator and the vehicle’s battery. These ancillary servicesinclude hydraulics (e.g. power steering and braking), electrics and electronics, and AC. In thisoperation mode, the ICE does not fulfill its main goal, i.e. providing power to the drivetrain inorder to initiate motion. However, even when the ICE is powering the drivetrain, the generatingaction is fulfilled at all times, thus indicating that never the full power generated by the ICE willbe transferred through the drivetrain towards the driven wheels of the vehicle. As this device isoperational whenever the ICE is running, it is important to select a machine with high efficiencyfor this purpose. For simplicity, here the pure generating mode is considered (irrespective ofdrivetrain power).

Power flow-wise, this operating mode is the exact opposite of the starter motor: mechanicalpower is generated by the ICE and transferred through the primary shaft to the EVT, andelectrical power exits the EVT through the electrical port, providing energy to the battery (seefig. 2.3). Here as well, the choice can be made to connect the ancillary equipment either directlyto the shaft of the ICE, or to an electric drive, connected to the battery. The latter optioneliminates the use of mechanical transmissions in the entire system, which - as will be discussedlater - is one of the advantages that could be realised by the EVT.

5


Pm

Pe

Figure 2.3: Generator mode of the EVT

Electric motor

The motoring mode of the EVT is characterised by a mechanical power output on the secondaryshaft. Here, the electric motor mode is considered, meaning that electrical power (from theelectrical energy storage device) is converted by the EVT into mechanical power exiting theEVT through the secondary shaft. Alternatively, this mechanical output power can be sourcedfrom a mechanical input power through the primary shaft. The latter mode will be consideredbelow (see sec. 2.1.1), the first mode will be discussed in the present section.

In the electric motor mode, electrical power is drawn from the battery and converted by the EVTinto mechanical power, which is then transferred through the secondary axis towards the wheelsof the vehicle (see fig. 2.4). This mode can thus be described as the electric vehicle workingof the EVT (note that in this operating mode, no power is flowing through the primary axis,such that the ICE can potentially be taken out of the picture. This feature could for examplebe taken advantage of in the case the vehicle runs out of fossil fuel.

Pm

Pe

Figure 2.4: Electric motor mode of the EVT

6


Regeneration

As indicated before, also the mechanical power in the secondary shaft should be able to flowin both directions. In (H)EVs, this operating mode is described as regenerative braking: thekinetic energy of the vehicle is converted (thus decreasing the vehicle’s speed) into electricalenergy (stored in the vehicle’s battery) instead of dissipating the energy by friction brakes. Thisis one of the key features that makes (H)EVs so interesting, and should definitely be includedin the EVT.

In regenerative mode, mechanical power flows through the secondary shaft towards the EVT,where it is transformed into electrical power, which becomes available at the electrical port (thuscharging the electrical energy storage unit). The power flow is indicated in fig. 2.5.

Pm

Pe

Figure 2.5: Regenerative mode of the EVT

Transmission

In this operating mode, the EVT takes over the role of the transmission system in a conventionalvehicle. The main component of the transmission system is the gear box, which is used to transfermechanical power from the input shaft to the output shaft at a different speed. With the use ofdifferent gear ratios, the ICE can operate in a given (limited) range of rotational speeds, whereasthe range of output speeds is determined by the selected gear. This allows the vehicle to driveat different speeds, even though the ICE operates at the same rotational speed, such that thedesign range of the ICE can be significantly reduced.

In the blackbox representation of the EVT, the transmission mode is characterised by an amountof mechanical power entering the system through the primary shaft, and (in the assumption of alossless EVT) the same amount of mechanical power leaving the system through the secondaryshaft (see fig. 2.6). Depending on the chosen gear ratio (discrete or continuous), the outputspeed and torque will differ with respect to the input speed and torque by a factor equal to thegear ratio.

7


Pm, out Pm, in

Figure 2.6: Transmission mode of the EVT

Conclusion

From the above discussion of the different operating modes of the EVT, it is immediately cleatthat this machine has the potential to replace multiple parts that are used in existing systems.The fact that many functionalities can be combined in the same machine, means that both sizeand weight of the complete system can potentially be decreased significantly.

2.2 System overview

In order to understand the role of the EVT in the HEV’s drivetrain, it is important to study thepresent technology: to understand the tasks the EVT will have to fulfill, and moreover, how thefunctionality is created by the present technology. This will give us insight in which drawbacksare solved by the EVT, and demonstrate its potential of replacing the existing technology.

The idea for the EVT system shows some clear correspondences with the series hybrid powertrain(see fig. 2.7). It is therefore instructive to start from this setup to explain the role of the EVTin HEVs.

In what follows, the terms generator and motor are used to indicate the electrical machinesthat are mainly operated (and thus designed) as such. However, as electrical machines have theintrinsic ability to operate in both ways, it is important to note that each machine can fulfilleither role, depending on the operating mode the EVT is in.

Figure 2.7: Series drivetrain for HEV’s1

8


In the series hybrid powertrain system, three main components can be distinguished: the ICE(fed from the fuel reservoir), the electric generator (feeding the battery), and the electric motor(fed from the battery). The series operation corresponds with subsequent working of the EVTin electric generator (section 2.1.1) and electric motor (section 2.1.1) mode. However, the EVTis not restricted to series operation: it also shares some features with parallel hybrid powertrainsystems (see fig. 2.8). As discussed above (section 2.1.1), the EVT must also allow mechanicalpower from the ICE to be transferred to the secondary shaft without intervention of an electricalenergy storage device. In currently available HEVs, this functionality is obtained by a paralleldrivetrain.

Figure 2.8: Parallel drivetrain for HEV’s2

It can thus be concluded that the EVT is equivalent with the combined hybrid powertrain system(also called the series-parallel hybrid powertrain system), which allows both series and paralleloperation, as shown in fig. 2.9.

Figure 2.9: Combined drivetrain for HEV’s3

2.2.1 Current technology

The powertrain systems discussed above each have practical implementations available on theautomotive market. Series hybrid powertrain systems are found for example in the Fisker Karma(released 2011) or the BMW i3 (released 2013), parallel systems are found in the BMW i8(released 2014), the Ferrari LaFerrari (released 2013), and the McLaren P1 (released 2013).Perhaps the most well-known example of HEVs is the Toyota Prius (released in Japan in 1997,

1source: http://en.wikipedia.org/wiki/Hybrid_vehicle_drivetrain2see footnote 13see footnote 1

9

http://en.wikipedia.org/wiki/Hybrid_vehicle_drivetrain


in the year 2000 in the rest of the world). For this vehicle, Toyota developed the so-called ToyotaHybrid System (THS), which is often used as the benchmarking system for R&D of new hybridsystems [2, 3]. The THS is classified as a combined system, allowing both series and paralleloperation of the system, hence a relevant benchmark for the considered EVT system.

In the THS (shown in fig. 2.10), a planetary gear set is used to split the mechanical powercoming from the ICE into two parts: one going directly towards the vehicle’s wheels, the otherbeing converted into electrical energy (which is stored in an electrical energy storage device) bya generator. In [4] the working of a mechanical planetary gear is explained, and the comparisonis made with a magnetic planetary gear. The goal of this paper was to demonstrate the possi-bility of replacing the mechanical planetary gear system by a magnetic planetary gearbox, thusresolving the main issues involved with using planetary gears.

Figure 2.10: Schematic representation of the Toyota Hybrid System [5]

Fig. 2.11 shows the lay-out of a planetary gear set: it is composed of a central sun gear, multiplesmall planet gears mounted on a planet carrier, and a ring gear (with internal gearing). Thepower splitting is obtained by varying the speed of the different components.

10


Figure 2.11: Mechanical planetary gear with three planets [4]

In [4] some advantages of planetary gears compared to parallel axis gears are listed: high powerdensity, large gear reduction in a small volume, multiple kinematic combinations, pure torsionalreactions and coaxial shifting. Planetary gears have some disadvantages too, among them: highbearing loads, inaccessibility and design complexity. Due to its mechanical nature, a numberof other disadvantages occur: mechanical contact between the gears requires lubrication, causesaudible noise, and - on the long term - wear of the gear teeth. It is concluded [4] that magneticgears can have comparable or better performance than mechanical gears, with the obviousbenefits of avoiding many mechanical drawbacks.

2.2.2 Emerging technology

Depending on whether or not permanent magnets are used in the EVT, the different topologiescan be split up in two main categories. In both cases, the principle of operation remains largelythe same: power is transferred between different parts of the machine through (electro)magneticfields instead of meshing gears. Independently of the category the EVT belongs to, a number ofadvantages can be listed compared to its mechanical counterpart.

• the absence of belt/chain connections

• no mechanical contact between rotating parts, thus eliminating

– the necessity to provide lubrication

– wear of the mechanical components

– friction losses in the machine

– audible noise caused by the gear meshing

• reduced mitigation of vibrations in the system

• inherent overload protection

Amongst the beneficial consequences of these advantages are: reduced maintenance of compo-nents, elimination of most failure-prone components of the mechanical system (belts and chains),

11


increased efficiency, and increased convenience of use.

From the listing of these advantages, it is obvious why the EVT is gaining interest in literature.In [4] the performance of a magnetic planetary gear is compared to the conventional mechanicalplanetary gear. In fig. 2.12 the lay out of the proposed magnetic gear is shown: as with theplanetary gear the system is composed of three concentrically rotating parts, which are nowequipped with permanent magnets instead of gear teeth to provide the transmission of powerfrom one part to another.

Figure 2.12: Magnetic planetary gear [4]

In [6], not only the interchangeability of the mechanical planetary gear with a magnetic planetarygear is studied, but the equivalency of the entire combined hybrid system using a planetary gearand an EVT is studied. In the latter paper, the relations between the two systems are determinedwhich have to be fulfilled in order to reach equivalency of both systems. In fig. 2.13, the intendedreplacement of the planetary gear with the EVT is graphically demonstrated. The replacementof the planetary gear set and the motor/generator 1 is represented by the double rotor machineinside the red dashed line. Both parts of the machine MG1 in fig. 2.13b are rotating, so that astator in the sense of a stationary machine part is not present.

12


(a) Planetary gear as used in THS

(b) Equivalent EVT

Figure 2.13: Equivalency of planetary gear and EVT in hybrid system [6]

2.3 Induction machine based EVT

2.3.1 Disclaimer

The EVT concept and name were firstly presented by Prof. Hoeijmakers in 2004 [7], based onearlier filed patents by himself. In this master thesis, the concept of Hoeijmakers is studied.Hence it is given most attention in the discussion of the induction machine based EVTs.

It is in no way the intention of this text to give a complete overview of all existing inductionmachine based EVTs.

2.3.2 Conceptualisation of the EVT

M. J. Hoeijmakers describes in his patents [8, 9] his version of the electric variable transmission.In [10] Hoeijmakers describes the basic idea behind the development of his version of the EVT.

The idea sets off with a consideration of the cascade system shown in fig. 2.14a. The cascadesystem presented here is nothing else than a series hybrid drivetrain. However, an unconventionalgenerator is used in which the squirrel-cage winding is on the stator and the electrical poweris withdrawn from the rotor (the rotor of a wound-rotor induction machine with sliprings).This particular choice of generator is made for the future development of the idea. In the firstinstance, the cascade system is studied purely as a continuously variable transmission (withoutbattery on the dc link).

Due to the nature of the cascade system, the entire power generated by the ICE flows throughthe converters. This means that they have to be designed for this rated power (as continuousload), which results in a high cost. Another disadvantage of the entire power flowing through

13


the converters, is that the entire power is subjected to the efficiency of the four converters (back-to-back ACDC- and DCAC-converters), resulting in a relatively low efficiency in practice [10].This statement has to be put in the perspective of the future development of the system, whichincreases the efficiency of the system. Although inverters reach efficiencies of 95% and higher[11], the efficiency resulting from this cascade system is lower than the efficiency of the EVTthat will ultimately be presented.

The efficiency can be increased significantly if the stator torque of the primary machine is directlyled to the rotor of the secondary machine. This is represented graphically in fig. 2.14b, wherethe stator of the primary machine is connected through an axis with the rotor of the secondarymachine. Due to this intervention, the primary machine now becomes a double rotor machine:its stator loses the property of being fixed and is hence called a rotor. The torque of the outgoingaxis is now composed of two contributions: partly originating from the interaction between theinner and outer rotor of the primary machine, partly originating from the interaction betweenthe stator and rotor of the secondary machine.

In this way, a power split device is obtained which transfers power either mechanically or elec-trically, resulting in a higher efficiency of the whole system. Also the power rating of the usedinduction machines and power electronic converters are reduced. For the system discussed in [7],the method for power rating of the components discussed in [10] results in an apparent powerrating for the induction machines and the power electronic converters of about 65% of the ratedpower of the ICE. As a result of the two paths existing for the power from the ICE to reach theload side, each path will - on average - transmit less than the rated power of the ICE. However,components with a lower power rating may transmit the entire power flow from the ICE duringshort periods. The duration of these short periods is determined by the overloading abilities ofthe different components.

(a) Cascade system (b) Combined system

Figure 2.14: Basic idea of the EVT system[10]

The next step in the evolution towards the ultimate design is the concentric arrangement of bothmachines: instead of connecting both machines through an axis, the parts of both machinesare attached directly to one another. By placing the two machines concentrically, the axiallength of the system is significantly reduced (see fig. 2.15). The resulting machine behaves astwo concentric, but independently operating induction machines. The idea of two concentricmachines has been implemented in various ways in the past as listed in [10]: combination of twodc machines (going back to a patent of 1935), and combinations of permanent magnet machines

14


or induction machines.

Figure 2.15: Two concentric induction machines [10]

As explained by Hoeijmakers, the interrotor yoke height can be strongly reduced if the inner andouter machines have the same slip frequency (including direction). By reducing the interrotoryoke height, the magnetic fields of the concentric machines are no longer separated. Due tosaturation of the interrotor yoke (in tangential direction), the magnetic field lines no longerclose their paths inside the interrotor yoke, but also cross the interrotor yoke. It is importantto note that the interrotor forms one whole both mechanically and electromagnetically. It isthis latter fact that makes the proposed EVT novel, allowing filing a patent application. Theobtained machine is shown in fig. 2.16.

The electromagnetic behaviour of the obtained EVT is totally different from two separate in-duction machines. Instead of having two electromagnetic devices, we now have one, in whichthere is also a direct interaction between the slipring armature rotor (inner rotor) and the sta-tor. In [10] a simple model of the EVT is proposed, and characteristic operating points arediscussed. In [12], the effect of electromagnetic coupling of both concentric machines due todecreasing interrotor yoke height is studied. The design of such a machine is presented, alongwith experimental research results.

Hoeijmakers concludes that the EVT can be useful as a continuously variable transmissionin a motor vehicle, providing also functionality as a starter motor and generator. Due to itscontinuous nature it allows the engine to work with better efficiency (increasing the fuel efficiencyby up to 25% in a case study [7]). Besides working as a CVT, it can also be used in hybridsystems where an electrical energy storage unit is connected to the electric port of the EVT.

In Hoeijmakers’ patents [8, 9] the invention claimed is summarized in a number of characteristicswhich define the proposed system:

15


Figure 2.16: Induction machine based EVT [10]

1. An electromechanical converter comprising:

• a primary shaft having a rotor mounted thereon;

• a secondary shaft having an interrotor mounted thereon; and

• a stator fixedly mounted to a housing of the electromechanical converter, wherein,viewed from the primary shaft in a radial direction, the rotor, the interrotor, and thestator are arranged concentrically relative to each other, and wherein the rotor andthe stator comprise one or more windings, and wherein the interrotor comprises onewhole both mechanically and electromagnetically, and the interrotor further comprisesa magnetic and an electric circuit, the magnetic circuit including a magnetic flux con-ducting cylinder and the electric circuit including a number of electric circuit-formingwindings extending in the flux conducting cylinder, and wherein the interrotor is ar-ranged as a conductor for the magnetic flux in a tangential and a radial direction sothat exertion of a direct torque between rotor and stator can occur upon magneticsaturation of the interrotor.

2. The electromechanical converter according to claim 1, characterized in that the interrotorcomprises an electric and a magnetic circuit, and the magnetic circuit comprises a cylinderhaving two sides, with both sides defining longitudinally extending grooves in which electriccircuit-forming shortcircuit windings extend.

3. The electromechanical converter according to claim 1, characterized by the interrotor beingformed by a magnetic flux conducting cylinder, and the electromechanical converter furthercomprises:

• permanently magnetic material applied on a first side of the interrotor; and

• longitudinally extending grooves associated with a second side of the interrotor inwhich an electrically accessible winding is provided.

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4. The electromechanical converter according to claim 1, characterized by the stator wind-ing and rotor winding being mutually connected with each other via one or more powerelectronic converters.

5. The electromechanical converter according to claim 4, characterized by said one or morepower electronic converters being electrically accessible via an electric gate.

6. The electromechanical converter according to claim 1, characterized in that the statorwinding and rotor winding are separately accessible through a power electronic converterand an electric gate.

7. An apparatus provided with an electromechanical converter according to claim 1.

8. An apparatus provided with an electromechanical converter according to claim 1, whereinthe apparatus is selected from the group consisting of an apparatus for starting a drivingcombustion engine and an apparatus for supplying electrical equipment.

9. The apparatus provided with an electromechanical converter according to claim 7, furthercomprising a system for the storage of energy.

2.3.3 Variants on the induction machine based EVT

Instead of using the integrated design with two concentrically arranged rotors, one can also usethe system described by fig. 2.14b. As discussed in [2] the two most occurring topologies ofEVTs are said to be using either two electrical machines or one double-rotor integrated machine.In both topologies, the choice can be made to use either permanent magnet based machines orinduction machines.

In the next section, a number of permanent magnet based variants of the EVT are proposed.

2.4 Permanent magnet based EVT

2.4.1 Disclaimer

It is in no way the intention of this text to give a complete overview of all existing permanentmagnet based EVTs. Some remarkable examples from literature are taken a closer look upon,to provide an idea of the state of the art machines that are being researched and developed.

From the overview of EVT researches it can be seen that permanent magnet EVTs are becomingmore attractive than other machine types due to their higher achievable torque and power density[3]. Different topologies of EVTs using permanent magnet based machines have been proposedand will be briefly discussed below.

