facolta di ingegneria` corso di laurea magistrale...

UNIVERSITA DEGLI STUDI DI ROMA

TOR VERGATA

FACOLTA DI INGEGNERIA

CORSO DI LAUREA MAGISTRALE IN INGEGNERIA

ENERGETICA E NUCLEARE

A.A. 2010/2011

Tesi di Laurea

Static controller for MAST-Upgrade scenario development and

simulation

RELATORE CANDIDATO

Dott. Daniele Carnevale Fabio Tocchi

CORRELATORI

Dott. Luigi Pangione

They’re not that different from you, are they? Same haircuts. Full of hormones, just

like you. Invincible, just like you feel. The world is their oyster. They believe they’re

destined for great things, just like many of you, their eyes are full of hope, just like

you. Did they wait until it was too late to make from their lives even one iota of

what they were capable? Because, you see gentlemen, these boys are now fertilizing

daffodils. But if you listen real close, you can hear them whisper their legacy to you.

Go on, lean in. Listen, you hear it? - - Carpe - - hear it? - - Carpe, carpe diem,

seize the day boys, make your lives extraordinary.

John Keating The dead poet society 1989

Contents

Acknowledgements 1

Abstract 2

1 Nuclear fusion 4

1.1 Nuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Lawson criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 The product neτE . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 The triple product neTτE . . . . . . . . . . . . . . . . . . . . 9

1.3 Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 MAST and MAST-Upgrade 15

2.1 MAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 MAST-Upgrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Movitation for MAST-Upgrade . . . . . . . . . . . . . . . . . 20

2.2.2 Key components of the funded stage 1 of the upgrade . . . . . 22

2.3 Super X divertor in MAST-Upgrade . . . . . . . . . . . . . . . . . . . 24

2.3.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.2 Poloidal field coils . . . . . . . . . . . . . . . . . . . . . . . . . 28

CONTENTS I

CONTENTS

3 Shape control 35

3.1 The control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Dynamic control . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Fiesta code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Passive currents . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Calculate the sensitivity matrix . . . . . . . . . . . . . . . . . 40

3.2.3 Shot simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.4 Relationship between flux and control problem during the shot 48

3.2.5 Control function . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.6 Using gaps as control parameters . . . . . . . . . . . . . . . . 51

3.3 Studying a new parameters set . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 Detecting parameters . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.2 Connection length . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.3 Minimum distance . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Control function: Newton-Raphson algorithm . . . . . . . . . . . . . 62

3.4.1 Algorithm description . . . . . . . . . . . . . . . . . . . . . . 62

3.4.2 Finding minima . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Newton-Raphson application . . . . . . . . . . . . . . . . . . . . . . . 64

3.5.1 Square function . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5.2 Parameter’s weights . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.3 Coil’s weights to avoid saturation . . . . . . . . . . . . . . . . 69

3.6 Gradient algorithm apllied as a dynamic coil weight . . . . . . . . . . 74

4 Simulations results 78

4.1 Configuration limits in MAST Upgrade SXD scenarios . . . . . . . . 78

CONTENTS II

CONTENTS

4.1.1 Changing plasma shape . . . . . . . . . . . . . . . . . . . . . 81

4.1.2 Scenario with high internal inductance . . . . . . . . . . . . . 83

4.1.3 Database creation . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Operative space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Comparison: gradient algorithm and simple inversion . . . . . . . . . 91

5 Conclusions and future developments 95

List of figures 97

Bibliography 101

CONTENTS III

Acknowledgements

I owe a great many thanks to a great many people who helped and supported me

during the writing of this thesis.

My deepest thanks to Lecturers, Prof. Daniele Carnevale and Doc. Luigi Pangione

the Guides of the project for guiding and correcting various documents of mine with

attention and care. They has taken pain to go through the project and make neces-

sary correction as and when needed.

I express my thanks to the the Prof. Luca Zaccarian of Universit´a di Roma Tor

Vergata, for extending his support.

My deep sense of gratitude to Geoff Fishpool and Grahm McArdle physicists in Cul-

ham centre fusion energy (CCFE) for their support and collaboration.

Thanks and appreciation to the helpful people at CCFE, for their support.

I would also thank my Institution and my faculty members without whom this project

would have been a distant reality.

I also extend my heartfelt thanks to my family and especially to my sister Leonilde

who has been always close to me for helping and making possible this project.

Lastly but not for this reason less important I would thank my friends in Rome and

in London for their support, they gave me unforgettable memories.

Abstract 1

Abstract

This thesis has been developed in the context of the scientific research on controlled

thermonuclear fusion. The final aim of the scientists devoted to this field is to achieve

the necessary knowledge to create a thermonuclear fusion reactor. This device would

allow commercial production of net usable power by a nuclear fusion process. This

source of energy, with respect to nuclear fission energy production, is cleaner and safer.

More specifically, this work has been realized through the collaboration between “Uni-

versita degli studi di Roma Tor Vergata” and “Culham Centre for Fusion Energy” that

runs MAST experiment, the Upgrade of this device has been considered for carrying

out this Master thesis. The professionals who have made this collaboration possi-

ble are the professors Luca Zaccarian and Daniele Carnevale from “Dipartimento di

Ing. Informatica, sistemi e produzione”, doctor Luigi Pangione and Graham McAr-

dle from “Culham Center for Fusion Energy”. The subject of this work has been

detecting the requirements for plasma controller for the Super X Divertor in MAST-

Upgrade, moreover has been explored the limit configurations on this new device. In

order to do so, a new parameters set has been chosen and a new control function has

been developed using the Newton-Raphson algorithm and the gradient algorithm as

well for solving the optimization problem with the final aim to use it also in the real

time control. The first chapter of this work represents a general introduction to the

physical principles of thermonuclear fusion, it also describes the purposes of magnetic

Abstract 2

Abstract

confinement and how this confinement is performed by tokamaks. There is also a

general overview of spherical tokamaks. Chapter 2 contains the description of MAST

and MAST-Upgrade, the last one will make the first plasma in 2015, the main features

and performance will be described, a comparison between them will be explained with

regards to the advantages that the Upgrade will bring to the fusion nuclear research.

It will be also described the Super X Divertor, that will be one of the most important

feature in MAST-Upgrade, the possible configurations which this device will allow

in the operating space will be illustrated. Chapter 3 reports the shape control issue

in Fiesta code simulator, the new parameters set, Newton Raphson and the gradient

algorithms used for the new control function will be described, with regards to the

performance and options that this tool offers. In the last chapters the results are

summarized and possible future developments and applications for the present work

are considered.

Abstract 3

Chapter 1

Nuclear fusion

In this chapter will be described the fusion nuclear process and Law-son criterion, a brief overview on the tokamak devices will be pre-sented as well.

1.1 Nuclear fusion

Nuclear fusion is, in a sense, the opposite of nuclear fission. Fission, which is a mature

technology, produces energy through the splitting of heavy atoms like uranium in

controlled energy chain reactions. Unfortunately, the by-products of fission are highly

radioactive and long lasting. In contrast, fusion is the process by which the nuclei

of two light atoms such as hydrogen are fused together to form a heavier (helium)

nucleus, with energy produced as by-product. This process is illustrated in Figure 1.1

where two isotopes of hydrogen (deuterium and tritium) combine to form a helium

nucleus plus an energetic neutron. In this reaction a certain amount of mass changes

form to appear as the kinetic energy of the products, in agreement with the equation

E = ∆mc2. Fusion produces no air pollution or greenhouse gases since the reaction

product is helium, a noble gas that is totally inert. The primary sources of radioactive

by-products are neutronactivated materials (materials made radioactive by neutron

bombardment) which can be safely and easily disposed of within a human lifetime,

4

Cap. 1 Nuclear fusion §1.1 Nuclear fusion

Figure 1.1: Fusion nuclear process.

in contrast to most fission by-products which require special storage and handling

for thousands of years. The primary challenge of fusion is to confine the plasma, a

state of matter similar to gas in which most of the particles are ionized, while it is

heated and its pressure increases to initiate and sustain fusion reaction. There are

three known ways to do so:

• Gravitational confinement: the method used by the stars. The gravitational

forces compress matter, mostly hydrogen, up to very large densities and tem-

peratures at the star-centers, igniting the fusion reaction. The same gravita-

tional field balances the enormous thermal expansion forces, maintaining the

thermonuclear reactions in a star, like the sun, at a controlled and steady rate.

Unfortunately huge gravitational forces, not available on Earth, are required.

• Inertial confinement: a fuel target, typically a pellet containing a mixture of

5

Cap. 1 Nuclear fusion §1.2 Lawson criterion

deuterium and tritium, is compressed and heated through high-energy beams of

laser light to initiate the nuclear fusion reaction. This method has not reached

the efficiency and the results that were expected in the 1970s but new approaches

and techniques are currently experimented in some research centers such as the

NIF (National Ignition Facility) in California and the Laser Megajoule in France.

• Magnetic confinement: hydrogen atoms are ionized, so that magnetic fields can

exert a force on them, according to the Lorentz law, and confine them in the

form of a plasma.

The magnetic confinement is the most promising technique and it is worth spending a

few words to describe it in more detail. In normal conditions the gas is unconfined and

free to move, if the gas is ionized and subject to a magnetic field the forces imposed by

the field cause the ions to travel along the magnetic fields lines with a radius known as

the Larmor radius. Ions and electrons have opposite charges, these particles move in

opposite directions along the field lines under the influence of an electric field. Since

positively charged ions are more massive than electrons, the positive ions rotate in a

much larger radius circle. The number of rotations per second at which the ions and

electrons rotate around the field lines are the ion cyclotron frequency and electron

cyclotron frequency, respectively. In fig.1.2 is showed the opposite trajectory between

ions and electrons in the magnetic field.

1.2 Lawson criterion

In nuclear fusion research, the Lawson criterion, first derived on fusion reactors (ini-

tially classified) by John D. Lawson in 1955 and published in 1957, is an important

general measure of a system that defines the conditions needed for a fusion reactor to

6


Figure 1.2: Opposite trajectory of ions and electrons in magnetic field.

reach ignition, that is, that the heating of the plasma by the products of the fusion

reactions is sufficient to maintain the temperature of the plasma against all losses

without external power input. As originally formulated the Lawson criterion gives a

minimum required value for the product of the plasma (electron) density ne and the

“energy confinement time” τE . Later analyses suggested that a more useful figure of

merit is the “triple product” of density, confinement time, and plasma temperature

T. The triple product also has a minimum required value, and the name ”Lawson

criterion” often refers to this inequality.

1.2.1 The product neτE

The confinement time τE measures the rate at which a system loses energy to its

environment. It is the energy content W divided by the power loss Ploss (rate of

energy loss):

τE =W

Ploss(1.2.1)

For a fusion reactor to operate in steady state, as magnetic fusion energy schemes

usually entail, the fusion plasma must be maintained at a constant temperature.

Thermal energy must therefore be added to it (either directly by the fusion products

or by recirculating some of the electricity generated by the reactor) at the same

rate the plasma loses energy (for instance by heat conduction to the device walls or

7


radiation losses like Bremsstrahlung). For illustration, the Lawson criterion for the

D-T (Deuterium-Tritium) reaction will be derived here, but the same principle can

be applied to other fusion fuels. It will also be assumed that all species have the same

temperature, that there are no ions present other than fuel ions (no impurities and

no helium ash), and that D and T are present in the optimal 50-50 mixture. In that

case, the ion density is equal to the electron density and the energy density of both

together is given by:

W = 3 · ne · kb · T (1.2.2)

where kb is the Boltzmann constant. The volume rate f (reactions per volume per

time) of fusion reactions is:

f = nD · nT · 〈συ〉 =1

4n2e〈συ〉 (1.2.3)

where σ is the fusion cross section, υ is the relative velocity, 〈〉 denotes an average

over the Maxwellian velocity distribution at the temperature T , nD is the deuterium

density and nT is the tritium density. The volume rate of heating by fusion is f times

Ech, the energy of the charged fusion products (the neutrons cannot help to keep the

plasma hot). In the case of the D-T reaction, Ech = 3.5MeV that the energy of the

α particles.

