An aspect of modelling and optimal control for Tokamaks EuroAd Workshop - Gael Selig... · An...
Transcript of 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
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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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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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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
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JET ITER
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JET
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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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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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Grad-Shafranov’s equation
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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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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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
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