2.4.2 Magnetic-geared EVT

In [5], the design of a magnetic gear is discussed as a replacement of the (mechanical) planetarygear used in the THS. The system is shown in fig. 2.17, and shows the integration of all ofthe components in fig. 2.10 into a single machine. Both the generator/motor, and the gearing

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functionality of the THS is encorporated in this permanent magnet based machine. The authordenotes the machine as a magnetic-geared electronic continously variable transmission (MG E-CVT). The advantage of the proposed system is the absence of carbon brushes and slip rings,as both of the wound parts of the machine are stationary. The downside of this is that anadditional rotating component is needed in between both rotors, to modulate the magneticfield. This modulation ring is equipped with ferromagnetic pieces, of which the working isexplained in the paper.

Figure 2.17: Magnetic-geared EVT [5]

In fig. 2.18, a cross section of this machine is shown schematically, together with indication ofits main components. The machine consists of two permanent magnet based electrical machineswith a magnetic gear in between them. Both rotors are equipped with a double layer of perma-nent magnets: one layer for interaction with its corresponding stator windings, and one layer tofulfill its role in the magnetic gear. Here, the modulating ring of the magnetic gear is connectedto the shaft of the ICE, while the outer rotor is connected to the driveline. The inner rotor isnot connected to an external shaft, and is used to transfer torque between the inner stator andthe magnetic gear.

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MGM/G1 M/G2

Segment ring

Stator of M/G2

Stator of M/G1

PM Rotor M/G1

PM Rotor M/G2

Figure 2.18: Schematic cross section: EVT with magnetic gear inside [13]

The proposed system has the advantages of small size and light weight, and additionally a hightorque density as a result of the permanent magnets used [5].

2.4.3 Double layer permanent magnet EVT

In fig. 2.19 the lay-out of the proposed permanent magnet based EVT is presented. Here theinterrotor is fitted with a double layer of permanent magnets, producing the magnetic fields inthe inner and outer air gap. In [3], the design procedure for the placement and dimensions ofthe permanent magnets is discussed. First, it is briefly discussed which benefits and drawbackscan be identified by changing the topology of the system (e.g. connecting the in- and outgoingshaft to the interrotor and rotor respectively). Next, the possibilities for placing the permanentmagnets on the different parts of the machine are evaluated. Finally, a number of design choiceswith respect to placement and dimensioning of the permanent magnets are presented.

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Figure 2.19: Permanent magnet based EVT [14]

Due to the presence of a double layer of permanent magnets, the inner and outer machineare magnetically decoupled. The similarities can be seen wit respect to the system with twoconcentric induction machines (see fig. 2.15). The proposed EVT can be seen as the permanentmagnet based counterpart of that system. Its cross section can be represented schematically asin fig. 2.20:

Stator

Rotor

InterrotorDouble layer PM

Figure 2.20: Schematic cross section: EVT with double layer of PMs [13]

2.4.4 Double-stator permanent magnet brushless machine EVT system

Another approach to the permanent magnet based EVT is presented by [15]. In recent years,the double-stator permanent magnet brushless (DS-PMBL) machine has been developed forwind power generation and integrated starter-generators, due to the fact that it can offer higherefficiency and higher power density than the other PMBL machines. Borrowing the idea of using

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a double-rotor machine to split the engine power into two paths, the double-stator machineshould be able to perform the same task while eliminating the use of carbon brushes and sliprings.

Figure 2.21: Double stator permanent magnet brushless machine [15]

As two of three composing elements of the machine are fixed, no torque can be directly trans-ferred to an outgoing axis. This system thus requires another electric motor (driven purelyelectrically) in order to operate as an EVT in HEV’s. The system overview is given in fig. 2.22

Figure 2.22: DS-PMBL machine as an EVT [15]

Due to the fact that no power can be transferred mechanically, an additional electrical powerflow is introduced, thus requiring an additional power electronic converter. However, in cruisingconditions the voltage generator in the stator windings can be used directly to power the elec-tric motor (due to design choices made on the generator). In this operating mode, the powerelectronic converters can be bypassed by a power switch, hence avoiding losses in them and

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increasing the overall efficiency of the system. In most cases, the power generated by the ICEwill be greater than the power required for cruising, hence the surplus can be used to charge thebattery via the inner stator windings. This situation is shown in the in fig. 2.23a.

(a) Cruising mode - power switch closed

(b) CVT mode - power switch open

Figure 2.23: Operating modes of the DS-PMBL EVT [15]

In the CVT mode as shown in fig. 2.23b, the mechanical power from the ICEPm1 is split intotwo components by the inner and outer stator windings. The first component Pd goes throughthe controlled rectifier and inverter to feed the induction motor. The other component Pe1 isbuffered in the battery. When Pd is less than the required mechanical output power Pm2, thebattery provides Pe2 to additionally feed the induction motor.

2.4.5 Single layer permanent magnet EVT

As the double layer PM-based EVT from fig. 2.19 represented the PM counterpart of the IM-based EVT shown in fig. 2.16, the single layer PM-based EVT (shown in fig. 2.24) is the PMcounterpart of the IM-based EVT from fig. 2.15.

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Figure 2.24: Single layer PM-based EVT [16]

In [16], a novel design of the interrotor is presented with only a single layer of PMs. Due tothis design choice, the inner and outer machines are no longer magnetically separated, but nowform a whole both mechanically and electrically (just as the EVT proposed by Hoeijmakers). Inthat paper, the design of this machine is elaborated, and special attention is paid to the studyof different combinations of number of slots on rotor and stator, and the number of magneticpoles on the interrotor.

Schematically, the cross section of the machine is represented in fig. 2.25

Stator

Rotor

InterrotorSingle layer PM

Figure 2.25: Schematic cross section: EVT with single layer of PMs [13]

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2.5 Switched reluctance based EVT

In [17] an EVT based on the switched reluctance (SR-EVT) principle is patented. Not belongingin any of the two main categories of EVTs, this third category is created to fit the presentedtopology. As with single rotor switched reluctance machines, this type of machine makes use ofvarying magnetic resistance (i.e. reluctance). The working of this type of machine is discussedin [18]. A cross section of the machine is shown in fig. 2.26:

Figure 2.26: Schematic cross section: EVT with single layer of PMs [13]

The rotor is manufactured from ferromagnetic material and is fitted with salient pieces. Theseenable the machine to work on the principle of switched reluctance. As discussed in the patent,the invention may be used in automotive applications and fulfill all of the functionalities requiredfrom an EVT.

2.6 EVT comparison

The present section gives an overview of the presented EVT topologies. Apart from the inherentadvantages of the EVT with respect to its mechanical counterpart (listed in sec. 2.2.2), eachof the proposed system has both advantages and drawbacks. To avoid too much repetitionof common advantages and drawbacks, these are treated first, independently of the proposedtopology.

Whether the topology is a IM- or a PM-based EVT, the presence of carbon brushes and slipringscan be identified as a drawback. The parts require maintenance (as the carbon brushes wear offdue to friction) and can cause sparks due to non-ideal commutation. Additionally, their presencecauses a (small) voltage drop.

As the magnetic field in IM-based EVTs is created by the windings themselves, the implemen-tation of field weakening (used at higher speeds to comply with the limited DC-bus voltage) iseasier implemented than with PM-based EVTs. In PM-based EVTs, additional windings mustbe installed to create a magnetic field opposing the one generated by the PMs in order to obtain

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field weakening. In [14], a method for implementing field weakening on a PM-based system isproposed and evaluated by simulation.

In IM-based EVTs, the interrotor also conducts currents, causing heat generation through joulelosses. When the concentric design is considered, an additional (compared to conventionalelectric machines) internal heat source is added. This additional source of heat changes thethermal problem as it is known in conventional electric machines, such that development of aneffective heat evacuation mechanism will be an important part of the design. With PM-basedEVTs, no interrotor currents are present, such that no additional heat source is introduced.This facilitates the design of the heat evacuation system.

The IM-based EVT will lead to higher joule losses due to the inherent slip between the squirrel-cage and the magnetic field, and due to the presence of a magnetisation current. Both of thesephenomena do not occur in PM-based EVTs, which operate as synchronous machines (thusno slip), and the magnetic field is created by the PMs, such that no magnetisation current isrequired.

PM-based EVTs reach higher torque densities and efficiencies[14], which is beneficial for the sizeand weight reduction of the EVT. Drawbacks of PM-based systems are the cost of rare earthmagnetic materials, their difficulty of machining, and the inevitable demagnetisation risk.

The MG E-CVT, as presented in sec. 2.4.2 avoids the use of sliprings. However, this comes atthe cost of the introduction of two additional concentrically rotating parts, which are requiredto fulfill the demanded functionality. Besides that, four layers of PMs have to be used, whichalso increases the cost of the machine. The DS PM brushless machine, presented in sec. 2.4.4,also avoids the use of slip rings. A double layer PM rotor is used, and an additional electricalmachine is needed to complete the EVT system.

The SR-EVT, presented in fig. 2.26, has an interrotor which is created from ferromagneticmaterial. Compared to the IM-based EVT, it has the advantage that no additional heat sourceis introduced. Whereas the PM-based EVT also overcomes this issue, the SR-EVT doesn’t makeuse of expensive PMs, and does not have the disadvantages relied to the use of PMs as discussedabove. Nevertheless, the proposed topology is also equipped with carbon brushes and sliprings,such that this disadvantage remains.

2.7 System control

In the considered application of the EVT (i.e. in the drivetrain of HEVs), two levels of controlcan be distinguished. Either the EVT as such can be considered, or the entire powertrain isconsidered. In the latter case, the entire powertrain including the ICE, EVT, electrical energystorage device, and the load (i.e. the vehicle) are considered. As is immediately clear, thishigher level control requires lower level control of the EVT itself, in order to work properly.

2.7.1 Powertrain control level

In the top level control, the goal is to satisfy the demands of the driver of the vehicle. However,due to the flexibility of the system (parallel power flows), some freedom is given to the controllerto fulfill these demands.

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There are three main areas that are taken into account by the controller while deciding upon astrategy to fulfill the driver’s demands:

• the operating point of the ICE and its efficiency

• the state of charge of the electrical energy storage device

• the combination of mechanical and electrical power flows through the EVT

In literature, a number of methods has been presented for modelling and optimising HEV systemcontrol. [2] presents a comparative study between THS and an EVT based HEV. Both aresubjected to the same control method using dynamic programming. Dynamic programming is amethod that is frequently applied to HEV to minimize the fuel consumption. This minimisationis performed offline, for a priori known driving cycles. The optimal control must give the shareof power between the ICE and the battery, and the operating point of the ICE. From this, thesystem is programmed to achieve the highest possible efficiency on the test driving cycle.

In [19] an integrated powertrain control is presented. First, the transmission ratio is optimised.Next, an energy management strategy is presented incorporating the complete hybrid func-tionality of the EVT (including pure electric propulsion). Simulation results are presented todemonstrate the feasibility of the control strategy, and to support the design process for opti-mal component specifications. The new european driving cycle is used to evaluate the energymanagement strategy. The paper concludes that the strategy, together with the presented sim-ulation environment offers an important tool for selecting and optimizing the total powertrainconfiguration.

In [20] and [14], the author proposes Energetic Macroscopic Representation (EMR) to model theentire vehicle system and an inversion-based control scheme is presented. [20] primarily discussesthe EMR modelling of the system. A simple rule-based strategy is used to check the modellingand control structure under a New European Driving Cycle (NEDC). In [14], more attentionis paid to the control itself. Additionally, field weakening is included in order to operate themachine at higher speeds. A simple energy management strategy has been developed in order tocheck the energetic modelling and the deduced control. Again, the NEDC is used as a referencedriving cycle to test the modelling and control. An interesting conclusion of the latter paperis that the developed model can be used both for PM-EVT and IM-EVT if the mathematicalmodel of the EVT is adapted accordingly.

Finally, in [21] a dSPACE hardware-in-the-loop simulation platform is provided for the EVTbased HEV powertrain. The model includes a load emulation system, a hybrid control system,and a driver model. The load model reflects influences such as road condition, drag, and inertia.The driver model takes the form of a PI controller tracking the reference speed. The platform isused to perform experiments, and it is concluded that the platform can track the given drivingcycles very well. Additionally, the platform is used to test a simple control strategy.

The above shows that the subject of EVT based HEV powertrain modelling and control is anactive research topic. However, in all of the above references, little to none attention is paid tothe control of the EVT itself.

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2.7.2 EVT control level

The principle of double layer PM-based EVT, as discussed in sec. 2.4.3, fig. 2.20, is also usedin wind energy conversion systems. For dual power flow wind energy conversion systems (DPF-WECSs) applications of the EVT, the speed of the inner rotor should be controlled precisely inorder to generate the constant voltage and frequency. The configuration of the system is shownin fig. 2.27:

Figure 2.27: Configuration of the DPF-WECS based on the PM-EVT [22]

However, the rotor position of the inner PMSM composed of the inner rotor and outer rotor isdifficult to be measured exactly due to the rotational stator and rotor. In order to avoid themeasurement, [22] proposes a sensorless control strategy based on the model reference adaptivesystem (MRAS). The MRAS based sensorless control strategy is applied to the inner rotorwhile the optimum tip speed ratio MPPT strategy is applied to the outer rotor. The controlstrategy is evaluated by simulation, and its effectiveness verified by experimental results. Thecontroller relies on PI control of the inner and outer rotor speed. The paper concludes that bothsimulations and experimental results confirm the effectiveness of the proposed control strategy.Moreover, this control strategy is not limited to the application in WECSs, but can also beeffective in other areas where the EVT is applied, such as HEVs.

In [23] and [13], field oriented control has been successfully applied to an induction machinebased EVT. Besides the control, a model is proposed for the machine as well. By controllingthe stator and rotor currents, the torques on rotor and interrotor can be controlled. Simulationsin a Matlab-Simulink environment have proven the effectiveness of the control method. Thiscontrol method will be studied further in chapter 5, where the control method will be used as areference for comparison with the proposed control strategy in this text.

The two proposed control methods differ in that the first concerns a double layer PM-basedEVT (of which the inner and outer machine are magnetically decoupled), whereas the secondconcerns the IM-based EVT as proposed by Hoeimakers (see sec. 2.3.2). As described earlier,the latter concerns a machine which is both mechanically and magnetically coupled.

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Chapter 3

Model based predictive control

In [24] a detailed description of the application of Finite Set Model Based Predictive Control(FS-MBPC) to power converters is presented. In literature, the alternative name Finite ControlSet Model Predictive Control (FSC-MPC) is also used. This paper presents a general overviewof several key aspects of the considered methodology, as well as a comparison with traditionalpower converter control techniques, in particular the combination of a linear PI controller andmodulation. The basic concepts, operating principles, control diagrams, and results are used toprovide a comparison between different control strategies. The traditional three-phase voltagesource inverter (VSI) has been chosen as a simple and comprehensive reference for the analysis.As this paper contains both a clear overview of the operating principle and its application to apower converter (in this case a VSI) - which is the intended use of the method in this thesis -the introduction, concept, and algorithm discussion in this chapter is largely based on it.

3.1 Introduction

Since the second half of the 20th century, power converters and drive control technology havebeen in continuous development. In particular, converter control techniques have been a veryactive research topic of power electronics. Linear controllers, together with modulation schemesand nonlinear controllers based on hysteresis comparators, have been analysed and developedthe most. This is the result of a technological limitation: control loops were to be implementedin analog form in the past. As technologies evolved, these control methods were adapted todiscrete time digital implementations - which are now established as the standard.

The advent of digital signal processing opened a new range of opportunities, allowing the de-velopment of more complex control techniques. Some of these techniques have been applied topower converters: fuzzy, adaptive, sliding mode, and predictive controls. The latter is a gen-eral control concept, and can take different implementation forms depending on the operatingprinciple and characteristics.

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CHAPTER 3. MODEL BASED PREDICTIVE CONTROL

3.2 Concept

In general terms, predictive control can be considered as any control algorithm that uses a modelof the system to predict its future behavior and selects the most appropriate control action basedon an optimality criterion. One of the earliest occurrences of this concept can be found in theso-called dead-beat control: this control scheme eliminates the classic linear controller by using apredictive model of the system. The model is used to calculate the required reference voltage inorder to reach the desired reference value for a certain variable (in most cases, i.e. the current).The predicted reference voltage is later generated by the converter via modulation stage. Thishas been applied for current control in a wide range of power electronic devices: inverters,rectifiers, active filters, and uninterruptible power supplies (UPSs).

One variant of the concept is MBPC. As the name suggests, the model fulfills a key role in thepredictive control algorithm: a model of the system is considered in order to predict the futurebehaviour of the variables over a given time frame (integer multiple of the sample time). Thesepredictions are evaluated based on an optimality criterion (e.g. their correspondence with thereference values), and then, the sequence that is most optimal among all predictions is chosen,determining the future control actions. Only the first action of this control sequence is applied,after which the algorithm is run again every sampling period.

The only output of the algorithm is the control action to be applied next: predictions of futurevalues of quantities, based on the model of the system, which are calculated during execution ofthe control algorithm are not registered. In every sampling period, the algorithm is carried outbased on the previously determined control action, and the instantaneous measurement of thesystem state variables instead of relying on the system behaviour predicted in the algorithm. Inthis way, a feedback loop is obtained, correcting the control method with the actual machinebehaviour.

The implementation of MBPC for power converters can take advantage of the inherent discretenature of power converters. As power converters have a finite number of switching states, theoptimisation problem can be simplified and reduced to the prediction of the system behaviouronly for these possible switching states. Then, each prediction is used to evaluate a cost function(in literature also known as quality or decision function), and consequently, the state withminimum cost is selected and applied. This method is entitled with Finite Set MBPC (FS-MBPC) since the number of possible control actions (switching states) is finite. This methodoccurs in literature with different names (finite alphabet MPC, or simply as predictive control)and has been successfully applied to a wide range of applications, as listed in [24].

3.3 Algorithm

In the considered paper, the operating principle of the control is explained starting from anidealised case: measurements, computations, and control actions are assumed to take placeinstantaneously. Although this situation deviates from reality, it is instructive to consider thisidealised case in order to understand the operating principle and the adaptations that are madein the non-ideal case.

