Magnetic control of a tokamak equilibrium

Jo Lister, Control for Fusion, Eindhoven, May 7 2008 1

Magnetic control of a tokamak equilibrium

With many thanks to all those with whom it has been such a pleasure to collaborate over the last few years

Especially to Sun Hee Kim, for the film on the talk

Keynote presentation for the 7-8 May workshop onControl for Nuclear Fusion, Eindhoven, the Netherlands

CRPP-EPFL, Lausanne, Switzerland

Jo Lister


What are we watching ?


Contents

I attempt to provide a rapid overview of the control of the plasma equilibrium over the last 35 years….

• What is an equilibrium – what is a tokamak for our purposes ?

• How do we control an equilibrium ?

• When does an equilibrium become unstable ?

• What work has been done over all these years, and why ?

• What work remains to be done for ITER ?

Warning: this is inevitably a very personalised view


Intro – Basics of a tokamak

We are confining plasma (ionised gas) at 200,000,000 degrees, using a strong magnetic field (toroidal).

The plasma is conducting and carries a large toroidal current.

That’s it, you now understand the tokamak

We need:• a strong toroidal field to magnetise the plasma

• a transformer to induce the plasma current

• a field to create an equilibrium force


Intro – An equilibrium

Unfortunately, the plasma will not just sit there – there are internal (gas) pressure and electromagnetic pressure – they both force the plasma to grow in radius

Force is of the order: Tesla * MAmp * 2R

Mass is of the order: 800m3 * 1020 (per m3) * 10-27 (kg/ion)

F = m a – the plasma will move very fast (sec) to find a force balance

2 modelling options are used - follow the acceleration

- assume the plasma is always in “instantaneous force balance”

(easier)

We already have 2 different physics modelling approaches

2

3

2

8ln

40

pi

pv

l

a

RI

RB

Shafranov Equation


Intro – We don’t have to react that fast

We do not have to react on this timescale, which would be very difficult

Why – Lenz’s law (1834)

“When a magnetic flux changes, currents are induced to counter the change”

The plasma sits in a metal vacuum vessel. If it moves, currents are induced in the vessel, and these create a magnetic field which counteracts the movement.

We do not have to control the plasma movement

We cannot control the plasma movement

XX

•

•


Intro - Equilibrium control is via imposed fields

ITER is large – powerful actuators

0 2 4 6 8 10 12-8

-6

-4

-2

0

2

4

6

8

g1

g2

g3

g4

g5

g6

R (m)

Z (

m)

PF coilsGaps (control targets)


Intro - Control of an equilibrium is mastered

TCV has a very wide range of magnetic configurations

What we are showing is isoflux contours, surfaces containing magnetic field lines

These are also contours of kinetic pressure

P = n T (ions plus electrons)

Equilibria are well understood


Intro – How are these shapes made in (e.g.) TCV ?

For today, let’s think of the coil currents as controlling the magnetic field configuration which pulls and pushes on the edge of the plasma


Intro – Back to the equilibrium model

The radial force balance, FR = Ip x Bv

Radial equilibrium is non-linear and time-varying

2

3

2

8ln

40

pi

pv

l

a

RI

RB

Non-linear

Time-varying

Any LTI model is locally linearised and approximate (film…)


Intro – What if our vertical field is not vertical ?

Radial force balance assumes a vertical field

In reality this field will not be homogeneous

To next approximation, assume it is gently curved

We have 2 cases, inward curvature and outward curvature


Intro – Unstable and stable curvature

Inward curvature

Created by a parallel inner current

Pulls the plasma current – stable vertically

•

Outward curvature

Created by an anti-parallel outer current

Pushes the plasma current – unstable vertically

X

More current Better tokamak


Intro – Instability timescale

Lenz’s law comes back again to help us –

- the metallic vacuum vessel does not let the flux get out, or in…

Although the system is unstable (destabilising force), the image currents generate an equal stabilising force (instantaneous force balance)

But…. The metallic vacuum vessel is resistive –image currents decay, and then the plasma will move, and induce more current, and move again….

