Model based approach to resistive wall magnetohydrodynamic...

Model based approach to resistive wallmagnetohydrodynamic instability control

Experimental modeling and optimal control for the reversed-field pinch

AGUNG CHRIS SETIADI

Doctoral ThesisStockholm, Sweden, 2017

TRITA-EE 2016:192ISSN 1653-5146ISBN 978-91-7729-228-9

KTH School of Electrical Engineering (EES)SE 10044 Stockholm

Sweden

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlagges tilloffentlig granskning for avlaggande av Teknologie doctorsexamen i elektroteknik torsda-gen den 9 Februari 2017 klockan 10.00 i Sal F3, Lindstedtsvagen 26, Kungliga TekniskaHogskolan, Stockholm.

c© Agung Chris Setiadi, January 9, 2017

Tryck: Universitetsservice US AB

i

Abstract

The primary objective of fusion research is to realize a thermonuclear fusion powerplant. The main method to confine the hot plasma is by using a magnetic field. Thereversed-field pinch is a type of magnetic confinement devicewhich suffers from varietyof magnetohydrodynamic (MHD) instabilities. A particular unstable mode that is treatedin this work is the resistive wall mode (RWM), which occurs due to the current gradi-ent in the RFP and has growth rates of the order of the magneticdiffusion time of thewall. Application of control engineering tools appears to allow a robust and stable RFPoperation. A model-based approach to stabilize the RWMs is pursued in this thesis. Theapproach consists of empirical modeling of RWMs using a class of subspace identificationmethodology. The obtained model is then used as a basis for a model based controller. Inparticular the first experimental results of using a predictive control for RWM stabilizationare obtained. It is shown that the formulation of the model based controller allows the userto incorporate several physics relevant phenomena along with the stabilization of RWM.Another use of the model is shown to estimate and compensate the inherent error field. Theresults are encouraging, and the methods appear to be generically useful as research toolsin controlled magnetic confinement fusion.

ii

Sammanfattning

Fusionsforskningens primara mal ar att forverkliga enny typ av kraftverk baseradepa termonuklear fusion. Den viktigaste metoden for att innesluta det heta plasmat aranvandandet av magnetfalt. ”Reverserat-falt pinch” (RFP) ar en typ av anlaggning for mag-netisk inneslutning av fusionsplasma som uppvisar ett flertal magneto-hydrodynamiskainstabiliteter. En specifik instabil mod som behandlas i detta arbete ar”resistiv-vagg”moden (RWM). Den orsakas av stromgradienten i RFPn och tillvaxer med en tidskonstantsom ar av samma storleksordning som magnetfaltets diffusionstid i det omgivande metall-skalet. Tillampning av verktyg fran reglerteknikomradet forefaller tillata en robust och sta-bil RFP drift. I detta arbete anvands ett modell-baserat tillvagagangssatt for kompenseringav RWM. Det innefattar empirisk modellering av RWM med anvandning av ”subspace”system-identifieringsmetoder. Den erhallna modellen anvands sedan som grund for enmodell-baserad regulator. De forsta experimentella resultaten fran modell-prediktiv kom-pensering av RWM har erhallits. I detta arbete har ocksa visats att formuleringen av denmodellbaserade regulatorn tillater anvandaren att integrera flera relevanta fysikaliska as-pekter forutom RWM. Ytterligare en anvandning av modellen ar for att gora uppskattningoch kompensering av avvikelser i anlaggningens magnetfalt, sa kallade falt-fel. Resultatenar uppmuntrande, och det forefaller som om de undersoktametoderna ar allmant anvand-bara som verktyg for forskning om magnetisk inneslutning av fusionsplasma.

List of Papers

This thesis is based on the work presented in the following papers listed chronologically:

I Graybox Modelling of EXTRAP T2R with Vacuum-Plasma Separation and OptimalControl Design of Resistive Wall ModesA.C. Setiadi, , P.R. Brunsell, F. Villone, S. Mastrostefano, L. FrassinettiSubmitted for publication.

II Reduced Order Modelling of Resistive Wall Modes in EXTRAP T2R Reversed-FieldPinchA.C. Setiadi, F. Villone, P.R. Brunsell, A. Polsinelli, S. Mastrostefano, L. Frassinetti43rd EPS Conference in Plasma Physics, 2016, Brussels, Belgium

III Improved model predictive control of resistive wall modes by error field estimator inEXTRAP T2RA.C. Setiadi, P.R. Brunsell, L. FrassinettiPlasma Physics and Controlled Fusion 58(2016) 124002

IV Design and operation of fast model predictive controller for stabilization of magne-tohydrodynamic mode in a fusion deviceA.C. Setiadi, P.R. Brunsell, L. Frassinetti2015 54th IEEE Conference on Decision and Control (CDC), Osaka, 2015, pp. 7347-7352.

V Implementation of model predictive control for resistive wall mode stabilization onEXTRAP T2RA.C. Setiadi, P.R. Brunsell, L. FrassinettiPlasma Physics and Controlled Fusion 57(2015) 104005

Publications with contribution from the author not included in this thesis :

VI A method for the estimate of the wall diffusion for non-axisymmetric fields usingrotating external fieldsL. Frassinetti, K.E.J. Olofsson, R. Fridstrom,A.C. Setiadi, P.R. Brunsell, F.A. Volpe,J. DrakePlasma Physics and Controlled Fusion 55(2013) 084001

iii

iv LIST OF PAPERS

VII Resistive Wall Mode Studies utilizing External Magnetic PerturbationsP.R. Brunsell, L. Frassinetti, F.A. Volpe, K.E.J. Olofsson, R. Fridstrom,A.C. SetiadiProceeding of the 25th IAEA Fusion Energy Conference, St. Petersburg, RussianFederation, 2014, Paper EX/P4-20

VIII A Technique for the Estimation of the Wall Diffusion TimeFrassinetti, L. and Fridstrom, R. and Olofsson, E. and Setiadi, A. C. and Brunsell,P. R. and Volpe, F. and Drake, J. R.American Physical Society, 54th Annual Meeting of the APS Division of PlasmaPhysics, October, 2012

Author’s contribution to the listed papers

For [PAPER I] to [PAPER V], I was the main author. I was responsible for performingsimulation, writing the control software, perform experiments and writing the manuscript.The CARIDDI model used in PAPER I and PAPER II is provided by mycollaboratorsfrom University of Cassino and Southern Lazio, Italy.

Acknowledgements

It has been a long and arduous process to write this thesis, which was only made possibleand memorable due to the help of many people along the way.

First of all, I would like to express my sincere gratitude to my supervisor Prof. PerBrunsell and co-supervisor Lorenzo Frassinetti for their continuous guidance, patience,and encouragement during my Ph.D. study.

Besides my supervisor, I would like to thank the rest of the Alfven Laboratory person-nel. Prof. James Drake has been very helpful with his insights in this research and hasbeen very kind to proof-read the manuscript. Many thanks to Hakan Ferm for his techni-cal expertise with EXTRAP T2R. I am grateful to Emma Geira forher help with all theadministration process. My sincere thanks also go to Fabio Villone from the University ofCassino who provided me with great support during my short visit in Cassino, Italy.

I thank my fellow Ph.D. colleagues: Alvaro, Armin, Emmi, Estera, Igor, Kristoffer,Pablo, Petter, Richard, Waqas and Yushan for the stimulating discussions during our fikas.I would also like to thank the ever-so-colorful Kampung Stockholm for all the fun we havehad.

Finally, I would like to dedicate this thesis to my family: myparents Bistok Sigaling-ging and Tumiar Naibaho and my siblings Tonggo, Grace and Armen for their endless loveand support throughout my life.

v

Contents

List of Papers iii

Acknowledgements v

Contents 1

1 Introduction 31.1 Thermonuclear Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Confinement concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Resistive Wall Modes 92.1 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 RFP Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Stabilization of Resistive Wall Modes . . . . . . . . . . . . . . .. . . . 12

3 Experimental Setup 153.1 EXTRAP T2R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Active Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Model-based approach of control design . . . . . . . . . . . . . .. . . . 17

4 Control Oriented Modeling of RWM 194.1 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Graybox Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Control of Resistive Wall Modes 335.1 Linear Quadratic Gaussian . . . . . . . . . . . . . . . . . . . . . . . . .345.2 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . .355.3 RWM control logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.5 On Error Field Estimation and Compensation . . . . . . . . . . .. . . . 41

1

2 CONTENTS

6 Summary and Discussion 476.1 Summary of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7 Conclusion 51

References 53

Chapter 1

Introduction

1.1 Thermonuclear Fusion

The demand for energy is expected to increase significantly in the near future due to theprojected increase of population and increase of the quality of life in particular in thedeveloping countries [1]. A major part of the energy demand is being supplied by burningfossil fuels [1]. Our consumption rate of fossil fuel does not appear to be sustainable [2].Furthermore, the emission associated with burning the fossil fuels enhances the greenhouseeffects and in turn has adverse effects on the global environment [3].

1980 1985 1990 1995 2000 2005 2010

0.9

1

1.1

1.2

1.3

1.4

1.5

x 105

year

TW

h

World Energy Demand

(a)

1990 1995 2000 2005 2010

1

1.5

2

2.5

3

3.5

4

4.5

5

x 104

year

TW

h

World Energy Consumption

liquid fuel

coal

natural gas

renewables

nuclear

(b)

Figure 1.1: World energy outlook obtained from the U.S. Energy Information Agency [4]

The projected increase of the energy consumption should be met with limiting the useof fossil fuels and increasing the use of alternative energysources. Several types of renew-able energies such as wind and solar power can contribute to providing the world energysupply. A continuously operating power plant, such as nuclear fission, would be beneficial

3

4 CHAPTER 1. INTRODUCTION

for the stability of energy supply. However, for nuclear fission, there are challenges whichrelate to the radioactivity of the nuclear waste and the riskof nuclear proliferation that stillneed to be solved.