The task of the algorithm can be described as the determination of an appropriate control

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action S(t) (usually the gate signals of the converter) that will drive a generic system variablex(t) as close as possible to a desired reference value x∗(t). Consider the behaviour of x(t) andits regularly sampled value x(tk) over a sample period Ts for a system with a finite number ofcontrol actions n, as shown in fig. 3.1. Since the control set is finite (n), they can be evaluatedtogether with the measured value x(tk) based on a prediction function fp, to predict all possiblesystem transitions xpi(tk+1) = fp(x(tk), Si), for i = 1, . . . , n. This prediction function is deriveddirectly from the discrete model and the parameters of the system. To evaluate each of thepossible switching states, a cost function can be defined, which is (usually) dependent on thedesired reference value (x∗(tk+1)) and the predictions (xpi(tk+1) for i = 1, . . . , n). A typicalexample for the cost function could be the absolute deviation of the prediction with respect tothe reference value (|x∗(tk+1) − xpi(tk+1)|). Remark that the reference value at the next timeinstance is needed to evaluate the cost function, thus requiring the reference to be known inadvance (or to be calculated by extrapolation techniques).

Figure 3.1: FS-MBPC operating principle: ideal theoretical case [24]

For example in fig. 3.1, the predicted value xp3(tk+1) is the closest to the reference x∗(tk+1),hence S3 is selected and applied at time t = tk. The same reasoning can be repeated in thefollowing time interval, resulting in S2 to be selected and applied at time t = tk+1.

However, it is not possible to achieve this theoretical case where variables can be measured,predicted, and controlled instantly. This non-ideality can be overcome by altering the timingof the control scheme towards a two-step-ahead prediction. The principle remains largely thesame, but is extended over an additional period. The main difference compared to the ideal caseis that the algorithm does not dictate the control action that should be applied at time t = tk(Sk), but the control action that should be applied at t = tk+1 (Sk+1) (the control action Sk isknown from execution of the algorithm in the previous period, or set to a default value in casethe first period is considered). The paragraphs below discuss the different steps of the controlalgorithm, based on fig. 3.2.

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Figure 3.2: FS-MBPC operating principle: implementation case [24]

It is important to keep in mind that each of the steps discussed below are performed in thecontroller in a time that is smaller than the sampling period Ts. The following figure showsgraphically how the timing of the controller is related to the real time flow. The explanation ofthe different steps in the algorithm below should enable the reader to understand the executionand timing of the control algorithm.

time!k! k+2!k+1!Ts!

apply Sk+1 !

estimation! prediction! optimisation!

estimated!xp(tk+1)!

predicted!xpi(tk+2)!

optimal!Sk+1!

control exec.!

apply Sk !

meas.!

Figure 3.3: Timing of the FS-MBPC algorithm execution with respect to real time

Estimation

The first step of the algorithm is called the estimation. During the previous execution of thealgorithm, the control action Sk to be applied at t = tk is determined. At t = tk a measurementof the system variables is taken (x(tk)) to serve as the initial conditions of the system model.Applying the control action Sk and the initial conditions x(tk), the system model can be used

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to estimate the evolution of the system as a result of these inputs. The output of the systemmodel is then the state of the system at t = tk+1 (the estimation).

Prediction

Consequently, the second step in the controller is executed. Here, the state of the system attk+2 (xpi(tk+2) for i = 1, . . . , n) is predicted based on the estimated system state xp(tk+1) andall n possible control actions Si (applied at tk+1). The result of this step is a range of n possiblestates xpi(tk+2) in which the system could be at t = tk+2, if the control action Si would beapplied at t = tk+1.

This is the step the algorithm is named after: the state of the system is predicted two stepsahead from the moment the measurements are taken.

Optimisation

The third and final step in the algorithm is the evaluation of these possible system states(similarly to the evaluation in the ideal case). Their correspondence with respect to a referencevalue is evaluated using a cost function: the control action Sm leading to the system statexpm(tk+2), which has the lowest associated cost, is then selected to be applied at t = tk+1

(Sk+1 = Sm).

The algorithm can be presented in a graphical way, alongside fig. 3.2, in the form of a flowchart: see end of this chapter.

3.4 Implementation

3.4.1 Model

The succes of the control algorithm depends largely on the quality of the system model, andthus the ability to correctly estimate and predict the evolution of the system over a sampleperiod. As indicated above the prediction step relies on the results of the estimation step.The correspondence between the real system state x(tk+1), measured at t = tk+1 and thepredicted system state xp(tk+1) is only dependent on the quality of the system model (giventhat no external disturbance corrupts the normal machine behaviour). Similarly, the result ofthe prediction step relies on the quality of the model.

From this, it is clear that the system model is crucial for the effective functioning of the controlalgorithm.

3.4.2 Control variables

The system to be controlled is the EVT, we assume the internal combustion engine and the loadto be deterministic (see sec. 4.7). From the system topology (fig. 2.16), we can derive which

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quantities can be considered inputs to the system, and which can be considered as outputs (thevariables to be controlled).

The gate signals of the inverters are considered as the control actions. In this set-up we will usetwo two-level voltage source inverters (2L-VSIs) to supply the stator and rotor windings. Theconverter consists of three phase legs, which can connect the corresponding load terminals to thepositive or the negative bars of the converter by controlling the power switches of a leg. For eachconverter 23 = 8 switching states are possible, making a total of 64(= 26) possible switchingstates if two converters are considered. The finite set of control actions Si (i = 1, . . . , n) asdiscussed in the above paragraph are thus the 64 different switching states. The output of thecontrol algorithm will thus determine which switching state should be applied in the next period.

Due to the nature of the considered system, the relevant output quantities are of mechanicalnature: the torques on the interrotor (T2) and the rotor (T3) are the relevant output quantities,and are thus desired to be controlled. As will be discussed later (sec. 4.2.10), the torque isdependent on both current and flux in the machine. If no other constraints are imposed onthe system, an infinite number of combinations of currents and fluxes can result in the samedesired torque. These additional degrees of freedom can be made use of to control additionalquantities of the machine. Consequently, as the magnetic behaviour of the machine is mainlygoverned by its saturation state, we chose to impose a constraint on the flux. Besides the torqueon the interrotor and rotor, also the stator flux Ψ1 will be considered as a control variable. Thereference value for the flux magnitude is fixed and chosen such that the machine is operatingclose to saturation (but still in the linear part of the material characteristic). In section 5.1.1,this particular choice of the flux reference will be discussed. The reference value for the interrotortorque T2 is considered variable, as this is the torque that will ultimately be desired by the driverof the HEV, and can thus vary from one moment to another. The reference value for the rotortorque T3 (a braking torque, since the rotor is driven by the ICE, acting as the rotor of anelectric generator) is also variable, such that the ICE generates the demand of mechanical powerin the most efficient possible way depending on the operating mode of the EVT (see section4.5).

3.4.3 Cost function

To evaluate the obtained predictions in order to determine the most favourable control actionan optimality criterion is required. As indicated in section 3.3 a cost function can be used toevaluate the correspondence between the predictions and the reference value(s) of the controlvariable(s). For example, the absolute deviation of the predicted value with respect to thereference value can be taken as a cost function:

Costi = |x∗(tk+1)− xpi(tk+1)| (3.1)

Where xpi and x∗ represent the ith prediction of control variable x (corresponding with thecontrol action Sk) and the reference value for the same control variable, respectively. As discussedabove, the control variables are: interrotor torque T2, rotor torque T3, and stator flux Ψ1. Thetotal cost related to a particular prediction, will thus be composed of three elements: it willconsider the correspondence of all three control variables with respect to their reference values.

Additional adaptations can be made to the individual cost functions:

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• assigning a greater cost to greater deviations by using exponential functions (e.g. squarederror)

• letting one control variable dominate the cost function if its deviation crosses a certainthreshold (thus ensuring recovery of the considered control variable)

• in order to control a variable within a small tolerance band without allowing any outliers,one can choose to assign an infinite cost to any control action leading to predictions outsidethis interval

• in order to limit a control variable to a given maximum, an infinite cost can be assigned toany control action leading to predictions above this maximum. Analogously, a minimumfor a control variable can be implemented

• an integrating term can be included in the cost function to penalise deviation of the averagevalue w.r.t. the reference

• etc . . .

Some of the above discussed cost function features are shown graphically in the following figure:

xpi(tk+1) – x*(tk+1)!0!

tolerance!band!

zero cost band!

Cost!

uppe

r lim

itat

ion!

Figure 3.5: Different cost functions

35


To avoid a nervous control, a tolerance band is included in the cost function: this means thatdeviations within a predetermined interval around the reference value will be tolerated (i.e. willresult in a zero cost). With this modification to the cost function, the switching frequency of theconverter will be lower, resulting in fewer switching losses. The rationale behind the toleranceband is that none of the possible control actions will result in exactly the desired reference value,such that even the control action resulting in a prediction very close to the reference value wouldresult in a cost (even though it is the best possible control action amongst the possibilities). Also,it could be more beneficial to maintain the same switching state throughout multiple periodsand to allow a small deviation from the reference value, compared to the additional switchinglosses that would occur if the reference value is tracked as closely as possible the whole time.The effect of including a tolerance band in the cost function on the switching frequency of theinverter and the quality of control is discussed in [25].

To result in the total cost of the considered control action, the sum of three individual costs(each corresponding with a certain control variable) is considered. However, as the order ofmagnitude of the flux and torque differ, one must rescale the costs in order to have comparablecosts for each of the control variables. In order to achieve this, relative errors are considered:the error is expressed relatively to a fixed reference value of the expected order of magnitude ofthe control variable.

Some of the control variables might require closer adherence to their reference values, such thattheir tolerance band is chosen narrower. Additionally, the individual costs in the summationmay be assigned a weighting factor to increase their relative weight in the total cost.

The above considerations ultimately lead to the following general form of the cost function:

CostSi = W1 · CostΨ1 (Ψ∗1(tk+1),Ψ1,pi(tk+1), toleranceΨ1)

+W2 · CostT2 (T ∗2 (tk+1), T2,pi(tk+1), toleranceT2)

+W3 · CostT3 (T ∗3 (tk+1), T3,pi(tk+1), toleranceT3)

(3.2)

In [26], an analysis of different cost functions is presented, besides a method for assessing controlquality, it discusses the possibility of including various aspects such as switching frequency,varying weight factors, and integrating action. This shows both the great strength of thiscontrol algorithm, but also it greatest difficulty.

A lot of freedom is given to the control designer: which cost function, which weight factors, andwhich tolerance bands to use, but also which quantities to include in, or exclude from the costfunction. The main difficulty in designing the control is finding the right balance between eachof the before mentioned aspects, resulting in a control that performs as it should. There areno prescripted rules or methods to determine the ”correct” parameters, such that tuning of thecost function requires careful consideration of the effect of each of the parameters. Amongstthe goals of this text is to provide qualitative insight in the effects of changing cost functionparameters on the control. Besides changing parameters of the cost function (see eq. 3.2),alternative control strategies will be considered in order to increase performance of the control.The proposed methods and their performance are discussed in chapter 5.

36


3.5 Advantages and disadvantages

In [24], a comparison between FS-MBPC and classic linear controllers is made. In particular,conceptual differences in the operating principle, controller design, and implementation areconsidered. Different operation modes and aspects are studied, and strategies to overcomedrawbacks are presented.

The conclusion to this comparison is very promising for the future of FS-MBPC. The controlmethod is conceptually very simple, yet powerful: it allows to control different converters andvarious variables without needing additional modulation techniques or internal cascaded controlloops. This is why the control method is very suitable for multiple input - multiple output(MIMO) systems. FS-MBPC is also praised for its flexibility to control different variables, anddealing with non-linearities, whilst complying with constraints and additional system require-ments. New terms can be introduced into the cost function dealing with other aspects thanthe control itself, leading thus to improved performance, efficiency, and safety, or even to meetstandards and operational limits. Below, a number of possibilities is listed:

• the switching frequency of the inverter can be influenced either indirectly or directly:

– indirectly by introducing tolerance bands to the regular cost functions (as discussedabove)

– directly by including a term in the cost function which helps to select the controlaction leading to the lowest switching frequency

• the losses can be minimised by adding an additional term considering the joule lossescaused by the control action

• the controller can be made self-learning and adjust the parameters of its model based oninformation it receives from the real machine

• etc . . .

The method also uses the discrete nature of power converters and microprocessors advanta-geously. The required high computation power - which was the limiting factor for the methodbefore - is provided by today’s standard DSPs and microcontrollers.

As indicated above, the model of the system plays a critical role in the functioning of thealgorithm. The quality of the model is determining the quality of the estimation and predictionstep, and thus the quality of the entire control algorithm. Here lays a weakness of the method:a trade-off has to be made between the quality of the model and the required computationtime to execute the estimation and prediction step. A more accurate model will require longercomputation time, which demands a longer sample period Ts, such that the control becomes lessaccurate in time. On the other hand, a fast model will allow smaller sample periods, but mayinvolve erroneous control actions due to reduced accuracy of the model.

As proposed in [24], part of the control algorithm can be executed offline to shorten the calcula-tion time. For example, the optimization problem could be replaced by a search tree (calculatedprior to the control execution), thus reducing the calculation time. Of course, this requirescareful consideration of the search tree, which is crucial for the successful operation of thecontrol algorithm. Another solution may be found in another variant on the concept, General-

37


ized Predictive Control (GPC), where the optimisation is solved analytically, obtaining a linearcontroller. Nevertheless, this makes it very difficult to include nonlinearities and constraints.

3.6 Field Programmable Gate Array (FPGA)

The Xilinx website1, provides a single-paragraph description of an FPGA, which is quoted here:

Field Programmable Gate Arrays (FPGAs) are semiconductor devices that are based arounda matrix of configurable logic blocks (CLBs) connected via programmable interconnects.FPGAs can be reprogrammed to desired application or functionality requirements aftermanufacturing. This feature distinguishes FPGAs from Application Specific IntegratedCircuits (ASICs), which are custom manufactured for specific design tasks. Although one-time programmable (OTP) FPGAs are available, the dominant types are SRAM (StaticRandom Access Memory) based which can be reprogrammed as the design evolves.

Basically, an FPGA provides a matrix of logical blocks of which the connections can be pro-grammed by the user in order to perform calculations. In- and output ports are configured inorder to enable communication with other devices. As the size of FPGAs increased, the imple-mentable digital circuits became more complex. In order to maintain an overview in the morecomplex designs, design of the circuit can be done using HDL (hardware description language),a higher programming language, which allows the user to focus on functional design. The designis made in a CAD program, which translates the HDL into lower programming language toconfigure the connections inside the FPGA.

SRAM is a memory technology which enables reprogramming. Data stored in this type ofmemory cannot be changed during execution of the program (hence it is static). However,the memory can be overwritten when the device is reprogrammed. RAM is a volatile memorytechnology, meaning that the memory is cleared when the supply voltage is removed. Due to itsreprogrammability, the FPGA is said to be programmable ’in the field’, hence its name.

In this thesis, a Xilinx Virtex-II Pro XC2VP30 will be used. The design of the program will beperformed through Xilinx System Generator 10.1, using Matlab/Simulink (R2007b) as a GUI.The System Generator is used to compile the CAD-generated model into a bitstream used toprogram the FPGA. For programming the FPGA, the software Project Navigator iMPACT(version 10.1.03) is used. The communication of the FPGA with the PC (to collect data aboutthe execution of the programmed system), is provided through a ChipScope. The software usedfor this purpose is ChipScope Pro Analyzer. The Xilinx System Generator, Project Navigator,and ChipScope are all part of the Xilinx Integrated Synthesis Environment (ISE) Design Suite.Anther useful program in the Design Suite is the Timing Analyzer, which is used to debug timingerrors in the model.

3.6.1 Advantages and drawbacks

The FPGA has the ability to perform simple calculations very fast. This enables complexmodels to be ran in short times. This property is very useful in the context of this thesis: acomputationally demanding control has to be ran fast enough such that it can be finished within

1http://www.xilinx.com/training/fpga/fpga-field-programmable-gate-array.htm

38

http://www.xilinx.com/training/fpga/fpga-field-programmable-gate-array.htm


the proposed sample time. An FPGA is very suitable for hardware-in-the-loop simulations, asit allows real time calculations: running the FPGA for 1 second will simulate the behaviour ofthe system during 1 second. This particular feature opens another area of application for theFPGA. As it is able to calculate in real time, it is possible to use the FPGA in a real setup, e.g.for control of a real machine.

As with other digital computers, the FPGA has a discrete time nature: its clock period can beprogrammed, and is chosen to be 10ns in this thesis. The difference with respect to modeling ina Matlab/Simulink environment for example, is that the FPGA does not command the timingof the calculations. Here lays one of three major drawbacks that come with the fast computationcapabilities.

The precise timing of each calculation has to be defined by the user. This is done by includingdelays in the path of a signal towards the calculation block. Once a calculation block is pro-grammed in the system, it will perform its task incessantly: at each clock pulse, it will processthe signals that are at its input ports according to its prescribed function. This shows imme-diately the importance of meticulously setting the timing of each of the signals paths leadingto each of the functional blocks in the system. Alternatively, the user can make use of enablesignals, which trigger the block to become active only when the enable is high (logical 1). Ofcourse this requires setting of the logic controller behind the enables, which in turn requiresknowledge of the timing of signal flows in the model.

The FPGA has the intrinsic ability to perform calculations in parallel, which is one of thereasons why such short computation times can be achieved. The user can benefit from thisparallel pipelining feature by choosing the timing and routing of the signals carefully. Withinsight in the flow of the signals, the user can optimise the computation time even further.

The second main drawback of the FPGA is that it only allows to perform simple calculations. Itspossibilities include: addition (and subtraction), multiplication with constants, multiplication oftwo varying signals, logical operations, sine (and cosine), and bitwise operations. Based on theprovided blocks, new functions can be generated. A block for divisions is available, but requiresa great amount of resources compared to the other operations. One of the main lacking elementsidentified during the development of the model, is a square root block.

