Advanced Tools for Three-Dimensional Modeling and Control of Thermonuclear Fusion Devices

Università degli Studi di Padova

Centro Interdipartimentale "Centro Ricerche Fusione"

Universidade de Lisboa

Instituto Superior Técnico (IST)

JOINT EUROPEAN RESEARCH DOCTORATE IN FUSION SCIENCE AND

ENGINEERING

Cycle XXIX

Advanced Tools for Three-Dimensional

Modeling and Control of Thermonuclear

Fusion Devices

Coordinator:

Prof. Paolo Bettini

Supervisors:

Dr. Tommaso Bolzonella

Prof. Paolo Bettini

Prof. Bernardo Brotas de Carvalho

Ph.D. Student: Leonardo Pigatto

Padova, January 2017

JOINT EUROPEAN RESEARCH DOCTORATE IN FUSIONSCIENCE AND ENGINEERING

Network Partners:

• Instituto Superior Técnico (IST) Lisboa, Portugal

• Università degli Studi di Padova, Italy

• Ludwig-Maximilians-Universität München, Germany

In collaboration with:

• Consorzio RFX, Italy

• IPP Garching, Germany

Leonardo Pigatto: Advanced Tools for Three-Dimensional Modeling andControl of Thermonuclear Fusion Devices, , January 2017

To my grandmother Teresa

To the loving memory of my grandmother Angela

F O R E W O R D

The research work presented in this thesis has been carried outat Consorzio RFX in Padova, within the framework of a Eu-ropean Research Doctorate in Fusion Science and Engineering.During the last few years I have been given the chance to de-velop professional skills and experience in research, thanks tothe people that I have met and worked with along the way. Themost obvious purpose of this thesis is to draw the conclusionsof a long journey that has nevertheless gone by very fast. As asecondary objective, I would like to convey my personal satis-faction for the work done and my passion for the subject. As asmall note on the text itself, the structure of the manuscript re-flects the evolution of my activities in time, with the exceptionof the introductory part in which I tried to summarize the mainbackground concepts in a neither lengthy nor simplistic way. Ihope I have found a good compromise that does not fall intoboredom. In the core chapters following the introduction, pa-pers and conference contributions where the content has beenpresented are explicitly mentioned. I have made this choice tohighlight where and when the results have been presented tothe community. I have also added a quotation at the beginningof each chapter, taken from some of my favorite classical andcontemporary authors. These quotes do not necessarily referto the content of the chapters as much as they are meant forentertainment and inspiration, which I often find to be lackingin scientific literature. I hope the reading of this thesis will beenjoyable as well as interesting.

Padova, January 2017

vii

Ergo vivida vis animi pervicit, et extraprocessit longe flammantia moenia mundiatque omne immensum peragravit mente animoque,unde refert nobis victor quid possit oriri,quid nequeat, finita potestas denique cuiquequanam sit ratione atque alte terminus haerens.

— Titus Lucretius Carus, De Rerum Natura

A B S T R A C T

This thesis represents the summary of the research activitiescarried out during a three-years Ph.D. project. The work isdivided into two parts, with the common feature of investi-gating the physical properties related to stability and controlof Magneto-Hydro-Dynamic modes in fusion relevant plasmas.One of the aims of the work is to better understand the interac-tion between such plasmas and a wide range of 3-dimensionalelectro-magnetic boundary conditions. This part of the researchhas been carried out on the RFX-mod device, where advancedcontrol strategies have been developed thanks to its state-of-the-art magnetic feedback system. A variety of interlaced problemshave been addressed, starting with the improvement of the vac-uum magnetic field spectrum through actuator-sensor decou-pling, compensation of broken or deactivated coils with sim-ple and real-time applicable strategies and multi-modal Resis-tive Wall Mode control with varying coil number and geometry.This has allowed to develop relevant control optimization tech-niques and knowledge for both the Reversed Field Pinch andTokamak configurations. The former is an excellent playgroundfor RWM studies, given the instability spectrum that is natu-rally developing. For the latter configuration instead, RWM sta-bility is considered to be one major milestone to be achievedalong the road to a commercial fusion reactor. The second partof the work is dedicated to this issue, and deals with the stabil-ity properties of Advanced Tokamak scenarios, with referenceto the JT-60SA experiment in particular. Studies to understandRWM physics in high β plasmas, where fluid rotation profilesand hot ions populations from Neutral Beams can play an im-portant role, have been carried out with the MARS-F/K linearMHD codes. If detailed physics such as kinetic effects is cou-pled to a simplified description of the passive/active structures

ix

on one side, on the other hand a simplified plasma can be cou-pled to a complex 3-D model of the structures to assess realisticactive control capabilities of a given system. Different tools areused and described for studying RWM damping physics, andto five a proof-of-principle for feedback control of such insta-bilities in Advanced Tokamak plasmas operating beyond theno-wall pressure limit.

S O M M A R I O

Questa tesi rappresenta la raccolta delle attività svolte duran-te i tre anni di un progetto di Dottorato di Ricerca. Il lavoro èstato diviso principalmente in due parti, con il comune deno-minatore di investigare le problematiche relative alla stabilitàe al controllo di instabilità Magneto-Idro-Dinamiche in plasmidi interesse fusionistico. Uno dei principali obiettivi di questolavoro è lo studio di come questi plasmi interagiscano con di-verse condizioni al contorno, strutturali ed elettro-magnetiche,con caratteristiche tridimensionali. Questa parte della ricercaè stata svolta sull’esperimento RFX-mod, dove è stato possibi-le sviluppare peculiari strategie di controllo grazie all’avanzatosistema di controllo attivo. Sono state affrontate varie problema-tiche tra loro interconnesse, a partire dallo sviluppo di tecnicheper il miglioramento del contenuto armonico dei campi magne-tici di vuoto tramite disaccoppiamento attuatori-sensori. Da ciòè stato sviluppato un metodo semplificato e applicabile in tem-po reale per la compensazione di attuatori rotti o disattivati,con il medesimo obiettivo di migliorare il contenuto armonicodei campi magnetici prodotti dal sistema di controllo reale. Aconclusione di questa parte il controllo multi-modale di modidi parete resistiva (RWM) è stato affrontato, dal punto di vistamodellistico e sperimentale. Le strategie sviluppate e gli stu-di effettuati sono rilevanti sia per la configurazione ReversedField Pinch sia per il Tokamak. Il primo è un ottimo terrenodi prova per studiare i modi RWM, per via dello spettro diinstabilità che naturalmente sviluppa. Per la seconda configu-razione invece, la stabilizzazione dei modi RWM è consideratouno dei principali obiettivi da raggiungere sulla strada versoun reattore a fusione commerciale. La seconda parte del lavoroè relativa proprio alla problematica della stabilità RWM nel-la configurazione Tokamak, in particolar modo negli scenari

x

avanzati in fase di sviluppo per l’esperimento JT-60SA. Una se-rie di studi è stata portata avanti con i codici MARS-F/K perdeterminare le proprietà dei modi RWM in plasmi ad alto β,nei quali i profili di rotazione e le popolazioni di ioni sovratermici provenienti dagli iniettori di neutri possono giocare unruolo importante. Da un lato una descrizione dettagliata delplasma, includendo gli effetti cinetici, è stata accoppiata a unmodello semplificato e bidimensionale delle strutture passive.D’altra parte una più semplice descrizione del plasma è stataconsiderata per l’accoppiamento con un modello dettagliato etridimensionale delle strutture attive e passive, in quest’ultimocaso è stato possibile sviluppare un modello di controllo attivoin catena chiusa dei modi RWM.

R E S U M O

Esta tese representa o resumo das atividades de pesquisa reali-zadas durante um projeto de doutorado de três anos. O traba-lho é dividido em duas partes com a característica comum deinvestigar as propriedades físicas relacionadas à estabilidadee ao controle dos modos magnetoidrodinâmicos em plasmasrelevantes para fusão. Um dos objetivos do trabalho é enten-der melhor as interações entre estes plasmas e um amplo in-tervalo de condições de contorno eletromagnéticas tridimensi-onais. Essa parte da pesquisa foi realizada no dispositivo RFX-mod onde avançadas estratégias de controle foram desenvolvi-das graças ao seu moderno sistema de feedback. Uma variedadede problemas entrelaçados foi abordada começando pela me-lhora do espectro de campo magnético no vácuo através dodesacoplamento atuador-sensor, compensação de bobinas da-nificadas ou desativadas com estratégias aplicáveis simples eem tempo real e controle de multimodal Resistive Wall Modecom numero de bobina e geometria variável. Isso permitiu odesenvolvimento de relevantes técnicas de controle e otimiza-ção e o conhecimento para ambas as configurações: ReversedField Pinch e Tokamak. O primeiro é um excelente “parque dediversões” para os estudos de RMW, dado o espectro de insta-bilidade que se desenvolve naturalmente. Para esta ultima con-figuração, no entanto, a instabilidade RMW é considera comoum dos marcos a ser alcançado na estrada para um reator defusão comercial. A segunda parte do trabalho é dedicada a essaquestão e trata das propriedades de estabilidade no cenário de

xi

Tokamaks avançados com referência, em particular, ao experi-mento JT-60SA. Estudos para compreender a física dos RMWem plasmas com alto β, em que o perfil de rotação do fluido eas populações superaquecidas de íons proveniente das partícu-las neutras podem desempenhar um papel importante, foramrealizados com os códigos MHD lineares MARS-F/K. Se porum lado detalhes físicos como efeitos cinéticos são acopladospara uma descrição simplificada das estruturas passiva/ativa,por outro lado, um plasma simplificado pode ser acoplado aum modelo complexo 3-D das estruturas para avaliar capaci-dades realistas de controle ativo de um determinado sistema.Diferentes ferramentas são usadas e descritas para o estudo dafísica de amortecimento dos RMW e para encontrar uma provade princípio para o controle de feedback de tais instabilidadesem Tokamaks avançados operando além do limite de pressãosem parede.

xii

C O N T E N T S

i introduction 1

1 thermonuclear fusion and plasma con-finement 3

2 introduction to equilibrium and stabil-ity 9

2.1 MHD Plasma Description 10

2.2 Toroidal Equilibrium 11

2.3 Linearization for Small Displacements 13

2.4 Ideal MHD Energy Principle 15

2.5 External Kink Instability and Resistive WallModes 17

2.5.1 Rotation Stabilization and Kinetic Ef-fects 18

2.5.2 Feedback Stabilization 20

2.6 Tokamak 21

2.7 Reversed Field Pinch 24

3 mars-k formulation : toroidal non-perturbative

mhd-kinetic hybrid plasma description 29

4 introduction to control theory 33

4.1 State variable representation of a linear time in-variant system 33

4.2 Feedback Control 36

5 magnetic confinement fusion experiments 39

5.1 Reversed Field eXperiment 39

5.1.1 RWM control with a reduced set of coils 42

5.2 JT-60SA 46

ii rfx-mod control system optimization 49

6 actuator-sensor decoupling on rfx-mod 51

6.1 The Coupling Problem 51

6.2 The Model 53

6.2.1 Dynamic Simulation Tool-Set 53

6.2.2 Validation of the Vacuum Model 58

6.3 Open Loop Operation With Decoupling Matri-ces 59

6.4 Closed Loop and Compensation 61

6.5 Conclusions 63

7 missing coil compensation on rfx-mod 65

xiii

xiv contents

7.1 Model 66

7.2 Magnetic Field Reconstruction Results 68

7.3 First Wall Temperatures in Compensted Shots 72

7.3.1 Currents with Actuator Compensation 75

7.3.2 Temperature Analysis 76

7.4 Conclusions 81

8 multi-modal rwm active control on rfx-mod 83

8.1 Reduced Sets of Active Coils 84

8.2 Multi-modal Control: Eigenvalue Studies 87

8.3 Conclusions 89

iii modeling of high β plasmas and resistive

wall mode stability 91

9 resistive wall modes in high beta fully

non-inductive scenarios 93

9.1 Scenario 5.1 Equilibrium 94

9.2 No-wall and Ideal-wall Stability Limits 96

9.3 Stability of fluid RWM: effects of plasma flowand flow profile 97

9.4 Kinetic contribution to RWM stability 99

9.5 Conclusions 104

10 active control of resistive wall modes on

jt-60sa 107

10.1 CarMa Model and Open-Loop Analysis 107

10.2 Closed-Loop Proof-of-Principle 110

10.3 Full Active System Implementation 113

10.3.1 Eigenvalue Analysis of the Full System 115

10.3.2 Time Domain Simulation 115

10.4 Conclusions 117

bibliography 125

L I S T O F F I G U R E S

Figure 1 Virtual shell 20

Figure 2 Tokamak configuration 22

Figure 3 Tokamak scheme 22

Figure 4 RFP scheme 25

Figure 5 Tokamak and RPF fields 25

Figure 6 State-Space system 34

Figure 7 Feedback loop diagram 36

Figure 8 Open-loop diagram 37

Figure 9 RFX-mod cutaway 40

Figure 10 RFX-mod copper shell 41

Figure 11 Supercoil concept 41

Figure 12 RFX-mod magnets 42

Figure 13 Active coil reconfigurations 44

Figure 14 RWM control with supercoils 45

Figure 15 JT-60SA cutaway 47

Figure 16 JT-60SA in-vessel 48

Figure 17 Decoupling matrices 54

Figure 18 Simulation diagrams 56

Figure 19 Shot 33863 57

Figure 20 Shot 33863 harmonics 59

Figure 21 THD analysis 60

Figure 22 Decoupling open-loop simulation 62

Figure 23 Decoupling closed-loop simulation 63

Figure 24 Open-loop compensation 67

Figure 25 Compensation matrix 68

Figure 26 Measured and simulated field 69

Figure 27 Single coil compensation 70

Figure 28 Compensation harmonic content 71

Figure 29 Compensation experimental test 73

Figure 30 Disabled coil step-like field 74

Figure 31 Disabled coil oscillating field 75

Figure 32 Compensation currents 77

Figure 33 Run temperature signal 78

Figure 34 Run temperature map 79

Figure 35 Run 2280 heating distribution 80

Figure 36 Run 2280 ∆T above 2 81

Figure 37 1×12 harmonic content 85

Figure 38 Reconfigurations THD 86

xv

Figure 39 Eigenvalues 4×3 vs 4×6 88

Figure 40 Reconfiguration field maps 88

Figure 41 Eigenvalues 1×8 89

Figure 42 Equilibrium profiles 95

Figure 43 q profile and boundary 96

Figure 44 Pressure scan 97

Figure 45 Rotation scan 98

Figure 46 Kinetic eigenvalue 100

Figure 47 Kinetic rotation scan 101

Figure 48 Eigenfuntion comparison 102

Figure 49 Non-adiabatic pressure map 102

Figure 50 Kinetic energy components 1 103

Figure 51 Kinetic energy components 2 104

Figure 52 JT-60SA coils and SP 108

Figure 53 JT-60SA coils and sensors 108

Figure 54 3D current distribution 109

Figure 55 m=2 normal displacement 110

Figure 56 Unstable modes 111

Figure 57 Unstable modes harmonics 111

Figure 58 Reference evaluation 112

Figure 59 Gain scan 113

Figure 60 Unstable modes - three arrays 114

Figure 61 Interpolation 116

Figure 62 Gain scan with all coils 117

Figure 63 Time simulation - gain scan 118

Figure 64 Time simulation - field and current 118

L I S T O F TA B L E S

Table 1 RFX-mod technical specifications. 40

Table 2 Main requirements of the JT-60SA RWMcontrol system 48

xvi

acronyms xvii

A C R O N Y M S

MHD Magneto-Hydro-Dynamic

XK External Kink

RFP Reversed Field Pinch

RFX-mod Reversed Field eXperiment

RWM Resistive Wall Mode

RP-RWM Resistive-Plasma-RWM

MARS MAgnetohydrodynamic Resistive Spectrum

EPs Energetic Particles

NBI Neutral Beam Injector

ITER International Thermonuclear Experimental Reactor

JT-60SA Japan Torus Super Advanced

SP Stabilizing Plate

DEMO DEMOnstration Power Station

MIMO Multiple-Input-Multiple-Output

PID Proportional-Integral-Derivative

THD Total Harmonic Distortion

AT Advanced Tokamak

ITB Internal Transport Barrier

CMC Clean-Mode-Control

N-NBI Negative Neutral Beam Injector

P-NBI Positive Neutral Beam Injector

ECRF Electron Cyclotron Radio-Frequency

ECCD Electron Cyclotron Current Drive

NTM Neoclassical Tearing Mode

Part I

I N T R O D U C T I O N

1T H E R M O N U C L E A R F U S I O N A N D P L A S M AC O N F I N E M E N T

"Unless you can make the world wag better than it does at present,King, your reign will be an endless series of petty battles."

— T.H. White, The Once and Future King

Standards of living and quality of life in our civilization areknown to be directly related to energy consumption, indicatorsof human development are often determined by the per capitaconsumption of electricity. Industrialized and developing coun-tries have ever-growing energy requirements to sustain foodproduction, industrial facilities, transport and communication.This energy must be produced with a reasonable efficiency anda convenient price. The world’s main energy resource in thelast centuries has been primary fossil fuels such as coal, oiland natural gases that are nowadays showing availability prob-lems. Furthermore the usage of these sources has caused se-vere issues in terms of greenhouse gases production, with doc-umented negative effects on the environment. The most noto-rious of these effects is surely global warming and climate is-sues in general, which require massive reduction of greenhousegases production in the future, up to 80% in the next decades.The price to pay for a greener energy is to gradually cease ex-ploitation of the already depleted oil reserves while not increas-ing the usage of largely available yet extremely polluting coalveins.

Conventional fission nuclear power has been vastly de-ployed and developed over the last few decades, representinga viable alternative to fossil fuels and allowing the rise of to-day’s world powers. This technology could possibly provideenergy for thousands of years once elements such as 239Pu and233U are considered beyond the standard 235U. Public opinionhowever has not been kind with fission technology for manyreasons: the atomic fear of Cold War era has resulted in pro-liferation concerns, storing for centuries dangerous and slowlydecaying radioactive wastes is a non-negligible and unsolvedissue, catastrophic scenarios such as Chernobyl or the more

3

4 thermonuclear fusion and plasma confinement

recent case of Fukushima have mined further development oftraditional nuclear energy in many countries.

Alternative energy sources that meet the requests of renew-ability and environmental sustainability are being researchedand increasingly deployed for electricity production. Solar andwind energy could be viable alternatives to fossil fuels in thenear term but still lack the efficiency and cost of production re-quirements to represent long term investments. Solar energy ismaybe the most important renewable resource and it is widelyused in industrialized countries both in photo-voltaic and ther-mal technologies. Today’s commercial solar panels howeverhave very low efficiencies, around 12 − 18%, and althoughgreat effort is being put into improving this aspect there areother reasons suggesting that solar power will still play a sup-portive role for many years: the lack of high energy densityrequired to operate large facilities and the high initial invest-ment and maintenance costs. Other minor resources such ashydroelectricity play an important role in specific geographicalareas and Italy relies on hydroelectric power for about 18% ofits national electricity production. While it is clearly incapableof sustaining massive urban and industrial complexes, hydro-electric energy could be a good solution to meet the request inpeak hours and play a supportive role.

In this complex background fusion power could play an im-portant role in providing a sustainable, solid and long termenergy source [1]. Nature itself seems to have chosen fusion topower stars: in these immense spatial objects fusion reactionsproduce the energy that has allowed for life on Earth to de-velop. The environment in which fusion occurs in nature how-ever, is hardly comparable to that on Earth; the laws of physicsgoverning fuel confinement, thus creating the conditions forthe reaction to take place, are radically different. Hypotheticalhuman-scale fusion plants could help reducing global warm-ing without many of the potential biological hazard of today’sfission technology. If the CO2 emission of a fusion reactor issomehow comparable to that of a traditional fission plant, thepotential danger coming from long living radioactive waste andoperational safety is greatly reduced. Of course risk reductiondoes not mean complete safety, a fusion reactor would still con-sume and produce radioactive material such as tritium, whichis the main fuel of today’s concepts, and rely on neutron acti-

thermonuclear fusion and plasma confinement 5

vated components; proper materials must be therefore chosenand developed that would not require hundreds of years of ab-solute containment.