Nuclear fusion is another alternative for future energy generation with virtually un-limited fuel supply [5]. The amount of energy that can be produced by nuclear fusion isenormous. For comparison, the amount of energy obtained by burning 106 ton of fossilfuels is roughly equal to the energy content of 0.8 ton of Uranium for fission and 0.14ton of Deuterium in fusion [6]. Energy generation from nuclear fusion in principle doesnot produce greenhouse gas unlike that of fossil fuels. Furthermore, the radioactive wasteproduced by nuclear fusion has a decay time significantly shorter compared to nuclearfission.

In nuclear fusion, two light atoms are combined to form a heavier atom. The samemechanism also occurs in the sun (and stars) where Hydrogen nuclei are fused to produceHelium and energy [7]. The energy produced is due to the mass difference between theproduct and the reactant and is determined by the Einstein’senergy relationshipE = ∆mc,whereE is the energy,∆m is the total difference of mass between the products and reactantsandc is the speed of light. Significant temperature is needed to achieve fusion. For thetype of fusion of interest the temperature region of 108 - 109 degrees is needed so that theirkinetic energy is sufficient to counter the electrostatic repulsive force. In this temperatureregion, atoms are stripped of the electrons and form a quasi-neutral gas called plasma.

Several candidates of fusion reactions are available for energy generation:

D+D → 3He+n+3.27 MeV

D+D → T+p+4.03 MeV

D+ 3He→ 4He+p+18.3 MeV

D+T → 4He+n+17.6 MeV

Among these fusion candidates, the Deuterium-Tritium (D-T) reaction has the largestcross-section, achieved at relatively low temperatures (∼ 15 keV). The cross-section of theD-T reaction and the amount of the energy it produces is the reason that the D-T reactionis considered as the most promising candidate for a future fusion reactor. The energyproduced by the D-T reaction is distributed among its products, α−particles(4He) andneutrons. The fraction of the energy that each of the products carries depends on the massratio of the products. In D-T reactions, around 80% of the energy produced is carried byneutrons while the rest are carried byα−particles. In magnetic confinement concepts, thismeans that the energy of theα−particles can be used as an additional source of heating tothe plasma, where the energy carried by the neutrons are to beextracted and converted toelectricity.

The use of D-T as the candidate for fusion still has some issues related to the fuels.Tritium is rare on earth. Furthermore, it is an unstable isotope of Hydrogen with a shortdecay time of 12.3 years. Thus to provide enough Tritium for future fusion power plant,the Tritium needs to be produced. A solution to this issue is to breed the Tritium in thereactor. One can use the neutron generated in the fusion reaction and Lithium to produce

1.2. CONFINEMENT CONCEPTS 5

Tritium according to the reactions:6Li +n→ 4He+T+4.8 MeV

7Li +n→ 4He+T+n−2.8 MeV

The abundance of fuel is another advantage of nuclear fusion. The D-T reaction re-quires Deuterium and Lithium to produce Tritium needed for the reaction. Deuterium is astable isotope of Hydrogen and available in sea water. Lithium occurs in nature as mineralsand can also be extracted from sea water. The amount of Deuterium and Lithium availablein sea water could sustain the use nuclear fusion power plantat the current level of worldenergy consumption for at least several million years [8].

1.2 Confinement concepts

Figure 1.2: Schematic of tokamak. Courtesy of EUROfusion [9].

The plasma can lose its energy through different means. In steady state conditions,the energy confinement timeτE is defined as the ratio of the total energy of the plasmawith the energy loss rate. The energy losses of the plasma canbe balance by externalheating. Formally the efficiency of a nuclear fusion power plant can be defined by theQ value which is the ratio of the energy produced by fusion with the energy provided bythe external heating. For a steady state reaction, it is desired to balance the energy losseswith the heat from theα−particles. The power balance conditions in the presence of theα−particles provide the triple product condition for ignition [10]:

nTτE > 3×1021 m−3keV

6 CHAPTER 1. INTRODUCTION

After the ignition, the external heating can be switched off, and the plasma can still besustained, although some power input will still be needed for control purposes or to drivecurrent in the plasma. Since for D-T reaction the highest cross section occurs at temper-ature∼ 15 keV, the criterion then provides the operating conditionof density and energyconfinement time for a nuclear fusion power plant to reach ignition.

A magnetic field can be used to confine the plasma and subsequently satisfy the crite-rion. This approach is called magnetic confinement fusion, and it is based on the fact thata charged particle will move helically along the magnetic field line. The simplest form ofmagnetic confinement is a cylindrical device. The confinement of the particle is provideddue to a magnetic mirror set up at both ends of the cylinder. However, for cylindrical con-figuration, there is always a fraction of the particles that can escape the magnetic mirror.For a simple mirror machine, these end losses are too severe,and the maximum Q valuethat can be achieved is too small as a candidate for future fusion reactor [6].

Another MCF concept is to use a toroidal configuration. In a toroidal configurationwhich consists of a purely toroidal field, the radial gradient of the magnetic field and thecurvature give rise to a vertical drift. This drift is chargedependent. Thus electron andions will move in opposite direction. This vertical drift gives rise to a vertical electricfield which in turn gives another drift in the radial direction. This radial drift is not chargedependent. Therefore it causes the whole plasma to move radially outward and eventuallyhit the vessel. The problem with the particle drift can be solved by adding a poloidalcomponent to the field. By adding a poloidal field, the field lines are made to be helical.Furthermore, the resulting drift will change periodically, and on average the net particledrift is canceled.

The fusion devices such as tokamak and reversed-field pinch (RFP) are some examplesthat utilize both toroidal and poloidal magnetic field to confine the plasma. The poloidalmagnetic field is generated by using the plasma as a secondarywinding of a transformer.The current that is driven toroidally in the plasma providesthe required poloidal field andohmically heat the plasma. One drawback of this approach is that it cannot run in a steadystate since transformer action depends on time-changing ofmagnetic flux. Thus to achievesteady state, a non-inductive current drive is required.

Tokamak is one of a MCF configuration for a fusion reactor. It is an axisymmetric toruswhere the toroidal magnetic field is significantly higher than the poloidal magnetic field. Aschematic of the main components for tokamak is shown in figure 1.2. Significant progresshas been achieved in term of fusion performance since the research started in 1950s [11].At the present day, tokamak is the most developed concept andthe leading candidate forthe future MCF power plant. Specifically, the highest recorded triple product has beenachieved in the tokamak with the device JT-60U [12]. Furthermore, D-T experiments havebeen performed in TFTR [13] and JET [14]. The D-T campaign in 1997 carried out inJET recorded a release of fusion power up to 16 MW [14]. Due to the success of tokamak,most of the research effort in fusion community is based on the tokamak configuration.Furthermore, the next generation research reactor ITER is based on tokamak configuration.ITER is expected to demonstrate the efficiency of energy production by fusion, and theexpected value of Q is 10.

An alternative MCF concept is the RFP. Like the tokamak, the RFP is an axisymmetric

1.2. CONFINEMENT CONCEPTS 7

toroidal configuration. The main difference is that in the RFP; the poloidal magnetic field isthe same order as the toroidal magnetic field. Furthermore, the toroidal field spontaneouslyreverses its direction toward the edge of the plasma. As a candidate for a future fusionreactor, the RFP has some advantages with respect to the tokamak. First, in principle, itcan use only ohmic heating to reach ignition. Second, the RFPas fusion reactor will notneed to use superconducting magnets to produce the toroidalfield. The RFP is, however,more prone to many instabilities which in turn leads to degraded confinement.

Both the tokamak and the RFP require an inductive current drive to produce the poloidalfield needed to confine the plasma. To this end, reaching a steady state operation is an issuethat needs to be solved for both the tokamak and the RFP. Thereis another configurationthat relies only on external coils to produce the required helical field. Such a configurationis called a stellarator [15] and in principle is ready to operate in steady - state. However,the stellarator requires high precision in the shape and alignment of the magnetic coils.Only after the availability of supercomputer, an optimization program could be used todetermine the shape and position of the magnetic coils that are now used in the newly builtWendelstein7X [16].

Chapter 2

Resistive Wall Modes

2.1 Magnetohydrodynamics

Fusion plasma is a very complex system which involves simultaneous interaction of∼ 1022

ions and electrons. The magnetohydrodynamics (MHD) model is often used to describethe fusion plasma. In the MHD model, the plasma is simplified to the extent that it isdescribed as a single conducting fluid. Thus as the name suggest, the MHD equations area coupling between the electrodynamics equation and the hydrodynamics equation. TheMHD equations are summarized below [6,17]:

ρdvdt

= J×B−∇p (2.1a)

σ(E+ v×B) = J (2.1b)

∇×E =−∂B∂ t

(2.1c)

∇×B = µ0J (2.1d)

∇ ·B = 0 (2.1e)

Equation (2.1b) is the Ohm’s Law. Due to the high temperature, fusion plasma is oftenassumed to have an infinite conductivity (σ → ∞). In such case, the ideal Ohm’s Law isused instead:

E+ v×B= 0 (2.2)

The ideal MHD equations are useful in describing the global properties of fusionplasma, in particular, those associated with the plasma equilibrium and stability of theequilibrium. The stationary solution of the set of MHD equations (2.1) defines the equilib-rium condition. For the static case (v = 0) one can obtain the force balance condition forideal plasma:

J×B = ∇p (2.3)

9

10 CHAPTER 2. RESISTIVE WALL MODES

The equation (2.3) represents the force balance between theplasma pressure and the mag-netic force. The force balance condition also implies that the magnetic field lines andcurrent lie on a set of a closed-nested surface on which the pressure is constant.