So the vertical instability now takes place on the timescale of the vacuum vessel image current decay time

TCV, DIII-D up to 3000 / second

ITER, around 14 / second

XX

• •


Introduction resumé (with apologies)

Plasma equilibrium is understood, good experimental and theory bases

Non-linear time-varying equations are available

There can be a positional instability

Complex equilibria have many coils which push and pull on the plasma

Forces are huge

Actuators are voltages on “Poloidal Field” coil circuits

Equilibrium has to be precise to stop 200,000,000 degree plasma from destroying the wall

Mastering the equilibrium presents an interesting control challenge


Progress – Pre-programmed control of radial position

Late 1960’s – first tokamaks, a few milliseconds

Short pulses in a vacuum vessel, equilibrium provided by Lenz’s law

Then try to make longer pulses, bigger currents

Need to add some vertical field to compensate the dissipating image currents

Programme capacitor banks to fire current into simple “PF” coils

Not what you want ? Do it again, keep adjusting the capacitor firing programme until you are happy – measure tokamak performance…

Feedforward control was enough for the first tokamaks


Progress – Feedback control of radial position

Scientists got both bored and more demanding (around 1973)

DITE tokamak – Hugill, Sheffield (UKAEA) developed closed loop control of the radial position

a) measurement

can’t measure radial position

can measure magnetic fields

can infer position from available data

b) actuator has to be a true power supply

more expensive, remember these were cheap experiments

c) controller

Shafranov equation gives the correction sign, keep adjusting gain

Stable equilibria were easily controlled


Progress – Control of shape

JET was a huge jump in size, and was designed to have a non-circular plasma, elongated vertically

shape control

vertical stability

But we had a full equilibrium equation (Grad-Shafranov) which tells us what the currents should be in the coils to get a chosen shape

This was enough to design the PF coil system (correctly) and to start (Noll) worrying about the vertical instability (late 1970’s early 1980’s)

Simple controllers were designed to change the shape, and were adjusted by hand by people who really understood the physics of shape control


Progress – Understanding the vertical instability

Keller (CRPP) and Noll (JET) derived simple models for INTOR with a filament plasma and a pair of filament conductors representing currents induced in the vacuum vessel if the plasma current moved vertically. Plasma mass was zero.

The result was a set of coupled equations giving a second-order system with one unstable pole and one stable pole.

This understanding is still intact – what we have done is to add lots of detail !

Simple models were/are adequate to derive stabilising controllers

X Power SupplyVessel circuit

A basic understanding of the vertical instability was necessary to stabilise the first elongated shaped plasmas, in the early 1980’s (you don’t “learn” stabilisation by trial and error …)


Progress – Refining vertical instability

Late 1980’s, in preparation for the Lausanne TCV tokamak, Lazarus (GA), Lister (CRPP) used a model with an eigenmode description of the vacuum vessel and an equilibrium vertical field. Analysis took the first up-down anti-symmetric eigenmode and explored controller design analytically. The DIII-D active PF coils were included in the analysis.

The role of different coils was studied. The role of the inboard coils was identified analytically (“hybrid control” planned for ITER). The role of PD controllers was also studied analytically.

The result was a proposed improvement to DIII-D stability, a record 2.5 elongation, to be broken only later by TCV.

Note this was perhaps the last serious work done without Matlab !

Simple models led to basic understanding to advance performance


Progress – Maximising vertical stability

TCV was built with coils inside the vacuum vessel to avoid the filtering action of the vessel image currents

Minimising the delays in the loop and taking care of the estimator of the vertical position led to stabilisation of growth rates up to 3000/sec (Hofmann, Dutch, Moret - CRPP)

Plasma elongations up to 2.95 were obtained and plasma currents up to 1MA demonstrated the gain in performance

The cost is control fragility


Progress – Defining the ITER challenges

Now mid-1990’s.

ITER is on the horizon, control of shape and position is clearly crucial since the required precision is high, the control powers and voltages have to be minimised and robustness has to be high

Effort was put into studying the shape and position control problem more experimentally, involving expertise from outside the tokamak experimental groups. TCV was encouraged to start collaborations on different topics with:

Imperial College (UK)

CREATE consortium (Italy)

EPFL automation

JAERI


Progress – Model-based controllers

In the 1980’s, work started on model based controllers, Examples of this are Portone (NET, Imperial College) and Garriba (JET).