A fusion reaction takes place when two light nuclei gainenough kinetic energy to overcome the Coulomb barrier andget in the strong nuclear force range; this force is responsiblefor atomic nuclei being tightly bound and it drives the fusionof two nuclei when the reaction is convenient in terms of en-ergy. The product of a fusion reaction is one or more heaviernuclei with slightly less mass that the sum of first two, thisdifference in mass accounts for the energy release. In tomor-row’s power plants the main fuel will probably be representedby deuterium (D) and tritium (T). The former is largely presenton Earth and can be extracted from natural water, the latterinstead is a radioactive isotope β−unstable with a half-life ofabout 12.6 years. Clearly tritium must be produced and cannotbe stored in long term facilities. A viable option for a reactorcould be the breeding of T on-site thanks to the interaction ofslowed down neutrons with a lithium mantle. Four differentnuclear reactions are considered important for fusion purposes[2] [3]:

D+ T −→4 He(3.5 MeV) +n(14.1 MeV)D+D −→ T(1.01 MeV) + p(3.02 MeV) 50%

D+D −→3 He(0.82 MeV) +n(2.45 MeV) 50%

D+3 He −→4 He(3.6 MeV) + p(14.7 MeV)

The D-T reaction is the easiest to obtain in terms of cross-sectionand required temperature, this is the reason why the first gen-eration of reactors, as well as experiments like the ITER project,will use a D-T mixture plasma confined in a toroidal cham-ber. If in space the enormous mass of stars balances thermalexpansion with gravitational attraction, on Earth other particleand energy confinement methods are clearly required. Thereare currently two ways of approaching this challenge: magneticconfinement and inertial confinement. The latter will not be con-sidered in the present work, let it suffice to say that it consists inheating to ignition temperatures solid pellets of fuel that gener-ate net power before falling apart in micro-explosions. This the-sis is set instead in the magnetic confinement framework wherethe primary objective is, as the name suggests, confining a hotplasma by means of external magnetic fields. Many techniques

6 thermonuclear fusion and plasma confinement

and devices have been developed in the past century to facethis challenge, some of the most promising have culminated inthe International Thermonuclear Experimental Reactor (ITER):the largest magnetic confinement experimental power plant cur-rently being built in Cadarache (France). The project is man-aged by an international collaboration and funded by EuropeanUnion (EU), India, Japan, China, Russia, South Korea and theUnited States. Much effort is being put into the ITER projectwhich would hopefully be the first reactor to achieve a pos-itive power gain, targeting operations at Q =

PoutputPinput

≈ 10.The power gain criterion is obviously fundamental for a powerplant to be successful, the minimum requirement being Q > 1.Another fundamental parameter that ought to be mentioned isthe β factor. Defined as the ratio of kinetic to magnetic pres-sure, β = p

B2/2µ0, it evaluates the effectiveness of plasma mag-

netic confinement. The achievable value of β is often limited byplasma instabilities.

High β values are desirable for robust confinement, this canbe expressed by introducing the triple product nτET , wheren is the particle density, τE the energy confinement time andT the temperature. This product can be used to estimate theso-called ignition condition, in which the power produced bythe plasma itself (through α particles) is sufficient to balancethe losses and sustain the reaction with non external heating.Among the many criteria that can be used for the ignition con-dition calculation, one of the first to be introduced is the Law-son criterion. Assuming that the total power leaving the plasma,including losses and fusion products, is converted to electricitywith efficiency η ∼ 30%, the condition for which fusion powercan heat the plasma by itself is:

Pb + PL = η (Pb + PL + Pf) (1)

where Pb represents radiation losses (mainly bremsstrahlung),PL the transport losses and Pf the power produced by fusionreactions. This condition can be translated into:

nτE >12T

η1−η〈σv〉Ef − 4αbT1/2

(2)

where 〈σv〉 = 〈σv〉(T) is the reactivity, Ef the energy producedby a single fusion reaction (17.6 MeV for D-T) and αb = 5.35×10−37Wm3keV−1/2 is a constant related to radiation losses.

thermonuclear fusion and plasma confinement 7

In the present thesis a series of studies on stability and con-trol of Magnetohydrodynamic instabilities is shown. The firstpart gives a general introduction to the main topics of the work.Starting with MHD theory, a short review of the studies car-ried out on instabilities such as the Resistive Wall Mode, withtheir relevance for high β operation, is given in Chapter 2. TheMARS-F/K codes are described in Chapter 3 while Chapter4 gives a useful introduction on control theory. This part ofthe manuscript is closed by Chapter 5, which describes themain experimental devices on which the work has been car-ried out. The second part of the thesis includes studies on theoptimization of the RFX-mod control system. With the mainfocus being active control of Resistive Wall Modes, the task ofactuator-sensor decoupling is tackled with a lightweight, real-time applicable strategy, as described in Chapter 6. The sameconcept is used to take into account the faulting or shuttingdown of sparse actuators. Chapter 7 describes how a matrix-based compensation method can help in reducing the harmonicdistortion of the magnetic field produced by a non-uniform coilarray. A study on the effects of missing coil compensation onfirst wall heating is also presented. The part concerning RFX-mod is closed by Chapter 8, with a numerical study aimingat assessing the multi-modal control capabilities of reduced ar-rays of active coils. The third and final part of the thesis is fo-cused on Resistive Wall Modes characterization and control inJT-60SA Advanced Tokamak, high β, fully non-inductive sce-narios. The characterization of Resistive Wall Mode physics,damping mechanisms, effect of fluid rotation and kinetic par-ticles, has been carried out with the MARS-F/K codes and isshown in Chapter 9. An important result is then shown in thefinal Chapter 10, with a numerical study on feedback control ofthe n=1 Resistive Wall Mode on JT-60SA.

2I N T R O D U C T I O N T O E Q U I L I B R I U M A N DS TA B I L I T Y

"Books only change the worldif the world is capable of digesting them."

— Luther Blissett, Q

A fusion relevant plasma is a hot fully ionized gas whose be-havior is dominated by long range electric and magnetic forcesthat make this plasma a very good conductor. The microscopicreason for good plasma conductivity lies in high temperatureand low density that cause long range Coulomb interactions,and thus resistance, to be very weak. A practical consequenceof conductivity is the shielding from electric fields whereasmagnetic fields penetrate making plasma confinement possible.Thermonuclear fusion plasmas have typical densities and tem-peratures of respective order n ∼ 1020 m−3 and T . 10 keV . De-scribing a plasma with a microscopic single particle approachwould be a tremendous task in terms of math or computa-tional requirements. Fluid models on the other hand presentless of a challenge and either two-fluid or single-fluid equa-tions can provide self consistent models that allow an accu-rate analysis of major issues in plasma physics such as equilib-rium and stability. This Chapter deals with the Magneto-Hydro-Dynamic (MHD) single-fluid model and specifically with its ap-plication to fusion relevant plasmas confined by means of mag-netic fields. This model can be derived both from the two-fluidmodel and from single particle guiding center theory, this is notreported here and can be found [4]. A complete derivation ofthe model from fundamental assumptions on the fluid an scalesof interest is given in [5], which will be followed for part of thisintroduction. In Section 2.1 the MHD model is introduced, thetoroidal equilibrium description is given in Section 2.2 with theGrad-Shafranov equation and introduction of the useful fluxfunctions. The linearized MHD equations are given in Section2.3, along with the plasma displacement definition and discus-sion of the ideal force operator as a foreword to stability. Theideal MHD energy principle is outlined in Section 2.4. A briefdiscussion on the ideal kink instability is presented in Section

9

10 introduction to equilibrium and stability

2.5, while a description of the Tokamak and RFP configurationscloses the Chapter with Sections 2.6 and 2.7 respectively.

2.1 mhd plasma description

In MHD description the plasma is considered as a single con-ducting fluid, thus coupling fluid-dynamics with the Maxwellequations. With physical quantities being averaged over phys-ically infinitesimal volumes, an ordering of length scales is as-sumed:

λ ∼ a0 << δ < L (3)

where λ is the mean free path in atomic collisions, a0 the atomicradius, δ the distance which is considered physically infinitesimaland L is the smallest relevant macroscopic distance to be con-sidered. This also implies an ordering or frequencies (i. e.timescales) which concentrates on low-frequency phenomena, there-fore V2/c2 << 1 where V is the characteristic fluid velocityand c the speed of light. The single-fluid equations of the MHDmodel can be written, in the Eulerian frame, as:MHD Model

∂ρ

∂t+∇ · ρV = 0 (4)

ρ

(∂V∂t

+ V · ∇V)

= −∇p+ 1

µ0(∇×B)×B +∇ ·Π (5)

∂ρ

∂t+ V · ∇p = −Γp∇ ·V + (Γ − 1)

[−∇ · q +Π · ∇V + ηJ2

](6)

∂B∂t

= ∇×(

V×B −η

µ0∇×B

)(7)

∇ ·B = 0 (8)

Where ρ is the mass density, p the scalar pressure, B the mag-netic field, Γ the adiabatic index, Π the stress tensor, q the heatflux and η the plasma resistivity. Specification of the last threeunknowns are required in order to close the system. For thework that will be presented in the following chapters the sys-tem can be simplified to the so-called ideal case, in which q, Πand η are neglected. The described plasma is therefore consid-ered a perfect electrical conductor with no viscosity of thermalconductivity. This is of course not true for a real plasma, sucha model can nevertheless give a good approximation of someproperties of hot and highly magnetized plasmas, as will be-

2.2 toroidal equilibrium 11

come clear throughout this introduction. Under the aforemen-tioned conditions, the above equations can be reduced to: Ideal MHD Model

∂ρ

∂t+∇ · ρV = 0 (9)

ρ

(∂V∂t

+ V · ∇V)

= −∇p+ 1

µ0(∇×B)×B (10)

∂ρ

∂t= −V · ∇p− Γp∇ ·V (11)

∂B∂t

= ∇× (V×B) (12)

∇ ·B = 0 (13)

2.2 toroidal equilibrium

Magnetic field lines in toroidal axisymmetric configurations lieon nested toroidal surfaces, these can be found as level surfacesof the solutions of the Grad-Shafranov equation [6]. Considera cylindrical coordinate system (R, z, φ) where R = 0 gives themajor axis of the torus and φ is the toroidal direction. The mag-netic field can thus be written as:

B = Bφeφ + Bp

where Bp is the field on the poloidal (R− z) plane. The compo-nents of the poloidal field can be used to define the poloidal fluxfunction ψ(R, z):BR = − 1

R∂ψ∂z

Bz =1R∂ψ∂R

=⇒ Bp =1

R

(∇ψ× eφ

)This function is the flux of the magnetic field through the sur-face enclosed between the magnetic axis and a generic magneticsurface. ψ is found to be constant along the magnetic field lines:

B · ∇ψ = 0 (14)

The very same procedure can be applied to the current densityto obtain a current flux function:JR = − 1

R∂f∂z

Jz =1R∂f∂R


at the same time Ampère’s law in its stationary form gives:

∇×B = µ0J =⇒

JR = − 1µ0

∂Bφ∂z

Jz =1µ01R

∂(RBφ)∂R

=⇒ f =RBφ

µ0

The f(R, z) plays the same role as ψ and again is constant alongthe current lines:

J · ∇f = 0 (15)

From the force balance Eq. 10 we can see that both B and Jare orthogonal with respect to the pressure gradient, that is tosay: the pressure is constant both on the magnetic surfaces andalong the current lines. The force balance can be expressed inthe previously introduced notation:

∇p = Jp × eφBφ + eφJφ ×Bp

=1

R

(∇f× eφ

)× eφBφ + eφJφ ×

1

R

(∇ψ× eφ

)= −

Bφ

R∇f+

Jφ

R∇ψ

Writing p and f as functions of the poloidal flux:

∇f(ψ) = df

dψ∇ψ (16)

∇p(ψ) = dp

dψ∇ψ (17)

the following expression for the toroidal component of the cur-rent is obtained:

Jφ = Rdp

dψ+Bφ

df

dψ

= Rdp

dψ+µ0Rfdf

dψ

The Grad-Shafranov equation is finally obtained comparing theprevious result with the toroidal component of the Ampère law:Grad-Shafranov

Equation

µ0Jφ =dBRdz

−dBz

dR

= −1

R

∂2ψ

∂z2−∂

∂R

(1

R

∂ψ

∂R

)

R∂

∂R

(1

R

∂ψ

∂R

)+∂2ψ

∂z2= −µ0R

2p ′(ψ) − µ20f(ψ)f′(ψ) (18)

2.3 linearization for small displacements 13

2.3 linearization for small displacements

Applying a linear perturbation to an otherwise stable equilib-rium simplifies the stability problem reducing its equations tothe form of an eigenvalue problem, with the eigenvalue beingthe growth rate of the perturbation. The idea is to slightly per-turb the plasma from its equilibrium state, examining the timedependence of the small amplitude perturbation. The proce-dure starts by splitting the problem into an equilibrium anda stability part, i. e.expanding all dependent variables as thesum of an equilibrium solution plus the perturbation. Assum-ing that the quantities of interest can be linearized about theirequilibrium values:

A (r, t) = A0 (r) + A1 (r, t)

where A1|A0| 1 is a small first order perturbation, which takes

the general form:

A1 (r, t) = A1 (r) e−ıωt (19)

The simplest definition of stability could be the following: ifany of the system’s eigenfunctions ω corresponds to exponen-tial growth (i.e. =(ω) > 0) the system is said to be exponentiallyunstable. In the given definition the assumption that modes arediscrete with distinguishable frequencies is implicit. With theassumption of a stationary plasma (i. e.V0 = 0) and by drop-ping all non-linear terms, the linearized MHD system can bewritten as follows: Linearized MHD

Model∂ρ1∂t

= −∇ · ρ0V1 (20)

ρ0∂V1

∂t= −∇p1 + J0 ×B1 + J1 + B0 (21)

∂p1∂t

= −V1 · ∇p0 − Γp0∇ ·V1 (22)

∂B1∂t

= ∇× (V1 ×B0) (23)

∇×B0 = µ0J1 (24)∇ ·B1 = 0 (25)

As a first step to simplify these equations we can extract thetime dependence by means of a normal mode expansion, i.e.rewriting all the perturbed observables as:

A1 (r, t) = A1 (r) e−ıωt


In the above equation ω is a complex frequency. A possible ap-proach to stability involves the study of its imaginary part, if=ω > 0 the perturbation grows exponentially and the systemis therefore unstable. If on the other hand =ω < 0 the pertur-bation is either exponentially damped of oscillates about anequilibrium position. The problem can be further simplified byintroducing the so-called plasma displacement ξ (r, t):Plasma

Displacement

V1 ≡∂ξ

∂t= −ıωξ (r) e−ıωt (26)

The previous linearized equations can now be written in termsof this new quantity:

ρ1 = −ξ · ∇ρ0 − ρ0∇ · ξp1 = −ξ∇p0 + Γp0∇ξQ ≡ B1 = ∇× (ξ×B0)

J1 =1

µ0∇× [∇× (ξ×B0)]

(27)

Where the time dependent factors e−ıωt cancel out in everyterm. There is one last equation remaining: the momentum con-servation represented by Eq. 21. Using 27 into 21 we obtain:Ideal MHD Force

Operator

ρ0∂2ξ

∂t2=1

µ0(∇×B0) +

1

µ0∇× [∇× (ξ×B0)] +

+1

µ0∇× [∇× (ξ×B0)]×B0 +∇ · (ξ · ∇p0 + Γp0∇ξ)

=1

µ0(∇×B0)× [∇× (ξ×B0)] +

1

µ0∇× [∇× (ξ×B0)]×B0+

+∇ (ξ∇p0 + Γp0∇ξ)≡ F (ξ)

(28)

Eq. 28 is to be solved as an eigenvalue problem where F iscalled force operator and has the property of being self-adjoint.This property, proof of which be found in [7], has importantconsequences such as the existence of an energy principle. Con-tinuing the analysis of the small displacements problem, or nor-mal modes approach, we suppose the desired solutions take theform:

ξn (r, t) = ξn (r) e−ıωnt

then Eq. 28 can be written as:Ideal MHD WaveEquation

2.4 ideal mhd energy principle 15

F (ξn) = −ω2nρ0ξn (29)

This has clearly the form of an eigenvalue problem where theeigenfunctions are ξn and ω2n the corresponding eigenvalues.Given the self-adjointness of the force operator F we can showthatω2n are forced to be real; for a generic eigenfunction ξn andits complex conjugate ξ∗n:∫

ξ∗n · F (ξn)dV =

∫ξn · F (ξ∗n)dV

This property can be used in Eq. 29 once integrated over vol-ume multiplying both members by ξ∗n:

ω2nρ0

∫ξ∗n · ξndV =

= −

∫ξ∗n · F (ξn)dτ =

= −1

2

∫ξ∗n · F (ξn)dV −

1

2

∫ξn · F (ξ∗n)dV

Therefore for the generic nth eigenvalue we can write:

ω2n = −1

2ρ0

∫ξ∗n · F (ξn)dV +

∫ξn · F (ξ∗n)dV∫

ξn · ξ∗ndV=

= −1

ρ0

<[∫ξn · F (ξ∗n)dV

]∫ξ2ndV

(30)

which is a real quantity. Therefore, due to the self-adjointnessof the force operator, its eigenvalues are real and the ideal MHDnormal modes are either purely oscillating or purely growing/-damped.

2.4 ideal mhd energy principle

The discussion on the force operator outlined in the previoussections suggested that, because of the self-adjointness prop-erty, the stability of a system could be assessed by using a vari-ational approach. An energy variational principle allows to doso without solving the plasma displacement equation (Eq. 28).The task of solving a stability problem requires writing an ex-pression for plasma potential energy and minimizing it usingplasma displacements ξ (r, t) as test functions. It can be shownthat the plasma potential energy can be written as a functionalof displacement vectors. By recalling the definition of plasma


displacement, Eq. 26, we define the total kinetic energy of thesystem as:

K =1

2

∫ρ0ξ

2dV

To find the potential energy we should examine how the ki-netic energy varies with time, remembering as well the self-adjointness of F, its linearity and the fact that the derivative ofplasma displacement ∂ξ∂t is also a solution to Eq. 29:

dK

dt=1

2

∫ρ0∂

∂t

(ξ2)dV =

∫∂ξ

∂t· ∂

2ξ

∂t2dV =

∫∂ξ

∂t· F (ξ)dV

=

∫ξ · F

(∂ξ

∂t

)dV

=1

2

∫∂ξ

∂t· F (ξ)dV +

1

2

∫ξ · F

(∂ξ

∂t

)dV

=1

2

∫∂ξ

∂t· F (ξ)dV +

1

2

∫ξ · ∂∂t

F (ξ)dV

=1

2

∫∂

∂t[ξ · F (ξ)]dV =

d

dt

[1

2

∫ξ · F (ξ)dV

]From this latter expression we see that the time variation of thetotal kinetic energy can be written as:

d

dt

[K−

1

2

∫ξ · F (ξ)dV

]=d

dt(K+ δW) = 0 (31)

this represents the total energy conservation and defines thevariation of the system potential energy with respect to equilib-rium as:

δW = −1

2

∫ξ · F (ξ)dV (32)

Calculating this energy variation allows to distinguish be-tween stable and unstable equilibra. If δW > 0 then everyplasma displacement ξn causes potential energy to increase andthe tested equilibrium is therefore stable because correspond-ing to a minimum of potential energy. Recalling the eigenvalueproblem introduced in the previous Section (Eq.29) and the dis-cussion on the eigenvalues ω2n being real, we can rewrite thepotential energy variation:

δW = −1

2

∫ξn · F (ξn)dV =

1

2ω2n

∫ρ0ξ

2ndV

2.5 external kink instability and resistive wall modes 17

The eigenvalues then take the following form:

ω2n =δW

12

∫ρ0ξ2ndV

From this last expression it appears clearly that the δW > 0

condition corresponds to a stable mode becauseω2n > 0 impliesωn ∈ R. On the other hand the opposite situation δW < 0 leadsto instability because ω2n < 0 implies a purely imaginary eigen-value ωn that causes exponential growth of the correspondingplasma displacement mode.

In order to apply the energy principle to specific problems,it has to be extended to include contributions from vacuum re-gions and plasma/vacuum boundaries. The total potential en-ergy of the perturbed system can then be written as [4]:

δW = δWF + δWS + δWV (33)

with the three components being fluid, surface of the plasma/-vacuum interface and vacuum respectively. These can be ex-pressed as follows:

δWF =1

2

∫VF

dV

[|Q|2

µ0− ξ∗⊥ · (J×Q) + Γp0|∇ · ξ|2 + (ξ⊥ · ∇p0)∇ · ξ∗⊥

](34)

δWS =1

2

∫SdS|n · ξ⊥|2n · ∇

[[p+

B2

2µ0

]](35)

δWV =1

2

∫VdV

1

µ0

∣∣∇× A∣∣2 (36)

2.5 external kink instability and resistive wall

modes

Extensive stability analysis of ideal MHD plasmas has allowedfor the identification of the External Kink (XK) instability asone of the major obstacles to the increasing of pressure in fu-sion experiments. This has been expressed in terms of the βNparameter since the discovery of the Troyon scaling law [8]. External Kink

βN = %βaBφ

Ip(37)

where a is the plasma minor radius, Bφ the toroidal magneticfield and Ip the plasma current in MA. According to this scalingthe maximum achievable βN was predicted to be constant for


all Tokamaks. With no external wall surrounding the plasma. Inthese conditions the growth rate of such instabilities is on theAlfvén time scale (τA = R/(Bφ/

√µ0ρ), with ρ being the plasma

density and R the major radius). This makes the XK mode par-ticularly dangerous, causing disruptions and limiting the ther-monuclear power of an eventual plant. Studies on the effect of aresistive wall surrounding the plasma have been carried out [9],proving that XKs cannot be stabilized by introducing walls withfinite electrical conductivity, while at the same time it is notpossible to enclose a reactor-relevant plasma in a superconduct-ing first wall. Nevertheless the resistivity of the wall material re-duces the growth rate of the XK to the time scale of the magneticperturbation penetration time in the wall itself (1/τW). The re-sultant instability is called Resistive Wall Mode (RWM) and itcan be considered a new declination of the XK. To summarizeResistive Wall

Mode the information on XKs stability, we can say that with no exter-nal wall (i. e.τW = 0) the plasma is unstable for βN > βno−wallN .No-Wall LimitIf an ideal wall is present instead (i. e.τW = ∞), the plasma isstable up to the so-called ideal-wall limit. When a finite wallIdeal-Wall Limitresistivity is introduced, the RWM branch appears, leading toa slower instability growing on wall time scales in the rangeβno−wallN < βN < βid−wallN . According to present understand-ing three factors are of key importance for determining thesestability limits: the current profile (i. e.the plasma internal in-ductance), the pressure profile (i. e.the pressure peaking factor)and the minimum value of the safety factor profile (i. e.qmin).