2.2 RFP Equilibrium

In the previous chapter, it has been mentioned that a Reversed-Field Pinch (RFP) is amagnetic confinement device for which the toroidal magneticfield and poloidal magneticfield are of the same order. Furthermore in the RFP the toroidal magnetic reverses itsdirection near the edge [18]. The RFP configurations have several physics implicationswhich will be discussed briefly in this section.

The simplest model of RFP equilibrium is the so-called force-free equilibrium. For thisequilibriumJ×B = 0 which implies that the current flows parallel to the magnetic field.Since in experiment the plasma is not pressure-less, a more realistic approach to describethe equilibrium is [18]:

∇×B0 = µ(r)B0+µ0B0×∇p

B20

(2.4)

where for a cylindrical geometry, the equilibrium reduced to a set of differential equation:

dBφ

dr= µBθ − µ0

Bφ

B20

dpdr

(2.5a)

1r

ddr

(rBθ ) = µBφ − µ0Bθ

B20

dpdr

(2.5b)

dpdr

=−χr

2µ0

(µB2

0

2Bθ−

Bφ

r

)

(2.5c)

The pressure profile used in the (2.5) gives Suydam’s necessary condition of stability whenχ < 1. The ratio of parallel current and magnetic field is denotedby µ and the profile iscommonly modeled according to [17–20]:

µ(r) =2a

Θ0

[

1− (ra)α]

(2.6)

Experimentally the equilibrium in usually characterized by the pinch parameterΘ and thereversal parameterF , which can be directly measured using the global magnetic diagnos-tics and defined as follows [18]:

Θ =Bθ (a)< Bφ >

(2.7a)

F =Bφ (a)

< Bφ >(2.7b)

HereBφ (a) andBθ (a) respectively denote the toroidal field and poloidal field at the edgeand< . > represents the volume average operator. A typical equilibrium profile for EX-TRAP T2R can be computed forΘ = 1.6 andF = −0.3 which is the typical experimental

2.2. RFP EQUILIBRIUM 11

BθBφ

(mT

)

r/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.05

0.1

0.15

0.2

0.25

Figure 2.1: Equilibrium profile of EXTRAP T2R computed withΘ = 1.6 andF =−0.3

value. The magnetic field profile can be seen in figure 2.1. Furthermore, the confinementin RFP is usually characterized by the value of poloidal betaβθ = 2µ0<p>

B2θ (a)

.

Given a cylindrical geometry, the safety factor profile can be defined as:

q(r) =rR

Bφ

Bθ(2.8)

A typical q profile for EXTRAP T2R is shown in figure 2.2, and theprofile suggests awide range of resonant modes can occur in the RFP plasma. Resonant surfaces appearin the plasma wheneverq(r) = −m

n and MHD modes that satisfy them andn conditionare said to be resonant. In EXTRAP the internally resonant modes are characterized byn< 0 and 0<−m/n< q(0) and the have the same helicity handedness as the equilibriuminside the reversal surface. The externally resonant modesare characterized byn > 0andq(a) < −m/n< 0 and they have helicity handedness opposite of the internalmodes.Resonant resistive MHD modesm= 0 andm= 1 are often rotating in EXTRAP T2R,and their field at the boundary wall can be suppressed provided that rotation frequencyexceeds the inverse wall diffusion time. In this case, the resistive wall behaves like anideally conducting wall for such modes. These resonant modes, withm= 0 andm= 1 aretypically tearing modes, and they play important roles in the sustainment of the toroidalfield through the dynamo process [18].

The non-resonant ideal unstable MHD modes can be stabilizedif there is a perfectlyconducting wall facing the plasma [6,17]. The growing magnetic perturbation due to insta-bility can induce current in the wall which in turns producesa magnetic field that cancels

12 CHAPTER 2. RESISTIVE WALL MODES

replacementssa

fety

fact

or(

q)

r/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 2.2: Safety factor profile of EXTRAP T2R computedΘ = 1.6 andF =−0.3 with

the perturbation. However since in experiments the wall always has finite resistivity, theresistive wall cannot stabilize them. Instead the modes areconverted into Resistive WallModes (RWM) and their growth times are comparable to the wallmagnetic diffusion time.Typically the growth rates of RWMs are much slower than the growth rates of the idealMHD modes and this allows them to be stabilized actively by feedback control.

2.3 Stabilization of Resistive Wall Modes

The RWM is the primary instability that is treated in this thesis. As mentioned before, theRWMs in RFP are current driven, and they are a potential causeof disruption whenever theplasma duration is longer than the wall diffusion time. In a tokamak, the RWM is pressuredriven and they play an important role in advance tokamak scenario since they limit theobtainable beta.

The RWM can be suppressed by passive or active stabilization. The plasma rotationcan provide a passive dissipative mechanism to the RWM [21].For the RFP plasma, it isunderstood that the plasma rotation must be of the order of the Alfven frequency for theplasma to have sufficient damping to stabilize the RWM. On theother hand, in a tokamak,the RWM stability has non-monotonic variation with the plasma rotation [21, 22]. It hasbeen shown in tokamak that the RWM could be stabilized also atlow plasma rotation due to

2.3. STABILIZATION OF RESISTIVE WALL MODES 13

kinetic effects not included in the MHD model [22,23]. However, various MHD activitiescan trigger the RWM thus it is desirable to also have methods for active stabilization ifneeded [24,25].

Active stabilization is achieved utilizing a controlled magnetic field. The original idea,dubbed the Intelligent Shell, was presented in reference [26]. The Intelligent Shell aimsto mimic an ideally conductive shell, by using an array of co-located actuator and sensorcoils. The sensor coils are used to measure the growing magnetic perturbation at a specificlocation on the shell, where the actuator coils can apply an external magnetic control fieldto suppress the growing mode.

Typically, a Proportional-Integral-Derivative (PID) type controller is used to determinethe necessary external field required to suppress RWM. The PID is a standard feedbackcontroller that does not require a model of the system. In theabsence of a model, there areseveral empirical rules of thumb available to tune the PID parameters [27]. In fact, the useof PID controller has been shown to be able to suppress multiple RWMs in RFP [28–31].

Recent results suggest that a model based controller might become necessary to im-prove the performance of RWM controller, especially for tokamak to reach beta close tothe ideal wall limit [25, 32]. Model-based control of the RWMis currently the frontierof RWM control algorithm research and has been successfullydemonstrated in tokamakdevices such as DIII-D [33, 34] and HBT-EP [35]. Some of the reported benefits of em-ploying model-based RWM control in tokamak experiments are: i) ability to discriminatethe transient noise such as ELM from the RWM [33], ii) optimization of control actuationrequired to stabilize RWM [35], iii) long-pulse high beta discharges [36,37].

Chapter 3

Experimental Setup

3.1 EXTRAP T2R

The results presented in this thesis are from experiments carried out on EXTRAP T2R [38].EXTRAP T2R is a medium sized reversed-field pinch operated atthe KTH Royal Instituteof Technology in Stockholm. The physical dimension of EXTRAP T2R is characterizedby its major radiusR= 1.24 m and minor radiusa = 0.183 m. The EXTRAP T2R vac-uum vessel is made of stainless steel equipped with molybdenum limiters. The vessel isenclosed with a double-layer copper shell(which constitutes the wall) with a traverse fieldpenetration time of around 6.3 ms. The EXTRAP T2R usually operating with plasma cur-rentIP = 50−120 kA, toroidal loop voltage in the range of 20−30 V. Table 1 provides thetypical parameters in EXTRAP T2R [39].

Figure 3.1: EXTRAP T2R reversed-field pinch

15

16 CHAPTER 3. EXPERIMENTAL SETUP

Table 3.1: Typical EXTRAP T2R Parameters

Plasma currentIp 50 - 120 kAToroidal loop voltageVφ 20 - 30 VMinor radiusa 0.183 mMajor radiusR 1.24 mWall timeτshell 6.3 msPinch parameterΘ 1.5 - 1.8Reversal parameterF −0.1 - −0.5Electron densityne 0.5 - 2.0 × 1019 m−3

Electron temperatureTe 200 - 300 eV

3.2 Active Control System

active coils

resistive shell

vacuum vessel

sensor coils

limiter


The experimental results obtained in this thesis mostly used the magnetic diagnosticsin EXTRAP T2R. The magnetic diagnostics can be divided into two sets: global and MHDdiagnostics. The global magnetic diagnostics are used to measure the plasma current,toroidal loop voltage as well as the poloidal and toroidal field at the edge and the averagetoroidal field [40].

The MHD diagnostics, which concerns the RWMs, is measured byusing an array ofsaddle coils. Figure 3.3 depicts the overview of EXTRAP T2R magnetic diagnostic coilsand a poloidal cross section of EXTRAP T2R is given in figure 3.2 which elaborates thecoils in more details. The coils used for RWM control are active and passive saddle coil.The passive sensor coils are placed inside the shell, and theactive actuator coils are placedoutside the shell. The arrays consist of a set of 4 poloidal and 32 toroidal coils evenly

3.3. MODEL-BASED APPROACH OF CONTROL DESIGN 17

distributed to cover the whole torus surface area. Furthermore, the coil signals are seriesconnected for the inboard and outboard coil to form an ”m=1” coil. The same connectionsare done for the top and bottom coil. The arrangement of the coils allowed the measure-ment of the radial field with poloidal mode number ofm= 1 and toroidal mode number−16≤ n≤ 15.