Not only the timing of the routing of the signals has to be set, each of the calculations is assigneda limited number of FPGA periods (multiple of 10ns) to complete its function. The durationof the calculation is indicated with the term ’latency’. The required latency of a calculationis determined by the nature of the calculation, and by the magnitude of the signals that areprocessed (increasing with the size of the signals). For each signal in the system, the user mustspecify a number of bits that may be used to represent the signal. This requires an estimation ofthe order of magnitude of each of the signals, in order to assign a number of bits to it, which canrepresent the value of the signal truthfully. Both the total length of the binary representationof the signal, and the location of the binary point can be chosen. In this way, not only its orderof magnitude is of importance, but also its (desired) level of accuracy. This shows the thirdmain drawback of the FPGA: it is up to the user to specify the assigned number of bits to eachsignal, thus introducing limitations on their values and precision. Inevitably, the introduction ofa limited precision involves rounding errors, which is the price to be paid for short computationtimes.

Along with the predetermination of signal timing (and resulting parallel pipelining), the a priori

39


limited number of bits assigned to represent each signal, is what makes the FPGA so fast. AMatlab/Simulink model for example, does not require these inputs from the user, but insteaddetermines these properties online. It also represents the signal’s values as precise as possible,such that computations are bound to take longer.

3.6.2 Useful features

A number of features may be used to facilitate the design of a model. Moreover, they mayenhance the readability for future users and provide flexibility to include additional blocks inthe model afterwards.

Parametric design

For convenience of use for the developer, and any future users of the same model, it is importantto include structure in model, and provide necessary comments in order to understand how themodel was built and how each of the signals flows through it. By using a parametric designand an according parameter file, all of the timing parameters (and others) can be grouped in aclarifying manner.

Multiplexing

For the implementation of the above discussed control algorithm, the model of the system isrequired twice: once in the estimation step, and once in the prediction step. Additionally,the model may be used a third time to emulate the machine itself. Instead of implementingthe same model two (or three times), one can make use of multiplexing. Multiplexing maybe used to select different values of a quantity during a given time interval. In this way, onlyone model of the machine is needed, as the input quantities for the estimation, prediction, andemulating step are applied consequently. By careful timing setting of the registration of theoutput of the machine model, the output of each of the steps may be registered and used forpost processing. Multiplexing can significantly reduce the amount of resources needed for theentire model, and can sometimes be a necessity when the limited resources of the consideredFPGA are all occupied.

Enabling

A useful tool to synchronise signal flow is to make use of the enable function. For example, enablesignals can be used to trigger a number of parallel calculations at the same time. Alternatively,they can be used to delay the advancement of signals to provide a time buffer for the blocksupstream. Time buffers are particularly useful when the intention exists to include additionalblocks later. A time buffer can then be made use of to assign computation time to these addedblocks, without changing the timing of the entire model.

40


2. FCS MPC generic control diagram.

3 FCS MPC i l ithFigure 3.4: FS-MBPC generic algorithm: flow chart[24]

41


42

Chapter 4

Modelling

4.1 The need for modelling

As indicated in section 3.4.1, having a model is the key element to be able to run the controlalgorithm. However, as a real machine to apply the control to is not available at the time thismaster thesis was written, we also require the machine model to simulate its behaviour whensubjected to the control.

The schematic representation of the setup, shown in fig. 4.1, shows a number of constitutingelements that require modelling: the controller, the power electronic converters, the machineitself, the electrical energy storage device (here shown as a battery), the ICE, and finally, theload (i.e. the wheels of the vehicle). From the explanation of the algorithm in sec. 3.3, it isclear that a model of the machine and the inverters must be known to the controller in order toexecute the control.

4.2 Machine

As discussed in chapter 2, we will be treating the control of an induction machine based EVT,as described by Hoeijmakers (see sec. 2.3.2). Due to the particular design choice of a thininterrotor (as explained in chapter 2), the stator and rotor are magnetically coupled, such thatthe machine can not be approached as two independent induction machines. In this section, wewill discuss the assumptions and mathematical equations that lead to the machine model.

4.2.1 Assumptions

With the help of a number of assumptions, the machine equations remain workable and allow tocreate a model. These assumptions (for a wound rotor induction machine) have been borrowedfrom [27] and allow to write the machine equations in a simple analytical form.

• a three phase symmetrical machine

• the windings are sinusoidally distributed (applicable to both stator and rotor windings)

43

CHAPTER 4. MODELLING

LOAD!

EVT!

ICE!

PEC 1!

PEC 3!

+!–!

BATT.!

REF.!

MEAS.!

CO

NT

RO

LLER!

Figure 4.1: Overview of the considered system

• there is no skin- or proximity effect

• there is no saturation

• there is no influence of the slots on the field

An additional assumption is be made concerning the interrotor. This assumption is borrowedfrom [28], where the interchangeability of a squirrel-cage rotor and a wound rotor is discussedin terms of equivalency of generated mmf. Hence, the following assumption should be added tothe above list:

• the squirrel cage is replaced by equivalent three phase symmetric and sinusoidally dis-tributed windings

The combination of these assumptions allows to treat the machine in an analytic manner, aswill be demonstrated in the next sections.

4.2.2 Reference frames

As a starter, it is instructive to consider the different reference frames that will be consideredin the following discussion. A clear understanding of the different (rotating) reference frames isneeded in order to understand the transformations that need to be applied in order to expressall the quantities in a common reference frame.

For simplicity, the considered reference frames in this section are drawn for a single pole-pairmachine (Np = 1), whereas the machine that will be modeled is a two pole-pair machine (Np =

44


2). This difference has no effect on the machine equations, only on the machine parameters andtorque calculation - where it has been taken into account.

Stator

In fig. 4.2, the stator is schematically represented. As in an ordinary induction machine, thestator is a fixed part of the machine (ω1 = 0). From fig. 4.1, it can be seen that the phasewindings of the stator are fed by one of the PECs. Its phase windings are at standstill (as theyare fixed to the machine housing), such that phase quantities are expressed in a reference frameat standstill.

stator!ω1=0!

α1"

β1!

d#q#

ωbt#

ωb#

Figure 4.2: Reference frames of the stator

By applying the Clarke transformation (see eq. 4.1 for the Clarke transformation applied on ageneral quantity F ) to these phase quantities, they can be expressed in another reference frame,denoted by α1β1. This reference frame is fixed to the stator, and as a consequence, stationary.Note that in the transition from phase quantities to Clarke components, the homopolar com-ponent has been omitted. In the following discussion of the machine equations, the homopolarcomponents of current and voltage will not be withheld; the explanation follows in sec. 4.3.4.

[FαFβ

]= TClarke

FaFbFc

=

2

3

[1 −1/2 −1/2

0√

3/2 −√

3/2

]FaFbFc

(4.1a)

FaFbFc

= T−1

Clarke

[FαFβ

]=

1 0

−1/2√

3/2

−1/2 −√

3/2

[FαFβ

](4.1b)

Furthermore, it is possible to express the obtained Clarke components in a rotating reference,named the Park reference frame, or alternatively, the qd-reference frame. In accordance with

45


the use of this reference frame in the course texts published at our department ([27][18][29][28]),a reference frame whereby the q-axis lags the d-axis is chosen. The qd-reference frame rotates ata predetermined speed ωb, such that the angle between the rotating and the stationary referenceframe is given by ωt at any time. The rotation speed of the qd-reference frame can be chosenarbitrarily. As a particular choice, one can choose the rotation speed of the air gap field (ωb = ω),thus obtaining an instantaneously synchronous reference frame. Another particular choice is arotational speed equal to zero (ωb = 0), thus resulting in an equivalency with the stationaryα1β1-reference frame.

Eq. 4.2 shows the transformation matrices to be applied to perform the transition of Clarke-to Park-components. The instantaneous angle between both reference frames is denoted by θ(=ωbt). By multiplying both transformation matrices in the correct oder, that is Trot ·TClarke, andT−1Clarke · T−1

rot , the transformation matrices for going from phase quantities to Park components,and back respectively, are obtained.

[FqFd

]= Trot

[FαFβ

]=

[cos(θ) sin(θ)−sin(θ) cos(θ)

] [FαFβ

](4.2a)

[FαFβ

]= T−1

rot

[FqFd

]=

[cos(θ) −sin(θ)sin(θ) cos(θ)

] [FqFd

](4.2b)

Rotor

The story for the rotor differs from that of the stator in that the rotor is not at standstill, butrotates (at rotational speed ω3). The phase windings of the rotor are fixed to the rotor, such thatthey rotate as well. This means that the phase quantities, applied by PEC3 through sliprings(see fig. 4.1), are not at standstill as was the case for the stator, but rotate along with the phasewindings. The Clarke components for the rotor are expressed in the α3β3-reference frame (asindicated in fig. 4.3), which is fixed to the rotor, and consequently rotates as well. Do notethat there is no relative speed between the phase windings and the considered Clarke referenceframe, such that the transformation between both (eq. 4.1) still holds.

46


rotor!

α3"

β3!

d"q"

(ωb - ω3)t"

ω3!

ωb"

Figure 4.3: Reference frames of the rotor

Here as well, the transition from the Clarke components in the α3β3-reference frame to a rotatingreference frame is possible. There is intentionally not made a notation difference for the qd-reference frame in fig. 4.2 and fig. 4.3, as a common qd-reference frame for both stator, rotor,and interrotor will be chosen. The difference in the transition from rotor Clarke components toPark components, compared to the stator, is the rotation angle between both reference frames.As in this case the α3β3-reference frame is rotating as well, it is the relative speed (ωb − ω3)between both frames that has to be considered. Thus, the instantaneous angle θ, which is usedto perform the rotation, according to eq. 4.2, is defined as: (ωb − ω3)t.

In the following, all machine equations will be developed in a common qd-reference frame, rotat-ing at an arbitrary rotational speed ωb, such that the generality of the equations is maintained.

4.2.3 Machine equations

The machine equations follow from the consideration of Faraday’s and Ohm’s law in the qd-reference frame, rotating a with rotational speed ωb. The equations are derived from [29].

V1q = R1I1q + pΨ1q − ωbΨ1d

V1d = R1I1d + pΨ1d + ωbΨ1q

(4.3a)

V2q = 0 = R2I2q + pΨ2q − (ωb − ω2)Ψ2d

V2d = 0 = R2I2d + pΨ2d + (ωb − ω2)Ψ2q

(4.3b)

V3q = R3I3q + pΨ3q − (ωb − ω3)Ψ3d

V3d = R3I3d + pΨ3d + (ωb − ω3)Ψ3q

(4.3c)

In which p is the Laplace operator, ωb the (electrical) rotational speed of the reference frame,ω2 and ω3 the (electrical) rotational speed of the interrotor and rotor respectively. The terms

47


on the right hand side of the equation represent the resistive voltage drop, the reactive voltagedrop caused by the inductance of the windings, and the back-emf respectively.

With the particular choice of ωb = ω (the electrical pulsation of the air-gap field), the speeddifferences can be written in a more condensed form, by using the definition of slip: ωb − ωi =ω − ωi = siω. The definition of the slip is given by:

si =ω − ωiω

(4.4)

By choosing ωb = 0, the machine equations are expressed in the stator Clarke reference frame(i.e. α1β1-reference frame), such that the indices qd can be replaced by α1β1 respectively.

By using the complex notation for flux and current vectors in the qd-reference frame,

Ψi = Ψiq + jΨid (4.5a)

Ii = Iiq + jIid (4.5b)

the fluxes are defined as:

Ψ1

Ψ2

Ψ3

=

L11 L12 L13

L21 L22 L23

L31 L32 L33

I1

I2

I3

(4.6)

Representing the flux vectors linked with the stator, interrotor, and rotor, respectively. As wehave made the assumption of absence of saturation (sec. 4.2.1), constant inductances are used.To ensure the validity of the model, attention has to be paid to the magnitude of the flux duringsimulations. The specific choice for the reference of the flux level and its control, is discussed inthe following chapter.

As rotor, interrotor, and stator are magnetically coupled, their flux linkage differs only by leakageflux.

4.2.4 Machine parameters

The machine dimensions and parameters have been provided by my counsellor, and will bemaintained throughout this entire text. To give an idea of the size of the machine, some of itsmain characteristics are listed: the considered machine is a four-pole (Np = 2) machine, withan axial length of 0.1m, and outer diameter of 0.24m. The machine parameters were calculatedfor a flux level leading to a flux density of 1.6T in the teeth of the machine. The latter value isa rule of thumb used to determine a magnetisation state for which the linearity assumption stillholds.

The inductances (expressed in H) of the machine were obtained by my counsellor through FEcalculations:

L11 L12 L13

L21 L22 L23

L31 L32 L33

=

0.0858 0.00391 0.08150.00391 0.000185 0.003840.0815 0.00384 0.0891

(4.7)

48


The resistances of stator, interrotor, and rotor, were calculated from knowledge of the windingparameters, and resulted in:

R1 = 0.5092Ω

R2 = 0.000596Ω

R3 = 1.25Ω

(4.8)

For the dynamics (see sec. 4.5 and sec. 4.7), the inertias for the interrotor (J2) and rotor (J3)are set to 0.1 and 0.5kgm2, respectively.

4.2.5 Differential equations

Combining eq. 4.6 and eq. 4.3, we can rewrite the latter in the following (matrix) form:

V = RI + LbdI

dt+ ωbG1I + ω2G2I + ω3G3I (4.9)

The matrices used in this expression are defined as follows:

V =

V1q

V1d

V2q

V2d

V3q

V3d

I =

I1q

I1d

I2q

I2d

I3q

I3d

R =

R1

R1

R2

R2

R3

R3

Lb =

L11 0 L12 0 L13 00 L11 0 L12 0 L13

L21 0 L22 0 L23 00 L21 0 L22 0 L23

L31 0 L32 0 L33 00 L31 0 L32 0 L33

G1 =

0 −L11 0 −L12 0 −L13

L11 0 L12 0 L13 00 −L21 0 −L22 0 −L23

L21 0 L22 0 L23 00 −L31 0 −L32 0 −L33

L31 0 L32 0 L33 0

G2 =

0 0 0 0 0 00 0 0 0 0 00 L21 0 L22 0 L23

−L21 0 −L22 0 −L23 00 0 0 0 0 00 0 0 0 0 0

G3 =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 L31 0 L32 0 L33

−L31 0 −L32 0 −L33 0

(4.10)

The differential equation in I (eq. 4.9) can be rewritten in the following form:

dI

dt= AI + Bu (4.11)

WithA = −L−1

b R, B = L−1b , and u = V − ωbG1I − ω2G2I − ω3G3I (4.12)

49


Note that some terms in equation for u can be calculated using eq. 4.6:

G1I =

−Ψ1d

Ψ1q

−Ψ2d

Ψ2q

−Ψ3d

Ψ3q

G2I =

00

Ψ2d

−Ψ2q

00

G3I =

0000

Ψ3d

−Ψ3q

(4.13)

4.2.6 Linearisation: Euler approximation

The machine equations have thus far been transformed into the differential (matrix) equation(eq. 4.11). Due to the discrete time nature of the FPGA (as indicated in sec. 3.6), the machineequations have to be written in a discrete time representation. The timestep (∆t) chosen forthe dicretisation of the equations can conveniently be chosen equal to the sample period (Ts) ofthe control algorithm.

The choice of the sampling time TS - and consequently of the discretisation step ∆t - is atrade-off between the accuracy of the first order Euler approximation of the derivative and thecalculation speed of the controller. By choosing Ts smaller, less computation time is given forthe controller, but the linearisation becomes more accurate. The optimisation of computationalspeed of the controller is outside the scope of this master thesis, such that a sufficiently large Tswill be chosen, to ensure that computation time is not a constraining factor for the controller(see sec. 7.1).

The first order Euler approximation for a derivative is characterised by following expression:

df(t)

dt=f(t+ ∆t)− f(t)

∆t(4.14)

From this, we can obtain the value of a quantity, at a time ∆t later than the considered moment,which is exactly what is needed in the control algorithm for the estimation and prediction step.

f(t+ ∆t) = f(t) +df(t)

dt∆t (4.15)

Applying eq. 4.15 to eq. 4.11 allows us to calculate the current at time t + ∆t, based on thecurrent, voltages, and pulsations at time t.

4.2.7 Implementation

Due to resource limitations of the FPGA, it was impossible to model the machine in a commonrotating reference frame, as this required too many sine/cosine blocks to perform all the necessarytransformations. Instead, the machine was modeled in a common stationary reference frame:the stator Clarke, α1β1-reference frame.

Due to the choice of this reference frame, a rotation of the rotor Clarke (α3β3) components isstill needed, as explained above. Fortunately, enough resources were available on the FPGA to

50


implement this transformation. For the chosen reference frame, the rotational speed ωb is zero,such that some terms in the above equations disappear.

As indicated above, the expression of components in a stationary qd-reference frame is the sameas expressing them in α1β1-reference frame, such that the indices in the above equations can beinterchanged accordingly.

In what follows, all quantities will be expressed in the common reference frame, which will benoted by αβ (the index 1 to indicate the stator Clarke reference frame is omitted henceforth).

4.2.8 Dynamics feedback loop

The electromagnetic torque (produced by the machine), the load torque, and the rotationalspeed are related through the equation of rotational motion:

Jdωmdt

= Tem − Tload (4.16)

In which J represents the inertia of the rotating part(s) connected to the shaft, Tem the elec-tromagnetic torque, and Tload the load torque. This physical law determines the resulting speedωm (and thus rotation θ by integration) from the generated and load torque. It is necessary toinclude this relation as an internal feedback loop in the machine model. In order to do this, theproduced electromagnetic torque has to be known.

From Lorentz’s law, the torque on interrotor and rotor can be calculated as follows:

T2 = 3

2Np(I2αΨ2β − I2βΨ2α)

T3 = 32Np(I3αΨ3β − I3βΨ3α)

(4.17)

Note that torque magnitude and sign are not only dependent on flux and current magnitude,but also their relative angle. The expression for torque contains four variables, such that forany three chosen values, the third can be calculated in order to result in the desired torque.As discussed in sec. 3.4.2, an infinite number of combinations of current and flux componentscan be chosen, such that these degrees of freedom can be made use of to implement additionalconstraints in the control.