2.5.1 Rotation Stabilization and Kinetic Effects

A first response to the challenge of understanding RWMs and go-ing towards stabilization has been given Bondeson and Ward in[10], proposing that low toroidal number modes could be fullystabilized by resistive walls in a plasma rotating at a fractionof the Alfvén speed. In this model plasma dissipation occursthrough ion Landau damping, transferring momentum fromthe plasma to the RWM which would be rotating with respect tothe wall, thus leading to stabilization. A critical rotation valueis found above which a stability window opens from low βNup to the ideal-wall limit. This critical rotation is found to beonly a few per cent of the Alfvén velocity, such values are eas-ily reached in present day Tokamaks relying on Neutral BeamInjector (NBI) heating and current drive. Both numerical andexperimental studies have been carried out for the verification

2.5 external kink instability and resistive wall modes 19

and investigation of RWM rotational stabilization [11]. From thetheoretical point of view it is worth mentioning introduction ofthe kinetic approach by Hu and Betti [12], with the inclusion ofthe potential energy δWk in the dispersion relation: RWM Kinetic

Dispersion RelationγτW = −

δW∞ + δWk

δWb + δWk(38)

where δW∞ is the no-wall fluid potential energy, while δWb

is the potential energy with an ideal wall at radius b. The ki-netic contribution can either be calculated using the fluid RWMeigenfunction or from a complete eigenfunction with the self-consistent inclusion of diamagnetic and magnetic drifts. Thislatter is the so-called self-consistent drift-kinetic approach intro-duced by the MARS-K code which will be described in Chapter3. The MARS code has been extensively used to study the criticalrotation and damping mechanisms for RWMs. In the first yearsof experimental activities on RWMs studies have been carriedout on JT-60 to assess mode characteristics in rotating plasmas[13] and the growth rate dependence on plasma-wall relativeposition [14]. The important relationship between RWMs anddisruptions has been studied on DIII-D [15] [16], with the ob-servation that the onset of the mode caused a plasma β col-lapse. Further experiments on JT-60U [17] and DIII-D [18] haveachieved lower stabilization thresholds with respect to previ-ous results with plasma rotation. Values lower that 0.3% of theAlfvén speed have been achieved by reducing neutral beamtorque and magnetic braking. The role of plasma resistivityon RWM stability has also been investigated as one possibledamping channel. The possibility of stabilizing RWMs with verysmall plasma rotation has been introduced [19] for the case inwhich a resistive plasma is considered. Further studies haveconfirmed the stabilizing role of resistivity [20] [21] for the so-called Resistive-Plasma-RWM (RP-RWM). The topic however isstill under investigation, recent theoretical work has applied anenergy principle [22] to modes interacting with resistive wallin a plasma with a resistive layer [23], finding a destabilizingcontribution to be introduced by resistivity. To conclude thisshort review we can say RWM stabilization is one of the cru-cial tasks for future fusion devices, particularly when aimingat steady state operation and high plasma pressure. For thisreason the problem of understanding the physical mechanismslying underneath these modes, including the possible damp-ing channels, has been challenged in theory and experiments[8][24][11][25][26]. Dealing with Resistive Wall Modes, which


although slowed down by passive structures are still strongideal instabilities, can be a complex task and has been donein different ways, mainly using passive and active techniques.Passive stabilization is usually sought with a combination ofplasma flow and different damping mechanisms such as vis-cosity or the so-called kinetic resonance with particle drift mo-tions. Plasma rotation has been theoretically found to have astabilizing effect on the RWM in Tokamak plasmas [27][10][28]and many studies are being carried out on the interaction withparticle drift motions [29][30][31]. The effect of precession driftresonance for trapped particles has been seen important in par-ticular [32][12][33].

2.5.2 Feedback Stabilization

According to the discussion so far, it is well known that anexternal resistive wall can provide transient stabilization forideal MHD modes by reducing instability growth rates. On timescales longer than the wall penetration time though, the resid-ual instability known as RWM is observed. In Reversed FieldPinch (RFP) as well as in Tokamak devices RWM stabilizationis crucial for fusion research, with the sophisticated control sys-tems above mentioned plasma stabilization by feedback controlstudies are being conducted [34] [35] [36][37]. An early system-atic study of RFP stabilization by feedback was accomplishedin 1989 by Bishop [38]. The idea was to cover the toroidal pinchwith a grid of both coils and magnetic sensors. This is the so-Virtual Shellcalled intelligent shell concept and the desired effect would beto freeze the total normal flux through each cell of the mesh. Aschematic picture from the original article is reported in Figure1. More recent research has challenged the aliasing problem in-

Figure 1: Section of a toroidal pinch showing the intelligent shellmesh [38].

troduced by the discrete nature of both the active coils and sen-sors in an intelligent shell. On RFX-mod, a particular algorithm

2.6 tokamak 21

has been developed, the so-called Clean-Mode-Control (CMC) Clean-Mode-Control[39] which applies real-time correction of the systematic error

affecting the measurements due to the aliasing of the sidebandharmonics produced by the discrete saddle sensors. With thisalgorithm, and with a flexible feedback system that will be de-scribed in Section 5.1, both the RWM and tearing modes can besimultaneously controlled [35][34]. In Tokamaks, when opera-tion beyond the no-wall pressure limit is desired and plasmarotation does not provide enough damping, the plasma mustbe stabilized with magnetic feedback. A large choice of feed-back schemes, magnetic signals to be used and active coils con-figurations is available for the purpose. The advantage of usingpoloidal sensor signals over radial for feedback has been stud-ied [40] and the usage of a combination of the two has beenfound preferable by Pustovitov [41]. It is foreseeable that fu-ture control techniques for fusion devices will take advantageof both passive and active stabilizing effects, in order to im-prove plasma performance.

2.6 tokamak

The word Tokamak stands for TOroidalnaya KAmera and MAg-nitnaya Katushka, Russian expressions for toroidal chamber andmagnetic coil respectively. Invented in the Soviet Union dur-ing the 1950s, its basic component is a toroidal chamber sur-rounded by coils that produce a toroidal magnetic field. In or-der to achieve an equilibrium configuration a poloidal field isalso needed and in Tokamak devices this latter is producedby a current flowing through the plasma in longitudinal di-rection. This combination of toroidal and poloidal field pro-duces field lines helically winding about the torus (Figure 2a).These field lines lie on the so-called magnetic surfaces (Fig-ure 2b), the contours of constant pressure, that are found tobe nested toroidal surfaces. Tokamak devices are character-ized by a toroidal field intensity many times greater than thepoloidal field. A time varying current flows in magnetizingwindings, along the toroidal direction, giving rise to a varia-tion of the magnetic flux and consequently to the plasma cur-rent. A schematic representation of a Tokamak device is givenin Figure 3 [42]. The toroidal discharge chamber is mountedon an iron core, this acts as a transformer and is used to in-duce the toroidal current. A third set of coils is used in Toka-maks to control plasma shape and position, the current in these


(a) (b)

Figure 2: (a) Representation of the helical magnetic field lines, (b)Nested magnetic surfaces [42].

Figure 3: Schematic representation showing the main components ofa Tokamak device .

2.6 tokamak 23

coils flows parallel to the plasma current. The discharge cham-bers where originally composed of two main casings: a stain-less steel inner vessel and a copper outer one. Contact betweenplasma and inner wall was reduced with the use of limiters [42]while in modern experiments the divertor technology is pre-ferred [3]. In modern Tokamaks a non-circular poloidal sectionhas also been adopted, the elongated D-shaped section allowshigher current and pressure values for a given magnetic field.This shape of the magnetic surfaces is obtained as one of thepossible solutions of the Grad-Shafranov equation, discussedin Sec. 2.2. Kink modes are the strongest ideal MHD instabili-ties and appear both as pressure driven and current driven inTokamaks. In the perfectly conducting plasma current modelall kink modes are stable if the safety factor q(r) is monotonicand q(a) > 1 [6]. According to more realistic models Tokamakplasmas are considered to be safely stable when q(a) > 2 [3],in order to avoid the development of sawtooth instabilities.

The Tokamak configuration has achieved many of the re-quirements of an actual reactor under a physical point of view.Nevertheless technological issues persist such as the need forsuperconducting and expensive magnets to generate the largetoroidal field. A major problem is also related to the strongtoroidal current: the concept previously illustrated in Figure3 uses a transformer to induce the plasma current and thismethod works perfectly for experimental pulsed devices. Anoperative reactor though would clearly be a steady state deviceand the toroidal current needs to be generated by other means. Bootstrap CurrentThe discovery of the bootstrap current, generated naturally byplasma radial transport [43], has given a viable solution to theproblem. The bootstrap current though is found to be propor-tional to the poloidal beta, high currents would therefore re-quire high pressure, beyond the MHD β limits. Severe MHDinstabilities would rise and spoil the discharge. The presence ofa resistive wall as plasma boundary would reduce the growthrate of these instabilities and allow feedback stabilization. TheAdvanced Tokamak (AT) [44][45][46] concept has been devel- Advanced Tokamakoped with the aim of sustaining fully non-inductive dischargeswith a combination of self-generated and external current drive.Given that the bootstrap current is found to be proportional tothe poloidal beta (βp = 2µ0〈p〉/〈B2p〉, with Bp the poloidal mag-netic field) and the requirement of operating at high toroidalbeta to sustain the fusion reaction, the resulting condition is to


operate at high normalized beta (βN). The following relation-ship holds between these three parameters:

βpβT ∼ 25

[(1+ E2

)2

](βN100

)2(39)

where E is the plasma elongation. It is presently thought thatan attractive reactor-like scenario should be able to provide atleast 75% of the total current through the bootstrap mechanism,leaving about 25% to be fed by external current drive. As aconsequence of the above relationship, high-bootstrap-fractionTokamaks will probably face the RWM issue. AT scenarios cantake advantage of the existence of Internal Transport Barriers(ITBs), which bring improved confinement with respect to H-mode operation. Furthermore, it has been seen that ITBs and AToperation give rise to high bootstrap current fractions becauseof the naturally hollow current profiles.

2.7 reversed field pinch

The basic idea behind the Reversed Field Pinch (RFP) configu-ration is, as for the Tokamak, plasma confinement with the so-called pinch effect: the passage of an unidirectional current inthe plasma creates an azimuthal magnetic field that constricts(pinches) the plasma resulting in magnetic confinement. Thetoroidal pinch has historically been the first system based onthe pinch effect, on which a great deal of research has beenaccomplished since the 1950s. Early pinches presented seriousstability issues due to kinks and other MHD modes. While inthe Soviet Union research led to the development of the Toka-mak configuration described in the previous section, Europeanand American experiments also explored other directions suchas the RFP configuration. Early Z-pinch and RFP devices are forinstance: Zeta-experiment and HBTX-I in the UK, ZT-I and ZR-IIin the US, Eta-Beta-I followed by Eta-Beta-II in Padova. A de-scription of these experiments is beyond the introductory aimsof this chapter and can be found in [47]. An RFP device is anaxis-symmetric toroidal system (Figure 4) in which plasma con-finement is obtained with a poloidal magnetic field Bθ, mostlyproduced by a current flowing in the plasma itself, combinedwith a toroidal field Bφ generated by a set of external coils. Likein the Tokamak configuration these two components form a he-lical total field. In the RFP case the two components are approx-imately of the same magnitude Bθ ∼ Bφ, while in a Tokamak

2.7 reversed field pinch 25

Figure 4: Scheme of a Reversed Field Pinch plasma ring with typicalfield profiles [48].

Figure 5: Radial field profiles for toroidal and poloidal componentsof the magnetic field. Dashed line represents Tokamak andsolid line RFP [49].


the profiles would be very different, as can be seen in Figure 5.The RFP configuration can be characterized by two parameters:Pinch and Reversal

Parameters

Θ =Bθ(a)

〈Bφ〉∝Ip

ΦtF =

Bφ(a)

〈Bφ〉(40)

respectively named pinch parameter and reversal parameter, whereBφ/θ(a) are the toroidal and poloidal fields measured at theboundary, 〈Bφ〉 is the toroidal field averaged on the poloidalsection and Φt is the toroidal magnetic flux. Compared to theTokamak case, ΘRFP ΘTOK. Therefore the main differencebetween the two configurations lies in the fact that the samemagnetic flux (generated by the central solenoid) gives a muchgreater current in an RFP with respect to a Tokamak. Further-more, RFP safety factor profiles are typically monotonicallydecreasing, and with respect to Tokamaks qRPF 1 qTOKbecause of the relatively higher plasma current.

q(r) =rBφ

RBθ(r)∝Bφ

Ip(41)

Another characteristic of RFP configurations is that, as a con-sequence of the above discussion, the toroidal magnetic fieldtends to change sign (i. e.reverse) at the plasma boundary, there-fore q(a) < 0. Such a system operates beyond the Kruskal-Shafranov limit, MHD stability is achieved with a combinationof sheared magnetic field (obtained due to the reversing of Bφ)and conducting shell at plasma boundary. Similarly to the Toka-mak case this external wall would stabilize XK modes if per-fectly conducting but a realistic situation involves walls withfinite resistivity, thus requiring some sort of feedback stabiliza-tion. These operation conditions also lead to the developmentof m=1 resistive kinks (i. e.a helical deformation of the plasma).These modes evolve non-linearly and give rise to a fluctuationinduced electric field Ef = −〈v× B〉 which enters the Ohm’slaw and sustains the RFP configuration by removing parallelcurrent gradients. This is the so-called dynamo. At low cur-Dynamorent, many of these resistive modes are simultaneously unstablewith different toroidal numbers n, this represents the so-calledMultiple Helicity regime. One early theoretical model for RFPequilibria has been obtained by Taylor [50]: starting from a per-fectly conducting plasma model, assuming the total magnetichelicity and magnetic flux conservation leads to a unique equi-librium state characterized by the aforementioned reversal and

2.7 reversed field pinch 27

pinch parameters. Taylor’s relaxation theory has found quitegood agreement with experimental results. The pinch parame-ter on the magnetic axis Θ0 is also important in the µ&p modeldescribing RFP equilibrium [51] [49]. This theory however, onlycaptures RFP physics qualitatively and clearer results are givenby the MHD model. The presence of Single Helicity states forinstance, has been predicted by MHD simulations [52]. Underthe experimental point of view, in large RFP devices a transi-tion from Multiple Helicity to the so-called Quasi Single Helic-ity states has been seen [53] [54] [55], persisting for the wholedischarge flat top at high (1 MA) current [56]. Rather recent de-velopments on RFX-mod have seen the experimental verificationof the self-organized helical state that develops at high current(> 1.5 MA) and is connected with the presence of an internaltransport barrier [57]. Improving optimization of the feedbackcontrol system has allowed for high current operation and forthe so-called quasi-single-helicity high persistence [58] [39] [59].

RFP fusion experiments have both advantages and disadvan-tages compared to Tokamak ones. For instance smaller currentsare be needed by the external coils to generate the modesttoroidal fields involved. On the other hand the achieved τE val-ues are low because of transport phenomena. Complex activecontrol systems are installed on modern RFP devices such asRFX-mod, in order to stabilize Resistive Wall Modes and thusimprove pulse duration. These control systems must be capa-ble of dealing with multiple instabilities which are simultane-ously present in a Reversed Field Pinch, this makes them per-fect benchmark environments also for Tokamak stability stud-ies in which more than one mode is unstable.

3M A R S - K F O R M U L AT I O N : T O R O I D A LN O N - P E RT U R B AT I V E M H D - K I N E T I C H Y B R I DP L A S M A D E S C R I P T I O N

"[...] Stars, hide your fires;let not light see my black and deep desires."

— William Shakespeare, Macbeth

The MAgnetohydrodynamic Resistive Spectrum (MARS) codesolves single-fluid linearized MHD equations in full toroidalgeometry. MARS-K [32] represents an extension of the MARS-Fcode [40] which adds drift-kinetic effects to the fluid plasma de-scription with toroidal flow and sound wave damping. The fullset of linearized equations, which is solved as an eigenvalueproblem with γ being the sought value, reads [60]:

(γ+ ınΩ)ξ = v + (ξ · ∇Ω)R2∇Φ (42)

(γ+ ınΩ) v = −∇ · p +∇×Q×B +∇×B×Q+

− ρ[2Ω∇Z× v + (v · ∇Ω)R2∇Φ

]+

−∇ · (ρξ)R2Ω2∇Z×∇Φ

(43)

(γ+ ınΩ)Q = ∇× (v×B) + (Q · ∇Ω)R2∇Φ (44)

p = p‖bb + p⊥

(I − bb

)(45)

Where Ω is the fluid rotation frequency along the toroidal an-gle Φ. It is worth noting that the toroidal flowΩ represents thebulk ion rotation, which is the sum of the equilibrium E× Bflow and the ion diamagnetic flow. These components are notseparable in the standard single-fluid description on which thishybrid formulation is based. The perturbed quantities ξ, v,p,Qrepresent respectively plasma displacement, perturbed velocity,pressure tensor and magnetic field. The assumed form of allperturbations is A(s, χ,Φ, t) = A(s, χ)e−iωt−inΦ, in a curvilin-ear flux coordinate system (s, χ,Φ) where s is the normalized s ≡√

(ψ−ψaxis)(ψedge−ψaxis)

radial coordinate and χ the generalized poloidal angle. B is theequilibrium magnetic field. ρ is the unperturbed plasma den-sity, I the unit tensor and b = B/|B| is the equilibrium magneticfield unit vector. (R,Φ,Z) is the cylindrical coordinate systemfor the torus. The MHD-kinetic equations above are solved in

29

30 mars-k formulation

the just mentioned magnetic flux surface based curvilinear co-ordinate system, this is not an orthogonal system in generalsince the radial and poloidal coordinates are not orthogonalto each other. The generalized poloidal angle in particular de-pends on the choice of the Jacobian (referring to the transforma-tion between contra-variant and covariant basis). In the workpresented within the following chapter, an equal-arc coordinatesystem is used, for which dl/dχ =

√gχχ is constant on a flux

surface, where gχχ = eχ · eχ is a metric element.

Drift-kinetic effects are included in the Equations 42-45

through the perturbed pressure tensor components p‖ and p⊥.These terms, which are calculated with the following integralsover the particle velocity space Γ , replace the standard adiabaticclosure 5/3P∇v (where P would be the equilibrium pressure):

p‖e−iωt+inΦ =

∑e,i

∫dΓMv2‖f

th1 +

∫dΓMv2‖f

h1 (46)

p⊥e−iωt+inΦ =

∑e,i

∫dΓM

1

2v2⊥f

th1 +

∫dΓM

1

2v2⊥f

h1 (47)

Where the summations run over electrons and ions, M is theparticle mass, v‖ and v⊥ the particle velocity components re-spectively parallel and perpendicular to the equilibrium mag-netic field. The perturbed distribution functions for thermalparticles (fth1 ) and fast ions (fh1 ) are derived for each specie fol-lowing the approaches in [61] and [62], with the MARS-K imple-mentation being described in [32].

In the work described in this thesis, Chapter 9 in particular, aslowing down equilibrium distribution is assumed for fast ionsin the energy space and isotropic in the pitch angle. Althoughthis is a rough approximation for additional heating hot ions,it has been chosen as a starting point for assessing the effectof fast particles Resistive Wall Mode stability in the consideredscenarios. Following [63] and [64], the slowing down equilib-rium fast ions distribution can be described as:

fh0 =

C

ε3/2k +ε

3/2c

0 < εk < εmax,

0 εk > εmax.(48)

Where

εc =

(3√π

4

)2/3(Mh

Mi

)(Mi

Me

)1/3Te (49)

mars-k formulation 31

While the parameter C can be calculated through the fast parti-cles density profile Nh =

∫fh0dΓ as:

C = NhM3/2h A,

A =

(4√2π

∫10

ε1/2k dεk

ε3/2k + ε

3/2c

)−1

=

=3

8√2π

1

ln(1+ ε

−3/2c

) (50)

Where in the above equations Mi, Me, Mh are the masses ofions, electrons and fast particles respectively, Te is the electrontemperature, εk ≡ εk/εmax and εc = εc/εmax. The particle max-imum energy εmax is generally a function of the minor radiusand determined from the fractions Ph/Pth, Nh/Nth where h Ph,th pressure

components for fastions and thermalparticles

Nh,th densitycomponents for fastions and thermalparticles

stands for hot and th for thermal particles.