The MHD diagnostics are used for real-time control application in EXTRAP T2R. TheEXTRAP T2R control hardware has been recently upgraded and the work presented in thisthesis includes the development of the new control system hardware and software. The de-velopment of the control system consists of a data management server for the experimentaldata and a feedback control algorithm library for EXTRAP T2R.


The hardware for the data acquisition is provided by D-tAcq [41] which consists of ahost PC and a data acquisition module. The host PC is a Linux based PC equipped witha 3.0 GHz 6-cores processor and 8 GB memory. The data acquisition modules consist ofseveral cards that are interconnected through a compact PCIcrate. The data acquisitionmodules provide 16 bit analog-to-digital (ADC) for 128 simultaneous channels and 64digital-to-analog (DAC) channels.

3.3 Model-based approach of control design

The plasma experiment performed in this thesis is carried out while invoking one of thealgorithms implemented in the host PC. To ensure a high confidence of the designed al-gorithm during implementation, a model-based framework isused as seen in figure 3.4.

18 CHAPTER 3. EXPERIMENTAL SETUP

CONTROL

ALGORITHM

MODEL

SYSTEM

SIMULATION

PCS

EXTRAP

T2R

EXPERIMENTAL

VALIDATION

ALGORITHM DESIGN

ALGORITHM REFINEMENT

ALGORITHM REFINEMENT

MODEL REFINEMENT

IMPLEMENTATION


The availability of a model, allows a simulation test to be performed before implementingthe algorithm in real-time thus increasing the confidence level of the algorithm before theexperimental validation.

The implemented approach in EXTRAP T2R is not entirely automatic, in the sensethat there is no automated code generation used. The controlalgorithms are written inC++, where a specific numerical library [42] is used to solve all the required mathematicaloperations in real-time. The main reason for using a custom made code is related to latencyin real-time. In our experience a custom made code can solve the required mathematicaloperation in a more efficient manner.

Chapter 4

Control Oriented Modeling of RWM

This section focuses on describing the process of obtainingthe required model for controldesign. To explain some basic concepts used in the thesis, the overview of the RWMfeedback control loop of EXTRAP T2R is shown in figure 4.1. Theexplanation of the

Σ Σ

ΣΣ

C

GA

G

−1

Controller

Power Amplifier

T2R

r

d

y

u wu

uv

wy

Figure 4.1: Overview of RWM feedback loop operation at EXTRAP T2R

symbols are :

G : The EXTRAP T2R

GA : The power amplifiers rack that are used to drive the actuatorcoils.

C : Feedback controller.

19

20 CHAPTER 4. CONTROL ORIENTED MODELING OF RWM

u : vector of voltages that are applied to the power amplifiers.

uv : vector of currents that are applied to the actuator coils.

y : vector of output signals which are the time-integrated voltages of the sensor coils.

r : vector of output reference values.

d : vector of extra inputs used for dither injection.

wu : vector of unmeasured input disturbances.

wy : vector of unmeasured output disturbances.

For the purpose of control design, EXTRAP T2R dynamics(bothshell dynamics andRWM) and power amplifier need to be modeled accurately. It canbe useful to find alinear representation of the system since the feedback control design theory for a linearsystem is well developed. However, a linear representationmight not truly represent thetrue dynamics of the plasma since in general, the MHD equation is nonlinear due to theresistivity of the plasma. However, under a certain condition, the linear representationmight be sufficient to represent the plasma dynamics. In the context of RWMs in the RFP,it is known that the eigenvalues of the RWMs depend on the equilibrium parametersΘ andF [21, 43, 44]. Furthermore, it is often assumed that the mode is rigid, in the sense thatthe eigenfunction of the RWM is not affected by external magnetic perturbation associatedwith the control actions. Due to this, a linear representation of RWM can be justified inRFP.

A physics based model is usually desired to predict the system behavior. For RWMseveral finite element codes are available such as CarMa [45,46], and VALEN [47]. Thesecodes can model the 3D walls surrounding the plasma and have been used as standard toolsfor many fusion devices [32,48,49]. The use of these finite element codes usually leads toa model with high order. Since most of the model-based control methods often embed themodel of the system in its structure, a complex model will lead to a sophisticated controllerthat can be difficult to be implemented in real-time. A model order reduction technique isoften necessary to obtain an optimal low order approximation of the finite element model.

One approach to obtaining a model is by using system identification. System identifi-cation is an empirical method to obtain a model [50]. The RWM in EXTRAP T2R alongwith its actuator and sensor dynamics can be captured in a single black-box model using asystem identification technique. In most cases, the system identification does not requirea physical understanding of the system to be able to predict system behavior. However, itrequires a carefully designed experiment in which the system is perturbed by some ran-dom signals, the d vector in figure 4.1, to expose the dynamic response of the system to thecontrol actuators.

4.1 System Identification

System identification can be seen as a classic supervised learning problem that aims togenerate a function that can predict output data given an input data. System identifica-

4.1. SYSTEM IDENTIFICATION 21

tion typically consists of several steps: experiment design, model parameters estimationand model validation. In this thesis, a class of system identification called the subspaceidentification method (SIM) is used. The SIM is used due to itsnumerical efficiency inestimating state space model and its ability to handle multivariate data in a closed loop.However, since state space representation is non-unique, the state variables obtained bySIM are difficult to interpret. Furthermore, the asymptoticstatistical property of SIM isnot well understood [51,52].

Conventional SIM method aims to represent a system in a typical discrete time statespace form:

G=

x(k+1) = Ax(k)+Bu(k)+Ke(k)

y(k) =Cx(k)+Du(k)+e(k)(4.1)

Equation (4.1) is often called innovation form, and the innovation terme(k) represents theresidual between the measured and predicted output. The innovation form can be recastinto a predictor form as follow:

G=

x(k+1) = AK x(k)+BKz(k)

y(k) =Cx(k)+Du(k)+e(k)(4.2)

with:

AK = A−KC; BK = (B K) ; B= B−KD; z(k) =(

u(k)y(k)

)

(4.3)

To obtain the system matrices, SIM method usually employs several stages of estima-tion. The first state of estimation is to obtain high order auto-regressive input with exoge-nous variables (ARX). The ARX is well known to be able to handle closed loop data if theorder is sufficiently high. In a closed-loop experiment, external noise can be correlated tothe input of the system due to feedback. Thus closed loop identification should take propercare of de-correlating of the external noise. First, we introduce several stacked notationswhich split the past and future data based on the user selected past horizon (np) and futurehorizon (nf ):

znp(k) =

z(k−np)...

z(k−1)

; znf (k) =

z(k+1)...

z(k+nf )

(4.4)

The stacked notation rules can be used to define stacked vector for the state, output, andinnovation term as well. Furthermore, the following matrices can be formed:

Ynp = [y(p+1) . . .y(Nb)] (4.5a)

Ynf =[

ynf (p) . . . ynf (Nb−nf +1)]

(4.5b)

Znp =[znp(p+1) . . . ynp(Nb)

](4.5c)

Znp,nf =[znp(p+1) . . . ynp(Nb−nf )

](4.5d)


with Nb denotes the length of data. The ARX estimator is a linear estimator that can bewritten using matrices in (4.5a) as:

Ynp = [Ξ(np) . . . Ξ(1) D]︸︷︷︸

Ξ

[Znp

Unp

]

(4.6)

where ˆ denotes the estimated quantity, and the parameters of ARX contain the Markovparameters and have the following structure :

Ξ(i) = [Ξu(i) Ξy(i)] (4.7)

with :

Ξu(i) =

D ; i = 0

CAi−1K B ; i ≥ 1

; Ξy(i) =

0 ; i = 0

CAi−1K K ; i ≥ 1

(4.8)

The ARX parameters can be found optimally by solving the least square problem :

argmin∥∥Ynp − ΞZnp

∥∥2

F(4.9)

The typical data equation for SIM is given as follows:

Yf = ΓK Znp,nf + HUnf + GYnf +Enf (4.10)

The data equation is obtained from the definition of prediction and assuming the influenceof the initial condition has vanished whennp is sufficiently long sinceAK is stable. ThematricesΓ andK is related to the observability and controllability matrix:

Γ =

CCAK

...

CAnf −1K

(4.11a)

K =[Anp−1BK . . . AKBK BK

](4.11b)

Another important relation in SIM is the following:

Γx(k) = ΓK znp(k) (4.12)

Several SIM methods aim to find the state sequence estimate from estimation of matrixΓK . There are several alternatives to obtain the state space realization that fits the data[53]. In SSARX [54], a following pre-estimates is used:

Ynf =Ynf − HUnf − GYnf = ΓK Znp,nf +Enf (4.13)

4.1. SYSTEM IDENTIFICATION 23

The pre-estimation step makes the future input-output terms and the innovation uncor-related. Then a Canonical Variate Analysis is performed betweenYnf andZnp,nf to obtainthe estimate ofΓK . The CVA procedures:

ˆΓK = (Ynf YTnf)(−1/2)Ynf Z

Tnp,nf

(Znp,nf ZTnp,nf

)(−1/2) (4.14)

Performing Singular Value Decomposition onΓK =UΣVT and using the relation (4.12),the estimate of state sequence can be obtainX =VT

nx(Znp,nf Z

Tnp,nf

)(−1/2)Znp,nf .Another approach is the so-called optimized Predictor Based Subspace Identification

[51, 55]. The PBSIDopt differ from SSARX in the way it obtainsthe state estimate [56].Instead of using CVA, the PBSIDopt directly obtain the statesequence estimate by solvinglow-rank approximation problem:

min∥∥∥ ˆΓK Znp,nf −ΓX

∥∥∥

F(4.15)

where the estimate ˆΓK is obtained directly from the ARX argument. The optimal low-rank approximation can be solve by employing the SVD onˆΓK Znp,nf =UΣVT . Finallythe state estimate can be obtain as:

Xp = Σ1/2nx VnT (4.16)

Note that the state sequence estimate obtained by SSARX and PBSIDopt is truncated upto ordernx. The truncation order is obtained by observing the decay rate of the singularvalues inΣ. After the state sequence has been obtained, the state spacematrices can beobtained by solving two least square problems based on the state space definition.