4.2.9 Measurements

It is important to keep in mind the construction of the machine: a wound stator, wound rotorwith slip rings, and a squirrel-cage interrotor. The use of a squirrel-cage interrotor has as aconsequence that interrotor currents cannot be measured (voltages are known and identicallyequal to zero). To create a representative model of the machine, these physical constraints shouldbe taken into account. Also note that flux is difficult to measure, such that flux measurementsare avoided or replaced by alternatives.

What we can measure on the physical machine, are the following quantities:

• stator voltages and currents

51


• rotor voltages and currents

• interrotor speed [and torque]

• rotor speed [and torque]

The measurement of torques has been put between brackets, as the devices which are usedto measure torque (torque transducers) are expensive and generally have limited bandwidth.Whenever possible, these devices are avoided.

From eq. 4.17, it follows that a number of quantities cannot be measured directly in order tocalculate the torques on rotor and interrotor: both of the current components I2α and I2β, andeach of the flux components are non-measurable quantities. Each of these quantities has tobe derived from measurable quantities, in order to be able to calculate the torque instead ofmeasuring it.

4.2.10 Flux estimation - torque calculation

As indicated, a number of calculations, based on measured quantities, have to be performed inorder to derive the unknown elements in the torque equation. To simplify the calculations, anadditional assumption is made: as the considered time step Ts is small, the speed and positionmeasured at t = tk are used throughout the execution of the control algorithm. This means thatthe speed and position is assumed to remain constant over the next two periods ( [tk, tk+1] and[tk+1, tk+2] )

The complex notation of the interrotor flux can be split into its components, using its definitionfrom eq. 4.6:

Ψ2 = Ψ2α + jΨ2β (4.18)

Ψ2α = L21I1α + L22I2α + L23I3α

Ψ2β = L21I1β + L22I2β + L23I3β

(4.19)

From eq. 4.3, and the definition of the stator flux according to eq. 4.6, one can write thefollowing to expressions:

V1α = L11I1α +dΨ1α

dt(4.20a)

Ψ1α = L11I1α + L12I2α + L13I3α (4.20b)

From eq. 4.20a, one can write an expression for the α-component of the stator flux:

Ψ1α =

∫ t

0(V1α −R1I1α) dt (4.21)

52


This means that the stator flux can be calculated from measurement of the stator voltage andcurrent (which are accessible from outside of the machine). Equations for the rotor flux can bederived analogously.

With this, the unknown interrotor current component I2α can be calculated by combining eqs.4.20b and 4.21:

I2α =Ψ1α − L11I1α − L13I3α

L12(4.22)

The calculation of the β-component of the interrotor current is completely analogous to thecalculation of the α-component, such that it is not repeated here. With the knowledge of theinterrotor currents, the interrotor flux can be calculated according to its definition (eq. 4.6).

Analogously, the interrotor currents can also be calculated from measurements of the rotorvoltage and current, resulting in a similar expression for the interrotor current:

I2α =Ψ3α − L31I1α − L33I3α

L32(4.23)

The interrotor flux components Ψ2α and Ψ2β follow from substitution of the obtained interrotorcurrent components I2α and I2β in eq. 4.19.

4.3 Inverter

To supply voltage to the stator and rotor, two inverters are used. For this purpose, we chosetwo-level voltage source inverters to transform DC voltage to AC voltage. PEC1 and PEC3

from fig. 4.1 are identical.

4.3.1 Topology

The topology of the two 2L-VSIs that are used to supply the machine is shown in the figurebelow.

+-

n

+-

VDC2

VDC2

Sa

Sa

Sb

Sb

Sc

Sc

C a b c

LastaFigure 4.4: Two level voltage source inverter (2L-VSI) topology [25]

53


Each of the inverters has three phase legs, equipped with two switched that determine whetherthe output is connected to the positive DC bus or the negative DC bus. There are thus 23 = 8possible output states for the three phase voltage at the output of the inverter. These eightvoltage vectors are represented in the αβ-reference frame in the figure below (fig. 4.5). Twoof these vectors coincide with the origin. This is the result of connecting each of the phaselegs to the same DC bus (either positive, or negative - hence the two coinciding outputs). Theremaining six vectors are called the active vectors. The selected voltage vector remains the samethroughout the entire sample period of the controller.

V1[100]

V2[110]

V3[010]

V4[011]

V5[001]

V6[101]

V0[000]

V7[111]

β

α

Figure 4.5: Voltage vectors for a three-phase 2L-VSI [25]

4.3.2 Assumptions

A number of assumptions made it possible to model the inverters. The main components of theinverter are its switches, such that the assumptions deal with these:

• immediate switching action

• no conduction losses

• no parasitic capacitance

• no voltage drop over switches or freewheel diodes

The assumption of immediate switching action also implies that complementary switches on onephase leg are switched simultaneously. In reality, some safety margin on the complementaryswitching signals is taken to avoid both switches of one phase leg to be conducting at the sametime. Simultaneous conduction of both switches would imply a short circuit of the DC bus,thus leading to intolerably high shoot-through currents. This safety margin on the timing ofthe complementary switching control signals is indicated with the name dead time. Duringthe dead time, it is assured that the previously conducting switch is extinguished, before thecomplementary switch is fired.

54


4.3.3 Implementation

There are six phase voltages to be applied, such that the output of both inverters can berepresented by a six-bit number. Each bit is assigned to a particular output phase: if the bitvalue is one, the applied voltage is VDC/2; if the bit value is zero, the applied voltage is −VDC/2.

The output voltages of the inverters are phase voltages. As described in sec. 4.2.2, sometransformations have to be performed in order to express the quantities in the common referenceframe decided upon in sec. 4.2.7.

PEC1 from fig. 4.1 is connected directly to the phase windings of the stator. As they arestationary, only the Clarke transform has to be applied on these quantities to express them inthe αβ-reference frame.

PEC3 from fig. 4.1 is connected through sliprings to the rotating phase windings of the rotor. Byapplying the Clarke transform to these quantities, the αβ-components are obtained in a referenceframe rotating at the speed of the rotor. In order to express the quantities in the commonreference frame, a rotation has to be performed over the instantaneous angle −θ = −ω3t. Theminus sign originates from the choice of the reference in fig. 4.3.

4.3.4 Homopolar components

As mentioned before, the homopolar voltage and current components will not be withheld inthe modelling of the machine. Combining fig. 4.4 and fig. 4.6 show the schematic of the inverterconnected to some phase windings. The phase windings of the considered machine are connectedin wye-formation.

O

a

bc!

Figure 4.6: Schematic representation of Y-connected phase windings

For the load (the phase winding), it is not the phase voltage vin that is important, but thevoltage vi0 across the load. The voltage across the load follows from:

vi0 = vin − v0n (4.24)

55


The resulting three-phase output voltage of the inverter may contain a homopolar component.From the schematics, an expression for the voltage v0n can be written:

v0n =van + vbn + vcn

3(4.25)

This is exactly the definition of the homopolar voltage component in the three-phase voltagesystem composed by van, vbn, and vcn. The homopolar voltage component in the three-phasevoltage system only changes the voltage of the mutual connection point 0 with respect to theneutral clamp of the DC bus (v0n), but has no effect on the voltage across each of the phasewindings (vin). As concerns currents: due to the wye-connection of the load, no homopolarcurrents can flow through the phase windings. In the short-circuited squirrel-cage of the inter-rotor, no homopolar current component can flow either. Additionally, homopolar currents andvoltages can never coexist in the machine due to the connection of the phase windings, suchthat no homopolar power is present.

The above reasoning defends the choice of omitting homopolar components in the machinemodel.

4.4 Controller

As can be seen in fig. 4.1 the input of the controller is a set of desired reference values for thecontrolled variables, and the output is a control signal, which is used to control the inverters. Asexplained in chapter 3 the machine model plays a key role in the control algorithm, namely inthe estimation and prediction step. The model is used (in accordance with eq. 4.15) to predictthe state (here the currents) at the next time instance.

To model the prediction step, a six bit counter is used to iterate over all 64 possible switchingstates, the corresponding output state of the machine model is then forwarded to an optimizationblock where the output is evaluated with an optimality criterion. The optimality criterion cantake different forms, as will be discussed in chapter 5.

4.5 Internal combustion engine

As the internal combustion engine only plays a secondary role in the topic of this research, littleattention is paid to its modelling. A very simple, yet acceptable, model of the ICE is used todescribe the primary axis’ (one of two mechanical ports of the EVT) behaviour.

It is assumed that the ICE keeps it rotational speed constant (at the desired reference value)independently of the torque it has to deliver. As the mechanical output power of the ICEis determined by the product of torque and rotational speed (Pm = TΩ), the assumption ofconstant rotational speed implies that the ICE can follow the torque demand at any time. Inother words: the ICE can control its power (at the given speed) to the desired value.

56


The dynamic equation of the rotor:

J3dΩ3

dt= Tem,3 − Tload,3 (4.26)

is thus overridden by imposing a constant speed of the rotor.

Not only is it acceptable that the output power of the ICE can be controlled within a certainrange to answer to the desired torque, also the inertia of the engine can prevent the rotationalspeed to change during the transient of the delivered torque. The operating speed of the ICE ischosen at 1500rpm (= 157rad/s), which is a realistic speed for a combustion engine.

4.6 Electrical energy storage device

The electrical energy storage device is not part of the core topic of this research either, such thatlittle attention will be paid to its modelling as well. The electrical storage device, for examplea battery, is represented by an infinitely strong voltage source. This means that the battery:

• keeps a constant voltage, independently of the load; and

• has no limitation on the deliverable (or receivable) current; hence

• has no limitation on the electrical power flow in both senses; and

• has unlimited storage capacity (can never be overcharged, or run out of power)

In the modelling of the EVT system, the battery is only considered as a voltage source to supplythe voltage inverters. The electrical power flows are not modeled explicitly.

4.7 Load

The load of the EVT is connected to the machine through the secondary axis. For simplicity, theload is modeled by a deterministic load, characterised by a speed dependency. For low vehiclespeeds, it is acceptable to assume a load torque which varies proportionally with speed. Forhigh speeds however, the drag from the air displacement (which is proportional to the velocitysquared) becomes dominant, and the assumption does not hold anymore. However, it is notthe intention of this work to model the load truthfully, such that the simple load model ismaintained.

The dynamic equation of the interrotor is given by:

J2dΩ2

dt= Tem,2 − Tload,2 (4.27)

In which the load torque is defined as follows:

Tload,2 = k · Ω2 (4.28)

57


The proportionality constant k is chosen such that in regime, the electromagnetic torque andthe speed equal their reference values:

k =T refem,2

Ωref2

(4.29)

58

Chapter 5

Control design

5.1 Simulink model

The Simulink model of the EVT was obtained from my counsellor. Field oriented control (FOC),as described in [13], is applied to the machine to determine the reference values of the currents(stator and rotor phase currents) to be realised by the inverters. A predictive current controller(model predictive control, MPC) is used to track the reference current with the inverters (two2L-VSIs). This control method is used as a reference for comparison with the proposed controlmethod. The state-space model (derived from the matrix equations, eq. 4.9, in sec. 4.2) andthe model of the loads have been maintained for simulating the MBPC method, whereas thecontroller has been reprogrammed.

All of the results shown in this chapter are obtained through simulations in Matlab/Simulink.Except when explicitly mentioned otherwise, the initial conditions for the simulations are asfollows:

• the machine is rotating: Ω3 = Ωref3 and Ω2 = Ωref

2

• initially, there is no flux: Ψ1 = 0

The same set of model parameters are maintained for all of the simulations in this chapter:

• DC-bus voltage: VDC = 500V

• speed reference: Ωref2 = 1000rpm and Ωref

3 = 1500rpm

• torque reference values: T ref2 = 20Nm and T ref3 = −30Nm

• stator flux reference: Ψref1 = 0.82Wb

5.1.1 Field oriented control

[13] describes how the equations for FOC are derived from the machine equations. To simplify theequations, a rotating qd-reference frame is chosen, which rotates at the speed of the magneticfield. As a result, an instantaneously synchronous reference frame is obtained, in which all

59

CHAPTER 5. CONTROL DESIGN

currents are DC-currents. With the expressions derived for the torques on interrotor and rotor,a control scheme that controls the torques on both the rotor and the interrotor is proposed.

The obtained torque equations are written as a function of four variables (the q− and d−componentsof the stator and rotor current). Two torque equations should be fulfilled by choosing four vari-ables, which leaves two degrees of freedom to the set of equations. The proposed FOC methodincludes control of the interrotor flux level Ψ2. A number of reasons to defend the choice tocontrol the flux level along with the torque is presented:

• Due to a fundamental law of electromagnetism, flux cannot be changed instantaneously.In order to have fast torque response without transient (waiting for the flux to settle toits new regime value), the flux needs to be held at a certain value.

• The magnitude of the flux determines the degree of saturation. As the model derived insec. 4.2 is based on the assumption of linearity, such that the machine parameters couldbe assumed constant. In order for the model to remain valid, the magnetisation state ofthe machine cannot differ much from the one used to determine the machine parameters.

• Torque is determined by the cross product of flux and current (according to Lorentz’s law)in the considered part of the machine. By increasing the flux level, the same torque valueis achieved with a smaller current. However, increasing the flux also implies increasing theflux-forming component of the current.

• The back-emf of the machine is determined by the product of electrical speed and fluxmagnitude (as can be seen from eq. 4.3). The voltage that can be applied to the terminalsof the machines however, is limited by the DC-bus voltage.

The latter of these arguments leads to implementing a field weakening strategy at higher speedsof the machine, in order to comply with the voltage limitation. As indicated in sec. 2.6, the IM-based EVT has the benefit of field weakening being easily implementable. Some more attentionwill be paid to field weakening in EVTs in sec. 6.2.

Instead of controlling the interrotor flux Ψ2 as in [13], here we will control the stator fluxΨ1. As the flux linkage with each of the parts of the machine only differ due to leakage, themagnetisation state of other parts can estimated by converting the magnitude of the flux in onepart to another with the according conversion factors (winding ratios). The proposed referencevalue for the stator flux level (0.82Wb) is verified to result in linear magnetic material behaviour.This choice for the flux level leads to a maximum magnetic flux density of 1.6T . As discussedbefore (sec. 4.2.4), this value for the flux density leads to linear magnetic behaviour. Moreover,as flux measurements are avoided, the magnitude of the flux has to be estimated. In sec. 4.2.10,a method is presented to calculate flux components from measurements of physically accessiblequantities (i.e. stator and rotor voltage and currents). The stator flux components follow fromeq. 4.21, whereas additional calculations have to be performed in order to obtain the interrotorflux components. The more intermediate steps between the measurement of quantities and theestimated value of the flux, the less accurate the estimation becomes.

Controlling the flux level takes up one of the two remaining degrees of freedom, such that thereis one left. The fourth degree of freedom is exploited to minimize the joule losses. This way acontroller is obtained which selects a set of four current components (I1q, I1d, I3q, and I3d) tobe realised by the inverters that:

• satisfy the demanded torque;

60


• keep the flux at a constant, predetermined level (except in the case of field weakening);and

• reduce the joule losses resulting from the chosen set of currents.

For comparison, the resulting lapse of the torques and stator phase currents is shown here:

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Figure 5.1: Resulting torque lapse from FOC

0 0.05 0.1 0.15 0.2 0.25−10

−5

0

5

10

time [s]

Sta

tor

phas

e cu

rren

ts [A

]

I1a

IIb

I1c

Figure 5.2: Resulting stator phase current lapse from FOC

For the FOC, the CSE values for T2 and T3 amounted to 0.9209 and 0.9067 respectively (totalCSE=1.8276). The joule losses amounted to 75J (calculated according to the method presentedin sec. 5.2.1).

5.2 Cost function analysis

In this section, an insight is provided in the influence of cost function parameters on the qualityof the control. In order to do so, a measure for control quality must be defined. The influence ofthe parameters is observed by varying them in a predetermined range, and studying the resultingcontrol quality.

In sec. 3.4.3, a discussion of different cost functions is presented. As a starter, the proposedMBPC of the torques on the interrotor and rotor (T2 and T3, respectively) is compared withthe FOC proposed in [13]. The latter also uses MPC to determine the gate signals of theinverters in order to track the reference current determined by the FOC. However, in the control

61


method proposed in this text, MBPC is applied directly to the mechanical output variables ofthe machine (i.e. the torques).

The general equation for the cost function (eq. 3.2) is chosen to take the following form:

CostSi = W1 · CostΨ1 (Ψ∗1(tk+1),Ψ1,pi(tk+1), toleranceΨ1)

+W2 ·(T2,pi(tk+1)− T ∗

2 (tk+1)

T ∗2 (tk+1)

)2

+W3 ·(T3,pi(tk+1)− T ∗

3 (tk+1)

T ∗3 (tk+1)

)2

(5.1)

The cost function implemented for the flux is represented graphically in the following figure:

Ψ1,pi(tk+1) – Ψ1*(tk+1)!0!

Cost!∞ ∞

Figure 5.3: Implemented flux cost function

As can be seen from the figure, the cost for the flux deviation is either zero or infinite, such thatthe weight factor W1 has no influence on the result of this cost term. For this reason, the weightfactor W1 will not be witheld in the further discussion.

Due to this particular choice of the cost function, it is made sure that the flux will be controlledclosely around its reference value, such that the control is indeed comparable with FOC (wherethe flux is controlled at a constant value all the time in the absence of field weakening). Bydefault, the weight factors are all set equal to 1: W2 = W3 = 1. The tolerance bands are definedfor relative deviations (relative w.r.t. the reference value), and are set by default at 2.5% (forboth positive and negative deviations). In this way, the total width of the default toleranceband is set to 5% (the same default tolerance band applies to both torque and flux control).

As the initial condition for the flux level is set at zero (sec. 5.1), the presented cost function forthe flux would result in an infinite cost from the start, thus upsetting the control algorithm. Inorder to avoid this, it is made sure that the flux level is built up to a level inside the toleranceband initially, before the control sets in. A similar strategy is applied in [13]. As the torquecontrol is not operational during this start-up phase, the error on the torques is high. To

62


eliminate this biasing of the control quality, the control quality assessment is only started afterthis start-up phase is complete.