The energy analysis module included in MARS-K allows fordeeper understanding of RWM physics. Components of thequadratic energy form [7] [4] are calculated for both fluid andkinetic energy perturbations. In the present work we will focuson the analysis of the kinetic part of the perturbed potentialenergy, which is obtained from the parallel and perpendicularcomponents of the pressure tensor when multiplying Equation43 by the plasma displacement and integrating over the plasmavolume:

δWk =1

2

∫Vp

[1

Bp⊥

(Q∗‖ +∇Bε

∗⊥

)+ p‖κ · ε∗⊥

]JdsdχdΦ (51)

This is in general a complex quantity and can be written as:δWk = δW

rek + iδWim

k . Substituting this expression in the gener-alized dispersion relation for Resistive Wall Modes with kineticeffects [28][65]:

γτ∗w = −δW∞ + δWk

δWb + δWk(52)

It is the straightforward to obtain, by taking the real part ofEquation 52, the stability condition:

δW∞δWb+δWrek (δW∞ + δWb)+ (δWre

k )2+(δWim

k

)2> 0 (53)

Where δW∞ = δWF + δWv,∞ and δWb = δWF + δWv,b. WithδWv,∞ and δWv,b being the vacuum energy without and withan ideal wall at minor radius b respectively. δWF is the fluid po-tential energy. It follows that the imaginary part of the kinetic

32 mars-k formulation

potential energy always brings a stabilizing effect, while thereal part can be either stabilizing or destabilizing. The globaleffect has to be evaluated taking into account all the particlepopulations and resonances included in the applied mode. Us-ing the MARS-K MHD-kinetic hybrid self-consistent formulation,the drift-kinetic potential energy can be written as reported in[32]:

δWk =ν√π

2B0

∑e,i

∫dΨPe,i

∫dεkε

5/2k e−εk·

·∑σ

[∫dΛ∑l

λαl τb|〈e−i(l+αnq)ωbt−inΦHL〉l|2] (54)

Where B0 is the on-axis magnetic field, Pe,i the equilibriumpressures for electrons and ions, εk = εk/T the normalizedparticle energy, σ = sign(v‖), Λ is the pitch-angle and λαl themode-particle resonance operator. Summations run over parti-cle species (e, i), poloidal Fourier harmonics, trapped and pass-ing particles. The last factor, HL, is the particle perturbed La-grangian [32]. The parameters ν, α and τb depend on the spe-cific population. In general α = 0, ν = 1/2 for trapped particleswith τb = τb/

√M/2εk being the normalized bounce period

and α = 1, ν = 1 for passing particles, with τb being the nor-malized transit period.

4I N T R O D U C T I O N T O C O N T R O L T H E O RY

"A world is supported by four things ...... the learning of the wise, the justice of the great,

the prayers of the righteous and the valor of the brave.But all these things are nothing,

without a ruler who knows the art of ruling."

— Frank Herbert, Dune

The ideas of control theory, and feedback in particular, arenowadays important for many physical applications and exper-iments. Let us consider a physical system whose output is thestudied observable variable, the word control identifies the ac-tions aimed at driving its time evolution towards the desireddirection. In the feedback approach the output variable is com-pared with an externally applied reference and driven as tomatch this reference. In the discussion that follows a theoreti-cal background on the study of linear systems is given. Both thecontinuous and discrete time situations are considered becausemany modern control applications make use of a digital com-puter to generate the control input reference. The digital com-puter together with a digital-to-analog converter form what isusually called a digital controller.

4.1 state variable representation of a linear time

invariant system

Let us consider a generic system, with the point of view of con-trol theory, as the mathematical tool that models the dynamicsof a given physical process. Three kind of variables are usedin the present description: input variables (ui) represent the ex-ternal forces that influence the dynamics, output variables (yj)are connected to the physical observable and state variables (xq)represent internal characteristics of the system. Knowing xq at agiven time and ui for that time onward, the output variables yjand the remaining states are completely determined. Time canbe considered both continuous, defining the continuous linearsystem ui(t), yj(t), xq(t), and discrete, thus defining a discretesystem in which the state variables are sequences depending on

33

34 introduction to control theory

discrete time: ui(tk), yj(tk), xq(tk)[66]. Every system can havemultiple state variable representations but all of them have thesame number of states, i.e. the number of elements (n) in thestate vector is fixed for each system and is referred to as theorder of the system. State variables for a generic system are usu-ally defined by first order differential equations:

xq(t) = fq [~x(t0), ~u(t0, t)]

Output variables on the other hand are determined by the statevector ~x(t) and the input vector ~u(t):

yj(t) = gj [~x(t), ~u(t)]

These equations can be put into vector form to give the mostgeneric representation of a continuous system:

~x(t) = f [~x(t), ~u(t)]~y(t) = g [~x(t), ~u(t)]

(55)

A block representation of this system is given in Figure 6. Inthe case of a generic discrete system with evenly spaced timesteps (tk − tk−1 = T ) Equation 55 becomes:

~x(k+ 1) = f [~x(k), ~u(k)]~y(k) = g [~x(k), ~u(k)]

(56)

This discrete system is not given for the the sake of curiosity, in

Figure 6: State variable representation of a generic system

many modern applications digital controllers are involved suchis the case on RFX-mod. Computer based simulations are an-other example, every routine used in the present work is basedon discrete blocks that appear similar to Equation 56. To bemore precise, systems in the aforementioned routines are con-sidered to be linear, hence Equations 55 and 56 can be respec-tively put into the simple form:

~x(t) = F~x(t) + G~u(t)

~y(t) = C~x(t) + D~u(t)(57)

4.1 state variable representation of a linear time invariant system 35

~x(k+ 1) = A~x(k) + B~u(k)~y(k) = C~x(k) + D~u(k)

(58)

where in the latter A is called the n× n state matrix, B is then×m input matrix, C the p×n output matrix and D the p×minput-output matrix. This is a useful representation for the RFX-mod control system, in this case the input vector contains 192

signals that are fed to the power supplies and the output vec-tor the 192 signals sampled by the saddle coil sensors coveringthe internal surface of the shell. Such a system is also called aMultiple-Input-Multiple-Output (MIMO) system to remark thedifference with the Single-Input-Single-Output case. A simpli- Multiple-Input-

Multiple-Output(MIMO)

fied version of this MIMO system is obtained by dropping theD matrix, since the input vector hardly has any direct effect onthe output.

~x = A~x+ B~u~y = C~x

A convenient method for studying nth order linear systemsis analysis in the frequency domain by means of the Laplacetransform, defined by the following operator [67]: Laplace Transform

L [x (t)] = x(s) =

∫∞0x(t)e−stdt (59)

The Laplace transform can be related to the Fourier transformby imposing s = iω, it gives nevertheless a better approach tothe linear system under study because of the evolution timestarting from t = 0 instead of t = −∞ and also because ofthe capability of reducing an nth order differential equation inthe time domain to an nth order algebraic equation in the sdomain. This characteristic of the Laplace transform is given bythe following properties:

L

[dnx

dtn

]= snx(s)

L

[∫x(t)dt

]=1

sx(s)

these allow to write the input and output of a linear system aspolynomials in s, in particular the transfer function of the systemcan be defined:

G(s) =y(s)

u(s)= C (sI − A)−1 B (60)

36 introduction to control theory

Transfer functions are extremely useful objects to study lin-ear systems, describing the input-output relationship and alsoyielding information on stability. This description of linear time-invariant systems has some important advantages: as previ-ously stated it allows to substitute differential equations withalgebraic equations, the output of the system is easily obtainedby multiplication while time evolution is calculated through in-verse Laplace transform, furthermore complex system can berepresented with simple block diagrams.

4.2 feedback control

A block diagram of a hypothetical closed-loop system is givenin Figure 7. The controlled output y(t) is required to follow theinput reference r(t), thus the controller is fed with the so-callederror signal e(t) = r(t) − y(t), that is multiplied by the transferfunction K which applies the desired control law. Using theLaplace transform the global transfer function is obtained:

y(s) = K(s)M(s)G(s)e(s)

e(s) = r(s) −H(s)y(s)

therefore:

y(s) =K(s)M(s)G(s)

1+K(s)M(s)H(s)G(s)r(s) =

K(s)M(s)G(s)

1+ L(s)r(s)

where L(s) = K(s)M(s)H(s)G(s) is called loop gain and deter-mines the dynamics of the whole system. An equivalent blockdiagram for the open-loop case is given in Figure 8.

Figure 7: Block diagram representing feedback control of a systemwith transfer function G(s). The controller has its own trans-fer function K(s) that describes its dynamics. M(s) and H(s)are the transfer functions of the actuators and sensors re-spectively.

4.2 feedback control 37

Figure 8: Block diagram representing the signal flow from the inputu(s) to the output y(s) obtained with the transfer functionG(s).

The regulator applied by K(s) can follow three different ap-proaches: proportional (Kp), integral (Ki), differential (Kd) or a com-bination of these. On RFX-mod all three the techniques are com-bined in a Proportional-Integral-Derivative (PID) controller, itstransfer function in the s domain appears as the following:

K(s) = Kp +Kis

+Kds (61)

where Kp, Ki and Kd are in general complex numbers externallyapplied by the user. Varying these parameters will modify thesystem’s eigenvalues thus affecting stability. For the sake of sim-plicity the RWM control simulations carried out in this work, aswell as the analyzed experiments, make use of a purely pro-portional gain in most of the cases. Just a few examples willbe given of mode control with both proportional and integralregulators.

5M A G N E T I C C O N F I N E M E N T F U S I O NE X P E R I M E N T S

"We have no need of other worlds. We need mirrors.We don’t know what to do with other worlds.

A single world, our own, suffices us;but we can’t accept it for what it is."

— Stanisław Lem, Solaris

Aim of this Chapter is to give an overview of the experimentaldevices on which the studies in the present thesis have been car-ried out. The first to be described in Section 5.1 if the RFX-modexperiment, operating in Padova up to 2015 and now being up-graded. The second part, Section 5.2 is dedicated to the descrip-tion of JT-60SA, which is being built in Naka (Japan) within theframework of the Broader Approach agreement between Japanand the European Union.

5.1 reversed field experiment

The Reversed Field eXperiment (RFX-mod) is a medium size (R= 2 m, a = 0.459 m) toroidal device that has been operating inPadova since 2004. Originally designed as an RFP experiment[68], it is capable of confining plasmas also in the Tokamak con-figuration [3]. It is equipped with a state-of-the-art system foractive control of MHD instabilities [69]. Such a system, operatingwith a cycle frequency from 2.5 to 5 kHz, is composed of 192

independently fed actuators (saddle coils arranged into a 4× 48grid), located on the outer surface of a stainless steel supportstructure at r=0.6 m, and a considerable set of magnetic sen-sors counting 192 bi-axial pick-up probes (measuring toroidaland poloidal B components) and 192 saddle sensors for the ra-dial component[70] . Each active coil can be fed a maximumcurrent of 400 A, providing a DC radial field of 46 mT at thevacuum vessel surface. The radial component of the magneticfield in particular, is measured by 192 saddle probes whosepositions correspond to the projection of the actuators on theouter surface of the vacuum vessel, at minor radius r=0.505 m.The high degree of flexibility of the control system, working in

39

40 magnetic confinement fusion experiments

a bi-dimensional Fourier space characterized by the poloidal(m) and toroidal (n) mode numbers, allows to switch on oroff each single coil. Many control schemes can thus be easilyimplemented[36] and the controller has the capacity to inde-pendently act on selected modes with different sets of coils. Arepresentation of RFX-mod with its most important active andpassive structures is given in Figure 9. The device is currentlyundergoing a major upgrade which will involve both the vac-uum vessel [71] and the active system. The main technical spec-ifications of the latest operating machine are reported in Table1. The copper shell which can be seen in Figure 9 is 3 mm

Minor radius (a) 0.427 mMajor radius (R) 2 mPlasma current 6 2 MA

β 10%Te 1 keV

Applied toroidal field < 0.55 TMagnetic circuit flux 15 Wb

Table 1: RFX-mod technical specifications.

Figure 9: Graphical representation of RFX-mod active coils (green),passive support structures (grey), copper shell (orange) andvacuum vessel (yellow).

thick and has a vertical field penetration time of τshell = 50 msthat allows feedback stabilization of MHD modes, given thatthe discharge duration time scale is one order of magnitudelonger. The shell has two gaps respectively along the poloidaland toroidal directions, an exploded view is given in Figure 10.These gaps in the shell are technically necessary but are also

5.1 reversed field experiment 41

a source of error fields produced by eddy currents. The torusis entirely covered with the 192 active saddle coils, Figure 12a,arranged in 4 toroidal arrays and 48 poloidal sections. The sad-dle coils are located outside the copper shell and mounted on astainless steel toroidal support structure. Each coil is indepen-dently fed, carries a maximum current of 400 A producing alocal maximum radial field of ∼ 50 mT . The magnetic measure-

Figure 10: Exploded view of the copper shell.

ments used for mode control are supplied by dedicated probes,for each component of the magnetic field (poloidal, toroidal,radial) an array of 192 sensors is present. The poloidal andtoroidal components are measured by pick-up probes while theradial field is obtained with saddle coil probes located undereach of the previously described active control coils. These sad-dle probes have therefore the same periodicity of the active onesand span the same poloidal and toroidal angles, measuring theaverage radial field over the probe area. The whole system iscontrolled by a PID controller that works both in the real space,applying gains to field signals, and in the (m,n) Fourier spacewhere gains are applied to Fourier harmonics ranging betweenn = [−23;+24] and m = [−1, 0,+1,+2]. This actuator systemproves to be very flexible to experimental needs, allowing boththe usage of a reduced set of coils and of the so-called super-coils, i. e.a set of neighboring coils fed in a way so that theyresemble a single bigger coil (Figure 11). Three more sets of

Figure 11: Graphical representation of two neighbouring coils usedas a single one.


coils contribute to generate the magnetic configuration. The 48Toroidal Field Coilstoroidal field coils (Figure 12c) wrap the whole torus and gen-erate a maximum 0.55 T toroidal magnetic field. These coils aregrouped in 12 blocks of 4 coils each independently fed. TheOhmic Heating

Windings magnetizing coils (Figure 12d) provide the time varying mag-netic flux that gives rise to the plasma current. A maximum of50 kA flowing in these coils produce a 15 Wb flux. A secondField Shaping Coilspoloidal field is generated by the so-called primary windingand it controls plasma position during the discharge (Figure12b).

(a) (b)

(c) (d)

Figure 12: (a) 192 actuators of the active control system, (b) field shap-ing winding, (c) 48 toroidal field coils, (d) Ohmic heatingwinding.

5.1.1 RWM control with a reduced set of coils

The number, shape and positioning of control coils has a directimpact on stability properties of RWMs. In principle varying theactive coil configuration can trigger different plasma responses,since the externally applied field influences both the feedbackcontrol chain and plasma behavior. Amplification of otherwisestable modes is in fact possible through the side-bands of theexternal field and depending on the target plasma this effect


might have important consequences on stability. A hypotheti-cal large and dense coil array covering the whole torus wouldintuitively be more powerful that a few sparse coils in termof control capabilities and side-band effect reduction. Very bigarrays though might be difficult to obtain because of realiza-tion and maintenance costs, this is just one of the reasons whytesting different control schemes is crucial in sight of the com-mercial exploitation of fusion energy. Changing active coils con-figuration is a very difficult task if working on the hardware,RFX-mod on the other hand allows software downgrading thecontrol scheme thanks to the real time controller. This flexiblesystem is also capable of applying different control sequencesto selected harmonics, for example in a hypothetical RFP con-figuration experiment the unstable m = 1 n = −6 harmoniccan be controlled with the desired set of coils and gains whilethe full array still works on other harmonics. Extensive stud-ies have been carried out in dedicated experimental sessions tocomprehend the effect of control downgrading on plasma sta-bility[36]. As previously mentioned, an interesting feature ofthe real time control system is the possibility of averaging thecurrents in two or more neighboring coils to make them act assingle larger super-coil, thus opening many other possible con-figurations. A set of modeling tools have been developed to testdifferent coil geometries, good agreement has been obtainedwith experimental data, as reported in Figure 14, which allowsfor predictive simulations. The same tools will be extended andused in the following parts of this thesis.


Figure 13: Rectified representation of some sample control configura-tions, on the right side the number of active coils alongthe toroidal (φ) and poloidal (θ) directions, the total num-ber of active coils, the number of supercoils and the areacoverage expressed as percentage.


Figure 14: 1 × 16 configuration is made of 16 super-coils along theouter toroidal array. Model results (blue) are compared toexperimental data (shot 33977, red). Amplitude of (1,-6)RWM (top) and coil currents for the same harmonics (bot-tom) are shown. Feedback starts at 0.16s.


5.2 jt-60sa

The superconducting device Japan Torus Super Advanced(JT-60SA) [72][73] is being built in Naka (Japan) and has animportant supporting mission for the development of fusionenergy: designed to achieve long pulses (100 s) and break-even equivalent plasmas, using superconducting poloidal andtoroidal field coils. It will help in both the exploitation of ITERand in solving key issues for the future DEMO devices [74]. JT-60SA will be able to explore plasma configurations with shapefactor up to S = q95Ip/(aBφ) ∼ 7 (where Bφ is the toroidalfield, Ip the plasma current in MA, a is the minor radius, q95the safety factor at 95% of the toroidal flux) and aspect ratiodown to A 2.5. The maximum plasma currents are 5.5 MAin a low aspect ratio, inductive, configuration and 4.6 MA inthe ITER-shaped configuration. The device will also pursuefully non-inductive steady-state operation with high β plasmasexceeding the no-wall ideal MHD stability limits. Additionalheating and current drive systems will provide up to 41 MWfor 100s, divided between 34 MW neutral beam injection (10

MW of 500 keV N-NBI and 24 MW of 85 keV P-NBI)1 and 7

MW of ECRF2. The off-axis N-NBI at 0.5 MeV beam energy inparticular, allows current profile tailoring for Advanced Toka-mak scenarios with fully non-inductive current drive. Figure 15

shows a cutaway of the JT-60SA device with the main externalheating and current drive systems. The most important goal ofJT-60SA in view of a DEMOnstration Power Station (DEMO) reac-tor concept is to demonstrate the sustainability of steady-stateoperation with integrated maximization of energy confinementfactor, normalized beta βN, bootstrap current fraction, plasmadensity normalized to the Greenwald density, fuel purity andradiation power. JT-60SA will in principle allow steady-state op-eration with the full exploitation of 10 MW/500 keV tangentialN-NBI current drive and 7 MW of Electron Cyclotron CurrentDrive (ECCD). In the most ambitious Advanced Tokamak sce-nario the 2.3 MA of plasma current will be fully non-inductivelydriven, with a normalized beta βN = 4.3.

JT-60SA will play a very important role in MHD stability andcontrol studies. It will explore the kinetic stabilization effect of

1 Negative Neutral Beam Injector (N-NBI), Positive Neutral Beam Injector(P-NBI)

2 Electron Cyclotron Radio-Frequency (ECRF)

5.2 jt-60sa 47

Figure 15: Cutaway view of the JT-60SA machine with external heat-ing and current drive systems

energetic particles from 500 keV N-NBIs, investigating RWM sta-bility for both ITER and DEMO relevant high βN plasmas. ECCDwill allow the validation of sawtooth control strategies for ITERhybrid scenarios, as well as real-time stabilization schemes ofm/n = 2/1 and 3/2 Neoclassical Tearing Modes (NTMs). Fur-thermore, JT-60SA will determine MHD stability boundaries byexploring RWM stabilization with active coils, sustaining longpulse high-βN plasmas. The feasibility of this kind of opera-tion depends on the controllability if disruptive instabilitiessuch as RWMs and NTMs. JT-60SA is equipped with an in-vesselStabilizing Plate (SP), made of SS316L and composed of twoplates (1 cm thick) connected by ribs, for a total thickness of9 cm. On the SP, Resistive Wall Mode Control Coils (RWMCC:3 poloidal × 6 toroidal) are located, as well as Fast PlasmaPosition control Coils (FPPC) and Error Field Correction Coils(EFCC: 3 poloidal × 6 toroidal). All these components areshown in Figure 16. As far as RWMs are concerned, one themain goals is to develop effective feedback control strategies,along with investigating the combined effect of plasma rotation.Flexibility in tailoring rotation profiles is provided by variouscombinations of NBIs. In addition, high energy N-NBIs will al-low to simulate burning plasmas where MHD instabilities arepredicted to interact with high energy particles. A significantdegree of flexibility in RWM studies is given by the feedback


Figure 16: 3D view of the SP, Fast Plasma Position control Coils(FPPC), Resistive Wall Mode control Coils (RWMC: 3

poloidal × 6 toroidal) and Error Field Correction Coils(EFCC: 3 poloidal × 6 toroidal)

control system, particularly the RWMCC power supply whichwill have a very fast dynamic performance. Furthermore eachcoil will be fed by an independent inverter [75]. The main rel-evant features of this system, for the subject of the followingChapters, are given in Table 2. The RWMCCs, carrying a maxi-mum current of 2.2 kAT each3, will be installed on the plasmaside of the SP. This will allow to minimize the shielding effectof the plate on the high frequency magnetic field modulationrequired by RWM control. With the mode control concept, stabi-lization studies on n=1,2 and 3 simultaneously will be possible.

Number of coils 3 poloidal × 6 toroidalNumber of turns per coil 8 turns

Max. peak current (sinusoidal)a 300 ABandwidth of the current at -3 dB 3 kHz

Maximum latencyb 50 µs

a Maximum peak current per coil turn.b Between the rise of a step reference and the change in output voltage.

