Real time reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile with the EQUINOX code Jacques Blum Cedric Boulbe Blaise Faugeras in collaboration with IRFM CEA and JET Universit´ e de Nice Sophia Antipolis Laboratoire J.-A. Dieudonn´ e Nice, France jacques.blum@unice.fr 7th Workshop on Fusion Data Processing Validation and Analysis Frascati, March 2012 JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 1 / 22
Introduction Equilibrium of a plasma : a free boundary problem Equilibrium equation inside the plasma, in an axisymmetric configuration : Grad-Shafranov equation Right-hand side of this equation is a non-linear source : the toroidal component of the plasma current density Goal Identification of this non-linearity from experimental measurements. Perform the reconstruction of 2D equilibrium and the identification of the current density in real-time. JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 2 / 22
Mathematical modelling of the equilibrium Grad-Shafranov Equation 3D MHD equilibrium + axisymmetric assump. : Grad-Shafranov eqn. 2D problem. State variable ψ ( r , z ) poloidal magnetic flux In the plasma 1 − ∆ ∗ ψ = rp ′ ( ψ ) + µ 0 r ( ff ′ )( ψ ) with ∆ ∗ . = ∂ ∂ r ( 1 ∂ r ) + ∂ ∂. ∂ z ( 1 ∂. ∂ z ) µ 0 r µ 0 r In the vacuum − ∆ ∗ ψ = 0 JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 3 / 22
Definition of the free plasma boundary Two cases outermost flux line inside the limiter (left) magnetic separatrix : hyperbolic line with an X-point (right) JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 4 / 22
Experimental measurements magnetic ”measurements” ψ ( M i ) = g i and 1 ∂ψ ∂ n ( N j ) = h j on ∂ Ω r on mesh boundary (experimental measurements if possible, or outputs from other codes : XLOC-FELIX (JET) and APOLO (ToreSupra)) interferometry and polarimetry on several chords � � n e ( ψ ) ∂ψ n e ( ψ ) dl = α m , ∂ n dl = β m r C m C m motional Stark effect f j ( B r ( M j ) , B z ( M j ) , B φ ( M j )) = γ j JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 5 / 22
Statement of the inverse problem State equation − ∆ ∗ ψ = λ [ r ψ ) + R 0 A ( ¯ r B ( ¯ ψ )]1 Ω p ( ψ ) in Ω R 0 ψ = g on ∂ Ω Least square minimization J ( A , B , n e ) = J 0 + K 1 J 1 + K 2 J 2 + J ǫ with j (1 ∂ψ ∂ n ( N j ) − h j ) 2 J 0 = � r ∂ψ � n e ∂ n dl − α i ) 2 J 1 = � i ( r C i � n e dl − β i ) 2 J 2 = � i ( C i � 1 � 1 � 1 ( ∂ 2 A ( ∂ 2 B ( ∂ 2 n e ψ 2 ) 2 d ¯ ψ 2 ) 2 d ¯ ψ 2 ) 2 d ¯ J ǫ = ǫ ψ + ǫ ψ + ǫ ne ψ ∂ ¯ ∂ ¯ ∂ ¯ 0 0 0 JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 6 / 22
Numerical method Finite element resolution Find ψ ∈ H 1 with ψ = g on ∂ Ω such that 1 � � λ [ r ψ ) + R 0 A ( ¯ r B ( ¯ ∀ v ∈ H 1 0 , µ 0 r ∇ ψ ∇ vdx = ψ )] vdx R 0 Ω Ω p with A ( x ) = � i a i f i ( x ) , B ( ψ ) = � i b i f i ( x ) , u = ( a i , b i ) Fixed point K ψ = Y ( ψ ) u + g K modified stiffness matrix, u coefficients of A and B , g Dirichlet BC Direct solver : ( ψ n , u ) → ψ n +1 ψ n +1 = K − 1 [ Y ( ψ n ) u + g ] JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 7 / 22
Numerical method Least-square minimization J ( u ) = � C ( ψ ) ψ − d � 2 + u T Au d : experimental measurements A : regularization terms Approximation J ( u ) = � C ( ψ n ) ψ − d � 2 + u T Au , with ψ = K − 1 [ Y ( ψ n ) u + g ] = � C ( ψ n ) K − 1 Y ( ψ n ) u + C ( ψ n ) K − 1 g − d � 2 + u T Au J ( u ) = � E n u − F n � 2 + u T Au Normal equation. Inverse solver : ψ n → u ( E nT E n + A ) u = E nT F n JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 8 / 22