The Lawson criterion is the requirement that the fusion heating exceed the losses:

f · Ech ≥ Ploss (1.2.4)

making the opportune substitutions:

1

4n2e〈συ〉 ·Ech ≥

3 · ne · kb · T

τE(1.2.5)

then:

ne · τE ≥ L ≡12

Ech

·kbT

〈συ〉(1.2.6)

8


The quantity T〈συ〉

is a function of temperature with an absolute minimum. Replacing

the function with its minimum value provides an absolute lower limit for the product

neτE . This is the Lawson criterion. For the D-T reaction, the physical value is at

least:

ne · τE ≥ 1.5 · 1020[s/m3] (1.2.7)

The minimum of the product occurs near T = 25 keV as showed in fig. 1.3.

Figure 1.3: The Lawson criterion, or minimum value of (electron density * energyconfinement time) required for self-heating, for three fusion reactions. For D-T, neτEminimizes near the temperature 25 keV (300 million kelvins).

1.2.2 The triple product neTτE

A still more useful figure of merit is the ”triple product” of density, temperature, and

confinement time, neTτE . For most confinement concepts, whether inertial, mirror,

or toroidal confinement, the density and temperature can be varied over a fairly

wide range, but the maximum pressure attainable is a constant. When that is the

case, the fusion power density is proportional to p2〈συ〉/T 2. Therefore the maximum

fusion power available from a given machine is obtained at the temperature where is

9

Cap. 1 Nuclear fusion §1.3 Tokamak

a maximum. Following the derivation above, it is easy to show the inequality:

ne · T · τE ≥12kBEch

T 2

〈συ〉(1.2.8)

For the special case of tokamaks there is an additional motivation for using the triple

product. Empirically, the energy confinement time is found to be nearly proportional

to n1/3/P 2/3. In an ignited plasma near the optimum temperature, the heating power

P is equal to the fusion power and therefore proportional to n2T 2. The triple product

scales as:

ne · T · τE ∝ ne · T · (n1/3e /P 2/3) ∝ ne · T ((n

1/3e /n2

eT2) ∝ T−1/3 (1.2.9)

Thus the triple product is only a weak function of density and temperature and

therefore a good measure of the efficiency of the confinement scheme. The quantity

T 2

〈συ〉is also a function of temperature with an absolute minimum at a slightly lower

temperature than T〈συ〉

. For the D-T reaction, the physical value is about:

ne · T · τE ≥ 1021[keV · s/m3] (1.2.10)

This number has not yet been achieved in any reactor, although the latest generations

of machines have come close. For instance, the TFTR (Tokamak Fusion Test Reactor

in Princeton New Jersey) has achieved the densities and energy lifetimes needed to

achieve Lawson at the temperatures it can create, but it cannot create those temper-

atures at the same time. ITER aims to do both. In fig. 1.4 the triple product for

three fusion reactions is reported.

1.3 Tokamak

The most promising device for magnetic confinement of plasma is the tokamak (Rus-

sian acronym for “Toroidal chamber with axial magnetic field”), a device shaped as

10


Figure 1.4: The fusion triple product condition for three fusion reactions.

a torus (or doughnut) that has been originally designed in Russia during the 1950s

by physicists Igor Tamm and Andrei Sakharov. The general structure of the device

is shown in Figure 1.5 The main problem with the magnetic confinement described

Figure 1.5: General structure of the tokamak device.

in the previous section is that the particles remain confined by the magnetic field

until the field lines end or dissipate, contrary to the desire of keeping them confined.

To solve this problem, the tokamak bends the field lines into a torus so that these

11


lines continue forever. The magnetic fields that create and confine the plasma in the

tokamak are generated by electric coils which can be located outside the chamber,

such in JET and most of the tokamak, or inside, as in MAST experiment. Since

the plasma is ionized and confined inside the toroidal chamber, it can be considered

as a coil circuit, the secondary side of a coupled circuit whose primary side is the

central solenoid. Figure 1.6 displays the currents and fields that are present inside

the tokamak. All existing tokamak are pulsed devices, that is, the plasma is main-

tained within the tokamak for a short time: from a few seconds to several minutes.

There is no agreement yet among fusion scientist on whether a fusion reactor must

operate with truly steady-state (essentially infinite length) pulses or just operate with

a succession of sufficiently long pulses. The main reason for this limitation is that,

in order to sustain constant values of plasma current, the derivative of the current

Figure 1.6: Currents and magnetic fields in a tokamak device.

on the central solenoid must be constantly ramping up (or down), rapidly reaching a

structural limit on the coil which cannot be exceeded. To avoid this limitation, differ-

12


ent methods to sustain the plasma current have been studied and introduced, such as

LH/ECRH antennas or neutral beams injectors (NBI), currently used at MAST. All

tokamak produce plasma pulses (also referred to as shots) with approximatively the

same sequence of events. Time during the discharge is measured relative to t=0: the

time when the physical experiment starts after all the preliminary operations. The

toroidal field coil current is brought up early to create a constant magnetic field to

confine the plasma when this is initially created. Just prior to t = 0 deuterium is

puffed into the interior of the torus and the ohmic heating coil (inner poloidal coils

in Figure 1.6) is brought to its maximum positive current, in preparation for pulse

initiation. At t=0 the primary coil is driven down to produce a large electric field

within the torus. This electric field accelerates free electrons, which collide with and

rip apart the neutral gas atoms, thereby producing the ionized gas or plasma. Since

plasma consists of charged particles that are free to move, it can be considered as a

conductor. Consequently, immediately after plasma initiation, the primary coil cur-

rent continues its downward ramp and operates as the primary side of a transformer

whose secondary is the conductive plasma. At the end of the downward ramp of the

primary coil the plasma current is gradually driven to zero and the shot moves towards

its conclusion. The separate time intervals in which the plasma current is increasing,

constant and decreasing are referred to, respectively, as ramp-up, flat-top and ramp-

down phase of the shot. At the moment the tokamak technology has reached a point

such as the quantity of energy produced by these devices is almost as much as the one

used in heating and confining the plasma. The next step is the construction and op-

eration of the proposed International Thermonuclear Experimental Reactor (ITER)

which, supported by an international consortium of governments, will provide major

advancements in fusion physics and constitute a testbed for developing technology

13


to support high fusion levels. In figure 1.7 tha main geometrical parameters such as

aspect ratio, elongation, triangularity, in a tokamak plasma are showed.

Figure 1.7: Geometric parameters in a tokamak plasma.

14

Chapter 2

MAST and MAST-Upgrade

In this chapter will be made a brief technical description on MASTand on MAST-Upgrade

2.1 MAST

Mega Amp Spherical Tokamak (MAST) is the fusion energy experiment, based at

Culham Centre for Fusion Energy since December 1999. Its main difference from a

classical tokamak is the shape: since the origin of tokamak in the 1950s, research is

mainly concentrated on machines that hold the plasma in a doughnut-shaped vacuum

vessel around a central column. MAST belongs to a different category of tokamak,

named spherical tokamak, which presents a more compact, cored apple shape and a

lower aspect ratio (Rasee figure 1.7). In Figure 2.1 is showed a typical plasma built

with a MAST experiment. Spherical tokamak hold plasmas in tighter magnetic fields

and could result in more economical and efficient fusion power for many reasons:

• plasmas are confined at higher pressures for a given magnetic field. The greater

pressure, the higher power output and the more cost-effective of the fusion

device.

15

Cap. 2 MAST and MAST-Upgrade §2.1 MAST

Figure 2.1: Example of plasma in a MAST experiment.

• The magnetic field needed to keep the plasma stable can be a factor up to ten

times less than in conventional tokamak, also allowing more efficient plasmas.

• Spherical tokamaks are cheaper, since they do not need to be as large as con-

ventional machines and superconducting magnets, which are very expensive, are

not required.

Spherical tokamaks, at the moment, are at a very early stage of development and they

will not be used for the first nuclear fusion power plants but they can be very useful

for component test facilities and they are providing insight into the way changes in the

characteristic of the magnetic field affect plasma behaviour. These informations have

been very useful for the development of ITER, the advanced experimental tokamak

which is being built in France. MAST, along with NSTX at Princeton, is one of

the world’s two leading spherical tokamak. Table 2.1 and Figure 2.2 give an idea of

its dimension, structure and technical specifications. A cross-section of the MAST

16


Plasma Vacuum vesselCurrent 1.3 [MA] Height 4.4 [m]Core up to Diameter 4 [m]temperature 23.000.000 ◦CPulse length up to 1 second Material Stainless steel 304LNPlasma 8 m3 Toroidal field 24 turns, 0.6 TeslaVolume 0.7 m radiusDensity 1020 particles/m3 Total mass 70 tonnes

of load assemblyDiameter approximatively 3 m Neutral beam 5 MW

heating power 75.000Volt

Table 2.1: Technical specifications of MAST experiment.

Figure 2.2: Section of MAST.

vessel and the position of the six PF (poloidal field) coils is shown in Figure 2.3. It is

important to understand the different purposes that the coils have.

17


Figure 2.3: Cross-section of the MAST vessel and position of the six PF coils.

• Start-up coil (P3): It is a capacitor bank used for the pre-ionization of the

plasma. It has no power supply or feedback, just a switch that starts the

discharging of the capacitor hence it cannot really be considered an actuator

from the plasma shape controller point of view.

• Vertical field/shaping coils (P4 and P5): Both coils contribute to the main

vertical field for radial position control. The shape and elongation depend both

on the plasma internal profile and on how the total vertical field current is

divided between P4 and P5. Each of them is driven by a bank which provide

the rapid initial vertical field rise and by power supplies (respectively SFPS

and MFPS), which provide controlled flat-top current. Both power supplies can

drive current in a single direction. The maximum value of the current is 17kA

18


for P4 and 18 kA for P5.

• Vertical position coil (P6): There are actually two coils in one can, each of them

with two turns. These coils provide the radial field for vertical position control.

Since the vertical dynamics are much faster than the time scale of the existing

MAST PCS, they are independently driven by a separate analogue controller.

In Figure 3.2.5 is showed how the coils currents values change during a real experiment

on MAST, and how the plasma current value goes up with the opposite slope of the

solenoid current variation, this is due to the Faraday-Neumann-Lenz’s Law (included

in the Maxwell equations).

Figure 2.4: Typical currents evolution in a real MAST experiment.

19

Cap. 2 MAST and MAST-Upgrade §2.2 MAST-Upgrade

2.2 MAST-Upgrade

2.2.1 Movitation for MAST-Upgrade

The development of a fusion power plant requires substantial advances in plasma

science and technology. Many of the issues will be significantly progressed by the

forthcoming ITER device, now under construction in Cadarache, France. There is

also expected to be continued development in research programmes using present

devices and numerical tools. The specific R&D needs and the required facilities with

EU participation have been reviewed in 2008 by an international panel of experts. In

this review MAST has been identified as one of the four devices that should continue

to operate until ITER operation and possibly beyond. This is because there are many

aspects which require research and development at power plant relevant parameters

that are either complementary to ITER, or go beyond ITER parameters and the

spherical tokamak (ST) line may help with this research with a future facility.