Table 2: Main requirements of the JT-60SA RWM control system

3 kilo-Ampere-Turn

Part II

R F X - M O D C O N T R O L S Y S T E MO P T I M I Z AT I O N

6A C T U AT O R - S E N S O R D E C O U P L I N G O NR F X - M O D

"The wise man will be as happy as circumstances permit, and if hefinds the contemplation of the universe painful beyond a point, he

will contemplate something else instead."

— Bertrand Russell, The Conquest of Happiness

1A series of techniques are presented that have been developedto optimize the output magnetic field of the feedback controlsystem on the RFX-mod reversed field pinch device. With the aimof minimizing the harmonic distortion and correcting localizederror fields, these methods should be lightweight for real-timeapplication and effective in improving the performance of a sys-tem that is routinely used for active control of magneto-hydro-dynamic plasma instabilities. The implementation of simple,linear algebra based, real-time optimization methods will be de-scribed along with proof of the sought beneficial effects. Focusof the work is set on a spurious harmonics reduction techniquebased on the decoupling of sensors and actuators, a descriptionof its derivation will be given together with the implementationin the control loop. A similar procedure for the compensationof faulted actuators will also be mentioned.

6.1 the coupling problem

In a control system such as the one described in Section 5.1,each of the 192 actuators will be coupled with the directly un-derlying saddle sensor and its neighbors. The magnetic fieldproduced by single active coils will therefore influence the sur-rounding measurements, leading to a source of error fields oncethe feedback loop is closed. Performance improvements can besought by making use of the coupling terms (i. e.mutual induc-tances) between actuators and sensors. Given the presence ofpassive structures in between, these coupling terms depend

1 The content of this Chapter is: Pigatto, L., et al. "Control System Optimiza-tion Techniques for Real-Time Applications in Fusion Plasmas: the RFX-mod Ex-perience" submitted for publication to IEEE-Transactions on Nuclear Science andPigatto, L., et al. 41St EPS Conference on Plasma Physics, Berlin. P5.080

51

52 actuator-sensor decoupling on rfx-mod

on frequency and on the local characteristics of the passives(e. g.cuts, gaps etc.). In the following, the term decoupling willbe used to define a family of techniques devoted to makingeach actuator aware of the existence of the neighboring sen-sors. Decoupling based optimization techniques have been de-veloped with different concepts, such as the so-called dynamicdecoupler [76] which relies on off-line pre-computed currentdistributions. In the present work we will concentrate on a dif-ferent, lightweight method that can be applied real-time takingaction on all the modes of the Fourier spectrum for a fixedfrequency. Dedicated measuring campaigns have been carriedout on RFX-mod, leading to the development of a full electro-magnetic model of the MHD modes active control system[77].Mutual coupling matrices between active coils can be derivedfor a given set of frequencies from the ratio between coil volt-age (Vi) and coil current (Ik) phasors. The resulting complexnumber can account for the effects of passive structures on themagnetic flux:

Vi(ω)

Ik(ω)= Zik(ω) = Zik(ω)ejφik(ω) = Rik + jωLik(ω) (62)

where the dot is used to represent complex matrices. The sameprocedure can be followed to measure the coupling terms be-tween actuators and sensors, only changing the voltage phasorwith the flux measured by each saddle sensor.

Φi(ω)

Ik(ω)= Mik(ω) (63)

Once the response of the saddle sensors has been measured forthe desired frequencies, full information on mutual couplingscan be used to build a full state-space electromagnetic model.This black-box model derived from experimental data can beused to develop simplified strategies for improving the qualityof harmonic content in output magnetic fields, taking into ac-count the effect of toroidal geometry and coil-sensor couplings.Another way to proceed in challenging the decoupling prob-lem, is making use of existing three-dimensional electromag-netic codes to obtain a first-principles description of the coilsand passive structures. This has been done for the RFX-mod de-vice with the CARIDDI code. From this second model, whichwill be described in Section 6.2, another state-space representa-

6.2 the model 53

tion of the system is obtained from which the following transferfunction can be calculated:

K(ω) =[C(Ijω−A)−1B+D

](64)

where standard state-space notation is considered with thematrices A,B,C,D. By inverting the transfer function K(ω) fora given frequency, a matrix is obtained that can be used to ac-count for the couplings and reduce the magnetic field harmonicdistortion. Various matrices have been obtained with both theaforementioned methods and applied in simulations (detailson the simulation tools are given is Section 6.2) and experi-ments. Some examples of these matrices are reported in Fig-ure 17, from which the different magnitude of coupling can beappreciated with varying frequency and between experimen-tal or model based calculations. Figure 17a and 17b representthe decoupling matrices obtained from experimental data forω = 0, 10 Hz, while Figure 17c is the inverse of the CarMatransfer function in the infinite frequency limit. This latter par-ticular case is also called D-matrix, since it is calculated from theonly surviving part of Eq.64 after the ω → ∞ limit. All thesematrices are normalized to their maximum value and they willbe used, as specified in Section 6.2, to recombine the currentreference pattern fed to active coils.

6.2 the model

Modeling tools have been fundamental for developing and test-ing decoupling strategies. The dynamic simulator will be hereintroduced along with the vacuum validation for one particularcase.

6.2.1 Dynamic Simulation Tool-Set

A dynamic simulator has been developed as a Matlab®Simulinkapplication to allow the simulation of a full control loop andthe test of the optimization techniques described in the presentpaper. The core of this flight simulator is a state-space represen-tation of the system (Plant) given by the CarMa code, which isintegrated with a complete representation of the digital controlsystem. This allows for a realistic description of the passive andactive structures with the effects of 3D geometry on plasma dy-namics. The active system can be simulated at its full potential


(a) (b)

(c)

Figure 17: (a) Experimental decoupling matrix calculated for ω = 0,i.e. step-like request. (b) Experimental decoupling matrixcalculated for ω = 10Hz. (c) CarMa derived decouplingmatrix representing the infinite frequency limit. The colorscales represent the actuator-sensor couplings, all matricesare normalized to their maximum value. Saddle sensor andactive coil numbers are given on the y-axis and x-axis re-spectively.

6.2 the model 55

or downgraded to a reduced number of coils with virtually anygeometry. The CarMa code itself is the result of coupling a 3Deddy current problem integral solution (given by the CARIDDIcode) with a 2D linearized MHD solution, which results fromthe MARS-F code[40]. Instantaneous plasma response to a givenperturbation is calculated on a coupling surface (S) and usedto evaluate the effect of the 3D structures on the plasma. Thecurrents induced by the plasma in these structures are com-puted using an equivalent surface current distribution on S,such that it generates the same magnetic field as the plasmajust outside S. The model used for the dynamic simulator inparticular, includes a fully 3D description of the resistive cop-per shell surrounding the vacuum vessel (3 mm thick withτV = 50 ms vertical field penetration time) as well as includingthe active system components. The copper shell itself is thefundamental component in plasma stability applications, sinceit slows down to its characteristic penetration time some ofthe fastest growing MHD instabilities (i.e. ideal kinks),thus al-lowing the possibility of feedback control[34]. The flexibility ofthe RFX-mod digital controller allows to apply different controlstrategies, the most common one being the CMC outlined in Sec-tion 2.5. This method allows independent action each harmoniccomponent of the discrete Fourier transform of the radial mag-netic field. A real-time algorithm is therefore applied to removethe aliasing error from the magnetic field harmonic componentmeasurement, caused by high order sidebands produced bythe 4× 48 discrete grid of saddle sensors. A block scheme ofthe system described above is given in Figure 18, both for theopen-loop case (Figure 18a, in which the sidebands cleaningis not applied) and closed-loop (Figure 18b). An exploded di-agram of the control system block can be seen in Figure 18c,in which the application of the decoupling based on staticmatrices is represented, as described in Section 6.1. The mainpart of the Plant block is the state-space representation of theCarMa model, which is implicitly preceded by the saddle coilcurrent controller. Finally a state-space filter has been added tothe Plant block, also in series with the main model, in order tomimic the effect of the stainless steel vacuum vessel, with a 5

ms time constant.


(a)

(b)

(c)

Figure 18: (a) Open loop diagram. (b) Closed loop diagram withclean-mode-control. (c) Exploded control system blockwith decoupler.

6.2 the model 57

(a)

(b)

Figure 19: Experimental and model time evolution of m=1, n=-6 har-monic component of the radial field (raw) (a) and saddlecoil currents (b). Simulation and data from shot 33863.


6.2.2 Validation of the Vacuum Model

The main goal to be achieved with the optimization proceduresdescribed here is the correct assessment of the system externalaction on a given plasma. In order to do so, all the experimen-tal sessions dedicated to the evaluation of mutual couplingshave been carried out in vacuum. The same has been done forthe modeling activity, a vacuum version of the CarMa modelhas been used and benchmarked against experimental data interms of harmonic content of the output magnetic field. Thevery good agreement of simulated B field amplitude and cur-rent with experimental data can be seen in Figure 19. Dur-ing this open-loop vacuum shot the active system has beenrequested a particular perturbation with m=1 and n=-6 modenumbers. The same shot can be reproduced as a dynamic sim-ulation. The amplitude of the (1,−6) component of the outputfield is shown for both cases in Figure 19a, where the raw datais used, indicating that the sideband cleaning procedure hasnot been applied for the particular case. Figure 19b shows thecurrent (1,−6) harmonic component, confirming that the de-veloped simulation tools can successfully represent the real vac-uum experimental setup. Figure 19 is not only showing the gen-eral agreement between simulation and measurements but alsothe capability of the software tools of reproducing a very pecu-liar configuration. Specifically, during shot 33863 a reduced coilarray has been implemented, with the (1,−6) magnetic pertur-bation generated by only 12 non-equispaced active coils alongthe outermost toroidal array. This leaves a characteristic patternin the produced sidebands which can be appreciated in Figure20. The amplitude of each harmonic is given in the bar-plot,normalized to the desired (1,−6), at a given time instant of theflat-top phase. It is worth mentioning that the harmonic pat-tern is given by the peculiar periodicity of the active array andit is correctly reproduced by the open-loop simulation. With anevenly distributed set of coils the expected toroidal sidebandswould have followed the relationship: n′ = n0 ± kN. Wheren0 is the desired harmonic (here n0 = −6), N the number ofcoils (here N = 12) and k an arbitrary integer. It follows that fortwelve equispaced coils the pattern would be n′ = +6,±18. Thecoil geometry in Figure 20 however, has a characteristic patternthat resembles that of two groups of six coils each. These areequispaced intra-group but not inter-group. The effect of this

6.3 open loop operation with decoupling matrices 59

superposition is that the sidebands of an N = 6 configurationare produced (i. e.as if six coils where used).

Figure 20: Comparison between model and experiment relative har-monic content of the radial magnetic field (raw) in shot33863. Amplitudes taken in steady phase at t=0.17 s

6.3 open loop operation with decoupling matrices

The first set of matrices generated from both experimental mea-surements (as described in Section 6.1) and the CarMa code(Section 6.2) has been applied for the open-loop generation ofmagnetic field perturbations of varying amplitude and phase.The Total Harmonic Distortion (THD) of the output fields has Total Harmonic

Distortionbeen used in order to evaluate the effectiveness in spurious har-monics reduction.

THD =

∑i B

2i −B

2target

B2target(65)


where the index i runs on all the harmonic components of thefield signal. An example of this total harmonic distortion analy-ses is reported in Figure 21, in which two different decouplingstrategies have been used for the generation of a step-like per-turbation. The red dashed line shows the reduction of spuriousharmonic content when the specific model-derived matrix isused, with respect to the blue solid line (identity matrix, i.e. nodecoupling applied). The green dotted line shows a degrada-tion of the harmonic content instead, this is due to the peculiarapplied matrix, obtained through an infinite frequency limitfrom Eq.64. Because of this limit, the so-called D matrix doesnot work well for a step-like request during the steady phase.On the other hand its effect can be appreciated during the fasttransient phase, as in Figure 21b, when the B field is increasingrapidly and the infinite frequency limit holds, during the first30 ms in particular. The smaller harmonic distortion obtained

(a) (b)

Figure 21: (a) Total harmonic distortion from simulations with D ma-trix, ω = 0 Hz matrix and identity matrix. (b) Zoom of thefirst 50 ms of the transient phase in which the effect of theD matrix, i.e. infinite frequency limit, can be appreciated.Amplitudes are normalized to the (1,−6) component.

with a given decoupling matrix is reflected in the amplitudeof the B field spurious harmonics, this can be appreciated fortwo different cases in Figure 22. For the step-like magnetic per-turbation (Figure 22a) a simple decoupling matrix helps in re-ducing the amplitude of the (m = 0, n = 6) component, thevery same result has been obtained in dedicated vacuum ex-periments[78]. Another matrix has been used with the dynamicsimulator for a 10 Hz perturbation, resulting in a reduced spuri-

6.4 closed loop and compensation 61

ous harmonics relative content (Figure 22b). It should be notedthat the matrices applied in the aforementioned reference cases(represented respectively in Figure 17a and 17b) are providingthe best correction during the steady phase of the perturba-tion, while during the fast transients the infinite frequency limitholds as previously discussed. It is also worth noting that Fig-ure 17b shows the amplitude oscillating around its mean valueat approximately 20 Hz. A complete explanation of this behav-ior is beyond the subject of the present Chapter, it is ultimatelycaused by the spatial variance of the system (e.g. toroidal ge-ometry, gaps, local features etc.) and can be explained with theUnified Signal Theory[79] formalism.

6.4 closed loop decoupling application and miss-ing coil compensation

Once the exact behavior of the active system has been assessedby generating open-loop perturbations, the same can be donewithin the feedback loop. While the results to be expectedare still related to reducing the harmonic pollution duringsteady phases, attention should also be paid to the possiblegain changes introduced by recombining the reference currentdistribution. The static matrix shown in Figure 17a has beenincluded in the feedback loop during dedicated experiments,allowing the verification of expected effects and validationof the modeling results. Figure 23 shows the amplitude of a(m = 1, n = −6) perturbation together with its poloidal side-bands. A reduction of the m = 0 component is found with thenon-diagonal decoupling matrix. Furthermore, since the ampli-tude of the requested n = −6 component is the same in bothcases, the gain substantial invariance with application of thisparticular decoupling matrix can be acknowledged. The impor-tance of this last matter lies in the fact that no change in thegain matrix has to be made if decoupling has to be switchedon or off during experimental campaigns. The idea of usinga static matrix to recombine the active system input currentdistribution has also been applied to a set of problems notstrictly related to decoupling sensors and actuators. Given thepossibility of implementing various active coil geometries, asmentioned in Section 6.2.2, a systematic study has been carriedout to investigate the effect of one or more faulted actuators onthe produced magnetic fields. This has led to the developmentof a set of matrices that while still aiming at the improvement


(a)

(b)

Figure 22: Poloidal harmonics produced for (1,−6) reference. (a)Step-like perturbation with no decoupling (dashed red)and with CarMa ω = 0 Hz matrix applied (solid blue).(b) For a 10 Hz perturbation the dedicated decoupler iscompared with the identity matrix case. As expected thedecoupling effect can be appreciated in the reduction ofthe (0, 6) and (1,+6) amplitudes, in this case a normaliza-tion to the maximum of the requested harmonic has beenapplied to highlight the effect.

6.5 conclusions 63

Figure 23: Step-like (m = 1, n = −6) perturbation generated ina closed-loop with purely proportional gain on the re-quested harmonic only. Comparison between shots 27294

and 27295.

of the harmonic content, are specifically designed for a systemwith sparse missing coils. The procedure, that can be found in[80] and in Chapter 7 of the present thesis, has been proven asa simple and fast option for faulted coil compensation that al-lows a local reconstruction of the magnetic field and reductionof spurious harmonics.

6.5 conclusions

This part of the work has summarized many efforts made forthe complete characterization of the RFX-mod active systemin terms of output magnetic field, in particular the actuator-sensor decoupling problem has been considered. Starting fromthe first experimental measurements the development of sim-ple, real-time applicable strategies to obtain almost monochro-matic fields has been described. While the (m = 1, n = −6)

perturbation has been mainly shown in the previous sections,the same can be done for any desired B harmonic. The impor-tance of a fast, reliable and extensively benchmarked modelingtool has to be highlighted. Each of the proposed decoupling ma-trices, as well as the whole missing coil compensation strategy,


has been developed with full dynamic simulations before test-ing in dedicated experiments or implemented in plasma shots.The effects of actuator-sensor decoupling has thus been inves-tigated and represented mainly by the reduction of spuriousharmonics when a particular one is requested. With the intro-duction of the total harmonic distortion the two limiting cases,ω = 0 and ω → ∞, have been identified. The high frequencybehavior during transient phases has been assessed with the im-plementation of the so-called D−matrix while two exampleshave been reported to show the m = 0 amplitude reductionwhen a specific decoupler is used. Finally, the closed-loop ap-plication has been reported with proof of the expected effect onthe B components and of the negligible influence of the currentmodification on the applied gains. The concept of sparse miss-ing coils compensation using a static matrix has been brieflymentioned.

7M I S S I N G C O I L C O M P E N S AT I O N O N R F X - M O D

"All human knowledge begins with intuitions, proceeds from thenceto concepts, and ends with ideas."

— Immanuel Kant, Critique of Pure Reason

1In many devices aiming at magnetic confinement of fusion rel-evant plasmas, feedback control of MHD instabilities by meansof active coils is nowadays mandatory to ensure the robustnessof high performance operational scenarios. As described in Sec-tion 5.1, actuators in this system are 192 active saddle coils en-tirely covering the plasma outer surface, while more than 600

magnetic sensors are included in the control loop, providingthe operator with a challenging coupling situation. The RFX-modactive saddle coils are mounted on a stainless steel supportstructure to cover the whole surface of the torus and organizedinto a 4 × 48 (poloidal × toroidal) array. The same geometryof these coils, which gives the uniform magnetic boundary, isreproduced on the outer side of the vacuum vessel where 192

saddle sensors measure the radial component of the magneticfield. Saddle coils will be numbered with two indices (i, j), theformer for poloidal and the latter for toroidal positions. Thishardware configuration allows the implementation of highlyflexible real time control strategies, action is carried out in thebi-dimensional Fourier space characterized by the poloidal (m)and toroidal (n) mode numbers. The coupling of the actuator-sensor system is a frequency dependent problem, intrinsically3D when considering the effect of non-uniformities of the ac-tive RFX-mod magnetic boundary such as the implementationof reconfigured arrays [36] or the fault of one or more activecoils. A decoupler should be in principle frequency dependentand thus realized as a convolution integral of the input signaland a time dependent function, the system impulse response.Given the difficulty of implementing a frequency and time de-pendent decoupler in real-time operation, the choice of usinga static matrix is by far more feasible. These matrices are cal-

1 Part of the content of this Chapter has been published in: Pigatto, L., et al."Optimal strategies for real-time sparse actuator compensation in RFX-mod MHDcontrol operations." Fusion Engineering and Design 96 (2015): 690-693.

65

66 missing coil compensation on rfx-mod

culated from actuator-sensor mutual inductances at a given fre-quency and have been successfully applied in both experimentsand simulations [77], as also reported in Chapter 6. The aim ofthe this part of the work is to study the local effect of a non-uniform boundary on the magnetic field produced by the ac-tive coils, introducing possible compensation methods whichtake the actuator-sensor coupling as a starting point.

A numerical plant model obtained from the CarMa code[81] is used for dynamic simulations in which different exper-imental setups and decoupling strategies can be tested[82][63].This model is described in Section 7.1. The baseline situationin RFX-mod is documented, where the Identity matrix is chosento represent the simplest case of mutual coupling matrix. Theproblem of missing or broken actuators is introduced and tack-led with dedicated compensation strategies. A detailed descrip-tion is given for a possible compensation concept which canbe applied in real-time operation thanks to its implementationstrategy, yielding very promising results in terms of local fieldreconstruction, as reported in Section 7.2. As it will becomeclear in the following, the problem of current request from ac-tuators is not negligible with compensation strategies that willbe tested. A study of these requests has been carried out and isreported in Section 7.3 along with modeling and experimentaldata as documentation of the local effect of missing actuatorson magnetic field geometry and localized first wall heating. Astatistical study is presented in order to evaluate how coil com-pensation affects this latter issue.