Although Garriba developed LQG controllers for JET, to my knowledge they were never used in anger.

…. But we just made a jump… where is the model ??

We have to convert the non-linear time-varying problem into an LTI model, specifically generating the common A,B,C,D(=0) matrices.

Various approaches have been tested, and survive.

Warning: LTI models are local approximations


Progress – Improving the models

Two classes of ab initio model were developed:

- deformable models for shape and position control

- rigid distribution models for position control, especially stabilisation

Model cross-comparison on TCV (CREATE model, RZIP model, Hofmann model) showed very close agreement, except for exceptionally shaped TCV equilibria (Hofmann, Dutch, Vyas, Lister – CRPP)

Unsurprisingly, they all agreed well with experimentally measured growth rates.

But… the models all depend on approximate engineering information on the construction of the tokamak and of the sensors, and also of the actual equilibrium when comparing with experiments.


Progress – Validating ab initio models

Before model-based controllers were designed for TCV, the ab initio model to be used (CREATE-L) was validated experimentally with CREATE (Vyas,Lister-CRPP, Villone-CREATE and Bandyopadhyay–India).

Square-wave and multi-harmonic stimulation was used

Agreement was good – whatever “good” means

Done in time and frequency domains.


Progress – H

One goal of this procedure on TCV was to demonstrate that the

superior performance of an H controller really could be obtained.

On the basis of the ab initio validated CREATE-L model, an H controller was designed (Pironti, Ariola – CREATE) and deployed on a digital control system on TCV built for this purpose.

After 1 failed attempt due to misunderstanding, the controller worked shadowing the standard controller and then replaced it for the following pulse.

Mission accomplished.

“So you will always use this controller ? Actually… no….Warning: what is the “best” controller


Progress – Black box identification

The interest of Imperial College increased and a PhD was undertaken to make a black box System Identification model of TCV (Coutlis).

Random Binary Sequence and Multi-Harmonic stimulations were used and a “good” result was obtained.

Experience with the technique was gained, but we already had a reasonable ab initio model.


Progress – Grey box identification

The accuracy of the RZIP model (rigid current distribution) surprised the ITPA and a project was launched to compare on JT-60U.

A PhD (Sharma-Imperial) used multi-harmonic frequency response experiments were compared with the ab initio RZIP model.

Tuning the 2000-parameter RZIP model used plasma-less tests

This taught us how insensitive the responses are to most engineering parameters, and how precisely other parameters were measured.


Progress – Linearisation of non-linear simulations ?

In mid-1990’s, the question had actually been raised – “is this non-linear problem actually linearisable?”

Ward (CRPP) performed the first multi-harmonic stimulation of a TSC non-linear simulation, and we demonstrated that the response only contained the frequencies driven – no harmonics.

The linearisation point had been made.


Progress – Full tokamak non-linear simulations

Today, 3 codes are used for full tokamak simulations of ITER

TSC, the oldest, including the plasma mass (Princeton)

DINA, mass-less (Moscow)

CORSICA, also mass-less (Livermore)

These codes can handle the effect of plasma heating and fuelling, as well as the full shape controller in a single self-consistent simulation.

But… they are no-longer light-weight – they take time to develop, test and run.

However, they are the only way of validating the results of other models


Progress – Optimising for actuator saturation

The primary non-linearity of modelling based on an LTI linearisation is not the plasma itself, but is the actuator saturation.

For ITER, this is simply due to money and to grid power limits.

Work in the 1990’s on this had failed to produce a simple approach and the question was re-visited early 2000’s.

Favez (CRPP) developed a formal proof of the optimal saturating actuator controller for the specific (but actually our own) case of a single unstable pole. Problems were found implementing it in DINA.

Liu (Chalmers) worked out how to make an estimator of the unstable state for another similar instability (RWM) and this method has to be tested in DINA.


Work needed for ITER – Minimise AC losses

Back to the ITER challenges.

Superconductors do not tolerate high levels of dB/dt across the filaments. A controller has to minimise these so-called AC losses.