Both methods have been used in this thesis to model the EXTRAPT2R. In EXTRAPT2R, the equilibrium parameters are highly reproducible and relatively constant during aplasma shot. The plasma operation condition and equilibrium parameters for 50 shots ofplasma experiment in which a dither signal is applied are shown in figure 4.2. In ditherexperiment, a nominal feedback controller(PID) is used to stabilize the RWM. Then adither signal is given,d in figure 4.1, to perturb the plasma. A typical dither signal forsystem identification is the Pseudo-Random Binary Signal [50].

The parameters ofnp,nf and ordernx will affect the accuracy of the model. As men-tioned before, the order of the systemnx can be chosen based on the observation of thesingular values decay. However, it is not so obvious to choose the optimalnp andnf [55],although in general highnp should be used such that it is consistent with the assumptionof the data equation. A common way to systematically choose the parameters (which haveinteger value) is using cross-validation technique.

The quality of the obtained model can be assessed by validating with real experimentaldata. It is essential that the data used for validation is different from that used to obtainthe model. A merit function is usually used to quantify the agreement of the model withreal data. One example of such a merit function is the variance-accounted-for (VAF) thatis defined as follows [57].

VAFi =

(

1−var(yexp

i − ymodeli

)

var(yexpi )

)

×100% (4.17)


VP(V

)

time (ms)

I P(k

A)

0 10 20 30 40 50 60 70 800

100

200

300

400

0

20

40

60

80

(a) Plasma currentIP and toroidal loop voltageVP

F

time (ms)

Θ

10 20 30 40 50 60 70

−0.4

−0.3

−0.2

1.4

1.5

1.6

1.7

(b) Pinch ParameterΘ and reversal parameterF

Figure 4.2: Plasma operating condition and the equilibriumparameters. The shaded arearepresent variation over 50 shots of plasma

4.2. MODAL ANALYSIS 25

The obtained VAF with the presented SIM method is shown in figure 4.3. The VAF valueindicates how well the simulated data agrees with experimental data. The VAF value of100 % represents perfect matching of experimental data.

PBSID

SSARX

output channel

VAF

(%)

10 20 30 40 50 60

80

82

84

86

88

90

92

94

96

98

Figure 4.3: VAF obtained with the SIM methods. Error bar represents the standard devia-tion of the VAF

4.2 Modal Analysis

Due to the black-box nature of the model obtained via SIM, interpretation of the modelparameters is non-trivial. The modal analysis suggests a method to distinguish the realphysical modes with mathematical modes. It is expected thatthe dominant physical modeswould be less sensitive to the data than the purely mathematical modes. This is the basicidea of bootstrap approach performed in [58–60]. Given a full set of data batches obtainfrom the experiment, a bootstrap re-sampling technique canbe used to synthesize some setof artificial batches of data. For each set of the synthesizedbatches, the SIM is performed,and the eigenmodes can be obtained from the model. The recurring eigenmodes are ex-pected to be correlated to the dominant physical modes. Furthermore, a Fourier analysiscan be performed on the eigenmodes to get some interpretation of the modes. In [58], theeigenvectors are mapped to the output and then transformed to Fourier space. In [61], akind of correlation function is introduced to quantify the correlation between the eigen-value and input-output helicity. In our experience, the latter approach seems to give betterclarity and provides an independent confirmation that the dynamics of RWM in EXTRAP


T2R can be explained by the linear cylindrical model [62] in the form:

τshellBm,n− γm,nτshellBm,n = Bextm,n (4.18)

whereτshell denotes the shell time constant of a Fourier mode andγm,n denotes the growth/decayrates. Note that for a vacuum case; the system is stable henceγm,n < 0. Some results ofthe modal analysis of the SIM model is given in figure 4.4, where the red contour indicatesa strong correlation of an eigenvalue with toroidal mode numbers. The interpretation isconsistent with the linear cylindrical model, where for vacuum case it represents the mag-netic diffusion time through the shell. The two branches represent the different diffusiontimes between the horizontal and vertical direction. The plasma case shows the broad spec-trum of RWMs in EXTRAP T2R, and makes a distinction between the stable and unstableRWMs.

4.3 Model Order Reduction

Model reduction is a method to obtain a low order approximation of a model. The simplestmodel reduction is done by transforming the state space intodiagonal form and arbitrarilyremoving the modes deemed unnecessary. However, one might expect that this methodis not optimal and often yields an unsatisfactory result since there is no clear directive onhow to select the discarded modes.

Another method is based on the so-called balanced realization of a system, which usesthe notion of controllability and observability to determine the minor modes. Looselyspeaking, a system is called controllable if there exists a control input that can transfer theinitial state to any arbitrary state in a finite time. Similarly, a system is called observableif the initial state can be determined uniquely from the output on finite time [63] . Thenotions of controllability and observability can be used todetermine which mode that isless important. The balanced truncation method aims to remove the modes that are mostdifficult to control and observe at the same time.

The balanced truncation method can be considered as the standard method for modelreduction. Most of the time, balanced truncation is the method used as a first attempt infinding a reduced order model. It has some useful properties such as stability preservationand upper bound guarantee of approximation error. However,the balanced truncation is notan optimal method. Another method named optimal Hankel [64–66]norm approximationexplicitly defines an optimality condition. More precisely, it aims to find a reduced ordermodel that minimizes the Hankel norm of the error system. Note that the Hankel norm ofa system G is given as:

||G||H = supu∈L2

∫ ∞0 y(t)2 dt∫ 0−∞ u(t)2 dt

(4.19)

which has a system interpretation as the amount of energy transferred from past input tofuture output. Similarly to balanced truncation, the optimal Hankel norm approximationpreserves stability and gives an error bound estimate.

Another approach is the so-called Krylov-based method which is usually referred as anapproximation by moment matching [65, 66]. The k-th moment of a systemG is defined

4.3. MODEL ORDER REDUCTION 27

γ 1,n

(ms−

1)

n−15 −10 −5 0 5 10 15

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

(a) vacuum case obtained with SSARX

γ 1,n

(ms−

1)

n−15 −10 −5 0 5 10 15

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

(b) vacuum case obtained with PBSIDopt

γ 1,n

(ms−

1)

n−15 −10 −5 0 5 10 15

−1.6−1.4−1.2−1

−0.8−0.6−0.4−0.2

00.20.4

(c) plasma case obtained with SSARX

γ 1,n

(ms−

1)

n−15 −10 −5 0 5 10 15

−1.6−1.4−1.2−1

−0.8−0.6−0.4−0.2

00.20.4

(d) plasma case obtained with PBSID

Figure 4.4: Experimental modal analysis of EXTRAP T2R with and without plasma. Theresult was obtained with 100 bootstrap resampling.

by the k-th derivative of the frequency response ofG at a specific frequency. The momentrepresents the coefficient of the Laurent series expansion of the systemG. It is equivalentwith the impulse response parameter when the Lauren series expansion is carried out atinfinite frequency. The Krylov-based method aims to find a reduced order model thatmatches the moment of full order model up to the k-th moment. When the full order systemis described in state space, this problem can be solved iteratively in an efficient manner. Theefficiency of computation is the main benefit of using Krylov method. However, Krylov-based method, in general, does not preserve the stability ofthe original system and no errorbound guarantee is given.

In fusion community, several application of model reduction can be found. In [67, 68]a controller was designed to control vertical instability based on a reduced order model.


imag

real ×107−18 −16 −14 −12 −10 −8 −6 −4 −2 0

×106

−4

−3

−2

−1

0

1

2

3

4

(a) complex plot of eigenvalues of the vacuum model

KrylovHankelBal.TruncCARIDDI

imag

real ×104−8 −7 −6 −5 −4 −3 −2 −1 0

−800

−600

−400

−200

0

200

400

600

800

(b) Complex plot of eigenvalues of the vacuum model zoomed inatlow frequency region

Figure 4.5: Comparison of eigenvalues of the CARIDDI model and reduced order models

4.4. GRAYBOX MODEL 29

Moreover, a model-based controller is proposed for ITER after the model is obtained fromVALEN [32] and reduced via balanced truncation. Similar approach is also carried out inNSTX [36,37].

The described methods have been used for EXTRAP T2R [PAPER II], to obtain areduced order of the shell. The shell, which is a stable system, can be obtained via finiteelement codes such as CARIDDI code [69]. The CARIDDI code usually yields a state-space model in the order or 103. The complexity of the model naturally comes from thediscretization of the complex 3D conducting structures which include gaps, ports, coilshape, etc. For EXTRAP T2R a state space of order 4285 seems toreasonably model theshell [70]. However, using such complex model for control design can be a problem sincethe standard model based control design such as Linear Quadratic Gaussian (LQG) yields acontroller that has the same order as the model. To this end, the model reduction techniqueis used to reduce the complexity of the model up to 97%. Figure4.5 shows the comparisonof the eigenvalues for the CARIDDI model and reduced order models. The reduced ordermodel seem to remove most of the high-frequency eigenvalues. The eigenvector of theCARIDDI model corresponds to the current distribution in the 3D structure. Figure 4.6shows the comparison of current distribution for the fast and slow eigenmodes. The sloweigenmode corresponds to current distribution with long spatial wavelength whereas thefast eigenmode corresponds to the short spatial wavelengthas seen in the erratic behaviorin the current distribution. These fast eigenmodes can be seen as the least controllable andleast observable modes, which are the first candidates of theeigenmodes to be removed inthe balanced truncation and optimal Hankel approximation.For the Krylov based method,the algorithm is set to match the Lauren series expansion at frequency 0, thus the removalof high frequency content is obtained by construction.