7.1 model

The same dynamic simulator introduced in Section 6.2 has beenused for the development and testing of the actuator compen-sation, given the strong relationship with the technique heredescribed with the one outlined in Chapter 6. The open-loopsystem is represented as a block diagram in Figure 24. Thecompensation block represents the matrix multiplication thatis used to redistribute the 192 current references on the sad-dle coils in order to take into account all the actuator-sensorcouplings and geometrical characteristics. Such a matrix canbe obtained by inversion of either an experimentally measuredmutual inductance matrix or the plant transfer function. The

7.1 model 67

Figure 24: Block diagram of open-loop dynamic simulator

latter can be obtained from the state-space representation ma-trices A,B,C,D:

K = C(iωI −A)−1B+D (66)

where ω is the frequency and I the identity matrix. The appli-cation of these matrices has already been seen useful for the re-duction of vacuum field harmonic distortion [77][78]. The twolimiting cases of zero and infinite frequency in particular can betaken as a starting point for the application of a similar strategyto a situation in which one or more actuators are either brokenor disabled. The basic idea is to generate, from the initial refer-ence currents Ir, a new current distribution Ir that allows thereconstruction of the desired field pattern Br

Br = S−1KIr = S

−1KIr (67)

where S represents the area of saddle sensors, K is calculatedfrom Eq. 66 and K is a new matrix in which the missing ac-tuators are accounted for. The new current distribution mustsatisfy:

Ir = K−1KIr (68)

The product K−1K is used as compensation block in the diagramrepresentation in Figure 24. The new matrix K can be calculatedstarting from either experimentally measured actuator-sensorcouplings [78] or by taking the model transfer function K. Theelements corresponding to missing actuators are then removedso to obtain a matrix with N − j non-zero columns, where Nis the total number of actuators and j are the missing ones. Apseudo-inverse is finally applied to obtain K−1. An example ofthe resulting matrix is given in Figure 25, where the ω = 0

case is represented in particular. The non-zero columns corre-sponding to missing actuators should be noted together withtheir null value on the diagonal, this means that the coil itselfdoes not contribute to the locally measured field. Instead, be-cause of the non-zero non-diagonal elements, the field initially


requested to missing coils will be compensated by the workingones with the matrix product in Eq. 68.

Figure 25: K−1K matrix for the case in which actuators(3,02)(3,09)(3,22) and (4,26) have been disabled. Rowsrepresent actuators and columns saddle sensors, eachpoint in the matrix is a coefficient for current redistribu-tion.

7.2 magnetic field reconstruction results

The full open-loop model of the system has been validated bycomparison with experimental values. Various poloidal posi-tions have been considered for comparison, an example is givenfor the i = 16 array in Figure 26. Ad-hoc calibration of the spe-cific shot has been introduced, consisting in a rescaling of thesimulated current reference to match the average of the mea-sured one. The resulting agreement with experimental valuescan be considered adequate for the purpose of evaluating theeffect of a missing coil on local fields. The effect of a miss-ing coil in the control system can be appreciated in Figure 27

(dashed line). The sensor corresponding to position (3,09) mea-sures the field produced by the neighboring coils only, whilethe left and right first neighbors see an increased radial field.This local error field can be corrected with the introductionof the strategy described above, leading to very good mitiga-tion of the missing coil effect. As can be noted from the graph-ical representation of the implemented matrix in Figure 25, allthe other active coils are involved in compensating the miss-ing ones, making the problem of assigning proper weights non-

7.2 magnetic field reconstruction results 69

Figure 26: Comparison between measured radial field (solid line) andcalculated radial field (dashed line) from input currentmodel for the 16th poloidal array.

negligible. The pseudo-inverse method described in Section 7.1allows for the coils closer to the missing one to have a majorrole in the task, thus avoiding non-realistic demands to furtherdistant and weakly coupled actuators. Interestingly enough, theapplied actuator compensation strategy does not drastically af-fect the final Fourier spectrum of the produced magnetic field.The effect of compensation on the harmonic content can be ap-preciated in Figure 28 where a comparison is given in termsof Fourier harmonic amplitudes normalized to the ideal all-active-actuators case. The most important information that canbe extracted from Figure 28 is that the compensation methoddoes not dramatically alter the desired harmonic content withrespect to the ideal situation, it can instead deliver a slight im-provement in terms of sideband amplitude reduction with re-spect to the not-compensated corresponding case. The price topay for the missing actuators though lies in the current redis-tribution that the strategy necessarily implies. Rescaling coilcurrent references could, according to the case, lead to reach-ing machine hardware and software operational limits. It mustthus be taken into account that the rescaled currents should stillbe compatible with such limits. In order to quantify the effect


Figure 27: Effect of (3,09) missing coil compensation strategy duringthe generation of a (1,-6) step-like vacuum perturbation.Simulated radial fields from surrounding sensors are rep-resented for three situations: all coils working (red solid),one inactive coil without compensation (blue dashes), localpseudo-inverse compensation (green dashes).

7.2 magnetic field reconstruction results 71

Figure 28: Fourier harmonic amplitudes, at t=0.2s, compared betweenthe applied compensation strategies and normalized to thedesired n=-6 amplitude in the ideal case without missingcoils. Different scales on the Y-axis are used to enhance thevisual effect of the comparison.


of the compensation method on the current fed to each coil thefollowing quantity has been evaluated:

| Ir − Ir |

Imax· 100 (69)

where Ir is the new current distribution, Ir the original one andImax the maximum regime current. The result of this operationhas shown that the current required in each coil in order tocompensate for a missing one is about 100% of the original cur-rent. This can be explained given the linearity of the system: ifthe absence of a current of magnitude I is to be compensated,other coils will see their reference increase of approximately −I.The compensation strategy described in Section 7.1 and repre-sented by the matrix in Figure 25 has been recently applied todedicated experiments. Figure 29, which represents the mostsignificant result from the techniques illustrated so far, showsthe compensation results for coils (3,02) (3,09) (3,22) and (4,26)compared to a simulation with all coils active and to the casein which they are simply disabled. It is worth noting that thelocal magnetic field under each missing coil (dashed blue line)has opposite sign if compared to what a working coil shouldhave produced (solid red line). This behavior is being correctedby the applied compensation strategy, thus recovering the cor-rect sign and reducing the localized error field (dotted greenline). Given the applicability of the required current distribu-tion, this technique allows to reach a satisfying compensationof the missing coils both in terms of harmonic content and localradial field.

7.3 first wall temperatures in compensted shots

As we have seen so far, the flexibility of the RFX-mod controlsystem can be exploited to model the fault of one or more coils.Evaluating the effects on localized error fields and the impor-tance of a compensation strategy for the case in which one ormore coils are actually broken. One missing saddle coil in anyposition along the torus can be thought, if no current flows init, as a hole in the plasma magnetic boundary and the corre-sponding saddle sensor would only measure the effect of thesurrounding working coils. For the purpose of visualizing thisconcept, one of the coils has been disabled for a single vac-uum discharge with an n = −6,m = 1 external perturbationrequested with f = 0Hz. The result can be seen in Fig.30. When

7.3 first wall temperatures in compensted shots 73

Figure 29: Shot 36660 (dashed blue) with coils (3,02) (3,09) (3,22) and(4,26) disabled compared to shot 36664 (green dots) inwhich the compensation strategy is applied. The solid line(red) represents a reference simulation in which all coilsare correctly working. In each case the system produces anm=1, n=-6 perturbation.


any actuator is for some reason disabled, the local magneticfield given by the surrounding coils will be significantly differ-ent from the target one. With the main contribution providedby the opposite saddle coils, the local field will not only besmaller than the desired one but it will also change sign. Thisbehavior becomes a phase difference in case of an oscillatingrequest, as shown in Fig.31 the field underlying coil (2, 38) hasopposite phase with respect to the target. The presence of these

Figure 30: Disabled coil example, radial field (upper) and current(bottom) are shown for the case of a correctly working sad-dle coil (solid red) and for a disabled coil (dotted blue).The coil in toroidal position 38 along the outboard arrayhas been manually disabled for this example.

defects in the magnetic field is reflected during operation in adegradation of plasma performance, mostly due to mode lock-ing and first wall heating issues. The hypothesis behind thiswork in particular, is that if it were possible to mitigate the ef-fects of missing saddle coils, this would leave a trace in the firstwall heating pattern. The flexibility of the RFX-mod control sys-tem is helpful for the idea of compensation, allowing maneuver-ing space for each single coil to be tuned according to the de-signed task: reproducing the magnetic field that was created bythe complete system and thus compensating for missing coils.Three experimental runs have been carried out with the specific


Figure 31: Radial field measured under disabled coil (2,38) for a 20

Hz perturbation requested. The target field is representedin solid red, the measured field by the dashed blue line.

target of evaluating the effect of coil compensation on the local-ized wall heating pattern. On the side of diagnostics, four setsof thermal sensors can be used [83] [84]. Thermocouples how-ever are not uniformly covering the toroidal surface, the mostnumerous set has therefore been chosen for the future analyses.This set gives a good coverage of the first 150 while yieldinga sparser but nevertheless sufficient sampling of the rest of thetorus. Experiments have been performed with plasma current(Ip) ranging from 0.9 MA to 1.65 MA. First objective of the cam-paign has been the creation of a shot database large enoughto evaluate the effects of compensation at different Ip values. Agood database has been prepared, in order to evaluate first wallheating, by alternating standard RFP shots with varying Ip andshots in which the compensation strategy was introduced.

7.3.1 Currents with Actuator Compensation

While the effect of the compensation strategy on localized errorfields has already been verified in both simulations and dedi-cated experiments, the problems of localized wall heating andoperational limits are still open. As a first step the operationallimit issue has been tackled. Since the compensation strategyimplies current reference recombination, the risk of feeding oneor more coils with currents above the allowed limits has beenforeseen, with the possibility of running into saturations. Fortu-nately these saturations were seen to be comparable both withand without the compensation matrix. In order to fully assessthe effect of compensation on the current distribution, the I2(t)signals have been considered (proportional to the total power)


for each active coil with/without compensation. All the avail-able shots have been sorted into two populations dependingon whether the compensation matrix was applied or not. Foreach active coil the maximum I2(t) is considered and normal-ized to the sum of maximums over all the coils, these values arethen summed over the shots for each population. The followingquantity has been implemented:

ΣI =∑shots

(I2(t)max,nI2(t)tot,n

)(70)

Where the n index stands for each coil along the four toroidalarrays. Even though the data manipulation is complex, it allowsto see whether or not the introduction of compensation signif-icantly modifies the power required by each coil with respectto the total power in each shot. The result shown in Fig.32 isenough to conclude that the compensation strategy, althoughchanging the current distribution, has no dramatic effect on thepower requirements. It is instead improving the situation insome areas such as the first 12 poloidal rings. Compensationcan thus be safely implemented with plasma current up to 1.5MA.

7.3.2 Temperature Analysis

Main objective of the experimental campaign is assessing thecapabilities and testing the effects of the compensation strategydeveloped for four broken actuators in the RFX-mod saddlecoil array. This strategy, described in the previous section, isbased on a recombination of the current reference that takesinto account the missing coils. Working actuators are askedto do part of the job of the broken ones, thus reducing local-ized error fields and possibly improving the uniformity of themagnetic boundary. To identify the possible relationship be-tween missing coil compensation and plasma-wall interaction,the available thermocouple sets have been used. With these sen-sors being located on the external surface of the vacuum vessel,a statistical approach has been chosen to overcome the thermalinertial effects in the evaluation of the localized wall heating.During the experimental campaign the compensation strategy,developed for the four missing coils in positions (3,02) (3,09)(3,22) and (4,26), has been applied alternately to the standardidentity matrix. A database of 57 useful shots is now availablefor data analysis, with plasma current ranging from 0.9 MA


1 5 9 13 17 21 25 29 33 37 41 450

0.2

0.4

ΣI

Toroidal array 1 - outboardId matrixCompensation

1 5 9 13 17 21 25 29 33 37 41 450

0.2

0.4

ΣI

Toroidal array 2 - upper

1 5 9 13 17 21 25 29 33 37 41 450

0.2

0.4

ΣI

Toroidal array 3 - internal

1 5 9 13 17 21 25 29 33 37 41 45

Coil

0

0.2

0.4

ΣI

Toroidal array 4 - bottom

Figure 32: Normalized I2(t) for each saddle coil. Along the fourtoroidal arrays, for each coil (index n) the maximum valueis calculated and normalized to the total I2(t) fo the samecoil. The four broken coils are visible as no current is flow-ing through them.


up to 1.65 MA. Both Matlab®and Python environments havebeen used for the purpose. Python routines in particular havebeen developed to run most of the analyses, allowing for fastdata-loading and database cross-checking.

Figure 33: Figure corresponding to run 2280 in which each shotis marked on the sensor signal. Compensated and not-compensated can be distinguished. Time is in secondsfrom midnight.

All the available shots have been analyzed with a set of ded-icated tools to extrapolate statistically useful information. Twosamples have been considered, containing compensated andnon-compensated shots, the identification of which has beenused to split thermocouples’ signal as reported in Fig.33. Eachshot has been completely characterized in terms of startingand ending time of each phase from initialization. A three di-mensional temperature map (poloidal, toroidal directions andtime) is therefore obtained for the chosen thermocouple setfrom which the general wall heating pattern during a high cur-rent RFX-mod run can be appreciated. Fig.34 shows part of theinformation contained in the aforementioned map, the poloidalangle has been compressed by selecting maximum values andthe toroidal heating distribution is shown. Once the single shotshad been isolated and characterized within the thermal signals,a way was sought to obtain information on when and how


Figure 34: Temperature map along the toroidal angle for run 2280.For each available toroidal position the maximum poloidalmeasurement is given. Temperature is given with the colorscale in Celsius degrees.

much temperature was increasing. For each shot a given num-ber of temporally equispaced ∆T values has been calculated.Two minutes is the chosen time difference between tempera-ture steps, allowing for the heating of the wall tiles but smallenough to give a satisfying resolution and number of samplesfor statistical analysis. These ∆Ts are graphically representedas error bars in Fig.33, three values are given for most of theshots. By a simple histogram of the two ∆T populations ob-tained as previously described, the first effect of missing coilcompensation on the heating pattern can be appreciated: theshifting of the mean ∆T towards lower values than can be seenin Fig.35 where both compensated and not-compensated his-tograms are shown, the mean values and standard deviationare (µId = 1.42; σId = 0.89) and (µcomp = 1.23; σcomp = 0.88)respectively.

To obtain a better picture of the compensation influence onthe heating pattern, as well as of the structure itself of this pat-tern, a geometrical element is very useful. This kind of infor-mation can be extracted from the available data thanks to thedistribution of thermocouples that gives a good coverage of the


Figure 35: ∆T histograms for the two populations of compensatedand not-compensated shots. Each histogram has 10 bins0.5 wide and is normalized to 1 because of the differentnumber of shots in the two populations.

toroidal angle, within the first 100 degrees in particular, as ithas already been stressed with Fig.34. The ∆T data obtainedfrom the above analysis has been filtered to keep the highestvalues only, the threshold of 2C has been chosen. After nor-malizing to the number of elements for each sample, in orderto give significant information on the occurrence of high tem-perature steps, the resulting value can be plotted against thetoroidal angle. The result can be appreciated in Fig.36, showinghow the percentage of ∆T > 2C is reduced in most toroidal po-sitions by implementing the compensation matrix. This effectis particularly clear within the first 100 degrees and can onlyarise from missing coil compensation since no other significantdifference between the shots had been introduced during theexperimental campaign. The assessment of missing coil com-pensation effects can be considered satisfying with this latterresult, which a clear demonstration of the benefits introducedby the method described in this paper as far as localized wallheating is concerned.

7.4 conclusions 81

Figure 36: Percentage of ∆T above 2 normalized to the total num-ber of calculated ∆T and plotted for each available toroidalposition.

7.4 conclusions

The missing coil compensation matrix had been developed[80] to answer the specific need of mitigating local errorfields caused by the fault of one or more active coils. Thesenon-uniformities of the magnetic boundary affect plasma per-formance as preferable spots for mode locking and intenseplasma-wall interaction. The compensation matrix has beentested during standard plasma operation, with a wide range ofcurrents, up to 1.65 MA. This testing has been successful andallows to state that the proposed strategy can be safely imple-mented with weak effects on the total power consumption, asexplained with the use of I2(t) data. The main objective of thecampaign though, has been achieved by statistical analysis ofthe thermal signals. During the highest current run in partic-ular, localized wall heating has been found to be slightly lessintense when compensation was used. A stronger effect couldbe seen in the geometrical distribution of the highest tempera-ture steps, which was positively affected by compensation overmost of the toroidal span. This work allowed to assess in aninnovative way what the fault of active saddle coils could imply.Since machine repairs are often expensive and time consuming,


simple strategies such as the one tested here could be helpfulin mitigating possible damages.

8M U LT I - M O D A L RW M A C T I V E C O N T R O L O NR F X - M O D

"In the consciousness of the infinite, the conscious subject has for hisobject the infinity of his own nature."

— Ludwig Feuerbach, The Essence of Christianity

1The destabilization of a single RWM and its control have beenextensively studied in existing tokamaks, but the same is nottrue for situations where more than one RWM are simultane-ously unstable. A spectrum of multiple unstable RWMs in factis expected in tokamak plasmas operating at βN values muchlarger than the no-wall limit, as is in the case of scenarios beingdeveloped in view of steady state DEMO reactors. A differentsituation can be found in RFP plasmas, described in Section 2.7.The RFP configuration is always characterized by the simulta-neous instability of multiple tearing mode (sustaining the so-called dynamo mechanism) and RWM spectra. For this reasonRFPs provides an ideal playground for the study of 3D RWMpassive and active stability. Physics and control of this unsta-ble system are made even more challenging by the intrinsic 3Dnature of the problem that involves both the role of the pas-sive structures details and of the 3D fields produced by ded-icated coil systems that might act as actuators in a feedbackcontrol loop. In the following we will report on recent experi-ments and modeling activities on RFP plasmas that have beenexplicitly tailored to study the effect of a realistic (i.e. composedby a reduced number of coils) RWM active control system andto suggest some of the main issues future tokamaks operatingat very high βN values could face when feedback controllingmultiple RWMs.

To properly model this complex physical system, quanti-tative comparison between numerical tools and experimental

1 The content of this Chapter has been published in: Bolzonella, T., et al."Physics and Control of External Kink Instabilities with Realistic 3D Boundaries:a Challenge for Modern Experiments and Modeling." Plasma and Fusion Research,Vol. 9 and Bolzonella, T., et al. 41st EPS Conference on Plasma Physics, Berlin.O4.132

83

84 multi-modal rwm active control on rfx-mod

data involves dynamic representation of toroidal plasma equi-librium and stability, 3D passive boundary, 3D active externalfield distribution and feedback control software algorithms.This task has been recently successfully accomplished by ap-propriate upgrading and tuning the CarMa code [81][32], thatwas first adapted to RFX-mod to reproduce the measured RWMpassive growth rates [85] and then coupled to a realistic rep-resentation of the RFX-mod control system. The new CarMaversion can now incorporate information on the stability ofmore toroidal harmonics. In this way the model can account forpossible toroidal couplings induced on the plasma by passiveor active currents flowing outside the plasma. The upgradedmulti-modal CarMa version has been then integrated in thefull dynamic simulator of RWM control already implementedfor RFX-mod experiments and presented in [82]. The model hasbeen extensively validated against experimental data includingthe plasma response of the most unstable mode, which in theRFX-mod case, for a standard RFP equilibrium, is the internallynon resonant m = 1, n = −6, where m and n stand for poloidaland toroidal mode number respectively.

8.1 reduced sets of active coils

ReconfiguredArrays Different configurations of active coils have been tested to inves-

tigate the minimum set of coils capable of stabilizing the wholegroup of RWMs. As introduced in Section 5.1 a wide range ofthese configurations are possible, the so-called super-coils inparticular allow to mimic the effect of a single bigger coil. Wecan understand the difference of multi-modal RWM control ex-periments (and modeling) with a reduced set of coils by com-paring the vacuum spectrum generated by the full system of192 coils and one example reduced configuration when tryingto produce one single harmonic. This comparison is done in Fig-ure 37, where the target harmonic for both systems is the (1,-6)and all amplitudes are normalized to the amplitude of that ofthe (1,-6). While the field generated by the full system is basi-cally monochromatic, the reduced array produces many otherspurious sidebands, both in poloidal and toroidal directions.To further quantify this effect and easily compare different con-figurations, we can use the Total Harmonic Distortion (THD)already introduced in Equation 65. This quantity is used in Fig-ure 38 to quantify the difference, in terms of harmonic content(i. e.amount of spurious harmonics), between the full 4×48 con-

8.1 reduced sets of active coils 85

Figure 37: Br spectrum for the (1,-6) target harmonic generated by thefull 4×48 array (blue) and by the 1×12 reduced configura-tion with single toroidal extension.


figuration with respect with two reduced ones: 1×12 with sin-gle toroidal extension and 1×12 with triple toroidal extension(i. e.twelve groups of thee coils to form super-coils along thetoroidal direction). The dynamical characteristics of the model

Figure 38: Total Harmonic Distortion for three active coil configu-rations: 4×48 (red), 1×12 triple (green) and 1×12 single(blue).

allow the estimation also of THD during transients, that for theRFX-mod typical timescales can be of the order of 50-100 ms. Thetime analysis highlights the importance of shape, showing thatthe case with larger coils in the toroidal direction (1×12 triple)produces a smaller amount of sidebands during the transientphase than the case with the same number of coils, but nar-rower (1×12 single).

To fully exploit the new multi-modal capabilities of themodel, open loop growth rates were evaluated and comparedwith experimental ones, when no feedback control is applied.From a control point of view, the open loop model is a dynamicsystem whose inputs are the saddle coil currents and whoseoutputs are the radial component of the magnetic field as mea-sured by the saddle probes mounted on the vacuum vesselouter surface. To identify the most unstable modes present inthe model, the Fourier transform of the outputs associated tothe main eigenvectors (i. e.the eigenvectors associated to thelargest unstable eigenvalues) has been evaluated. The eigen-function exhibiting the higher relative content in the selected(m, n) harmonic has been then selected. Under this assumptionit is possible to correlate harmonic components and eigenval-

8.2 multi-modal control : eigenvalue studies 87

ues and to perform a direct comparison with the results of 2DFourier analyses on the experimental saddle sensors measuringthe radial component of the magnetic field.