5.2.1 Control quality assessment

Many measures exist to assess the adherence of a variable to its desired value: (relative) averageerror, (relative) average absolute error, standard deviation of error/absolute error, etc. . . . Here,the cumulative squared error (CSE) is used as a quality measure:

CSEx =

∫ [x∗(t)− xm(t)

]2dt (5.2)

This measure was chosen for two reasons:

1. the expression for the error is positive semidefinite, such that each error either increasesthe measure or keeps it the same;

2. by squaring the deviation, large errors affect the quality measure more than small ones.

Do note that the proposed quality measure does not take the tolerance band included in thecost function into account, such that a control action that results in a zero cost (thus managesto keep the variable within the tolerance band around its reference value) still causes the CSEto increase when the variable does not remain in its tolerance band.

For the end-user of the machine, only the external quantities are of interest, such that the controlquality assessment only deals with the torques on interrotor and rotor. The interrotor torquedirectly influences the ride smoothness, whereas the rotor torque can transmit vibrations to theICE. The only condition on the flux (which is an internal quantity) that should be respected isthat it should not cause a magnetization state that differs much from the state for which themodel was developed (as discussed in sec. 5.1.1). By selecting an appropriate cost function forthe flux (such as the one shown in fig. 5.3), it is made sure that the resulting flux does not causea magnetisation state different from the one used to develop the model.

Additionally, the quality of the control can be assessed by its ability to follow a step in thereference. This part of the control quality is discussed further, see chapter 6.

As efficiency is also of importance, the joule losses will be observed as well. In sec. 5.4 somemore attention is paid to enhancing the efficiency of the control by reducing the joule losses. Toevaluate the total dissipated energy during the simulation, the joule losses are integrated. Thejoule losses in all three components of the machine are considered:

Pj = R1I21 +R2I

22 +R3I

23 (5.3)

5.2.2 Qualitative influence of parameters

By varying the parameters of the cost function (eq. 5.1), a qualitative insight in the effect ofthe cost function of the control quality is presented. As the control method presented doesnot directly control physical quantities, but rather controls a user-defined cost function that

63


reflects the adherence of physical quantities to their respective reference values (or restrictions),the influence of the cost function parameters can not be explained by means of equations. Aheuristic approach is needed in order to achieve an understanding of the control method and itsindirect effect on controlling the desired quantities.

To this extent, the current section presents an exhaustive set of figures, each reflecting thequalitative influence of the varying parameters on the control quality. As proposed in sec. 5.2.1,the quality of control will be assessed through the CSE values of the torques on interrotor androtor (T2 and T3 respectively). For each set of varying parameters, four quality optimisationswill be performed: twice for each of the torques separately (optimising CSET2 and CSET3) ,and a third time for both of the torques at the same time (optimising CSET2 + CSET3). Theforth optimisation determines the control parameters leading to the minimal joule losses.

As only the cost terms for T2 and T3 result in finite costs, it is sufficient to vary only one ofthe weight factors W2 and W3 to investigate the effect of changing their relative values. Theinterval in which the parameters are varied are a priori determined. The weight factor W2 willbe varied from 10−1 = 0.1 to 101 = 10, while W3 is kept constant at its default value (W3 = 1).With this choice, all cost functions are evaluated where the relative weight CostT2 and CostT3

to the total cost varies from 10 times as weak to 10 times as strong.

The intervals in which the tolerances on the torques are varied, are set from 0% to 21%. Theupper limit for the flux tolerance is chosen based on the achieved performance of the torquecontrol through FOC (see fig. 5.1). Here, the deviation amounts on average to about 20%.Choosing the upper limit for the torque tolerance equal to 21%, means that the performanceobtained with FOC is tolerated by this cost function. To avoid excessive usage of the percentagesymbol, tolerances will further be expressed as absolute values.

In order not to overcharge the chapter with figures, the lapses of machine variables are not shownfor each optimisation case. Only some remarkable lapses will be included to support qualitativeinsight.

Varying only W2

To start off, the influence of varying only W2 will be studied. With this simple parametricvariation, the intention is to develop an understanding in the applied optimisation strategy thatwill be applied.

The figure below illustrates how the different CSE values vary when W2 is varied.

As can be seen from the figure, CSET2 is a monotonously decreasing function of W2, whereasCSET3 is a monotonously increasing unction of W2. Thus, an optimum W2, for which the totalCSE is minimal exists. From this, a first heuristic can be derived: by increasing the relativeweight factor of a certain variable, it will be controlled closer to its reference value.

64


−1 −0.5 0 0.5 10

1

2

3

log(W2) [−]

CS

E [−

]

CSET2

+CSET3

CSET2

CSET3

Figure 5.4: Influence of varying W2 on CSE

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control for W2 = 0.1

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(b) Torque control for optimal W2(= 0.6310)

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(c) Torque control for W2 = 10


As an illustration of these results, the lapses for torque on rotor and interrotor are provided in

65


the figures above. The most extreme cases of relative weight, as well as the optimal solution areshown.

The weight factor W2 resulting in the minimal CSE is 0.6310, which results in CSET2 = 0.2163and CSET3 = 0.1712 (totaling at 0.3875). The control method thus gives better results as totorque control quality than was the case for FOC. However, the losses resulting for these controlparameters amounted to 234J , indicating that the control method is less efficient.

Indeed, the observed stator phase currents are observed to have larger amplitude (25A comparedto around 10A) than for FOC. The high peak currents in the beginning of the simulation arethe result of the control demanding the flux to be built up as quickly as possible. As mentionedbefore, the quality assessment and the joule losses are only calculated once the flux is builtup, such that this transient is not included. Other strategies where the current amplitude ismaintained at a constant level are possible as well.

0 0.05 0.1 0.15 0.2 0.25−200

−100

0

100

200

300

time [s]

Sta

tor

phas

e cu

rren

ts [A

]

I1a

IIb

I1c

Figure 5.6: Resulting stator phase currents for optimal W2

Although the controller does not control current directly, the phase currents automatically adopta sinusoidal waveform. In order to produce torque, a rotating magnetic field should be created,thus implying that the phase currents must form a set of sinusoidal currents.

Below, the resulting flux lapse for this control method is shown: the flux is built up very quickly,and consequently maintained at a constant level. Each of the flux values lies within the toleranceband, as the choice of the cost function does not allow any values outside this interval.

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

Figure 5.7: Resulting flux for optimal W2

66


Varying tolerance on torques T2 and T3

Varying two parameters and keeping the others constant at their default values allows to drawcontour graphs showing the dependency of a variable as function of these two parameters. Thecontour graphs for each of the optimisation parameters are shown in the figures. These figuresallow to develop a understanding in the effect of each of the parameters on the control quality.

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]

CSE T2 minimisation

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) CSE T2

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]

CSE T3 minimisation

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.5

1

1.5

2

2.5

3

3.5

(b) CSE T3

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]

Total CSE minimisation

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(c) CSE = CSE T2 + CSE T3

Figure 5.8: Influence of varying tolerance on T2 and T3 on CSE

Fig. 5.8a shows contours that are mainly vertical, indicating that mainly the x-axis parameter(tolerance on T2) has influence on the resulting control quality CSET2. For small tolerances onT2 however, increasing the tolerance on T3 has a beneficial effect on the control quality on T2.

Fig. 5.8b shows contours with a negative slope ∆y∆x (for large tolerances on T3, indicating that

reducing the tolerance onT2 increases the control quality on T3 (and vice versa). The effect isopposite for small tolerances on T3, where increasing the tolerance of T2 has a beneficial effect.

Also note that the CSE values for T3 are significantly larger than those for T2, indicating thatgenerally, the control on T2 is better than on T3. This is also reflected in fig. 5.8c, in whichthe contours correspond largely with those in fig. 5.8b. This figure shows that increasing bothtolerances at a time, results in overall lower quality of the torque control. This was expectedfrom the definition of the cost functions and the quality measure (which does not take tolerancesinto account, as discussed in sec. 5.2.1).

The table below gives the numerical results of the optimisation. For each optimisation criterion,the optimal parameter values are given, as well as the resulting CSE values. This way ofpresenting the optimisation results will be maintained throughout the discussion of results.

Table 5.1: Optimisation results for varying tolerance on T2 and T3

Opt. criterion tol T2 tol T3 CSET2 CSET3

CSE 0.03 0.045 0.1487 0.2366

CSET2 0 0.21 0.05444 1.5609

CSET3 0.21 0.015 0.6844 0.06205

loss 0 0.12 0.09372 0.5757

67


The following figure shows the resulting losses for each combination of parameters. Althoughthe contours take remarkable shapes, a qualitative observation can be made. Choosing eitherone of the tolerances small results in lower losses. This indicates that a tighter control on thetorques results in lower joule losses. The optimal (w.r.t. joule losses) pair of parameters, resultin losses of 154J, which is lower than the previously found optimum (by only varying weightfactor W2).

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]

Total loss minimisation

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

200

250

300

350

400

450

500

550

600

Figure 5.9: Influence of varying tolerance on T2 and T3 on joule losses

Varying W2 and tolerance on torque T3

The previous section showed the beneficial effect of reducing tolerance on the torques in orderto result in better torque control. In this section and the next, one of the torque tolerances willbe kept constant, while the other, and the relative weight of both torque costs, is varied.

log(W2) [−]

Tol

eran

ceT

3 [−]

CSE T2 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

(a) CSE T2

log(W2) [−]

Tol

eran

ceT

3 [−]

CSE T3 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(b) CSE T3

log(W2) [−]

Tol

eran

ceT

3 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Figure 5.10: Influence of varying tolerance on T3, and W2 on CSE

Fig. 5.10a shows that either increasing the weight factor W2 or the tolerance on T3 results inbetter control of T2. This can be understood intuitively by consideration of the cost function: alarge weight factor W2 will penalise errors on T2 more, such that they will be corrected quicker.By increasing the tolerance band on T3, a larger range of control possibilities is allowed. Ifa control variable lies inside its tolerance band (e.g. exactly equal to the reference value) theprediction values of this control variable will be more likely to fall within the tolerance band

68


as well, such that a large number of control actions will lead to a cost equal to zero for thiscontrol variable. This allows for the optimisation step to choose from a large number of controlactions in order to minimise the cost of the other control variable (in this case: torque T2). Thecontroller will only have to step in to correct T3 when its value is near (or outside) its tolerancelimits.

However, the parameter set optimising the control on T2 indicates that a small tolerance onT3 should be chosen (see table 5.2). Due to the weight factor W2 being equal to 10, the costresulting for T3 laying outside the tolerance interval is relatively very small compared to thecost resulting for T2, such that the above reasoning holds. This can be seen from the fact thatalthough the tolerance on T3 is quite small, its corresponding CSE value is high, indicating thatthe values for T3 lay outside the tolerance band frequently.

This reasoning also explains the obtained contour graph for CSET3 shown in fig. 5.10b. Notethat again, the values for CSET3 are significantly higher compared to CSET2, such that theoptimisation of the control on both torques (fig. 5.10c) corresponds largely with optimisingcontrol on torque T3.

Table 5.2: Optimisation results for varying tolerance on T3, and W2

Opt. criterion W2 tol T3 CSET2 CSET3

CSE 0.7943 0.03 0.1702 0.2021

CSET2 10 0.03 0.05220 1.337

CSET3 0.10 0.015 0.4862 0.06155

loss 0.1 0.015 0.2867 0.1655

The contour graph for the losses again shows remarkable contours. Nevertheless, a trend canbe seen for increasing losses with decreasing parameters (W2 and tolerance on T3). This is thesame qualitative influence of the parameters on the control of T2, such that these two measuresare observed to be related. The parameter set optimising the losses results in 185J of losses.

log(W2) [−]

Tol

eran

ceT

3 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

200

250

300

350

400

Figure 5.11: Influence of varying tolerance on T3, and W2 on joule losses

69


Varying W2 and tolerance on torque T2

Following the discussion in the previous section, now the tolerance on torque T2 is varied. Thediscussion is quite similar to the one in the previous section, and so are the obtained results.

log(W2) [−]

Tol

eran

ceT

2 [−]

CSE T2 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.1

0.2

0.3

0.4

0.5

0.6

(a) CSE T2

log(W2) [−]

Tol

eran

ceT

2 [−]

CSE T3 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(b) CSE T3

log(W2) [−]

Tol

eran

ceT

2 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6



Fig. 5.12a is the dual version of fig. 5.10b, although the x-axis has been mirrored. Indeed:increasing W2 decreases the relative weight of the cost resulting from torque T3. On the y-axis,the duality is obvious. Although these figures show the same contour pattern, the obtained CSEvalues differ significantly. The CSET2 values in fig. 5.12a are approximately a factor 3 lowerthat the CSET3 values in fig. 5.10b. This indicates that for a dual choice of parameters, torqueT3 is less well controlled than T2. It can thus be concluded that control parameters W2 andtolerance on T3 should be chosen more in favour of the control on T3 (that is: lower W2 andsmaller tolerance on T3) to achieve a CSET3 level compared to selecting W2 and tolerance onT2 to achieve the same CSE level (now CSET2).



CSE 0.6310 0 0.2094 0.1629

CSET2 10 0.015 0.05239 1.363

CSET3 0.1 0.045 0.4850 0.06038

loss 3.981 0.015 0.06473 0.6196

The contour graph for losses now shows even more capricious contours. However, the dualrelation w.r.t. the previous section can be observed: losses decreasing for decreasing toleranceon T2 and increasing weight factor W2. As in the previous section, the losses and torque qualityshare the same qualitative influence parameters. The set of parameters minimising losses resultsin joule losses of 193J.

70


log(W2) [−]

Tol

eran

ceT

2 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

200

250

300

350

400

450


Varying W2 and tolerance on flux Ψ1

To complete the overview of varying parameters, this section evaluates the effect of changingthe tolerance on the flux control, and the weight factor W2 on the control. As the flux leveldetermines the saturation state of the machine, and the machine model was developed with theassumption of linear magnetic behaviour, the flux level may not exceed the reference value forthe flux too much. To this extent, the interval in which the tolerance is varied is chosen smallercompared to the intervals in which the tolerance on torques have been varied. The choice ofthe cost function for flux (see fig. 5.3) does not allow choosing the tolerance equal to zero, asthe control method would then fail. For these reasons, the flux tolerance will be varied in theinterval of 1 to 5%.

Fig. 5.4 show the variation of the different CSE values along the line toleranceΨ1 = 0.025. Ascould be seen from that figure, the variation of CSET3 is stronger than the one for CSET2,which is also reflected in the figures below.

log(W2) [−]

Tol

eran

ceΨ

1 [−]

CSE T2 minimisation

−1 −0.5 0 0.5 10.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) CSE T2

log(W2) [−]

Tol

eran

ceΨ

1 [−]

CSE T3 minimisation

−1 −0.5 0 0.5 10.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.5

1

1.5

2

2.5

3

3.5

(b) CSE T3

log(W2) [−]

Tol

eran

ceΨ

1 [−]


−1 −0.5 0 0.5 10.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.5

1

1.5

2

2.5

3

3.5

4


Figure 5.14: Influence of varying tolerance on Ψ1, and W2 on CSE

Reducing the tolerance on the flux (compared to its default value of 0.025) decreases the controlquality on T2. Moreover, the effect is more significant for low values of W2. Compared to thedefault value, increasing the flux tolerance does not alter the profile of CSET2 to W2 significantly.

71


The effect on the control quality of T3 is similar, with the difference that now the effect is moresignificant for high values of W2. From the optimisation results in the table below, it followsthat increasing the tolerance on flux, increases the control quality on the torques. As discussedabove, increasing the tolerance band on a variable results in more control actions leading to azero cost. This effect is even stronger for the flux control, as any control action leading to aflux level exceeding the tolerance limits is a priori excluded. Increasing the tolerance band thusgives the optimization step a wider range of control actions to choose from in order to optimisecontrol on the torque.

Table 5.4: Optimisation results for varying tolerance on Ψ1, and W2

Opt. criterion W2 tol Ψ1 CSET2 CSET3

CSE 0.6310 0.05 0.07884 0.1534

CSET2 10 0.05 0.06582 1.783

CSET3 0.1 0.05 0.4417 0.06250

loss 3.162 0.0125 0.1855 0.8368

The tolerance on the flux control appears to have little influence on the joule losses. The figuresabove show the effect of the weight factor on the torque control. Comparison of these withthe figure below shows that it is the quality of the control on T2 which affects the joule losses(the worse CSET2/the lower W2, the higher the joule losses). The set of parameters leading tominimal losses results in joule losses of 122J, which is significantly better than the efficienciesobtained above.

log(W2) [−]

Tol

eran

ceΨ

1 [−]


−1 −0.5 0 0.5 10.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

150

200

250

300

350

400

Figure 5.15: Influence of varying tolerance on Ψ1, and W2 on joule losses

5.3 Omitting flux control

In FOC, the reference value for the flux is fixed, unless it must be changed in order to fulfill fieldweakening at high speeds. As magnetic flux is an internal quantity, its level is not of (direct)importance for the end-user of the machine. Torque on the other hand, is an external quantityof the machine, and is of direct importance for the end-user. In order to increase the quality ofthe torque control, another control strategy is proposed which does not control the flux level ata predetermined flux level. Nevertheless, the magnetisation state of the machine must still bemaintained linear in order for the model to maintain its validity (see sec. 4.2). In order to do

72


so, the implemented cost function maintains the form of eq. 5.1, but the cost function for theflux is adapted according to fig. 5.16.

The effect of the proposed control method on the torque control quality, and the flux is discussedbelow. In the next chapter (chapter 6), the response analysis of the system using this controlstrategy (extended with a functionality described in the next section) is studied. The red linein the figures of flux lapses now doesn’t represent a reference value anymore, it represents thelevel to which the flux is limited by the controller.

Ψ1,pi(tk+1) – Ψ1*(tk+1)!0!