For example the prototype fusion power plant, known as DEMO, will go beyond ITER

divertor heat loads with five times higher heating power normalised to plasma radius.

It will also require the development of technology in a much more aggressive neutron

environment, for which new components need to be tested (particularly in relation to

how their heat handling properties can be maintained in a high neutron environment).

It will further require the development of techniques for quasi-continuous operation

(on the scale of weeks or more, rather than hours), with fully self-sustaining plasmas

(still largely relevant even if a long pulse, e.g 1 day, DEMO is adopted), strong

current drive, tritium breeding, and self-reconditioning, at performance levels where

new instabilities may need to be controlled from the energetic fusion products. Thus a

new facility has been proposed to help speed the path towards DEMO and improve its

20


design a Component Test Facility (CTF). The principal here is that by focussing more

narrowly on particular aspects of nuclear technology (especially breeding blankets)

that need to be validated at meaningful dimensions, one can go further and faster

than is achievable through the integrated approach of ITER, thereby providing a

complementary capability. In particular the emphasis with a CTF is to focus on a

high neutron high heat flux environment in a driven machine (i.e. substantially heated

and controlled by external systems) in order to understand the optimisation of device

components for a power plant and the ramifications of high heat flux and long pulse

quasi-continuous operation.

Detailed studies at Culham and elsewhere have shown that the Spherical Tokamak

(ST) is a particularly promising candidate for such a device. A major advantage of

an ST based CTF would be that the tritium consumption would be low enough to

avoid the necessity for tritium breeding, hence avoiding the reliance for continued

operation on the main components being tested. It is in this context that MAST

as one of the two major STs has its strongest role. Therefore a 10 year milestone

was included in the EFDA R&D programme to assess the feasibility of an ST-CTF.

However, for MAST to continue to play a strong role in the international fusion R&D

and to strengthen the physics basis of an ST-CTF an upgrade to the current facility

is required. This view was endorsed by the EU review panel corroborated by a 2nd

review of an international panel setting the vision for the 20 year fusion strategy in

the UK. The importance of the MAST upgrade programme with respect to DEMO

research has been further increased by the inclusion of a novel divertor concept. The

unique open design of the MAST vessel is instrumental in this novel concept, which

may provide a solution to the divertor power challenge in DEMO (and also in an

STCTF).

21


2.2.2 Key components of the funded stage 1 of the upgrade

The budget for MAST Upgrade released by CCFE’s funding agency the Engineering

and Physical Sciences Research Council (EPSRC), though generous in a time of finan-

cial stringency, will not fund the whole upgrade programme. Hence, a first affordable

stage has to be defined that allows a substantial advance on as many key physics

issues as possible, but also facilitates future cost effective upgrades towards the full

MAST upgrade. It is clear that with this first stage it is not possible to address all

of the physics issues. The key elements of the MAST-U stage 1 with their physics

functions and key hardware components are listed below:

• Upgrade of the toroidal field from Irod = 2.2MA (Current value in the toroidal

coils) 1 (B0 = 0.63T Toroidal field) to Irod = 3.2MA(B0 = 0.91T ).

Physics: The confinement scaling in ST’s seems to scale more strongly with

B0(toroidal field) than in conventional tokamaks. Hence, access to hotter plas-

mas with more efficient current drive will be possible. The higher B0 also al-

lows operation at elevated q0 (security factor helping to avoid detrimental low

n(plasma density) MHD(magneto-hidro-dynamic)).

Hardware: New long pulse TF(Toroidal Field) power supply 133 kA, 5s capa-

bility; New centre rod with new sliding joint design.

• Closed, pumped divertor with Super X Divertor, now on SXD capability.

Physics: The new divertor allows the assessment of the SXD concept for tar-

get power load reduction for future devices. The pumping capability enables

stationary operation at low density for efficient current drive and reduced ν

(particles velocity). The closed divertor design keeps the main vessel density

1The values of the magnetic field are calculated at R=0.7 m.

22


low leading to higher pedestal temperatures.

Hardware: Two new divertor chambers with 8 divertor coils each and 7 new coil

power supplies and CFC(carbon fiber carbon) and Graphite target plates; Two

cryogenic pumps and a new cryogenic plant to distribute 4.5K (He).

• Flexible 7.5 MW heating system with three co-current injectors, two of which

are mounted off-axis at Z = 0.65m.

Physics: The new heating system allows studies of off-axis current drive with

access to fully non-inductive scenarios at Ip > 1MA for several current redis-

tribution times. The higher heating power capability will give access to regimes

with lower ν , higher ρ (Larmor radius) and higher βN (ratio of plasma and mag-

netic pressures). The flexibility will allow modification of the fast-ion density

and q-profile (security factor) for detailed studies of fast-ion driven instabilities.

Hardware: New double beam box with Rtan = 0.9m; Jacked up beam box at

Rtan = 0.8m (possibility to move to on-axis port with short outage).

• Increased flux swing from 0.7 Wb to 1.6 Wb

Physics: Access to higher Ip ≤ 2MA for lower ν with a flux swing that enables

long pulse operation at Ip > 1MA. The higher current operation will also

provide better confinement in particular of the off-axis beams increasing the

heating capability.

Hardware: New solenoid; Chilled centre column; New centre tube.

• Upgrade of the poloidal field coil set for higher shaping capability and sustained

high κ (elongation factor) operation.

Physics: This will allow operation at higher κ < 2.7, optimising the bootstrap

fraction, and higher δ < 0.7 (Triangularity factor) to increase edge stability for

23

Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade

higher pedestal pressures. The new coil set also allows more independent control

of κ and δ and the inner gap.

Hardware: New high field side pusher coil; New high field side X-point control

coils (part of the divertor set); Passive stabilising plates for vertical position

control.

The excellent diagnostic set of MAST will be maintained as much as possible with

upgrades in the diagnostics for the divertor area and fast-ion physics. All the power

supplies will enable long pulse operation of several current redistribution times, τR.

For the MAST-U scenarios 0.5s < τR < 1s. Also the internal coil set for ELM

mitigation, neoclassical toroidal viscosity (NTV) studies and error field correction

will be maintained with 12 coils below the mid-plane and up to 4 coils above the

mid-plane. This flexible set can also be used as TAE antennas (as now). Independent

variation of the plasma rotation will only be possible in this stage using magnetic

braking due to NTV using the internal coil set. However, in vessel structures and port

modifications to enable a quick cost effective implementation of a mid-plane counter

current beam line are foreseen to control the plasma rotation without the application

of 3D magnetic fields. Furthermore, it will be possible to convert the jacked-up beam

to mid-plane injection during a normal engineering break to maintain the highest

flexibility in the heating and current drive system. In figure 2.5 the MAST-U section

design is showed.

2.3 Super X divertor in MAST-Upgrade

MAST-Upgrade is intrinsically a double-null divertor tokamak. The well-matched pair

of divertors, one upper, one lower, are therefore an intrinsic part of the overall physics

programme and the engineering design of the machine. The physics programme re-

24


Figure 2.5: Cross-section of MAST Upgrade. Only the lower of each pair of coils isshown (denoted as L).

quires that the divertors are able to provide the power handling and particle control

for experiments that focus on the behaviour of the core, confined plasma, that is that

they act as conventional divertors. In addition the divertors must provide the oper-

ational flexibility and range of diagnostics that are necessary in order to carry out

experiments that increase understanding and guide the design of divertors that are

needed in future, higher power, fusion devices. In this latter respect, the SXD con-

figuration, is the main focus. The MAST-Upgrade divertor should allow substantial

progress in understanding the roles of:

• divertor design in both steady state and transient power handling;

25


• connection length in determining scrape off layer (SOL) transport and edge-

localized modes (ELM) properties ;

• divertor in determining up-stream conditions affecting the core plasma (e.g.

H-mode access, ELM stability);

• main-chamber neutral density in the performance of a double-null spherical

tokamak.

2.3.1 Layout

The layout of the divertor may be considered in four respects:

• the location and capability of the poloidal field coils that define the magnetic

geometry

• the material surfaces onto which particles and power arrive from the plasma

• the pumping and fuelling capabilities

• the diagnostics that are built into, and monitor, the divertor area

It is important to have a clear understanding of the overall set of poloidal field coils,

in order to understand the rationale for the choice and location of the divertor coils.

Hence, the full set of coils is shown in Figure 2.6, together with typical “conventional”

and “Super-X” divertor configurations. The numbering of the P coils relates to their

use in the present configuration of MAST, and in particular neither P2 nor P3 will

be present in the Upgrade.

It is not possible entirely to isolate the effect that an individual coil has on the

magnetic field equilibrium, since the length-scales involved mean that the influence of

the poloidal field coils is felt significantly throughout the whole plasma. However, each

26


Figure 2.6: Location of poloidal field coils, and main plasma-facing surfaces, and twoexample equilibria for MAST-Upgrade on the left with a conventional divertor, onthe right with a SXD.

of the coils has primary functions, and these are explored below. The discussion is in

terms of up/down symmetric, connected double-null (CDN) equilibria, in which the

coils themselves are also considered in essentially up/down symmetric pairs (except for

the two large coils, PC and P1, that are designed to be essentially up/down symmetric

in themselves, and the anti-symmetric P6 coil pair that provides the control of the

plasma vertical position).

27


2.3.2 Poloidal field coils

Central Solenoid (P1)

P1 is the central solenoid of MAST-Upgrade, and hence its principle role is to vary

the flux that links the plasma current for, the standard, induction of the plasma

current. However, MAST-Upgrade does not have any significant quantity of magneti-

cally susceptible material to “channel” this flux in any way. Hence varying the current

in the P1 coil varies the “fringing” magnetic field through the vacuum vessel. This

means that varying the P1 current will vary the shape of the magnetic field surfaces,

particularly in the divertor regions.

Radial position and core chape (P4, P5)

The P4 and P5 coils provide the main vertical field that resists the “hoop” force on

the plasma. Hence they provide the major part of the control of the radial location

of the plasma centre. However, it is important to note that the plasma current and

radial position control in a spherical tokamak are strongly coupled, and hence the P1,

P4 and P5 coils are used in concert for the combined control of current and radial

position. The difference in current between the P4 and P5 coils allows the shape of

the core, confined plasma, on the large-radius side, to be varied. In the context of the

divertor, enhanced current in P4 is used to tailor the trajectory of the SOL towards

the divertor entrance, and to balance the effect of the divertor coils in the formation

of the Super-X configurations.

Centre-column clearance (PC)

The PC coil has a relatively clear influence allowing control of the gap between the

plasma equilibrium and the centre-column of the machine, and as a result making

the separatrix of the plasma less convex on that side. There is also some consequent

28


effect on the plasma triangularity and X point location.