8.2 multi-modal control : eigenvalue studies

Multi-Modal RWM

ControlDifferent configurations of active coils have been tested to inves-tigate the minimum set of coils capable of stabilizing the wholegroup of RWMs, in particular trying to improve our knowledgeon the rigidity of the plasma response. We use this term todefine the possibility of controlling a single mode even by a re-duced set of coils without triggering the growth of other modesoriginally stable. On the contrary, the mode rigidity can be de-fined as the possibility of controlling that single global modeby applying a local control action.

As an application of the aforementioned model, a particu-lar configuration has been implemented, composed of 3 fullpoloidal arrays of 4 coils evenly spaced along the toroidal direc-tion (4×3). This has been used on the unstable harmonics m = 1,n = -6, -5, -4, -3 and compared to the less sparse configurationwith 6 full poloidal arrays (4×6). In Figure 39 the open-loopRWM eigenvalues are compared to the closed-loop result for thetwo coil geometries. This eigenvalue analysis points out that the4×3 configuration cannot stabilize all the four harmonics simul-taneously, in particular the coupling of the m = 1, n = |3| withthe m = 1, n = |6| ones makes the system unstable. This cou-pling cannot be avoided with the sparse 4×3 configuration andit might as well be triggered by the produced sidebands. On theother hand, the 4×6 configuration, which doubles the numberof coils, allows for the complete stabilization of the system. Abetter picture of what a reconfigured coil array is doing, can begrasped by comparing the output magnetic field with the full4×48 configuration. This can be seen in Figure 40, where thefield pattern of a (1,-3) perturbation produced by the full sys-tem is compared with the (1,-3) component of the field used tocontrol the four aforementioned harmonics. The same kind ofanalysis has been done for a different configuration, with eightsingle coils along the outer toroidal array (1×8), applied to twodifferent sets of unstable harmonics. The results are reported inFigure 41, it is clear that while this control system is able to con-trol the two most unstable modes, the same is not true whenalso (1,-4) and (1,-3) harmonics are included in closed-loop. Not


Figure 39: Open-loop (red) growth rates compared to the applicationof two coil configurations: 4×3 (blue) and 4×6 (green).

(a) (b)

Figure 40: Magnetic field pattern, as measured by the 192 saddle sen-sors, of a (1,-3) perturbation generated by the complete sys-tem (a) and by the reduced configuration 4×6 when actingon n = -6, -5, -4, -3(b)

8.3 conclusions 89

only in the latter case one of the modes, the (1,-4), is not con-trolled because of the periodicity of the active coil system, butalso the control of the (1,-5) is lost due to the coupling of themode to other control fields. It has to be finally noted that alsothe (1,-3) is destabilized by the closed loop action of the controlsystem in the last case.

Figure 41: Open loop (red) growth rates are compared to two differ-ent control schemes using 1×8 reconfigured coil system.

8.3 conclusions

The multi-modal version of the dynamic model for the studyof RWM stability on RFX-mod has been implemented and exten-sively validated. Results are encouraging the possibility of itsuse as a predictive tool to assist the experimental investigationson the control of RWMs by means of a limited number of ac-tive coils. This case can be representative of some control issuestypical of multi-modal control in the presence of reduced num-ber/size of actuators. In particular, it is worth noting the factthat unwanted sidebands generated during fast transients byreduced actuator configurations can couple the control exter-nal field with other, non-controlled, harmonics. This couplingcan be virtuous, in such a way that the spurious harmonicsof the external field interact destructively with the plasma andprovide stabilization. On the other hand this interaction canbe constructive and lead to the destabilization of otherwise(marginally) stable modes by the Resonant Field Amplification


mechanism. We suggest that this might also be the case of fu-ture tokamak devices operating at high values of normalizedbeta, where more than one pressure driven RWM might ap-proach the stability threshold.

Part III

M O D E L I N G O F H I G H β P L A S M A S A N DR E S I S T I V E WA L L M O D E S TA B I L I T Y

9R E S I S T I V E WA L L M O D E S I N H I G H B E TAF U L LY N O N - I N D U C T I V E S C E N A R I O S

"Instead of wanting to be like this or that, make yourself into a silent,immovable giant. That’s what the mountain [Mount Fuji] is."

— Eiji Yoshikawa, Musashi

1As already introduced in Section 2.5, RWM stabilization is oneof the crucial tasks for future fusion devices, particularly whenaiming at steady state operation and high plasma pressure. Forthis reason the problem of understanding the physical mech-anisms lying underneath these modes, including the possibledamping channels, has been challenged in theory and exper-iments. Dealing with Resistive Wall Modes, which althoughslowed down by passive structures are still strong ideal insta-bilities, can be a complex task and has been done in differentways, mainly using passive and active techniques. Passive sta-bilization is usually sought with a combination of plasma flowand different damping mechanisms such as viscosity or the so-called kinetic resonance with particle drift motions. Plasma ro-tation has been experimentally seen to have a stabilizing effecton the RWM in tokamak plasmas and many studies are beingcarried out on the interaction with particle drift motions. The ef-fect of precession drift resonance for trapped particles has beenseen important in particular and it will therefore be the first oneto be considered here. Other contributions, as bounce resonancefrom trapped particles or transit resonance from passing ones,should be included in the future for a complete kinetic descrip-tion and to assess the relative magnitude in reactor-relevantscenarios.

A series of stability studies has been carried out in thepresent work with the 2D MHD codes MARS-F/K [40][32], whichhave been describe in Chapter 3, focusing on the most unstablen=1 RWM. A plasma representative of reference Scenario 5.1(high βN, fully non-inductive current drive, Single Null) is con-sidered. The specific equilibrium will be described in Section

1 The content of this Chapter has been presented in: Pigatto, L., et al. "ResistiveWall Mode stability in JT-60SA high βN Scenarios", 43rd EPS Conference onPlasma Physics, 4–8 July 2016, Leuven, Belgium.

93

94 rwm in jt-60sa high beta scenarios

9.1. As a first step for the assessment of the stability proper-ties of this scenario the ideal kink operational limits have beencomputed and compared to previously available data (Section9.2). The Resistive Wall Mode instability is then consideredfrom both the fluid and kinetic [86] point of view. A simplerotation profile is introduced in Section 9.3, then the effect ofdrift kinetic contributions on the mode eigenvalue is assessedand combined with the aforementioned plasma flow (Section9.4). Conclusions and possible future developments are drawnin Section 9.5.

9.1 scenario 5 .1 equilibrium

The reversed magnetic shear Scenario ]5 belongs to those thatshould establish the base for high βN, fully non-inductive cur-rent drive operation on JT-60SA. In particular it has been de-veloped to study the MHD limits brought on by Resistive WallModes and Neoclassical Tearing Modes. In this part of the the-sis we challenge the task of understanding RWM physics in oneof the possible Scenario ]5 equilibria. The implemented equilib-rium in particular has been obtained from an integrated model-ing simulation performed with the CRONOS suite of codes [87]which used the transport model CDBM [88] for simulating heattransport whereas the density profile has been prescribed. Thegoodness of this transport model and in general of CRONOSfor performing JT-60SA extrapolations was verified with JETand JT-60U data [89]. The boundary parameters, and in par-ticular the pedestal top pressure, have been obtained from thereference Scenario ]5.1 [90]. The resulting scenario represents aversion with lower input power developed using 17 MW of NBIand 7 MW of ECRH, therefore leaving sufficient power man-agement flexibility for mode and profile control purposes. Thenormalized β value is 3.6, with q95 ∼ 6.5. The original equilib-rium has been smoothed in its internal profiles ( dpdψ and f dfdψwhere p is the pressure and f the poloidal current flux func-tion, as defined in Eq. 16-17) and plasma boundary, in orderto avoid numerical issues and increase the resolution close tothe x-point. This latter point itself has been effectively removed,since it cannot be treated by the used codes, in a way such thatthe safety factor profile is left almost unaffected. The result-ing plasma boundary and q-profiles are shown in Figure 43, itis worth noting that the smoothing procedure gives negligible

9.1 scenario 5 .1 equilibrium 95

(a) (b)

(c) (d)

Figure 42: Relevant radial profiles for the implemented equilibrium.Plasma current components (a), electron and hot ions den-sity (b), total and supra-thermal pressure components (c),ion and electron temperature (d).


changes, from the MHD point of view, in the original equilib-rium.

(a) (b)

Figure 43: (a) Plasma boundary, first wall (i. e.stabilizing plate) andvacuum vessel contours. (b) Safety factor profile before(blue) and after (red dashed) smoothing of plasma bound-ary and internal profiles.

9.2 no-wall and ideal-wall stability limits

Identification of no-wall and ideal-wall limits is important forthe assessment of the operational space in which Resistive WallMode stability is relevant, and the whole setting of this workconsequently. As a first step in understanding the stability prop-erties of the aforementioned equilibrium the no-wall and ideal-wall β limits have been assessed with a pressure scan. The re-sults predicting βnw = 1.96 and βiw = 5.40 (Figure 44) are con-sistent with those obtained for similar equilibria [91]. It shouldbe stressed that only one wall has been implemented in thisstudy (dashed in Figure 43b), as a continuous 2D contour thatdoes not take into account the real geometrical characteristicsof the stabilizing plate such as gaps or non-uniformities. Theplasma-ideal-wall distance has been scanned and set as to allowconsistency with previous results. This however leads to an op-timistic ideal wall limit; comparison with fully 3D simulationsis foreseen in order to establish an equivalent effective wall po-sition and obtain more realistic βiw estimation. With a βN = 3.6the reference equilibrium is in between the calculated limits, inthe so-called wall stabilized region, where the ideal wall stabi-lizes the external kink. Since any realistic wall has a finite resis-tivity this mode is not fully stabilized but slowed down fromAlfvénic time scale to the wall penetration characteristic time.

9.3 stability of fluid rwm : effects of plasma flow and flow profile 97

In this case the computed Alfvén time is τA ≈ 9.66 × 10−7 s.In the following work a 2D thin wall has been implementedwhich does not take into account the real three-dimensionalcharacteristics of the stabilizing plate. The time constant of thissimplified boundary has been calculated with a vacuum MARS-F run to be τW ≈ 10 ms. All the results related to RWMs are pre-sented with a normalization to τw, but can be rescaled to anycalculated wall time for comparison with other codes and data.Since the Resistive Wall Mode instability can be influenced byboth plasma flow and kinetic properties of the plasma, these re-lationships and their effect on mode dynamics are investigatedin the following sections.

0 1 2 3 4 5 6 7

βN

0

0.02

0.04

0.06

0.08

0.1

0.12

γ τ

A

βno wall

∼ 1.96β

id wall ∼ 5.40

Figure 44: No-wall and Ideal-wall stability limits calculated with con-stant plasma current and ideal wall position b/a = 1.119

9.3 stability of fluid rwm : effects of plasma flow

and flow profile

Stabilization of the most unstable n=1 RWM is explored withplasma rotation, an ion acoustic Landau damping term is in-cluded in the momentum equation, in particular as a viscousterm. The perturbed viscous force can be expressed as:

−∇−→Π = ρκ‖|k‖vth,i|

−→v ‖ (71)

This term is scaled with the strength parameter κ‖ which hasbeen set here to 1.5 in order to achieve a significant damping.


While the NBI systems on JT-60SA allow great flexibility in the de-sign of plasma flow, efforts are ongoing to obtain self-consistentrotation profiles [92]. In order to make the discussion as generalas possible, two simplified cases have been implemented here.A parabolic rotation profile has been assumed and comparedwith the uniform rotation case. The effect of viscous dampingon the RWM can be seen in Figure 45 where the rotation at theinnermost q=2 magnetic surface is scaled. Normalization to theτw factor is applied to the eigenvalues, while the flow is normal-ized to the Alfvén frequency (ωA) is applied. A strong dampingωA =

B0R0√µ0miZn

=

1.06× 106s−1of the growth rate (γτw) is given by the uniform rotation profilein particular. Even though the normalized rotation frequency atthe q=2 surface is used in the Figure 45 plots, a very differentbehavior is found for the two considered situations. This proba-bly indicates that the whole global rotation profile has an effectof the fluid RWM stability, thus opening to the possibility of pro-files tailored on the control requirements. The presence of two

0 0.02 0.04 0.06 0.08 0.1

Ωq=2

/ ωA

0

5

10

15

Re(γ

τ w

)

(a)

0 0.02 0.04 0.06 0.08 0.1

Ωq=2

/ ωA

0

1

2

3

4

5

6

7

8

Im(γ

τw

)

(b)

Figure 45: Eigenvalue scaling with toroidal plasma flow for uniform(dashed blue) and parabolic (red) rotation profiles. (a) Realpart, (b) imaginary part.

q=2 rational surfaces in the plasma, due to the reversed safetyfactor profile, as well as the complex nature of the n=1 mode,can be responsible for the importance of the flow profile alongthe whole minor radius. As a result of this purely fluid rota-tion scan we can confidently say that mode stabilization is notachieved, although the growth rate can be significantly reduced.It is worth noting that reasonable plasma flow velocity shouldbe used for mode stabilization, which has been estimated beingof the order of 0.03vA for a fast rotating case [93]. It is thereforemandatory to investigate further damping mechanisms whichcan be both active and passive. In the following sections some

9.4 kinetic contribution to rwm stability 99

of the most relevant effects in terms of passive stabilization areintroduced in the model, while the subject of active stabiliza-tion will be addressed in future work.

9.4 kinetic contribution to rwm stability

As already remarked in the introductory sections the behaviorof RWMs will be influenced by drift kinetic effects [33] from boththermal and energetic particle populations from NBIs, these con-tributions have been included in the stability analysis. Ther-mal ions and electrons have been introduced as a first step,assuming a simple Maxwellian distribution and focusing onthe resonance between the mode and the precession motion oftrapped particles banana orbits. This will be named 2-species inthe following. Energetic Particles (EPs) from normal NBI injec-tion have been then introduced, i. e.the 3-species case. For EPs aslowing-down distribution is assumed in particle energy spaceand isotropic distribution in particle pitch-angle [63]. For bothcases the kinetic contribution to the pressure tensor is scaledthough the αD parameter, as shown in Figure 46. The depen-dence of the eigenvalue on the kinetic contribution is reportedfor both the two particle species and three particle species (in-cluding EPs) case, without plasma flow. Figure 46 shows howthe n=1 eigenvalue is slightly modified, in its real part, by thekinetic contribution of thermal particles. A more significantmodification is seen when the energetic population is included,this is due to the further non-adiabatic contribution to the per-turbed kinetic potential energy. It should be mentioned that thesmall effect that is found in Figure 46 is probably given by thelarge growth rate and frequency of this particular RWM. Thebase equilibrium which have been considered here is quite farfrom the no-wall limit, giving rise to a strong instability thatreduces the effect of kinetic damping. All the components ofthe potential energy can be evaluated at various levels of detail,here we choose to focus on the drift kinetic contributions andtheir relationship with plasma rotation. The effect of rotationis being assessed in presence of kinetic effects with the sameprocedure followed in the purely fluid case, using a smallerparallel viscosity coefficient this time (κ‖ = 0.1). This meansthat in the following kinetic simulations rotation is to be con-sidered mainly given by E× B flow. The uniform rotation pro-file has been assumed for this proof-of-principle study, rotationamplitude has been scanned up to the stability threshold for


0 0.2 0.4 0.6 0.8 1

20

25

30

35

Re

(γ τ

w)

0 0.2 0.4 0.6 0.8 1

αD

-10

0

10

20

Im(γ

τw

)

th. i+e-

th. i+e- & EPs

Figure 46: Dependence of the eigenvalue on the drift-kinetic contri-bution with two populations (thermal ions and electrons,blue solid line) and three populations of particles (thermalions and electrons + energetic particles, red dashed line).The parameter αD represents scaling from fluid descrip-tion (αD = 0) to full kinetic contribution to the pressuretensor (αD = 1). No plasma flow is applied.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Ωq=2

/ ωA

-10

0

10

20

30

40

Re

(γ τ

w)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Ωq=2

/ ωA

0

10

20

30

40

Im(γ

τw

)

Figure 47: Toroidal rotation scan for the kinetic RWM in the 2 species(blue-circles, thermal ions and electrons) and 3 species (red-squares, thermal + EPs) cases. A uniform rotation profile isassumed.

both the 2-species and 3-species case. As Figure 47 reports, astrong damping is obtained for both cases, interestingly enougha slightly higher critical frequency is found for the 3-species.These two frequencies are found to be ω2spec ≈ 0.0126ωA andω3spec ≈ 0.0169ωA for 2-species and 3-species respectively. Inorder to characterize the effect of rotation and kinetic parti-cles on the mode structure, the perturbed plasma displacementalong the radial coordinate has been reported for the purelyfluid simulation compared to the 3-species case with rotation(Ω = 0.012ωA). Figure 48 shows this comparison, highlightinga smaller displacement, with respect to the fluid case, withinthe plasma as well as the increased effect of rational surfaces.In Figure 49 we report the parallel and perpendicular compo-nents of the pressure tensor in Eq. 45, which reflect modal struc-ture with all its components. Detailed analysis of the potentialenergy contributions has been done to characterize mode sta-bility while varying rotation, all the calculated components ofδWk are normalized by the plasma inertia associated with ra-dial displacement. If the electron contribution to δWk can beneglected, the population of thermal ions is found to be much


0 0.2 0.4 0.6 0.8 1

s≡ψp

1/2

0

0.2

0.4

0.6

0.8

1

|ξ·∇

s|

0

1

2

3

4

Figure 48: Comparison of RWM eigenfunctions between fluid (solid)and kinetic theory (dashed) including thermal and ener-getic particles. Example with Ω = 0.012ωA, the q=m/nsurfaces are marked with dashed black lines. Fourier har-monics of the perturbed radial displacement are given inthe equal-arc coordinate system introduced in Chapter 3.

(a) (b)

Figure 49: Example of parallel (a) and perpendicular (b) kinetic pres-sure components for 2 species case with toroidal rotationΩ = 0.01ωA


more relevant, as we can appreciate from the bottom plots inFigure 50. It is worth noting that the thermal ions contribution

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.02

0

0.02

0.04

Re(d

Wk)

Adiabatic contr. from Passing Ions and EPs

Passing EPs

Passing th. Ions

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.04

-0.02

0

0.02

Im(d

Wk)

(a)

0 0.005 0.01 0.015 0.02Ω

q=2 / ω

A

-0.01

0

0.01

0.02

Re(

dWk)

Adiabatic contr. from Trapped Ions and EPs

Trapped EPsTrapped th. Ions

0 0.005 0.01 0.015 0.02Ω

q=2 / ω

A

-0.04

-0.02

0

0.02

Im(d

Wk)

(b)

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-1

0

1

Re

(dW

k)

×10-4 Non-Adiabatic Contrib. Therm. e

-

2 species

3 species

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

2

4

6

8

Im(d

Wk)

×10-4

(c)

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.01

0

0.01

Re

(dW

k)

Non-Adiabatic Contrib. Therm. Ions

2 species

3 species

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.015

-0.01

-0.005

0

Im(d

Wk)

(d)

Figure 50: Real and imaginary parts of drift kinetic energy compo-nents: for the 3-species case adiabatic contributions frompassing (a) and trapped (b) particles. Non-adiabatic partsfrom thermal electrons (c) and ions (d) are compared forboth cases. Energy components normalized to plasma iner-tial associated with radial displacement.

has the same evolution in the rotation scan for both 2-speciesand 3-species. For the latter case the EPs component of δWk isseparately shown in Figure 51a. An increasing stabilizing effectis seen at Ω ∼ 0.002ωA, explaining the damping in Figure 45

around the same frequency. This in particular is the value cor-responding to the ion diamagnetic drift frequency. Figure 51bshows all the components of δWk added together for both the2-species and 3-species case, highlighting a stronger dampingeffect introduced by EPs in the imaginary part of δWk. As pre-viously mentioned, this effect shows a peak close to the iondiamagnetic drift frequency, but then decreases with scaling ro-tation.


0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.03

-0.02

-0.01

0

Re

(dW

k)

Non-Adiabatic Contrib. EPs

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-2.5

-2

-1.5

-1

Im(d

Wk)

×10-3

(a)

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.04

-0.02

0

0.02

Re

(dW

k)

Non-Adiabatic contr. Precession Resonance - All Species

0 0.005 0.01 0.015 0.02

Ωq=2

/ ωA

-0.015

-0.01

-0.005

0

Im(d

Wk)

(b)

Figure 51: (a) Real and imaginary part of non-adiabatic contributionfrom EPs to the perturbed potential energy. (b) Real andimaginary part of the total perturbed kinetic energy, con-tributions from all populations are added together for the2-species and 3-species case.