4.4 Graybox Model

Graybox model stems from the idea to combine physics-based model (”white-box”) modeland empirical model (”black-box”). In EXTRAP T2R, it is envisioned that Graybox ap-proach might give some improvement compared to using white-box or black-box alone.

In [PAPER I], a clear separation of the plasma model and shellmodel is made wherethe plasma is modeled using SIM and the shell is modeled from the CARIDDI code. Thecascade interconnection (see figure 4.7) of the model is based on physics understanding thatthe measured field is a resulting contribution of both the plasma and shell. Furthermore, thefield produced by the plasma is driven by the field produced by the shell (not necessarilythe same as the exerted field by the magnetic coils) [62].

The proposed gray-box approach is envisioned to have several benefits. The use ofCARIDDI code should incorporate the 3D effects of the conducting shell and and clearinterpretation of the state variables. Furthermore, the use of system identification use theexperimental data to compensate the approximation error arising from using model reduc-tion for the shell model.

In the gray-box approach of [PAPER I], the cascade model can be used to reconstructthe field from the vacuum and plasma separately. The cascadedmodel contains a separate


(m)

(m)(m)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−0.2

0.2

(a) Current distribution of the slow eigenmode of the CARIDDI model

(m)

(m)(m)

−1

−0.50

0.5

1

−1

−0.5

0

0.5

1

−0.2

0.2

(b) Current distribution of the fast eigenmode of the CARIDDI model

Figure 4.6: Comparison of slow and fast eigenmodes of the CARIDDI model

4.4. GRAYBOX MODEL 31

u uV yV

yP

+ y

GA GV

GP

G

Figure 4.7: Block diagram of interconnection between amplifier, vacuum response andplasma response

vacuum and plasma structure that can be exploited in the control design process. Anotherimportant application of a model is state estimation, in which a real-time measurement isused to reconstruct the state variables. The state variables might belong to some physicalquantity that is not directly available by real-time diagnostics [71].

Chapter 5

Control of Resistive Wall Modes

Model-based control has been used successfully in many existing fusion devices, and it islikely to be used in future fusion devices such as ITER [32, 71]. One particularly matureapplication of model-based control in fusion is the controlof equilibrium and vertical sta-bility that have been used effectively in many devices [72].A control model can describethe effects of inputs (including internal, self-driven dynamics) to the plasma response. Oneway to interpret a model-based control approach is to see it as an ”inverse problem” inwhich we want to find the appropriate input given the desired plasma response. Advancedcontrol theory provides a variety of solutions to accomplish this ”inverse problem” whichare not limited to the linear model that is used in this thesis. Some examples of model-based control design methodology have been considered in this thesis [73–75]:

1. Linear Quadratic Gaussian (LQG) in which the controller includes an optimal stateobserver and a state feedback regulator. The observer is typically designed to min-imize the influence of noise, where the regulator is designedto minimize measurederror or actuator power. This type of controller has been widely proposed for RWMstabilization in the tokamak configuration [32, 36, 76], andnumerical results shownincrease plasma performance achieved by using LQG when compared to a simplePID controller.

2. Model Predictive Control (MPC) [77] in which the controller predicts the futureresponse of the system in real time and calculates an optimized control response. Theoptimized control response can be found while satisfying a well-defined constraint.

A brief description of both approaches will follow.

33

34 CHAPTER 5. CONTROL OF RESISTIVE WALL MODES

5.1 Linear Quadratic Gaussian

Traditional Linear Quadratic Gaussian control assumes a linear system with the additionof the process noisewd and measurement noisewn:

G=

x(k+1) = Ax(k)+Bu(k)+wd(k)

y(k) =Cx(k)+Du(k)+wn(k)(5.1)

For simplicity we setD= 0; extension to the non-zeroD is straight forward [63]. The exactvalue of the noise does not need to be known, only its statistical property is needed. Theprocess and measurement noise are usually assumed to be a zero-mean Gaussian processwith static covariancesRwd andRwn.

The LQG control problem is to find optimal control inputu which minimizes the fol-lowing cost function:

J = E

[

limT→∞

1T

ΣTk=0(x(k)

TQx(k)+u(k)TRu(k))

]

(5.2)

whereQ andR are positive definite constant weighting matrices andE is the expectationoperator.

The solution to the LQG control problem is turned out to be thecombination of thedeterministic (noiseless) linear quadratic regulator andoptimal state estimation problem.The solution to the linear quadratic regulator problem is a simple state feedback lawu(k) =−KLQRx(k). WhereKLQR is a constant matrix that is found from solving the algebraicRiccati equation:

ATSA−S− (ATSB)(BTSB+R)−1(BTSA)Q= 0 (5.3a)

K = (BTSB+R)−1(BTSA) (5.3b)

Note that the state feedback requires the state informationx(k) which is not always directlymeasured. This leads to the second part of the LQG which is to construct an optimal stateestimator in the form of:

x(k+1) = Ax(k)+Bu(k)+K f (y−Cx(k) (5.4)

where the optimal observer gainK f is chosen such that the covariance of the state estima-tion errorE

[(x(k)− x(k))T(x(k)− x(k)

]is minimized. The solution in finding optimalK f

is the well-known Kalman filter [78,79], which in the steady state limit is found by solvingthe following Riccati equation:

P= APAT +Rwd− (APCT)(Rwn+CPCT)−1(CPAT) (5.5a)

K f = (APCT)(Rwn+CPCT)−1 (5.5b)

Kalman filter is widely used in many applications due to its ability to obtain the stateestimation from a noisy measurement. The use of Kalman filteris also has become morecommon in fusion community [35,80].

5.2. MODEL PREDICTIVE CONTROL 35

5.2 Model Predictive Control

MPC utilizes the model of a system to predict the future response. For a model in the formof discrete-time state space illustration of the prediction is straightforward. Consider thestate-space withe(k) = 0:

x(1) = Ax(0)+Bu(0) (5.6)

which is the one step ahead response of the system at initial conditionx(0) andu(0). Theresponse atk= 1 is:

x(2) = Ax(1)+Bu(1) (5.7)

Substituting (5.6) to (5.7) can remove the dependency ofx(1) in (5.7). This procedure canbe generalized for arbitrary finite time horizonsNp:

x(1) = Ax(0)+Bu(0)

x(2) = A2x(0)+ABu(0)+ABu(1)

x(3) = A3x(0)+A2Bu(0)+ABu(1)+Bu(2)

...

x(Np) = ANpx(0)+ANp−1Bu(0)+ . . .+ABu(Np−2)+Bu(Np−1)

or written in a matrix form :X := Φx(0)+ΓU (5.8a)

with :

X :=

x(1)x(2)

...x(Np)

; U :=

u(0)u(1)

...u(Np−1)

; Φ :=

AA2

...ANp

(5.8b)

Γ :=

B 0 . . . 0AB B . . . 0...

..... .

...ANp−1B ANp−2B . . . B

(5.8c)

The set of equations (5.8) can be used to predict the future response of a system for a finitetime window provided the initial conditions and the sequence of inputs are available. Theaim of MPC essentially is to find the sequence of inputs that optimize the future responseof the system given an initial condition and set of constraints. Formally stated:

U∗ = argminJ = x(Np)TPx(Np)+

Np−1

∑i=0

(x(i)TQx(i)+u(i)TRu(i)

)(5.9a)


subject to :

x(k+1) = Ax(k)+Bu(k),∀k= 0, ...,Np−1 (5.9b)

u(k) ∈ U,∀k= 0, ...,Np−1 (5.9c)

x(k) ∈ X,∀k= 0, ...,Np (5.9d)

When the constraint is the upper and lower limit of actuator,the optimization problem canbe written as a convex optimization problem:

J =12

θ THθ +θ TFx s.t: θmin ≤ θ ≤ θmax (5.10)

with all the necessary matrices are given by:

θ =

u(0)u(1)

...u(Np−1)

; H = 2(Ψ+ΓTΩΓ

); Fx = 2ΓTΩΦx(0); (5.11)

Ω = diag(Q,Q, . . . ,P); Ψ = diag(R,R, . . . ,R); (5.12)

Φ =

AA2

...ANp

; Γ =

B 0 . . . 0AB B . . . 0...

..... .

...ANp−1B ANp−2B . . . B

(5.13)

Here,diag(.) denotes the diagonal block operator. The matricesP,Q,R are user-definedpositive definite matrices that describe the trade-off of the cost function term [81, 82]. Inpractice, the implementation of MPC uses the so-called receding control horizon scheme.In receding horizon control, the optimization is solved at one time to obtain the futuresequence of optimal input. However only the input in the sequence that corresponds to thenext time step is applied to the system and the process is repeated for every time instance.

The MPC control action is similar to LQG in the sense that it requires the state infor-mation. A Kalman filter can be used in combination with MPC if the state of the system isnot directly measure. A fundamental difference between MPCand LQG is that the MPCsolves the optimization problem using a moving time window while LQG uses a fixedwindow. The use of a moving time window includes the ability of MPC to satisfy a set ofconstraints on plant variables.