Cost!∞

Figure 5.16: Implemented flux limiting cost function

No flux control - Varying only W2

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

log(W2) [−]

CS

E [−

]

CSET2

+CSET3

CSET2

CSET3


The figure above shows that due to the absence of explicit flux control, the control quality onthe torques remains approximately the same, independent of the relative weight of the cost ofT2 and T3. To demonstrate this, the resulting lapses of the torques are shown for both extremaof the spectrum of W2. The respective lapses of the flux are shown as well. It is clear thatthe weight factor W2 has little effect on the control. This is expected to be reflected in thecontour graphs by horizontal contours (parallel to the W2-axis). The weight factor W2 doesseem to have a positive effect on the lapse of the flux: despite the absence of flux control, the

73


flux automatically reaches a regime value. For a large weight factor, the flux remains closer toits regime value.

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control for W2 = 0.1

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(b) Torque control for W2 = 10


74


0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(a) Flux lapse for W2 = 0.1

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux lapse for W2 = 10

Figure 5.19: Influence of varying W2 on flux

This can be understood intuitively from the fact that now only two variables are controlled.When one variable is controlled to within its tolerance band, a large number of control actionswill lead to predictions of that variable that will also be within the tolerance band, thus resultingin a cost zero (which is not affected by the weight factor). This shows that it is only the costassociated with the second control variable which matters for the optimisation step. The relativeweight factor for the costs associated with the different control variables only has a meaningwhen both costs differ from zero. Whenever one of the costs is zero (at least for a large portionof the control actions) the cost associated with the other variable is only compared with costassociated with the same control variable, such that this is not affected by its weight factor.The roles of the variables might be switched in the next execution of the control algorithm, suchthat the control alternates between controlling one variable and the other.

The weight factor W2 resulting in the minimal CSE is 4.467, which results in CSET2 = 0.01495and CSET3 = 0.04251 (totaling at 0.05746). This is of little significance here of course, as thisis a very flat minimum. However, this control strategy performs very badly with respect toefficiency. For the optimal value of W2, the joule losses amount to 3154J. This is reflected in thelapse for the stator phase currents, which now have an amplitude of about 50A. The disturbancesin the currents are caused by flux spikes as can be seen in the figures above showing the lapseof flux. With flux control, the set of currents resulting in the desired torque, must also resultin the desired flux level (a boundary condition). When no explicit flux control is applied, thisconstraint is not present anymore, and any set of currents resulting in the desired torque valueare allowed (given that the satisfy the limitation on the flux level).

75


0 0.05 0.1 0.15 0.2 0.25−200

−100

0

100

200

time [s]

Sta

tor

phas

e cu

rren

ts [A

]

I1a

IIb

I1c

Figure 5.20: Resulting stator phase currents for optimal W2

No flux control - Varying tolerance on torques T2 and T3

Comparing the contour graphs below with the corresponding ones including flux control showsthat the scales of CSE values are now significantly larger. Although the minimisation showsthat the control quality can be higher when tolerances are chosen smaller, the control qualitydrops more when tolerances are chosen larger.

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]

CSE T2 minimisation

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(a) CSE T2

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]

CSE T3 minimisation

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1

2

3

4

5

6

(b) CSE T3

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]


0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1

2

3

4

5

6

7

8


Figure 5.21: Influence of varying tolerance on T2 and T3 on CSE

Again, the CSE values for T3 are significantly higher than those for T2, such that generaloptimisation of the torque control corresponds largely with optimising the control on T3.

Table 5.5: Optimisation results for varying tolerance on T2 and T3

Opt. criterion tol T2 tol T3 CSET2 CSET3

CSE 0 0 0.003619 0.005453

CSET2 0 0.1950 0.0007394 4.096

CSET3 0.1950 0 0.8396 0.0004673

loss 0.12 0 0.2215 0.005392

The table above shows a remarkable result for the CSET3 value resulting from the set of param-eters to optimise CSET2. When a large tolerance is chosen on a control variable, the behaviour

76


of the control changes w.r.t. when flux is controlled. The figure below shows the lapse of thetorques for this particular set of control parameters.

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Figure 5.22: Resulting torque lapse for CSET2 optimisation

A large tolerance band is assigned to T3, such that relatively large deviations are allowed. Whena predicted value exceeds one of the tolerance limits, the relative deviation is large, such that thecorresponding cost is high as well. The results in the controller choosing a control action leadingto a value of the variable within the tolerance band (with zero cost). When there is a largenumber of possible control actions leading to a zero cost for one variable, the cost of the othervariable will dominate. When a control variable is near one of its tolerance limits, it will onlybe controlled not to exceed the limit, such that the possibility exists that the variable remainsclose to the limit and is each time controlled to just within the tolerance band. This is the casein the control of T3 in the figure below. Whenever inside the tolerance band, T3 simply followsfrom the control action that is selected to control T2. As a result, it might happen that valuesfor T3 reach and stick with the lower tolerance limit. The flux control that was applied in theprevious section thus has a positive influence on the average deviation of the control variables.

ToleranceT

2

[−]

Tol

eran

ceT

3 [−]


0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1000

1500

2000

2500

3000

3500

4000

Figure 5.23: Influence of varying tolerance on T2 and T3 on joule losses

The figure above shows the influence of changing the control parameters on the joule losses.Due to the additional degree of freedom given to the choice of current components, the currentcomponents are chosen only to fit the demanded torque control as well as possible. As it seems,this results in selecting higher values for the currents, which thus results in increased joule losses.The losses are observed to decrease for increasing tolerance on T2 and T3. However, a regionwith much lower losses can be found for very low values of tolerance on T3. Optimising for

77


efficiency leads to a set of parameters resulting in 593J joule losses. Although still very high,the losses are strongly reduced compared to the previous case. The resulting CSE values forT2 and T3 are respectively equal to, and significantly smaller than their values in the previouscase, showing that both efficiency and control quality can be improved by choosing the rightparameters.

No flux control - Varying W2 and tolerance on torque T3

As expected, the contour graphs below show that the weight factor W2 has little effect on thecontrol (this can be seen from the horizontal contours). Remarkable here are the obtainedCSET2 values: they are significantly lower than in the above discussed cases, and moreover,much smaller than CSET3 values.

log(W2) [−]

Tol

eran

ceT

3 [−]

CSE T2 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(a) CSE T2

log(W2) [−]

Tol

eran

ceT

3 [−]

CSE T3 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

(b) CSE T3

log(W2) [−]

Tol

eran

ceT

3 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.2

0.4

0.6

0.8

1

1.2

1.4



The results in the table below show that the control quality on the torques is much higher thanin the previous case.



CSE 0.2512 0 0.01492 0.001293

CSET2 10 0.06 0.004713 0.4631

CSET3 0.1259 0 0.01577 0.001051

loss 0.1 0 0.04132 0.004318

When optimising the parameter set for losses, the minimal joule losses amount to 2106J. Thefigure below show how the joule losses vary upon changing the cost function parameters: a smallregion with joule losses significantly smaller than in the rest of the graph can be identified. Com-parison with fig. 5.24a show that the region corresponding with lowest joule losses correspondswith the region of worst control quality on T2.

78


log(W2) [−]

Tol

eran

ceT

3 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

2200

2400

2600

2800

3000

3200

3400

3600

3800


No flux control - Varying W2 and tolerance on torque T2

Remarkable in the following set of figures is that, in contrast with all of the previous consid-erations - now the values for CSET2 are significantly higher than those for CSET3. This is aconsequence of keeping the tolerance on torque T3 at its default value, whereas the tolerance ontorque T2 is increased. As previously seen, the result of setting larger tolerances on a quantitymay lead to it being controlled closely to either one of the limits. As a result, its value deviatesall (or most of) the time from its reference value, causing the control quality to be low. In theprevious section, where the flux was controlled, this effect was less prominent.

As explained in the beginning of this section, the control method applied to controlling twovariables alternates control actions to correct either one of these two variables. The introductionof a third control variable (and even more explicit for a tightly controlled variable such as the fluxlevel) leads to the other variables to be controlled more evenly distributed within the toleranceband, resulting in a better average value of the variable.

log(W2) [−]

Tol

eran

ceT

2 [−]

CSE T2 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(a) CSE T2

log(W2) [−]

Tol

eran

ceT

2 [−]

CSE T3 minimisation

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.035

0.04

0.045

0.05

0.055

0.06

0.065

(b) CSE T3

log(W2) [−]

Tol

eran

ceT

2 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Figure 5.26: Influence of varying tolerance on T2 and W2 on CSE

79




CSE 0.2512 0.015 0.003842 0.03059

CSET2 10 0 0.0009872 0.05399

CSET3 0.2515 0.015 0.003842 0.03059

loss 0.3981 0.03 0.02204 0.04137

As in the previous case, the joule losses are observed to be inversely proportional to the controlquality on T2. Nevertheless, some regions with relatively low joule losses are seen for relativelylow tolerances on T2. The set of parameters, optimised for losses, result in joule losses of 3241J.

log(W2) [−]

Tol

eran

ceT

2 [−]


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

3300

3400

3500

3600

3700

3800

3900

4000

4100

4200

Figure 5.27: Influence of varying tolerance on T2 and W2 on joule losses

5.4 Efficiency optimisation

5.4.1 Control extensions

The joule losses resulting from the proposed control method far exceed the losses resulting fromFOC, even when the cost function parameters were optimised to result in an as low as possiblejoule loss. Moreover, when flux control is omitted, the qualitative influence of the cost functionparameters on joule losses become less evident and coherent. As the flexibility of the controlalgorithm allows for expanding the optimization procedure, a new strategy is presented whichimproves the efficiency of the control method.

More insight in this additional optimisation can be gained by considering a single rotor inductionmachine. When FOC (as presented in [29]) is applied to an induction machine, the statorcurrent consists of two components (in quadrature): a flux forming component Isφ, and a torqueforming component Isτ . The magnitude of the torque is determined by the product of flux andtorque, such that it is proportional to the product of both current components. The proposedminimisation of the joule losses thus corresponds to a bound extremum problem:

minimise

I2sτ + I2

sφ (5.4)

80


with the boundary conditionIsτIsφ = cst. = f(T ref ) (5.5)

The problem can be represented graphically as in the figure below: minimising joule lossescorresponds with controlling both current components such that neither one is (much) largerthan the other.

Isφ

Isτ

Figure 5.28: Bound extremum problem for optimising efficiency [25]

For the EVT however, two torque equations must be fulfilled, and four current componentsmust be chosen. A four dimensional surface is determined by the different possibilities for thecurrent components, such that it is impossible to represent this graphically. The additional twodimensions are the flux and torque forming components of the rotor current. Nevertheless, thebasic idea behind the optimization remains the same. However, in the proposed control method,not the current components, but instead the torque is directly controlled. The optimisation ofjoule losses is thus carried out indirectly by selecting the control action leading to minimal joulelosses out of all the control actions that lead to satisfactory torque control. This method ensuresgood quality of the torque control before it optimises losses. Moreover, the control method onlylooks two steps ahead, such that the optimisation is not performed on the long term. These tworeasons show that the proposed method will not reach the most efficient solution on the longterm, but will still reduce the losses compared to the previous section where no flux control isapplied.

This method is implemented by extending the optimization step such that all low costs areforwarded to a second optimisation step. The threshold for deciding whether or not a cost islow must be defined a priori. The proposed threshold is:

Costthres = 0.052 · W2 +W3

2(5.6)

which can be interpreted as tolerating a relative torque deviation of 0.05 (the square originatesfrom the use of a squared cost function, eq. 5.1). The threshold is scaled with the average valueof weight factors, such that deviation of neither of the torques is favoured.

81


If no control actions are found which fulfill the criterion of low cost, the original optimisationis maintained. In that case, the optimisation simply chooses the control action resulting inminimal cost. This feature was included such that the system would give priority to followingthe (changing) torque reference values, rather than minimising losses.

The optimisation of the cost function parameters will still be performed based on the qualityof the torque control. The only difference with respect to the previous cases is that the costfunction parameters are optimised to result in the best possible CSE, with minimising joulelosses as a boundary condition. As before, also the cost function parameters leading to theminimal loss will be determined.

5.4.2 Qualitative influence of parameters

Efficiency optimisation - Varying only W2

The influence of this added functionality is first evaluated on a simple case, where only theweight factor W2 is changed. The applied control method includes control of the flux level.This case is considered in order to demonstrate that the effect of the efficiency optimisationonly affects the resulting joule losses without significantly deteriorating the control quality onthe torques. The results should be compared to the case Varying only W2 in sec. 5.2.2. Asa reminder: the joule losses corresponding with the value for W2 resulting in the best controlquality on the torque amounted to 234J .

−1 −0.5 0 0.5 10

1

2

3

log(W2) [−]

CS

E [−

]

CSET2

+CSET3

CSET2

CSET3


Comparing the figure above to the corresponding figure in sec. 5.2.2 shows that the controlquality is not significantly affected by the proposed extension of the control algorithm. Forcompleteness, the lapses of torques and flux are shown in the figures below. Visual inspectionof these figures and the corresponding ones from the section mentioned before, confirms thatindeed the control is not affected by the additional step in the control method.

82


0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Figure 5.30: Torque lapse for optimal W2

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

Figure 5.31: Flux lapse for optimal W2

The weight factor W2 resulting in the minimal CSE is 0.7079, which results in CSET2 = 0.2018and CSET3 = 0.1836 (totaling at 0.2054). For these control parameters, the joule losses amountto 225J, which is indeed less than the original value (234J). By maintaining the quality of thecontrol, the addition of the optimisation of joule losses, only has limited effect.

Efficiency optimisation - No flux control - Varying only W2

In this section, the flux will no longer be controlled, such that the results obtained here shouldbe compared with the case No flux control - Varying only W2 in sec. 5.3. Comparison of thefigure below with the corresponding figure in sec. 5.3 shows that here as well, the additionaloptimisation step does not affect the control quality significantly. To add to the comparison,both the resulting torque and flux lapse are shown for the optimal value of W2 (optimisation forcontrol quality).

83


−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

log(W2) [−]

CS

E [−

]

CSET2

+CSET3

CSET2

CSET3


0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Figure 5.33: Torque control for optimal W2

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

Figure 5.34: Flux lapse for optimal W2

Originally, the joule losses resulting from selecting the optimal value of W2 amounted to 3154J.Here, the weight factor W2 resulting in the minimal CSE is 1, which results in CSET2 = 0.01778and CSET3 = 0.04471 (totaling at 0.06249). The joule losses corresponding with this value ofW2 amount to 1700J, which corresponds to a reduction of 46% compared to when the losses arenot optimised. The effect of the additional optimisation step is observed to have greater effectwhen the flux is not controlled. Nevertheless, the resulting joule losses still far exceed the lossesresulting when flux is controlled (1700J compared to 234J).

The reason for this is that by controlling the flux level, the current components must satisfy anadditional equation. Consequently, fewer control actions lead to low (or zero) cost levels. As a

84


result, the optimisation for the losses can select an optimal control action from a smaller subsetof switching states compared to when flux is not controlled.

Efficiency optimisation - No flux control - Varying cost function parameters

As demonstrated in the previous two cases, the addition of the optimisation step for the joulelosses does not (significantly) affect the control quality. For this reason, only the effect on thejoule losses is studied in this section, without repeating the effect of the parameters on thecontrol quality.

The following tables provides an overview of the effect of the added optimisation step on theresulting joule losses. For each case of varying parameters, the parameter set optimising lossesis considered.

Table 5.8: Reduction of joule losses by added optimisation step

Joule losses [J] with flux control without flux control no flux control, with loss opt.

tol. T2 and T3 154 593 440

W2/ tol. T3 185 2106 1073

W2/ tol. T2 193 3241 732

A remarkable effect of the optimisation can be observed when tolerances are chosen large.Torques are proportional to the current components, such that a large absolute value of torquerequires larger values of the current. As a result, the losses corresponding with higher torquesare higher. When a tolerance band is added in the cost function, the controller is given somefreedom to select the resulting value of the variable. The first step of the optimisation (min-imising the cost function) forwards a number of control actions leading to values of the controlvariables within their respective tolerance bands. The second step in the optimisation selectsthe control action leading to minimisation of the joule losses (hence minimisation of the currentcomponents). As a result, the second optimisation step will select those control actions whichlead to the lowest (absolute) values of the torques. The smallest acceptable torque correspondswith the tolerance limit closest to zero.

This is shown in the following figures, in which the resulting lapses for torque and flux are shownfor choosing tolerances on T2 and T3 both equal to 21%.

0 0.05 0.1 0.15 0.2 0.25−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Figure 5.35: Loss minimisation with tolerance bands - torque lapse

85


0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

Figure 5.36: Loss minimization with tolerance bands - flux lapse

Efficiency optimisation - No flux control - Overall parameter optimisation

To conclude this chapter, a final optimisation procedure is performed in which all three param-eters (W2 and tolerances on both torques T2 and T3) are varied. As the number of possiblecombinations increases fast when evaluating three parameters, the intervals in which the param-eters will be varied are chosen smaller: W2 will be varied between 1 and 10, the tolerances onthe torques will be varied from 0 to 5%. The results of this optimisation cannot be representedgraphically in contour graphs as before, but its sections for choosing either one of the parametersconstant (at its default value) can be found in the contour graphs above.

Optimising the set of parameters for control quality gives: W2 = 1.2589, tolerance on T2 = 1%,tolerance on T3 = 0.5%. The obtained CSE values are: CSET2 = 0.0021 and CSET3 = 0.0056.The joule losses resulting from this set of parameters amount to 3512J.

Alternatively, optimising for losses gives: W2 = 1.9953, tolerance on T2 = 5%, tolerance onT3 = 3%. The obtained CSE values are: CSET2 = 0.0618 and CSET3 = 0.0696. The joulelosses resulting from this set of parameters amount to 732J.

5.5 Conclusions

The control algorithm (MBPC) does not directly control the control variables, but controls thecost resulting from the values of these control variables. This disconnection between the controlmethod and the control variables has the disadvantage that the control behaviour cannot beexplained by equations. However, the cost function used to reflect the correspondence betweencontrol variables and their reference values is user defined, such that priorities in the controlcan be set. Observations of the control for changing parameters can be observed and lead to aqualitative understanding of their influence on the control.

The above considerations show that the proposed control method is effective: the torques onboth rotors are successfully controlled. Compared to FOC, better control quality is obtained.However, the joule losses resulting from the control method are observed to be higher than withFOC.