Algorithm. EQUINOX One equilibrium reconstruction : Fixed-point iterations : ◮ Inverse solver : ψ n → u n +1 ◮ Direct solver : ( ψ n , u n +1 ) → ψ n +1 ◮ Stopping condition || ψ n +1 − ψ n || < ǫ || ψ n || A pulse in real-time : Quasi-static approach : ◮ first guess at time t = equilibrium at time t − δ t ◮ limited number of iterations Normal equation : ≈ 10 basis func. → small ≈ 20 × 20 linear system Tikhonov regularization parameters unchanged K = LU and K − 1 precomputed and stored once for all Expensive operations : update products C ( ψ ) K − 1 and C ( ψ ) K − 1 Y ( ψ ) JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 9 / 22
Algorithm verification : twin experiments Method Functions A and B given. Generate ”measurements” with direct code Test the possibility to recover the functions by solving the inverse problem Noise free experiments. Magnetics only. With a well-chosen regularization parameter ε , A and B are well recovered. Averaged current density and q profiles are not very sensitive to ε . Experiments with noise. Magnetics only and mag+polarimetry. Averaged current density and q profiles are less sensitive to noise than A and B . With polarimetry A and B are better constrained. JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 10 / 22
Average over magnetic surfaces ∂ gdv = 1 � � gds � gdl � dl < g > = |∇ ρ | = / ∂ V V ′ B p B p V S C ρ C ρ ρ a coordinate indexing the magnetic surfaces JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 11 / 22
Definitions Current density averaged over magnetic surfaces r 0 < j ( r , ¯ ψ ) 0 < 1 > = λ A ( ¯ ψ ) + λ r 2 r 2 > B ( ¯ ψ ) r Safety factor q For one field line ” q = ∆ φ 2 π ”. q = − 1 ∂ F φ ∂ψ = − 1 ∂ V ∂ψ f < 1 r 2 > = 1 � B φ dl 4 π 2 2 π 2 π rB p C JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 12 / 22
Noise free twin experiment. Magnetics only. Identified A and B, and relative error for different ε JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 13 / 22
Noise free twin experiment. Magnetics only. Mean current density, safety factor and relative error for different ε JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 14 / 22
1% noise twin exp. Magnetics only. Mean ± stand. dev. (200 exp.) identified A and B for ε = 0 . 01 , 0 . 1 , 1 JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 15 / 22
Same for mean current density and safety factor JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 16 / 22
1% noise twin exp. Mag. and polar. Mean ± stand. dev. (200 exp.) identified A and B for ε = 0 . 01 , 0 . 1 , 1 JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 17 / 22
Same for mean current density and safety factor JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 18 / 22
Tore Supra - Magnetics and polarimetry JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 19 / 22
JET - Magnetics and polarimetry JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 20 / 22
Conclusion Real-time equilibrium reconstruction and identification of the current density. EQUINOX Robust identification of the averaged current density profile and of the safety factor Ref : Blum, Boulbe and Faugeras. Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time, JCP 231 (2012) 960-980 . JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 21 / 22
Perspectives New standalone version of EQUINOX = FELIX + EQUINOX. ◮ Direct use of magnetic measurements ◮ Compute boundary conditions using ⋆ PF coils modelization ⋆ toroidal harmonics ◮ Substitute for FELIX - Apolo Makes possible future real-time control of current profile JB CB BF (Universit´ e de Nice) Real time equilibrium reconstruction March 2012 22 / 22
Tore Supra. Magnetics and polarimetry.
Jet 68694. Magnetics only.
Jet 68694. Magnetics and polarimetry.
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