Fast Gradient MPC

The computational cost of MPC is relatively high compared toother model-based controlmethods. The feasibility of implementing MPC depends on therequired time to solve theoptimization problem relative the time scale of the system.Originally, MPC was developedfrom an industrial application with slow time scale [83–85]. However, recent advancement

5.3. RWM CONTROL LOGIC 37

in optimization theory [86–88] and increase of computational power has allowed the ap-plication of MPC at MegaHertz rate [89]. Fast Gradient algorithm can be used to solve theMPC optimization problem in an efficient manner:

Algorithm 1 Fast Gradient MPC

Define : imax, θ0, θ0 = θ0, 0<√

µ/L ≤ a0 < 1Compute : ai , bi , ∀i from a2

i+1 = (1−ai+1)a2i +

µL ai+1 and

bi = (ai(1−ai))/(a2i +ai+1)

1: for i = 0 to imax do2: ∇Ji = Hθi +Fx

3: θi+1 = max

θmin,min

θi −1L ∇Ji ,θmax

4: θi+1 = θi+1+bi(θi+1−θi)5: end for

Finding the parametersL andµ is crucial for fast gradient algorithm. In the context ofMPC problem (5.9), where theH is a positive definite and symmetric matrix, the parame-tersL andµ are the maximum and minimal eigenvalue ofH.

The underlying idea of the fast gradient method is to drop thestrict condition of form-ing the relaxation sequence as in the conventional, traditional method and make use of anestimate sequence. In [87,88] an estimate on the upper boundof the residual to the optimalvalue is provided, which in turn can be used to obtain the required maximum iterationimax

that guarantee a particular value residual upper bound.In some cases, the convergence of the fast gradient MPC couldbe too slow for it to be

implemented. There are some adjustments that can be used to improve the convergence;such as the use of a warm starting point (use the previous result as an initial guess of thenext iteration), numerical conditioning and generalization of L to matrix instead of scalarvalue [90]. In our experience combining these methods with amodest maximum numberof iterations, is sufficient to achieve good performance by MPC.

5.3 RWM control logic

For RWM control, there are several types of control-logic have been proposed [91,92]. Wewill try to summarize some of the most common control logic types and show that theycan be seen as different weighting matrices in the LQG and MPC. For simplicity, the directfeedthrough term is assumed to be 0 (D = 0) in the state space model.

The most basic control logic for RWM is perhaps the Intelligent Shell, in which thecontroller aims to suppress the raw value of the measured field. Following from the outputequation, the IS control logic can be achieved by setting theQ matrix as:

Q=CTQISC (5.14)

Thus the state penalty in the cost function essentially becomes the output penalty withweightQIS.


Another commonly used control logic is the so-called Raw Mode Control(RMC), wherethe feedback value is the field in Fourier space. Initially the Raw Mode Control(RMC)[29, 29] logic employs an online spectral analyzer to resolve the mode number. However,this control logic can also be implemented simply by modification of weighting matrices.This follows from the relation thatyf (k) = WDFTy(k), whereyf is the output in Fouriermode andWDFT is a Discrete Fourier Transform (DFT) matrix. The weightingmatricesfor RMC control logic then:

Q=CTW†f QRMCWfC (5.15)

where† denotes the conjugate transpose andQRMC is a diagonal matrix for which its di-agonal entry corresponds to a weighting value of a specific mode number. The state costis then transformed to the 2-norm of the weighted Fourier transformed output signals. Theuse of RMC control logic allows incorporating physics information such as the exclusionof axisymmetric mode and error field mode number in the control objective[PAPER V].

If the case of cascade model such as in [PAPER I], a separationof plasma responsefrom the measured field is possible through weight modification. The state variables ofthe shell and plasma can be estimated separately in real-time using the state estimator. Tothis end, the field contribution of the shell and plasma can bereconstructed. By usingweight modification, the contribution of the shell can be excluded in the cost function.Such an approach is similar to the Explicit Mode Control (EMC) logic. The EMC iscommonly used in the tokamak, where the plasma response is obtained by subtraction ofthe measured field with the modeled vacuum response and used to measure the resonancefield amplification factor by the RWM [93]. A similar approachlike the EMC is used inthe RFP [94], to show that the RWM can be unlocked from the wallby inducing rotationexternally.

5.4 Experimental Results

RWM Stabilization

Here are shown the results of RWM stabilization using the approaches mentioned above.Both LQG and MPC are able to stabilize the RWMs as can be seen infigure 5.1 andfigure 5.2, where the RMC control logic is used. The plasma termination does not seemto be attributed to RWM instability. A possible cause of plasma termination is due tothe incomplete suppression of the field. The residual field can interact with the rotatingtearing modes and cause it to lock to the wall. Furthermore since the tearing modes inEXTRAP T2R have a high mode number that cannot be resolved by the coils, some ofthese modes might appear as aliased modes which affect the performance of the modecontroller. Furthermore, the plasma current power supply in EXTRAP T2R sets the limitof the duration of the discharge.

5.4. EXPERIMENTAL RESULTS 39

(mT

)

time (ms)

shot 25940to

roid

alm

ode

(n)

0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

−15

−10

−5

0

5

10

15

(a) The obtained output of EXTRAP T2R with active LQG; shownfor different Fourier toroidal modes

(A

)

time (ms)

shot 25940

toro

idal

mod

e(n

)

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5

−15

−10

−5

0

5

10

15

(b) The obtained actuator current of EXTRAP T2R with active LQG;shown for different Fourier toroidal modes

Figure 5.1: Plasma shot obtained with the designed LQG in EXTRAP T2R. The y-axisshows different toroidal mode numbers and the color-codingshows the amplitude of themode as it evolves over time.


time (ms)

(mT

)

shot 25331to

roid

alm

ode

(n)

0 20 40 600

0.05

0.1

0.2

−15

−10

−5

0

5

10

15

(a) The obtained output of EXTRAP T2R with active MPC; shownfor different Fourier toroidal modes

(A

)

time (ms)

shot 25331

toro

idal

mod

e(n

)

0 20 40 600

0.5

1

1.5

2

−15

−10

−5

0

5

10

15

(b) The obtained actuator current of EXTRAP T2R with active MPC;shown for different Fourier toroidal modes

Figure 5.2: Plasma shot obtained with the designed MPC in EXTRAP T2R. The y-axisshows different toroidal mode numbers and the color-codingshows the amplitude of themode as it evolves over time.

5.5. ON ERROR FIELD ESTIMATION AND COMPENSATION 41

Extension to output tracking

The initial formulation of the LQG and MPC focused on finding the controller that opti-mally stabilizes the system. In other words the aim is to achieve near zero amplitude forthe unstable modes. For possible physics experiments it canbe desirable to control thespectrum so that one or more modes do not have zero amplitude but have a controlled am-plitude and rotation. In this case where it is desired to track an arbitrary reference, somemodification of the controller formulation needed. One alternative to achieve good track-ing performance is to introduce an extra state variable which corresponds to the integral oferror between the output and reference.

In EXTRAP T2R the reference, the vector r in figure 4.1, is usually given in term ofamplitude or phase in Fourier domain. For a simple reference, this can be computed inreal-time by employing an inverse DFT operation. For a more complex reference, an off-line computation of the reference can be done to reduce the real-time computation load.Some of the results with output tracking can be seen in figure 5.3 and figure 5.4. For thiscampaign, the LQG based on a gray box model is used [PAPER I].

The output tracking capability of the controller in EXTRAP T2R has several physicsstudy applications. (i) The controller can be used to generate a static resonant magneticperturbation at the edge of the plasma that couples to the TMs. The purpose is to studythe locking-unlocking mechanism of TMs [95,96].(ii)The feedback control can be used togenerate rotating magnetic perturbation that interacts with a stable RWM. The interactionis observed in the modulated amplitude of the field and can be used to assess the intrinsicerror field [97].

5.5 On Error Field Estimation and Compensation

Error field always exists in all magnetic confinement devices. Any small geometrical im-perfection can lead to error field. Error field can interact with stable RWM and leads togreater error field and braking of plasma rotation. This effects is called resonant field am-plification and has been observed in many devices. Error fieldcan also be a trigger ofmarginally stable RWM that leads to disruption [98].

One possible approach of suppressing the error field is by applying a pre-programmedperturbation [99]. The pre-programmed perturbation can beobtained by dedicated exper-iments to estimate the error field. The most common way to estimate error field is byapplication of a slowly increasing external perturbation at different toroidal phase to gen-erate a locked mode [100–102]. Another approach makes use ofa rotating perturbationthat excites a stable RWM; the error field is the assessed fromthe non-uniformity of therotation [97]. More recently, an adaptive controller is proposed to compensate the errorfield in DIII-D tokamak [103].

The suppression of error field sometimes overlaps with RWM control since they sharethe same actuators. In fact, the control logic such as RMC andIS are expected to suppressthe error field to some extent since they do not discriminate the field based on the plasmacontribution or vacuum contribution. The conventional approach is to set the feedbackgain to a very high value to suppress the static error field. However, the dynamics of


(mT

)

time (ms)

shot 25975

toro

idal

mod

e(n

)

10 15 20 25 30 35 40 45

0.02

0.04

0.08

0.1

0.12

−15

−10

−5

0

5

10

15

(a) The obtained output of EXTRAP T2R with LQG; shownfor different Fourier toroidal modes

(A)

time (ms)

shot 25975

toro

idal

mod

e(n

)

10 15 20 25 30 35 40 45

0.20.40.6

1

1.41.61.82

−15

−10

−5

0

5

10

15

(b) The obtained actuator current of EXTRAP T2R LQG;shown for different Fourier toroidal modes

(mm

)

∠θ

(πra

d)

∠φ (π rad)

shot 25975 (t = 32 ms)

0 0.5 1 1.5

−0.8

−0.6

−0.4

0

0.2

0.4

0.6

0.8

0

0.5

1

1.5

(c) Last closed flux surface of EXTRAP T2R with a sustainedn= 6 mode

Figure 5.3: Tracking of the unstable RWMn= 6 using LQR.


