An aspect of modelling and optimal control for Tokamaks EuroAd Workshop - Gael Selig... · An...

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An aspect of modelling and optimalcontrol for Tokamaks

Gaël Selig

CEA-IRFM, CadaracheUniversité de Nice - Sophia Antipolis

9th Euro AD Workshop — 22 October 2009

An aspect of modelling and optimal control for Tokamaks 9th Euro AD Workshop — 22 October 2009 1 / 29

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1 Introduction - Nuclear fusion

2 About Tokamaks

3 Modelling of plasma equilibrium

Plasma equilibrium

Grad-Shafranov’s equation

Equilibrium codes

4 Optimal control of tokamak plasmas

5 Possible usage of AD


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Introduction - Nuclear fusion

Nuclear FusionLight atoms fuses to form heavier atoms

Energy is released due to the mass defect :E = (mr −mp)c2

Studied for energy production

PlasmaTo overcome the Columbian repulsion, atoms must havesufficient kinetic energy⇒ Heating

The matter at these temperatures (107 – 108 k ) is in astate of plasma : ions and free electrons


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Reaction studied

CriterionHigh probability of reaction at the lowest temperature

Elements abundant everywhere on Earth

100 101 102 103

temperature (keV)

10-27

10-26

10-25

10-24

10-23

10-22

10-21

react

ivit

y<

σv>

(m

s )

D-TD-DD-He3

3-1

D-T reaction : 2H + 3H −→ 4He + n (17.6 MeV )


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Plasma confinementThe plasma must be confined to make the atoms fuse

Types of confinement :

Gravitational : Stars

Inertial : Matter studies

Magnetic : Tokamaks

Magnetic confinement

Aim : create a “magnetic box” to trap the plasma’scharged particles

Most advanced device : Tokamak


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2 About Tokamaks


Plasma equilibrium


Equilibrium codes




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About Tokamaks

TokamakMost promising devices for the development of fusionpower plants

Russian concept (1960’s)

Toroidal device

Uses external coils to produce a toroidal field

Induces a current in the plasma to create a poloidal field


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Tokamak principle


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Some Tokamak experiments

JETLargest Tokamak in the world

Fusion record : 16 MW

Located in Culham, England

Tore SupraUses superconducting coils

Duration record : > 6 min

Located in Cadarache, France

ITERWorld wide collaboration

Aim : 500 MW of fusion power with 50 MW of heating

Is being built in Cadarache, FranceAn aspect of modelling and optimal control for Tokamaks 9th Euro AD Workshop — 22 October 2009 9 / 29

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JET ITER


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JET


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2 About Tokamaks


Plasma equilibrium


Equilibrium codes




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Plasma modelling

AimSimulate a tokamak

Compare simulations with experiments

Predict (scenarios), control and optimise the experiments

ToolsMathematical models of the physical phenomenonsinvolvedIntegrated modelling codes :

Take into account coupled phenomenonsFast enough for full scenario studiesSeveral models availableEuropean framework : ITM


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Example : the CRONOS suite


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Physical models

ModelsNewton’s laws

Maxwell’s laws

MagnetoHydroDynamics (MHD)Most simple mathematical model

Considers the plasma as a single charged fluid

Introduces fluid variables : n,T ,v ,p . . .

System of 13 equations


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Plasma equilibrium

Equilibrium

Plasma state when Laplace forces (~j×~B) compensatesthe pressure forces

Allows to know the magnetic configuration inside theplasma

Hypotheses

Axial symmetry : studies in the (r ,z) plane

Isotropic pressure

MHD equilibrium equation

~j×~B = ~∇p


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Flux function

Definition

ψ(r ,z)≡Z

D(r ,z)Bz(r , z) drdz

f (r ,z)≡ r BT (r ,z)

D(r, z)Z

×(r, z)


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Properties

The magnetic field can be deduced from ψ and f~j and~B lie on iso-ψ surfaces (called magnetic surfaces)

p and f are constant on iso-ψ ⇒ p(ψ), f (ψ)

In usual plasmas, magnetic surfaces are a set of nestedtorus


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Property (Maxwell + axial symmetry)

Lψ(r ,z) = j(r ,z), ∀(r ,z) ∈ R∗×R

Lψ≡− ∂

∂r ( 1µr

∂ψ

∂r )− ∂

∂z ( 1µr

∂ψ

∂z ) : elliptic operatorj : toroidal current density

Grad-Shafranov’s equation in the plasma

Lψ(r ,z) = r∂p∂ψ

(ψ(r ,z)

)+

12µ0r

∂f 2

∂ψ

(ψ(r ,z)

)Properties

p and f are unknowns : solutions of the transportequations

p and f are non linear in ψ.


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Plasma boundary

DefinitionInnermost magnetic surface that intersect the wallsΓp = M ∈ Ω/ψ(M) = ψbψb = max(supDL

ψ,supDXψ)

DL : Set of possible limiter pointsDX : Set of X-points (saddle points)

Possible limited or diverted shape


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Free boundary plasma equilibrium problem

Given (Vi , p,f ,µ), find ψ,Γp such that :

Lψ = r∂p∂ψ

+1

2µ0r∂f 2

∂ψin the plamsa

Lψ =Vi

Ri Si− ni

Ri S2i

ZΩBi

∂ψ

∂tdS in the coils

Lψ =−σv

r∂ψ

∂tin conductive structures

Lψ = 0 in air and vacuum

With boundary conditions :

ψ = 0 on (Oz) and when ‖(r ,z)‖→ ∞


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Equilibrium codes

AimSolves the previous system

Exemple : the CEDRES++ equilibrium codeUses a Finite Elements Method

Non linearities treated with Newton or Picard algorithms

Will be coupled with the CRONOS suite to get p and f

Written in C++


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Example : Limiter equilibrium

−10 −5 0 5 10 15 20 25

−10

−5

0

5

10


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Example : X-point equilibrium

−10 −5 0 5 10 15 20 25

−10

−5

0

5

10


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2 About Tokamaks


Plasma equilibrium


Equilibrium codes




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Optimal control of tokamak plasmas

AimTo fit given parameters (from a scenario) all along theexperiment⇒ minimisation of a cost function : minx∈D J(x)

Example of cost function

Distance from a given plasma intensity Ipg(t) :Z T

0(Ipg(t)− Ip(t))2 dt

Voltages value in the coilsZ T

0∑

iVi(t)2 dt


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Cost function

Distance from a given plasma boundaryΓg(t) :

Z T

0

(ZΓg(t)

(ψ(r ,z, t)−ψb(r ,z, t))2 drdz

)dt

Γg(t0)

Γ(t0)


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Optimal control problem

Optimal control problem

Find Vi(t) such that :

Vi(t) = min

ViJ(Vi ,ψ)

Under the constraint of verifying the equilibrium problem

J(Vi,ψ,Γg) ≡ w1

Z T

0

(ZΓg(t)

(ψ−ψb)2 (r ,z, t) drdz

)dt

+w2

Z T

0(Ipg(t)− Ip(t))2 dt

+w3

Z T

0∑

iVi(t)2 dt


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Possible usage of AD

Automatic DifferentiationComputes the adjoint state

Allows to get ∇J

For real-time control : allows to get a linearised model

LimitationsC++ : object oriented

Code not written for AD


An aspect of modelling and optimal control for Tokamaks EuroAd Workshop - Gael Selig... · An...

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