Vertical position (P6)

The P6 coils are connected to produce a dominantly radial field, for controlling the

vertical position of the plasma on timescales ranging from steady-state to the fast

response that controls the unstable vertical position of the elongated plasma. The

slow steady state control of the vertical position of the plasma is important for the

divertor, in providing the precise control of the balance of power between the upper

and lower divertors. This equilibrium control is usually described by the parameter

∆Rsep, this being the radial distance at the outer mid-plane of the plasma between

the magnetic separatrix that passes through the lower X-point and the one that passes

through the upper X point (Figure 2.7). In a connected double-null configuration this

distance is negligibly small, since the X points are considered to be “connected” by

a single separatrix. The length-scale with respect to which this separation may be

neglected is open to debate, but is of the order of 1cm i.e. of the order of both the

ion larmor radius and the power fall off length

Conventional divertor (PX, D1, D2, D3, DP)

The dominant contributions to the control of the X-point height, Zx, radius, Rx, and

flux expansion, Ex, in the vicinity of the X-point comes from the PX, D1 and DP

coils. Control of the inner strike point is also implicit in the combination of fields

from these coils, and the fringing field from the P1 solenoid. However, assuming that

the control is optimised for the three quantities Zx, Rx, Ex, there is little residual

freedom to independently control the inner strike point with these coils. In connected

double-null, the experimental evidence is that the power loading to the inner strike

zones is low (less than or order of 10%). Hence the design includes multiple high-field

29


Figure 2.7: Definition of Rsep.

side gas puffs, which together with the parasitic variation in the inner strike zone with

the swing of the solenoid, allow control of the power loading to the inner strike zone

through a combination of partial detachment and/or modest sweeping. The D2 and

D3 coils are located to allow control of the radius of the outer strike point, ROSP ,

(i.e. the radius of interception of the separatrix with the surface), and of the area

expansion of the outer strike, EOSP . A significant development in the understanding

of the way in which MAST-Upgrade will operate has come about through study of

the interaction between the divertor coils and the fringing field from the solenoid, as

its flux is varied.

30


Super-X Divertor (D5, D6, D7)

The coils from the previous sections were all included in the 2008 design of MAST-

Upgrade, with space being left within the upper and lower sections of the vacuum

vessel for potential future experimental divertor developments. Since then, a choice

has been made on the nature of the (first) divertor development to pursue namely the

SXD configuration.

The basic idea of the SXD is to maximise Rdiv, where Rdiv = ROSP is the radius

of the outer divertor strike. In addition, the magnetic line length L from the SOL

midplane to the divertor plate can also be increased by decreasing the poloidal field

in the long divertor leg, at large R and large Rdiv also reduces the parallel heat flux

q‖. The ability to pull the divertor leg out to large radius and to expand out the

flux within the divertor chamber is provided by the fields from the D5, D6 and D7

coils, with D5 coil being particularly important in pulling out the leg. D6 and D7 are

largely needed to screen out the field from the main-chamber coils, particularly P4,

in order to achieve poloidal fields in the divertor chamber that are typically an order

of magnitude lower than the fields at the low-field mid-plane of the confined plasma.

In practice this Super-X capability of the divertor enhances MAST-Upgrade in two

important respects. Firstly it provides a test-bed for this, as-yet untried, potential

solution of the ST-CTF/DEMO divertor power-loading problem. Secondly, it provides

a level of risk mitigation in the event that the power-fall-off lengths in MAST-Upgrade

itself turn out to be as small as some predictions suggest (see below).

Single-null operation

In principle there is a wide variety of plasma geometries that can be generated in

MAST-Upgrade, as summarised in figure 2.8. Where reasonable, consideration has

31


been given to modest operational capability in a single-null configuration. It is recog-

nised that a single-null configuration might allow interesting modes of operation,

allowing access to certain extremes of the core performance operating space, and ar-

guably providing closer support of future machines with single-null configurations.

However, single-null operation provides direct connection along the SOL between the

very different inner and outer divertor strike zones. In the case of the very unalike

Super-X leg and highly loaded low radius inner leg of a single null ST configuration,

this complicates the understanding of the interplay between the upstream and target

conditions. As potentially the first machine to explore the Super-X configuration, and

with a view to the support of ST-CTF double-null machine designs, MAST-Upgrade

retains predominantly a double null configuration with both ends of a single SOL be-

ing connected to a Super-X divertor leg. The position-sensing, control and coil power

supply systems will be designed and integrated to achieve the required accuracy and

stability of control in order to maintain the connected double null configuration, in

order to maintain ∆Rsep ≤ 1 mm. The normal operation of MAST-Upgrade will

have the ion-grad B drift direction towards the lower divertor, with the expectation

therefore being that there will be roughly a 40%/60% upper/lower divertor power

load under attached conditions, in connected double null. Control of the vertical po-

sition should allow this power asymmetry to be removed, for certain experiments, by

breaking the up/down magnetic symmetry. Reversal of the toroidal field will allow

operation with the upper divertor receiving the majority of the power. In figure 2.9

the cut-out of the SXD is shown

32


Figure 2.8: Schematic summary of magnetic configuration possibilities for MASTUpgrade. Configurations with asymmetric configurations are shown with grey back-ground.

33


Figure 2.9: Cut-out section of the SXD.

34

Chapter 3

Shape control

This section discusses the plasma shape control challenges forMAST Upgrade in terms of the strategy being followed to reachthe end goal, rather than attempting to describe already what thefinal algorithm design will look like. The plasma control require-ments for MAST Upgrade are significantly more complex than theyare for MAST, due mainly to the new divertor and the considerableincrease in the number of coil currents and shape parameters to becontrolled.

3.1 The control problem

Any control problem usually begins with a set of requirements, expressed in terms

of parameters that need to be controlled, the accuracy and quality of that control,

and the actuators available to perform that control, subject to their limitations. At

this stage in the project, some basic requirement have been defined in terms of the

position and shape accuracy that is expected, in particular for divertor control.

When the requirements for physics performance are defined, they usually start in

terms of actual parameters of interest to the physicist, e.g. the amount of plasma-

wall interaction, divertor closure, SOL thickness, or proximity to a stability limit.

These are impractical to use as control parameters because they would be too dif-

ficult to measure or calculate in real time. Therefore an abstraction is made into

35

Cap. 3 Shape control §3.1 The control problem

simple parametric descriptions of the plasma shape (e.g. major radius, elongation,

triangularity, etc.) that is required to sustain the physics scenario of interest. De-

pending on the controller implementation, dimensionless shape parameters such as

elongation and triangularity may be expressed as minor radius, minor height and ra-

dial displacement of the plasma top, or as a set of gaps around the boundary. There

is usually a non-linear mapping from the desired physics performance parameters to

these geometric parameters, and there is even a non-linear mapping between such

representations (e.g. elongation κ = ba).

The principal control algorithm to be used on MAST also uses a set of discrete points

on the plasma boundary to represent the whole boundary contour, but it further ab-

stracts these geometric parameters into flux units. The reasoning behind this is that a

flux-based controller can have a wider scope of linear operation. This is because both

the magnetic signals used to determine the flux value and the coil current required

to correct it have a much more direct and linear relationship. The flux at a defined

point is always single-valued and unambiguous; therefore control of a flux parameter

can be more reliable than control of a geometric parameter whose definition may only

exist in certain scenarios.

In all of these abstractions however, one must be mindful of the indirect relation-

ship and non-linear mapping between the control parameters and the original physics

values that they represent. This is important to avoid pathological cases where the

controller appears successful in controlling its target value (e.g. flux at strike point)

but it is missing the underlying physics value (e.g. power load on strike point).

36

Cap. 3 Shape control §3.2 Fiesta code

3.1.1 Dynamic control

Returning to the discussion of control requirements and defining what is considered

sufficient, note that a control design process is usually a balance between the amount

of variation allowed in the control parameters and the amount of work done by the

actuators. The most work on this topic have been magnetostatic, i.e. without con-

sidering dynamics. As such, the coil locations and required currents are well defined

for the static scenarios, but for dynamic control one also needs a description of the

time domain requirements. Fortunately, only the vertical position is unstable, so most

control timing requirements are based on how fast it need to migrate from one plasma

state to another, rather than being tied to the timescale of an unstable mode. In

general it is considered that new control requirements, such as divertor control, will

be on the same timescale as the existing MAST plasma dynamics, so the power sup-

plies could use similar technology to what is already employed. The maximum voltage

headroom needed in the power supplies will be determined by the circumstances where

the most rapid changes of state are required, such as startup, plasma current ramp

up down, and recovery from disturbances such as loss of H mode. These have been

used to define the initial power supply specifications, but dynamical simulation is still

desirable to refine these answers.

3.2 Fiesta code

The principal tool for predictive equilibrium modelling on MAST is Fiesta, a magneto-

static free boundary Grad Shafranov solver written in MATLAB. Much of the analysis

of the operating space of MAST, has been performed either by manual manipulation

of the inputs to this code, in particular the PF currents, or with the assistance of a

37


few “iterative feedback”, loops internal to Fiesta. These iterate the coil currents in

an attempt to constrain the plasma boundary contour based on simple flux mapping.

The algorithm used in the latter case is essentially a least squares fit solution of an

over determined problem, which often needs further hand-optimisation. Because it

does not consider transients it can reproduce a shot as a sequence of snapshots.

Fiesta started life as a simple forward equilibrium solver, but since then has been

expanded into a toolbox for dealing with many equilibrium related problems. It can

do the forward problem with a range of control methods. Once an equilibrium has

been calculated, there are extensive facilities for calculating things like the q profile,

global quantities (β, the internal inductance li etc), signals from a range of sensors,

field line following, and so on. The programming style is object oriented. Essentially,

data are kept inside “objects”, which are like glorified “classes”. There are two major

classes, the fiesta configuration (config) and the fiesta equilibrium (equil). Other data

classes are used to build these, for example the constructor for a fiesta configuration

requires a fiesta coilset, but once built the two major objects contain all the data

which was used to build them so any other objects become redundant. It is therefore

possible safely save just these two items to save a complete calculation. The fiesta

configuration contains the description of the tokamak (coils, limiters, etc) plus the

grid of the points on which the equilibrium is to be calculated. Importantly, it also

contains all necessary Green’s functions. The fiesta equilibrium contains the descrip-

tion of the plasma (current density, coil currents) plus the results of the equilibrium

calculation shape. In the pure “forward” problem, the plasma current density profile

is specified in terms of the basis function, the coefficients and the total current, and

the boundary conditions are specified in terms of the coil currents. Once the equilib-

rium is obtained it could be possible then calculate the signals from various sensors,

38


flux loop, magnetic field, pressure, and so on. In figure 3.1 an example of fiesta equil

plot is reported.

Figure 3.1: Plot example of Fiesta equil object.

39


3.2.1 Passive currents

The magnetostatic modelling reported above has thus far not addressed the impact of

passive currents on the control dynamics. In general terms, the passive structure acts

like a low pass filter on the coil currents. In particular, the PF and divertor coils are

enclosed in metal cans as a vacuum boundary, and the solenoid is in close proximity to

the centre tube of the vessel. However, the effect may not be as pronounced as it first

appears. This is because although the passive current in the coil case (or centre tube)

opposes the direction of wire current in the coil (or solenoid), it also has the side effect

of reducing the apparent AC inductance of the coil. If considering the response to a

voltage step from the power supply, part of the low pass filtering effect of the passive

structure is offset by the inrush current resulting from the momentary reduction of the

effective coil inductance. This “filtering” effect also helps mitigate the current ripple

from the power supply. Fiesta incorporates an ’RZIP’ function, which can calculate

a linear response model based on the defined passive structure and semi rigid plasma

model (where only R, Z and Ip can change). It is planned to use this generated model

to simulate coil current changes for a given scenario evolution, then incorporate the

simulated resulting passive structure currents back into Fiesta as a set of virtual PF

coils. The aim of this is to develop a full plasma discharge simulation, accounting for

passive currents, to confirm the voltage requirements for the power supplies.

