facolta di ingegneria` corso di laurea magistrale...
TRANSCRIPT
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
Cap. 1 Nuclear fusion §1.2 Lawson criterion
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
Cap. 1 Nuclear fusion §1.2 Lawson criterion
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
Cap. 1 Nuclear fusion §1.2 Lawson criterion
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
Cap. 1 Nuclear fusion §1.3 Tokamak
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
Cap. 1 Nuclear fusion §1.3 Tokamak
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
Cap. 1 Nuclear fusion §1.3 Tokamak
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
Cap. 1 Nuclear fusion §1.3 Tokamak
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
Cap. 2 MAST and MAST-Upgrade §2.1 MAST
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
Cap. 2 MAST and MAST-Upgrade §2.1 MAST
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
Cap. 2 MAST and MAST-Upgrade §2.1 MAST
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
Cap. 2 MAST and MAST-Upgrade §2.2 MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.2 MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.2 MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
• 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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
Figure 2.8: Schematic summary of magnetic configuration possibilities for MASTUpgrade. Configurations with asymmetric configurations are shown with grey back-ground.
33
Cap. 2 MAST and MAST-Upgrade §2.3 Super X divertor in MAST-Upgrade
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.2 Fiesta code
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.3 Studying a new parameters set
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.5 Newton-Raphson application
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
Cap. 3 Shape control §3.6 Gradient algorithm apllied as a dynamic coil weight
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
Cap. 3 Shape control §3.6 Gradient algorithm apllied as a dynamic coil weight
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
Cap. 3 Shape control §3.6 Gradient algorithm apllied as a dynamic coil weight
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
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
Figure 4.2: Standard scenario, the yellow colour represents the plasma current distri-bution.
80
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
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
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
Figure 4.3: Plasma shape changed moving four parameters together.
82
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
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
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
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
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
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
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
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
Cap. 4 Simulations results §4.1 Configuration limits in MAST Upgrade SXD scenarios
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
Cap. 4 Simulations results §4.2 Operative space
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.
Figure 4.12: Operative area with Hi Li simulations at 1.1 MA of plasma current.
89
Cap. 4 Simulations results §4.2 Operative space
Figure 4.13: Operative area with Hi Li simulations at 1.2 MA of plasma current.
Figure 4.14: Operative area with Hi Li simulations at 1.3 MA of plasma current.
90
Cap. 4 Simulations results §4.3 Comparison: gradient algorithm and simple inversion
Figure 4.15: Operative area with Hi Li simulations at 1.4 MA of plasma current.
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
Cap. 4 Simulations results §4.3 Comparison: gradient algorithm and simple inversion
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
Cap. 4 Simulations results §4.3 Comparison: gradient algorithm and simple inversion
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
Cap. 4 Simulations results §4.3 Comparison: gradient algorithm and simple inversion
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
LIST OF FIGURES LIST OF FIGURES
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
LIST OF FIGURES LIST OF FIGURES
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
4.12 Operative area with Hi Li simulations at 1.1 MA of plasma current. . 89
4.13 Operative area with Hi Li simulations at 1.2 MA of plasma current. . 90
4.14 Operative area with Hi Li simulations at 1.3 MA of plasma current. . 90
4.15 Operative area with Hi Li simulations at 1.4 MA of plasma current. . 91
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
LIST OF FIGURES LIST OF FIGURES
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
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brosiana, 2006.
[3] J.Wesson , “Tokamaks”, Oxford University press USA, 2011.
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102