9.5 conclusions

The ideal and no-wall β limits have been calculated for the n=1

ideal kink in the fully non-inductive scenario 5.1, future com-parison with 3D calculations will allow a more detailed descrip-tion and the effective 2D wall will be positioned. The n=1 RWMwill be unstable in the scenario under investigation with a close-fitting resistive wall. Stability studies with fluid rotation havebeen carried out by implementing two simple profiles, noneof which provides full mode stabilization. This means that theRWM cannot be stabilized by fluid effects only, kinetic contribu-tions must be taken into account. The uniform rotation case inparticular is found to give a stronger damping, suggesting thatthe global rotation profile should be taken into account overthe single q=2 surface. An extended fluid analysis of the prob-lem is possible with different, detailed rotation profiles and a3D description of passive structures which would allow includ-ing new energy terms. Further physics is being implementedin the description step by step, starting with the kinetic con-tribution of thermal and EPs. Future work will consider moreadvanced distribution models for the latter. Mode evolution inboth these cases is still under investigation but giving promis-ing results when energetic particles are included in particular.A further important future step will be the study of RWMs withdifferent toroidal numbers, which can be simultaneously un-stable in the considered scenarios. Active stabilization studies,

9.5 conclusions 105

which could prove to be necessary, are currently ongoing witha simplified physical model not including kinetic effects. Thepresent work represents a summary of some of the first effortsin characterizing RWM stability in JT-60SA high β plasmas. Theresults achieved so far are encouraging the possibility of stableoperation though improvement is necessary and foreseen, withreference to the kinetic profiles and energetic particles distribu-tions in particular.

10A C T I V E C O N T R O L O F R E S I S T I V E WA L LM O D E S O N J T- 6 0 S A

"True heroism is minutes, hours, weeks, year upon year of the quiet,precise, judicious exercise of probity and care, with no one there to

see or cheer. This is the world."

— David Foster Wallace, The Pale King

1As illustrated in Chapter 2, it is well known that in the case ofAdvanced Tokamak (AT) plasmas, the so-called no-wall stabil-ity threshold of ideal external kink modes can be reached at βNvalues that strongly limit the operational space of such scenar-ios. For this reason stabilizing effects are studied, both passive(e. g.as in Chapter 9) and active (e. g.external coils). This lattermethod is the topic of the present Chapter, in which a dedi-cated CarMa model [94] is applied to RWM feedback control ina JT-60SA AT scenario.

10.1 carma model and open-loop analysis

A simplified plasma description, given by the MARS-F code,has been coupled to a detailed representation of the stabilizingplate and active coils. A 3D view of these structures is given inFigure 52. In the following, the error field correction coils willnot be used and have been therefore excluded from the model.The results that will be described in the present work refer tothe equilibrium configuration reported in Chapter 9, scaled toβN ∼ 2.7. In the model the most unstable eigenvalues can beselected and identified as RWMs from their growth rates andfrom the comparison with the 2D problem solved by MARS-F.Synthetic magnetic probes have been added to the 3D model todocument e. g.how the most unstable RWM would appear. Froma total of 108 magnetic probes available in JT-60SA for real timemeasurements, 39 will be placed on the stabilizing plate closeto the plasma on the low field side. A view the sensors posi-tions is given in Figure 53. It is worth noting that the magnetic

1 Part of this Chapter has been presented in: Bolzonella, T., et al. "Securing HighβN JT-60SA Operational Space by MHD Stability and Active Control Modelling",26th IAEA Fusion Energy Conference, 17–22 October 2016, Kyoto, Japan.

107

108 active control of resistive wall modes on jt-60sa

(a) (b)

Figure 52: 3D representation of the Stabilizing Plate (a) and of theactive coils (b), both Error Field Correction Coils and Re-sistive Wall Mode control Coils.

probes are shifted with respect to the center of the active coilsof 30, since they are placed underneath the toroidal field mag-nets. In Figure 54 the 3D current density pattern corresponding

(a) (b)

Figure 53: 3D view of the active coils and sensors, side (a) and top(b).

to the unstable n=1 eigenfunction is shown, as computed bythe CarMa code. A more detailed description of the couplingprocedure and of the features of 3D structures together withsome preliminary results can be found in [94]. A coil current in-put model has been built with the CarMa code output. This al-lows for straightforward implementation of the desired controlschemes with the implicit assumption that the current controlloop is capable of meeting the requests. A more detailed dis-cussion concerning the current requirements will be given inthe following sections. An open-loop version (i. e.without anyactive external action) of the model has been used to evaluate

10.1 carma model and open-loop analysis 109

Figure 54: 3D view of the current density pattern corresponding tothe unstable n=1 eigenfunction, computed by the CarMacode [94].

free-mode evolution, spatial distribution and harmonic content.Interestingly enough, due to the 3D features of the stabilizingplate, which exhibits holes with no periodical symmetry, theexpected unstable RWM is actually split in two modes, withdifferent imaginary part (phase) and slightly different growthrates. These latter are found to be γ1 = 551s−1 and γ2 = 402s−1.When moving to a possible active control strategy, as a first at-tempt a modal control scheme has been chosen, focusing onthe n=1 harmonic of the tangential component of the magneticfield. The poloidal harmonic content of such a mode has in gen-eral many components, with an internal kink structure (m=1) inthe core and higher mode numbers towards the plasma bound-ary. One of the most significant however, is the m=2 harmonicthat characterizes the RWM component of the mode, located be-tween the two q=2 surfaces and close to the edge. This is shownin Figure 55 with the m=2 harmonic of the perturbed plasmadisplacement as calculated by MARS-F. As a first step, only theuppermost and lowermost active coil arrays (along the toroidaldirection) have been implemented. This choice has been madefor practical reasons that will be explained in the following sec-tion. Figure 56 shows the spatial distributions of the magneticfield tangential component in the poloidal plane as measuredby the uppermost and lowermost toroidal arrays of sensors, lo-cated onto the stabilizing plates (6 probes each in toroidal di-rection); please note that the tangential component is to be un-derstood as the field component tangential in a poloidal sectionto the plane where the sensor is installed (i.e. mainly, but notexactly, poloidal in a physical reference frame). The splitting of


Figure 55: Normal component of the perturbed plasma displacement,m=2 harmonic, calculated by MARS-F.

the most unstable mode into two different parts is clearly visi-ble from both the implemented sensor arrays. A largely prevail-ing n=1 harmonic content can be appreciated in both eigenfunc-tions by a DFT analysis of the same measurements, as shownin Figure 57.

10.2 closed-loop proof-of-principle

As previously mentioned, in order to begin the closed-loopanalysis of the problem, the model has been reduced to arraysof equispaced sensors and coils. In particular the upper andlowermost sensor arrays have been used, coupled to the topand bottom sets of actuators. This has been done in order toapply a modal control scheme with a straightforward DiscreteFourier Transform (DFT) of the magnetic field. Further compli-cations to take into account more sensors and all the active coilswill be introduced later. Another preliminary step has beendone in verifying the vacuum field produced by the coils witha known sinusoidal reference input. This is shown in Figure58, where a cosine-like signal is fed to the actuators, which pro-duce a magnetic field shifted in phase of about π/2. This shiftwill be taken into account in the following eigenvalue analysesof the closed-loop system. As already stated, a modal controlscheme has been chosen, focusing on the n=1 harmonic of thetangential component of the magnetic field. This is proposed inagreement with experimental results obtained by the same ap-

10.2 closed-loop proof-of-principle 111

Figure 56: Spatial pattern of the two most unstable eigenvalues ob-tained from the open loop model. The tangential compo-nent of the field (here in arbitrary units) is shown againstthe toroidal angle corresponding to each sensor. The n=1

dominant harmonic component is clearly visible .

Figure 57: The spatial Fourier transform along the toroidal direc-tion shows the dominant n=1 component in both unsta-ble eigenvectors, both seen by the upper (blue) and lower(yellow) sensor arrays. The dominant n=1 components arehighlighted by orange boxes.


Figure 58: Cosine input signal (blue) and vacuum magnetic field pro-duced by the upper active coils (red) as measured by 6

probes of the uppermost toroidal array. The position of thecoils is marked by the black dotted lines.

proach in different experimental devices [95] that showed howthis choice guarantees to reduce the coupling between the actu-ators and the feedback signal. Following the feedback controlimplementation strategy sketched in Chapter 6, the DFT of thesignals from the upper and lower sensor arrays is calculatedand the control relevant harmonics selected. A correspondingharmonic signal is obtained at the controller output, whose in-verse transform is then evaluated along the toroidal coordinate,taking into account the coil periodicity (in this case equal tothat of the sensors) and the angular shift with respect to thesensors. The choice of selecting only the upper and lower coilarrays allows for a one-to-one sensor-actuator correspondencewith constant angular shift. A set of references for the upperand lower coil currents is generated, assuming an ideal currentcontrol scheme. The result of a first gain scan using only the up-permost coils is reported in Figure 59. It is interesting to notethat stability of both unstable modes is achieved, even with a re-duced configuration and ideal proportional controller. Furtherstudies are then necessary to include the other actuators and as-sess the feasibility in terms of current of the control strategiesmade possible by the independent coils.

10.3 full active system implementation 113

Figure 59: Gain scan on the two unstable eigenmodes using the uppercoil array.

10.3 full active system implementation

In order to fully exploit the capabilities of the active system,the equatorial coil array has been implemented, coupled to aset of magnetic sensors located on the Stabilizing Plate alongthe equatorial length. This latter however is found to be com-posed of 5 magnetic sensors only, compared to the 6 constitut-ing the upper and lower arrays. In order to recover a one-to-oneactuator-sensor relationship and perform DFT analyses, an in-terpolation has been implemented. The necessity for this inter-polation can be understood from Figure 60 where the open-loopspatial pattern is shown for both unstable modes and the threearrays. It clearly appears that the equatorial array is lacking themeasurement at Φ = 205. In order to interpolate the missingmeasurement two more sensors have been used, located justabove and below the desired position. It should be noted thatthese new sensors do not belong to the so-called uppermostand lowermost arrays, but they are different probes located onthe Stabilizing Plate closest to the missing one position. Prelimi-nary analyses of the open-loop system have allowed to identifythe interpolation over the poloidal angle as the best choice, interms of the capability of the system to measure a given ref-erence field from each of the active coils. In particular, for theposition corresponding to Φ = 250, the equatorial magneticfield has been reconstructed by using the first upper and lower


0 50 100 150 200 250 300 350-1

-0.5

0

0.5

1

Bt [A

.U.]

Eigenvalue = 551 s-1 Bt Upper

Bt Equatorial

Bt Down

0 50 100 150 200 250 300 350

φ [°]

-1

-0.5

0

0.5

1

Bt [A

.U.]

Eigenvalue = 402 s-1

Figure 60: Spatial pattern of the two most unstable eigenvalues ob-tained from the open loop model with upper (red dashed),equatorial (blue dotted) and lower (black dot-dash) arrays.The tangential component of the field (here in arbitraryunits) is shown against the toroidal angle correspondingto each sensor.

10.3 full active system implementation 115

available measurements. This procedure is shown in Figure 61,where the measured patterns are given in dashed red line andthe interpolated field in blue dotted line for both eigenmodes.

10.3.1 Eigenvalue Analysis of the Full System

A set of preliminary analyses with a dedicated vacuum modelhave been carried out in the same way outlined in the previoussection. This vacuum version allows a full spatial and dynamiccharacterization of the magnetic field produced by the actuatorsin the presence of non-axisymmetric passive structures. Alsofollowing the same track, an eigenvalue study of the systemhas been performed to evaluate the critical gain required formode stabilization. A constant phase shift has been applied tothe action of each coil in order to maximize its effect. Thesephases have been calculated during the open-loop preliminaryanalyses with a helical reference signal. The gain scan reportedin Figure 62 shows that the unstable modes are fully stabilizedwith a proportional gain of 7× 106 A/T. The applicability ofsuch a gain cannot be assessed with instruments such as eigen-value analyses and requires challenging the problem from thepoint of view of time simulations.

10.3.2 Time Domain Simulation

The closed-loop system represented has been implemented asa Matlab®Simulink model to obtain a dynamic simulator forRWM control studies. This allowed the development of simula-tion tools similar to those shown in Chapters 6 and 7. A gainscan similar to that shown in the eigenvalue study has beenperformed around the supposed critical value. The time simula-tions are confirming the previously calculated stabilizing gain,as it can be appreciated in Figure 63 where the amplitude ofthe n=1 harmonic is plotted for the three sensor arrays. As forthe previous eigenvalue analyses, the tangential magnetic fieldis used as feedback variable for the time simulations. An ini-tial free evolution phase, lasting 10 ms, has been set to allowthe desired mode to develop from the random initial condi-tion (generated t = 0 s). The feedback loop is then closed withthree different gains: 4× 106 A/T, 6× 106 A/T and 8× 106 A/T.The stabilizing threshold is found in between the last two val-ues. It is worth noting that two different dynamics are found inthe closed-loop part of the simulation. The mode undergoes a


0 50 100 150 200 250 300 350-1

0

1

Bt [A

.U.]

Upper sensors

Measured

Interpolated

0 50 100 150 200 250 300 350-1

0

1

Bt [A

.U.]

Equatorial sensors

0 50 100 150 200 250 300 350

φ [°]

-1

0

1

Bt [A

.U.]

Lower sensors

(a)

0 50 100 150 200 250 300 350-1

0

1

Bt [A

.U.]

Upper sensors

0 50 100 150 200 250 300 350-1

0

1

Bt [A

.U.]

Equatorial sensors

0 50 100 150 200 250 300 350

φ [°]

-0.5

0

0.5

Bt [A

.U.]

Lower sensors

(b)

Figure 61: Image of the two unstable eigenmodes on the three sensorarrays: γ1 = 551s−1 (a) and γ2 = 402s−1 (b). The actualavailable measurements are given by red circles and theinterpolated field by blue diamonds.

10.4 conclusions 117

104 105 106 107

Gain [A/T]

-100

0

100

200

300

400

500

600

Gro

wth

rat

e [s

-1]

Figure 62: Eigenvalue study on the critical proportional gain requiredto stabilize the two modes with all the available active coils.

fast initial phase (0.01 s – 0.012 s) during which its amplituderapidly decreases. Once the faster components have decayedthe growth rate stabilizes around the expected eigenvalue. Thisbehavior is consistent with results obtained with similar con-trol strategies [82]. The full output of the time simulation hasbeen considered to investigate the current required by the ap-plied control scheme. For the most stabilizing case (Kp = 8×106A/T) the simulated magnetic field (tangential component), asmeasured by all the sensors, and the current flowing in eachactive coil is shown in Figure 64. The resulting Ampere-turnsare compatible with the rated current of JT-60SA power supplies,as described in Section 5.2, and Table 2 in particular.

10.4 conclusions

As a result of the above discussion we can conclude that Re-sistive Wall Mode active stabilization has been achieved in arepresentative of JT-60SA Scenario 5.1 plasma, with a model in-cluding realistic 3D description of the passive and active struc-tures. The core of this model is represented by the CarMa code.A series of tools have been developed that allow user-friendlyimplementation of different control schemes. Both eigenvalueanalyses of the closed-loop system and full time simulations


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.5

1

|Bt,D

FT| [

T]

×10-4 Upper Array

n=1 - Kp=4x106 A/T

n=1 - Kp=6x106 A/T

n=1 - Kp=8x106 A/T

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.5

1|B

t,DF

T| [

T]

×10-4 Equatorial Array

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02t [s]

0

0.5

1

|Bt,D

FT| [

T]

×10-4 Lower Array

Figure 63: Amplitude of the n=1 component calculated from the tan-gential field measurements on each sensor array. Three dif-ferent proportional gains are applied after a 10 ms lastingfree evolution.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-2

-1

0

1

2

Bt [T

]

×10-4 B Tangential

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02t [s]

-1000

-500

0

500

1000

I [A

/turn

]

I Saddle Coils

Figure 64: Tangential field (upper plot) and coil current (lower plot)for each sensor and each active coil, example taken fromthe highest gain case Kp = 8× 106 A/T.

10.4 conclusions 119

can be performed thanks to the dedicated dynamic simulatorwhich has been described in the present work. A purely pro-portional control strategy has been implemented, with an idealcurrent controller. Future work will involve the development ofa voltage input system, with realistic evaluation of the currentloop and implementation of a full PID controller. This improve-ment will allow increasing the level of detail in the simulationsand expanding the range of applicable control schemes. Theindependence of the actuators is inherent in the model and itwill be exploited in the future to test different control strategies,the possibility of acting with different sets of coils on differentmodes is also an important possibility to be exploited given thepossible multi-modal nature of the instability spectrum in highβ plasmas.

A F T E RW O R D

The work carried out during the three years of the Ph.D. pro-gram has been thoroughly described in the chapters of this the-sis, and the main results already summarized in dedicated con-clusion sections. As a global overview, this work has addresseda series of relevant topics for the exploitation of present andfuture fusion experimental devices.

With the main focus being set on ideal MHD instabilities andrealistic magnetic boundaries, the first part has been devoted tofine tuning and optimization of the RFX-mod control system,developing a series of techniques that are simple enough to beimplemented in real-time operation yet bringing noticeable ef-fects. The missing coil compensation in particular has been ap-plied in experiments referring to different topics, and it can begenerally implemented to virtually any device with a set of ac-tuators and a given degree of flexibility. The modeling activityon multi-modal RWM feedback control in RFP plasmas has ex-tended previous work and investigated the coupling phenom-ena between different poloidal harmonics. A similar approachcould be applied to a completely different range of plasmas, thehigh βN Advanced Tokamak scenarios, in which the q profilesand high pressure would make multiple n RWMs unstable.

The second part of the thesis has investigated RWM physicsin this kind of plasmas, with reference to the JT-60SA exper-iment which is under construction in Japan. A study of thekinetic effects in mode damping physics has been carried out,highlighting that the drift-kinetic contribution to stability, al-though small for βN values close to the ideal-wall limit, is never-theless important and possibly fundamental for RWM control.Feedback stabilization has also been considered with a proof-of-principle study. In this work the basis has been given forfuture analyses of RWM active control on JT-60SA. In the nearfuture developments on this side are foreseen, with improve-ment of the implemented dynamic simulator and studies onpossible control schemes. The obtained results are encouragingthat strong instabilities such as RWMs could be controlled infuture devices with a combination of plasma rotation (coupled

121


to drift-kinetic effects) and magnetic feedback. This representsa major step towards the development of high β operationalscenarios.

A C K N O W L E D G M E N T S

During my time at Consorzio RFX as a Ph.D. student, andbefore that while working at my M.Sc. thesis, I have found sup-port and friendship in many people. In particular I thank Dr.Tommaso Bolzonella for being my mentor, for his guidance andadvice that has played an important role in developing my pro-fessional profile. I thank Prof. Paolo Bettini for his literally con-stant support, by both proposing new things and backing myown ideas. I would also like to thank Dr. Giuseppe Marchiori,with whom I enjoy working and whose advice is always pre-cious. For the always entertaining chats, and for the commentson part of this thesis, I thank Dr. Marco Veranda. Aside frommy research activities, I have been involved in the organizationof scientific events, which has proven to be a very instructive ex-perience. For this opportunity, and for her commitment, I thankMargherita Basso.

For teaching me so much, introducing me to the part of mywork that perhaps I enjoy the most, for always answering myquestions from literally anywhere around the world, for hisguidance, I want to thank Dr. Yueqiang Liu.

Of the many things done in the past three years, most Ihave shared with the colleagues and friends of my doctoratecycle; for this I thank Tautvydas J. Maceina, Yangyang Zhang,Carlo Baltador, Marco Gottardo and Alberto Maistrello. Alongwith them I also thank all the friends who have somehow con-tributed to the office life: Matteo Zaupa, Matteo Vallar, DanieleAprile, Flavia Raggi, Daniele Maggio, Ondrej Kudlácek, OisinMcCormack, Ferdinando Gasparini, Gustavo Grenfell (thanksfor translating the abstract into Portuguese!), Mattia Dan,Michele Fadone, Dario Friso, Marco Barbisan, Marco Zanini,Pietro Vincenzi, Nicolò Visonà, Vadim Yanovskiy, Nisarg Patel.

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B I B L I O G R A P H Y

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[7] I.B. Bernstein, E.A. Frieman, M.D. Kruskal, and R.M. Kul-srud. «An energy principle for hydromagnetic stabilityproblems.» In: Proceedings of the Royal Society of LondonA: Mathematical, Physical and Engineering Sciences. Vol. 244.The Royal Society, 1958. Chap. 1236, pp. 17–40.

[8] F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato, andS. Succi. «MHD-limits to plasma confinement.» In: PlasmaPhysics and Controlled Fusion 26.1A (1984), p. 209.

[9] D. Pfirsch and H. Tasso. «A theorem on MHD-instabilityof plasmas with resistive walls.» In: Nuclear Fusion 11.3(1971), p. 259.

[10] A. Bondeson and DJ Ward. «Stabilization of externalmodes in tokamaks by resistive walls and plasma rota-tion.» In: Physical Review Letters 72.17 (1994), p. 2709.

[11] M.S. Chu and M. Okabayashi. «Stabilization of the Exter-nal Kink and the Resistive Wall Mode.» In: Plasma Phys.Control. Fusion 52 (2010), p. 123001.

[12] B. Hu and R. Betti. «Resistive wall mode in collisionlessquasistationary plasmas.» In: Physical Review Letters 93.10

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[13] S. Takeji, S. Tokuda, T. Fujita, T. Suzuki, A. Isayama, S. Ide,Y. Ishii, Y. Kamada, Y. Koide, and T. Matsumoto. «Resis-tive instabilities in reversed shear discharges and wall sta-bilization on JT-60U.» In: Nuclear Fusion 42.1 (2002), p. 5.

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