3.2.2 Calculate the sensitivity matrix

Keeping in mind that the final controller has to run on a real time machine, it is

possible to assume that locally there is a simple linear and static mapping between

the currents in the coils and the parameters to be controlled in steady state. This

40


relationship can be expressed as:

∆P =M ·∆I (3.2.1)

Where ∆P is the desired variation of the parameters vector (i.e. external radius,

inner radius, X point, gaps) and ∆I is the vector of the coils currents variation. It is

important to remark that the relationship 3.2.1 is based on the important hypotesis

of a linear dependence. It is possible to consider the equation 3.2.1 as a line equation

where the sensitivity matrix M is exactly the slope. After this preliminary remarks

it is possible to explane how the sensitivity matrix M is calculated. Applying the

perturbations (variations in the coils currents) around the given equlibrium point,

the variations in the plasma parameters are obtained. The base equilibrium is a

scenario with parameters values and coil currrents setted manually by the phisicists.

Basically each perturbation is applied for each coil with a logaritmic range value in two

directions around the base value (in positive and in negative direction), for each step

we have got a new equilibrium and if it is acceptable (i.e. plasma did not touch the

divertor baffle) Fiesta calculates the new parameters values. In pseudo-code language

is described such as:

% Logaritmic perturbations

Coils Number = 12

I0 = Equilibrium current coil vector

for K = 1 : Coils number

for n =1 : 75

I(K) = I0(K) + ln (n)

evaluate Delta Parameters % by Grad Shapranov solver

if Equilibrium is admissable % Plasma did not touch the walls

41


store the data

else

break

end

I(K) = I0(K) - ln (n)

evaluate Delta Parameters % by Grad Shapranov solver

if Equilibrium is admissable % Plasma did not touch the walls

store the data

else

break

end

end

end

The new equilibrium is calculated solving the Grad Shafranov equation.

The Grad Shafranov equation is the equilibrium equation in ideal magnetohydrody-

namics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal

plasma in a tokamak. This equation is a two-dimensional, nonlinear, elliptic par-

tial differential equation obtained from the reduction of the ideal MHD equations to

two dimensions, often for the case of toroidal axisymmetry (the case relevant in a

tokamak). Interestingly the flux function ψ is both a dependent and an independent

variable in this equation:

∆⋆ψ = −µ0R2 dp

dψ−

1

2

dF 2

dψ(3.2.2)

42


where µ0 is the magnetic permeability, p(ψ) is the pressure, F (ψ) = RBφ and the

magnetic field and current are given by

~B =1

R∇ψ × eφ +

F

Reφ (3.2.3)

µ0~J =

1

R

dF

dR∇ψ × eφ −

1

R∆⋆ψeφ (3.2.4)

The elliptic operator is given by

∆⋆ψ = Rδ

δR

(1

R

δψ

δR

)

+δ2ψ

δZ2(3.2.5)

The nature of the equilibrium, whether it be a tokamak, reversed field pinch, etc... is

largely determined by the choices of the two functions F (Ψ) and p(Ψ) as well as the

boundary conditions.

After this procedure all the data related to the new equilibria are stored. At this point

starts the linear analysis that is, for each coil and for each parameter find the largest

linear space by a linear fitting, the output value is one element belonging to M and it

represents the linear relationship between parameters and one coil current variations

or, the slope of the equation 3.2.1. In Figure 3.2 a linear fit is showed, it is related

to the relationship between the parameter outer radius and the central solenoid P1,

it is possible to see that the linear range is included between −400A and +400A and

the behaviour is almost linear for 99.5%, this percentage is calculated on all the 151

perturbations applied to the coil P1. The slope of the line showed in the Figure 3.2 is

exactly one coefficient belonging to M and located in the row of the parameter outer

radius and in the column of the coil P1. Each element belonging to M is measured in

[m/A]. The linear fitting can be expressed in pseudo-code by:

Linear Fitting

%Step 2: Calculate the linear coefficient

43


Figure 3.2: Linear fit between outer radius and P1 coil.

for each coil

for each parameter

arrange data

find the biggest linear space

linear fit

store the slope value

end

end

44


The linear relation is:

Parameters vector(m×1)︷︸︸︷

Outer radiusInner radius

X point radiusX point zGap 1

=

Sensitivity matrix(m×n)︷︸︸︷

m11 m12 · · · m1,Coils

m21 m22 · · · m2,Coils

m31 m32 · · · m3,Coils

m41 m42 · · · m4,Coils

mParam,1 mParam,2 · · · mParam,Coils

·

Coils vector(n×1)︷︸︸︷

Ip1Ip4Ip5IpcIpxIdpId1Id2Id3Id5Id6Id7

There is only one coil that is not used to evaluate the matrix M that is P6. The

reason is that P6 purpose is to stabilize vertically and it will be made also on MAST-

Upgrade by a real time controller with hardware completely dedicated. In Figure 3.3

an example of the matrix M plot is showed, looking at the fifth row that is related to

one gap it is possible to see how this parameter is higly influenced by the coil D6, it

means that if the same current variation is applied to D6 and to D2 will be obtained a

larger change of the parameter position with the first one rather than with the second

one. This is also intuitive seeing the fig 3.4 that shows how the gap considered (blue

line) has the closest coil that is right D6 (brown square exactly above it), for this

reason the influence of D6 coil on that gap is much stronger than the others coils.

3.2.3 Shot simulation

As explained above Fiesta is a magneto-static code, for this reason it is completely

released of time measurement. In a real MAST experiment the solenoid swing hap-

pens in a finite time (0.45 s flat top time). Fiesta can reproduce an experiment as a

snapshots sequence considering constant the plasma current value and the user can

45


Figure 3.3: Sensitivity matrix plot, it is possible to see how the fifth line (gap) ishighly influenced by D6 coil.

assign the range value related to the solenoid current variation, if the user sets this

range in a reasonable value the linear relationship expressed by the sensitivity ma-

trix has been considered valid otherwise the simulation could generate wrong results.

Usually the shot is simulated starting from 0 A (the ramp-up before is considered

for prepare the plasma scenario and reaching the flap-top current) and arriving until

−30 kA (solenoid current value), a reasonable range related to each single step that

separates two consecutive snapshots is 500 A, in this way the experiment is simulated

as a sequence of 60 snapshots. For each step of the solenoid current variation the

function control is called to move the control coils values for restoring the parame-

ters values, in such way the plasma’s shape is maintained during the simulation. In

tokamak devices the solenoid current variation is necessary for increasing the plasma

current to the target value (1 MA for MAST). In fig 3.5 is reported one plot of the

46


Figure 3.4: Boundary of plasma plot shows the closest relationship between the gap(blue) and the coil D6 (exactly above it).

plasma evolution shape during a simulated shot, the blue line is the base equilibrium

and the red line is the last snapshot, all the other colours represent the snaphshots

included between the first one and the last one in gradual order.

47


Figure 3.5: Plasma shape evolution during the simulated shot, blue line startingpoint(base equilibrium), red line last snapshot.

3.2.4 Relationship between flux and control problem during

the shot

One idea has been developed to detect the realationship between the variation of

magnetic flux in the plasma boundary during a simulated shot respect to the variation

of current in P1 (central solenoid). After several simulations has been highlighted the

linear relation between these two parameters in agreement with the hipotesis made.

This result has been the starting point for developing further concepts with the aim

to find a simplified vector for manage the control of plasma during the shot. In fig 3.6

is showed the linear dependence between P1 current and the flux. This vector should

48


belong to sensitivity matrix and especially could be the line related to magnetic flux

multiplied for a corrective factor α.

Figure 3.6: Linear relationship between flux and solenoid current during the swing.

3.2.5 Control function

At any step of the simulated shot the control function is called to restore the base

equilibrium. When the current in P1 (central solenoid) changes the parameters change

position as well, at this point Fiesta evaluate these parameters displacements and here

takes place the control function. It uses the matrix M to determinate the control coils

currents variations indispensable for moving back the parameters position to the base

equilibrium value. The matrix M can be splitted in two parts: the first column that

represents the linear relationship between the controlled parameters and P1 coil, will

call it as MSol , the second part is composed by all the other columns belonging to

M and expressing the relationship between the controlled parameters and the control

coils, will call it as MCoils. Parameters variation ∆P is given by :

∆PSol =MSol ·∆ISol, (3.2.6)

49


where ∆ISol is the current variation value in P1 for each step.The control system has

to counteract the changes Sol induced by ∆ISol to let unchanged the plasma position,

i.e.:

∆PCoils = −∆PSol, (3.2.7)

Where ∆PCoils is given by:

∆PCoils =MCoils ·∆ICoils. (3.2.8)

Where ∆ICoils represents the variation that must be applied to remaining coils in order

to compensate the variation ∆PSol. Now adding the equation 3.2.6 to the equation

3.2.8 and making the opportune sostitutions it yelds:

0 =MCoils ·∆ICoils +MSol ·∆ISol, (3.2.9)

and then the coils current values for restoring the parameters positions are:

∆ICoils = −MyCoils ·MSol ·∆ISol (3.2.10)

In some cases the matrix MCoils is not square, in this cases will be calculated the

Moore-Penrose pseudo-inverse. The Moore-Penrose pseudoinverse is a matrix B of

the same dimensions as transpose M⊤Coils satisfying four conditions:

1. MCoils · B ·MCoils =MCoils ;

2. B ·MCoils · B = B ;

3. MCoils · B is Hermitian ;

4. B ·MCoils is Hermitian .

Applying the new currents values ∆ICoils Fiesta carries out the new equilibrium with

the new parameter values, at this point is possible to evaluate the error between the

50


parameters calculated after using the control function and the parameters values in

the base equilibrium, if it is not admissable (this is up to the user decide the maximum

error tolerance) an additional loop to select a new ∆ICoils is performed to decrease

the error to the desired value. The loop is made assigning the error tolerance value

Tol and if the maximum error value is greater than the tolerance it will be reduced

in this way:

∆Presized =Tol · ParametersError Vector

Max Error Value(3.2.11)

After applying ∆Presized to the equation 3.2.8 is possible to calculate ∆ICoils such as:

∆ICoils =M−1Coils ·∆Presized (3.2.12)

as made above the new equilibrium is generated and if the maximum error value is still

greater than the tolerance it will resized again until obtaining the loop convergence,

if it will not converge the equilibrium with minimum error value will be used for

continuing the simulation.

3.2.6 Using gaps as control parameters

The gap is a virtual line created in order to control one point of the plasma. The point

controlled is given by the intersection between the gap and a given flux curve. It could

be the separatrix or another flux curve calculated with the Scrape-Off Layer (SOL)

setted with the base equilibrium. The term Scrape-Off Layer refers to the plasma

region characterized by open field lines (commencing or ending on a material surface).

With limiter plasmas, this region is the region outside the Last Closed Flux Surface

(LCFS). With divertor plasmas, this region is the region outside the separatrix. In

divertor plasmas, the SOL absorbs most of the plasma exhaust (particles and heat)

and transports it along the field lines to the divertor plates. Hence, this region is of

51


prime importance for future reactors. Transport in the SOL is very different from

transport in the confined plasma due to the open field lines: it is predominantly

convective (rather than diffusive). Typically, the density decays exponentially away

from the LCFS. This does not mean that the detailed transport is easy to understand

or model: intermittency is very strong in the SOL region. From a physical point

of view locks one point of the plasma is not the best compromise because it is a

constraint for plasma flux curve which is not free to follow its natural curvature, for

this reason a new parameters set will be studied forward in the same chapter. Further

some situations cannot be solved using the gaps, this is the case in SXD scenarios, in

some of these one null point of the magnetic field moves inside the divertor chamber

while the solenoid swing, so the plasma confinement is lost as it would tend to rotate

around the coil that generates the null point. In Figure 3.7 a typical example of

this problem is reported, it has been simulated with a SXD scenario with 1.0 MA of

plasma current and 2.5 cm of Scrape-Off Layer.