(mT

)

time (ms)

shot 25977

toro

idal

mod

e(n

)

10 15 20 25 30 35 40 450

0.02

0.04

0.08

0.1

0.12

−15

−10

−5

0

5

10

15

(a) The obtained output of EXTRAP T2R with active MPC;shown for different Fourier toroidal modes

(A)

time (ms)

shot 25977

toro

idal

mod

e(n

)

10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−15

−10

−5

0

5

10

15

(b) The obtained actuator current of EXTRAP T2R with activeMPC; shown for different Fourier toroidal modes

(mm

)

∠θ

(πra

d)

∠φ (π rad)

shot 25977 (t = 30 ms)

0 0.5 1 1.5

−1

−0.5

0

0.5

1

0

0.5

1

1.5

(c) Last closed flux surface of EXTRAP T2R with a sustainedn= 10 mode

Figure 5.4: Tracking of the unstable RWMn= 10 using LQR.


DOB

u y

wu

wu

G

G−1

FDOB

− +

+

Figure 5.5: Block diagram of disturbance observer

the plasma and noise might set a limit on the value of the gain.Thus the separation oferror field compensator and RWM control can be beneficial to achieve proper error fieldcompensation without compromising the RWM control stability.

From control engineering perspective, error field can be seen as a disturbance or un-known input to the system which can affect the state estimation process. The disturbanceobserver(DOB) and unknown input observer(UIO) are some of the common tools that havebeen adapted to estimate error field in EXTRAP T2R[PAPER III]. A DOB is illustrated infigure 5.5, where from the block diagram the estimated disturbance relation can be writtenas:

d(k) = FDOB[G−1Gu(k)+ G−1Gd(k)−u(k)

](5.16)

The idea of DOB is to use an inverse of a model to reconstruct the input from the measuredoutput, and by comparing the reconstructed input to the actual input, the unknown dis-turbance can be estimated. The performance of the DOB is characterized by two factors.First is the accuracy of the inverse modelG−1, which is related to the quality of model thatwe have. The inverse plant can be an issue when dealing with non-minimum phase [63]system. Extension to non-minimum phase system can be seen in[104,105]. Furthermore,the DOB is limited to a system which has the same number of inputs and outputs. Thesecond factor is the filterFDOB which ideally should be set to 1. However, a static filtermight amplify sensor noise or make the DOB cannot be implemented due to the inverseplant dynamics. In most cases, the filter is set to be a low passfilter in which the cut-offfrequency is related to the expected frequency of the disturbance.

The UIO approach is slightly different than DOB. In UIO, an assumption of the dis-turbance dynamic is required. Error field is assumed to be static, and the state observer is


modified to incorporate the error field dynamic:

x(k+1) = Ax(k)+B(u(k)+d(k))+Ke(k)

y(k) =Cx(k)(5.17a)

xEF(k+1) = AEFxEF(k)+KUIOe(k)

d(k) =CEFxEF(k)(5.17b)

The DOB and UIO can provide an online and off-line estimationof error field. Fig-ure 5.6 shows multiple error field modes can be estimated simultaneously and some ofthe results are consistent with the previously estimated error field in EXTRAP T2R [97].The online estimation of error field can be coupled with the existing RWM controller andprovides a dedicated error field compensator.

time (ms)

[mT

]shot 25609

toro

idal

mod

e(n

)

15 20 25 30 35 400

0.5

1

1.5

−15

−10

−5

0

5

10

15

(a) DOB

time (ms)

[mT

]

shot 25609

toro

idal

mod

e(n

)

15 20 25 30 35 400

0.5

1

1.5

−15

−10

−5

0

5

10

15

(b) UIO

UIODOB

toroidal mode (n)

ampl

itude

(mT

)

−15 −10 −5 0 5 10 150

0.5

1

(c) time average of the estimated amplitude

UIODOB

toroidal mode (n)

phas

e(r

ad)

−15 −10 −5 0 5 10 15

−π

0

π

(d) time average of the estimated phase

Figure 5.6: Estimated amplitude and phase of error field withDOB and UIO

Chapter 6

Summary and Discussion

The main scientific results are summarized in this section. The primary focus of this thesisis to present a model based approach to stabilizing the RWM instability. The contributionsof this thesis can be categorized as modeling of RWM, controlsystem design and errorfield estimation/compensation.

6.1 Summary of Papers

Paper I: Graybox Modelling of EXTRAP T2R with Vacuum-PlasmaSeparation and Optimal Control Design of Resistive Wall Modes

In this paper, we combine the use of the CARIDDI model and the system identificationto model the EXTRAP T2R RWM dynamics. The shell and plasma response are mod-eled separately, and a cascade interconnection is proposed. The shell is modeled by theCARIDDI code whereas the plasma is modeled by system identification. The resultingmodel is used to design an LQG controller where it is shown that the gray box modelallows the use of different control logic through weight modification. The designed con-troller is tested experimentally against different plasmaequilibrium parameters.

Paper II: Reduced Order Modelling of Resistive Wall Modes inEXTRAPT2R Reversed-Field Pinch

In this paper, the finite element code CARIDDI is used to modelEXTRAP T2R. CARIDDIis a 3D code which solves the integral formulation of the eddycurrent problem. Theresulting model is obtained as a state space model with over 4000 states. The complexityof the model prevents it to be used in real-time. Thus the paper aims to obtain low orderapproximations of the CARIDDI model. Several model order reduction techniques suchas balanced truncation, optimal Hankel norm approximation, and Arnoldi method wereapplied to obtain low order models, and the results are compared.

47

48 CHAPTER 6. SUMMARY AND DISCUSSION

Paper III: Improved model predictive control of resistive wall modes by errorfield estimator in EXTRAP T2R

This paper proposed to use the empirical model obtained by system identification to es-timate the intrinsic error field in EXTRAP T2R. From a controlengineering perspective,error field can be seen as an unknown input to the system, and there exist a well-knownconcept to estimate it. In particular, two concepts were tested which are the DOB andUIO. Both methods simultaneously determine multiple harmonics of the error field andsome modes are consistent the apriori known error field. The DOB and UIO can also usein a real-time setting and coupled with the RWM controller thus improving overall fieldsuppression.

Paper IV: Implementation of model predictive control for re sistive wall modestabilization on EXTRAP T2R

This paper shows the first experimental result of using MPC tostabilize RWMs. The useof MPC is made possible due to the upgrade of the EXTRAP T2R control hardware. TheMPC was designed based on an empirical model which is obtained by the SSARX subspaceidentification method. The performance of the MPC is compared with a conventionalPID controller. Despite the computational complexity the MPC is shown to be feasibleto be implemented in EXTRAP T2R through the use of the fast gradient method. Theexperimental result shows that the MPC outperforms the conventional PID controller insuppressing the radial magnetic field perturbation.

Paper V: Design and Operation of Fast Model Predictive Controller forStabilization of Magnetohydrodynamic Modes in a Fusion Device

In this work, the design process of the MPC is discussed in more detail. The system identi-fication process of obtaining a control-oriented model and validation of the model is shown.The mode control logic extension for MPC is discussed in moredetail. It is shown that themode control logic in MPC can be done simply by modification ofthe weighting matricesthus avoiding additional computational load which can arise if an online DFT Analyzer isused instead. The feature of mode control logic in MPC is discussed, in particular, how totreat the error field in EXTRAP T2R.

6.2 Discussion

This thesis shows that tools from control theory can be useful in solving some of thephysics issues in fusion research in particular with RWMs stabilization. The availabil-ity of a model thus can provide a mechanism to design a controller with high confidenceto succeed. To this end, the availability of a real-time capable model with good predic-tive quality is crucial. The empirical modeling through SIMhas shown to be sufficientin capturing the RWM dynamics, but there are still some issues that need to be addressedrelated to the uncertainty of the model. This uncertainty can be related to the validity of the

6.2. DISCUSSION 49

main assumption used for linear representation: constant equilibrium condition and moderigidity. A natural extension of the SIM used in this thesis is to try to model the process asa linear parameter varying (LPV) system. For a LPV system thesystem dynamics dependon some scheduling parameters. Recently, the PBSIDopt has been extended to model LPV.Thus it might be interesting to use the method to model the RWMin EXTRAP T2R anduse the equilibrium parameters as the scheduling parameters [106–108].

Some of the MPC features have not yet been fully explored, in particular, the abilityto handle state and input constraints. The input constrainthandling feature of MPC can beuseful when attempting to stabilize the RWM with a subset of actuator coils, which is morerelevant to a tokamak case. In a tokamak, the coils usually donot cover the whole torus.Thus the sideband effect can play a crucial role. Optimal actuator selection is still an openproblem which needs to be considered when attempting to stabilize the RWM with subsetsof coils.

The state constraint has not been used in this thesis. The state constraint has the poten-tial to formulate the ”forbidden” region which the MPC should try to avoid. Formulatingphysics requirement in term of the state is difficult, especially when using empirical modelsince the state variables do not necessarily have a physicalinterpretation. The combinationof finite element model, model reduction, and SIM simultaneously potentially can help incharacterizing physics-relevant state constraint.

Chapter 7

Conclusion

A model-based approach to stabilize resistive wall mode is pursued in this thesis. Theapproach relies on the availability of a real-time capable model that focuses on predictivequalities. The control oriented model can be obtained through i)a first principle model andemploying a model reduction technique, ii)empirical modeling with system identification,and iii) combination of the first principle and empirical modeling.

It is shown that the model-based controller allows the user to incorporate several physicsrequirements related to RWM stabilization. Another application of the model is to estimateand compensate the inherent error field in real-time. The results obtain in this thesis arebased on the experiments performed on the EXTRAP T2R RFP, however, the presentedalgorithms are generic and in principle can be adapted for other type of configuration suchas a tokamak.

51

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