Favez and Schaerz (CRPP) showed that small improvements could be made at little cost, but that the major AC losses are simply linked to the operation of the tokamak, not to the details of the controller. However, a method to ensure that there are no wasted AC losses was demonstrated.


Work needed for ITER – Magnetics sensors

Over 1500 magnetic sensors are foreseen for ITER……

Faraday rotation of laser light – Total current



Total current – inductive Rogowski

coils



Field outside the metallic vessel, in

both directions

B

B

Partie interne de la chambre à vide

Partie externe de la chambre à vide



Field in the “divertor” region, inside the

vessel



Field in the “divertor” region, inside the

vessel

Hal

o R

ogow

ski c

oils

“Halo” currents flowing into the metallic vessel



Flux loops inside the vacuum vessel


Work needed for ITER – Equilibrium reconstruction

We can only “infer” the equilibrium from magnetic measurements, sometimes reinforced by some non-magnetic measurements.

“Reconstructing” (inferring) the equilibrium essentially resides in a least-squares expression of an optimisation problem, collectively known as “EFIT” but actually a set of different implementations.

Some interest was generated in non-linear mapping via neural networks (Lister – CRPP), but this has not been applied in anger.

ITER will need a collectively optimised set of the existing techniques.


Work needed for ITER – Guarantee measurements

We have assumed that the measurements of shape and position are available…

Measurements of the magnetic field around the plasma face multiple technical challenges, including:

• effect of radiation on the signal cable (RIEMF)

• effect of temperature changes on the signal cable (TIEMF)

• effect of magnetic material (ferritic inserts) around the tokamak

effect of non-axisymmetric plasma effects on the shape estimation

• reliability to protect a 5B€ investment

• long-pulse drift-free integration of induced voltages

All these effects make magnetics a serious challenge


Work needed for ITER – Tolerate/identify errors

ITER will be unlike previous experiments in cost per failure, requiring unprecedented levels of reliability during long 1000’s of second pulses.

Errors of 5% in one sensor lead to an unacceptable error in the estimated plasma shape and position (Moreau – CEA)

Significant R&D will be needed on identifying and removing dubious sensors from the estimation of the equilibrium.

Err

or

(mm

)

020

020

020

020

020

020

Gap

#1G

ap#2

Gap

#3G

ap#4

Gap

#5G

ap#6

0 20 40 60 80 100 120#coil

B tangent B normal

Error reaches20mm(Spec. < 10mm)!


Work needed for ITER – Full loop simulations

Finally, the successful operation of ITER will require a higher level strategic control to optimise the references for the shape control.

The full tokamak non-linear simulators will be the only appropriate place to develop and demonstrate these ideas.

Developing a high-level toolkit for tokamak control methods has also attracted increasing attention at DIII-D and at C-MOD tokamaks

Most of this work is done in Matlab, but EU work (Integrated Tokamak Modelling) is pushing for a non-Matlab framework.


Work needed for ITER – DINA-CH full simulator

Kim-CRPP


Apologies…

The control problem has been addressed by many groups, since all tokamaks need control.

The work used as examples is representative, and heavily biased.

Work on formal equilibrium control has also been notably carried out on:

PBX, see Okabayshi

DIII-D, see Humphreys, Walker et al.

JET Extreme Shape Controller project, see Villone, Crisanti, Lomas et al.,

CREATE, see Albanese, Coccorese, Rubinacci, Villone, Pironti

C-MOD, see Hutchinson et al

JT-60U, see Yoshino, Kurihara et al.

MAST, see McArdle

EAST, KSTAR, see Humphreys and Walker et al.

ITER… see above, plus Portone, Gribov, Mitrishkin, Kavin, Kessel

……………………and many others…………….


Conclusions

The understanding of axisymmetric equilibrium control is on a sure footing today

Lots of preparation work has already been done in many groups

ITER will be very demanding

• robustness against errors, failures

• precision

• harsh environment

• long timescale

There is plenty of work to do ITER will provide plenty of control challenges

– get busy…. Thank you for your attention


So now you see what you saw !

Shape varying with time Gap reference targets, vertical instability

DINA-CH full tokamak simulation

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Magnetic control of a tokamak equilibrium

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