52


Figure 3.7: Case of wrong control using gaps.

53

Cap. 3 Shape control §3.3 Studying a new parameters set

3.3 Studying a new parameters set

3.3.1 Detecting parameters

The SXD in Mast-Upgrade is one of its most important features, having additional

coils for controlling the plasma in the SXD area, it is possible to control more param-

eters. With the aim to increase the performances of the controlled system, several

kind of new parameters have been detected and tested as well in order to replace the

gaps which seem to give some troubles in several scenarios as explained above.

3.3.2 Connection length

One idea has been trying to consider the connection length as a parameter. The

connection length in the scrape of layer (SOL), now on L, is defined as the distance

along a field line from the mid-plane to the first material surface that the field line

strikes. Hence, it is a first approximation to the distance that plasma particles travel,

parallel to the magnetic field, before arriving at a surface. This clearly ignores the

reality of cross-field drifts, collisions, turbulence etc. However it provides a well-

defined parameter for quantifying the character of a given magnetic field geometry,

with respect to providing the background against which the processes of transport

parallel as well as perpendicular to the magnetic field will combine to produce a given

scrape off layer, and hence exhaust of heat and particles from the plasma. With the

magnetic field being dominated by the toroidal component, it is hardly surprising

that the L is strongly dependent upon the toroidal field in MAST-Upgrade. The

relationship between the L and expansion is illustrated in the following way.

54


Figure 3.8: Parameters characterising the relationship between L and flux expansionin the Super-X configuration.

Figure 3.8 shows a thickness of scrape-off layer at the outer-mid-plane of ∆rm.

Given the vertical (poloidal) field, Bv , and radius, rm , at the midplane, the poloidal

flux for this thickness is:

∆Ψ ≈ ∆rm · 2πrmBv (3.3.1)

For simplicity, in the Super-X divertor chamber the poloidal field is assumed to be

essentially radial and constant, Br , at a radius, r , and extending over a height, ∆z,

55


the same flux will be given by

∆Ψ = ∆z · 2πrBr (3.3.2)

Equating the two fluxes, in the standard way, gives the radial field as,

Br =∆rm∆z

·rmr

· Bv (3.3.3)

The field line trajectory in the Super-X region is defined in terms of the ratio of the

radial increment, δr , to the toroidal increment, rδφ , through the ratio of the radial

and toroidal fields i.e.

δr

rδφ=Br

Bφ(3.3.4)

In addition radial variation of the toroidal field can normalised to the value of the

toroidal field at the midplane, Bφm, hence

δr

rδφ=

rBr

Bφmrm(3.3.5)

In this simplified geometry, the increment in the distance along the field line, s , is

given by

δs =√

δr2 + (rδφ)2 (3.3.6)

Substituting, and taking the limit as δr → 0 , we have

δs

δr=

√

1 +

(∆z

∆rm·Bφm

Bv

)2

(3.3.7)

Hence in the assumed geometry, the rate at which the L increases with radius is a

constant, determined by the ratio of the poloidal and toroidal fields at the mid-plane

(i.e. the field-line angle at the mid-plane) and the ratios of the thickness of the SOL

at the mid-plane (that we wish to include in the SXD chamber), and the height of

that chamber. Consequently, if we wish to control the plasma surface interactions on

56


the top and bottom of the chamber, the mean amount of connection length extension

in the SXD chamber is constrained. In the case of significant flux expansion and

consequently a square-root of a number much larger than unity, the mean L increment,

∆L , from the SXD chamber of radial extent, ∆r , is approximated as:

∆L ≈∆z

∆rm·Bφm

Bv·∆r (3.3.8)

Taking the example of 3cm of mid-plane SOL entering a 0.4m high SXD chamber,

of radial extent 0.4m, with ratio of fields at the mid-plane of 1.4, based on a 1MA,

fulltoroidal- field MAST-Upgrade case, then the mean increment in L from the Super-

X chamber is 7.5m this becomes 22m if only 1 cm of SOL is considered. In reality,

the magnetic field is non-uniform and there is the capability to increase the L on a

chosen flux surface, by introducing an X-point into the chamber. The example in the

figure 3.8 shows that there are X-points close to both the top and the bottom of the

chamber. The lower X-point is between the D5 and D3 coils, whilst the most relevant

upper X-point lies beneath the D6 coil. In fig 3.9 is reported the course of the particle

and its length is just L.

57


Figure 3.9: Connection length.

After applying this new parameter value to the base equilibrium a new sensitivity

matrix has been calculated, the results have not been so encouraging because the row

in the sensitivity matrix related to the L has a different order of magnitude compared

to the others parameters such as external radius, inner radius, X point radius. In

fig 3.10 is showed the different order of value related to the parameter L (CL) in

the matrix M. Furthermore some simulations have been run and the results were not

satisfactorily due to numerical issues in the matrix inversion. Another problem is that

the calculation of the L is too sensitive, so a little error in the coils currents would

58


means a not acceptable error on the L value.

Figure 3.10: Dimensional differents between the L and the other parameters in thematrix M.

3.3.3 Minimum distance

An other idea has been to control the minimum distance between the isoflux curves

and the SXD’s surfaces. In other words it means to create a death zone which the

plasma must avoid when it flows through the SXD. The values of these parameters

in the sensitivity matrix are admissable compared with the others. This new set is

made by four parameters: the first one to avoid the contact between plasma and the

divertor “nose”, the second one to avoid the impact between plasma and the upper

surface in the SXD, the third one and the fourth one to protect the lower surfaces

in the SXD. Compared to the gaps these parameters have the advantage that the

flux curve can shift alongside the line that separate that death-zone, in such way the

59


plasma flux curves are not constrained to pass for a controlled point how happens

with gaps, this is suitable from a physical point of view for permitting to the flux

curves to follow as more as possible their natural courses. Infact the plasma can shift

along the line parallel to the divertor surfaces and setted at the minimum distance

calculated in Fiesta simulator. In fig 3.11 is showed one shot simulated obtained

controlling the minimum distance paramters, is possible to see how the plasma shifts

in parallel direction respect to the divertor surfaces.

60


Figure 3.11: Shot simulation controlling the minimum distance in the SXD

61

Cap. 3 Shape control §3.4 Control function: Newton-Raphson algorithm

3.4 Control function: Newton-Raphson algorithm

3.4.1 Algorithm description

A number of optimization problems, estimation, solution to linear and non linear map

inversion can be reformulated as a zeros finding problem of an appropriate function.

Here is briefly illustrated the Newton-Raphson iterative algorithm used to find a zero

x⋆ of a function f(x), i.e., f(x⋆) = 0.

This method has been described by I. Newton in the 17th century, and J. Raphson

developed its recursive version, successively refined by T. Simpson to solve non linear

equations. Similar techniques have been studied in the same century also in Japan

and much earlier by Babylonians for calculating square roots. The main idea, for x

and f scalar, consists in approximating a differentiable function f with its first order

Taylor expansion as:

f(x⋆) = f(x) + f ′(x)(x⋆ − x) + o((x⋆ − x)2) ≈ f(x) + f ′(x)(x⋆ − x) (3.4.1)

where f ′(x) is the first derivative of f(x). It is then possible to get an estimate (correct

value if f is a linear function) of x⋆ solving (3.4.1) and noting that f(x⋆) = 0 yelds:

x⋆ ≈ x−f(x)

f ′(x)(3.4.2)

assuming f ′(x) 6= 0. Reitering this approximation yelds the Newton-Raphson recur-

sive method in the scalar case:

xk+1 = xk −f(xk)

f ′(xk), (3.4.3)

The issue becomes more complicated when dealing with x ∈ ℜn and f(·) : ℜn → ℜm.

case m = n: denoting the Jacobian matrix of f(x) as Jf(x) = δf(x)/δx, assumed

invertible

xk+1 = xk − Jf (xk)−1f(xk) (3.4.4)

62

Cap. 3 Shape control §3.4 Control function: Newton-Raphson algorithm

case m ≥ n : denoting with Jf(x)† = (Jf(x)

⊤Jf(x))−1Jf(x)

⊤ as the pseudo-inverse

(Penn Rose or generalized least square inverse matrix) of the Jacobian matrix Jf(x),

with Jf (x)⊤Jf(x) invertible, then

xk+1 = xk − Jf (xk)†f(xk). (3.4.5)

It is straightforward to note that if, during the iterations, xk is such that f ′(xk) = 0

or close by (equivalently the matrix Jf(x) or Jf(x)⊤Jf (x) are not invertible or bad

conditioned), this method can not be used. However, it is extremely appealing since,

at least locally, the convergence is quadratic (the number of accurate digits roughly

doubles during iterations) under some assumption as shown next in the scalar case.

3.4.2 Finding minima

The NR algorithm can be used to find the stationary points (and then minima or

maxima) in the scalar case of a two time differentiable function h(x) simply selecting

f(x) = h′(x) and the system (3.4.3) becomes

xk+1 = xk −h′(xk)

h′′(xk). (3.4.6)

This formula can also be derived by another consideration: extend the Taylor expan-

sion (3.4.1) of h(x) at the second order namely

h(x⋆) ≈ h(x) + h′(x)(x⋆ − x) +1

2h′′(x)(x⋆ − x)2 (3.4.7)

that is, h(x) is approximated by a second order polynomial h(x+∆) ≈ (a+b∆+c∆2,

∆ = x⋆−x, whose stationary point can be evaluated solving d(a+ b∆+ c∆2)/d∆ = 0

yielding ∆ = −b/2c, namely x⋆ = x− h′(x)/h′′(x).

Note that if h(x) is quadratic, the stationary point is evaluated exactly by (3.4.6) in

one step; this is the dual property compared with the zero found in one step when

63

Cap. 3 Shape control §3.5 Newton-Raphson application

f(x) is linear using the eq.(3.4.3). The following assumption is needed to re-formulate

(3.4.6) in case x ∈ ℜn that is right the case of study. x ∈ ℜn and h(·) : ℜn → ℜ is

two times differentiable and there exists a stationary point x⋆ such that ∇h(x) |⋆x= 0

and the hessian Hh(x) = δ2h(x)/δx2 is invertible. Then, the algorithm for finding

stationary points when x ∈ ℜn is:

x(k+1) = xk −Hh(xk)−1∇h(xk)

⊤. (3.4.8)

3.5 Newton-Raphson application

3.5.1 Square function

In the case of study one square function has been created in order to apply Newton-

Raphson for dectecting the currents values necessary to restore the equilibrium during

the solenoid swing:

h(x) = (∆P +MCoils · x)⊤ ·WP · (∆P +MCoils · x) (3.5.1)

Where ∆P is the vector containing the parameters displacements evaluated as the

subtraction between the parameters positions calculated by Fiesta after applying the

step current on P1 and the parameters positions in the base equilibrium, MCoils is

the matrix containing the columns related to the control coils, x is the unknown that

is the coil currrents variations for restoring the plasma shape and lastly WP is an

identity matrix which can be multiplied by opportune weights. The objective is to

find the x, the controlled coils current variation, that minimize the cost function h(x)

(stationary point: minima). Moreover x ∈ ℜn and h(·) : ℜn → ℜ then the equation

(3.4.8) has been used. The xk+1 generated by eq. ( 3.5.2) minimizes the function

(3.5.1), so that it is possible to restore the value of the parameters positions to the

base equilibrium. The power of Newton-Raphson algorithm is that with quadratic

64


functions such as the eq. (3.5.1) the convergence is obtained immediately after only

one step, so the control function does not need any iterations. This is an important

goal because it could be used on the real time control as well. In this case the solution

x is given by:

xk+1 = xk − h′(xk)/h′′(xk), (3.5.2)

where the Hessian h′′(xk) is:

h′′(x) = 2 · (M⊤Coils ·WP ·MCoils) (3.5.3)

The hessian is a square symmetric matrix with dimensions n×n where n is the control

coils number (11). If the matrix is singular, will be calculated the pseudo-inverse as

in the previous control.