Introducing tolerance bands causes the control function to correct the value to within the tol-erance interval, without taking into account the average value. As a result, the risk exists that

86


a variable is controlled close to one of the limits of its tolerance band rather than around itsreference value. This causes the average value of the variable to deviate from the reference value.This is observed particularly for T3 when the flux level is controlled, but also for T2 when theflux level is not controlled. This effect is even increased when besides the cost, also the joulelosses are optimised.

Omitting the flux control results in better control quality of the torques. However, the cor-responding joule losses are significantly increased. The introduction of an additional step foroptimising the losses effectively reduces the joule losses. However, this does not happen to theextent that the resulting joule losses reach the same (or better) values compared to when fluxis controlled (a fortiori not lower than the joule losses obtained with FOC).

The correlation between the cost function parameters of the cost corresponding with T2 (W2,tolerance on torque T2) and the effect on the control quality of T3 is observed not to be sym-metrical with the correlation between the cost function parameters of the cost correspondingwith T3 (W3, tolerance on torque T2) and the effect on the control quality of T2. Increasing theweight of, and tightening the tolerance on the control of T2 deteriorates the control quality onT3 more than the other way around. This shows that, in order to control T3 to the same controlquality level as T2, relatively more weight and tighter tolerances must be assigned to the costcorresponding with the control of T3.

87


88

Chapter 6

Response analysis

In this chapter, the behaviour of the control upon reference step changes will be observed. Inthe first section, the influence of torque steps (both in the interrotor and rotor torque) will beconsidered, whereas the second section shows the influence of interrotor and rotor speed changes.All of the results intros chapter have been obtained through simulations in Matlab/Simulink.

6.1 Torque step

The overall optimisation of control parameters in the previous chapter leads to the followingcontrol parameter set:

• W2 = 1.2589

• tolerance on T2 = 1%

• tolerance on T3 = 0.5%

The flux is not explicitly controlled (only limited as explained in sec. 5.3) and the optimisationof joule losses is included in the optimisation step of the algorithm (as explained in sec. 5.4).

0 0.2 0.4 0.6 0.8 1−40

−20

0

20

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

Figure 6.1: Response of torque control to dynamic reference

The figure above shows the response of the control upon a dynamic torque reference. It isobserved that the control variables follow the reference values very well. Each step in the

89

CHAPTER 6. RESPONSE ANALYSIS

reference value is accurately tracked by the variable without any transient behaviour. Evenwhen the steps in both torque references occur simultaneously, the control manages to track thechanges perfectly.

As listed in sec. 5.1.1, one of the reasons to control the flux at its maximal value is to ensure thedynamic response properties of the control. However, the figure above demonstrates that evenwithout explicitly controlling the flux level, the torque control accurately follows the dynamicreference values.This shows that for MBPC, it is not necessary to control the flux level in orderto obtain good dynamic behaviour.

6.2 Speed step

In FOC, field weakening is applied such that the back-emf does not exceed the maximal voltage(VDC/2) that can be applied to the terminals. Back-emf is determined by the product of electricalpulsation and flux level, such that upon increasing the speed, the flux level will have to be reduced(weakened) in order to limit its value. As the EVT has two electrical ports to which the voltageis supplied, two such limitations exist. The expressions for the back-emf in the stator and rotorcan be rewritten to provide insight in the meaning of these limits. From eq. 4.3, combined withthe definition of slip:

E1 = ωΨ1 = (s2ω + ω2)Ψ1 (6.1)

E3 = s3ωΨ3 = (s2ω + ω2 − ω3)Ψ3 (6.2)

The back-emf of the stator is proportional to the interrupter speed, whereas the back-emf ofthe interrotor is proportional to the relative speed between the interrotor and the rotor. At alltimes, the most constraining limit must be taken into account. In FOC, the reference value forthe flux is chosen fixed and is only varied when one of the above limits is reached and exceeded.It is the control of the flux level which could require the controller to generate an impossiblecontrol action (requiring a voltage exceeding the maximum achievable value). In the proposedcontrol method however, flux is not explicitly controlled. When the flux level is not controlled,its value just follows from the choice of control actions that have been selected to control thetorques. The controller thus only considers the possible control actions, all leading to feasiblecontrol outputs, and thus automatically fulfilling the constraints imposed by equations 6.1 and6.2.

For the sake of obtaining clear figures (especially of the flux lapse) the parameters of the costfunction have been chosen as follows: weight factors W2 = W3 = 1, tolerance on torque T2 =10%, tolerance on torque T3 = 2.5%. In what follows, different steps in the speeds have beenapplied to study the influence of the working point on the regime level for the flux.

90


0 0.2 0.4 0.6 0.8 1−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux lapse

0 0.2 0.4 0.6 0.8 150

100

150

200

250

time [s]

Mec

hani

cal s

peed

s [r

ad/s

]

Ω2

Ω3

(c) Speed lapse

Figure 6.2: Effect of step increase in Ω3

By applying a step increase in the rotor speed, and maintaining the interrotor speed constant,the relative speed between interrotor and rotor is increased. With flux control (at the defaultreference value), it is verified to be possible to maintain the flux level at this reference valuefor the operating speeds encountered in this lapse. This shows that neither of the limits eq.6.1 and 6.2 are reached. However, not controlling the flux explicitly shows the flux settling ata lower regime level. Upon increasing the relative speed between the two rotors, the flux isautomatically weakened. It appears that - although the limits set by eq. 6.2 are not reached -the flux is automatically decreased in order to remain far enough from the limit.

In the following, a step is applied in the reference speed for the interrotor. Due to the inertia ofthe load attached to the interrotor, the variation of the speed would occur slowly. In order tosee the effect on a smaller time scale, the inertia of the load has been reduced for the followingset of figures (fig. 6.3 and 6.4). Its value is now set to J2 = 0.005kgm2.

91


Fig. 6.3 shows the effect of reducing the relative speed between both rotors. Again, it is verifiedthat it is possible to control the flux to its default reference value for the occurring speeds, suchthat neither of the limits are reached. Its behaviour is exactly as in fig. 6.2: reducing the relativespeed between both rotors causes an increase in the settling value for the flux level.

0 0.2 0.4 0.6 0.8 1−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux lapse

0 0.2 0.4 0.6 0.8 10

50

100

150

200

time [s]

Mec

hani

cal s

peed

s [r

ad/s

]

Ω2

Ω3

(c) Speed lapse

Figure 6.3: Effect of step increase in Ω2 - reducing relative speed

Fig. 6.4 shows the combined effect of increasing the absolute value of the interrotor speed, andthe relative speed between both rotors. With flux control (at the default reference value), it isverified to be possible to maintain the flux level at this reference value for the operating speedsencountered in this lapse. This shows that neither of the limits eq. 6.1 and 6.2 are reached, suchthat strictly speaking, field weakening is not required. Nevertheless, a behaviour consistent withthe theory of field weakening is observed: the increase in the absolute and relative speed causesthe flux to settle at a lower level.

92


0 0.2 0.4 0.6 0.8 1−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux lapse

0 0.2 0.4 0.6 0.8 160

80

100

120

140

160

time [s]

Mec

hani

cal s

peed

s [r

ad/s

]

Ω2

Ω3

(c) Speed lapse

Figure 6.4: Effect of step increase in Ω2 - increasing absolute speed

Next, we want to observe the effect of a gradual speed change over a longer period of time, tobetter see the effect of the speed values on the settling value of the flux. To realise this gradualchange in interrotor speed, the inertia of the interrotor is now chosen larger: J2 = 0.5kgm2. Thesimulation parameters are chosen such that the interrotor speed starts at 0 and varies towardstwice the rotor speed. This means that in the beginning and at the end of the simulationthe relative speed between both rotors is equal in magnitude, but opposite in sign, whereasthe absolute value of the interrotor speed increases throughout the entire simulation. Theresults of this simulation are shown in fig. 6.5. At first, the relative speed between the rotorsdecreases, and the flux level is observed to increase: this is in accordance with eq. 6.2. Whenthe interrotor speed becomes equal to the rotor speed, the flux is observed to suddenly increase,and temporarily being controlled at its maximal value. In the following part of the curve, boththe absolute interrotor speed as the relative speed between both rotors increases. Inversely towhat is expected the flux level is observed to increase.

93


0 5 10 15−100

0

100

200

300

time [s]

Mec

hani

cal s

peed

s [r

ad/s

]

Ω2

Ω3

(a) Speed lapse

0 5 10 150

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux lapse

Figure 6.5: Effect of continuous variation of speed Ω2

As mentioned before, the occurring operating points have been verified not to reach either ofthe limitations (eq. 6.1 and 6.2), such that it is not necessary to apply field weakening. In theintroduction of this section, it is explained how the flux simply follows the control actions thatare chosen. In such a way that automatically a flux level will settle that satisfies the limitationson the back-emf. Although this settling value of the flux level is affected by the speeds ofinterrotor and rotor, its behaviour is found not to be consistent.

In chapter 5, a drawback of the applied control method (MBPC) has already been addressed:the control does not directly control quantities, it controls a cost resulting from these quantities.This cost is a user defined measure and can include many different quantities. The use ofthis control algorithm removes the possibility for understanding the behaviour of the controlledquantities through formulas. This is observed again here: although we know that the controlmethod will result in the torques being controlled and all boundary conditions being fulfilled, itdoes not allow to explain the behaviour of those quantities that are not controlled (in this case,the flux).

Despite the fact that the speed undergoes a transient, and such does the flux, the MBPC methodkeeps controlling the torque without any disturbance.

94

Chapter 7

Experimental validation

One of the main issues of the proposed control method is the required computational effort.Using an FPGA for implementing the controller allows for executing the control on a realmachine. However, as the real machine is not available, the EVT will be emulated on the FPGAas well. This allows for performing hardware-in-the-loop simulations in order to experimentallyverify the feasibility of applying the control method in real time.

The controller is programmed such that it should be applicable to a real machine whenever itwould become available. This chapter thus gives experimental results of the control method,applied to an emulated EVT.

7.1 FPGA model

The FPGA model was built using the equations determined in chapter 4, such that it is mathe-matically equal to the model used in Simulink in the previous chapter. The model was createdaccording to the principles listed in sec. 3.6.2 from chapter 3. As explained in sec. 3.6, theFPGA imposes limitations on the precision and orders of magnitude of values of within themodel. As a result, the FPGA model is numerically not identical to the Simulink model. Nev-ertheless, the model has been built such that the errors on the values are limited not to havean influence on the machine behaviour. As the orders of magnitudes of each of the signals inthe model are user defined, the model parameters must not deviate a lot (more than an orderof magnitude) in order for it to remain valid.

For the discretization of the machine equations in chapter 4, a first order Euler approximationis used. In Simulink, the machine is modeled with a state-space block, and is then solved by aSimulink solver: the default solver for simulations, a variable step solver (ODE45), is used. Onlythe solver for emulating the EVT is different, in both cases the controller uses the first orderEuler approximation to calculate the estimation and prediction values. Despite the differencein solving method, they do not result in a different outcome.

The sample time Ts is maintained at 50µs. The FPGA model of the controller, including theemulation of the EVT, required 210 delays, which result in a computation time of 2.1µs, as theclock period of the FPGA is set at 10ns, showing that more than enough time is available forexecution of the control algorithm. Fig. 3.3 thus shows a wrong proportion of the timing for this

95

CHAPTER 7. EXPERIMENTAL VALIDATION

particular case. Nevertheless, it may be argued that the sample time can be reduced in order toimprove the accuracy of the discretisation. By reducing the sample time however, the maximalswitching frequency is increased. For example, reducing the sample time Ts from 50 to 2.1µsresults in the maximal frequency going from 50kHz to 467.2kHz. A trade-off must be madebetween switching losses and accuracy, taking into account the maximal switching frequency ofthe inverter switches.

The cost function used to test the method is one that maintains control of the flux, and does notoptimise joule losses. Moreover, the cost function implemented takes a slightly different formfor the convenience of programming in the FPGA:

CostSi = W1 · |(Ψ1,pi(tk+1)−Ψ∗1(tk+1)|

+W2 · |(T2,pi(tk+1)− T ∗2 (tk+1)|

+W3 · |(T3,pi(tk+1)− T ∗3 (tk+1)|

(7.1)

As there is no such thing as infinite values in an FPGA, it is impossible to implement a costfunction as in fig. 5.3, which is the limit case of eq. 7.1 for W1 =∞. By choosing W1 sufficientlylarger than W2 and W3, a control with the same characteristics is obtained. As before, tolerancesare added to the cost functions: 5% and 2.5% on both torques and the flux respectively. Theweight factors in the cost function are set as follows: W1 = 100, W2 = W3 = 1.

7.2 Results

7.2.1 Optimisation step

The FPGA allows easily to follow the process of the optimistion step. To gain a better under-standing in what happens exactly in this step, the figures below show the predicted values oftorques and flux, and the corresponding cost values. Out of the 64 possible switching states,the 39th results in the lowest cost and is thus selected to be applied as the next control action.As can be seen, many of the switching states result in predictions for the control values thatare within a small margin of the reference value. In this case, the deviation of T3 and statorflux Ψ1 corresponding with switching state 39 lays within the tolerance band, thus resulting in azero cost. Torque T2 is approximately 15% off its reference value, thus resulting in a cost. Thisshows that tight control of one variable is sacrificed in order to minimize the total cost.

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0 10 20 30 40 50 60 70−60

−40

−20

0

20

40

T [

Nm

]T

2

T3

T2,ref

T3,ref

Switching state

39

(a) Torque predictions

0 10 20 30 40 50 60 700.8

0.81

0.82

0.83

Switching state

Sta

tor

fluxΨ

1[W

b]

39

(b) Flux predictions

0

10

20

30

40

0 10 20 30 40 50 60 70

Co

st

Switching state

39

(c) Cost evaluations

Figure 7.1: Iterating all possible switching states and their result on control variables

7.2.2 Experimental verification of the control method

The operation of the optimisation step has been discussed in the section above. In the figuresbelow, the resulting lapses of torque and flux are presented as an experimental verification ofthe control method. Indeed, the proposed control method does control both torques and theflux as desired.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux control

Figure 7.2: Validation of control strategy on FPGA

7.2.3 Torque step response

With the same set of cost function parameters that succeeded in controlling the EVT, theperformance of the control upon changing the reference values is verified. Figures 7.3 and 7.4show the lapses of torque and flux when a step in the interrotor torque is applied. Both figuresprove the effectiveness of the control. Due to the nature of the controller, the control variablefollows the change in reference without any transient, showing the strong dynamic capabilitiesof the control method.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux control

Figure 7.3: Response of control on torque step (up)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−40

−20

0

20

40

time [s]

T [N

m]

T2

T3

T2,ref

T3,ref

(a) Torque control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

time [s]

Sta

tor

flux

Ψ1 [W

b]

(b) Flux control

Figure 7.4: Response of control on torque step (down)

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100

Conclusion

This master thesis provides an overview of the different EVT topologies that have been presentedin literature. Based on their build form, they can be divided into three categories: beingeither induction machine based, permanent magnet based, or switched reluctance based. Moreattention is paid to the conceptualisation of the induction machine based EVT, as this was thefirst to be documented as such in literature. Moreover, it is this type of EVT that is studiedin this thesis. Advantages and drawbacks of the different topologies are presented, along withpossible applications, and control methods for the EVT.

A model of the EVT has been developed, both in Simulink and FPGA including all the func-tionalities needed to implement the MBPC method.

The flexibility of the MBPC method has been exploited to include new functionalities. Thecontrol method presented omits explicit flux control which has the benefit that field weakeningis automatically applied when necessary. MBPC is observed to result in much better control ofthe torques compared to FOC. However, the losses resulting from the presented control methodfar exceed those resulting from FOC. To reduce the joule losses, the algorithm is extended witha second optimisation step, choosing the control action leading to minimal joule losses. Thisextension has the desired effect of reducing the joule losses while maintaining the torque controlquality at a very high level. Nevertheless the resulting joule losses remain significantly highercompared to those resulting from FOC.

One of the main concerns about MBPC is the high computational effort required to execute thecontrol. The control method has been implemented in FPGA and successfully applied to controlan emulated EVT.

Future work

FOC with variable flux reference

FOC lists a number of reasons to control the flux to its maximal value, and only reducing itwhen the speed becomes too high (field weakening). One of these reasons (sec. 5.1.1) is to ensurethe dynamic behaviour of the control is good. Nevertheless, the control method presented inthis text (MBPC without explicitly controlling the flux) shows that even without keeping theflux at its maximal value, the control exhibits very good dynamic behaviour without any sortof transients whatsoever.

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However, it has been observed that the proposed control method increases the joule losses toan unacceptable level (compared to the joule losses resulting from applying FOC to the sameEVT). Listed as one of the challenges of the EVT (see sec. 2.6) is the thermal design aspect ofthe EVT to evacuate the generated heat. Due to the concentric design, the heat generated byboth rotors is trapped inside the stator housing, thus making it harder to evacuate. One way to(partly) overcome this, is by limiting the joule losses to a minimum. The exact opposite is truefor the proposed control method, such that the benefits of the increased control effectiveness isovershadowed by the increase in joule losses.

In order to control the flux value to its maximal value, the flux forming component of the currentmust be kept at its maximum value all the time. Fig. 5.28 shows that keeping the flux formingcurrent component constant generally does not result in the minimal joule losses.

These observations lead to the presentation of a new control strategy which includes the benefitsof both FOC and MBPC. Instead of keeping the reference value for the flux constant, it shouldbe determined by an additional control loop. The obtained reference value must then be used bythe controller to control flux level and torques according to the MBPC method. The additionalcontrol loop must carry out the optimisation problem of fig. 5.28 in order to choose the referencevalue for the flux that minimizes joule losses. Of course, this control loop must also take fieldweakening into account whenever required.

Adding integrator in cost function

One of the results of assigning tolerance bands to control variables was that it was controlledclose to either one of the limits to this tolerance band, rather than being controlled around itsreference value. As a result, the average value of the control variable differs from the referencevalue. By including an integrator (e.g. by a moving average) of the control error, an averageerror will be penalised. This way, the control method is expected to control its values withinthe tolerance band, and additionally control its average value towards the reference value.

However, adding the average error as a new term to the cost function increases its complexity:additional cost function parameters are introduced, and have to be optimised with respect tothe others.

102

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