Moreover h′(x) is the Jacobian and it is given by:

h′(x) = 2 · (M⊤Coils ·WP ·∆P ) + 2 · (M⊤

Coils ·WP ·MCCoils · x). (3.5.4)

To conclude the previous control function used to make a loop for resizing the error

value and it was unsuitable for using in the real time control, because the calculation

time was too long, instead Newton-Raphson converges qiuckly in only one step for

this reason is suitable for real time control in terms of frequency of data acquisition.

3.5.2 Parameter’s weights

In some scenarios could be necessary to constrain or release one or more parameters

which otherwise will generate problems during the scan simulation. Looking at the

function 3.5.1 there is the matrix WP , that is the weight parameters matrix that

is setted as an identity matrix by default. Changing the values on WP matrix’s

diagonal it is possible to give more or less weight in the cost function to one or more

65


parameters. This is an important tool because it permits to have much flexibility

in the control shape. Applying the values < 1 the parameters will be less weighted,

otherwise with values > 1 they will be more weighted. This method is useful in

several conditions where the control function is not enough accurate for restoring

the base equilibrium shape. In the Figure 3.12 is showed the error values evolution

related to the parameters using the Newton-Raphson control function during the shot

simulation, on this simulation no weights have been applied. The Figure 3.13 shows

Figure 3.12: Parameters error using Newton-Raphson controller without any weightsapplied.

the same simulation, but this time one weight has been applied on the gap 2 (red

gap in fig.3.14), the weight applied is 5 so the parameter has been weighted more,

then the error value is lower than the previous case. In the Figure 3.15 is the same

66


Figure 3.13: Parameters error using Newton-Raphson controller with weight 5 appliedon the gap 2 for constraining it.

simultion with a weight value of 10 applied on the same gap (red gap in fig.3.14), even

in this case the gap is more weighted than in the previous case. The input value that

is changed for permitting to constrain the parameters is the current value in the coil,

as showed in figure 3.16 where is reported the delta currents between for each step of

the swing between the case with no weight applied and that one with a weight of 10.

It is possible to see that the main variations are related to the coils D2 and D3 that

are closer to the gap2 (figure 3.14). Instead in the Figure 3.17 is applied a weight

value lower than one, in this case is 0.5 and how is possible to see the parameter gap 6

(red gap in figure 3.18) is less weighted than in the case without weight (Figure3.12).

In the last picture Figure 3.19 is applied a weight on the same parameter (gap 6 red

67


Figure 3.14: Gap red is the gap 2 where the weight is applied.

gap 3.18) of 0.1, it is intuitive that the parameter is less weighted than in the previous

case. The input value that is changed for permitting to weighting less the parameters

68


Figure 3.15: Parameters error using Newton-Raphson controller with weight 10 ap-plied on the gap 2 for constraining it.

is the current value in the coil, as showed in figure 3.20

3.5.3 Coil’s weights to avoid saturation

The commissioning of MAST-Upgrade has been already made, so the coils set will

have operating space setted by the designers. It means that they will have intrinsic

and structural limit related to the current value which every coil will be able to reach.

The previous control in the simulator (eq. 3.2.10) does not know the saturation levels

and in some simulation could happen that the coils currents are selected above such

level. In order to avoid this issue, a tool has been created in order to manage this

situation applying dynamic gains to avoid the coils saturations. It has been made

69


Figure 3.16: Delta Currents during the shots between the case without any weightapplied and the case with weight of 10 applied.

adding to the square function 3.5.1 another quadratic function given by:

x⊤ ·WC · x (3.5.5)

Where x is still the unknow that has to be applied for controlling the plasma shape and

WC is the coils weight matrix that is a zeros matrix setted by default. Changing the

zeros values on the diagonal such as on the parameters weights matrix, it is possible

to reduce or to enlarge the currents values in every single coil during the simulation

scan. The function 3.5.1 to be minimized becomes:

h(x) = (∆P +MCoils · x)⊤ ·WP · (∆P +MCoils · x) + x⊤ ·WC · x (3.5.6)

Where the second term is equal to zero if the simulation does not need to use weights

on the coils.

70


Figure 3.17: Parameters error using Newton-Raphson controller with weight 0.5 ap-plied on the gap 6 for relaxing it.

Basicly has been implemented a function that can apply different weight values during

the simulated shot considering that in MAST Upgrade there are some coils which are

symmetric and others that cannot change the current’s sign. At the beginning the

matrix WC has been created in order to use costant values during the scan, this is not

dynamic and the weights have to be chosen after one test scan. The issue has been

improved apllying the variation of the weight for each step of the scan. It has been

realized choosing the range weights values that is 0 when is not needed the weight and

100 when the coil has to be locked. The function implemented can check the position

of the currents respect to the saturation limit for each coil, if it is beetween the 80%

and 90% will be apllied a linear intepolation between 0 and 100 and the result will be

71


Figure 3.18: Red gap is the gap 6 where the weight is applied.

the weight to be used for that coil, moreover if it goes over 90% the weight applied

is the maximum that is 100, otherwise if the coil is under 80% no weight is applied.

If the coil current at the beginning of the scan is already close to the saturation zone

72


Figure 3.19: Parameters error using Newton-Raphson controller with weight 0.1 ap-plied on the gap 6 for relaxing it.

so the function will check the coil’s direction at each step before to apply the weight,

because if it is moving away from the saturation no weight will be applied. In figure

3.21 is showed an example of currents evolution scan in one scenario with 1 MA of

plasma current and 25 mm of SOL, in this simulation no weight has been applied and

the coil PC goes over the structural limit (Black lines) because at the beginning of

the simulation it stays already close to the limit saturation that is -12 kA. The same

scan has been made but this time using the dynamic weights, in figure 3.22 is showed

how the coil PC is locked during the scan and the values evolution is always within

the structural limit (Black lines).

73

Cap. 3 Shape control §3.6 Gradient algorithm apllied as a dynamic coil weight

Figure 3.20: Delta Currents during the shots between the case without any weightapplied and the case with weight of 0.1 applied.

Figure 3.21: Coils currents evolution during a simulated scan without any weightapplied, is showed how the coil PC crosses the structural limit (Black line).

3.6 Gradient algorithm apllied as a dynamic coil

weight

An improvement of the dynamic coil weights for avoiding the saturation has been

developed with the aim to use a continue function such as an hyperbolic for choosing74


Figure 3.22: Coils currents evoluiton during a simulated scan with dynamic weightapplied on the coil PC, is possible to see that the values are within the structurallimit (Black lines) for all the snapshots of the scan.

the weights at each step of the scan. The starting function has been the equation 3.5.6,

the weight matrix in the previous case contained always a constant value completely

released from the unknow x. In this case the weight matrix WC depends nonlinearly

on x. More specifically each element of the diagonal matrix is an hyperbolic function

dependent of x. The function has been chosen by considering the boundary conditions

that are requested from the control problem. The function needed has to grow quicly

when the coil values are getting close to the saturation limits, in this way the weight

applied is high, on the other hand when the coil values are close to the central value

the function has to be close to zero, in this way no weight will be applied. After this

considerations the function chosen is an Hyperbolic function such as Y = a · X20.

where X has a range values like −1 < X < 1 and a is the maximum weight value that

is 100. The shape of the function is showed in figure 3.23 The unknow x is contained

75


Figure 3.23: Hyperbolic function shape used for applying the dynamic weights on thecoils.

inside the X in the following form:

X =(ICoil + x− Iavg)

(Imax − Iavg). (3.6.1)

Where ICoil is the coil current value at each step, x is the unknow as the ∆Icoil to

be applied for restoring the parameters positions, Iavg is the central value of the coil

range defined as:

Iavg =(Imax + Imin)

2. (3.6.2)

Imax is the maximum current value that the every single coil can supply, Imin instead

is the minimum limit of the coil.

Now the function 3.5.6 to be minimized becomes:

h(x) = (∆P +MCoils · x)⊤ ·WP · (∆P +MCoils · x) + γ · (X⊤ ·WC(X) ·X) (3.6.3)

76


Where γ is a gain value that has to be setted in function of the SOL used in the

scan simulation. Unfortunately the function is not quadratic anymore so in order to

minimize it has been chosen an alternative method that is the gradient algorithm.

When the control function is called is applied the Newton-Raphson method just on

the first term of the function 3.6.3, in this way a first estimate is calculated, after

is used a loop for the gradient method because the convergence is not immediate in

order to avoid the coil saturation and moving the current value as close as possible to

the central levels.

It does not require the computation of the Hessian, so it is simpler, but usually it

converges slowly around the zero and if the gain γ is not chosen carefully, there might

be persistently oscillations that prevent the estimate to converge. The goodness of the

method is that it is global and usually converges quickly when the estimation error

is large. The algorithm is very simple, is needed to set d = 1 for finding a maximum,

d = −1 otherwise, then the direction of the maximum growth for is evaluated h(x) is

evaluated in such recursive way:

x+ = x+ d · γ · ∇h(x)⊤ (3.6.4)

where ∇h(x) is the the gradient of the funciton h(x) (3.6.3).

77

Chapter 4

Simulations results

In this chapter will be reported the simulation analysis runned usingthe gradient algorithm, more specifically will be explored the limitconfigurations with a kind of scenario with high internal inductance,the operating space of the machine with that scenario in terms ofcoils currents limits.

4.1 Configuration limits in MAST Upgrade SXD

scenarios

This task has been developed in order to understand which are the limit configurations

in terms of the structural coil limits. The simulations has been runned using one

scenario with high internal inductance, now on HILi. In figure 4.1 the equilibrium

is showed. The intial value of the parameters was not the right one for running the

simulated scans. So first of all has been built one tool which permits to move the

parameter positions to the target desired, the plasma boundary has been created with

the aim to maximize the area expansion in the SXD and to match it with a standard

configuration. The standard scenario has a lower internal inductance as is showed in

yellow colour in figure 4.2. The HILi scenario is more complicated to be controlled

because the plasma current is more concentrated in the core and for this reason is less

sensible to the control coils.

78

Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios

Figure 4.1: High internal inductance scenario, the yellow colour represents the plasmacurrent distribution.

79


Figure 4.2: Standard scenario, the yellow colour represents the plasma current distri-bution.

80


4.1.1 Changing plasma shape

The tool for matching the HILi scenario to the standard one has been created for

changing the plasma shape and moving it to any target desired. It is also possible

to move more parameters together in the same time. Basically the function used

for controlling the parameter positions is still the control function with the Newton-

Raphson algorithm 3.4.8. It is very useful because the user can match two different

plasma configurations to the same shape for testing the performances during the

simulated scan. It is also possible to apply the weights on coils and parameters such

as in the shot simulation (chapter 3). The input instructions for using this tool are

simply the target coordinates of the parameters that the user desires to move. In fig.

4.3 is showed an example of plasma boundary shape evolution moving four paramers

together: outer radius, X point radius, X point height and two gaps in the SXD area.

In fig 4.4 is reported the parameters displacements related to the parameters which

are moving toward the configuration desired.

81


Figure 4.3: Plasma shape changed moving four parameters together.

82


Figure 4.4: Parameters displacement during the shape changing.

4.1.2 Scenario with high internal inductance

The HILi scenario has been modified for matching the standard one, during the

plasma parameters changing has been highlighted that some parameters had almost

parallel values of the sensitivity matrix’ s row. It means that is impossible move one

parameter and kepping controlled (locked) the parameter with the similar row in M.

They have to be moved together, this relation is showed in figure 4.5 where is possible

to see how moving the X point height has to be not controlled also the gap (red) on

divertor nose.

83


Figure 4.5: Displacement of gap4 and X point height due to the parallel values in thesensitivity matrix.

More specifically in the figure 4.6 is illustrated the parallelism between the two

parameters related to the rows of the sensitivity matrix.

84


Figure 4.6: Parallel vectors in the sensitivity matrix related to the gap4 and the Xpoint height.

4.1.3 Database creation

The simulations with the gradient algorithm has been runned on forty HILi equilibria

with different SOL (from 2.5 cm to 6 cm by step of 0.5 cm) and different plasma current

(from 1.0 MA to 1.4 MA by step of 0.1 MA). The starting position obtained with the

HILi scenario in respect with the standard one is showed in figure 4.10, the X point

has an upper position otherwise the SOL would touch the divertor nose.

85


Figure 4.7: Starting position of the Hi Li scenario (brown) compared to the standardone(red).

The simulated scans has been runned from 0 A to -50 kA in the solenoid current.

In figure 4.8 an example of plasma boundary evolution during the simulated shot is

86


showed, it has been runned with a SOL of 4.5 cm and a plasma current value of

1.0 MA. In figure 4.9 is showed the current coils evolution and in the figure 4.9 are

reported the parameter’s error. For all of these graphs the blue line is the starting of

the shot, the red line is the end.

Figure 4.8: Boundary evolution during one simulation runned with Hi Li scenariowith SOL 4.5 cm and plasma current of 1.0 MA.

87

Cap. 4 Simulations results §4.2 Operative space

Figure 4.9: Current coil variations during the simulated scan.

Figure 4.10: Parameter errors during the solenoid swing.

4.2 Operative space

With all the results of the simulation runned has been created an operative space for

the device in terms of coil current limits. In the figures (4.11,4.12,4.13,4.14,4.15) are

88


showed in blue the admissable operative areas for several SOL and different plasma

currents. It is possible to see that increasing the plasma current the operative space

area becomes smaller and this is due to the current increasing in the control coils that

go over the structural limits.

Figure 4.11: Operative area with Hi Li simulations at 1.0 MA of plasma current.


89




90

Cap. 4 Simulations results §4.3 Comparison: gradient algorithm and simple inversion


4.3 Comparison: gradient algorithm and simple

inversion

In this chapter is showed the comparison of the control function used for restoring

the parameter positions. The comparison has been done between the simple inversion

of the sensititvity matrix (equation: 3.2.10) and the controller based on the gradient

algorithm (equation: 3.6.4). It is possible to see how the second one is more accurate,

infact the figure 4.16 shows just the comparison between the error on the parameters

during the simulation (shot with Hi Li scenario SOL= 3.5 cm and plasma current 1.0

MA) and it is easy to understand that the error is less with the gradient algorithm

rather than with the simple inversion. In the figures 4.17, 4.18 are showed the param-

eter errors evolution during the simulation, the first one with simple inversion, the

second one with the gradient algorithm. Moreover the gradient algorithm permits to

use the weights on the coil currents and on the parameters errors as showed in the

chapter 3. In fig 4.19 is reported the plasma boundary evolution and the gap positions

91


during the simulation with the gradient algorithm.

Figure 4.16: Parameter errors comparation between the controller based on the simpleinversion and the gradient algortihm.

92


Figure 4.17: Parameter errors during the simulation with the controller based on thegradient algorithm.

Figure 4.18: Parameter errors during the simulation with the controller based on thesimple inversion.

93


Figure 4.19: Plasma boundary evolution during the shot with the gradient algorithmcontroller and parameters positions.

94

Chapter 5

Conclusions and futuredevelopments

The present work is the result of the collaboration between Universita di Roma Tor

Vergata and the Culham Centre Fusion Energy (CCFE) and should be considered

in the framework of the thermonuclear fusion research. The thesis has addressed

the problem of the shape control in the upgrade of the tokamak MAST that will be

ready to run experiments on 2015. The purpose of the work has been to develop a

magneto-static controller in MATLAB environment. It has been implemented using

the Newton-Raphson and the gradient algorithm. The controller has been added on

Fiesta code, a magnetostatic free boundary Grad-Shafranov solver written in MAT-

LAB. The possibility of using the weights for avoiding the coil saturations as well as

on the parameter errors makes the operative space more flexible and wider. Moreover

the calculation time for solving the control shape is suitable with the clock of the real

time controller in a standard tokamak. It has also been detected a new parameter

set to be controlled during the shot simulations. The new database runned with an

high internal inductance scenario in several configurations related to plasma current

and SOL has highlighted a wide operative space. The limit configuration are mainly

related with this kind of scenario to the lower limit of the coil DP. The cost function

95

Cap. 5 Conclusions and future developments

to be minimized requires an accurate choice of the gain γ. The database created is

being used for finding out by a regressive analysis the linear relation for controlling

the plasam shape in the SXD. Future developments could be extending the controller

to the real time control and optimizing the paramters set detected.

96

List of Figures

1.1 Fusion nuclear process. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Opposite trajectory of ions and electrons in magnetic field. . . . . . . 7

1.3 The Lawson criterion, or minimum value of (electron density * energy

confinement time) required for self-heating, for three fusion reactions.

For D-T, neτE minimizes near the temperature 25 keV (300 million

kelvins). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 The fusion triple product condition for three fusion reactions. . . . . 11

1.5 General structure of the tokamak device. . . . . . . . . . . . . . . . . 11

1.6 Currents and magnetic fields in a tokamak device. . . . . . . . . . . . 12

1.7 Geometric parameters in a tokamak plasma. . . . . . . . . . . . . . . 14

2.1 Example of plasma in a MAST experiment. . . . . . . . . . . . . . . 16

2.2 Section of MAST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Cross-section of the MAST vessel and position of the six PF coils. . . 18

2.4 Typical currents evolution in a real MAST experiment. . . . . . . . . 19

2.5 Cross-section of MAST Upgrade. Only the lower of each pair of coils

is shown (denoted as L). . . . . . . . . . . . . . . . . . . . . . . . . . 25

97

LIST OF FIGURES LIST OF FIGURES

2.6 Location of poloidal field coils, and main plasma-facing surfaces, and

two example equilibria for MAST-Upgrade on the left with a conven-

tional divertor, on the right with a SXD. . . . . . . . . . . . . . . . . 27

2.7 Definition of Rsep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 Schematic summary of magnetic configuration possibilities for MAST

Upgrade. Configurations with asymmetric configurations are shown

with grey background. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.9 Cut-out section of the SXD. . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 Plot example of Fiesta equil object. . . . . . . . . . . . . . . . . . . . 39

3.2 Linear fit between outer radius and P1 coil. . . . . . . . . . . . . . . 44

3.3 Sensitivity matrix plot, it is possible to see how the fifth line (gap) is

highly influenced by D6 coil. . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Boundary of plasma plot shows the closest relationship between the

gap (blue) and the coil D6 (exactly above it). . . . . . . . . . . . . . 47

3.5 Plasma shape evolution during the simulated shot, blue line starting

point(base equilibrium), red line last snapshot. . . . . . . . . . . . . . 48

3.6 Linear relationship between flux and solenoid current during the swing. 49

3.7 Case of wrong control using gaps. . . . . . . . . . . . . . . . . . . . . 53

3.8 Parameters characterising the relationship between L and flux expan-

sion in the Super-X configuration. . . . . . . . . . . . . . . . . . . . . 55

3.9 Connection length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.10 Dimensional differents between the L and the other parameters in the

matrix M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.11 Shot simulation controlling the minimum distance in the SXD . . . . 61

98


3.12 Parameters error using Newton-Raphson controller without any weights

applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.13 Parameters error using Newton-Raphson controller with weight 5 ap-

plied on the gap 2 for constraining it. . . . . . . . . . . . . . . . . . . 67

3.14 Gap red is the gap 2 where the weight is applied. . . . . . . . . . . . 68

3.15 Parameters error using Newton-Raphson controller with weight 10 ap-

plied on the gap 2 for constraining it. . . . . . . . . . . . . . . . . . . 69

3.16 Delta Currents during the shots between the case without any weight

applied and the case with weight of 10 applied. . . . . . . . . . . . . 70

3.17 Parameters error using Newton-Raphson controller with weight 0.5 ap-

plied on the gap 6 for relaxing it. . . . . . . . . . . . . . . . . . . . . 71

3.18 Red gap is the gap 6 where the weight is applied. . . . . . . . . . . . 72

3.19 Parameters error using Newton-Raphson controller with weight 0.1 ap-

plied on the gap 6 for relaxing it. . . . . . . . . . . . . . . . . . . . . 73

3.20 Delta Currents during the shots between the case without any weight

applied and the case with weight of 0.1 applied. . . . . . . . . . . . . 74

3.21 Coils currents evolution during a simulated scan without any weight

applied, is showed how the coil PC crosses the structural limit (Black

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.22 Coils currents evoluiton during a simulated scan with dynamic weight

applied on the coil PC, is possible to see that the values are within the

structural limit (Black lines) for all the snapshots of the scan. . . . . 75

3.23 Hyperbolic function shape used for applying the dynamic weights on

the coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

99


4.1 High internal inductance scenario, the yellow colour represents the

plasma current distribution. . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Standard scenario, the yellow colour represents the plasma current dis-

tribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Plasma shape changed moving four parameters together. . . . . . . . 82

4.4 Parameters displacement during the shape changing. . . . . . . . . . 83

4.5 Displacement of gap4 and X point height due to the parallel values in

the sensitivity matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6 Parallel vectors in the sensitivity matrix related to the gap4 and the X

point height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Starting position of the Hi Li scenario (brown) compared to the stan-

dard one(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.8 Boundary evolution during one simulation runned with Hi Li scenario

with SOL 4.5 cm and plasma current of 1.0 MA. . . . . . . . . . . . . 87

4.9 Current coil variations during the simulated scan. . . . . . . . . . . . 88

4.10 Parameter errors during the solenoid swing. . . . . . . . . . . . . . . 88

4.11 Operative area with Hi Li simulations at 1.0 MA of plasma current. . 89





4.16 Parameter errors comparation between the controller based on the sim-

ple inversion and the gradient algortihm. . . . . . . . . . . . . . . . . 92

4.17 Parameter errors during the simulation with the controller based on

the gradient algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

100


4.18 Parameter errors during the simulation with the controller based on

the simple inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.19 Plasma boundary evolution during the shot with the gradient algorithm

controller and parameters positions. . . . . . . . . . . . . . . . . . . . 94

101

Bibliography

[1] Ioannis K. Argyros , “Convergence and application of Newton-type Iterations”,

Springer, 2008 .

[2] ,E.Pennestrı - F.Cheli “Cinematica e dinamica dei sistemi multibody”, Am-

brosiana, 2006.

[3] J.Wesson , “Tokamaks”, Oxford University press USA, 2011.

[4] Hendrik Meyer, “The MAST Upgrade Conceptual and Physics design”, 2010.

[5] Nome Autore, “Nome del libro”, Nome Editore, Anno di Pubblicazione.

[6] Nome Autore, “Nome del libro”, Nome Editore, Anno di Pubblicazione.

102

facolta di ingegneria` corso di laurea magistrale...

Documents